Wave propagation in 3D orthorhombic media

Next: RTM of BP 2007 Up: Examples Previous: Wave propagation in TTI

Wave propagation in 3D orthorhombic media


c11 Figure 14. The perturbed $C_{11}$ coefficient in the orthorhombic test. All other stiffness coefficients are perturbed in a similar manner.


wavec-0,wavecs Figure 15. Wavefield snapshot of wave propagation in orthorhombic media taken at : (a) qP-wave, (b) Coupled qS-waves.

To demonstrate 3D wave propagation in tilted orthorhombic media, we use the classic model from Schoenberg and Helbig (1997), which characterizes a TI medium with vertical fractures. The density-normalized orthorhombic stiffness matrix is (Schoenberg and Helbig, 1997):

$\displaystyle \left[ \begin{array}{cccccc} 9 & 3.6 & 2.25 & 0 & 0 & 0 3.6 & ... ...& 0 0 & 0 & 0 & 0 & 1.6 & 0 0 & 0 & 0 & 0 & 0 & 2.182 \end{array} \right]$

(34)

To introduce spatial heterogeneity, we apply a moderate perturbation to the stiffness coefficients that is a function of $\mathbf{x}$ , as demonstrated by Figure 14 for the case of $C_{11}$ . The model is further rotated $45 ^{\circ}$ counterclockwise about the

axis (azimuth angle) and $45 ^{\circ}$ counterclockwise about the

axis (dip angle).

We employ the exact phase velocity in orthorhombic media (Tsvankin, 1997):

$\displaystyle V^2(\mathbf{x},\mathbf{k})\vert\mathbf{k}\vert^2 = 2\sqrt{\frac{a^2/3-b}{3}}\cos \left( \frac{\upsilon}{3}+n\frac{2\pi}{3} \right)-\frac{a}{3} \; ,$

(35)

where

corresponds to the P wave and

corresponds to the shear waves, and

		$\displaystyle \upsilon = \arccos \left( -\frac{2(a/3)^3 -ab/3 + c}{2\sqrt{((a^2/3-b)/3)^3}} \right) \; (0 \leq \upsilon \leq \pi) \; ,$	(36)
		$\displaystyle a = -(G_{11}+G_{22}+G_{33}) \; ,$	(37)
		$\displaystyle b = G_{11}G_{22} + G_{11}G_{33} + G_{22}G_{33} - G_{12}^2 - G_{13}^2 - G_{23}^2 \; ,$	(38)
		$\displaystyle c = G_{11}G_{23}^2 + G_{22}G_{13}^2 + G_{33}G_{12}^2 - G_{11}G_{22}G_{33} - 2G_{12}G_{13}G_{23} \; ,$	(39)
		$\displaystyle G_{11} = c_{11}\hat{k}_x^2 + c_{66}\hat{k}_y^2 + c_{55}\hat{k}_z^2 \; ,$	(40)
		$\displaystyle G_{22} = c_{66}\hat{k}_x^2 + c_{22}\hat{k}_y^2 + c_{44}\hat{k}_z^2 \; ,$	(41)
		$\displaystyle G_{33} = c_{55}\hat{k}_x^2 + c_{44}\hat{k}_y^2 + c_{33}\hat{k}_z^2 \; ,$	(42)
		$\displaystyle G_{12} = (c_{12} + c_{66})\hat{k}_x\hat{k}_y \; ,$	(43)
		$\displaystyle G_{13} = (c_{13} + c_{55})\hat{k}_x\hat{k}_z \; ,$	(44)
		$\displaystyle G_{23} = (c_{23} + c_{44})\hat{k}_y\hat{k}_z \; .$	(45)

To incorporate tilting into the orthorhombic anisotropy, we replace the original wavenumber components , and with $\hat{k}_x$ , $\hat{k}_y$ , and $\hat{k}_z$ , which are wavenumbers evaluated in the rotated coordinate system aligned with the symmetry axis:

$\begin{displaymath}\begin{array}{*{20}c} \hat{k}_x=k_x\cos{\phi}+k_y\sin{\phi}\;... ...eta}-k_y\cos{\phi}\sin{\theta}+k_z\cos{\theta}\;, \end{array}\end{displaymath}$

(46)

where $\phi$ is the azimuth angle representing horizontal rotation (the angle between the original

axis and the rotated one) and $\theta$ is the dip angle measured from vertical. Figure 15 demonstrates the wavefield snapshots taken at $t=1\;s$ for three wave modes: quasi-P-wave and the coupled quasi-S-waves. Note that the quasi-S-waves are propagated separately using solutions from equation 35 and then summed together, since the two modes do not decouple easily in an orthorhombic medium.

Next: RTM of BP 2007 Up: Examples Previous: Wave propagation in TTI

2016-11-16