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 3D 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.
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
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
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 3
Event 1
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.