Rho filter

Notice the dark halo around the reconstruction in Figure 3. It was suppressed in Figure 5 by the subroutine halfdif(). Recall that slant-stack inversion (see IEI for example) requires an $\vert\omega\vert$ filter. Without doing any formal analysis I guessed that the same filter would be helpful here because the dark halo has a strong spectral component at $\omega=0$which would be extinguished by an $\vert\omega\vert$ filter. Because of the close relation to wave propagation and causality, I found it appealing to factor $\vert\omega\vert$ into a causal $\sqrt{-i\omega}$ part and an anticausal $\sqrt{ i\omega}$ part. I applied a causal $\sqrt{-i\omega}$ after generating the (t,x)-space and an anticausal $\sqrt{ i\omega}$before making the $(\tau,v^{-2})$-space. I implemented the causality by taking the square root of a Fourier domain representation of causal differentiation, namely (1-Z)^1/2. The listing of this half-order differentiator is straightforward.

Notice also that vspray() includes a scaling variable named scale. I did not develop a theory for this scale factor, but if you omit it, amplitudes in the reconstructions will be far out of amplitude balance with the input.