How to Locate Epicenters
Earthquake location is one of the most basic and important activities in seismology. Hypocentral location is a complex topic that involves technical difficulties such as the reliability of absolute time at seismographs, observational difficulties such as the picking of arrival times, and theoretical difficulties such as error estimates in nonlinear inverse theory. Even so, earthquake location is such a fundamental topic that we want students to do exercises in epicenter location. While there are many resources available for the (S-P) method of epicentral determination, this column discusses a new addition to the EduQuakes online lab manual which shows students how to use just P waves to locate an epicenter.
The classic earthquake location method found in nearly all introductory seismology books is the (S-P) method. The simplest case is for local earthquake location where the epicenter-to-station distances are less than two hundred kilometers, hypocentral depth is less than 15 km, and we already know the crustal P and S wave velocities. Then we can use a flat-Earth map with a uniform x-y grid, and the observed (S-P) time at a single station translates directly into the epicentral distance, which can be drawn as a circle about the station location. With three stations (not all three on a single line), the near intersection of three circles gives the epicentral location. The simplicity and geometric construction aspects of this (S-P) method make it ideal for student exercises. The map-based geometric construction replaces the iterative matrix inversion problem that professional seismologists solve to locate earthquakes. Furthermore, the geometric construction also illustrates the effects of errors. There are many good exercises and problem sets that use the (S-P) method. (For links to online earthquake location resources, go to the EduQuakes online Lab Manual at http://AAMC.geo.lsa.umich.edu/eduQuakes/eduQuakesLab.html. Please send e-mail to me if I have left out a good resource.)
After mastering the (S-P) method, students are surprised to learn that most epicenters are not located this way. When I tell students that catalog hypocenters are mostly based on P-wave arrival times, the inevitable question is, "How does that work?" The students might eventually believe that it can work as an iterative solution to a matrix problem for travel-time residuals, but what we really need is a direct geometric construction technique similar to the (S-P) method.
Before computers, seismologists used clever geometric techniques for earthquake location. In this exercise, I use a classic geometric technique for epicentral location based on P arrival times. There are two levels of consideration. The first level simply uses the order of P-wave arrivals at the stations to define a subregion that contains the epicenter. The second level makes quantitative use of the travel-time difference between P-wave arrivals at any two stations. To do the geometric construction, first draw a line between the two stations, then draw the perpendicular bisector. Clearly, if the P arrival times were identical at both stations, then the epicenter is constrained to lie on the perpendicular bisector. More generally, the travel-time difference between the two stations is multiplied by the crustal velocity, which yields the epicentral distance difference. Hence, we are presented with the geometry problem of finding the curve in the plane such that rays drawn from some point on the curve to the two stations always have the same constant distance difference-students of geometry will recognize this problem as the description of a hyperbola where the station locations are the two foci. While hyperbolas are not as easy to draw as circles, the exercise gives helpful advice on calculating and drawing free-hand hyperbolas (though it is more fun to use a ruler and string to draw a proper hyperbola!). Ponder the number of stations required to locate the epicenter: Two seismic stations give just one hyperbola, while three seismic stations give three hyperbolas. Thus, it seems that three stations (not collinear) are adequate to determine the epicenter whether we use the (S-P) times or the P arrival times. Also, just as for the (S-P) method, students can see how pick errors and crustal velocity uncertainty impart uncertainties to the epicentral location.
The (S-P) method will always be the first introduction to how seismologists can locate earthquakes. But we can move one step beyond that-and get closer to the routine procedures of seismologists-by showing students how P arrival times also determine epicenters with a map-based geometric construction. Virtually all of the advantages of the (S-P) method are retained-we just need to teach students how to draw hyperbolas rather than circles!
SRL encourages guest columnists to contribute to "EduQuakes." Please contact Larry Ruff with your ideas. His e-mail address is firstname.lastname@example.org.
Posted: 12 March 2001