Deconvolution

One of the reasons that multiple predictions must be adaptively subtracted from the data is that the wavelet in the data is squared during the convolution. To address the problem, we introduce deconvolution to both the conventional Guitton et al. (2006); Lee et al. (1991) and the multiple prediction imaging conditions.

To normalize the frequency content and provide a sharper image, the imaging condition in equation 3 becomes  

$$i_z({\bf x},{\bf h})= \sum_{{\bf x}_s}\sum_\omega \frac{ ... ...-h;{\bf x}_s,\omega) D_z^*({\bf x}-h;{\bf x}_s,\omega)\gg}\;,$$ (16)

where the denominator is smoothed across horizontal coordinates for stability. In this implementation, the smoothing operator is a triangle function with a base five samples wide. Equation 9 can similarly be normalized/deconvolved and implemented  

$$m_z({\bf x},{\bf h})= \sum_{{\bf x}_s}\sum_\omega \frac{ ... ... x}_s,\omega) D_z^*({\bf x}-{\bf h};{\bf x}_s,\omega)\gg }\;.$$ (17)

It could be argued that the deconvolutional imaging condition for multiples above should be normalized by a smoothed version of the power spectrum of U instead of D. However, in the interest of parallel construction, we have implemented it as above. If the wavelet used to model D accurately represents the amplitude and frequency content of the data, the choice of U or D as divisor will be of little importance. The benefit of a deconvolution imaging condition is to collapse the wavelet that events are convolved with as much as possible. Consequently, the image-space volumes $i_z({\bf x},h)$ and $m_z({\bf x},h)$ share approximately the same bandwidth and can be more easily subtracted from each other to provide an image space consisting of only primaries. If the data are not zero-phase before implementing IS-SRMP, deconvolution will not address phase-roll of the wavelet within the data introduced by squaring to compute the multiple prediction.