An extension of Backus-Gilbert theory to nonlinear inverse problems

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Abstract
The inverse problem where one wants to estimate a continuous model with infinitely many degrees of freedom from a finite data set is necessarily ill-posed. Although some examples exist of exact nonlinear inversion schemes for infinite data sets, there exists apart from data-fitting procedures no theory for nonlinear inversion that takes into account that a real data set is finite. A nonlinear perturbation theory is presented for the optimal determination of a model from a finite data set which generalizes Backus-Gilbert theory for linear inverse problems to include nonlinear effects. The extent to which the reconstructed model resembles the true model is described by linear and nonlinear resolution kernels. In this way, it is possible to evaluate to what degree the reconstructed model resembles the true model. A statistical analysis reveals the effects of errors in nonlinear inversion. The most dramatic effect is that if the data have no bias, the reconstructed model may suffer from a bias due to the nonlinearity of the problem. The theory is extended for the special case of infinite data sets which are of mathematical interest. As an example, it is shown that the Newton-Marchenko method for the inverse problem of the 3D Schrodinger equation requires a redundant data set, even if the nonlinearities are taken into account.

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