The Uniqueness of Single Data Function, Multiple Model Functions, Inverse Problems Including the Rayleigh Wave Dispersion Problem

Abstract

We prove that the problem of inverting Rayleigh wave phase velocity functions \(c\left( k \right)\), where \(k\) is wavenumber, for density \(\rho \left( z \right)\), rigidity \(\mu \left( z \right)\) and Lame parameter \(\lambda \left( z \right)\), where \(z\) is depth, is fully non-unique, at least in the highly-idealized case where the base Earth model is an isotropic half space. The model functions completely trade off. This is one special case of a common inversion scenario in which one seeks to determine several model functions from a single data function. We explore the circumstances under which this broad class of problems is unique, starting with very simple scenarios, building up to the somewhat more complicated (and common) case where data and model functions are related by convolutions, and then finally, to scale-independent problems (which include the Rayleigh wave problem). The idealized cases that we examine analytically provide insight into the kinds of nonuniqueness that are inherent in the much more complicated problems encountered in modern geophysical imaging (though they do not necessarily provide methods for solving those problems). We also define what is meant by a Backus and Gilbert resolution kernel in this kind of inversion and show under what circumstances a unique localized average of a single model function can be constructed.