Numerical solutions of p_h

To implement prestack phase-shift migration efficiently in v(z) media, we need to construct a table of $\tau (p_x,t)$ using ray tracing, where $\tau$ is the vertical two-way traveltime and t is the two-way zero-offset traveltime. Due to the lateral homogeneity assumption, this table is constructed once and is applicable everywhere. The stationary point is evaluated by finding  

$${\rm max} \left\{ T \right\},$$ (17)

where  

$$T = 0.5 [\tau(p_x+p_h,t) + \tau(p_x-p_h,t)] + 2 p_h h,$$ (18)

and

$$\tau(p_x,t) = \int_0^t p_{\tau}(p_x,\zeta) d\zeta,$$

where $p_{\tau}$ is the zero-offset normalized dispersion factor, shown just above equation (8).

To find the maximum of equation (18), I use a method described by Buchanan and Turner (1992). This is a robust algorithm used to find a maximum or minimum of a function numerically without the need to resort to evaluating derivatives (similar in principal to the bisection method used to evaluate roots of functions). Other methods, based on the finding the roots of the derivative of equation (18) with respect to p_h, will most likely fail due to the sensitivity of these derivatives to typical numerical errors associated with ray tracing. Another table, $\tilde{T}(p_x,t)$, consisting of T for the solutions of equation (17) for given p_x, time, and offset is, therefore, constructed. Again, such a table is applicable everywhere in the medium due to the lateral homogeneity assumption considered here. The cost of a prestack migration of any offset is about 10$\%$higher than that for a zero-offset algorithm, where p_h=0 is known, and thus we do not need to evaluate it. This relatively small additional cost reduces even further when large data sets are migrated and the cost of precomputing the traveltime tables become insignificant.