0    3D TTI Decoupled Equations

Rewriting equations 3.6 using the relation $\hat{k}^2_r + \hat{k}^2_z = k^2_r + k^2_z$ yields

$$\left\{ \begin{array}{ll} \omega^2=V^2_{p_0} \bigg[ (k^2_r + k^2_... ...at{k}^4_z}{k^2_r+k^2_z} \bigg] \;\;\;\;\;\mbox{(SV wave)} \end{array} \right. .$$ (01)

Plugging equations 3.8 into equations A.1, we get the approximated P wave and SV wave dispersion relations for TTI media in 3D case

$$\left\{ \begin{array}{ll} \omega^2 = V^2_{p_0}\bigg ( A_{\script... ...style 13}k_xk_z+B_{\scriptscriptstyle 23}k_yk_z - H \bigg ) \end{array}\right.,$$ (02)

where the differential operator $H$ is defined as

$$H = A_{\scriptscriptstyle 1111}\displaystyle\frac{ k^4_x }{k^2_r+k^... ...k^2_z} + A_{\scriptscriptstyle 1113}\displaystyle\frac{ k^3_xk_z }{k^2_r+k^2_z}$$

$$+ A_{\scriptscriptstyle 1222}\displaystyle\frac{ k_xk^3_y }{k^2_r... ...2_z} + A_{\scriptscriptstyle 1122}\displaystyle\frac{ k^2_xk^2_y }{k^2_r+k^2_z}$$

$$+ A_{\scriptscriptstyle 1133}\displaystyle\frac{ k^2_xk^2_z }{k^2... ...z} + A_{\scriptscriptstyle 1233}\displaystyle\frac{ k_xk_yk^2_z }{k^2_r+k^2_z}.$$

Here $A$ and $B$ are coefficients related to Thomsen's anisotropy parameters ($\epsilon$ and $\delta$ ), dip and azimuth ($\theta$ and $\phi$ ), and the ratio of SV wave and P wave velocities ($V^2_{s_0}$ /$V^2_{p_0}$ )

$$A_{\scriptscriptstyle 11} = (1+2\epsilon)-2(2\epsilon-\delta)\sin^2\theta\cos^2\phi,$$

$$A_{\scriptscriptstyle 22} = (1+2\epsilon)-2(2\epsilon-\delta)\sin^2\theta\sin^2\phi,$$

$$A_{\scriptscriptstyle 33} = (1+2\epsilon)-2(2\epsilon-\delta)\cos^2\theta,$$

$$A_{\scriptscriptstyle 12} = -2(2\epsilon-\delta)\sin^2\theta\sin2\phi,$$

$$A_{\scriptscriptstyle 13} = 2(2\epsilon-\delta)\sin2\theta\cos\phi,$$

$$A_{\scriptscriptstyle 23} = 2(2\epsilon-\delta)\sin2\theta\sin\phi,$$

$$B_{\scriptscriptstyle 11} = \frac{V^2_{s_0}}{V^2_{p_0}} + 2(\epsilon-\delta)\sin^2\theta\cos^2\phi,$$

$$B_{\scriptscriptstyle 22} = \frac{V^2_{s_0}}{V^2_{p_0}} + 2(\epsilon-\delta)\sin^2\theta\sin^2\phi,$$

$$B_{\scriptscriptstyle 33} = \frac{V^2_{s_0}}{V^2_{p_0}} + 2(\epsilon-\delta)\cos^2\theta,$$

$$B_{\scriptscriptstyle 12} = 2(\epsilon-\delta)\sin^2\theta\sin2\phi,$$

$$B_{\scriptscriptstyle 13} = -2(\epsilon-\delta)\sin2\theta\cos\phi,$$

$$B_{\scriptscriptstyle 23} = -2(\epsilon-\delta)\sin2\theta\sin\phi,$$

$$A_{\scriptscriptstyle 1111} = 2(\epsilon-\delta)\sin^4\theta\cos^4\phi,$$

$$A_{\scriptscriptstyle 2222} = 2(\epsilon-\delta)\sin^4\theta\sin^4\phi,$$

$$A_{\scriptscriptstyle 3333} = 2(\epsilon-\delta)\cos^4\theta ,$$

$$A_{\scriptscriptstyle 1112} = 4(\epsilon-\delta)\sin^4\theta\sin2\phi\cos^2\phi,$$

$$A_{\scriptscriptstyle 1113} = -4(\epsilon-\delta)\sin2\theta\sin^2\theta\cos^3\phi,$$

$$A_{\scriptscriptstyle 1222} = 4(\epsilon-\delta)\sin^4\theta\sin2\phi\sin^2\phi,$$

$$A_{\scriptscriptstyle 2223} = -4(\epsilon-\delta)\sin2\theta\sin^2\theta\sin^3\phi,$$

$$A_{\scriptscriptstyle 1333} = -4(\epsilon-\delta)\sin2\theta\cos^2\theta\cos\phi,$$

$$A_{\scriptscriptstyle 2333} = -4(\epsilon-\delta)\sin2\theta\cos^2\theta\sin\phi,$$

$$A_{\scriptscriptstyle 1122} = 3(\epsilon-\delta)\sin^4\theta\sin^22\phi,$$

$$A_{\scriptscriptstyle 1133} = 3(\epsilon-\delta)\sin^22\theta\cos^2\phi,$$

$$A_{\scriptscriptstyle 2233} = 3(\epsilon-\delta)\sin^22\theta\sin^2\phi,$$

$$A_{\scriptscriptstyle 1123} = -6(\epsilon-\delta)\sin2\theta\sin^2\theta\sin2\phi\cos\phi,$$

$$A_{\scriptscriptstyle 1223} = -6(\epsilon-\delta)\sin2\theta\sin^2\theta\sin2\phi\sin\phi,$$

$$A_{\scriptscriptstyle 1233} =$$ (03)

The corresponding decoupled P wave and SV wave equations in the time-wavenumber domain for 3D TTI media are

$$\left\{ \begin{array}{ll} \displaystyle\frac{1}{V^2_{p_0}} \disp... ...iptstyle 23}k_yk_z - H \bigg ) P_{\scriptscriptstyle SV} \end{array} \right. .$$

Figure A.1 shows a snapshot of an impulse response at time $t=0.4~s$ in a 3D homogeneous TTI medium with $V_{p_0}=3000~m/s$ , $\epsilon =0.24$ , $\delta =0.1$ , $\theta =45^\circ$ and $\phi=15^\circ$ using the above TTI decoupled P wave equation.

For the case of HTI, $\theta$ is 90 degrees and $\cos\theta$ becomes zero. In this case, equation A.3 is simplified with coefficients $A_{\scriptscriptstyle 3333},A_{\scriptscriptstyle 1333}$ and $A_{\scriptscriptstyle 2333}$ all becoming zero.

For the case of tilted elliptical isotropy, $\epsilon = \delta$ and many coefficients in equations A.2 and A.3 go to zero, for examples, all $A_{\scriptscriptstyle ijkl}$ and $B_{\scriptscriptstyle 12}, B_{\scriptscriptstyle 13}$ and $B_{\scriptscriptstyle 23}$ .

Figure: Wavefield snapshots at time $t=0.4~s$ in a 3D homogeneous TTI medium with $V_{p_0}=3000~m/s$ , $\epsilon =0.24$ , $\delta =0.1$ , $\theta =45^\circ$ and $\phi=15^\circ$ . A point source is located in the center at $X=Y=Z=1.5~km$ . (a), (b) and (c) are 2D $Y$ -$X$ , $Z$ -$X$ and $Z$ -$Y$ slices across the source location, respectively.