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
Transcript
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 SchoolSouth 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
Purpose of the researchOn 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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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 researchTwo 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 literatureI 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 projectI 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.
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.
MethodologySince 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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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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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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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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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
Figure 7 – filtered seismometer data – 0.3-1Hz frequency band, Y-direction
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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-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.
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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 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
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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/DiscussionIn 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.
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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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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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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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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ConclusionsIt 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.
