Adaptive wavelets for analyzing dispersive seismic waves

Our primary objective is to develop an efficient and accurate method for analyzing time series with a multiscale character. Our motivation stems from the studies of the physical properties of marine sediment (stiffness and density) derived from seismic acoustic records of surface/interface waves along the water-seabed boundary. These studies depend on the dispersive characteristics of water-sediment surface waves. To obtain a reliable retrieval of the shear-wave velocities, we need a very accurate time-frequency record of the surface waves. Such a time-frequency analysis is best carried out by a wavelet-transform of the seismic records. We have employed the wavelet crosscorrelation technique for estimating the shear-wave propagational parameters as a function of depth and horizontal distance. For achieving a greatly improved resolution in time-frequency space, we have developed a new set of adaptive wavelets, which are driven by the data. This approach is based on a Karhunen-Loeve (KL) decomposition of the seismograms. This KL decomposition allows us to obtain a set of wavelet functions that are naturally adapted to the scales of the surface-wave modes. We demonstrate the superiority of these adaptive wavelets over standard wavelets in their ability to simultaneously discriminate the different surface-wave modes. The results can also be useful for imaging and statistical data analysis in exploration geophysics and in other disciplines in the environmental sciences.