How Mathematics Improves The World | Finding Where an Earthquake Began From Waves Recorded Far Away

How Mathematics Improves The World | Finding Where an Earthquake Began From Waves Recorded Far Away

An earthquake starts where nobody can watch it happen.

Kilometres underground.

Inside opaque rock.

On a fault surface whose rupture may race through the crust faster than any human observer could follow.

Then the ground moves in cities, mountains and coastlines far away.

Seismometers record wiggles.

From those wiggles, scientists must answer:

Where did the earthquake begin?
How deep was it?
When exactly did rupture start?
How large was it?
Which fault may have moved?

The first three questions form an inverse problem.

We know when waves reached instruments.

We do not know the hidden source that launched them.

Mathematics works backwards.

Quick Read

An earthquake radiates seismic waves outward from its hypocentre, the point inside Earth where rupture begins. The point on the surface directly above it is the epicentre. Seismic stations record when different wave types arrive. P waves travel faster than S waves, so the gap between their arrivals provides information about distance to the source.

The familiar classroom method draws a circle around each station using the inferred distance. Three circles intersect near the epicentre. Real earthquake location is more sophisticated. Earth is three-dimensional, wave speed varies with rock type and depth, measurements contain error, and the source has four fundamental unknowns: latitude, longitude, depth and origin time. Modern location software therefore predicts arrival times from a trial hypocentre, compares them with observed arrivals, calculates residuals and iteratively adjusts the source to minimise mismatch—commonly using least squares or related robust methods.

The U.S. Geological Survey describes this explicitly: start with a guessed location, depth and origin time, calculate travel times through a velocity model, compare with observed wave arrivals, solve for small corrections and repeat until the fit stabilises. USGS also notes that routine locations are never exact because both arrival picks and Earth-velocity models contain uncertainty.

One-sentence answer: Mathematics improves the world by turning differences in seismic-wave arrival times across a network into an estimate of an invisible earthquake’s position and depth, allowing scientists and emergency systems to map what happened beneath the ground even though the source itself could never be directly observed.

Epicentre and Hypocentre Are Not the Same Thing

The hypocentre, also called the focus, is the three-dimensional point where rupture begins.

The epicentre is the point on Earth’s surface directly above the hypocentre.

If an earthquake starts 12 km underground, plotting only the epicentre throws away depth.

Depth matters.

A shallow earthquake can produce much stronger shaking near the source than a deeper earthquake of similar magnitude because seismic energy travels a shorter path to the surface.

Depth can also help identify which fault plane or tectonic structure is involved.

The real location problem is therefore four-dimensional:

x coordinate;
y coordinate;
depth z;
origin time t₀.

P Waves Arrive First

P waves are compressional body waves.

They alternately compress and dilate rock in the direction of propagation.

They travel faster than S waves.

That is why P stands for primary.

USGS gives an intuitive comparison.

Lightning arrives before thunder because light travels faster than sound.

P waves arrive before S waves because their seismic velocity is higher.

The farther the station is from the earthquake, the larger the P–S arrival-time separation generally becomes under a given Earth model.

S Waves Arrive Later

S waves shear material perpendicular to their travel direction.

They are slower than P waves.

They do not propagate through liquids in the same way P waves do, which is one reason seismic-wave behaviour helped reveal that Earth’s outer core is liquid.

For local earthquake location, the key quantity is often the difference:

Δt = t_S − t_P

If the P and S arrivals are close together, the earthquake is probably nearby.

If the gap is larger, it is farther away.

The Classroom Geometry: Three Circles

Suppose Station A estimates the earthquake is 100 km away.

Draw a circle of radius 100 km around A.

The earthquake lies somewhere on that circle.

Station B estimates 150 km.

Draw another circle.

The two circles intersect at two possible locations.

Station C adds a third circle.

Ideally, the three intersect at one point.

That point is the epicentre.

USGS uses this triangulation explanation for teaching because it makes the core geometry visible.

Real location software goes beyond circles because depth and heterogeneous Earth structure matter.

Travel-Time Curves: Distance Is Hidden Inside Arrival Time

Seismologists build travel-time curves showing how long P and S waves take to reach stations at different source distances and depths.

Because Earth is layered, these curves are not perfectly straight.

Wave speed changes with depth.

Rays bend according to seismic refraction.

Some phases reflect from interfaces.

Others pass through the mantle and core.

USGS publishes P and S–P travel-time tables that show how time differences grow with distance for assumed source depth.

A timing measurement becomes distance only after a velocity model gives it meaning.

The Earth Model Is Part of the Location

If all crustal rock transmitted P waves at exactly 6 km/s, location would be easier.

It does not.

Sedimentary basins are slower.

Dense crystalline rocks are faster.

Velocity changes with depth.

Fault zones can alter travel paths.

So a seismic location requires a velocity model.

USGS states directly that earthquake location needs a mathematical model of crustal velocities beneath the network to calculate predicted travel times.

The earthquake is inferred relative to a model of the Earth.

Forward Problem: If the Earthquake Were Here, When Would the Wave Arrive?

The forward problem is conceptually straightforward.

Choose a candidate hypocentre and origin time.

Use the velocity model to calculate travel time from source to each station.

Add origin time.

Predicted arrival:

t_predicted,i = t₀ + T(x, y, z → station i)

If the source hypothesis is correct and the velocity model perfect, predicted and observed arrivals match.

They rarely match exactly.

Residuals: The Difference Between Predicted and Observed Time

For station i:

r_i = t_observed,i − t_predicted,i

If r is positive, the observed wave arrived later than predicted.

If negative, earlier.

One residual may be a picking error.

A systematic pattern across many stations may mean the trial source is in the wrong place or the velocity model is biased.

The location algorithm asks:

How should I move the source in x, y, z and origin time so the residuals become smaller?

Linearise the Nonlinear Problem

Travel time depends nonlinearly on source coordinates.

But if the current trial hypocentre is reasonably close, small changes can be approximated linearly with derivatives.

For each station:

r_i ≈ (∂T/∂x)Δx + (∂T/∂y)Δy + (∂T/∂z)Δz + Δt₀

Collect equations from many stations.

Write them as a matrix system:

G Δm ≈ r

where Δm contains corrections to source coordinates and origin time.

Linear algebra turns dozens or hundreds of station residuals into one update.

Least Squares: Find the Correction That Best Fits All Stations

Measurement noise means no correction perfectly satisfies every station.

Least squares chooses Δm that minimises:

Σ r_i²

or a weighted version where more reliable picks receive greater weight.

USGS describes routine earthquake location this way: solve a system of linear equations by least squares, adjust the hypocentre and repeat.

Many imperfect clocks become one best-fitting source.

Iteration: Guess, Compare, Correct, Repeat

Step 1: choose an initial hypocentre.

Step 2: predict arrival times.

Step 3: calculate residuals.

Step 4: solve for a correction.

Step 5: update the source.

Step 6: repeat until corrections and residual changes become small.

This is numerical optimisation in a geological setting.

The earthquake location is not “read” from the seismogram.

It is solved iteratively from many records.

Origin Time: The Clock We Never Observed Directly

No instrument sits at the hypocentre waiting to timestamp rupture start.

Origin time is inferred simultaneously with position.

Move the source farther from every station and predicted travel times increase.

Move origin time earlier and predicted arrivals shift earlier.

Distance and origin time can trade off.

A well-distributed network helps separate them.

The invisible earthquake contains an invisible timestamp as well as an invisible coordinate.

Station Geometry Matters

Ten stations all north of the earthquake do not constrain the source as well as ten distributed around it.

This is analogous to GNSS dilution of precision.

If stations surround the source, arrival times constrain position from many directions.

If the source lies outside the network, depth and horizontal location become more correlated.

Observatory design is therefore a geometry problem.

Where instruments are placed influences what earthquakes can be located well.

Depth Is Usually Harder Than Epicentre

Surface stations surround an epicentre horizontally more easily than they surround a hypocentre vertically.

All instruments sit above the source.

Depth can therefore be poorly constrained, especially if the nearest station is far away.

Small errors in velocity model can translate into significant depth errors.

Dense local networks improve depth resolution because nearby arrival times respond strongly to source depth.

Depth Phases: The Surface Creates a Second Clock

For larger distant earthquakes, depth can be estimated using phases such as pP and sP.

A pP phase travels upward from the source, reflects from Earth’s surface near the epicentre, then continues as a P wave to the station.

The time difference pP − P depends strongly on source depth.

USGS notes that pP and sP depth phases provide accurate focal-depth information when they can be identified.

A reflection from the surface becomes a depth ruler.

Picking Arrivals: Where Exactly Did the P Wave Begin?

A seismogram is noisy.

Wind shakes instruments.

Traffic creates vibrations.

Previous earthquakes leave coda waves.

The first P-wave motion may emerge gradually rather than as a sharp spike.

Analysts or algorithms “pick” the arrival time.

A pick may have uncertainty of hundredths, tenths or more seconds depending on signal quality.

At 6 km/s, a 0.1-second timing error corresponds to roughly 600 metres of travel distance before geometry and ray-path effects are considered.

Timing precision becomes spatial precision.

Automatic Picking: Let Algorithms Find the First Motion

Modern seismic networks receive continuous data from many stations.

Human analysts cannot inspect every trace instantly.

Automatic algorithms detect sudden changes in signal statistics and identify candidate P and S arrivals.

Classic methods compare short-term and long-term signal averages.

Modern systems increasingly use machine learning trained on large libraries of analyst-picked seismograms.

Automation makes location fast.

It creates new failure modes.

A false pick can generate a false event.

Multiple stations and association algorithms provide cross-checks.

Association: Which Arrivals Belong to the Same Earthquake?

During an aftershock sequence, dozens of earthquakes may occur within minutes.

Station A detects a P wave at 12:01:03.

Station B detects one at 12:01:08.

Station C detects several overlapping phases.

Which picks belong together?

Event association tests whether a common source can explain a set of arrivals under the velocity model.

The network first has to decide that an earthquake exists before locating it.

Robust Statistics: One Bad Station Should Not Move the Earthquake

Ordinary least squares squares residuals.

One enormous residual can dominate the fit.

But a station may have a clock problem.

A phase may be mispicked.

The velocity model may be poor along one ray.

Robust estimators downweight outliers.

Quality-control rules reject impossible residuals.

A resilient location algorithm should use many observations without letting one corrupted one decide the source.

Uncertainty Ellipsoids: The Hypocentre Is a Region, Not a Pixel

Earthquake catalogues print latitude, longitude and depth with decimals.

That presentation can create false confidence.

The true source location has uncertainty.

Least-squares covariance analysis can estimate how residual error maps into uncertainty in source parameters.

The uncertainty is often anisotropic.

Horizontal position may be precise to hundreds of metres while depth is uncertain by several kilometres.

The honest mathematical object is an error ellipsoid or probability distribution.

The catalogue point is a summary.

Relative Location: Nearby Earthquakes Share the Same Earth Errors

Suppose two aftershocks occur one kilometre apart.

The same station receives both waves along almost the same path.

Any velocity-model error along that path affects both similarly.

Subtract their arrival times.

Much of the shared path error cancels.

Relative relocation methods exploit this.

USGS notes that locating clusters of nearby events relative to one another can dramatically sharpen earthquake patterns and reveal faults invisible in routine catalogues.

Absolute location may be uncertain.

Relative geometry can still be exquisite.

Double-Difference Location: Subtract the Residuals

If two nearby earthquakes i and j are recorded at the same station k, compare the difference in observed arrival times with the difference predicted by the model.

The double-difference residual is roughly:

(t_ik − t_jk)_obs − (t_ik − t_jk)_calc

Common station and path effects cancel substantially.

Thousands of event pairs can then be solved together.

A diffuse cloud of hypocentres can collapse onto a narrow fault plane.

Subtraction creates resolution.

Cross-Correlation: Compare Whole Waveforms, Not Just Hand Picks

Nearby earthquakes often produce nearly identical waveform shapes at one station.

Cross-correlate the two traces.

The lag at maximum correlation gives a differential arrival time much more precise than independent manual picks.

Milliseconds become useful.

At seismic velocities, milliseconds correspond to metres of path difference.

Signal processing sharpens geology.

Velocity Tomography: Earthquakes Help Improve the Earth Model That Locates Earthquakes

If one region consistently produces late arrivals, perhaps waves travel more slowly there.

If another produces early arrivals, perhaps it is faster.

Collect travel-time residuals from many earthquakes and stations.

Invert them for three-dimensional velocity structure.

Update the Earth model.

Relocate earthquakes.

Repeat.

The earthquakes improve the model used to locate later earthquakes.

This connects naturally to Seeing Inside Without Cutting Open, where seismic tomography reconstructs Earth’s hidden interior from wave travel.

Magnitude Is a Different Inference

Location asks where and when the source began.

Magnitude asks how large the earthquake source was.

Seismologists measure waveform amplitudes, periods and spectra, correct for distance and instrument response, then calculate a magnitude scale appropriate to the event.

Moment magnitude relates to seismic moment:

M₀ = μ A D

where μ is rock rigidity, A rupture area and D average slip.

Location and magnitude use the same seismograms for different hidden properties.

Magnitude Is Not Intensity

Magnitude describes the earthquake source.

Intensity describes shaking at a location.

One magnitude has many intensities.

A city near the fault may shake strongly.

A distant city weakly.

Soft sediments can amplify shaking.

Buildings respond differently.

This is why locating an earthquake is only the beginning of hazard assessment.

Earthquake Early Warning: Location Is Used While the Waves Are Still Travelling

P waves travel faster than the more damaging S and surface waves.

A station close to the source detects the earthquake first.

Algorithms estimate location and magnitude rapidly.

If a city farther away has not yet received strong shaking, a warning can arrive seconds to tens of seconds earlier depending on distance and system geometry.

That is earthquake early warning.

It is not earthquake prediction.

The earthquake has already begun.

Mathematics races the slower seismic waves.

Prediction and Early Warning Must Not Be Confused

USGS states clearly that scientists cannot currently predict the exact time, location and magnitude of future earthquakes.

Hazard models estimate probabilities over years or decades.

Early warning detects rupture after it starts and warns places that strong shaking has not yet reached.

Location algorithms determine where an event already underway began.

Three different jobs.

Three different meanings of prediction.

Network Latency: A Fast Location Needs Fast Data

A perfect algorithm is useless if station data takes five minutes to arrive.

Modern networks transmit continuous seismic data in near real time.

Clock synchronisation is essential.

A station timestamp error of one second can create kilometres of apparent travel-time error.

GNSS timing often synchronises remote stations.

The earthquake-location system therefore depends on the same global timing infrastructure used by telecommunications and power grids.

Station Density: More Sensors Reveal Smaller Earthquakes

A magnitude 1 earthquake may be visible only at nearby stations.

A magnitude 7 is recorded globally.

Dense local networks detect smaller events and locate them more precisely.

This matters for:

fault mapping;
volcano monitoring;
induced seismicity;
aftershock studies;
geothermal systems.

Observation density changes what geology becomes visible.

Induced Seismicity: Tiny Location Differences Can Matter

Fluid injection, mining, reservoir impoundment and geothermal operations can alter subsurface stresses.

When earthquakes occur near such operations, precise locations help determine whether events cluster along a particular fault or depth interval.

A location error of five kilometres can blur an important relationship.

Relative relocation can resolve structures hundreds of metres or less apart.

Mathematical precision changes causal interpretation.

Volcano Monitoring: Earthquake Migration Can Reveal Moving Magma

Volcanoes generate swarms of small earthquakes as magma and fluids move through rock.

Locate each event precisely.

Plot depth through time.

A cluster may migrate upward.

Another may trace a dike intrusion sideways.

Seismicity becomes a moving image of hidden subsurface processes.

The individual earthquakes are dots.

The pattern is the geologic story.

Machine Learning Can Find More Earthquakes—But the Catalogue Must Stay Physical

Deep-learning pickers can identify subtle seismic phases human analysts miss.

Template matching finds repeated tiny earthquakes hidden in noise.

Catalogues become dramatically more complete.

But machine-learning detection can create false events if noise resembles training patterns.

The answer is not to reject automation.

It is to require network consistency, waveform evidence and physical travel-time plausibility.

A classifier can propose an arrival.

Seismology has to prove that the arrivals fit one source.

A Classroom Thought Experiment: Three Phones Hear a Clap

Place three phones with audio recorders around a room.

Synchronise them as closely as possible.

Clap once from an unknown location.

The sound reaches each phone at a different time.

Multiply arrival-time differences by the speed of sound.

Estimate distance differences.

Can students locate the clap?

Now move one phone.

Geometry changes.

Now add echo.

Arrival picking becomes harder.

The classroom has recreated the structure of earthquake location using sound instead of seismic waves.

A Second Thought Experiment: P and S With Two Walking Speeds

Two students leave the same hidden origin simultaneously.

Student P walks 2 m/s.

Student S walks 1 m/s.

At a station 10 m away:

P arrives at 5 seconds.

S arrives at 10.

Gap: 5 seconds.

At 20 m:

P arrives at 10.

S at 20.

Gap: 10.

The farther the source, the larger the arrival gap.

Students feel the travel-time curve before seeing it.

Primary Mathematics: Earthquake Location Begins With Distance and Time

Primary students already know:

distance;
time;
speed;
circles;
maps;
coordinates;
graphs.

The classroom triangulation method uses exactly these ideas.

One station gives a radius.

Three give an intersection.

The advanced inverse problem grows from a compass and map.

Secondary Mathematics: The Source Becomes a System of Equations

Secondary students add:

simultaneous equations;
vectors;
coordinate geometry;
least squares;
statistics;
functions;
iteration.

Arrival residuals become equations.

The velocity model becomes a function.

Station geometry becomes linear algebra.

Uncertainty becomes covariance.

Advanced Mathematics: Seismology as Inverse Theory

Modern earthquake location draws on:

linear algebra;
nonlinear optimisation;
least squares;
ray theory;
partial differential equations;
probability;
signal processing;
Bayesian inference;
inverse problems;
machine learning.

The source is hidden.

Wave arrivals are observed.

The Earth model connects them.

That is the architecture of inverse science.

Why This Improves the World

1. It tells emergency systems where the earthquake occurred

Rapid location helps identify affected regions and which faults may be involved.

2. It enables earthquake early warning

Fast P-wave detection and rapid location can warn places that strong shaking has not reached yet.

3. It maps hidden faults

Precise clusters of hypocentres reveal fault planes and subsurface structures.

4. It monitors volcanoes and induced seismicity

Changes in earthquake location through time reveal moving fluids and stress changes underground.

5. It improves Earth models

Travel-time residuals can be inverted to refine crustal velocity structure, which in turn improves later locations.

6. It teaches the difference between direct observation and inference

No one sees the hypocentre. Its location is an evidence-based estimate supported by many wave arrivals and an explicit physical model.

What Mathematics Does Not Do

Earthquake location does not predict when a future earthquake will happen.

It does not make the Earth velocity model exact.

It does not make every phase pick correct.

It does not guarantee depth is as precise as horizontal location.

It does not identify the active fault automatically when several faults are nearby.

It does not turn a catalogue coordinate into ground truth without uncertainty.

And a fast automatic first location should not be confused with the more carefully reviewed location that may follow as additional data arrives.

Frequently Asked Questions

How do scientists locate an earthquake?

They measure seismic-wave arrival times at many stations, use an Earth velocity model to predict travel times from trial sources, then solve iteratively for the source latitude, longitude, depth and origin time that best match the observations.

What is the difference between an epicentre and a hypocentre?

The hypocentre is the point underground where rupture begins. The epicentre is the point on Earth’s surface directly above it.

Why are P and S waves useful for location?

P waves travel faster than S waves. The difference between their arrival times grows with source distance under a given velocity model, providing strong information about how far the station is from the earthquake.

Why are at least three stations used in classroom triangulation?

One station gives a circle of possible epicentres. Two circles generally intersect at two points. A third resolves the ambiguity. Real three-dimensional earthquake location typically uses many more stations and also solves for depth and origin time.

Can earthquakes be predicted?

Scientists can estimate long-term earthquake probabilities and rapidly detect earthquakes after they begin, but USGS states that exact prediction of a future earthquake’s time, place and magnitude is not currently possible.

Sources and Further Reading

U.S. Geological Survey, How Do Seismologists Locate an Earthquake?, explaining iterative hypocentre adjustment and least-squares fitting of arrival times.
U.S. Geological Survey, The Science of Earthquakes, including P/S wave timing and the educational triangulation method.
U.S. Geological Survey, Determining the Depth of an Earthquake, explaining depth phases such as pP and sP.
U.S. Geological Survey, Everything Is Relative: Precise Earthquake Location, on improved relative locations and velocity-model uncertainty.
U.S. Geological Survey, Generalized Bancroft Algorithm for Locating Earthquakes with P- and S-Wave Arrival Times, 2025, highlighting mathematical similarities between earthquake location and GPS positioning.

Continue Through eduKateSG

Continue with How Mathematics Works. This article connects directly to Giving Every Place on Earth an Address, because GPS and earthquake location both recover position from signal travel time, and to Seeing Inside Without Cutting Open, where the same seismic waves are used to reconstruct Earth’s interior rather than the earthquake source.

Final Thought: The Earthquake Is Found After It Has Already Left

The fault slips.