Wavefront synthesis using wave stacks

A wave stack is any stack over a common shot or geophone gather in which the moveout is independent of time Schultz and Claerbout (1978). It synthesizes a particular wavefront of raw data because Huygens' principle permits the synthesis of arbitrarily-shaped downward propagating wavefronts from the superposition of many spherical wavefronts. In other words, a line-source can be simulated by firing many smaller, closely-spaced point sources simultaneously or sequentially (Figure ). Unlike the common midpoint stack, wave stacks have the important property of being the sampling of a wavefield and, as such, permit the treatment of formerly difficult or impossible problems with the use of the wave equationSchultz and Claerbout (1978).

 

Huygens
Figure 3 A line source can be simulated by firing many point sources simultaneously or sequentially so as to result in a plane wavefront.

One of the most attractive features of the wave stack method is computational efficiency. Stacking along common geophone gathers significantly reduces the size of the data space to be migrated and produces a stacked section which has the same size as a common-offset section. Compared to shot profile migration, which requires the migration of all shot profiles to obtain a global picture of the subsurface image, the use of a wave stack reduces the size of the data space to be migrated by the order of the number of shots as common-offset migration dose.

Another advantage of the wave stack is that it provides a way to get an interpretable image in terms of angle-dependent reflection coefficients by controlling the incidence angle of the wavefront synthesized to a reflector.

The wave stack can be formulated by means of the linear property of the forward model operator, equation (). Since the forward operator in equation () is independent of time, a linear combination of the recorded trace can be expressed by the combination of source functions as follows :  

$$\sum_{j=1}^M \gamma_j {\bf g}_j(z_0) = F(z_0,z_0) \sum_{j=1}^M \gamma_j {\bf s}_j(z_0) ,$$ (3)

where M is the number of source functions used for wave stacking and $\gamma_j$ is any complex value. The ${\bf \gamma}$ is called a synthesis operator because it synthesizes a wavefront by scaling the amplitude and shifting the phase of each source function. Therefore, we can control the wavefront by adjusting $\gamma_j$ appropriately. Equation () can be simplified as  

$${\bf \bar g}(z_0) = F(z_0,z_0) {\bf \bar s}(z_0) ,$$ (4)

where ${\bf \bar g}(z_0)$ is called synthesized stack and ${\bf \bar s}(z_0)$ is the synthesis operator because it is equal to ${\bf \gamma}$ if we assume each shot is an impulse. The following sections explain how to determine $\gamma_j$according to the shape of the wavefront at the surface and subsurface.