Wavelet compression in marine seismic data

Marine seismic data is often modeled as a convolution of a source wavelet and a reflectivity sequence representing the propagation medium in the Earth at a common midpoint between source and receiver. In such a model, the reflectivity sequence associated with each source-receiver pair is a compressed version of a hypothetical reflectivity sequence that corresponds to a zero-offset reference source-receiver pair placed at the same midpoint. The process of normal moveout correction to align the data to the zero-offset reference trace, stretches both the reflectivity sequences and source wavelet simultaneously. While stretching the compressed reflectivity sequences is desirable, stretching the wavelet, which has not been compressed before, will cause considerable loss in resolution. If an inverse filter is applied to shorten the length of (i.e. compress) the wavelet prior to the normal moveout correction, then the amount of stretch encountered by the wavelet will be reduced. If the filter can compress the wavelet to a perfect spike, then the amount of stretch will be reduced to zero. This suggests that the zero-offset trace along with any single non-zero offset trace could provide two nonlinear simultaneous equations that can be solved to determine the deconvolution filter needed to reduce the wavelet to a spike. Once this is done, the reflectivity sequence will be recovered and the propagation geometry will be delineated. In this paper the author shows that this filter can be determined, theoretically, without a priori knowledge of the wavelet and with no assumptions in regards to its statistical properties. In practice, however, more than two traces will be needed to minimize the effects of noise and random errors. This deconvolution problem is formulated as an optimization problem