Lyapunov Exponents for 2-D Ray Tracing Without Interfaces

Abstract

The Lyapunov exponents quantify the exponential divergence of rays asymptotically, along infinitely long rays. The Lyapunov exponent for a finite 2-D ray and the average Lyapunov exponents for a set of finite 2-D rays and for a 2-D velocity model are introduced. The equations for the estimation of the average Lyapunov exponents in a given smooth 2-D velocity model without interfaces are proposed and illustrated by a numerical example. The equations allow the average exponential divergence of rays and exponential growth of the number of travel-time branches in the velocity model to be estimated prior to ray tracing.

Seismic travel time inversion for 3D structures regularized with Sobolev norms

“…An attempt to quantify the exponential divergence of rays with respect to the complexity of the model and to formulate explicit criteria enabling the construction of models suitable for ray tracing was made by Klimeš [14]. In his paper, he observed that, for an isotropic medium and a model parametrization with bi/tricubic splines, the average Lyapunov exponent may be approximated by the square root of the Sobolev norm of order 2 (appendix A) of the corresponding velocity model [15,1]:…”