Normalization for variable fold

``Mathematical derivations of integral operators assume that physical quantities such as time, source-receiver midpoint, azimuth and offset are continuous. In practice, the resulting algorithms are simply then applied to the discretely sampled seismic field data'' Beasley and Klotz (1992). However, a fundamental problem exists in the implementation of integrals as discrete summations; the integrands must be regularly and adequately sampled. A way of thinking that can be very helpful takes us back to the definition of definite 3D integrals, and their volume approximations. This suggests that we should consider a multiplicity concept to avoid summing over redundant traces within bin elements. A straightforward extension of the NMO-Stack compensation for variable fold, based on dividing the summed data by a function of the multiplicity, cannot be used for our application since traces within local bins may have different azimuths and offsets. The summed trace will then suffer severe amplitude and phase distortions.

What we propose as a new development in our imaging sequence is a normalization procedure which includes all the traces within each CMP bin in the discrete summation while it normalizes each input trace by the local fold of its corresponding bin. This solution is accurate for the case in which all the traces within the CMP bin share the same offset and azimuth and provides a good approximation for the case of small variations in offset and azimuth. Application of this normalization procedure leads to an approximation of the survey subdivision by a single-fold 3D subset that is better suited for prestack processing when the wave equation operator is implemented as a discrete integral summation.

 

model
Figure 4 Synthetic reflectivity model of a flat reflector. The white circles are zones of high reflection coefficient of 2.5 in a constant background of unit reflectivity.