Questions: Earthquake Location and Hypocenter Determination

Question 1

A seismologist records P and S wave arrivals at a single station following an earthquake, with a P–S arrival time gap of 25 seconds. What can be determined from this measurement alone?

• A. The exact latitude and longitude of the epicenter
• B. The depth of the hypocenter below the surface
• C. The approximate distance from the station to the earthquake source
• D. The magnitude and seismic moment of the earthquake

Answer: (C)

Because P and S waves travel at known speeds, the time gap between their arrivals is proportional to the distance traveled. A single station gives a distance estimate — placing the source somewhere on a sphere of that radius — but provides no directional information. At least three stations are needed to triangulate a 3D location. Magnitude requires wave amplitude, not just arrival timing.

Question 2

The double-difference relocation method achieves sub-kilometer precision for earthquake clusters primarily because:

• A. It uses a much denser network of seismometer stations than conventional methods
• B. It applies a probabilistic Bayesian framework instead of least-squares inversion
• C. Nearby earthquakes share nearly identical ray paths, so velocity model errors cancel in the difference
• D. It measures absolute rather than differential arrival times, reducing timing uncertainty

Answer: (C)

The core insight is that two earthquakes occurring close together send waves along nearly the same ray paths to any given station. Velocity model errors affect both wave trains almost identically, so when you difference the arrival times of the two events, those errors cancel out. The remaining signal reflects only the small spatial separation between the two hypocenters, enabling very precise relative locations even with an imperfect velocity model.

Question 3

The epicenter of an earthquake is the underground point where fault rupture initiates.

• T. True
• F. False

Answer: False

The hypocenter (or focus) is the point underground where rupture initiates. The epicenter is the point on Earth's surface directly above the hypocenter. This distinction matters in hazard assessment: the epicenter may be far from the actual rupture zone if the fault is deep or has significant along-strike extent.

Question 4

Recording P and S wave arrivals at only two seismometer stations is insufficient to uniquely determine the three-dimensional location of an earthquake hypocenter.

• T. True
• F. False

Answer: True

Locating a hypocenter requires solving for four unknowns: three spatial coordinates (x, y, z) plus origin time. Each P–S pair from a station constrains distance but not direction. Two stations provide two distance spheres that intersect in a circle, not a point — the location remains ambiguous. In practice, at least three stations (providing multiple distance spheres whose intersection narrows to one or two points) are needed, and four or more are required to solve the full four-unknown system reliably.

Question 5

Why does Geiger's method use an iterative algorithm to locate earthquake hypocenters rather than solving directly for the source coordinates in a single calculation?

Model answer: The relationship between hypocenter coordinates and observed arrival times is nonlinear — it depends on ray tracing through a velocity model with curved ray paths. Geiger's method linearizes this relationship around a trial solution, solves the linear system for a correction, applies the correction, and repeats. Direct solution would require inverting a nonlinear system, which has no general closed-form solution.

This is a general feature of inverse problems in geophysics: the forward model (computing predicted arrivals from a given location) is straightforward, but inverting it to find location from arrivals requires iterative refinement. The quality of convergence depends on how good the initial guess is and how well-conditioned the problem is — which is why network geometry and velocity model accuracy matter so much.