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Introducing the residual moveout parameter

The previous section showed how the slowness change can be associated with the kinematic change of the image. The remaining task is to determine how the image's kinematics should change by evaluating the objective function. Ideally we would like to know the true depth of the reflector, so that we can guarantee that the correct shift direction of the image is extracted. Unfortunately, we usually do not have the true depth of the reflector, and have to find the shift direction by relying only on the flatness criterion. A first attempt that directly maximizes the angle stack after shifting the initial image would be

$\displaystyle \bold J(s) = \frac{1}{2} \sum_{x}\sum_{z} \int dz_w {\left[ \int d\gamma \, I(z+z_w+b,\gamma,z,x;s_0) \right]}^2 .$ (4)

The corresponding derivative over $ b$ for a fixed $ (\gamma,z,x)$ is

$\displaystyle \frac{\partial{J}}{\partial{b(\gamma,z,x)}}\vert _{b=0} = \frac{1...
...gamma \, I(z+z_w,\gamma,z,x;s_0) \right] \dot{I}(z+z_w,\gamma,z,x;s_0) \Big\} .$ (5)

Apparently, this objective function is very susceptible to the cycle-skipping problem: for a fixed $ (\gamma,z,x)$ , if the image $ I(z+z_w,\gamma,z,x;s_0)$ and the angle-averaged image $ \int d\gamma \, I(z+z_w,\gamma,z,x;s_0)$ become out of phase, the derivative of eq. (5) will point to the wrong shift direction.

To prevent cycle-skipping, we need a way to detect the global shape of the ADCIGs. Almomin (2011) propose to measure the relative shift the traces at each angle with respect to some reference trace by picking cross-correlation peak. Here we use the residual moveout (RMO) parameters so that the objective function knows whether the angle gather is curving up or curving down.

As shown in Biondi (2003) Chap 11, in the case of constant velocity error, the residual moveout of an ADCIG gather is

$\displaystyle \theta(\gamma) = z_{\rho 0} \frac{\rho-1}{\cos{\alpha_x}} \frac{\sin^{2}{\gamma}}{\cos^2{\alpha_x}-\sin^2{\gamma}} ,

where $ \rho = s/s0,\; \alpha_x$ and $ \gamma$ are the dip angle and reflection angle respectively, and $ z$ is the true reflector depth, $ z_{\rho 0} = z/\rho$ . If we assume the dip is small, then the expression above can be further simplified to

$\displaystyle \theta(\gamma) = z_{\rho 0}(\rho-1)\tan^2{\gamma}. $

Therefore we introduce the moveout parameter $ \alpha$ and the moveout function $ g(\gamma) = \tan^2{\gamma}$ . The objective function we want to maximize is the angle stack-power of the inital image after applying the residual moveout:

$\displaystyle \bold J = \frac{1}{2} \sum_{x}\sum_{z} \int dz_w {\left[ \int d\gamma \,
I(z+z_w+\alpha g(\gamma),\gamma,z,x;s_0) \right]} ^2 .

The derivative is

$\displaystyle \frac{\partial J}{\partial s} \vert _{s=s_0} =
\sum_{x}\sum_{z} ...,z,x;s_0)g(\gamma)\frac{\partial{\alpha}}{\partial{s}}) \right]
\right\} .

$\displaystyle A(z_w;z,x,s_0) =$   $\displaystyle \int d\gamma \, I(z+z_w,\gamma;z,x,s_0) ,$  
$\displaystyle B(z_w;z,x,s_0) =$   $\displaystyle \int d\gamma \, \dot{I}(z+z_w,\gamma;z,x,s_0)g(\gamma),$  


$\displaystyle \frac{\partial J}{\partial s}$ $\displaystyle = \sum_{x} \sum_{z} \frac{\partial{J}}{\partial{\alpha}} \frac{\partial \alpha}{\partial s}$ (6)
  $\displaystyle = \sum_{x} \sum_{z} \{ \int dz_w A(z_w;z,x,s_0)B(z_w;z,x,s_0) \} \frac{\partial{\alpha}}{\partial{s}}.$    

We find there are two ways to derive the $ \frac{\partial{\alpha}}{\partial{s}}$ relation (see appendix B):
  1. We can link $ \Delta s$ to $ \Delta\alpha$ by defining an auxilary objective function; we call this the direct operator.

  2. we can convert the perturbation of $ \alpha$ to the shift parameter $ b$ perturbation at each angle, through a weighted least-squares fitting formula; thus $ \Delta \alpha \rightarrow \Delta b \rightarrow \Delta s$ ; and as shown previously, we know how to calculate $ \frac{\partial{b}}{\partial{s}}$ . We call this the indirect operator.

The sensitivity kernel $ \frac{\partial{\alpha}}{\partial{s}}$ calculated using the direct operator and the indirect operator are shown in figure 2, as with the Toldi operator (Toldi, 1985), the characteristic shape of such a sensitivity kernel is a center lobe, with two side lobes with oppositite polarity, which reaffirms the well known fact that velocity perturbation at the center and the side-end lateral position will change the curvature of ADCIGs towoard opposite directions. Yet the overall average is positive, which would give the correct update in case of a bulk shift slowness error.

sensKer1w.bf2 sensKer1w.bf1m
Figure 2.
Sensitivity kernel $ \partial {\alpha }/\partial {s}$ calulated using the direct operator (a) and the indirect operator (b), using constant background velocity and a flat reflector.
[pdf] [pdf] [png] [png]

Now if we review this method on eq. (6), the success of this method simply relies on the proper behavior of the two components in eq. (6): $ \frac{\partial{J}}{\partial{\alpha}}$ needs to correctly detect the curvature of the ADCIGs so that the inversion will choose a moveout direction that properly flatten the gathers; $ \frac{\partial{\alpha}}{\partial{s}}$ needs to properly convert the curvature perturbation to the update in slowness space.

In cases where velocity error is big, the actual curvature of the gather may be poorly represented the $ \frac{\partial{J}}{\partial{\alpha}}$ term. To further improve the robustness and precondition the gradient, the analytic expression of $ \frac{\partial{J}}{\partial{\alpha}}$ in eq. (6) is replaced by a numerical approach. First a semblance panel of $ J(\alpha)$ will be calculated. To ensure that the derivative at $ \alpha=0$ can determine the correct curvature that maximizes the semblance value, a Gaussian derivative rather than a simple $ (-1,1)$ finite-difference derivative is applied. The width of the Gaussian can be reduced in later iterations.

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