Inverting the Stokes and Vening Meinesz integrals using the wavelet transform

Abstract

A wavelet transform algorithm is used for inverting Stokes’s integral and evaluating the inverse Vening Meinesz integral. Orthogonal wavelet base functions are used. For inverting the Stokes integral, a set of equations is formed and solved using a preconditioned conjugate gradient method. The full solution with all equations requires a large computer memory; therefore, multiresolution properties of the wavelet transform are used to divide the full solution into parts. High compression levels are achieved by using global wavelet thresholding. The singularity level of the kernel is studied and the compression levels depend on the singularity properties of the kernels. Global thresholding achieved a 85% compression level in the case of the Stokes kernel and 97% in the case of the Vening Meinesz kernel without a loss in accuracy. These compression levels lead to large savings in computer memory and the ability to work with sparse matrices, which increases the computations’ speed. Hard thresholding is used in the compression of the matrices; however, soft thresholding is used for denoising of the data because of its smoothing properties. Conclusions and recommendations are given with respect to the suitability, accuracy, and efficiency of this method.