Depth Limits in Body Wave Inversions

The Wiechert-Herglotz integral provides a direct means of calculating a seismic velocity-depth (v-d) curve from a continuous ray parameter-distance (p-Δ) curve. There is a straightforward method by which this approach can be extended to determine a velocity-depth envelope for any set of T - Δ or p - Δ data. The method requires the construction, in the p-Δ plane, of an envelope around the data. This envelope is defined by the scatter of the data points, the uncertainties in the measurement of the points, and various assumptions regarding the behaviour of the curve. One such assumption might be whether or not all arrivals have been detected at some distance. Within this envelope, two paths of integration are defined such that (a) their integrals over p give satisfactory travel times; and (b) the depth increments determined by the two paths are extremal. Thus there is a closed form solution for the velocity depth envelope given p-Δ and T - Δ envelopes.