Generalized Nonlinear Inverse Problems Solved Using the Least Squares Criterion

TLDR: In this article, a general definition of the nonlinear least squares inverse problem is given, where the form of the theoretical relationship between data and unknowns may be general (in particular, nonlinear integrodierentia l equations).

Abstract: We attempt to give a general definition of the nonlinear least squares inverse problem. First, we examine the discrete problem (finite number of data and unknowns), setting the problem in its fully nonlinear form. Second, we examine the general case where some data and/or unknowns may be functions of a continuous variable and where the form of the theoretical relationship between data and unknowns may be general (in particular, nonlinear integrodierentia l equations). As particular cases of our nonlinear algorithm we find linear solutions well known in geophysics, like Jackson’s (1979) solution for discrete problems or Backus and Gilbert’s (1970) a solution for continuous problems.