Geometry-based deconvolution of geophysical data

A seismic trace can be modeled as a convolution of a source wavelet and the reflectivity sequence of the Earth at a common depth point midway between the source and receiver. In a common depth point gather, the reflectivity sequence associated with each source-receiver pair is a compressed version of the zero-offset reflectivity sequence at that point. The process of normal moveout correction to align the traces to the zero-offset trace, stretches both the reflectivity sequences and wavelet simultaneously. While stretching the compressed reflectivity sequences is desirable, stretching the wavelet is not. If an inverse operator is applied to spike the wavelet prior to the normal moveout correction, the amount of stretch encountered by the wavelet will be reduced. If the wavelet has been perfectly spiked, the amount of stretch will be reduced to zero. This suggests that the zero-offset trace along with a non-zero offset trace could provide two nonlinear simultaneous equations that can be solved to determine the deconvolution filter. That is, a filter to spike the wavelet before normal moveout. Additional traces, however, will be needed to minimize the effect of noise and random errors. In the present paper the author formulates this deconvolution problem and shows that it can be solved using a constrained least square minimization technique. The solution process can be shown to reduce to an eigenvalue problem. A generalization of this work to data is also discussed where the stretch in the reflectivity sequences is introduced in the recorded data because of the geometry of the propagation medium relative to the array data acquisition system. This geometry-based deconvolution approach could be very effective in numerous remote sensing applications.