Two problems in WKBJ theory: the interpolation of sampled velocity profiles and the use of frequency‐dependent, complex velocities

Summary Phase‐integral (WKBJ) methods are widely used in seismology and other branches of physics. Both uniform and non‐uniform methods require that integrals such as be computed along a contour in the complex p ‐plane. When velocity is specified at isolated depths {z i i = 1, N } any method of computing these integrals makes use, either explicitly or implicitly, of some interpolation scheme, τ and the turning point z p are well defined by such a scheme only if the p ‐contour passes through the points 1/v i . Such a contour forces the turning point z p to intersect the Re(z)‐axis at those values of z for which velocity is specified. Before passing through 1/v N and after passing through 1/v 1 the p‐contour has only the usual constraint of being such that the motion integrals are convergent. In models for which Q varies with depth the effects of attenuation can be correctly calculated only by making velocity a complex function of frequency as well as z. Then the points 1/v i , through which the p‐contour must pass, are frequency‐dependent and so a slightly different p‐contour must be used for each frequency. These results are applied to the spectral WKBJ method where a correct treatment of sampled velocity functions at first appears to complicate the theory but is then seen to simplify it. The restrictions on the p‐contour are easily satisfied because the points through which it must pass are always the reciprocals of known velocities; also, if the generalized Filon method of quadrature is used for the p‐integral then there is no need to deform the contour over the saddle points of the integrand. The tunnelling problem is more straightforward with a sampled velocity function than with an analytic velocity function because a sampled velocity with a low‐velocity zone always divides naturally into three zones in which velocity is mono‐tonic; in each zone the solution is a sum of well‐understood functions with a single turning point. The problem of reflection at a discontinuity is simplified because, unlike an analytic velocity function, a sampled velocity function can always be made to remain monotonic when analytically continued across a discontinuity.