General reflection geometry

To understand how velocity errors influence the results of shot-profile migration, we consider the simple case of a reflector of dipping angle $\theta$, located at (x_t,z_t) of a homogeneous medium of velocity v_t. As shown in Figure 2, a shot is at surface location x_s and a receiver is on the surface with coordinate x=x_r. We define $\alpha$ to be the angle of the incident ray to the vertical and $\beta$ to be the angle of the reflected ray to the vertical. Suppose we migrate shot records with velocity v_m, and we want to determine the location of the image of the reflector on the migrated sections. For a fixed shot location, the travel time $\tau(x,z)$ of a reflection event is a function of the location of observation.

Now we apply Goldin's three-step method. The first step is to find the travel time of the reflection event observed by a receiver at x=x_r on the earth's surface.  

$$\tau_0(x_r) = \tau(x_r,0) = {z_t \over v_t}\left({1 \over \cos \alpha}+ {1 \over \cos \beta}\right).$$ (1)

where

$$\cos \alpha = {z_t \over \sqrt{(x_t-x_s)^2+z^2_t}}, \ \ \ \ \cos \beta = {z_t \over \sqrt{(x_r-x_t)^2+z^2_t}}.$$

The next step is to find $\tau(x,z)$ by solving Eiconal equation

$$\left({\partial \tau \over \partial x}\right)^2+ \left({\partial \tau \over \partial z}\right)^2={1 \over v^2_m}$$ (2)

with equation (1) as the boundary condition. In a case of homogeneous media, the problem can be solved analytically.  

$$\tau(x,z)=\tau(x_r,0)-{1 \over v_m}\sqrt{(x-x_r)^2+z^2}.$$ (3)

It can be shown that (Goldin, 1982)  

$$\begin{array} {lll} x & = & x_r-\lambda v_m \tau_x \\ z & = & \lambda \sqrt{1-v^2_m \tau^2_x},\end{array}$$ (4)

where

$$\tau_x(x,z)={\sin \beta \over v_t}$$

and $\lambda$ is a parameter to be determined.

The last step is imaging. Let the travel time of the reflection event observed at point (x,z) be equal to the travel time from the shot to this point.  

$$\tau(x,z)={1 \over v_m} \sqrt{(x-x_s)^2+z^2}.$$ (5)

Combining equations (1), (3) and (5), we have relation  

$${1 \over v_m}\sqrt{(x-x_s)^2+z^2} = {z_t \over v_t} \left({1... ...{1 \over \cos \beta}\right) -{1 \over v_m}\sqrt{(x-x_r)^2+z^2}.$$ (6)

To determine the $\lambda$ in equation (4), we replace x and z in equation (6) with equation (4). In Appendix A, I show that  

$$\lambda ={z_t \over \gamma \cos \beta}[1+(\gamma^2-1) {\cos^2 ({1 \over 2}(\beta -\alpha)) \over \cos \alpha \cos \beta }]$$ (7)

and  

$$\begin{array} {lll} x & = & x_t + z_t \tan \beta - \lambda \... ...ta \\ z & = & \lambda \sqrt{1-\gamma^2 \sin^2 \beta}\end{array}$$ (8)

where

$$\gamma = {v_m \over v_t}.$$

Thus, for given location (x_t,z_t) of a reflector and the relationship between $\alpha$ and $\beta$, we can compute the location (x,z) of the migrated image.

We consider, as an example, a plane reflector of dipping angle $\theta$, 

$$z_t=z_0+x_t \tan \theta,$$ (9)

that is illuminated by a shot at x=x_s. The relation between $\alpha$ and $\beta$ follows Snell's law,  

$$\phi = \beta - \theta =\alpha + \theta$$ (10)

where $\phi$ is the open angle between the incident (or reflected) ray and the normal of the reflector. Substituting this relation and equation (9) into equations (7) and (8) offers us formulas to compute the coordinates of the migrated image. In Figure 3 are plotted the migrated images for two different $\gamma$ values. When $\gamma < 1$, the migrated image is stretched out and curves upwards; when $\gamma \gt 1$, the migrated image shrinks and curves downwards. In Figure 3 is also drawn a set of migration ellipses. As expected, the migrated images are the envelopes of those migration ellipses.

 

envbed
Figure 3 When the reflected energy from a plane reflector in a homogeneous medium is migrated with a velocity of (a) -10% error; (b) +10% error, the image of the reflector, defined by the envelope of the migration ellipses, is distorted from the actual image of the reflector. The dipping angle of the reflector is $15^\circ$.