Interior radiances in optically deep absorbing media—I Exact solutions for one-dimensional model

The exact solutions are obtained for a one-dimensional model of a scattering and absorbing medium. The results are given for both the reflected and transmitted radiance for any arbitrary surface albedo as well as for the interior radiance. These same quantities are calculated by the matrix operator method. The relative error of the solutions is obtained by comparison with the exact solutions as well as by an error analysis of the equations. The importance of an accurate starting value for the reflection and transmission operators is shown. A fourth-order Runge-Kutta method can be used to solve the differential equations satisfied by these operators in order to obtain such accurate starting values. Except for extremely large values of the optical thickness of a layer, the reflection and transmission operators calculated from accurate starting values obtained by the Runge-Kutta method are orders of magnitude (a factor of 10¹¹ better in a typical case at unit optical depth) more accurate than those obtained by the use of the single scattering approximation for an optically thin layer. The relative error in the reflection and transmission operators is less than 10^-12 and 10^-8 respectively up to an optical thickness of 32,768 when calculated by this procedure, while the relative error in the interior radiance is less than 10^-8 at all points within a layer of optical thickness 32,768. It is shown that flux conservation is a poor test of the accuracy of a numerical method, since flux is conserved to all orders for a conservative medium when the doubling method is used, no matter how inaccurate the starting values may be.