implementation of depth-focusing

Estimating whether the scattering wavefield is focused or not is difficult for simultaneously imaging primaries and multiples with depth-focusing. The wavefield extrapolation is carried out in the depth domain, and picking the image amplitude must be implemented in the time domain, since the traveltime from the source to the scattering point is not necessarily known. Assuming that the macro velocity model is reliable, the horizontal positions of the focused scattering points are correct. The wavefield extrapolation depth determines the focused depth, which is also correct under the assumption. The obvious method is to use the amplitude of the focused scattering wavefield. When a scattered wavefield is focused, the amplitude at the focused point is maximized. During the process of wavefield extrapolation, the amplitude of the wavefield at every point fluctuates. Therefore, the amplitude itself can not be used as an indication. Other attributes should be used, such as the envelope of the amplitudes, the derivative of the envelope, and so on. Hence, several extrapolated wavefields should be saved, including the current extrapolated layer and its adjacent layers. This helps to avoid picking the wrong focused amplitude.

Another method is to estimate the radius of curvature of the wavefront of the scattering wave. MacKay and Abma (1993) present a method that, in the CMP geometry, uses the following formula:  

$$R \approx \frac{\left(X^{2}-\triangle t^{2}V_{r}^{2}\right)}{2\triangle t V_{r}},$$ (1)

where X is the offset, V_r is the medium velocity, and $\triangle t$ is the time difference between the two-way vertical traveltime and the observed traveltime. However, this formula is not suitable here, because the time difference is unknown. For depth-focusing imaging, the source position is not a concern, and the traveltime between the source and the scattering point is not explicitly used. I propose the following method to estimate the radius of curvature of the scattered wavefield. Assuming that the macro velocity is correct, and with the help of ray-tracing, the radius of curvature of the scattered wavefield can be estimated with the following formula:

 

R = V_r t_scatter, (2)

where t_scatter is the traveltime from the scatterer to the receivers, V_r is the medium velocity, and $\triangle t= t-t_{s}=t_{scatter}$, where t is the observed two-way traveltime and t_s is the traveltime from the source to the scatter point. t_s may include the traveltime of the multiples. According to equation 2, the radius of curvature of the scattered wavefield can be estimated with the extrapolated wavefield. Some ideas in Jager et al. (2001) suggest how to estimate the radius of curvature of the scattered wavefield.