# Slant-stack transformation

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

Two steps typically are used in synthesizing plane waves by summing amplitudes in the offset domain along slanted paths (Figure 6.3-8). First, a linear moveout (LMO) correction is applied to the data through a coordinate transformation defined by Claerbout[1]

 ${\displaystyle \tau =t-px,}$ (4a)

where p is the ray parameter, x is the offset, t is the two-way traveltime, and τ is the intercept time at p = 0. Next, the data are summed over the offset axis by

 ${\displaystyle S(p,\tau )=\sum _{x}P(x,\tau +px),}$ (4b)

where, S(p, τ) represents a plane wave with ray parameter p = sin θ/v. By repeating the LMO correction for a range of p values and performing the summation in equation (4b), a complete slant-stack gather is constructed. A slant-stack gather, in practice, alternatively is referred to as a τ − p gather; it consists of all the dip components within the specified range of p values in the original offset data.

The mapping from the t − x domain to the τ − p domain is reversible [2]. First, apply inverse linear moveout (LMO) correction to the data in the τ − p domain by

 ${\displaystyle t=\tau +px.}$ (5a)

Then, sum the data in the τ − p domain over the ray parameter p axis to obtain

 ${\displaystyle P(x,t)=\sum _{p}S(p,t-px).}$ (5b)

To restore amplitudes properly, rho filtering is applied before inverse mapping. This is accomplished by multiplying the amplitude spectrum of each slant-stack trace by the absolute value of the frequency. Rho filtering is equivalent to differentiating the wavefield before the summation that is involved in the integral formulation of migration (migration principles).

There is a distinction between slant stack and the exact plane-wave decomposition of a wavefield. Treitel [3] mathematically analyzed the plane-wave decomposition process and distinguished between conventional slant stack described here and what may be called the proper slant stack. A conventional slant stack yields an exact plane-wave decomposition when we deal with line sources; a proper slant stack yields an exact plane-wave decomposition when we deal with point sources. A proper slant stack is generated using the same steps that we just described for a conventional slant stack except that a convolution of the LMO-applied wavefield by a filter operator is performed before summation. This operator corrects for 3-D effects by converting a wavefield that was obtained from a point source into a wavefield that was obtained from a line source. As far as kinematics is concerned, the two types of slant stacking are equivalent. They differ only in treatment of amplitudes (Treitel, personal communication).

A schematic description of the plane-wave mapping given by equations (4a, 4b) is shown in Figure 6.3-9. We start by summing amplitudes in the offset domain along the horizontal path p = 0. This line intersects the reflection hyperbola in the vicinity of apex A. Thus, point A maps onto point A′ on the τ − p plane, corresponding to the trace with p = 0. By tilting the line of summation, the hyperbola is intersected at location B, which maps onto B′. Note that a major contribution to summation along the slanted path comes in the area of the tangential point B. This zone of tangency is called the Fresnel zone.

The Fresnel zone gets broader for higher velocities and deeper events. In fact, the summation over the offset axis after linear moveout correction can be confined to the Fresnel zone. The steepest necessary path of summation is along p = 1/v, which is the asymptote to the hyperbola. This path corresponds to rays that are 90 degrees to the vertical. The energy along the asymptote maps to C′ on the p-axis. By using the mapping described previously — linear moveout correction followed by summation over the offset axis for a range of p-values (equations 4a, 4b), the hyperbolic trajectories in the t − x domain are mapped to elliptical trajectories in the τ − p domain. In reality, we never record a hyperbola with infinite extent nor a zero-offset trace. Therefore, the elliptical path in the τ − p domain never is complete from A′ to C′.

Figure 6.3-10 illustrates the τ − p transform of more than one hyperbolic event in the t − x domain. Subcritical reflections A and D (those with an angle of incidence smaller than the critical angle) map into the region of lower p-values, while supercritical reflections C (wide-angle reflections) map into the region of higher p values. Ideally, a linear event in the offset domain, such as a refraction arrival B, maps to a point in the slant-stack domain. Conversely, a linear event in the slant-stack domain maps to a point in the offset domain.

Figure 6.3-11 shows a shot gather that contains predominantly water-bottom and peg-leg multiples. The horizontal axis in the τ − p domain, in this example, is horizontal phase velocity 1/p. Besides the water-bottom reflection W, there are two distinct primaries, P1 and P2. Multiple reflections map along the elliptical trajectories that converge at p = (1/1500) s/m, the inverse of the water velocity.

A shot gather containing linear events is shown in Figure 6.3-12. Note the strong amplitudes on the τ − p gather that correspond to the guided waves observed in the offset data. Again, the horizontal axis in the τ − p domain is horizontal phase velocity 1/p. In both field data examples shown in Figures 6.3-11 and 6.3-12, the τ − p gathers were constructed using only positive p-values. Thus, the backscattered energy, for instance in the shot gather shown in Figure 6.3-12, is not represented in the τ − p gather.

## References

1. Claerbout, 1978, Claerbout, J. F., 1978, Snell waves: Stanford Exploration Project Report No. 15, 57–72. Stanford University.
2. Thorson, 1978, Thorson, J., 1978, Reconstruction of slant stacks: Stanford Exploration Project Report No. 14, 83–91. Stanford University.
3. Treitel et al. (1982), Treitel, S., Gutowski, P. R., and Wagner, D. E., 1982, Plane-wave decomposition of seismograms: Geophysics, 47, 1375–1401.