The SSSS scheme: a method for calculating multiple scattering of electromagnetic radiation by a collection of sparsely spaced spherical scatterers of Mie-scattering size based on first principles

Carynelisa Haspel; Idan Cohen; Idan Cohen

doi:10.1364/AO.465701

Applied Optics
Vol. 61,
Issue 29,
pp. 8563-8577
(2022)
•https://doi.org/10.1364/AO.465701

The SSSS scheme: a method for calculating multiple scattering of electromagnetic radiation by a collection of sparsely spaced spherical scatterers of Mie-scattering size based on first principles

Carynelisa Haspel and Idan Cohen

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Carynelisa Haspel and Idan Cohen, "The SSSS scheme: a method for calculating multiple scattering of electromagnetic radiation by a collection of sparsely spaced spherical scatterers of Mie-scattering size based on first principles," Appl. Opt. 61, 8563-8577 (2022)

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Table of Contents Category
- Atmospheric and Oceanic Optics
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About this Article
History
- Original Manuscript: June 1, 2022
- Revised Manuscript: August 4, 2022
- Manuscript Accepted: August 30, 2022
- Published: October 3, 2022

Abstract

We present a method for calculating multiple scattering of electromagnetic radiation by a collection of sparsely spaced spherical scatterers (SSSS) of Mie-scattering size based on first principles rather than radiative transfer theory. In this respect, our methodology is conceptually similar to the superposition $T$-matrix method. However, our implementation, which we call the SSSS scheme, differs in a number of respects. Overall, the SSSS scheme is simpler, it is better suited numerically to sparse spacing, and the computer memory required is only linearly dependent on the total number of scatterers. We suggest that the SSSS scheme would be particularly useful for examining the effects of different spatial configurations of drops within water clouds in Earth’s atmosphere and would also be useful in other fields of research in which the exact configuration of a collection of sparsely spaced Mie-sized scatterers is important.

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Data availability

The SSSS scheme and the output shown in this study can be obtained from the authors by request.

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Cloud Label	Monodisperse (M) or Polydisperse (P)	Cloud Dimensions $[m] \times [m] \times [m]$	$N_{d r o p t o t}$
M1	M	$0.001 \times 0.001 \times 0.01$	3
M2	M	$0.01 \times 0.01 \times 0.01$	300
M3	M	$0.1 \times 0.1 \times 0.01$	30,000
M4	M	$1.0 \times 1.0 \times 0.001$	300,000
P1	P	$0.001 \times 0.001 \times 0.01$	3
P2	P	$0.01 \times 0.01 \times 0.01$	300
P3	P	$0.1 \times 0.1 \times 0.01$	30,000
P4	P	$1.0 \times 1.0 \times 0.001$	300,000

Cloud	$k_{e x t}$ [ $m^{- 1}$ ]	$Δ z$ [m]	$τ_{e x t}$	$ϖ$	$g$	Delta-Eddington $R$
M1	$1.913 \times 10^{- 1}$ ^a	0.01	$1.913 \times 10^{- 3}$	0.9999995 ^b	0.8535837 ^b	$8.027 \times 10^{- 5}$
M2	$1.913 \times 10^{- 1}$	0.01	$1.913 \times 10^{- 3}$	0.9999995	0.8535837	$8.027 \times 10^{- 5}$
M3	$1.913 \times 10^{- 1}$	0.01	$1.913 \times 10^{- 3}$	0.9999995	0.8535837	$8.027 \times 10^{- 5}$
M4	$1.913 \times 10^{- 1}$	0.001	$1.913 \times 10^{- 4}$	0.9999995	0.8535837	$8.026 \times 10^{- 6}$
P1	$1.352 \times 10^{- 1}$	0.01	$1.352 \times 10^{- 3}$	0.9999995	0.8589660	$5.440 \times 10^{- 5}$
P2	$1.440 \times 10^{- 1}$	0.01	$1.440 \times 10^{- 3}$	0.9999995	0.8633849	$5.591 \times 10^{- 5}$
P3	$1.485 \times 10^{- 1}$	0.01	$1.485 \times 10^{- 3}$	0.9999995	0.8633030	$5.770 \times 10^{- 5}$
P4	$1.487 \times 10^{- 1}$	0.001	$1.487 \times 10^{- 4}$	0.9999995	0.8633337	$5.775 \times 10^{- 6}$

Cloud	SSSS Scheme $R$
M1	$5.561 \times 10^{- 11}$
M2	$5.685 \times 10^{- 9}$
M3	$5.568 \times 10^{- 7}$
M4	$5.567 \times 10^{- 6}$
P1	$3.778 \times 10^{- 11}$
P2	$3.828 \times 10^{- 9}$
P3	$3.971 \times 10^{- 7}$
P4	$3.969 \times 10^{- 6}$

Cloud	Orders of Scattering to Converge	Run Time to Convergence
M1	5	0 s
M2	7	1 s
M3	10	211 s
M4	9	18,300 s
P1	5	0 s
P2	6	1 s
P3	9	17,156 s
P4	13	$\sim 22 d a y s$

Cloud Label	Mean % Diff; Full $P (θ_{s c a}, ϕ_{s c a})$	Max % Diff; Full $P (θ_{s c a}, ϕ_{s c a})$	$(θ_{{s c a}_{max d i f f}}, ϕ_{{s c a}_{max d i f f}})$	Mean % Diff; $P (θ_{s c a})$	Max % Diff; $P (θ_{s c a})$	$θ_{{s c a}_{max d i f f}}$
M1	2%	323%	(43°,170°)	0.2%	7%	90°
M2	7%	4865%	(164°,123°)	0.5%	12%	89°
M3	7%	2985%	(34°,87°)	0.7%	27%	90°
M4	2%	579%	(166°,221°)	0.1%	2%	90°
P1	3%	315%	(1°,331°)	0.3%	7%	180°
P2	5%	3313%	(4°,100°)	0.4%	10%	90°
P3	5%	453%	(17°,348°)	0.4%	10%	91°
P4	2%	206%	(104°,332°)	0.1%	2%	180°

Cloud Label	Mean % Diff; Full $P (θ_{s c a}, ϕ_{s c a})$	Max % Diff; Full $P (θ_{s c a}, ϕ_{s c a})$	$(θ_{{s c a}_{max d i f f}}, ϕ_{{s c a}_{max d i f f}})$	Mean % Diff; $P (θ_{s c a})$	Max % Diff; $P (θ_{s c a})$	$θ_{{s c a}_{max d i f f}}$
M1	0.02%	2%	(106°,335°)	0.002%	0.09%	90°
M2	0.2%	50%	(91°,268°)	0.008%	0.4%	89°
M3	0.3%	48%	(65°,61°)	0.02%	0.7%	91°
M4	0.1%	60%	(92°,269°)	0.006%	0.5%	91°
P1	0.02%	1%	(1°,331°)	0.004%	0.08%	180°
P2	0.08%	9%	(98°,9°)	0.005%	0.4%	90°
P3	0.2%	28%	(90°,1°)	0.01%	0.6%	91°
P4	0.1%	13%	(91°,268°)	0.005%	0.02%	91°

Abstract

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Applied Optics