Transform-domain sparsity regularization for inverse problems in geosciences

We have developed a new regularization approach for estimating unknown spatial fields, such as facies distributions or porosity maps. The proposed approach is especially efficient for fields that have a sparse representation when transformed into a complementary function space (e.g., a Fourier space). Sparse transform representations provide an accurate characterization of the original field with a relatively small number of transformed variables. We use a discrete cosine transform (DCT) to obtain sparse representations of fields with distinct geologic features, such as channels or geologic formations in vertical cross section. Low-frequency DCT basis elements provide an effectively reduced subspace in which the sparse solution is searched. The low-dimensional subspace is not fixed, but rather adapts to the data. The DCT coefficients are estimated from spatial observations with a variant of compressed sensing. The estimation procedure minimizes an l(2)-norm measurement misfit term while maintaining DCT coefficient sparsity with an l(1)-norm regularization term. When measurements are noise-dominated, the performance of this procedure might be improved by implementing it in two steps-one that identifies the sparse subset of important transform coefficients and one that adjusts the coefficients to give a best fit to measurements. We have proved the effectiveness of this approach for facies reconstruction from both scattered-point measurements and areal observations, for crosswell travel-time tomography, and for porosity estimation in a typical multiunit oil field. Where we have tested our sparsity regularization approach, it has performed better than traditional alternatives.