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.
Numerous methods have been developed in order to reduce inversion time or physical memory required. In general, the most computational efficiency has been achieved through integral transforms by storing only significant coefficients of those functions. This not only increases the speed of solving the inverse problem, but also enables larger datasets to be inverted on a computer with limited resources. The Fourier transform (Cordell, 1992; Pilkington, 1997) accelerates problems based on the convolution theorem. The separable wavelet transform (Li and Oldenburg, 1999, 2003) has a compression property by winnowing small coefficients that do not significantly change the function after the inverse transform. These two techniques have been the principal approaches for accelerating this class of inverse problems in geophysics. The main limitation of inverting in the Fourier domain is the requirement of a grid of data on a plane. The principal drawback of wavelets is the entire construction of the dense coefficient matrix prior to applying the transform. However, the wavelet transform can be improved upon by reducing the size of the dense coefficient matrix which must be calculated prior to utilizing it. Ridsdill-Smith (2000) used wavelets along profiles in order to find edges and reduce the number of parameters. Octree-based methods have been used for 3D inversion of electromagnetic data (Ascher and Haber, 2001). Further more, grid refinement (e.g. multi-grid methods) based on hierarchical mesh structures can increase the efficiency of non-linear problems (e.g. Haber et al., 2007). We create an octree based mesh prior to inversion and use it throughout because of linearity of the problem. For a quantitative comparison of techniques, we will define the ratio of total coefficients in the dense matrix to the number of stored coefficients (i.e. after the wavelet transform and thresholding) known as the compression ratio. Computation time is inversely proportional to this ratio.
