Data analysis for impulsive sources with resonant g.w. detectors

Pia AstoneROG collaboration

Villa Mondragone International school ofGravitation and Cosmology

7-10 September 2004

http://www.roma1.infn.it/rog

M ; T ; Q

The Eq of geodetic deviation is the basis

for all the experiments to detect

g.w.

Use of powerful signal analysis tools(adaptive matched filters) to extract thesignal from noise (non-stationary)

Thermal noise T= 3 K, DL/L = 10-17 Thermal noise T=300 mK, DL/L = 3

10-18

NI 200 days

AU 221 days

EX 1036 days

NA 831 days

AL 852 days

ON times of resonant detectors from 1 Jan 1997 up to13 Jun 2003. --all (almost) parallel--

IGEC 1997-2000

+ data during S1, S2.

+

+

The expected signal h is a short pulse ( a few ms).

The expected value on Earth, if 1% of a solar mass is converted into

g.w. in the GC,is of the order of 10

Burst events for a resonant detector: a millisecond pulse, a signal made by a few millisecond cycles, or a signal sweeping in frequency through the detector resonances.

E.g.: a stellar gravitational collapse, fall of a body into a BH, the last stable orbits of an inspiraling NS or BH binary, its merging and final ringdown.

-18

The signal: detector response usig(t) [m] to a short (g.w.) pulse

Excitation= impulse force applied to the bar (mx oscillator)

Energy absorbed by the detector:

System transfer function:

Vsig(t) = usig(t)

0 s 140

8.5 s 10.5

here the decay time is

1= 1/ 1(if - = = 1)

The noise

Total noise spectrum, at the transducer output, is due to the Brownian noise of the two mechanical oscillators (narrowband) + wide band noise due to electronics:

In addition to this noise, due to known effects, we may havesome excess noise of non stationary nature, due to unknownnoise sources

[V / Hz]2

(the mechanical dissipations correspond to noise force generators with 2-sided spectral density Sfx= 4kTexmx , Sfy= 4kTeymy . The spectral densities of the associateddisplacement are )

SNR= signal to noise ratio

SNR is defined as the ratio of the square amplitude of the signalto the noise variance

• the detector is a linear system• the g.w. signal is added to the noise

• We normally consider the signal (in the detector bandwidth) a delta function, that is we suppose to know the waveform;• If we suppose the noise to be Gaussian and stationary,the problem is classical: the detection of a signal of known shape in Gaussian noise THIS IS OPTIMALLY SOLVED USING A MATCHED FILTER

Block diagram ofantenna and matched filter Fx(j)

So N(j)

Wux(j)

1 / N(j) (FxWux / N)*

matched filter sectionSt()

whitening matched

The matched filter

complex conjugate of the FFT of the antenna response to the pulse

noise power spectrum at the antenna output

• it has the property that the output SNR is maximized• it is linear and non-causal

(where and = fo for impulsive input )

The matched filter for impulsive input (a part from constants)

inverse filter smoothing filter (“spectral gain”)

Transfer function of the smoothing filter = frequency domain signal at the filter output

The bandwidth after the filter

The inverse filter cancels the dynamics of the antenna;the smoothing filter minimizes the contribution of the wideband noise and thus limits the bandwidth.

The bandwidth after the filter is much larger than the mechanicalbandwidth of the antenna oscillator: in fact the antenna responds inthe same way to an excitation due to a g.w. burst and to the browniannoise, and thus the bandwidth is limited only by the wideband noise

The ratio fx

defines the bandwidth

The SNR after the matched filter

SNR here is written in terms of the input force,but it can be shown that it can always be writtenin terms of the spectral gain.

In particular, to reason in terms of h ...

The SNR after the matched filter

...we define Sh () as the “noise spectral density”, [1/Hz], the noisespectrum at the detector input. We realize that G() = 1/Sh ().If the Fourier transform of the input force is hg then it is possible toshow that:

(h gSh ()

2

The sensitivity to bursts (1):

( strain sensitivity)

SNRm-------- =

SNRo

Te

------Teff

The improvement in SNRo obtained by filtering the

data (SNRm) can be expressed in terms of a reduction

of the equivalent temperature Te to the effective temperature Teff

The sensitivity to bursts (2):

h = 7.97 10 Sqrt(Teff) for 1 ms burst-18

The signal g(t) at the filter output

bandwidth:

minus mode only

plus mode only

filter input:Vsig(t) = usig(t)

filter output:g(t) [K]

0 seconds 50

Simulated, inthe absence ofnoise

decay time: 1= 1/ 1

decay time:

3= 1/ 3

Real data: the arrival of a cosmic ray shower on NAUTILUS

Unfiltered

data (V2)

The signal after

filtering (kelvin)

NAUTILUS1999

Sqrt(T/MQ)

cooled at100 mK

Note thatthe bandwidth depends ONLY on the

transducer and amplifier

Calibrationsignal

AN EXAMPLE OF STRAIN SENSITIVITY

880 Hz 980

The year 2001Explorercooled at 2.6 K

Nautiluscooled at 1.3 K

Phys Rev Letters 91 111101 (2003)

880 Hz 980

10 Hz

880 Hz 980

The year 2003Explorercooled at 4.2 K

Nautiluscooled at 2 K

880 Hz 980

Auriga is also broadbandnow (since Dec. 2003)

Problems in the detection ofg.w. pulses:

• low SNRs and rarity of the events• ignorance of their shape• the non stationary noise of the detector• the presence of many spurious events

There are different ways of implementing the filter procedures

➔ different ways of taking data (high frequency sampling, aliased sampling, lock-ins);➔ frequency or time domain procedures;➔ use of adaptive or non-adaptive procedures;➔ definition of the threshold and procedure to extract the events;➔ procedure to extract and use the events features, also to recognize spurious events.

Periodogram

From the 0-2500 Hz the high sensitivity band of the antenna is extracted (40 Hz)

Power Spectrum of the Nautilus data (5 kHz Acquisition Frequency)

In theory the best spectral estimation is obtained using as much data as possible. But various scenarious of non stationary noise are possible:

● Spurious peaks in the spectra;● “Short” time disturbances in the unfiltered data; ● “Long” time disturbances in the unfiltered data.

The adaptive algorithm is the method to

estimate a new spectrum from the data.

To construct the matched filter we need the power spectrum:given the presence of non stationarities in the system we do notuse the theoretical prediction for the noise, but we estimatethe noise from the data

A power spectrum and thecorresponding filter tranfer function

900 Hz 925 900 Hz 927.5

Power spectrum Filter transfer function

To face with these problems, we have implemented three different method to estimate the spectra and hence to build up the filters.

WHOLE CLEAN

ADAPTED (or varying memory)

WHOLE

CLEAN

ADAPT

The matched filter : Wux*(j ) S()

OUTPUT

CHANNELS

Adaptive filters use the actual noise spectrum, estimated from the data, to evaluate the filter transfer function.

The periodogram Pi is used to estimate the spectrum Si : a new periodogram is evaluatedevery 105 s. The time constant for thespectrum is 1 hour.

THE EVALUATION OF THE SPECTRUN IN NON-STATIONARY NOISE IS THE CRUCIAL STEP OF THE FILTERING PROCEDURE

The DAGA2_HF noise estimatorsfor matched filters on non-stationary noise

P. Astone, S. D'Antonio, S. Frasca, M. A. Papa

All the procedures use the same recursive equation to evaluate the new spectrum, but they differ for the value of the time constant and for the criterion to accept a periodogram in the average:

Whole: is fixed to 1 hour and all the periodograms are used Clean: is fixed to 1 hour but only “good” periodograms are used Adapted: varies according to the variance of the periodogram and all the periodograms are used

The CLEAN MATCHED filter

The problem of The problem of “short” time disturbances“short” time disturbances can be resolved by can be resolved by eliminating the correspondent periodogram in the spectral eliminating the correspondent periodogram in the spectral

estimation.estimation.The choice of the periodogram is done evaluating the integral of the periodogram S and comparing it with its expected value Sp. The periodogram is eliminated if its integral differs from Sp (evaluated over a long time period) by more than one standard deviation.

The spectrum obtained with this procedure does not have high excess noise, respect to expectation.

This filter gives good results for short disturbances but is not a good filter

when data are noisier for long time period!

0.00 100

2.00 10-2

4.00 10-2

6.00 10-2

8.00 10-2

1.00 10-1

23 23.5 24 24.5 25

Teff[K]

Hours

clean

adapted

0.0

0.10

0.20

0.30

0.40

0.50

0.60

23 23.5 24 24.5 25

Unfiltered data

Hours

one minute average

V^2

The integral of the periodograms is over the threshold for about 40 min and the clean filter does not adjourn itself…

23 2 hours of data 25

Presence of “long” time disturbances in the data

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

900 905 910 915 920 925 930 935

spectrumof

adapted

V^2/Hz

Hz

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

900 905 910 915 920 925 930 935

spectrumof

clean

V^2/Hz

Hz

10-9

10-7

10-5

10-3

10-1

101

103

900 905 910 915 920 925 930 935

adapted and clean filters

V^2/Hz

cleanadapted

The ADAPTED is able to recover the disturbance due to higher noise around the minus mode (the white noise doesn’t change).The transfer functions are very different: the adapted, well adapted to the actual noise characteristics, has a lower gain around the minus mode, where the disturbances acted, compared to the gain of the clean filter, not well adapted to meet the new situation.

Spectrum estimated by the ADAPTED

Spectrum estimated by the

CLEAN

The CLEAN does not use the periodograms whose integral is over the threshold: the spectral estimation is not degraded.

The WHOLE uses all the periodograms the spectra estimation is degraded.

The CLEAN filter is better than the WHOLE, when the disturbance is over

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

7.6 7.8 8 8.2 8.4

08-0ct-2001

Teff[K]

Hours

clean

whole

0.020

0.040

0.060

0.080

0.10

0.12

0.14

0.16

7.6 7.8 8 8.2 8.4

08-oct-2001

V^2

Hours

7.5 8 8.5 hours

Presence of “short” time disturbances in the data

The calibration signals will be one channel of the ROG acquisition system, DAGA2-HF

- The best filter is the one that, properly normalized, gives the lower Teff

- Calibration signals, added to the noise of the detector, will be used to compare the filters, to evaluate the experimental efficiencies of detection and all the event parameters.

Now, let's show some data........

0.5 mK 8

Explorer 2001209 daysmedian=2.2 mK

Explorer 2003258 daysmedian=2.5 mK

2 mK=3.6 *10-19

0.5 mK 8

0.5 mK 8

Nautilus 2001161 daysmedian=2.8 mK

Nautilus 2003185 daysmedian=1.6 mK

0.5 mK 8

2 mK=3.6 *10-19

Data of unprecedentedsensitivity and veryhigh duty cycles andoverlapping times

The threshold mechanism: how to extract the events

The threshold changes with time, to follow the changes of the output noise : ADAPTIVE THRESHOLD= memory of the autoregressive average (10 minutes)

;

The threshold mechanism: how to extract the events

The threshold is set on the CRITICAL RATIO CR (which is easily related to the SNR)

CR

CR = 6 corresponds to SNR (energy)= 19.5

When the signal goes above the threshold an “event” begins. The eventends when the data go below the threshold for a time > “dead time”,which has been set depending on the apparatus, noise, expected signals.

threshold

time

amplitude

Definition of event

Any event is identified by various parameters (energy, CR, duration, nmax...)

Nautilus 2001: the bandwidth was of the

order of 1 Hz or smaller

Time dispersion: it is a function of the bandwidth (after filter) and SNR

-0.25 s 0.25

-0.1 s 0.1

Nautilus 2003: the bandwidth was of the order of 10 Hz

g(t)

g(t)

Coincidences: the pulse correlationCoincidences among events of different detectors are the basic pointof the search for bursts, in a network of detectors.

Coincidences are done within a given “window”, which can be fixed or may vary, according to the event time uncertainties.

To compare coincidences with the expected background we normallyuse the pulse correlation (the Poisson statistics is not good in caseof non-stationary rates, which is our case). The pulse correlation is a non-parametric procedure, introduced by J. Weber.

The background rate is evaluated by adding, N times, a bias time tothe events of one detector. The mean value c of C(is then compared to C(0)

Coincidences: the pulse correlationBecause of non-stationarities, the shape of C() may not be uniform,thus the evaluation of the chance probability of getting C(0) deservessome cares. We can use:

the Poisson probability with parameter c; the histogram of C(), evaluating how many times C() > C(0)

But, both methods have to be used with care. A deep study ofthe features in the time delay histogram is needed

seconds

Coincidences: the time delay histogram

Allegro-Explorer : Jun-Dec 1991 (180 days) Phys. Rev D 59, 1999

Explorer-Nautilus-Niobe : Dec 1994-Oct 1996 (Explorer-Nautilus: 57 days; Explorer-Niobe: 56

days)Astrop. Phys. 10, 1999IGEC 1997-1998 : Phys. Rev.

Letters, 85, 2000

Explorer-Nautilus 1998 : CQG, 18, 2001

Explorer-Nautilus 2001 : CQG, 19, 5449 (2002)

IGEC 1997-2000: IGEC 1997-2000: PRD 68, 022001 PRD 68, 022001 (2003)(2003)

Coincidence analyses done among resonant

detectors:

Coincidences with Astro-Particle detectors

http://igec.lnl.infn.it

IGEC search for burst g.w.in the years 1997-2000PRD 68,022001 (2003)

new upper limit on the rate of g.w. bursts

Net observation times(1997-2000 data-New IGEC protocol)

• 1 detector: 1322 days1.2 detectors: 713 days2.3 detectors: 178 days3.4 detectors: 29 days4.5 detectors: 0 days

The total span of the time of the analysis is 4 years=1460 days and:

– the time coverage is 90%, over 4 years

Upper limit on the rate r of g.w.,with the IGEC detectors

95% CL

90% CL

10

100

r/yr

Burst signals for a bar detector:we use to model them as 'delta' signals

➔G. w. from the core collapse: Muller catalog1. G. w. from neutron stars at different evolutionary stages

(Ferrari, Miniutti, Pons astro-ph/0210581 and CQG 20, S841 presented at GWDAW2002 in Kioto by V. Ferrari): hot joung stars: damped sinusoids with f(t) and (t) cooled stars: damped sinusoids with f and , for the QNMs

( 'moderate' ; 'small' -->the spectrum becomes 'flat')

2. G. w. from the Ringdown of BHs: damped sinusoids (s M/M0 f ~ 12kHz/(M/M0))

Burst signals for a bar detector:we use to model them as 'delta' signals

➔ Is this reasonable, given the actual bandwidth ? ➔ Which sources are suitable to do coincidences within a

network of bars and interferometers ? Approaches:

➔ Analytical➔ Simulations, adding fake signals to the noise of the

detectors

..in progress..

Use of Energy filters and Antenna pattern

● The sensitivity of each detector varies with time● The sensitivities of the various detectors are different● The same signal generates events with energies different for

each detector (due to the noise effect, related to SNR)

selection algorithm based on the event energies CQG, 18 (2001)

Practical problems of coincidence analysis:

A&A,398 (2003)

Event SNR

Differentialprobability

SNR=signal to noiseratio of the signal

(here: 10, 20, 50)

The x-axis gives thesignal to noise ratio

of the eventThe y-axis gives the

differential probabilityfor the SNRof the event

The basics of the use of energy filters: signals (from the source) and events (what we observe after

filtering)

SNR of the threshold

Probabilityof detection

SNR=signal to noiseratio of the signal

(here: 10, 20, 50)

The x-axis gives thesignal to noise ratio

of the thresholdThe y-axis gives the

probabilityof detection for a given

SNR-t

The basics of the use of energy filters: probability of detection as a function of

the threshold

<b>=0.57

GD GC

Explorer and Nautilus:coincidences in the year2001Rog:CQG 19, 5449 (2002)

P.A., G. D' Agostini,S. D' AntonioCQG 20 (2003)

sidereal time, in hours

Review critically how our beliefs are modified by the actual observation -> Bayesian analysis P. Astone,G. D'Agostini,S. D'Antonio CQG 20 (2003)

Ingredients of the inference are:

-->the data; -->the knowledge of the detectors; -->hypotheses on the underlying physics;

-->the physical quantity with respect to which we are uncertain is the g.w. rate on Earth, r, and the model responsible for g.w. emission;-->we are rather sure about b, but not about the number which will actually be observed;-->what is certain is the number nc of coincidences;

On times for the coincident observation withthe Explorer and Nautilus detectors

2001:126 days

2003:160 days

Nautilus 2004: 180 days with Teff < 3 mK

Explorer 2004: 60 days with Teff < 3 mK

Web sites, of resonant detectors:• Allegro gravity.phys.lsu.edu• Auriga www.auriga.lnl.infn.it• Explorer, www.lnf.infn.it/esperimenti/rog Nautilus• Niobe www.gravity.phys.edu.au