1. A Determination of a Mathematical Model for the Relationship between
LIGO-Hanford Interferometer Sensitivity and Global Seismic Activity
Brian W. Huang
Saint Joseph’s High School
South Bend, IN
Teacher: Mr. John Wojtowicz
Supervising Scientist: Dr. Thomas Loughran
Abstract
The objective of this research project is to investigate the relation between earthquakes as
detected by the seismometers at LIGO-Hanford (Laser Interferometer Gravitational Wave
Observatory at Hanford, Washington) and the sensitivity of the interferometers to gravitational
waves. To achieve this goal, I collected interferometer sensitivity data from the LIGO-Hanford
e-log (electronic logbook) and compared them to confirmed earthquakes readings on the
seismometers at the Hanford observatory at LIGO. An earthquake reading is labeled as a
“confirmed earthquake reading” when spikes of activity appearing in the seismometer reading
coincide with the predicted arrival of earthquake waves. This investigation revealed some
initially promising correlation and a direction for further research

2. Purpose of the research
On any given day, approximately 8,000 earthquakes occur around the world1
. These
earthquakes range from barely noticeable tremors to exceptionally powerful events. This LIGO
project hosted by the Notre Dame QuarkNet Center focuses on the effects of these seismic
disturbances on the interferometers at the LIGO observatory at Hanford, Washington. The
purpose of the LIGO-Hanford Observatory (LHO) is to detect gravitational waves.
The LHO interferometers are built to measure very small fluctuations in space-time by
measuring the changes in the distance between two points. This is done by first splitting a single
laser beam and firing it down two 2km long vacuum tubes that have been placed perpendicular to
each other. Each of the two laser beams is reflected at the end of a vacuum tube and bounce
back and forth between the two end mirrors before it returns to the beam splitter where the two
beams are finally recombined. Figure 12
offers a simple illustration of how the interferometer
works. If the two distances are precisely the same, the two lasers interfere constructively so that
all the light is reflected completely to the laser source. If the two distances vary, however, some
of the light will be reflected to where it can be detected by the photodetector. Since gravitational
waves actually stretch space, altering the length of the vacuum tubes, some interference would
occur such that some light would reach the photodetector. These waves, however, are so weak
that the interferometers at the LHO are built to be sensitive enough to detect changes in distance
under the width of an atom.
Figure 1 – The LIGO-Hanford Interferometers
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3. Seismic activity, though, can bend the vacuum tubes, changing the distance between the
mirrors and the light source, mimicking the effects of a gravitational wave. Thus, seismic
activity can significantly inhibit the functionality of interferometers. LIGO researchers currently
utilize an isolation and suspension system to filter most seismic activity3
. But, because present
isolating techniques are unable to completely block out earthquake waves of sufficient
magnitude and proximity, it is of significant interest to the scientists at LIGO to be able to adjust
for them or, at least, be able to determine if data has been affected by seismic noise.
As such, our objective is to model the sensitivity of the LIGO detectors at Hanford to
seismic activity, showing how close and of what magnitude an earthquake must be in order to
affect the accuracy of interferometer measurements of gravitational waves at the LHO.
Rationale for the research
Two LIGO interferometers have been built in relatively quiet seismic zones, one at
Livingston, Louisiana and the other at Hanford, Washington, in order to minimize disturbances
in the interferometer data due to earthquakes. Since the earthquakes that interfere with the
interferometers are generally localized, a comparison of data from two interferometers located in
completely different areas could serve to identify local disturbances (i.e., anything that is not a
gravitational wave such as nearby traffic and earthquakes). With only two interferometers,
however, there exist certain locations around the world such that earthquake waves originating
from there would arrive at the interferometers at the same time, thus registering as a false
positive reading.
The outcome of this research project would allow LIGO researchers to identify
earthquakes that could interfere with the LIGO interferometers without depending on a possibly
faulty comparison between two detectors. Furthermore, if the mathematical model were capable
of predicting the effects of earthquakes on the interferometers accurately enough, it would be
possible for LIGO researchers to receive warning of a disturbance before it arrives. This would
enable the researchers to counteract the disturbance or specifically filter it. At the very least, this
would enable LIGO scientists to identify data that were corrupted by seismic activity.
Pertinent scientific literature
I utilized data based on the ISAP-91 Earth Reference Model (B. Kennett & E. Engdahl,
1991). The IASP-91 Earth Reference Model is a table of earthquake wave travel times derived
from data on hundreds of earthquakes and the arrival times of their waves. This table was used
by USGS in their Earthquake Travel Time Calculator which I used to approximate, within the
error of one to two minutes, the time at which earthquake waves would reach the LHO.
Prior work and contributions of others on this project
I worked with three other members during the summer of 2008 evaluating how seismic
disturbances affect the LIGO seismometers at Hanford, Washington. Together, we determined a
range of frequencies on which seismometers would detect earthquakes at the Hanford
observatory. Having determined this, we then took a set of data on earthquakes from various
years from the USGS earthquake database and created a map (Figure 2) that located around 150
earthquakes that were detected at the LHO.
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Huang, Brian W.

4. Figure 2 – LIGO-Hanford Earthquake Sensitivity Map
To the best of my knowledge, there has been no prior effort to model the relationship
between seismic disturbances detected by both USGS and LIGO and interferometer sensitivity.
Methodology
Since the goal of the project was to create a mathematical model for the relationship
between LHO interferometer sensitivity and seismic activity, it would have been most reasonable
to set up an experiment where all variables in earthquake data, such as earthquake location,
magnitude, depth, and distance, could be held constant while one was allowed to vary. This
would have allowed the effects of each variable to be isolated and followed, making the
determination of a mathematical model very easy. This, unfortunately, was impossible, since the
data on the sensitivity of the LHO interferometers on the LHO e-lab was incomplete, with entire
expanses of time in which no sensitivity was recorded or discrepancies in graph formatting.
Thus, it was necessary to collect data based on when sensitivity was consistently available. The
year 2007 particularly stood out in this regard, with only a few short gaps in sensitivity data
throughout the year and in December.
Thus, in order to collect a random sample of data on earthquakes detectable at the LHO
in 2007, I took the entire list of earthquakes that occurred in the year 2007 and found the ones of
the largest magnitude for each week (Monday to Sunday). I acquired the appropriate data from
the United States Geological Survey (USGS) Earthquake Data Search Engine4
. The USGS
database provides accurate information on date, time, latitude and longitude, magnitude, and
depth. Using the provided earthquake coordinates, magnitude, and depth, I then employed the
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Huang, Brian W.

5. Haversine formula5
for approximating the distance between two points on a sphere. The formula
is given as follows
d = arccos{sin(long1) · sin(long2) + cos(long1) · cos(long2) · cos(lat1 – lat2)} · R
where R is 6371, lat1 and long1 are the latitude and longitude, respectively, of one point, and lat2
and long2 are, respectively, the latitude and longitude of the other. Thus, for an earthquake that
occurred on May 6, 2007 at 21:11:52.50 in the evening at the coordinates (-19.4,-179.35) with a
magnitude of 6.5 on the Richter scale at a depth of 676km, and given that the LHO is located at
(46.46, -119.41), the distance from the earthquake’s epicenter to the LHO is 7877.5264km.
I also utilized the USGS Earthquake Travel Time Calculator6
to calculate the range of
times at which the various waves of an earthquake would be felt at the LHO. The USGS
Earthquake Travel Time Calculator utilizes the IASP-91 Earth Reference Model published in
1991 by Kennett and Engdahl7
to generate a list of seismic waves and their arrival times at a
given site. Table 1 is a list of earthquake waves and their calculated arrival times for our
example earthquake. Earthquake waves are labeled according to how they propagate. Note that
P waves and S waves are usually the largest or most intensely felt waves since the others are
rebounded or reflected versions of the P and S waves. For example, a PP wave is free surface
reflection of a P wave leaving a source downwards and a PKiKP is a P wave that was reflected
from the inner core boundary.
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Huang, Brian W.

6. Table 1 – Earthquake Wave Arrival Times
Using this information, I then used the Bluestone Data Analysis tool8
hosted by the LIGO
I2U2 e-Lab to check within a margin of about thirty minutes whether the earthquake was truly
detectable at the LHO. The procedure for this process was developed by my research group in
summer of 2008. We, by examining several earthquakes on all frequency filters, determined that
the 0.1-0.3Hz and the 0.3-1Hz frequency bands were the optimal range of frequencies to
characterize earthquake activity. Frequencies lower or higher than this band did not identify any
earthquakes missed by the 0.1-1Hz frequency band. As such, we concluded that only the 0.1-
0.3Hz and the 0.3-1Hz frequency bands were necessary to identify earthquake activity
Thus, I took seismometer data from both of the aforementioned frequency bands to check
for signs of any earthquake activity. The data collected at the LHO is separated into three
orthogonal directions, labeled X, Y, and Z, similar to the three-dimensional Cartesian coordinate
system, in order to cover all ranges of motion. Figures 3-8 show examples of data obtained from
Bluestone from each filter that is believed to be an earthquake. Figures 3, 4, 5, 6, 7, and 8 are
examples of data from seismometers at the LHO acquired by the Bluestone program. They cover
time from 21:00 to 23:00 once May 6, 2007. Figures 3, 4, and 5 display data from the 0.1-0.3Hz
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Huang, Brian W.

7. frequency band and are the X, Y, and Z directions, respectively. Figures 6, 7, and 8 display data
from the 0.3-1Hz frequency band and are the X, Y, and Z directions, respectively.
Figure 3 – filtered seismometer data – 0.1-0.3Hz frequency band, X-direction
Figure 4 – filtered seismometer data – 0.1-0.3Hz frequency band, Y-direction
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Huang, Brian W.

8. Figure 5 – filtered seismometer data – 0.1-0.3Hz frequency band, Z-direction
Figure 6 – filtered seismometer data – 0.3-1Hz frequency band, X-direction
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Huang, Brian W.

9. Figure 7 – filtered seismometer data – 0.3-1Hz frequency band, Y-direction
Figure 8 – filtered seismometer data – 0.3-1Hz frequency band, Z-direction
There is clear activity around and after 21:23, as predicted by the USGS Earthquake
Travel Time Calculator (Table 1) providing strong evidence that the activity seen is indeed the P-
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Huang, Brian W.

10. wave of the May 6, 21:11:52.50 earthquake. At the same time, however, there is activity around
22:13 which is not predicted by Table 1. Thus, consulting the USGS database again, we find that
another earthquake occurred at 22:01:08.92 on the same day at the coordinates (-19.41,-179.32)
with a magnitude of 6.1 on the Richter scale at a depth of 688km. Entering this data into the
USGS Earthquake Travel Time Calculator again, we discover that the earthquake’s P-wave
arrives at 22:12:33, almost exactly where the second spike appears. Thus, we can confirm that
the May 6, 21:11:52.50 earthquake was truly detected at LIGO-Hanford.
Having confirmed that earthquakes were detected at the LHO, it was now possible to test
their effect on the sensitivity of the interferometers. The sensitivity given by the I2U2 e-Log
(Figure 9) is the calculated range of LIGO interferometer sensitivity measured in megaparsecs.
That is, it predicts how far away a source of gravitational waves must be in order for the LHO
Interferometers to detect it. The red and blue lines represent the sensitivity of the LHO
interferometers while the green represents the interferometer at LIGO-Livingston. By
calculating the changes in the sensitivity of the LIGO interferometers, it is possible to determine
how they are affected by earthquakes. This is the main goal of the project, to determine how
earthquakes might detrimentally affect the sensitivity of the LHO Interferometers to far-away
sources of gravitational waves.
Figure 9 – Sensitivity Data
Note, however, that the time axis on the graphs here are slightly unconventional as they
count backwards by the hour from the current time. Also, the T0 entry gives the time and date in
the form dd/mm/yyyy hh:mm:ss where hours are given in the 24-hr format. Thus, the -6th
hour
refers to 21:09:47 on May 6, 2007.
The May 6, 21:11:52.50 earthquake would thus register, using the predictions from the
USGS Earthquake Travel Time Calculator which were confirmed by the Bluestone seismometer
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Huang, Brian W.

11. graphs, between -5:46:29 and -5:18:03. Looking at Figure 10, there is a significant change in
sensitivity near those very times.
There are, however, two different readings for the sensitivity of the interferometers at the
LHO and, by the same token, two different baselines for the sensitivity trends. That is, H1
sensitivity usually hovers around 14 to 15MPc while H2 is usually around 6 to 7MPc. To
compensate for this, I only considered the changes in sensitivity, measured in percents of the
baseline reading (15.5 for H1 and 7.25 for H2), rather than the actual values of the sensitivity
readings.
Also, since more detailed data is not available, I fount it more useful to record the largest
drop in sensitivity within the range of times at which waves from the May 6, 21:11:52.50
earthquake would be arriving since it was nearly futile to discern the effects of individual waves.
This, though weakening the connection of the sensitivity readings to the actual data, made it
possible to explore the greatest effect of earthquakes on interferometer sensitivity.
Thus, the two sensitivity readings for the May 6, 21:11:52.50 earthquake would be
recorded as -0.2129 and -1.03448 for H1 and H2 respectively. Finally, to find a composite
change in sensitivity, I took the square root of the sum of the squares of the changes in sensitivity
for H1 and H2. Thus, the final recorded change in sensitivity for the May 6, 21:11:52.50
earthquake would be 1.056163984.
Results/Discussion
In order to determine how earthquakes affect the sensitivity of the LIGO interferometers,
it was necessary to examine how each attribute of the earthquakes influenced sensitivity. The
three variables are distance, magnitude, and depth. First, I mapped sensitivity against the inverse
of the distance from the earthquake epicenter to the LHO since common sense suggests that
earthquakes occurring farther from LIGO-Hanford would have a lesser effect on the
interferometers located there than those with epicenters nearer to LIGO-Hanford. The strange
resulting graph (Figure 10), with an extremely low R2
value (a measurement of how well two
variables correlate) of .0317, seems to suggest that either there was at least one confounding
variable (likely magnitude or depth) or that the two variables were uncorrelated. Note that the
fact that distance in the denominator is squared would not affect the overall trend of the data.
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12. Figure 10 – Graph of Sensitivity against Inverse Distance
Doing the same with sensitivity against energy yielded similar inconclusive results. The
graph of sensitivity and energy (Figure 11) had an R2
value of .1918, a closer fit, but nonetheless
still a very weak one. As such, while it was encouraging and showed some promise, I lead to
the same conclusion: that either there existed a confounding variable or simply that energy had
no influence on sensitivity. Note that the Richter scale operates on a base 32 logarithmic scale;
thus, in order to obtain energy, I had to raise magnitude to a power of 32.
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13. Figure 11 – Graph of Sensitivity against Energy
Depth, however, was a different matter from distance or energy. There was no obvious
way in which it would affect interferometer sensitivity. It would have negligible influence on the
distance from the earthquake epicenter to LIGO-Hanford and no imaginable influence on energy.
The graph of sensitivity against depth (Figure 12) confirmed this, yielding another low R2
value,
a .0466.
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Huang, Brian W.

14. Figure 12 – Graph of Sensitivity against Depth
Depth does, however, affect what kind of phases of rock the earthquake waves would
travel through. Since I had no knowledge of how this might affect the intensity of the earthquake
waves, however, I ignored it for the time being. Instead, I focused on how energy and distance
alone might affect sensitivity. Thus, I tried graphing different combinations of the variables to
see how the two might combine to influence sensitivity.
First, I tried dividing energy by distance squared (Figure 13). This resulted in a
particularly odd graph due to the presence of two extreme outliers in the data. These points
turned out to be particularly large earthquakes with magnitudes over 8 on the Richter scale. This
explained the outrageously high values on the x-axis and, taking into account that sensitivity
could only drop a maximum of 15.5 MPc for H1 and 7.25 MPc for H2, I concluded that it was
reasonable to drop the two points from the graph and re-plot it. The result was Figure 14 which,
in fact, models the data even more poorly than the plot with the outliers. In fact, there is not even
an apparent upward trend
Since even this basic ratio, which modeled the idea that drops in sensitivity would
increase with greater earthquake magnitude and epicenter proximity, failed to model a slightly
smooth upward correlation, I concluded that there was either no correlation between the energy
and proximity of an earthquake to the LHO and the sensitivity of the LHO interferometers, or
that there was another confounding variable.
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15. Figure 13 – Graph of Sensitivity against Energy/Distance (including outliers)
Figure 14 – Graph of Sensitivity against Energy/Distance (not including outliers)
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Huang, Brian W.

16. Conclusions
It is intuitive that stronger seismic disturbances would cause LHO interferometer
sensitivity to drop more than a weaker disturbance would. Nevertheless, the data does not show
that kind of trend. This could be a result of a multitude of reasons, all worthy of further
investigations.
First and foremost is the issue of earthquake epicenter depth. I was not able to implement
depth into my initial equation since I had neither sufficient data to test depth while holding other
variables constant, nor a proper understanding of how depth might factor into earthquake wave
propagation. It is certain, however, that depth would factor into the equation in some way since
the path in which a wave propagates depends heavily upon its medium and at a sufficient depth,
the very epicenter of an earthquake would be located in a different medium. This gap in
understanding, however, will require the assistance of a trained seismologist specializing in
earthquakes wave propagation.
Another explanation for the disappointing results is that the interferometers at the LHO
could be more sensitive in one direction than another. The arms at the LHO were built
orthogonally so that they would be able to measure the direction of any gravitational wave. Even
so, it is possible that there are directions in which the interferometers are less sensitive. Indeed,
the sensitivities of the LHO interferometers measured in mega-parsecs are often not equal.
The apparent absence of a correlation between the LHO sensitivity data and seismic
disturbances could also be due to improper modeling. In further study, I would like to explore
how depth might factor into the pattern.
Of course, the possibility remains that the LHO interferometers are already well enough
protected from seismic interference and that the instances in which sensitivity dropped at the
very same time an earthquake affected the area were simply coincidences. I believe, however,
that this is quite unlikely since there are so many of these instances.
The simplest solution to this problem, however, and a more concrete method of
determining a mathematical model for the relationship between LIGO-Hanford interferometer
sensitivity and global seismic activity, is to simply gather more data once the LHO has collected
more sensitivity data. This solution, though, may be the most difficult to realize since the LHO
is currently undergoing a luminosity upgrade (improving its sensitivity) and may not be back
online for quite some time.
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17. References
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18. 1
USGS: FAQ- Measuring Earthquakes
2
http://www.ligo-wa.caltech.edu/ligo_overview/ligo_overview.html
3
http://www.ligo.caltech.edu/advLIGO/scripts/summary.shtml
4
http://neic.usgs.gov/neis/epic/epic_global.html
5
U. S. Census Bureau Geographic Information Systems FAQ, What is the best way to calculate the
distance between 2 points?
6
http://neic.usgs.gov/neis/travel_times/compute_tt.html
7
Kennett, B.L.N. & Engdahl, E.R., 1991. Travel times for global earthquake location and phase
identification, Geophys. J. Int., 105, 429–465.
8
http://tekoa.ligo-wa.caltech.edu/tla/work_flow.php