Kinematic relations

Let us suppose that a data gather P(t,x_r) is recorded in an area that has a subsurface structure Q(x,z) and slowness model s(x,z). If this gather is migrated with a different slowness model $\hat{s}(x,z)$, the resulting image $\hat{Q}(\hat{x},\hat{z})$ will be a distortion of the true image of the subsurfaces.

Residual depth migration is the transformation from image $\hat{Q}(\hat{x},\hat{x})$ to image Q(x,z). The kinematic operator that effects this transformation is defined by the mapping functions between the image locations (x,z) and $(\hat{x},\hat{z})$. In my last report, I showed that these mapping functions are expressed implicitly in the following pair of equations:  

$$\left\{ \begin{array} {l} \tau(x,z;x_s)+\tau(x,z;x_r)=\hat{\... ...\partial x_r}}+ \hat{p}(\hat{x},\hat{z};x_r).\end{array}\right.$$ (1)

where x_s is the surface location of shot and x_r is the surface location of receiver. The terms $\tau$ and $\hat{\tau}$ are traveltimes associated with slowness models s(x,z) and $\hat{s}(\hat{x},\hat{z})$, respectively. The terms p and $\hat{p}$ represent surface horizontal slownesses defined as follows  

$$\left\{ \begin{array} {lllll} p(x,z;x_c) & = & \displaystyle... ...l x_c} & = & \hat{s}(x_c,0) \sin \hat{\alpha}\end{array}\right.$$ (2)

where x_c=x_s or x_r. The angles $\alpha$ and $\hat{\alpha}$ are between ray directions and the vertical at the surface.

The partial derivative of x_s with respect to x_r depends on the type of a data gather. For post-stack data,

$$x_s=x_r\ \ \ \ \Longrightarrow \ \ \ \ {\partial x_s \over \partial x_r} = 1.$$

For pre-stack data with common shot geometry,

$$x_s=\hbox{constant}\ \ \ \ \Longrightarrow \ \ \ \ {\partial x_s \over \partial x_r} = 0,$$

with constant offset geometry,

$$x_s=x_r-\hbox{offset}\ \ \ \ \Longrightarrow \ \ \ \ {\partial x_s \over \partial x_r} = 1.$$

Therefore, equation (1) is generally applicable.

For each image location (x,z) on Q(x,z), equation (1) implicitly tells us the corresponding image location $(\hat{x},\hat{z})$ on $\hat{Q}(\hat{x},\hat{z})$. We want to solve $(\hat{x},\hat{z})$ in this equation as functions of (x,z). For general slowness models, this nonlinear equation set has to be solved using a numerical method. Searching is a straightforward method, in which, for each x_r and (x,z), all points around (x,z) are checked to find the $(\hat{x},\hat{z})$ that satisfies equation (1). However, this algorithm is time-consuming, especially when the image dimensions are large.

I have developed an algorithm that solves equation (1) using finite-difference techniques. The ideas are similar to those used in the finite-difference calculation of traveltimes (Van Trier, 1990). The solution of equation (1) is known on the surface. We can extrapolate this solution in depth once we know the derivative of the solution with respect to the depth.