# Linearized Inversion of (Teleseismic) Data

## Summary

This paper consists of two parts. First, I show how the inverse problem for teleseismic waves can be linearized. The second part deals with linear inversion methods, and is applicable to geophysical data in general.

The two principles of stationarity in seismology, Fermat’s Principle and Rayleigh’s Principle, can be used to establish a linearized relationship between perturbations in model parameters on the one hand, and the resulting perturbations in travel times or dispersion data on the other hand. Tables of formulae are presented for the calculations of the direct problem for Love and Rayleigh dispersion and for travel times, with flat and spherical Earth models and different interpolation rules between model points. Integral kernels for inversion of these data are also catalogued.

Backus-Gilbert theory for the inversion of linearized relationships between model and data perturbations in general is briefly reviewed and extended to the case of discretized models. Interpolation functions may be used to span a subspace of the Hilbert space of all possible Earth models. This leads to a discrete set of equations of much smaller size than the original Backus-Gilbert formulation, thus allowing for the simultaneous inversion of much larger sets of data. Orthonormalization of the basis of interpolation functions leads to simplifications in the theory, a more logical approach to discrete inversion, and allows the resolution calculations of Backus and Gilbert to be carried over from the domain of piecewise continuous models to be discretized case. The method presented here has considerable computational advantages over earlier methods.

## Keywords

Travel Time Rayleigh Wave Love Wave Linear Inversion Simultaneous Inversion## Preview

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