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Similarly to the derivation for
, we have first,
![$\displaystyle \psi_{4,0(N)} = \frac{48t^3_{0(N)}a_{1111(N)}-3t_{0(N)}a^2_{11(N)}}{a^4_{11(N)}}~,$](img163.png) |
(50) |
which can be calculated for interval
in the
-th layer using,
![$\displaystyle \Psi_{4,0(N)} = \psi_{4,0(N)}- \psi_{4,0(N-1)}~.$](img165.png) |
(51) |
Subsequently, interval
in the
-th layer can be computed from,
![$\displaystyle A_{1111(N)} = \frac{3T_{0(N)}A^2_{11(N)}+A^4_{11(N)}\Psi_{4,0(N)}}{48T_{0(N)}^3}~,$](img167.png) |
(52) |
where
. 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
by considering
and
instead of
and
. To derive the corresponding expression for
, we follow an analogous procedure ,which leads to
![$\displaystyle \psi_{2,2(N)} = \frac{8t^3_{0(N)}a_{1122(N)}-t_{0(N)}a_{11(N)}a_{22(N)}}{a^2_{11(N)}a^2_{22(N)}}~.$](img174.png) |
(53) |
Using
![$\displaystyle \Psi_{2,2(N)} = \psi_{2,2(N)} - \psi_{2,2(N-1)}~,$](img176.png) |
(54) |
in the
-th layer can be computed as
![$\displaystyle A_{1122(N)} = \frac{T_{0(N)}A_{11(N)}A_{22(N)}+A^2_{11(N)}A^2_{22(N)}\Psi_{2,2(N)}}{8T_{0(N)}^3}~.$](img178.png) |
(55) |
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Next: Comparison with known expressions
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Previous: Formulas for interval NMO
2017-04-14