# Physical aspects of slant stacking

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

We now examine the physical aspects of constructing a slant-stack gather. Each trace in this gather represents a plane wave that propagates at a certain angle from vertical. In reality, when a dynamite source explodes, the energy propagates at all angles (Figure 6.3-1). The reflected energy arrives at different receiver groups at different angles because of the offset between source and receiver locations. The farther the offset or the shallower the reflecting interface, the more oblique the angle of the upcoming wavefront.

As an aid in defining a scheme for constructing slant-stack gathers, first consider how plane waves can be generated. Imagine a line of point sources as shown in Figure 6.3-2. Assume that this line of sources is activated so that all points on the line are excited simultaneously, and each point generates a spherical wavefield. Some distance from the surface, the spherical wavefronts superimpose and result in a plane wave that travels vertically downward. This plane wave reflects from an interface and is recorded by a receiver at the surface. (Actually, there are source types, such as Geoflex and Primacord, which approximate short line sources.)

Instead of a vertically traveling plane wave, a plane wave that travels at a desired angle from vertical can be generated using the same line of point sources as illustrated in Figure 6.3-3. To do this, the point sources must be activated in succession starting at one end of the line with an equal time delay between them. When a particular point source is activated, the wavefront generated from the previous source location already will have traveled a certain distance into the earth. When all the spherical wavefronts generated by the various sources superimpose, the result is a tilted plane wavefront. This plane wave then propagates, reflects from an interface, and is recorded by a receiver at the surface.

The amount of tilt of the wavefront, defined by the angle of propagation of the plane wave, can be controlled. Consider the raypath-wavefront geometry in Figure 6.3-4. By the time the wavefront generated at source location S1 reaches point A in the subsurface, the point source at location S2 should be excited so that the desired angle is attained. Define the distance between S1 and S2 as Δx, and the medium velocity with which the waves travel as v. If it takes Δt time for the wavefront to go from S1 to A, using the triangle S1AS2, then the dip angle θ of the plane wave is given by

 ${\displaystyle \sin \theta ={\frac {v\Delta t}{\Delta x}}.}$ (1)

The active source location must therefore travel with speed given by

 ${\displaystyle {\frac {\Delta x}{\Delta t}}={\frac {v}{\sin \theta }}}$ (2)

along the horizontal direction, and the point source at location S2 must be excited so that we can catch the wavefront at S1 as it reaches point A on the wavefront in the subsurface. The velocity (v/sin θ) with which the source location must move is called the horizontal phase velocity.

From the experiments illustrated by Figures 6.3-2 and 6.3-3, note that a plane wave propagating at an angle from the vertical can be generated by:

1. Placing a line of point sources at the earth’s surface.
2. Exciting the point sources in succession with a time delay.
3. Superimposing the responses that are in the form of spherical wavefronts.

The superimposed response is recorded on a single receiver (Figure 6.3-3). This response is in the form of a plane wave that is reflected from an interface. Superposition means summing over the shot axis for a given receiver location. Using the reciprocity principle, summation also can be performed over the receiver axis for a given shot location.

We just discussed how a common-shot gather as a single wavefield can be decomposed into its plane-wave components. By replacing the shot axis in Figure 6.3-4 with the receiver axis, the raypath geometry in Figure 6.3-5 results. The time delay associated with the plane wave that travels at angle θ from the vertical is given by

 ${\displaystyle \Delta t={\frac {\sin \theta }{v}}\Delta x.}$ (3a)

Snell’s law says that the quantity sin θ/v, which is the inverse of the horizontal phase velocity, is constant along a raypath in a layered medium (Figure 6.3-6). This constant is called the ray parameter p. Equation (3a) then is rewritten as

 ${\displaystyle \Delta t=p\Delta x.}$ (3b)

The angle of propagation of the plane wave is controlled by adjusting the p value. If the ray parameter p is specified, then the ray can be traced in a horizontally layered earth model with a known velocity function v(z). Setting p = 0 corresponds to a plane wave that travels vertically.

Given the ray parameter p and the velocity function v(z) for the layered earth, the family of raypaths associated with a particular p value can be traced as shown in Figure 6.3-7. A plane wave that travels in a layered earth is called a Snell wave [1]. This type of plane wave changes its direction of propagation at each layer boundary according to Snell’s law (Figure 6.3-6). For a single p value, note that the signal is recorded at many offsets (Figure 6.3-7).

In general, receivers at all offsets record plane waves of many p values. To decompose a shot gather into its plane-wave components, all the trace amplitudes in the gather must be summed along several slanted paths, each with a unique time delay defined by equation (3b).

As long as there is no dip, the traveltimes in a common-shot and a common-midpoint gather are indistinguishable. Since a CMP gather is not a single wave-field, plane-wave decomposition would not seem to apply to CMP gathers. However, the equivalence of CMP gathers and common shot gathers in a horizontally layered earth provides a rationale for applying plane-wave decomposition to both types of gathers.

## References

1. Claerbout, 1978, Claerbout, J. F., 1978, Snell waves: Stanford Exploration Project Report No. 15, 57–72. Stanford University.