Linearization of Markov chain formalism for vector radiative transfer in a plane-parallel atmosphere/surface system

Feng Xu; Anthony B. Davis; Suniti V. Sanghavi; John V. Martonchik; David J. Diner

doi:10.1364/AO.51.003491

Applied Optics
Vol. 51,
Issue 16,
pp. 3491-3507
(2012)
•https://doi.org/10.1364/AO.51.003491

Linearization of Markov chain formalism for vector radiative transfer in a plane-parallel atmosphere/surface system

Feng Xu, Anthony B. Davis, Suniti V. Sanghavi, John V. Martonchik, and David J. Diner

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Feng Xu, Anthony B. Davis, Suniti V. Sanghavi, John V. Martonchik, and David J. Diner, "Linearization of Markov chain formalism for vector radiative transfer in a plane-parallel atmosphere/surface system," Appl. Opt. 51, 3491-3507 (2012)

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Abstract

The Markov chain formalism for polarized radiative transfer through a vertically inhomogeneous atmosphere is linearized comprehensively with respect to the aerosol and polarizing surface properties. For verification, numerical results are compared to those obtained by the finite difference method. We demonstrate the use of the linearized code as part of a retrieval of aerosol and surface properties for an atmosphere overlying a black and Fresnel-reflecting ocean surface.

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$Δ x_{q} / x_{q}$	$1 E - 3$	$1 E - 4$	$1 E - 5$	$1 E - 6$	$1 E - 7$	$1 E - 8$	$1 E - 9$
$d I / d m_{r}$	$3.312 E + 01$	$1.852 E + 01$	$5.230 E + 00$	$2.064 E + 00$	$1.189 E - 02$	$1.714 E - 04$	$5.308 E - 04$
$d I / d m_{i}$	$3.357 E - 06$	$3.785 E - 05$	$7.895 E - 05$	$2.862 E - 03$	$2.146 E - 02$	$2.845 E - 01$	1.902 $E + 00$
$d I / d r_{m}$	$5.756 E - 04$	$5.754 E - 06$	$3.794 E - 07$	$1.493 E - 06$	$3.797 E - 05$	$1.982 E - 04$	$1.915 E - 03$
$d I / d S$	$3.177 E - 04$	$3.177 E - 06$	$2.912 E - 08$	$3.562 E - 07$	$2.756 E - 06$	$4.532 E - 05$	$8.408 E - 04$
$d I / d N_{a}$	$2.829 E - 04$	$2.829 E - 06$	$1.362 E - 08$	$7.952 E - 07$	$3.207 E - 07$	$6.686 E - 06$	$6.268 E - 05$
$d I / d a$	$1.702 E - 08$	$3.720 E - 10$	$6.123 E - 09$	$2.791 E - 08$	$2.166 E - 07$	$1.119 E - 05$	$2.251 E - 05$
$d I / d b$	$2.742 E - 06$	$2.904 E - 08$	$6.186 E - 08$	$8.040 E - 08$	$9.888 E - 08$	$9.760 E - 05$	$6.039 E - 05$
$d I / d k$	$1.033 E - 04$	$1.033 E - 06$	$2.047 E - 08$	$6.960 E - 09$	$4.471 E - 07$	$1.696 E - 06$	$2.693 E - 05$
$d I / d ξ$	$1.158 E - 07$	$1.175 E - 09$	$5.155 E - 10$	$6.275 E - 09$	$2.369 E - 07$	$4.298 E - 07$	$1.856 E - 05$
$d I / d m_{S}$	$4.673 E - 05$	$4.674 E - 07$	$5.585 E - 09$	$5.408 E - 09$	$1.645 E - 07$	$9.770 E - 07$	$7.968 E - 06$
$d I / d v$	$3.036 E - 05$	$2.921 E - 07$	$7.417 E - 08$	$2.840 E - 06$	$5.941 E - 06$	$6.168 E - 05$	$1.359 E - 03$

Parameter	Case 6	Case 7	Case 8	Case 10
$m_{r}$	1.380	1.380	1.380	1.380
$m_{i}$	0.000	0.000	0.000	0.000
$r_{m}$ (μm)	0.099	0.100	0.100	0.099
$S$	2.725	2.718	2.717	2.728
$τ_{a}$	0.050	0.100	0.200	0.400
$k^{*}$	17	22	20	19
$F (x_{k^{*} + 1})$	$2.114 E - 5$	$5.522 E - 5$	$1.721 E - 4$	$1.873 E - 4$

Parameter	Case 6	Case 7	Case 8	Case 10
$m_{r}$	1.435	1.382	1.415	1.443
$m_{i}$	0.000	0.000	0.000	0.000
$r_{m}$ (μm)	0.103	0.097	0.128	0.141
$S$	2.539	2.732	2.461	2.308
$τ_{a}$	0.044	0.100	0.183	0.344
$k^{*}$	50	50	50	50
$F (x_{k^{*} + 1})$	$7.021 E - 2$	$3.098 E - 3$	1.361E0	3.652E0

Parameter	Case 6	Case 7	Case 8	Case 10
$m_{r}$	1.380	1.380	1.380	1.699
$m_{i}$	0.000	0.000	0.000	0.009
$r_{m}$ (μm)	0.095	0.099	0.100	0.099
$S$	2.750	2.727	2.720	1.938
$τ_{a}$	0.050	0.100	0.200	0.268
$v$ ( $m / s$ )	4.999	5.001	5.000	6.516
$k^{*}$	19	23	20	50
$F (x_{k^{*} + 1})$	$1.864 E - 4$	$1.919 E - 4$	$2.570 E - 5$	$6.196 E + 1$

Parameter	Case 6	Case 7	Case 8	Case 10
$m_{r}$	1.380	1.465	1.467	1.464
$m_{i}$	0.000	0.001	0.001	0.000
$r_{m}$ (μm)	0.095	0.239	0.221	0.208
$S$	2.750	1.975	2.021	2.056
$τ_{a}$	0.050	0.080	0.161	0.320
$v$ ( $m / s$ )	4.999	5.132	5.260	5.488
$k^{*}$	23	50	50	50
$F (x_{k^{*} + 1})$	$1.984 E - 4$	$5.390 E - 1$	$9.824 E - 1$	1.162E0

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Applied Optics