Efficient 3D inversion of magnetic data via octree mesh discretization, space-filling curves, and wavelets

Inversion of potential-field data 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 inversion require an immense amount of CPU power. These two factors pose a significant challenge to solving large-scale 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 octree model discretization and wavelet transforms on a re-ordered parameter set. The adaptive mesh discretizes the model region by starting with large cells and splitting the region into smaller cells for localized anomalies to maintain resolution. Hilbert space-filling curves and similar ordering of the reduced parameter set produce higher compression of the coefficient matrix to form its sparse representation in the 1D wavelet domain. This combination can reduce the storage requirement by 100 to 1000 times and, therefore also speeds up the computation of the inversion by the same amount. As a result, problems can now be solved that were computationally prohibitive. We present the algorithm and illustrate its effectiveness with synthetic and field examples.