Introduction

The inversion problem in transversely isotropic media consists of the estimating 5 elastic constants for each point of the model space, using the recorded wavefield. For this particular symmetry, the dispersion relation for the SH wave is exactly elliptical, while the other two wave types obey a dispersion relation that depends on all 5 elastic parameters. Using an elliptic approximation for the dispersion relations of P and SV waves can considerably simplify the inversion problem (Levin, 1978, Muir and Dellinger, 1985) because the number of independent parameters per wave type is reduced to 2. This method also provides a fast way of testing the validity of the axial-symmetry, by comparing the independently estimated parameters of SH and SV waves.

Karrenbach (1989) proposed a practical scheme to invert cross-well data for the case of a homogeneous, transversely isotropic medium, using the elliptic approximation and traveltime information for each component. I applied this homogeneous inversion scheme to a three-component cross-well dataset recorded by Western Geophysical. Unfortunately, it was not possible to detect a coherent SH arrival. Consequently, we were able to obtain only four elastic parameters from the estimated elliptic parameters. As an extension to the homogeneous inversion, I have derived a scheme for the more general case of a horizontally layered model and tested it in a synthetic dataset. In this scheme, the model is represented by a superposition of a basis set of symmetric and anti-symmetric square functions. When applied to the cross-well data, the layered inversion scheme generates a stable solution with associated traveltimes that show a better fit to the data than the homogeneous results. However, the insufficient amount of data used compromises the reliability of a higher resolution estimation. Nevertheless, the results obtained with the first few iterations (corresponding to the low frequency components of the model) can be still considered a reliable improvement in the resolution of the model.