Determination of source and receiver coordinates by stationary phase method

Once the residual depth moveouts and the reflector positions are measured, they will be used to update the velocity. The residual depth moveouts should be backprojected along the ray paths that are specular to the reflector points. But how do we determine the ray paths? To compute the ray paths, the source and receiver positions are needed, and can be obtained by a stationary phase method during the migration process. According to the stationary phase theory, the source and receiver coordinates of the specular ray can be determined by implementing the following weighted integration in the migration:

$$x_s(x)=\frac{\int x_s(\xi)w(x,\xi)\dot{U}(\tau_s+\tau_g)d\xi} {\int w(x,\xi)\dot{U}(\tau_s+\tau_g)d\xi},$$ (11.1)

and

$$x_g(x)=\frac{\int x_g(\xi)w(x,\xi)\dot{U}(\tau_s+\tau_g)d\xi} {\int w(x,\xi)\dot{U}(\tau_s+\tau_g)d\xi},$$ (11.2)

where $x_s(\xi)~,~x_g(\xi)$ are the trace source and receiver coordinates and x_s(x) , x_g(x) are the stationary source and receiver coordinates for the image point x; $w(x,\xi)$ is the migration weight function and $\dot{U}(t)$ is the time derivated data.

I use this method to obtain the stationary source and receiver coordinates in the prestack migration of the overthrust data with an inaccurate migration velocity. Figure a is a common image gather for the CDP point at 13,100 m. Figures b and c show the corresponding source and receiver coordinates obtained with the stationary phase method. To verify the accuracy of the estimated source and receiver coordinates, the source coordinate is subtracted from the receiver coordinate for each image point. The difference between the source coordinate and the receiver coordinate should be about equal to the offset. Figure d shows the consistency.

  

Figure: (a). The common image gather for the CDP point at 13,100 m; (b). the source coordinates of the stationary points; (c). the receiver coordinates of the stationary points and (d). the offset of the stationary points.