The Basics of Earthquake Location

1. The Basics of Earthquake Location William Menke Lamont-Doherty Earth Observatory Columbia University

2. The basic data in earthquake location is Arrival Time, t The time of day that a wave from the earthquake arrives at a seismograph station

3. The distinction between Arrival Time: time of day something arrives And Travel Time: the length of time spent traveling Is very important in earthquake location!

4. Arrival Time ≠Travel TimeQ: a car arrived in town after traveling for an half an hour at sixty miles an hour. Where did it start?A. Thirty miles awayQ: a car arrived in town at half past one, traveling at sixty miles an hour. Where andwhen did it start?A. Are you crazy?

5. An earthquake location has 4 Parameters x, y (epicenter) z (depth)t (origin time)Together, (x, y, z) are called the hypocenter. The fact that origin time is an unknown adds complexity to the earthquake location problem!

6. Suppose you contour arrival timeon surface of earth Earthquake’s (x,y) is center of bulls-eye but what about its depth?

7. Since origin time unknownwe have not marked it on time axis Deep Earthquake’s depth related to curvature of arrival time at origin Shallow

8. Fundamental data:arrival time tpi of wavesfrom earthquake p to station i Wave could be either P wave or S wave. Both are used.

9. Fundamental RelationshipArrival Time = Origin Time + Travel Timetpi = tp + Tpi Traveltime Tpi along ray connecting earthquake p with station q can be calculated using ray theory ray earthquake p with origin time tp

10. Locating an earthquakerequires knowing theearth’s seismic velocity structureaccuratelyso that traveltime can be calculatedbetween stations and hypothetical hypocenters

11. examplevelocitystructure(Iceland)in this caseassumed to vary only with depth

12. Basic Principle Best estimates of the hypocentral parameters and origin time are the ones that best predict the arrival times at all the stations. Usually, “best predicts” means minimizing the least-squares prediction error, E: Ep = Si [ tpiobserved – tpipredicted ]2 where tpipredicted = tppredicted + Tpipredicted and where Tpipredicteddepends on (xp, yp, zp)

13. The mathematical problem is to find the hypocentral parameters, xppredicted=(xp, yp, zp)predicted and origin time, tppredicted that give the best fit (which is to say, minimize the error) But the problem is that the traveltime varies in a complicated, non-linear way with the hypocentral parameters, xppredicted

14. The usual solution is to use an iterative method: Step 1: Guess a set of hypocentral parameters, h=(xp, yp, zp, tp) = (xp, , tp) and use it to predict the traveltime Step 2: Determine how much the arrival time would change if the guess were changed by a small amount, dh =(dx,dt). Step 3: Use that information to attempt to find a slightly different h that reduces the error, E. Do steps 2 and 3 over and over again, hoping that eventually the error will become acceptably small.

15. It turns out that Step 2 is incredibly easy. A small change in origin time, dt, simply shifts the arrival times by the same amount, dt = dt. The effect of a small change in location depends on the direction of the shift. A change dx along the ray direction shifts the time by dt=dx/v. But a change perpendicular to the ray has no effect. This is Geiger’s Principle, and illustrated in the next slide.

17. Step 3 is pretty easy too. The trick is to realize that the equation that says the observed and predicted traveltimes are equal is now linear in the unknowns: tpiobs = tpipre = tp + Tpipre = tpguess + dt + Tpipre(xpguess) + (t/v)dx Or by moving two terms to the left: tpiobs - Tpipre(xpguess) - tpguess = dt + (t/v)dx

18. The methodology for solving a linear equation in the least-squares sense is very well known. It requires some tedious matrix algebra, so we wont discuss it here. But is routine.

19. but with any method, a key question is … What can go wrong? here are some possibilities …

20. Too few data … Since there are four unknowns,you must have at least four arrival time measurements. Any fewer, and you cannot locate the earthquake.

21. Bad Station Geometry … But P and S waves from each of two stations won’t do it, because there is a left-right ambiguity earthquake here? station 1 station 2 or here?

22. When the stations are all to one side of the stations, the rays all leave the source in roughly the same direction and location trades off with origin time Another poor geometry … station 2

24. shallow and late deep and early Depending upon ray geometry, this trade-off can also involve depth and origin time

25. Recently, a new earthquake location method has been developedthat instead of locating a single earthquake on the basis of its arrival times (as above)locates groups of earthquakes on the basis of the differencein their arrival times

26. This method is often called theDouble-Difference Methodthe following figures illustrate its power

27. Earthquakes in Long Valley Caldera, California located with traveltimes Note amorphous clouds of earthquakes, no evidence of faults. Courtesty of Felix Walhhauser, LDEO

28. Earthquakes in Long Valley Caldera, California located with the double-difference method Note many earthquakes fall on lines, so there is clear evidence of faults! Courtesty of Felix Walhhauser, LDEO

29. The basic data in the double-difference method is the differential arrival time between two different earthquakes observed at the same station: Dtpqi = tpi - tqi

30. But that is another story …