Note: this is my thesis for my master's degree in physics at the University of Texas at El Paso; I studied in a joint program with the University of Texas at Brownsville, where I was located and participated in the UTB Physics Dept. LIGO data analysis. This project was under the supervision and instruction of Soumya Mohanty and was completed in May 2004.
Interferometric detectors such as LIGO are now being used in searches for astrophysical gravitational waves. One search seeks to test for possible gravitational waves bursts correlated with external astrophysical triggers (e.g. gamma ray bursts). This study compares various statistical tests for signal detection in an externally triggered burst search and identifies optimal detection strategies. Initially, several tests were compared for two co-located co-aligned detectors. These tests, including the likelihood ratio test and various components of the likelihood ratio statistic, were compared analytically for ideal noise (stationary Gaussian) and with Monte Carlo simulations for both ideal noise and a realistic noise model (mixed Gaussian). The cross-correlation test was found to significantly outperform the likelihood ratio test in cases involving low signal-to-noise ratios and non-Gaussian noise. This motivated construction of a generalized test using the likelihood ratio method to obtain a statistic from which only the optimally-weighted cross-correlation terms were retained. This was done for networks of two or more detectors. In the two-detector case the test does not converge to cross-correlation for a pair of mis-aligned detectors. Expressions were obtained for cross-correlation term weightings in networks of three or more detectors, and a computer code was written to compare these weightings in a multi-interferometer network of real detectors (including LIGO, VIRGO, GEO, and TAMA). Results from this code are presented, and directions for further development are discussed.
Link to full text of Detection strategies for a multi-interferometer triggered search in .pdf form. NOTE: this is a LARGE file (2.2 MB).
My project involved data analysis related to the LIGO project. Einstein's theory of general relativity predicts gravitational waves, which may be thought of as ripples in space-time, ripples in the basic curvature of the universe. (This is discussed more here.) Such gravitational waves are produced by rapid changes in the most intense gravitational fields by such things as merging black holes, colliding neutron stars, and supernovae. Gamma ray bursts, intense bursts of high energy radiation lasting only a few seconds but coming from distant galaxies, are perhaps associated with events that also produce significant gravitational waves. (These are discussed more here.)
Gravitational waves produce exceedingly tiny stretching and compression of the space (and matter) they encounter. (This is illustrated more here.) International projects have produced observatories in hopes of detecting these waves. One design uses a laser interferometer: a laser beam is split to travel two perpendicular paths to end mirrors, which reflect the beam back. The recombined beam contains information on the relative length of the two paths (arms), which would change microscopically if a gravitational wave went by. (This is discussed more here.) The U.S. project is the Laser Interferometer Gravitational-Wave Observatory, or LIGO, and has three detectors at two locations. Other projects are VIRGO in Italy, GEO in Germany, TAMA in Japan, and a future one in Australia called AIGO.
The signals produced by passing gravitational waves would be so small as to be easily lost in the static-like noise of the detectors. We use statistical methods to compare the signals from more than one detector at a time in hopes of canceling out this noise. There are several groups working on methods that target particular types of sources of gravitational waves. The sub-group I worked with was looking for any gravitational waves coinciding with a gamma ray burst detected by Earth-orbiting satellites. My project is concerned with this type of search, where we know the direction of the gravitational wave (because of the detection of the associated gamma ray burst).
In my project, I first considered a simple case: two detectors, identical, at the same location, oriented the same way. I used mathematical derivations and computer simulations to compare three simple statistical methods for combining their results. When I considered a simple simulation of realistic noise, the best method was cross-correlation, a particular statistical combination of the data from the two detectors.
Next, I needed to develop a method for combining the results from the various real detectors using cross-correlation. Suppose we want to use three, four, or more detectors together: since cross-correlation is done with two detectors at a time, we want the combination of each possible paired cross-correlation that gives the best chance of detecting a gravitational wave. This depends on where the source is in the sky because the detectors are not equally sensitive in all directions, they are located at different places on Earth and in different orientations, and each detector is slightly different in construction giving different sensitivities.
Using a statistical method, I found mathematically optimal (or theoretically ideal, within limits) formulas to weigh each paired cross-correlation before adding them together. To illustrate these formulas, I wrote a computer program which shows the particular detector pair that contributes the most out of a set of three, four, or more detectors chosen from the real detectors (VIRGO, GEO, TAMA, or one of the three LIGO detectors). I'm just a novice at this, however: there are many simplifying assumptions I used to get this problem to something manageable; our goal is for future research to replace these simplifications with more detailed treatment.
© 2004 by Wm. Robert Johnston.
Last modified 1 September 2004.
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