Theoretical comparison between time-distance functions

The Kolmogorov impulse response is essentially a large impulse at zero lag (in time and space) with a small amplitude signal corresponding to the diving waves. Both components are band-limited, so we can write  

$$B(Z)=W(Z) \left[ 1 + \epsilon F(Z) \right],$$ (11)

where F(Z) is the causal function of interest, $\epsilon$ is simply a scalar indicating the small amplitude of that term, and W(Z) is a minimum phase band-limited seismic wavelet.

The crosscorrelation process produces the autocorrelation of equation ():  

$${\bar B} B = {\bar W} W \left[ 1 + \epsilon F + \epsilon {\bar F} + \epsilon^2 {\bar F} F \right].$$ (12)

This function contains $\epsilon F$, the function we are interested in studying; however there are two major differences.

Firstly, ${\bar B} B$ also contains the additional terms $\epsilon {\bar F}$ and $\epsilon^2 {\bar F} F$. We can discard the first of these terms, $\epsilon {\bar F}$ since it is anti-causal, and the second term contains $\epsilon^2$ so will be much smaller than the signal of interest.

The second difference between equations () and () is the wavelet. The Kolmogorov wavelet is minimum phase, whereas the crosscorrelation wavelet ${\bar W} W$ is zero-phase. The amplitude spectrum of the crosscorrelation wavelet will also be the square of the Kolmogorov wavelet.

Thus the principal advantage of the Kolmogorov result is that it has a broader bandwidth than the crosscorrelation. Whereas the Kolmogorov result has the same amplitude spectrum as the original data, the amplitude spectrum of the crosscorrelation impulse response is equal to the power spectrum of the original data.