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In wave-equation migration, as for example reverse-time migration, the image is computed from the back-propagated receiver wavefield, $ {{P}_{g} }\left({t},\vec x,x_{s};{s}\right)
$ , and the forward-propagated source wavefield, $ {{P}_{s} }\left({t},\vec x,x_{s};{s}\right)
$ , where $ {t}$ is the recording time, $ \vec x=z\vec z_0+x\vec x_0$ is the model-coordinate vector, $ x_{s}$ is the source position at the surface, and $ {s}\left(\vec x\right)
$ is the slowness model.

The prestack image, $ I_{h}\left(\vec x,x_{h}\right)
$ , is computed as the zero lag of the temporal cross-correlation between the spatially-shifted back-propagated receiver wavefield and forward-propagated source wavefield as (Rickett and Sava, 2002):

$\displaystyle I_{h}\left(\vec x,\vec{x_{h}}\right) \left[ {{P}_{s} }\left({t},\...
...-\vec{x_{h}},x_{s}\right) {{P}_{s} }\left({t},\vec x+\vec{x_{h}},x_{s}\right) ,$ (A-1)

where $ \vec{x_{h}}=x_{h}\vec x_0$ is the half subsurface offset, which in this paper I will assume to be horizontal, but it does not need to be in the general case (Biondi and Symes, 2004).

The prestack image as a function of subsurface offset can be transformed to an image as a function of reflection aperture angle, $ I_{\gamma }\left(\vec x,\gamma \right)
$ by using a linear operator $ {\bf\Gamma}$ (Sava and Fomel, 2003). In matrix notation, if $ {\bf I}_{h}$ is a $ N_{\vec x}N_{h}\times 1$ matrix and $ {\bf I}_{\gamma }$ is a $ N_{\vec x}N_{\gamma }\times 1$ matrix, the image transformation from subsurface offset into the angle domain can be expressed as:

$\displaystyle {\bf I}_{\gamma }= {\bf\Gamma}{\bf I}_{h}.$ (A-2)

I introduce an objective function that maximizes the flatness of the angle-domain image along the aperture-angle axis at all spatial locations $ \vec x$ . This objective function aims at maximizing image flatness not directly as a function of the slowness, but indirectly through the application of an angle-domain moveout operator $ \mathcal M_{\gamma }$ , which depends on the column vector of $ N_{{\mu}}=N_{\vec x}$ moveout parameters $ {\boldsymbol \mu}_{\vec x}$ .

I define the application of the moveout operators $ \mathcal M_{\gamma }$ to a prestack image computed by equations 1 and 2 with a background slowness $ \bar {s}$ , as

$\displaystyle { I_{\gamma }\left(\vec x+ \vec{{\zeta}}\left({\boldsymbol \mu}_{...
...ldsymbol \mu}_{\vec x}\right] I_{\gamma }\left(\vec x,\gamma ;\bar {s}\right) ,$ (A-3)

where $ \vec{{\zeta}}\left({\boldsymbol \mu}_{\vec x}\right)
{\zeta}\left({\boldsymbol \mu}_{\vec x}\right)
\vec z_0$ are the moveout shifts, assumed here to be simple depth shifts. The operator $ \mathcal M_{\gamma }$ is linear with respect to the input image, but it is nonlinear with respect to the vector of moveout parameters $ {\boldsymbol \mu}_{\vec x}$ . In matrix notation, the application of the moveout operator to the background image $ {\bf\bar I_{\gamma }}$ can be expressed as $ \mathcal M_{\gamma }\left[{\boldsymbol \mu}_{\vec x}\right]{\bf\bar I_{\gamma }}
$ .

I further define the stacking operator $ {\bf S}_{\gamma }$ that sums the image along the aperture-angle axis $ \gamma $ . I can now introduce the objective function that measures the flatness of the image as:

$\displaystyle {J}\left( {\boldsymbol \mu}_{\vec x}\left({\bf {s}}\right) \right...
...u}_{\vec x}\left({\bf {s}}\right) \right]{\bf\bar I_{\gamma }} \right\Vert^2_2,$ (A-4)

where $ {\bf {s}}$ is the slowness vector. This objective function is not a direct function of $ {\bf {s}}$ , but it depends on it indirectly through the moveout parameters $ {\boldsymbol \mu}_{\vec x}$ . The dependency of the moveout parameters from the slowness function is not defined explicitly; these parameters are defined as the solutions of $ N_{\vec x}$ independent fitting problems, one for each spatial location in the image.

The fitting problems maximize the zero lag of the cross-correlation between the prestack image computed for a realization of the slowness vector $ {\bf {s}}$ and the moved-out image computed with the background slowness $ {\bar {\bf {s}}}$ . For the sake of keeping the notation as compact as possible, I combine the $ N_{\vec x}$ independent fitting problems into one by defining the following objective function:

$\displaystyle {J_{\rm F}}\left( {\boldsymbol \mu}_{\vec x}\left({\bf {s}}\right...
...I_{\gamma }} , {\bf I}_{\gamma }\left({\bf {s}}\right) \right\rangle_{\gamma },$ (A-5)

where with the notation $ \langle {\bf x}, {\bf y}\rangle_{\gamma }$ I indicate the ensemble of inner products between the image vectors $ {\bf x}$ and $ {\bf y}$ which spans only the aperture-angle axis $ \gamma $ ; the result of these inner products is a vector of length $ N_{\vec x}$ . The stacking operator $ {\bf S}_{\vec x}$ sums the elements of this vector to combine the objective functions into one.

The vector of moveout parameters is therefore the solutions of the following maximization problem:

$\displaystyle \max_{{\boldsymbol \mu}_{\vec x}} {J_{\rm F}}\left( {\boldsymbol \mu}_{\vec x}\left({\bf {s}}\right) \right) .$ (A-6)

For velocity estimation in the angle domain, an effective parametrization of the moveout is the "curvature" $ {\mu_{\vec x}}$ , that defines the following moveout equation

$\displaystyle {\zeta}\left({\mu}_{\overline{x}}\right) ={\mu_{\vec x}}\tan^2 \gamma .$ (A-7)

Notice that when the slowness $ {\bf {s}}$ is equal to the background slowness $ {\bar {\bf {s}}}$ , the corresponding best-fitting moveout parameters $ {\bar{{\mu}}}_{\vec x}$ are obviously the ones corresponding to no moveout; that is, $ {
{\zeta}\left({\bar{{\mu}}}_{\vec x}\right)
=0}$ , and consequently $ {\bar{{\mu}}}_{\vec x}=0$ .

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Next: Gradient of the objective Up: Biondi: Wave-equation MVA Previous: Introduction