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 c_ijkl for orthorhombic media can be represented in a compressed two-index notation (the so-called ``Voigt recipe'') as follows:

$$\begin{eqnarray} C = \left( \begin{array} {cccccc} c_{11}& c_{12} & c_{13} & 0 &... ... c_{55} & 0 \\ 0 & 0 & 0 & 0 & 0 & c_{66}\end{array}\right) \,\,.\end{eqnarray}$$ (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 $\eta$ parameter. I will list the nine parameters needed to characterize orthorhombic media below. However, for convenience in later derivations, I will replace the $\epsilon$ parameters with $\eta$ 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:  

  $$v_v \equiv \sqrt{c_{33} \over \rho} \, \, \, (\rho \, \, {\rm is \, the \, density}) \, ,$$ (2)

• the vertical velocity of the S-wave polarized in the x₁-direction:  

  $$v_{s1} \equiv \sqrt{c_{55} \over \rho} \, ,$$ (3)

• the vertical velocity of the S-wave polarized in the x₂-direction:  

  $$v_{s2} \equiv \sqrt{c_{66} \over \rho} \, ,$$ (4)

• the horizontal velocity of the S-wave polarized in the x₃-direction:  

  $$v_{s3} \equiv \sqrt{c_{44} \over \rho} \, ,$$ (5)

• the NMO P-wave velocity for horizontal reflectors in the [x₁,x₃] plane:  

  $$v_1 \equiv \sqrt{\frac{c_{13}(c_{13}+2c_{55})+c_{33}c_{55}}{ \rho(c_{33} - c_{55})}} \, ,$$ (6)

• the NMO P-wave velocity for horizontal reflectors in the [x₂,x₃] plane:  

  $$v_2 \equiv \sqrt{\frac{c_{23}(c_{23}+2c_{66})+c_{33}c_{66}}{ \rho(c_{33} - c_{66})}} \, ,$$ (7)

• the $\eta$ parameter in the [x₁,x₃] symmetry plane:  

  $$\eta_1 \equiv \frac{c_{11} (c_{33}-c_{55})}{2c_{13}(c_{13}+2c_{55})+2c_{33}c_{55}} - \frac{1}{2} \, ,$$ (8)

• the $\eta$ parameter in the [x₂,x₃] symmetry plane:  

  $$\eta_2 \equiv \frac{c_{22} (c_{33}-c_{44})}{2c_{23}(c_{23}+2c_{44})+2c_{33}c_{44}} - \frac{1}{2} \, ,$$ (9)

• the $\delta$ parameter in the [x₁,x₂] plane (x₁ plays the role of the symmetry axis):  

  $$\delta \equiv \frac{(c_{12}+c_{66})^2 - (c_{11} - c_{66})^2} {2 \, c_{11} \, (c_{11}-c_{66})} \, .$$ (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:  

$$V_1 \equiv v_1 \sqrt{1+2 \eta_1},$$ (11)

and the horizontal velocity in the x₂ direction:  

$$V_2 \equiv v_2 \sqrt{1+2 \eta_2},$$ (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 $\gamma$ as follows  

$$\gamma \equiv \sqrt{1+2 \delta}.$$ (13)

Thus, for isotropic media $\gamma=1$.