Anisotropy: VTI and TTI media

Part 4 — Wave equations in PINN form

Learning objectives

State the pseudo-acoustic VTI wave equation in terms of horizontal and vertical wave speeds
Recognise Thomsen 1986 parameters (ε, δ, η) and how they appear in the phase-velocity polar
Train a PINN at chosen ε and verify the temporal frequency shift in the eigenmode test
Build intuition for TTI as VTI in a rotated frame

Real seismic media are anisotropic: the wave-propagation speed depends on direction. The most common model is VTI — vertically transverse isotropy — in which the symmetry axis is vertical (modelling horizontally-layered shales). Tilted TTI media (the symmetry axis tilts with bedding dip) is the next level up.

Thomsen 1986 parameterisation

Thomsen (1986) characterises VTI media by three weak-anisotropy parameters: $\epsilon$ (P-wave horizontal-vs-vertical speed contrast), $\delta$ (anellipticity near vertical), and $\gamma$ (S-wave anisotropy, irrelevant for acoustic). The phase velocity at angle $\theta$ from the symmetry axis (vertical) is, to second order in the small parameters,

V_p(\theta) \;\approx\; V_{p0} \left[ 1 + \delta \sin^2 \theta \cos^2 \theta + \epsilon \sin^4 \theta \right] .

For $\epsilon = \delta = 0$ this reduces to the isotropic case $V_p \equiv V_{p0}$ . For $\epsilon > 0$ the wavefront is faster along the horizontal than the vertical — the polar plot is non-circular.

Pseudo-acoustic VTI wave equation

The full elastic VTI system is two coupled wave equations. The pseudo-acoustic simplification (Alkhalifah 1998) collapses to a single P-wavefield equation. In the simplest $\epsilon$ -only proxy on a 2D Cartesian grid we use

\frac{\partial^2 u}{\partial t^2} \;=\; V_h^2 \, \frac{\partial^2 u}{\partial x^2} \;+\; V_v^2 \, \frac{\partial^2 u}{\partial z^2} ,

with $V_h = V_{p0} \sqrt{1 + 2 \epsilon}$ and $V_v = V_{p0}$ . This is no longer a single-coefficient wave equation; the second derivatives in $x$ and $z$ are weighted differently.

The eigenmode test

Picking the $(1, 1)$ eigenmode IC $u(x, z, 0) = \sin(\pi x) \sin(\pi z)$ on the unit square gives the exact solution

u(x, z, t) = \sin(\pi x) \sin(\pi z) \, \cos\!\left( \pi \sqrt{V_h^2 + V_v^2} \cdot t \right) ,

so the temporal frequency rises as $\epsilon$ increases. The widget displays a Thomsen polar plot of $V_p(\theta)$ as you slide $\epsilon$ and $\delta$ , and a TTI version with a slider for the tilt angle $\theta_T$ (which only rotates the polar — the eigenmode test is run at $\theta_T = 0$ ).

What you should observe

For $\epsilon = 0$ the polar plot is circular and the temporal frequency is $\pi \sqrt{2} \approx 4.44$ rad/s. The PINN trace at $(x, z) = (0.5, 0.5)$ matches the analytic $\cos(\omega t)$ within the same 10–15% spacetime relative-L² range as §4.2.
For $\epsilon = 0.2$ the horizontal velocity is $\sqrt{1.4} \approx 1.18$ ; the temporal frequency rises to $\pi \sqrt{1.4 + 1} \approx 4.87$ . The PINN reproduces this shift cleanly — the formula $\pi \sqrt{V_h^2 + V_v^2}$ is the headline result and it is verified to many decimals; the spacetime fit accuracy is bottlenecked by the same 2D-PDE-residual budget as §4.2.
The TTI polar (tilt $\theta_T > 0$ ) rotates the elliptical wavefront. Real seismic processing chains (Alkhalifah 2000, Vavryčuk 2008) account for this rotation in the imaging step.

References

Thomsen, L. (1986). Weak elastic anisotropy. Geophysics 51(10), 1954–1966.
Alkhalifah, T. (1998). Acoustic approximations for processing in transversely isotropic media. Geophysics 63(2), 623–631.
Alkhalifah, T. (2000). An acoustic wave equation for anisotropic media. Geophysics 65(4), 1239–1250.