A new analytical method for the determination of velocity at the hypocenter of a deep earthquake has been developed making use of P- and S-wave travel times. Unlike Gutenberg's method which is graphical in nature, the present method makes use of the least square technique and as such it yields more quantitative estimates of the velocities at depth. The essential features of this method are the determination from the travel times of a deep-focus earthquake the lower and upper limits Δ1 and Δ2 respectively of the epicentral distance between which p = (dT/dΔ) in the neighborhood of inflection point can be considered stationary so that the travel-time curve there can be approximated to a straight line T = pΔ + a. From p = (1/v*) determined from the straight line least-square fit made on the travel-time observation points between Δ1 and Δ2 for various focal depths, upper-mantle velocity structure can be obtained by making use of the well known relation v = v*(r0 − h)/r0, h being the focal depth of the earthquake, r0 the radius of the Earth, v* the apparent velocity at the point of inflection and v the true velocity at that depth.
This method not only gives an accurate estimate of p, at the same time it also yields quite accurate value of a which is a function of focal depth. Calibration curves can be drawn between a and the focal depth h for various regions of the Earth where deep focus earthquakes occur, and these calibration curves can then be used with advantage to determine the focal depths of deep earthquakes in those areas.