Characteristic Length and Time Scales of the Highly Forward Scattering of Photons in Random Media

Background: Elucidation of the highly forward scattering of photons in random media such as biological tissue is crucial for further developments of optical imaging using photon transport models. We evaluated length and time scales of the photon scattering in three-dimensional media. Methods: We employed analytical solutions of the time-dependent radiative transfer, M-th order delta-Eddington, and photon diffusion equations (RTE, dEM, and PDE). We calculated the fluence rates at different source-detector distances and optical properties. Results: We found that the zeroth order dEM and PDE, which approximate the highly forward scattering to the isotropic scattering, are valid in longer length and time scales than approximately 10 / μ t ′ and 40 / μ t ′ v , respectively, where μ t ′ is the reduced transport coefficient and v the speed of light in a medium. The first and second order dEM, which approximate the highly forward-peaked phase function by the first two and three Legendre moments, are valid in the longer scales than approximately 4.0 / μ t ′ and 6.3 / μ t ′ v ; 2.8 / μ t ′ and 3.5 / μ t ′ v , respectively. The boundary conditions less influence the length scales, while they reduce the times scales from those for bulk at the longer length scale than approximately 4.0 / μ t ′ . Conclusion: Our findings are useful for constructions of accurate and efficient photon transport models. We evaluated length and time scales of the highly forward scattering of photons in various kinds of three-dimensional random media by analytical solutions of the radiative transfer, M-th order delta-Eddington, and photon diffusion equations.

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Analytical Solutions of Three-Dimensional Contaminant Transport Models with Exponential Source Decay

Ground Water ◽

10.1111/gwat.12564 ◽

2017 ◽

Vol 56 (1) ◽

pp. 96-108 ◽

Cited By ~ 3

Author(s):

Ombretta Paladino ◽

Arianna Moranda ◽

Marco Massabò ◽

Gary A. Robbins

Keyword(s):

Contaminant Transport ◽

Analytical Solutions ◽

Three Dimensional ◽

Transport Models

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Three Dimensional Radiative Effects in Passive Millimeter/Sub-Millimeter All-sky Observations

Remote Sensing ◽

10.3390/rs12030531 ◽

2020 ◽

Vol 12 (3) ◽

pp. 531

Author(s):

Vasileios Barlakas ◽

Patrick Eriksson

Keyword(s):

Radiative Transfer ◽

Root Mean Square ◽

Three Dimensional ◽

Mean Square ◽

Photon Transport ◽

Ice Cloud ◽

Modeling Uncertainties ◽

3D Effects ◽

Mean Square Errors ◽

Satellite Instruments

This study was conducted to quantify the errors prompted by neglecting three-dimensional (3D) effects, i.e., beam-filling and horizontal photon transport effects, at millimeter/sub-millimeter wavelengths. This paper gives an overview of the 3D effects that impact ice cloud retrievals of both current and proposed (Ice Cloud Imager) satellite instruments operating at frequencies of ≈186.3 and ≈668 GHz. The 3D synthetic scenes were generated from two-dimensional (2D) CloudSat (Cloud Satellite) observations over the tropics and mid-latitudes using a stochastic approach. By means of the Atmospheric Radiative Transfer Simulator (ARTS), three radiative transfer simulations were carried out: one 3D, one independent beam approximation (IBA), and one-dimensional (1D). The comparison between the 3D and IBA simulations revealed a small horizontal photon transport effect, with IBA simulations introducing mostly random errors and a slight overestimation (below 1 K). However, performing 1D radiative transfer simulations results in a significant beam-filling effect that increases primarily with frequency, and secondly, with footprint size. For a sensor footprint size of 15 km, the errors induced by neglecting domain heterogeneities yield root mean square errors of up to ≈4 K and ≈13 K at 186.3 GHz and 668 GHz, respectively. However, an instrument operating at the same frequencies, but with a much smaller footprint size, i.e., 6 km, is subject to smaller uncertainties, with a root mean square error of ≈2 K at 186.3 GHz and ≈7.1 K at 668 GHz. When designing future satellite instruments, this effect of footprint size on modeling uncertainties should be considered in the overall error budget. The smallest possible footprint size should be a priority for future sub-millimeter observations in light of these results.

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Hybrid model for photon propagation in random media based on the radiative transfer and diffusion equations

10.1117/12.2019914 ◽

2013 ◽

Author(s):

H. Fujii ◽

Y. Hoshi ◽

S. Okawa ◽

T. Kosuge ◽

S. Kohno

Keyword(s):

Radiative Transfer ◽

Hybrid Model ◽

Random Media ◽

Diffusion Equations ◽

Photon Propagation ◽

And Diffusion

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Inhomogenous diffusion equations in plasma physics and fluid dynamics

Canadian Journal of Physics ◽

10.1139/p78-006 ◽

1978 ◽

Vol 56 (1) ◽

pp. 23-29

Author(s):

C. S. Lai

Keyword(s):

Fluid Dynamics ◽

Plasma Physics ◽

Analytical Solutions ◽

Diffusion Coefficients ◽

Three Dimensional ◽

Diffusion Equations ◽

Diffusion Characteristics ◽

Self Similar Solution ◽

Self Similar ◽

Inhomogeneous Diffusion

Using the method of self-similar solution of partial differential equations, analytical solutions for the two- and three-dimensional inhomogeneous diffusion equations with the diffusion coefficients D ~ rm are obtained. The solutions found can be useful in studying the diffusion characteristics of some fluids and plasmas.

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