Constant-Q wavelet estimation via a nonstationary Gabor spectral model
Jeff P. Grossman, Gary F. Margrave, Michael P. Lamoureux, Rita Aggarwala
As a seismic wave propagates through the earth, its amplitude attenuates over time and frequency due to microscopic processes such as internal friction. Thus, on one hand, the earth is an anelastic medium; on the other hand, Hooke.s law, which is normally used in the derivation of the wave equation, applies only to perfectly elastic media. Despite that fact, seismic attenuation can be modelled macroscopically over typical seismic bandwidths via an exponential amplitude decay in both time and frequency, at a rate determined by a single dimensionless quantity, Q.
Current seismic deconvolution methods, based on the stationary convolutional model, attempt to estimate, and subsequently filter out, the embedded causal wavelet. We present a nonstationary seismic model, expressed in the time-frequency Gabor domain, in which (1) the embedded causal wavelet is represented as the product of a stationary seismic signature with a nonstationary exponential decay; and (2) a nonstationary impulse response for the earth is tractable.
By least-squares fitting our model to the Gabor-transformed seismic trace, we robustly determine both a unique Q-value and an estimate of the seismic signature, and thus an estimate of the nonstationary causal wavelet. Using these estimates to obtain a smoothed version of the seismic trace in the time-frequency domain, a least squares nonstationary minimum phase deconvolution filter is constructed. The preliminary results, coded in MATLAB, look very promising. It is hoped that in future work, the residual error in this least-squares approximation will provide a good measure of the ambient random noise, up to the accuracy of our model, and hence a method by which to improve the signal-to-noise ratio.