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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:
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.