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:

(1) |

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*x*-direction:_{1}(3) - the vertical velocity of the
*S*-wave polarized in the*x*-direction:_{2}(4) - the horizontal velocity of the
*S*-wave polarized in the*x*-direction:_{3}(5) - the NMO
*P*-wave velocity for horizontal reflectors in the [*x*,_{1}*x*] plane:_{3}(6) - the NMO
*P*-wave velocity for horizontal reflectors in the [*x*,_{2}*x*] plane:_{3}(7) - the parameter in the [
*x*,_{1}*x*] symmetry plane:_{3}(8) - the parameter in the [
*x*,_{2}*x*] symmetry plane:_{3}(9) - the parameter in the [
*x*,_{1}*x*] plane (_{2}*x*plays the role of the symmetry axis):_{1}(10)

To ease some of the derivations later in this paper I will also use the horizontal velocity in the *x _{1}* direction:

(11) |

(12) |

(13) |

8/21/1998