Anisotropic parameters and P‐wave velocity for orthorhombic media

Abstract

Although orthorhombic (or orthotropic) symmetry is believed to be common for fractured reservoirs, the difficulties in dealing with nine independent elastic constants have precluded this model from being used in seismology. A notation introduced in this work is designed to help make seismic inversion and processing for orthorhombic media more practical by simplifying the description of a wide range of seismic signatures. Taking advantage of the fact that the Christoffel equation has the same form in the symmetry planes of orthorhombic and transversely isotropic (TI) media, we can replace the stiffness coefficients by two vertical (P and S) velocities and seven dimensionless parameters that represent an extension of Thomsen's anisotropy coefficients to orthorhombic models. By design, this notation provides a uniform description of anisotropic media with both orthorhombic and TI symmetry. The dimensionless anisotropic parameters introduced here preserve all attractive features of Thomsen notation in treating wave propagation and performing 2-D processing in the symmetry planes of orthorhombic media. The new notation has proved useful in describing seismic signatures outside the symmetry planes as well, especially for P‐waves. Linearization of P‐wave phase velocity in the anisotropic coefficients leads to a concise weak‐anisotropy approximation that provides good accuracy even for models with pronounced polar and azimuthal velocity variations. This approximation can be used efficiently to build analytic solutions for various seismic signatures. One of the most important advantages of the new notation is the reduction in the number of parameters responsible for P‐wave velocities and traveltimes. All kinematic signatures of P‐waves in orthorhombic media depend on just the vertical velocity [Formula: see text] and five anisotropic parameters, with [Formula: see text] serving as a scaling coefficient in homogeneous media. This conclusion, which holds even for orthorhombic models with strong velocity anisotropy, provides an analytic basis for application of P‐wave traveltime inversion and data processing algorithms in orthorhombic media.