PLANE WAVES & SNELL WAVES

Plane waves have the desirable property that they can be downward continued simply by time shifting in negative increments. In the Fourier domain this is particularly simple since this operation is limited to multiplication by a time shift operator. Unfortunately wave fields produced by shots, vibroseis, or airguns produce spherical wave fields and cannot be as simply downward continued. Plane wave theory applied to reflection seismic work should not be abandoned though; any wave field can be decomposed into a collection of plane waves of differing orientations in space, analogous to the the way Fourier synthesis can be used to represent an arbitrary functionClaerbout (1984). Since seismic data is not recorded as plane wave sets, the problem exists of how to decompose the seismic wave field into constitute plane waves. One approach is to consider the angle that energy arrives at the receivers in a given shot gather. Assuming that the near surface velocity is not drastically variable, energy arriving with the same angle at regular time intervals along a shot gather can be associated with a plane wave. The angle of incidence at the surface can be simply related to the slowness, or the Snells parameter, of the incident energy by the equation:

$$\begin{eqnarray} \frac{sin(\theta)}{v} &=& \frac{\partial t_{0}}{\partial h}\end{eqnarray}$$ (1)

where $\theta$ is the angle of the ray with respect to the vertical, v is the velocity of the shallow subsurface, t₀ is the time at which the wave is incident on the surface and h is the offset. The right side of this equation is simply the dip of events in shot gather coordinates. It follows from this relation of plane waves to dips in shot gathers that the dip information obtainable by slant stacks of shot gathers can be a key tool in plane wave decompositionClaerbout (1984).

Discussion of plane waves in media of spatially variant velocity, such as the Earth, is at best a crude approximation. For the special case of a media of stratified velocities, a model which can be considered in many cases a good approximation of the Earth, plane waves have a simple derivative. A plane wave in a media of constant velocity incident at an acute angle to a media of stratified velocities will be distorted upon entering the region of stratified velocities. In the stratified media, the former plane wave will retain memory of its former existence and maintain a distinctive character, one important characteristic being the value of the velocity of the wavefront along a horizontal plane, such as the Earths surface. Waves with these characteristics are known as Snells waves. The inverse of the horizontal velocity of the wavefront is known as Snells parameter, or the slowness, which is the linear shift factor encounter in the slant stack transformation equation, $\tau=t-ph$, where p is Snells parameter, $\tau$ is the retarded time, t is the travel time in the shot gather and h is the offset in the shot gather. The important characteristic of Snell waves to practitioners is not only that the Snells parameter is constant but also that it is an observableClaerbout (1984).

Shot gathers are an observation of a propagating wave event and can be modeled directly using the wave equation. Slant stacks applied to shot gathers reveal information about a the physical state of the propagating wave field for values of Snells parameter and time. Unfortunately slant stacks of common shot gathers can be plagued by aliasing effects when applied to sparsely sampled data due to the presence of conflicting dips. On the other hand, common midpoint gathers are much less prone to having severely conflicting dips because all the hyperbolic events are aligned along the zero offset axis. Common midpoint gathers are not a coherent representation of a wave field, rather a cmp gather is a fragmentation of many separate shot events and cannot be used to evaluate characteristics of the wavefield such as Snells parameter.