Fast inversion of large-scale magnetic data using wavelet transforms and a logarithmic barrier method

In this paper wavelet transforms and a logarithmic barrier method are applied to the inversion of large-scale magnetic data to recover a 3-D distribution of magnetic susceptibility. The fast wavelet transform is used, along with thresholding the small wavelet coefficients, to form a sparse representation of the sensitivity matrix. The reduced size of the resultant matrix allows the solution of large problems that are otherwise intractable. The compressed matrix is used to carry out fast forward modelling by performing matrix-vector multiplications in the wavelet domain. The reduction in CPU time is directly proportional to the compression ratio of the matrix. A second important feature of the algorithm used here is the use of an interior-point method of optimization to enforce positivity constraints. In this approach, the positivity is incorporated into the inversion by a sequence of non-linear optimizations approximated by truncated Newton steps. At the heart of the algorithm, a linear system of equations is solved. The conjugate gradient technique has been used as the basic solver to take the advantage of the efficient forward modelling offered by the sparse matrix representation. Overall, the combination of wavelet transforms, interior point optimization and conjugate gradient solutions readily allows us to solve magnetic inverse problems that have a few hundred thousand parameters and tens of thousands of data.