General formulas for coefficients

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

General formulas for coefficients

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

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

Table 1. Nonzero half-offset and one-way traveltime derivatives with respect to and 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 [ , ] 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

and

denote NMO velocities squared associated with

and

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

to derivatives of

according to equation 10. Therefore, we can conveniently express both derivatives in terms of the derivatives of vertical slowness

(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 [

] plane whereas the other with [

] 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.

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

2017-04-14