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| Modeling, migration, and inversion in the generalized source and receiver domain | |
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By using the Born approximation to the two-way wave equation, the primaries can be modeled by a linear operator as follows:
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(A-1) |
where
is the modeled data for a single frequency with source and receiver located at
and
on the surface;
and
are the Green's functions connecting
the source and receiver, respectively, to the image point
in the
subsurface; and
denotes the reflectivity at image point .
In Equation 1, we assume and are infinite in extent and independent of each other.
For a particular survey, however, we do not have infinitely long cable and infinitely many sources;
thus we have to introduce an acquisition mask matrix to limit the size of the modeling. We define
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(A-2) |
For the marine acquisition geometry,
is similar to a band-limited diagonal matrix;
for Ocean Bottom Cable (OBC) or land acquisition geometry, where all shots share the same receiver array,
is a rectangular matrix.
Figure 1 illustrates the acquisition mask matrices for these two typical geometries in 2-D cases.
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acquisition-mask
Figure 1. Acquisition mask matrices for different geometries in 2-D cases. Greys denote ones while whites denote zeros.
The left panel shows the acquisition mask matrix for a typical marine acquisition geometry;
the right panel shows the acquisition mask matrix for a typical OBC or land acquisition geometry. [NR]
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To find a model that best fits the observed data, we can minimize the following data-misfit function in the least-squares sense:
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(A-3) |
The gradient of the above objective function gives the conventional shot-profile migration algorithm:
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(A-4) |
where denotes the real part of a complex number and means the complex conjugate;
is the weighted residual defined as follows:
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(A-5) |
The gradient or migration is only a rough estimate of the model
; to get a better recovery of the model space, the inverse of the Hessian,
the second derivatives of the objective function, should be applied to the gradient:
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(A-6) |
The Hessian can be explicitly constructed by taking the second-order derivatives of the objective function with respect to the model parameters
as follows (Tang, 2008; Plessix and Mulder, 2004; Valenciano, 2008):
where is a neighbor point around the image point in the subsurface.
Valenciano (2008) demonstrates that the Hessian can be directly computed using the above formula; however it requires storing a large number of Green's functions,
which is inconvenient for dealing with large 3-D data set. Tang (2008) shows that with some minor alteration of Equation 7,
an approximate Hessian can
be efficiently computed using the phase-encoding method.
However, Tang (2008) focuses more on the algorithm development, and the physics behind the Hessian by phase-encoding has not been carefully discussed.
In this companion paper, I complete the discussion of the actual physics behind using phase-encoding methods,
such as plane-wave phase encoding and random phase encoding, to obtain the Hessian. In the subsequent sections,
I start with the modeling equation in the encoded source, encoded receiver and simultaneously encoded source and receiver domains.
I show that the corresponding imaging Hessian in the generalized source and receiver domain is the same as
those phase-encoded Hessians discussed in Tang (2008).
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| Modeling, migration, and inversion in the generalized source and receiver domain | |
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Next: encoded sources
Up: Modeling, migration, and inversion
Previous: introduction
2009-04-13