In the double spherical harmonics method, which is employed for one-dimensional problems, the spectral radiation intensity is presented as two separate series on the Legendre functions in forward and backward hemispheres; i.e., at = μ > 0 and μ < 0 (Davison, 1957). This expansion is sometimes preferable due to discontinuity of *I*_{λ} angular dependence on the free surface at μ = 0, whereas the angular dependences of *I*_{λ} are continuous at μ > 0 and μ < 0. The double spherical harmonics method (*DP*_{n} approximation) was originally suggested by Yvon (1957) and employed first by Yvon (1957), Ziering and Schiff (1958), and Stewart and Zweifel (1958) to the problems of neutron physics. Solutions for cylinders and spheres were first reported by Drawbaugh and Noderer (1959) (see also additional comments by Schmidt and Gelbard, 1966). Long before Yvon’s sudy (1957), Sykes (1951) discussed the discontinuity of radiation intensity at μ = 0 and suggested using the so-called double Gaussian quadrature, which is equivalent to the double spherical harmonics (Case and Zweifel, 1967).

Following earlier studies conducted by Dombrovsky (1972, 1974a,b), we consider also the first approximation of the double spherical harmonics method, *DP*_{1}, which presents the spectral radiation intensity as two linear functions of angles in two hemispheres. It is clear that this presentation of the angular dependence combines the advantages of *P*_{1} and *DP*_{0} approximations. Here, we will not give the derivation of the *DP*_{1} equations, but only the result for one-dimensional radiative transfer problems at linear anisotropic scattering:

One should review some of the alternative attempts that employed the original idea by Krook (1955) on the moment method applied to separate solid angle subregions. Mengüç and Iyer (1988) used this procedure to formulate the *DP*_{1} approximation for a medium with linear-anisotropic scattering. They also gave the formulation for the octuple spherical harmonics (*OP*_{1}) approximation for two-dimensional, axisymmetric cylindrical enclosures with a solid angle divided into eight subdomains. Iyer and Mengüç (1989) developed a similar hybrid model for two-dimensional rectangular enclosures by combining a four-flux method (which should not be confused with the generalization of the Kubelka-Munk two-flux model discussed in the article Two-flux Approximation) with the *P*_{1} approximation. It was shown in studies conducted by Mengüç and Iyer (1988) and Iyer and Mengüç (1989) that the computational models suggested work fairly well for particular problems considered by these authors. At the same time, the formal dividing of the solid angle cannot take into account the real angular structure of the radiation field. Therefore, this approach is not expected to work well with more complex problems. Remember that the classic spherical harmonics method is based on the angular expansion around the local radiation flux direction. In other words, it takes into account the local angular structure of the radiation field.

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