On initialization, error and flux conservation in the doubling method

Abstract Truncation errors and flux conservation errors in the doubling method are examined. The error properties of five different initial-layer approximations are compared as a function of initial-layer size, layer optical depth, single-scattering albedo, and phase function asymmetry parameter. The “diamond” initial-layer approximation is found to be orders of magnitude more accurate than the others for fixed initial-layer size, or of equivalent accuracy starting from a very much larger initial layer. The commonly used single-scattering initialization is shown to lead to serious flux conservation errors. Analytic error estimates, based upon a new derivation of the single-scattering initialization directly from doubling, are shown to be useful when the layer optical depth is on the order of 10 or less. Finally, questions of round-off error, calculation of an “exact” answer using Richardson extrapolation, and computational efficiency are all addressed briefly.