Traveltime approximations for transversely isotropic media with an inhomogeneous background

Appendix D: Expansion in 3D

The eikonal equation for $P$-waves in a TI medium with a tilt in the symmetry axis satisfies the following relation,

$$a_4 v_t^4 \left(\frac{\partial \tau }{\partial z}\right)^4+a_3 v_... ...l \tau }{\partial z}\right)^2+a_1 v_t \frac{\partial \tau }{\partial z}+a_0=0,$$ (45)

where

$$a_0 = 2 v^2 v_t^2 \eta \sin ^2\theta \cos ^2\theta \left(\cos\phi \frac... ...partial \tau }{\partial x}-\sin\phi \frac{\partial \tau }{\partial y}\right)^2$$

$$+ 2 v^2 v_t^2 \eta \sin ^2\theta \left(\sin\phi \frac{\partial \ta... ...partial \tau }{\partial x}-\sin\phi \frac{\partial \tau }{\partial y}\right)^2$$

$$- v^2 (2 \eta +1) \left(\sin\phi \frac{\partial \tau }{\partial x}... ...tial \tau }{\partial x}-\sin\phi \frac{\partial \tau }{\partial y}\right)^2+1,$$ (46)

$$a_1 = -\frac{2}{v_t} \sin\theta \cos\theta \left(\cos\phi \frac{\parti... ..._t \sin\phi \frac{\partial \tau }{\partial y}-1\right) \right. \right. \right.$$

$$+ \left. \left. \left. v_t \left(\frac{\partial \tau }{\partial x}\... ...cos ^2\theta \sin ^2\phi+\cos ^2(\phi )\right) \right. \right. \right. \right.$$

$$+ \left. \left. \left. \left. \sin ^2\theta \sin\phi\right)\right)-v_t\right)+v^2 (2 \eta +1)\right),$$ (47)

$$a_{2} = \frac{1}{4} \left(v^2 \eta \left(-4 \sin ^2\theta (3 \cos (2\th... ...rtial y}\right)^2\right)+8 \sin ^2\theta (3 \cos (2\theta )+2) \right. \right.$$

$$\left. \left. \sin (2\phi ) \frac{\partial \tau }{\partial x} \f... ...frac{\partial \tau }{\partial y}\right)^2\right)\right)-4 \cos ^2\theta\right)$$

$$- \frac{v^2 (2 \eta +1) \sin ^2\theta}{v_t^2},$$ (48)

$$a_3 = \frac{v^2 \eta \sin (4\theta ) \left(\cos \phi \frac{\partial \tau }{\partial x}-\sin\phi \frac{\partial \tau }{\partial y}\right)}{v_t},$$ (49)

$$a_{4} = \frac{v^2 \eta \sin ^2\theta \cos ^2\theta}{v_t^2}.$$ (50)

To develop equations for the coefficients of a traveltime expansion in 3D from a background elliptical anisotropy with a vertical symmetry axis I use vector notations ($n_{x}$ and $n_{y}$) to describe the tilt angles, where the components of this 2D vector describe the projection of the symmetry axis on each of the $x-z$ and $y-z$ planes, respectively. As a result,

$$n_{x} = \sin\theta \cos\phi,$$ (51)

and

$$n_{y} = \sin\theta \sin\phi.$$ (52)

Using these two equations to solve for $\sin\theta$ and $\sin\phi$ and plugging them into equation D-1 yields an eikonal for TTI media in terms of $n_x$ and $n_{y}$. Thus, inserting the following trial solution

$$\tau(x,y,z) \approx \tau_{0}(x,y,z) +\tau_{\eta}(x,y,z) \eta+\tau_{\eta_{2}}(x,y,z) \eta^{2}+\tau_{n_{x}}(x,y,z) n_{x}+ \tau_{n_{y}}(x,y,z) n_{y},$$ (53)

where $\eta$, $n_{x}$, and $n_{y}$ are independent parameters and small, into the eikonal equation yields an extremely long equation. Again, setting the coefficients of the independent parameters ($\eta$, $n_{x}$, and $n_{y}$) to zero in the equation gives the eikonal equation for elliptical anisotropy with vertical symmetry axis. On the other hand, the coefficients of the first power of the independent parameters yield:

$$v^2 \frac{\partial \tau _0}{\partial x} \frac{\partial \tau _{\e... ... \frac{\partial \tau _0}{\partial z} \frac{\partial \tau _{\eta }}{\partial z} = v^2 \left( \left( v_t^2 \left(\frac{\partial \tau _0}{\partial z}... ... x}\right)^2+\left(\frac{\partial \tau _0}{\partial y}\right)^2\right)\right),$$

$$v^2 \frac{\partial \tau _0}{\partial y} \frac{\partial \tau _{n_... ...2 \frac{\partial \tau _0}{\partial z} \frac{\partial \tau _{n_x}}{\partial z} = -\left(v^{2}-v_t^{2}\right) \frac{\partial \tau _0}{\partial x} \frac{\partial \tau _0}{\partial z},$$

$$v^2 \frac{\partial \tau _0}{\partial y} \frac{\partial \tau _{n_... ...2 \frac{\partial \tau _0}{\partial z} \frac{\partial \tau _{n_y}}{\partial z} = -\left(v^{2}-v_t^{2}\right) \frac{\partial \tau _0}{\partial y} \frac{\partial \tau _0}{\partial z},$$ (54)

corresponding to $\eta$, $n_{x}$, and $n_{y}$, respectively.

The coefficient of the $\eta^{2}$ term, for higher accuracy in $\eta$, is given by

$$2 v^2 \frac{\partial \tau _0}{\partial x} \frac{\partial \tau _{... ...\partial \tau _0}{\partial y} \frac{\partial \tau _{\eta }}{\partial y}\right)$$

$$+ 4 v_t^2 v^2 \frac{\partial \tau _0}{\partial z} \left(\left(\fra... ...partial \tau _0}{\partial y} \frac{\partial \tau _{\eta }}{\partial y} \right)$$

$$- v_t^2 \left(\frac{\partial \tau _{\eta }}{\partial z}\right)^2.$$ (55)

These first-order PDEs, when solved, provide traveltime approximations using equation D-9 for 3D TI media in a generally inhomogeneous elliptical anisotropic background.

For a homogeneous medium simplification, the traveltime is given by the following analytical relation in 3-D elliptical anisotropic media:

$$\tau_{0}(x,y,z) = \sqrt{\frac{x^2+y^{2}}{v^{2}}+\frac{z^2}{v^{2}_{t}}},$$ (56)

which satisfies the eikonal equation B-2 in 3D. Using equation D-12, I evaluate $\frac{\partial \tau _{0}}{\partial x}$, $\frac{\partial \tau _{0}}{\partial y}$ and $\frac{\partial \tau _{0}}{\partial z}$ and insert them into equations D-10 to solve these first-order linear equations to obtain

$$\tau_{\eta}(x,y,z) = -\frac{v_t^4 \left(x^2+y^2\right)^2 \sqrt{\frac{x^2+y^2}{v^2}+\frac{z^2}{v_t^2}}}{\left(v^2 z^2+v_t^2 \left(x^2+y^2\right)\right)^2},$$

$$\tau_{n_{x}}(x,y,z) = \frac{\left(v_t^2-v^2\right) x z}{v^2 v_t^2 \sqrt{\frac{x^2+y^2}{v^2}+\frac{z^2}{v_t^2}}},$$

$$\tau_{n_{y}}(x,y,z) = \frac{\left(v_t^2-v^2\right) y z}{v^2 v_t^2 \sqrt{\frac{x^2+y^2}{v^2}+\frac{z^2}{v_t^2}}},$$ (57)

respectively.

I now evaluate $\frac{\partial \tau _{\eta}}{\partial x}$, $\frac{\partial \tau _{\eta}}{\partial y}$, and $\frac{\partial \tau _{\eta}}{\partial z}$ and use them to solve equation D-11. After some tedious algebra, I obtain

$$\tau_{\eta_{2}}(x,y,z) = \frac{3 v_t^6 \left(x^2+y^2\right)^... ...right)}{2 \left(v^2 z^2+v_t^2 \left(x^2+y^2\right)\right)^4}.$$ (58)

The application of Pade approximation on the expansion in $\eta$, by finding a first order polynomial representation in the denominator, yields a TI equation that is accurate for large $\eta$ (Alkhalifah, 2010), as well as small tilt, given by

$$\tau(x,y,z) \approx \frac{1}{v^2 v_t^2 \sqrt{\frac{x^2+y^2}{v^2}+\frac{z^2}{v_t^2}} ... ...\eta +1) \left(x^2+y^2\right)+v_t^4 (3 \eta +2) \left(x^2+y^2\right)^2\right)}$$

$$\left(2 v^6 z^5 (z- \sin\theta \left( x \cos\phi+y \sin \phi \ri... ...^4 z^3 \left(\left((6 \eta +2) \left(x^2+y^2\right)-z^2\right) \right. \right.$$

$$\left. \left. (\sin\theta \left( x \cos\phi+y \sin \phi \right))... ... \eta +1) \left(x^2+y^2\right)\right)-v_t^4 v^2 z \left(x^2+y^2\right) \right.$$

$$\left. \left(\left((3 \eta +2) \left(x^2+y^2\right)-4 z^2 (3 \et... ...os\phi+y \sin \phi \right)-z (13 \eta +6) \left(x^2+y^2\right)\right) \right.$$

$$\left. +v_t^6 \left(x^2+y^2\right)^2 \left(x^2 (\eta +2)+z (3 \e... ... \sin\theta \left( x \cos\phi+y \sin \phi \right)+y^2 (\eta +2)\right)\right).$$ (59)

Traveltime approximations for transversely isotropic media with an inhomogeneous background