A13B-0915: A Fast and Accurate Two Orders of Scattering Model to
Account for Polarization in Trace Gas Retrievals From Satellite
Measurements
Vijay Natraj1,*, Robert J.D. Spurr2, Hartmut Bösch3 and Yuk L. Yung1
1
MC 150-21, Division of Geological and Planetary Sciences, California Institute of
Technology, 1200 E California Blvd, Pasadena, CA 91125, USA
2
RT Solutions Inc., 9 Channing St., Cambridge, MA 02138, USA
3
Jet Propulsion Laboratory, California Institute of Technology, MS 183-601, 4800
Oak Grove Dr, Pasadena, CA 91109, USA
Introduction
Satellite measurements have been playing a major role in weather and climate research
for the fast few decades. For most applications, interpretation of such measurements
requires an accurate modeling of the atmosphere and surface. Typically, trace gas
retrieval algorithms are simplistic because of computer resource and time considerations.
In particular, they neglect polarization effects due to the surface, atmosphere and
instrument. This can cause significant errors in retrieved trace gas column densities,
particularly in the ultraviolet (UV) and near infrared (NIR) spectral regions because of
appreciable scattering by air molecules, aerosols and clouds. On the other hand, it has
been shown [1], for example, that retrieving the sources and sinks of CO2 on regional
scales requires the column density to be known to 1-2 ppm (0.3-0.5%) precision. Clearly,
radiative transfer (RT) models need to be developed to achieve such precisions, while
working at speeds necessary to meet operational needs.
Proposed Solution: Two Orders of Scattering to Compute Polarization
Multiple scattering is known to be depolarizing. It follows, then, that the major
contribution to polarization comes from the first few orders of scattering. Ignoring
polarization leads to two kinds of errors. The first kind is errors due to the neglect of the
polarized components of the Stokes vector. The second kind is the errors in the intensity
itself from not accounting for polarization. The simplest (and fastest) approximation for
polarization would clearly be single scattering. However, for unpolarized incident light
(such as sunlight), single scattering does not account for polarization effects on the
intensity. On the other hand, three (and higher) orders of scattering, while giving highly
accurate results, involve nearly as much computation as a full multiple scattering
calculation. It would thus appear that two orders of scattering is optimal.
We calculate the reflection matrix for the first two orders of scattering (2OS) in a
vertically inhomogeneous, scattering-absorbing medium. We take full account of
polarization, and perform a complete linearization (analytic differentiation) of the
reflection matrix with respect to both the inherent optical properties of the medium and
the surface reflection condition. Further, we compute a scalar-vector correction to the
total intensity due to the effect of polarization; this correction is also fully linearized. The
intensity correction is meant to be combined with a scalar intensity calculation (with all
orders of scattering included) to approximate the intensity with polarization effects
included. An approximate spherical treatment is given for the solar and viewing beam
attenuation, enabling accurate computations for the range of viewing geometries
encountered in practical radiative transfer applications.
The following equation summarizes the approach:
 I   I sca   I cor 

 
  
 Q   0   Q2OS 
U    0   U 
  2OS 
  
V   0  V2OS 
where Isca and Icor refer to the scalar intensity with polarization neglected but all orders of
scattering accounted for and the scalar-vector intensity correction computed using our
approach, respectively; the subscript 2OS indicates results calculated using the 2OS
model.
Scenarios to Test Proposed Technique
We use the spectral regions to be measured by the Orbiting Carbon Observatory (OCO)
mission [2] to test the 2OS model. Six different locations and two different seasons have
been considered (see Fig. 1 for geographical location map), with four different aerosol
loadings (0.01, 0.05, 0.1, 0.2) for each of the above.
Ny Alesund (79 N, 12E)
Park Falls (46 N, 90.3 W)
Lauder (45 S, 170 E)
Surface CO2, July 1, 12 UT
South Pacific (30 S, 210 E)
Algeria (30 N, 8 E)
Darwin (12 S, 130 E)
Figure 1: Geographical Location of Test Sites
The details of the geometry, surface type and tropospheric aerosol type [3] for the various
scenarios are summarized in Table 1. The stratospheric aerosol has been assumed to be a
75% solution of H2SO4 with a modified gamma size distribution [4].
Solar Zenith Angle
Surface Type
Aerosol Type (Kahn Grouping)
(degrees)
Algeria Jan1
57.48
Desert
Dusty Continental (4b)
Algeria Jul 1
21.03
Desert
Dusty Continental (4b)
Darwin Jan 1
23.24
Deciduous
Dusty Maritime (1a)
Darwin Jul 1
41.44
Deciduous
Black Carbon Continental (5b)
Lauder Jan 1
34.22
Grass
Dusty Maritime (1a)
Lauder Jul 1
74.20
Frost
Dusty Maritime (1b)
Ny Alesund Apr 1
80.77
Snow
Dusty Maritime (1b)
Ny Alesund Jul 1
62.43
Grass
Dusty Maritime (1b)
Park Falls Jan 1
72.98
Snow
Black Carbon Continental (5a)
Park Falls Jul 1
31.11
Conifer
Dusty Continental (4b)
South Pacific Jan 1
24.62
Ocean
Dusty Maritime (1a)
South Pacific Jul 1
58.84
Ocean
Dusty Maritime (1b)
Table 1: Scenario Details
Residuals
The spectral residuals have been plotted for two scenarios, Algeria Jul 1 (aerosol od 0.01)
and Ny Alesund Apr 1 (aerosol od 0.2), which represent the best and worst case,
respectively (see Figs. 2 and 3). The blue, green and red lines in the top panel refer to the
vector, scalar, and 2OS results, respectively. The middle and bottom panels show the
radiance errors using scalar and 2OS models, respectively. For the former case, the low
solar zenith angle and relatively high surface albedos combine to reduce the polarization.
The latter case is one of high solar zenith angle and a surface that is extremely bright in
the O2 A band and extremely dark in the CO2 bands. This explains the high continuum
polarization in the CO2 bands. Nevertheless, the 2OS model improves the residuals by an
order of magnitude in the worst case and more than two orders of magnitude in the best
case. The rms residuals are plotted for all the scenarios in Figs. 4 and 5.
Figure 2: Spectral Residuals for Algeria Jul 1 (Aerosol OD 0.01) Scenario
Figure 3: Spectral Residuals for Ny Alesund Apr 1 (Aerosol OD 0.2) Scenario
Figure 4: RMS Residuals (Scalar)
Figure 4: RMS Residuals (2OS)
Sensitivity Studies
It is more instructive to understand the effect of the approximation on the errors in the
retrieved CO2 column. These errors can be assessed by performing a linear error analysis
study [5,6]. Forward model errors are typically systematic and result in a bias in the
retrieved parameters x. This bias can be expressed as:
x  G F
where G is the gain matrix that represents the mapping of the measurement variations
into the retrieved vector variations and F is the error in the modeling made by the scalar
(or 2OS) approximation.
F 
I  Q I calc

2
2
where Icalc is the calculated quantity, and is equal to Isca for the scalar model and Isca +
Icor – Q2OS for the 2OS model. All the quantities are vectors over the detector pixels.
The measurement and smoothing errors and the error due to the scalar and 2OS
approximations are summarized in Table 2. Clearly, the errors due to the 2OS
approximation are smaller or of the same order of magnitude than the noise and
smoothing errors even in the worst case (in most cases, it is at least an order of magnitude
smaller). On the other hand, the reverse situation is true if polarization is ignored.
Scenario
Measurement
Scalar Error
2OS Error
Smoothing Error
Error (ppm)
(ppm)
(ppm)
(ppm)
Algeria Jul 1
0.32
0.42
0.0027
0.25
Ny Alesund Apr 1
4.26
136.37
4.13
4.11
Table 2: Summary of Errors
Conclusions
Sensitivity studies were performed to evaluate the errors resulting from using a novel
approximation to compute polarization in simulations of backscatter measurements of
spectral bands by space-based instruments such as that on OCO. It was found that the
errors in the top of the atmosphere (TOA) radiance were less than 0.1% in most cases.
The computation time was two orders of magnitude less than that for an exact vector
computation. A linear error analysis study of simulated measurements from the OCO
absorption bands shows that errors due to the 2OS approximation are much lower than
the smoothing and measurement noise errors. This is in contrast to the observation that
the retrieval error budget may be dominated by polarization if a scalar approximation to
the total intensity is used. The 2OS model is also an order of magnitude faster than a full
multiple scattering scalar radiative transfer computation, making it feasible to be
implemented in operational retrieval algorithms as an adjunct model to deal with
polarization effects.
References
1.
Rayner, P.J. and O’Brien, D.M., GRL 28(1), 175-178, 2001
2.
Crisp, D., et al., Adv. Space Res. 34(4), 700-709, 2004
3.
Kahn et al., JGR 106(D16), 18219-18238, 2001
4.
http://www.eumetcal.org/euromet/english/satmet/s2400/s240009d.htm
5.
Rodgers, C.D., Inverse methods for atmospheric sounding: theory and practice,
2000
6.
Natraj, V., et al., JQSRT 103(2), 245-259, 2007
Related Presentations
A13B-0896 - Nair et al., poster, this session
A21F-0898 - Bösch et al., poster, Tuesday 8-10 am
B42A-08 - Crisp, D., talk, Thursday 12.05 pm