The Inversion of Poisson’s Integral in the Wavelet Domain

A wavelet transform algorithm combined with a conjugate gradient method is used for the inversion of Poisson’s integral (downward continuation), used in airborne gravimetry applications. The wavelet approximation is dependent on orthogonal wavelet base functions. The integrals are approximated in finite multiresolution analysis subspaces. Mallat’s algorithm is used in the multiresolution analysis of the kernel and the data. The full solution with all equations requires large computer memory, therefore, the multiresolution properties of the wavelet transform are used to divide the full solution into parts at different levels of wavelet multiresolution decomposition. Global wavelet thresholding is used for the compression of the kernel and because of the fast decrease of the kernel towards zero, high compression levels are reached without significant loss of accuracy. Hard thresholding is used in the compression of the kernel wavelet coefficients matrices. A new thresholding technique is introduced. A first-order Tikhonov regularization method combined with the L-curve is used for the regularization of this problem. First, Poisson’s integral is inverted numerically with the full matrix without any thresholding. The solution is obtained using the conjugate gradient method after 28 iteration steps with a root mean square error equal to 5.58 mGal in comparison to the reference data. Second, the global hard thresholding solution achieved a 94.5% compression level with less than 0.1 mGal loss in accuracy. These high compression levels lead to large savings in computer memory and the ability to work with sparse matrices, which increases the computational speed.