CO2 Line Broadening by Pressure and Profile Retrieval from Near Infrared Observations
Run-Lie Shia, Le Kuai, John T. Trauger and Yuk L.Yung
*** put in affiliations etc./ use style of Applied Optics
Abstract
Analytic expressions are derived for the transmittance and reflectance of sunlight and
their Jacobians for an absorption line with Lorentz broadening. The Rodgers information
analysis is applied to calculate the information content and the degree of freedom for a simple
atmospheric model to investigate the feasibility of retrieving the profile of CO2. The results have
implications for the design of future space instruments with high spectral resolution and high
signal noise ratio to obtain global scale information on CO2 vertical distribution that is important
for inferring the sources, sinks and transport of CO2.
1. Introduction
CO2 is the most important anthropogenic greenhouse gas. Its increase in the atmosphere
is the main cause for the rise of the globally average surface temperature (IPCC, 2007). The
prediction for climate change at the end of this century is dire if the trend of CO2 is not contained.
To quantitatively determine the global sources and sinks of CO2, satellite remote sensing is a
most powerful tool. Current measurements of the reflected solar near infrared (NIR) radiation
from space, e.g. GOSAT (Yokota et al., 2009, Saitoh et al., 2009, Sato et al., 2009) and future
mission OCO-2 (Crisp et al. 2012), are used to retrieval the column averaged CO2 mixing ratio,
XCO2. Measurements of solar NIR transmittance are also obtained at surface stations (TCCON,
Wunch et al., 2010a; Wunch et al., 2010b). Although the global distributions of XCO2 are quite
useful for constraining CO2 sources and sinks, the vertical distribution of CO2, especially in the
lower and the middle troposphere, is needed for understanding the transport of CO2 and the
carbon cycle.
In this study we use a simple model of one CO2 line in the NIR region with Lorentz
pressure broadening to derive the transmittance of the sunlight at the surface and the reflectance
at the top of the atmosphere. The derivatives of these quantities with respect to the CO2
concentration, the Jacobians derived in Appendix A, are then used to calculate the information
content (IC) and the degree of freedom ( d s ) using the Rodgers method (Rodgers, 2000). For
high signal noise ratios and spectral resolution, measurements over one line provide sufficient
information for the retrieval of the CO2 vertical distribution.
This paper is divided into five sections. A brief discussion the Rodgers information
analysis is presented in the section 2. The IC and d s in the measurements over a NIR CO2 line
with and without the Lorentz line broadening are calculated for different S/N in section 3 to
demonstrate the feasibility of the CO2 profile retrieval. A test case of using TCCON data for
vertical retrieval is presented in section 4. A brief discussions on the application of this study is
presented in section 5.
2. Rodgers information analysis
In the theory of optimal estimation, the degree of freedom ( d s ) measures the number of
independent variables that can be retrieved and the Shannon information content ( H ) is the total
amount of information (in bits) gained from the measurements (Rodgers, 2000). In Appendix A
the analytic formulas for the transmittance at the surface and reflectance at the top of the
atmosphere (ToA), and their Jacobians K are derived for a CO2 line with Lorentz broadening.
The normalized Jacobian matrix ( K˜ ) is calculated from the Jacobian ( K ), the measurement
error covariance ( S ) and the a priori covariance matrix ( Sa ) using the following equation
(Rodgers 2000).
K  S1/2 KSa1/2
(1)
*** all these formulas must be reduced somewhat in size / font 12 is standard
H and d s are defined in the Eqns (2) and (3)
1
ln(1  i2 )

2 i
(2)
d s   i2 / (1  i2 )
(3)
H
i
where i are the singular values of K˜ . In the cases studied in the following section, a diagonal a
priori covariance matrix for CO2 with the error of 3% is used.
3. IC and d s in the transmittance of NIR solar radiance
A CO2 line at 6243.9 cm-1 is selected with the spectral resolution of 0.02 cm-1 and 30
channels to cover the line for numerical calculations. The line strength is S =1.5310-23 cm and
the Lorentz line broadening constant A =7.210-5 cm-1 mb-1. (*** need ref to Hitran) The model
atmosphere is evenly divided into 100 layers in the pressure coordinate from the surface, Ps =
1000 mb, to the top of the atmosphere, Pt = 0 mb.
3. 1. Transmittance at the surface
To simplify the calculations, in the cases studied in this subsection the solar zenith angle
is zero, i.e.
  1.
A. Without line broadening
The case without line broadening is calculated assuming the absorption cross section of
CO2 at all levels has the line broadening equal to that at Ps /3. The transmittance and the
Jacobian for this case are calculated using the formulas in Appendix A and plotted in Fig. 1a and
1b, respectively. The transmittances for the 30 channels, which cover 8 times of the line width at
the surface (0.072 cm-1), are plotted in Fig. 1a. The transmittances range from 1.0 to 0.3. In Fig.
1b the Jacobian for the 30 channels are plotted. They are all vertical lines. This means the
transmittances contain no information on the vertical distribution of CO2. Adding CO2 at
different altitudes would have same effect on the transmittance at the surface. The only variable
that affects the transmittances in this case is the total amount of CO2. This means d s is always
less than or equal to 1. From the point of view of numerical calculations, this is because all the
columns of Jacobian, K are linearly dependent so the scaled Jacobian, K only has one nonzero
eigenvalue, no matter how large the signal to noise ratio (SNR) is, what the a priori is or how
large the dimension of the state vector is. Eqn. (4) would only generate ds, which is  1.Table 1
shows that better SNR can increase the information content significantly, but cannot make d s
greater than 1.
Figure. 1 (a) and (b): Transmittance at the surface and its Jacobian, respectively, for the case
without line broadening for 30 channels. (c) and (d): Same as (a) and (b) for the case with line
broadening. (e) and (f): Same as (c) and (d) for Reflectance at the top of the atmosphere and its
Jacobian. (g) and (h): Same as (c) and (d), but with 150 channels).
Table 1. H and d s of the transmittance of the solar NIR radiance at the surface vs. the signal
noise ratio for the no line broadening atmosphere. The S/N for the baseline is 1000 and the
spectral resolution 0.02 (1/cm)
SNR (1000)
0.5
1.
2.
10.
100.
IC
0.763
1.55
2.48
4.78
8.10
ds
0.653
0.883
0.968
1.00
1.00
B. Lorentz line broadening
The transmittance and the Jacobian of the case with the line broadening are plotted in Fig.
1c and 1d, respectively. Although the transmittances in cases without and with line broadening
are quite similar (Fig. 1a and 1c), their Jacobians are quite different (Fig. 1b and 1d). In the case
with line broadening, the plot shows that the Jacobian for each channel is not a vertical line and
there is vertical information in the measured transmittances because the location of CO2 does
make differences in the measurements. Now K has more than one non-zero singular value.
Accordingly, d s become larger than 1 for large SNR and also H becomes larger (Table 2).
Table 2. Same as Table 1, but with the Lorentz line broadening.
SNR (1000)
0.5
1.
2.
10.
100.
IC
0.810
1.90
3.62
9.83
23.1
ds
0.794
1.40
2.02
3.27
4.69
C. The effect of increasing spectral resolution
With finer spectral resolutions, more channels can be measured over one line. This would
increase the information content and the degree of freedom in retrievals. A new case with 5 times
finer spectral resolution than the previous cases (150 channels) is tested with the same line
broadening. The transmittance and Jacobian for this case are plotted in Fig. 1g and 1h. The
transmittances of the two cases with the different spectral resolutions and the same line
broadening (Fig. 1c and 1g) are similar, except the minimum of the transmittances for the case
with the finer spectral resolution is smaller because the corresponding channel is closer to the
line center. In all the cases in this paper the line center is intentionally not measured because at
the center other line broadenings would play more important role than the Lorentz line
broadening in the upper atmosphere. The Jacobians in Fig. 1h are similar to that in Fig. 1d, just
with 5 times more lines.
Comparing Table 2 and 3, the increasing of the spectral resolution significantly increases
the information content and the degree of freedom.
Table 3. Same as Table 2, but with the spectral resolution = 0.004 cm-1
SNR (1000)
0.5
1.
2.
10.
100.
IC
2.23
4.27
7.16
17.0
37.6
ds
1.63
2.47
3.32
5.09
7.24
3. 2 Reflectance at the top of the atmosphere
In order to simplify the discussions, the case studied in this subsection has the zero
incident solar zenith angle and reflective zenith angle, i.e.
reflectivity
i  r  1 and the surface
  1.
Although the reflected NIR solar radiance at the top of atmosphere (TOA) is smaller than
the radiance measured at the surface (Fig. 1c and 1e) because of the doubling of the optical path,
there is still enough vertical information contained in the measurements of reflectance at TOA.
The Jacobians for the transmittance and reflectance seems comparable for most channels (Fig. 1d
and 1f) owning to the double role of each CO2 plays, once in the incident path and once in the
reflected path. Also, the vertical information in the reflectance case (Fig. 1f) is more smoothly
distributed than in transmittance case (Fig. 1d), which have more information in the upper
atmosphere. The results of the numerical calculations of the reflectance case are listed the Table
4 for different S/N. They are comparable to the transmittance case (Table 2).
Table 4. H and d s of the reflectance at TOA vs. the signal noise ratio for the Lorentz line
broadening atmosphere. The SNR for the baseline is 1000 and the spectral resolution is 0.02 cm-1.
SNR (1000)
0.5
1.
2.
10.
100.
IC
0.978
2.07
3.70
9.87
23.6
ds
0.845
1.35
1.94
3.34
4.86
4. Vertical profile retrieval of CO2 from observations of TCOON
In the cases studied in section 3 the CO2 absorption line is simplified. Only the Lorentz
line broadening is included. As a comparison, the transmittances at the surface and their
Jacobians for the same spectral line are calculated using the GFIT (KL, please give Ref. ????),
which is the radiative transfer model for TCCON retrieval. The results are plotted in Fig. 2. They
are quite comparable to that of the simplified line case in Fig. 1c and 1d.
Kuai et al., (2011) calculated IC and d s for the TCCON resolution of 0.014 cm-1 and
SNR = 1000 for the TCCON spectrum and find out IC=6.80 and d s =2.72. These values are
higher than the corresponding numbers for the simplified line case in Table 2 (the baseline case
with SNR = 1000) because many more lines are included in the TCCON retrieval. It seems there
is information in TCCON to constrain the vertical distribution of CO2 in the atmosphere. Kuai et
al. (2011) retrieved three partial column CO2 density instead the one XCO2.
Figure 2. Left: Computed transmittance of the CO2 line at v0 = 6243.9 cm-1 viewed at cosine
(SZA) = 1.0. The spectral resolution of the measurement points is 0.014 cm-1. In the CO2 profile,
blue denotes absorption in the wings, while red denotes absorption near the center of the line.
The spectral resolution of a PEPSIOS instrument is indicated for comparison. Right: Jacobian
corresponding to the measurement points in the spectral profile at left, using the same color code.
5. Discussions
The measurements of the NIR solar radiation can provide information on CO2 vertical
distribution through the line broadening. Although TCCON observations can be used to retrieve
2-3 CO2 partial column densities, the measurements of the current and the near future satellite
remote sensing instruments like GOSAT and OCO-2 do not contain sufficient information on the
vertical distribution of CO2, due to their lower SNR and coarse spectral resolution as compared
with TCCON.
Although, the instruments like those used in COSAT and OCO-2 can be improved to
provide information on the vertical distribution of CO2 by improving SNR and/or using finer
spectral resolution, the possibility of using new instruments, which cover a few CO2 lines with
much higher SNR and spectral resolution should be considered too. From Tables 2 and 3, it is
quite feasible to use the measurement from those new instruments to retrieve the profile of CO2
in the atmosphere. If the new instruments can have 10 times of SNR ratio and 5 times the
spectral resolution than TCCON, their measurements would provide enough information ( H =17)
for more than 5 degrees of freedom for profile retrievals of CO2.
The versatile PEPSIOS was originally designed and built at JPL/Caltech for the study of
trace constituents (HD, O2, CO) in planetary spectra at visible and NIR wavelengths from
ground-based telescopes (Trauger and Lunine, 1983). It has the spectral resolution of 0.014 cm-1,
which is indicated in Fig. 2, and the SNR ratio of about 1000. An updated PEPSIOS with a
modern NIR sensor and data acquisition system, this instrument can be configured for remote
sensing from satellites with the spectral resolution and SNR ratio capable of profile retrieval of
CO2.
Appendix A: Transmittance and Reflectance Derivations
1. At the surface
Assuming the Doppler line broadening is negligible, the absorption coefficient of a Lorentz
line can be expressed as
kn = S × f (n - n 0 ) = S ×
a
1
p a 2 + (n - n 0 )2
(1)
 is the frequency of the NIR line and the line broadening at
pressure p is a = A × p , A = a s / ps , ps is the surface pressure. For the zenith angle
  0 , the total optical depth of the atmosphere,   can be calculated as the integration of
where
S
is the line strength,
dtn = kn ×n ×dz,
t n = ò 0¥ kn × n × dz = ò 0p kn × c ×
s
where
z
is the altitude,
dp
,
mg
n is the number density of, e.g. CO2, c is the mixing ratio of CO2, is
the molecule mass of the atmosphere and is the gravitational constant.
Using (1),
tn =
(2)
Sc p
a
dp,
ò0 2
2
pmg a + (n - n 0 )
s
Sc p
Ap
ò
= pmg 0 (Ap)2 + (n - n )2 dp
0
s
Sc
1
= 2pmgA ò 0 p 2 +[(n - n )/ A]2 dp
0
ps
2
Sc
ps2 +[(n - n 0 )/ A]2
=
ln
2pmgA
[(n - n 0 )/ A]2
æ ps2 +[(n - n 0 )/ A]2 ö
= r × lnç
÷
2
è [(n - n 0 )/ A] ø
æ a s2 + (n - n 0 )2 ö
= r × lnç
÷
2
è (n - n 0 ) ø
where
rº
(3)
Sc
ScPs
ScPs
SN
=
=
=
2pmgA 2pmgAPs 2pmga s 2pa s
The transmittance at the surface, which is like the measurements of TCCON, is
Tn = e
-t n
æ (n - n 0 )2 ö
=ç 2
2÷
è a s + (n - n 0 ) ø
r
(4)
The calculations of IC and d s only use the finite difference of Jacobians. In the finite
difference form,
(5)
where
is the size of the grid box in the pressure coordinate.
The transmittance
= Õ Tn ,i
(6)
i
where
.
To calculate the Jacobian for the
i th grid box,
dTn = Tn (tn + dtn ,i )- T(tn )
= e-(t n +dt n ) - e-t n » -e-t n × dtn ,i = -Tn × dtn ,i
,i
where
(7)
(8)
The Jacobian is defined as
= -Tn ×
Using the finite difference form of
2r × ai × a s
ai2 + (n - n 0 )2
(9)
Tn , Eqn. (6)
= (e-dt n -1)×Tn » -Tn × dt n ,i
,i
(10)
The two Jacobians derived from the analytical and finite difference form of the
transmittance, Eqns. (10) and (7) are exactly the same.
For cases where
m º cos(q ) is not equal 1
æ ps2 +[(n - n 0 )/ A]2 ö
t = × lnç
÷
m è [(n - n 0 )/ A]2 ø
m
n
r
r /m
Tnm = e
-t nm
æ [(n - n 0 )/ A]2 ö
=ç 2
2÷
è ps +[(n - n 0 )/ A] ø
æ (n - n 0 )2 ö
=ç 2
2÷
è a s + (n - n 0 ) ø
r /m
(11)
ai a s
m (ai )2 + (n - n 0 )2
2r
jnm,i = -Tnm
2.
(12)
At the top of the atmosphere
This is similar to the GOSAT/OCO-2 measurements, with
reflect zenith angles, and
m i and m r for incident and
r for the surface reflectivity. The transmittance at the top of the
atmosphere can be expressed as
mi
Tn = Tnm × r ×Tnm = r × e-(t n
i
r
+t nm r )
dTn = -Tn (dtnm,i + dtnm,i )
i
r
æ (n - n 0 )2 ö
Tn = r × ç 2
2÷
è a s + (n - n 0 ) ø
jn ,i = -Tn
r (1/ m i +1/ m r )
2rai a s
1 1
×(
+ )
ai2 + (n - n 0 )2 m i m r
(13)
(14)
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