SUMMARY
Tomography is one of the cornerstones of geophysics, enabling detailed spatial descriptions of otherwise invisible processes. However, due to the fundamental ill-posedness of tomography problems, the choice of parametrizations and regularizations for inversion significantly affect the result. Parametrizations for geophysical tomography typically reflect the mathematical structure of the inverse problem. We propose, instead, to parametrize the tomographic inverse problem using a geologically motivated approach. We build a model from explicit geological units that reflect the a priori knowledge of the problem. To solve the resulting large-scale nonlinear inverse problem, we employ the efficient Ensemble Kalman Inversion scheme, a highly parallelizable, iteratively regularizing optimizer that uses the ensemble Kalman filter to perform a derivative-free approximation of the general iteratively regularized Levenberg–Marquardt method. The combination of a model specification framework that explicitly encodes geological structure and a robust, derivative-free optimizer enables the solution of complex inverse problems involving non-differentiable forward solvers and significant a priori knowledge. We illustrate the model specification framework using synthetic and real data examples of near-surface seismic tomography using the factored eikonal fast marching method as a forward solver for first arrival traveltimes. The geometrical and level set framework allows us to describe geophysical hypotheses in concrete terms, and then optimize and test these hypotheses, helping us to answer targeted geophysical questions.
Piecewise Partially Separable Functions and a Derivative-free Algorithm for Large Scale Nonsmooth Optimization