Atmospheric Temperature and Pressure Measurements from
the ACE-MAESTRO Space Instrument
by
Caroline Rebecca Nowlan
A thesis submitted in conformity with the requirements
for the degree of Doctor of Philosophy
Graduate Department of Physics
University of Toronto
c 2006 by Caroline Rebecca Nowlan
Copyright Abstract
Atmospheric Temperature and Pressure Measurements from the ACE-MAESTRO
Space Instrument
Caroline Rebecca Nowlan
Doctor of Philosophy
Graduate Department of Physics
University of Toronto
2006
A method is developed and tested for determining atmospheric pressure and temperature from space using spectral absorption by the A and B bands of molecular oxygen as
measured by the MAESTRO (Measurement of Aerosol Extinction in the Stratosphere
and Troposphere Retrieved by Occultation) space instrument in solar occultation mode.
MAESTRO is the UV-visible-near-infrared dual grating spectrometer component of the
Atmospheric Chemistry Experiment (ACE) scientific payload, and was launched in August 2003 on the SciSat satellite to investigate atmospheric processes affecting the stratospheric ozone distribution. On-orbit measurements of pressure and temperature are desirable for accurate retrievals of other atmospheric constituents from the instrument, and
as independent data products. The constant mixing ratio of molecular oxygen is exploited in these retrievals to derive atmospheric density profiles from oxygen absorption,
and the density profiles are then converted to pressure and temperature using hydrostatic
balance and the ideal gas law.
A highly accurate fast forward model is developed using a fast-line-by-line approach
for modelling the high spectral resolution oxygen absorption lines, and a correlated-k
technique is used to calculate analytic weighting functions for the retrieval of density. A
global fitting algorithm is developed to simultaneously fit all spectra from one occultation.
Retrieval characterization shows a small amount of information is added by the inclusion
ii
of the weaker B band to a retrieval using the strong A band absorption, and tests with real
data show the B band retrievals alone are less sensitive to instrument characterization
uncertainties and perform better than A band retrievals at altitudes below 30 km. The
new algorithm is applied to 230 occultation observations collected in the Arctic winter,
Arctic spring, and tropical regions, and during the Eureka 2004 and 2005 ACE validation
campaigns in the high Arctic. Comparisons with profiles from the ACE-FTS (Fourier
Transform Spectrometer), also on SciSat, coincident radiosondes, and meteorological
analyses yield promising results, with biases between MAESTRO’s and other profiles
generally within 2-5% in pressure and less than 5 K in temperature, and RMS differences
between 2-10% in pressure and 5-10 K in temperature.
iii
Acknowledgements
Any accomplishments in this project are thanks in large part to my co-supervisor Tom
McElroy. He is the originator of the MAESTRO instrument, has been heavily involved
in all aspects of the mission and science, and has been my primary scientific guide. I
am extremely grateful to my other co-supervisor, Jim Drummond, who has given me
much help and many excellent opportunities during my Ph.D., and whose knowledge
and leadership I admire greatly. I would also like to thank my committee members
Ted Shepherd and Jerry Mitrovica for their comments on this thesis work, and Dylan
Jones and Kelly Chance, who took the time to read this thesis and who sat on my final
examination committee.
I would like to acknowledge several organizations and foundations who provided financial support for both MAESTRO and my personal survival as a graduate student:
the Canadian Space Agency, the Meteorological Service of Canada, the Natural Sciences
and Engineering Research Council of Canada, the Canadian Foundation for Climate and
Atmospheric Sciences, Zonta International, the Leonard and Kathleen O’Brien Humanitarian Trust, and the Walter C. Sumner Foundation.
Several members of the MAESTRO team at the University of Toronto have made
contributions to this work. Clive Midwinter gave me a considerable amount of assistance
during my early time working with the instrument, and later on was always accessible
and good-humoured whenever I needed help. Denis Dufour led the pre-flight instrument
characterization, and provided characterization data for this research and many discussions as my MAESTRO grad student compatriot. Jason Zou, Jay Kar, and Florian
Nichitiu extracted and provided seemingly endless amounts of data at my (numerous)
requests.
David Barton, Bob Hall, Akira Ogyu, and Aaron Ullberg at the Meteorological Service of Canada helped in significant ways over my years on this project, particularly
during three ballooning field campaigns, which in the end would not fit into this wordiv
count-limited thesis, but were highly enjoyable and productive experiences during my
Ph.D. Thanks also to Shawn Turner at the MSC who was very helpful during the initial
development of the O2 radiative transfer model.
The ACE-FTS team at the University of Waterloo have also been a key part of the
MAESTRO project. Ray Nassar provided the a priori profile data used in this thesis
work and Chris Boone provided the pointing data and FTS profiles. They, in addition
to Sean McLeod and Kaley Walker, have also been the providers of prompt and patient
responses to my many questions.
Meteorological data from ECMWF and NCEP was provided by the Data Support
Section at NCAR. Rebekah Martin at the U of T came to my rescue to help extract the
data into a usable format.
Outside of work, my friends, both from the physics department and elsewhere, distracted me when needed (and occasionally when probably not needed!) with lunches in
Chinatown, bike rides, and road trips, or joined me when working on my back-up career
as a fiddle player.
Most of all, I would like to thank my parents Bill and Juliet Nowlan and brother
James for all their encouragement in every aspect of my life, and Jerome Klassen, whose
love and support has sustained me through to the end of this thesis.
v
Contents
1 Introduction
1
1.1
Occultation Measurements . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
The MAESTRO Instrument . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3
Motivation for Pressure and Temperature Retrievals . . . . . . . . . . . .
7
1.4
Pressure and Temperature Measurement Methodology . . . . . . . . . . .
12
1.5
The A, B, and γ Bands of O2 . . . . . . . . . . . . . . . . . . . . . . . .
13
1.6
Remote Sounding of O2 from Space . . . . . . . . . . . . . . . . . . . . .
16
1.7
Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
1.8
Personal Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2 MAESTRO Forward Model
21
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.2
Instrument Characterization . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.2.1
Pixel-Wavelength Registration . . . . . . . . . . . . . . . . . . . .
22
2.2.2
Slit Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.2.3
Field-of-View . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
Geometry and Ray-Tracing Model . . . . . . . . . . . . . . . . . . . . . .
26
2.3.1
Atmospheric Grid Model . . . . . . . . . . . . . . . . . . . . . . .
26
2.3.2
Refraction and Ray-Tracing . . . . . . . . . . . . . . . . . . . . .
28
The Radiative Transfer Model . . . . . . . . . . . . . . . . . . . . . . . .
30
2.3
2.4
vi
2.5
2.6
2.4.1
Radiative Transfer in the Atmosphere . . . . . . . . . . . . . . . .
30
2.4.2
Exoatmospheric Model . . . . . . . . . . . . . . . . . . . . . . . .
32
2.4.3
Application to MAESTRO . . . . . . . . . . . . . . . . . . . . . .
33
2.4.4
O2 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
2.4.5
Forward Model Output . . . . . . . . . . . . . . . . . . . . . . . .
46
Approximation Errors and Forward Model Resolution Requirements . . .
46
2.5.1
Line-by-Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
2.5.2
Fast-Line-by-Line . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
2.5.3
Correlated-k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
2.5.4
Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
3 Profile Retrievals
57
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
3.2
Inversion of Atmospheric Profiles . . . . . . . . . . . . . . . . . . . . . .
58
3.3
Constrained Linearized χ2 Minimization Method . . . . . . . . . . . . . .
60
3.4
Global Fit Applied to MAESTRO Data . . . . . . . . . . . . . . . . . . .
63
3.5
A Priori and First-Guess Profiles . . . . . . . . . . . . . . . . . . . . . .
66
3.6
Calculating the Weighting Function Matrix . . . . . . . . . . . . . . . . .
67
3.7
Calculating the Smoothing Constraint
. . . . . . . . . . . . . . . . . . .
69
3.8
Testing for Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
3.9
Simulated Retrievals . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
3.9.1
Atmospheric Cases . . . . . . . . . . . . . . . . . . . . . . . . . .
73
3.9.2
Retrievals on Noise-Free Spectra . . . . . . . . . . . . . . . . . . .
74
3.9.3
Retrievals on Noisy Spectra . . . . . . . . . . . . . . . . . . . . .
78
3.10 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
vii
4 Expected Retrieval Performance
89
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
4.2
Retrieval Characterization . . . . . . . . . . . . . . . . . . . . . . . . . .
90
4.2.1
Weighting Functions . . . . . . . . . . . . . . . . . . . . . . . . .
92
4.2.2
Gain Function Matrix . . . . . . . . . . . . . . . . . . . . . . . .
93
4.2.3
Averaging Kernels . . . . . . . . . . . . . . . . . . . . . . . . . .
95
Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
4.3
4.4
4.3.1
Smoothing Error . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.3.2
Model Parameter Error . . . . . . . . . . . . . . . . . . . . . . . . 101
4.3.3
Forward Model Error . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.3.4
Retrieval Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.3.5
Total Error Estimates . . . . . . . . . . . . . . . . . . . . . . . . 114
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5 Retrievals from Satellite Data
118
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.2
Input Data for p-T Retrievals . . . . . . . . . . . . . . . . . . . . . . . . 119
5.3
5.4
5.5
5.2.1
Pointing Information . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.2.2
Spectral Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Analysis Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.3.1
MAESTRO Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.3.2
Comparison Datasets . . . . . . . . . . . . . . . . . . . . . . . . . 125
Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.4.1
Pre-Processing: Pixel-Wavelength Assignment . . . . . . . . . . . 128
5.4.2
Instrument Slit Function . . . . . . . . . . . . . . . . . . . . . . . 130
5.4.3
Spectral Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.4.4
Analysis Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Pressure and Temperature Results . . . . . . . . . . . . . . . . . . . . . . 141
viii
5.6
5.5.1
Sample Profile Retrievals . . . . . . . . . . . . . . . . . . . . . . . 141
5.5.2
Characterization and Error Analysis . . . . . . . . . . . . . . . . 143
5.5.3
Multiple Profile Comparisons . . . . . . . . . . . . . . . . . . . . 151
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6 Conclusions
6.1
166
Summary of Results and Achievements . . . . . . . . . . . . . . . . . . . 166
6.1.1
Forward Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.1.2
Retrieval Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.1.3
Satellite Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . 168
6.1.4
Implications for MAESTRO Data Products . . . . . . . . . . . . 169
6.2
Suggestions for Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.3
Recommendations for Future Missions . . . . . . . . . . . . . . . . . . . 171
Bibliography
172
A List of Acronyms
187
B Model Parameter Weighting Functions
190
ix
List of Tables
1.1
MAESTRO data products . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
Spectral ranges and vibrational transitions of O2 bands measured by MAESTRO 37
2.2
Forward model resolutions . . . . . . . . . . . . . . . . . . . . . . . . . .
49
3.1
Parameters modelled and retrieved in A and B band spectra . . . . . . .
65
3.2
Evolution of χ2 m and d2i toward retrieval convergence . . . . . . . . . . .
71
3.3
Atmospheric cases used for simulations . . . . . . . . . . . . . . . . . . .
74
4.1
Forward model parameters and uncertainties . . . . . . . . . . . . . . . . 103
5.1
Occultations used in Chapter 5 analysis . . . . . . . . . . . . . . . . . . . 123
x
6
List of Figures
1.1
Solar occultation geometry.
. . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2
SciSat occultation latitudinal coverage for 2004 . . . . . . . . . . . . . .
4
1.3
MAESTRO spectrometer optics . . . . . . . . . . . . . . . . . . . . . . .
7
1.4
Simulated transmission through the atmosphere at a tangent height of 30
km for the spectral region of the MAESTRO instrument . . . . . . . . .
1.5
8
Transmission through the atmosphere for the A, B, and γ bands of molecular oxygen for tangent heights of 10, 30, and 60 km . . . . . . . . . . .
15
2.1
MAESTRO IR slit function . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.2
MAESTRO slit and solar disk. . . . . . . . . . . . . . . . . . . . . . . . .
25
2.3
MAESTRO VIS field-of-view weightings in elevation. . . . . . . . . . . .
25
2.4
Path of a refracted ray through a three-layer atmosphere. . . . . . . . . .
26
2.5
Atmospheric model grid with M retrieval levels and N optical levels. . .
27
2.6
Ray path and turning angles through a refracting atmosphere . . . . . .
29
2.7
Incident and outgoing radiation through a path of length L . . . . . . . .
31
2.8
Exoatmospheric ATLAS 3 solar reference spectrum . . . . . . . . . . . .
33
2.9
Sample O2 cross-sections in the A and B bands . . . . . . . . . . . . . .
38
2.10 O2 A band cross-sections between 770 and 772 nm at 0 and 30 km, and
their cumulative distribution functions . . . . . . . . . . . . . . . . . . .
41
2.11 Modelled optical depths for tangent altitudes of 10, 30, and 60 km . . . .
47
2.12 Errors in FLBL-calculated transmission . . . . . . . . . . . . . . . . . . .
52
xi
2.13 Errors in correlated-k -calculated transmission . . . . . . . . . . . . . . .
53
2.14 Transmission errors for the FLBL, direct correlated-k, and table correlatedk approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
3.1
MAESTRO retrieval algorithm flow diagram. . . . . . . . . . . . . . . . .
72
3.2
Retrieval using FLBL for Arctic spring case on simulated noise-free spectra. 76
3.3
Retrieval using correlated-k for Arctic spring case on simulated noise-free
spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4
77
Retrieval using FLBL-modelled optical depths and correlated-k -calculated
weighting functions for Arctic spring case on simulated noise-free spectra.
79
3.5
Retrieval for Arctic spring case on simulated data with noise. . . . . . . .
81
3.6
Retrieval for mid-latitude summer case on simulated data with noise. . .
82
3.7
Retrieval for mid-latitude winter case on simulated data with noise. . . .
83
3.8
Retrieval for tropical spring case on simulated data with noise. . . . . . .
84
3.9
Retrieval for Antarctic winter case on simulated data with noise. . . . . .
85
3.10 Tropospheric and lower stratospheric temperature retrieval for Arctic spring
case on simulated data with noise. . . . . . . . . . . . . . . . . . . . . . .
87
4.1
Typical weighting functions from the Kx matrix for ln nO2 retrieval. . . .
94
4.2
Typical gain functions from the Gy matrix for ln nO2 retrieval. . . . . . .
96
4.3
Typical averaging kernels for A and B band retrievals. . . . . . . . . . .
97
4.4
FWHM of ln nO2 averaging kernels. . . . . . . . . . . . . . . . . . . . . .
98
4.5
Model sensitivity Kb to a 0.0015 cm−1 perturbation in O2 line positions.
105
4.6
Model sensitivity Kb to a 2.0% perturbation in O2 line strengths. . . . . 106
4.7
Errors in density, pressure, and temperature from O2 line parameter database uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.8
Errors in density, pressure, and temperature from uncertainty in slit function110
xii
4.9
Errors in density, pressure, and temperature from uncertainty in satellite
pointing angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.10 Forward model difference ∆f from the correlated-k approximation. . . . . 112
4.11 Forward model errors in density, pressure, and temperature from correlatedk approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.12 Forward model errors in density, pressure, and temperature from FLBL
approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.13 Random errors in density, pressure, and temperature from measurement
noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.14 Total errors in density, pressure, and temperature from model parameter
error and random noise error . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.1
Corrected spectra for occultation ss3004 output from SCALE V32.0 software122
5.2
Locations of occultations used in Chapter 5 analysis . . . . . . . . . . . . 124
5.3
MAESTRO and FTS tangent altitudes for occultation ss3004 with a priori
temperature profile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.4
Exoatmospheric spectrum high frequency component used in the ss3004
pixel-wavelength registration . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.5
The original first-guess MAESTRO slit function fitted to the infrared diode
laser line shape during pre-flight characterization, and slit functions derived from deconvolution of the A and B band spectra. . . . . . . . . . . 132
5.6
Observed and modelled optical depths for the A and B band at low tangent
altitudes, and their residuals offset by 0.5, for occultation ss3004. . . . . 134
5.7
Observed and modelled optical depths for the A and B band at high
tangent altitudes, and their residuals offset by 0.02, for occultation ss3004. 135
5.8
Occultation sr7891 (Arctic winter): Pressure and temperature profiles
from the ACE-FTS, a priori /CMC, NCEP, ECMWF, and MAESTRO . 144
xiii
5.9
Occultation ss4831 (Arctic summer): Pressure and temperature profiles
from the ACE-FTS, a priori /CMC, NCEP, ECMWF, and MAESTRO . 145
5.10 Occultation sr3733 (tropics): Pressure and temperature profiles from the
ACE-FTS, a priori /CMC, NCEP, ECMWF, and MAESTRO . . . . . . . 146
5.11 Occultation ss3004 (Eureka 2004): Pressure and temperature profiles from
the ACE-FTS, a priori /CMC, NCEP, ECMWF, a nearby radiosonde, and
MAESTRO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.12 Occultation ss8413 (Eureka 2005): Pressure and temperature profiles from
the ACE-FTS, a priori /CMC, NCEP, ECMWF, a nearby radiosonde, and
MAESTRO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.13 Averaging kernels of ln nO2 for occultation ss3004, shown for altitudes from
5 to 75 km, spaced every 5 km. . . . . . . . . . . . . . . . . . . . . . . . 149
5.14 Slit function shape and stray light error in a) density, b) pressure, and c)
temperature for occultation ss3004. . . . . . . . . . . . . . . . . . . . . . 150
5.15 Expected systematic, random, and total errors in a) density, b) pressure,
and c) temperature for occultation ss3004. . . . . . . . . . . . . . . . . . 152
5.16 Arctic winter occultations: mean and RMS differences between MAESTRO
and comparison profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.17 Arctic winter occultations: mean and RMS differences between MAESTRO
or FTS and meteorological analyses, on constant pressure levels . . . . . 157
5.18 Arctic summer occultations: mean and RMS differences between MAESTRO
and comparison profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
5.19 Arctic summer occultations: mean and RMS differences between MAESTRO
or FTS and meteorological analyses, on constant pressure levels . . . . . 158
5.20 Tropical occultations: mean and RMS differences between MAESTRO and
comparison profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
xiv
5.21 Tropical occultations: mean and RMS differences between MAESTRO or
FTS and meteorological analyses, on constant pressure levels . . . . . . . 159
5.22 Eureka 2004: mean and RMS differences between MAESTRO and comparison profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.23 Eureka 2004: mean and RMS differences between MAESTRO or FTS and
meteorological analyses, on constant pressure levels . . . . . . . . . . . . 160
5.24 Eureka 2005: mean and RMS differences between MAESTRO and comparison profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5.25 Eureka 2005: mean and RMS differences between MAESTRO or FTS and
meteorological analyses, on constant pressure levels . . . . . . . . . . . . 161
5.26 Mean temperature profiles and their standard deviations for Eureka 2004
and 2005 occultations within 500 km of Eureka campaign site, retrieved
from MAESTRO, the ACE-FTS, and coincident radiosondes . . . . . . . 162
5.27 Mean differences between ECMWF and NCEP geometric altitudes and
temperatures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
B.1 Model sensitivity Kb to a 2.5% perturbation in O2 air-broadened half-widths.190
B.2 Model sensitivity Kb to a 2.5% perturbation in O2 A-band self-broadened
half-widths and a 10% perturbation in O2 B-band self-broadened half-widths.191
B.3 Model sensitivity Kb to a 15% perturbation in O2 line width temperature
dependency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
B.4 Model sensitivity Kb to a 0.1 nm perturbation in the slit function width.
192
B.5 Model sensitivity Kb to a 0.002◦ perturbation in pointing angle. . . . . . 193
xv
Chapter 1
Introduction
Over the last half-century the introduction of space-based instrumentation to measure
the terrestrial atmosphere has enabled global measurements of atmospheric constituents.
Atmospheric remote sounding from space has played a crucial role in the modern study of
the Earth’s climate and humanity’s influence on that climate. Canada has been active for
some time in space-based atmospheric sounding with instruments on large international
platforms, but had not launched a fully Canadian mission for sounding of the Earth’s
atmosphere in over thirty years. The launch of the SciSat satellite and its Atmospheric
Chemistry Experiment (ACE) scientific payload [1] on August 12, 2003 marked Canada’s
return to the deployment of all-Canadian small scientific satellite missions for atmospheric
studies. ACE is a mission for measuring vertical profiles of atmospheric trace gases and
other constituents with the goal of increasing our understanding of the chemical and
dynamical factors affecting the global distribution of ozone, particularly over northern
high latitudes. SciSat carries two primary instruments: the ACE-FTS Fourier transform
spectrometer (measuring between 750 and 4400 cm−1 with a 0.02 cm−1 spectral resolution), and the MAESTRO (Measurement of Aerosol Extinction in the Stratosphere and
Troposphere Retrieved by Occultation) UV-visible-near-infrared spectrometer. The goal
of the research in this thesis is to develop and test a method of retrieving atmospheric
1
Chapter 1. Introduction
2
pressure and temperature profiles from MAESTRO’s on-orbit measurements of molecular
oxygen. While pressure and temperature profiles are measured by several other satellites,
the method implemented here is unique in its application to a low-resolution spectrometer measuring visible and near-infrared solar radiation in two separate oxygen bands and
with high vertical resolution.
1.1
Occultation Measurements
The primary measurement mode of both the ACE-FTS and MAESTRO is solar occultation. Solar occultation spectra are collected during satellite sunrise and sunset when the
Sun-satellite vector passes through the Earth’s atmosphere. Figure 1.1 shows a sunrise
solar occultation measurement sequence of three rays. Occultation spectra contain spectral features that are characteristic of atmospheric constituents’ absorption and scattering
of solar radiation along the solar ray’s path. Atmospheric density changes exponentially
with altitude, causing the spectral features of a particular ray to be heavily weighted
to the tangent height zt , where the ray passes closest to the Earth’s surface. A typical MAESTRO measurement sequence consists of roughly 60 measurements at tangent
heights between 0 and 100 km. Each occultation sequence also contains exoatmospheric
high-sun solar reference measurements where no atmospheric features are present in the
spectra. These reference spectra allow occultation instruments to be self-calibrating,
with the advantage that measurements are much less susceptible to instrument characterization drifts and solar cycle changes over time. Occultation geometry also allows
the retrieval of atmospheric profiles with high vertical resolution as spectra are heavily
weighted to the tangent layer. However, the main disadvantage to occultation measurements is their lack of global coverage across latitude. SciSat’s 650 km altitude and
74◦ inclination circular orbit results in approximately fifteen sunrises and fifteen sunsets
per day. Figure 1.2 shows the latitudinal coverage for SciSat occultations during 2004.
3
Chapter 1. Introduction
The latitude-longitude location of a sunset or sunrise is determined by the Earth-Sunspacecraft geometry at the time of the occultation. The locations of SciSat occultations
for a particular day tend to occur in two latitude bands, typically with about 15 sunsets
in one hemisphere and 15 sunrises in the other, but with full longitudinal coverage over
each latitude band. SciSat’s orbital parameters have been optimized to ensure Arctic
and Antarctic measurements occur during each region’s respective spring season when
ozone destruction is at its maximum.
Exoatmospheric
Ray 3
Ray 2
Ray 1
Sun
zt
SCISAT
Figure 1.1: Solar occultation geometry.
1.2
The MAESTRO Instrument
The MAESTRO instrument consists of two independent spectrometers. One spectrometer measures UV and visible spectra from 280 to 550 nm with a resolution of approximately 1 nm, while the second measures visible and near-infrared spectra from 525 to
1010 nm with a resolution of approximately 2 nm. These spectrometers will henceforth
be referred to as the UV and VIS spectrometers respectively. Figure 1.3 shows the optical
4
Chapter 1. Introduction
80
60
40
latitude
20
0
-20
-40
-60
sunrise
sunset
-80
Jan
Apr
Jul
Oct
Jan
month
Figure 1.2: SciSat occultation latitudinal coverage for 2004. While this figures includes all
possible satellite sunrises and sunsets during this year, data was not necessarily collected
for every occultation.
Chapter 1. Introduction
5
design of the MAESTRO instrument. The main optical components of each spectrometer are a lens, a slit, a concave holographic diffraction grating, and a 1024-pixel Reticon
randomly addressable photodiode array detector. During an occultation, the MAESTRO
input port receives a beam of radiation picked off from the FTS input beam after the
solar pointing mirror. The light then travels through a series of input fore-optics in
MAESTRO before reaching the two independent spectrometers contained within the full
instrument. Each lens focuses light onto a slit, which defines the spectrometer’s field-ofview and vertical resolution. Light is then diffracted from the holographic grating onto
the photodiode array where the intensity of radiation at various wavelengths is detected
by the array of 1024 individual detector pixels. The MAESTRO spectrometer design is
based strongly on its precursor instrument, the Composition and Photodissociative Flux
Measurement (CPFM) instrument [2], which has flown on over one hundred high-altitude
aircraft experiments to measure ozone columns and photodissociative J-values of ozone
and NO2 [3], and in several high-altitude balloon campaigns as the SunPhotoSpectrometer instrument.
MAESTRO has the capability to measure vertical profiles of several atmospheric
constituents that are radiatively active in the spectral range of the two spectrometers.
Table 1.1 gives these species and their expected accuracy as listed in the MAESTRO
preliminary science study [4]. Figure 1.4 is a spectral simulation produced using the
MODTRAN radiative transfer model [5] showing the individual species’ contributions to
the total absorption for a tangent height of 30 km over the spectral range of the UV
and VIS spectrometers. Several other species including OClO and BrO are observable
under certain conditions, but are quite weak absorbers and not obviously visible in this
transmission spectrum.
6
Chapter 1. Introduction
Table 1.1: MAESTRO data products
Product
Altitude (km)
Accuracy
50 - 80
10%
20 - 50
3%
10 - 20
10%
8 - 10
15%
40 - 50
15%
20 - 40
10%
10 - 20
15%
8 - 10
25%
10 - 50
10−3 O.D.
8 - 10
0.01 O.D.
Aerosol Wavelength Dependence
15 - 30
0.005 O.D. per 100 nm
Water Vapour
8 - 20
100 ppm
OClO
8 - 20
20%
BrO
8 - 20
20%
O3
NO2
Aerosol Extinction
BrO column
20%
Chapter 1. Introduction
7
Figure 1.3: MAESTRO spectrometer optics (courtesy of C. Midwinter).
1.3
Motivation for Pressure and Temperature
Retrievals
This research develops a method to measure vertical profiles of pressure and temperature
(‘p-T’) from orbit. While some p-T profiles are available from other observing systems
including satellites and balloon-borne radiosondes, and databases of global pressure and
temperature forecasts and analyses, on-orbit p-T measurements from MAESTRO are
highly desirable. Simultaneous measurements of p-T with other constituents ensure geolocated measurements through the entire measurement altitude range with a consistent
instrument field-of-view. Accurate p-T measurements, and their associated atmospheric
densities, are required for reliable constituent retrievals and solar ray path calculations,
and are interesting as another independent data product.
8
Chapter 1. Introduction
1
0.9
0.8
transmission
0.7
0.6
0.5
0.4
Total
H2 O
0.3
O2
O3
0.2
NO2
aerosols
Rayleigh scattering
0.1
0
300
400
500
600
700
800
900
1000
wavelength (nm)
Figure 1.4: Simulated transmission through the atmosphere at a tangent height of 30
km for the spectral region of the MAESTRO instrument. The dotted line shows the
approximate cross-over point of the two spectrometers.
Chapter 1. Introduction
9
Several molecular species observed by MAESTRO have temperature-dependent absorption cross-sections. The Hartley-Huggins band of ozone (< 350 nm) shows strong
temperature dependency [6, 7]. The absorption cross-section of NO2 also shows temperature dependency (for example, see [8] and [9]). Recent experiments have quantified
the temperature dependence of the absorption cross-sections of BrO [10] and SO2 [11],
although their temperature dependencies are not yet included in the HITRAN spectroscopic database [12] which is used for many atmospheric radiative transfer applications,
including this work.
Temperature measurements are also highly desirable for studies of polar stratospheric
clouds (PSCs). PSCs are ice clouds that can form in winter when Antarctic and Arctic temperatures are very low, and affect the ozone distribution in two main ways [13].
Firstly, these clouds can act as catalytic sites for the activation of chlorine, which can
lead to ozone destruction. Secondly, PSCs sequester stratospheric HNO3 from the gaseous
state into particles, thereby affecting the nitrogen balance and therefore the chlorine balance of the stratosphere, leading to further ozone loss. There are two types of PSC.
Type I PSCs contain nitric acid and occur at temperatures less than 197 K, while Type
II PSCs contain primarily water and are much less common as they occur at temperatures
approaching the frost point of water in the lower stratosphere (185-191 K) [14]. Measurements of stratospheric temperatures are vital for understanding PSC observations,
both as a check for the PSC measurement itself and for use in any attempt to categorize
PSC types.
As will be discussed in detail in Section 2.3, refraction is of major importance in
calculations of radiative transfer in solar occultation geometry. As air becomes denser
lower in the atmosphere, its refractive index increases, causing a solar ray to bend and a
different ray path and tangent height to be measured than would be in vacuum. Measurements of density are necessary to determine an accurate index of refraction throughout
the measurement altitude range.
Chapter 1. Introduction
10
Rayleigh molecular scattering is directly dependent on atmospheric density and the
accurate modelling of the Rayleigh component of the spectrum is essential for proper
retrievals of aerosols and ozone. Rayleigh scattering is significant in the region of the O3
Chappuis band, and has a larger extinction contribution than O3 above the stratospheric
O3 peak. Aerosol extinction at most tangent heights is much weaker than Rayleigh
extinction, but its broad spectral feature may be difficult to distinguish from that of
Rayleigh scattering, and its accurate retrieval is highly dependent on a proper estimate
of Rayleigh scattering.
Often atmospheric profiles are expressed in terms of volume mixing ratio; however,
most observations are in fact a measurement of constituent number density. A measurement of the total number density of air at a given altitude is required to calculate the
fractional mixing ratio of a constituent.
Long-term stratospheric temperature trends, particularly over the poles, are also of
interest for understanding the chemistry and climate interactions which affect the ozone
distribution. Over the past several decades, a decrease in temperature in the polar
lower stratosphere has been observed (e.g., [15, 16, 17] and references therein), and is
attributed to several factors including changing ozone, greenhouse gas, and water vapour
concentrations, and solar effects [13]. While the proposed lifetime of the ACE mission
is only two years and so prevents the mission from observing trends on the scale of
decades, its retrieved temperature profiles could become part of a larger dataset of many
radiosonde and satellite observations for use in stratospheric temperature trend analysis.
In addition to these key motivations for on-orbit p-T retrievals, it is also interesting to
compare MAESTRO results with p-T profiles measured by the FTS instrument. This is
the first time an instrument measuring in the UV-visible-NIR and an instrument measuring in the infrared have been placed on the same platform to view the same beam of solar
radiation from space. In addition, the two instruments also employ different methods for
Chapter 1. Introduction
11
retrieving pressure and temperature profiles, allowing for a direct comparison of the FTS
technique (which uses CO2 ) and the MAESTRO technique discussed in this dissertation.
There are several major databases of global meteorological analyses available for determining the p-T profiles at any latitude and longitude on a given day, including those
of the National Centers for Environmental Prediction (NCEP), the United Kingdom Met
Office (MetO), the Canadian Meteorological Centre (CMC), and the European Centre
for Medium-range Weather Forecasts (ECMWF), which could be used for MAESTRO
processing instead of on-orbit retrievals. These meteorological analyses provide pressure
and temperature profiles derived from assimilation of observations and model forecasts.
While these analyses are quite reliable in the troposphere and where radiosonde observations are frequent, the temperature profiles can have biases of several degrees in the
stratosphere (see for example [18] and references cited in [19]) with significant implications for the polar stratospheric studies that are the focus of the ACE mission (see for
example [19, 20, 21]). The models also have a limited altitude range, with maximum
altitudes ranging from 10 hPa (about 30 km) to 0.4 hPa (about 55 km), depending on
the analysis, and poor vertical resolution in the stratosphere. It is hoped that on-orbit
MAESTRO p-T retrievals can be used for improving the quality of other MAESTRO data
products over those retrieved using meteorological analysis profiles, and ideally could also
be used for assessing the performance of these analyses in the stratosphere.
The pressure and temperature requirements for MAESTRO were determined by the
desired uncertainty contribution to the O3 and NO2 mixing ratio profiles. A 1% error
contribution in mixing ratio from the p-T analysis can be achieved with a 1% uncertainty in the density. This implies a requirement of roughly 1% in pressure or absolute
temperature. Ideally MAESTRO retrievals would be able to achieve a 1% uncertainty in
pressure and a 2-3 K uncertainty in temperature.
Chapter 1. Introduction
1.4
12
Pressure and Temperature Measurement
Methodology
Vertical profiles of pressure and temperature can be derived through the hydrostatic
equation and ideal gas law from measurements of the total atmospheric density at different altitudes. The measurement of total density can be performed from remote sounding
instruments by retrieving the density of a gas with a known mixing ratio in altitude.
Infrared sounders have traditionally used measurements of the well-mixed gas CO2 for
this purpose. The only well-mixed gas with features appearing in MAESTRO spectra
is molecular oxygen. O2 has a constant mixing ratio of 0.20947 [22] through most of
the atmosphere (up to a height of approximately 85 km [23]). There are three strong
absorption bands of O2 that appear in MAESTRO spectra in the red (γ and B bands)
and near-infrared (A band). Measurements of the strength of these absorption bands are
used in this work to derive profiles of oxygen number density in the atmosphere.
In remote sounding, the parameter observed is an indirect measurement of the desired state, and is often a complicated function of that state. In the case of MAESTRO,
the observed parameter is the radiation transmitted through the atmosphere as a function of wavelength in the spectral regions of the oxygen bands. The conversion of the
measured signal to a density profile requires the application of inverse theory. The inverse problem in atmospheric remote sounding consists of finding the most likely state
of the atmosphere from the remote measurements of radiation, given any available information on the relationship between the atmospheric state and the measured radiative
signal. The method developed here for retrieving p-T uses measurements of the differential absorption between an exoatmospheric solar reference and occultation spectra at
various tangent heights. Modelled optical depths (the negative natural logarithms of the
transmissions), produced using a forward model that describes the atmospheric radiative
transfer and instrument effects on the signal, are fit to measured optical depths using
13
Chapter 1. Introduction
a spectral fitting inversion algorithm to retrieve a profile of oxygen number density for
each occultation.
The retrieved partial density of oxygen as a function of altitude z is then used to
calculate the total air density ρ as a function of altitude using the known mixing ratio of
O2 . The pressure p is calculated using the hydrostatic equation
dp = −g(z)ρ(z)dz
(1.1)
where g is the acceleration of gravity altitude at z. The pressure and density at each
altitude can be used to calculate a temperature T for a gas of molecular weight Mr with
the ideal gas law
ρ(z) =
Mr p(z)
RT (z)
(1.2)
where R is the gas constant per mole. If MAESTRO were a high spectral resolution
instrument, it would be possible to make an independent retrieval of T based on the
relative absorption line strengths within a band. Although the direct temperature inversion is not possible due to the instrument’s low resolution, the temperature dependency
of the O2 measurement is still included in the later calculations and any temperature
information from the shape of the measured band does contribute to the final solution.
1.5
The A, B, and γ Bands of O2
The three strong bands of molecular oxygen in MAESTRO spectra result from the elec3 −
tronic transition b1 Σ+
g ←X Σg , with each band resulting from a different vibrational
transition (v ′←v ′′ ). They all appear on the VIS spectrometer: the strong A band centred at 762 nm (0←0), the slightly weaker B band at 690 nm (1←0), and the much
weaker γ band at 630 nm (2←0). Each band’s high resolution structure depends on
the rotational transitions within each vibrational transition. The O2 signal strength is
primarily determined by the amount of oxygen in the viewing path of the instrument,
Chapter 1. Introduction
14
although the absorption features of the spectral lines are also dependent on pressure and
temperature. The relative placements of the three bands and their surrounding absorbers
are visible in Figure 1.4 for a tangent height of 30 km.
The A band is the strongest O2 feature in the MAESTRO VIS spectra, and lies
in a region where other species absorb and scatter only weakly. These features make
it particularly attractive for remote sounding of pressure and temperature. However,
many absorption lines in the A band saturate completely in the long optical paths of
solar occultation geometry, causing a highly nonlinear absorption change as a function
of tangent height and necessitating very accurate absorption modelling.
The B band is also a prominent feature in MAESTRO spectra, although it lies in
a region more affected by ozone absorption, and slightly more affected by Rayleigh and
aerosol scattering than the A band. The B band experiences less saturation than the A
band at low tangent heights, although it is not detectable at very high tangent altitudes
given the instrument’s signal-to-noise ratio.
The γ band is the weakest O2 feature measurable by MAESTRO and is generally only
visible in spectra collected at the lowest tangent altitudes in the troposphere. In addition,
its placement in the strong Chappuis band of ozone absorption makes the determination
of a baseline for the band quite difficult. Preliminary assessment of MAESTRO spectra
from on-orbit shows that only a very faint γ band feature is visible at the very lowest
tangent altitudes, and the band is often difficult to distinguish from instrument noise
when Rayleigh scattering removes much of the signal at the lowest tangent heights. As
a result, this dissertation will focus only on the use of the A and B bands for pressure
and temperature remote sounding.
While the atmospheric O2 bands each have hundreds of individual rotational lines and
structure at high spectral resolution, only one broad feature per band is detected by the
low-resolution MAESTRO spectrometer. Figure 1.5 shows the transmission of radiation
through the atmosphere for the A, B and γ bands for several tangent heights, modelled
15
Chapter 1. Introduction
at high resolution and then smoothed to the MAESTRO VIS spectrometer resolution.
A transmission of one signifies total transmission through the atmospheric path, while a
transmission of zero signifies complete absorption. Saturation is visible in several lines
for all the spectra except the very weak γ band at 60 km. The contributions of the
relatively weak branches of the less abundant O2 isotopes become important in the A
and B band absorptions at the 10 and 30 km heights when the branches of the most
abundant isotope have saturated.
transmission
transmission
transmission
10 km
30 km
60 km
γ band
B band
A band
1
1
1
0.5
0.5
0.5
0
625
630
635
0
680
690
700
0
1
1
1
0.5
0.5
0.5
0
625
630
635
1
690
700
630
635
0
770
780
760
770
780
760
770
780
1
0.5
0.5
0
625
0
680
1
760
0.5
0
680
690
700
0
wavelength (nm)
Figure 1.5: Transmission through the atmosphere for the A, B, and γ bands of molecular
oxygen for tangent heights of 10, 30, and 60 km, at high spectral resolution (grey) and
the MAESTRO spectral resolution (black).
Chapter 1. Introduction
1.6
16
Remote Sounding of O2 from Space
Several other missions have exploited the constant vertical mixing ratio of O2 for remote sounding of the atmosphere. These include several limb-viewing instruments such
as MLS (Microwave Limb Sounder) [24], MAS (Millimeter Wave Atmospheric Sounder)
[25], and Odin-SMR (Sub-Millimetre Radiometer) [26], which use oxygen emission in the
microwave for retrievals of pressure and temperature profiles from limb-viewing geometry, as well as several missions that use the A band for surface and cloud-top pressure
retrievals.
Yamamoto and Wark [27] were the first to suggest using reflected sunlight in the A
band to measure air columns for determining cloud-top height. Their work was further
developed in several theoretical studies for both cloud-top height and pressure retrievals
and surface pressure retrievals (e.g., [28, 29, 30, 31]). Several recent satellite-based sensors, including the Global Ozone Monitoring Experiment (GOME) [32], the SCanning
Imaging Absorption SpectroMeter for Atmospheric CHartographY (SCIAMACHY) [33],
the POLarization and Directionality of the Earth’s Reflectances (POLDER) instrument
[34, 35], the Modular Optoelectronic Scanner (MOS) [36], and the MEdium Resolution Imaging Spectrometer MERIS (MERIS) [37] have carried sensors in the A band
for measurements of cloud-top height and surface pressure. The planned Orbiting Carbon Observatory (OCO) mission [38] will use the A band to determine total O2 column
measurements for deriving high-accuracy CO2 mixing ratios.
While the use of the A band for cloud-top remote sounding has been fairly extensive,
the use of O2 for temperature and pressure remote sounding has been less common. The
limb-scanning High Resolution Doppler Imager (HRDI) on the Upper Atmosphere Research Satellite (UARS) used the A band to derive mesospheric and lower thermospheric
temperatures by measuring the A band emission [39]. Dayglow emission has also been
used to measure temperatures in the mesosphere by sounding rockets [40, 41]. An early
solar absorption measurement by Matsuzaki et al. [42] provided rocket observations of
Chapter 1. Introduction
17
the rotational profile of the A band to derive atmospheric temperature at a tangent
altitude of 21 km.
The retrieval of pressure and temperature profiles from the A band has been attempted with two solar occultation instruments: ILAS and SAGE III. The Improved
Limb Atmospheric Spectrometer (ILAS) was launched on the Japanese Advanced Earth
Observing Satellite (ADEOS) in August 1996 and operated until June 1997. ILAS measured absorption spectra in the A band from 753 to 784 nm over 1024 individual pixels.
Pre-launch simulated measurements for ILAS showed that temperature could theoretically be retrieved to within 2 K below 40 km [43]. The tangent height of an ILAS
measurement is retrieved using a technique that combines independent information from
the on-board sun-edge sensor with information from the A band when a forecast p-T
profile is assumed [44]. These tangent heights are then used in an onion-peeling retrieval
algorithm for p-T retrievals. The ILAS retrievals from actual occultation data have
stated uncertainties below 40 km of 4 K and 4% in temperature and pressure, respectively [45]. Comparisons with HALogen Occultation Experiment (HALOE) coincident
occultations showed differences ranging from ±5 K to ±10 K in the southern hemisphere
and a significant cooling bias for ILAS measurements of up to 20 K in the northern
hemisphere. The most significant error source for the ILAS retrievals is considered to be
wavelength position uncertainty in the application of the instrument line shape function
to the forward model [45]. Temperature retrievals have also been attempted with the
follow-on instrument, ILAS-II. These temperature profiles show biases ranging from 10
to 35 K when compared with HALOE and SABER (Sounding of the Atmosphere using
Broadband Emission Radiometry) [46].
The Stratospheric Aerosol and Gas Experiment (SAGE) III on the Meteor-3M satellite
uses the O2 A band for on-orbit p-T retrievals. SAGE III measures in a similar spectral
region to MAESTRO, from 290 to 1030 nm, and at a similar low spectral resolution of 1
to 2 nm. In addition, both SAGE III and MAESTRO spectra have a signal-to-noise ratio
Chapter 1. Introduction
18
of approximately 3000. SAGE III has 14 CCD channels spaced approximately every 1 nm
in the A band region from 759 to 771 nm for making spectral measurements with a 1.4 nm
resolution. The feasibility of using a low-resolution spectrometer for A band p-T retrievals
was originally demonstrated using simulated measurements by the SAGE III team [47,
48, 49]. In these simulations, the error sources from measurement noise, aerosol clearing,
background ozone cross-sectional uncertainties, O2 spectroscopic database uncertainties,
and their Emissivity Growth Approximation forward model for radiative transfer [50]
were combined to produce expected temperature errors of 2 K and pressure errors of
2%, from 0 to 85 km. The challenges of the retrieval are demonstrated by Pitts and
Thomason [51] where one of the preliminary SAGE III temperature retrievals differs by
as much as 15 K from a nearby profile measured by the HALOE (Halogen Occultation
Experiment) instrument on the UARS satellite. The SAGE III team has found that
systematic biases in the measurements (particularly from an attenuator plate etalon
problem in their instrument) can induce serious errors in the retrievals, and that spectral
transmissions must be extremely well-known for reliable retrievals (M.C. Pitts, personal
communication, 2005). However, they consider their current temperature retrievals to be
producing reasonable results below 60-70 km, although pressure retrievals are still biased
slightly (M.C. Pitts, personal communication, 2006).
Neither ILAS nor SAGE III have sensors able to measure in the region of the O2
B band. In fact, few studies have used the B band for space-based remote sounding,
although Kuze and Chance [31] discussed its potential for cloud-top height retrievals and
Daniel et al. [52] showed that including the B band and the weak O2 -O2 feature at 477
nm could decrease uncertainties in A band cloud parameter retrievals by more than 50%,
depending on cloud properties. HRDI has also used the Doppler shift in the rotational
lines of the B and γ bands to determine winds in the stratosphere [53].
In addition to its use of O2 for cloud-top estimation when making nadir measurements, the SCIAMACHY instrument also uses the O2 B band and CO2 1550-1590 nm
Chapter 1. Introduction
19
band for tangent height corrections in its solar occultation measurement mode [54, 55].
SCIAMACHY is a spectrometer on the European Envisat satellite, making measurements
from 240 to 2380 nm at spectral resolutions between 0.26 and 1.48 nm, depending on the
spectral channel. O2 A and B band measurements are made by Channel 4 (604-805 nm)
at a spectral resolution of 0.48 nm. An a priori climatology profile is used to calculate
an O2 reference density profile and the tangent heights are adjusted to fit the O2 B band.
This allows corrections to the tangent altitudes provided by satellite orbital parameters
down to the troposphere [55]. The B band radiative transfer model for SCIAMACHY
uses a line-by-line model [54]. (The line-by-line method will be discussed in detail in
Chapter 2.)
1.7
Structure of the Thesis
The retrieval of pressure and temperature from MAESTRO remote sounding measurements requires a forward model of instrument characterization, solar ray optical geometry,
and radiative transfer through the atmosphere in the region of the A and B bands, a
retrieval algorithm for inverting remote spectral measurements into retrieved profiles,
and data for testing the application of the developed algorithms. In this thesis I will
discuss all aspects of the development of the pressure and temperature retrievals for the
MAESTRO instrument, from modelling and retrieval to data analysis.
This thesis is divided into six chapters. Chapter 1 has provided an overview of the
MAESTRO instrument, the motivations for p-T measurements, the basic methodology
for p-T retrievals, and a discussion of previous work using O2 for remote sounding from
space. Chapter 2 discusses the development of the radiative transfer and geometry forward model, as well as the details of instrument characterization functions used in this
analysis. Chapter 3 provides details on the global fitting retrieval algorithm developed
for MAESTRO p-T retrievals, which can also be used universally for any other species
Chapter 1. Introduction
20
retrieval from the instrument. The retrieval algorithm is also implemented in several
case study simulations to examine the reliability of the algorithms for retrieving known
solutions. Chapter 4 explores the error sources for retrieval uncertainties and provides a
formal retrieval characterization. Chapter 5 presents retrievals using on-orbit data from
the satellite instrument. The algorithms and results are summarized and evaluated in
Chapter 6.
1.8
Personal Contribution
A large satellite project like ACE involves contributions from a number of people. The
research described in this thesis describes some of my personal contribution to the ACEMAESTRO project. I developed, from scratch, the complete forward model and retrieval
algorithms described in this thesis, often with suggestions from Tom McElroy of the
Meteorological Service of Canada (MSC), and have implemented these algorithms to
retrieve the first p-T profiles from the MAESTRO satellite data. Several ACE team
members provided the data used as input for the MAESTRO p-T retrievals. The a priori
data was provided by Ray Nassar at the University of Waterloo, the FTS pointing and
profiles were provided by Chris Boone at the University of Waterloo, and the MAESTRO
spectra were provided by Tom McElroy at MSC.
Chapter 2
MAESTRO Forward Model
2.1
Introduction
An accurate forward model of the atmospheric radiative transfer, viewing geometry, and
instrument characteristics of a remote sounding measurement is necessary for deriving
the desired atmospheric state parameters. The MAESTRO forward model is used to
produce modelled spectral observations, given knowledge of the state of the atmosphere
and instrument. In the MAESTRO p-T retrieval algorithm, the forward model is iterated
with adjustments to the atmospheric pressure and temperature profile until a minimum
difference is found between the modelled and observed optical depth spectra for one occultation. This chapter will discuss the development of this forward model, including the
instrument characteristics that must be considered, how the path and refraction of a solar
ray through the atmosphere are modelled, and the radiative transfer model for modelling
the absorption of molecular oxygen and other species in the O2 A and B band spectral ranges. The development of an atmospheric radiative forward model often involves
making compromises between accuracy and computation time. The approximations that
have been developed for a computationally efficient model will also be discussed in some
depth.
21
Chapter 2. MAESTRO Forward Model
2.2
22
Instrument Characterization
Instrument characterization is an essential part of making an accurate measurement. A
radiative transfer model is used to model the high-resolution absorption of solar radiation
between the Sun and satellite; however, the instrument characteristics will determine
how this radiation is expressed on a detector. Ideally, the spectra would be fitted in
‘instrument space’ where the forward model predicts raw detector counts. However, in
practise, spectra are often fitted in ‘calibrated spectral space’ where the observed raw
detector counts have been first corrected for most instrumental effects. The eventual
goal of MAESTRO data analysis is to apply the instrument model as part of the forward
model and retrieve against the observed raw detector counts. However, this is an ongoing
activity and the analysis presented here assumes observed counts are corrected for most
instrumental effects. Nevertheless, there are several instrument characteristics which are
not used to correct raw spectra, but are actually critical components of the forward
model.
2.2.1
Pixel-Wavelength Registration
The photodiode array on which the incoming solar radiation is dispersed is composed
of a row of 1024 individual detector pixels, each about 0.1 inches high and 0.001 inches
wide. While the radiative transfer may be modelled at any resolution, it must eventually
be transfered onto the measurement’s pixel grid. Pre-flight mercury lamp laboratory
tests and on-orbit solar Fraunhofer tests are used to match calibrated reference lines
with known wavelengths to the detector pixels on which they appear. This allows a
wavelength, λ, to be assigned to each pixel using a polynomial such as
λp = ap3 + bp2 + cp + d
(2.1)
where p is the pixel number (from 0 to 1023). The VIS spectrometer coefficients measured during a pre-flight characterization lamp test were: a = −2.1524 × 10−8 , b =
Chapter 2. MAESTRO Forward Model
23
−8.6224 × 10−6, c = 0.52626, and d = 514.23 nm, representing a nearly linear pixelwavelength dispersion relationship. This pixel-wavelength registration is used in the
forward model calculations for the transformation from wavelength to pixel space, although pixels will still be expressed in terms of a wavelength axis. It is important to
note that when dealing with real data collected on-orbit, the structure of the instrument changes with temperature and for each occultation the pre-flight pixel-wavelength
registration is adjusted using solar Fraunhofer features as a wavelength reference. The
on-orbit wavelength registration is discussed in detail in Section 5.4.1.
2.2.2
Slit Function
Two other instrumental effects that must be considered in the forward model are the
spectral resolution and shape of the slit function. The spectral resolution is a measure
of the ability of the spectrometer to resolve two nearby wavelengths. The slit function
of MAESTRO is determined by the application of a monochromatic source (a laser) to
the entrance aperture of the instrument, and the spectral resolution is defined as the
full-width-at-half-maximum (FWHM) of the response. Figure 2.1 shows the slit function
for the MAESTRO VIS spectrometer near the O2 A and B bands. The shape of this
theoretical slit function was calculated by Denis Dufour [56] by fitting Gaussian curves to
each side of a slit function measured using an infrared diode laser with a wavelength near
830 nm during pre-flight characterization at the University of Toronto. The slit function’s
shape and resulting spectral resolution are determined by several factors including the
resolving power of the grating, the dimensions and locations of the entrance slit and
photodiode array, and aberration and magnification of the image [57]. The asymmetric
spread in the image at shorter wavelengths seen in Figure 2.1 is caused by the offaxis placements of the slit, diffraction grating, and detector array. The low-resolution
MAESTRO modelled observations are produced by convolving the slit function and the
24
Chapter 2. MAESTRO Forward Model
high-resolution spectra from the forward model, and therefore proper characterization of
the slit function is necessary for an accurate retrieval.
1
0.9
normalized intensity units
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
relative wavelength (nm)
Figure 2.1: MAESTRO IR slit function, fitted from data measured with the infrared
diode laser during pre-flight characterization.
2.2.3
Field-of-View
The instrument field-of-view (FOV) must also be simulated in the forward geometry
model. MAESTRO’s FOV is determined by the size and shape of the input slit and is
the main factor that determines the vertical resolution of the measurements. Pre-flight
laboratory tests used the satellite’s suntracker to scan a laser source and found the VIS
slit was fairly rectangular, with dimensions of 0.037◦ in elevation and 1.075◦ in azimuth,
defined at the full-width-at-half-maximum [58]. The azimuthal width of the slit is twice
as wide as the solar disk, and is not considered in the calculation; however, the elevation
height of the slit is a small fraction of the solar disk, and covers 1 to 2 km of atmosphere
25
Chapter 2. MAESTRO Forward Model
as viewed from SciSat. The slit’s half-maximum dimensions as compared to the solar disk
are shown in Figure 2.2. This FOV elevation range measured in the pre-flight testing
is a necessary component of the geometry model. The FOV shape used in this work is
shown in Figure 2.3. It is the average of several laser scans over the slit’s height in the
elevation direction, taken at different places along the width of the slit. The negative
angles represent the direction of the slit toward the centre of the Earth, while the positive
angles are on the side of the slit pointed toward space.
0.037o
1.075o
Figure 2.2: MAESTRO slit and solar disk.
1
weighting normalized to peak
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
degrees relative to centre of slit
Figure 2.3: MAESTRO VIS field-of-view weightings in elevation.
Chapter 2. MAESTRO Forward Model
26
ztangent
Sun
SCISAT
Figure 2.4: Path of a refracted ray through a three-layer atmosphere.
2.3
Geometry and Ray-Tracing Model
The MAESTRO solar beam trajectory model consists of a spherical shell model, where a
pressure, temperature, and density are assigned to each shell. As a ray travels through the
atmosphere, it passes through a number of these shells, each with a different path length
contribution to the ray. In addition to providing a method of calculating individual path
lengths for all altitudes through which the ray travels, the spherical shell model provides
the means of calculating atmospheric refraction as a ray passes from one shell to another
with a different index of refraction. Figure 2.4 shows a ray-trace of a typical ray through
a three-layer spherical shell atmosphere, with refraction.
2.3.1
Atmospheric Grid Model
There are two grids applied in this spherical shell model. The fine ‘optical grid’ defines
the shells for the optical radiative transfer calculations of air density and refraction, and
has a typical grid spacing on the order of tens or hundreds of metres, while the coarser
27
Chapter 2. MAESTRO Forward Model
‘retrieval grid’ is used for the retrieval of constituents and has a grid spacing corresponding
roughly to the vertical resolution of MAESTRO. As the vertical resolution of MAESTRO
is approximately 1.5 to 2 km, the grid spacing in this model is set to 1 km. While it is
possible to retrieve on the fine optical grid, this would be quite computationally timeconsuming and would not add information if the optical grid’s spacing is much less than
the instrument’s vertical resolution. Retrieval on a grid with spacing much greater than
the instrument’s vertical resolution would be computationally faster, but the resulting
profiles would be oversmoothed as finer features in the profile would not be detectable.
A schematic of the retrieval and optical grid is shown in Figure 2.5. The altitude and
pressure values correspond to the grid point, while the temperature is the average value
between two retrieval grid points, and the density is a weighted average value determined
by calculating the air column mass between two grid points from their corresponding
pressures.
zrM, prM
zoN, roN
zr4, pr4
Tr3, rr3
zr3, pr3
Tr2, rr2
zr2, pr2
Tr1, rr1
zr1, pr1
zo5, ro5
zo4, ro4
zo3, ro3
zo2, ro2
zo1, ro1
Figure 2.5: Atmospheric model grid with M retrieval levels and N optical levels.
The atmospheric model is forced to obey both the hydrostatic balance equation and
the ideal gas law. The first-guess atmosphere is determined using a first-guess tempera-
28
Chapter 2. MAESTRO Forward Model
ture profile that is combined with the ideal gas law to calculate a pressure at each grid
point, which in turn is used to calculate an average density in the layer. The reverse
process is applied after a retrieval iteration to calculate pressure and temperature from
the retrieved density profile. In order to calculate the pressures from densities, the firstguess pressure at the top of the atmosphere (assumed to be 100 km) is fixed as a reference
pressure. An inaccurate reference pressure induces errors in the very highest altitudes
(above about 80 km). However, the retrieval is not expected to produce useful results at
these altitudes due to non-constant O2 mixing ratios and little O2 signal above 85 km.
2.3.2
Refraction and Ray-Tracing
Atmospheric refraction plays a significant role in occultation measurements and refraction
must be included when calculating the ray path. Without refraction, the path length of
a ray through an optical shell is determined by simple geometry; however, the addition
of refraction causes the calculation to become significantly more complex. The index of
refraction of air n in each shell is first calculated using the empirical equation developed
by Edlén [59]
"
#
29498.1
255.4
n(λ) = 1 + a 64.328 +
× 10−6
1 +
146 − λ2
41 − λ12
(2.2)
where λ is the wavelength of light in µm and a is a scaling factor that depends on pressure
and temperature with
a=
p T0
p0 T
(2.3)
and where p0 = 1013 hPa and T0 = 288.15 K. The refractive path lengths are calculated
using the principle of refractive invariance (e.g., see Born and Wolf [60]) in a spherically
symmetric atmosphere:
nz z sin φz = constant,
(2.4)
where nz is the index of refraction at radius z and φz is the zenith angle between the
vertical vector and the ray vector at this radius.
29
Chapter 2. MAESTRO Forward Model
Figure 2.6 shows the relationship between various tangent heights and angles in an
occultation measurement. The true tangent height zt is defined at the minimum ray
height, while there also exists an apparent tangent height za defined by the satellite
pointing angle α and a geometric tangent height zg defined by the true solar position at
solar zenith angle θ. In a non-refracting atmosphere these tangents heights are all equal.
The MAESTRO geometry model is designed to propagate multiple rays through the
atmosphere for each measurement to account for FOV effects. In the MAESTRO model,
rays are propagated through the atmosphere at several solar zenith angles centred at
the satellite pointing angle, α, and weights are assigned according to the rays’ positions
within the FOV. The path lengths are calculated for every optical shell travelled by each
ray as it turns through the atmosphere, and stored for later transmission calculations.
apparent Sun
apparent path
q
true refracted path
zt
geometric path
true Sun
re
zg
za
a
S
rs
Figure 2.6: Ray path and turning angles through a refracting atmosphere (after Vervack
et al. [61]).
Chapter 2. MAESTRO Forward Model
2.4
2.4.1
30
The Radiative Transfer Model
Radiative Transfer in the Atmosphere
There are three ways the atmosphere can interact with radiation: emission, scattering,
and absorption. Figure 2.7 shows the change in intensity for radiation passing through
a path of length L. Following the notation of Liou [62], the change in intensity after
passing through a segment dl can be written as
I(λ) − J(λ) =
−dI(λ)
σ(λ)n(l)dl
(2.5)
where I(λ) is the radiance, J(λ) is a source term representing emission and scattering
into the beam, n(l) is the number density of the material (in units of molecules per unit
volume), and σ(λ) is the cross-section (in units of area per molecule) of the material.
The cross-section is a function, almost always laboratory-determined, that characterizes
an atmospheric constituent’s wavelength-dependent absorption, emission, or scattering
features. In the case of solar occultation measurements, the radiative transfer equations
are greatly simplified using the assumption that the incident radiation and absorption is
so intense that any source terms are negligible (including molecular Rayleigh scattering
into the beam, which is negligible for the small field-of-view of the instrument as compared
to the direct radiation). In the absence of source terms, the equation can now be reduced
to
−dI(λ)
.
σ(λ)n(l)dl
I(λ) =
(2.6)
For radiation travelling along a path L, this equation becomes
IL (λ) = I0 (λ)e−τ (λ)
(2.7)
where
τ (λ) =
Z
0
L
σ(λ)n(l)dl,
(2.8)
with the possibility that σ(λ) may be dependent on the state (i.e., temperature or pressure) of a segment of atmosphere dl. It is this optical depth τ (λ) that will be fitted in
31
Chapter 2. MAESTRO Forward Model
the retrieval, when the optical depth is calculated using the forward model and compared
to that calculated from the observations. As long as I0 is also measured in addition to
I, the absolute intensity calibration of the instrument is then not required as τ , or the
transmission I(λ)/I0 (λ), can be fitted directly.
I(l)
I(l)+ dI(l)
I0(l)
IL(l)
0
L
dl
Figure 2.7: Incident and outgoing radiation through a path of length L (after Liou [62]).
The optical depth can also be written as
τ (λ) = ln [I0 (λ)] − ln [I(λ)],
(2.9)
and in terms of signal S on an instrument’s detector (assuming infinite spectral resolution), as
τ (λ) = ln [S0 (λ)] − ln [S(λ)],
(2.10)
provided that the instrument’s responsivity is constant between the reference and occultation spectra. This expression is used to determine the optical depth from the observed
spectra using the measured exoatmospheric reference spectrum S0 (λ) and the occultation
spectra S(λ). It should also be noted that the optical depth τ (λ) is in fact the sum of
the optical depths of all constituents. For example, it could contain information from N
atmospheric gases, aerosols, and molecular (Rayleigh) scattering which is a prominent
Chapter 2. MAESTRO Forward Model
32
feature of UV-visible spectra, in which case it would be expressed as
τ (λ) = τRayleigh (λ) + τaerosols (λ) + τgas1 (λ) + τgas2 (λ) + . . . + τgasN (λ).
(2.11)
While the molecular and Rayleigh terms have predictable wavelength-dependent behaviour as their cross-sections are known, the size distribution of aerosols in the optical path
is likely unknown, although aerosol scattering tends to appear as a broadband feature in
the spectra. The aerosol optical depth spectrum is modelled in the MAESTRO model
as a series of terms representing a second-order polynomial.
2.4.2
Exoatmospheric Model
The forward model requires an exoatmospheric model spectrum at a higher resolution
than the measurements for two reasons. First, a solar reference of known wavelengths
is required for fitting the measured solar reference spectrum from MAESTRO to solar
Fraunhofer features in the reference to improve the pixel-wavelength registration for each
occultation. Second, the forward model also requires a value for the modelled intensity
incident on the atmosphere, I0 (λ). As described in Section 2.4.1, this is not strictly
necessary when optical depth is fitted in the retrieval. However, as the slit function is
more correctly applied to the high-resolution I and I0 (and not to τ ), the best forward
model must accurately account for any gradients in the solar baseline I0 which could
affect at what point a line will saturate relative to other lines in the band.
The solar reference spectrum used for this work is the composite spectrum constructed
from data collected during the ATLAS 3 mission on the space shuttle [63]. Figure 2.8
shows the reference solar structure in the spectral regions of the O2 A and B bands at
the modelled spectrum resolution of 0.5 nm and also smoothed to the MAESTRO resolution by convolution with the VIS spectrometer slit function. The ATLAS 3 spectrum
is a composite spectrum derived using a high-resolution model spectrum [64] in combination with spectra collected from the 1-nm resolution SOLSPEC (SOLar SPECtrum)
33
Chapter 2. MAESTRO Forward Model
instrument between 400 and 870 nm, and the 20-nm resolution SOSP (SOlar SPectrum)
instrument at wavelengths greater than 870 nm [65].
1600
1550
high resolution
MAESTRO resolution
irradiance (mW/m2 /nm)
1500
1450
1400
1350
1300
1250
1200
1150
1100
680
700
720
740
760
780
800
wavelength (nm)
Figure 2.8: Exoatmospheric solar reference spectrum in the region of O2 A and B bands
at 0.5 nm spectral resolution, and smoothed to the MAESTRO 2 nm spectral resolution.
2.4.3
Application to MAESTRO
The major components of the forward model have been discussed, and it is necessary to
further develop an expression for the forward calculation that accounts for the physical
effects that determine the signal measured at the detector. The generalized radiative
transfer equations will now be applied to the MAESTRO forward model.
Assume the intensity of light at the instrument has arrived through the atmosphere
from a particular direction with solar zenith angle θ, azimuth angle φ, and wavelength
λ, and is expressed as I(λ, θ, φ). The total flux at the detector can be expressed as
f=
Z Z Z Z
θ
φ λ
a
I(λ, θ, φ)dadλdφdθ
(2.12)
Chapter 2. MAESTRO Forward Model
34
where da is an increment of the entrance area of the slit. If a particular pixel p has
a responsivity of R(λp ), representing the relationship of the measured detector counts
to the incoming flux, and a slit function defined by F (λ − λp ), and the FOV weights a
particular ray measurement corresponding to a particular time centred about a reference
angle (θt , φt ) by W (θ − θt , φ − φt ), then the signal S on a pixel p from a measurement t
can be expressed as
S(λp , θt , φt ) = R(λp )
Z Z Z Z
θ
φ λ
a
I(λ, θ, φ)W (θ − θt , φ − φt )F (λ − λp )dadλdφdθ.
(2.13)
As the MAESTRO slit views the entire solar disk in the azimuthal direction, the azimuthal dependence is a function of θ only and depends on the solar disk’s horizontal
extent at a given θ. Thus W (θ − θt , φ − φt ) can be replaced by the weighting G(θ − θt )
and the integral over φ can be removed. In addition, the integral over a can be removed
and replaced by a function A which is the same for a observed high-sun spectrum S0
and an occultation spectrum S, as long as the slit is illuminated the same way for both
exoatmospheric and occultation reference measurements. This simplifies the equation for
the detector signal to
S(λp , θt ) = R(λp )A
Z Z
θ
λ
I(λ, θ)G(θ − θt )F (λ − λp )dλdθ.
(2.14)
Finally this equation can be written in terms of the exoatmospheric spectrum I0 and optical depth τ , and discretized over the Nr model rays lying within the FOV and expressed
as
S(λp , θt ) = R(λp )A
Nr
X
G(θr − θt )
r=1
Z
λ
I0 (λ, θr )e−τ (λ,θr ) F (λ − λp )dλ.
(2.15)
The total optical depth τ (λ, θr ) is the sum of the optical depths of Nc altitude-dependent
constituent absorbers and Nb spectrum background absorbers. Each of the Nc molecular
absorbers has altitude-dependent absorption which is calculated as a function of the
cross-section σch , number density nch , and path length Lh for each optical cell h, along
Nh individual shells. This allows the total optical depth for one ray to be expressed as
τ (λ, θr ) =
Nh
Nc X
X
c=1 h=1
σch (λ)nch Lhr ,
(2.16)
Chapter 2. MAESTRO Forward Model
35
Aerosol scattering is modelled in a similar manner using effective ‘cross-sections’ σb
which represent the spectral characteristics of the aerosols’ extinction. The product of
the number density and path length (nL) in Equation 2.16 is modelled using an effective
‘slant column’ Xb which is representative of the total slant column absorber amount per
unit area. Although the aerosol extinction is altitude-dependent, for this forward model
where aerosol is merely an interfering species in a narrow spectral window, the scattering
is assumed to be uncorrelated between spectra and each spectrum’s effective slant column
can be adjusted independently.
2.4.4
O2 Modelling
The O2 A, B, and γ bands have well-defined structures and, unlike several other atmospheric constituents in the UV-Visible-NIR regions of the spectrum, are not broad
absorbers in spectral space. During a typical occultation measurement, many absorption
lines saturate in the O2 bands (particularly in the strong A band), even at very high
tangent altitudes. This saturated absorption for certain wavelengths for many possible
ray paths introduces a highly nonlinear problem: how to model the band over its entirety (required to compare to the low MAESTRO spectral resolution), but still keep
computation time reasonable? If the lines did not saturate, a smoothed cross-section
spectrum could be used. Initial attempts at the MAESTRO forward model used this
method unsuccessfully. The nonlinear O2 bands require a full high-resolution calculation
of the absorption cross-sections, a high-resolution calculation of transmission, convolution with the slit function to instrument resolution, and conversion to apparent optical
depth on the pixel grid. The following sections discuss the cross-section database for O2 ,
the line-by-line high-resolution calculations, and the fast line-by-line and correlated-k approximations implemented to improve computation speed for the MAESTRO O2 forward
model.
Chapter 2. MAESTRO Forward Model
36
O2 Spectroscopic Database
Every molecule has characteristic spectral absorption features. These spectral features
occur where the molecule absorbs photons whose energies match the differences between
the molecule’s quantized energy states. The O2 bands measured by MAESTRO in the
3 −
red and near-infrared result from the electronic transition b1 Σ+
g ←X Σg between the
ground electronic state and the second excited state. Each individual band results from a
transition between vibrational levels of these two electronic states, and the finest structure
internal to each band is because these transitions occur between the very closely-spaced
rotational levels. Their absorption structure is also strongly dependent on the pressure
and temperature of the absorbing O2 .
The standard reference for molecular line parameters of atmospheric importance is
HITRAN (HIgh-resolution TRANsmission molecular absorption database) [66] which
contains the line position ν0 , strength S, air-broadened half-width γair , self-broadened
half-width γself , and temperature dependence of the width n for over one million spectral
lines of molecules present in the atmosphere. The work in this thesis is performed using
the HITRAN 2004 database [66]. The spectral ranges of the A, B, and γ bands and
their vibrational transitions as noted in the HITRAN 2004 edition are listed in Table
2.1. Three isotopes of O2 have been measured in these bands and recorded in HITRAN:
16
O2 , 16 O18 O, and 16 O17 O. While the 16 O2 isotope is by far the most abundant, the other
isotopes’ lines tend not to saturate except at very low tangent heights in occultation and
so their absorption gradients are often more measurable with the long path lengths of
solar occultation geometry.
The HITRAN parameters are based on a combination of theory and laboratory measurements. The calculations used to produce the O2 database parameters are summarized in Gamache et al. [67]. The most recent calculations use A band experimental
measurements from Brown and Plymate [68] and Camy-Peyret et al. [69], and B band
measurements from Giver et al. [70].
37
Chapter 2. MAESTRO Forward Model
Table 2.1:
Spectral ranges and vibrational transitions of O2 bands measured by
MAESTRO
Spectral Range by Isotope
16
O2
16
O18 O
v ′ ←v ′′
16
O17 O
A band
759.58-778.38 nm
759.60-770.68 nm
759.58-770.96 nm
0←0
B band
686.91-699.23 nm
688.72-696.03 nm
687.92-691.94 nm
1←0
γ band
627.86-636.14 nm
630.87-633.32 nm
2←0
Line-by-Line Modelling
Although absorption lines are defined by the frequencies between quantized energy levels, these lines are broadened by several processes: natural (lifetime) broadening (relatively small in atmospheric cases), collisional broadening caused by interaction with other
molecules, and Doppler broadening caused by the difference in relative motion between
a molecule and a photon. Line-by-line (LBL) radiative transfer modelling involves the
calculation of high-resolution absorption cross-section lineshapes for particular pressures
and temperatures from given frequency, strength, and half-width parameters. A detailed
discussion of lineshape calculations can be found in Liou [62], Goody and Yung [71], and
other textbooks on radiative transfer. The MAESTRO forward model uses a Matlab
toolbox developed at the University of Toronto for LBL cross-section calculations [72].
Figure 2.9 shows sample O2 A and B band cross-sections calculated by the LBL for a
typical pressure-temperature combination at a 30 km altitude.
Line frequency parameters are usually presented in units of wavenumber (cm−1 ) which
suits discussions of energy transitions. However, the standard in UV-visible remote sensing is to use units of wavelength and will be followed in this work. Early calculations in
the line-by-line model use wavenumbers, but switch to units of wavelength after crosssections are calculated.
38
Chapter 2. MAESTRO Forward Model
A band
-50
-50
-55
-55
log(σ)
log(σ)
B band
-60
-60
-65
-65
-70
-70
685
690
695
wavelength (nm)
700
760
765
770
775
780
wavelength (nm)
Figure 2.9: Sample O2 cross-sections in the A and B bands calculated using an LBL code
for a typical shell at 30 km altitude where p = 13 hPa and T = 225 K.
Chapter 2. MAESTRO Forward Model
39
The ideal forward model would perform LBL calculations for every temperature and
pressure in a model; however, LBL calculations are very time-consuming with current
computer power so most radiative transfer models use approximations when calculations
must be made over a broad spectral range and for p-T combinations over an extended
altitude range. Some common approximations include band models (e.g., [73, 74, 75]), the
emissivity growth approximation ([50, 76]), which is used in the SAGE III O2 retrievals,
and the correlated-k distribution method ([77, 78]). The following two sections discuss
the two approximations that have been applied in the MAESTRO forward model. As
will be discussed, LBL calculations are still required as input to the approximations.
Fast-Line-by-Line Modelling
At each iteration of the retrieval a new set of cross-sections must be calculated for the
current guess of pressure and temperature profile combinations on the altitude grid.
A pure LBL calculation can be very time-consuming when performed for many p-T
combinations, so a fast-line-by-line (FLBL) algorithm has been implemented using the
two-dimensional interpolation method described by Turner [79].
The application of the FLBL method involves pre-computing a cross-section table
for a range of p-T combinations on a pre-defined wavenumber grid. This can be quite
computationally intensive, but need only be done once and can be used for any O2
retrieval. The cross-sections are stored as log(σ) as a function of log(p/p0 ) and T /T0 for
each wavenumber. The interpolation procedure converts an input p and T to log(p/p0 )
and T /T0 and uses a two-dimensional interpolation to compute the new log(σ). Both
available memory and accuracy requirements determine the number of p-T pairs to use
in the pre-computed table. These requirements will be explored further in Section 2.5.
The pressure limits of the tables are set at 0.001 and 1060 hPa, and the temperature
limits are set at 180 and 320 K. On the rare occasion that a pressure or temperature
exceeds these limits, the cross-section is calculated at the table limit.
40
Chapter 2. MAESTRO Forward Model
Correlated-k Modelling
While the FLBL method could be used for MAESTRO retrievals, an even faster method
is developed here using a correlated-k technique in combination with the FLBL. The
correlated-k distribution method for radiative transfer is implemented in the MAESTRO
forward model to reduce the time required for slit function convolution with highresolution spectra. The k refers to the absorption coefficient of a molecule. This work
follows the convention of UV-visible-NIR remote sounding by using the absorption crosssection σ (units of cm2 /molecule) rather than the absorption coefficient, but the term
correlated-k will continue to be used.
The correlated-k method is based on the fact that the spectral transmittance for a
given spectral interval is independent of the order of σ’s in that interval, which allows the
replacement of an integration over high-resolution wavelength space with an integration
over σ space. The original development of correlated-k for inhomogeneous paths can be
found in Lacis and Oinas [77] and Goody et al. [78].
To illustrate the concept of correlated-k, consider a slant column integrated along a
R
path of number density n, u = n(z)dz. The transmission over a spectral interval λmin
to λmax can be written as
Tλ̄ (u) =
1
λmax − λmin
Z
λmax
λmin
e−σλ u dλ =
Z
0
∞
e−σu f (σ)dσ
(2.17)
where f (σ) is the normalized probability distribution function. The cumulative distribution function is then defined as
g(σ) =
Z
σ
0
f (σ)dσ,
(2.18)
where g(0) = 0, g(σ → ∞) = 1, and dg(σ) = f (σ)dσ, which allows the transmission to
be expressed as
Tλ̄ (u) =
Z
1
0
for M discrete spectral subintervals.
e−σ(g)u dg ∼
=
M
X
j=1
e−σ(gj )u ∆gj
(2.19)
41
Chapter 2. MAESTRO Forward Model
In actual practise the calculation of the cumulative probability distribution function g
is computationally quite simple; a set of σ values with equal λ spacing is simply sorted in
increasing order and assigned to a g grid between 0 and 1 with equal spacing. Figure 2.10
shows cross-sections in the 2 nm window centred at 771 nm, for two typical atmospheric
-56
-56
-58
-58
-60
-60
-62
-62
log(σ)
log(σ)
pressure-temperature combinations, and their cumulative distribution functions.
-64
-64
-66
-66
-68
-68
-70
-70
-72
770
770.5
771
771.5
wavelength (nm)
772
-72
0
0.5
1
g
Figure 2.10: a) O2 A band cross-sections between 770 and 772 nm for a typical atmosphere
at 0 km (P = 1013.25 hPa, T = 288.15 K) (black) and 30 km (P = 11.97 hPa, T = 226.51
K) (grey); b) their cumulative distribution functions digitized over M = 50 points.
The application of this theory to an inhomogeneous path requires the basic assumption that for a given spectral interval the k-distributions (or σ-distributions in our case)
are correlated in wavelength space. The validity of this assumption in the limits of weak
and strong lines is demonstrated by Lacis and Oinas [77] and Goody et al. [78]. With
this assumption, Equation 2.19 can be generalized to calculate the transmission for this
42
Chapter 2. MAESTRO Forward Model
spectral interval over all Nh paths and expressed as
Tλ̄ ∼
=
M
X
j=1

exp −
Nh
X
h=1

σh (gj )uh  ∆gj .
(2.20)
However, the original Equation 2.15 for the MAESTRO forward model is more complex, and contains terms for the low-resolution slit function shape and the medium resolution exoatmospheric spectrum within an integral over wavelength space. Stam et al.
[80] developed an extended correlated-k method that considered the instrument spectral
response function and the solar function to simulate GOME (The Global Ozone Monitoring Experiment) polarization measurements in the O2 A band. The method is applied
here to the occultation case.
Equation 2.15 gives the general expression for signal on a MAESTRO pixel. In
order to simplify the expression for demonstration of the extended correlated-k method,
consider only the signal S ′ on a pixel p that contains the convolution integral in λ space
over the spectral interval from λmin to λmax , but not the instrument responsivity or
summation over multiple rays:
′
S (λp ) =
λmax
Z
λmin
I0 (λ)e−τ (λ) F (λ − λp )dλ.
(2.21)
If the slit function F (λ − λp ) is normalized, and a new wavelength scale is defined as
λ′ = λ − λmin so that
Z
λmax
λmin
F (λ − λp )dλ =
Z
0
1
F (λ′ − λ′p )dλ′ = 1,
(2.22)
′
(2.23)
the signal can now be written as
S ′ (λp ) =
Z
0
1
I0 (λ′ )e−τ (λ ) F (λ′ − λ′p )dλ′.
In order to express S ′ in terms of correlated-k, F and I0 must be removed from the
integral. This is done by substituting the variable x(λ′ ) for λ′ using
dx =
1
I0 (λ′ )F (λ′ − λ′p )dλ′
Cp
(2.24)
43
Chapter 2. MAESTRO Forward Model
and
x(λ′ ) =
1
Cp
Z
λ′
0
I0 (y)F (y − yp )dy.
(2.25)
The constant Cp is calculated by requiring x(1) = 1 and is defined by
Cp =
Z
0
1
I0 (y)F (y − yp )dy.
(2.26)
After this substitution, Equation 2.23 becomes
′
S (λp ) = Cp
Z
1
0
e−τ (x) dx.
(2.27)
According to the correlated-k distribution method, the signal S ′ can now be represented
as
S ′ (λp ) = Cp
M
X
e−τ (gj ) ∆gj .
(2.28)
j=1
For N multiple absorbers [77] this generalizes to
S ′ (λp ) = Cp
M1 X
M2
X
···
j1 =1 j2 =1
M
N
X
e−τ (gj1 ) e−τ (gj2 ) · · ·e−τ (gjN ) ∆gj1 ∆gj2 · · ·∆gjN .
(2.29)
jN =1
The general Equation 2.15 for S in terms of correlated-k for one altitude-dependent
absorber is now expressed as
S(λp , θt ) = R(λp )ACp
Nr
X
r=1
G(θr − θt )
M
X
j=1

exp −
Nh
X
h=1

σh (gj )nh Lhr  ∆gj
(2.30)
and can also be generalized to include multiple absorbers as in Equation 2.29.
The MAESTRO algorithm that applies these equations can be summarized as follows:
1. Calculate series of log(σ) from the FLBL for every p-T combination on a coarse
altitude grid (similar resolution to the retrieval grid, otherwise there are too many
cross-sections to hold in memory) in wavenumbers, then interpolate using cubic
interpolation to a grid in equally-spaced wavelength space.
2. Interpolate the slit function to the same grid spacing as log(σ), then normalize and
apply to an equally spaced x-axis between 0 and 1 as in Equation 2.22.
For each pixel p:
Chapter 2. MAESTRO Forward Model
44
3. Centre the slit function over pixel p’s corresponding wavelength to determine range
of wavelengths that have an effect on the signal on pixel p.
4. Normalize the log(σ) wavelength x-axis λ in this range to lie on an evenly-space
grid λ′ between 0 and 1.
5. Calculate the constant of integration Cp using Equation 2.26 by convolving the
exoatmospheric spectrum in this range with the slit function and taking the sum
of all values.
6. Calculate x by calculating the cumulative sum of the convolved exoatmospheric
spectrum and slit function using Equation 2.25 and substitute one-for-one for the
λ′ grid.
7. Change x to evenly-spaced grid and reinterpolate log(σ) onto this grid.
8. Calculate g by sorting σ and keeping on the same x-axis.
9. Save smooth function σ(g) over M points determined using Gaussian-quadrature
weights.
In order to make the forward model even faster, the correlated-k coefficients are calculated for several gridded pressure-temperature combinations to form a four-dimensional
table for a set of given pressures, temperatures, model wavelengths, and Gaussianquadrature weights. The correlated-k coefficients for a given atmospheric level, at each
wavelength and Gaussian-quadrature weight, are then interpolated using cubic interpolation for the pressure and temperature at that level, as in the FLBL table interpolation.
The table-interpolated correlated-k approximation can provide significant savings in computation time. While an FLBL provides a fast method of calculating cross-sections, the
correlated-k method eliminates the need for high-resolution spectral convolutions with
the slit function as the slit function information is now inherent in the correlated-k constant of integration C and stretch of the cross-section wavelength axes. A typical A
Chapter 2. MAESTRO Forward Model
45
band forward model calculation performed in Matlab for 60 spectra, and their differential matrices required for the retrieval, takes about 20 minutes using the correlated-k
table approximation on a laptop with an Intel Centrino 1.5 GHz processor and 1 Gbyte of
RAM. A similar high spectral resolution calculation using the FLBL takes approximately
3.5 hours.
Modelling Other Constituents in the O2 Bands
Ozone absorption, water vapour, Rayleigh scattering, and aerosol scattering also contribute to the transmittance of radiation in the A and B bands. The wavelengthdependent extinction of these constituents must also be calculated in the forward model.
O3 absorption in the visible and near-infrared is calculated in this model using
Chappuis-Wulf band cross-sectional data collected by the SCanning Imaging Absorption spectroMeter for Atmospheric CHartographY (SCIAMACHY) pre-flight model from
Bogumil et al. [11], who measured O3 cross-sections at five temperatures between 203 and
293 K. Although the Chappuis band cross-sections are largely temperature-independent
in the vicinity of the band’s absorption peak near 600 nm, the combined Chappuis and
Wulf-band cross-sections in the regions of the O2 bands do have some temperature dependency. The Bogumil et al. measurements show as large as a 10% difference between
cross-sections measured at different temperatures. When required, the temperaturedependent O3 cross-sections can be calculated using a quadratic polynomial interpolation
as expressed by Paur and Bass [7] and recommended by Orphal [81].
Water vapour absorption can also interfere in the background spectrum of the O2
bands, but only in a very moist troposphere, and primarily in the long-wavelength wing
of the B band. Like O2 , H2 O has high spectral resolution absorption. When required, the
correlated-k method is also applied to H2 O absorption in this forward model, with M =
100 correlated-k cumulative distribution function divisions. The H2 O spectral database
used in this work is from HITRAN 2004, where H2 O lines intensities and positions are
Chapter 2. MAESTRO Forward Model
46
based on work by Mérienne et al. [82] and Coheur et al. [83], and line assignments are
from Tolchenov et al. [84].
Rayleigh and aerosol scattering also produce background extinction in the A and B
bands. Rayleigh scattering is modelled using an effective cross-section proportional to
the Rayleigh approximate 1/λ4 dependence, as described in Liou [62], and is dependent
on the atmospheric density profile. Aerosol scattering is modelled using constant and
linear effective cross-section offset terms. When simulated aerosol measurements are
produced in Chapters 2 and 3, they are calculated using aerosol extinction values from
the stratospheric background aerosol model implemented in the MODTRAN model [5].
2.4.5
Forward Model Output
The result of these forward model calculations is a set of optical depths for a given
atmospheric temperature and pressure profile. The optical depths are functions of the
instrument characteristics, ray-tracing calculations, O2 model, and other background
absorbers and scatterers.
Figure 2.11 shows sample modelled optical depth spectra at instrument resolution
with a data point at the centre of each pixel, for three sample tangent heights. The optical depth contributions from O2 , O3 , aerosols, and Rayleigh scattering are also shown.
These optical depths were modelled using the fast correlated-k table method. The errors
induced in the forward model outputs from the correlated-k method and other approximations will be examined in the following section.
2.5
Approximation Errors and Forward Model
Resolution Requirements
An assessment of the accuracy of the FLBL and correlated-k approximations and an
analysis of the forward model parameter resolutions are required. The forward model
47
Chapter 2. MAESTRO Forward Model
A band
B band
0.03
optical depth
60 km
0.03
O2
0.01
optical depth
30 km
0
680
optical depth
685
690
695
700
0.02
0.01
0
750
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
680
10 km
O3
aerosol
Rayleigh
total
0.02
685
690
695
700
0
750
4
4
3
3
2
2
1
1
0
680
685
690
695
wavelength (nm)
700
0
750
760
770
780
760
770
780
760
770
780
wavelength (nm)
Figure 2.11: Modelled optical depths for tangent altitudes of 10, 30, and 60 km, showing
contributions from individual species to total optical depth.
Chapter 2. MAESTRO Forward Model
48
parameters include those used for the LBL generation of cross-sections, the creation of
FLBL tables for cross-section interpolation, the application of the correlated-k distribution method, and the setup of ray-tracing geometry calculations. The choice of the
resolution of these forward model parameters is necessarily a compromise between computational time and accuracy. It may be possible to increase the accuracy of the model
by increasing the resolution of certain parameters with no or only slight increases in computational time; however, increasing the resolution of other parameters may have serious
consequences for computational efficiency.
Table 2.2 summarizes the various forward model input parameters for which resolutions must be chosen, whether or not increasing each parameter’s resolution affects the
computation speed, and the final parameter values chosen for the forward model. While
a thorough analysis of the choice of a parameter’s resolution should involve an ensemble
test of the resolution’s effect on the retrieval results, in practise this is often very computationally time-consuming. This initial examination of the forward model parameters will
involve simulations and direct comparison of spectral errors, rather than errors induced
in the retrieved product. Some of the parameter resolution analyses in this section will
involve a more qualitative discussion on the compromises made between computational
efficiency and accuracy where appropriate, while others will be studied using simulated
transmission calculations. All simulated observations are performed using the 1976 U.S.
Standard Atmosphere [85] pressure and temperature profiles for the atmospheric model.
The spectra are forward modelled at high spectral resolution, then convolved with the
slit function to produce low-resolution simulated MAESTRO spectra. The analysis presented in this section is expressed in terms of the transmission’s absolute percent error.
Although the optical depth τ is used for fitting observations and modelled spectra in
the MAESTRO retrievals, in these cases it is more intuitive to consider percent errors in
transmission (e−τ ), as regions of little O2 absorption will show reasonable percent errors
when they are considered to have a transmission of one rather than an optical depth of
49
Chapter 2. MAESTRO Forward Model
zero. The following sections examine how the parameter resolutions were selected, and
also determine which parameters need to be examined further in the later chapters on
retrievals.
Table 2.2: Forward model resolutions
Category
Parameter
LBL
Wavenumber resolution
FLBL
Correlated-k
Geometry
2.5.1
Loss of Speed?
Chosen Value
some
0.005 cm−1
Wing cut-off
no
300 cm−1
Interpolation tolerance
no
0.01%
Np pressures in table
some
20
Nt temperatures in table
some
10
M cdf divisions
yes
100
Np pressures in table
yes
30
Nt temperatures in table
yes
10
optical grid spacing
yes
100 m
Nr rays in FOV
yes
5
Line-by-Line
The line-by-line code must be run to create cross-sections for the construction of the
FLBL tables. In addition to the wavenumber range, HITRAN database version, and
molecule name, the resolution of three parameters must be supplied to the Matlab LBL
code used for the cross-section calculations. These are the wavenumber resolution for the
calculation, the interpolation error tolerance which is required by this particular code for
Chapter 2. MAESTRO Forward Model
50
its calculations, and a wing cut-off parameter that defines where to end the calculation of
a line’s contributions from its centre. As the LBL cross-sections are calculated once and
saved in FLBL tables that are reused for every retrieval, they have little computational
cost. The interpolation accuracy and wing cut-off values only affect the total time of
FLBL table generation, but not the time for table interpolation. As such, their resolutions
can be set arbitrarily high. The lowest interpolation error tolerance accepted by the LBL
software is 0.01%, and this value is used in the model. The default wing cut-off value
for the LBL program is 20 cm−1 ; however, increasing the wing cut-off produces spectra
inconsistent with those produced using the the default value, and a 300 cm−1 cut-off is
chosen here to allow the wing effects of each line to be calculated across the O2 bands,
which are roughly 300 cm−1 wide.
Unlike the interpolation error tolerance and the wing cut-off, the wavenumber spectral resolution does have some effect on table size and therefore interpolation time and
memory requirements. The wavenumber resolution must be chosen based on the ability
to resolve individual O2 lines. To resolve a line, at least four to five points are required.
The typical Doppler half-widths at half-maximum for the A and B bands are on the
order of α = 0.011 cm−1 for 180 K and α = 0.015 cm−1 for 320 K. A spectral resolution
of ∆ν = 0.005 cm−1 has been chosen as it will resolve these lines and also produce tables
that are small enough to be loaded in Matlab on a computer with 1 Gbyte of RAM.
A comparison of transmission spectra computed with ∆ν = 0.005 cm−1 and a higher
resolution reference computed with ∆ν = 0.001 cm−1 showed a maximum percent error
in transmission over all wavelengths and tangent altitudes of 0.003% for both the A and
B bands relative to the ∆ν = 0.001 cm−1 spectra.
2.5.2
Fast-Line-by-Line
The number of pressures and temperatures used in the construction of the FLBL tables
will affect the accuracy of the FLBL method. Although the size of the table will have
Chapter 2. MAESTRO Forward Model
51
some effect on computation time, the selection of these values is primarily limited by the
available memory for the storage of one high spectral resolution table for fast look-up. An
analysis was performed using different numbers of pressures and temperatures to create
the FLBL tables. Fidelity of the interpolations is not much improved by increasing
the number of temperatures (Nt ) used to create the tables; however, increasing the
number of pressures (Np ) used in the table production does improve the accuracy of
table-interpolated cross-sections when compared to LBL output. Figure 2.12 shows the
expected absolute percent errors in transmission from spectra calculated using FLBLinterpolated cross-sections with Nt = 10, as compared to an LBL. Figures 2.12a and 2.12b
show the maximum absolute percent errors over all wavelengths for Np = 20, for each
tangent height in the simulation. Figure 2.12c shows these errors averaged over tangent
altitudes for several possible values of Np . Np = 20 was chosen for table construction as
improvements in accuracy are minimal with further increases in Np . These figures also
serve to illustrate the relative uncertainties in transmission spectra between the FLBL
and the LBL.
2.5.3
Correlated-k
The correlated-k modelling requires three major parameter choices: the number of divisions for accurate integration of the cumulative distribution function M, and the number
of pressures, Np , and temperatures, Nt , to include in the tables for correlated-k cumulative distribution function interpolation (similar to FLBL table construction). The
rapidly changing high-resolution O2 cross-sections require several divisions to accurately
represent the cumulative distribution function’s shape. However, increasing the value of
M comes at a significant computational cost. Figures 2.13a and 2.13b show the maximum absolute transmission percent error for all wavelengths in the band, at each tangent
height for a value of M = 100, where the spectra have been modelled with the correlatedk method using the FLBL to provide input cross-sections. M = 100 was chosen for
52
Chapter 2. MAESTRO Forward Model
A band
mean transmission percent error
100
tangent altitude (km)
tangent altitude (km)
B band
(a)
80
60
40
20
0
0
100
(b)
80
60
40
20
0
0.005 0.01 0.015 0.02
0
0.005
0.01
transmission percent error
transmission percent error
100
(c)
A band
B band
10−2
10−4
5
10
15
20
25
number of pressures
Figure 2.12: a, b) Absolute percent errors in transmission for Np = 20 in FLBL table
relative to LBL reference; c) Average over all tangent heights for maximum transmission
errors relative to LBL, for several values of Np .
53
Chapter 2. MAESTRO Forward Model
correlated-k integration as increasing M above this value provides little improvement
in the correlated-k accuracy, as is shown in Figure 2.13c where the mean error over all
tangent altitudes is plotted for several values of M. As these figures indicate, the error
induced by the correlated-k approximation is much more significant than that from the
LBL wavenumber resolution choice or FLBL interpolation. Most of the error is caused by
the assumption of cumulative distribution function correlation between cross-sections for
all p-T combinations, which is not necessarily true. The effect of this error contribution
from the correlated-k approximation on p-T retrieval results will be assessed in Chapter
4.
mean transmission percent error
100
A band
tangent altitude (km)
tangent altitude (km)
B band
(a)
80
60
40
20
0
0.2
0
0.4
0.6
100
60
40
20
0.8
transmission percent error
102
(b)
80
0
0
0.5
1.5
2
transmission percent error
(c)
A band
B band
100
10−2
1
0
20
40
60
80
100
120
140
160
180
200
M divisions for cdf integration
Figure 2.13: a, b) Absolute percent error in transmission for M = 100 cumulative distribution function (cdf) divisions in correlated-k approximations relative to LBL reference;
c) Average over all tangent heights for maximum transmission errors relative to LBL, for
several values of M.
54
Chapter 2. MAESTRO Forward Model
Tests on the number of temperatures Nt required for accurate correlated-k table
interpolation showed that Nt = 10 produced very small transmission errors. On the
other hand, Np = 30 gridded pressures were required in the table to produce reliable
interpolated cumulative distribution functions.
Figure 2.14 presents a comparison of the wavelength-dependent transmission percent
errors induced by the FLBL, direct correlated-k, and interpolation table correlated-k
approximations as compared to the LBL base case, for a sample 30 km tangent altitude.
The errors induced by the correlated-k are much more significant than those induced
by the FLBL. Interestingly, the use of the table correlated-k method can occasionally
increase errors in the spectra, but more often lowers the errors produced by the direct
correlated-k method. This is due to a reduction in the contributions of some badly
transmission error, (approximation - LBL)/LBL × 100%
correlated layers to the solution when interpolation is applied.
A band
B band
0.02
0.12
0.01
0.1
0
0.08
-0.01
0.06
-0.02
0.04
-0.03
0.02
-0.04
0
-0.05
-0.02
-0.06
FLBL
correlated-k
correlated-k table
-0.07
-0.08
680
685
690
695
wavelength (nm)
700
FLBL
correlated-k
correlated-k table
-0.04
-0.06
750
760
770
780
wavelength (nm)
Figure 2.14: Transmission errors for the FLBL, direct correlated-k, and table correlated-k
approximations as compared to the LBL base case, for a tangent height of 30 km.
Chapter 2. MAESTRO Forward Model
2.5.4
55
Geometry
The choice of an optical grid spacing for the forward model will have consequences for
atmospheric refraction and optical mass calculations. Increasing the optical grid is very
costly on both computer time and memory. The optical grid spacing of 100 metres
is selected based on simulations of occultation ray traces which were compared to those
done at a 1 metre resolution optical grid. These ray traces found less than a 1 metre error
in estimated tangent height and less than a 0.2% error in total air slant column amount
along a ray’s path for a tangent height of 5 km. In addition, increasing the optical grid
to a resolution higher than 100 metres becomes prohibitively computationally expensive.
Five rays are used to fill the instrument’s FOV for ray propagation. When compared
with a simulation run using 20 rays per measurement, the resulting percent difference
in transmission in both bands is less than 0.01% above 20 km where refraction is less
significant, and always less than 0.05% over all altitudes. Increasing the number of
rays used in the propagation adds significant time to the computations. The use of five
rays can achieve good accuracy while keeping calculation time within the scope of this
research.
2.6
Discussion
This chapter has developed the mathematics for the implementation of the forward model
and discussed the instrument characteristics that must be included in the model, the construction of the geometry and refractive ray tracing program, and atmospheric radiative
transfer as applied to the O2 bands. A fast-line-by-line model has been developed for
O2 forward modelling which can be used in pressure-temperature retrievals. The speed
of the retrieval can be improved significantly by applying a modified correlated-k technique with interpolation tables using cross-section input from the fast-line-by-line. The
fast-line-by-line introduces minimal errors in simulated spectra, while the errors from the
Chapter 2. MAESTRO Forward Model
56
correlated-k approximation are more considerable, although the maximum transmission
errors across the A and B bands are still generally smaller than 0.5%. The errors in
the spectra caused by the correlated-k method are by far the most significant of the
forward modelling errors, and the effect of this approximation on the retrieved profiles
of pressure and temperature must be examined further in Chapter 4. In addition to the
forward model approximation errors discussed in this chapter, there are also uncertainties
in spectroscopic cross-section parameters. A discussion of these forward model errors will
be given in Chapter 4.
Chapter 3
Profile Retrievals
3.1
Introduction
The measurement of pressure and temperature profiles from the MAESTRO instrument
requires a retrieval algorithm to invert from detector signal to profile amounts. This
algorithm involves the minimization of a cost function which is representative of the
squared difference between the observed and forward-modelled signals. The current standard for limb sounding retrievals is the global fit method, as developed by Carlotti [86],
which allows for a direct inversion from spectral measurements to constituent amounts
by simultaneously fitting all spectral channels at all tangent heights for one sequence of
limb measurements. This chapter discusses the general aspects of atmospheric profile
retrievals, the generalized constrained linearized inversion chosen for this problem, the
application of a global fit retrieval to MAESTRO data, the calculation of the weighting
functions required for the retrievals, and the choice of retrieval method parameters for
determining retrieval convergence and applying prior knowledge constraints on the atmospheric state. The retrieval algorithm will also be tested by performing retrievals for
several atmospheric cases using simulated observations.
57
Chapter 3. Profile Retrievals
3.2
58
Inversion of Atmospheric Profiles
In atmospheric remote sounding, the retrieval algorithm is used to derive the desired
atmospheric state from an indirect measurement of that state. Following the notation
of Rodgers [87], consider a measurement vector y with m elements y1 , y2, ..., ym , and
with measurement errors ǫ1 , ǫ2 , ..., ǫm . The parameter for which one wants to solve is
usually a continuous function (i.e., a temperature profile, trace gas concentration with
altitude, or a map of ozone distribution over the globe). However, the continuous solution
function must be discretized into a state vector x of a length n deemed suitable for the
problem, with elements x1 , x2 , ..., xn . The forward function f relates the state vector x
and the forward function parameters b (for example, the radiative transfer parameters
and instrument characteristics described in Chapter 2) to the measurement vector y, and
contains the true physics of the observation. The forward model F(x) is the function used
in the retrieval and is similar to the forward function, but contains any approximations
to the true physics. The measurement, forward function, and forward model are related
by
y = f(x, b) + ǫ ≈ F(x, b) + ǫ.
(3.1)
For conciseness, the retrieval algorithm mathematics in this chapter are derived using
the notation common in the literature, where the forward model is assumed to be precise
enough to express the relationship as
y = F(x) + ǫ,
(3.2)
where the model parameters are inherent in the forward model. The forward model
parameter and forward function approximations will be explored further in Chapter 4.
While Equations 3.1 and 3.2 express the transfer function from state to measurement,
the desired solution is the state vector. This is the inverse problem. The best estimated
solution of the true state x is the retrieved state x̂. Its relationship to the measurement
Chapter 3. Profile Retrievals
59
and model can be expressed as
x̂ = R(y, b̂, xa , c)
(3.3)
where R is the transfer function from measurement to state, b̂ is the estimate of the
forward function parameters (as opposed to the true parameters b) and includes all the
known physics about the measurement, xa contains the prior knowledge of the state
vector, and c contains the retrieval method parameters, which are anything that may
affect the retrieved state but are not part of the forward model, such as convergence
criteria.
There are several types of inverse methods commonly used to derive an atmospheric
state x from remote sounding measurements y. An in-depth discussion of these methods can be found in Rodgers [87, 88]. An appropriate method must be chosen for the
MAESTRO retrievals. The inverse problem is best solved by the minimization of a cost
function that represents the squared difference between the forward model calculated using the estimated solution, and the observation weighted by its estimated measurement
errors.
As previously discussed, the dependence of O2 signal on p-T is a nonlinear problem. In
atmospheric remote sounding, the nonlinear inverse problem is most often solved using
a nonlinear iterative estimator, where the problem is linearized about a current guess
of the state vector x and solved iteratively until the cost function converges (to some
pre-defined convergence criteria). This can be performed on an individual spectrum-byspectrum basis with an independent fit for each spectrum, or as part of a global fitting
algorithm where all spectra are fitted simultaneously. If performed on an individual
spectrum basis, a slant column amount of a trace gas is retrieved for each spectrum
and a separate algorithm is required to transform the observations to a profile. Onion
peeling is a technique often used in retrieving limb-sounding profiles. The onion peeling
method involves retrieving layer-by-layer from the highest tangent altitude and using the
retrieved layers above in the calculation of those below. This has the disadvantage that
Chapter 3. Profile Retrievals
60
the error from each layer propagates to those below, and any prior information below a
layer cannot be used in the retrieval of that layer. The Chahine relaxation method [89]
is occasionally applied to nonlinear retrievals instead of a global fit method, but formal
error analysis is problematic for this method [88].
The current MAESTRO retrieval method for O3 and NO2 profiles developed at the
Meteorological Service of Canada involves a two-step process where the total slant column
amount along the path of each observation is retrieved using a nonlinear iterative retrieval,
and the slant columns are converted to profiles using Chahine’s method. The retrieval
of a total slant column for each tangent height is unsuitable for O2 retrievals as the
O2 cross-sections are much more highly dependent on pressure and temperature than
those of O3 and NO2 and require altitude-dependent calculations. In addition, each solar
ray’s path depends on the density profile, and therefore on the retrieved p-T profile. In
order to allow the optimal retrieval of p-T profiles, while permitting a formal retrieval
characterization and error analysis, global spectral fitting using a nonlinear iterative
estimator is applied for the MAESTRO p-T retrievals. The theoretical development of
the retrieval is described in Section 3.3, while its application to the MAESTRO data is
explained in Section 3.4.
3.3
Constrained Linearized χ2 Minimization Method
The retrieval method developed here uses an iterative linearized χ2 minimization to
minimize the squared difference between the observation y, with covariance matrix Sǫ
(derived from the error vector ǫ, and whose calculation for MAESTRO will be described
in Section 3.4), and a model F(x) as expressed by
χ2 m = [y − F(x)]T Sǫ −1 [y − F(x)].
(3.4)
Chapter 3. Profile Retrievals
61
The nonlinear problem is made linear at each iteration i by linearizing the problem about
the current guess xi using
y − F(xi ) =
∂F(xi )
(x − xi ) + ǫ = K(x − xi ) + ǫ.
∂x
(3.5)
where K is referred to as the weighting function and contains elements Kij = ∂Fi (x)/∂xj .
Supplementary information about the atmospheric state can be used in the retrieval in
addition to the information provided by the measurements. The two additional constraint methods applied here are optimal estimation, which can be used to constrain the
solution vector to a priori knowledge of the state (for example, a forecast or external
measurements) within its known statistical uncertainties, and a smoothing constraint
that mathematically forces a degree of smoothness on the solution.
Optimal estimation (e.g. [87, 88]) is often applied in contemporary remote sensing
inverse problems. This method allows the incorporation of prior knowledge of the state
vector x into the solution. The a priori state vector xa contains the best estimate of the
state vector before the measurements are considered. Its uncertainties are contained in
the covariance matrix Sa . Sa must be well-characterized if a priori information is to be
included, as the degree to which the solution is constrained to the a priori is determined
by the relative magnitudes of the elements of Sa and Sǫ . When a priori information is
applied, the retrieval algorithm attempts to minimize the expression
χ2 a = [x − xa ]T Sa −1 [x − xa ]
(3.6)
simultaneously with other constraints.
Retrievals with measurement noise can suffer from oscillations in vertical profiles. In
order to suppress these oscillations, a smoothing constraint can be applied. The retrievals
discussed here use a common type of smoothing constraint originally developed simultaneously by Twomey [90] and Tikhonov [91]. This smoothing constraint is a secondderivative type where the displacement of a retrieved point xv is minimized relative to its
two neighbours using the term (xv−1 − 2xv + xv+1 ). Other smoothing constraints could
62
Chapter 3. Profile Retrievals
involve such constraints as those derived from a priori climatological constraints, or first
order-smoothing that constrains the profile’s slope, or combinations of various smoothing
constraints. The second-order derivative smoothing was chosen to constrain curvature of
the profile, and can be used effectively in the case where the a priori is not well-known
for constraining the values directly.
This displacement is represented by the expression
χ2 s =
where L is the matrix

L=











1 −2
X
1
(Lx)2
0
(3.7)
···
..
.
0 1 −2 1
.. . . . . . . . .
.
.
.
.
.
0 ···
0
1
0
..
.
0
−2 1






,





(3.8)
if x is defined on an evenly-spaced grid. This form of the matrix is applied only to those
retrieved values of x which are altitude-dependent and so have layers correlated by the
smoothing requirement. While elimination of these oscillations is somewhat artificial, in
that the result of including the smoothing term is the reduction of vertical resolution
by effectively averaging over several layers, this suppression of unphysical densities is
necessary so that refraction calculations for the next iteration remain stable.
The cost function J, which includes the squared difference between the model and
measurements, the a priori knowledge, and the departure of a retrieved point from its
neighbours, is expressed at iteration i + 1 as
Ji+1 = χ2 m + χ2 a + γi+1 χ2 s
= [y − F(xi )]T Sǫ −1 [y − F(xi )] + [xi+1 − xa ]T Sa −1 [xi+1 − xa ] + γi+1
(Lxi+1 )2
(3.9)
P
where γi+1 is a scalar that weights the relative contribution of the smoothing term.
To minimize J, Equation 3.9 is differentiated with respect to the retrieved parameter
vector x and linearized about the current guess xi using Equation 3.5, and the resulting
Chapter 3. Profile Retrievals
63
expression ∂J/∂x is set to zero. Subsequently solving for x produces
xi+1 = xi + (KTi Sǫ −1 Ki + Sa −1 + γH)−1 [KTi Sǫ −1 (y − F(xi )) − Sa −1 (xi − xa ) − γi+1 Hxi ],
(3.10)
where partial differentiation of the smoothing term results in the matrix H = LT L. This
expression is iterated successively until the cost function has converged to a specified
convergence criterion between iterations. The error covariance matrix of the final optimal
solution x̂ is
Ŝ = (KT Sǫ −1 K + Sa −1 + γH)−1 .
(3.11)
The following sections will explore the implementation of this constrained linear retrieval to the MAESTRO data, including the construction of the a priori state parameters
xa and Sa , the choice of the first-guess atmospheric state x0 at which to begin the iterative retrieval, the construction of the measured and modelled observation vectors and
covariances for the global fit retrieval, the calculation of the weighting function matrix,
K, the choice of the smoothing parameter, γ, and the criteria for achieving retrieval
convergence.
3.4
Global Fit Applied to MAESTRO Data
Section 3.3 discussed a generalized nonlinear inverse method with a priori and smoothing
constraints. This section will examine the practicalities of implementing this retrieval
method for deriving p-T from the MAESTRO measurements.
A global fit retrieval involves simultaneous fitting of all spectra from a single occultation to determine the most likely state parameters. In the MAESTRO global fit retrieval,
a series of Nt measured optical depth spectra, each with Np pixels, are assembled into a
single measurement vector y with length Nt × Np . Similarly, a corresponding series of
modelled optical depth spectra are assembled into a modelled vector F(x) with length
Nt × Np . The A and B bands can be fitted separately or in combination. If fitted in
Chapter 3. Profile Retrievals
64
combination, the vectors y and F(x) will each have dimensions Nt × (NpA + NpB ). The
vectors y and F(x) are now effectively treated as a single spectrum. The error covariance
matrix Sǫ , constructed to apply to this single vector, is a square matrix with Nt × Np
rows and Nt × Np columns. Measurement error is assumed to be uncorrelated between
pixels. Therefore, Sǫ consists of the square of the optical depth measurement errors ǫ on
its diagonal, and zeros elsewhere.
The retrieval algorithm developed here is general enough that it can be applied to
either the UV or VIS spectrometer for any constituent or spectral range. However, these
p-T retrievals focus only on the narrow O2 band regions for improved computational
efficiency. The retrievals are performed for the A band from 752.6 nm to 780.3 nm
and the B band from 681.9 nm to 699.3 nm. Table 3.1 lists the parameters that are
simultaneously fitted in a MAESTRO O2 A or B band retrieval. The altitude-dependent
number density constituents each have an element at each retrieval grid interval in the
x solution vector. The offset and wavelength parameters are weighting factors that are
retrieved on a per spectrum basis, and are represented by one element of x for each
spectrum. In practise, it is convenient to retrieve the natural logarithm of the number
densities, ln n, and a scaled value of the other parameters to ensure that the retrieved
parameters are within just a few orders of magnitude of each other to avoid unstable
matrix inversions. In addition, the number density of air changes exponentially with
altitude and therefore ln n is a preferable space for any interpolation over altitude.
Although the main feature in these spectral regions is O2 , the other parameters in
Table 3.1 must also be considered. If the altitude profile of O3 and its absolute crosssections are well-known, the profile can be fixed and used to model O3 background
absorption in the O2 bands. SAGE III uses this method to clear O3 from the O2 A
band region. However, in the MAESTRO retrievals, O3 slant column amounts are preretrieved for individual spectra using the strong Chappuis band between 550 and 680
nm with cross-sections from Bogumil et al. [11] collected at 243 K, instead of retrieving
65
Chapter 3. Profile Retrievals
O3 profiles. This approach ensures the total amount of O3 in the path for a particular
observation remains constant, so that the O3 background absorption is independent of
any changes in the ray’s path resulting from a different retrieved p-T profile.
Although simulations in this chapter are for a dry atmosphere, water vapour absorption becomes significant, particular in the B band, under very moist tropospheric
conditions. H2 O features can be modelled using the correlated-k approximation and fitted during the retrieval to remove its background spectral effects. Rayleigh scattering is
dependent on density and is therefore inherently tied to the p-T profile. It is remodelled
at each iteration according to the new density profile. The aerosol background effect is retrieved simultaneously with O2 by constant and linear offset terms across each band. The
λshif t is a wavelength shift adjusted at each iteration to optimize the pixel-wavelength
assignments of the observed spectra.
Table 3.1: Parameters modelled and retrieved in A and B band spectra
Parameter
Altitude-Dependent?
High Spectral
Retrieved?
Resolution?
ln nO2
Yes
Yes
Yes
O3 slant column
No
No
Pre-retrieved
Rayleigh scattering
Yes
No
No
Aerosols (constant offset)
No
No
Yes
Aerosols (linear offset)
No
No
Yes
λshif t
No
No
Yes
ln nH2 O (case-specific)
Yes
Yes
Yes
Chapter 3. Profile Retrievals
3.5
66
A Priori and First-Guess Profiles
The a priori atmospheric states used in the MAESTRO p-T retrievals are density profiles
derived from atmospheric profiles assembled at the Science Operations Centre at the
University of Waterloo for each ACE occultation for the sub-tangent location of the
occultation measurement with a 30 km tangent altitude as described by Boone et al.
[92] and Nassar and Bernath [93]. Values from the surface to 10 hPa (approximately
30 km) are based on data from the Global Environmental Multiscale (GEM) model
analyses of the Canadian Meteorological Centre (CMC) [94]. These analyses are produced
on 28 vertical layers and are interpolated to the ACE 30 km sub-tangent location in
space and time. The CMC analyses used in this a priori are produced 12 to 18 hours
after the reference time using a data assimilation scheme that incorporates a model and
observations including those from a radiosonde network and information from operational
nadir temperature sounders. The a priori profile from just below the stratopause to the
top of the atmosphere is determined using the US Naval Research Laboratory, Mass
Spectrometer Incoherent Scatter Radar Extended Model (NRL-MSISE-00) [95]. The
middle stratosphere is not covered by either model. The temperature profile in this
altitude region is determined by linear interpolation of the temperature as a function of
ln p. These a priori temperature profiles are used in the MAESTRO code to determine
pressure and density profiles for a hydrostatically-consistent atmosphere that obeys the
ideal gas law.
As the uncertainties in the ACE a priori profiles have not been quantified above 30 km
(in this region the ACE profiles are merely interpolated, and in any case the atmospheric
variability is large), the a priori covariance matrix Sa is difficult to determine for p-T
retrievals. As a result, the inverse method developed for MAESTRO uses a modified
optimal estimation technique where the solution is only weighted to the a priori profile
in the lower atmosphere. The elements of Sa that represent uncertainties in retrieval
layers below the lowest tangent height are set to very small values, while those above
Chapter 3. Profile Retrievals
67
are set to artificially large values so that the solution at higher altitudes is independent
of the a priori, while that below the lowest tangent is constrained tightly by the a
priori. A cross-over region of 5 km is used to connect the lower a priori CMC profile
with the higher profile derived with no a priori information. The uncertainties used in
the covariance matrix for the cross-over region are from the CMC estimates of 2 K in
temperature (J. Morneau, personal communication, 2005). The top pressure at 100 km
is also fixed to an a priori pressure so that a reference air column above the atmosphere
may be determined for the calculation of pressure from density. Errors in this reference
air column can propagate down into the layers below, but become insignificant below
about 80 km.
At each iteration, the retrieval requires a current guess for the state vector xi about
which to calculate a new solution. A first-guess profile x0 is required for use in the first
iteration of the retrieval algorithm. While in principle the first-guess used in the retrieval
should not affect the solution, an unrealistic first-guess profile could in theory cause the
retrieval algorithm to find an incorrect local minimum or unphysical solution to the
nonlinear minimization problem. In addition, a first-guess close to the solution should
cause the iterator to converge more quickly. The a priori profile provides a convenient
first-guess profile and is used as a first-guess in these retrievals.
3.6
Calculating the Weighting Function Matrix
The retrievals require a weighting function matrix K, which is the derivative of the
forward model, with elements Kij = ∂Fi (x)/∂xj . There are two methods for calculating
the K matrix: analytically, or by perturbing xj and determining the effect on Fi . While
not every problem allows K to be expressed analytically, it is preferable to find an analytic
expression for K as model perturbations can be very time-consuming, and the solution
68
Chapter 3. Profile Retrievals
can have numerical errors if the perturbation is too small or nonlinearity errors if the
perturbation is too large [87].
The MAESTRO weighting function calculations use an analytic approach. One advantage of the correlated-k method is that the signal expressed in Equation 2.30 can
be differentiated analytically. Following Equation 2.10, a weighting function can be expressed as Kiv = ∂τi /∂xv = ∂(− ln Si )/∂xv . The partial derivatives taken with respect
to number density of molecular absorbers which have no cross-sectional p-T dependence,
or taken with respect to background absorber amounts, are calculated analytically from
the expressions for the signal in Equations 2.29 and 2.30.
Equation 2.30, the simplified correlated-k expression for one absorber, can be used for
the differential calculations with respect to x = ln nO2 , where the cross-sections, optical
grid densities, and path lengths are all dependent on the density profile. In this case, the
partial differential for a pixel and one retrieval layer v is
Nr
M
X
X
∂τ
∂(ln S)
1
∂τj
e−τj
Kv =
=−
= R(λ)AC
G(θr − θt )
∆gj
∂(ln nv )
∂(ln nv )
S
∂(ln nv )
r=1
j=1
(3.12)
where
τj =
Nh
X
σh (gj )nh Lhr (θt )
(3.13)
h=1
and
Nh
X
∂τj
∂σh (gj )
∂nh
∂Lhr
=
nh Lhr + σh (gj )
Lhr + σh (gj )nh
∂(ln nv ) h=1 ∂(ln nv )
∂(ln nv )
∂(ln nv )
"
#
(3.14)
where the term involving the σ differential can be expanded to
∂σh (gj )
∂σh (gj ) ∂T
∂σh (gj ) ∂p
=
+
.
∂(ln nv )
∂T ∂(ln nv )
∂p ∂(ln nv )
(3.15)
The terms ∂T /∂ ln nv and ∂p/∂ ln nv can be calculated analytically, while the ∂σ/∂T and
∂σ/∂p terms can be calculated by differentiating the cubic interpolator used to determine
σh (gj ) from the tables of correlated-k coefficients.
The elements of the K-matrix for the wavelength shift are calculated by fitting a
quadratic to a pixel’s optical depth τp and its two nearest pixel neighbours as a function
Chapter 3. Profile Retrievals
69
of λ using τp = aλ2 + bλ + c, and then taking the analytic derivative of that quadratic
∂τp /∂λ = 2aλ + b for the differential.
3.7
Calculating the Smoothing Constraint
An optimal smoothing constraint weighting γ must be chosen so that any major oscillations in the solution are suppressed without oversmoothing of the retrieved profile. The
selection process developed for this retrieval involves successively adjusting γ, retrieving a temporary p-T profile for each attempt at γ, and calculating the new modelled
measurement vector F(x) for each γ by minimizing the cost function J. The γ value is
chosen according to the L-curve technique as outlined by Hansen [96], which optimizes
a regularization parameter γ to ensure smoothness while maintaining a nearly-minimum
measurement χ2 m (+χ2 a if applicable). This method involves plotting χ2 m + χ2 a versus
χ2 s for a range of values of γ. If γ is very small, the solution is not highly constrained
by smoothing and minimizing the cost function is equivalent to minimizing χ2 m + χ2 a ;
decreasing γ causes no additional changes to χ2 m + χ2 a but will increase χ2 s . This results in a nearly vertical plot of χ2 m + χ2 a versus χ2 s . However, if γ is large enough,
χ2 m is increased, minimizing the cost function corresponds to minimizing χ2 s alone, and
the plot is nearly horizontal. The final plot resembles an L for many values of γ, and
the optimum smoothing constraint weighting for the retrieval iteration, γi+1 , is taken to
occur at the corner of the L-curve at the point of maximum curvature. The result for
an optimal γ determined by this method is then used in the retrieval to calculate the
final solution vector x. This procedure is implemented at each iteration of the retrieval
in order to select γ.
Although a covariance matrix representing relative smoothing weightings could be
applied for relaxing the smoothing at particular altitudes (like in the mesosphere where
larger errors may be acceptable and the atmosphere may be expected to be less smooth),
Chapter 3. Profile Retrievals
70
the retrievals presented in this thesis assume a constant smoothing correlation at all
altitudes.
3.8
Testing for Convergence
The method used in this algorithm for testing retrieval convergence is based on that
outlined by Rodgers [87] for minimization of the state’s cost function. The cost function
representing the squared difference between the retrieved state x̂ and the true state vector
x with length n, scaled by the estimated error Ŝ, is defined at its minimum as
χ2 x = (x̂ − x)T Ŝ−1 (x̂ − x) ≈ n,
(3.16)
in which case the difference in the cost function from iteration i to iteration i + 1 is
defined as
d2i = (xi+1 − xi )T Ŝ−1 (xi+1 − xi ).
(3.17)
During convergence the error estimate for a particular iteration tends toward the converged error estimate as
(KiT Sǫ −1 Ki + Sa −1 + γH)−1 → Ŝ.
(3.18)
Substitution into Equation 3.10 allows d2i to be written as
d2i ≈ (xi+1 − xi )T [Ki T Sǫ −1 (y − F(xi )) − Sa −1 (xi − xa ) − γHxi ]
(3.19)
The algorithm used for the p-T retrievals follows the recommendation of Rodgers [87] to
stop the retrieval when d2i is an order of magnitude smaller than n.
Table 3.2 shows a sample convergence for a retrieval of an O2 number density profile of
100 altitude shells, performed using 98 simulated occultation spectra with realistic noise,
each with 55 pixels across the A band. Note the increase in χ2 m between iterations 6
and 7. This is a typical occurrence in nonlinear iterative retrievals with constraints when
the iterator is searching close to the cost function’s minimum; however, convergence has
71
Chapter 3. Profile Retrievals
been reached by this point and small increases in the cost function between iterations
have insignificant implications on the solution. The magnitude of χ2 m can also be used
to check for correct convergence by comparison with the number of degrees of freedom of
the measurement. The degrees of freedom for n = 100 state parameters and m = 98 × 55
measurements is m − n = 5290, with a standard deviation of
q
2(m − n) = 103 for
normalized error statistics. χ2 m has fallen within the range of the measurement degrees
of freedom plus its standard deviation by iteration 4, so correct convergence can be
assumed in the final solution.
Table 3.2: Evolution of χ2 m and d2i toward retrieval convergence for typical retrieval case
with simulated data.
Iteration
χ2 m
d2i
0
4.933 × 106
-
1
7.9274 × 104
5.970 × 106
2
9193
1.648 × 105
3
5704
3772
4
5374
330.9
5
5318
60.67
6
5309
16.34
7
5310
6.587
A summary of the retrieval algorithm is illustrated in Figure 3.1, which shows the
various major steps in the retrieval as the iterator moves toward convergence. Although
the choice of smoothing constraint is shown as only one step in the diagram, recall that
this procedure actually involves several sub-iterations of the forward model to choose γ
before iteration i + 1 is allowed to continue.
72
Chapter 3. Profile Retrievals
1st guess p-T profile
Calculate p-T profile from new xi
Calculate forward modelled observation F(xi)
Update xi with xi+1
Calculate xi+1
Calculate di2
Optimize g
Calculate Jacobian Ki
YES
d i2 £
n
10
NO
Calculate p-T profile from xi
Stop
Figure 3.1: MAESTRO retrieval algorithm flow diagram.
Chapter 3. Profile Retrievals
3.9
73
Simulated Retrievals
The retrieval algorithm developed in this chapter must be tested to ensure it is stable
and obtains reasonable results for a variety of possible atmospheric cases. This section
describes pressure-temperature retrievals obtained from simulated observations of noisefree and noisy data for several atmospheric cases likely to be encountered by MAESTRO
measurements. Unless otherwise stated, simulated spectra have been produced using the
MAESTRO geometry model with LBL radiative transfer calculations. The modelled A
band spectra were calculated for 55 individual pixels between 752.6 and 780.3 nm, while
the B band spectra were calculated for 34 pixels between 681.9 and 699.3 nm. The
spectra are simulated at tangent heights between 2 km and 99 km, every 1 km. The
retrieval is performed on a 1 km-spaced retrieval grid, from 0 to 100 km. The retrievals
from simulated data are plotted from 0 to 85 km, as retrievals above this altitude range
are not anticipated to be reliable due to very low measurement signal and non-constant
O2 mixing ratios.
The figures in the following sections show simulation results of retrieved ln p (where
p is in units of Pascals) and temperature, and their variations from the true atmosphere,
where ∆p = (pretrieved − ptrue )/ptrue × 100% and ∆T = Tretrieved − Ttrue . The left-hand
panels show the retrieved pressure and temperature profiles. The right-hand panels show
the deviation of the retrieved pressures and temperatures from the ‘true’ profiles used to
create the simulated spectra. In most cases, retrievals are performed for individual A and
B band spectral windows and also for the two bands combined into a single measurement
vector.
3.9.1
Atmospheric Cases
The five atmospheric cases listed in Table 3.3 were chosen for simulated retrieval cases
and are used as ‘true’ atmospheres to produce simulated observations. These profiles were
74
Chapter 3. Profile Retrievals
chosen from the ACE a priori database to represent the wide variability of atmospheric
conditions measured by MAESTRO over latitude and season. As the ACE mission focuses
on the Arctic winter and springtime stratosphere for the study of Arctic ozone depletion,
the Arctic springtime atmosphere is used as the baseline atmosphere for the retrieval
simulations that do not examine several cases.
The ‘true’ atmosphere is also used as the a priori atmosphere in order to fix the
bottom densities and the top reference pressure to the forecast. Although the a priori
atmosphere will be used as the first-guess in retrieving profiles from real on-orbit data
(see the discussion in Section 3.5), the 1976 U.S. Standard atmosphere [85] is used here as
the first-guess atmosphere to show the independence of the retrieval from the first-guess.
Naturally, this independence is limited as the use of a distinctly unphysical first-guess
could prevent the retrieval from converging.
Table 3.3: Atmospheric cases used for simulations
Region
Season
Latitude
Longitude
Date
Occultation Number
Arctic
Spring
79.8 N
80.5 W
03-Mar-2004
ss3004
Mid-latitude
Summer
45.9 N
46.0 W
26-Jul-2004
ss5138
Mid-latitude
Winter
46.8 N
151.4 W
04-Dec-2004
ss7060
Tropics
Spring
3.8 N
82.4 W
24-Apr-2004
sr3749
Antarctic
Winter
77.0 S
59.0 W
25-Aug-2004
ss5581
3.9.2
Retrievals on Noise-Free Spectra
The retrieval is tested here on simulated noise-free spectra for the Arctic spring baseline
case. Retrievals on noise-free spectra do not produce noise-induced oscillations in the
retrieved profile, so the smoothing constraint γ has been set to zero for these retrievals.
Chapter 3. Profile Retrievals
75
Fast-Line-by-Line Retrieval
An FLBL retrieval on noise-free spectra simulated using the FLBL should produce a
retrieved profile equal to the true profile. Any errors in this retrieval indicate a bias in
the retrieval algorithm itself. Figure 3.2 shows the results from a retrieval using the FLBL,
both to model the atmospheric optical depths and to calculate weighting functions, on
noise-free spectra simulated using the FLBL. The errors in this retrieval are insignificant
with maximum errors for both the A and B bands over all altitudes less than 0.005%
in pressure and 0.05 K in temperature. This indicates little bias in the retrieval method
itself. These small errors are primarily caused by the use of interpolation. The combined
A and B band retrieval is excluded here due to excessive memory requirements and
computational time; however, inferrence of errors from the individual A and B band
retrievals implies insignificant errors for the combined retrievals.
Correlated-k Retrieval
Figure 3.3 shows the effect of using the correlated-k approximation to retrieve p-T from
noise-free spectra simulated using the high-resolution LBL for the Arctic spring case.
The pressure error induced by the correlated-k approximation is different for the A and
B bands. The A band error peaks near 20 km, while the B band error continues to grow
to the surface, but is constrained by the influence of the a priori at altitudes below the
lowest measurement and those 5 km above the lowest measurement.
Combined FLBL and Correlated-k Method
Although the errors induced by the correlated-k table method are less than the p-T uncertainty requirements, they can be reduced significantly by using the FLBL for calculating
the optical depths while only using the correlated-k method for calculating weighting
functions. Figure 3.4 shows a retrieval performed using this method and its associated
errors. The errors induced by the use of the correlated-k approximation for only calculat-
76
Chapter 3. Profile Retrievals
80
80
60
altitude (km)
altitude (km)
70
40
true
1st guess
A band
B band
20
0
-5
0
60
50
40
30
20
10
5
10
0
-5
15
-3
80
80
70
70
60
60
50
40
30
30
10
10
250
temperature (K)
5
40
20
200
3
50
20
0
150
1
∆ p (%)
altitude (km)
altitude (km)
ln p (Pa)
-1
300
0
-10
-5
0
5
10
∆ T (K)
Figure 3.2: Retrieval using FLBL for Arctic spring case on simulated noise-free spectra.
77
Chapter 3. Profile Retrievals
80
80
60
40
true
1st guess
A band
B band
A+B
20
0
-5
altitude (km)
altitude (km)
70
50
40
30
20
10
5
0
60
10
0
-5
15
-3
80
80
70
70
60
60
50
40
30
30
10
10
250
temperature (K)
5
40
20
200
3
50
20
0
150
1
∆ p (%)
altitude (km)
altitude (km)
ln p (Pa)
-1
300
0
-10
-5
0
5
10
∆ T (K)
Figure 3.3: Retrieval using correlated-k for Arctic spring case on simulated noise-free
spectra.
Chapter 3. Profile Retrievals
78
ing the weighting functions are very small with maximum pressure errors of 0.005% and
maximum temperature errors of 0.1 K, for both bands over all altitudes. This retrieval
method requires a small increase in computational time for high-resolution optical depth
calculations, but much larger memory requirements than the full correlated-k method,
particularly for combined A and B band retrievals. However, the significant reduction of
errors in the final p-T product makes this a desirable modelling method for p-T retrievals.
3.9.3
Retrievals on Noisy Spectra
The retrieval algorithm is tested in this section on the five atmospheric cases described
in Table 3.3. The simulated noise-free spectra are created using these ‘true’ atmospheres,
and then artificial noise is applied to the spectra using the detector noise model described
below.
Detector Noise Model
The detector noise is modelled by calculating the signal-to-noise ratio (SNR) using the
readout noise and detector shot noise. Although the detector’s output is in terms of
counts from an A/D (analogue-to-digital) converter, the true detector measurement is of
electrons and the SNR calculations must be performed in terms of electrons. This noise
model assumes e = 2000 electrons/count (C.T. McElroy, personal communication, 2005).
The shot noise is defined as the square-root of the total signal. Therefore the SNR for C
detector counts can be expressed as
Ce
SNR = q
.
(Creadout e)2 + Ce
(3.20)
Pre-launch characterization tests on MAESTRO found Creadout ≈ 7 counts for both
detectors at all temperatures, while analysis of on-orbit data showed significant noise
improvement with Creadout ≈ 4 counts. Using this noise model, 25000 counts on the
MAESTRO detector (half-scale of a 16-bit A/D converter minus an estimated dark cur-
79
Chapter 3. Profile Retrievals
80
80
60
40
true
1st guess
A band
B band
A+B
20
0
-5
altitude (km)
altitude (km)
70
50
40
30
20
10
5
0
60
10
0
-5
15
-3
80
80
70
70
60
60
50
40
30
30
10
10
250
temperature (K)
5
40
20
200
3
50
20
0
150
1
∆ p (%)
altitude (km)
altitude (km)
ln p (Pa)
-1
300
0
-10
-5
0
5
10
∆ T (K)
Figure 3.4: Retrieval using FLBL-modelled optical depths and correlated-k -calculated
weighting functions for Arctic spring case on simulated noise-free spectra.
Chapter 3. Profile Retrievals
80
rent of roughly 5000 counts) would translate to an SNR of 4683:1. The pixels’ signalto-noise ratios are used with Gaussian-distributed random numbers to apply simulated
noise to exoatmospheric and occultation spectra for these simulations.
Retrievals
Figures 3.5 to 3.9 show p-T retrievals for the five atmospheric cases with simulated noisy
measurements. These were all performed using the FLBL for the forward model and the
correlated-k table approximation for the weighting function calculations.
The smoothing constraint weighting γ has been chosen optimally for each retrieval.
Therefore, in the combined retrievals, the errors are sometimes larger than those of the A
band retrieval alone as the combined retrieval may apply a larger smoothing constraint
to remove a large oscillation induced by the B band. These simulated retrievals are
not intended as a direct comparison of errors between the three methods (this will be
presented in Chapter 4), but rather as a check that the algorithm can successfully retrieve
p-T under a variety of conditions using any of the methods.
In all five cases, the pressure retrievals with the A band are generally able to follow
the true pressure profile to within 1%, up to about 80 km. The B band pressure retrievals
are less successful at high altitudes, but are generally within 1% of the true value up to
approximately 40 km. This is not surprising, as the strongest feature in the B band is
at only 3% absorption at tangent altitudes near 40 km. At altitudes above about 60
km, it becomes impossible to distinguish the B band feature from the detector noise.
The corresponding A band temperature retrievals are usually within 2 K of the true
atmosphere up to about 70 to 80 km. The B band temperatures generally achieve 2 K
of the true temperatures up to approximately 40 km.
These sample case studies also show combined A and B band retrievals, where the
observations from both bands have been assembled into a single vector. At first glance,
the results from these retrievals look quite similar to those of the A band alone, although
81
Chapter 3. Profile Retrievals
80
80
60
40
true
1st guess
A band
B band
A+B
20
0
-5
altitude (km)
altitude (km)
70
0
60
50
40
30
20
10
5
10
0
-5
15
-3
80
80
70
70
60
60
50
40
30
30
10
10
250
temperature (K)
5
40
20
200
3
50
20
0
150
1
∆ p (%)
altitude (km)
altitude (km)
ln p (Pa)
-1
300
0
-10
-5
0
5
∆ T (K)
Figure 3.5: Retrieval for Arctic spring case on simulated data with noise.
10
82
Chapter 3. Profile Retrievals
80
80
60
40
true
1st guess
A band
B band
A+B
20
0
-5
altitude (km)
altitude (km)
70
50
40
30
20
10
5
0
60
10
0
-5
15
-3
80
80
70
70
60
60
50
40
30
30
10
10
250
temperature (K)
5
40
20
200
3
50
20
0
150
1
∆ p (%)
altitude (km)
altitude (km)
ln p (Pa)
-1
300
0
-10
-5
0
5
10
∆ T (K)
Figure 3.6: Retrieval for mid-latitude summer case on simulated data with noise.
83
Chapter 3. Profile Retrievals
80
80
60
40
true
1st guess
A band
B band
A+B
20
0
-5
altitude (km)
altitude (km)
70
50
40
30
20
10
5
0
60
10
0
-5
15
-3
80
80
70
70
60
60
50
40
30
30
10
10
250
temperature (K)
5
40
20
200
3
50
20
0
150
1
∆ p (%)
altitude (km)
altitude (km)
ln p (Pa)
-1
300
0
-10
-5
0
5
10
∆ T (K)
Figure 3.7: Retrieval for mid-latitude winter case on simulated data with noise.
84
Chapter 3. Profile Retrievals
80
80
60
40
true
1st guess
A band
B band
A+B
20
0
-5
altitude (km)
altitude (km)
70
0
60
50
40
30
20
10
5
10
0
-5
15
-3
80
80
70
70
60
60
50
40
30
30
10
10
250
temperature (K)
5
40
20
200
3
50
20
0
150
1
∆ p (%)
altitude (km)
altitude (km)
ln p (Pa)
-1
300
0
-10
-5
0
5
∆ T (K)
Figure 3.8: Retrieval for tropical spring case on simulated data with noise.
10
85
Chapter 3. Profile Retrievals
80
80
60
40
true
1st guess
A band
B band
A+B
20
0
-5
altitude (km)
altitude (km)
70
0
60
50
40
30
20
10
5
10
0
-5
15
-3
80
80
70
70
60
60
50
40
30
30
10
10
250
temperature (K)
5
40
20
200
3
50
20
0
150
1
∆ p (%)
altitude (km)
altitude (km)
ln p (Pa)
-1
300
0
-10
-5
0
5
10
∆ T (K)
Figure 3.9: Retrieval for Antarctic winter case on simulated data with noise.
Chapter 3. Profile Retrievals
86
there are some deviations at high altitudes due to the choice of different smoothing
constraints. In fact, in these simulations, adding the B band to the A band retrieval does
little to reduce the differences between the true and retrieved atmospheres. However, as
will be seen in Chapters 4 and 5, the B band adds further information to the retrieval,
which results in a reduction in the retrieval uncertainties at certain altitudes.
Large-scale structure is apparent in the retrieval error for all figures. These features
are not directly indicative of the behaviour of the retrieval algorithm, and are unique for
each atmospheric case. Broad features in the errors, such as those shown in Figure 3.5 at
approximately 28 and 55 km in the B band temperature, are a result of oversmoothing
of the retrieved profile relative to the true profile. The locations of the error peaks
are generally at locations of sharp features in the true profile (be it density, pressure,
or temperature), where the solution may have significantly more information from its
smoothing constraint than from noisy measurements tangent near those altitudes and is
unable to follow precisely the gradients in the profile.
While errors are variable for both bands at higher altitudes, the retrieval does appear
to be able to detect fine structure at low altitudes with good reliability. This is evidenced
in Figure 3.10, where the plot of retrieved tropospheric and lower stratospheric temperatures from Figure 3.5 has been expanded. Both the A, B, and combined retrievals easily
detect the Arctic tropopause near 7 km. Evidence of the enhanced B band smoothing
constraint is visible above 11 km where the B band does not follow the true atmosphere
as well as the A band and combined retrievals.
3.10
Discussion
This chapter has discussed the development and implementation of the algorithm created
for retrieving pressure and temperature profiles from the MAESTRO space instrument.
Retrievals of pressure and temperature require an algorithm more sophisticated than
87
Chapter 3. Profile Retrievals
20
20
18
16
altitude (km)
14
12
10
8
18
16
14
altitude (km)
true
A band
B band
A+B
12
10
8
6
6
4
4
2
2
0
200
220
240
temperature (K)
260
0
-2
-1
0
1
2
∆T (K)
Figure 3.10: Tropospheric and lower stratospheric temperature retrieval for Arctic spring
case on simulated data with noise.
that currently in use for operational trace gas retrievals from MAESTRO, and so an
iterative linearized global fit χ2 minimization is applied for these retrievals, with analytic
weighting function calculations. The retrieval is also constrained in two ways. First,
optimal estimation is applied at the bottom of the profile, where measurements are
lacking, to constrain the profile to an a priori operational meteorological analysis profile.
The lack of meteorological information about the atmosphere in the upper stratosphere
prevents the use of optimal estimation at all altitudes of the profile (i.e., the covariance
of the a priori is large and in this region the ACE measurement is expected to provide
the information.) Second, a Twomey-Tikhonov smoothing constraint is applied to ensure
smoothness from layer-to-layer. Retrieval convergence is determined using the method
outlined in Rodgers [87] for minimization of the state’s cost function.
Retrievals were performed on simulated observations of several different atmospheres,
using both noise-free measurements and measurements with simulated noise. Noise-free
measurements show insignificant bias in the retrieval algorithm itself. These simulations
Chapter 3. Profile Retrievals
88
also show the use of correlated-k for fast forward modelling introduces small errors into
the retrieval using the A band, and slightly larger errors from the B band, although
both remain under 1% in pressure and 1 K in temperature. In the case where a very
fast algorithm is not required, the errors from correlated-k can be virtually eliminated
by using the FLBL to forward model the atmosphere, but using the correlated-k method
for analytic weighting function calculations. Retrievals on observations with simulated
noise show that the A band retrieval alone is able to retrieve profiles from the surface
to about 80 km with pressure errors less than 1% and temperature errors generally less
than 2 K. The B band alone is able to retrieve up to about 40 km within these same
error estimates. Similar to the A band retrievals, the combined A and B band retrievals
are able to retrieve accurate profiles up to about 80 km. A formal characterization of the
uncertainties of these retrievals is presented in Chapter 4.
Chapter 4
Expected Retrieval Performance
4.1
Introduction
This chapter presents a characterization and error analysis of the expected performance
of the p-T retrieval algorithm. Characterization of how the solution state is related to
the true state and an assessment of error sources and their contribution to the total
error are necessary for proper use of a data product and for comparisons with models
and other instruments with different vertical resolutions. Rodgers [87, 97] developed a
widely-used formal method for retrieval characterization and error analysis. This chapter
applies Rodgers’s formalism to the MAESTRO p-T retrievals.
It should be noted that the retrieval characterization and error analysis may vary
among retrieved profiles, and a single set of characterization parameters and their error
estimates are not valid for every profile. This is because each occultation will have particular noise estimates resulting from detector measurement parameters (including number
of spectra co-added and detector readout timing), a different p-T profile, and different
amounts of background interference from O3 , aerosols, Rayleigh scattering, and H2 O. In
addition, the relative weight of the smoothing constraint, γ, affects the vertical resolution and error estimates of the solution. As can be inferred from Equation 3.11, a large
89
Chapter 4. Expected Retrieval Performance
90
smoothing constraint will reduce the error estimates on the solution. This is explained
by the reduction of the vertical resolution from increased smoothing. The reduced resolution results in an effective averaging over several layers, so that the error estimate
now represents an uncertainty in the averaged function. A complete data product therefore includes a profile accompanied by each occultation’s characterization functions and
retrieval error estimates.
Although not every case will have the same error estimates or characterization functions, a formal characterization and error analysis of a typical retrieval case can be
instructive for understanding the solution and expected errors from the MAESTRO p-T
retrievals. A retrieval characterization and error analysis is presented in this chapter
using the baseline reference Arctic spring case from Chapter 3. The characterization calculations are performed using a constant smoothing constraint weighting, γ, for both the
A and B bands (required for a direct comparison of relative vertical resolutions). In order
to demonstrate the characterization and error analysis free from any contribution of the a
priori, the retrievals in this chapter employ a maximum likelihood approach where the a
priori constraint is excluded from the error and characterization calculations. However,
the profiles are tied to an a priori pressure at the top of the atmosphere. The a priori is
also used as a linearization vector in the mathematical descriptions for consistency with
Rodgers’s development [87], although any similar profile could be used for linearization.
4.2
Retrieval Characterization
The retrieval characterization developed here closely follows the method and notation of
Rodgers [87]. Consider again Equation 3.1, which expresses the observation vector y in
terms of the forward function f, and Equation 3.3, which represents the operation of a
transfer function R to convert the measurement y to the retrieved state x̂. Combining
Chapter 4. Expected Retrieval Performance
91
these two equations results in
x̂ = R[f(x, b) + ǫ, b̂, xa , c],
(4.1)
where b represents the model parameters, b̂ represents the best estimate of those parameters, xa is the a priori state vector, and c represents any other retrieval method
parameters. The ‘true’ forward model f, which represents the real physics, can be replaced by the actual forward model F, which may represent the best estimate of these
physics or an approximation for numerical efficiency, plus a term ∆f that represents the
difference between the true and actual forward model, resulting in
x̂ = R[F(x, b) + ∆f(x, b) + ǫ, b̂, xa , c].
(4.2)
Linearization about a reference state (xa , b̂) gives the expansion
x̂ = R[F(xa , b̂) + Kx (x − xa ) + Kb (b − b̂) + ∆f(x, b) + ǫ, b̂, xa , c],
(4.3)
where the weighting function discussed in Section 3.6 is Kx = ∂F/∂x and the forward
model sensitivity to the model parameters is Kb = ∂F/∂b. Further linearizing of the
retrieval method about y results in
x̂ = R[F(xa , b̂), b̂, xa , c] + Gy [Kx (x − xa ) + Kb (b − b̂) + ∆f(x, b) + ǫ]
(4.4)
where
Gy =
∂R
∂x̂
=
,
∂y
∂y
(4.5)
with Gy representing the sensitivity of the retrieved state to a change in the measurement
y. The averaging kernel matrix A is often used to express the smoothing effect of the
instrument and retrieval on a true profile, and represents the sensitivity of each layer’s
retrieved state to the true state at each layer. The averaging kernel matrix is defined as
A = Gy Kx =
∂x̂
.
∂x
(4.6)
Chapter 4. Expected Retrieval Performance
92
Further examination of several of these characterization parameters allows a look at
the sensitivity of the measured signal to the atmospheric state, and of the retrieval to
the observing system, the forward model, and the true state of the atmosphere. The
weighting function Kx , instrument gain function Gy , and averaging kernel A for the p-T
retrievals are examined in Sections 4.2.1 to 4.2.3. Although the forward model sensitivity
matrix Kb could also be considered part of the retrieval characterization, its usefulness
is associated with the estimation of model error and as a result is discussed with model
parameter error analysis in Section 4.3.2.
4.2.1
Weighting Functions
The O2 weighting function matrix Kx (K in Section 3.6, where its calculation is discussed
at length) is determined at each retrieval iteration. The weighting function matrix Kx =
∂F/∂x represents the sensitivity of the observed signal to the atmospheric state.
Figure 4.1 shows weighting functions for the base case retrieval for the A and B bands,
for four sample pixels in each band. For clarity, the weighting functions are only plotted
for tangent altitudes between 5 and 75 km, at 10 km intervals. Each curve in a plot
represents the sensitivity of that pixel’s total optical depth signal to the natural logarithm
of the O2 number density over all altitudes. The sharp peaks in the weighting functions
are characteristic of occultation measurements, where the occultation geometry causes
absorption to be heavily weighted to the tangent layer, with some contribution from layers
above, and no contribution from layers below the tangent. In this figure, pixels 333 and
338 (B band) and pixels 480 and 485 (A band) represent wavelengths near to the centre
of the bands, while pixels 329, 343, 475, and 490 represent wavelengths in the wings of
the bands. The central pixels are often more heavily influenced by density changes than
the wing pixels. The wing pixels are often influenced by the effect of population changes
in the wings of the O2 rotational spectrum due to the dependence of temperature on
density. This is evidenced in the shape of the weighting functions of pixels 343 and 490,
Chapter 4. Expected Retrieval Performance
93
which are dissimilar to the sharply-peaked functions characteristic of most occultation
observations. In these cases, a local increase in density in a tangent layer effectively
lowers the local temperature and therefore the resulting high energy band population in
the wings of the band, so that the optical depth in the long wavelength region of the
band is reduced as the density increases. Mathematically, this effect is represented by
the ∂T /∂ ln nv term in Equation 3.15, which describes the sensitivity of an absorption
cross-section to changes in density.
4.2.2
Gain Function Matrix
The gain matrix Gy = ∂x̂/∂y represents the sensitivity of the retrieved state to a ∆y
change in the measurement. This could also be considered as the retrieved state’s sensitivity to measurement error. Taking the partial derivative of Equation 3.10 with respect
to y produces
Gy = (KT Sǫ −1 K + Sa −1 + γH)−1 KT Sǫ −1 .
(4.7)
The gain matrix must be determined at each iteration calculation, as it is a subset of
Equation 3.10.
Figure 4.2 shows sample elements of the gain matrix for the retrieval of ln nO2 . Each
gain function curve in a plot represents the contribution of a pixel’s signal from one
observation tangent height to the retrieved O2 number densities at all altitudes. For
clarity, the gain functions are shown only for measurements with tangent altitudes from
5 km to 75 km, spaced every 10 km. The general increase in gain function magnitude with
increased tangent altitude demonstrates the retrieval’s greater sensitivity to small optical
depth changes (and therefore noise) in measurements at high tangent altitudes where the
true signal is weak. The spread of the functions gives some indication of the effect of
the smoothing constraint as several retrieval layers are sensitive to one measurement.
Although the gain functions alone are not particularly useful for interpretation of the
retrieval, they are necessary for the error analysis (to be performed in Section 4.3) and for
94
Chapter 4. Expected Retrieval Performance
A band
B band
80
80
60
60
pixel 329
λ = 685.7 nm
40
40
20
20
0
-0.1
0
0.1
0.2
0.3
80
altitude (km)
0
1.5
2
pixel 480
λ = 762.5 nm
0.1
0.2
0.3
0
-0.5
0
0.5
1
1.5
2
80
60
60
pixel 338
λ = 690.3 nm
40
pixel 485
λ = 765.0 nm
40
20
20
0
0.1
0.2
0.3
80
0
-0.5
0
0.5
1
1.5
2
80
60
pixel 343
λ = 692.9 nm
40
20
0
-0.1
1
20
80
40
0.5
40
20
60
0
60
pixel 333
λ = 687.7 nm
40
0
-0.1
0
-0.5
80
60
0
-0.1
pixel 475
λ = 760.0 nm
pixel 490
λ = 767.5 nm
20
0
0.1
0.2
Kx (∆τ /∆ ln nO2 )
0.3
0
-0.5
0.5
1
1.5
0
Kx (∆τ /∆ ln nO2 )
2
Figure 4.1: Typical weighting functions from the Kx matrix representing the sensitivity
of the signal to the atmospheric profile of ln nO2 , shown for sample pixels on the detector
and sample tangent altitudes from 5 km to 75 km, spaced every 10 km. The most intense
features are, from the lowest altitude, tangent at 5 km (blue), 15 km (green), and 35 km
(red).
Chapter 4. Expected Retrieval Performance
95
the calculation of the averaging kernels, which are required for a thorough understanding
of the performance of the retrieval and observing system.
4.2.3
Averaging Kernels
Figure 4.3 shows averaging kernels obtained for separate A and B band retrievals of
ln nO2 , as computed using Equation 4.6. Each curve in the figure represents the effect
on the retrieved O2 number density profile of a delta perturbation in one layer of the
true atmospheric state. For clarity, the figure shows only the averaging kernels for layers
between 5 and 75 km spaced at 5 km intervals. Averaging kernels are useful for assessing the smoothing effect and vertical resolution of the observing system and retrieval
method. The sharply-peaked averaging kernels shown in Figure 4.3 are characteristic of
occultation measurements, where the geometry of the measurement is conducive to high
vertical resolution. The peak of the averaging kernel tends to occur at the altitude of the
perturbed true atmospheric state layer where the retrieved profile is heavily weighted to
the true profile by tangent ray geometry, and then falls off quickly.
Although often referred to in this text as the ‘retrieved’ pressure and temperature, it
is important to remember that pressure and temperature are in fact derived quantities
of the number density retrievals. The averaging kernels presented in Figure 4.3 are
for the retrieved parameter ln nO2 . As pressure and temperature are functions of the
entire density profile by the hydrostatic equation and ideal gas law, a change in true
pressure or temperature at a given layer implies an entire hydrostatic readjustment of
the atmosphere, and the retrieved temperatures and pressures at any layer are affected by
this adjustment. A simple temperature averaging kernel may be determined to first order;
by only considering the local differentials in the ideal gas law, the temperature averaging
kernel is identical to that of density for a fixed pressure. A complete averaging kernel
calculation for a derived pressure quantity might be best done with perturbation methods.
However, the two primary uses for these averaging kernels are for a relative assessment of
96
Chapter 4. Expected Retrieval Performance
A band
B band
80
80
60
60
40
40
pixel 475
20 λ = 760.0 nm
pixel 329
20 λ = 685.7 nm
0
-3
-2
-1
0
1
2
3
80
80
60
60
40
40
pixel 333
20 λ = 687.7 nm
altitude (km)
0
-3
0
-3
-2
-1
0
1
2
3
0
-3
80
60
60
40
40
pixel 338
20 λ = 690.3 nm
-2
-1
0
1
2
3
0
-3
80
60
60
40
40
pixel 343
20 λ = 692.9 nm
-2
0
1
2
3
-2
0
1
2
3
0
1
2
3
-1
1
2
0
Gy (∆ ln nO2 /∆τ )
3
-1
pixel 485
20 λ = 765.0 nm
80
0
-3
-1
pixel 480
20 λ = 762.5 nm
80
0
-3
-2
-1
0
1
2
Gy (∆ ln nO2 /∆τ )
-2
-1
pixel 490
20 λ = 767.5 nm
3
0
-3
-2
Figure 4.2: Typical gain functions from the Gy matrix representing the sensitivity of the
retrieved parameter ln nO2 to the signal, shown for sample pixels on the detector and
sample tangent altitudes from 5 km to 75 km, spaced every 10 km.
97
Chapter 4. Expected Retrieval Performance
the A, B, and combined A and B band retrievals, and for later use in comparisons with
the FTS where smoothing is required, and the density averaging kernels are sufficient for
these purposes.
altitude (km)
B band
A band
A and B bands
80
80
80
70
70
70
60
60
60
50
50
50
40
40
40
30
30
30
20
20
20
10
10
10
0
0
0.5
A
1
0
0
0.5
A
1
0
0
0.5
1
A
Figure 4.3: Typical averaging kernels for A and B band retrievals of ln nO2 , shown for
altitudes from 5 to 75 km, spaced every 5 km.
Although not necessarily a rigorous measurement of vertical resolution (see Rodgers
[87] for the various ways in which resolution may be defined), the full-width-at-halfmaximum (FWHM) of the peak of the averaging kernel is commonly used to express the
vertical resolution of an occultation measurement. The averaging kernels for the A and
B bands in Figure 4.3 show the effects of the smoothing constraint on the retrieval, as
the noise in the measurements grows at high altitudes and the smoothing constraint is
weighted more heavily. The FWHM at half-maximum of the averaging kernels for the
three retrieval types is plotted in Figure 4.4. Using an estimated FOV with a vertical
FWHM of 0.037◦ , A band averaging kernels for this particular smoothing constraint
98
Chapter 4. Expected Retrieval Performance
weighting have about a 2.5 km resolution at the lowest tangent altitudes below about 20
km where the resolution is determined primarily by the instrument’s FOV and satellite
viewing geometry. Their FWHM increases to 5 km by an altitude of 35 km, and broadens
further at higher altitudes. B band averaging kernels are broader than A band retrievals
at similar altitudes as the smoothing constraint compensates for lower signal, although
retrieval resolutions at the lowest altitudes (below 10 km) are very similar to those of
the A band, but become poorer with increasing altitude and reach a FWHM of 5 km by
25 km. The addition of the B band to the A band retrieval does add some resolution,
particularly at lower tangent heights (below 15 km) where it lowers the FWHM-defined
resolution by as much as 1 km. It should also be noted that these are not absolute
numbers and are only for a typical value of γ. The resolutions may be improved by using
a smaller γ, but this will have implications for the estimated errors.
80
altitude (km)
70
60
50
40
30
20
A band
B band
A and B
10
0
0
5
10
15
FWHM of averaging kernel (km)
Figure 4.4: FWHM of ln nO2 averaging kernels.
The number of degrees of freedom for the retrieval, defined as the trace of the A
matrix, can also be used to understand the overall non-altitude-specific vertical resolution,
while the degrees of freedom for the retrieval of each altitude layer is the corresponding
diagonal element of A. In the example represented by Figure 4.3, the A band retrieval
Chapter 4. Expected Retrieval Performance
99
of ln nO2 has 23 degrees of freedom, the B band retrieval has 14, and the combined A
and B band retrieval has 24 degrees of freedom for the retrieval, between 0 and 100 km.
Between 0 and 35 km, where the retrieval is most sensitive, the A band retrieval has
15 degrees of freedom, the B band retrieval has 9 degrees of freedom, and adding the B
band to the A band retrieval increases the degrees of freedom by about 0.2.
4.3
Error Analysis
A retrieved p-T profile must be accompanied by its error estimates in order for the data
product to be fully understood. In this section, the contributions of different types of
error to the retrieved profiles are assessed. The formal error analysis developed here
once again follows the formalism of Rodgers [87] and follows from the characterization
formalism developed in Section 4.2.
Assuming no bias in the retrieval method (so that xa = R[F(xa , b̂), b̂, xa , c]), Equation 4.4 can be used to express the error between the retrieved state x̂ and the true state
x by expressing their difference as
x̂ − x = (A − In )(x − xa ) → smoothing error
+Gy Kb (b − b̂)
→ model parameter error
+Gy ∆f(x, b)
→ forward model error
+Gy ǫ.
→ retrieval noise
(4.8)
A covariance matrix is often used to describe the error represented by each of these
terms. The square-roots of the diagonal elements of a covariance matrix represent the
uncorrelated errors of the solution vector x̂, while the off-diagonal elements represent
error correlation between different retrieved values. The most significant contribution
to off-diagonal elements in the MAESTRO retrievals results from the smoothing that
causes correlation between layers. The averaging kernels represent much of this correlation. In general the correlation is low between layers in occultation, and as the
Chapter 4. Expected Retrieval Performance
100
off-diagonal elements present additional challenges in their graphical representation for
easy interpretation, and for simplification of the analysis and comparisons between A,
B, and combined retrievals, only the uncorrelated diagonal errors are presented in this
chapter.
This section aims to quantify errors associated with the retrieval for the A, B, and
A and B band combined retrievals. A direct comparison between these three retrievals
is difficult. Finding the ideal smoothing constraint weighting as in Section 3.7 is an
option, but then the method with the least information (the B band) will always be oversmoothed and its errors will appear much smaller than those of the other two methods.
Fixing the smoothing constraint to be constant for all three retrievals, as in Section
4.2.3, has a similar effect; B band errors from the diagonal of the covariance matrix
will consistently appear smaller than the other retrievals as the relative weight of the
smoothing causes a higher degree of correlation between levels. Instead, an alternate
method of choosing the smoothing parameter γ is used for these analyses based on the
expected number of degrees of freedom for the retrieval (see Steck [98] for example). In
the approach applied in the following sections, the value for γ is successively iterated until
the averaging kernel matrix A produces the expected number of degrees of freedom for
the retrieval. In these cases, the expected degrees of freedom for the retrieval is calculated
from the averaging kernels shown in Figure 4.3 for the A and B band combined retrieval.
Whatever method is used to present errors, it should be emphasized that these errors
must be considered with the off-diagonal error elements and the averaging kernels that
represent their correlations in order for the retrieval to be properly understood.
4.3.1
Smoothing Error
The smoothing error term represents the uncertainty introduced in the retrieval by the
observing system smoothing the true profile x, which most likely has features at a higher
resolution than the estimated solution. The covariance matrix of the smoothing error
Chapter 4. Expected Retrieval Performance
101
term is
SS = (A − In )Se (A − In )T .
n
where Se = ǫ (x − x̄)(x − x̄)T
o
(4.9)
[87]. It is a covariance matrix of an ensemble of states
about the mean state. The formulation of Se requires knowledge of the statistics of the
fine structure of the true profile. As is the case for the MAESTRO retrievals where
statistical information on the fine structure of the a priori profile of p-T is difficult to
estimate (at least at high altitudes), these statistics are not always well-known, and
therefore the smoothing error is often unquantifiable. As Rodgers [87] recommends, it is
instead more useful to consider the retrieved profile as an estimate of a smooth version
of the true profile, rather than as an estimate of the true state with a smoothing error.
4.3.2
Model Parameter Error
The error covariance for the forward model parameter error is
Sf = Gy Kb Sb KTb GTy ,
n
(4.10)
o
where Sb = ǫ (b − b̂)(b − b̂)T . Sf is a covariance matrix with diagonal elements
representing estimates of uncertainties in the model parameters. Table 4.1 lists the major
parameters used in the forward model and their estimated uncertainties. The sources of
the uncertainty estimates are described in a section relevant to each parameter.
Systematic and random model parameter uncertainties require slightly different treatment. While the uncertainties in density are derived using the formal error analysis
method, systematic and random density errors propagate differently to the derived pressure and temperature quantities. Random errors in pressure and temperature are calculated using standard error analysis propagation. However, a systematic model parameter
error (for instance, all the line strengths of oxygen could be too large) would likely cause
all density errors to be of the same sign. Therefore, all pressure errors would also be
of the same sign, and as the temperature is dependent on the pressure gradient across
Chapter 4. Expected Retrieval Performance
102
a layer, the temperature errors have little dependency on absolute pressure error and
more on the gradient of the error. The systematic errors presented here for pressure and
temperature are calculated by re-deriving pressures and temperatures with the density
profile error estimate. Systematic temperature errors are plotted, but should be used
with caution; they may be highly variable due to their high dependency on the pressure
error gradient.
The sensitivity of the forward model to the forward model parameters is described by
the Kb matrix. In addition to offering insight into the sensitivity of the modelled signal
to the forward model parameters, the elements of Kb are also necessary for the retrieval
error analysis using Equation 4.10. The elements of Kb in this work are calculated by
perturbing a given parameter by its expected uncertainty in both the positive and negative directions and calculating the average change in optical depth from that uncertainty
perturbation.
O2 Line Parameters
The sensitivity of the forward model to the O2 line parameters contained in the HITRAN
database is determined by perturbation of the line parameters by their estimated uncertainties, and recalculation of the forward model. The uncertainties for the A band are
taken from those estimated by Brown and Plymate [68], which are 0.0015 cm−1 in line
position, 2.0% in absolute line strength, 2.5% in the air and self-broadened half-widths,
and 15% in the temperature dependency of the width. These uncertainties are considered
to represent possible systematic biases in all the database lines in one band.
Unfortunately, uncertainties in the HITRAN 2004 B band line parameters are not
well-quantified. Neither the HITRAN 2004 database nor Giver et al. [70] give estimates
of the absolute accuracies of most of the B band line parameters, although Giver et al.
cite a band intensity precision of 1.5% and half-width precision exceeding 2 to 3%, and
the HITRAN database lists air-broadened half-width uncertainties between 2 and 5%
103
Chapter 4. Expected Retrieval Performance
Table 4.1: Forward model parameters and uncertainties
Uncertainty
Parameter
A Band
B Band
O2 line position (S)
0.0015 cm−1
0.0015 cm−1
O2 line strength (S)
2.0%
2.0%
O2 air-broadened half-width (S)
2.5%
2.5%
O2 self-broadened half-width (S)
2.5%
10%
O2 width T-dependence (S)
15%
15%
O3 cross-sections (S)
5%
3%
O3 slant column (R)
3%
3%
Slit function width (S)
0.1 nm
0.1 nm
Pointing (R)
0.002◦
0.002◦
R = random, S = systematic
Chapter 4. Expected Retrieval Performance
104
for most major B band lines. As a result, the A band uncertainties are also applied
to the B band for the line position, line strength, and temperature dependence of the
half-width for this analysis. The A band air-broadened half-width uncertainty of 2.5% is
also applied to the B band analysis, as it is consistent with the uncertainty range given
by HITRAN for the B band. The B band self-broadened half-width data of Giver et al.
and Barnes and Hays [99] differs by approximately 10%. This difference is taken as the
estimated uncertainty in the self-broadened half-width for the p-T error analysis. The
2.0% A band line strength uncertainty is also applied in this analysis to the B band,
and work by Newnham and Ballard [100] showing 1% agreement in B band intensities
with Giver et al. [70] indicates that at least the line strengths of the B band are likely
well-characterized.
Although the model parameter error estimates require the calculation of the Kb matrices for all model parameter uncertainties, not all these matrices are discussed in this
dissertation. However, in order to illustrate the sensitivities represented by the Kb matrices, sample elements of these matrices are presented in Figures 4.5 and 4.6 for the
first two parameters listed in Table 4.1 (O2 line position and line strength uncertainties,
respectively). Sample model parameter functions from the remaining Kb matrices can
be found in Appendix B.
Figure 4.5 illustrates sample elements of the Kb matrix representing the sensitivity
of the optical depth signal to a systematic 0.0015 cm−1 perturbation in the line centres.
Pixels 329, 333, and 475 occur on the short wavelength end of the band, and therefore a
positive perturbation in the wavenumber, which translates to a shift in the band toward
the negative wavelength direction, causes a positive change in the optical depth. Pixels
338, 342, 485, and 490 experience the opposite effect; if the band moves to the left in
wavelength space, with a positive wavenumber perturbation, then the optical depths on
these pixels decrease. Pixel 480 occurs nearly at the peak of the A band absorption, and
the optical depth on this pixel experiences a combination of the two effects, depending on
105
Chapter 4. Expected Retrieval Performance
the shape of the absorption at different altitudes. Figure 4.6 illustrates the sensitivities
of the modelled optical depths to a 2% increase in the O2 cross-section line strengths. As
expected, an increase in the strength of the O2 lines always produces a positive change
in the optical depth.
A band
B band
100
100
pixel 329
λ = 685.7 nm
50
0
-1
0
1
altitude (km)
pixel 333
λ = 687.7 nm
50
-1
0
1
pixel 338
λ = 690.3 nm
50
-1
0
1
0
0
1
pixel 480
λ = 762.5 nm
50
0
-1
0
1
pixel 485
λ = 765.0 nm
50
0
-1
0
1
100
100
50
-1
100
100
0
0
100
100
0
pixel 475
λ = 760.0 nm
50
pixel 343
λ = 692.9 nm
-1
1
0
Kb × 104 (∆τ /∆b)
pixel 490
λ = 767.5 nm
50
0
-1
0
1
Kb × 104 (∆τ /∆b)
Figure 4.5: Model sensitivity Kb to a 0.0015 cm−1 perturbation in O2 line positions.
The expected errors from each O2 spectroscopic line parameter calculated from Equation 4.10 are presented in Figure 4.7. Errors from the uncertainties in line position are
negligible, and even if they were larger, the uncertainties would be mostly removed by
the λshif t parameter determined in the retrieval. Uncertainties in the spectroscopic line
strengths, air-broadened half-widths, and temperature dependence of the half-widths are
the main contributors to errors in pressure and temperature.
The A band error, and the combined retrieval which is heavily weighted to the A
band, both show features with maxima near 18 and 42 km. These are due to the varying
sensitivities of the A band absorption lines as they saturate, with two peaks occurring
106
Chapter 4. Expected Retrieval Performance
A band
B band
100
100
pixel 329
λ = 685.7 nm
50
0
0
0.005 0.01 0.015 0.02
altitude (km)
pixel 333
λ = 687.7 nm
50
0
0
100
0.005 0.01 0.015 0.02
pixel 338
λ = 690.3 nm
50
0
0.005 0.01 0.015 0.02
0
0.02
0.04
0.06
0.08
50
0
pixel 480
λ = 762.5 nm
50
0
0
100
0.02
0.06
0.08
pixel 485
λ = 765.0 nm
50
0
0.04
0
0.02
0.04
0.06
0.08
100
100
0
0
100
100
0
pixel 475
λ = 760.0 nm
50
pixel 343
λ = 692.9 nm
0.005 0.01 0.015 0.02
Kb (∆τ /∆b)
50
0
0
pixel 490
λ = 767.5 nm
0.02 0.04 0.06
Kb (∆τ /∆b)
0.08
Figure 4.6: Model sensitivity Kb to a 2.0% perturbation in O2 line strengths.
where the lines from each of the two most abundant isotopes saturate and the next most
abundant isotope becomes most sensitive to any perturbation in the model parameters.
A recent study by Yang et al. [101] using ground-based measurements of the O2
A band also suggests the spectroscopy of the A band requires improvement. Recent
laboratory measurements by Tran et al. [102] have determined the line-mixing and
collision induced absorption for the O2 A band. These effects are not currently modelled
in HITRAN. The authors suggest they may account for the discrepancies between the
modelled absorption and measurements from Yang et al. [101]. In a manuscript in
preparation, Tran and Hartmann [103] use simulated ILAS measurements to explore the
implications of not including the line-mixing and collision induced absorption in retrievals
of pressure and temperature, and find that systematic errors of up to 2 K in temperature
and 2% in pressure could occur in ILAS retrievals.
107
Chapter 4. Expected Retrieval Performance
80
(a)
60
(b)
60
80
40
40
20
20
20
0
1
80
2
(d)
60
0
0
1
80
2
(e)
60
0
-2
40
20
20
20
0
1
80
2
(g)
60
0
0
1
80
2
(h)
60
0
-2
40
20
20
20
1
80
2
0
1
80
(j)
60
0
2
(k)
60
0
-2
40
20
20
20
0
1
80
2
(m)
60
0
0
1
80
2
(n)
60
0
-2
40
20
20
20
0
1
∆n (%)
2
0
0
1
∆p (%)
2
0
-2
A band
B band
A and B
2
(o)
60
40
0
0
80
40
2
(l)
60
40
0
0
80
40
2
(i)
60
40
0
0
80
40
0
(f)
60
40
2
0
80
40
0
(c)
60
40
0
altitude (km)
80
0
∆T (K)
2
Figure 4.7: Errors in density, pressure, and temperature from O2 line parameter database
uncertainties in a, b, c) line positions; d, e, f) line intensities; g, h, i) air-broadened halfwidths; j, k, l) self-broadened half-widths; and m, n, o) temperature dependencies of
half-widths, using A band, B band, and A and B band combined retrievals.
Chapter 4. Expected Retrieval Performance
108
O3 Cross-Sections and Slant Columns
Ozone is a major absorber in the regions of both the O2 A and B bands and proper
modelling of its absorption is required for removing the O3 signal from the background of
the O2 bands. The SAGE III retrievals clear the O3 spectral feature using a previouslyretrieved O3 profile before retrieving O2 from the A band. The MAESTRO retrievals
also pre-retrieve O3 but use slant column retrievals, although the pre-retrieval of O3
profiles is also optional in the retrieval code and will be a more valid approach when the
temperature dependence of the O3 cross-sections becomes better quantified in the region
of the O2 bands.
Errors in O3 cross-sections and O3 pre-retrieved slant columns that could contribute
to errors in the retrieved O2 profile are included in Table 4.1. The estimated errors
for slant column are from those estimates for O3 profile uncertainty originally presented
in Table 1.1. The estimate for uncertainties in the B band cross-sections are from the
estimates of Bogumil et al. [11], who cite a maximum uncertainty of 3.1% in these crosssections, and whose estimate has also been confirmed by Orphal [81]. The estimate for A
band cross-sections is more complicated; data collected by Bogumil et al. in the region
of the A band show uncertainties on the order of 5%, while the absolute cross-sections
vary by about 10% between measurements collected at 203 and 293 K. Several studies
[104, 105, 106, 107, 108] have attempted to better quantify the absolute cross-sections and
temperature dependencies of the O3 absorption in the A band region, but uncertainties
of 5 to 10% remain.
These error estimates are significant in the case where the aerosol background absorbers are fixed and cleared from the O2 signal. For example, just a 3% error in the O3
cross-sections can cause up to a 10 K error in stratospheric temperatures derived using
the B band where O3 is the major absorption feature, and a 5% cross-section error can
cause 1 to 2 K errors in temperatures retrieved from A band measurements. However, one
of the major benefits of simultaneously retrieving an ‘aerosol’ offset in these retrievals is
Chapter 4. Expected Retrieval Performance
109
that the absolute uncertainties in O3 cross-sections and slant columns propagate almost
entirely to uncertainties in the retrieved offset values, which are adjusted in the fit to
remove all traces of absolute errors in background absorbers. As such, it is necessary to
recognize the requirement of error characterization for O3 background-induced errors in
the retrieval case where background absorbers are fixed, but those error calculations do
not pertain to the retrieval method presented in this work.
Slit Function
The slit function is an integral part of the forward model; uncertainty in the slit function
FWHM in the O2 band regions can have a major impact on the retrieved solution. The
slit function was only measured at discrete wavelengths during pre-flight characterization
(the 830 nm slit function is used in these calculations), but the width can change across
the detector (see for example [109]). Figure 4.8 shows the errors in pressure and temperature associated with an uncertainty of 5% in the slit function width, estimated from
examination of the relative resolutions of the ATLAS 3 solar exoatmospheric reference
with the slit function applied and the reference spectrum observed on-orbit. Errors in
this simulation tend to be larger in the stratosphere, where the temperature is high and
the wings of each band are heavily populated. The model parameter sensitivity is high
in this region to any perturbation affecting the width of the band.
Pointing
Uncertainty in the pointing angle of the MAESTRO FOV will change the estimated
ray path, and therefore the O2 number density measured in the p-T retrievals. Figure
4.9 shows the error expected from a 0.002◦ uncertainty in the apparent solar zenith
angles, which corresponds to a 100 metre estimated tangent height uncertainty from
the pointing data derived during the FTS analysis (C. Boone, personal communication,
2005). The pointing uncertainty is considered to be a random error here, but there may
110
Chapter 4. Expected Retrieval Performance
80
80
80
altitude (km)
(a)
(b)
(c)
60
60
60
40
40
40
20
20
20
0
0
1
∆n (%)
2
0
0
1
∆p (%)
2
A band
B band
A and B
0
-2
0
∆T (K)
2
Figure 4.8: Errors in a) density, b) pressure, and c) temperature from uncertainty in slit
function width, using A band, B band, and A and B band combined retrievals.
also likely be a systematic component. The pointing uncertainty appears to be the most
significant error source for p-T retrievals of all the model parameter errors considered in
this chapter, although as a random error it is most significant for density and temperature
error estimates. A systematic error in the pointing would affect pressure estimates to a
greater extent.
80
80
80
altitude (km)
(a)
(b)
(c)
60
60
60
40
40
40
20
20
20
0
0
1
∆n (%)
2
0
0
1
∆p (%)
2
0
0
A band
B band
A and B
1
2 3 4
∆T (K)
5
Figure 4.9: Errors in a) density, b) pressure, and c) temperature from uncertainty in
satellite pointing angle using A band, B band, and A and B band combined retrievals.
Chapter 4. Expected Retrieval Performance
4.3.3
111
Forward Model Error
The forward model error given in Equation 4.8 is represented by
Gy ∆f = Gy [f(x, b) − F(x, b)] ,
(4.11)
where f is the true forward model describing the true physics of the measurement, and
F is the actual forward model, which contains any approximations to the true model.
Unfortunately F is often the best guess of the true physics and f is not known exactly,
which means that the only forward model error calculable is the error that is induced by
any approximation of f by F for computational efficiency. In the case of these MAESTRO
retrievals, the only forward model approximations where the ‘true’ physics is well-known
are the correlated-k and fast-line-by-line (FLBL) approximations. Forward model error
is unlike forward model parameter error in that it need not be treated as a regular source
of error. If forward model parameter error for any given retrieval is well-characterized,
the final retrieved profile can in theory be adjusted by this error to remove its effects.
The forward model difference ∆f from the correlated-k approximation was determined in Section 2.5.3, where the transmission calculated using the correlated-k table
method was compared to that calculated with the LBL. As seen in Figure 2.13, the
transmission errors induced by the correlated-k approximation are not greater than approximately 0.6% for the B band and 1.5% for the A band. Sample values of ∆f for the
correlated-k approximation are shown in Figure 4.10, where the differences between the
LBL calculation and the correlated-k approximation are expressed in terms of absolute
optical depth. In general, the error in the optical depth appears to increase near the
surface of the Earth. This is due in part to the exponentially increasing air density which
results in larger optical depth measurements and hence greater absolute errors in optical
depth, but also to the loss of some correlation between layer cross-sections as increased
pressure broadening in the wings of strong lines overshadows the absorption of weak lines
in those wings. The loss of correlation is most apparent in the increased sensitivity to
112
Chapter 4. Expected Retrieval Performance
the correlated-k approximation in the 10 to 30 km region (depending on wavelength),
where the radiative transfer changes from a high- to low-pressure regime with increasing
altitude and weak lines become visible in high-resolution cross-sections.
The resulting errors in the retrieved pressure and temperature profiles are shown in
Figure 4.11. These are quite small (generally within 1 K) for the A band and combined
retrieval up to about 60 km, and for the B band retrieval up to about 40 km. Note
that although errors for a correlated-k retrieval were also shown in Figure 3.3, the errors
presented here are for the case with noise and the smoothing constraint has removed errors
near the bottom of the profile, while large noise in high measurements has increased errors
at high altitudes. The forward model differences ∆f for the FLBL relative to the LBL
are not illustrated here, but are extremely small. If the FLBL is used to forward model
the O2 radiative transfer, the forward model errors shown in Figure 4.12 result. These
errors are negligible at all altitudes, and demonstrate the high accuracy of the FLBL
MAESTRO forward model.
A band
B band
100
100
50
altitude (km)
0
-2
100
0
0
50
0
-2
4
0
-15
100
2
6
2
4
6
pixel 338
λ = 690.3 nm
50
0
-2
100
50
pixel 333
λ = 687.7 nm
50
0
-2
100
pixel 329
λ = 685.7 nm
0
2
4
6
pixel 343
λ = 692.9 nm
2
4
0
∆f × 103 (τLBL − τck )
6
50
0
-15
100
50
0
-15
100
50
0
-15
pixel 475
λ = 760.0 nm
-10
-5
0
5
0
5
0
5
pixel 480
λ = 762.5 nm
-10
-5
pixel 485
λ = 765.0 nm
-10
-5
pixel 490
λ = 767.5 nm
-10
0
-5
∆f × 103 (τLBL − τck )
5
Figure 4.10: Forward model difference ∆f from the correlated-k approximation.
113
Chapter 4. Expected Retrieval Performance
80
80
80
altitude (km)
(a)
(c)
(b)
60
60
60
40
40
40
20
20
20
0
-5
0
∆n (%)
5
0
-5
0
∆p (%)
5
0
-5
A band
B band
A and B
0
∆T (K)
5
Figure 4.11: Forward model errors Gy ∆f in a) density, b) pressure, and c) temperature
from correlated-k approximation using A band, B band, and A and B band combined
retrievals.
80
80
80
altitude (km)
(a)
(b)
(c)
60
60
60
40
40
40
20
20
20
0
-0.02
0
∆n (%)
0
0.02 -0.02
0
∆p (%)
0
0.02 -0.05
A band
B band
A and B
0
∆T (K)
0.05
Figure 4.12: Forward model errors Gy ∆f in a) density, b) pressure, and c) temperature
from FLBL approximation using A band, B band, and A and B band combined retrievals.
4.3.4
Retrieval Noise
The covariance matrix for the retrieval noise is given by
Sm = Gy Sǫ GTy ,
(4.12)
114
Chapter 4. Expected Retrieval Performance
where Sǫ is the detector noise covariance matrix estimated as described in Section 3.9.3,
for noise uncorrelated between pixels, with error terms on the matrix diagonal and zero
elsewhere.
The estimated uncertainties from retrieval noise are shown in Figure 4.13. As expected, the B band error from noise is considerable at high altitudes. The expected
A band error is much less at high altitudes, while the combined retrieval experiences a
minimally improved error estimate over the A band.
80
80
80
altitude (km)
(a)
(b)
(c)
60
60
60
40
40
40
20
20
20
0
0
1
∆n (%)
2
0
0
1
∆p (%)
2
0
0
A band
B band
A and B
1
2 3 4
∆T (K)
5
Figure 4.13: Random errors in a) density, b) pressure, and c) temperature from measurement noise using A band, B band, and A and B band combined retrievals.
4.3.5
Total Error Estimates
The total error estimate is determined by the square-root of the sum of the squares of
the forward model parameter errors calculated in Section 4.3.2 and the random noise
errors from Figure 4.13. The systematic and random estimates are treated similarly,
although systematic temperature errors must be treated with some caution. These errors
contribute less than 0.5 K at most altitudes, but are highly dependent on the gradients in
systematic pressure errors. The resulting total error estimates are shown in Figure 4.14.
Again, as with the individual model parameter errors, the total error estimate shows
several persistent features, including two peaks in the A band errors caused by line
115
Chapter 4. Expected Retrieval Performance
saturation of the two most abundant isotopes and somewhat improved error estimates
from the addition of the B band. Temperature errors for the combined retrieval are
generally about 2 K, while pressure errors are less than 1%.
80
80
80
altitude (km)
(a)
(b)
(c)
60
60
60
40
40
40
20
20
20
0
0
1
2
∆n (%)
3
0
0
1
∆p (%)
2
0
0
A band
B band
A and B
1
2 3 4
∆T (K)
5
Figure 4.14: Total errors in a) density, b) pressure, and c) temperature from model
parameter error and random noise error for A band, B band, and A and B band combined
retrievals.
4.4
Discussion
This chapter presented a formal retrieval characterization for a typical retrieval from
MAESTRO. The weighting functions for the retrieval of ln nO2 show peaked features
typical of occultation geometry, and usually show positive responses in optical depth
to positive changes in ln nO2 , although a density change can also induce temperature
responses with the weighting functions reflecting changing band population distribution.
The gain matrices are not necessarily useful alone, but are used in the calculation of the
averaging kernels and error estimates. The averaging kernels presented in this chapter
use a fixed smoothing weight for A, B, and combined retrievals, and show varying vertical
resolutions with altitude. At low altitudes (less than 10 km for the B band, and less than
20 km for the A band), the averaging kernel widths are about 2 km, and are determined
mostly by the instrument’s FOV with some influence from smoothing. These widths grow
Chapter 4. Expected Retrieval Performance
116
with altitude as the O2 signal becomes smaller and the contribution of the smoothing
constraint to the solution increases.
This chapter also presented a formal error analysis for a typical p-T retrieval. This
analysis was performed using 24 degrees of freedom for the retrieval, as calculated using
the averaging kernels of the combined retrieval determined during retrieval characterization. As the statistics of the fine structure of the a priori are not well-known, the
retrievals developed in these chapters are considered to be an estimate of a smooth version
of the true profile, and smoothing error is not determined explicitly. Model parameter error calculations in this section include the error contributions from O2 spectroscopic line
parameters, slit function width uncertainty, and pointing uncertainty. Although there
are significant uncertainties in O3 cross-sections, the error resulting from these uncertainties is not quantified here as the aerosol offset parameter will adjust to compensate
for these errors. The uncertainties in the line parameters of O2 have different contributions to the total error budget, with uncertainties from the spectroscopic line strengths,
air-broadened half-widths, and half-width temperature dependency having the greatest
contributions to the total error. The uncertainty in the slit function width is also a significant error source. The pointing uncertainty of 0.002◦ is found to produce the greatest
contribution to the total error. In the case where the correlated-k approximation is used,
the associated errors are small (less than 1 K) for the A band at altitudes less than 60
km, and for the B band at altitudes less than 40 km. Even at higher altitudes, these
errors remain below 3 K. The forward model errors are negligible in the case where the
FLBL approximation is used in the forward model. The retrieval noise is less than 2 K
in temperature and 0.5% in pressure for all altitudes for the A band, and small in the B
band until its rapid growth at altitudes greater than about 35 km.
The total error estimates reflect the individual model parameter errors and the retrieval noise error. The B band errors are similar to the A band and combined errors
at altitudes below about 50 km. (Recall that in these error calculations the A and B
Chapter 4. Expected Retrieval Performance
117
bands have been forced to produce identical averaging kernels, so that the B band errors
show similar correlations between layers to the A band errors.) The combined retrieval
is heavily weighted to the A band, but its errors are always smaller as it receives some
contribution from the B band signal. Below 15 km and above 40 km, the combined
retrieval outperforms both bands, and shows an improvement over the A band retrieval
of up to 0.25% in pressure and 0.5 K in temperature. If the noise performance of the
instrument on-orbit is similar to that simulated in these retrievals, these results suggest
the B band may be used by itself below about 50 km, although the combined retrieval
produces the best results at all altitudes. This approach would result in error estimates
less than 2 K in temperature and 1% in pressure at most altitudes.
Chapter 5
Retrievals from Satellite Data
5.1
Introduction
In this chapter, the O2 retrievals are applied to real data collected by the MAESTRO
satellite instrument from space. The chapter first introduces the ACE-FTS satellite
pointing retrievals that provide the tangent height registration for the MAESTRO retrievals, and discusses the MAESTRO spectra used as input to the retrievals. The
MAESTRO datasets chosen for these analyses are described and the comparison datasets
of instrument measurements and meteorological analyses are introduced. The processing
of MAESTRO spectra is discussed, with particular emphasis on several analysis issues
that were found to be particularly significant for the O2 band retrievals. Several sample
retrievals are shown, and characterization and error analysis are presented for a typical
retrieval. An ensemble of MAESTRO p-T retrievals is compared with p-T profiles from
the FTS CO2 retrievals, coincident radiosonde observations, and CMC (the lower section
of the a priori ), NCEP (National Centers for Environmental Prediction), and ECMWF
(European Centre for Medium-range Weather Forecasts) operational analyses.
Where appropriate, the results from a typical single MAESTRO occultation analysis are illustrated using data from the sunset occultation ss3004. This occultation was
118
Chapter 5. Retrievals from Satellite Data
119
collected on March 3, 2004 with a 30 km sub-tangent point at (79.8◦ N, 80.5◦ W) at a
distance of 109.5 km from Eureka, Nunavut, Canada (80.0◦ N, 85.9◦ W), the site of the
2004, 2005, and 2006 ACE Arctic Validation Campaigns [110].
5.2
Input Data for p-T Retrievals
The MAESTRO p-T retrievals require pointing information and spectra from one occultation as input for each profile retrieval. These data are described in the following two
sections.
5.2.1
Pointing Information
The ACE-FTS and MAESTRO share a common input beam and pointing acquired from
a suntracker that operates at 1.55 µm. In theory, highly accurate instrument pointing
information should be accessible from information provided by several sensors on the
satellite, including the suntracker, a star tracker, and a fine sun sensor, when combined
with the time stamp on each spectrum and the satellite’s position. However, several
problems with pointing sensor calibration and a lack of accurate co-registration of time
stamps between the spectrometers, satellite bus, and pointing sensors, have thus far
prevented the use of satellite sensor information in pointing determination. As a result,
the current FTS CO2 p-T retrievals are also used to retrieve satellite pointing with time,
and this pointing information is used for determining MAESTRO apparent solar zenith
angles for each spectrum. Therefore, the MAESTRO p-T retrievals presented here are
not entirely independent of those of the FTS in that to proceed they require the FTS
p-T retrieval results. Once reliable satellite sensor data for pointing becomes available,
it will be used as input to the MAESTRO p-T retrievals. The retrievals presented in this
chapter use tangent altitudes derived with Version 2.2 of the FTS processing software.
Chapter 5. Retrievals from Satellite Data
120
The FTS retrievals of p-T profiles and tangent altitudes involve a significant amount
of analysis, as described at length by Boone et al. [92]. A summary is provided here for
a thorough understanding of MAESTRO pointing and for the comparisons with the FTS
p-T profiles that will be presented later. The approach of Boone et al. employs a modified global fit with a Levenberg-Marquardt [111, 112] nonlinear least-squares estimator
to fit high-resolution (≈ 0.02 cm−1 ) absorption lines of CO2 in small microwindows of
the spectra. Although the MAESTRO absorption measurements do contain some temperature information in the shape of the absorption features, the high spectral resolution
FTS observations contain significant information in both the absolute and the relative
strengths of the lines as the FTS is able to resolve individual absorption lines. The
absolute line strengths within an absorption band are most sensitive to the amount of
CO2 in the path (and therefore the air density or mixing ratio if not assumed constant),
while the rotational lines within that band provide most of the temperature information
through their relative strengths.
Boone et al. use different approaches for different altitude ranges. Above 43 km,
they initially assume geometric tangent heights from satellite and solar position, retrieving temperature and forcing pressure to obey hydrostatic equilibrium. Above 70 km,
the CO2 mixing ratio cannot be assumed constant and is also retrieved, in addition to
temperature. The lower altitude processing retrieves pressure and temperature, and uses
an analytical expression to calculate tangent altitude from the p-T profile using the hydrostatic equation. These results are on a relative grid, which is then shifted to match
tangent altitudes retrieved between 12 and 20 km with the p-T profile fixed to the a
priori CMC analysis. FTS pointing is also derived between 5 and 12 km, where tangent
altitudes are determined entirely by the a priori and p-T is not retrieved.
The use of the same solar input beam by the FTS and MAESTRO instruments offers
an unprecedented opportunity for comparing infrared and UV-visible-NIR atmospheric
retrievals, not least for the relatively new O2 solar occultation p-T retrieval technique,
Chapter 5. Retrievals from Satellite Data
121
which will benefit greatly from the ability to compare each profile with one derived
by the more established CO2 retrievals. This will become particularly interesting once
independent satellite pointing information is available.
5.2.2
Spectral Data
This analysis uses spectra output from the software SCALE (Version 32.0), developed by
C.T. McElroy at the Meteorological Service of Canada for correcting raw data from the
MAESTRO spectrometers. The complete set of spectra from occultation series ss3004
is shown in Figure 5.1. In this figure, the grey curves represent the 60 spectra with
tangent heights less than 100 km, and the black envelope spectrum is the observed
extraterrestrial spectrum (tangent at 145 km) used in the analysis of this occultation.
During a measurement sequence, the photodiode detector is integrated for a length of
time determined by the expected atmospheric transmission at each tangent height, and
so the integration times change between spectra. In addition, each individual pixel is
also integrated for a slightly different amount of time from the other pixels in the array
due to the time required for detector readout, and two or three spectra are generally
co-added by the on-board computer processing. The SCALE software corrects for these
differing pixel integration times to calculate detector counts per unit time. A modelled
correction for stray light is also applied in the software. These spectra are not calibrated
in absolute intensity. In Figure 5.1, the instrument responsivity is still visible in the fall-off
of intensity of all spectra at pixels less than 50, and the four broad peak features between
pixels 50 and 500 are evidence of detector etaloning. However, occultation measurements
do not require absolute calibration as the exoatmospheric spectrum is used as an intensity
reference. The collection of each occultation spectrum takes approximately 0.16 seconds.
During this time, the tangent height of the observation changes slightly, from about 80
metres near 10 km, up to as much as 400 metres in the upper stratosphere. The reference
tangent of the observation is assigned to the mid-point of the observation.
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Chapter 5. Retrievals from Satellite Data
10
9
B band
A band
detector counts / µs
8
7
6
5
4
3
2
1
0
0
100
200
300
400
500 600
pixel
700
800
900
1000
Figure 5.1: Corrected spectra for occultation ss3004 output from SCALE V32.0 software.
The grey spectra are the occultation measurements, while the black spectrum represents
the observed extraterrestrial spectrum used in the analysis.
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Chapter 5. Retrievals from Satellite Data
5.3
5.3.1
Analysis Dataset
MAESTRO Data
As of February 1, 2006, spectra from 5611 MAESTRO occultations with simultaneous
FTS pointing and p-T retrievals had been processed at the University of Toronto, collected during the period between launch and September 30, 2005. Due to time constraints
(each retrieval takes approximately 20 minutes), a sub-selection of occultations is analyzed. The analysis presented in this chapter examines MAESTRO p-T retrievals from
230 occultations collected during 2004 and the spring of 2005, divided into the latitudinal
and seasonal groups as described in Table 5.1. ACE occultations are referenced spatially
by the surface latitude-longitude point of a 30 km tangent measurement.
Table 5.1: Occultations used in Chapter 5 analysis
Region
Season
Start Date
End Date
Latitude Range
No. Occs.
Arctic
winter
27-Jan-2005
21-Feb-2005
65◦ N to 90◦ N
64
Arctic
summer
05-Jul-2004
15-Jul-2004
65◦ N to 90◦ N
56
Tropics
all
01-Jan-2004
31-Dec-2004
10◦ S to 10◦ N
73
Eureka 2004
spring
22-Feb-2004 10-Mar-2004 <500 km from Eureka
25
Eureka 2005
spring
22-Feb-2005 08-Mar-2005 <500 km from Eureka
12
The Arctic latitudes are chosen for comparisons as the northern high latitudes are the
main focus of the ACE mission, and the Arctic is a region where ACE measurements are
quite dense. Both the winter and summer seasons are examined at latitudes greater than
65◦ N. For contrast, an additional dataset of tropical occultations between 10◦ S and 10◦
N is analyzed. The tropical atmosphere does not exhibit as much seasonal variability
and therefore this dataset is not divided by season.
Occultations falling within 500 km of the Eureka ACE Arctic Validation Campaign
site in 2004 [110] and 2005 are also analyzed. Ozonesondes, carrying radiosondes for in
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Chapter 5. Retrievals from Satellite Data
situ p-T measurements, were launched nearly daily to correspond with ACE overpass
occultations, and provided p-T and ozone profile measurements coincident with ACE
observations.
The locations of the occultations analyzed in this work are presented in Figure 5.2.
The location of an occultation with season and whether it is a sunrise or sunset is determined by the satellite’s orbit as described by Figure 1.2.
180◦ W
90◦ N
120◦ W
60◦ W
0◦
60◦ E
120◦ E
180◦ E
75◦ N
60◦ N
45◦ N
30◦ N
15◦ N
0◦
◦
15 S
30◦ S
45◦ S
60◦ S
75◦ S
90◦ S
Arctic winter, sunrise
Arctic winter, sunset
Arctic summer, sunset
tropics, sunrise
tropics, sunset
Eureka spring, sunset
Eureka site
Figure 5.2: Locations of occultations used in Chapter 5 analysis, showing Arctic winter,
Arctic summer, tropical, and Eureka campaign occultations, as well as the location of
the Eureka measurement station.
Chapter 5. Retrievals from Satellite Data
5.3.2
125
Comparison Datasets
The various datasets used in the MAESTRO p-T comparisons are described in the following sections. Where model output is used, it is interpolated to the 30 km sub-tangent
measurement’s latitude and longitude.
ACE-FTS
The most interesting comparison dataset for all MAESTRO species is that of the ACEFTS instrument on SciSat. The two instruments share the same solar input beam and the
ACE mission presents the first opportunity for direct comparison of atmospheric infrared
and UV-visible-NIR instrumentation from space.
The CO2 p-T retrieval method of the FTS was described in Section 5.2.1. As a result of
the combined pointing and p-T retrieval method, each FTS temperature profile between
5 and 12 km is fixed to the a priori profile. The p-T profile between 12 and 20 km is
also somewhat weighted to the a priori as a tangent height shift is referenced to tangent
heights calculated using the a priori. The vertical resolution of an FTS p-T profile is
determined by the instrument’s scan time and field-of-view (FOV). The FTS instrument
requires approximately two seconds to complete a scan and collects a spectrum every 2
to 6 km (depending on orbital angle relative to the Sun). In combination with its circular
1.25 mrad diameter FOV, this provides a best-case vertical resolution of 3 to 4 km.
Both MAESTRO and FTS profiles presented in this work are on 1-km output grids.
However, both MAESTRO and the FTS make measurements on unevenly-spaced tangent
altitudes. The p-T values between MAESTRO tangent altitudes are determined during
the retrieval process and are heavily influenced by the smoothing constraint, while those
from the FTS are interpolated. Typical tangent altitudes are presented in Figure 5.3 for
sample occultation ss3004, superimposed on a plot of the a priori temperature profile.
A typical MAESTRO occultation series has approximately 60 spectra collected between
0 and 100 km, while the FTS typically records about 25 spectra.
126
Chapter 5. Retrievals from Satellite Data
100
MAESTRO
FTS
90
80
altitude (km)
70
60
50
40
30
20
10
0
180
190
200
210
220 230 240
temperature (K)
250
260
270
280
Figure 5.3: MAESTRO and FTS tangent altitudes for occultation ss3004 with a priori
temperature profile.
Current FTS p-T retrievals are not provided with associated averaging kernels or
rigorous error estimates. The FTS profiles have not been extensively validated, although
an early validation effort of ACE (v1.0) and HALOE (v19) comparisons by McHugh et al.
[113] found mean temperature differences up to 2 K, with ACE colder than HALOE below
40 km and warmer above, and RMS differences of about 2 K through the stratosphere
and as high as 4-10 K above 50 km.
A Priori
The a priori profiles used for the ACE mission were described in Section 3.5. As the
upper stratospheric component of an a priori profile consists only of interpolated values
between the Canadian Meteorological Centre (CMC) profile and the climatological NRLMSISE profile, the most useful comparisons in this section are those below 10 hPa (about
30 km) where the CMC operational analyses provides the z-p-T estimates on 28 vertical
Chapter 5. Retrievals from Satellite Data
127
levels. The a priori profile derived from these CMC profiles is provided by the ACE
Science Operations Centre on a 1-km vertical grid.
NCEP
The National Centers for Environmental Prediction (NCEP) Climate Prediction Center
(CPC) produces daily stratospheric meteorological analyses on southern and northern
hemispheric 65 × 65 polar grids, for eight pressure levels at 70, 50, 30, 10, 5, 2, 1, and
0.4 hPa. The profiles used in these comparisons are interpolated to the ACE 30 km
sub-tangent location.
ECMWF
The European Centre for Medium-Range Weather Forecasts (ECMWF) produces meteorological analyses using data assimilation of radiosondes and satellite data, with the
current system described by Klinker et al. [114]. The data used here are from the
ECMWF TOGA (Tropical Ocean and Global Atmosphere) global surface and upper air
daily analyses, on a twice daily global 2.5◦ × 2.5◦ grid, for 22 pressure levels at the
surface, 1000, 925, 850, 700, 500, 400, 300, 250, 200, 150, 100, 70, 50, 30, 20, 10, 7, 5, 3,
2, and 1 hPa. The profiles used in these comparisons are interpolated to the ACE 30 km
sub-tangent location from the closest profiles coincident in time.
Radiosondes
Radiosondes are small balloon-borne platforms that are used to measure several in situ
atmospheric parameters as a function of altitude, including profiles of temperature and
pressure. During balloon ascent, radiosonde temperature measurements are collected
using a resistance thermistor. Pressure measurements are made by an aneroid barometer,
which consists of a partially evacuated metallic capsule that expands with decreasing
pressure as the balloon ascends. The radiosondes used in these comparisons were on
Chapter 5. Retrievals from Satellite Data
128
ozonesonde payloads, and typically ascended to altitudes corresponding to 10 hPa (about
30 km). The vertical resolution of these radiosonde p-T profiles is on the order of 50
metres.
5.4
Processing
This section describes the MAESTRO data processing, several issues that arose during
spectral fitting, and sample results from the retrieval.
5.4.1
Pre-Processing: Pixel-Wavelength Assignment
As discussed in Section 2.2.1, each pixel corresponds to a small region of the spectrum and
must be assigned to that region’s central wavelength. In the laboratory, emission lamps
of known reference spectra were used to assign a pixel-wavelength dispersion by fitting
the polynomial described by Equation 2.1. This nearly-linear laboratory dispersion was
used in Chapters 3 and 4 for creating simulated observations. However, the wavelength
dispersion is very sensitive to any changes in the spectrometer after launch and any
thermal changes which cause shifts in the instrument alignment from orbit-to-orbit. As
such, the laboratory analysis is not entirely valid on orbit, and the pixel-wavelength
dispersion is adjusted at each occultation using solar Fraunhofer lines.
The pixel-wavelength pre-processing algorithm fits the wavelengths of a MAESTRO
high-sun spectrum between 525 and 950 nm on the VIS spectrometer to those from
the 0.5-nm resolution ATLAS 3 solar reference spectrum [63] discussed in Section 2.4.2.
While the reference spectrum is calibrated in absolute intensity, the observed spectrum
(i.e., the black curve in Figure 5.1) is not. In order to fit the two spectra, a high-pass filter
is applied to each spectrum so that only the high frequency spectral information remains.
These resulting spectra are then normalized to each other in intensity. Figure 5.4a shows
the wavelengths for the reference and modelled spectra for occultation ss3004, using
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Chapter 5. Retrievals from Satellite Data
the anticipated approximately linear dispersion relationship. An iterative least-squares
minimizing routine is then used to determine new coefficients for a high-order pixelwavelength polynomial relationship, until a minimum in the residuals between the two
spectra is achieved. As is visible in Figure 5.4a, the first-guess nearly-linear dispersion
works well for wavelengths shorter than about 700 nm. However, at wavelengths greater
than 700 nm, the true wavelengths vary from the linear dispersion by a significant amount,
with a discrepancy as large as 15 nm near 950 nm. The spectra using the new wavelength
dispersion, described by a 7th order polynomial that allows for the sharp wavelength
deviations after 700 nm, are shown in Figure 5.4b. The wavelength accuracy of the new
normalized intensity
normalized intensity
MAESTRO wavelength assignment is estimated to be approximately 0.01 nm.
5
0
-5
model
observation
a
-10
550
600
650
700
750
800
wavelength (nm)
850
900
950
5
0
-5
model
observation
b
-10
550
600
650
700
750
800
wavelength (nm)
850
900
950
Figure 5.4: Exoatmospheric spectrum high frequency component used in the ss3004 pixelwavelength registration as a function of a) the first-guess wavelengths; and b) the best-fit
wavelengths.
Chapter 5. Retrievals from Satellite Data
5.4.2
130
Instrument Slit Function
Understanding MAESTRO’s instrument slit function behaviour across the detector continues to be a significant challenge for MAESTRO data analysis, as it was for the SunPhotoSpectrometer, MAESTRO’s precursor instrument (see for example [109]). While
the retrieval of low spectral resolution absorbers including O3 is not highly dependent on
a very accurate slit function, proper knowledge of the slit function is critical for retrieving
species with sharp absorption features such as O2 , as demonstrated in Section 4.3.2 for an
uncertainty in the line shape width. When the preliminary spectral fits of the O2 bands
were examined, it became apparent that the residuals between the modelled and observed
optical depths had a consistent shape, for all altitudes and every analyzed occultation.
These initial fits suggested that not only was slit function width an uncertainty, but there
was also a systematic error in the shape of the slit function.
While pre-flight laboratory measurements of the MAESTRO slit function were performed at the University of Toronto in March 2003, their results have been found to be
insufficient for a very accurate analysis of the instrument slit function. At the time, a
more thorough characterization was desired but not possible, mainly due to the short
period (one month and a half) allotted for all pre-flight characterization. Laboratory
measurements of the MAESTRO instrument slit function were performed at three points
across the VIS detector using three diode lasers with wavelengths of roughly 530 nm
(green laser), 670 nm (red laser) and 830 nm (infrared laser). Of the two lasers used in
the regions of the O2 bands (the red and infrared), only data from the infrared laser slit
function was usable. The asymmetric Gaussian fitted to this slit function was presented
earlier in Figure 2.1 and has been used for all previous analysis in this thesis.
An improved instrument slit function has been determined for each band by deconvolution from an observed spectrum using a high-resolution modelled spectrum. Ideally,
the slit would be deconvolved using the observed MAESTRO high sun spectrum and a
reference spectrum such as the higher spectral resolution exoatmospheric reference spec-
Chapter 5. Retrievals from Satellite Data
131
trum from ATLAS 3. However, the lack of strong solar features in the regions of the A
and B bands and the lack of an absolute intensity reference for MAESTRO makes a slit
function deconvolution from the solar spectrum very difficult. Improved slit functions
for the A and B band have been deconvolved in this work from atmospheric occultation
spectra with low tangent altitudes, where the O2 signal is strong and the true p-T profile
is anticipated to have very small deviations from the a priori analysis. The B band slit
function is derived from a subset of B band measurements with tangents near 5-6 km
in dry atmospheres where water vapour absorption is not visible in the B band region.
The A band slit function is derived from a subset of A band spectra with tangents near
15-17 km, where the signal is strong but errors associated with a near-saturated A band
core can be ignored.
The improved slit functions are shown with the original first-guess slit function in
Figure 5.5. Although quite similar to the original asymmetric Gaussian function, both
the A and B band slit functions show an unexpected feature on the short wavelength
side of the slit function. The A band slit function is slightly wider than the others,
with a FWHM of 2.14 nm, while the B band slit function has a FWHM of 1.93 nm
and the first guess slit function has a FWHM of 1.91 nm. The feature on the left of the
deconvolved slit functions is consistent with a similar feature observed in the slit function
of MAESTRO-B, the clone instrument of MAESTRO used for high-altitude balloon field
campaigns.
5.4.3
Spectral Fitting
After pixel-wavelength pair assignment, the retrieval proceeds as described in Chapter
3. The O3 slant column background amount for each spectrum is first derived using a
Chappuis band fitted between 550 nm and 680 nm. A combined retrieval is then used to
simultaneously fit the A band from 752.6 nm to 780.3 nm, the B band from 681.9 nm to
699.3 nm, and two independent parameters for constant and linear offsets that represent
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Chapter 5. Retrievals from Satellite Data
1
first guess
A band, deconvolved
B band, deconvolved
0.9
arbitrary intensity units
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-5
-4
-3
-2
1
2
-1
0
relative wavelength (nm)
3
4
5
Figure 5.5: The original first-guess MAESTRO slit function fitted to the infrared diode
laser line shape during pre-flight characterization, and slit functions derived from deconvolution of the A and B band spectra.
Chapter 5. Retrievals from Satellite Data
133
aerosols in each band. A two-pixel-wide triangular filter is applied to the observations
and model to reduce possible pixel-to-pixel noise that could possibly result from any
unexpected detector issues on a pixel-to-pixel level, such as uneven distribution of signal
to one detector pixel when photons arrive at the detector boundary or small uncertainties
in the pixel-wavelength registration.
Sample optical depth fits are shown in Figures 5.6 and 5.7. These spectral fits were
performed on MAESTRO spectra with the stray light correction removed, and using
the improved slit functions shown in Figure 5.5. Figure 5.6 shows the observed and
modelled spectra for five sample tangent heights between 4.7 and 33.2 km, and the
residuals between the observed and modelled spectra, offset by an optical depth of 0.5.
Because these signals are so large, it is impossible to show high tangent altitude fits
on the same linear scale, and so four high-altitude spectral fits have also been plotted
separately in Figure 5.7, with residuals offset by an optical depth of 0.02.
The fits are generally very good, with residuals usually within the noise estimates.
The residuals in the A band in Figure 5.6 do show some small structures at low tangent
altitudes where the band is nearly saturated, which are most likely from remaining uncertainties in the shape and width of the slit function, the lack of a stray light correction
(see Section 5.4.4), or errors in readout timing within the SCALE software. The residuals
at high altitudes are also within the expected noise limits, although there are often small
persistent errors associated with the 8-pixel group detector readout error (to be discussed
in Section 5.4.4).
5.4.4
Analysis Issues
In addition to the uncertainties encountered in the slit function described in Section
5.4.2, several other analysis issues arose during the MAESTRO processing. The following
sections discuss instrument noise, pointing, and several instrument characterization issues
and their treatment in the MAESTRO analysis. Although corrections have been applied
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Chapter 5. Retrievals from Satellite Data
B band
A band
7
6
7
observation
model
residual
6
4
5
4.7 km
3
7.8 km
2
optical depth
optical depth
5
4
3
4.7 km
2
7.8 km
12.4 km
1
0
-1
20.4 km
33.2 km
690
695
685
wavelength (nm)
12.4 km
1
20.4 km
0
-1
33.2 km
770
760
wavelength (nm)
780
Figure 5.6: Observed and modelled optical depths for the A and B band at low tangent
altitudes, and their residuals offset by 0.5, for occultation ss3004.
135
Chapter 5. Retrievals from Satellite Data
A band
0.1
0.1
0.08
0.08
0.06
33.2 km
observation
model
residual
0.04
0.02
0
41.4 km
53.5 km
78.1 km
-0.02
optical depth
optical depth
B band
0.06
0.04
33.2 km
0.02
41.4 km
53.5 km
0
78.1 km
-0.02
685
695
690
wavelength (nm)
760
770
wavelength (nm)
780
Figure 5.7: Observed and modelled optical depths for the A and B band at high tangent
altitudes, and their residuals offset by 0.02, for occultation ss3004.
Chapter 5. Retrievals from Satellite Data
136
to account for several of these problems, they are still most likely the dominant sources
of uncertainty in the retrieved MAESTRO p-T profiles.
Expected versus Actual Signal-to-Noise
Chapter 3 presented the results of several retrievals on MAESTRO data simulated with
realistic noise. While the noise counts used in these simulations are the same as those
observed in these MAESTRO spectra, the signal observed is often not as large as expected. The dynamic range of the detectors is approximately 25000 counts, but many
spectra often only use 10000 counts of the dynamic range. However, spectra are often
produced from co-adding two or three quick successive reads and in these cases there is
some reduction in the expected noise, and while noise levels may be slightly higher than
in simulation, they should still be small enough to allow the retrieval of reasonable p-T
profiles. Noise estimates used to produce the Sǫ measurement error covariance matrix are
derived using the number of co-added spectra, the shot noise estimate, and the random
noise counts estimate.
Pointing
The MAESTRO p-T retrievals currently rely on pointing derived by the FTS, and therefore a poor FTS p-T retrieval usually results in an unreliable MAESTRO retrieval. Most
MAESTRO density profiles, retrieved using either the A or B bands, show an irregularity
near 12 km, the FTS pointing retrieval cross-over point between where p-T is fixed to
the a priori (5 to 12 km) and where the FTS tangent altitudes are shifted (12 to 20 km)
to match those retrieved using the a priori.
Although the density and pressure profiles contained in FTS v2.2 profiles are relatively
smooth, there appears to be an irregularity near 12 km when the FTS altitude versus
temperature profiles are used to produce hydrostatically-consistent pressure and density
Chapter 5. Retrievals from Satellite Data
137
profiles using the MAESTRO hydrostatic code, which may explain the poor MAESTRO
results near 12 km.
MAESTRO could also be used to retrieve pointing, by keeping the p-T profile constant
and retrieving tangent altitudes. However, for the purpose of this work, which aims to
assess MAESTRO’s ability to retrieve p-T, the pointing is assumed to be sufficient. The
persistent feature at 12 km (and another often present near 20 km, to be discussed later)
does, however, prevent the use of the smoothing constraint determination introduced in
Section 3.7 as the algorithm will over-smooth the entire profile in its attempt to remove
the 12 km feature. In order to avoid this over-smoothing, the method of determining the
smoothing constraint weighting γ from the expected degrees of freedom for the retrieval
is applied to the retrievals presented in this chapter, as described in Section 4.3.
There could also be a small offset between the FTS and MAESTRO pointing due to
errors in relative time registration, which would cause sunset and sunrise errors to be
opposite in sign. A small offset was observed between sunset and sunrise residuals and a
timing correction was applied to reduce this offset.
Stray Light Correction
The SCALE processing software described in Section 5.2.2 that corrects the raw detector
data for integration time and dark counts is also used to apply a correction factor for
stray light inside the instrument. The strength of this stray light correction was originally
scaled to ensure that when the modelled stray counts were removed from the spectra, the
core of a typical A band measurement at a low tangent altitude would reach the zerotransmission baseline of the detector. This assumption cannot be used if the A band itself
is to be retrieved. The original stray light-corrected observations from SCALE produced
optical depths that were larger than the first guess optical depths by as much as 20% for
tangent altitudes near 5 km. This caused the A band retrieval algorithm to consistently
retrieve excessive density amounts at low altitudes where the stray light correction had
Chapter 5. Retrievals from Satellite Data
138
the most significant impact on optical depth. The B band’s optical depths are small and
not affected by stray light errors to the same extent as those of the A band.
The results were improved somewhat when the stray light correction was removed
entirely; however, A band optical depth spectra with no stray light correction were then
slightly smaller than expected. It became evident that an improved stray light correction
was required. The refinement of the stray light correction remains an outstanding issue
for the entire MAESTRO analysis and is not addressed in this work; rather, an estimated
stray light uncertainty is used to calculate the resulting systematic error introduced into
the profile.
Readout Time Correction Errors
During the data processing for this work it became apparent that the corrections applied
in the SCALE software which compensate for different pixel integration times on each
pixel were incorrect. These errors are associated with the time it takes for detector
readout. The MAESTRO detector is randomly addressable; a pixel may be integrated
independently and read out independently from the other pixels in the array. For short
integration times, when the light is very bright, each pixel is read individually and then
the entire array is reset to avoid pixel saturation and current leakage from other pixels.
For very long integration times, all pixels on the array are integrated simultaneously and
then each pixel is read out in sequence, with no pixel resets between reads.
The majority of MAESTRO measurements in occultation use mid-range integration
times and a hybrid readout system. The pixels are divided into groups, with the number
of pixels in each group dependent on the expected light intensity. During a measurement,
the pixels in a group are read out in sequence, and then the entire array is reset. The
SCALE program corrects for pixel integration times, but must also account for the difference in readout times, as the last pixel in a group is integrated for longer than the first
pixel. When MAESTRO spectra were first examined before launch, the corrected spectra
Chapter 5. Retrievals from Satellite Data
139
showed a surprising slope in detector counts within a readout group after a simple integration time plus readout time correction was applied. An empirical correction for the
problem was applied in SCALE. To this day the source of the problem remains unclear.
Although this correction may be sufficient for the O3 and aerosol retrievals which are the
focus of the MAESTRO mission, the O2 retrievals are much more sensitive to spectral
errors on the pixel-to-pixel level, and some remaining readout residuals were discovered
during the course of this work. They were found to be particularly strong in spectra
collected in groups of 8 pixels. The current inability to properly correct for readout time
affects all optical depth spectra as the exoatmospheric reference spectra are collected in
8-pixel groups, and particularly those spectra at higher altitudes (above about 40 km)
which are collected using this mode and whose O2 signals are weak.
Although not the cause of the previously mentioned readout errors in the A and B
bands, the readout error associated with the tangent height changing during a single
spectral observation should be mentioned at this point. Although integration times vary
significantly between spectra over an occultation, the number of co-added spectra (usually
two or three) and the number of pixels per readout group changes so that each spectral
observation takes approximately 0.16 seconds. The tangent height can change during this
time from tens of metres in the lower troposphere to as much as 400 metres in the upper
stratosphere, so that the pixel read first at one of the detector sees a different tangent
height from the pixel read last at the opposite end. This effect is not currently accounted
for in the O2 or the operational MAESTRO processing, and is difficult to determine as
spectra are co-added on-board. In the narrow A and B band retrievals its effect will be
insignificant across wavelength, but may have some small effect in absolute pointing that
should eventually be considered.
Chapter 5. Retrievals from Satellite Data
140
A Band Wing Residuals
During the course of data processing, it became clear that the A band retrievals were
consistently poor in the stratosphere, but only when the A band spectra at tropospheric
and lower stratospheric tangents were also included in the global fit. This was linked to
the residuals that remain in the wings of the A band spectral fits, which are most likely
due to remaining uncertainties in the slit function shape and to uncertainties in the stray
light response which will have varying effects over the band’s optical depth.
The weighting functions for the A band in the stratosphere are often more significant
in the wings of the band than in the centre of the band where many absorption lines have
saturated. Often, adding density to a stratospheric layer increases the core absorption,
but causes a more significant decrease in the band wings as the temperature is decreased
and band population changes. This returns to the discussion of weighting functions in
Section 4.2.1, where local density changes within a tangent layer were affecting temperature and therefore the band distribution as shown in pixels 343 and 490 of Figure 4.1.
Although the scale in Figure 4.1 prevents the full illustration of the stratospheric weighting functions, at stratospheric altitudes the A band wing pixels can be more sensitive to
temperature changes induced by changing density (i.e., the ∂T /∂ ln nv term in Equation
3.15 is large) than the central pixels are to changes in total density. This is seen to a
much lesser extent in the B band. This effect can cause the stratospheric layers to adjust
their densities to add or remove A band optical depth in the wings of lower tangent measurements which pass through the stratospheric layers, but are not necessarily tangent
within the stratosphere. Measurements tangent in these stratospheric layers have less
O2 signal and are less constrained by estimated measurement error than those at lower
tangent altitudes. The B band experiences neither these stratospheric adjustments to
tropospheric measurements nor the saturation of the low-resolution signal to the same
extent as the A band, and also shows better fitting residuals at lower altitudes. In these
retrievals, the measurement error vector for the A band is set artificially high below 30
141
Chapter 5. Retrievals from Satellite Data
km, so that the B band is primarily responsible for retrieval results below 30 km. Above
30 km, the retrievals use the combined A and B band approach.
5.5
Pressure and Temperature Results
In this section, sample MAESTRO p-T measurements are presented, with updated averaging kernels and error analysis applied to the real on-orbit MAESTRO data. Profiles
from the analysis dataset presented in Section 5.1 are compared with profiles from the
ACE-FTS instrument, the a priori profiles, radiosondes where available, and operational
analyses from NCEP and ECMWF.
Where comparisons are presented, the pressure differences are percent differences
calculated by
pdif f =
pM AEST RO − pother
1
(p
+ pother )
2 M AEST RO
× 100%,
(5.1)
and similarly for density differences, while the temperature differences are
Tdif f = TM AEST RO − Tother .
5.5.1
(5.2)
Sample Profile Retrievals
Figures 5.8 to 5.12 show single profile retrievals from each group listed in Table 5.1.
Pressure and temperature profiles from all comparison sources are plotted as well as
the differences between MAESTRO and the comparison profiles. While the complete a
priori profile is presented in the pressure and temperature plots to demonstrate the firstguess profile, only the CMC portion of the profile is used in the difference calculations as
the interpolated profile above the CMC analysis does not provide a reliable estimate of
the atmospheric profile. These comparisons are shown on their provided altitude grids.
Differences have been calculated using interpolation, with no smoothing of the profiles
to match other profiles. The minimum tangent height of the MAESTRO occultation,
Chapter 5. Retrievals from Satellite Data
142
zmin , has also been indicated for understanding the influence of the a priori in the lower
altitude region of the retrievals.
Several typical features of the retrieved MAESTRO profiles are demonstrated in these
figures. The MAESTRO mesospheric temperature profiles are much smoother than those
retrieved by the FTS. This is a function of the use of the smoothing constraint parameter
and the estimated noise levels in MAESTRO’s high-tangent-altitude spectra. Reducing
the strength of the smoothing constraint can allow apparently higher vertical resolution
high-altitude measurements; however, improvements in MAESTRO calibration, particularly in the 8-pixel group readout error, are required if these low-O2 signal spectra are to
provide improved resolution at higher altitudes.
Both Figures 5.8 and 5.9 show two persistent features present in many, but not all,
MAESTRO retrievals. A smaller-than-expected signal in both the A and B bands, usually affecting four to five spectra, is often seen in the region from 20 to 25 km as compared
to the MAESTRO optical depths modelled using a fixed a priori or FTS-retrieved profile.
This feature is often accompanied by a too-large signal near 30 km. Although only 5-10%
in optical depth amplitude, this low signal can cause a sharp underestimation in retrieved
density with altitude, which translates to sharp anomalies in pressure and temperature
at these levels. Densities in other layers may then adjust to compensate for the effect this
density underestimation has on other modelled measurements. This feature is common,
but varies in magnitude between occultations. It appears to have little dependency on
instrument detector readout mode, latitude, or time, and remains the least understood
feature of the MAESTRO O2 data. The Arctic summer dataset, where the beta angle
(the angle between the satellite orbital plane and the sun) varies significantly over time
(between 2 and 40 degrees over ten days) suggests there may be some dependency on
beta angle, although the data contains too much scatter to draw definitive conclusions.
A similar problem is often encountered near 12 km, as observed in all the sample profiles,
Chapter 5. Retrievals from Satellite Data
143
except Figure 5.10 where the lowest observation is at 18 km. The 12 km anomaly was
discussed previously in Section 5.4.4.
On the whole, MAESTRO is able to follow the other atmospheric profiles reasonably
well, with pressure differences generally within 5% and temperatures differences within
10 K at altitudes less than 60 km.
Also of interest in the figures in this section are the variations in the NCEP and
ECMWF stratospheric analyses. At altitudes near 40-50 km, these two meteorological
analyses can vary relative to each other by as much as 10% in pressure and 10 K in
temperature when compared on the constant geometric altitude grid, demonstrating the
need for accurate p-T measurements in the upper stratosphere.
5.5.2
Characterization and Error Analysis
Chapter 4 provided a characterization and error analysis for the simulated occultation
ss3004. In this section, updated averaging kernels and an error analysis are provided using
the actual MAESTRO on-orbit measurements from occultation ss3004, whose retrieved
pressure and temperature profiles are presented in Figure 5.11.
Figure 5.13 shows the ln nO2 averaging kernels for occultation ss3004. As described
in Section 5.4.4, the errors on the A band are set artificially high; therefore, the B band
is responsible for most of the retrieved information below 30 km. This causes the effect
visible in Figure 5.13 where the peaks of the averaging kernels decrease with altitude to
30 km, but then increase again above 30 km, where the A band is added to the retrieval.
Overall, the averaging kernels are similar to those of the simulated occultation presented
in Figure 4.3, although the slightly higher noise of the measurement and ss3004’s sparse
measurement points at high tangent altitudes (only about every 5 km above 50 km)
cause somewhat broader averaging kernels and a non-uniformity in the averaging kernel
shape where some retrieved layers are highly dependent on a measurement while others
are more dependent on smoothing.
144
Chapter 5. Retrievals from Satellite Data
FTS
a priori/CMC
NCEP
ECMWF
MAESTRO
sr7891: 29-Jan-2005 (65.5◦ N, 30.7◦ W)
80
80
70
(a)
60
altitude (km)
altitude (km)
70
50
40
30
50
40
30
20
10
10
0
5
ln p (Pa)
15
10
-10
70
altitude (km)
50
40
30
5
10
(d)
60
50
40
30
20
20
150
0
∆ p (%)
70
(b)
60
10
-5
80
80
altitude (km)
60
20
-5
(c)
←zmin = 12 km
200
250
temperature (K)
10
300
-20
-10
0
∆ T (K)
10
20
Figure 5.8: Occultation sr7891 (Arctic winter): Pressure and temperature profiles from
the ACE-FTS, a priori /CMC, NCEP, ECMWF, and MAESTRO showing a) pressure;
b) temperature; c) percent difference with MAESTRO in pressure; and d) difference with
MAESTRO in temperature.
145
Chapter 5. Retrievals from Satellite Data
FTS
a priori/CMC
NCEP
ECMWF
MAESTRO
ss4831: 05-Jul-2004 (65.0◦N, 80.7◦ E)
80
80
70
(a)
60
altitude (km)
altitude (km)
70
50
40
30
50
40
30
20
10
10
0
5
ln p (Pa)
10
-10
15
70
altitude (km)
50
40
30
5
10
(d)
60
50
40
30
20
20
150
0
∆ p (%)
70
(b)
60
10
-5
80
80
altitude (km)
60
20
-5
(c)
←zmin = 9.4 km
200
250
temperature (K)
300
10
-20
-10
0
∆ T (K)
10
20
Figure 5.9: Occultation ss4831 (Arctic summer): Pressure and temperature profiles from
the ACE-FTS, a priori /CMC, NCEP, ECMWF, and MAESTRO showing a) pressure;
b) temperature; c) percent difference with MAESTRO in pressure; and d) difference with
MAESTRO in temperature.
146
Chapter 5. Retrievals from Satellite Data
FTS
a priori/CMC
NCEP
ECMWF
MAESTRO
sr3733: 22-Apr-2004 (3.7◦ S, 49.5◦ W)
80
80
70
(a)
60
altitude (km)
altitude (km)
70
50
40
30
50
40
30
20
10
10
0
5
ln p (Pa)
10
-10
15
70
altitude (km)
50
40
30
5
10
(d)
60
50
40
30
20
←zmin = 18 km
10
10
150
0
∆ p (%)
70
(b)
60
20
-5
80
80
altitude (km)
60
20
-5
(c)
200
250
temperature (K)
300
-20
-10
0
∆ T (K)
10
20
Figure 5.10: Occultation sr3733 (tropics): Pressure and temperature profiles from the
ACE-FTS, a priori /CMC, NCEP, ECMWF, and MAESTRO showing a) pressure; b)
temperature; c) percent difference with MAESTRO in pressure; and d) difference with
MAESTRO in temperature.
147
Chapter 5. Retrievals from Satellite Data
FTS
a priori/CMC
NCEP
ECMWF
radiosonde
MAESTRO
ss3004: 03-Mar-2004 (79.8◦ N, 80.5◦ W)
80
80
70
(a)
60
altitude (km)
altitude (km)
70
50
40
30
50
40
30
20
10
10
0
5
ln p (Pa)
15
10
-10
70
altitude (km)
50
40
30
10
5
(d)
60
50
40
30
20
20
150
0
∆ p (%)
70
(b)
60
10
-5
80
80
altitude (km)
60
20
-5
(c)
←zmin = 4.8 km
200
250
300
temperature (K)
10
-20
-10
0
∆ T (K)
10
20
Figure 5.11: Occultation ss3004 (Eureka 2004): Pressure and temperature profiles from
the ACE-FTS, a priori /CMC, NCEP, ECMWF, a nearby radiosonde, and MAESTRO
showing a) pressure; b) temperature; c) percent difference with MAESTRO in pressure;
and d) difference with MAESTRO in temperature.
148
Chapter 5. Retrievals from Satellite Data
FTS
a priori/CMC
NCEP
ECMWF
radiosonde
MAESTRO
ss8413: 05-Mar-2005 (80.0◦ N, 81.6◦ W)
80
80
70
(a)
60
altitude (km)
altitude (km)
70
50
40
30
50
40
30
20
10
10
0
5
ln p (Pa)
15
10
-10
70
altitude (km)
50
40
30
10
5
(d)
60
50
40
30
20
20
150
0
∆ p (%)
70
(b)
60
10
-5
80
80
altitude (km)
60
20
-5
(c)
←zmin = 4.8 km
200
250
300
temperature (K)
10
-20
-10
0
∆ T (K)
10
20
Figure 5.12: Occultation ss8413 (Eureka 2005): Pressure and temperature profiles from
the ACE-FTS, a priori /CMC, NCEP, ECMWF, a nearby radiosonde, and MAESTRO
showing a) pressure; b) temperature; c) percent difference with MAESTRO in pressure;
and d) difference with MAESTRO in temperature.
149
Chapter 5. Retrievals from Satellite Data
80
PSfrag
70
altitude (km)
60
50
40
30
20
10
0
-0.1
0
0.1
0.2
A
0.3
0.4
Figure 5.13: Averaging kernels of ln nO2 for occultation ss3004, shown for altitudes from
5 to 75 km, spaced every 5 km.
The error analysis originally summarized in Figure 4.14 is repeated here for occultation ss3004 to include the additional error terms considered in Section 5.4.4 and the
actual measurement error from the satellite measurements for the combined A and B
band method used in these retrievals.
The additional systematic errors introduced by uncertainty in the slit function shape
and the stray light correction are illustrated in Figure 5.14. When the deconvolved A
and B band slit functions from Figure 5.5 are stretched to the same FWHM, their shapes
are nearly identical, with the exception of the right-hand side of the slit function, which
appears to fall off much more sharply for the B band. Although it is possible that the slit
shape may change across the detector, the difference in the two slit functions represents
the best knowledge of the uncertainty in the slit function shape. The Kb matrix for the
slit function shape is calculated for the A band by applying the B band’s deconvolved
150
Chapter 5. Retrievals from Satellite Data
slit function (as shown in Figure 5.5), stretched to the A band slit’s FWHM, and vice
versa to calculate the B band Kb matrix.
The stray light error is calculated using the spectra uncorrected by stray light as
compared to those with the original SCALE software stray light correction applied. Recall
the retrievals below 30 km exclusively use the B band; if the A band were used in the
retrievals at low altitudes, the stray light error estimates would be significantly larger.
80
80
80
altitude (km)
(a)
(c)
(b)
60
60
60
40
40
40
20
20
20
0
0
1
∆n (%)
2
0
0
1
∆p (%)
2
0
-2
slit function shape
stray light
0
∆T (K)
2
Figure 5.14: Slit function shape and stray light error in a) density, b) pressure, and c)
temperature for occultation ss3004.
Figure 5.15 shows the total error budget for occultation ss3004, with the individual
contributions from systematic and random error. The systematic error includes uncertainty contributions from the spectroscopic parameters, the slit width discussed in
Section 4.3.2, and the additional uncertainties from the slit function shape and stray
light uncertainties. The random error includes the measurement noise estimates and
pointing uncertainties in this combined A and B band retrieval. The systematic error
is approximately constant in altitude for density and pressure, while the random error
increases with altitude (due to increased measurement noise at high tangent altitudes).
The systematic error dominates the pressure error at approximately 2% over all altitudes.
The calculated systematic error is much smaller than the random error for temperature,
but this may be somewhat misleading when applied to real data. As the temperature is
Chapter 5. Retrievals from Satellite Data
151
based on the gradient of the pressure, any sharp gradients in pressure error will cause
sharp temperature errors. For example, the systematic pressure error increases by about
0.4% near 30 km, which results in a peak in the systematic temperature error of about
2.5 K near that altitude.
The additional pointing and readout time uncertainties discussed in Section 5.4.4 are
not included in this error budget. The error resulting from a 100 metre pointing uncertainty was discussed in Section 4.3.2, and at this time there is no additional quantifiable
uncertainty source from pointing; however, it should be noted that until independent
satellite pointing information is available, MAESTRO retrievals processed using FTS
pointing should be used with caution, and particularly those where FTS profiles show
irregularities. The readout error uncertainty is difficult to quantify. The contribution of
the readout error also appears to vary between occultations taken with different observing
command tables and between spectra within a single table with different row numbers
but exactly the same measurement conditions (i.e., integration time, pixels per group,
etc.). A small realistic perturbation applied to visibly reduce these 8-pixel readout features in a relatively flat region of the ss3004 exoatmospheric spectrum resulted in changes
in retrieved temperatures less than 0.5 K and retrieved pressures between -0.2 and 0.2%;
however, even this correction failed to remove some of the residuals associated with these
errors within the band. This indicates the readout timing error contribution may be less
than those from other error sources, but it continues to be a source of uncertainty that
requires improved detector modelling.
5.5.3
Multiple Profile Comparisons
Comparisons between MAESTRO profiles and the various comparison datasets are shown
in Figures 5.16 through 5.25. Figures 5.16, 5.18, 5.20, 5.22, and 5.24 illustrate the differences between the retrieved number density profiles and the derived pressure and temperature profiles as functions of height as compared to the FTS, CMC, and radiosonde
152
Chapter 5. Retrievals from Satellite Data
80
80
80
altitude (km)
(a)
(c)
(b)
60
60
60
40
40
40
20
20
20
0
0
1 2 3 4 5
∆n (%)
0
0 1 2 3 4 5
∆p (%)
0
0
systematic
random
total
5
10
∆T (K)
15
Figure 5.15: Expected systematic, random, and total errors in a) density, b) pressure,
and c) temperature for occultation ss3004.
profiles where available. As the NCEP and ECMWF analyses are provided on constant
pressure surfaces, with estimated geopotential heights (converted to geometric heights
here) and temperatures, these meteorological comparisons are presented separately in
Figures 5.17, 5.19, 5.21, 5.23, and 5.25 on a pressure-altitude grid, with their approximate corresponding altitudes on the right-hand axis. This allows the coarsely-spaced
meteorological analyses to be compared without interpolating the analyses, and separates these comparisons from the constant grid comparisons for ease of readability. In all
figures in this section, the dashed lines represent the mean differences and the solid lines
represent the root-mean-square (RMS) differences between MAESTRO and the comparison profiles. The dotted lines in Figures 5.16, 5.18, 5.20, 5.22, and 5.24 represent the
errors in the mean of the difference between MAESTRO and the FTS. To avoid overcrowding within the figures, the error in the mean is only shown in MAESTRO/FTS
comparisons, but is generally about 0.5% in pressure and 0.5 K in temperature at low
altitudes, and increases to about 2% in pressure and 2 K in temperature by 80 km. Although some MAESTRO occultations have measurements tangent as low as 4 km, the
mean and RMS differences are only presented at altitudes where there are at least five
profile measurements, and in no case is profile data included below the lowest tangent
Chapter 5. Retrievals from Satellite Data
153
altitude. With the exception of the tropical occultation dataset where only about nine
profiles have measurements tangent below 10 km, the majority of profiles in each dataset
have measurements tangent below 10 km. The comparisons for Eureka 2004 and 2005
are calculated using fewer profiles than the other datasets (25 and 12 respectively) and
the reliability of these results may suffer from a lack of data.
Ideally the mean difference represented by the dashed lines in these figures would be
zero. Any deviation from zero indicates a bias in the results. In addition, the extent
of the biased mean is often mirrored in the RMS difference (see for example Figure
5.18 where the bias difference appears to be the dominating contribution to the RMS
temperature difference peaks between MAESTRO and the FTS and CMC near 20 and
30 km). Although the mean differences represent biases in the retrieved profiles, the
RMS differences are indicators of the general accuracy of the retrieval. While the results
presented here are RMS differences, and not RMS errors, the CMC, NCEP, and ECMWF
analyses can be considered better validated than MAESTRO, particularly below 30 km
where the analyses have been validated against radiosonde data.
Although pressure and temperature are the products of interest, the number density
differences are also plotted between MAESTRO and the FTS and CMC. As number
density is the retrieved product and pressure and temperature are derived from that
retrieval, features in the number density differences may be more indicative of retrieval
performance.
Several general features can be observed among the MAESTRO and FTS/CMC comparison plots. At first glance, the high-altitude density and pressure mean differences
appear quite different between the different datasets. This is a result of the shape of the
logarithmic density profile. If high-altitude O2 signals are weak, the smoothing constraint
tends to push the retrieval toward a linear ln n profile, even though the true ln n profile
is not entirely linear and has a shape depending on latitude and season (for example, see
panel (a) in Figures 5.8 through 5.12 for the various resulting ln p profiles). The smooth-
Chapter 5. Retrievals from Satellite Data
154
ing constraint is not very significant in this respect below altitudes of about 50 to 60 km
in these measurements. An underestimation or overestimation of density at high altitudes
may propagate into a pressure bias at low altitude, although the exponential increase of
atmospheric density usually prevents these errors from significant propagation.
The difference between MAESTRO and FTS densities at 12 km can be observed in
each comparison. Also, the mean and RMS differences between MAESTRO and the CMC
and the FTS temperature profiles appear almost identical. There are some significant
biases in the mean differences below 30 km that are due entirely to measurement (the
smoothing constraint is very weak here) from the unexplained biases in signal near 20 to
30 km, and the overcompensation of the density retrieval for those errors at other levels.
In individual profiles, these errors are very sharp (again, see Figures 5.8 and 5.9), but
as they occur at somewhat varying altitudes, they may be removed in the mean density
profile, although broader resulting biases are apparent in several of the mean densities
and temperatures.
According to the RMS differences, generally the MAESTRO pressures agree with the
FTS and CMC profiles, to within 2 to 5% below about 50 km, and the temperatures agree
to within 5 to 10 K over most altitudes. The largest RMS differences appear in the two
2005 winter/spring datasets (Arctic winter and Eureka 2005). The profiles from these
two datasets were collected during the Arctic winter and spring between 27-January-2005
and 08-March-2005. The Arctic winter dataset contains occultations at several latitudes,
both inside and outside of the polar vortex. In 2004, the Eureka campaign site lay
outside of the polar vortex for the duration of the campaign [110], while in 2005, the
site lay inside the vortex until February 25 and on the edge until March 7 (K. Walker,
personal communication, 2006). The mean of the MAESTRO and FTS profiles and
their standard deviations derived from occultations within 500 km of the Eureka groundbased ACE validation campaign site are shown in Figure 5.26 for the 2004 and 2005
campaigns. The temperature profiles collected during the Eureka 2004 campaign indicate
Chapter 5. Retrievals from Satellite Data
155
little variability in the atmospheric temperature profile during this polar spring, while
those from the Eureka 2005 campaign showed considerably more variability, and a much
different temperature profile. The atmospheres in the 2003-2004 and 2004-2005 Arctic
winters behaved very differently. On average, the 2003-2004 Arctic lower stratosphere
winter was remarkably warm but with unusually cold middle stratospheric temperatures
after February [115], while the 2004-2005 winter was the coldest Arctic winter on record
but with significant variations over time [116]. Several FTS temperature profiles from the
two 2005 datasets show small oscillations and spikes which are most likely also indicative
of poor tangent height retrieval, which usually results in poor MAESTRO retrievals.
Retrievals under these conditions may suffer from a geometric characteristic of limb
measurements; although ACE measurements are assigned to the 30 km sub-tangent point,
both the sub-tangent track of an occultation and an individual ray’s path may cover a
significant amount of the Earth’s surface, and may travel partially in and out of the
vortex.
The MAESTRO comparisons with the NCEP and ECMWF analyses are similar to
those with the FTS and CMC. The agreement between the FTS and the analyses tends to
be better than that between MAESTRO and the analyses, although MAESTRO generally
shows agreement within 0.5 km in geometric height and 5-10 K in temperature on the
pressure grid. NCEP and ECMWF temperatures are usually within 2-5 K of the FTS
retrievals, although their means can diverge from the FTS by as much as 10 K in the
Arctic winter cases at high altitudes, and from each other by several degrees.
The large uncertainties between stratospheric meteorological analyses are illustrated
in Figure 5.27, which shows the mean differences between ECMWF and NCEP geometric altitudes and temperatures for the pressure levels common between the two analyses
and for the datasets used in the MAESTRO analysis. Although there is fairly good
agreement below 35 km (less than 2 K biases), the analyses show biases of about 5 K
above 35 km, and as high as 20 K in the case of the Eureka 2004 dataset. The remaining
Chapter 5. Retrievals from Satellite Data
156
biases between these two widely-used stratospheric analyses highlight the importance of
independent ACE-FTS and MAESTRO measurements in the upper stratosphere. In addition, since the errors in MAESTRO stratospheric p-T profiles are likely mostly random,
these measured MAESTRO profiles may eventually assist in the assessment of biases in
the upper stratospheric analyses that may be introduced by the assimilation of radiances
from meteorological satellite instruments into the analysis models.
157
altitude (km)
Chapter 5. Retrievals from Satellite Data
80
80
80
70
70
70
60
60
60
50
50
50
40
40
40
30
30
30
20
20
20
10
10
10
FTS
-10
0
10
∆ n (%)
-10
0
10
∆ p (%)
CMC
-10
0
10
∆ T (K)
Figure 5.16: Arctic winter occultations: mean (dashed) and RMS (solid) differences between MAESTRO and comparison profiles, and error in MAESTRO/FTS mean (dotted).
1
51
10
34
100
17
1000
-1
-0.5
0
0.5
∆ z (km)
1
-20
-10
0
10
∆ T (K)
approximate altitude (km)
pressure (hPa)
MAESTRO - NCEP
MAESTRO - ECMWF
FTS - NCEP
FTS - ECMWF
0
20
Figure 5.17: Arctic winter occultations: mean (dashed) and RMS (solid) differences
between MAESTRO or FTS and meteorological analyses, on constant pressure levels.
158
altitude (km)
Chapter 5. Retrievals from Satellite Data
80
80
80
70
70
70
60
60
60
50
50
50
40
40
40
30
30
30
20
20
20
10
10
10
FTS
-10
0
10
∆ n (%)
-10
0
10
∆ p (%)
CMC
-10
0
10
∆ T (K)
Figure 5.18: Arctic summer occultations: mean (dashed) and RMS (solid) differences
between MAESTRO and comparison profiles, and error in MAESTRO/FTS mean (dotted).
1
51
10
34
100
17
1000
-1
-0.5
0
0.5
∆ z (km)
1
-20
-10
0
10
∆ T (K)
approximate altitude (km)
pressure (hPa)
MAESTRO - NCEP
MAESTRO - ECMWF
FTS - NCEP
FTS - ECMWF
0
20
Figure 5.19: Arctic summer occultations: mean (dashed) and RMS (solid) differences
between MAESTRO or FTS and meteorological analyses, on constant pressure levels.
159
altitude (km)
Chapter 5. Retrievals from Satellite Data
80
80
80
70
70
70
60
60
60
50
50
50
40
40
40
30
30
30
20
20
20
10
10
10
FTS
-10
0
10
∆ n (%)
-10
CMC
-10
0
10
∆ p (%)
10
0
∆ T (K)
Figure 5.20: Tropical occultations: mean (dashed) and RMS (solid) differences between
MAESTRO and comparison profiles, and error in MAESTRO/FTS mean (dotted).
1
51
10
34
100
17
1000
-1
-0.5
0
0.5
∆ z (km)
1
-20
-10
0
10
∆ T (K)
approximate altitude (km)
pressure (hPa)
MAESTRO - NCEP
MAESTRO - ECMWF
FTS - NCEP
FTS - ECMWF
0
20
Figure 5.21: Tropical occultations: mean (dashed) and RMS (solid) differences between
MAESTRO or FTS and meteorological analyses, on constant pressure levels.
160
altitude (km)
Chapter 5. Retrievals from Satellite Data
80
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80
70
70
70
60
60
60
50
50
50
40
40
40
30
30
30
20
20
20
10
10
10
-10
Figure 5.22:
0
10
∆ n (%)
Eureka 2004:
-10
0
10
∆ p (%)
FTS
CMC
radiosonde
-10
0
10
∆ T (K)
mean (dashed) and RMS (solid) differences between
MAESTRO and comparison profiles, and error in MAESTRO/FTS mean (dotted).
1
51
10
34
100
17
1000
-1
Figure 5.23:
-0.5
0
0.5
∆ z (km)
Eureka 2004:
1
-20
-10
0
10
∆ T (K)
approximate altitude (km)
pressure (hPa)
MAESTRO - NCEP
MAESTRO - ECMWF
FTS - NCEP
FTS - ECMWF
0
20
mean (dashed) and RMS (solid) differences between
MAESTRO or FTS and meteorological analyses, on constant pressure levels.
161
altitude (km)
Chapter 5. Retrievals from Satellite Data
80
80
80
70
70
70
60
60
60
50
50
50
40
40
40
30
30
30
20
20
20
10
10
10
-10
Figure 5.24:
0
10
∆ n (%)
Eureka 2005:
-10
0
10
∆ p (%)
FTS
CMC
radiosonde
-10
0
10
∆ T (K)
mean (dashed) and RMS (solid) differences between
MAESTRO and comparison profiles, and error in MAESTRO/FTS mean (dotted).
1
51
10
34
100
17
1000
-1
Figure 5.25:
-0.5
0
0.5
∆ z (km)
Eureka 2005:
1
-20
-10
0
10
∆ T (K)
approximate altitude (km)
pressure (hPa)
MAESTRO - NCEP
MAESTRO - ECMWF
FTS - NCEP
FTS - ECMWF
0
20
mean (dashed) and RMS (solid) differences between
MAESTRO or FTS and meteorological analyses, on constant pressure levels.
162
Chapter 5. Retrievals from Satellite Data
Eureka 2004
80
altitude (km)
70
60
50
40
30
MAESTRO
FTS
radiosondes
20
10
180
200
220
240
260
temperature (K)
280
300
Eureka 2005
80
altitude (km)
70
60
50
40
30
MAESTRO
FTS
radiosondes
20
10
180
200
220
260
240
temperature (K)
280
300
Figure 5.26: Mean (solid) temperature profiles and their standard deviations representing
variability (dashed) for Eureka 2004 and 2005 occultations within 500 km of Eureka
campaign site, retrieved from MAESTRO, the ACE-FTS, and coincident radiosondes.
163
1
51
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100
-1
-0.5
zECM W F
0
0.5
1
− zN CEP (km)
-20 -10 0
10 20
TECM W F − TN CEP (K)
Arctic winter
Arctic summer
tropics
Eureka 2004
Eureka 2005
approximate altitude (km)
pressure (hPa)
Chapter 5. Retrievals from Satellite Data
17
Figure 5.27: Mean differences between ECMWF and NCEP geometric altitudes and
temperatures.
5.6
Discussion
This chapter presented the results from applying the O2 A and B band retrievals for pressure and temperature determination to real data collected from the MAESTRO satellite
instrument. The input data for instrument pointing from the FTS was discussed, as well
as the spectral input from the MAESTRO instrument. The data processing was presented together with a description of pixel-wavelength assignments, spectral fitting, the
method used to determine improved slit functions for the analysis, and several significant
analysis issues that were discovered during the O2 retrievals. Sample retrieved profiles
were shown with sample retrieval characterization and error analysis for one profile, and
results were presented from comparing 230 MAESTRO profiles with FTS, a priori /CMC,
NCEP, ECMWF, and radiosonde data.
Chapter 5. Retrievals from Satellite Data
164
The O2 p-T retrievals are very sensitive to instrument characterization and noise
parameters. As a result, several problems in MAESTRO instrument characterization
were discovered during the data processing. A very significant problem in the data
processing was the lack of knowledge of the slit function shape. Using the original guess
for a slit function, as measured in the laboratory pre-flight, caused significant residuals
between the modelled and observed optical depths. Improved slit function shapes were
derived by deconvolution from the A and B band observations at low tangent altitudes.
As a result, retrieval results are unfortunately not completely independent of the slit
function determination. However, the A and B band slit function shapes are consistent
with each other, and at the very least an improved slit function has been provided for
the retrieval of other atmospheric constituents from MAESTRO. Another significant
problem with the MAESTRO analysis was the stray light correction originally applied in
the raw data processing and included in the input to these p-T analyses. The stray light
correction has been removed from the spectra for these retrievals, and the associated
uncertainty in the spectra is applied in the error analysis. The spectral data also shows
some biases related to inaccurate correction of pixel readout times, and a slightly higher
level of noise than that used in the simulated retrievals of Chapters 3 and 4. Residuals
in the wings of the A band fit remain, most likely still due to an uncertainty in the A
band slit function, and as a result the retrievals are performed using only the B band
below 30 km and a combined A and B band retrieval above 30 km. A significant problem
with the O2 retrievals is an apparent forward model underestimation of the signal near
30 km and a larger overestimation of the signal near 20-25 km. The density profile
adjusts itself accordingly, which often results in poor retrievals near 20-30 km. The
source of this error has not yet been discovered, and could be a problem with instrument
characterization or pointing. Although the FTS pointing retrievals are likely quite good,
without independent pointing information it is difficult to separate possible pointing
problems from MAESTRO characterization or O2 spectroscopic issues.
Chapter 5. Retrievals from Satellite Data
165
Retrieved MAESTRO profiles generally follow the FTS retrievals and other comparisons reasonably well, although there remain significant biases in the retrieved profiles,
and the RMS differences between MAESTRO and the comparison profiles are somewhat larger than predicted by error analysis, with RMS pressure differences ranging
from about 2-10% and temperature differences from 5-10 K. The RMS differences remain highly influenced by large biases at certain altitudes. The updated error analysis
for occultation ss3004 demonstrates the theoretical pressure and temperature errors that
should be achievable with MAESTRO as being less than 2% in pressure and 2-5 K below
about 45 km, and increasing to 10 K by 80 km.
Chapter 6
Conclusions
6.1
Summary of Results and Achievements
The goal of this research was to develop and test a method for retrieving atmospheric
pressure and temperature profiles from the MAESTRO satellite instrument using the
absorption of molecular oxygen. This was achieved through the development of forward model and retrieval algorithms and tests on simulated and real satellite data. To
our knowledge, this work presents the first assessment of using the oxygen B band for
retrieving pressure and temperature profiles from an occultation instrument, and also
includes an analysis of the A band and combined A and B band retrievals for pressure
and temperature measurements applied to MAESTRO.
6.1.1
Forward Model
One of the main challenges of modelling the high spectral resolution O2 bands for these
low-resolution retrievals was to develop a high-accuracy forward model that could model
the many saturated absorption lines of O2 , but would be computationally fast enough
to allow the calculation of analytical weighting functions for the retrieval. A correlatedk technique was applied to the solar occultation case that allowed the inclusion of the
166
Chapter 6. Conclusions
167
slit function information within the correlated-k distribution functions, thereby negating
the need for slit function convolutions within the algorithm and drastically increasing the
computational speed. The addition of a fast-line-by-line code to the forward model, while
using the correlated-k method to calculate weighting functions, allowed the forward modelling error from radiative transfer approximations to be negligible. Using these radiative
transfer approximations, a complete forward model was constructed for MAESTRO that
includes radiative transfer, instrument characteristics, and occultation geometry. The
forward model is highly accurate and its limitations are due only to the uncertainties in
the input data. The forward model is general enough that it can be used for any region
of the spectrum and for either MAESTRO spectrometer.
6.1.2
Retrieval Algorithm
A global fitting algorithm was developed to allow the retrieval of an O2 density profile
from a series of MAESTRO occultation spectra using either the A, B, or combined
band retrievals. The density profile is then used to derive the pressure and temperature
profiles through the hydrostatic equation and ideal gas law. This retrieval algorithm
uses a nonlinear least-squares iterative estimator with modified optimal estimation and
a smoothing constraint to reduce noise-driven oscillations at high altitudes where the O2
signal is weak.
Retrievals performed using the retrieval algorithm and simulated observations with
realistic measurement noise indicate that combined A and B band retrievals should be
able to determine pressures to within 1% and temperatures to within 2 K of the true
profile over altitudes up to 80 km.
A formal characterization and error analysis was also performed using the retrieval
algorithm and simulated MAESTRO data. The error analysis of model and noise error
showed that model parameter error tends to dominate in the lower atmosphere, while
measurement noise dominates at high altitudes where the O2 signal is weak. Although
Chapter 6. Conclusions
168
uncertainties in the spectroscopy of O2 and slit function width contribute significantly
to the pressure error budget, random uncertainties in the pointing tend to dominate the
model parameter temperature error budget. Error analysis with simulated data shows
that densities and pressures should be able to be retrieved to within 1% and temperatures
to within 2 K over most altitudes.
The retrieval algorithm is general enough that it can be applied to retrieve all MAESTRO
constituents through all spectral regions and for both spectrometers. In the future, it
will most likely be implemented as the MAESTRO global fitting algorithm for retrievals
of all observed constituents.
6.1.3
Satellite Data Analysis
The pressure and temperature retrievals have also been tested using data acquired by
the satellite instrument on-orbit. Comparisons of MAESTRO profiles with the ACEFTS, radiosondes, and meteorological analyses from the Canadian Meteorological Centre
(CMC), National Centers for Environmental Prediction (NCEP), and European Centre
for Medium-range Weather Forecasts (ECMWF) yield encouraging results. Biases in the
measured pressures are usually within 2-5%, and biases in temperature less than 5 K,
while RMS differences are usually 2-10% in pressure and 5-10 K in temperature. An
updated formal error analysis indicates a 2% uncertainty in pressure and 2-10 K (25 K below 45 km and 5-10 K above) uncertainty in temperature should be achievable
with current model parameter uncertainty estimates. The discrepancies between the
predicted temperature errors and the observed temperature differences are due mostly to
remaining biases in the retrieved profiles. Several instrument characterization problems
were discovered during the course of the retrievals, some of which remain outstanding
issues in MAESTRO processing. These include uncertainties in slit function shape, stray
light, and readout timing correction errors. Satellite pointing is derived from the FTS
retrievals and any errors in the pointing will have a significant negative impact on the
Chapter 6. Conclusions
169
MAESTRO results. An additional discrepancy appears in many occultations where the
O2 signal appears over- and underestimated in peaks near 20-30 km, resulting in retrieval
errors near these altitudes. These possible data anomalies are not accounted for in the
error analysis and are the main contributors to bias in the lower atmospheric retrievals.
This work has added MAESTRO to the three occultation instruments, ILAS, ILAS-II,
and SAGE III, which use the A band to retrieve p-T profiles from space. Both ILAS and
SAGE III results have indicated how challenging this retrieval can be. The MAESTRO
results have demonstrated that reasonable profiles can be derived from an even lower
spectral resolution occultation instrument, and have added the use of the B band for
retrievals. Improvements in pointing and instrument characterization can only add to
the reliability and usefulness of the results.
The B band has proven to be more useful than originally anticipated. While it is
much weaker than the A band and in a region of higher ozone absorption and aerosol
and Rayleigh scattering, its core does not saturate to the same extent as that of the
A band, and it is less sensitive to some of the instrument characterization issues still
affecting the A band retrievals at low tangent altitudes. As a result, the B band was
used to retrieve at altitudes lower than 30 km, and a combined A and B band retrieval was
used at higher altitudes. Without the inclusion of the B band, it would have been nearly
impossible with current instrument knowledge to determine pressure and temperature
over all altitudes with much reliability.
6.1.4
Implications for MAESTRO Data Products
While pressure and temperature may be interesting products by themselves, the emphasis
of the mission is on the retrieval of ozone, NO2 , and aerosols. Even with the preliminary
densities retrieved in this work, the O2 density retrievals would allow mixing ratios to
be determined with errors within 2-5% for other gaseous species (not including uncertainties from the cross-sections or their temperature dependencies which would be need
Chapter 6. Conclusions
170
to be added to any error budget). From the results presented in Chapter 5, O2 bands
could theoretically be used to derive satellite pointing altitudes within 0.5 km over most
trospospheric and stratospheric altitudes using a fixed pressure-temperature profile if the
FTS spectrometer were to fail but the suntracker continued to operate. This redundancy
in pointing retrieval is particularly important at the moment; as of early 2006, with the
POAM (Polar Ozone and Aerosol Measurement) III, SAGE III, and HALOE missions
all recently ending, ACE is the only primarily solar occultation payload in orbit.
6.2
Suggestions for Future Work
Spectroscopy
There remain uncertainties in the spectroscopic parameters of the O2 A and B bands
which would benefit from more laboratory analysis of O2 spectroscopy. The B band
spectroscopic constants contained in HITRAN were collected over thirty years ago and
there is little information on uncertainty estimates in the B band parameters, while the
recent results from Yang et al. [101] and Tran et al. [102] show that even the spectroscopy
of the A band is still not sufficiently characterized.
Instrument Characterization and Pointing
A major difficulty in the current MAESTRO analysis is the remaining uncertainty in
instrument characterization. The calibration of the raw spectra is still under revision.
New corrections are currently being tested and raw spectra re-calibrated, and MAESTRO
retrievals need to be reprocessed using any revisions. In addition, there exists some
currently unexamined data from pre-flight characterization tests that could be useful for
developing an improved stray light model for the instrument. Ideally the MAESTRO
retrievals will eventually use pointing information derived from satellite sensors and not
Chapter 6. Conclusions
171
from the FTS, so that its retrievals will become completely independent of those of the
FTS.
Validation
While these MAESTRO results have been compared with the ACE-FTS, radiosondes, and
three meteorological analysis, there are several instruments that have made measurements
that could be used for pressure-temperature comparisons when coincident observations
occur. These include HALOE, SAGE III, and GPS missions, among others. Comparisons
with SAGE III would be particularly interesting as spectra could also be compared.
Algorithms
The eventual goal is to migrate these retrieval algorithms to be used operationally in a
global fit of all constituents during each occultation.
6.3
Recommendations for Future Missions
The importance of instrument characterization cannot be emphasized enough for future
versions of MAESTRO or similar instruments. A proper characterization of the slit
function is particularly important, both for understanding how the slit function width
changes across the detector and for the absolute shape of the slit function. Laboratory
measurements of O2 in a cell made by the instrument pre-flight, perhaps with simultaneous measurements using a high spectral resolution FTS, could provide data which
would be useful for understanding instrument characterization, spectroscopic and forward
modelling issues, and any changes in the instrument after launch.
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Appendix A
List of Acronyms
ACE
Atmospheric Chemistry Experiment
A/D
Analogue/Digital
ADEOS
Advanced Earth Observing Satellite
AMSU
Advanced Microwave Sounding Unit
ATLAS
Atmospheric Laboratory for Applications and Sciences
CCD
Charge-Coupled Device
CMC
Canadian Meteorological Centre
CPC
Climate Prediction Center
CPFM
Composition and Photodissociative Flux Measurement
ECMWF
European Centre for Medium-range Weather Forecasts
FLBL
Fast-Line-By-Line
FOV
Field-Of-View
FTS
Fourier Transform Spectrometer
FWHM
Full-Width-at-Half-Maximum
GEM
Global Environmental Multiscale
GOME
Global Ozone Monitoring Experiment
GPS
Global Positioning System
187
Appendix A. List of Acronyms
HALOE
HALogen Occultation Experiment
HITRAN
HIgh-resolution TRANsmission molecular absorption database
HRDI
High Resolution Doppler Imager
ILAS
Improved Limb Atmospheric Spectrometer
IR
Infrared
LBL
Line-By-Line
MAESTRO
Measurement of Aerosol Extinction in the Stratosphere and
188
Troposphere Retrieved by Occultation
MAS
Millimeter Wave Atmospheric Sounder
MERIS
MEdium Resolution Imaging Spectrometer
MLS
Microwave Limb Sounder
MODTRAN
MODerate spectral resolution atmospheric TRANsmittance
algorithm and computer model
MOS
Modular Optoelectronic Scanner
MSC
Meteorological Service of Canada
NCEP
National Centers for Environmental Prediction
NIR
Near-Infrared
NRL-MSISE
Naval Research Laboratory, Mass Spectrometer Incoherent Scatter
Radar Extended Model
OCO
Orbiting Carbon Observatory
O.D.
Optical Depth
p-T
pressure-temperature
POAM
Polar Ozone and Aerosol Measurement
POLDER
POLarization and Directionality of the Earth’s Reflectances
PSC
Polar Stratospheric Cloud
RMS
Root-Mean-Square
SABER
Sounding of the Atmosphere using Broadband Emission Radiometry
Appendix A. List of Acronyms
SAGE
Stratospheric Aerosol and Gas Experiment
SCIAMACHY
SCanning Imaging Absorption SpectroMeter for Atmospheric
CHartographY
SMR
Sub-Millimetre Radiometer
SNR
Signal-to-Noise Ratio
SOLSPEC
SOLar SPECtrum
SOSP
SOlar SPectrum
TOGA
Tropical Ocean and Global Atmosphere
UARS
Upper Atmosphere Research Satellite
UV
Ultraviolet
VIS
Visible
189
Appendix B
Model Parameter Weighting
Functions
A band
B band
100
100
pixel 329
λ = 685.7 nm
50
0
0
0.005 0.01 0.015 0.02
altitude (km)
pixel 333
λ = 687.7 nm
50
0
0
100
0
0.02
0.04
0.06
0.08
0
0.005 0.01 0.015 0.02
0
0
0
100
0.02
0.005 0.01 0.015 0.02
0
0.04
0.06
0.08
pixel 485
λ = 765.0 nm
50
100
50
pixel 480
λ = 762.5 nm
50
pixel 338
λ = 690.3 nm
50
0
0
100
100
0
pixel 475
λ = 760.0 nm
50
0
0.02
0.04
0.06
0.08
100
pixel 343
λ = 692.9 nm
50
0.005 0.01 0.015 0.02
Kb (∆τ /∆b)
0
0
pixel 490
λ = 767.5 nm
0.02 0.04 0.06
Kb (∆τ /∆b)
0.08
Figure B.1: Model sensitivity Kb to a 2.5% perturbation in O2 air-broadened half-widths.
190
Appendix B. Model Parameter Weighting Functions
B band
pixel 329
λ = 685.7 nm
50
0
0.005 0.01 0.015 0.02
altitude (km)
pixel 333
λ = 687.7 nm
50
0
0
100
0.005 0.01 0.015 0.02
pixel 338
λ = 690.3 nm
50
0
0.005 0.01 0.015 0.02
0
0
0.005 0.01 0.015 0.02
50
0
pixel 480
λ = 762.5 nm
50
0
0
100
pixel 485
λ = 765.0 nm
50
0
0.005 0.01 0.015 0.02
0
0.005 0.01 0.015 0.02
100
100
0
pixel 475
λ = 760.0 nm
50
100
100
0
A band
100
100
0
191
pixel 343
λ = 692.9 nm
0.005 0.01 0.015 0.02
Kb (∆τ /∆b)
50
0
0
pixel 490
λ = 767.5 nm
0.005 0.01 0.015 0.02
Kb (∆τ /∆b)
Figure B.2: Model sensitivity Kb to a 2.5% perturbation in O2 A-band self-broadened
half-widths and a 10% perturbation in O2 B-band self-broadened half-widths.
192
Appendix B. Model Parameter Weighting Functions
B band
A band
100
100
pixel 329
λ = 685.7 nm
50
0
0
0.005 0.01 0.015 0.02
altitude (km)
100
pixel 333
λ = 687.7 nm
0
0
100
0.005 0.01 0.015 0.02
pixel 338
λ = 690.3 nm
50
0
0.005 0.01 0.015 0.02
0
0.02
0.04
0.06
0.08
pixel 343
λ = 692.9 nm
50
0
pixel 480
λ = 762.5 nm
50
0
0
100
0.02
0.04
0.08
pixel 485
λ = 765.0 nm
50
0
0.06
0
0.02
0.04
0.06
0.08
100
100
0
0
100
50
0
pixel 475
λ = 760.0 nm
50
0.005 0.01 0.015 0.02
Kb (∆τ /∆b)
pixel 490
λ = 767.5 nm
50
0
0
0.02 0.04 0.06
Kb (∆τ /∆b)
0.08
Figure B.3: Model sensitivity Kb to a 15% perturbation in O2 line width temperature
dependency.
A band
B band
100
100
pixel 329
λ = 685.7 nm
50
0
0
0.05
0.1
altitude (km)
pixel 333
λ = 687.7 nm
50
0
0.05
0.1
100
pixel 338
λ = 690.3 nm
0
0.05
0.1
100
0.05
0.1
0.15
0.2
pixel 480
λ = 762.5 nm
50
0
0
0.05
0.1
0.15
0.2
pixel 485
λ = 765.0 nm
50
0
0
0.05
0.1
0.15
0.2
100
pixel 343
λ = 692.9 nm
50
0
0
100
50
0
0
100
100
0
pixel 475
λ = 760.0 nm
50
0
0.05
Kb (∆τ /∆b)
0.1
pixel 490
λ = 767.5 nm
50
0
0
0.05 0.1 0.15
Kb (∆τ /∆b)
0.2
Figure B.4: Model sensitivity Kb to a 0.1 nm perturbation in the slit function width.
193
Appendix B. Model Parameter Weighting Functions
A band
B band
100
100
pixel 329
λ = 685.7 nm
50
0
0
0.01
0.02
0.03
altitude (km)
pixel 333
λ = 687.7 nm
50
0
0.01
0.02
0.03
pixel 338
λ = 690.3 nm
50
0
0.01
0.02
0.03
100
50
0
0
0.05
0.1
pixel 480
λ = 762.5 nm
50
0
0
0.05
0.1
100
100
0
0
100
100
0
pixel 475
λ = 760.0 nm
50
0
pixel 343
λ = 692.9 nm
0.01
0.02
Kb (∆τ /∆b)
0.03
pixel 485
λ = 765.0 nm
50
0
0
100
0.1
pixel 490
λ = 767.5 nm
50
0
0.05
0
0.05
0.1
Kb (∆τ /∆b)
Figure B.5: Model sensitivity Kb to a 0.002◦ perturbation in pointing angle.