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Seismic imaging · A course with experiments

L2 · Chapter 5

Error ellipses for earthquake locations, and what they leave out

The same matrix that moves a trial hypocentre also says how far random pick errors can push the answer. It gives a covariance matrix, and the covariance gives an ellipse that should contain the true epicentre with a chosen probability. That promise can be tested by counting. It holds when the pick errors are the only errors. A speed model that is a few per cent wrong breaks it, and the ellipse itself gives no sign of that.

Sandbox Physics ·

Pick errors become a covariance matrix

Assume every arrival time carries an independent error with the same standard deviation. Near the solution the location is a linear function of the picks, so its errors are Gaussian too. Their covariance is the pick variance times the inverse of the normal matrix from the last chapter. When picks differ in quality, each row is weighted by its own variance; the location Lab reads S picks twice as loosely as P picks.

Klein’s HYPOINVERSE-2000 manual writes the same matrix in Appendix 4 through the eigenvalues and eigenvectors of the problem. For the pick variance it uses the reading error RDERR squared plus a coefficient ERCOF times the squared RMS residual of the event. With ERCOF set to one, an event that fits badly gets a larger ellipse.

The diagonal entries are variances of origin time, the two horizontal coordinates and depth. The off-diagonal entries are trade-offs, such as the one between depth and origin time. Dropping origin time leaves a 3 by 3 block for the hypocentre, and the two horizontal rows and columns form the 2 by 2 block for the epicentre.

C=σ2(GTG)−1=w2 VS−2VT,w2=RDERR2+ERCOF⋅RMS2\mathbf C=\sigma^2\left(\mathbf G^{\mathsf T}\mathbf G\right)^{-1}=w^2\,\mathbf V\mathbf S^{-2}\mathbf V^{\mathsf T},\qquad w^2=\mathrm{RDERR}^2+\mathrm{ERCOF}\cdot\mathrm{RMS}^2

From the covariance to an ellipse

The eigenvectors of the 2 by 2 epicentre block give the directions of the ellipse axes, and the square roots of its eigenvalues give the standard errors along them. Those standard errors draw the 1-sigma ellipse. To reach 95 per cent, every axis is multiplied by a factor that depends on how many coordinates the region covers.

For one coordinate the 95 per cent factor is the familiar 1.96. For two coordinates it is 2.45, and for three it is 2.80. The probability inside the 1-sigma region also falls with dimension: 68.3 per cent on a line, 39.3 per cent inside an ellipse and 19.9 per cent inside an ellipsoid.

Klein’s manual states that the hypocentre has a 32 per cent chance of lying inside the 1-sigma ellipsoid and that a factor of 2.4 gives 95 per cent. Neither matches a three-dimensional Gaussian, for which the values are 19.9 per cent and 2.80. The factor 2.4 is close to the two-dimensional value. The location Lab uses the two-dimensional factor for its epicentre ellipse.

k95=χν, 0.952:ν=1→1.96,  ν=2→2.45,  ν=3→2.80;P1σ=68.3%, 39.3%, 19.9%k_{95}=\sqrt{\chi^2_{\nu,\,0.95}}:\quad \nu=1\to1.96,\ \ \nu=2\to2.45,\ \ \nu=3\to2.80;\qquad P_{1\sigma}=68.3\%,\ 39.3\%,\ 19.9\%

Count how often the ellipse catches the truth

A 95 per cent ellipse makes a testable claim: over many events with independent pick noise, about 95 in 100 true epicentres should fall inside their own ellipses. The location Lab locates a stream of seeded synthetic quakes from the same source and counts.

For the ring network with S picks and 0.1 s of pick noise, 944 of 1000 true epicentres fall inside. With 1000 events the count itself scatters by about 0.7 percentage points, so 94.4 per cent agrees with the promise.

A wrong speed model moves the answer, not the ellipse

Now tell the locator that the rock is 5 per cent faster than it is, and keep everything else. The same picks are relocated. The mean epicentre error grows from 0.31 km to 0.42 km and the mean depth moves from 9.99 to 9.62 km. The ellipse barely changes, because it depends on the geometry and the assumed pick noise, not on the speed error. Only 849 of 1000 true epicentres now fall inside.

The fit gets only a little worse: the mean RMS residual rises from 0.138 s to 0.173 s. A locator can absorb much of a speed error by moving the source and shifting the origin time. That is why the manuals are careful. Lee and Lahr note that the statistical standard errors may not represent actual error limits, and Klein writes that a true error ellipse would need the uncertainty of the crustal model, which is nearly impossible to know.

Worked example

Turn one covariance into an ellipse

Take an epicentre covariance block with an east–east variance of 0.040 square kilometres, a north–north variance of 0.010 and a covariance of 0.010.

Half the trace is 0.025. Half the difference of the diagonal is 0.015, and the square root of 0.015 squared plus 0.010 squared is 0.0180. The eigenvalues are therefore 0.0430 and 0.00697 square kilometres, and the standard errors along the axes are 0.2074 and 0.0835 km.

Multiply both by 2.448, the square root of 5.991, for a 95 per cent epicentre ellipse: the half-axes are 0.508 km and 0.204 km. The long axis is turned from east towards north by half the angle whose tangent is 0.020 over 0.030, 16.8 degrees.

λ±=0.025±0.0152+0.0102=0.0430, 0.00697 km2;a95=2.448λ±=0.508, 0.204 km;θ=12arctan⁡0.0200.030=16.8∘\lambda_{\pm}=0.025\pm\sqrt{0.015^2+0.010^2}=0.0430,\ 0.00697\ \mathrm{km^2};\quad a_{95}=2.448\sqrt{\lambda_\pm}=0.508,\ 0.204\ \mathrm{km};\quad \theta=\tfrac12\arctan\frac{0.020}{0.030}=16.8^\circ

An experiment to think through

Predict, change, explain

  1. In the location Lab, start from Ring network: a tight ellipse and note the coverage readout. Predict what happens to it when you raise the error in the assumed speed to 5 per cent.
  2. Raise the speed error slider to 5 per cent, or choose Wrong speed model.
  3. Explain why the ellipse size hardly changes while the coverage falls.
Locate quakes and test their error ellipses

The Labs use flat, isotropic, elastic or acoustic rock with made-up but realistic speeds; none of them reproduces a real survey. Each Lab states its model and limits in its notes and can export the data behind its readouts.

Try explaining it yourself

1. What factor turns the 1-sigma epicentre ellipse into a 99 per cent ellipse?

Read an explained answer

For two coordinates the probability inside k standard errors is one minus the exponential of minus k squared over two. Setting it to 0.99 gives k equal to the square root of minus two times the natural log of 0.01, which is 3.03.

k99=−2ln⁡0.01=3.03k_{99}=\sqrt{-2\ln 0.01}=3.03

2. The Lab’s coverage readout counts 40 events. If the ellipse is correct, how much can the count scatter by chance?

Read an explained answer

The count is binomial. Its standard deviation in per cent is the square root of 0.95 times 0.05 over 40, about 3.4 percentage points. A reading of 90 per cent is within two standard deviations; with 1000 events the scatter shrinks to 0.7 points.

3. Is 849 inside out of 1000 consistent with a correct 95 per cent ellipse?

Read an explained answer

No. The expected count is 950 with a standard deviation of about 6.9, so 849 lies about 15 standard deviations below. The ellipse is too small for the actual errors.

Sources to read alongside

  1. W. H. K. Lee and J. C. Lahr, HYPO71 (revised): a computer program for determining hypocenter, magnitude, and first motion pattern of local earthquakes, U.S. Geological Survey Open-File Report 75-311 (1975)

    Section 4.1 sets out Geiger’s method: expand the arrival-time residuals to first order in the origin time and the three coordinates (Eq. 2), solve the least-squares normal equations (Eqs. 3 to 5), update the trial hypocentre (Eq. 6) and repeat. The introduction warns that small residuals and standard errors do not guarantee an accurate hypocentre.

  2. F. W. Klein, User’s guide to HYPOINVERSE-2000, a Fortran program to solve for earthquake locations and magnitudes, U.S. Geological Survey Open-File Report 02-171 (2002)

    Appendix 4 builds the 4 by 4 covariance matrix from the eigenvalues and eigenvectors of the location problem, scaled by the variance of the arrival times, and takes the error ellipsoid from its 3 by 3 spatial block. It notes that the ellipse scales with the assumed timing error, and that a true error estimate would also need the uncertainty of the crustal model, which is nearly impossible to know.

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