The parabolic cylinder function in elastodynamic problems
P. F. Daley
When comparing the synthetic response obtained using a zero-order, high-frequency approximation method with that obtained from a highly numerically accurate method, such as the Alekseev-Mikhailenko Method (AMM) (Mikhailenko, 1985), the largest disparities usually appear near critical reflection points of a ray. This difference between the two methods occurs in both the pre- and post-critical offset range near the critical (branch) point. The reason this occurs is that the zero-order, high-frequency method takes only the saddle point contribution corresponding to the reflected arrival and neglects any effects from the branch point which gives rise to the critically refracted (head-wave) arrival. This problematic area has been discussed in several texts on seismology, notably, Brekhovskikh (1960) and Cerveny and Ravindra (1970), where it is shown that higher order approximations for both the reflected arrival and the associated critically refracted arrival, which increases the accuracy of the high frequency solution, may be written in terms of the frequencydependent parabolic cylinder functions (PCF).
Assuming a band-limited source wavelet, computation time is increased, as all frequency points where the spectrum of the wavelet is non-zero should be included. Another obstacle is the numerical computation of the parabolic cylinder function required in the higher order high-frequency solution. The particular PCF related to this problem is of order -3/2 . (In actuality, the PCF of order 1/2 is also required, but may be obtained from the PCF of order -3/2, as they are both linearly dependent solutions of an ordinary differential equation.).