Nonparametric estimation of the heterogeneity of a random medium using compound Poisson process modeling of wave multiple scattering

In this paper, we present a nonparametric method to estimate the heterogeneity of a random medium from the angular distribution of intensity of waves transmitted through a slab of random material. Our approach is based on the modeling of forward multiple scattering using compound Poisson processes on compact Lie groups. The estimation technique is validated through numerical simulations based on radiative transfer theory.

Figure 1

Position of the problem. A plane wave or a narrow beam are normally incident on a slab of random heterogeneous material with thickness $z$ expressed in transport mean-free-path units. The transmitted intensity probability distribution is $I\left(\theta \right)$. The gray level fluctuations reflect the inhomogeneity of the random medium.

Figure 2

Comparison between Monte Carlo simulations (dots) and CPP model for the intensity transmitted through a slab of random medium with Heynyey-Greenstein phase function. The four curves correspond to increasing slab thickness $H=25/8,25/4,25/2,25\ell$, with $\ell$ as the mean free path. The agreement is very good in a cone around the forward direction and degrades as the scattering angle approaches grazing angles.

Figure 3

Comparison between Monte Carlo simulations (dots) and CPP model for the intensity transmitted through a slab of random medium with Gaussian phase function. The four curves correspond to increasing slab thicknesses $H=25/16,25/8,25/4,25/2\ell$, with $\ell$ as the mean free path. The agreement is very good in a cone around the forward direction and degrades as the scattering angle approaches grazing angles.

Figure 4

Distribution of the number of scattering events for the intensity transmitted through a slab of Gaussian random medium with thickness $H=3.125\ell$. Squares: $\left(0.8<\mu <1\right)$; dots: $\left(0.3<\mu <0.5\right)$. Solid line: Poisson distribution with parameter $\Lambda =3.125$. Dashed line: Gaussian distribution with mean $\mu =7$ and standard deviation $\sigma =2$.

Figure 5

Transmitted intensity profile through a slab of thickness $H={\ell }^{\ast }/2$ and anisotropy parameter $g=0.98$. Squares: Gaussian random medium. Solid line: HG random medium. Dashed line: diffusion approximation. The difference between the Gaussian and HG medium is only noticeable close to the forward direction $\left(\mu =1\right)$.

Figure 6

Variance of the Legendre coefficients empirical estimator for different phase functions: Gaussian (squares), exponential (triangles), and Henyey-Greenstein (circles) with anisotropy parameter $g=0.995$.

Figure 7

Cumulative distribution function of the intensity transmitted through a slab of random material with thicknesses $H=6.25\ell$ (dashed line), $12.5\ell$ (solid line), and anisotropy parameter $g=0.98$. Top: exponential medium. Bottom: “Kolmogorov” medium.

Figure 8

Estimated Legendre coefficients of the heterogeneity power spectrum for two slabs of random material with thicknesses $H=6.25\ell$ (dots), $12.5\ell$ (squares), and anisotropy parameter $g=0.98$. Top: exponential medium. Bottom: Kolmogorov medium. The dashed line shows the exact Legendre coefficients of the distributions.