Sensitivity of estimated elastic moduli to completeness of wave type, measurement type, and illumination apertures at a receiver in multicomponent VSP data

Abstract

Inversion of phase slowness and polarization vectors measured from multicomponent vertical seismic profile data can yield estimates of all 21 density-normalized elastic moduli for anisotropic elastic media in the neighborhood of each 3C geophone. Synthetic test data are produced by direct evaluation of the Christoffel equation, and by finite-difference solution of the elastodynamic equations. Incompleteness of the data, with respect to illumination (polar and azimuth angle) apertures (qP and/or qS) wave types, wave-propagation directions, and the amount of data (e.g., with or without horizontal slowness components), produces solutions with variations in quality, as revealed by the distribution of model parameter correlations. In a good solution, with all parameters well constrained by the data, the correlation matrix is diagonally dominant. qP-only and qS-only solutions typically produce complementary distributions in their correlation matrices, as they are orthogonal in their sampling of the medium with respect to polarization. The elastic moduli become less independent as the data apertures decrease. If the other input data are relatively complete, the horizontal components of the slowness vector are not needed as the information they contain is redundant. The main consequence of omitting horizontal slowness components is slower convergence. When modest amounts of random noise are added to the slowness and polarization data, in otherwise adequately sampled apertures, the solution is still very close to the correct model, but with larger residual variance.