Moveout-based wave-equation migration velocity analysis

from slowness perturbation to the change in image kinematics

In this section we present the formula that links the slowness perturbation to the shifts of the ADCIGs. Starting from the initial slowness model $s_0(z,x)$ , we first define the pre-stack common-image gather in the angle domain as $I(z,\gamma,x;s_0)$ , where $\gamma$ is the reflection angle. If we choose a different slowness $s(z,x)$ , the new image $I(z,\gamma,x;s)$ will be different from $I(z,\gamma,x;s_0)$ in terms of both kinematics and amplitude. If, as is commonly done, we focus on the kinematic change, then a way to characterize this kinematic change is to define a shift parameter $b$ at each image location, $b(\gamma,z,x)$ , such that if we apply this shift parameter to the initial image, the resulting image $I(z+b,\gamma,x;s_0)$ will agree with the new image $I(z,\gamma,x;s)$ in terms of kinematics. This is indicated by the maximum point of the auxilary objective function:

$$J_{aux}(b) = \int_{-L/2}^{L/2} dz_w \int d\gamma \, I(z+z_w+b,\gamma;z,x,s_0) I(z+z_w,\gamma;z,x,s) \;\;$$ (1)

Note that in order to handle multiple events, we use a local window of length $L$ along the depth axis. For the rest of the paper, the integration bound for variable $z_w$ is always $[-L/2,L/2]$ , and each $J_{aux}$ is defined within that window around image point (x,z). $I(z+z_w,\gamma;z,x) = I(z+z_w,\gamma,x)$ represents a windowed version of the entire image.

This methodology is borrowed from Luo and Schuster (1991) who tried to find the relation of the travel-time perturbation to the slowness change. Then $\frac{\partial{b}}{\partial{s}}$ can be found using the rule of partial derivatvies for implicit functions (please refer to the appendix A):

$$\frac{\partial{b}}{\partial s} = -\frac{ \int dz_w \, \dot{I}(z+z... ...ma;z,x,s_0)\frac{\partial{I(z+z_w,\gamma;z,x,s)}}{\partial{s}}}{E(\gamma,z,x)},$$ (2)

in which $E(\gamma,z,x) = \int dz_w \ddot{I}(z+z_w+b,\gamma;z,x,s_0)I(z+z_w,\gamma;x,s)$ . $\dot{I}$ and $\ddot{I}$ indicate the first and second derivatives in $z$ (depth). In practice, eq. (2) will be greatly simplified if we evaluate this expression at $s=s_0$ , in other words $b=0$ . In fact, this will always be the case if we update the intial slowness $s_0$ after each iteration. The simplified relation becomes

$$\frac{\partial{b}}{\partial s}\vert _{s=s_0} = -\frac{ \int dz_w ... ...ma;z,x,s_0)\frac{\partial{I(z+z_w,\gamma;z,x,s)}}{\partial{s}}}{E(\gamma,z,x)},$$ (3)

and the $\frac{\partial{I(z+z_w,\gamma;z,x,s)}}{\partial{s}}$ term is indeed the wave-equation image-space tomographic operator. Each part in eq. (3) has clear physical implications: the $E$ term acts as an energy term to normalize the amplitude of the back-projected image; the back-projected image, $\dot{I}(z+z_w,\gamma;z,x,s_0)$ is built based on the initial image; it also has a first-order $z$ derivative that introduces a proper $90^\circ$ phase shift, ensuring a well behaved slowness update from the tomographic operator.

sensKer1w.zref,sensKer1w.Tilt.zref
Figure 1. Slowness sensitivity kernel at incident angle $\gamma =30^{\circ }$ for a flat reflector (a) and a dipping reflector (b).

For a simple illustration of eq. (3), the slowness sensitivity kernel is calculated, by back-projecting a shift perturbation $\Delta b(\gamma,x)$ that has one single spike at $\gamma = 30^{\circ}, x=0$ . A uniform background velocity of 2000 m/s is used. Figure 1(a) shows the sensitivity kernel if the reflector is flat, and figure 1(b) shows the sensitivity kernel with a dipping reflector (dip angle = $20^{\circ}$ ). As is clearly shown in these two plots, this operator will project the slowness perturbation along the corresponding wave path based on the location and reflection angle of the image shift.

Moveout-based wave-equation migration velocity analysis