The Effect of the Noise and the Regularization in Inverse Problems: Geophysical Implications

The solution of geophysical inverse problems has an intrinsic uncertainty that is mainly caused by noise in data, incomplete data sampling and simplified physics. This paper analyzes the roles of noise in data and that of the regularization for nonlinear inverse problems. We prove that noise deforms the topography of the cost function non-homogeneously, generally decreasing the regions of low misfits. As a result of this deformation, finding the global optimum by direct search methods becomes a more difficult task. Nevertheless, noise acts similarly to a regularization when local optimization methods are used. Tikhonov’s regularization transforms the linearized hyper-quadric of equivalence from an elliptical cylinder to a very oblong ellipsoid in the directions that originally spanned the kernel of the linearized forward operator in absence of regularization, and also deforms anisotropically the regions of equivalence. Prior models in the regularization term serves to inform the components of the solution that locally belongs to the kernel of the Jacobian. Unfortunately regularization does not cause the disappearance of the nonlinear equivalent models. Thus a full nonlinear uncertainty analysis is still needed.