The role of the Born approximation in nonlinear inversion

A perturbation analysis of nonlinear inversion is presented. As a prototype of nonlinear inversion, inverse scattering with the Marchenko equation is considered from a perturbative point of view. It is shown that inverse scattering methods using the Marchenko equation implicitly reconstruct the potential by removing the nonlinearities from the data, and by performing a Born inversion of the resulting linear component in the data (the first Born approximation). This is illustrated with a one-dimensional example. This interpretation of the mechanism of inverse scattering algorithms clarifies the 'miracle of Newton', and has profound consequences for both the theoretical and the practical aspects of inverse scattering, in particular for the stability of inverse scattering schemes.

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