USArray surface-wave tomography of the western US

§10.2 located individual microseismic events in a frac stage (km-scale, single-event inversions). §10.3 takes a SCALE JUMP: invert a continental-scale 2-D phase-velocity map $c(x, y)$ from inter-station travel times measured between AMBIENT-NOISE cross-correlations of dozens to hundreds of broadband stations.

What is ambient-noise tomography?

Earth never stops shaking. Ocean microseisms, wind-driven ground motion, anthropogenic noise — they all produce a continuous low-amplitude wavefield recorded at every seismometer. Cross-correlate the ambient noise time-series at two stations A and B over months of data, and the cross-correlation function approximates the GREEN'S FUNCTION between A and B (Shapiro-Campillo 2004). Pick the surface-wave arrival time on this function at chosen periods → you have $t_{AB}(T)$ without ever needing an active source.

Inverting these inter-station travel times for the spatially-varying phase velocity $c(x, y, T)$ at each period $T$ is AMBIENT-NOISE TOMOGRAPHY (ANT). It revolutionised regional seismology in the 2000s — with USArray Transportable Array (TA) deployment across the contiguous US (2007-2014), entire-continent maps became possible.

The inverse problem

For each station pair $(A, B)$ with separation $d_{AB}$ and observed phase travel time $t_{AB}$, the FORWARD model is the path integral of slowness:

Discretise the region into $N_x \times N_y$ cells; the path integral becomes a weighted sum $t_{AB} = \sum_k G_{AB,k} , s_k$ where $G_{AB,k}$ is the path length of the ray AB through cell $k$. Stack over all pairs to get $\mathbf{t} = G \mathbf{s}$.

Inverse: solve for the slowness perturbation $\delta \mathbf{s} = \mathbf{s} - \mathbf{s}_{\mathrm{ref}}$ via Tikhonov-regularised least squares:

where $L$ is a 5-point Laplacian smoothing operator. The widget below solves this via gradient descent on the penalised objective.

Try it

Setup mimics a scaled-down USArray TA deployment in the western US:

• 60 broadband stations distributed over a 1500 × 1000 km region (~70 km mean spacing — typical for TA).
• Truth phase-velocity map at 20 s period (typical Rayleigh-wave dispersion in the western-US lithosphere) with 3 anomalies modelled after real geological features:
• — A SLOW HOT SPOT (Yellowstone-like): −8% velocity, σ = 180 km
• — A FAST RIDGE (Sierra-Nevada-like): +5% velocity, σ = 150 km
• — A MILD SLOW ZONE (Basin-and-Range-like): −4% velocity, σ = 200 km
• Inter-station travel times for ~1200 station pairs (those with separation 200-800 km, the productive range for surface-wave tomography), with σ_pick = 0.5 s noise.
• Inverse: 600-cell slowness map, Tikhonov-regularised LS, 150 iterations of gradient descent (λ = 8).

Three panels:

• Truth map: red = slow, blue = fast, grey lines = ray paths, white dots = stations. The anomalies are visible as red and blue patches.
• Recovered map: same colormap, after Tikhonov inversion. Compare to truth: well-resolved anomalies recover with correct sign and ~50-90% of true amplitude; poorly-resolved areas (low ray density) get smoothed away by the prior.
• Ray-coverage density: how many rays pass through each cell. High coverage → good recovery; low coverage → smoothed-over. This is the production "resolution proxy" used by tomographers to flag which areas of a published map are trustworthy.

Below the panels, a per-anomaly summary reports recovered amplitude as a percentage of true amplitude. A good run recovers all 3 anomalies with correct sign and >40% amplitude.

Why straight-ray (and where PINN-eikonal helps)

The widget uses STRAIGHT-RAY path integrals — the simplest forward model. This is acceptable when the velocity perturbation is SMALL (here, $\le 8$): Fermat's principle says rays follow paths that extremise travel time, and for small perturbations the ray bend is second-order. For STRONG perturbations (e.g., subduction-zone slabs at +20% velocity), straight-ray under-estimates the anomaly amplitude — because the ray actually bends AROUND the fast region, taking less of the perturbation than a straight line implies.

PRODUCTION FIX: replace straight-ray with PINN-eikonal solver (§7.2-§7.3). The PINN computes the actual curved-ray travel times $T(x_a, y_a \to x_b, y_b; \mathbf{c})$ as a network forward pass — fast, differentiable, and exact within the network's training regime. Smith et al 2021 ("EikoNet for surface-wave tomography") and Bin Waheed et al 2021 use this recipe at scale.

Production challenges

• Multi-period inversion. Each period $T$ probes a different depth range (long periods penetrate deeper). Real ANT inverts maps at $T = 8, 12, 18, 25, 35, 50$ s simultaneously, then converts the periods to depth via local 1-D Vs(z) inversions per cell.
• Anisotropy. Rayleigh and Love waves at the same period have DIFFERENT speeds in anisotropic media. Inverting both jointly constrains 2θ azimuthal anisotropy + radial anisotropy — see Lin et al 2009 USArray work.
• Source heterogeneity. Ambient noise isn't perfectly diffuse; ocean swells dominate certain azimuths. Cross-correlation Green's functions are biased toward dominant noise directions. Production ANT applies SOURCE-ILLUMINATION corrections.
• Pick uncertainty. Real picks have σ ≈ 0.5-2 s depending on inter-station distance and SNR. Production codes propagate this into FORMAL POSTERIOR uncertainty (§9.6 bootstrap).

What §10.4 will do

§10.4 zooms back in to RESERVOIR-SCALE imaging: AVO (Amplitude-Versus-Offset) analysis from PINN-augmented FWI. Where §10.1 inverted velocity, §10.4 inverts ELASTIC properties (V_p, V_s, ρ) jointly using the dependence of reflection amplitudes on incidence angle.

References

• Shapiro, N.M., Campillo, M., Stehly, L., Ritzwoller, M.H. (2005). High-resolution surface-wave tomography from ambient seismic noise. Science 307(5715), 1615–1618. The foundational ANT paper.
• Lin, F.C., Ritzwoller, M.H., Townend, J., Bannister, S., Savage, M.K. (2007). Ambient noise Rayleigh wave tomography of New Zealand. Geophys. J. Int. 170(2), 649–666. Production ANT methodology.
• Lin, F.C., Ritzwoller, M.H., Snieder, R. (2009). Eikonal tomography: surface wave tomography by phase front tracking across a regional broad-band seismic array. Geophys. J. Int. 177(3), 1091–1110. Eikonal-tomography theory used in USArray analyses.
• Bin Waheed, U., Haghighat, E., Alkhalifah, T., Song, C., Hao, Q. (2021). PINNeik: Eikonal solution using physics-informed neural networks. Computers & Geosciences 155, 104833. PINN-eikonal forward operator usable inside ANT.
• Smith, J.D., Azizzadenesheli, K., Ross, Z.E. (2021). EikoNet: Solving the Eikonal equation with deep neural networks. IEEE Trans. Geosci. Remote Sens. 59(12), 10685–10696. PINN-eikonal usable as the forward operator in production tomography.