Traveltime approximations for transversely isotropic media with an inhomogeneous background

Appendix A: Expansion in $\theta$

To derive a traveltime equation in terms of perturbations in $\theta$, we first establish the form for the governing equation for TI media given by the eikonal representation. The eikonal equation for $p$-waves in TI media in 2D (for simplicity) is given by

$${v^2} (1+2 \eta) \,{\left(\cos\theta \frac{\partial \tau}{\partial x} + \sin\theta \frac{\partial \tau}{\partial z}\right)^2 } +$$

$${{{v_t}}^2}\,{\left( \cos\theta \frac{\partial \tau}{\partial z}-... ...{\partial x} +\sin\theta \frac{\partial \tau}{\partial z} \right)^2} \right)=1.$$ (25)

To solve equation A-1 through perturbation theory, we assume that $\theta$ is small, and thus, a trial solution can be expressed as a series expansion in $\sin\theta$ given by

$$\tau(x,z) \approx \tau_0(x,z) +\tau_1(x,z) \sin\theta+ \tau_2(x,z) \sin^{2}\theta,$$ (26)

where $\tau_0$, $\tau_1$ and $\tau_2$ are coefficients of the expansion given in units of traveltime, and, for practicality, terminated at the second power of $\sin\theta$. Inserting the trial solution, equation A-2, into equation A-1 yields a long formula, but by setting $\sin\theta=0$, I obtain the zeroth-order term given by

$${v^2} (1+2 \eta) \,{\left(\frac{\partial \tau_{0}}{\partial x}\ri... ... \eta {v^2} \,{ \left(\frac{\partial \tau_{0}}{\partial x}\right)^2} \right)=1,$$ (27)

which is the eikonal formula for VTI anisotropy. By equating the coefficients of the powers of the independent parameter $\sin\theta$, in succession, we end up first with the coefficients of first-power in $\sin\theta$, simplified by using equation A-3, and given by

$$v^2 \frac{\partial \tau _1}{\partial x} \left( (2 \eta +1) \frac... ...ial \tau _0}{\partial x}\right)^2 \frac{\partial \tau _0}{\partial z}\right) =$$

$$2 v^2 v_t^2 \eta \left( \frac{\partial \tau _0}{\partial x} \lef... ...v_t^2 \frac{\partial \tau _0}{\partial x} \frac{\partial \tau _0}{\partial z},$$ (28)

which is a first-order linear partial differential equation in $\tau_1$. The coefficient of $\sin\theta^{2}$, with some manipulation, has the following form

$$2 v^2 \frac{\partial \tau_2}{\partial x} \left( (2 \eta +1) \fra... ...al \tau _0}{\partial x}\right)^2 \frac{\partial \tau _0}{\partial z}\right) =$$

$$v^2 (2 \eta +1) \left(\frac{\partial \tau _0}{\partial x}\right)... ...{\partial x}\right)^2+v_t^2 \left(\frac{\partial \tau _0}{\partial z}\right)^2$$

$$+4 v^2 v_t^2 \eta \frac{\partial \tau _1}{\partial x} \left(\fra... ...au _1}{\partial x}\right)^2 \left(\frac{\partial \tau _0}{\partial z}\right)^2$$

$$+12 v^2 v_t^2 \eta \frac{\partial \tau _0}{\partial x} \frac{\pa... ...al x} \frac{\partial \tau _1}{\partial z} \frac{\partial \tau _0}{\partial z}$$

$$-v^2 (2 \eta +1) \left(\frac{\partial \tau _1}{\partial x}+\frac... ...a +1) \frac{\partial \tau _0}{\partial x} \frac{\partial \tau _1}{\partial z},$$ (29)

which is again a first-order linear partial differential equation in $\tau_2$ with an obviously more complicated source function given by the right-hand side. Though the equation seems complicated, many of the variables of the source function (right-hand side) can be evaluated during the evaluation of equations A-3 and A-4 in a fashion that will not add much to the cost.

Traveltime approximations for transversely isotropic media with an inhomogeneous background