Pressure Line Broadening and Profile Retrievals in Near IR Remote
Sensing
Abstract
The global scale information of CO2 vertical distribution is important in inferring the
global distribution of the sources and sinks and the large scale transport of CO2, e.g. N-S
exchange. Current generation of the satellite remote sensing instruments on GOSAT and OCO
and the surface observations in TCCON stations measure the absorption of the solar near infrared
radiation by CO2 in the atmosphere. All of them are set to retrieve the column averaged CO2
mixing ratio, XCO2. Due to the line broadening, especially the pressure line broadening, those
measurements content the information of the CO2 vertical distribution. A simple model of the
atmosphere, in which a CO2 line has only the Lorentz line broadening, is used to investigate the
feasibility of retrieving CO2 vertical distribution. The Rodgers’ information analysis method is
applied those measurements to find out their information content and degree of freedom. The
partial CO2 column density in the different regions of the atmosphere can be retrieved
simultaneously using TCCON data. It is more difficult to find the vertical information of CO2
from the observations GOSAT and OCO than TCCON because of weak signals, lower spectral
resolution and smaller signal noise ratio. The possibility of using instruments from with higher
spectral resolution, higher signal noise ratio and only over a few spectral lines to retrieve the
XCO2 and vertical information is discussed.
1. Introduction
TCCON [www.tccon.caltech.edu, Wunch et al., 2010a; Wunch et al., 2010b]
OCO
GOSAT Saitoh et al. [2009]
Global change, CO2 retrieval and Carbon cycle. Importance of the vertical distribution of
CO2 in constrain C sources and sinks from satellite observations
2. Degree of Freedom and information content
The degree of freedom (DoF) roughly measures the number of independent quantities,
which can be retrieved from measurements and the Shannon information content is the total
amount of information (in bits) gained by the retrieval process (Rodgers, 2000). In Appendix A
the analytic formulas of the Jacobian K is derived for a CO2 atmosphere with the Lorentz line
broadening due to the pressure. The following equations are used to calculate the information
content (H) degrees of freedom (ds), the singular values (i) of the normalized Jacobian matrix
( K˜ ) and the averaging kernel matrix (A) from the Jacobian (K), the measurement error
covariance (Se) and the a priori covariance matrix (Sa) (Rodgers 2000).
-
1
1
K˜ = Sx 2 KSa2 ,
(1.1)
In - A = (K T Sx-1K + Sa-1)Sa-1 = SˆSa-1,
(1.2)
1
1
ln(1+ l2i ) = - ln In - A ,
å
2 i
2
(1.3)
H=
ds = å l2i /(1+ l2i ) = tr(A),
i
(1.4)
3. The transmittance at the surface and the reflected radiance at ToA of NIR solar spectrum with
and without the Lorentz line broadening
A CO2 line of 6243.9 1/cm is selected with the spectral resolution of 0.02 1/cm and 30
channels to cover whole line for numerical calculations. The model atmosphere is evenly divided
into 100 layers in the pressure coordinate. For a comparison a case without line broadening is
calculated too. The CO2 absorption cross section at all levels in the later case is equal to the
cross section at Ps/3 in the case with the line broadening.
3. 1 The transmittance at the surface
A. Without line broadening
The transmittance, the Jacobian and the average kernel of the case without the line
broadening are plotted in Fig. 1a, 1b and 1c, respectively. The later two are vertical lines, which
mean measurements contain no vertical information. Adding certain amount of CO2 has the
same effect to the transmittance no matter which altitude in the atmosphere it is added. From the
point of view of information analysis this means the DoF is always less than one. Usually, it is
the total column CO2. Because all the columns of K are linear dependent the K-tilda only has
one nonzero eigenvalue no matter how large the signal noise ratio is, which a priori is used or
which resolution of the atmosphere model is used. Table 1 and 2 show those factors can changes
information content significantly, but they can make DoF greater than 1.
Table 1. The information content and DoF vs. the signal noise ratio for the no line broadening
case at the surface
Table 2. The information content and DoF vs. the diagonal a priori covariance matrix Sa for the
no line broadening case at the surface
In the scaling retrieval the CO2 at all the model levels can be retrieved, only the scaling
factor is retrieved from measurements. The vertical profile information is totally from the a priori.
Although we use the atmosphere with constant mixing ratio, the same conclusions can be
reached for atmosphere with no-constant CO2. In this case, the Jacobian are not vertical lines,
but are the lines with the same profile and linearly dependent to each other.
However, the information analysis of the real observations like TCCON and GOSAT
generates great than one DoFs. There is vertical information contained in those observations due
to the CO2 line broadening.
B. Lorentz line broadening
The transmittance, the Jacobian and the average kernel of the case with the line
broadening are plotted in Fig. 1d, 1e and 1f, respectively. By comparing Fig. 1d and 1e with Fig
1b and 1c, the Jacobian and the average kernel of this case clearly show that there is vertical
information in the measured transmittances. And numerical calculations show that the number of
the non-zero eigenvalues is great than 1. Accordingly, the DoF is great than one for moderate
accurate measurements and increases with the signal noise ratio.
Table 3. The information content and DoF vs. the signal noise ratio for the line broadening case
at the surface
Table 4. The information content and DoF vs. the diagonal a priori covariance matrix Sa for the
line broadening case at the surface
3. 2 The reflected NIR solar radiance at ToA
The ratio of the reflected NIR solar radiance at ToA over the original solar radiance
usually is much smaller than the transmittances measured at the surface because of the double of
the optical path and the surface reflectivity. There is still vertical information contained in the
measurements. The results of the numerical calculations are listed in the following tables to show
the effects of Sa, Se and the surface reflectivity.
Table 5. The information content and DoF vs. the signal noise ratio for the line broadening case
at ToA
Table 6. The information content and DoF vs. the diagonal a priori covariance matrix Sa for the
line broadening case at ToA
Table 7. The information content and DoF vs. the surface reflectivity for the line broadening case
at ToA
3. 3 Atmosphere model resolution
The numerical results listed above are calculated using an atmosphere model of 100
yayers. The DoFs in all the cases are less than 5. Therefore, to retrieve all 100 components of the
state vector, the CO2 mixing ratio, most vertical information would still come from the a priori it
it is well established or the retrieved state would be unphysical if the a priori can’t provide
enough constrain. In order to retrieve the vertical information from the measurements the
dimension of the state vector should not be too large than the DoF. Table 8 shows how the
information content and DoF change with the model resolution
Table 8. The information content and DoF vs. the model resolution for the line broadening case
at the surface
4. The vertical profile retrievals of CO2 from observations of TCOON
Although the above calculated DoF for the cases similar to TCCON and GOSAT/OCO
are only 2.5 and 1.3, respectively, DoF of measurement is larger (Kwai et al., 2011) because
many more lines are covered in the real measurements than our cases where only one line is
considered. There is enough information for profile retrieval at least for TCCON. The following
case is a three partial column CO2 retrieval of TCCON:
(Le)
5. One line retrieval of CO2 vertical information using finer spectral resolution and higher S/N
ratio
The information content and DoF increase with the signal noise ratio and also with the
spectral
6. Discussions
Pressure coordinate vs. altitude coordinate
TCCON data have vertical information, but barely in OCO or GOSAT
Appendix A The feasibility of TCCON profile retrieval
1. Absorption coefficient of Lorentz line (assuming the Doppler line broadening is
small) at pressure p
kn = S × f (n - n 0 ) = S ×
where
a
1
p a 2 + (n - n 0 )2
a = A × p , A = as / ps
and
(1)
ps is the surface pressure.
2. Optical depth
dtn = kn ×n ×dz,
(2)
t n = ò 0¥ kn × n × dz = ò 0p kn × c ×
s
dp
,
mg
where
CO2. Using (1),
tn =
Sc p
a
dp,
ò0 2
2
pmg a + (n - n 0 )
s
Sc p
Ap
= pmg ò 0 (Ap)2 + (n - n )2 dp
0
s
Sc
1
p
2
= 2pmgA ò 0 p 2 +[(n - n )/ A]2 dp
0
s
Sc
ps2 +[(n - n 0 )/ A]2
=
ln
2pmgA
[(n - n 0 )/ A]2
c is the mixing ratio of
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
t n = r × lnç
÷
2
è (n - n 0 ) ø
Here
rº
(3)
Sc
ScPs
ScPs
SN
=
=
=
2pmgA 2pmgAPs 2pmga s 2pa s
In the finite difference form:
(4)
where
is the size of the grid box in the pressure coordinate.
3. Transmittance
Tn = e
-t n
æ (n - n 0 )2 ö
=ç 2
2÷
è a s + (n - n 0 ) ø
r
(5)
The finite difference form
= Õ Tn ,i
i
where
(6)
4. 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.
5. Including the solar zenith angle q
m º cos(q )
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
jnm,i = -Tnm
6. OCO case with
(11)
ai a s
m (ai )2 + (n - n 0 )2
2r
(12)
m i , m r and r (surface reflectivity)
Tn = T × r ×T = r × e
mi
n
r /m
mr
n
-(t nm i +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)
7. Discussion
If the line broadening is independent of the pressure, J n ,i is constant in the
pressure dimension. That’s why it is so difficult to use transmittance to retrieval
vertical information.
For the Lorentz line profile,
Jn ,i has a minimum at pi =
(n - n 0 )
, i.e. n
A
at the half width of this pressure. So the higher the layer is the minimum of the
Jacobian is closer to the line center. TCCON has about 150 measurements along
one line. Thus, it is possible to resolve the CO2 vertical distribution using TCCON
data.
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