In Earth's mantle, seismic velocity generally increases with depth. Consider two seismic rays launched from the same earthquake: one at a steep downward angle and one at a shallow angle nearly horizontal. Which ray penetrates deeper before turning back toward the surface?
AThe shallow-angle ray, because it is nearly parallel to the velocity gradient and travels farther before refracting upward
BThe steep-angle ray, because it has a smaller ray parameter and penetrates to greater depth before turning
CBoth rays turn at the same depth because the turning depth depends only on the velocity structure, not the launch angle
DThe shallow-angle ray turns immediately near the surface because low-angle rays cannot refract into high-velocity material
The ray parameter p = sin(θ)/v is conserved along the ray path. A steeper ray (larger θ from horizontal at launch) has a smaller p value. The turning point occurs where the ray becomes horizontal (θ = 90°), meaning sin(θ)/v = 1/v_turn = p. A smaller p corresponds to a larger v_turn — a deeper layer with higher velocity — so steep rays (small p) turn deeper. Shallow-angle rays (large p) hit the turning condition at shallower, lower-velocity depths. This is the inverse of intuition: it's the steep ray, not the shallow one, that penetrates deeper.
Question 2 Multiple Choice
A seismologist observes a gap in a travel-time curve — a range of epicentral distances where no direct seismic arrivals are recorded. What does this shadow zone most likely indicate about Earth's interior?
AThe earthquake was too small to generate seismic waves detectable at those distances
BA low-velocity zone at depth deflects rays away from those distances, leaving a gap in direct-wave coverage
CSeismic attenuation absorbs all energy before the waves can travel that far through the mantle
DThe seismometers at those distances were not operating when the earthquake occurred
A low-velocity zone (LVZ) bends rays toward steeper angles (they speed up entering the LVZ, which by Snell's law bends them away from horizontal). This refracts direct rays into a narrower distance range, leaving a gap — a shadow zone — at intermediate distances. The classic example is Earth's outer core: P-wave velocity drops sharply at the core-mantle boundary, creating a P-wave shadow zone between ~105° and ~140°. Similarly, the asthenosphere LVZ creates subtle effects in teleseismic travel times. Shadow zones are diagnostic of velocity decreases with depth.
Question 3 True / False
The ray parameter p = sin(θ)/v is conserved along a seismic ray path, which causes rays to curve continuously as they travel through rock where velocity changes smoothly with depth.
TTrue
FFalse
Answer: True
Conservation of the ray parameter is the seismic analog of Snell's law generalized to continuously varying media. At every point along the ray, sin(θ)/v must equal the same constant p. As v increases with depth, sin(θ) must increase to maintain p constant — meaning θ increases toward 90° (horizontal). This progressive bending continues until the ray reaches its turning point, then reverses on the way back up. The result is curved ray paths through Earth rather than straight lines, which is why seismic rays from shallow earthquakes follow arcs through the interior.
Question 4 True / False
Seismic rays travel in straight lines through Earth's interior and primarily change direction abruptly when they encounter sharp velocity discontinuities like the Moho or core-mantle boundary.
TTrue
FFalse
Answer: False
Rays curve continuously wherever velocity changes smoothly with depth — which describes most of Earth's mantle. Sharp discontinuities produce abrupt direction changes (refraction and reflection following Snell's law), but gradual velocity gradients cause continuous bending. In reality, most seismic rays follow smooth curves that arc through the mantle, turning at depth before returning to the surface. Assuming straight-line propagation would yield completely wrong travel times and epicenter locations.
Question 5 Short Answer
How does the shape of the travel-time curve — arrival time plotted against epicentral distance — encode information about Earth's velocity structure, and how can it be inverted to recover velocity as a function of depth?
Think about your answer, then reveal below.
Model answer: The travel-time curve's slope at any distance gives the ray parameter p = dt/dΔ of the ray arriving at that distance. Since each ray samples a specific depth range determined by its turning point, the curve encodes which velocities are encountered at which depths. A gradual smooth curve indicates velocity increasing steadily with depth. A discontinuity in the curve indicates a sharp velocity jump (like the Moho). A triplication (multiple arrivals at the same distance) indicates a rapid velocity increase. The Herglotz-Wiechert formula inverts the p(Δ) function directly into a v(z) profile. Modern tomography extends this by fitting travel times from many earthquakes and stations to build 3D velocity models.
The inverse problem — recovering Earth structure from surface observations — is the core of seismic tomography. The travel-time curve is the bridge between observable quantities (arrival times) and the target (velocity structure). Every shape feature of the curve has a physical interpretation in terms of what the interior is doing to the rays.