Next: 3-D theory Up: Sava: Image decomposition Previous: Introduction

# 2-D theory

The derivation in this section follows the one of Fomel (1996). Assuming that a reflection event in the extrapolated wavefield is described by the function t(z,s,r), we find from the Snell's law the following derivatives:
 (1)

 (2)
where s,r stand for the source and receiver spatial coordinates, v is the wave velocity, is the dip angle, and is the reflection angle (Figure ).

 local Figure 1 A sketch of reflection rays in an arbitrary-velocity medium.

The traveltime derivative with respect to the depth of the observation surface z has contributions from the two branches of the reflected ray, as follows:
 (3)
Equation 3 corresponds to the well-known double-square-root equation Claerbout (1985). This equation simply reflects the fact that the traveltime increases with increasing depth of the reflector.

Transforming Equations (1-3) to the midpoint and half-offset coordinates, we obtain
 (4)

 (5)

 (6)
At a fixed image location x, we can transform the derivatives of t(z,x,h) to the derivatives of z(t,x,h) by applying the implicit function theorem. Using Equations (5-6), we obtain
 (7)
and using Equations (4-6), we obtain
 (8)

 sls Figure 2 Angle decompositions. Panel (a) corresponds to slant-stack angle-decomposition in the z-h plane. Panel (b) corresponds to slant-stack angle-decomposition in the z-x plane. In both cases, a dipping segment maps at a particular angle.

Sava and Fomel (2003) use Equation (7) to compute angle-domain common image gathers for images obtained by wavefield extrapolation. This formula can be implemented either as a slant-stack in the space domain, or as a radial-trace transform in the Fourier domain. In both cases, we obtain at every point in the image the reflection strength as a function of scattering angle, independent of the reflector dip.

Similarly, we could employ formula (8) for another decomposition of the migrated image function of the structural dip . As for equation (7) which represents a slant-stack in the x-h plane, equation (8) represents a slant-stack in the x-z plane (Figure ).

 mag Figure 3 A sketch of a generic reflection experiment and its associated angle decomposition. The left panel corresponds to the x-z location of the reflection point in the image. In the 2-D case, at every point in image, the reflectivity is described by two angles, and .

Next: 3-D theory Up: Sava: Image decomposition Previous: Introduction
Stanford Exploration Project
7/8/2003