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General formulas for coefficients

Considering equations 9-14 in the case of an orthorhombic medium, only certain derivatives of half-offset $ h_i$ and $ t$ with respect to ray parameters $ p_1$ and $ p_2$ are nonzero at zero offset. The non-zero terms are listed in Table 1.

Derivatives of $ h_1$ Derivatives of $ h_2$ Derivatives of $ t$
$ h_{1,1} $ $ h_{2,2} $ $ t_{,11} $ $ t_{,1111} $
$ h_{1,111} $ $ h_{2,222} $ $ t_{,22} $ $ t_{,2222} $
$ h_{1,122} $ $ h_{2,112} $ $ t_{,1122} $  

Table 1. Nonzero half-offset and one-way traveltime derivatives with respect to $ p_1$ and $ p_2$ in the case of aligned orthorhombic layers.

As a consequence, equation 7 reduces to the expressions given below and derived previously by Al-Dajani et al. (1998). Considering the case where the [$ x_1$ ,$ x_3$ ] plane corresponds to one of the vertical symmetry planes of an orthorhombic medium, Al-Dajani et al. (1998) show that, for pure modes,

$\displaystyle a_{ij} x_i x_j = a_{11}x^2_1 + a_{22}x^2_2 = \frac{x^2_1}{v^2_2} + \frac{x^2_2}{v^2_1}~,$ (35)

where $ v^2_2$ and $ v^2_1$ denote NMO velocities squared associated with $ x_1$ and $ x_2$ directions, respectively. The quartic term can be expressed as

$\displaystyle a_{ijkl} x_i x_j x_k x_l = a_{1111} x^4_1 + a_{1122} x^2_1 x^2_2 + a_{2222} x^4_2~,$ (36)

where, in the notation of the previous section, the exact expressions of these coefficients are given by
$\displaystyle a_{11}$ $\displaystyle =$ $\displaystyle t_0 \frac{t_{,11}}{h^2_{1,1}}~,$ (37)
$\displaystyle a_{22}$ $\displaystyle =$ $\displaystyle t_0 \frac{t_{,22}}{h^2_{2,2}}~,$ (38)

and
$\displaystyle a_{1111}$ $\displaystyle =$ $\displaystyle \frac{t^2_{,11}}{16 h^4_{1,1}} + \frac{t_0(t_{,1111} h_{1,1}-4 t_{,11} h_{1,111})}{48 h^5_{1,1}}~,$ (39)
$\displaystyle a_{1122}$ $\displaystyle =$ $\displaystyle \frac{t_{,11}t_{,22}}{8 h^2_{1,1} h^2_{2,2}} - \frac{t_0}{24 h^3_{1,1} h^3_{2,2}}\bigg[4(t_{,11}h_{2,112}h_{2,2} + t_{,22}h_{1,122}h_{1,1})$  
    $\displaystyle 2(t_{,11}h_{1,122}h_{2,2}+t_{,22}h_{2,112}h_{1,1}) - 3t_{,1122}h_{1,1}h_{2,2} \bigg]$ (40)
$\displaystyle a_{2222}$ $\displaystyle =$ $\displaystyle \frac{t^2_{,22}}{16 h^4_{2,2}} + \frac{t_0(t_{,2222} h_{2,2}-4 t_{,22} h_{2,222})}{48 h^5_{2,2}}~.$ (41)

Equations 37-41 appear somewhat cumbersome, we can simplify them further by relating the derivatives of $ t$ to derivatives of $ h_i$ according to equation 10. Therefore, we can conveniently express both derivatives in terms of the derivatives of vertical slowness $ Q$ (equation 9) thanks to implicit differentiation. By using the notation

$\displaystyle \psi_{i,j} = \sum\limits^N_{n=1} D_{(n)} \frac{\partial^{(i+j)} Q_{(n)}}{(\partial p_1)^{i} (\partial p_2)^{j}}~,$ (42)

we rewrite equations 37-41 in a simpler form as follows:
$\displaystyle a_{11}$ $\displaystyle =$ $\displaystyle -\frac{t_0}{ \psi_{2,0}}~,$ (43)
$\displaystyle a_{22}$ $\displaystyle =$ $\displaystyle -\frac{t_0}{\psi_{0,2}}~,$ (44)
$\displaystyle a_{1111}$ $\displaystyle =$ $\displaystyle \frac{1}{16 \psi_{2,0}^2} + \frac{t_0 \psi_{4,0}}{48 \psi_{2,0}^4}~,$ (45)
$\displaystyle a_{1122}$ $\displaystyle =$ $\displaystyle \frac{1}{8}\left(\frac{1}{\psi_{2,0}\psi_{0,2}} + \frac{t_0\psi_{2,2}}{\psi_{2,0}^2\psi_{0,2}^2}\right)~,$ (46)
$\displaystyle a_{2222}$ $\displaystyle =$ $\displaystyle \frac{1}{16 \psi_{0,2}^2} + \frac{t_0 \psi_{0,4}}{48 \psi_{0,2}^4}~.$ (47)

Alternatively, equations 43-47 can also be derived directly from equations 25-26. They provide an easy way to adding layers by a simple accumulation in the $ \psi_{i,j}$ parameter. Note that equations 45 and 47 have a similar functional form because one is associated with [$ x_1$ ,$ x_3$ ] plane whereas the other with [$ x_2$ ,$ x_3$ ] plane and they are equivalent to the corresponding expressions of VTI media once the derivatives are evaluated. However, equation 46 has a different form from equations 45 and 47. In light of this result, the previous approach of approximating the effective value of $ a_{1122}$ by VTI averaging formula, or equivalently by averaging equations 45 and 47 (Vasconcelos and Tsvankin, 2006; Al-Dajani et al., 1998), is no longer needed.


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Next: Formulas for interval NMO Up: Coefficients of traveltime expansion Previous: Coefficients of traveltime expansion

2017-04-14