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We consider the problem of recovering material parameters in a transverselyisotropic medium from the qP and qSV waves' travel times, given the axis ofisotropy and the material parameters associated to the qSH wave speed. Theoperators obtained from the pseudolinearization argument are of parabolic type,and so we discuss inverting operators whose symbols are of parabolic type. Wepresent stability estimates for recovering either one parameter from one wavespeed or two parameters from two wave speeds with the remaining parameterseither known or with a known functional relationship, and these estimatesprovide injectivity among parameters that differ on sets of small width.

Partial Global Recovery in the Elastic Travel Time Tomography Problem for Transversely Isotropic Media