Physical Oceanography, Vol. 15, No. 3, 2005
NUMERICAL METHOD FOR THE SOLUTION OF THE EQUATION OF RADIATION
TRANSFER. REFLECTION AND TRANSMISSION COEFFICIENTS FOR AN
OPTICALLY THIN PLANE-PARALLEL LAYER
E. B. Shibanov
We study the problem of the choice of initial approximation for the reflection and transmission
coefficients in numerical methods based on the principle of “interaction.” The disadvantages of
the approximation of single scattering are demonstrated and the regularities of propagation of
light in media with strongly anisotropic scattering are analyzed. Semianalytic expressions pro-
posed for the evaluation of the initial approximation enable one to determine the characteristics
of the light field in plane-parallel media with relative errors of about
within the framework
of the algorithm of “adding” of layers.
The data on the spectral and angular characteristics of the light field are used to study the biophysical state
of marine ecosystems. The radiation recorded by an optical sensor is formed as a result of scattering and absorp-
tion of light by the substances contained in water. For the description of the relationship between the intensity of
scattering of light in a given direction and the absorption and scattering properties of a substance, it is customary
to apply the equation of radiation transfer. The numerical methods used for the solution of this equation make it
possible to take into account the basic specific features of transformation of radiation in the ocean–atmosphere
system and solve the equation of radiation transfer for the actual conditions of illumination and scattering indi-
The numerical method for the solution of the equation of radiation transfer in the ocean regarded as basic in
the author’s algorithm is ideologically close to the methods of “adding” and “doubling” [1, 2] and the method of
matrix operator . Despite the fact that these methods were developed about 30 years ago, they were applied
solely in atmospheric optics. At present, the extensive application of personal computers and the increase in the
accuracy of operations with floating point open a possibility of realization of numerical schemes earlier regarded
as unstable. Nevertheless, up to now, the main methods used by foreign researchers for the numerical solution of
the transport equations in the sea are the Monte-Carlo method and the method of invariant imbedding .
Clearly, each method has its own disadvantages. However, the very fact of existence and extensive appli-
cation of the Monte-Carlo method proves that the other methods encounter serious problems. Note that the
method of invariant imbedding is based on the principle of invariance, whereas the method of matrix operator
and the methods of “adding” and “doubling” are based on the principle of “interaction” , which yields the
principles of invariance. The principles of invariance made it possible one to deduce analytic expressions for the
reflection coefficients in the case of simple scattering indicatrices [6, 7]. However, the numerical algorithm is
not a sequence of equivalent transformations and, in author’s opinion, the main problem of the indicated methods
is the problem of solution of the transfer equation for a thin layer with sufficiently high accuracy. The aim of
the present work is to justify the criterion of accuracy and develop a method for the solution of the transfer equa-
tion for an optically thin layer with arbitrary scattering indicatrix.
Marine Hydrophysical Institute, Ukrainian Academy of Sciences, Sevastopol.
Translated from Morskoi Gidrofizicheskii Zhurnal, No.
62–72, May–June, 2005. Original article submitted December 2, 2003;
revision submitted March 15, 2004.
192 0928-5105/05/1503–0192 © 2005 Springer Science+Business Media, Inc.