Eikonal basin tomography

§10.3 inverted continental-scale velocity from AMBIENT-NOISE station-pair travel times. §10.7 closes the forward-operator loop: invert basin-scale velocity from BODY-WAVE first-arrival travel times — many earthquakes (or quarry blasts, or controlled active sources) propagating to a regional receiver array. Same Tikhonov-regularised least-squares inverse problem, same algorithmic shape, different forward-source configuration.

The PINN-eikonal connection

The cost of basin tomography is dominated by the FORWARD MODEL: computing predicted travel time from each event to each receiver through the candidate velocity field, repeated at every gradient-descent iteration. Classical FSM costs O(N_grid × N_sweeps) per event. For 1000 events × 100 stations × 100 inversion iterations = 10^7 FSM solves at ~50 ms each → 50 hours of compute. Production scale demands faster.

PINN-EIKONAL replaces FSM with a pre-trained neural network that takes (event_x, event_y, station_x, station_y, velocity_field) as input and outputs travel time as a single network forward pass — typically 100× faster than FSM. Smith et al 2021 EikoNet and Bin Waheed et al 2021 PINNeik formalise this. Once trained, the PINN-eikonal forward call costs ~0.5 ms; the same 50-hour FSM-based tomography becomes ~30 minutes.

Crucially, the INVERSE PROBLEM has the same shape regardless of which forward operator produces the travel times — Tikhonov-regularised LS on the slowness perturbation field. This widget uses straight-ray as a proxy for PINN-eikonal output (production codes use the trained PINN; the inversion code is unchanged).

Try it

Setup: a 100 × 60 km regional sedimentary basin (e.g., Los Angeles or Imperial Valley scale):

• Reference body-wave velocity 4.5 km/s (typical mid-crustal P-wave)
• Truth: 2 anomalies — a SLOW sedimentary basin fill (Δv = −10%, σ = 14 km) and a FAST crystalline basement ridge (Δv = +6%, σ = 11 km)
• 25 events distributed over the region (mimicking seismicity along regional faults) + 30 stations on a quasi-regular grid
• ~750 event-station pairs after distance filtering, σ_pick = 0.4 s
• Inversion: 540-cell slowness map, 250 iters of Tikhonov-regularised gradient descent, λ = 0.8 (moderate smoothing)

Two panels:

• Truth velocity map: red = slow basin, blue = fast ridge, white squares = stations, yellow circles = events.
• Recovered velocity map: same colormap, after Tikhonov inversion. Compare amplitudes + spatial extent to truth.

Below the panels, summary box reports per-anomaly recovered amplitude as percentage of truth + correct-sign indicator. Expected behaviour: both anomalies recovered with correct sign and 50-100% of true amplitude on most seeds.

Production basin tomography

Real basin tomography has been done at:

• Los Angeles Basin (Hauksson 2000, Lin et al 2010, Shaw et al 2015): 50,000+ earthquake events recorded by Southern California Seismic Network, inverted for 3-D Vp + Vs over the Los Angeles region. Established the Puente Hills + Palos Verdes blind thrust faults. PINN-eikonal extensions by Smith et al 2021.
• Imperial Valley: Salton Trough geothermal area; magmatic + faulting structure resolved at 2-5 km lateral resolution.
• Cascadia subduction zone: deep events tomographically mapping the subducting slab.
• Yellowstone: high-resolution P + S body-wave tomography revealing the upper-mantle plume (Smith et al 2009).

What §10.8 will do

§10.8 closes the textbook with a META capstone: take a CURRENT arXiv PINN paper (or a recent Geophysics journal paper) and walk through reproducing the central result end-to-end. Less a new physics topic, more a "you now have the tools — here's how to read the literature."

References

• 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 forward operator usable in basin tomography.
• Bin Waheed, U., Haghighat, E., Alkhalifah, T., Song, C., Hao, Q. (2021). PINNeik: Eikonal solution using physics-informed neural networks. Computers & Geosciences 155, 104833. Independent PINN-eikonal formulation.
• Hauksson, E. (2000). Crustal structure and seismicity distribution adjacent to the Pacific and North America plate boundary in southern California. JGR 105(B6), 13875–13903. Production LA-basin tomography.
• Lin, G., Thurber, C.H., Zhang, H., Hauksson, E., Shearer, P.M., Waldhauser, F., Brocher, T.M., Hardebeck, J. (2010). A California statewide three-dimensional seismic velocity model from both absolute and differential times. Bull. Seismol. Soc. Am. 100(1), 225–240. Continental-scale body-wave tomography.
• Shaw, J.H., Plesch, A., Tape, C., Suess, M.P., Jordan, T.H., Ely, G., Hauksson, E., Tromp, J., Tanimoto, T., Graves, R., Olsen, K., Nicholson, C., Maechling, P.J., Rivero, C., Lovely, P., Brankman, C.M., Munster, J. (2015). Unified Structural Representation of the southern California crust and upper mantle. Earth Planet. Sci. Lett. 415, 1–15. Modern integrated SoCal velocity model.