A Generalized Linear Transport Model for Spatially Correlated Stochastic Media

Abstract

We formulate a new model for transport in stochastic media with long-range spatial correlations where exponential attenuation (controlling the propagation part of the transport) becomes power law. Direct transmission over optical distance r(s), for fixed physical distance s, thus becomes (1 + r(s)/a)~", with standard exponential decay re covered when a -*■ oo. Atmospheric turbulence phenomenology for fluctuating optical properties rationalizes this switch. Foundational equations for this generalized trans port model are stated in integral form for d = 1,2,3 spatial dimensions. A deterministic numerical solution is developed in d = 1 using Markov Chain formalism, verified with Monte Carlo, and used to investigate internal radiation fields. Standard two-stream theory, where diffusion is exact, is recovered when a = oo. Differential diffusion equa tions are not presently known when a < oo, nor is the integro-differential form of the generalized transport equation. Monte Carlo simulations are performed in d = 2, as a model for transport on random surfaces, to explore scaling behavior of transmit tance T when transport optical thickness rt » 1. Random walk theory correctly predicts T a rt ĩm,I',1,a/2) in the absence of absorption. Finally, single scattering theory in d = 3 highlights the model's violation of angular reciprocity when a < oo, a desirable prop erty at least in atmospheric applications. This violation is traced back to a key trait of generalized transport theory, namely, that we must distinguish more carefully between two kinds of propagation: one that ends in a virtual or actual detection and the other in a transition from one position to another in the medium.

A spectral approach for solving the nonclassical transport equation

“…Nevertheless, there are several important applications [6,7] in which the free-path distribution has a large tail, decaying algebraically as s → ∞:…”

“…This leads to the Beer-Lambert law, with particle flux decreasing as an exponential function of s. However, a nonexponential attenuation law for the particle flux occurs in certain heterogeneous media in which the scattering centers in the system are spatially correlated [2,[8][9][10][11][13][14][15][16][17][18][19]. The theory of nonclassical particle transport [1,[3][4][5][6][7] was developed to address this class of problems. In particular, it assumes that Σ t is a function of both Ω and s, which requires an extended phase space that includes the free-path s as an extra independent variable.…”

Coupled Retrieval of Liquid Water Cloud and Above‐Cloud Aerosol Properties Using the Airborne Multiangle SpectroPolarimetric Imager (AirMSPI)

“…Additionally, cloud horizontal inhomogeneity, cloud geometrical thickness, cloud top roughness, pixel domain size, and solar and viewing zenith angle are also major factors that can bias COD and droplet size retrievals due to 3-D radiative transfer (RT) effects (e.g., Chambers et al, 1997;Davis et al, 1997;Horváth & Davies, 2004;Iwabuchi & Hayasaka, 2002;Liang & Di Girolamo, 2013;Loeb & Coakley, 1998;Loeb & Davies, 1996;Marshak et al, 1995Marshak et al, , 2006Oreopoulos et al, 2000;Peers et al, 2015;Várnai & Marshak, 2001;Zhang et al, 2012;Zuidema & Evans, 1998). While progress is still being made toward mitigating (e.g., Cairns et al, 2000;Davis & Xu, 2014;Xu, Davis, & Diner, 2016) or correcting (e.g., Marshak et al, 1998) the effects of scene heterogeneity when using 1-D RT, in this paper, we focus on analysis of stratocumulus cloud layers that occur persistently in marine boundary layers off the west coast of Africa using data collected by the Airborne Multiangle SpectroPolarimetric Imager (AirMSPI). 3-D RT effects on COD retrievals for such clouds are relatively well understood, and their impact on the droplet size retrievals using polarimetric data is minor, as discussed later.…”

Radiative Transfer in Half Spaces of Arbitrary Dimension

“…There are both practical and theoretical reasons to consider transport problems in general dimension. The 1D rod has long been a useful domain for transport research and education [5,6,7,8]. The same can be said for the 2D "Flatland" domain [9], where it is possible to visualize the entire lightfield with 2D images [10].…”

“…Either kernel can be exhibited in R d under isotropic scattering by selecting the free-paths between collisions from a family of distributions involving Bessel-K functions, transformed propagator in R d is a diffusion mode to a power p [kernel in Eq. (22) arises when p = 1, which happens in Flatland[43] via p c (s) = sK 0 (s) (D 8). and in 3D via[43] p c (s) = e −s s. (D.9)The MacDonald kernel in Eq.…”