Wave-equation migration velocity analysis by residual moveout fitting

Theory

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):

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

$${\bf I}_{\gamma }= {\bf\Gamma}{\bf I}_{h}.$$ (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

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

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

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

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

$${\zeta}\left({\mu}_{\overline{x}}\right) ={\mu_{\vec x}}\tan^2 \gamma .$$ (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 ${\bar{{\mu}}}_{\vec x}=0$ .

Wave-equation migration velocity analysis by residual moveout fitting