A perturbative analysis of non-linear inversion

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-linf. 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. Key words: inverse scattering, non-linear inversion.