Kinematic relations

Suppose a data gather P(t,x_r) is recorded in an area that has a subsurface structure Q(x,z) and a velocity model v(x,z). Based on the stationary-phase approximations, this gather can be modeled by a Kirchhoff integral:  

$$P(t,x_r)=\int_{x,z}Q(x,z)A(x,z,x_s)A(x,z,x_r)\delta(t-\tau(x,z;x_s)-\tau(x,z;x_r))dxdz$$ (1)

where A(x,z;x_c) and $\tau(x,z;x_c), x_c=x_s$ or x_r, are an amplitude function and a traveltime function respectively. The traveltime function can be found by solving the Eikonal equation:  

$${\tau_x}^2+{\tau_z}^2=v^{-2}(x,z)$$ (2)

with initial condition $\tau(0,x_c,x_c)=0$.

If this gather is migrated with a different velocity model $\hat{v}(x,z)$, the image obtained will be distorted from the true image of the subsurfaces.  

$$\hat{Q}(\hat{x},\hat{z})=\int_{t,x_r}P(t,x_r) \hat{A}(\hat{x... ...u}(\hat{x},\hat{z};x_s)-\hat{\tau}(\hat{x},\hat{z};x_r)) dtdx_r$$ (3)

where $\hat{\tau}(x,z;x_c), x_c=x_s$ or x_r is the traveltime function with the velocity model $\hat{v}(x,z)$.

Residual migration is a transformation from image $\hat{Q}(\hat{x},\hat{x})$to image Q(x,z). Therefore, we want to find out the relationship between (x,z) and $(\hat{x},\hat{z})$ that defines the kinematic operators of this transformation. Substituting P(t,x_r) from equation (1) into equation (3) yields The image of a single scatterer at (x,z) can then be found to be:  

$$\begin{array} {lll} \hat{Q}_0(\hat{x},\hat{z}) & = & \displa... ...\hat{z};x_s) -\hat{\tau}(\hat{x},\hat{z};x_r))dx_r}.\end{array}$$ (4)

It is well known that the kinematics of the summation operator that does full migration is defined by the trajectory of the reflection event from a scatterer. Similarly, the kinematics of the summation operator that does residual migration can be determined from $\hat{Q}_0(\hat{x},\hat{z})$, or more specificly from the argument of the $\delta$-function in equation (4). Let

$$h(x_r)=\tau(x,z;x_s)+\tau(x,z;x_r)-\hat{\tau}(\hat{x},\hat{z};x_s) -\hat{\tau}(\hat{x},\hat{z};x_r),$$

$$G(x_r)=A(x,z;x_s)A(x,z;x_r)\hat{A}(\hat{x},\hat{z};x_s) \hat{A}(\hat{x},\hat{z};x_r).$$

Recall the properties of $\delta$-function: where $h(x^\ast)=0$.Clearly, $\hat{Q}_0(\hat{x},\hat{z})$ has extremal values when

$$\left\{ \begin{array} {l} h(x_r)=0 \\ h^\prime(x_r)=0. \end{array}\right.$$

Therefore the relationship between (x,z) and $(\hat{x},\hat{z})$,is implicitly expressed by a pair of equations:  

$$\left\{ \begin{array} {l} \tau(x,z;x_s)+\tau(x,z;x_r)=\hat{\... ...x_r}}+ \hat{\tau}_{x_r}(\hat{x},\hat{z};x_r).\end{array}\right.$$ (5)

The partial derivative of x_s with respect to x_r is determined by the type of the data gather. For common shot gathers,

$$x_s=\hbox{constant}\ \ \ \ \hbox{and} \ \ \ \ {\partial x_s \over \partial x_r} = 0.$$

For constant offset sections,

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

For each point (x,z), equation (5) defines a curve in $(\hat{x},\hat{z})$. This curve is exactly the kinematics of the residual-migration operator at point (x,z). For general velocity models, the traveltimes must be computed by some numerical methods. These methods generate traveltime tables rather than continuous functions. Therefore, it is natural to solve equation (5) numerically. A straight forward method is searching. For each x_r and (x,z), all points around (x,z) are checked to find the $(\hat{x},\hat{z})$ that satisfies equation (5). But the this algorithm is time-consuming when the dimensions of images are large. Motivated by the results of the finite-difference calculation of traveltimes (Van Trier, 1990), I begin to explore the possibility to calculate residual-migration operators with finite-difference techniques.