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.

Latest epicentre error $0.0613\,\mathrm{km}$

95 % ellipse, long half-axis $0.6\,\mathrm{km}$

RMS pick residual $0.101\,\mathrm{s}$

True epicentres inside their ellipse $95\,\%\ \text{of}\ 40$

Locate each quake, then check its ellipse

P pick noise (s)

\sigma_{\mathrm P}

0.1

Error in the assumed speed (%)

\delta v/v

0

True depth (km)

z_0

10

S picks

Drag the gold source or any station; every move relocates the same seeded quakes.

More settings

Noise seed

s

3601

Model, units and sources

$\Delta\mathbf m=\left(\mathbf G^{\mathsf T}\mathbf G+\lambda\operatorname{diag}\mathbf G^{\mathsf T}\mathbf G\right)^{-1}\mathbf G^{\mathsf T}\mathbf r,\qquad \mathbf C=\sigma^2\left(\mathbf G^{\mathsf T}\mathbf G\right)^{-1}$

The map is 100 km square; the depth section on the right shares its east–west axis.

Read the course chapter behind this Lab

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

Earthquake Location from Arrival Times and Its Error Ellipse

BackgroundAn 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 mattersFour 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 misconceptionIf the epicentre is sharp, the depth is too.

Better mental modelDepth 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

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.
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.
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.

Earthquake Location from Arrival Times and Its Error Ellipse

Earthquake Location from Arrival Times and Its Error Ellipse

Core mathematical model

Travel time in a uniform half-space

One Geiger step

95 % epicentre ellipse

Common difficulties

Depth is the hardest coordinate

Run the experiment

Check the ellipse honestly

Move the stations to one side

Break the speed model

Related experiments and articles

Seismic waves and imaging, from rock boundaries to tomography