Wide-azimuth angle gathers for wave-equation migration |
In this section, we derive the formula for the moveout function characterizing reflections in the extended domain. The purpose of this derivation is to find a procedure for angle decomposition, i.e. a representation of reflectivity as a function of reflection and azimuth angles.
An implicit assumption made by all methods of angle decomposition is that we can describe the reflection process by locally planar objects. Such methods assume that (locally) the reflector is a plane, and that the incident and reflected wavefields are also (locally) planar. Only with these assumptions we can define vectors in-between which we measure angles like the angles of incidence and reflection, as well as the azimuth angle of the reflection plane. Our method uses this assumption explicitly. However, we do not assume that the wavefronts are planar. Instead, we consider each (complex) wavefront as a superposition of planes with different orientations. In the following, we discuss how each one of these planes would behave during the extended imaging and angle decomposition. Thus, our method applies equally well for simple and complex wavefields characterized by multipathing.
We define the following unit vectors to describe the reflection geometry and the conventional and extended imaging conditions:
With these definitions, the (planar) source and receiver wavefields
are given by the expressions:
Similarly, we can rewrite the extended imaging condition using the
planar approximation of the source and receiver wavefields using the
expressions:
We can eliminate the space variable
by substituting equation 6
in equation 8 and equation 7 in equation 9:
So far, we have not assumed any relation between the vectors characterizing the source and receiver planes, and . However, if the source and receiver wavefields correspond to a reflection from a planar interface, these vectors are not independent of one-another, but are related by Snell's law which can be formulated as
Substituting Snell's law into the system 12-13,
and after trivial manipulations of the equations, we obtain the
system:
Figures 1-3 illustrate the process involved in the extended imaging condition and describe pictorially its physical meaning. Figure 1 shows the source and receiver planes, as well as the reflector plane together with their unit vector normals. Figure 2 shows the source and receiver planes displaced by the space lag vector contained in the reflector plane, as indicated by equation 18. The displaced planes do not intersect at the reflection plane, thus they do not contribute to the extended image at this point. However, with the application of time shifts with the quantity , i.e. a translation in the direction of plane normals, the source and receiver planes are restored to the image point, thus contributing to the extended image, Figure 3.
eicawfl
Figure 1. The reflector plane (of normal ), together with the source and receiver planes (of normals and , respectively). The figure represents the source/receiver planes in their original position, i.e. as obtained by wavefield reconstruction. |
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eicxshift1
Figure 2. The reflector plane (of normal ), together with the source and receiver planes (of normals and , respectively). The figure represents the source/receiver planes displaced with the space-lag constrained in the reflector plane. |
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eictshift1
Figure 3. The reflector plane (of normal ), together with the source and receiver planes (of normals and , respectively). The figure represents the source/receiver planes displaced with space-lag and time-lag . The space and time-lags are related by equation 17. |
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Wide-azimuth angle gathers for wave-equation migration |