Questions: Seismic Tomography and Velocity Imaging

Question 1

Seismic waves from a deep earthquake arrive at a recording station 2 seconds earlier than predicted by the reference Earth model. What does this travel-time anomaly most likely indicate about the mantle along that ray path?

• A. The mantle is hotter than average, causing molecules to vibrate faster and waves to travel more quickly
• B. The mantle is colder, denser, or compositionally distinct (e.g., a subducting slab), producing higher seismic velocity
• C. The recording station's clock is slightly fast, introducing a systematic error
• D. The earthquake was shallower than estimated, shortening the ray path

Answer: (B)

Seismic velocity increases with rock stiffness (elastic moduli) and decreases with temperature and partial melt. Cold subducting slabs are stiffer than the surrounding mantle and appear as fast (negative residual) anomalies in tomographic images. Hot material — mantle plumes, mid-ocean ridges — is slower and appears as positive (late-arriving) anomalies. The common misconception is answer A: higher temperature means faster molecular motion thermally, but in solids, higher temperature *reduces* seismic velocity by softening the rock. Seismic velocity is governed by elastic moduli and density, not molecular thermal speed.

Question 2

Why must seismic tomography use regularized inversion rather than directly solving the system of linear equations for the velocity model?

• A. Because ray paths are curved, making the relationship between travel time and velocity inherently nonlinear
• B. Because the system is simultaneously overdetermined (thousands of redundant measurements) and underdetermined (poor ray coverage in some regions), requiring damping and smoothing constraints to produce a stable, geologically reasonable solution
• C. Because travel-time measurements have random errors that make linear algebra inapplicable
• D. Because the forward problem (predicting travel times from a velocity model) cannot be expressed as a matrix equation

Answer: (B)

The mathematical challenge is uneven coverage. In well-sampled regions (e.g., under seismically active subduction zones), the system is overdetermined — many crossing rays constrain the velocity well. In poorly sampled regions (e.g., deep mantle under remote ocean areas with few stations), the system is underdetermined — a near-infinite family of velocity models could fit the data equally well. Without regularization (damping toward a smooth or reference model), the inversion amplifies noise in under-sampled cells. Regularization finds the solution that fits the data while remaining physically plausible — the 'minimum structure' model.

Question 3

Seismic tomography and medical computed tomography (CT) scanning share the same fundamental mathematical principle: both recover a 3D spatial property by inverting integral measurements along paths through the medium.

• T. True
• F. False

Answer: True

Both techniques solve essentially the same inverse problem. In medical CT, X-rays travel through the body and attenuation is integrated along each ray; the inverse problem recovers tissue density. In seismic tomography, seismic waves travel through the Earth and travel time is integrated along each ray; the inverse problem recovers seismic velocity. The analogy is why the term 'tomography' (from the Greek for 'slice') was borrowed from medical imaging. The key mathematical structure — Radon transform / back-projection / regularized inversion — is the same in both fields.

Question 4

The spatial resolution of a seismic tomographic image is uniform throughout the model, because the same number of earthquakes and stations contribute equally to all regions.

• T. True
• F. False

Answer: False

Resolution is entirely dependent on ray coverage — the density and angular diversity of ray paths crossing each region. Well-instrumented continental areas with many nearby earthquakes produce high-resolution images (kilometer scale for crustal studies). Remote oceanic regions with few seismographs and few local earthquakes are poorly sampled; the tomographic image there is highly smoothed and uncertain. Resolution tests (like 'checkerboard tests,' where a synthetic velocity model is recovered to see how well small-scale anomalies are retrieved) are a standard way to map where a tomographic model can be trusted.

Question 5

What are travel-time residuals in seismic tomography, and what information do they encode about Earth structure?

Model answer: A travel-time residual is the difference between the observed arrival time of a seismic wave at a station and the predicted arrival time from a reference (1D) Earth model. A negative residual (early arrival) means the wave traveled faster than average — it passed through colder, denser, or compositionally distinct material with higher seismic velocity (e.g., a subducting slab). A positive residual (late arrival) means the wave was slowed — it traversed hotter, partially molten, or fluid-saturated material (e.g., a mantle plume or magma chamber). Each residual represents an integral of the velocity anomaly along the entire ray path, not a point measurement. By assembling thousands of residuals from crossing paths, seismologists set up a linear system whose solution is the 3D velocity model that best explains all the observed residuals simultaneously.

The key insight is that residuals are path integrals — they contain information about every point the ray passed through, not just one location. This is why you need many crossing rays to separate the contributions of different Earth regions, and why sparse coverage leads to ambiguous models. A student who says 'residuals tell you where it's fast or slow' without explaining the path-integral nature misses the core reason tomography is an inverse problem rather than a direct measurement.