Theory of interval traveltime parameter estimation in layered anisotropic media

Formulas for interval quartic coefficients

Similarly to the derivation for $A_{ii}$ , we have first,

$$\psi_{4,0(N)} = \frac{48t^3_{0(N)}a_{1111(N)}-3t_{0(N)}a^2_{11(N)}}{a^4_{11(N)}}~,$$ (50)

which can be calculated for interval $\Psi_{4,0(N)}$ in the $N$ -th layer using,

$$\Psi_{4,0(N)} = \psi_{4,0(N)}- \psi_{4,0(N-1)}~.$$ (51)

Subsequently, interval $A_{1111(N)}$ in the $N$ -th layer can be computed from,

$$A_{1111(N)} = \frac{3T_{0(N)}A^2_{11(N)}+A^4_{11(N)}\Psi_{4,0(N)}}{48T_{0(N)}^3}~,$$ (52)

where $T_{0(N)} = t_{0(N)}-t_{0(N-1)}$ . Equation 52 is similar to the Dix-type formula proposed by Tsvankin and Thomsen (1994) for the VTI case. Similar expressions can be derived for $A_{2222}$ by considering $\Psi_{0,4}$ and $A_{22}$ instead of $\Psi_{4,0}$ and $A_{11}$ . To derive the corresponding expression for $A_{1122}$ , we follow an analogous procedure ,which leads to

$$\psi_{2,2(N)} = \frac{8t^3_{0(N)}a_{1122(N)}-t_{0(N)}a_{11(N)}a_{22(N)}}{a^2_{11(N)}a^2_{22(N)}}~.$$ (53)

Using

$$\Psi_{2,2(N)} = \psi_{2,2(N)} - \psi_{2,2(N-1)}~,$$ (54)

$A_{1122(N)}$ in the $N$ -th layer can be computed as

$$A_{1122(N)} = \frac{T_{0(N)}A_{11(N)}A_{22(N)}+A^2_{11(N)}A^2_{22(N)}\Psi_{2,2(N)}}{8T_{0(N)}^3}~.$$ (55)

Theory of interval traveltime parameter estimation in layered anisotropic media