Advances in Earthquake Location and Tomography

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Advances in Earthquake LocationandTomography

• William Menke
• Lamont-Doherty Earth Observatory
• Columbia University

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Waves from earthquake first arrived in Palisades
NY at 150032 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
theseismic velocity structureof the
earthaccuratelythe scalar fields Vp(x) and
Vs(x)(which are strongly correlated)
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Arrival Time ?Travel TimeQ a car
arrived in town after traveling for an half an
hour at sixty miles an hour. Where did it
start? A. Thirty miles awayQ 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 Whats 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 Earths Crust
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Compressional Velocity just below Crust
Overall model has 1.3?106 tetrahedra
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Variations in Traveltime due to 3D earth structure
seismometer
earthquake
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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
Earthquakes (x,y) is center of bullseye
but what about its depth?
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Earthquakes depth related to curvature of
arrival time at origin
Deep
Shallow
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Earthquakes in Long Valley Caldera, California
located with absolute traveltimes
Courtesty of Felix Walhhauser, LDEO
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Earthquakes in Long Valley Caldera, California
located with differential traveltimes
Courtesty of Felix Walhhauser, LDEO
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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
minimum
mean
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.
ray
wavefront
The pattern of differnetial traveltime is more
complicated
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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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2) Use cross-correlation to measure differential
arrival times
Very accurate DTs !
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Issue Statistical Correlations in Data
DTpqi Tpi Tqi DTrqi Tri Tqi Then even if
errors in Ts uncorrelated, errors in DTs
will be strongly correlate. Covariance/variance1/
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Furthermore, relationships exist between
different data DTpqi DTpri DTqri
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Issue How does the statistics of
cross-correlation enter in to the problem?
Monte-Carlo simulations Differential arrival
times as calculated by cross-correlation are less
correlated than implied by the formula covariance
variance 1/2
formula
simulation
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What is the practical advantageof using
differential arrival timesto 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 differentia
l arrival time data When the data are
noise Or the 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)
differential data
differential data
absolute data
absolute 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 data
absolute 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
• and
• when 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
times or differential arrival times
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A numerical test
11 stations 50 earthquakes on fault
zone Heterogeneity near fault zone only
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True earthquake locations And fault zone
heterogenity ( ?1 km/s)
Reconstructed earthquake locations And 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
.. Dont assume that something is unimportant,
just because youve 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 2
Event 3
Event 1
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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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