Convergence of traveltime power series for a layered VTI medium

Abstract

For a multiply reflected SH-wave or a multiply reflected and converted qP/qSV-wave in a layered VTI medium, traveltime and offset can be expressed by power series in horizontal slowness. As for a stack of isotropic layers, these can be used to express traveltime as a series of even powers of offset. The existence and behavior of this power series depend on the derivative of offset with respect to horizontal slowness, formally expressed as traveltime multiplied by NMO-velocity squared. When the NMO-velocity squared is different from zero, the power series always exists for sufficiently small offset. The NMO-velocity squared is always positive for SH-waves and for qP-waves in media with normal polarization, for which there are no triplications in traveltime as function of offset. For qSV-wave propagation insome layers, the NMO-velocity squared may be positive, zero, or negative. When it is positive, the power series exists, but there may be an off-axis triplication in traveltime. When the NMO-squared velocity is zero, the power series does not exist, and there is an incipient triplication on the vertical axis. When the NMO-velocity squared is negative, there is a triplication in traveltime on the vertical axis. The power series exists and represents the first-arrival traveltime branch. Numerical examples show that when the value of the NMO-velocity squared is small, the power series for a qSV-wave can be used only for very small values of offset. The series of traveltime squared in powers of offset al-ways exists when the series for traveltime exists; therefore, it must have the same or larger region of convergence.