Enhancing gravitational-wave burst detection confidence in expanded detector networks with the BayesWave pipeline
EEnhancing gravitational-wave burst detection confidence in expanded detectornetworks with the BayesWave pipeline
Yi Shuen C. Lee ∗ School of Physics, The University of Melbourne, Victoria 3010, Australia
Margaret Millhouse † and Andrew Melatos ‡ OzGrav, The University of Melbourne, Victoria 3010, Australia
The global gravitational-wave detector network achieves higher detection rates, better parame-ter estimates, and more accurate sky localisation, as the number of detectors, I increases. Thispaper quantifies network performance as a function of I for BayesWave , a source-agnostic, wavelet-based, Bayesian algorithm which distinguishes between true astrophysical signals and instrumentalglitches. Detection confidence is quantified using the signal-to-glitch Bayes factor, B S , G . An analyticscaling is derived for B S , G versus I , the number of wavelets, and the network signal-to-noise ratio,SNR net , which is confirmed empirically via injections into detector noise of the Hanford-Livingston(HL), Hanford-Livingston-Virgo (HLV), and Hanford-Livingston-KAGRA-Virgo (HLKV) networksat projected sensitivities for the fourth observing run (O4). The empirical and analytic scalingsare consistent; B S , G increases with I . The accuracy of waveform reconstruction is quantified usingthe overlap between injected and recovered waveform, O net . The HLV and HLKV network recovers and of the injected waveforms with O net > . respectively, compared to with theHL network. The accuracy of BayesWave sky localisation is ≈ times better for the HLV networkthan the HL network, as measured by the search area, A , and the sky areas contained within and confidence intervals. Marginal improvement in sky localisation is also observed with theaddition of KAGRA. I. INTRODUCTION
The Laser Interferometer Gravitational-Wave Obser-vatory (LIGO) [1–3] has completed three observingruns, O1 [4, 5], O2 [5, 6] and O3 [7] between 2015 to2020, including joint searches with Italian partner, Virgo[8], in the final month of O2 and the whole of O3. InApril 2019, Advanced LIGO commenced its third ob-serving run in collaboration with Advanced Virgo as athree-detector network: the Hanford-Livingston-Virgo(HLV) network. The Kamioka Gravitational Wave De-tector (KAGRA) [9–11] also began observing in Febru-ary 2020 [7].With access to these upgraded instruments, there isa burgeoning interest in detecting short-duration grav-itational wave (GW) signals by combining data frommulti-detector networks. These signals typically havedurations of milliseconds up to a few seconds, withthe most common sources being compact binary coales-cences (CBCs) such as black hole or neutron star merg-ers, along with other potential sources like core-collapsesupernovae (SNe) of massive stars [12], pulsar glitchesof astrophysical origin [13] and cusps in cosmic strings[14]. In addition to these known sources, it is also plausi-ble to detect transient signals of unknown astrophysicalorigin.Searches for generic GW transients, or burst searches,require the ability to distinguish such signals from anynoise artefacts present in the detector data. Hence, it is ∗ [email protected] † [email protected] ‡ [email protected] crucial to understand the noise properties of the detec-tor data. Results from the initial LIGO-Virgo scienceruns revealed non-stationary and non-Gaussian detectornoise, which includes short-duration noise transients de-noted by the term ‘glitches’ [15–17]. If not accounted forproperly, these features could resemble GWs and conse-quently limit the ability to detect low-amplitude signals.Since CBC signals come from known and well-studiedsources, such signals are accurately modelled in most re-gions of parameter space and therefore can be detectedwith high confidence using matched-filter searches [18–20]. Other GW bursts signals, on the other hand, mayoriginate from either complex or unanticipated sources.Given the stochastic nature and complexity of the po-tential sources (e.g. core collapse supernovae), there areno robust models available to date to assist with thesearches of generic burst signals, making it challengingto distinguish them from other non-Gaussian featureslike glitches in the detector data, as well as to accu-rately reconstruct the underlying signal waveform.There are a number of unmodelled burst searchesperformed in LIGO and Virgo data [21, 22]. In thiswork we look at an unmodelled search algorithm called BayesWave [23–25], which was proposed to enable thejoint detection and characterisation of GW bursts andinstrumental glitches.
BayesWave reconstructs bothsignals and glitches as a sum of sine-Gaussian wavelets,where the number of wavelets and their parameters aredetermined via a reversible-jump Markov chain MonteCarlo (RJMCMC) algorithm. Bayesian model selectionis then used to determine the likelihood of an event be-ing a true signal, or a noise artefact.Previous studies have quantified the performance of
BayesWave in recovering simulated waveforms fromsimulated noise with a two-detector network (HL net- a r X i v : . [ g r- q c ] F e b work) [26, 27]. However, with Virgo joining GWsearches alongside the HL-network in O2 and O3, KA-GRA coming online towards the end of O3, and futuredetectors like LIGO-India in the planning stages [28],the network of GW detectors is expanding rapidly. Ex-panding detector networks will increase the likelihoodof detecting more events with higher confidence. Theseimprovements are evident in previous studies and willbe elaborated further in Section II.In this paper, we aim to evaluate BayesWave ’s per-formance in searching for GW bursts from detectordata beyond the HL-network. We achieve this by using
BayesWave to recover injected signals from simulatednoise with the HLV and the HLKV detector networks,and comparing the outcomes with those of the HLnetwork. We quantify the performance of
BayesWave based on the following metric: (i) Bayes factor betweensignal and glitch models, (ii) overlap (match) betweeninjected and recovered waveforms, and (iii) accuracy ofrecovered sky location. In Section III, we provide a de-tailed overview of the
BayesWave algorithm. We de-rive the analytic scaling relation of the signal-to-glitchBayes factor in Section IV. We then discuss the methodsof injecting simulated waveforms into simulated detectornoise samples in Section V, followed by comparisons andanalyses of the metrics mentioned above: Bayes Factorin Section VI A, overlap in Section VI B and sky local-isation in Section VI C. Finally, we present a summaryof the results along with their implications in SectionVII.
II. BENEFITS OF EXPANDING DETECTORNETWORKS
Increasing the number of operational ground-baseddetectors has several major benefits for GW astronomy,including a higher rate of detection of GW transients,and better characterisation of those signals. Here wediscuss some of the benefits of adding new detectors tothe existing network.
A. SNR and search volume
One major advantage of a larger detector network isthe ability to confidently detect quieter events. Thestrain amplitude, s ( i ) in detector i of the network con-sists of a signal, h ( i ) (if present) and detector noise, n ( i ) which can be expressed as s ( i ) = h ( i ) + n ( i ) . (1)The squared matched-filter signal-to-noise ratio (SNR)of signal h ( i ) s in detector i is then given by [29]SNR i = (cid:16) h ( i ) s (cid:12)(cid:12)(cid:12) h ( i ) s (cid:17) (2)where ( . | . ) on the right-hand-side of the expression isthe noise-weighted inner product. We define the noise- weighted inner product between two arbitrary wave-forms h a ( t ) and h b ( t ) as [30] ( h a | h b ) = (cid:90) ∞ ˜ h a ∗ ( f ) ˜ h b ( f ) + ˜ h a ( f ) ˜ h b ∗ ( f ) S n ( f ) df. (3)where ˜ h ( f ) is the Fourier-transformed waveform, ˜ h ∗ ( f ) is its complex conjugate and S n ( f ) is the one-sidedpower spectral density (PSD) of stationary, Gaussiandetector noise.For a network with I detectors, the overall networkSNR is given by [26]SNR net = I (cid:88) i =1 SNR i (4)According to Equation 4, adding more detectors to thenetwork increases the SNR of all detected GW signals.This enables detection pipelines to estimate waveformparameters more accurately [31]. With improved pa-rameter estimates, more accurate models can be con-structed to represent the detected waveform [32].In addition, the SNR of GW signals scales with lumi-nosity distance, D L as [33]SNR i ∝ D L . (5)By combining Equations 4 and 5 and assuming coherentsearches, the overall SNR for a network of I detectorswith equal sensitivities is given by SNR net ∝ √I /D L .Assuming that GW sources are uniformly distributedacross the sky, an I -detector network can detect √I times further and up to √I more sources compared toa single detector network since the search volume scalesas V ∝ D L . B. Sky coverage
The sensitivity of a detector towards a particular skylocation is determined by the antenna pattern in thatgiven direction. Adding more detectors to the networkat different geographical locations and orientations in-creases the sensitivity of the network to a wider region ofthe sky (increased sky coverage), consequently increas-ing the detection rate and volume along those directions[34].Reference [35] presented a visual comparison be-tween the network antenna pattern across the wholesky between a three-detector (HLV) network and a four-detector (HLKV) network, where ‘K’ denotes KAGRA.As expected, results show that both networks are moresensitive to some regions in the sky than others. How-ever, the HLKV network has higher overall network an-tenna power pattern and an overall increase in sky cov-erage is also reflected in the expansion of regions withrelatively higher sensitivity.
C. Observing time
Adding detectors to the existing network also in-creases the duty cycle where two or more detectors arefunctional and simultaneously observing. This conse-quently increases the chances of the detectors pickingup a coherent astrophysical signal and leading to higherdetection rates [34].
D. Sky localisation
Sky localisation of a GW source is of vital importancefor locating and identifying any existing electromagneticcounterparts to the GW event [36]. Ground-based GWdetectors are nearly omnidirectional, so with a single de-tector we are not able to impose a strict constrains to thesky location of a GW event. Nevertheless, sky localisa-tion of GW signals improves significantly with multipleinterferometers. The times of arrival at two detectorsconstrain the position of the source to an error ellipse inthe sky map. Thus, having more detectors will reducelocalisation volume by imposing stricter constraints tothe location of the sources, improving the accuracy oflocating the source in the sky [36, 37]. .To sum up the points above, the advantages of havingmore detectors in the network include: (i) improvementin SNR and increased search volume, (ii) alignment-dependent sky coverage, (iii) increased rates of detec-tion, and (iv) improved sky localisation.
III.
BAYESWAVE
OVERVIEW
BayesWave is a Bayesian data analysis algorithmthat detects transient features in a stretch of detec-tor data and identifies whether they are an astrophys-ical signal or instrumental noise.
BayesWave recon-structs non-Gaussian features in the data using a sumof sine-Gaussian (also called Morlet-Gabor) wavelets.The number of wavelets and their respective param-eters are sampled using a trans-dimensional MarkovChain Monte Carlo algorithm, otherwise known as theReversible-Jump Markov Chain Monte Carlo (RJM-CMC). The RJMCMC is implemented to allow for ad-justable number of wavelets and hence variable modeldimensions.
BayesWave outputs posterior distributionsand Bayesian evidences for three separate models: (i)Gaussian noise only, (ii) Gaussian noise with glitchesand (iii) Gaussian noise with GW signal. The modelevidences are then used for Bayesian model selectionbetween the three scenarios.
A. Wavelet Frames
BayesWave uses a sum of sine-Gaussian (also calledMorlet-Gabor) wavelets to reconstruct non-Gaussianfeatures (either signals or glitches) in the detector data. Even though Sine-Gaussian wavelets form a non-orthogonal frames , their shape is variable in time-frequency plane and can optimally reconstruct a tran-sient GW signal with no a priori assumption on thesignal source or morphology.The number of wavelets used in the reconstructionis marginalised via the RJMCMC, where signals withcomplex structure in time-frequency plane will use morewavelets in the reconstruction. Previous studies [26, 38]have shown that the number of wavelets scales linearlywith SNR such that N ≈ γ + β SNR (6)where γ and β are constants which depend on waveformmorphology. The results from Ref. [26] show that β andhence N increase with waveform complexity. For bi-nary black hole (BBH) waveforms, the typical numbersare γ = 5 . and β = 0 . for sine-Gaussian waveletreconstructions [38].In BayesWave , each wavelet in the time domain hasfive intrinsic parameters t , f , Q, A, φ which representcentral time, central frequency, quality factor, ampli-tude and phase offset respectively. These intrinsic pa-rameters can be expressed as a single parameter vector λ = { t , f , Q, A, φ } and the mathematical representa-tion of a sine-Gaussian wavelet is given by Ψ( t ; t , f , Q, A, φ ) = Ae − ∆ t /τ cos(2 πf ∆ t + φ ) (7)with τ = Q/ (2 πf ) and ∆ t = t − t [26].The glitch model in BayesWave is independent be-tween detectors owing to the fact that noise artefacts areuncorrelated across different detectors. Hence, the setof glitch model parameters must contain the respectiveparameters for each individual detector across the net-work. The complete set of glitch model parameters fora network of detectors comprising Hanford, Livingston,and Virgo (HLV) can be written as [26] θ G = { λ H ∪ λ L ∪ λ V } (8)with λ i = { λ i ∪ λ i ∪ · · · ∪ λ iN G i } where the numericalsubscripts indicate a single wavelet used in the glitchmodel and N G is the total number of wavelets in theglitch model. The superscripts indicates the i -th detec-tor in the network.In contrast to the glitch model, the signal model iscommon across all detectors in the network. As a re-sult, signal models should have a single set of intrinsicwavelet parameters λ ⊕ = { λ ∪ λ ∪ · · · ∪ λ N S } , along Discrete wavelets can form orthogonal bases for signal or glitchrepresentations, but projecting the signal wavelets onto eachdetector requires the time translation operator which is com-putationally expensive. Despite the lack of orthogonality, sine-Gaussian wavelets are flexible in shape and have an analyticFourier representation. Hence the analysis can be done in thefrequency domain without the need of a time-translation oper-ator. Further details can be found in Section 3 of [24]. with a set of extrinsic parameter Ω = { α, δ, ψ, (cid:15) } whichsequentially describes the right ascension (RA), declina-tion (dec), polarisation angle and ellipticity of the GWsignal. The sky location (RA, dec) and polarisation an-gle of a source determine antenna beam patterns of thedetector network, as well as provide information on theamplitude and the arrival-time delay of the signal ineach detector [39]. Ellipticity defines the relative phaseand amplitude of the plus and cross polarisations, h + and h × respectively with h × = (cid:15)h + e iπ/ . The elliptic-ity parameter, (cid:15) takes values between to with thelower and upper bounds denoting linear to circular po-larisations respectively [24]. Altogether a complete setof signal model parameters is given by [26] θ S = { λ ⊕ ∪ Ω } . (9) BayesWave produces posterior distributions of theparameters described above. Each draw from the poste-rior contains a unique set of wavelet parameters (and ex-trinsic parameters for the signal model), which are thensummed to produce a posterior on the waveform, h ( t ) .By using this basis of sine-Gaussian wavelets, h ( t ) is re-constructed with no a priori assumption on the sourceof the GW signal. B. Model Selection
In addition to waveform reconstruction,
BayesWave performs model selection between the signal and glitchhypotheses described above. The ratio of model evi-dences, otherwise known as the Bayes factor, is the keyto model selection in Bayesian inference as it assessesthe plausibility of two different models, M α and M β ,parameterised by their respective parameter sets (cid:126)θ α and (cid:126)θ β . In other words, it quantifies which model is bettersupported by the data. The model evidence (also calledthe margainalised evidence) is given by p ( (cid:126)s |M α ) = (cid:90) p ( (cid:126)θ α |M α ) p ( (cid:126)s | (cid:126)θ α , M α ) d(cid:126)θ α (10)where (cid:126)s is the observed data, M α is the model, and (cid:126)θ α isthere parameter vector for model M α . The prior prob-ability of parameters (cid:126)θ α before the data are observed isgiven by p ( (cid:126)θ α |M α ) , and p ( (cid:126)s | (cid:126)θ α , M α ) is the likelihoodof obtaining the observed data (cid:126)s , given the model M α .Hence, the Bayes factor between models M α and M β ,parameterised by their respective parameter vectors (cid:126)θ α and (cid:126)θ β is B α,β ( (cid:126)s ) = p ( (cid:126)s |M α ) p ( (cid:126)s |M β ) . (11) B α,β ( (cid:126)s ) > implies that model M α is more stronglysupported by the data than model M β . To reduce com-putational costs, the BayesWave algorithm calculatesmodel evidence using thermodynamic integration [40].
BayesWave calculates the Bayes factor between thesignal model (i.e. the data contains a real astrophysical signal), and the glitch model (i.e. the data containsan instrumetnal glitch). In Section IV we discuss howthe signal-to-glitch Bayes factor scales with SNR, thenumber of wavelets used in the MCMC, and the numberof detectors in the network.
C. Overlap
In addition to distinguishing between signals andglitches,
BayesWave also produces a posterior distri-bution of the wavelet-expanded waveforms, h ( t ) tomatch the true waveform, h s ( t ) . One way to quantifythe agreement or similarity between h ( t ) and h s ( t ) isthrough the overlap, O . Reconstructed waveforms in BayesWave are analogous to waveform templates, hencethe overlap between reconstructed models and the in-jected waveform can be computed the same way as theoverlap in matched-filtering.The normalised overlap between the two waveformscan be written as [32] O = ( h | h s ) (cid:112) ( h | h ) ( h s | h s ) (12)where ( . | . ) is the noise-weighted inner product as de-fined in Equation 3. Since Equation 12 is normalised, O takes values between − to . When O = 1 , thereis a perfect match between the injected and recoveredwaveform; O = 0 implies that there is no match at alland O = − implies a perfect anti-correlation.Equation 12 only applies to a single detector. Anetwork overlap, O net is required to fully evaluate BayesWave ’s performance in recovering waveforms fromall the detectors combined. In order to define the net-work overlap, we sum each factor in Equation 12 overall I detectors in the network such that O net = (cid:80) I i =1 (cid:16) h ( i ) (cid:12)(cid:12)(cid:12) h ( i ) s (cid:17)(cid:114)(cid:80) I i =1 (cid:0) h ( i ) (cid:12)(cid:12) h ( i ) (cid:1) (cid:80) I i =1 (cid:16) h ( i ) s (cid:12)(cid:12)(cid:12) h ( i ) s (cid:17) , (13)where h ( i ) and h ( i ) s denote the recovered waveform andwaveform present in detector i respectively. IV. ANALYTIC BAYES FACTOR SCALING
In this work, we aim to understand the behavior ofthe Bayes factor between signal and glitch models fornetworks comprising different numbers of GW detectors.Hence it is in our interest to analytically understand theconditions of model selection. We want to know underwhat circumstances a model is favoured over another.
A. Occam Penalty
A key to understanding Bayes factor behavior whenusing a trans-dimensional model as
BayesWave does, isthe role of the Occam penalty.The parameter value at which the posterior distribu-tion peaks is known as the maximum a posteriori (MAP)value, denoted as (cid:126)θ
MAP . For high SNR events, the inte-grand of model evidence in Equation 10 peaks sharplyin the vicinity of the MAP. Following the Laplace-Fisherapproximation, the integral can be estimated as p ( (cid:126)s |M ) (cid:39) p ( (cid:126)s | (cid:126)θ MAP , M ) p ( (cid:126)θ MAP |M )(2 π ) D/ √ det C. (14)where p ( (cid:126)s | (cid:126)θ MAP , M ) is the MAP likelihood; p ( (cid:126)θ MAP |M ) is the prior evaluated at the MAP parameter values; D is the dimension of the model; and det C is the deter-minant of the full covariance matrix for the N waveletsused in waveform reconstruction. If the covariance ma-trix for a single wavelet is C n , then we have det C = N (cid:89) n =1 det C n , (15)assuming minimal overlap between the wavelet param-eter spaces. Since the Laplace-Fisher approximation isassociated with the MAP likelihood, the covariance ma-trix can be approximated as the inverse of the FisherInformation Matrix (FIM), Γ [41]. A comprehensive dis-cussion of the FIM and its relation to wavelet parameterjump proposal is presented in Appendix A.By definition, det C measures the variance of thelikelihood. Thus, √ det C quantifies the characteris-tic spread of the likelihood function. The product of √ det C and (2 π ) D/ , which account for the dimension-ality of the model, can then be used as a measure for thevolume of the uncertainty ellipsoid (posterior volume), ∆ V M for a given model M [26, 42, 43]. Assuming uni-form priors for all wavelet parameters, one can also write p ( (cid:126)θ MAP |M ) = 1 /V M where V M represents the total pa-rameter space volume. Hence, the last three factors ofEquation 14 can collectively be interpreted as the frac-tion of the prior occupied by the posterior distribution,such that the model evidence is now given by p ( (cid:126)s |M ) (cid:39) p ( (cid:126)s | (cid:126)θ MAP , M ) ∆ V M V M . (16)where ∆ V M /V M is the “Occam penalty factor”.Following equations 11 and 16, the Bayes factor be-tween two models can be re-expressed as B α,β ( (cid:126)s ) = Λ α,β ( (cid:126)s ) ∆ V α V α V β ∆ V β (17)where the ratio of MAP likelihoods is given by Λ α,β ( (cid:126)s ) = p ( (cid:126)s | (cid:126)θ MAP ,α ) p ( (cid:126)s | (cid:126)θ MAP ,β ) . (18)Equation 17 suggests that the Bayes factor is depen-dent on the likelihood ratio and the ratio of the Oc-cam penalty factors. The Occam factor penalises modelsthat require an unnecessarily large parameter space vol-ume to fit the data by suppressing the model evidence. Note that Occam penalty is not an intentionally addedcomponent to the Bayes factor, rather it is inherentlyimposed as a result of using the Bayes Theorem.As a heuristic explanation as to how the Occampenalty aids in BayesWave ’s ability to distinguish be-tween signals and glitches, recall that signal models ( S )for each detector share the same intrinsic parametersand four extrinsic parameters. Since there are five in-trinsic parameters ( t , f , Q, A, φ ) per wavelet, the di-mension of signal models scales as D S ∼ N + 4 (19)where N is the number of wavelets. Glitch models ( G ),on the other hand, have no extrinsic parameters but theglitch model of each detector is described by a uniqueset of intrinsic parameters. Assuming that signal andglitch models use the same number of wavelets such that N G = N S = N (see Appendix B), the dimension ofglitch models scales as [24] D G ∼ N I . (20)One therefore has D G > D S for I ≥ . This implies thatthe total parameter space volume for the glitch modelis larger than that of the signal model (i.e. V G > V S ). Ifboth models fit the data equally well (i.e. Λ S , G ≈ and ∆ V S ≈ V G ), then by Occam’s razor we should expectto see a selection bias towards the signal model as I increases. In other words, Equation 17 gives B S , G ( (cid:126)s ) = Λ B S , G ( (cid:126)s ) ∆ V S ∆ V G V G V S > (21)with increasing I .In Section IV B, we use the Laplace approximation tothe Bayesian evidence to derive an analytic scaling ofthe Bayes factor. B. Dependence of Bayes factor on number ofdetectors
In Ref. [26], Littenberg et al. put forth an analyticscaling of the log signal-to-glitch Bayes factor, ln B S , G ,in an effort to fully understand BayesWave ’s ability torobustly distinguish astrophysical signals from instru-mental glitches. They showed that the primary scalingof the Bayes factor goes as ln B S , G ∝ N ln( SNR net ) (22)where N is the number of wavelets used in the recon-struction, which is related to the signal morphology andSNR as described in Equation 6. The dependence ofBayes factor on N (and therefore the complexity of thesignal in time frequency plane) differentiates BayesWave from other unmodelled searches whose detection statis-tics scale primarily with SNR. The scaling found inRef. [26] assumes a network comprising two GW detec-tors. Here we extend this work to an arbitrary numberof detectors I . We begin with the Laplace approximation of model evidences for the signal and glitch models. From equa-tion 14, we find ln p ( d | S ) (cid:39) SNR net − N S − N S ln( V λ ) + N S (cid:88) n =1 ln (cid:32) ¯ Q n SNR net ,n (cid:33) + D Ω √ det C Ω V Ω (23) ln p ( d | G ) (cid:39) SNR net − I (cid:88) i =1 N G i N G i ln( V λ ) − N G i (cid:88) n =1 ln (cid:32) ¯ Q n SNR i,n (cid:33) (24)with ¯ Q n ≡ (2 π ) / √ Q n π . V λ is the prior volume of intrin-sic parameters and N xi is the total number of waveletsfor model x . The subscript i refers to detector i in thenetwork and n labels an individual wavelet from the setof wavelets for a given model. For instance: SNR i,n isthe SNR of wavelet n in the i -th detector . In the lasttwo terms of Equation 23, D Ω =4, C Ω and V Ω denotethe dimension, covariance matrix and the prior volumeof extrinsic parameters respectively. The full derivationfrom Equation 14 to Equations 23 and 24 can be found in Section III(A) of Ref. [26].To simplify the expressions for these evidences, we fol-low the same assumptions used in Ref [26], and whichare detailed further in Appendix B. One simplifyingassumption we highlight here again is that the num-ber of wavelets used in the signal model will be ap-proximately the same as the glitch model, and so weset N S = N G ≡ N (i.e. the N in Equation 22).Upon implementing the assumptions in Appendix B,the theoretical log Bayes factor between the signal andglitch model for a network of I detector(s) is given by ln B S , G (cid:39) ln p ( d | S ) − ln p ( d | G ) : ln B S , G (cid:39) ( I − (cid:34) N N ln( V λ ) − N (cid:88) n =1 ln (cid:0) ¯ Q n (cid:1) + 5 N ln (cid:18) SNR net √ N (cid:19)(cid:35) − I N ln( I ) + (cid:18) √ det C Ω V Ω (cid:19) . (25)The equation shows explicit dependence of the Bayesfactor on network SNR, number of wavelets and numberof detectors. We pay close attention to the scaling ln B S , G ∝ I N ln SNR net (26)which now has an extra scaling factor of I compared toEquation 22.The dependence on the number of wavelets used im-plies that the signal model is favoured over the glitchmodel with increasing waveform complexity (higher N ).In other words, a more complex waveform is more likelyto be classified as a signal [26]. This analytic resultagrees with the discussion in Section IV A where if twomodels fit the data equally well, the less complex modelwill be selected to represent the waveform. The pro-portionality ln B S , G ∝ I suggests that for signals withequal SNR and N , the Bayes factor should increase if weincrease the number of detectors in the network. Again, Each individual wavelet used in signal or glitch model recon-struction has an amplitude which can be converted into to SNR.For details, see [24]. this result agrees with the discussion in Section IV A;including more detectors in the network increases thedimensionality of the glitch model and thus the signalmodel will be even more strongly preferred.
V. INJECTION DATA SET
To empirically test the Bayes factor scaling given byEquation 25, as well as investigate the effect on wave-form reconstructions with detector networks of differentsizes, we inject a set of simulated BBH signals into simu-lated detector noise and recover them using
BayesWave .While
BayesWave is a flexible algorithm that can de-tect a variety of signals from different sources, we useBBH waveforms as our test bed because they are well-understood sources, and have previously been used tostudy the performance of BayesWave [26, 32, 45].In this work, we use tools from the LIGO AnalysisLibrary [46] to inject a set of non-spinning binary blackholes (BBH) with equal component masses of M (cid:12) .We use the phenomenological waveform IMRPhenomD tomodel spinning but non-precessing binaries using a com-bination of analytic post-Newtonian (PN), effective-one-
Figure 1. Projected LIGO, Virgo and KAGRA strain noise(i.e. amplitude spectral density), √ S n as a function of fre-quency for the fourth observing run, O4. The data used togenerate the noise curves above are retrieved from [44]. body (EOB) and numerical relativity (NR) methods[47, 48]. The GW sources are distributed isotropicallyacross the sky, and the inclinations ι are distributeduniformly in arccos ι . SNR net is distributed uniformlyin SNR net ∈ { , } where this SNR is calculated froma network comprising the HL detectors.We inject 150 BBH signals into Gaussian noisecoloured by the projected PSD of LIGO, Virgo and KA-GRA for the fourth observing run, O4, as given in theLIGO, Virgo and KAGRA Observing Scenario [44]. Thenoise curves are shown in Figure 1.We then recover the injected signals with BayesWave in three different scenarios: (i) Running only on Hanfordand Livingston (HL) data (a two detector network), (ii)Running on the Hanford, Livingston, and Virgo (HLV)data (a three detector network) and (iii) Running on the Hanford, Livingston, KAGRA and Virgo (HLKV) data(a four detector network). All three detector configura-tions use the exact same injection data set.In the two following sections, Sections VI A and VI B,we analyse two figures of merit: (i) Bayes factor and(ii) the overlap. By comparing these quantities betweenthe HL and HLV networks, we can evaluate the perfor-mance of
BayesWave in recovering the injected wave-forms from detector networks of different sizes. As anextension to previous studies on sky localisation withexpanded detector networks, we also compare the accu-racy of BayesWave in recovering the sky location fromdetector networks of different sizes in Section VI C.
VI. RESULTSA. Recovered Bayes factors
After analysing the injections described in Section V,we use the model evidences calculated by
BayesWave to understand the impact of GW detector network sizeon the log signal-to-glitch Bayes factor, ln B S , G . Forall the analyses in this paper, we only injections thathave been identified as inconsistent with Gaussian noise(this can be either a signal or glitch) by BayesWave .Injections indicated to be consistent with the Gaussiannoise model ( N ) by BayesWave are removed from thedata set, since it would be meaningless to evaluate theirrespective signal and glitch model evidences. In otherwords, injections with ln B S , N error bars encompassingvalues below zero are removed from the data set. Thewidths of ln B S , N error bars are given by [24] ∆[ln B S , N ] = (cid:112) { ∆[ln p ( d | S )] } + { ∆[ln p ( d | N )] } (27)where ∆[ln p ( d | M )] is the uncertainty for the logarith-mic evidence of model M . A total of 14 data points areremoved under this constraint. These events are all lowSNR net injections.The top left panel of Figure 2 shows ln B S , G as a func-tion of SNR net for the HL, HLV and HLKV networks.All three networks show a clear trend of increasing BayesFactor with increasing network SNR as expected. Ourresults also show that the HLKV injections have thehighest SNR overall, agreeing with Equation 4 whichindicates that increasing I increases SNR net . Further-more, we can see that injections at comparable SNRs arerecovered with higher ln B S , G in the HLV network thanthe HL network. In other words, even after accountingfor the increased SNR, we observe further enhancementin detection confidence for an expanded detector net-work, suggesting that ln B S , G is related to I , and notjust the SNR of the signal as predicted by Equation 25. The top right panel of Figure 2 shows the mediannumber of wavelets used in the BayesWave reconstruc-tion, N versus the injected SNR in the respective de-tector networks, SNR net . The median here refers to themedian of posterior distribution for N . We see that N increases systematically with SNR net in both the HLand HLV networks. This is expected since the detec-tors are able to pick up more complex features of thewaveform at high SNR. At low SNR (SNR (cid:46) ) thereis a slight deviation from the linear trend described byEquation 6 between N and SNR in both detector net-works. This is primarily due to the prior on the num-ber of wavelets. This prior is determined empiricallyfrom runs in LIGO data after O1, and peaks around N = 3 [25]. N also depends on waveform morphologyand complexity [26, 38]. Injecting the same set of BBHwaveforms into all three detector configurations resultin similar trends between N and SNR net . Figure 2. Top left panel shows the log signal-to-glitch Bayes Factor ln B S , G of BBH injection recoveries versus networksignal-to-noise ratio, SNR net . Each data point represents one BBH injection. Top right panel shows the median numberof wavelets used in signal model reconstruction for each injection, N versus SNR net . Bottom panel shows ln B S , G versus N , and the three colour bars indicates the network SNR of each data point in the corresponding detector network. In thetop panels, the horizontal axis corresponds three different network SNRs: (i) for the blue dot data points it corresponds toSNR net of the HL network, (ii) for the orange star data points it corresponds to SNR net of the HLV network, (iii) for thegreen cross data points it corresponds to SNR net of the HLKV network. Equation 25 shows that ln B S , G also scales with thenumber of wavelets used in the reconstruction. Hencewe also show empirically how the dimensionality of sig-nal model (i.e. the number of wavelets) also contributesto the increase in ln B S , G for different I . We show this inthe bottom panel of Figure 2 by plotting ln B S , G versus N . Colour bars indicate the SNR net of each data point.For all three detector configurations, ln B S , G generallyincreases with N , as predicted by Equation 25. At lowSNRs (i.e. SNR < ), detector networks recover thewaveform with N ≤ and ln B S , G ≤ because lowSNR injections have low amplitude features which areharder to reconstruct resulting in lower detection con-fidence. It is clear for injections recovered with N > that ln B S , G in the HLKV network are generally higherthan that of the HL and HLV networks at comparable N and SNR net . This again emphasizes the point thatthe Bayes factor scales with I .A more thorough investigation of the relation betweenthe empirical and analytic Bayes factor can be foundin Appendix C, where we use a simplified injection setof single sine-Gaussian wavelets. By recovering sine-Gaussian wavelets with sine-Gaussian wavelets, Equa-tion 6 reduces to N = 1 . The results show that the empirical scaling of the Bayes factor with I agrees withthe analytical scaling in Equation 25 to a good approx-imation.In summary, we show by comparing Bayes Factors be-tween the HL, HLV and HLKV networks that expandingdetector networks increases detection confidence. Ourempirical results are consistent with the analytic re-sults discussed Section IV, viz. ln B S , G ∝ I N ln SNR net .Heuristically, this can be understood via Occam’s ra-zor: if coincident identical glitches are unlikely in twodetectors, they are even more unlikely in three or moredetectors. Therefore when identical waveforms are de-tected simultaneously across larger networks, they havea higher likelihood of being a signal. B. Recovered Waveform Overlap
In the previous section, we showed that for a set ofBBH waveforms, ln B S , G increases with a larger numberof detectors in the network, meaning with more detec-tors our confidence in detection is strengthened. In thissection, we quantify the accuracy of BayesWave in wave-form recovery by comparing the overlap (also sometimes
Figure 3. Median overlap between the injected and recov-ered waveform, O net of the HL (blue dot) and HLV (orangestar) network, as a function of SNR net . The horizontal blueline indicates O net = 0 . and the vertical blue line indicatesSNR net > . called the match) between the injected and recoveredwaveforms for the HL, HLV and HLKV detector net-works. The network overlap, O net is given by Equation13. For the rest of this paper, any mention of overlaprefers to the network overlap.Figure 3 shows the median overlap, O net as a func-tion of network SNR, where O net of all three detectornetworks show positive correlation with their respectivenetwork SNR. This observation is consistent with previ-ous results, which show that network overlap scales withSNR [38, 45]. To illustrate how waveform reconstruc-tion improves with SNR, Figure 4 shows the injectedwaveform (black), the detector data (blue) and the credible interval of the recovered waveform (red) for twoevents in the HLKV network. The top and bottompanels show the waveforms for the injection recoveredwith the smallest overlap ( O min = 0 . ) and largestoverlap ( O max = 0 . ) of the whole injection data setrespectively. The event with the smallest overlap hasSNR net = 11 . and was recovered with ln B S , G = 9 . ,while the event with the largest overlap has SNR net =52 . and was recovered with ln B S , G = 218 . This isconsistent with the observed trend between overlap andnetwork SNR in Figure 3. The similar trend betweenoverlap and network SNR between all three detectorconfigurations indicates that waveform reconstructionfidelity is not directly related to the number of detec-tors in the network.However as noted earlier, increasing the number ofdetectors does increase the network SNR. By compar-ing the percentage of waveforms recovered with overlapabove a given threshold for all three detector configu-rations, we show that having an additional detector al-lows us to better reconstruct the signal waveform. Thethreshold is arbitrarily defined here to be O net > . and is indicated by the horizontal blue line in Figure3. We found that of the injections were recoveredwith O net > . for the HL network, for the HLVnetwork and for the HLKV network. While the inclusion of additional detector(s) doesnot have an extra benefit in the same way it does forthe Bayes factor as shown in the previous section, itnonetheless allows us to better reconstruct the signalwaveform due to increased SNR. However, the improve-ment is less significant upon the addition of KAGRA,since it is less sensitive compared to Virgo as shownin Figure 1 and therefore the increase in SNR is lesscompared to when Virgo is added to the network. Theoverall results also show that BayesWave is able to re-construct waveforms reasonably well with all three de-tector configurations for injections with SNR net > asindicated by the vertical blue line in Figure 3. C. Sky localisation
Expanding detector networks improves sky localisa-tion of GW events, as has been shown by various stud-ies on coherent network detections e.g [34] [36] and [49];see Section II. In this section, we compare the accuracyof
BayesWave in locating the source with the HL andHLV networks. We use two separate measures: (i) skyarea enclosed within the and credible intervals(CI) and (ii) search area, A .For every injection, BayesWave produces posteriordistributions for the sky location (in the form of rightascension and declination) of the GW signal. We firstlook at the sky area enclosed within and credi-ble intervals (CIs) of the posterior distribution of sourcelocation. In the left panel of Figure 5, we show the plotfor sky area enclosed within the
CI versus networkSNR for each injection, and similarly for the the
CIon the right panel. For all three detector configurations,we note that the area within the and
CIs mea-sured in square degrees (deg ) fundamentally reduceswith increasing network SNR due to improved accuracyin arrival time differences [36]. However, both sky areasare generally an order of magnitude smaller for the HLVnetwork compared to the HL network. Upon addition ofthe KAGRA detector, we observe further reduction inthe sky area, but not as drastic as that between the HLand HLV networks since KAGRA is less sensitive thanVirgo. The areas enclosed within both and CIsreduces with increasing I due to the additional arrivaltime differences which further constrain the location ofeach source. These results reiterate that accuracy of skylocalisation improves at fixed CI as I increases.We also compare the inferred sky location with thetrue injected location of the source. We introduceanother metric - the search area, A , the hypothet-ical sky area observed by a detector before it cor-rectly points towards the true location. To define thisquantity mathematically, we first denote the posteriorprobability density function (PDF) of sky location as p sky ( φ, θ ) . If the true location of the source is ( φ t , θ t ) and p = p sky ( φ t , θ t ) , then all points within A shouldhave p sky ≥ p . Mathematically, we write [32, 50] A = (cid:90) H [ p sky ( φ, θ ) − p ] d Ω (28)0 Figure 4. The top panel shows, for an injection with SNR net = 11 . and O = 0 . , the injected waveform (black), thedetector data (blue) and the credible interval of the recovered waveform (red) for each detector in the HLKV network.Similarly in the bottom panel but for an injection with SNR net = 52 . and O = 0 . . where H is the Heaviside step function and d Ω is thesurface area element on the celestial sphere i.e. d Ω =cos δdθdφ . In Figure 6 we plot the search area, A againstnetwork SNR for both the HL and HLV networks. TheHLVK search area is slightly smaller than the HLVsearch area, which in turn is significantly smaller thanthe HL search area, consistent with Figure 5.Overall, we see that sky localisation improves remark-ably when a detector of high-sensitivity is added to thenetwork. If a less sensitive detector is added, the im-provements are small but not negligible. VII. CONCLUSION
The aim of this study is to compare the performanceof
BayesWave in recovering GW waveforms from de-tector networks of different sizes. We derive an analyticscaling for the Bayes factor between the signal and glitchmodels, B S , G . We then inject a set of simulated BBHsignals of fixed masses at different SNRs into simulatedO4 detector data of the HL, HLV and HLKV network.We quantify BayesWave ’s performance in signal iden-tification with B S , G and the performance in waveformreconstruction with overlap, O net . We also compare theaccuracy of sky localisation between the two networks.We find that events of similar injected SNR analysed1 Figure 5. The left panel shows the sky area enclosed within the credible interval (CI) in square degrees versus thenetwork SNR of the corresponding detector network. Similarly on the right panel, except for the
CI.Figure 6. Search area, A (Equation 28) versus network SNRfor the HL (blue dots) and HLV (orange stars) networks. using the HLV and HLKV network have higher ln B S , G than those using the HL network. This agrees with the-oretical prediction of the Bayes factor scaling: ln B S , G ∝ I N ln SNR net . (29)Previous work [26] demonstrated that BayesWave isunique amongst GW umodelled burst searches inthat the so-called “complexity” of the signal in time-frequency plane plays a crucial role in the detectionstatistic, rather than just the signal’s strength. This isunderstood through the factor of N in Equation 29: asignal with more complex structure needs more waveletsto accurately reconstruct the waveform. In this work,we expose another novel feature of the BayesWave al-gorithm: the detection statistic is also influenced by thenumber of detectors i.e. the factor of I in Equation 29.Events of similar injected SNR (SNR net ) analysed usinglarger detector networks have higher ln B S , G , indicatingdetection confidence increases more than we would ex-pect purely from the increase in SNR net .The network overlap, O net , between the injected andrecovered waveforms increases with SNR net . We alsoshow that of the HLKV network, of the HLVnetwork and of the HL network injections have O > . . Since larger detector networks can detect sig-nals at higher SNR, they pick up more details of thetrue waveform. Thus, BayesWave can reconstruct thewaveforms more accurately.Finally in Section VI C, we quantify accuracy of skylocalisation with the sky area enclosed within the and credible intervals (CI). We find that both areasdecrease with increasing SNR net and are generally an or-der of magnitude smaller for the HLV networks than theHL network. The reduction of sky area is less significantupon the addition of the KAGRA detector due to its lowsensitivity compared to Virgo. The search area, A alsodecreases with increasing SNR net and increasing numberof detectors. The overall results suggest that increasingthe number of detectors at different geographical loca-tions improves sky localisation, consistent with previousanalyses [34, 36, 49].With the global detector network growing in size,the outlook for improving detection confidence with un-modelled burst searches is promising. Prospective workalong the lines of the research presented in this papermay include injecting different waveform morphologiesto compare detection confidence between detector net-works of different sizes. We also recommend looking intoquantifying and comparing the outcomes of BayesWave in recovering simulated signals from more realistic de-tector noise (i.e. in the presence of glitches) betweendifferent detector configurations.In summary,
BayesWave shows significant improve-ments in terms of waveform recovery and parameter es-timation when working with a larger detector network.This promising result suggests that with more detectorsjoining the global network in the future, we will be ableto reconstruct generic GW burst signals more accuratelyusing
BayesWave making detections with higher Bayesfactor and hence with higher confidence.2
ACKNOWLEDGEMENTS
Parts of this research were conducted by the Aus-tralian Research Council Centre of Excellence for Grav-itational Wave Discovery (OzGrav), through projectnumber CE170100004. The authors are grateful forcomputational resources provided by the LIGO Labo-ratory and supported by National Science FoundationGrants PHY-0757058 and PHY-0823459. We thankBence Bécsy for his helpful comments.
Appendix A: Fisher Information Matrix
Each wavelet has its Fisher Information Matrices(FIMs), Γ written in terms of its five intrinsic parame-ters { t , f , Q, ln A, φ } Γ =
SNR π f (1+ Q ) Q − πf Q f − Qf − f − Qf Q Q − f Q − πf . (A1)FIMs contain information on local curvature of the like-lihood of wavelet parameters which accelerates conver-gence by proposing jumps in the MCMC algorithm to-wards regions of higher likelihood [24]. BayesWave usesFIMs to update wavelet parameters by drawing propos-als from a multivariate Gaussian distribution q ( y | x ) = det Γ(2 π ) exp (cid:18) −
12 Γ ij ∆ x i ∆ x j (cid:19) (A2) where ∆ x i = x i − y i denotes the displacement in intrin-sic parameter i before and after the update. Appendix B: Assumptions for Bayes Factor Scaling
Laplace approximations for the logarithmic signal ( S )and glitch ( G ) model evidences are given by Equations23 and 24 respectively. In order to see how B S , G scaleswith the waveform parameters, we make some assump-tions to simplify the two logarithmic evidences. In thiswork we use the same assumptions as in Ref. [26].Loud signals typically have optimal extrinsic param-eters across the detector network, so the SNR in eachdetector will be approximately equal such thatSNR i,n ≈ SNR net ,n √I (B1)where SNR i,n is the SNR of the n -th wavelet in detector i . We use a further simplifying assumption that the SNRof each wavelet is the sameSNR net ,n ≈ SNR net √ N , (B2)which has been empirically validated. We assume thatthe glitch model in each detector uses similar reconstruc-tion parameters as the signal model, and as such thequality factors of all wavelets are approximately equal: Q G i,n ≈ Q S n ≡ Q (B3)and similarly, N G ≈ N S ≡ N. (B4)Recall that N G indicates the number of wavelets usedin the glitch model for a single detector, so for an I -detector network, the total number of wavelets used inglitch models across the entire network is I N .Equations 23 and 24 can be simplified to ln p ( d | S ) (cid:39) SNR net − N − N ln( V λ ) + N (cid:88) n =1 ln (cid:0) ¯ Q n (cid:1) − N ln (cid:18) SNR net √ N (cid:19) + (cid:18) √ det C Ω V Ω (cid:19) (B5) ln p ( d | G ) (cid:39) SNR net − I (cid:34) N N ln( V λ ) − N (cid:88) n =1 ln (cid:0) ¯ Q n (cid:1) + 5 N ln (cid:18) SNR net √ N I (cid:19)(cid:35) . (B6) Appendix C: Scaling of Bayes factor with I Our results in Section VI A show that ln B S , G scaleswith SNR, N , and I . As per Equation 6, N itself de-pends on both the SNR of the signal, and the waveformmorphology. In order to specifically test the scaling of ln B S , G with I alone, we inject a set of sine-Gaussian wavelets as coherent signals into detector noise for theHL, HLV and HLKV network and then recover themusing BayesWave . Because sine-Gaussian wavelets arethe basis of reconstruction for
BayesWave , the numberof wavelets used is N = 1 , with no dependence on SNR.The dataset used this analysis is a set of 150 singlesine-Gaussian wavelets. The parameters of each wavelet3 Figure 7. Log signal-to-glitch Bayes factor, ln B S , G of sine-Gaussian wavelet recoveries versus network signal-to-noiseratio, SNR net . The solid lines with colours corresponding tothe data symbols are analytic predictions of ln B S , G given byEquation C2. are randomly drawn from the following distributions: t ∈ [1 . , . s (where t = 1 s is the center of the analysiswindow), f ∈ [32 , Hz, Q ∈ [0 . , and φ ∈ [0 , π ] . The SNR of the signals are drawn randomlyfrom a uniform distribution and SNR ∈ [10 , , andthe amplitude is then found viz. A = SNR (cid:115) √ πf S n ( f ) Q (C1)(see [38] for details). As we are injecting a coher-ent signal, we also require four extrinsic parametersas described in Section III A. These parameters arealso drawn randomly from uniform distributions suchthat α ∈ [0 , π ] , δ ∈ [ − π / , π / ] , ψ ∈ [0 , π ] and (cid:15) ∈ [ − . , . .In Figure 7, we plot ln B S , G of each injection against SNR net for the HL, HLV and HLKV network injections.We note that ln B S , G increases with SNR net and is gen-erally higher for networks with greater I , as predictedfrom Equation 26. Since N in this case does not de-pend on SNR, we can be certain that the differences in ln B S , G between the different detector configurations atcomparable SNR net are entirely due to I .In order to compare the analytic and empirical scal-ing of ln B S , G with I , we fit analytic approximation of ln B S , G for each detector network with a generalised ex-pression ln B S , G ≈ ( I −
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