I derive analytical expressions giving traveltime-offset curves for multilayered weakly azimuthally isotropic and anisotropic media on terms of the elastic properties of each layer. This method is based upon an approximate skewed hyperbolic moveout formula involving three measured bulk velocities for each reflector. The primary benefits of this technique are: (1) it allows for fast traveltime computation; (2) it makes possible an extremely rapid estimation of the interval elastic parameters; and (3) it provides physical insight into the wave propagation in anisotropic media.I establish a traveltime inversion algorithm which allows the estimation of the five elastic constants together with the orientation of the axis of symmetry for each layer. In the isotropic limit, this algorithm reduces to the conventional one used for determining the interval velocities from stacking velocity measurements in isotropic media. The inversion technique is applied to surface seismic measurements and VSP field surveys. In both cases, the method provides very good estimates of these six parameters, even in highly anisotropic materials. The results also show that the orientation of the horizontal axis of symmetry, for a given azimuthally anisotropic layer at depth, can be obtained using only quasi-P-wave information. This technique can be easily incorporated into conventional velocity analysis algorithms. Interval anisotropic properties can then be estimated with little additional effort.