The computational procedure for a new method of computing high-frequency body-wave seismograms for a spherically symmetric earth model is presented. The seismograms are accurate provided that the WKBJ approximation can be used for the transformed response and the asymptotic form of the Legendre function is applicable. The method remains valid when geometrical ray theory breaks down at caustics and for head waves. The computation of the seismograms is extremely easy and rapid as both inverse transforms are evaluated analytically. Seismograms are presented to show how, for a discrete time series, the method is insensitive to small features in the model. The method is illustrated for a reflected signal plus a head wave and for turning rays in three very different upper mantle models.