Wavelet evaluation of the Stokes and Vening Meinesz integrals

The wavelet transform is a powerful tool in evaluating some singular geodetic integrals. Due to its localization properties in both the time (space) and frequency (scale) domains, and because the kernels of some geodetic integrals have singular points and decay smoothly and quickly away from the singularities, many wavelet transform coefficients of the kernels become zeros or negligible, and only a small number of wavelet transform coefficients are significant. It is thus possible to significantly compress the kernels of these integrals on a wavelet basis by neglecting the zero coefficients and the small coefficients below a certain threshold. Therefore, wavelets provide a convenient way of efficiently evaluating these integrals in terms of fast computation and savings of computer memory. A modified algorithm for the wavelet evaluation of Stokes' integral is presented. The same modified algorithm is applied to the evaluation of the Vening Meinesz integral, whose kernel has a stronger singularity than does Stokes' kernel. Numerical examples illustrate the efficiency and accuracy of the wavelet methods.Acknowledgments.The author express their sincere thanks to Dr. Salamonwicz for providing his PhD thesis. E-mail correspondence between the authors and Dr. Barthelmes and Dr. Benciolini contributed to the work. R. Benciolini and the other two anonymous reviewers are thanked for their constructive comments. Support for this research was provided by research grants to Dr. Sideris from the Natural Sciences and Engineering Reserch Council of Canada (NSERC) and the Geomatics for Informed Decisions (GEOIDE) Network of Centres of Excellence. The MATLAB Wavelet Toolbox package was used as the platform to develop the software in this project.