Next: The dispersion relation Up: Alkhalifah: Wave equation for Previous: Introduction

# Anisotropic media parameters in orthorhombic media

Unlike in VTI media, where the model is fully characterized by 5 parameters, in orthorhombic media we need 9 parameters for full characterization. However, like in VTI media, not all parameters are expected to influence P-wave propagation to a detectable degree. Therefore, alternative parameter representation is important to simplify the problem to a level where key parameter dependencies are recognizable.

The stiffness tensor cijkl for orthorhombic media can be represented in a compressed two-index notation (the so-called Voigt recipe'') as follows:
 (1)
In VTI media, c11=c22, c13=c23, c44=c55, c12=c11-2 c44, and thus the number of independent parameters reduce from 9 to 5. With additional constraints given by the isotropic model the number of independent parameters will ultimately reduce to two.

Significant progress, however, can be made by combining the stiffnesses in such a way that will simplify analytic description of seismic velocities. Tsvankin (1997) suggested a parameterization similar to what Thomsen (1986) used for VTI media and to what Alkhalifah and Tsvankin (1995) added for processing purposes, namely the parameter. I will list the nine parameters needed to characterize orthorhombic media below. However, for convenience in later derivations, I will replace the parameters with parameters and use slightly different notations than those given by Tsvankin (1997). In summary, these nine parameters are related to the elastic coefficients as follows:

• the P-wave vertical velocity:    (2)
• the vertical velocity of the S-wave polarized in the x1-direction:    (3)
• the vertical velocity of the S-wave polarized in the x2-direction:    (4)
• the horizontal velocity of the S-wave polarized in the x3-direction:    (5)
• the NMO P-wave velocity for horizontal reflectors in the [x1,x3] plane:    (6)
• the NMO P-wave velocity for horizontal reflectors in the [x2,x3] plane:    (7)
• the parameter in the [x1,x3] symmetry plane:    (8)
• the parameter in the [x2,x3] symmetry plane:    (9)
• the parameter in the [x1,x2] plane (x1 plays the role of the symmetry axis):    (10)
This notation preserves the attractive features of Thomsen parameters in describing velocities, and traveltimes. They also provide a simple way to measure anisotropy, since the dimensionless parameters in the new representation equal zero when the medium is isotropic. The notation used above, also, simplify description of time-related processing equations.

To ease some of the derivations later in this paper I will also use the horizontal velocity in the x1 direction:
 (11)
and the horizontal velocity in the x2 direction:
 (12)
Both horizontal velocities are given in terms of the above parameters and thus do not add to the number of independent parameters required to represent orthorhombic media. Finally, I will use as follows
 (13)
Thus, for isotropic media .

Next: The dispersion relation Up: Alkhalifah: Wave equation for Previous: Introduction
Stanford Exploration Project
8/21/1998