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Inversion of the spherical layer-matrix

DEPARTMENT OF GEOLOGICAL SCIENCES UNIVERSITY OF SOUTHERN CALIFORNIA, LOS ANGLES, CALIFORNIA

Abstract

In their formulation of wave problems in a spherically layered medium, Phinney and Alexander have arrived at a layer-matrix M which forms the fundamental building block of their solution. In actual application of the theory, the inversion of M is needed for each assumed spherical shell. Since numerical inversion of the matrix M may introduce undesired accumulation of errors, an analytical inverse matrix M^–1 is obtained. Using Wronskians and recurrence relations of the spherical Bessel functions, it is shown that the inverse matrix M^–1 can be simplified enough to insure an improvement in economy and accuracy in machine computations. Some useful properties of the inverse matrix M^–1 are discussed which reduce the amount of machine time even further.

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