Asymptotic radiance and polarization in optically thick media: ocean and clouds

George W. Kattawar; Gilbert N. Plass

doi:10.1364/AO.15.003166

Applied Optics
Vol. 15,
Issue 12,
pp. 3166-3178
(1976)
•https://doi.org/10.1364/AO.15.003166

Asymptotic radiance and polarization in optically thick media: ocean and clouds

George W. Kattawar and Gilbert N. Plass

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George W. Kattawar and Gilbert N. Plass, "Asymptotic radiance and polarization in optically thick media: ocean and clouds," Appl. Opt. 15, 3166-3178 (1976)

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Abstract

Deep in a homogeneous medium that both scatters and absorbs photons, such as a cloud, the ocean, or a thick planetary atmosphere, the radiance decreases exponentially with depth, while the angular dependence of the radiance and polarization is independent of depth. In this diffusion region, the asymptotic radiance and polarization are also independent of the incident distribution of radiation at the upper surface of the medium. An exact expression is derived for the asymptotic radiance and polarization for Rayleigh scattering. The approximate expression for the asymptotic radiance derived from the scalar theory is shown to be in error by as much as 16.4%. An exact expression is also derived for the relation between the diffusion exponent k and the single scattering albedo. A method is developed for the numerical calculation of the asymptotic radiance and polarization for any scattering matrix. Results are given for scattering from the haze L and cloud C3 distributions for a wide range of single scattering albedos. When the absorption is large, the polarization in the diffusion region approaches the values obtained for single scattered photons, while the radiance approaches the value calculated from the expression: phase function divided by (1 + kμ), where μ is the cosine of the zenith angle. The asymptotic distribution of the radiation is of interest since it depends only on the inherent optical properties of the medium. It is, however, difficult to observe when the absorption is large because of the very low radiance values in the diffusion region.

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Corrections

George W. Kattawar and Gilbert N. Plass, "Asymptotic radiance and polarization in optically thick media: ocean and clouds; erratum," Appl. Opt. 16, 1130-1130 (1977)
https://opg.optica.org/ao/abstract.cfm?uri=ao-16-5-1130

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k	Rayleigh vector theory	Rayleigh scalar theory	Isotropic
0.05	0.99917	0.99917	0.99917
0.10	0.99665	0.99665	0.99666
0.20	0.98647	0.98651	0.98652
0.30	0.96900	0.96916	0.96924
0.40	0.94337	0.94389	0.94418
0.50	0.90818	0.90948	0.91024
0.60	0.86114	0.86388	0.86562
0.70	0.79826	0.80340	0.80710
0.80	0.71178	0.72054	0.72819
0.90	0.58166	0.59497	0.61132
0.95	0.47840	0.49330	0.51862
0.99	0.32195	0.33434	0.37406
0.999	0.21069	0.21812	0.26288
0.9999	0.15498	0.15959	0.20193
1 — 10⁻⁶	0.10106	0.10324	0.13785
1 — 10⁻⁸	0.07493	0.07619	0.10464
1 — 10⁻¹⁰	0.05953	0.06035	0.08432
1 — 10⁻²⁰	0.02936	0.02951	0.04279
1 — 10⁻⁴⁰	0.01458	0.01463	0.02155
1 — 10⁻⁶⁰	0.00970	0.00972	0.01440
1 — 10⁻⁸⁰	0.00726	0.00728	0.01082
1 — 10⁻⁹⁰	0.00645	0.00646	0.00962

			Radiance			Radiance		Polarization
k	ω₀	θ	Vector	Scalar	θ	Vector	Scalar	Polarization
0.3	0.96900	180°	1.000	1.0000	0°	5.385 — 1	5.385 — 1	0
		160°	9.715 — 1	9.736 — 1	20°	5.442 — 1	5.454 — 1	3.446 — 3
		140°	8.978 — 1	9.050 — 1	40°	5.623 — 1	5.667 — 1	1.228 — 2
		120°	8.054 — 1	8.171 — 1	60°	5.953 — 1	6.039 — 1	2.251 — 2
		100°	7.174 — 1	7.310 — 1	80°	6.464 — 1	6.586 — 1	2.930 — 2
		90°	6.794 — 1	6.927 — 1	90°	6.794 — 1	6.927 — 1	3.024 — 2

0.5	0.90818	180°	1.0000	1.0000	0°	3.333 — 1	3.333 — 1	0
		160°	9.339 — 1	9.396 — 1	20°	3.369 — 1	3.389 — 1	9.844 — 3
		140°	7.825 — 1	7.998 — 1	40°	3.491 — 1	3.568 — 1	3.566 — 2
		120°	6.250 — 1	6.508 — 1	60°	3.750 — 1	3.905 — 1	6.666 — 2
		100°	5.033 — 1	5.307 — 1	80°	4.229 — 1	4.459 — 1	8.792 — 2
		90°	4.583 — 1	4.842 — 1	90°	4.583 — 1	4.842 — 1	9.091 — 2

0.8	0.71178	180°	1.0000	1.0000	0°	1.111 — 1	1.111 — 1	0
		160°	7.853 — 1	7.955 — 1	20°	1.113 — 1	1.127 — 1	2.591 — 2
		140°	4.705 — 1	4.936 — 1	40°	1.129 — 1	1.185 — 1	9.794 — 2
		120°	2.794 — 1	3.064 — 1	60°	1.197 — 1	1.313 — 1	1.932 — 1
		100°	1.836 — 1	2.080 — 1	80°	1.388 — 1	1.572 — 1	2.648 — 1
		90°	1.568 — 1	1.784 — 1	90°	1.568 — 1	1.784 — 1	2.754 — 1

0.93	0.52525	180°	1.0000	1.0000	0°	3.627 — 2	3.627 — 2	0
		160°	5.352 — 1	5.425 — 1	20°	3.601 — 2	3.650 — 2	3.738 — 2
		140°	2.124 — 1	2.238 — 1	40°	3.568 — 2	3.759 — 2	1.458 — 1
		120°	1.006 — 1	1.117 — 1	60°	3.674 — 2	4.080 — 2	3.005 — 1
		100°	5.854 — 2	6.772 — 2	80°	4.226 — 2	4.889 — 2	4.261 — 1
		90°	4.844 — 2	5.637 — 2	90°	4.844 — 2	5.637 — 2	4.452 — 1

0.99	0.32195	180°	1.0000	1.0000	0°	5.025 — 3	5.025 — 3	0
		160°	1.369 — 1	1.383 — 1	20°	4.943 — 3	4.993 — 3	4.802 — 2
		140°	3.469 — 2	3.610 — 2	40°	4.767 — 3	4.961 — 3	1.931 — 1
		120°	1.398 — 2	1.521 — 2	60°	4.724 — 3	5.139 — 3	4.160 — 1
		100°	7.488 — 3	8.458 — 3	80°	5.292 — 3	5.976 — 3	6.126 — 1
		90°	6.083 — 3	6.911 — 3	90°	6.083 — 3	6.911 — 3	6.439 — 1

0.999	0.21069	180°	1.0000	1.0000	0°	5.002 — 4	5.002 — 4	0
		160°	1.550 — 2	1.560 — 2	20°	4.896 — 4	4.930 — 4	5.352 — 2
		140°	3.496 — 3	3.594 — 3	40°	4.648 — 4	4.779 — 4	2.187 — 1
		120°	1.347 — 3	1.431 — 3	60°	4.497 — 4	4.775 — 4	4.830 — 1
		100°	7.003 — 4	7.657 — 4	80°	4.932 — 4	5.393 — 4	7.277 — 1
		90°	5.657 — 4	6.214 — 4	90°	5.657 — 4	6.214 — 4	7.676 — 1

l — 10⁻⁵	0.12237	180°	1.0000	1.0000	0°	5.000 — 6	5.000 — 6	0
		179°	6.160 — 2	6.160 — 2	1°	5.000 — 6	5.000 — 6	1.417 — 4
		170°	6.486 — 4	6.492 — 4	10°	4.968 — 6	4.972 — 6	1.423 — 2
		160°	1.568 — 4	1.574 — 4	20°	4.875 — 6	4.894 — 6	5.754 — 2
		140°	3.453 — 5	3.509 — 5	40°	4.574 — 6	4.649 — 6	2.379 — 1
		120°	1.302 — 5	1.350 — 5	60°	4.341 — 6	4.501 — 6	5.358 — 1
		100°	6.642 — 6	7.018 — 6	80°	4.677 — 6	4.942 — 6	8.219 — 1
		90°	5.348 — 6	5.669 — 6	90°	5.348 — 6	5.669 — 6	8.697 — 1

1 — 10⁻²⁰	0.02936	180°	1.0000	1.0000	0°	5.000 — 21	5.000 — 21	0
		179.99°	6.579 — 13	6.579 — 13	0.01°	5.000 — 21	5.000 — 21	1.497 — 8
		179.9°	6.566 — 15	6.566 — 15	0.1°	5.000 — 21	5.000 — 21	1.500 — 6
		179°	6.565 — 17	6.565 — 17	1°	5.000 — 21	5.000 — 21	1.500 — 4
		170°	6.485 — 19	6.486 — 19	10°	4.964 — 21	4.965 — 21	1.507 — 2
		160°	1.563 — 19	1.564 — 19	20°	4.858 — 21	4.863 — 21	6.112 — 2
		140°	3.405 — 20	3.418 — 20	40°	4.510 — 21	4.528 — 21	2.554 — 1
		120°	1.261 — 20	1.272 — 20	60°	4.204 — 21	4.242 — 21	5.856 — 1
		100°	6.232 — 21	6.409 — 21	80°	4.451 — 21	4.513 — 21	9.142 — 1
		90°	5.076 — 21	5.150 — 21	90°	5.076 — 21	5.150 — 21	9.702 — 1

1 — 10⁻⁹⁰	0.00645	180°	1.0000	1.0000	0°	5.000 — 91	5.000 — 91	0
		179.99°	6.579 — 83	6.579 — 83	0.01°	5.000 — 91	5.000 — 91	1.515 — 8
		179.9°	6.566 — 85	6.566 — 85	0.1°	5.000 — 91	5.000 — 91	1.518 — 6
		179°	6.565 — 87	6.565 — 87	1°	5.000 — 91	5.000 — 91	1.518 — 4
		170°	6.483 — 89	6.484 — 89	10°	4.963 — 91	4.963 — 91	1.526 — 2
		160°	1.562 — 89	1.562 — 89	20°	4.855 — 91	4.856 — 91	6.191 — 2
		140°	3.394 — 90	3.397 — 90	40°	4.496 — 91	4.500 — 91	2.593 — 1
		120°	1.252 — 90	1.255 — 90	60°	4.175 — 91	4.183 — 91	5.969 — 1
		100°	6.252 — 91	6.271 — 91	80°	4.402 — 91	4.415 — 91	9.355 — 1
		90°	5.016 — 91	5.032 — 91	90°	5.016 — 91	5.032 — 91	9.935 — 1

k	Rayleigh vector theory	Rayleigh scalar theory	Isotropic
0.05	0.99917	0.99917	0.99917
0.10	0.99665	0.99665	0.99666
0.20	0.98647	0.98651	0.98652
0.30	0.96900	0.96916	0.96924
0.40	0.94337	0.94389	0.94418
0.50	0.90818	0.90948	0.91024
0.60	0.86114	0.86388	0.86562
0.70	0.79826	0.80340	0.80710
0.80	0.71178	0.72054	0.72819
0.90	0.58166	0.59497	0.61132
0.95	0.47840	0.49330	0.51862
0.99	0.32195	0.33434	0.37406
0.999	0.21069	0.21812	0.26288
0.9999	0.15498	0.15959	0.20193
1 — 10⁻⁶	0.10106	0.10324	0.13785
1 — 10⁻⁸	0.07493	0.07619	0.10464
1 — 10⁻¹⁰	0.05953	0.06035	0.08432
1 — 10⁻²⁰	0.02936	0.02951	0.04279
1 — 10⁻⁴⁰	0.01458	0.01463	0.02155
1 — 10⁻⁶⁰	0.00970	0.00972	0.01440
1 — 10⁻⁸⁰	0.00726	0.00728	0.01082
1 — 10⁻⁹⁰	0.00645	0.00646	0.00962

			Radiance			Radiance		Polarization
k	ω₀	θ	Vector	Scalar	θ	Vector	Scalar	Polarization
0.3	0.96900	180°	1.000	1.0000	0°	5.385 — 1	5.385 — 1	0
		160°	9.715 — 1	9.736 — 1	20°	5.442 — 1	5.454 — 1	3.446 — 3
		140°	8.978 — 1	9.050 — 1	40°	5.623 — 1	5.667 — 1	1.228 — 2
		120°	8.054 — 1	8.171 — 1	60°	5.953 — 1	6.039 — 1	2.251 — 2
		100°	7.174 — 1	7.310 — 1	80°	6.464 — 1	6.586 — 1	2.930 — 2
		90°	6.794 — 1	6.927 — 1	90°	6.794 — 1	6.927 — 1	3.024 — 2

0.5	0.90818	180°	1.0000	1.0000	0°	3.333 — 1	3.333 — 1	0
		160°	9.339 — 1	9.396 — 1	20°	3.369 — 1	3.389 — 1	9.844 — 3
		140°	7.825 — 1	7.998 — 1	40°	3.491 — 1	3.568 — 1	3.566 — 2
		120°	6.250 — 1	6.508 — 1	60°	3.750 — 1	3.905 — 1	6.666 — 2
		100°	5.033 — 1	5.307 — 1	80°	4.229 — 1	4.459 — 1	8.792 — 2
		90°	4.583 — 1	4.842 — 1	90°	4.583 — 1	4.842 — 1	9.091 — 2

0.8	0.71178	180°	1.0000	1.0000	0°	1.111 — 1	1.111 — 1	0
		160°	7.853 — 1	7.955 — 1	20°	1.113 — 1	1.127 — 1	2.591 — 2
		140°	4.705 — 1	4.936 — 1	40°	1.129 — 1	1.185 — 1	9.794 — 2
		120°	2.794 — 1	3.064 — 1	60°	1.197 — 1	1.313 — 1	1.932 — 1
		100°	1.836 — 1	2.080 — 1	80°	1.388 — 1	1.572 — 1	2.648 — 1
		90°	1.568 — 1	1.784 — 1	90°	1.568 — 1	1.784 — 1	2.754 — 1

0.93	0.52525	180°	1.0000	1.0000	0°	3.627 — 2	3.627 — 2	0
		160°	5.352 — 1	5.425 — 1	20°	3.601 — 2	3.650 — 2	3.738 — 2
		140°	2.124 — 1	2.238 — 1	40°	3.568 — 2	3.759 — 2	1.458 — 1
		120°	1.006 — 1	1.117 — 1	60°	3.674 — 2	4.080 — 2	3.005 — 1
		100°	5.854 — 2	6.772 — 2	80°	4.226 — 2	4.889 — 2	4.261 — 1
		90°	4.844 — 2	5.637 — 2	90°	4.844 — 2	5.637 — 2	4.452 — 1

0.99	0.32195	180°	1.0000	1.0000	0°	5.025 — 3	5.025 — 3	0
		160°	1.369 — 1	1.383 — 1	20°	4.943 — 3	4.993 — 3	4.802 — 2
		140°	3.469 — 2	3.610 — 2	40°	4.767 — 3	4.961 — 3	1.931 — 1
		120°	1.398 — 2	1.521 — 2	60°	4.724 — 3	5.139 — 3	4.160 — 1
		100°	7.488 — 3	8.458 — 3	80°	5.292 — 3	5.976 — 3	6.126 — 1
		90°	6.083 — 3	6.911 — 3	90°	6.083 — 3	6.911 — 3	6.439 — 1

0.999	0.21069	180°	1.0000	1.0000	0°	5.002 — 4	5.002 — 4	0
		160°	1.550 — 2	1.560 — 2	20°	4.896 — 4	4.930 — 4	5.352 — 2
		140°	3.496 — 3	3.594 — 3	40°	4.648 — 4	4.779 — 4	2.187 — 1
		120°	1.347 — 3	1.431 — 3	60°	4.497 — 4	4.775 — 4	4.830 — 1
		100°	7.003 — 4	7.657 — 4	80°	4.932 — 4	5.393 — 4	7.277 — 1
		90°	5.657 — 4	6.214 — 4	90°	5.657 — 4	6.214 — 4	7.676 — 1

l — 10⁻⁵	0.12237	180°	1.0000	1.0000	0°	5.000 — 6	5.000 — 6	0
		179°	6.160 — 2	6.160 — 2	1°	5.000 — 6	5.000 — 6	1.417 — 4
		170°	6.486 — 4	6.492 — 4	10°	4.968 — 6	4.972 — 6	1.423 — 2
		160°	1.568 — 4	1.574 — 4	20°	4.875 — 6	4.894 — 6	5.754 — 2
		140°	3.453 — 5	3.509 — 5	40°	4.574 — 6	4.649 — 6	2.379 — 1
		120°	1.302 — 5	1.350 — 5	60°	4.341 — 6	4.501 — 6	5.358 — 1
		100°	6.642 — 6	7.018 — 6	80°	4.677 — 6	4.942 — 6	8.219 — 1
		90°	5.348 — 6	5.669 — 6	90°	5.348 — 6	5.669 — 6	8.697 — 1

1 — 10⁻²⁰	0.02936	180°	1.0000	1.0000	0°	5.000 — 21	5.000 — 21	0
		179.99°	6.579 — 13	6.579 — 13	0.01°	5.000 — 21	5.000 — 21	1.497 — 8
		179.9°	6.566 — 15	6.566 — 15	0.1°	5.000 — 21	5.000 — 21	1.500 — 6
		179°	6.565 — 17	6.565 — 17	1°	5.000 — 21	5.000 — 21	1.500 — 4
		170°	6.485 — 19	6.486 — 19	10°	4.964 — 21	4.965 — 21	1.507 — 2
		160°	1.563 — 19	1.564 — 19	20°	4.858 — 21	4.863 — 21	6.112 — 2
		140°	3.405 — 20	3.418 — 20	40°	4.510 — 21	4.528 — 21	2.554 — 1
		120°	1.261 — 20	1.272 — 20	60°	4.204 — 21	4.242 — 21	5.856 — 1
		100°	6.232 — 21	6.409 — 21	80°	4.451 — 21	4.513 — 21	9.142 — 1
		90°	5.076 — 21	5.150 — 21	90°	5.076 — 21	5.150 — 21	9.702 — 1

1 — 10⁻⁹⁰	0.00645	180°	1.0000	1.0000	0°	5.000 — 91	5.000 — 91	0
		179.99°	6.579 — 83	6.579 — 83	0.01°	5.000 — 91	5.000 — 91	1.515 — 8
		179.9°	6.566 — 85	6.566 — 85	0.1°	5.000 — 91	5.000 — 91	1.518 — 6
		179°	6.565 — 87	6.565 — 87	1°	5.000 — 91	5.000 — 91	1.518 — 4
		170°	6.483 — 89	6.484 — 89	10°	4.963 — 91	4.963 — 91	1.526 — 2
		160°	1.562 — 89	1.562 — 89	20°	4.855 — 91	4.856 — 91	6.191 — 2
		140°	3.394 — 90	3.397 — 90	40°	4.496 — 91	4.500 — 91	2.593 — 1
		120°	1.252 — 90	1.255 — 90	60°	4.175 — 91	4.183 — 91	5.969 — 1
		100°	6.252 — 91	6.271 — 91	80°	4.402 — 91	4.415 — 91	9.355 — 1
		90°	5.016 — 91	5.032 — 91	90°	5.016 — 91	5.032 — 91	9.935 — 1

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Equations (42)

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