Microphysical characterization of free space optical link due to hydrometeor and fog effects

Saverio Mori; Frank S. Marzano

doi:10.1364/AO.54.006787

Applied Optics
Vol. 54,
Issue 22,
pp. 6787-6803
(2015)
•https://doi.org/10.1364/AO.54.006787

Microphysical characterization of free space optical link due to hydrometeor and fog effects

Saverio Mori and Frank S. Marzano

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Saverio Mori and Frank S. Marzano, "Microphysical characterization of free space optical link due to hydrometeor and fog effects," Appl. Opt. 54, 6787-6803 (2015)

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Abstract

Free space optics (FSO) channel availability is affected by atmospheric water particles, which may introduce severe path attenuation. A unified microphysically oriented atmospheric particle scattering (MAPS) model is proposed and described to simulate particle scattering effects on FSO links. Atmospheric particles, such as raindrops, graupel particles, and snowflakes, together with fog droplets, are considered. Input data to characterize liquid and frozen water particle size distribution, density, and refractivity are derived from available literature data and measurements. Scattering, absorption, and extinction coefficients as well as the asymmetry factor are numerically simulated for each particle class and then parametrized with respect to particle water content, fall rate, and visibility, spanning from visible to infrared wavelengths. Both single- and multiple-scattering effects are discussed and quantified by using a radiative transfer model for small-angle approximation. MAPS simulations confirm that fog layers are those causing the largest power extinction on FSO links, but also several decibels of attenuation can be attributed to snow and rain conditions. Multiple-scattering effects, especially due to fog droplets, heavy rain, and dry snowflakes, typically tend to reduce the total attenuation by increasing the received power. An estimate of these effects, parameterized to single-scattering extinction, is proposed for near-infrared FSO link design.

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Equations (49)

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Class	$ρ_{p \min} \leq ρ_{p} \leq ρ_{p \max}$ [ $g \cdot {cm}^{- 3}$ ]	$T$ [K]	$r_{\min} \leq r \leq r_{\max}$ [mm]	$W_{p \min} \leq W_{p} \leq W_{p \max}$ [ $g \cdot m^{- 3}$ ]	$r_{e \min} \leq r_{e} \leq r_{e \max}$ [mm]	$μ_{\min} \leq μ_{e} \leq μ_{\max}$ [adim]	$N_{e \min} \leq Ne \leq N_{e \max}$ [ ${mm}^{- 1} m^{- 3}$ ]	$Λ_{e \min} \leq Λ_{e} \leq Λ_{e \max}$ [adim]
LR, light rain	1.0–1.0	298	0.3/3.5	0.0/0.1	0.1/0.6	0.0/0.1	$9.86 \cdot 10^{- 2} / 2.43 \cdot 10^{6}$	3.00/3.10
MR, moderate rain	1.0–1.0	298	0.3/3.5	0.0/0.6	0.4/0.8	0.0/0.6	$8.32 \cdot 10^{1} / 1.18 \cdot 10^{5}$	3.00/3.60
HR, heavy rain	1.0–1.0	298	0.3/3.5	0.0/5.0	0.6/1.3	0.0/0.8	$2.29 \cdot 10^{1} / 1.94 \cdot 10^{5}$	3.00/3.80
G/SH, graupel	0.4–0.4	273	0.5/2.5	0.0/2.5	0.8/2.5	0.0/0.0	$2.26 \cdot 10^{0} / 4.66 \cdot 10^{4}$	3.00/3.00
DS, dry snow	0.01–0.2	273	0.5/7.5	0.0/1.0	0.7/1.0	0.0/0.0	$1.20 \cdot 10^{1} / 8.48 \cdot 10^{5}$	3.00/3.00
WS, wet snow	0.01–2.0	273	0.5/7.5	0.0/1.0	0.7/1.0	0.0/0.0	$2.24 \cdot 10^{1} / 1.10 \cdot 10^{6}$	3.00/3.00
AF, advection fog	1.0–1.0	298	0.0/0.04	0.0/0.4	0.019/0.021	2.9/3.1	$8.31 \cdot 10^{7} / 2.86 \cdot 10^{11}$	5.90/6.10
RF, radiation fog	1.0–1.0	298	0.0/0.04	0.0/0.025	0.001/0.006	1.0/5.0	$2.45 \cdot 10^{9} / 6.00 \cdot 10^{15}$	4.01/8.00

Wavelength [nm]	Class	$a_{k e p}$	$b_{k e p}$	$a_{k s p}$	$b_{k s p}$	$a_{k a p}$	$b_{k a p}$	$a_{g p}$	$b_{g p}$
785.00	Light Rain	$+ 4.83 E + 00$	$+ 1.00 E + 00$	$+ 4.82 E + 00$	$+ 1.00 E + 00$	$+ 4.09 E - 03$	$+ 1.01 E + 00$	$+ 4.27 E + 00$	$+ 1.00 E + 00$
	Mod. Rain	$+ 4.56 E + 00$	$+ 9.94 E - 01$	$+ 4.56 E + 00$	$+ 9.94 E - 01$	$+ 5.46 E - 03$	$+ 1.00 E + 00$	$+ 4.04 E + 00$	$+ 9.94 E - 01$
	Heavy Rain	$+ 3.14 E + 00$	$+ 1.01 E + 00$	$+ 3.13 E + 00$	$+ 1.01 E + 00$	$+ 5.53 E - 03$	$+ 9.98 E - 01$	$+ 2.78 E + 00$	$+ 1.01 E + 00$
	Graupel	$+ 3.78 E + 00$	$+ 1.02 E + 00$	$+ 3.77 E + 00$	$+ 1.02 E + 00$	$+ 7.48 E - 03$	$+ 1.01 E + 00$	$+ 3.37 E + 00$	$+ 1.02 E + 00$
	Dry Snow	$+ 3.13 E + 01$	$+ 1.00 E + 00$	$+ 3.13 E + 01$	$+ 1.00 E + 00$	$+ 2.18 E - 03$	$+ 1.00 E + 00$	$+ 3.11 E + 01$	$+ 1.00 E + 00$
	Wet Snow	$+ 6.54 E + 00$	$+ 9.79 E - 01$	$+ 6.51 E + 00$	$+ 9.78 E - 01$	$+ 2.40 E - 02$	$+ 1.03 E + 00$	$+ 5.79 E + 00$	$+ 9.74 E - 01$
	Adv. Fog	$+ 1.56 E + 02$	$+ 1.01 E + 00$	$+ 1.56 E + 02$	$+ 1.01 E + 00$	$+ 5.64 E - 03$	$+ 1.00 E + 00$	$+ 1.35 E + 02$	$+ 1.01 E + 00$
	Rad. Fog	$+ 1.13 E + 03$	$+ 1.01 E + 00$	$+ 1.13 E + 03$	$+ 1.01 E + 00$	$+ 7.96 E - 03$	$+ 1.05 E + 00$	$+ 9.44 E + 02$	$+ 1.01 E + 00$
850.00	Light Rain	$+ 5.00 E + 00$	$+ 9.98 E - 01$	$+ 4.99 E + 00$	$+ 9.98 E - 01$	$+ 9.29 E - 03$	$+ 1.02 E + 00$	$+ 4.42 E + 00$	$+ 9.98 E - 01$
	Mod. Rain	$+ 4.37 E + 00$	$+ 9.73 E - 01$	$+ 4.36 E + 00$	$+ 9.72 E - 01$	$+ 1.12 E - 02$	$+ 1.00 E + 00$	$+ 3.87 E + 00$	$+ 9.73 E - 01$
	Heavy Rain	$+ 3.05 E + 00$	$+ 1.00 E + 00$	$+ 3.04 E + 00$	$+ 1.00 E + 00$	$+ 1.12 E - 02$	$+ 9.99 E - 01$	$+ 2.70 E + 00$	$+ 1.00 E + 00$
	Graupel	$+ 3.51 E + 00$	$+ 1.00 E + 00$	$+ 3.49 E + 00$	$+ 1.00 E + 00$	$+ 1.10 E - 02$	$+ 9.98 E - 01$	$+ 3.13 E + 00$	$+ 1.00 E + 00$
	Dry Snow	$+ 3.13 E + 01$	$+ 1.00 E + 00$	$+ 3.13 E + 01$	$+ 1.00 E + 00$	$+ 5.52 E - 03$	$+ 1.00 E + 00$	$+ 3.11 E + 01$	$+ 1.00 E + 00$
	Wet Snow	$+ 6.61 E + 00$	$+ 9.87 E - 01$	$+ 6.57 E + 00$	$+ 9.87 E - 01$	$+ 4.08 E - 02$	$+ 1.02 E + 00$	$+ 5.87 E + 00$	$+ 9.84 E - 01$
	Adv. Fog	$+ 1.54 E + 02$	$+ 1.00 E + 00$	$+ 1.54 E + 02$	$+ 1.00 E + 00$	$+ 1.28 E - 02$	$+ 1.00 E + 00$	$+ 1.34 E + 02$	$+ 1.00 E + 00$
	Rad. Fog	$+ 7.67 E + 02$	$+ 9.56 E - 01$	$+ 7.67 E + 02$	$+ 9.56 E - 01$	$+ 1.25 E - 02$	$+ 9.88 E - 01$	$+ 6.21 E + 02$	$+ 9.59 E - 01$
1064.00	Light Rain	$+ 4.70 E + 00$	$+ 1.01 E + 00$	$+ 4.68 E + 00$	$+ 1.01 E + 00$	$+ 2.37 E - 02$	$+ 9.85 E - 01$	$+ 4.16 E + 00$	$+ 1.01 E + 00$
	Mod. Rain	$+ 4.82 E + 00$	$+ 1.02 E + 00$	$+ 4.78 E + 00$	$+ 1.02 E + 00$	$+ 3.75 E - 02$	$+ 9.97 E - 01$	$+ 4.25 E + 00$	$+ 1.02 E + 00$
	Heavy Rain	$+ 3.16 E + 00$	$+ 9.70 E - 01$	$+ 3.12 E + 00$	$+ 9.70 E - 01$	$+ 3.81 E - 02$	$+ 9.98 E - 01$	$+ 2.78 E + 00$	$+ 9.70 E - 01$
	Graupel	$+ 3.56 E + 00$	$+ 9.81 E - 01$	$+ 3.47 E + 00$	$+ 9.81 E - 01$	$+ 8.91 E - 02$	$+ 9.88 E - 01$	$+ 3.12 E + 00$	$+ 9.81 E - 01$
	Dry Snow	$+ 3.13 E + 01$	$+ 1.00 E + 00$	$+ 3.13 E + 01$	$+ 1.00 E + 00$	$+ 4.48 E - 02$	$+ 1.00 E + 00$	$+ 3.11 E + 01$	$+ 1.00 E + 00$
	Wet Snow	$+ 6.70 E + 00$	$+ 1.00 E + 00$	$+ 6.53 E + 00$	$+ 1.01 E + 00$	$+ 1.69 E - 01$	$+ 9.95 E - 01$	$+ 5.93 E + 00$	$+ 1.01 E + 00$
	Adv. Fog	$+ 1.55 E + 02$	$+ 1.00 E + 00$	$+ 1.55 E + 02$	$+ 1.00 E + 00$	$+ 4.20 E - 02$	$+ 1.00 E + 00$	$+ 1.34 E + 02$	$+ 1.00 E + 00$
	Rad. Fog	$+ 6.95 E + 02$	$+ 9.76 E - 01$	$+ 6.95 E + 02$	$+ 9.76 E - 01$	$+ 4.27 E - 02$	$+ 9.98 E - 01$	$+ 5.32 E + 02$	$+ 9.76 E - 01$
1550.00	Light Rain	$+ 4.95 E + 00$	$+ 9.96 E - 01$	$+ 3.68 E + 00$	$+ 9.95 E - 01$	$+ 1.26 E + 00$	$+ 9.96 E - 01$	$+ 3.42 E + 00$	$+ 9.95 E - 01$
	Mod. Rain	$+ 4.81 E + 00$	$+ 1.03 E + 00$	$+ 3.35 E + 00$	$+ 1.04 E + 00$	$+ 1.45 E + 00$	$+ 1.02 E + 00$	$+ 3.15 E + 00$	$+ 1.04 E + 00$
	Heavy Rain	$+ 3.06 E + 00$	$+ 1.04 E + 00$	$+ 1.95 E + 00$	$+ 1.04 E + 00$	$+ 1.11 E + 00$	$+ 1.02 E + 00$	$+ 1.85 E + 00$	$+ 1.04 E + 00$
	Graupel	$+ 3.62 E + 00$	$+ 9.73 E - 01$	$+ 1.93 E + 00$	$+ 9.73 E - 01$	$+ 1.69 E + 00$	$+ 9.73 E - 01$	$+ 1.88 E + 00$	$+ 9.73 E - 01$
	Dry Snow	$+ 3.14 E + 01$	$+ 1.00 E + 00$	$+ 2.59 E + 01$	$+ 1.00 E + 00$	$+ 5.48 E + 00$	$+ 1.00 E + 00$	$+ 2.58 E + 01$	$+ 1.00 E + 00$
	Wet Snow	$+ 6.76 E + 00$	$+ 1.02 E + 00$	$+ 3.94 E + 00$	$+ 1.02 E + 00$	$+ 2.82 E + 00$	$+ 1.02 E + 00$	$+ 3.78 E + 00$	$+ 1.02 E + 00$
	Adv. Fog	$+ 1.57 E + 02$	$+ 1.00 E + 00$	$+ 1.54 E + 02$	$+ 1.00 E + 00$	$+ 2.93 E + 00$	$+ 1.00 E + 00$	$+ 1.34 E + 02$	$+ 1.00 E + 00$
	Rad. Fog	$+ 7.61 E + 02$	$+ 8.97 E - 01$	$+ 7.58 E + 02$	$+ 8.97 E - 01$	$+ 2.58 E + 00$	$+ 9.57 E - 01$	$+ 6.29 E + 02$	$+ 8.97 E - 01$
10000.00	Light Rain	$+ 4.02 E + 00$	$+ 9.47 E - 01$	$+ 2.09 E + 00$	$+ 9.47 E - 01$	$+ 1.93 E + 00$	$+ 9.47 E - 01$	$+ 2.06 E + 00$	$+ 9.47 E - 01$
	Mod. Rain	$+ 4.60 E + 00$	$+ 9.88 E - 01$	$+ 2.40 E + 00$	$+ 9.88 E - 01$	$+ 2.20 E + 00$	$+ 9.88 E - 01$	$+ 2.36 E + 00$	$+ 9.88 E - 01$
	Heavy Rain	$+ 3.27 E + 00$	$+ 9.83 E - 01$	$+ 1.70 E + 00$	$+ 9.83 E - 01$	$+ 1.56 E + 00$	$+ 9.83 E - 01$	$+ 1.68 E + 00$	$+ 9.83 E - 01$
	Graupel	$+ 3.60 E + 00$	$+ 9.91 E - 01$	$+ 1.88 E + 00$	$+ 9.91 E - 01$	$+ 1.72 E + 00$	$+ 9.91 E - 01$	$+ 1.85 E + 00$	$+ 9.91 E - 01$
	Dry Snow	$+ 3.16 E + 01$	$+ 1.00 E + 00$	$+ 1.59 E + 01$	$+ 1.00 E + 00$	$+ 1.57 E + 01$	$+ 1.00 E + 00$	$+ 1.59 E + 01$	$+ 1.00 E + 00$
	Wet Snow	$+ 6.75 E + 00$	$+ 9.87 E - 01$	$+ 3.53 E + 00$	$+ 9.88 E - 01$	$+ 3.23 E + 00$	$+ 9.86 E - 01$	$+ 3.45 E + 00$	$+ 9.87 E - 01$
	Adv. Fog	$+ 1.84 E + 02$	$+ 1.00 E + 00$	$+ 1.14 E + 02$	$+ 1.00 E + 00$	$+ 6.99 E + 01$	$+ 1.00 E + 00$	$+ 1.08 E + 02$	$+ 1.00 E + 00$
	Rad. Fog	$+ 2.30 E + 02$	$+ 9.97 E - 01$	$+ 1.04 E + 02$	$+ 9.94 E - 01$	$+ 1.14 E + 02$	$+ 9.93 E - 01$	$+ 6.13 E + 01$	$+ 9.76 E - 01$

Wavelength [nm]	Class	$a_{0}$	$a_{1}$	$a_{2}$	$a_{3}$	$a_{4}$	$a_{5}$	RMSE
785	Light rain	$+ 6.41 E - 12$	$- 4.30 E - 12$	$+ 1.04 E - 12$	$- 8.84 E - 14$	$+ 1.00 E + 00$	$+ 1.03 E - 16$	$+ 1.52 E - 16$
	Mod. rain	$- 6.00 E - 06$	$+ 4.01 E - 07$	$- 1.06 E - 04$	$- 1.20 E - 03$	$+ 1.00 E + 00$	$+ 2.70 E - 07$	$+ 1.02 E - 07$
	Heavy rain	$+ 1.21 E - 05$	$- 5.71 E - 04$	$+ 1.09 E - 02$	$- 1.14 E - 01$	$+ 8.48 E - 01$	$+ 3.45 E - 02$	$+ 7.18 E - 03$
	Graupel	$+ 7.68 E - 05$	$- 2.29 E - 03$	$+ 2.73 E - 02$	$- 1.78 E - 01$	$+ 8.94 E - 01$	$+ 1.58 E - 02$	$+ 3.60 E - 03$
	Dry snow	$+ 3.86 E - 07$	$- 5.24 E - 05$	$+ 2.83 E - 03$	$- 6.63 E - 02$	$+ 8.47 E - 01$	$+ 6.45 E - 02$	$+ 1.31 E - 02$
	Wet snow	$+ 2.94 E - 04$	$- 6.12 E - 03$	$+ 5.17 E - 02$	$- 2.43 E - 01$	$+ 9.30 E - 01$	$+ 7.67 E - 03$	$+ 1.76 E - 03$
	Adv. fog	$+ 1.94 E - 07$	$- 3.56 E - 05$	$+ 2.40 E - 03$	$- 6.59 E - 02$	$+ 9.56 E - 01$	$+ 1.15 E - 01$	$+ 6.20 E - 02$
	Rad. fog	$+ 7.88 E - 07$	$- 7.80 E - 05$	$+ 3.40 E - 03$	$- 7.12 E - 02$	$+ 8.58 E - 01$	$+ 5.44 E - 02$	$+ 1.23 E - 02$
850	Light rain	$+ 8.54 E - 12$	$- 6.63 E - 12$	$+ 2.02 E - 12$	$- 2.69 E - 13$	$+ 1.00 E + 00$	$- 1.76 E - 16$	$+ 1.42 E - 16$
	Mod. rain	$- 6.11 E - 06$	$- 1.11 E - 06$	$- 1.10 E - 04$	$- 1.28 E - 03$	$+ 1.00 E + 00$	$+ 2.35 E - 07$	$+ 8.57 E - 08$
	Heavy rain	$+ 1.43 E - 05$	$- 6.52 E - 04$	$+ 1.20 E - 02$	$- 1.18 E - 01$	$+ 8.39 E - 01$	$+ 3.53 E - 02$	$+ 7.30 E - 03$
	Graupel	$+ 1.07 E - 04$	$- 2.93 E - 03$	$+ 3.22 E - 02$	$- 1.93 E - 01$	$+ 9.00 E - 01$	$+ 1.39 E - 02$	$+ 3.14 E - 03$
	Dry snow	$+ 3.87 E - 07$	$- 5.20 E - 05$	$+ 2.79 E - 03$	$- 6.50 E - 02$	$+ 8.30 E - 01$	$+ 7.11 E - 02$	$+ 1.42 E - 02$
	Wet snow	$+ 2.89 E - 04$	$- 6.06 E - 03$	$+ 5.15 E - 02$	$- 2.43 E - 01$	$+ 9.27 E - 01$	$+ 7.96 E - 03$	$+ 1.84 E - 03$
	Adv. fog	$+ 1.95 E - 07$	$- 3.55 E - 05$	$+ 2.38 E - 03$	$- 6.52 E - 02$	$+ 9.43 E - 01$	$+ 1.18 E - 01$	$+ 5.85 E - 02$
	Rad. fog	$+ 2.32 E - 06$	$- 1.61 E - 04$	$+ 4.90 E - 03$	$- 8.03 E - 02$	$+ 8.43 E - 01$	$+ 5.16 E - 02$	$+ 1.09 E - 02$
1064	Light rain	$+ 9.55 E - 12$	$- 6.55 E - 12$	$+ 1.77 E - 12$	$- 2.18 E - 13$	$+ 1.00 E + 00$	$- 1.40 E - 16$	$+ 1.05 E - 16$
	Mod. rain	$- 8.95 E - 06$	$+ 9.65 E - 07$	$- 1.60 E - 04$	$- 1.84 E - 03$	$+ 1.00 E + 00$	$+ 4.66 E - 07$	$+ 1.83 E - 07$
	Heavy rain	$+ 1.57 E - 05$	$- 7.06 E - 04$	$+ 1.26 E - 02$	$- 1.20 E - 01$	$+ 8.24 E - 01$	$+ 3.82 E - 02$	$+ 7.61 E - 03$
	Graupel	$+ 1.11 E - 04$	$- 3.01 E - 03$	$+ 3.29 E - 02$	$- 1.94 E - 01$	$+ 8.95 E - 01$	$+ 1.45 E - 02$	$+ 3.19 E - 03$
	Dry snow	$+ 3.85 E - 07$	$- 5.05 E - 05$	$+ 2.67 E - 03$	$- 6.14 E - 02$	$+ 7.81 E - 01$	$+ 8.92 E - 02$	$+ 1.73 E - 02$
	Wet snow	$+ 2.78 E - 04$	$- 5.90 E - 03$	$+ 5.07 E - 02$	$- 2.41 E - 01$	$+ 9.26 E - 01$	$+ 8.13 E - 03$	$+ 1.91 E - 03$
	Adv. fog	$+ 1.82 E - 07$	$- 3.34 E - 05$	$+ 2.25 E - 03$	$- 6.11 E - 02$	$+ 8.89 E - 01$	$+ 1.42 E - 01$	$+ 5.55 E - 02$
	Rad. fog	$+ 6.12 E - 06$	$- 3.39 E - 04$	$+ 7.75 E - 03$	$- 9.59 E - 02$	$+ 8.15 E - 01$	$+ 4.97 E - 02$	$+ 1.01 E - 02$
1550	Light rain	$+ 4.21 E - 12$	$- 3.91 E - 12$	$+ 1.45 E - 12$	$- 2.45 E - 13$	$+ 1.00 E + 00$	$- 4.23 E - 16$	$+ 1.28 E - 16$
	Mod. rain	$+ 8.36 E - 11$	$- 3.74 E - 10$	$+ 6.45 E - 10$	$- 4.85 E - 10$	$+ 1.00 E + 00$	$- 8.60 E - 12$	$+ 3.54 E - 12$
	Heavy rain	$+ 5.74 E - 06$	$- 2.90 E - 04$	$+ 5.91 E - 03$	$- 6.67 E - 02$	$+ 9.38 E - 01$	$+ 1.57 E - 02$	$+ 3.67 E - 03$
	Graupel	$- 6.49 E - 07$	$+ 5.85 E - 06$	$- 5.22 E - 05$	$- 2.87 E - 04$	$+ 1.00 E + 00$	$+ 1.58 E - 05$	$+ 6.02 E - 06$
	Dry snow	$+ 4.19 E - 07$	$- 4.40 E - 05$	$+ 2.05 E - 03$	$- 4.79 E - 02$	$+ 8.43 E - 01$	$+ 6.98 E - 02$	$+ 1.50 E - 02$
	Wet snow	$- 8.64 E - 07$	$+ 4.34 E - 06$	$- 4.81 E - 05$	$- 5.04 E - 04$	$+ 1.00 E + 00$	$+ 4.41 E - 06$	$+ 1.78 E - 06$
	Adv. fog	$+ 1.53 E - 07$	$- 2.85 E - 05$	$+ 1.94 E - 03$	$- 5.30 E - 02$	$+ 8.04 E - 01$	$+ 1.66 E - 01$	$+ 4.84 E - 02$
	Rad. fog	$+ 7.11 E - 07$	$- 6.96 E - 05$	$+ 2.98 E - 03$	$- 6.01 E - 02$	$+ 7.13 E - 01$	$+ 1.07 E - 01$	$+ 1.95 E - 02$
10,000	Light rain	$+ 7.33 E - 12$	$- 5.06 E - 12$	$+ 1.38 E - 12$	$- 1.71 E - 13$	$+ 1.00 E + 00$	$+ 3.26 E - 17$	$+ 1.19 E - 16$
	Mod. rain	$+ 3.61 E - 10$	$- 1.56 E - 09$	$+ 2.62 E - 09$	$- 1.93 E - 09$	$+ 1.00 E + 00$	$- 3.35 E - 11$	$+ 1.29 E - 11$
	Heavy rain	$+ 6.19 E - 06$	$- 3.04 E - 04$	$+ 5.95 E - 03$	$- 6.20 E - 02$	$+ 9.34 E - 01$	$+ 1.66 E - 02$	$+ 3.60 E - 03$
	Graupel	$+ 1.42 E - 06$	$- 4.84 E - 05$	$+ 1.91 E - 04$	$- 7.85 E - 03$	$+ 1.00 E + 00$	$- 4.65 E - 05$	$+ 1.84 E - 05$
	Dry snow	$+ 5.02 E - 07$	$- 4.63 E - 05$	$+ 1.68 E - 03$	$- 3.12 E - 02$	$+ 8.73 E - 01$	$+ 5.45 E - 02$	$+ 1.09 E - 02$
	Wet snow	$+ 6.67 E - 07$	$- 3.50 E - 05$	$+ 1.05 E - 04$	$- 7.60 E - 03$	$+ 1.00 E + 00$	$- 5.85 E - 06$	$+ 2.30 E - 06$
	Adv. fog	$+ 2.11 E - 08$	$- 5.32 E - 06$	$+ 5.01 E - 04$	$- 1.87 E - 02$	$+ 7.58 E - 01$	$+ 1.45 E - 01$	$+ 2.27 E - 02$
	Rad. fog	$- 1.57 E - 07$	$- 1.11 E - 05$	$- 6.44 E - 07$	$- 3.80 E - 03$	$+ 1.00 E + 00$	$+ 1.22 E - 06$	$+ 2.41 E - 07$

Class	$ρ_{p \min} \leq ρ_{p} \leq ρ_{p \max}$ [ $g \cdot {cm}^{- 3}$ ]	$T$ [K]	$r_{\min} \leq r \leq r_{\max}$ [mm]	$W_{p \min} \leq W_{p} \leq W_{p \max}$ [ $g \cdot m^{- 3}$ ]	$r_{e \min} \leq r_{e} \leq r_{e \max}$ [mm]	$μ_{\min} \leq μ_{e} \leq μ_{\max}$ [adim]	$N_{e \min} \leq Ne \leq N_{e \max}$ [ ${mm}^{- 1} m^{- 3}$ ]	$Λ_{e \min} \leq Λ_{e} \leq Λ_{e \max}$ [adim]
LR, light rain	1.0–1.0	298	0.3/3.5	0.0/0.1	0.1/0.6	0.0/0.1	$9.86 \cdot 10^{- 2} / 2.43 \cdot 10^{6}$	3.00/3.10
MR, moderate rain	1.0–1.0	298	0.3/3.5	0.0/0.6	0.4/0.8	0.0/0.6	$8.32 \cdot 10^{1} / 1.18 \cdot 10^{5}$	3.00/3.60
HR, heavy rain	1.0–1.0	298	0.3/3.5	0.0/5.0	0.6/1.3	0.0/0.8	$2.29 \cdot 10^{1} / 1.94 \cdot 10^{5}$	3.00/3.80
G/SH, graupel	0.4–0.4	273	0.5/2.5	0.0/2.5	0.8/2.5	0.0/0.0	$2.26 \cdot 10^{0} / 4.66 \cdot 10^{4}$	3.00/3.00
DS, dry snow	0.01–0.2	273	0.5/7.5	0.0/1.0	0.7/1.0	0.0/0.0	$1.20 \cdot 10^{1} / 8.48 \cdot 10^{5}$	3.00/3.00
WS, wet snow	0.01–2.0	273	0.5/7.5	0.0/1.0	0.7/1.0	0.0/0.0	$2.24 \cdot 10^{1} / 1.10 \cdot 10^{6}$	3.00/3.00
AF, advection fog	1.0–1.0	298	0.0/0.04	0.0/0.4	0.019/0.021	2.9/3.1	$8.31 \cdot 10^{7} / 2.86 \cdot 10^{11}$	5.90/6.10
RF, radiation fog	1.0–1.0	298	0.0/0.04	0.0/0.025	0.001/0.006	1.0/5.0	$2.45 \cdot 10^{9} / 6.00 \cdot 10^{15}$	4.01/8.00

Wavelength [nm]	Class	$a_{k e p}$	$b_{k e p}$	$a_{k s p}$	$b_{k s p}$	$a_{k a p}$	$b_{k a p}$	$a_{g p}$	$b_{g p}$
785.00	Light Rain	$+ 4.83 E + 00$	$+ 1.00 E + 00$	$+ 4.82 E + 00$	$+ 1.00 E + 00$	$+ 4.09 E - 03$	$+ 1.01 E + 00$	$+ 4.27 E + 00$	$+ 1.00 E + 00$
	Mod. Rain	$+ 4.56 E + 00$	$+ 9.94 E - 01$	$+ 4.56 E + 00$	$+ 9.94 E - 01$	$+ 5.46 E - 03$	$+ 1.00 E + 00$	$+ 4.04 E + 00$	$+ 9.94 E - 01$
	Heavy Rain	$+ 3.14 E + 00$	$+ 1.01 E + 00$	$+ 3.13 E + 00$	$+ 1.01 E + 00$	$+ 5.53 E - 03$	$+ 9.98 E - 01$	$+ 2.78 E + 00$	$+ 1.01 E + 00$
	Graupel	$+ 3.78 E + 00$	$+ 1.02 E + 00$	$+ 3.77 E + 00$	$+ 1.02 E + 00$	$+ 7.48 E - 03$	$+ 1.01 E + 00$	$+ 3.37 E + 00$	$+ 1.02 E + 00$
	Dry Snow	$+ 3.13 E + 01$	$+ 1.00 E + 00$	$+ 3.13 E + 01$	$+ 1.00 E + 00$	$+ 2.18 E - 03$	$+ 1.00 E + 00$	$+ 3.11 E + 01$	$+ 1.00 E + 00$
	Wet Snow	$+ 6.54 E + 00$	$+ 9.79 E - 01$	$+ 6.51 E + 00$	$+ 9.78 E - 01$	$+ 2.40 E - 02$	$+ 1.03 E + 00$	$+ 5.79 E + 00$	$+ 9.74 E - 01$
	Adv. Fog	$+ 1.56 E + 02$	$+ 1.01 E + 00$	$+ 1.56 E + 02$	$+ 1.01 E + 00$	$+ 5.64 E - 03$	$+ 1.00 E + 00$	$+ 1.35 E + 02$	$+ 1.01 E + 00$
	Rad. Fog	$+ 1.13 E + 03$	$+ 1.01 E + 00$	$+ 1.13 E + 03$	$+ 1.01 E + 00$	$+ 7.96 E - 03$	$+ 1.05 E + 00$	$+ 9.44 E + 02$	$+ 1.01 E + 00$
850.00	Light Rain	$+ 5.00 E + 00$	$+ 9.98 E - 01$	$+ 4.99 E + 00$	$+ 9.98 E - 01$	$+ 9.29 E - 03$	$+ 1.02 E + 00$	$+ 4.42 E + 00$	$+ 9.98 E - 01$
	Mod. Rain	$+ 4.37 E + 00$	$+ 9.73 E - 01$	$+ 4.36 E + 00$	$+ 9.72 E - 01$	$+ 1.12 E - 02$	$+ 1.00 E + 00$	$+ 3.87 E + 00$	$+ 9.73 E - 01$
	Heavy Rain	$+ 3.05 E + 00$	$+ 1.00 E + 00$	$+ 3.04 E + 00$	$+ 1.00 E + 00$	$+ 1.12 E - 02$	$+ 9.99 E - 01$	$+ 2.70 E + 00$	$+ 1.00 E + 00$
	Graupel	$+ 3.51 E + 00$	$+ 1.00 E + 00$	$+ 3.49 E + 00$	$+ 1.00 E + 00$	$+ 1.10 E - 02$	$+ 9.98 E - 01$	$+ 3.13 E + 00$	$+ 1.00 E + 00$
	Dry Snow	$+ 3.13 E + 01$	$+ 1.00 E + 00$	$+ 3.13 E + 01$	$+ 1.00 E + 00$	$+ 5.52 E - 03$	$+ 1.00 E + 00$	$+ 3.11 E + 01$	$+ 1.00 E + 00$
	Wet Snow	$+ 6.61 E + 00$	$+ 9.87 E - 01$	$+ 6.57 E + 00$	$+ 9.87 E - 01$	$+ 4.08 E - 02$	$+ 1.02 E + 00$	$+ 5.87 E + 00$	$+ 9.84 E - 01$
	Adv. Fog	$+ 1.54 E + 02$	$+ 1.00 E + 00$	$+ 1.54 E + 02$	$+ 1.00 E + 00$	$+ 1.28 E - 02$	$+ 1.00 E + 00$	$+ 1.34 E + 02$	$+ 1.00 E + 00$
	Rad. Fog	$+ 7.67 E + 02$	$+ 9.56 E - 01$	$+ 7.67 E + 02$	$+ 9.56 E - 01$	$+ 1.25 E - 02$	$+ 9.88 E - 01$	$+ 6.21 E + 02$	$+ 9.59 E - 01$
1064.00	Light Rain	$+ 4.70 E + 00$	$+ 1.01 E + 00$	$+ 4.68 E + 00$	$+ 1.01 E + 00$	$+ 2.37 E - 02$	$+ 9.85 E - 01$	$+ 4.16 E + 00$	$+ 1.01 E + 00$
	Mod. Rain	$+ 4.82 E + 00$	$+ 1.02 E + 00$	$+ 4.78 E + 00$	$+ 1.02 E + 00$	$+ 3.75 E - 02$	$+ 9.97 E - 01$	$+ 4.25 E + 00$	$+ 1.02 E + 00$
	Heavy Rain	$+ 3.16 E + 00$	$+ 9.70 E - 01$	$+ 3.12 E + 00$	$+ 9.70 E - 01$	$+ 3.81 E - 02$	$+ 9.98 E - 01$	$+ 2.78 E + 00$	$+ 9.70 E - 01$
	Graupel	$+ 3.56 E + 00$	$+ 9.81 E - 01$	$+ 3.47 E + 00$	$+ 9.81 E - 01$	$+ 8.91 E - 02$	$+ 9.88 E - 01$	$+ 3.12 E + 00$	$+ 9.81 E - 01$
	Dry Snow	$+ 3.13 E + 01$	$+ 1.00 E + 00$	$+ 3.13 E + 01$	$+ 1.00 E + 00$	$+ 4.48 E - 02$	$+ 1.00 E + 00$	$+ 3.11 E + 01$	$+ 1.00 E + 00$
	Wet Snow	$+ 6.70 E + 00$	$+ 1.00 E + 00$	$+ 6.53 E + 00$	$+ 1.01 E + 00$	$+ 1.69 E - 01$	$+ 9.95 E - 01$	$+ 5.93 E + 00$	$+ 1.01 E + 00$
	Adv. Fog	$+ 1.55 E + 02$	$+ 1.00 E + 00$	$+ 1.55 E + 02$	$+ 1.00 E + 00$	$+ 4.20 E - 02$	$+ 1.00 E + 00$	$+ 1.34 E + 02$	$+ 1.00 E + 00$
	Rad. Fog	$+ 6.95 E + 02$	$+ 9.76 E - 01$	$+ 6.95 E + 02$	$+ 9.76 E - 01$	$+ 4.27 E - 02$	$+ 9.98 E - 01$	$+ 5.32 E + 02$	$+ 9.76 E - 01$
1550.00	Light Rain	$+ 4.95 E + 00$	$+ 9.96 E - 01$	$+ 3.68 E + 00$	$+ 9.95 E - 01$	$+ 1.26 E + 00$	$+ 9.96 E - 01$	$+ 3.42 E + 00$	$+ 9.95 E - 01$
	Mod. Rain	$+ 4.81 E + 00$	$+ 1.03 E + 00$	$+ 3.35 E + 00$	$+ 1.04 E + 00$	$+ 1.45 E + 00$	$+ 1.02 E + 00$	$+ 3.15 E + 00$	$+ 1.04 E + 00$
	Heavy Rain	$+ 3.06 E + 00$	$+ 1.04 E + 00$	$+ 1.95 E + 00$	$+ 1.04 E + 00$	$+ 1.11 E + 00$	$+ 1.02 E + 00$	$+ 1.85 E + 00$	$+ 1.04 E + 00$
	Graupel	$+ 3.62 E + 00$	$+ 9.73 E - 01$	$+ 1.93 E + 00$	$+ 9.73 E - 01$	$+ 1.69 E + 00$	$+ 9.73 E - 01$	$+ 1.88 E + 00$	$+ 9.73 E - 01$
	Dry Snow	$+ 3.14 E + 01$	$+ 1.00 E + 00$	$+ 2.59 E + 01$	$+ 1.00 E + 00$	$+ 5.48 E + 00$	$+ 1.00 E + 00$	$+ 2.58 E + 01$	$+ 1.00 E + 00$
	Wet Snow	$+ 6.76 E + 00$	$+ 1.02 E + 00$	$+ 3.94 E + 00$	$+ 1.02 E + 00$	$+ 2.82 E + 00$	$+ 1.02 E + 00$	$+ 3.78 E + 00$	$+ 1.02 E + 00$
	Adv. Fog	$+ 1.57 E + 02$	$+ 1.00 E + 00$	$+ 1.54 E + 02$	$+ 1.00 E + 00$	$+ 2.93 E + 00$	$+ 1.00 E + 00$	$+ 1.34 E + 02$	$+ 1.00 E + 00$
	Rad. Fog	$+ 7.61 E + 02$	$+ 8.97 E - 01$	$+ 7.58 E + 02$	$+ 8.97 E - 01$	$+ 2.58 E + 00$	$+ 9.57 E - 01$	$+ 6.29 E + 02$	$+ 8.97 E - 01$
10000.00	Light Rain	$+ 4.02 E + 00$	$+ 9.47 E - 01$	$+ 2.09 E + 00$	$+ 9.47 E - 01$	$+ 1.93 E + 00$	$+ 9.47 E - 01$	$+ 2.06 E + 00$	$+ 9.47 E - 01$
	Mod. Rain	$+ 4.60 E + 00$	$+ 9.88 E - 01$	$+ 2.40 E + 00$	$+ 9.88 E - 01$	$+ 2.20 E + 00$	$+ 9.88 E - 01$	$+ 2.36 E + 00$	$+ 9.88 E - 01$
	Heavy Rain	$+ 3.27 E + 00$	$+ 9.83 E - 01$	$+ 1.70 E + 00$	$+ 9.83 E - 01$	$+ 1.56 E + 00$	$+ 9.83 E - 01$	$+ 1.68 E + 00$	$+ 9.83 E - 01$
	Graupel	$+ 3.60 E + 00$	$+ 9.91 E - 01$	$+ 1.88 E + 00$	$+ 9.91 E - 01$	$+ 1.72 E + 00$	$+ 9.91 E - 01$	$+ 1.85 E + 00$	$+ 9.91 E - 01$
	Dry Snow	$+ 3.16 E + 01$	$+ 1.00 E + 00$	$+ 1.59 E + 01$	$+ 1.00 E + 00$	$+ 1.57 E + 01$	$+ 1.00 E + 00$	$+ 1.59 E + 01$	$+ 1.00 E + 00$
	Wet Snow	$+ 6.75 E + 00$	$+ 9.87 E - 01$	$+ 3.53 E + 00$	$+ 9.88 E - 01$	$+ 3.23 E + 00$	$+ 9.86 E - 01$	$+ 3.45 E + 00$	$+ 9.87 E - 01$
	Adv. Fog	$+ 1.84 E + 02$	$+ 1.00 E + 00$	$+ 1.14 E + 02$	$+ 1.00 E + 00$	$+ 6.99 E + 01$	$+ 1.00 E + 00$	$+ 1.08 E + 02$	$+ 1.00 E + 00$
	Rad. Fog	$+ 2.30 E + 02$	$+ 9.97 E - 01$	$+ 1.04 E + 02$	$+ 9.94 E - 01$	$+ 1.14 E + 02$	$+ 9.93 E - 01$	$+ 6.13 E + 01$	$+ 9.76 E - 01$

Abstract

Cited By

Figures (14)

Tables (3)

Equations (49)

Applied Optics