Numerical simulation and prediction of atmospheric aerosol extinction using singular value decomposition

The remote sensing problem of polydispersed aerosols in the single scattering approximation is a classical example of the first kind Fredholm integral equation. Assuming that the prediction errors due to arbitrarily small perturbations in the complex aerosol refractive index or the upper radius bounds are negligible, one can form the signal-to-noise ratio (SNR) of the kernel matrix in terms of the singular value of the kernel matrix and the number of measurement wavelengths. The smoothness of the kernel matrix and the information potentialities vary, depending on the choice of a combination of sounding channels. The optimal choice is the one that provides the largest SNR. A numerical simulation with 11 samples of possible combinations is conducted in order to demonstrate that the prediction of aerosol extinction measurements using singular value decomposition is comparable with reference values. If two similar prediction results (e.g. one with SNR 2.019 and the other 2.132) are obtained, the higher value is apparently better, however, a drawback in this case is that the prediction errors increase with the increasing number of sounding channels used. In conclusion, it is noted that the information on the smoothness and potentialities of the kernel matrix has to be factorized in order to increase the success rate of the prediction. Fortran source code is available by its authors upon request.