Nonlinear inversion of geodetic and geophysical data : Diagnosing nonlinearity

This paper addresses the problem of diagnosing nonlinearity in the nonlinear inversion of geodetic and geophysical data. Measures of nonlinearity are proposed that can be used to assess the amount of nonlinearity in nonlinear models and to test whether a linear (ized) model is a sufficient approximation.After the introductory section, which gives a brief overview of the various problems associated with nonlinear inversion, section two discusses three simple measures of nonlinearity that can be used as a first step in analyzing the amount of nonlinearity of the model. In section three we show that in the problem of nonlinear inversion one has to reckon with two different types of nonlinearity. First of all there is the nonlinearity of the parameter-curves which obviously depends on the chosen parametrization. Secondly there is the nonlinearity related to the curvature of the manifold. It is intrinsic in the sense that it is independent of the choice of parametrization. The two types of nonlinearity are described using concepts from differential geometry.In section four we discuss the geometry of nonlinear least-squares inversion. In this section it is also shown how the above mentioned two types of nonlinearity affect the first moments of the nonlinear least-squares estimators. Finally in section five a strategy is proposed for diagnosing the significance of nonlinearity in nonlinear models.