# Impulse response of the velocity-stack operator

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

An isolated spike in the offset domain (Figure 6.4-11a) maps to the velocity domain using equation (10a) along a curved trajectory (Figure 6.4-12a). Solve equation (9b) for v to obtain the equation for this trajectory in the velocity domain:

 ${\displaystyle u(v,\tau )=\sum _{h}d(h,t={\sqrt {\tau ^{2}+4h^{2}\!/\!v^{2}}}),}$ (10a)

 ${\displaystyle t^{2}=\tau ^{2}+{\frac {4h^{2}}{v^{2}}},}$ (9b)

 ${\displaystyle v={\frac {2h}{\sqrt {t^{2}-\tau ^{2}}}}.}$ (20)

The curvature is greater for a spike situated at far offset than a spike situated at near offset (Figures 6.4-11b and 6.4-12b). Also, the curvature is greater for a spike situated at an early time on a given offset than a spike situated at a late time on the same offset (Figures 6.4-11c and 6.4-12c).

Inverse transformation of the conventional velocity-stack gathers (Figures 6.4-12a,b,c) back to the offset domain does not reproduce the isolated spikes (Figures 6.4-13a,b,c). Instead, the amplitudes are smeared across each of the CMP gathers. The amplitude smearing is worse for spikes situated at near offsets (Figure 6.4-13b) and late times (Figure 6.4-13c).

Figures 6.4-14a,b,c show the velocity-stack gathers based on the Radon transform of equation (16) associated with the isolated spikes in Figures 6.4-11a,b,c. Inverse mapping these velocity-stack gathers, in contrast with the results obtained from inverse mapping of the conventional velocity-stack gathers (Figures 6.4-13a,b,c), yields a fairly good focusing of energy to the isolated spike locations (Figures 6.4-15a,b,c).

 ${\displaystyle \mathbf {u} =(\mathbf {L^{T\ast }L} )^{-1}\mathbf {L^{T\ast }d} ,}$ (16)

How is the velocity-stack processing affected by irregularities in the data? Refer to the CMP gather in Figure 6.4-11d. It contains a trace with a monofrequency signal, another trace with polarity reversed, a dead trace, and another trace with a dead zone. The conventional velocity-stack gather is shown in Figure 6.4-12d, and the CMP gather reconstructed from it is shown in Figure 6.4-13d. Compare this figure with Figure 6.4-11d and note the differences along the reflection hyperbolas in amplitude and curvature where the anomalous traces are located. Also, note the smearing of the monofrequency signal over a large range of traces away from the original trace location. The velocity-stack gather based on the Radon transform associated with the CMP gather in Figure 6.4-11d is shown in Figure 6.4-14d. The CMP gather reconstructed from it (Figure 6.4-15d) shows less smearing of the monofrequency signal. As with the conventional velocity-stack gather, however, the dead trace and the trace with a dead zone in the original CMP gather (Figure 6.4-11d) again have been replaced with nonzero amplitudes, and traveltimes have been distorted along the reflection hyperbolas where the anomalous traces are located.

The transform parameters of practical importance are the velocity range and the velocity increment used in constructing velocity-stack gathers. The velocity range should span the velocities associated with primary and multiple reflections. A good practice for the choice of velocity increment is such that the number of traces in velocity space is set equal to the traces in the offset space.