Tomographic traveltime inversion for linear inhomogeneity and elliptical anisotropy

A velocity model is described in which we assume that velocity increases linearly with depth and varies elliptically with propagation direction. That is, we consider a linearly inhomogeneous elliptically anisotropic model. The variation of velocity with depth is given in terms of parameters a and b, and the elliptical anisotropy is given in terms of parameter x. An analytical traveltime expression is then derived to account for the direct traveltime between an offset source and a receiver in a well; such as in a vertical seismic profile (VSP) setting. A method of inverting traveltime observations to estimate parameters a, b and x is derived. The application of this method is exemplified using a data set from the Western Canada Basin. The parameter estimation also includes a statistical analysis. In the above case, we obtain a good agreement between the field data and the model. Furthermore, the inclusion of elliptical anisotropy is validated by showing that an isotropic model is outside of the confidence interval for x. Once a, b and x are known, a further application is considered; namely, we use the model to calculate the possible reflection points, collectively referred to as the zone of illumination, for a VSP experiment with a given source-receiver geometry. Such modelling is useful for both data analysis and survey design. Two computer codes are given using Maple®. The first code is for the estimation of the parameters and the second one is for the calculation of the zone of illumination