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Kirchhoff migration methods

Kirchhoff methods using surface-related coordinates are well understood and have been used for decades (, ). The Kirchhoff equation that is the basis for Kirchhoff migration simply states that the wavefield at a given point U(x,y,z,t) is a summation of waves propagating from other points at earlier times. When applied to migration in the surface-related domain, the Kirchhoff equation becomes the simple act of summing over the hyperbolas that are generated along the offset axis with different curvature depending on v.

When we turn to the subsurface-related angle domain, its kinematics make constructing a Kirchhoff type (summation) migration algorithm straightforward. () described how to generate angle domain common image gathers (ADCIGs) in the depth domain by simply summing over all dip vectors 12#12 for each point in the subsurface. Their methodology for creating an ADCIG from data recorded at the surface (d(s,r,t)) can be simply written as:

13#13 (4)
where 14#14 is the ADCIG for reflection angle 5#5at subsurface position 3#3. The summation is over normal vectors 12#12 (summing over all dips at position 3#3). The traveltimes used in this angle domain Kirchhoff method vary with 12#12. A weighting operator 15#15 that is dependent on the subsurface position and the dip and reflection angle at that position is applied to the data. This weighting operator handles issues such as amplitudes, that do not deal with the presence or absence of events. I refer you to (, ) for more detailed information on angle-domain Kirchhoff methods.