Using analytic derivatives to assess the impact of phase function Fourier decomposition technique on the accuracy of a radiative transfer model

Fourier decomposition of the phase function is essential to decouple the azimuthal component of the radiative transfer equation for multiple scattering calculations. This decomposition can be carried out by means of a direct numerical method based on the definition of the Fourier transform (numFT), or by an expansion of the phase function in terms of spherical Legendre polynomials (sphFT). numFT requires interpolation of the phase function between discrete angles, leading to spurious errors in the final computations. This error is difficult to quantify by means of intensity-only computations, since it is hard to determine the absolute accuracy of any given approach. We show that a linearization (analytic computation of derivatives) of the intensity with respect to parameters governing the phase function can be compared against results using the finite difference method, thereby providing a self-consistency test for characterizing and quantifying the error. We have applied this approach to two linearized versions of the Matrix Operator Method, which are identical in all respects except that one uses numFT while the other uses sphFT. In both cases, we compute the derivatives of the intensity with respect to aerosol parameters governing scattering in the simulated atmosphere. Comparison of the derivatives against their finite difference estimates shows a reduction of error by several orders of magnitude when Legendre polynomials are employed. We have also examined the effect of the angular resolution of the phase function on the error due to the numFT technique. A general reduction of error is seen with increasing angular resolution, indicating that interpolation is indeed the major error source. Also, we have pointed out a related source of error in numFT computations that occurs when Fourier decomposition is carried out on the composite phase function of a layer consisting of more than one scatterer. We conclude that an expansion of the phase function in terms of Legendre polynomials eliminates the need for interpolation, making it the more suitable method for linearization of radiative transfer, especially with respect to aerosol properties. -- Highlights: ► Compare numerical and Legendre polynomial based Fourier transforms (FT). ► Differences found in both radiance and Jacobian computations. ► Examine Jacobians given strongly peaked aerosol phase functions. ► Identify interpolation as the main source of error for numerical FT. ► Quantify interpolation error by testing linearized Jacobians vs. FD method