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