---
title: "Earthquake Location from Arrival Times and Its Error Ellipse Interactive Lab | Sandbox Physics"
description: "How do arrival times at a few stations pin down where an earthquake started, and how sure can you be?"
canonical_url: "https://sandboxphysics.com/labs/earthquake-location/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/earthquake-location/"
last_modified: "2026-10-31"
---

FR36-B · Geiger’s method and confidence regions

# Earthquake Location from Arrival Times and Its Error Ellipse

**How do arrival times at a few stations pin down where an earthquake started, and how sure can you be?** Each station gives one arrival time, and the source position and start time that best explain all of them is the location. The answer is found by repeated linear corrections. Its error ellipse is tight when stations surround the quake and stretches when they sit on one side, but it ignores errors in the assumed wave speed.

Seeded earthquakes ring a small seismic network. Each one is located from noisy P and S picks by iterated least squares, with a 95 % error ellipse. Drag the stations or the source and see the ellipse stretch.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

A uniform half-space with P speed 6 km/s and S speed 3.46 km/s (the P speed divided by the square root of 3); straight rays; flat surface. Picks are true travel times plus seeded Gaussian noise, with S picks twice as noisy as P picks (assumed). Location uses Geiger’s linearized least squares with Levenberg–Marquardt damping, as in HYPO71 and HYPOINVERSE. The ellipse comes from the linearized covariance and the chi-square quantile for two degrees of freedom, so it ignores model error: the wrong-speed preset shows what happens then. The grid-search map is an independent check of the same misfit.

Latest quake: picks against its fitted travel-time curves

- P picks and model curve
- S picks and model curve

Running check: how often the ellipse holds the truth

- simulated events
- nominal 0.95, with a band of two standard errors

Grid-search misfit around the latest estimate

- Geiger estimate and its 95 % ellipse
- grid-search minimum
- true epicentre

More readouts and the model record

- **Estimated depth**
- **True depth**
- **Depth, 95 % half-width**
- **Ellipse short half-axis**
- **Ellipse long-axis azimuth**
- **Origin-time error**
- **Geiger iterations**
- **Picks used**
- **Grid search minus Geiger, distance**
- **Assumed P speed**

The record saves every pick, estimate, covariance and whether the truth fell inside its ellipse.

[Continue the course: seismic imaging from waves to inversion](https://sandboxphysics.com/stories/seismic-imaging/)

Physics tutorial

## Earthquake Location from Arrival Times and Its Error Ellipse

**Background** An earthquake starts at one point and one moment. P and S waves spread from there and reach each station at a time set by the distance and the wave speed. A seismologist reads these arrival times, called picks, from the records.

**Why it matters** Four unknowns describe the start: east, north, depth and origin time $t_0$. Every pick adds one equation, but travel time depends on distance through a square root, so the equations are not linear.

Start with the essentials

- **Focus question**: How do arrival times at a few stations pin down where an earthquake started, and how sure can you be?
- **One-sentence intuition**: Linearizing about a trial location turns the problem into least squares for a small correction; repeating converges in a few steps. The same matrix gives the covariance and hence the error ellipse, which is only as good as the speed model.

### Core mathematical model

#### Travel time in a uniform half-space

$$
t_i=t_0+\frac{\sqrt{(x_i-x)^2+(y_i-y)^2+z^2}}{v}
$$

Station $i$ sits at the surface and $v$ is the P or S speed.

#### One Geiger step

$$
\mathbf G\,\Delta\mathbf m\approx\mathbf r,\qquad \Delta\mathbf m=\left(\mathbf G^{\mathsf T}\mathbf G+\lambda\operatorname{diag}\mathbf G^{\mathsf T}\mathbf G\right)^{-1}\mathbf G^{\mathsf T}\mathbf r
$$

$\mathbf G$ holds the derivatives of each travel time with respect to the four unknowns and $\mathbf r$ the residuals. The damping $\lambda$ grows only when a full step would raise the misfit, which keeps early steps short when the trial point is far off.

#### 95 % epicentre ellipse

$$
\mathbf C=\sigma^2\left(\mathbf G^{\mathsf T}\mathbf G\right)^{-1},\qquad \Delta\mathbf x^{\mathsf T}\mathbf C_{xy}^{-1}\Delta\mathbf x\le\chi^2_{2}(0.95)=5.99
$$

The horizontal block of the covariance defines an ellipse; its half-axes are 2.45 times the square roots of its eigenvalues.

### Common difficulties

#### Depth is the hardest coordinate

**Typical misconception** If the epicentre is sharp, the depth is too.

**Better mental model** Depth trades off against origin time when all stations are far away. S picks or a station close to the epicentre break the trade-off; try P picks only: depth goes loose.

### Run the experiment

1. 01

   #### Check the ellipse honestly

   Start from Ring network: a tight ellipse and watch the coverage chart fill up.

   **What to observe:** About 95 in every 100 true epicentres fall inside their own ellipse, inside the sampling band.

2. 02

   #### Move the stations to one side

   Choose Stations on one side, or drag stations to the left edge yourself.

   **What to observe:** The ellipse stretches along the line from the network to the quake: distance is constrained less well than direction.

3. 03

   #### Break the speed model

   Choose Wrong speed model, then move the speed error back to zero.

   **What to observe:** With a 5 % speed error the ellipses still look tight, but fewer than 95 % contain the truth. The ellipse only measures pick noise.

Physical conclusion

Location is least squares on arrival times; the ellipse is honest about noise and silent about model error.

## Related experiments and articles

Seismic imaging · Reading path

### Seismic waves and imaging, from rock boundaries to tomography

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Locating an earthquake from P and S arrival times**](https://sandboxphysics.com/stories/seismic-imaging/locating-an-earthquake/)
2. [Article **Error ellipses for earthquake locations, and what they leave out**](https://sandboxphysics.com/stories/seismic-imaging/error-ellipses-and-model-error/)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)