Elastic and anisotropic FWI

Part 6 — Full-Waveform Inversion

Learning objectives

Explain why acoustic FWI systematically mis-fits elastic reflection data at non-zero offsets
Describe Aki–Richards R_PP and R_PS reflection coefficients as functions of angle
Identify the three principal elastic parameters (Vp, Vs, ρ) and their cross-talk in elastic FWI
Summarise what anisotropic (VTI/TTI) FWI adds and when it is required

Everything in §6.1–6.3 assumed the acoustic wave equation: one parameter (velocity) per pixel, P-waves only, no shear. Real rocks support shear and real seismic data contains mode-converted and shear-wave arrivals. Running acoustic FWI on elastic data leaves a systematic residual that cannot be reduced by any velocity update — because the missing events are not in the synthetic data at all. Elastic FWI is the physics upgrade: invert for V_p, V_s, and ρ simultaneously using the elastic wave equation to simulate full vector-wavefield physics.

1. The missing physics — mode conversion at oblique incidence

At a flat interface between two elastic layers, an incident P-wave produces four arrivals: reflected P, transmitted P, reflected mode-converted S (SV), and transmitted SV. The Zoeppritz equations give the exact amplitudes; Aki and Richards' linearisation is the small-contrast approximation used in production:

R_{PP}(\theta) \approx \tfrac{1}{2}\!\left(\tfrac{\Delta\alpha}{\alpha} + \tfrac{\Delta\rho}{\rho}\right) + \tfrac{1}{2}\,\tfrac{\Delta\alpha}{\alpha}\,\tan^2\theta - 2\!\left(\tfrac{\beta}{\alpha}\right)^{\!2}\!\sin^2\theta\,\left(2\tfrac{\Delta\beta}{\beta} + \tfrac{\Delta\rho}{\rho}\right)

where $\alpha = V_p$ , $\beta = V_s$ . The PP reflection has a normal-incidence intercept $\frac{1}{2}(\Delta\alpha/\alpha + \Delta\rho/\rho)$ and an AVO gradient that scales with the $V_s$ contrast. The mode-converted PS coefficient is zero at normal incidence and peaks near 30–45°:

R_{PS}(\theta) \approx -\frac{\tan\phi}{\sin\theta}\,\Bigl[4\!\left(\tfrac{\beta}{\alpha}\right)^{\!2}\!\sin^2\theta\,\tfrac{\Delta\beta}{\beta} + \tfrac{1 - 2(\beta/\alpha)^2 \sin^2\theta}{2}\,\tfrac{\Delta\rho}{\rho}\Bigr]

with Snell's law $\sin\phi = (\beta/\alpha)\sin\theta$ giving the S reflection angle. R_PS depends on $\Delta V_s$ and $\Delta\rho$ only; it carries no information about $\Delta V_p$ .

Left panel: a ray diagram for the user's current angle. Incident P (teal), reflected P (yellow, thickness and brightness proportional to |R_PP|), and mode-converted reflected S (red dashed, same scaling for |R_PS|). Right panel: R_PP(θ) and R_PS(θ) across 0–60°, with a vertical cursor at the current θ. Slide the angle toward zero — R_PS vanishes. Slide toward 30° — R_PS is a substantial fraction of R_PP. The bottom info strip reports the percentage.

The takeaway for FWI: a real trace recorded at 2 km offset from a reflector at 2 km depth samples reflection angles near 30°. At that angle, $|R_{PS}|$ can range from ~10 % of $|R_{PP}|$ for mild sand–shale contrasts up to ~40 % for strong contrasts with sizeable shear-wave changes (the widget's default 15/10/8 % contrasts sit around the upper end at 40 %). An acoustic synthetic has no PS — so whatever fraction the earth actually produces is a systematic residual the acoustic FWI cannot fit. The gradient tries to reduce the residual by tweaking V_p, but the residual is physically unrelated to V_p; tweaking anything will only distort the model without reducing the misfit. The symptom: false structures along the illumination direction, correlated with the source-receiver azimuth.

3. Elastic FWI — three parameters per pixel

Elastic FWI solves the elastic wave equation forward and adjoint (at 3–4× the cost of acoustic) and inverts for three parameters per grid cell instead of one: $V_p(x,z), V_s(x,z), \rho(x,z)$ . The gradient now has three components per pixel; the Hessian is $3N \times 3N$ and its condition number is worse than the acoustic case because the three parameters trade off in the data:

V_p and ρ trade off in R_PP's intercept (they appear as the sum $\Delta V_p/V_p + \Delta\rho/\rho$ ). Only amplitude-AVO information separates them.
V_s and ρ trade off in the PS coefficient in a similar way.
V_p and V_s both affect the AVO gradient, but V_s dominates.

To mitigate parameter cross-talk, elastic FWI uses re-parameterisations that decorrelate the gradients:

Impedance: $I_p = V_p \rho$ , $I_s = V_s \rho$ — constrains the AVO intercept/gradient separately.
Vp/Vs ratio + V_p: $V_p/V_s$ is sensitive to fluid content and lithology and is often more stable than the individual velocities.
Poisson’s ratio + V_p: similar motivation; Poisson is well-constrained from AVO.

4. Anisotropic FWI — adding Thomsen parameters

Real sedimentary rocks, especially shales, exhibit VTI anisotropy: the vertical velocity differs from the horizontal. The velocity surface is a rotational ellipsoid described by four Thomsen parameters:

$V_{p0}$ : P-wave velocity along the symmetry (vertical) axis.
$V_{s0}$ : S-wave velocity along the symmetry axis.
$\epsilon$ : P-wave anisotropy (typically 0.05–0.2 for shales). Controls the difference between horizontal and vertical P velocity.
$\delta$ : near-offset P-wave anisotropy. Controls NMO velocity.

A VTI FWI inverts for $V_{p0}, V_{s0}, \epsilon, \delta$ (and ρ) at every pixel — five parameters per grid cell. The numerical cost climbs to 5–7× acoustic. The payoff: in shaley overburdens (most of the North Sea, Gulf of Mexico deep water, Middle Eastern carbonates with shale drapes), anisotropic effects contribute 5–20 % of the velocity field. Isotropic FWI in shaley media pushes the apparent velocity toward the NMO value, distorting depths by 5–10 %. Anisotropic FWI fixes this.

TTI (tilted transverse isotropy) adds a dip angle $\alpha$ and azimuth $\phi$ of the symmetry axis — seven parameters per pixel. Required for dipping shales and for thrust-belt settings where the shale fabric is rotated. Orthorhombic anisotropy (nine-ish parameters) is needed when the rock has two orthogonal anisotropy directions (fractured reservoirs, horizontally transversely isotropic + VTI combined). FWI at these complexities is on the cutting edge of production.

5. When acoustic FWI is good enough

Short offsets only. If all angles are below 15°, R_PS is tiny and acoustic FWI is fine.
Fluid-filled reservoir rocks. Within a reservoir where the formation behaves close to fluid-like (high V_p/V_s), mode conversion is weak.
Isotropic overburden with large $V_s$ contrast. If the overburden is isotropic to within a few percent, the V_s signal is diffusive enough that acoustic inversion works for V_p.

6. When elastic/anisotropic is mandatory

Shaley overburden above a reservoir target. Anisotropy is 10–20 %, must be included.
Multi-component (OBN, OBC) data. Hydrophone and three-component geophones record both P and S; ignoring shear is throwing away half the data.
Fractured reservoirs. Fracture orientation imprints on azimuthal anisotropy. Orthorhombic FWI extracts that.
AVO and rock-physics inversion. If the end product is a fluid/lithology estimate, you need the full elastic parameter set.

**The one sentence to remember**

Acoustic FWI makes the assumption "R_PS = 0" and distorts V_p to fit mode-converted energy it cannot represent; elastic and anisotropic FWI add the missing parameters, costing 3–7× more per iteration but producing physically meaningful V_p, V_s, ρ (and Thomsen parameters) instead of a scrambled V_p.

Where this goes next

§6.5 closes Part 6 with the QC question: given a migrated FWI model, how do you know it is right? The answer is synthetic-vs-recorded comparison — propagate through the final model, compare to data, and accept only when the comparison is tight enough across frequency, offset, and azimuth.

References

Virieux, J., Operto, S. (2009). An overview of full-waveform inversion in exploration geophysics. Geophysics, 74, WCC1.
Tarantola, A. (1984). Inversion of seismic reflection data in the acoustic approximation. Geophysics, 49, 1259.
Pratt, R. G. (1999). Seismic waveform inversion in the frequency domain, Part 1. Geophysics, 64, 888.
Etgen, J., Gray, S. H., Zhang, Y. (2009). An overview of depth imaging in exploration geophysics. Geophysics, 74, WCA5.