Open AccessA perturbative analysis of non‐linear inversion
Author(s) -
Snieder Roel
Publication year - 1990
Publication title -
geophysical journal international
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.302
H-Index - 168
eISSN - 1365-246X
pISSN - 0956-540X
DOI - 10.1111/j.1365-246x.1990.tb05569.x
Subject(s) - inversion (geology) , inverse problem , inverse , inverse scattering problem , inverse theory , linear model , mathematics , generalized inverse , linear system , mathematical analysis , algorithm , computer science , geology , geometry , seismology , surface wave , statistics , tectonics , telecommunications
SUMMARY In most geophysical forward problems, the data are related in a non‐linear way to the model. Similarly, the inverse problem constitutes a non‐linear mapping from the data to the model. In general, it is poorly understood which parts of the data contribute to the reconstruction of the model, and how the non‐linearities are handled in the inversion. A perturbative treatment of non‐lin***inversion clarifies these issues. This perturbative treatment is first applied to ***inverse scattering algorithm which has an exact solution, and is then generalized for a very wide class of non‐linear inverse problems. It is shown that only the linear component in the data (the first Born approximation) contributes to the reconstruction of the model, and that the non‐linear components in the data are being subtracted in the inversion. These arguments are generalized for a very wide class of non‐linear inverse problems. The analysis can be used to derive non‐linear inversion schemes. As an example an algorithm is derived which performs non‐linear traveltime tomography without iteratively shooting or bending rays. The analysis has profound implications for the stability of non‐linear inversions.