(1) Department of Geology and
Geophysics and Minnesota Supercomputing
Institute, University of Minnesota, MN USA
(2) Dept.
Physique, University of Montreal, Montreal, Quebec, Canada
(3) Department of Geophysics,
Faculty of Mathematics and Physics, Charles University, Prague, Czech
Republic
Wavelets are numerical filters which can be set to zoom in and out both in the position and the magnification axis. Up to now, most wavelet applications in geophysics have been restricted to one-dimensional wavelets, either to time-series or a one-dimensional spatial track. We will demonstrate that multidimensional wavelets , in 2-D or 3-D, can shed greater light on geophysical problems of multidimensional nature. We will devote our attention to three class of problems which involve extremely large data-sets. They include (1) 2-D thermal convection with high Rayleigh number for both finite and infinite Prandtl numbers (2) 3-D seismic tomography (3) large data-set from Synthetic Aperture Radar (SAR). Wavelets are usually visualized using the scalogram representation, which is a display along the axis showing the magnification or the scales and the axes representing the position. In 2-D image for isotropic wavelets the scalogram has three axes, the two position axes and the scale axis.
We have constructed 3-D scalogram for thermal convection at very high Rayleigh number, up to 10^8 for finite Pr number ranging from 7 characteristic of water to Pr= 40 characteristic of cream and the same for infinite Pr, characteristic of the mantle. We have followed this strategy for detailed data-mining of the complex structure. We first zoom into a particular section of the 2-D image and then construct a local scalogram by conducting a multi-dimensional integration locally. We then then take two-dimensional cross-sections by either keeping one of the position axis constant or by keeping the scale axis constant. In the first instance, we can pick out the distribution of scales along a given axis, but still able to have some peripheral vision with respect to the second axis. In the second instance, we can focus on a feature with a particular scale, such as thin plumes developed in high Rayleigh number convection or thin vorticity tendrils developed in the shear flow of finite Pr convection.
However, these scalograms , or local spectra, generate one more dimension for visualization, thus leading to an enormous amount of information to digest and to difficult problems in visualization. We have also developed another method to assimilate and synthesize the local spectra by using two non-redundant proxy quantities, the spatial distribution of the local maxima along the scale axis, E-max and the associated local wavenumber k-max. This proxy, low-dimensional parameter approach, allows us to work with the same spatial dimensionality as the original data set. They allow us to characterize the different quasi-equilbrium states in thermal convection. In the case of 3-D seismic tomography of the mantle, this process allows us to discern interesting features, which may be related to plume-like object under Iceland.
We have also used wavelet analysis of zoomed-in area of the Mojave desert to an automatic pattern recognition of anthropomorphic features, such as a railroad track with a length-scale of O (10 m), which is buried inside a large SAR interferogram containing more than 140 million pixels. This local zoomed-in method with wavelets can also be applied to planetary surfaces with complexity, such as the Jovian satellites.
Not available.