Wavelet evaluation of the terrain correction integral

Abstract

A computational scheme using the wavelet transform is employed to evaluate the terrain correction integral. The efficiency of the wavelet multiresolution analysis as an alternative approach to the well-established Fast Fourier Transform (FFT) is studied. The integral is approximated in finite multiresolution analysis subspaces, using a two-dimensional wavelet algorithm. An orthogonal wavelet base function is used to build the wavelet algorithm. The characteristics of the base function and its effect on the results are investigated. This algorithm solves the problem in a point-wise procedure. The role of the kernel singularity in the wavelet multiresolution analysis is studied. Wavelet thresholding is used for the compression of the kernel, and hard and soft thresholding are tested. Global thresholding is compared to level direction-wise thresholding. Because of the fast decreasing kernel, high compression levels are achieved without loss of accuracy. A numerical example is given to clarify the use of this procedure in the numerical evaluation of the terrain correction integral. Conclusions and recommendations are given with respect to the suitability, accuracy, and efficiency of this method.