Direct arrival times and slownesses from wide-aperture walkaway vertical seismic profile data acquired in a layered anisotropic medium can be processed to give a direct estimate of the phase slowness surface associated with the medium at the depth of the receivers. This slowness surface can, in turn, be fit by an estimated transversely isotropic medium with a vertical symmetry axis (a ''TIV'' medium). While the method requires that the medium between the receivers and the surface be horizontally stratified, no further measurement or knowledge of that medium is required. When applied to data acquired in a compacting shale sequence (here termed the ''Petronas shale'') encountered by a well in the South China Sea, the method yields an estimated TIV medium that fits the data extremely well over 180¿ of propagation angles sampled by 201 source positions. The medium is strongly anisotropic. The anisotropy is significantly anelliptic and implies that the quasi-shear mode should be triplicated for off-axis propagation. Estimated density-normalized moduli (in units of km2/s2) for the Petronas shale are A11=6.99¿0.21, A33=5.53¿0.17, A55=0.91¿0.05, and A13=2.64¿0.26. Densities in the logged zone just below the survey lie in the range between 2200 and 2400 kg/m3 with an average value close to 2300 kg/m3. ¿ American Geophysical Union 1994