Abstract

Several aspects of the behavior of Fredholm integral equations are examined in this paper. It is shown that collocation methods are better in general than least squares methods in linear approaches. The amplification of random noise inherent to the numerical inversion of the equation puts an upper limit to the information content of an ill-conditioned system. An estimation based on the magnitude of SNR is proposed for a system that lacks statistical information to determine the information content and to reconstruct the solution profile. To reduce the numerical instability of matrix inversion, some specific kernel transformations are discussed. Illustrative examples are given and compared to results of other approaches. An alternative linear approach that orthonormalizes the kernels is also proposed. The linear approach was then employed in solving the radiative transfer equation with temperature-independent kernels. The necessary variable separation in linear inversions was examined. Iteration refinement was found necessary to accommodate the strong nonlinearity of high temperature sensing.

© 1975 Optical Society of America

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