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July/August 2001

Thanks to the many readers who have sent their comments and suggestions to me over the past year. I plan to use your suggestions when I return to those topics in future columns. This issue's column features a rapid return to the subject of earthquake location, which generated an unusual level of correspondence. Here, I mention two contributions. First, Prof. José Pujol (University of Memphis) has created graphics related to the classic geometric techniques of earthquake location, and links to his figures can be found via the EduQuakes Web page. Second, Prof. Cinna Lomnitz (UNAM) has provided a guest column that further discusses the problems of hypocenter determination. Here is the quite engaging column from Cinna Lomnitz.

How Not to Locate Epicenters

Guest Columnist:
Cinna Lomnitz
Department of Geophysics

Earthquake location is not just one of the most basic and important activities in seismology, it is still an unsolved problem. Ruff ("EduQuakes", SRL 72, p. 197, 2001) lifts the lid off an ancient bucket of worms and provides an enticing glimpse of its contents. The geometric technique for epicenter location is the way to go; it is absolutely travel-time free. The stations are numbered in ascending order of arrival. For each sequential pair one traces the great circle which bisects the arc between stations. The resulting connected polygon is the epicenter and its area is the epicentral error. It is easy to show that this method permits an epicentral location to any desired accuracy by increasing the number of observations.

Can travel-time free methods be further refined? They can, by introducing the assumption that the travel-time function is continuous and nondecreasing with distance. This leads to some tricky Kolmogorov-Smirnov statistics on the sphere, a feast for an enterprising graduate student. Gutenberg did it by fitting a sine function to the minimum travel-time residuals as a function of event-to-station azimuth. But Jeffreys (a genius at mathematics) hated Gutenberg's method because it showed that regression on all the travel times is essentially incorrect. Since Jeffreys did not read seismograms, he never realized that late readings are more probable than early ones. He once got "steamed up" because I dared to suggest such a thought.

What about the estimation of focal depth? The messy method based on joint nonlinear regression merely succeeds in smearing errors all over time and space. But suppose that a travel-time free epicentral solution has been obtained. What now? Well, now would be a good time to discard time-domain regression altogether, since the sensitivity of travel times to depth decreases beyond a distance equal to the depth. We need plenty of data in the epicentral region. Even so, our focal depth estimate can only be as good as the Earth model.

Teleseismic data are only marginally useful. Systematic errors in focal depth are due to a combination of poor statistics and inadequate Earth models. In California, where the situation is better than elsewhere, focal depths are routinely overestimated by around 5 km. One fares considerably worse along subduction coasts because of the lopsided distribution of the data: Nearly all the observations are in the landward quadrants, and the epicentral polygon becomes elongated normal to the coast. The resulting loss of accuracy contaminates the error in focal depth. The number of stations rapidly grows with distance through the teleseismic range, and the distant stations can overpower the near stations in a statistical analysis. Statistical estimation on the sphere can be quite tricky (see book by Mardia).

Finally, let me mention the sticky matter of distance-dependent accuracy. Many colleagues are apt to feel offended when you argue that a distant P arrival ought to carry less weight than a near P arrival. But the fact is that teleseismic P-wave amplitudes and frequencies are closer to thebackground noise. Of course, most of our observations are teleseismic for global seismicity. All these difficulties are expressed by the prevailing philosophy and practice of the global data centers, where shallow earthquakes are assigned a default focal depth.

Using travel times to estimate earthquake locations is clearly not the ideal procedure; it is like estimating a distance along a rough path by the time it takes to walk it. To make matters worse, the travel-time tables are estimated from epicenters which were calculated from earlier versions of these same tables. We all hope that this cumbersome bootstrap procedure converges ... good luck!

SRL encourages guest columnists to contribute to "EduQuakes." Please contact Larry Ruff with your ideas. His e-mail address is

Posted: 12 March 2001