next up previous [pdf]

Next: Gaussian anomaly example Up: 3-D RMO WEMVA method Previous: 2-D Theory Review

Extension to 3-D

As we shift from 2-D seismic surveys to 3-D ones, two extra dimensions (cross-line axis $ y$ and subsurface azimuth $ \phi$ ) are added to our seismic image. Therefore we denote the prestack image as $ I(z,\gamma,\phi,x,y)$ . Assuming there are $ m$ values for $ \phi$ , then $ \phi \in \{\phi_i:i=1,2,...,m\}$ . Now the maximum-stack-power objective function 1 can be generalized to

$\displaystyle J(s) = \frac{1}{2} \sum_{x,y}\sum_{z} {\left[\sum_{\phi_i} \sum_{\gamma} \; I(z,\gamma,\phi_i,x,y;s) \right]}^2,$ (11)

in which we stack the gather along both $ \gamma$ and $ \phi$ axes.

The next step is to choose a proper residual moveout parameterization for the 3-D ADCIGs, in which the moveout is a surface (defined by $ (\gamma ,\phi )$ ) rather than a curve (of $ \gamma$ ). There are certainly more than one way to design such parameterization. As an initial attempt, we choose a straightforward approach, in which we separate the moveout surface into individual curves by azimuth $ \phi$ . For each azimuth angle $ \phi_i$ , we assign the curvature parameter $ \rho_i$ and the static shift parameter $ \beta$ , respectively. Notice that all the curves share the same $ \beta$ parameter, because the center of the move-out surface at ( $ \gamma=0,\forall\phi_i$ ) is shared by all curves.

Under this parameterization, the 3-D counterpart of objective function 3 would be

$\displaystyle J_{Sm}(\rho(s)) = \frac{1}{2} \sum_{x,y} \sum_{z} \sum_{\phi_i}\f...
...um_{z_w} \sum_{\gamma} I^2(z+z_w+\rho_i\tan^2{\gamma},\gamma,\phi_i,x,y,;s_0)},$ (12)

where $ \mathbf{\rho}$ becomes a vector, $ \rho = \{\rho_i:i=1,2,...,m\}$ . The gradient formula 4 now turns into

$\displaystyle \frac{\partial{J_{S_{m}}}}{\partial{s}} = \sum_{\rho_i}\frac{\partial{J_{S_{m}}}}{\partial{\rho_i}} \frac{\partial{\rho_i}}{\partial{s}}.$ (13)

Because each $ \phi_i$ is treated separately, we can compute $ {\partial{J_{Sm}}}/{\partial{\rho_i}}$ in exactly the same way as we do in the 2-D case.

Analogously, we can define an auxiliary objective function for each image point that uncovers the $ s \leftrightarrow \rho$ relationship:

$\displaystyle J_{\mathrm{aux}} = \sum_{z_w} \sum_{\gamma}\sum_{\phi_i} \, I(z+z...
...(\rho_i \tan^2{\gamma} + \beta),\gamma,\phi_i,x,y;s_0)I(z,\gamma,\phi_i,x,y;s).$ (14)

Using the same trick of finding partial derivatives for implicit functions, equation 6 is generalized as

$\displaystyle \renewedcommand{arraystretch}{1.5} \left\{ \begin{array}{c} \frac...
...\ \frac{\partial{J_{\mathrm{aux}}}}{\partial{\beta}} = 0 \end{array} \, \right.$ (15)

We differentiate equation 15 with respect to $ s$ :

$\displaystyle \renewedcommand{arraystretch}{1.5} \begin{bmatrix}\frac{\partial^...
... \frac{\partial^2{J_{\mathrm{aux}}}}{\partial{\beta}\partial{s}} \end{bmatrix}.$ (16)

We can calculate the $ m+1$ by $ m+1$ Jacobian matrix elements and the right-hand side based on equation 14:

$\displaystyle \frac{\partial^2{J_{\mathrm{aux}}}}{\partial{\rho_i}^2} =$ $\displaystyle \sum_{z_w} \sum_{\gamma} \, \ddot{I}(z+z_w,\gamma,\phi_i,x,y;s_0) \tan^4{\gamma} I(z+z_w,\gamma,\phi_i,x,y;s)$    
$\displaystyle \frac{\partial^2{J_{\mathrm{aux}}}}{\partial{\rho_i}\partial{\rho_j}} \equiv$ $\displaystyle \; 0 \qquad \qquad \qquad \forall \quad i \ne j$    
$\displaystyle \frac{\partial^2{J_{\mathrm{aux}}}}{\partial{\rho_i}\partial{\beta}} =$ $\displaystyle \sum_{z_w} \sum_{\gamma} \, \ddot{I}(z+z_w,\gamma,\phi_i,x,y;s_0) \tan^2{\gamma} I(z+z_w,\gamma,\phi_i,x,y;s)$    
$\displaystyle \frac{\partial^2{J_{\mathrm{aux}}}}{\partial{\beta}^2} =$ $\displaystyle \sum_{z_w} \sum_{\gamma} \sum_{\phi_i} \, \ddot{I}(z+z_w,\gamma,\phi_i,x,y;s_0) I(z+z_w,\gamma,\phi_i,x,y;s)$    
$\displaystyle \frac{\partial^2{J_{\mathrm{aux}}}}{\partial{\rho_i}\partial{s}} =$ $\displaystyle \sum_{z_w} \sum_{\gamma} \, \dot{I}(z+z_w,\gamma,\phi_i,x,y;s_0) \tan^2{\gamma} \frac{\partial{I(z+z_w,\gamma,\phi_i,x,y;s)}}{\partial{s}}$    
$\displaystyle \frac{\partial^2{J_{\mathrm{aux}}}}{\partial{\beta}\partial{s}} =$ $\displaystyle \sum_{z_w} \sum_{\gamma} \sum_{\phi_i}\, \dot{I}(z+z_w,\gamma,\phi_i,x,y;s_0) \frac{\partial{I(z+z_w,\gamma,\phi_i,x,y;s)}}{\partial{s}}.$ (17)

Denoting matrix $ \mathbf{F}=\{F_{i,j}\}$ to be the inverse of the Jacobian, then

$\displaystyle \renewedcommand{arraystretch}{1.5} \begin{bmatrix}\frac{\partial{...
... \frac{\partial^2{J_{\mathrm{aux}}}}{\partial{\beta}\partial{s}} \end{bmatrix}.$ (18)

Finally, plugging equation 18 and 17 back into the model gradient expression 13, we get

$\displaystyle \frac{\partial{J}}{\partial{s}} = -\sum_{z_w} \sum_{\gamma} \sum_...
...partial{s}} (G_{i}\tan^2{\gamma}+G_{m+1}) \dot{I}(z+z_w,\gamma,\phi_i,x,y;s_0),$ (19)

in which

$\displaystyle \begin{bmatrix}G_1 & G_2 & \cdots & G_m & G_{m+1} \end{bmatrix} =...
...ts & \vdots \\ F_{m,1} & F_{m,2} & \cdots & F_{m,m} & F_{m,m+1} \end{bmatrix} .$ (20)

In practice, there are some caveats when taking the inverse of the Jacobian matrix. The Jacobian can be ill-conditioned when all elements in one row or column are close to zero. For example, if the image point is not illuminated from a certain azimuth direction $ \phi_j$ , i.e. $ I(z,\gamma,\phi_j,x,y) = 0$ , then the $ j_{th}$ row and column of the Jacobian would be zero. In order to avoid numerical overflow under this circumstance, we pre-exclude those azimuth angles with poor illumination energy from the Jacobian, and we invert a subset of the original Jacobian that contains only image gathers at well-illuminated azimuth angles.

next up previous [pdf]

Next: Gaussian anomaly example Up: 3-D RMO WEMVA method Previous: 2-D Theory Review