Advances in Earthquake Location
and
Tomography
William Menke
Lamont-Doherty Earth Observatory
Columbia University
Outline
Part 1: Advantage of using differential
arrival times to locate earthquakes
Part 2: Simultaneous earthquake
location and tomography
Part 3: In depth analysis of the special
case of unknown origin time
Part 1
Advantage of using differential arrival
times to locate earthquakes
that was the recent
Gulf of Mexico earthquake,
by the way …
Locating an earthquake
requires knowing the
seismic velocity structure
accurately
What’s the best way to represent 3 dimensional
structure
Best for what?
compatibility with data sources
ease of visualization and editing
facilitating calculation
Overall
organization into
interfaces
Small-scale
organization into
tetrahedra
Linear interpolation
within tetrahedra
implying rays that
are circular arcs
seismometer
earthquake
Location Errors:
= 0.5 degree = 55 km = 30 miles
Note: this preliminary calculation used data from a
limited number of stations
Two parallel approaches
work to improve earth model
design earthquake location techniques
that are as insensitive to model as possible
Waves from earthquake first arrived
in Palisades NY at 15:00:32 on Sept
10, 2006
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?
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
differential arrival time = difference in arrival times
mean origin time
cancels out
T = arrival time
TT = travel time
To = Origin Time
(start time of
earthquake)
Station i
Very accurate DT’s !
A technical question for Applied Math types …
Are differential arrival
times as calculated by
cross-correlation less
correlated than implied by
the formula
They seem to be.
If so, the this is another
advantage of using the
method
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
Patterns of differential arrival time
B
A
B
B
C
C
C
A
B
A
A
C
C
C
Can you guess the orientation of the two sources in these six cases?
This pattern an be
seen in actual data,
in this case from a
pair of earthquakes
on the San Andreas
Fault
A
B
C
Boxes: differential
arrival times
observed at
particular stations
Shading: theoretical
calculation for bestfitting locations of the
earthquake pair
Another
example …
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”
Part 2
Simultaneous earthquake location and
tomography
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)
Reconstructed earthquake locations
And fault zone heterogenity, using
noise free differential data
Note the amplitude of the “signal” is only 1 ms, so
noise might be a problem.
Reality Check: How big is the Signal?
How much better are the data fit?
When the earth structure is allowed to vary
compared with holding a simple, layered
earth structure fixed?
Answer: 0.7 milliseconds, for a dataset that
has traveltimes of a few seconds
Need very precise measurements!
Part 3
Is Joint Tomography/Earthquake
Location
Really Possible ?
Study a simplified version of the
problem
In depth analysis of the special case of
unknown origin time
but known location
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!
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
Part 1: Earthquake location with differential data is
the way to go!
Part 2: Simultaneous tomography / earthquake
location possible with differential data, but
requires high-precision data.
Part 3: Coupled Tomography/Location is extremely
nonunique and extremely likely to fool you.