Advances in Earthquake Locationand

Tomography

William Menke

Lamont-Doherty Earth Observatory

Columbia University

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

10, 2006

that was the recent Gulf of Mexico earthquake,

by the way …

Locating an earthquakerequires knowing the

“seismic velocity structure*”of the earthaccurately

*the scalar fields Vp(x) and Vs(x)

(which are strongly correlated)

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?

Big Issue: Representing 3 dimensional structure

What’s the best way?

compatibility with data sources

ease of visualization and editing

embodies prior knowledge

e.g. geological layers

facilitating calculation

Overall organization into interfaces

Small-scale organization into tetrahedra

Linear interpolation within tetrahedra implying rays that are circular arcs

Thickness of Earth’s Crust

Compressional Velocity just below Crust

Overall model has 1.3106 tetrahedra

seismometer earthquake

Variations in Traveltime due to 3D earth structure

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

Geometrical Ideas

What are the important characteristics of arrival time data that allow earthquakes to be located ?

(Careful thinking is more important than furious scribbling of formula … )

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

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

differential arrival time = difference in arrival times

1)

Very accurate DT’s !

2) Use cross-correlation to measure differential arrival times

Issue: Statistical Correlations in Data

DTpqi = Tpi – Tqi

DTrqi = Tri – Tqi

Then even if errors in T’s uncorrelated, errors in DT’s will be strongly correlate.

Covariance/variance=1/2 Furthermore, relationships exist between different data

DTpqi = DTpri – DTqri

Monte-Carlo simulations:

Differential arrival times as calculated by cross-correlation are less correlated than implied by the formula

covariance:variance = 1/2

Issue: How does the statistics of cross-correlation enter in to the problem?

formulasimulation

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”

Tomography:

Use

To reconstruct

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

Seems to work !

Reality Check: How big is the Signal?

How much better are the data fit?

When the earth structure is allowed to vary

compared with

using a simple, layered earth structure

and keeping it fixed?

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

Need very precise measurements!

What are the other key issues in

Joint Tomography/Earthquake Location

Study a simplified version of the problem

In depth analysis of the special case of unknown origin time

but known location

Cautionary Tale …..

Don’t assume that something is unimportant, just because you’ve eliminated it from the problem !

Since you solve for m first, and use infer x with the formula

Then if there is more than one m that solves the problem, there is more than one x, too.

So we must address the issue of whether the solution for m is unique.

This cute little matrix can be explicitly triangularized by Gaussian elimination. (What a wonderful linear algebra homework problem!). Just one row, the last, is zero, so its rank is indeed Q-1.

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!

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

Earthquake location with differential data works extremely well, for good reasons. But properly assessing errors in locations requires further work.

Simultaneous tomography / earthquake location possible with differential data, but:

- requires high-precision data.

- has an inherent nonuniqueness that and extremely likely to fool you, but that can be assessed by direct calculation.