Inversion of seismic travel times using the Tau method

Abstract

Summary A powerful technique of solving the inverse problem of seismology, the Tau method, has been described recently by Bessonova et al. Limits of the function τ(p) = T(p) - pX(p), where p is the ray parameter, T the travel time and X the epicentral distance are mapped into limits in the velocity–depth plane. The function τ(p) is estimated from observed times and distances of body wave data using the fact that τ(p) is the singular solution of Clairaut's equation with free term T(X). A new method of inverting seismic data using the function τ(p) has been developed in which τ(p) limits are assumed to be piecewise second order polynomials in p. As errors in travel time—distance data become large, interpretation of the Clairaut equation, for the purpose of τ(p) estimation, may become ambiguous. The alternate procedure for τ(p) estimation is as follows: for each branch of the travel-time curve, T observations are fitted to a family of second order polynomials in X. The families of curves are then mapped into the T@) plane. The Tau method is illustrated by inverting T(X) data recorded by the University of Alberta on Project Early Rise. The resolving power, as a function of separation of observation points of the Tau method is examined by inverting τ(p) envelopes calculated from exact velocity—depth functions.