Questions: Seismic Ray Tracing and Wave Path Geometry

Question 1

In a region where seismic velocity increases continuously with depth, what happens to the path of a seismic ray emitted from a surface source at an oblique angle?

• A. It travels in a straight line at the takeoff angle, since there are no discrete boundaries to cause bending
• B. It bends increasingly toward the vertical as it descends into faster material
• C. It curves in a broad arc, bending away from the vertical as it descends, eventually returning to the surface
• D. It reflects at depth and returns along the same path it descended

Answer: (C)

Where velocity increases with depth, the ray parameter p = sin(θ)/v must remain constant. As v increases, θ (the angle from vertical) must also increase to keep p constant — the ray bends away from vertical (toward horizontal). In a continuous velocity gradient this produces a smooth arc. Eventually θ approaches 90°, the ray is traveling horizontally, and it then curves back upward — returning to the surface without any discrete reflection boundary.

Question 2

Two seismic rays leave the same earthquake source. Ray A has a small ray parameter p; Ray B has a large ray parameter p. Which statement correctly describes their behavior?

• A. Ray A leaves at a shallow angle, stays near the surface, and arrives at a nearby station
• B. Ray B leaves steeply, penetrates deep into the mantle, and arrives at a distant station
• C. Ray A leaves steeply, penetrates deeply, and arrives at a distant station; Ray B leaves at a shallow angle and arrives nearby
• D. Both rays follow the same path because they originate from the same source and travel through the same velocity structure

Answer: (C)

p = sin(θ)/v; a small p means a small takeoff angle θ (steep departure from vertical). A steeply departing ray penetrates deeply before curving back, traveling greater horizontal distance and arriving at a far station. A large p means a large takeoff angle (shallow departure), so the ray stays in shallow crust and arrives nearby. This is counterintuitive — steep launch angle means greater distance traveled, the opposite of a projectile in uniform gravity.

Question 3

A seismic shadow zone — a region of the surface that receives no direct seismic waves from a distant earthquake — forms because seismic velocity increases sharply at a boundary, refracting most rays away from that region.

• T. True
• F. False

Answer: False

Shadow zones form where velocity decreases (or drops suddenly), not increases. When rays enter a region of lower velocity, they refract toward the vertical (the low-velocity zone acts like a lens that bends rays inward), leaving a gap on the surface where no direct rays emerge. The classic example is the P-wave shadow zone caused by the sudden drop in velocity at the core-mantle boundary, where seismic energy enters the liquid outer core and bends, creating a shadow zone between about 103° and 143° from the earthquake source.

Question 4

The ray parameter p = sin(θ)/v remains constant along the entire path of a seismic ray traveling through a layered or continuously varying Earth velocity structure.

• T. True
• F. False

Answer: True

This is the seismic expression of Snell's law applied to the complete ray path. At every point along the ray, sin(θ)/v is the same constant p. At discrete boundaries, the angle changes precisely to preserve this ratio across the boundary. In a continuous gradient, the angle changes continuously for the same reason. The ray parameter is the ray's fundamental identity: it determines the takeoff angle, the depth of penetration, and the surface arrival distance.

Question 5

Explain why seismic rays traveling through an Earth with velocity increasing with depth follow curved paths that return to the surface, rather than continuing downward indefinitely.

Model answer: The ray parameter p = sin(θ)/v is conserved along the ray path. As velocity v increases with depth, θ (the angle from vertical) must increase to keep p constant. As θ grows toward 90°, the ray becomes horizontal. Since θ continues increasing past 90° (the ray is now pointing upward), the ray curves back toward the surface. In a continuous velocity gradient, this produces a smooth arc without requiring any reflection boundary.

The ray is not reflected — it is continuously refracted by the velocity gradient. This distinction matters: the curved paths are head-wave-like refraction phenomena, not reflections. The steeper the initial angle (smaller p), the deeper the ray penetrates before curving back, since it needs a higher velocity (deeper depth) to accumulate enough change in θ to reverse direction.