New insights into the seismic time term method for heterogeneous upper mantle slowness structures

The seismic time term method of Gardner (Geophysics 4:247–259, 1939 ) has become popular in the context of seismic P_n studies, since it provides a possibility to estimate not only the Earth’s crustal thickness, but also the P -wave slowness of the uppermost mantle. In the last decades an extended form of this Time Term Method has been extensively used in order to predict a heterogeneous isotropic or anisotropic upper mantle slowness function. One of the main goals of this study is to formalize the mathematical construction for which the Time Term Equations are indeed linearized versions of the Travel Time Equations for such a complex laterally varying slowness function. For this purpose, an alternative definition of Time Terms is given that allows to consider them as parts of the exact travel time equation. In the case of constant upper mantle slowness functions these terms coincide with the classical Time Terms of Gardner. As a consequence, an alternative derivation of the extended Time Term Method can be given for heterogeneous (isotropic) upper mantle slowness functions.