Local Determination of Weak Anisotropy Parameters from qP-wave Slowness and Particle Motion Measurements

We propose an algorithm for local evaluation of weak anisotropy (WA) parameters from measurements of slowness vector components and/or of particle motions of qP waves at individual receivers in a borehole in a multi-azimuthal multiple-source offset VSP experiment. As a byproduct the algorithm yields approximate angular variation of qP-wave phase velocity. The formulae are derived under assumption of weak but arbitrary anisotropy and lateral inhomogeneity of the medium. The algorithm is thus independent of structural complexities between the source and the receiver. If complete slowness vector is determinable from observed data, then the information about polarization can be used as an independent additional constraint. If only the component of the slowness along the borehole can be determined from observations (which is mostly the case), the inversion without information about polarization is impossible. We present several systems of equations which can be used when different numbers of components of the slowness vector are available. The SVD algorithm is used to solve an overdetermined system of linear equations for WA parameters for two test examples of synthetic multiazimuthal multiple-source offset VSP data. The system of equations results from approximate first-order perturbation equations for the slowness and polarization vectors of the qP wave. Analysis of singular values and of variances of WA parameters is used for the estimation of chances to recover the sought parameters. Effects of varying number of profiles with sources and of noise added to ‘‘observed’’ data are illustrated. An important observation is that although, due to insufficient data, we often cannot recover all individual WA parameters with sufficient accuracy, angular phase velocity variation can be recovered rather well.