A weighting Monte Carlo method for modelling the optical radiation field in the ocean-atmosphere System

Abstract

The paper deals with a new Monte Carlo algorithm for the calculation of the field of optical radiation reflected and refracted by the sea surface. The algorithm allows one to take into account the effects of shadowing and re-reflecting of the radiation by the surface elements. In contrast to the known Monte Carlo methods the algorithm does not contain a cumbersome procedure of numerical construction of the random surface realizations. The paper provides the corresponding local estimates for the radiation intensity field in the ocean-atmosphere system.When elaborating and optimizing the sea remote sensing methods, and also when solving quite a number of fundamental and applied problems of the sea optics, one faces a problem of calculating the electromagnetic field of the radiation reflected and refracted by the sea surface. The mathematical essence of the problem is that one has to calculate certain functionals of the solution to the radiation transfer equation. The functionals are given on a random field which is an undulating sea surface.Evidently the Monte Carlo method was first applied to such a problem in [16]. In the work the simplest model was considered. The model assumes the air-water boundary to be a random surface formed by a set of elementary surface elements whose centers lie in a plane and whose normals are distributed according to a prescribed single-point distribution function. Calculations of the radiation field were made on the basis of the direct modelling of the radiation transfer process. Later on in [3] more effective weighting algorithms were proposed for the same model. Thus the results of calculations [3,16] were obtained without taking account of the effects of shadowing and re-reflecting of the radiation by the surface elements. To take account of these effects, two approaches were proposed in [9]. According to the first approach, one approximately constructs random realizations of the surface deviations in respect to a mean level, and then models the trajectories of radiation quanta (photons). Though the corresponding calculation algorithms (see [8,11]) are rather simple to realize, but their realization costs much computational time. The present work considers the second, more efficient, approach that is in essence a variant of the expectation method.

<title>Statistical modeling of stochastic problems of the atmosphere and ocean optics</title>

“…In the second approach, which is based on the method of mathematical expectations, in order to construct Ncomponent random trajectory, realizations of surface elevations are constructed only at N-points computed in a certain way [14] . At the same time, at the points of photon reaching a random interface, the selection is made of random realizations of normals to the surface.…”

Local estimates in Monte Carlo method for the ocean-atmosphere system with a random interface

“…The problem is to calculate the mean value 7 with respect to realizations of s of the functional / = (p,/) of the solution to the integral transport equation [2,3] /(x) = K/(x) + ψ(χ) (1.2) where χ = (r, ω) e Χ = Η χ Ω is the phase space of the coordinates r e H and the directions co e Ω\ /(χ) is the collision density; Κ is an integral operator of the second kind with a generalized kernel &(x',x) that includes the above boundary conditions on the surfaces S and 5 Q [2]. The function ψ(\) is the density of first collisions of rays from radiation sources.…”

“…Radiation sources localized in space along with the solar radiation flux that is incident on the surface S H are considered as radiation sources. Note that, as in [2], the space H is the union of points of the three-dimensional space and the surfaces 5, S Q , and S H . It should be noted that although the facet model uses nonexistent planes whose normals are not perpendicular to them, the model provides a good approximation and is widely used for practical computations.…”

“…We substitute the explicit expression for the kernel /c(x',x) [2] in the integral, integrate with respect to 5, and derive where η = (r -r')/ 1 r -r'| , r = r' -ω^-ω*^)^, ^ is the distance between the point r' and the plane S in the direction -co r (-co*,s), and y(r,cu',co) = g(r,co',co) a(r) 2π 0<z</z = 0.…”

“…Further, we use the refraction operator [2] to turn from the intensity of radiation outgoing from water to the intensity of radiation incoming from below to the interface, i.e. where χ = (r,co), r e 5, and ω = -<u r (-co*,s).…”