Seismological Applications of Orthogonal Function Expansions

摘要

A method is presented to represent very complicated transients by only a few numbers rather than by curves or by the set of coordinate values necessary to draw the curves. The representation used is an expansion in terms of a set of orthonormalized exponential functions. Procedures necessary to generate these functions, find the expansion coefficients, and use these coefficients in further analysis are described in detail. Two sets of procedures are developed, one which is exact for analog computations, and one which is exact for numerical calculations. It is shown that these procedures are easy to apply in practice. Section II of the report is concerned with sets of continuous orthonormal functions and is oriented toward analog computation. Section III is devoted to developing sets of orthonormal sequences. These are orthonormal functions which are defined on the set of integers and are used for numerical calculations using sampled data. Special emphasis is placed on the class of transients called teleseismic P-waves.