Advances in Earthquake Location and Tomography
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Advances in Earthquake Locationand
Tomography
William Menke
Lamont-Doherty Earth Observatory
Columbia University
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Waves from earthquake first arrived in Palisades NY at 15:00:32 on Sept
10, 2006
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that was the recent Gulf of Mexico earthquake,
by the way …
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Locating an earthquakerequires knowing the
“seismic velocity structure*”of the earthaccurately
*the scalar fields Vp(x) and Vs(x)
(which are strongly correlated)
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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?
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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
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Overall organization into interfaces
Small-scale organization into tetrahedra
Linear interpolation within tetrahedra implying rays that are circular arcs
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Thickness of Earth’s Crust
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Compressional Velocity just below Crust
Overall model has 1.3106 tetrahedra
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seismometer earthquake
Variations in Traveltime due to 3D earth structure
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Location Errors: = 0.5 degree = 55 km = 30 miles
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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 … )
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Suppose you contour arrival timeon surface of earth
Earthquake’s (x,y) is center
of bullseye
but what about its depth?
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Earthquake’s depth related to
curvature of arrival time at
origin
Deep
Shallow
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Courtesty of Felix Walhhauser, LDEO
Earthquakes in Long Valley Caldera, California located with absolute traveltimes
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Courtesty of Felix Walhhauser, LDEO
Earthquakes in Long Valley Caldera, California located with differential traveltimes
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How does differential arrival time vary spatially?
Depends strongly on this angle
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In a 3 dimensional homogeneous box …
maximum
meanminimum
If you can identify the line AB, then you can locate earthquakes
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as long as you have more than two earthquakes
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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
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The same idea works …
p q
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differential arrival time = difference in arrival times
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1)
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Very accurate DT’s !
2) Use cross-correlation to measure differential arrival times
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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
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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
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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
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Geometry of the numerical experiment …
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Effect of noisy data(10 milliseconds of measurement error)
absolute data
absolute data
differential data
differential data
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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
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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”
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Tomography:
Use
To reconstruct
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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
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A numerical test
11 stations
50 earthquakeson fault zone
Heterogeneitynear fault zone only
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True earthquake locationsAnd fault zone heterogenity( 1 km/s)
Reconstructed earthquake locationsAnd fault zone heterogenity, using noise free differential data
Seems to work !
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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!
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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
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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.
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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.
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Station 1 2 3 4
Event 1
Event 2
Event 3
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If you can …
Then that structure is indistinguishable from a perturbation in origin time!
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If you can …
Then that structure is indistinguishable from a perturbation in origin time!
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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 !
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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 !
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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.
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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.
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