We will see that this amounts to accurate frequency-domain interpolation. First, consider an impulse function at time t 0 . 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. So the first proposed improvement to the Stolt migration program is to multiply in the frequency domain by ,then interpolate, and finally multiply by .
Linear interpolation is almost the easiest form of interpolation.
The sinc function of frequency, when brought back into the time domain is a rectangle function of time. 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.