The Effect of Small, Aspherical Perturbations on Travel Times and a Re-examination of the Corrections for Ellipticity

Summary We use the stationarity of the Fermat ray path to develop theoretical expressions that relate small, aspherical perturbations in velocity to small perturbations in travel times. For the Earth's ellipticity of figure we derive a compact expression for the perturbation in travel time. If the ellipticity is hydrostatic and is the dominant perturbation, then anomalies in travel times are linear constraints on the radial gradient of velocity. Otherwise, the anomalies are constraints on the (unknown) aspherical perturbations. We are led to an inverse problem in either case. Using recently derived models of the Earth we present calculations of the effect of ellipticity. If the effect of focal depth is neglected in the calculations, then mislocations in both epicentre and origin time can result. Differences between our calculations and those predicted by the approximate formula 6t = (h+H)f(A) are as large as 0-25 s for P at 90". Nowadays, seismologists are prone to attach significance to anomalies as small as 0.10 s. Consequently, we advocate that the effect of focal depth be considered and that traditional approximations be replaced by the more accurate calculations tabulated in this report. Theoretical development The effect of the Earth's ellipticity of figure on travel times was first studied more than 40 years ago by Comrie and by Gutenberg and Richter. Bullen (1965, p. 173176) has presented a concise theoretical derivation of the effect. Recent improvements in our knowledge of the mechanical structure of the Earth make it desirable not only to re-examine the effect of ellipticity, but also to consider other small, aspherical perturbations as well. We follow the derivation of Bullen A source is located a (r,,, ,!lo, &) in spherical polar co-ordinates, and a receiver at