Simultaneous constraining of model and data smoothness for regularization of geophysical inverse problems

Abstract

Summary In this paper, a new approach is introduced to solve ill-posed linear inverse problems in geophysics. Our method combines classical quadratic regularization and data smoothing by imposing constraints on model and data smoothness simultaneously. When imposing a quadratic penalty term in the data space to control smoothness of the data predicted by classical zero-order regularization, the method leads to a direct regularization in standard form, which is simple to be implemented and ensures that the estimated model is smooth. In addition, by enforcing Tikhonov's predicted data to be sparse in a wavelet domain, the idea leads to an efficient regularization algorithm with two superior properties. First, the algorithm ensures the smoothness of the estimated model while substantially preserving the edges of it, so, it is well suited for recovering piecewise smooth/constant models. Second, parsimony of wavelets on the columns of the forward operator and existence of a fast wavelet transform algorithm provide an efficient sparse representation of the forward operator matrix. The reduced size of the forward operator makes the solution of large-scale problems straightforward, because during the inversion process, only sparse matrices need to be stored, which reduces the memory required. Additionally, all matrix–vector multiplications are carried out in sparse form, reducing CPU time. Applications on both synthetic and real 1-D seismic-velocity estimation experiments illustrate the idea. The performance of the method is compared with that of classical quadratic regularization, total-variation regularization and a two-step, wavelet-based, inversion method.