We present imaging results in geophysics based on using multidimensional Gaussian wavelets as a filter in a 2-D Cartesian domain. Besides decomposing the field into various distinct lengthscales, we have also constructed the 2-D maps describing the spatial distributions of the maximum of the wavelet-transformed L2-norm Emax (x, y) and its corresponding local wavenumber kmax (x, y), where x and y are the Cartesian coordinates. For geoid anomalies, using a wavelet filter extending to 90 degrees, we have discerned the distinct outlines of convergent and divergent tectonic zones and have conducted a quantitative comparison of the short-wavelength gravitational anomalies at those wavelengths between two different geographical locations. We have also compared the wavelet results with a nonlinear bandpass filter in the spectral domain where a Gaussian filter with the logarithm of the degree/acting as the argument has been employed. A wavelet solution, with a length-scale corresponding to 256 degrees, would need a filter with over 400 spherical harmonies centering around / = 157 for an optimal spatial fit. The computational effort with the bandpass filter technique greatly exceeds those associated with wavelets. We have also shown the ability of the wavelets to analyze the vastly different scales present in high Rayleigh number convection and the mixing of passive heterogeneities driven by thermal convection. Wavelets will be a useful tool for rapid analyzing of the large multidimensional fields to be captured in many other geophysical endeavors, such as the upcoming gravity satellite missions and satellite radar interferometry images.
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We are grateful for numerous discussions with our Canadian colleague Dr. Stephen Y. Bergeron concerning many issues pertaining to wavelets and other matters, and also Cathy Hier for having stimulated us to look at scalograms in multidimensional convection. Other people we would like to acknowledge are Oleg Vasilyev, Claudia Piromallo, Fabien Dubuffet, Gordon Erlebacher, Shige Maruyama and Tetsu Seno for various discussions. This research has been supported by the DOE program in Complex Fluids, the geophysics program of the NSF, the NSF International program for Japan-U.S. and a NATO grant.
- Multidimensional wavelets