Model regularization for linear inverse problems in exploration geophysics
Keywords:
seismic methods, inverse problem, regularization, greedy algorithms, Radon transformAbstract
Exploration geophysics allows us to characterize the subsurface using measurements typically acquired at the surface. The physical properties of the medium can be inferred from the effects observed in the data. This approach in science is known as solving an inverse problem. The inverse problem can be formulated as an optimization problem, where parameters are determined to minimize a specific cost function. Many inverse problems in exploration geophysics are either linear or can be studied through a linear approximation. For an inverse problem to be well-posed, three conditions must be met: the solution must exist, it must be unique, and it should vary continuously with the model parameters (i.e., it should be stable). Unfortunately, these conditions are never all satisfied simultaneously. Generally, the solutions sought are either non-unique and/or unstable, which means the model to be inverted must be regularized. Regularization can be explicit or implicit. The former is achieved by introducing a penalty term in the cost function. The latter can be achieved by early stopping of iterative algorithms, discarding outliers, or using more modern greedy algorithms. Examples of these latter techniques are presented here using the Radon transform.
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