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theory

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]
sdm
[pdf] [png]

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:

$\displaystyle {\bf m} = {\bf H}_0^{\dagger}{\bf L}_0^{*}{\bf d}_{\rm obs},$     (1)

where $ ^{*}$ and $ ^{\dagger}$ denote adjoint and pseudo inverse, respectively; $ {\bf m}$ is the seismic image; $ {\bf L}_0$ is the Born modeling operator computed using initial velocity $ {\bf v}_0$ , whose adjoint $ {\bf L}_0^{*}$ is the well-known depth migration operator; $ {\bf H}_0$ is the Hessian operator (Tang, 2009; Plessix and Mulder, 2004; Valenciano, 2008); $ {\bf d}_{\rm obs}$ is the observed surface data.

It is important to note that the seismic image $ {\bf m}$ 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 $ {\bf H}_0$ is impossible to obtain in practice due to its size and computational cost, we therefore approximate it by a diagonal matrix as follows:

$\displaystyle {\bf H}_0 \approx {\rm diag}\{ {\bf H}_0 \}.$     (2)

Substituting equation 2 into equation 1 yields
$\displaystyle {\bf m} = {\rm diag}\{ {\bf H}_0 \}^{-1} {\bf L}_0^{*}{\bf d}_{\rm obs}.$     (3)

Equation 3 is also widely known as normalized or amplitude-preserving migration (Tang, 2009; Rickett, 2003; Plessix and Mulder, 2004).

We obtain a target image $ {\bf m}_{\rm target}$ by applying a selecting operator $ {\bf S}$ to the initial image as follows:

$\displaystyle {\bf m}_{\rm target} = {\bf S} {\bf m},$     (4)

where the selecting operator $ {\bf S}$ can be simply a windowing operator. A new data set $ {\widetilde {\bf d}}_{\rm obs}$ can then be simulated as follows:
$\displaystyle {\widetilde {\bf d}}_{\rm obs} = {\widetilde {\bf L}}_0 {\bf m}_{\rm target},$     (5)

where $ {\widetilde {\bf L}}_0$ is the Born modeling operator computed using the same initial velocity $ {\bf v}_0$ , but with different acquisition configuration. The wavefield propagation can be restricted to regions with inaccurate velocities, and the modeled data can be collected at the top of the target region. The target-oriented modeling strategy makes the new data set much smaller than the original one. The new data set can be imaged using the migration operator, i.e., the adjoint of $ {\widetilde {\bf L}}$ , as follows:
$\displaystyle {\widetilde {\bf m}} = {\widetilde {\bf L}}^{*} {\widetilde {\bf d}}_{\rm obs}.$     (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:

$\displaystyle J = \vert\vert{\bf D}{\widetilde {\bf m}}\vert\vert^2,$     (7)

where $ {\bf D}$ is the subsurface-offset-domain differential semblance operator (DSO) (Shen and Symes, 2008; Shen, 2004), which is simply a multiplication of the subsurface offset. DSO optimizes the velocity model by penalizing energy at non-zero subsurface offset, utilizing the fact that subsurface-offset gathers should focus at zero subsurface offset if migrated using an accurate velocity model. We evaluate the gradient of equation 7 using the adjoint-state method (Shen and Symes, 2008; Sava and Vlad, 2008; Tang et al., 2008), and use gradient-based methods to optimize the velocity model.


next up previous [pdf]

Next: Field-data examples Up: Tang and Biondi: Target-oriented Previous: introduction

2010-11-26