Advances in Earthquake Locationand

Tomography

William Menke

Lamont-Doherty Earth Observatory

Columbia University

Part 1: Advantage of using differential arrival times to locate earthquakes

Part 2: Simultaneous earthquake location and tomography

Part 3: In depth analysis of the special case of unknown origin time

Outline

Part 1

Advantage of using differential arrival times to locate earthquakes

that was the recent Gulf of Mexico earthquake,

by the way …

Locating an earthquakerequires knowing the

seismic velocity structure

accurately

What’s the best way to represent 3 dimensional structure

Best for what?

compatibility with data sources

ease of visualization and editing

facilitating calculation

Overall organization into interfaces

Small-scale organization into tetrahedra

Linear interpolation within tetrahedra implying rays that are circular arcs

seismometer earthquake

Location Errors: = 0.5 degree = 55 km = 30 miles

Note: this preliminary calculation used data from a limited number of stations

Two parallel approaches

work to improve earth model

design earthquake location techniques that are as insensitive to model as possible

Waves from earthquake first arrived in Palisades NY at 15:00:32 on Sept

10, 2006

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 did it start?

A. Are you crazy?

Suppose you contour arrival timeon surface of earth

Earthquake’s (x,y) is center

of bullseye

but what about its depth?

Earthquake’s depth related to

curvature of arrival time at

origin

Deep

Shallow

Courtesty of Felix Walhhauser, LDEO

Earthquakes in Long Valley Caldera, California located with absolute traveltimes

Courtesty of Felix Walhhauser, LDEO

Earthquakes in Long Valley Caldera, California located with differential traveltimes

differential arrival time = difference in arrival times

T = arrival time

TT = travel time

To = Origin Time (start time of earthquake)

mean origin time cancels out

Station i

Very accurate DT’s !

A technical question for Applied Math types …

Are differential arrival times as calculated by cross-correlation less correlated than implied by the formula

They seem to be.

If so, the this is another advantage of using the method

How does differential arrival time vary spatially?

Depends strongly on this angle

In a 3 dimensional homogeneous box …

maximum

meanminimum

If you can identify the line AB, then you can locate earthquakes

as long as you have more than two earthquakes

In a vertically-stratified earth, rays are bent back up to the surface, so both Points A and B are on the surface.

The pattern of differnetial traveltime is more complicated …

ray

wavefront

The same idea works …

p q

Patterns of differential arrival time

C

CC

C

C

B

C

B BA

A A

B

A

Can you guess the orientation of the two sources in these six cases?

This pattern an be seen in actual data, in this case from a pair of earthquakes on the San Andreas Fault

Boxes: differential arrival times observed at particular stations

Shading: theoretical calculation for best-fitting locations of the earthquake pair

C

A

B

Anotherexample …

What is the practical advantageof using differential arrival times

to locate earthquakes

My approach is toexamine the statistics of location errorsusing numerical simulations

Compare the result of usingabsolute arrival time data

Anddifferential arrival time data

Whenthe data are noise

Orthe earth structure is poorly known

Geometry of the numerical experiment …

Effect of noisy data(10 milliseconds of measurement error)

absolute data

absolute data

differential data

differential data

Effect of near surface heterogeneities(1 km/s of velocity variation with a scale length of 5 km)

absolute data

differential data differential

dataabsolute data

Both absolute locations and relative locations of earthquakes are improved by using differential arrival time data

when arrival times are nosily measured andwhen near-surface earth structure is poorly

modeled

Relative location errors can be just a few meters even when errors are “realistically large”

Part 2

Simultaneous earthquake location and tomography

simultaneous earthquake location and tomography?

Many earthquakes with unknown X, Y, Z, To

Unknown velocity structure

Solve for everything

Using either

absolute arrival timesor

differential arrival times

A numerical test

11 stations

50 earthquakeson fault zone

Heterogeneitynear fault zone only

True earthquake locationsAnd fault zone heterogenity( 1 km/s)

Reconstructed earthquake locationsAnd fault zone heterogenity, using noise free differential data

Note the amplitude of the “signal” is only 1 ms, so noise might be a problem.

Reality Check: How big is the Signal?

How much better are the data fit?

When the earth structure is allowed to vary

compared with holding a simple, layered

earth structure fixed?

Answer: 0.7 milliseconds, for a dataset that has traveltimes of a few seconds

Need very precise measurements!

Part 3

Is Joint Tomography/Earthquake Location

Really Possible ?

Study a simplified version of the problem

In depth analysis of the special case of unknown origin timebut known location

Station 1 2 3 4

Event 1

Event 2

Event 3

If you can …

Then that structure is indistinguishable from a perturbation in origin time!

Case of sources near bottom of the model

This velocity perturbation causes constant travel time perturbation for a station on the surface anywhere in the grey box for the event at but zero traveltime perturbation for all the sources at !

Case of sources near top of model

This velocity perturbation causes constant travel time perturbation for a station on the surface anywhere in the grey box for the event at but zero traveltime perturbation for all the sources at !

But you can always find such structures!

And they often look ‘geologically interesting’

Yet their presence of absence in an area cannot be proved or disproved by the tomography.

Summary

Part 1: Earthquake location with differential data is the way to go!

Part 2: Simultaneous tomography / earthquake location possible with differential data, but requires high-precision data.

Part 3: Coupled Tomography/Location is extremely nonunique and extremely likely to fool you.