Advances in Earthquake Location and Tomography William Menke

Advances in Earthquake Location and 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 earthquake requires knowing the “seismic velocity structure*” of the earth accurately *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. 3 106 tetrahedra

Variations in Traveltime due to 3 D earth structure seismometer earthquake

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 time on surface of earth Earthquake’s (x, y) is center of bullseye but what about its depth?

Deep Shallow Earthquake’s depth related to curvature of arrival time at origin

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 Courtesty of Felix Walhhauser, LDEO

How does differential arrival time vary spatially? Depends strongly on this angle

In a 3 dimensional homogeneous box … maximum minimum mean 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. y ra t ron vef wa The pattern of differnetial traveltime is more complicated …

The same idea works … p q

differential arrival time = difference in arrival times

1)

2) Use cross-correlation to measure differential arrival times Very accurate DT’s !

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

Issue: How does the statistics of crosscorrelation enter in to the problem? Monte-Carlo simulations: simulation formula Differential arrival times as calculated by crosscorrelation are less correlated than implied by the formula covariance: variance = 1/2

What is the practical advantage of using differential arrival times to locate earthquakes My approach is to examine the statistics of location errors using numerical simulations Compare the result of using absolute arrival time data And differential arrival time data When the data are noise Or the earth structure is poorly known

Geometry of the numerical experiment …

Effect of noisy data (10 milliseconds of measurement error) differential data absolute data

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

Both absolute locations and relative locations of earthquakes are improved by using differential arrival time data when arrival times are nosily measured and when 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 times or differential arrival times

A numerical test 11 stations 50 earthquakes on fault zone Heterogeneity near fault zone only

True earthquake locations And fault zone heterogenity ( 1 km/s) Seems to work ! Reconstructed earthquake locations And fault zone heterogenity, using noise free differential data

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 2 Event 1 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.