Rightfulness of summation cut-offs in the albedo problem with Gaussian fluctuations of the density of scatterers

The one-dimensional version of the radiative transfer problem (i.e. the so-called rod
model) is analysed with a Gaussian random extinction function σ(x). Then the optical
length X =

L
0 dx σ(x) is a Gaussian random variable. The transmission and reflection
coefficients, T (X) and R(X), are taken as infinite series. When these series (and also
when the series representing T 2(X), R2(X), R(X)T (X), etc.) are averaged, term by
term, according to the Gaussian statistics, the series become divergent after averaging.
As it was shown in a former paper by the authors (in Acta Physica Slovaca (2003)), a
rectification can be managed when a ‘modified’ Gaussian probability density function is
used, equal to zero for X < 0 and proportional to the standard Gaussian probability
density for X >0. In the present paper, the authors put forward an alternative, showing
that if the m.s.r. of X is sufficiently small in comparison with ¯X, the standard Gaussian
averaging is well functional provided that the summation in the series representing the
variable T m−j(X)Rj(X) (m = 1, 2, . . ., j = 1, . . . , m) is truncated at a well-chosen finite
term. The authors exemplify their analysis by some numerical calculations.