Improving resolution of NMO stack using shaping regularization
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Common midpoint (CMP) stacking is one of the major steps in seismic data processing. Traditional CMP stacking sums a combination of normal moveout (NMO) corrected traces across a CMP gather to produce a single trace with a higher signal-to-noise (S/N) ratio than that of individual traces within the gather. Several problems arise with the assumptions and principles of conventional NMO and stack. NMO correction causes undesirable distortions of signals on a seismic trace known as "NMO stretch", which lowers the frequency content of the corrected reflection event at far offsets. This violates the assumption of a uniform distribution of phase and frequency of seismic reflections across the corrected gather. Common procedures to eliminate this stretching effect involve muting all of the samples with severe distortions. This causes a decrease in fold and can destroy useful far-offset information essential for amplitude variation with offset (AVO) analysis. Inaccuracy in stretch muting with residual "stretching" effects produces a lower amplitude and lower resolution stack. I present two methods that eliminate the effects of "NMO stretch" and restore a wider frequency band by replacing conventional NMO and stack with a regularized inversion to zero offset. The resulting stack is a model that best fits the data using additional constraints imposed by the method of shaping regularization. Shaping regularization implies a mapping of the input model to a space of acceptable models. The shaping operator is integrated in an iterative inversion algorithm and provides an explicit control on the estimated stack. I use shaping regularization to achieve a stack that has a denser time sampling and contains higher frequencies than the conventional stack. In the first approach, I define the backward operator of shaping regularization using the principles of conventional NMO correction and stack. In the second approach, I introduce a recursive stacking scheme using plane-wave construction in the backward operator of shaping regularization. The advantage of using recursive stacking along local slopes in the application to NMO and stack is that it avoids "stretching" effects caused by NMO correction and is insensitive to non-hyperbolic moveout in the data. Numerical tests demonstrate each algorithm's ability to attain a higher frequency stack with a denser temporal sampling interval compared to those of the conventional stack and to minimize stretching effects caused by NMO correction. I apply both methods to two 2-D marine datasets from the North Sea and achieve noticeable resolution improvements in the stacked sections compared with that of conventional NMO and stack. By treating NMO and stack as an iterative inversion using shaping regularization, resolution is enhanced by utilizing signal from different offsets and minimizing stretching effects to reconstruct a high resolution stack.