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# 3-D theory

Equations (7) and (8) are derived with the assumption of a 2-D earth. As pointed out by Tisserant and Biondi (2003), the Fourier-domain angle-gather transformation
 (9)
is not valid in general in 3-D and needs to be corrected for the crossline dip component of the structure.

Tisserant and Biondi (2003) show that we can make this 3-D correction by re-writing the angle-gather transformation as
 (10)
where is the acquisition azimuth, is the absolute value of the offset wavenumber, and kmx,kmy are the components of the midpoint wavenumber. If we introduce the notation and , we can write
 (11)
and represent the projection angles of the normal to the reflection plane onto the Cartesian coordinate system. Equation (11) gives the relation between the four angles describing the reflectivity decomposition in 3-D, namely the structural dip angles (), the recording azimuth (), and the scattering angle ().

The 3-D decomposition algorithm is based on the following scheme:

• Begin from a 3-D migrated image

 R(z,hx,hy,mx,my). (12)

Transform it to polar coordinates in the hx-hy plane  (13)
where, by definition, and .
• Decompose the image by dip and dip azimuth, or equivalently by the projections of the normal to the reflection plane on the Cartesian axes ():  (14)
where and .
• At every , decompose the image function of scattering angle  (15)
using Equation (11).
The four angles describing reflectivity are just one particular and convenient choice. Any other convenient quartet of angles can be used, although such alternatives yield no new information.

Next: Examples Up: Sava: Image decomposition Previous: 2-D theory
Stanford Exploration Project
7/8/2003