Inversion of normal moveout for monoclinic media1

Abstract

Multiple vertical fracture sets, possibly combined with horizontal fine layering, produce an equivalent medium of monoclinic symmetry with a horizontal symmetry plane. Although monoclinic models may be rather common for fractured formations, they have hardly been used in seismic methods of fracture detection due to the large number of independent elements in the stiffness tensor. Here, we show that multicomponent 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.In order to facilitate the inversion procedure, we introduce a Thomsen‐style parametrization for monoclinic media that includes the vertical velocities of the P‐wave and one of the split S‐waves and a set of dimensionless anisotropic coefficients. Our notation, defined for the coordinate frame associated with the polarization directions of the vertically propagating shear waves, captures the combinations of the stiffnesses responsible for the normal‐moveout (NMO) ellipses of all three pure modes. The first group of the anisotropic parameters contains seven coefficients (ε(1,2), δ(1,2,3) and γ(1,2)) analogous to those defined by Tsvankin for the higher‐symmetry orthorhombic model. The parameters ε(1,2), δ(1,2) and γ(1,2) are primarily responsible for the pure‐mode NMO velocities along the coordinate axes x1 and x2 (i.e. in the shear‐wave polarization directions). The remaining coefficient δ(3) is not constrained by conventional‐spread reflection traveltimes in a horizontal monoclinic layer. The second parameter group consists of the newly introduced coefficients ζ(1,2,3) which control the rotation of the P‐, S1‐ and S2‐wave NMO ellipses with respect to the horizontal coordinate axes. Misalignment of the P‐wave NMO ellipse and shear‐wave polarization directions was recently observed on field data by Pérez et al.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 P‐, S1‐ and S2‐waves. 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 PS1‐ and PS2‐waves. Numerical tests show that our method yields stable estimates of all relevant parameters for both a single layer and a horizontally stratified monoclinic medium.