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We have addressed the seismic impedance inversion problem, which is often ill posed because of inaccurate and insufficient data. The approach taken is based on dictionary learning and sparse representation. By shifting a patch window of fixed size on all well-log data, a large number of small overlapping patches are generated. Then regarding these patches as a training set and using K-singular value decomposition algorithm, we obtain a dictionary that describes the common features of subsurface models within the current survey area. On the basis of the assumption that the subsurface geology has similarity and lateral continuity to some extent, the dictionary is used to approximate each model via sparse representations over the learned dictionary. In particular, we impose the sparse representations as additional constraints into the inversion procedure, leading to a well-defined objective function that can not only fit the observed seismic data but also honor the features of the well-log data. We adopt a coordinate descent strategy to solve this objective function. Meanwhile, to enforce lateral continuity in the inverted models, we use an additional stage in which we use the nonlocal similarity information that is extracted from seismic data as spatial coherent prior knowledge to refine the estimated models. Compared with several traditional impedance inversion methods, our algorithm can produce solutions of much higher quality qualitatively and quantitatively.

Amplitude variation with offset (AVO) inversion is a typical ill-posed inverse problem. To obtain a stable and unique solution, regularization techniques relying on mathematical models from prior information are commonly used in conventional AVO inversion methods (hence the name model-driven methods). Due to the difference between prior information and the actual geology, these methods often have difficulty achieving satisfactory accuracy and resolution. We have developed a novel data-driven inversion method for the AVO inversion problem. This method can effectively extract useful knowledge from well-log data, including sparse dictionaries of elastic parameters and sparse representation of subsurface model parameters. Lateral continuity of subsurface geology allows for the approximation of model parameters for a work area using the learned dictionaries. Instead of particular mathematical models, a sparse representation is used to constrain the inverse problem. Because no assumption is made about the model parameters, we consider this a data-driven method. The general process of the algorithm is as follows: (1) using well-log data as the training samples to learn the sparse dictionary of each elastic parameter, (2) imposing a sparse representation constraint on the objective function, making the elastic parameters be sparsely represented over the learned dictionary, and (3) solving the objective function by applying a coordinate-descent algorithm. Tests on several synthetic examples and field data demonstrate that our algorithm is effective in improving the resolution and accuracy of solutions and is adaptable to various geologies.

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