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![]() | Target-oriented wavefield tomography: A field data example | ![]() |
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Our method can be formulated under the framework of seismic data mapping (SDM) (Bleistein and Jaramillo, 2000; Hubral et al., 1996), where the idea is to transform the original observed seismic data from one acquisition configuration to another with a designed mapping operator. SDM can be summarized into two main steps as illustrated in Figure 1: (1) apply the (pseudo) inverse of the designed mapping operator to the original data set to generate a model; (2) apply the forward mapping operator to the model to generate a new data set with different acquisition configuration than the original one. This idea has been widely used in the area of seismic data interpolation and regularization. For example, in Radon-based interpolation methods (Trad et al., 2002; Sacchi and Ulrych, 1995), Radon operator is used as the mapping operator to regularize the data; the azimuth moveout (AMO) (Biondi et al., 1998) uses dip moveout (DMO) as the mapping operator to transform the data from one azimuth to another.
sdm
Figure 1. Flow diagrams of seismic data mapping. [NR] |
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In our method, we use wave-equation-based Born modeling or demigration as the mapping operator to peform data mapping.
With an initial velocity model, seismic prestack images can be obtained using the pseudo inverse of
the Born modeling operator as follows:
It is important to note that the seismic image
has to be parameterized as a function
of both spatial location and some prestack parameter, such as the subsurface offset, reflection angle, etc.,
in order to preserve the velocity information for later velocity analysis (Tang and Biondi, 2010).
In this paper, we use the subsurface offset as our prestack parameter.
The significance of the Hessian operator in equation 1 is that
its pseudo inverse removes the influence of the original acquisition
geometry in the least-squares sense and the resulting image is independent from the original data.
However, the full Hessian
is impossible to obtain in practice due to its size and computational cost,
we therefore approximate it by a diagonal matrix as follows:
We obtain a target image
by applying a selecting operator
to the initial image as follows:
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(4) |
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(5) |
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(6) |
We pose our velocity analysis problem as an optimization problem defined
in the image domain, where the objective function to minimize is defined as follows:
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![]() | Target-oriented wavefield tomography: A field data example | ![]() |
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