Application of Mathematics in Real-World Usage · Guide 39 · BTT Mathematics Hub
Earthquake Mathematics combines logarithms, waves, geometry and inverse problems. A seismometer records motion at one location; scientists infer an event’s size and source location from multiple measurements whose travel times, amplitudes and directions are not directly visible in the raw trace.
This is a Mathematics teaching guide, not earthquake prediction, hazard assessment or emergency guidance. Every station geometry and simplified wave speed below is fictional unless explicitly identified as a general scientific relationship. Real earthquake locations use calibrated velocity models, many stations and uncertainty analysis.
Magnitude is logarithmic
USGS explains that common earthquake magnitude scales are logarithmic. A whole-number increase corresponds to about ten times the measured wave amplitude under comparable conditions.
A magnitude difference ΔM=2 therefore corresponds to an amplitude ratio10²=100 in the simplified comparison.
Energy grows faster than amplitude
USGS gives the commonly used approximation that each magnitude unit corresponds to about32 times the radiated energy.
A two-unit difference therefore implies approximately32²=1,024 times the energy. Using the more exact teaching relationship10^(1.5ΔM), ΔM=2 gives1,000.
Magnitude and shaking intensity are different quantities
Magnitude describes earthquake size at the source. Local shaking at a site depends additionally on distance, depth, geology and other effects.
One event therefore has one preferred magnitude estimate but can produce different observed intensities at different locations.
Wave speed connects distance and travel time
For constant speed v over path length d, travel time is t=d/v.
In a fictional uniform medium, a P wave moving at6km/s takes20s to travel120km.
P waves arrive before S waves because they travel faster
USGS notes that P waves arrive before S waves. The gap between their arrivals generally grows with source distance.
In a simplified medium with vP=6km/s and vS=3.5km/s, a source84km away gives tP=14s and tS=24s, so S−P=10s.
S−P time can estimate distance in a constant-speed model
If Δt=tS−tP, then Δt=d(1/vS−1/vP). Hence d=Δt/(1/vS−1/vP).
With Δt=10s,vP=6 andvS=3.5km/s, d=84km.
Real Earth travel times use curves rather than one constant speed
USGS publishes P and S-P travel-time tables because seismic waves change speed and path through Earth’s layered interior.
The constant-speed calculation above is therefore a teaching approximation for the algebra, not a substitute for real travel-time models.
Wavelength is speed divided by frequency
For a sinusoidal wave, λ=v/f.
A model wave moving at6km/s with frequency2Hz has wavelength3km.
One station gives distance but not direction in the circle model
If a station determines that an epicentre lies5km away, the possible map locations form a circle of radius5km around that station.
A single radius therefore cannot identify one point without directional information.
Two station circles can leave two possible intersections
Place stations at(0,0) and(6,0), each with estimated epicentral distance5. Solving x²+y²=25 and(x−6)²+y²=25 gives x=3 and y=±4.
Two stations reduce the possibilities to two points in this symmetric case.
A third station selects the common intersection
If a third station’s distance circle passes through(3,4) but not(3,−4), the upper point is the unique three-circle intersection in the ideal model.
USGS describes this circle-intersection method as the basic geometric idea behind triangulating an epicentre from several stations.
Measurement error turns a point intersection into an uncertainty region
If each estimated station distance has uncertainty, the circles become annular bands rather than infinitely thin curves.
Their overlap can form an area. Modern location methods therefore use optimisation and residuals rather than expecting perfect circle intersections.
Travel-time residuals measure model mismatch
A residual can be observed arrival time minus predicted arrival time.
If observed P arrival is25.4s and model predicts24.9s, residual is+0.5s. A collection of residuals helps test whether a proposed source location and velocity model fit the data.
Root-mean-square residual summarises several timing errors
For residuals0.3,−0.4 and0.0s, RMS=√[(0.09+0.16+0)/3]≈0.289s.
RMS removes sign cancellation by squaring before averaging.
Logarithmic inversion recovers magnitude differences
If an ideal amplitude ratio is100, the corresponding simplified magnitude difference is log10(100)=2.
If an energy ratio is1,000 under E ratio=10^(1.5ΔM), then ΔM=log10(1000)/1.5=2.
A complete earthquake calculation distinguishes source, path and station
State the magnitude type, station geometry, wave phase, travel-time model, units, and whether the calculation concerns source size, propagation or location.
The return path is rupture source → seismic waves → travel path → station arrivals/amplitudes → inverse calculation → source estimate with uncertainty.
Practice: twenty earthquake Mathematics questions
- Magnitude difference1: find ideal amplitude ratio.
- Magnitude difference2: find ideal amplitude ratio.
- Using32 per magnitude unit, find approximate energy ratio for ΔM=2.
- Using10^(1.5ΔM), find energy ratio for ΔM=2.
- At6km/s, how long does120km take?
- For d=84km,vP=6km/s, find P travel time.
- For d=84km,vS=3.5km/s, find S travel time.
- Find S−P gap from questions6–7.
- Use Δt=10s,vP=6,vS=3.5 to recover distance.
- At v=6km/s,f=2Hz, find wavelength.
- A station estimates epicentral distance5km. What geometric locus is implied?
- Stations at(0,0),(6,0), both radius5: find x-coordinate of intersections.
- Find y-coordinates.
- Observed arrival25.4s, predicted24.9s: find residual.
- Residuals.3,−.4,0: find RMS.
- An amplitude ratio100 corresponds to what ideal magnitude difference?
- An energy ratio1000 corresponds to what ΔM under10^(1.5ΔM)?
- Why does one station not determine epicentre direction in the circle model?
- Why do real earthquake travel-time calculations use tables/curves instead of one constant speed?
- Why does measurement error produce an uncertainty region rather than one exact intersection?
Worked answers
- 10.
- 100.
- About1,024.
- 1,000.
- 20s.
- 14s.
- 24s.
- 10s.
- 84km.
- 3km.
- A circle of radius5km around the station.
- x=3.
- y=±4.
- +0.5s.
- About0.289s.
- 2.
- 2.
- Because every direction at the same radius is still possible.
- Because Earth is layered and seismic speeds/paths vary with depth and material.
- Because uncertain distances correspond to bands of possible radii whose overlaps occupy an area.
Sources and connected applications
For magnitude and energy scaling, see USGS: Earthquake Magnitude, Energy Release, and Shaking Intensity. For P/S arrivals and epicentre location, see USGS: The Science of Earthquakes and USGS: Earthquake Travel Times.
Continue with Heat Transfer, Thermal Resistance, Insulation and Cooling; Structural Mechanics, Loads, Stress, Strain, Deflection and Safety Factors; and Surveying, Triangulation, Levelling, Traverse Closure and Error. Return to the BTT Mathematics Hub.