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Velocity continuation (Fomel, 2003c) is the imaginary
process of a continuous transformation of seismic time-migrated
images as they are propagated through different migration velocities. In the most general terms, the
kinematics of velocity continuation can be described by an equation of
the Hamilton-Jacobi type
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(1) |
where
is the location of a time-migrated
reflector with time-domain coordinates
imaged with spatially constant time-migration velocity . The particular form
of function in equation (1) depends on the acquisition
geometry of the input data. For the case of common-offset 2D
velocity continuation for data with half-offset ,
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(2) |
and equation (1) corresponds to the characteristic
equation of the image propagation process which describes a propagation of the time-migrated image in
velocity (Fomel, 2003c). Time-domain imaging can be performed effectively by extrapolating
images in velocity and estimating velocity of the best
image (Fomel and Landa, 2014; Fomel, 2003b; Larner and Beasley, 1987).
As shown by Fomel (2003a), it is possible to extend the
formulation of a wave propagation process from the usual
time-and-space coordinates to the phase space consisting of time,
space, and slope. Applying a similar approach to
equation (1), we first employ the Hamilton-Jacobi
theory (Evans, 2010; Courant and Hilbert, 1989) to write the corresponding system of
ordinary differential equations for the characteristics (velocity
rays), as follows:
where stands for , the gradient or slope of time-migrated wavefield energy.
If the image
is decomposed in slope components
so that
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(6) |
we can then look for an equation that would adequately describe a continuous
transformation of . To preserve the geometry of the
transformation, it is sufficient to require that transports
along the characteristics described by
equations (3-5). Applying partial derivatives
and the chain rule, we arrive at the
equation analogous to the Liouville equation (Engquist and Runborg, 2003):
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(7) |
Equation (7) describes, in the most general form, the
process of oriented velocity continuation, image propagation
in velocity in the coordinates of time-space-slope. It is a linear
first-order partial differential equation of convection type which operates in the phase space.
Subsections
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| Diffraction imaging and time-migration velocity analysis using oriented velocity continuation | |
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Next: Common-offset oriented velocity continuation
Up: Decker et al.: Oriented
Previous: Introduction
2017-04-20