Analytical solution to the optical transfer function of a scattering
medium with large particles

Eleanor P. Zege; Alexander A. Kokhanovsky

doi:10.1364/AO.33.006547

Applied Optics
Vol. 33,
Issue 27,
pp. 6547-6554
(1994)
•https://doi.org/10.1364/AO.33.006547

Analytical solution to the optical transfer function of a scattering medium with large particles

Eleanor P. Zege and Alexander A. Kokhanovsky

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Eleanor P. Zege and Alexander A. Kokhanovsky, "Analytical solution to the optical transfer function of a scattering medium with large particles," Appl. Opt. 33, 6547-6554 (1994)

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Abstract

A new analytical expression for the optical transfer function of multiple-scattering media such as clouds, mists, and dust aerosols is given in terms of their microphysical characteristics. The geometrical optics approximation is used to find local optical parameters of a scattering medium, including the simple approximation of the phase function, which is the key to the solution of the problem considered here. The optical transfer function is taken within a small-angle approximation of the radiative transfer theory. A comparison with Monte Carlo data shows a fairly satisfactory accuracy of our analytic formulas.

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Equations (48)

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N	f(a)	M _j	Z _j
Gamma distributionb	${(\frac{μ}{a_{0}})}^{μ + 1} \frac{a^{μ} exp (- μ a / a_{0})}{Γ (μ + 1)}$	${(\frac{a_{0}}{μ})}^{j} \frac{Γ (μ + 1 + j)}{Γ (μ + 1)}$	P(μ + 1 + j, Δ),
			$Δ = \frac{μ ω^{*}}{2 ρ_{0}}$ ,
			ρ₀ = ka ₀
Log-normal distributionc	$\frac{exp (- 0.5 σ^{- 2} {ln}^{2} a / a_{m})}{a σ {(2 π)}^{1 / 2}}$	${(a_{m})}^{j} exp (\frac{σ^{2} j^{2}}{2})$	$\frac{1 + erf (x)}{2}$ ,
			$x = \frac{1}{σ \sqrt{2}} {ln \frac{ω^{}}{2 ρ^{}} - j σ^{2}}$ ,
			$ρ^{*} = \frac{2 π a_{m}}{λ}$

n	α	β
1.33	3.0	4.7
1.4	2.7	3.6
1.53	2.4	2.4
1.6	2.3	2.0

	σ(θ)

θ (deg)	Mie Theory	Eq. (A10)	Δ (%)
0	2.81 × 10⁻⁴	2.53 × 10⁻⁴	9.9
1	1.98 × 10⁻⁴	1.77 × 10⁻⁴	10.7
2	7.56 × 10⁻³	7.05 × 10⁻³	6.8
3	2.22 × 10⁻³	2.16 × 10⁻³	2.4
4	8.18 × 10⁻²	8.06 × 10⁻²	1.4
5	4.48 × 10⁻²	4.38 × 10⁻²	2.2
6	3.08 × 10⁻²	3.01 × 10⁻²	2.3
7	2.36 × 10⁻²	2.33 × 10⁻²	1.5
8	1.93 × 10⁻²	1.93 × 10⁻²	< 1
9	1.66 × 10⁻²	1.67 × 10⁻²	< 1
10	1.47 × 10⁻²	1.49 × 10⁻²	−1.4
15	9.78 × 10⁻¹	1.03 × 10⁻²	−5.3
25	5.22 × 10⁻¹	5.42 × 10⁻¹	−3.7
35	2.74 × 10⁻¹	2.32 × 10⁻¹	15

N	f(a)	M _j	Z _j
Gamma distributionb	${(\frac{μ}{a_{0}})}^{μ + 1} \frac{a^{μ} exp (- μ a / a_{0})}{Γ (μ + 1)}$	${(\frac{a_{0}}{μ})}^{j} \frac{Γ (μ + 1 + j)}{Γ (μ + 1)}$	P(μ + 1 + j, Δ),
			$Δ = \frac{μ ω^{*}}{2 ρ_{0}}$ ,
			ρ₀ = ka ₀
Log-normal distributionc	$\frac{exp (- 0.5 σ^{- 2} {ln}^{2} a / a_{m})}{a σ {(2 π)}^{1 / 2}}$	${(a_{m})}^{j} exp (\frac{σ^{2} j^{2}}{2})$	$\frac{1 + erf (x)}{2}$ ,
			$x = \frac{1}{σ \sqrt{2}} {ln \frac{ω^{}}{2 ρ^{}} - j σ^{2}}$ ,
			$ρ^{*} = \frac{2 π a_{m}}{λ}$

n	α	β
1.33	3.0	4.7
1.4	2.7	3.6
1.53	2.4	2.4
1.6	2.3	2.0

Abstract

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Figures (4)

Tables (3)

Equations (48)

Applied Optics