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

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

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!

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

An earthquake location has 4 Parameters x, y (epicenter) z (depth)  (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!

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

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

Fundamental data: arrival time t pi of waves from earthquake p to station i Wave could be either P wave or S wave. Both are used.

Fundamental Relationship Arrival Time = Origin Time + Travel Time t pi =  p + T pi Traveltime T pi along ray connecting earthquake p with station q can be calculated using ray theory ray earthquake p with origin time  p

Locating an earthquake requires knowing the earth’s seismic velocity structure accurately so that traveltime can be calculated between stations and hypothetical hypocenters

example velocity structure (Iceland) in this case assumed to vary only with depth

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: E p =  i [ t pi observed – t pi predicted ] 2 where t pi predicted =  p predicted + T pi predicted and where T pi predicted depends on (x p, y p, z p )

The mathematical problem is to find the hypocentral parameters, x p predicted =(x p, y p, z p ) predicted and origin time,  p predicted 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, x p predicted

The usual solution is to use an iterative method: Step 1: Guess a set of hypocentral parameters, h=(x p, y p, z p,  p ) = (x p,,  p ) 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,  h =  x, . 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.

It turns out that Step 2 is incredibly easy. A small change in origin time, , simply shifts the arrival times by the same amount,  t = . The effect of a small change in location depends on the direction of the shift. A change  x along the ray direction shifts the time by  t=  x/v. But a change perpendicular to the ray has no effect. This is Geiger’s Principle, and illustrated in the next slide.

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: t pi obs = t pi pre =  p + T pi pre =  p guess +  + T pi pre (x p guess ) + (t/v)  x Or by moving two terms to the left: t pi obs - T pi pre (x p guess ) -  p guess =  + (t/v)  x

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.

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

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.

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? or here? station 2 station 1

Another poor geometry … 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 station 2

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

Recently, a new earthquake location method has been developed that instead of locating a single earthquake on the basis of its arrival times (as above) locates groups of earthquakes on the basis of the difference in their arrival times

This method is often called the Double-Difference Method the following figures illustrate its power

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

Courtesty of Felix Walhhauser, LDEO 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!

The basic data in the double-difference method is the differential arrival time between two different earthquakes observed at the same station: Dt pqi = t pi - t qi

But that is another story …