Controlling Stolt migration wraparound

As with the phase-shift algorithm, the key to reducing the computational artifacts of the Stolt algorithm is to suppress the time-domain wraparound. We will see that this amounts to accurate frequency-domain interpolation.

First, consider an impulse function at time t₀. Its Fourier transform is $\exp ( -i \omega t_0 )$.If t₀ is large, then the Fourier transform is a rapidly oscillating function of $\omega$.Rapidly oscillatory functions are always difficult to interpolate. It is better to shift backward the time function, thereby smoothing its frequency function, then to interpolate the frequency function, and finally to undo the shift. Given seismic data on the time interval 0<t<T, the frequency function will be smoother if the data is shifted to an interval -T/2<t<T/2. So the first proposed improvement to the Stolt migration program is to multiply in the frequency domain by $\exp ( i \omega T/2 )$,then interpolate, and finally multiply by $\exp ( -i \omega T/2 )$.

Linear interpolation is almost the easiest form of interpolation. On the other hand, Fourier transform theory suggests interpolation with the sinc function (by definition ${\rm sinc} \, u \,=$ $(\sin\,u)/u$). The sinc function of frequency, when brought back into the time domain is a rectangle function of time. Take this rectangle function to be nonzero on the interval -T/2<t<T/2. Recall that the fast (inverse) Fourier transform algorithm sums at uniform intervals in the frequency domain. This implicitly assumes zero between sample points, which in turn assumes that the time-domain function is periodic outside the given time interval. Now take the rectangle function of time to be the multiplier in the time domain that converts the periodic time function to the observed transient one. This multiplication in the time domain is equivalent to a convolution in the frequency domain with the appropriate sinc function. Convolution of the continuous sinc function with the given discrete-interval frequency function is really interpolation. Unfortunately, the sinc function extends infinitely down the frequency axis. Worse yet, it decays slowly. So some approximation or truncation of the sinc is used. Bill Harlan showed that tapered sinc functions achieve satisfactory accuracy more cheaply than zero padding. It seems, however, that the best approach is both to zero pad and to use some sinc-like interpolation. A definitive study of interpolation is that of Rosenbaum and Boudreaux [1981].