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LASER INTERFEROMETER GRAVITATIONAL WAVE OBSERVATORY

- LIGO -CALIFORNIA INSTITUTE OF TECHNOLOGY

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Document Type LIGO-T020191-00 December 3, 2002

Efficiency Calculation in the S1 Burst Analysis

L. Cadonati, A. Weinstein

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California Institute of Technology Massachusetts Institute of TechnologyLIGO Project - MS 51-33 LIGO Project - MS 20B-145

Pasadena CA 91125 Cambridge, MA 01239Phone (626) 395-2129 Phone (617) 253-4824Fax (626) 304-9834 Fax (617) 253-7014

E-mail: [email protected] E-mail: [email protected]

WWW: http://www.ligo.caltech.edu/

Processed with LATEX on 2002/12/05

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1 Introduction

The efficiency of an Event Trigger Generator (ETG) to a particular waveform with peak amplitudeh0 can be determined by simulation within LDAS. In this note, we describe the procedure used tocalculate efficiencies in the S1 burst analysis (results in [2]).The waveform is added to the raw (noisy AS Q) data such that it corresponds to optimal

response with respect to source direction and polarization. The result is that the efficiency is 0for very small h0, 1 for very large h0, and smoothly rising from 0 to 1 in between. (This is theefficiency for detection during unvetoed live-time). This turn-on in efficiency is well modeled by asigmoid in log h0:

ε(h0) =1

1 + e(log h0−b)/a(1)

It is specific to a given waveform, ETG algorithm, and data epoch.In our study, we added gaussian and sine gaussian waveforms to the S1 playground, and

calcunated efficiency curves for optimal polarization and incident direction. The efficiency curvesat individual interferometers have then been averaged over the sky (antenna pattern) and combinedto get an average efficiency curve for triple coincidence events, to be used in the construction ofupper limit versus strength plots presented in the S1 burst report [2].

2 Simulated waveforms

There are several morphologies one could think of using, but at this stage we are not ready for aproper interpretation of anything but a family of simplified waveforms:

1. Gaussians (broad-band):

sga(t) = h0 e−(t−t0)2/τ2

(2)

With τ = 1.0, 2.5, 5.0, 7.5, 10, 20, 50 and 100 ms. Spectra and waveforms for these signalsare shown in figure 1. Our search is not sensitive to signals below 150 Hz, thus only the twonarrowest signals (1.0 and 2.5 ms) are detected. Spectra and waveforms for these signals areshown in figure 1.

2. Sine-gaussians (narrow-band):

ssg(t) = h0 sin(2πf0t)e−(t−t0)2/τ2

(3)

where f0τ = 2 so that Q ∼ 2πf0/√2 ∼ 9. Eigth waveforms have been simulated, with central

frequency: f0 = 100, 153, 235, 361, 554, 850, 1304, 2000Hz. Spectra and waveforms forthese signals are shown in figure 2.

3 Efficiency curves

3.1 Method

Waveforms of a given amplitude, incident from the zenith with optimal polarization have beeninjected in the dataconditionAPI, at known times. Their detection efficiency in triple coincidencewas obtained with the following recipe:

1. inject the same signal, with maximum amplitude h0 at the same GPS time, in the threeinterferometer data streams.

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Figure 1: Gaussian injections: spectra (a) and waveforms (b).

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Figure 2: Sine-Gaussians injections: spectra (a) and waveform of the 554Hz signal (b).


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2. apply the same data selection criteria used in the pipeline analysis [2]. For S1:

• TFCLUSTER: only events overlapping the 100-3000 Hz band; power threshold = 15;

• SLOPE: amplitude threshold IFO dependent: 6000 at L1, 10000 at H1, 12800 at H2

3. select clusters of events in triple coincidence (0.5 sec window for TFCLUSTER, 0.05 sec forSLOPE) that pass the frequency cut (cluster frequency ranges at the 3 interferometers arerequired to overlap or to be separated by less than 80 Hz);

4. select events taking place within 0.5 sec around the (known) injection time.

The ratio of the number of events thus retrieved to the number of injected signals gives the combineddetection efficiency for all three interferometers:

εall(h0) =Ndet(h0, h0, h0)

Nin.

This quantity is useful only for interferometers with the same orientation - not too bad an as-sumption for the three LIGO interferometers, but wrong if we want to include GEO600 in thepicture.The single interferometer efficiencies εj(h0) have been calculated with the same procedure, but

rather than using simulations with the same amplitude at all interferometer, we chose an amplitudeh1 such that εall(h1) = 1 and then used the coincidence between simulations of amplitude h0 atone site and h1 at the other two:

ε0(h0) =Ndet(h0, h1, h1)

Nin, ε1(h0) =

Ndet(h1, h0, h1)

Nin, ε2(h0) =

Ndet(h1, h1, h0)

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In order to put error bars on the efficiencies, we can treat the detected injections as successes ina set of binomial trials. In general, if we perform N trials, with success probability p, and measuren successess, the best estimate for the success probability and its variance are:

p̂ = nN

σ̂(p̂) =

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(4)

In our case, for a given waveform, N = 84 is the number of injected events and amplitude, nj(h0)is the number of detected events at interferometer j, with amplitude h0.

N = 84 number of injected eventsnj(h0) = number of detected events at interferometer jεj(h0) = nj(h0)/N detection efficiency

σj(h0) =√

εj(h0)(1− εj(h0))/N error

(5)

This procedure is repeated for each signal amplitude, so that we can build εj(h0) curves, to befitted with sigmoids (eq. 1).


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Figure 3: Efficiency and power vs peak strain for simulated Gaussians with τ = 1.0ms (see text fordescription).

3.2 Results

Figure 3 shows the results obtained with gaussians at τ = 1ms, for tfcluster (a) and slope (b).The points in the top figure show the efficiencies versus amplitude for L1 (red), H2 (blue) andH2 (green). The black dots mark εall(h0), the efficiency to detect events with optimal orientationand polarization at the three sites. The lower plot shows the power (for tfcluster) or amplitude(for slope) at which the simulated events are detected. The dependency of power/amplitude onthe signal strain is a power law (linear in these log-log plots) when the signal has relatively highdetection efficiency. The curve flattens as the detection efficiency approaches zero and we startpicking up background events. if we look, for instance, at the H2, tfcluster data in figure 3-a,the transition between detection of simulated events and background is between 7.5 × 10−18 and1.0 × 10−17. For h0 ≤ 7.5 × 10−18, our procedure detects background events and nH2(h0) nevergoes to zero. We refer to hmin as the point where the efficiency data and the power vs strain curvesindicate a transition between detection of simulated events and background.The efficiency points at h0 > hmin have been fit with sigmoids. In practice, we fit only where

there is evidence that we are not detecting background. In each case, the fit result is shown, inthe efficiency plots (as the top plot in fig. 3-a) as a continuous curve. The dotted curve is theextrapolation of the sigmoid to the region not included in the fit.Similar plots for the 2.5 ms gaussian and the sine gaussian simulations are shown in figures 4-9.

The sigmoid parameters are summarized in table 1.


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Table 1: parameters for the sigmoids in figures 3-9. The functional form is shown in eq. 1

TFCLUSTER L1 H1 H2 Triple Coincidencesignal a b a b a b a bGA τ=1.0ms 0.08080 -17.4631 0.10747 -16.9919 0.10074 -17.0476 0.05646 -16.9172GA τ=2.5ms 0.08374 -16.9038 0.07462 -16.4698 0.09777 -16.2368 0.09073 -16.2284SG f0=153Hz 0.08193 -17.8851 0.08258 -17.4294 0.11020 -17.1564 0.09786 -17.1503SG f0=235Hz 0.07881 -18.1210 0.08219 -17.6230 0.08414 -17.7230 0.06492 -17.5856SG f0=361Hz 0.02897 -18.0141 0.08487 -17.7218 0.03553 -17.6140 0.05134 -17.5971SG f0=554Hz 0.03933 -17.8708 0.05405 -17.7790 0.07430 -17.5428 0.06018 -17.5116SG f0=850Hz 0.03885 -17.6147 0.07822 -17.6745 0.05207 -17.3663 0.05185 -17.3564SG f0=1304Hz 0.02690 -17.2687 0.07467 -17.2736 0.06835 -17.1454 0.05157 -17.1021SG f0=2000Hz 0.02603 -16.7540 0.08742 -16.8477 0.05178 -16.6657 0.03751 -16.6553SLOPE L1 H1 H2 Triple Coincidencesignal a b a b a b a bGA τ=1.0ms 0.02342 -17.1035 0.05056 -16.8690 0.04380 -16.5291 0.04380 -16.5291GA τ=2.5ms 0.02943 -16.2824 0.03637 -15.7993 0.04653 -15.6688 0.04653 -15.6688SG f0=235Hz 0.05829 -17.4526 0.05763 -17.1217 0.11049 -16.8754 0.09963 -16.8710SG f0=361Hz 0.03645 -17.6228 0.05980 -17.4811 0.09167 -17.0895 0.09167 -17.0895SG f0=554Hz 0.02385 -17.7917 0.03100 -17.7172 0.04649 -17.3122 0.04650 -17.3122SG f0=850Hz 0.03535 -17.8380 0.04585 -17.7800 0.05486 -17.4962 0.05486 -17.4962SG f0=1304Hz 0.03926 -17.6462 0.05071 -17.5470 0.04559 -17.4583 0.03924 -17.4390SG f0=2000Hz 0.04316 -16.9183 0.04074 -16.9308 0.05967 -16.7608 0.05967 -16.7608

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Figure 4: Gaussians, τ = 2.5ms


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Figure 5: Sine Gaussians, Q ∼ 9, 235 Hz

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4 Network analysis

We have so far discussed direct measurements of the single interferometer and combined efficiencyfor waveforms with the same optimal polarization, at the zenith. However, our true detectionefficiency depends on the source direction and polarization. We can assume that the sources aredistributed isotropically, with only one polarization component, which is oriented randomly. Inthat case, we can easily calculate the resultant detection efficiency averaged over source directions(θ, φ) and polarization ψ through the well-known detector antenna function R(θ, φ, ψ), as follows:

εave(h0) =

∫

d cos θ dφ dψ [ε(h0R(θ φψ))] (6)

Examples of the average efficiency versus h obtained in this way is shown in figure 10 (1 msgaussians, tfcluster) and figure 11 (1 ms gaussians, slope). For each of the three LIGO interferom-eters, the dashed lines are εj(h0), the measured efficiency at optimal direction/polarization. Thecontinuous lines represent the single interferometer efficiency averaged over the antenna pattern.The average value for R(θ φψ) is 0.37.It is straightforward to generalize this to multiple detector coincidences, by evaluating:

εcoinc(h0) =

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d cos θ dφ dψ [εa(h0Ra(θ φψ)) εb(h0Rb(θ φψ)) ...] (7)

where a, b, ... label different detectors. We can add efficiencies for 2, 3, 4, or more detectorsin coincidence. We can use Earth-centered co-rotating coordinates for the locations and orienta-tions of the different detectors, and for the source direction and polarization. For example, theblack continuous line in figure 10 (11) represents the tfcluster (slope) efficiency curve for gaussianswith τ = 1.0ms for triple coincidences of the LIGO detectors during S1. Detector locations andorientations are taken from the LAL Software Documentation [1].These curves must be re-drawn for different waveform morphologies, different ETG algorithms

and thresholds, and different data epochs / calibrations.The principal advantage of this procedure is that detailed simulations in LDAS are only required

for single detectors, with optimal source orientation and polarization; the remainder of the averagingover source directions and polarizations, and coincidence efficiencies, are handled through simplenumerical integrations.However, we have made some crucial assumptions:

• There is no loss of coincidence efficiency due to the time delay between bursts in differentdetectors; the coincident time window must be kept large compared with the source-direction-dependent delay times.

• There is no loss of coincidence efficiency due to any post-coincidence cuts, such as requir-ing coherence between the burst waveforms, or their amplitudes or frequency bands, in thedifferent detectors.

• We assume that the burst waveforms come in only one polarization (or, more generally, bothpolarizations have the same waveform, with fixed amplitude ratio). If, contrary-wise, burstshave two polarizations with different waveforms, then the resultant excitation of the IFOdifferential mode will be some source-direction-dependent combination of the two waveforms.This can only be dealt with by going back to the LDAS simulations; and the parameter spacethat must be explored via those simulations becomes much larger.

While the first two assumptions are reasonable, the third is still an open issue.


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5 Upper limit versus strength

The top plots in figures 12 and 13 show the combined, average efficiency for the gaussian (dashed)and sine gaussian (continuous) simulations we analyzed,using the tfcluster or slope ETG.A first rate-versus-strength plot can be obtained by dividing the rate upper limit by the effi-

ciency: the result is shown in the lower plots of figures 12 and 13.The rate upper limit used in these plots is the one obtained with the standard Feldman-Cousins

approach [3] and quoted in [2]. These are preliminary results and are subject to be a factor 2 worsefor tfcluster, if we adopt alternative methods for the upper limit estimate [4]. Moreover, once theamplitude cuts will be introduced in the analysis, the best upper limit for efficiency=1 will be closerto 2.44/LiveTime ∼ 1/day.

References

[1] http://www.lsc-group.phys.uwm.edu/lal/lsd.pdf - Table 9.1

[2] S1 Burst Group Report on the Search for Gravitational Wave Bursts of Unknown Origin,using S1 data from the three LIGO interferometers - LIGO-T020187-03-Z

[3] G.J. Feldman and R.D. Cousins, Phys Rev D57 (1998) 3873

[4] L. Cadonati Background and Upper Limits in the S1 Burst Analysis - LIGO-T020190-00-Z


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