Fast solution of geophysical inversion using adaptive mesh, space-filling curves and wavelet compression

Abstract

Summary Many geophysical inverse problems involve large and dense coefficient matrices that often exceed the limitations of physical memory in commonly available computers. The repeated multiplications of such matrices to vectors during processing or inversion require an immense amount of computing power. These two factors pose a significant challenge to solving large-scale inverse problems in practice and can render many realistic problems intractable. To overcome these limitations, we develop a new computational approach for this class of problems by combining an adaptive quadtree or octree model discretization and wavelet transforms on reordered parameter sets. The adaptive mesh discretizes the model region according to the required resolutions based on localized anomalies. Hilbert space-filling curves and similar ordering of the reduced parameter set then enable a higher compression of the coefficient matrix by forming its sparse representation in the 1-D wavelet domain. This combination can reduce the storage requirement by 100 to 1000 times and, therefore, also speeds up the computation during the processing stage by the same factor. As a result, problems can now be solved that were computationally prohibitive. We present the algorithm and illustrate its effectiveness with an example from equivalent source construction in potential-field processing.