The rho filter

In practical work, the rho filter is often ignored because it can be absorbed into the rest of the filtering effects of the overall data recording and processing activity. However, the rho filter is not inconsequential. The integrations in the slant stack enhance low frequencies, and the rho filter pushes them back to their appropriate level. Let us inspect this filter. The rho filter has the same spectrum as does the time derivative, but their time functions are very different. The finite-difference representation of a time derivative is short, only $\Delta t$ in time duration. Because of the sharp corner in the absolute-value function, the rho filter has a long time duration. The Hilbert kernel -1/t has a Fourier transform $i\,\sgn( \omega )$.Notice that $\vert \omega \vert = (-i \omega ) \,\times\, i\,\sgn ( \omega )$.In the time domain this means that d/dt (-1/t) = 1/t², so $\ \it\hbox{rho} (t) = 1/t^2+\delta (t)$.

An alternate view is that the rho filter should be divided into two parts, with half going into the forward slant stack and the other half into the inverse. Then slant stacking would not cause the power spectrum of the data to change. An interesting way to divide the $\vert \omega \vert$ is $\vert \omega \vert = \sqrt{-i \omega}\,\sqrt{i \omega }$.The expression $\sqrt{-i \omega }$ has a causal time function and $\sqrt{i \omega}$ has an anticausal one.

In practice, slant stack is not so cleanly invertible as 2-D FT, so various iteration and optimization techniques are often used.

EXERCISES:

1. Assume that v(z)=const and prove that the width of a Fresnel zone increases in proportion to $\sqrt{t}$.
2. Given v(z), derive the width of the Fresnel zone as a function of t.