ON INTERPOLATION FUNCTIONS IN TRAVELTIME TOMOGRAPHY

Abstract

Grid representation of 3-D media and linear interpolation functions have been widely used in recent seismic traveltime tomographic studies. Either velocity or slowness is used as a model parameter in such studies, which are by definition mutually reciprocal. Field representations in grid-based models are, however, mutually inconsistent when velocity and slowness respectively are used, as widely used interpolation functions do not hold the reciprocal relationship between velocity and slowness fields. As a consequence these two field representations lead to two tomographic results from one set of data, and the difference is systematic and cannot be disregarded. We propose a criterion for evaluating this systematic difference in the tomographic results. We first derive a general analytical expression for interpolation functions for traveltime tomographic studies. This interpolation scheme uses norm estimation in distance measurement, and equates grid representation with block representation in the limit. The systematic difference in the two representations could be avoided by carefully formulating the tomographic problem.