Inversion of normal moveout for monoclinic media

Multiple vertical fracture sets, possibly combined with horizontal fine layering, produce an equivalent medium of monoclinic symmetry with a horizontal symmetry plane. Here, we show that multi component wide azimuth reflection data (combined with known vertical velocity or reflector depth) or multi-azimuth walkaway $VSP$ surveys provide enough information to invert for all but one anisotropic parameters of monoclinic media. To facilitate the inversion procedure, we introduce a Thomsen style parametrization that includes the vertical velocities of the $P$-wave and one of the split $S$ waves and a set of dimensionless anisotropic coefficients. This notation captures the combinations of the stiffnesses responsible for the normal moveout $(NMO)$ ellipses of all three pure modes. Our parameter-estimation algorithm, based on NMO equations valid for any strength of the anisotropy, is designed to obtain anisotropic parameters of monoclinic media by inverting the vertical velocities and NMO ellipses of the waves $P$, S$_1$ and S$_2$. A Dix type representation of the NMO velocity of mode-converted waves makes it possible to replace the pure shear modes in reflection surveys with the waves PS$_1$ and PS$_2$. Numerical tests show that our method yields stable estimates of all relevant parameters for both a single layer and a horizontally stratified monoclinic medium.