The problem of recovering density-normalized elastic moduli of a transversely isotropic anisotropic medium from data consisting of qP or qSV phase velocities measured in multiple directions is addressed. Previous studies have used linear fitting methods with approximate forms of the dispersion relation. Here, it is shown that with algebraic manipulation, and a prior estimate of the squared shear velocity along the symmetry axis (A55), it is possible to use simple linear methods with an exact alternative form of the anisotropic dispersion relation. The method is demonstrated with an application to data from a walkaway vertical seismic profile (VSP) experiment and then used as a tool to address several questions raised by that experiment. It is shown that given data with realistically achievable accuracy, the prior estimate of A55 cannot be improved by optimizing the fit to qP data. It is shown that a near perfect fit by a transversely isotropic medium with a vertical symmetry axis (TIV) model to qP data collected in a single vertical plane does not rule out azimuthal anisotropy. Finally, it is shown that a variation of the method, combined with an algorithm suggested by Hood and Schoenberg, suggests a practical way to determine from walkaway VSP data, all the parameters of an orthorhombic medium formed by adding vertical fractures to a transversely isotropic medium. ¿ American Geophysical Union 1994