Seismic Tomography

By the end of this lecture, you will be able to:

• [LO-12.1] Formulate a tomographic inverse problem as and explain what each term represents.
• [LO-12.2] Recognise that tomographic inversions are ill-posed, and describe the role of regularisation (damping, weighting).
• [LO-12.3] Interpret a global or regional tomographic image — slabs, plumes, LLSVPs, ULVZs — and connect the Cascadia Juan de Fuca slab to this framework.

Prerequisites: PREM and phase identification (Lec 11); linear algebra (matrix inverse, transpose); travel time as integral of slowness (Lec 4, Lec 11).

Stand in Seattle. What is 60 km below your feet?

The Juan de Fuca plate descends eastward beneath Seattle, then under the Cascade arc. The melt zone feeds Rainier, St Helens, and the rest of the arc — east of Seattle.

Every number above came from a travel-time residual measured in fractions of a second.

The motivating question

We cannot drill there. We cannot see there. Yet the picture on the previous slide is built to kilometre resolution from broadband seismograms at surface stations.

How?

Lecture 11 answered this for the 1-D Earth.
Lecture 12 does it for the 3-D Earth — the real one.

The geometric core idea

• One source: anomaly is localised along a cone.
• Two sources: cones intersect → 2-D location.
• Many sources × many stations: full 3-D localisation.

Consequence: coverage controls resolution

• Well-imaged regions: Japan, California, western Europe (dense rays, diverse azimuths).
• Poorly-imaged regions: oceans, Antarctica, Southern Hemisphere.

Always ask: where are the earthquakes? Where are the stations?

Every published tomogram must include a resolution test (checkerboard).

The forward problem, discretised

Divide Earth into cells; assume slowness constant per cell.

= path length of ray through cell . Most entries are zero — most rays don't cross most cells.

In matrix form:

• : travel times (length ) | : slownesses (length ) | : sensitivity matrix.

The simplest tomography — 2×2 cells, 4 rays

(a) Geometry. (b) True model: one slow cell. (c) Damped inversion with 1% noise: anomaly localised; amplitude smeared.

The toy system explicitly

is rank 3, not rank 4.
A uniform slowness change fits any data: the null space is nonzero.
Every real tomography has a null space. Regularisation picks what to fill it with.

But rays bend — Snell's law makes non-linear

When varies, the ray bends → the path lengths change → depends on itself.

Born (linearised) approximation around a reference model :

Iterate: solve for , update model, retrace rays, repeat.

Full-waveform inversion (FWI) replaces ray tracing entirely:

• Simulate the 3-D elastic wavefield numerically (SPECFEM3D)
• Compute adjoint kernels — volumetric sensitivity to every parameter
• Match the entire waveform, not just the first arrival
• Doubles resolution; captures finite-frequency "banana-doughnut" effects

The inverse problem — why least squares needs help

Ordinary least squares:

is often singular or near-singular → noise explodes.

Damped least squares (key equation):

trades resolution for stability. Too small → noise. Too large → washed out.

Two philosophical cautions

1. The inverse problem has no unique solution.
Many models fit the data within error. Regularisation selects one. The published model is a choice, not a fact.

2. Resolution is not uniform.
Near array edges, and where rays don't bottom, anomalies are smeared and attenuated.
Always report resolution tests.

Reading a global mantle tomogram

Blue (fast): cold slabs. Red (slow): hot plumes, wedge melts. LLSVPs at CMB. ULVZs, D″.

Global mantle tomography: what to recognise

• Blue slabs descending through the upper mantle. Some stagnate at 660 km; some penetrate to the CMB (Farallon).
• Red plumes rising from the lower mantle under hotspots (Hawaii, Iceland, Yellowstone, Afar).
• LLSVPs — Pacific + Africa — cover ~25% of the CMB.
• ULVZs — 10–50 km thick, , partial melt or Fe-enriched.
• D″ — post-perovskite phase transition a few hundred km above the CMB.

Seismic vs. medical tomography

Same math. Different geometry is why seismic tomography is harder.

Ambient-noise tomography (Shapiro et al. 2005) — treat every station as a virtual source → fixes the geometry problem regionally.

Cascadia anchor — why this matters for PNW hazard

• Slab geometry → width of the locked megathrust → maximum magnitude of the next Cascadia earthquake.
• Hydrated slab-top low-velocity layer → controls episodic tremor and slip (ETS) observed every ~14 months under the Olympic Peninsula (Rogers & Dragert 2003).
• Mantle-wedge melt volume → long-term volcanic hazard from Rainier, St Helens, and the rest of the Cascades.

Every tomographic picture is also a hazard-assessment tool.

Combining body waves and surface waves

Joint inversion stacks both blocks in one :

In Cascadia: body waves anchor the slab below 100 km; ambient-noise surface waves resolve the upper 50 km.

Resolution: where can we trust the image?

A region the data cannot constrain ≠ a region with zero anomaly.

Two ingredients of poor resolution

• Sparse sources — earthquakes only on plate boundaries
• Sparse stations — oceans, polar regions, intraplate continents

The fast diagnostic: ray density map

Count non-zero entries in each column of (or sum path lengths through each cell).
A directional version checks azimuthal coverage — 1000 N–S rays ≠ good lateral resolution.

The formal answer: the resolution matrix

• Diagonal → fraction of true anomaly recovered at cell
• Off-diagonal → smearing between cells
• A row of is the point-spread function at that cell

Often too large to form explicitly — diagonal estimated stochastically.

Empirical resolution: the checkerboard test

1. Build a synthetic model of alternating fast/slow blocks of known size
2. Forward-model with the same
3. Add realistic noise
4. Invert with the same damping
5. Compare recovered to input

Where the pattern recovers → trust the inversion at that scale.
Where it smears → do not interpret features at that scale.

Repeat at multiple checkerboard wavelengths → scale-dependent resolution.
Probabilistic / Bayesian methods replace this with full posterior uncertainties.

Reading checklist for any tomographic image

Always ask four questions:

1. Where are the sources and stations? (Show the ray density.)
2. What checkerboard test was done, at what scale? (Show recovery.)
3. What damping was used? (L-curve justification.)
4. Are uncertainties reported? (Bayesian posterior > single model.)

A paper that does not address all four is incomplete. Anomaly amplitudes are lower bounds at best.

Research horizon (2021–2026)

Travel-time → full-waveform

• GLAD-M25 and successors (Lei et al. 2020; Tromp 2020 Nat Rev Earth Env) — adjoint FWI doubles resolution
• Finite-frequency kernels: sensitivity is a banana-doughnut, not a ray delta-function
• Multiscale FWI incorporating multiple period bands simultaneously (Fichtner group, ETH, 2021–2024)

ML augmentation

• SeisBench (Woollam et al. 2022, SRL) — unified ML picker API; one to two orders of magnitude more picks from continuous data
• Neural-network tomographic solvers: promising, but must be validated against conventional inversions before geological interpretation

New data sources

• Ambient-noise FWI: cross-correlation wavefields now used in full adjoint inversion — no earthquakes needed for crustal imaging
• DAS: fibre-optic cables as dense seismic arrays; PNW shallow structure (UW, 2022–2024)

Probabilistic inversion

• HMC / variational Bayes approaches: sample the full model posterior → honest uncertainty estimates on slab geometry, LLSVP boundaries

AI Literacy — critical tool use (LO-7)

Three places ML sits in the tomography pipeline — and fails:

1. ML phase pickers. Miss emergent arrivals; confound S/P on vertical; degrade out-of-domain. Always spot-check against manual picks.
2. ML tomographic solvers. Can hallucinate structure not in the data. Validate against conventional least squares on the same data.
3. AI literature summaries. Can invent papers, authors, figures. Every citation must be DOI-verified.

Prompt to try

"Compare the Cascadia tomographic models of Schmandt & Humphreys (2010) and Bodmer et al. (2018). What slab-geometry differences do they report?"

Now verify:

• Do both papers exist? (DOI resolves?)
• Does each paper actually discuss slab geometry?
• Are the numerical differences the assistant reports traceable to figures/tables?

If any claim fails verification, assume the whole answer is suspect.

Concept checks

1. Write for a 3×3 grid with 3 horizontal + 3 vertical rays. Find a null-space .
2. Damping recovers a slab with . At , recovered amplitude is . What does this say about the true amplitude? About resolution?
3. In Fig. fig-cascadia: where on the seafloor was the subducted crust created? Will the wedge melt zone be a positive or negative anomaly, and at what order of magnitude?

Summary

• Why: To see beneath our feet in 3-D, from surface seismograms alone.
• What: , inverted with damped least squares.
• Caveats: Non-unique; resolution non-uniform; regularisation is a choice.
• Applications: Cascadia slab geometry ties directly to megathrust and volcanic hazard.
• Next: These same equations reappear in earthquake location (Lec 13) and moment-tensor inversion (Lec 14).

Further reading

• Aster, Borchers & Thurber 2018, Parameter Estimation and Inverse Problems, 3rd ed. (UW Libraries.)
• Shapiro et al. 2005, Science — ambient-noise tomography. (Open access.)
• Schmandt & Humphreys 2010, EPSL — Cascadia body-wave tomography.
• Bodmer et al. 2018, GRL — Juan de Fuca buoyancy and megathrust segmentation.
• Lei et al. 2020, GJI — GLAD-M25 global full-waveform inversion. (Open access.)
• Lowrie & Fichtner 2020, Ch. 3.6–3.7. (UW Libraries — primary text.)