Velocity analysis using $AB$ semblance

Appendix B: Automatic velocity picking from semblance scans

The problem of automatic picking of velocities from semblance scans has been considered by many authors (Adler and Brandwood, 1999; Arnaud et al., 2004; Siliqi et al., 2003). The approach taken in this paper is inspired by the suggestion of Harlan (2001) to look at velocity picking as a variational problem. According to Harlan (2001), an optimally picked velocity trend $v(t)$ in the semblance field $\alpha(t,v)$ corresponds to the maximum of the variational integral

$$P_1[v(t)] = \int\limits_{t_{min}}^{t_{max}} \alpha\left(t,v(t)\right)\,d t\;.$$ (13)

I take the variational formulation further by considering its analogy to the ray tracing problem. The first-arrival seismic ray is a trajectory corresponding to the minimum traveltime. The trajectory corresponding to an optimal velocity trend should minimize an analogous measure defined in the space of the velocity scan $\{t,v\}$ . I use the variational measure

$$P_2[v(t)] = \int\limits_{t_{min}}^{t_{max}} \exp[-\alpha\left(t,v(t)\right)]\,\sqrt{\lambda^2+\left[v'(t)\right]^2}\,d t\;.$$ (14)

where $\lambda$ is a scaling parameter. According to variational theory (Lanczos, 1966), an optimal trajectory can be determined by solving the eikonal equation

$$\left(\frac{\partial T}{\partial v}\right)^2 + \frac{1}{\lambda^2}\,\left(\frac{\partial T}{\partial t}\right)^2 = \exp[-2\,\alpha\left(t,v\right)]$$ (15)

with a finite-difference algorithm. The quantity in the right hand side of equation (B-3) plays the role of squared slowness. Small slowness corresponds to high semblance and attracts ray trajectories in a ``wave guide''. After obtaining a finite-difference solution, the picking trajectory can be extracted by tracking backward along the traveltime gradient direction. An analogous approach has been used in medical imaging in the method of virtual endoscopy (Deschamps and Cohen, 2001). To remove random oscillations, I smooth the picked trajectory using the method of shaping regularization (Fomel, 2007).

Velocity analysis using $AB$ semblance