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Suppose a data gather P(t,xr) is recorded in an area
that has a subsurface structure Q(x,z) and a velocity model v(x,z).
Based on the stationary-phase approximations,
this gather can be modeled by a Kirchhoff integral:
| |
(1) |
where A(x,z;xc) and or xr, are an amplitude
function and a traveltime function respectively. The traveltime function
can be found by solving the Eikonal equation:
| |
(2) |
with initial condition .
If this gather is migrated with a different velocity model , the
image obtained will be distorted from the true image of the subsurfaces.
| |
(3) |
where or xr is the traveltime function
with the velocity model .
Residual migration is a transformation from image to image Q(x,z). Therefore, we want to find out the relationship
between (x,z) and that defines the kinematic operators
of this transformation.
Substituting P(t,xr) from equation (1) into equation
(3) yields
The image of a single scatterer at (x,z) can then be found to be:
| |
(4) |
It is well known that the kinematics of the summation operator that does
full migration is defined by the trajectory of the reflection event from
a scatterer.
Similarly, the kinematics of the summation operator that does residual
migration can be determined from , or more
specificly from the argument of the -function in equation
(4). Let
Recall the properties of -function:
where .Clearly, has extremal values when
Therefore the relationship between (x,z) and ,is implicitly expressed by a pair of equations:
| |
(5) |
The partial derivative of xs with respect to xr is determined by the
type of the data gather. For common shot gathers,
For constant offset sections,
For each point (x,z), equation (5) defines a curve in
. This curve is exactly the kinematics of the
residual-migration operator at point (x,z).
For general velocity models, the traveltimes must be computed by
some numerical methods. These methods generate traveltime tables
rather than continuous functions. Therefore, it is
natural to solve equation (5) numerically. A straight forward
method is searching. For
each xr and (x,z), all points around (x,z) are checked to find
the that satisfies equation (5).
But the this algorithm is time-consuming when the dimensions
of images are large. Motivated by the results of the finite-difference
calculation of
traveltimes (Van Trier, 1990), I begin to explore the possibility to calculate
residual-migration operators with finite-difference techniques.
Next: Finite-difference method
Up: RESIDUAL MIGRATION OPERATOR
Previous: Computational costs
Stanford Exploration Project
1/13/1998