CO2 Profile Retrievals Using a Single Absorption Line in the Near Infrared
Run-Lie Shia1, King-Fai Li2,3, Le Kuai4, Michael R. Line1, John T. Trauger4 and Yuk
L.Yung1
1
Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA,
USA
2
Department of Applied Mathematics, University of Washington, Seattle, WA, USA
3
University Corporation for Atmospheric Research, Boulder, CO, USA
4
Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena,
CA, USA
To be submitted to J.Q.S.R.T.
1
Abstract
Current satellite instruments observing the atmospheric carbon dioxide (CO2) have been mostly
designed for measuring the total column CO2 abundance using broad CO2 absorption bands. Here
we investigate the required spectral resolution (  ) and spectral noise level ( S/N ) of an
instrument for measuring the CO2 concentrations at multiple atmospheric layers using only one
CO2 absorption line, which will provide important information about the CO2 transport in the lower
atmosphere. A simplified analytic radiative transfer model of the CO2 transmission in the reflected
near-infrared solar spectrum and the Bayesian optimal estimation are used to calculate the gain in
the degrees of freedom and the information contents against the a prori knowledge of the CO2
variability. For a retrieval of two CO2 partial columns below and above 6 km, the required 
and S/N are 0.01 cm–1 and 1000, respectively; for a retrieval of three CO2 partial columns in the
layers 0–2.3 km, 2.3–9.8 km, and above 9.8 km, the required  and S/N are 0.002–0.004 cm–1
and 1700–2400, respectively. These requirements are a few times better than those of the operating
satellite instruments. The uncertainty of the retrieved CO2 is 2% near the surface and 1% elsewhere.
2
1. Introduction
Atmospheric carbon dioxide (CO2) is the most important anthropogenic greenhouse gas, whose
increase in the atmosphere is the main cause of the rise in the globally averaged surface
temperature after the Industrialization [1]. Satellite measurements have been providing global CO2
distributions that are necessary for estimating natural and anthropogenic sources/sinks of CO2 in
order to contain the growth of the atmospheric CO2 and the related climate change. Current
spaceborne instruments, such as the Greenhouse Gases Observing Satellite (GOSAT) [2-4] and
the Orbiting Carbon Observatory-2 (OCO-2) [5], have been specifically designed to measure the
column-averaged CO2,  CO2 , at an accuracy better than 0.3% or 1.5 ppmv for monitoring the
surface CO2 flux [6]. These instruments, however, still cannot resolve the vertical variations of
CO2 in the troposphere.
The CO2 vertical profile contains rich information about the combined effects of surface fluxes
and transport on various spatiotemporal scales [7-9]. For example, the CO2 vertical gradient at low
latitudes are related to the interhemispheric transports due to the imbalanced biogenic sources in
the northern and southern hemispheres [10, 11]. These knowledge thus help diagnose the chemistry
transport models that have been used for inverse modeling of the CO2 fluxes [12-15]. The major
difficulty of measuring the CO2 vertical profile from space is that the CO2 vertical variation is
small, usually less than 3% or 10 ppmv except in the boundary layers (≤ 1 km) near urban cities
where diurnal changes can be as high as 100 ppmv during rush hours [16], and that the weighting
functions (defined in Section 2) for the tropospheric CO2 usually has a broad vertical peak of ~few
km, losing the sensitivity to resolve the CO2 concentrations in finer layers. Furthermore, the
3
spectral resolutions and the spectral noise level of the current spaceborne instruments do not satisfy
the requirements for a profile retrieval. To overcome these difficulties, a spectral instrument of
high signal-to-noise ratio (S/N) and high spectral resolution is desired. The present study aims to
demonstrate the feasibility of retrieving CO2 vertical profile at multiple atmospheric levels by
quantifying the S/N ratio and the spectral resolution that are required to retrieval CO 2
concentrations at more than one levels using a Bayesian approach.
A profile retrieval is facilitated by the fact that the molecular absorptions of radiation is dependent
on the ambient pressure, i.e. the effect of the pressure broadening. In principle, a single absorption
line contains all necessary information of the vertical variations of CO2 along its line shape: The
line center is more sensitive to the CO2 concentrations at high altitudes (e.g. stratosphere) where
the pressure broadening is small while the line wing is more sensitive to the CO2 concentration
near the surface where the pressure broadening is stronger. The line strength is also dependent on
the ambient temperature but the effect on the line shape is smaller than that of the pressure
broadening. Ideally, the CO2 vertical profile can be retrieved by modeling the CO2 absorptions
from the line center to the line wing. This idea of retrieving the CO2 concentration using a single
absorption line has recently been employed by the Active Sensing of CO2 Emissions over Nights,
Days, and Seasons (ASCENDS) team in developing a spaceborne lidar instrument [17, 18]. The
ASCEND instrument has been designed to retrieve the CO2 concentrations at eight points on a
target CO2 absorption line in the solar NIR spectrum that are chosen to convey the CO 2 vertical
variations near the surface and at a few altitudes in the troposphere and the stratosphere [18].
However, since the weighting functions of these eight frequencies are very broad as discussed
earlier, only the column average  CO2 will be derived from the ASCENDS measurements. The
4
information analysis (Section 3) also suggests that the ASCENDS measurements cannot be used
to determine the CO2 concentrations at more than one atmospheric level using its current setting.
We will discuss how the instruments should be improved to unveil the details of the CO2 vertical
variations.
While theoretically possible, the “one-line” retrieval must be accompanied with a careful selection
of the absorption line such that the spectral region around the selected absorption line should, for
example, have minimal absorptions by other species and minimal scattering effects by, e.g., clouds
and aerosols. On the other hand, “multi-line” retrievals with measurements covering a broad
spectral range in the NIR with multiple CO2 absorption lines would avoid these problems by
simultaneously retrieving all important species and any scattering agents. The use of a large
number of frequency channels also help reduce the sample error. For example, Kuai et al. [19]
have already demonstrated the possibility of retrieving the CO2 vertical profile at 2–3 levels using
the whole 6230 cm-1 CO2 band measured by the ground-based instrument of the Total Carbon
Column Observing Network (TCCON) [20, 21], which has a higher spectral resolution and S/N
than a typical spaceborne instrument. However, multi-line retrievals may be subject to too many
degrees of freedom; a lot of information in a broad band spectrum may not be related to CO2 but
these irrelevant information can easily deteriorate the accuracy of the resultant CO2 retrieval. The
central idea of the present work is to develop a retrieval strategy to estimate the CO2 vertical profile
using minimal spectral information.
This rest of the paper is organized as follows. Section 2 describes the theoretical tools for the
instrument design, including an analytic radiative transfer model of a single absorption line in a
5
reflected solar NIR transmission spectrum, the analytic expressions of the weighting functions for
the absorption line, and a Bayesian information analysis of the statistical contents that can be
retrieved from the measurement of the single absorption line. For simplicity, a nadir view setting
and a perfect reflection at the surface will be assumed. Section 3 presents the results of the
information analysis, quantifying the S/N ratio and the spectral resolution of a ground-based
instrument that are required to retrieval CO2 concentrations at multiple levels. In most of the
calculations, only the absorption of CO2 will be considered. Section 4 discusses other factors that
may be important in the instrument design and presents an error analysis, characterizing the effects
from other absorbing species such as water vapor and the effects due to the temperature.
2. Theory
2.1 A Simplified Spectral Absorption Model
The transmission spectrum around an absorption line at a transition frequency  0 will be
considered and the infrared emission will be ignored. The absorption coefficient at a nearby
frequency  is given by
k  p, T   S T  f ( , 0 , p, T )
where S (T ) is the line strength per molecule and f ( , 0 , p, T ) the absorption line profile at
pressure p and temperature T . The line strength is a function of temperature and the total internal
partition function Q(T ) is given by Eq. (A11) of Rothman et al. [22]. f ( , 0 , p, T ) is chosen to
6
be the Voigt profile. The absorption line center may not be  0 exactly but may be weakly
dependent on the pressure; see Eq. (A14) in Rothman et al. [22]. This effect, however, will be
ignored in the simplified model. As a result, f ( , 0 , p, T ) is given by
f  , 0 , p, T  
H  x, y 
 D T  
where H  x, y  is the Voigt function defined by
H  x, y  
y


et

y2   x  t 

2
2
dt ,
x    0   D T  , and y   L  p, T   D T  .  D T    0 2RT Mc 2 is the Doppler width,
where R is the ideal gas constant, c the speed of light, and M is the molar mass of the absorbing
species, i.e. CO2;  L  p, T    air
n
p  Tref 

 is the air-broadened line width, where  air is the
pref  T 
value at some reference pressure pref and temperature Tref , and n is the coefficient of the
temperature dependence. Note that the effect of self-broadening has also been ignored in the
expression of  L  p, T  ; see Eq. (A12) of Rothman et al. [22]. Air-broadening is dominant in the
lower atmosphere while the Doppler broadening is dominant in the upper atmosphere. All
parameters used in the simplified model have been obtained from HITRAN 2008 [23].
7
For a downwelling light path, the optical depth   z  at the frequency  and a height z is given
by an integral of the absorption coefficient over height:   z    1  k  p  z  , T  z    z  dz ,

z
where   z  is the number density of the absorbing species and   cos  ,  being the viewing
zenith angle. When  increases,   z  also increases due to the longer optical path. As a result,
the accuracy of the measurement increases when  increases. Since this study aims to examine
the minimum requirements of the instrument design for a prescribed vertical distribution of the
absorbing species, it is desired to examine the transmittance corresponding to the shortest optical
path. Therefore,   0 or   1 will be taken. The integral in z can be transformed into an
integral in pressure p using the hydrostatic equation dp   Mg air dz N A , where  air is the
number density of air, g is the gravitation acceleration, and N A is the Avogadro’s number.
Ignoring the vertical dependence of g , the optical depth can be rewritten as
  p  
NA p
k  p, T  p     p  dp ,
Mg 0
where    air is the volume mixing ratio. The transmittance of the downwelling light path is
simply given by T  p   e  p  .

For spectral measurements made on ground, the transmittance is given by T  ps   e  ps  , where

ps is the surface pressure. For a nadir spectral measurements made in space, the photon passes the
same optical path twice when   1 and the same optical depth applies. If the surface reflectivity
8
is 1, then the total transmittance observed in space is simply the square of the downwelling
transmittance at the surface: T  T  ps   T  ps   e2  ps  .

2.2 The Weighting functions
In the theory of remote sensing, the weighting function of a measurement is defined as the Jacobian
of the transmittance (or radiance when the infrared emission is important) with respect to the
abundance of the target species at some altitude. Since the volume mixing ratio to be differentiated
is involved in an integral, singular perturbation has to be used to derive the weighting function.
Consider a small change   p  in the volume mixing ratio at the pressure level p  . For a singular
perturbation,   p  is expressed in term of the Dirac-δ function:   p      p  p  , where
 is a perturbation quantity with dimensions (volume mixing ratio)×(pressure). For a
downwelling light path, the corresponding change in the optical depth is
  p  
NA
Mg

p
0
k  p, T  p     p  dp
 NA
k  p, T  p   , p  p

  Mg

0,
otherwise

where T  is the temperature at the pressure level p  . The change in the transmittance is then given
by  T  T . For ground-based measurements,  T  ps     N AT  ps  k  p, T  p   Mg .
Thus, the weighting function is given by
9
K  p  
N AT  ps  k  p, T  p  
 T  ps 
.


Mg
This expression is only for infinitesimal layer thickness. For computational purposes with finite
model layers, the perturbation  is given by   p p , where p  is the thickness of the
atmospheric layer at pressure p  . Therefore, the weighting function computed numerically is given
by
N AT  ps  k  p, T  p  
 T  ps 
K  p  

p

Mg

For a nadir measurement with   1 and the surface reflectivity being 1,
K  p   
2 N AT k  p, T  p  
Mg
p .
(1)
2.3 Information Content Analysis
Abshire et al. [17] retrieve the vertical profile of the absorbing species directly from the measured
spectrum without any constraints on the final estimate. Their approach essentially assumes that
either there only is one vertical profile that fits the measured spectrum or, if multiple solutions
exist, the initial guess of the vertical profile must be close to the generic one in order to guarantee
10
a convergence. This retrieval algorithm has also been adopted for the spectral measurements by
the NASA’s Atmospheric InfraRed Sounder (AIRS). Another common retrieval method is the
optimal estimation using Bayesian statistics [24]. In this formulism, prior information of the
quantity being retrieved are used to constrain the possible range of the final estimate. Shannon
information analysis can then be applied to quantify the gain in the a posteriori knowledge and
hence the number of CO2 partial columns that can be retrieved. The optimal estimation will be
used in this study.
Let m be the number of frequency channels, q the number of atmospheric model layers, Tobs an
m1 vector of the reflected transmittance observed in the space, Tcal an m1 of the calculated
transmittance, S  an m m covariance error matrix of the measurement (usually diagonal), χ true
a q  1 vector of the generic vertical profile to be measured, χ a a q  1 vector of the a priori
estimate of the vertical profile, S a an q  q a priori covariance matrix of vertical profile, and K
an m  q matrix of the weighting function defined in Eq. (1) . Only χ true is unknown. An estimate
χ̂
of
χ true
is
obtained
by
minimizing
the
cost
function
   Tobs  Tcal  S1  Tobs  Tcal    χ true  χ a  S a1  χ true  χ a  . When minimized, the expectation
T
T
value of  is equivalent to m (the number of measurements). The resultant χ̂ is given by

χˆ  χ a  G  Tobs  Kχ a  , where G  K T S1K  S a1

1
K T S1 is the gain matrix.
Another form of χ̂ is  I  A  χ a  GTobs , where I is the identity matrix and A  GK is the
averaging kernel. A is a q  q square matrix characterizing the rate of change of χ̂ with respect
11
to χˆ true , i.e. χˆ χ true . The first dimension (row) is for the measurement space and the second
S a1 , G  K 1 and A  I .
dimension (column) is for the model space. In the limit K T S1K
Then χˆ χ true  I and χˆ  K 1Tobs , meaning that there is no influence from the a priori
information. More generally, the relative contribution of the measurement and the a priori
information to the estimate χ̂ is characterized by the sum along the columns of A , i.e.

j
A ij .
If the sum at a particular level is close to 1, then the estimation of χ̂ at that level is mainly
contributed from the measurement; if, on the other hand, the sum is close to 0, then no information
has been gained from the measurement and the best estimate of χ̂ at that level is from the a priori.
In the Information Theory, the degrees of freedom for signal d s defined as the expectation value
of  χ true  χ a  S a1  χ true  χ a  is a statistical measure of how many elements in χ can be inferred
T
from the observations; the rest of the elements in χ will have to be estimated from the a priori.
The mathematical expression of d s is given by
ds  
i
i2
1  i2

1
2
1
2
a
where i are the singular values of the reduced Jacobian matrix K  S KS . Then the species
concentrations can be determined for at most d s vertical layers; the altitudes of the layer
boundaries can be defined such that the data density, defined as the trace of A , are evenly
distributed [25]. For example, if d s  3 , the altitudes  1 and  2 of the top of the first two layers
above the ground are defined such that

0 zk  1
Akk 

 1  zk  2
Akk 
A
zk  2
kk
 1 . Finally, the Shannon
12
information content, H , is the total amount of information (in bits) gained from the measurements
against the a priori knowledge. It is given by H   ln(1  i2 ) 2 .
i
With the above theoretical tools, one can now easily show that a vertical-dependent pressure
broadening (via the Lorentz profile half-width  L ), the temperature dependence of Q T  , and
the vertical dependence of S a are the three critical factors for a profile retrieval. Consider a
hypothetic case where there is no pressure/temperature dependence in k . Then K is a constant
at all heights. Assume that S  is diagonal. If S a is furthermore independent of the vertical, then
all columns of K in this case are linearly dependent and there is only one non-zero i . So d s
cannot be greater than 1, meaning that only the averaged CO2 column can be determined from the
measurements.
3. Results
The simplified radiative transfer model is applied to the temperature-insensitive absorption line at
6359.96 cm-1 selected by Abshire et al. [17]. For this line, S0  1.744 1023 cm 1 molecule cm 2 ,
 air  0.0746 cm1 , n  0.67 . Around 6359.96 cm-1, there are a few isotopic CO2 absorption lines,
as shown in Abshire et al. [17], which will be ignored in this single absorption line model for
simplicity. The 1976 US standard model atmosphere is used in the transmittance calculation
(Figure 1a). In this model atmosphere, the tropopause is located at 11 km and there are 11 layers
below the tropopause, each layer being 1 km thick. The CO2 concentration is assumed to be 379
ppmv at all levels.
13
A meaningful profile determination requires d s  2 . A few factors determine d s : (1) The number
of measurements or channels m (via the reduction of the standard error), (2) the S/N ratio (via the
increase of S 1 ), (3) the absolute magnitude of the Jacobian K , and (4) the a priori covariance S a
of the tropospheric CO2. Although the absorption at the line wings are more sensitive to the CO2
variations near the surface, the frequency range of the channels should be within a few factors of
 L from the line center  0 because the absorption are too weak at those wings and influence from
other species may become important. In this study, the range  0  3 L will be adopted. m is then
determined by the spectral resolution  of the instrument. In addition, to retrieve the vertical
profile of the absorbing species, the number of model layers to be used in the radiative transfer
calculation should not exceed m .
The typical settings of a state-of-the-art Fourier-transform spectroscopic (FTS) instrument, such
as that employed by TCCON, have   0.00753 cm 1 and S/N  400 . Suppose that the satellite
instrument has a coarser resolution of   0.02 cm 1 (compared to 0.2–0.3 cm–1 of the GOSAT
and OCO-2 instruments) and a similar S/N ratio of 400 . Assume that the channels are chosen
symmetrically about  0 . If a spectral width 3 L , then m  23 . Figure 1b shows the spectrum of
this absorption line; the selected channels at the resolution   0.02 cm 1 are explicitly shown
(pink/brown dots). Figure 1c shows a contour plot of the weighting function K  p  given in Eq.
(1) except the derivative has been taken as ln  so that the weighting function becomes
dimensionless. The saturated absorption at the line center  0 has very weak weightings on the CO2
variations anywhere below 20 km [in terms of the absolute values of K 0  p  ]. The peak at the
14
tropopause (11 km) at this frequency implies that the line center is useful to retrieval the tropopause
CO2 concentration but not useful for retrieving the surface CO2 concentration. To aid the visual,
the vertical dependence of the weighting function at  0 is shown in Figure 1d (dark brown;
following the color coding in Panel b), with the maximum value normalized to –1. In the wing
regions, e.g. where   0  2 L , the unsaturated absorption allows a higher weighting to the
surface CO2 concentration (dark cyan lines in Figure 1d) but the absolute values of K  p  at
these frequencies are also small, implying that the CO2 retrievals using the wing absorptions may
be subject to large errors. In between the wings and the line center is thus a frequency region of
the largest weightings to the surface CO2 concentration, which occurs at    0   L .
To proceed with the information content analysis, S  is assumed to be diagonal and has a value
400 2 for all channels. The specification of S a is more involved. Based on Figure 4 of Kuai et al.
[26], the vertical decorrelation length of the CO2 variability is ~5 km in the troposphere. Assuming
a 3% and 1% CO2 variability in the lower troposphere and in the stratosphere respectively, S a has
a symmetric form of
 z  z  2

  z  z

5 km

S a  z , z    2%  e
 1%  e 5 km .




2
(2)
Figure 2a shows the contour plot of S a . With these quantities, the resultant d s is only 1.26, and the
resultant H is 2.78 (c.f. Figure 2b). Therefore only the column-averaged CO2 can be measured
15
from this instrument setting. This measurement is most sensitive to the CO2 abundance below 6.6
km (because

zk  6.6 km
A kk  1 ).
If the spectral resolution is twice finer   0.01 cm 1 (while S/N  400 ), such that m is
increased to 45, then d s and H increase slightly to 1.46 and 3.47 respectively, and the columnaverage is more sensitive to CO2 below 5.5 km. On the other hand, if the S/N ratio is improved
more than twice, e.g. S/N  1000 (while   0.02 cm 1 ), then the resultant d s and H are 1.8
and 4.8 respectively. Now, if the spectral resolution and the S/N ratio are both improved such that
  0.01 cm 1 and S/N  1000 , then d s is 2.04, and H is 5.78. Thus the CO2 concentration can
be determined in two layers, one for the tropospheric column between 0–4.2 km and one for the
whole column above 4.2 km. The spectral resolution  and the S/N ratio can be further varied
to see the effects on d s and H . Figure 2b shows d s (solid contour lines) and H (color shades) as
functions of  and the S/N ratio with a spectral width 3 L . The slopes of the contour lines
demonstrate that the dependence on the S/N ratio is stronger than that on  .
Suppose that a retrieval of CO2 abundance in three bulk atmospheric layers, or equivalently d s  3 ,
is desired. Then an optimal choice of the spectral resolution and the S/N ratio may be
  0.004 cm 1 ( m  111) and S/N  2400 . This is equivalent to a TCCON instrument that has
a spectral resolution twice finer and an S/N ratio 6 times better than the current one. Another choice
would be a further reduction of the spectral resolution to   0.002 cm 1 while the S/N ratio is
~4 times better with S/N  1700 . In both cases, the bulk layers correspond to 0–2.3 km, 2.3–9.8
16
km, and above 9.8 km. Thus, the lower tropospheric CO2, the upper tropospheric CO2, and the
stratospheric column can be distinguished in this setting.
To further gain resolution to the surface CO2, d s  4 will be required, whence the bulk layers
correspond to 0–1.5 km, 1.5–7.0 km, 7.0–14.7 km, and above 14.7 km. But the required S/N ratio
would be even higher, with 6,000 for   0.002 cm 1 , or 4,100 for   0.001 cm 1 .
For an instrument setting like those for ASCENDS, i.e.   0.05 cm 1 ( m  9 ) and S/N  1300 ,
the degrees of freedom d s and the information entropy H are 1.48 and 3.98, respectively. If
d s  2 are desired while keeping only nine channels, the ASCENDS instrument has to achieve a
spectral noise level such that S/N  2800 .
4. Discussions
4.1 Implications on the instrumental designs
The above results show that even with only one absorption line, it is possible to distinguish three
partial CO2 columns (in the lower troposphere, the upper troposphere, and the stratosphere) with
an instrument having a spectral resolution and a spectral noise level that are a few times better than
TCCON’s. Note that in the above calculations, the width of the spectral window 3 L has been
chosen only for a nadir view configuration. Since other viewing zenith angles would lead to a
larger optical path (by a factor sin 1  ), a larger spectral window would be required in those cases
17
in order to cover the spectral wings. The surface reflectivity have also been assumed to be unity
and the optical path would be smaller otherwise.
On the other hand, the required spectral resolution and the S/N ratio may seem to be quite
demanding for a spaceborne instrument; they both have to be ~10 times better than those of
GOSAT and OCO-2. But with the advance in technology, such an instrument may be realized in
the near future. A possible candidate may be a lidar instrument similar to the one being used by
the ASCENDS team. An advantage of using a lidar type is its high S/N ratio. However, to achieve
d s  3 , the required number of frequency channels have to be more than 100, which presents a
great challenge of assembling more than 100 laser emitters in a small satellite platform. Even more
laser frequencies will be needed for longer optical paths. Another candidate would be an FTS like
the ones currently operating on GOSAT, OCO-2, and TCCON, or a grating spectrometer like the
one operating on the Atmospheric Infrared Spectrometer (AIRS). These spectrometers provide
large number of frequency channels as an advantage but the trade-off is their relatively low S/N.
Improving the spectral noise level would be the major challenge for this kind of instruments. The
third candidate may be the versatile PEPSIOS, originally designed and built for the study of trace
constituents (HD, O2, CO) in planetary spectra at visible and NIR wavelengths from ground-based
telescopes [27]. It is well suited for the measurement of line profiles at high spectral resolutions
with high throughput and improved suppression of the spectral wings. It is a relatively compact
instrument, and can be configured for remote sensing from satellites with the spectral resolution
and S/N ratio required to allow for profile retrievals of CO2.
18
The neglect of continuum absorptions in the present work could lead to significant errors in the
lower tropospheric CO2, especially near the surface, because such a retrieval critically depends on
the absorptions at the wings. Here continuum absorptions are referred to as the absorption features
that are much broader than the absorption lines and are not related to the target species. They may
be results of aerosol scatterings, inhomogeneous surface reflectivity (in the case of satellite
measurements), uncertainties of instrument calibrations, and etc. Some retrieval algorithms (e.g.
TCCON’s) simply treats the continuum absorptions as a background baseline of non-unity
transmittance and fits the baseline and the absorption lines simultaneously while some
sophisticated algorithms (e.g. GOSAT’s and OCO-2’s) retrieve simultaneously the species
concentrations, temperature, aerosol depths, surface properties, and etc. In both strategies, a broad
spectral range covering multiple absorption lines are required such that the continuum can be
determined from the “window” regions where absorptions due to the target species are known to
be weak. For a one-line retrieval, two or more spectral channels in the window regions can be built
to estimate the continuum baseline (e.g. using the Angstrom relation for the case of aerosol
scatterings).
4.2 Error budgets
The above formulism considers the CO2 absorption only. Uncertainties of other factors may
contribute the retrieval errors of the CO2 profile. Following Kuai et al. [26], the a posteriori
covariance matrix for the retrieval error  x of a state vector x is given by
T
Sˆ  xˆ   I  A x  S x  I  A x   G xS G Tx  G x K T ST K TT G Tx  G x K LS L K TLG Tx .
19
The first term on the right hand side,  I  A x  S x  I  A x  , is the smoothing error, where S x and
T
A x are the a priori covariance and the averaging kernel for x , respectively; the second term,
G xS G Tx , is the measurement error, where G x is the gain matrix for x ; the third term,
G x K T ST K TT G Tx , is the temperature effect; the four term, G x K LS L K TL G Tx , is the spectroscopic error.
Consider a simple retrieval model in which both the absorptions by CO2 and water vapor (H2O)
are fitted simultaneously. The state vector x is composed of two state vectors for CO2 and H2O
mixing ratios:
 χ CO2 
x
,
 χ H2O 
where χ CO2 is a q1  1 vector and χ H 2O is a q2 1 vector. This means that CO2 and H2O can be
retrieved at different sets of atmospheric layers. The weighting matrix is K  K CO2
K H2O  ,
where K CO2 and K H 2O are the m  q1 and m  q2 weighting matrices for CO2 and H2O
respectively. If the variations of CO2 and H2O are assumed to be independent, then S x is block
diagonal:
S CO2
Sx  
 0
0 
,
S H2O 
20
where S CO2 and S H 2O are the q1  q1 and q2  q2 a priori covariance matrices for CO2 and H2O
respectively. After some manipulation,
G 
Gx   1 
G 2 
and
A12 
A
A x   11

 A 21 A 22 
where
G1  G CO2  K TCO2 S1K H2O K TH2OS1
G 2  G H2O  K TH2OS1K CO2 K TCO2 S1
A11  A CO2  K TCO2 S1K H2O K TH2OS1K CO2

 G

K
A12  G CO2  K TCO2 S1K H2O K TH2OS1 K H2O
A 21
H2O
 K TH2OS1K CO2 K TCO2 S1
A 22  A H2O  K
in
which

T
H2O
1
S K CO2 K
T
CO2
CO2
1
S K H2 O

1
G CO2  K TCO2 S1K CO2  S CO
K TCO2 S1 ,
2


G H2O  K TH2OS1K H2O  S H12O K TH2OS1 ,
A CO2  G CO2 K CO2 , and A H2O  G H2O K H2O are the usual quantities for retrievals of either CO2 or
 Sˆ
Sˆ 12 
H2O alone. Rewrite Sˆ  xˆ   11
 , where
Sˆ 21 Sˆ 22 
21
T
T
Sˆ 11   I  A11  S CO2  I  A11   A12S H2O A12
 G1 S G1T  G1 K T ST K TT G1T  G1 K LS L K TLG1T .
Ŝ11 is the a posteriori covariance matrix for the retrieval error  χ CO of χ CO ; the sum of the first
2
2
two terms is the total smoothing error for χ CO2 due to the simultaneous retrieval of CO2 and H2O.
For a retrieval of CO2 alone, the smoothing error would be  I  A CO2  S CO2  I  A CO2  . Thus its
T
difference from the total smoothing error can be interpreted as the smoothing error due to the prior
information of H2O.
The spectroscopic error term G1 K LS L K TL G1T represents a systematic bias of the retrieval. Kuai et
al. [26] found that this term is –5 ppmv and is primarily due to the uncertainty of the O2 cross
section. The contributions of other error terms will be discussed below, assuming   0.01 cm 1
and S/N = 1000 such that d s  2 (Figure 3a). Using other values of  and S/N ratios (and hence
different d s ) may produce slight differences in the measurement errors and errors due to the H2O
and temperature variability, but the conclusions remain the same.
4.2.1 Smoothing error due to a priori CO2 covariance
Using Eq. (2) for S CO2 , the smoothing error due to a priori CO2 covariance,
I  A  S I  A 
T
CO2
CO2
CO2
, is 0.7% above 10 km, 1% at 5 km, and 2% at the surface (Figure 3a;
22
blue solid line). Comparing with the a priori variance (Figure 3a; blue dashed line), the smoothing
error represents an uncertainty reduction by one-third after the measurements.
4.2.2 Measurement error
The a posteriori CO2 uncertainty due to the measurement error, G1 S G1T , is the next largest source
of uncertainty. For S/N = 1000, this term is ~0.5% at the surface, ~0.2% at 5 km, and ~0.3% above
10 km (Figure 3a; orange line).
4.2.3 Smoothing error due to a priori H2O covariance
For illustration purpose, only the total H2O column is retrieved in this study; i.e. q2  1 . K H 2O (an
m1 vector) is calculated first by putting the H2O vertical profile from the US standard
atmosphere into the radiative transfer model to obtain the 2-dimensional weighting matrix, which
is then averaged along the vertical dimension. S H 2O is assumed to be 100%. The estimated
contribution
to
the
 I  A11  SCO  I  A11 
T
2
a
posteriori

CO2
uncertainty,


T
 A12S H2O A12
 I  A CO2 S CO2 I  A CO2
defined

T
as
the
difference
, is only ~0.1% at the surface
and is negligible above 5 km (Figure 3a; red line).
4.2.4 Systematic error due to a priori temperature covariance
23
The a priori covariance matrix for temperature ST below 60 km is based on the one used for the
Aura TES temperature retrievals [28], which can be approximated by a third-order Mórlet wavelet
2
 1  z  z 2 
z  z  2 

 3  z  z  

ST  z, z   0.7 
 ,
 cos 
 exp   
75
km
s
2
s

zz 




  zz  
(3)
 z  z  2 
where szz  2exp 
 is the scale of the Mórlet wavelet. szz is ~ 2 km below 40 km but
 40 km 
it grows abruptly between 40–60 km. ST is close to zero when z  z
szz and it is positive along
the diagonal for z  z   szz 6 . There is a narrow band of weak negative covariance for
 szz 6  z  z   szz 2 next to the positive diagonal, characterizing an anticorrelation between
two layers ~2 km apart in the troposphere (Figure 3b). The resultant a posteriori CO2 uncertainty
due to the a priori temperature covariance, G1 K T ST K TT G1T , is also ~0.1% at the surface and is
negligible above 5 km (Figure 3a; green line).
Therefore, the largest source of error is from the uncertainty of the prior CO2 variability (via the
smoothing error). A better choice of the a priori profile would be one of the keys for improving
the Bayesian approach. Note that this error source does not exist in retrievals using direct methods
like the Gauss-Newton algorithm, which do not apply the a priori constraints.
Acknowledgements
We thank Dr. J. Margolis for his valuable suggestions on the noise-resolution relation of
instrument, Dr. V. Natraj for helpful comments and GOSAT retrieval data and Dr. S. Newman for
her suggestions to improve the writing. This research is supported in part by the Orbiting Carbon
24
Observatory 2 (OCO-2) project, a NASA Earth System Science Pathfinder (ESSP) mission and
Project JPL.1382974 to the California Institute of Technology. KFL is supported by the NASA’s
Jack Eddy Fellowship organized by the University Corporation for Atmospheric Research,
Boulder, CO.
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Figure 1. (a) The pressure and temperature profile used in the radiative transfer calculations;
adapted from the 1976 US Standard Atmosphere. (b) The simulated CO2 absorption feature around
6359.967 cm–1. The CO2 concentration is assumed to be a constant of 379 ppmv at all altitudes.
The pink/brown dots are the channels selected at a resolution of 0.02 cm–1. The wavenumber span
is restricted to within ±3 air-broadening widths  L . (c) The weighting function K corresponding
to the absorption line shown in Panel b. K is given by Eq. (1) except that the derivative is taken
with respect to ln  instead of  so that the weighting functions are dimensionless. (d) The
weighting functions of the selected channels in Panel b (same color coding) as functions of altitude.
The maximum values have been normalized to –1 for comparisons.
29
Figure 2. (a) The empirical a priori CO2 covariance matrix approximated by the analytic
expression in Eq. (2). (b) The degrees of freedom d s (contour lines) and the information content
H (color shades) of the proposed instrument as functions of the spectral resolution  and the
signal-to-noise ratio S/N.
30
Figure 3. (a) The a posteriori uncertainty of retrieved CO2 contributed from the smoothing error
I  A  S I  A 
T
CO2
CO2
CO2
related to the a priori CO2 variability (blue solid line), the
measurement error G1 S G1T a prori CO2 variability (orange line), the smoothing error
 I  A11  SCO  I  A11 
T
2



T
 A12S H2O A12
 I  A CO2 S CO2 I  A CO2

T
related to the a priori H2O
covariance (red line), and the systematic error G1 K T ST K TT G1T related to the a priori temperature
variance (green line). The diagonal of the a prori CO2 covariance matrix S CO2 is shown for
reference (blue dashed line). (b) The a priori temperature covariance matrix approximated by the
Mórlet wavelet in Eq. (3).
31