# Enhancing gravitational-wave burst detection confidence in expanded detector
networks with the BayesWave pipeline

Yi Shuen C. Lee<EMAIL_ADDRESS>School of Physics, The
University of Melbourne, Victoria 3010, Australia  Margaret Millhouse
<EMAIL_ADDRESS>OzGrav, The University of Melbourne, Victoria
3010, Australia  Andrew Melatos<EMAIL_ADDRESS>OzGrav, The
University of Melbourne, Victoria 3010, Australia

###### Abstract

The global gravitational-wave detector network achieves higher detection
rates, better parameter estimates, and more accurate sky localisation, as the
number of detectors, $\mathcal{I}$ increases. This paper quantifies network
performance as a function of $\mathcal{I}$ for BayesWave, a source-agnostic,
wavelet-based, Bayesian algorithm which distinguishes between true
astrophysical signals and instrumental glitches. Detection confidence is
quantified using the signal-to-glitch Bayes factor,
$\mathcal{B}_{\mathcal{S},\mathcal{G}}$. An analytic scaling is derived for
$\mathcal{B}_{\mathcal{S},\mathcal{G}}$ versus $\mathcal{I}$, the number of
wavelets, and the network signal-to-noise ratio, SNR${}_{\text{net}}$, which
is confirmed empirically via injections into detector noise of the Hanford-
Livingston (HL), Hanford-Livingston-Virgo (HLV), and Hanford-Livingston-KAGRA-
Virgo (HLKV) networks at projected sensitivities for the fourth observing run
(O4). The empirical and analytic scalings are consistent;
$\mathcal{B}_{\mathcal{S},\mathcal{G}}$ increases with $\mathcal{I}$. The
accuracy of waveform reconstruction is quantified using the overlap between
injected and recovered waveform, $\mathcal{O}_{\text{net}}$. The HLV and HLKV
network recovers $87\%$ and $86\%$ of the injected waveforms with
$\mathcal{O}_{\text{net}}>0.8$ respectively, compared to $81\%$ with the HL
network. The accuracy of BayesWave sky localisation is $\approx 10$ times
better for the HLV network than the HL network, as measured by the search
area, $\mathcal{A}$, and the sky areas contained within $50\%$ and $90\%$
confidence intervals. Marginal improvement in sky localisation is also
observed with the addition of KAGRA.

##  I Introduction

The Laser Interferometer Gravitational-Wave Observatory (LIGO) LIGO2015 ;
2009RPPh…72g6901A ; Harry_2010  has completed three observing runs, O1
2017PhRvD..95d2003A ; 2019PhRvX…9c1040A , O2 2019PhRvD.100b4017A ;
2019PhRvX…9c1040A  and O3 2020NatRP…2..222G  between 2015 to 2020, including
joint searches with Italian partner, Virgo 2015CQGra..32b4001A , in the final
month of O2 and the whole of O3. In April 2019, Advanced LIGO commenced its
third observing run in collaboration with Advanced Virgo as a three-detector
network: the Hanford-Livingston-Virgo (HLV) network. The Kamioka Gravitational
Wave Detector (KAGRA) 2019CQGra..36p5008A ; 2020arXiv200505574A ;
2013PhRvD..88d3007A  also began observing in February 2020 2020NatRP…2..222G .

With access to these upgraded instruments, there is a burgeoning interest in
detecting short-duration gravitational wave (GW) signals by combining data
from multi-detector networks. These signals typically have durations of
milliseconds up to a few seconds, with the most common sources being compact
binary coalescences (CBCs) such as black hole or neutron star mergers, along
with other potential sources like core-collapse supernovae (SNe) of massive
stars 2011LRR….14….1F , pulsar glitches of astrophysical origin
2014ApJ…787..114S  and cusps in cosmic strings 2005PhRvD..71f3510D . In
addition to these known sources, it is also plausible to detect transient
signals of unknown astrophysical origin.

Searches for generic GW transients, or burst searches, require the ability to
distinguish such signals from any noise artefacts present in the detector
data. Hence, it is crucial to understand the noise properties of the detector
data. Results from the initial LIGO-Virgo science runs revealed non-stationary
and non-Gaussian detector noise, which includes short-duration noise
transients denoted by the term ‘glitches’ 2008CQGra..25r4004B ;
2015CQGra..32k5012A ; 2012CQGra..29o5002A . If not accounted for properly,
these features could resemble GWs and consequently limit the ability to detect
low-amplitude signals.

Since CBC signals come from known and well-studied sources, such signals are
accurately modelled in most regions of parameter space and therefore can be
detected with high confidence using matched-filter searches
2016CQGra..33u5004U ; 2019arXiv190108580S ; 2012PhRvD..86b4012H . Other GW
bursts signals, on the other hand, may originate from either complex or
unanticipated sources. Given the stochastic nature and complexity of the
potential sources (e.g. core collapse supernovae), there are no robust models
available to date to assist with the searches of generic burst signals, making
it challenging to distinguish them from other non-Gaussian features like
glitches in the detector data, as well as to accurately reconstruct the
underlying signal waveform.

There are a number of unmodelled burst searches performed in LIGO and Virgo
data 2017PhRvD..95j4046L ; cWB . In this work we look at an unmodelled search
algorithm called BayesWave 2015PhRvD..91h4034L ; 2015CQGra..32m5012C ; BWIII ,
which was proposed to enable the joint detection and characterisation of GW
bursts and instrumental glitches. BayesWave reconstructs both signals and
glitches as a sum of sine-Gaussian wavelets, where the number of wavelets and
their parameters are determined via a reversible-jump Markov chain Monte Carlo
(RJMCMC) algorithm. Bayesian model selection is then used to determine the
likelihood of an event being a true signal, or a noise artefact.

Previous studies have quantified the performance of BayesWave in recovering
simulated waveforms from simulated noise with a two-detector network (HL
network) 2016PhRvD..94d4050L ; 2016PhRvD..93b2002K . However, with Virgo
joining GW searches alongside the HL-network in O2 and O3, KAGRA coming online
towards the end of O3, and future detectors like LIGO-India in the planning
stages 2013IJMPD..2241010U , the network of GW detectors is expanding rapidly.
Expanding detector networks will increase the likelihood of detecting more
events with higher confidence. These improvements are evident in previous
studies and will be elaborated further in Section II.

In this paper, we aim to evaluate BayesWave’s performance in searching for GW
bursts from detector data beyond the HL-network. We achieve this by using
BayesWave to recover injected signals from simulated noise with the HLV and
the HLKV detector networks, and comparing the outcomes with those of the HL
network. We quantify the performance of BayesWave based on the following
metric: (i) Bayes factor between signal and glitch models, (ii) overlap
(match) between injected and recovered waveforms, and (iii) accuracy of
recovered sky location. In Section III, we provide a detailed overview of the
BayesWave algorithm. We derive the analytic scaling relation of the signal-to-
glitch Bayes factor in Section IV. We then discuss the methods of injecting
simulated waveforms into simulated detector noise samples in Section V,
followed by comparisons and analyses of the metrics mentioned above: Bayes
Factor in Section A, overlap in Section B and sky localisation in Section C.
Finally, we present a summary of the results along with their implications in
Section VII.

##  II Benefits of Expanding Detector Networks

Increasing the number of operational ground-based detectors has several major
benefits for GW astronomy, including a higher rate of detection of GW
transients, and better characterisation of those signals. Here we discuss some
of the benefits of adding new detectors to the existing network.

###  A SNR and search volume

One major advantage of a larger detector network is the ability to confidently
detect quieter events. The strain amplitude, $s^{(i)}$ in detector $i$ of the
network consists 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 1994PhRvD..49.2658C

$\text{SNR}_{i}^{2}=\left(h^{(i)}_{s}\,\middle|\,h^{(i)}_{s}\right)$ (2)

where $(.|.)$ on the right-hand-side of the expression is the noise-weighted
inner product. We define the noise-weighted inner product between two
arbitrary waveforms $h_{a}(t)$ and $h_{b}(t)$ as keppel…2009

$\left(h_{a}\,\middle|\,h_{b}\right)=\int_{0}^{\infty}\frac{\tilde{h_{a}}^{*}(f)\tilde{h_{b}}(f)+\tilde{h_{a}}(f)\tilde{h_{b}}^{*}(f)}{S_{n}(f)}\,df.$
(3)

where $\tilde{h}(f)$ is the Fourier-transformed waveform, $\tilde{h}^{*}(f)$
is its complex conjugate and $S_{n}(f)$ is the one-sided power spectral
density (PSD) of stationary, Gaussian detector noise.

For a network with $\mathcal{I}$ detectors, the overall network SNR is given
by 2016PhRvD..94d4050L

$\text{SNR}_{\text{net}}^{2}=\sum_{i=1}^{\mathcal{I}}\text{SNR}_{i}^{2}$ (4)

According to Equation 4, adding more detectors to the network increases the
SNR of all detected GW signals. This enables detection pipelines to estimate
waveform parameters more accurately 2017CQGra..34q4003G . With improved
parameter estimates, more accurate models can be constructed to represent the
detected waveform 2017ApJ…839…15B .

In addition, the SNR of GW signals scales with luminosity distance, $D_{L}$ as
2014arXiv1409.0522C

$\text{SNR}_{i}\propto\frac{1}{D_{L}}.$ (5)

By combining Equations 4 and 5 and assuming coherent searches, the overall SNR
for a network of $\mathcal{I}$ detectors with equal sensitivities is given by
$\text{SNR}_{\text{net}}\propto\sqrt{\mathcal{I}}/D_{L}$. Assuming that GW
sources are uniformly distributed across the sky, an $\mathcal{I}$-detector
network can detect $\sqrt{\mathcal{I}}$ times further and up to
$\sqrt{\mathcal{I}^{3}}$ more sources compared to a single detector network
since the search volume scales as $V\propto D_{L}^{3}$.

###  B Sky coverage

The sensitivity of a detector towards a particular sky location is determined
by the antenna pattern in that given direction. Adding more detectors to the
network at different geographical locations and orientations increases the
sensitivity of the network to a wider region of the sky (increased sky
coverage), consequently increasing the detection rate and volume along those
directions 2011CQGra..28l5023S .

Reference 2012arXiv1202.4031M  presented a visual comparison between the
network antenna pattern across the whole sky between a three-detector (HLV)
network and a four-detector (HLKV) network, where ‘K’ denotes KAGRA. As
expected, results show that both networks are more sensitive to some regions
in the sky than others. However, the HLKV network has higher overall network
antenna power pattern and an overall increase in sky coverage is also
reflected in the expansion of regions with relatively higher sensitivity.

###  C Observing time

Adding detectors to the existing network also increases the duty cycle where
two or more detectors are functional and simultaneously observing. This
consequently increases the chances of the detectors picking up a coherent
astrophysical signal and leading to higher detection rates 2011CQGra..28l5023S
.

###  D Sky localisation

Sky localisation of a GW source is of vital importance for locating and
identifying any existing electromagnetic counterparts to the GW event
2018ApJ…854L..25P . Ground-based GW detectors are nearly omnidirectional, so
with a single detector we are not able to impose a strict constrains to the
sky location of a GW event. Nevertheless, sky localisation of GW signals
improves significantly with multiple interferometers. The times of arrival at
two detectors constrain the position of the source to an error ellipse in the
sky map. Thus, having more detectors will reduce localisation volume by
imposing stricter constraints to the location of the sources, improving the
accuracy of locating the source in the sky 2018MNRAS.479..601D ;
2018ApJ…854L..25P . .

To sum up the points above, the advantages of having more detectors in the
network include: (i) improvement in SNR and increased search volume, (ii)
alignment-dependent sky coverage, (iii) increased rates of detection, and (iv)
improved sky localisation.

##  III BayesWave Overview

BayesWave is a Bayesian data analysis algorithm that detects transient
features in a stretch of detector data and identifies whether they are an
astrophysical signal or instrumental noise. BayesWave reconstructs non-
Gaussian features in the data using a sum of sine-Gaussian (also called
Morlet-Gabor) wavelets. The number of wavelets and their respective parameters
are sampled using a trans-dimensional Markov Chain Monte Carlo algorithm,
otherwise known as the Reversible-Jump Markov Chain Monte Carlo (RJMCMC). The
RJMCMC is implemented to allow for adjustable number of wavelets and hence
variable model dimensions. BayesWave outputs posterior distributions and
Bayesian evidences for three separate models: (i) Gaussian noise only, (ii)
Gaussian noise with glitches and (iii) Gaussian noise with GW signal. The
model evidences are then used for Bayesian model selection between the three
scenarios.

###  A Wavelet Frames

BayesWave uses a sum of sine-Gaussian (also called Morlet-Gabor) wavelets to
reconstruct non-Gaussian features (either signals or glitches) in the detector
data. Even though Sine-Gaussian wavelets form a non-orthogonal
frames111Discrete wavelets can form orthogonal bases for signal or glitch
representations, but projecting the signal wavelets onto each detector
requires the time translation operator which is computationally expensive.
Despite the lack of orthogonality, sine-Gaussian wavelets are flexible in
shape and have an analytic Fourier representation. Hence the analysis can be
done in the frequency domain without the need of a time-translation operator.
Further details can be found in Section 3 of 2015CQGra..32m5012C ., their
shape is variable in time-frequency plane and can optimally reconstruct a
transient GW signal with no _a priori_ assumption on the signal source or
morphology.

The number of wavelets used in the reconstruction is marginalised via the
RJMCMC, where signals with complex structure in time-frequency plane will use
more wavelets in the reconstruction. Previous studies 2016PhRvD..94d4050L ;
2018PhRvD..97j4057M  have shown that the number of wavelets scales linearly
with SNR such that

$N\approx\gamma+\beta\text{ SNR}$ (6)

where $\gamma$ and $\beta$ are constants which depend on waveform morphology.
The results from Ref. 2016PhRvD..94d4050L  show that $\beta$ and hence $N$
increase with waveform complexity. For binary black hole (BBH) waveforms, the
typical numbers are $\gamma=5.6$ and $\beta=0.066$ for sine-Gaussian wavelet
reconstructions 2018PhRvD..97j4057M .

In BayesWave, each wavelet in the time domain has five intrinsic parameters
$t_{0},f_{0},Q,A,\phi_{0}$ which represent central time, central frequency,
quality factor, amplitude and phase offset respectively. These intrinsic
parameters can be expressed as a single parameter vector
${\bf\it\lambda}=\\{t_{0},f_{0},Q,A,\phi_{0}\\}$ and the mathematical
representation of a sine-Gaussian wavelet is given by

$\Psi(t;t_{0},f_{0},Q,A,\phi_{0})=Ae^{-\Delta t^{2}/\tau^{2}}\cos(2\pi
f_{0}\Delta t+\phi_{0})$ (7)

with $\tau=Q/(2\pi f_{0})$ and $\Delta t=t-t_{0}$ 2016PhRvD..94d4050L .

The glitch model in BayesWave is independent between detectors owing to the
fact that noise artefacts are uncorrelated across different detectors. Hence,
the set of glitch model parameters must contain the respective parameters for
each individual detector across the network. The complete set of glitch model
parameters for a network of detectors comprising Hanford, Livingston, and
Virgo (HLV) can be written as 2016PhRvD..94d4050L

${\bf\it\theta_{\mathcal{G}}}=\\{{\bf\it\lambda}^{H}\cup{\bf\it\lambda}^{L}\cup{\bf\it\lambda}^{V}\\}$
(8)

with
${\bf\it\lambda}^{i}=\\{{\bf\it\lambda}_{0}^{i}\cup{\bf\it\lambda}_{1}^{i}\cup\cdots\cup{\bf\it\lambda}_{N^{\mathcal{G}}_{i}}^{i}\\}$
where the numerical subscripts indicate a single wavelet used in the glitch
model and $N^{\mathcal{G}}$ is the total number of wavelets in the glitch
model. The superscripts indicates the $i$-th detector in the network.

In contrast to the glitch model, the signal model is common across all
detectors in the network. As a result, signal models should have a single set
of intrinsic wavelet parameters
${\bf\it\lambda}^{\oplus}=\\{{\bf\it\lambda}_{0}\cup{\bf\it\lambda}_{1}\cup\cdots\cup{\bf\it\lambda}_{N^{\mathcal{S}}}\\}$,
along with a set of extrinsic parameter
${\bf\it\Omega}=\\{\alpha,\delta,\psi,\epsilon\\}$ which sequentially
describes the right ascension (RA), declination (dec), polarisation angle and
ellipticity of the GW signal. The sky location (RA, dec) and polarisation
angle of a source determine antenna beam patterns of the detector network, as
well as provide information on the amplitude and the arrival-time delay of the
signal in each detector 2004ARNPS..54..525C . Ellipticity defines the relative
phase and amplitude of the plus and cross polarisations, $h_{+}$ and
$h_{\times}$ respectively with $h_{\times}=\epsilon h_{+}e^{i\pi/2}$. The
ellipticity parameter, $\epsilon$ takes values between $0$ to $1$ with the
lower and upper bounds denoting linear to circular polarisations respectively
2015CQGra..32m5012C . Altogether a complete set of signal model parameters is
given by 2016PhRvD..94d4050L

${\bf\it\theta_{\mathcal{S}}}=\\{{\bf\it\lambda}^{\oplus}\cup{\bf\it\Omega}\\}.$
(9)

BayesWave produces posterior distributions of the parameters described above.
Each draw from the posterior contains a unique set of wavelet parameters (and
extrinsic parameters for the signal model), which are then summed to produce a
posterior on the waveform, $h(t)$. By using this basis of sine-Gaussian
wavelets, $h(t)$ is reconstructed with no a priori assumption on the source of
the GW signal.

###  B Model Selection

In addition to waveform reconstruction, BayesWave performs model selection
between the signal and glitch hypotheses described above. The ratio of model
evidences, otherwise known as the Bayes factor, is the key to model selection
in Bayesian inference as it assesses the plausibility of two different models,
$\mathcal{M}_{\alpha}$ and $\mathcal{M}_{\beta}$, parameterised by their
respective parameter sets $\vec{\theta}_{\alpha}$ and $\vec{\theta}_{\beta}$.
In other words, it quantifies which model is better supported by the data. The
model evidence (also called the margainalised evidence) is given by

$p(\vec{s}|\mathcal{M}_{\alpha})=\int
p(\vec{\theta}_{\alpha}|\mathcal{M}_{\alpha})p(\vec{s}|\vec{\theta}_{\alpha},\mathcal{M}_{\alpha})d\vec{\theta}_{\alpha}$
(10)

where $\vec{s}$ is the observed data, $\mathcal{M}_{\alpha}$ is the model, and
$\vec{\theta}_{\alpha}$ is there parameter vector for model
$\mathcal{M}_{\alpha}$. The prior probability of parameters
$\vec{\theta}_{\alpha}$ before the data are observed is given by
$p(\vec{\theta}_{\alpha}|\mathcal{M}_{\alpha})$, and
$p(\vec{s}|\vec{\theta}_{\alpha},\mathcal{M}_{\alpha})$ is the likelihood of
obtaining the observed data $\vec{s}$, given the model $\mathcal{M}_{\alpha}$.
Hence, the Bayes factor between models $\mathcal{M}_{\alpha}$ and
$\mathcal{M}_{\beta}$, parameterised by their respective parameter vectors
$\vec{\theta}_{\alpha}$ and $\vec{\theta}_{\beta}$ is

$\begin{split}\mathcal{B}_{\alpha,\beta}(\vec{s})&=\frac{p(\vec{s}|\mathcal{M}_{\alpha})}{p(\vec{s}|\mathcal{M}_{\beta})}.\end{split}$
(11)

$\mathcal{B}_{\alpha,\beta}(\vec{s})>1$ implies that model
$\mathcal{M}_{\alpha}$ is more strongly supported by the data than model
$\mathcal{M}_{\beta}$. To reduce computational costs, the BayesWave algorithm
calculates model evidence using thermodynamic integration 2009PhRvD..80f3007L
.

BayesWave calculates the Bayes factor between the signal model (i.e. the data
contains a real astrophysical signal), and the glitch model (i.e. the data
contains an instrumetnal glitch). In Section IV we discuss how the signal-to-
glitch Bayes factor scales with SNR, the number of wavelets used in the MCMC,
and the number of detectors in the network.

###  C Overlap

In addition to distinguishing between signals and glitches, BayesWave also
produces a posterior distribution of the wavelet-expanded waveforms, $h(t)$ to
match the true waveform, $h_{s}(t)$. One way to quantify the agreement or
similarity between $h(t)$ and $h_{s}(t)$ is through the overlap,
$\mathcal{O}$. Reconstructed waveforms in BayesWave are analogous to waveform
templates, hence the overlap between reconstructed models and the injected
waveform can be computed the same way as the overlap in matched-filtering.

The normalised overlap between the two waveforms can be written as
2017ApJ…839…15B

$\mathcal{O}=\frac{\left(h\,\middle|\,h_{s}\right)}{\sqrt{\left(h\,\middle|\,h\right)\left(h_{s}\,\middle|\,h_{s}\right)}}$
(12)

where $(.|.)$ is the noise-weighted inner product as defined in Equation 3.
Since Equation 12 is normalised, $\mathcal{O}$ takes values between $-1$ to
$1$. When $\mathcal{O}=1$, there is a perfect match between the injected and
recovered waveform; $\mathcal{O}=0$ implies that there is no match at all and
$\mathcal{O}=-1$ implies a perfect anti-correlation.

Equation 12 only applies to a single detector. A network overlap,
$\mathcal{O_{\text{net}}}$ is required to fully evaluate BayesWave’s
performance in recovering waveforms from all the detectors combined. In order
to define the network overlap, we sum each factor in Equation 12 over all
$\mathcal{I}$ detectors in the network such that

$\mathcal{O_{\text{net}}}=\frac{\sum_{i=1}^{\mathcal{I}}\left(h^{(i)}\,\middle|\,h_{s}^{(i)}\right)}{\sqrt{\sum_{i=1}^{\mathcal{I}}\left(h^{(i)}\,\middle|\,h^{(i)}\right)\sum_{i=1}^{\mathcal{I}}\left(h_{s}^{(i)}\,\middle|\,h_{s}^{(i)}\right)}},$
(13)

where $h^{(i)}$ and $h_{s}^{(i)}$ denote the recovered waveform and waveform
present in detector $i$ respectively.

##  IV Analytic Bayes Factor Scaling

In this work, we aim to understand the behavior of the Bayes factor between
signal and glitch models for networks comprising different numbers of GW
detectors. Hence it is in our interest to analytically understand the
conditions of model selection. We want to know under what circumstances a
model is favoured over another.

###  A Occam Penalty

A key to understanding Bayes factor behavior when using a trans-dimensional
model as BayesWave does, is the role of the Occam penalty.

The parameter value at which the posterior distribution peaks is known as the
maximum a posteriori (MAP) value, denoted as $\vec{\theta}_{\text{MAP}}$. For
high SNR events, the integrand of model evidence in Equation 10 peaks sharply
in the vicinity of the MAP. Following the Laplace-Fisher approximation, the
integral can be estimated as

$p(\vec{s}|\mathcal{M})\simeq
p(\vec{s}|\vec{\theta}_{\text{MAP}},\mathcal{M})p(\vec{\theta}_{\text{MAP}}|\mathcal{M})(2\pi)^{D/2}\sqrt{\text{det}C}.$
(14)

where $p(\vec{s}|\vec{\theta}_{\text{MAP}},\mathcal{M})$ is the MAP
likelihood; $p(\vec{\theta}_{\text{MAP}}|\mathcal{M})$ is the prior evaluated
at the MAP parameter values; $D$ is the dimension of the model; and $\det C$
is the determinant of the full covariance matrix for the $N$ wavelets used in
waveform reconstruction. If the covariance matrix for a single wavelet is
$C_{n}$, then we have

$\det C=\prod^{N}_{n=1}\det C_{n},$ (15)

assuming minimal overlap between the wavelet parameter spaces. Since the
Laplace-Fisher approximation is associated with the MAP likelihood, the
covariance matrix can be approximated as the inverse of the Fisher Information
Matrix (FIM), $\Gamma$ doi:10.2307/3315678 . A comprehensive discussion of the
FIM and its relation to wavelet parameter jump proposal is presented in
Appendix A.

By definition, $\det C$ measures the variance of the likelihood. Thus,
$\sqrt{\det C}$ quantifies the characteristic spread of the likelihood
function. The product of $\sqrt{\det C}$ and $(2\pi)^{D/2}$, which account for
the dimensionality of the model, can then be used as a measure for the volume
of the uncertainty ellipsoid (posterior volume), $\Delta V_{\mathcal{M}}$ for
a given model $\mathcal{M}$ 2016PhRvD..94d4050L ; 2017LRR….20….2R ;
littenberg…2009 . Assuming uniform priors for all wavelet parameters, one can
also write $p(\vec{\theta}_{\text{MAP}}|\mathcal{M})=1/V_{\mathcal{M}}$ where
$V_{\mathcal{M}}$ represents the total parameter space volume. Hence, the last
three factors of Equation 14 can collectively be interpreted as the fraction
of the prior occupied by the posterior distribution, such that the model
evidence is now given by

$p(\vec{s}|\mathcal{M})\simeq
p(\vec{s}|\vec{\theta}_{\text{MAP}},\mathcal{M})\frac{\Delta
V_{\mathcal{M}}}{{V_{\mathcal{M}}}}.$ (16)

where $\Delta V_{\mathcal{M}}/V_{\mathcal{M}}$ is the “Occam penalty factor”.

Following equations 11 and 16, the Bayes factor between two models can be re-
expressed as

$\mathcal{B}_{\alpha,\beta}(\vec{s})=\Lambda_{\alpha,\beta}(\vec{s})\frac{\Delta
V_{\alpha}}{V_{\alpha}}\frac{V_{\beta}}{\Delta V_{\beta}}$ (17)

where the ratio of MAP likelihoods is given by

$\displaystyle\Lambda_{\alpha,\beta}(\vec{s})$
$\displaystyle=\frac{p(\vec{s}|\vec{\theta}_{\text{MAP},\alpha})}{p(\vec{s}|\vec{\theta}_{\text{MAP},\beta})}.$
(18)

Equation 17 suggests that the Bayes factor is dependent on the likelihood
ratio and the ratio of the Occam penalty factors. The Occam factor penalises
models that require an unnecessarily large parameter space volume to fit the
data by suppressing the model evidence. Note that Occam penalty is not an
intentionally added component to the Bayes factor, rather it is inherently
imposed as a result of using the Bayes Theorem.

As a heuristic explanation as to how the Occam penalty aids in BayesWave’s
ability to distinguish between signals and glitches, recall that signal models
($\mathcal{S}$) for each detector share the same intrinsic parameters and four
extrinsic parameters. Since there are five intrinsic parameters
($t_{0},f_{0},Q,A,\phi_{0}$) per wavelet, the dimension of signal models
scales as

$D_{\mathcal{S}}\sim 5N+4$ (19)

where $N$ is the number of wavelets. Glitch models ($\mathcal{G}$), on the
other hand, have no extrinsic parameters but the glitch model of each detector
is described by a unique set of intrinsic parameters. Assuming that signal and
glitch models use the same number of wavelets such that
$N^{\mathcal{G}}=N^{\mathcal{S}}=N$ (see Appendix B), the dimension of glitch
models scales as 2015CQGra..32m5012C

$D_{\mathcal{G}}\sim 5N\mathcal{I}.$ (20)

One therefore has $D_{\mathcal{G}}>D_{\mathcal{S}}$ for $\mathcal{I}\geq 2$.
This implies that the total parameter space volume for the glitch model is
larger than that of the signal model (i.e. $V_{\mathcal{G}}>V_{\mathcal{S}}$).
If both models fit the data equally well (i.e.
$\Lambda_{\mathcal{S},\mathcal{G}}\approx 1$ and $\Delta
V_{\mathcal{S}}\approx V_{\mathcal{G}}$), then by Occam’s razor we should
expect to see a selection bias towards the signal model as $\mathcal{I}$
increases. In other words, Equation 17 gives

$\mathcal{B}_{\mathcal{S},\mathcal{G}}(\vec{s})=\Lambda_{B_{\mathcal{S},\mathcal{G}}}(\vec{s})\frac{\Delta
V_{\mathcal{S}}}{\Delta
V_{\mathcal{G}}}\frac{V_{\mathcal{G}}}{V_{\mathcal{S}}}>1$ (21)

with increasing $\mathcal{I}$.

In Section B, we use the Laplace approximation to the Bayesian evidence to
derive an analytic scaling of the Bayes factor.

###  B Dependence of Bayes factor on number of detectors

In Ref. 2016PhRvD..94d4050L , Littenberg et al. put forth an analytic scaling
of the log signal-to-glitch Bayes factor,
$\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}$, in an effort to fully understand
BayesWave’s ability to robustly distinguish astrophysical signals from
instrumental glitches. They showed that the primary scaling of the Bayes
factor goes as

$\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}\propto
N\ln(\text{SNR}_{\text{net}})$ (22)

where $N$ is the number of wavelets used in the reconstruction, which is
related to the signal morphology and SNR as described in Equation 6. The
dependence of Bayes factor on $N$ (and therefore the complexity of the signal
in time frequency plane) differentiates BayesWave from other unmodelled
searches whose detection statistics scale primarily with SNR. The scaling
found in Ref. 2016PhRvD..94d4050L  assumes a network comprising two GW
detectors. Here we extend this work to an arbitrary number of detectors
$\mathcal{I}$.

We begin with the Laplace approximation of model evidences for the signal and
glitch models. From equation 14, we find

$\ln
p\left(d\,\middle|\,\mathcal{S}\right)\simeq\frac{\text{SNR}_{\text{net}}^{2}}{2}-\frac{5N^{\mathcal{S}}}{2}-N^{\mathcal{S}}\ln(V_{\lambda})+\sum_{n=1}^{N^{\mathcal{S}}}\ln\left(\frac{\bar{Q}_{n}}{\text{SNR}_{\text{net},n}^{5}}\right)+\frac{D_{\Omega}}{2}+\ln\frac{\sqrt{\det
C_{\Omega}}}{V_{\Omega}}$ (23)

$\ln
p\left(d\,\middle|\,\mathcal{G}\right)\simeq\frac{\text{SNR}_{\text{net}}^{2}}{2}-\sum_{i=1}^{\mathcal{I}}\left[\frac{5N_{i}^{\mathcal{G}}}{2}+N_{i}^{\mathcal{G}}\ln(V_{\lambda})-\sum_{n=1}^{N_{i}^{\mathcal{G}}}\ln\left(\frac{\bar{Q}_{n}}{\text{SNR}_{i,n}^{5}}\right)\right]$
(24)

with $\bar{Q}_{n}\equiv(2\pi)^{\nicefrac{{5}}{{2}}}\frac{\sqrt{2}Q_{n}}{\pi}$.
$V_{\lambda}$ is the prior volume of intrinsic parameters and $N^{x}_{i}$ is
the total number of wavelets for model $x$. The subscript $i$ refers to
detector $i$ in the network and $n$ labels an individual wavelet from the set
of wavelets for a given model. For instance: $\text{SNR}_{i,n}$ is the SNR of
wavelet $n$ in the $i$-th detector222Each individual wavelet used in signal or
glitch model reconstruction has an amplitude which can be converted into to
SNR. For details, see 2015CQGra..32m5012C .. In the last two terms of Equation
23, $D_{\Omega}$=4, $C_{\Omega}$ and $V_{\Omega}$ denote the dimension,
covariance matrix and the prior volume of extrinsic parameters respectively.
The full derivation from Equation 14 to Equations 23 and 24 can be found in
Section III(A) of Ref. 2016PhRvD..94d4050L .

To simplify the expressions for these evidences, we follow the same
assumptions used in Ref 2016PhRvD..94d4050L , and which are detailed further
in Appendix B. One simplifying assumption we highlight here again is that the
number of wavelets used in the signal model will be approximately the same as
the glitch model, and so we set $N^{\mathcal{S}}=N^{\mathcal{G}}\equiv N$
(i.e. the $N$ in Equation 22). Upon implementing the assumptions in Appendix
B, the theoretical log Bayes factor between the signal and glitch model for a
network of $\mathcal{I}$ detector(s) is given by
$\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}\simeq\ln
p\left(d\,\middle|\,\mathcal{S}\right)-\ln
p\left(d\,\middle|\,\mathcal{G}\right)$:

$\displaystyle\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}\simeq(\mathcal{I}-1)\left[\frac{5N}{2}+N\ln(V_{\lambda})-\sum_{n=1}^{N}\ln\left(\bar{Q}_{n}\right)+5N\ln\left(\frac{\text{SNR}_{\text{net}}}{\sqrt{N}}\right)\right]-\frac{5}{2}\mathcal{I}N\ln(\mathcal{I})+\left(2+\ln\frac{\sqrt{\det
C_{\Omega}}}{V_{\Omega}}\right).$ (25)

The equation shows explicit dependence of the Bayes factor on network SNR,
number of wavelets and number of detectors. We pay close attention to the
scaling

$\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}\propto\mathcal{I}N\ln\text{SNR}_{\text{net}}$
(26)

which now has an extra scaling factor of $\mathcal{I}$ compared to Equation
22.

The dependence on the number of wavelets used implies that the signal model is
favoured over the glitch model with increasing waveform complexity (higher
$N$). In other words, a more complex waveform is more likely to be classified
as a signal 2016PhRvD..94d4050L . This analytic result agrees with the
discussion in Section A where if two models fit the data equally well, the
less complex model will be selected to represent the waveform. The
proportionality $\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}\propto\mathcal{I}$
suggests that for signals with equal SNR and $N$, the Bayes factor should
increase if we increase the number of detectors in the network. Again, this
result agrees with the discussion in Section A; including more detectors in
the network increases the dimensionality of the glitch model and thus the
signal model will be even more strongly preferred.

##  V Injection data set

Figure 1: Projected LIGO, Virgo and KAGRA strain noise (i.e. amplitude
spectral density), $\sqrt{S_{n}}$ as a function of frequency for the fourth
observing run, O4. The data used to generate the noise curves above are
retrieved from 2018LRR….21….3A .

To empirically test the Bayes factor scaling given by Equation 25, as well as
investigate the effect on waveform reconstructions with detector networks of
different sizes, we inject a set of simulated BBH signals into simulated
detector noise and recover them using BayesWave. While BayesWave is a flexible
algorithm that can detect a variety of signals from different sources, we use
BBH waveforms as our test bed because they are well-understood sources, and
have previously been used to study the performance of BayesWave
2016PhRvD..94d4050L ; 2017ApJ…839…15B ; 2020arXiv200309456G .

In this work, we use tools from the LIGO Analysis Library (lalsuite, ) to
inject a set of non-spinning binary black holes (BBH) with equal component
masses of $30M_{\odot}$. We use the phenomenological waveform IMRPhenomD to
model spinning but non-precessing binaries using a combination of analytic
post-Newtonian (PN), effective-one-body (EOB) and numerical relativity (NR)
methods 2016PhRvD..93d4006H ; 2016PhRvD..93d4007K . The GW sources are
distributed isotropically across the sky, and the inclinations $\iota$ are
distributed uniformly in $\arccos{\iota}$. ${\rm SNR}_{\rm net}$ is
distributed uniformly in SNR${}_{\text{net}}\in\\{10,50\\}$ where this SNR is
calculated from a network comprising the HL detectors.

We inject 150 BBH signals into Gaussian noise coloured by the projected PSD of
LIGO, Virgo and KAGRA for the fourth observing run, O4, as given in the LIGO,
Virgo and KAGRA Observing Scenario 2018LRR….21….3A . The noise curves are
shown in Figure 1.

We then recover the injected signals with BayesWave in three different
scenarios: (i) Running only on Hanford and 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
configurations use the exact same injection data set.

In the two following sections, Sections A and B, we analyse two figures of
merit: (i) Bayes factor and (ii) the overlap. By comparing these quantities
between the HL and HLV networks, we can evaluate the performance of BayesWave
in recovering the injected waveforms from detector networks of different
sizes. As an extension to previous studies on sky localisation with expanded
detector networks, we also compare the accuracy of BayesWave in recovering the
sky location from detector networks of different sizes in Section C.

##  VI Results

###  A Recovered Bayes factors

Figure 2: Top left panel shows the log signal-to-glitch Bayes Factor
$\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}$ of BBH injection recoveries versus
network signal-to-noise ratio, SNR${}_{\text{net}}$. Each data point
represents one BBH injection. Top right panel shows the median number of
wavelets used in signal model reconstruction for each injection, $N$ versus
SNR${}_{\text{net}}$. Bottom panel shows
$\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}$ versus $N$, and the three colour
bars indicates the network SNR of each data point in the corresponding
detector network. In the top panels, the horizontal axis corresponds three
different network SNRs: (i) for the blue dot data points it corresponds to
SNR${}_{\text{net}}$ of the HL network, (ii) for the orange star data points
it corresponds to SNR${}_{\text{net}}$ of the HLV network, (iii) for the green
cross data points it corresponds to SNR${}_{\text{net}}$ of the HLKV network.

After analysing the injections described in Section V, we use the model
evidences calculated by BayesWave to understand the impact of GW detector
network size on the log signal-to-glitch Bayes factor,
$\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}$. For all the analyses in this
paper, we only injections that have been identified as inconsistent with
Gaussian noise (this can be either a signal or glitch) by BayesWave.
Injections indicated to be consistent with the Gaussian noise model
($\mathcal{N}$) by BayesWave are removed from the data set, since it would be
meaningless to evaluate their respective signal and glitch model evidences. In
other words, injections with $\ln\mathcal{B}_{\mathcal{S},\mathcal{N}}$ error
bars encompassing values below zero are removed from the data set. The widths
of $\ln\mathcal{B}_{\mathcal{S},\mathcal{N}}$ error bars are given by
2015CQGra..32m5012C

$\Delta[\ln\mathcal{B}_{\mathcal{S},\mathcal{N}}]=\sqrt{\\{\Delta[\ln
p\left(d\,\middle|\,\mathcal{S}\right)]\\}^{2}+\\{\Delta[\ln
p\left(d\,\middle|\,\mathcal{N}\right)]\\}^{2}}$ (27)

where $\Delta[\ln p\left(d\,\middle|\,\mathcal{M}\right)]$ is the uncertainty
for the logarithmic evidence of model $\mathcal{M}$. A total of 14 data points
are removed under this constraint. These events are all low
SNR${}_{\text{net}}$ injections.

The top left panel of Figure 2 shows
$\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}$ as a function of
SNR${}_{\text{net}}$ for the HL, HLV and HLKV networks. All three networks
show a clear trend of increasing Bayes Factor with increasing network SNR as
expected. Our results also show that the HLKV injections have the highest SNR
overall, agreeing with Equation 4 which indicates that increasing
$\mathcal{I}$ increases $\mathrm{SNR}_{\mathrm{net}}$. Furthermore, we can see
that injections at comparable SNRs are recovered with higher
$\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}$ in the HLV network than the HL
network. In other words, even after accounting for the increased SNR, we
observe further enhancement in detection confidence for an expanded detector
network, suggesting that $\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}$ is related
to $\mathcal{I}$, and not just the SNR of the signal as predicted by Equation
25.

The top right panel of Figure 2 shows the median number of wavelets used in
the BayesWave reconstruction, $N$ versus the injected SNR in the respective
detector networks, SNR${}_{\text{net}}$. The median here refers to the median
of posterior distribution for $N$. We see that $N$ increases systematically
with SNR${}_{\text{net}}$ in both the HL and HLV networks. This is expected
since the detectors are able to pick up more complex features of the waveform
at high SNR. At low SNR (SNR $\lesssim 15$) there is a slight deviation from
the linear trend described by Equation 6 between $N$ and SNR in both detector
networks. This is primarily due to the prior on the number of wavelets. This
prior is determined empirically from runs in LIGO data after O1, and peaks
around $N=3$ BWIII . $N$ also depends on waveform morphology and complexity
2016PhRvD..94d4050L ; 2018PhRvD..97j4057M . Injecting the same set of BBH
waveforms into all three detector configurations result in similar trends
between $N$ and SNR${}_{\text{net}}$.

Equation 25 shows that $\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}$ also scales
with the number of wavelets used in the reconstruction. Hence we also show
empirically how the dimensionality of signal model (i.e. the number of
wavelets) also contributes to the increase in
$\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}$ for different $\mathcal{I}$. We
show this in the bottom panel of Figure 2 by plotting
$\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}$ versus $N$. Colour bars indicate
the SNR${}_{\text{net}}$ of each data point. For all three detector
configurations, $\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}$ generally increases
with $N$, as predicted by Equation 25. At low SNRs (i.e. SNR$<15$), detector
networks recover the waveform with $N\leq 3$ and
$\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}\leq 50$ because low SNR injections
have low amplitude features which are harder to reconstruct resulting in lower
detection confidence. It is clear for injections recovered with $N>3$ that
$\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}$ in the HLKV network are generally
higher than that of the HL and HLV networks at comparable $N$ and
SNR${}_{\text{net}}$. This again emphasizes the point that the Bayes factor
scales with $\mathcal{I}$.

A more thorough investigation of the relation between the empirical and
analytic Bayes factor can be found in Appendix C, where we use a simplified
injection set of single sine-Gaussian wavelets. By recovering sine-Gaussian
wavelets with sine-Gaussian wavelets, Equation 6 reduces to $N=1$. The results
show that the empirical scaling of the Bayes factor with $\mathcal{I}$ agrees
with the analytical scaling in Equation 25 to a good approximation.

In summary, we show by comparing Bayes Factors between the HL, HLV and HLKV
networks that expanding detector networks increases detection confidence. Our
empirical results are consistent with the analytic results discussed Section
IV, viz.
$\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}\propto\mathcal{I}N\ln\mathrm{SNR}_{\mathrm{net}}$.
Heuristically, this can be understood via Occam’s razor: if coincident
identical glitches are unlikely in two detectors, they are even more unlikely
in three or more detectors. Therefore when identical waveforms are detected
simultaneously across larger networks, they have a higher likelihood of being
a signal.

Figure 3: Median overlap between the injected and recovered waveform,
$\mathcal{O}_{\text{net}}$ of the HL (blue dot) and HLV (orange star) network,
as a function of SNR${}_{\text{net}}$. The horizontal blue line indicates
$\mathcal{O}_{\text{net}}=0.8$ and the vertical blue line indicates
SNR${}_{\text{net}}>15$.

Figure 4: The top panel shows, for an injection with
SNR${}_{\text{net}}=11.61$ and $\mathcal{O}=0.52$, the injected waveform
(black), the detector data (blue) and the $90\%$ 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${}_{\text{net}}=52.72$ and
$\mathcal{O}=0.98$.

###  B Recovered Waveform Overlap

In the previous section, we showed that for a set of BBH waveforms,
$\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}$ increases with a larger number of
detectors in the network, meaning with more detectors our confidence in
detection is strengthened. In this section, we quantify the accuracy of
BayesWave in waveform recovery by comparing the overlap (also sometimes called
the match) between the injected and recovered waveforms for the HL, HLV and
HLKV detector networks. The network overlap, $\mathbb{O}_{net}$ is given by
Equation 13. For the rest of this paper, any mention of overlap refers to the
network overlap.

Figure 3 shows the median overlap, $\mathbb{O}_{net}$ as a function of network
SNR, where $\mathbb{O}_{net}$ of all three detector networks show positive
correlation with their respective network SNR. This observation is consistent
with previous results, which show that network overlap scales with SNR
2018PhRvD..97j4057M ; 2020arXiv200309456G . To illustrate how waveform
reconstruction improves with SNR, Figure 4 shows the injected waveform
(black), the detector data (blue) and the $90\%$ credible interval of the
recovered waveform (red) for two events in the HLKV network. The top and
bottom panels show the waveforms for the injection recovered with the smallest
overlap ($\mathbb{O}_{\text{min}}=0.52$) and largest overlap
($\mathbb{O}_{\text{max}}=0.98$) of the whole injection data set respectively.
The event with the smallest overlap has SNR${}_{\text{net}}=11.6$ and was
recovered with $\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}=9.66$, while the
event with the largest overlap has SNR${}_{\text{net}}=52.72$ and was
recovered with $\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}=218.0$ This is
consistent with the observed trend between overlap and network SNR in Figure
3. The similar trend between overlap and network SNR between all three
detector configurations indicates that waveform reconstruction fidelity is not
directly related to the number of detectors in the network.

However as noted earlier, increasing the number of detectors does increase the
network SNR. By comparing the percentage of waveforms recovered with overlap
above a given threshold for all three detector configurations, we show that
having an additional detector allows us to better reconstruct the signal
waveform. The threshold is arbitrarily defined here to be
$\mathbb{O}_{\text{net}}>0.8$ and is indicated by the horizontal blue line in
Figure 3. We found that $81\%$ of the injections were recovered with
$\mathbb{O}_{\text{net}}>0.8$ for the HL network, $86\%$ for the HLV network
and $87\%$ for the HLKV network.

While the inclusion of additional detector(s) does not have an extra benefit
in the same way it does for the Bayes factor as shown in the previous section,
it nonetheless allows us to better reconstruct the signal waveform due to
increased SNR. However, the improvement is less significant upon the addition
of KAGRA, since it is less sensitive compared to Virgo as shown in Figure 1
and therefore the increase in SNR is less compared to when Virgo is added to
the network. The overall results also show that BayesWave is able to
reconstruct waveforms reasonably well with all three detector configurations
for injections with SNR${}_{\text{net}}>18$ as indicated by the vertical blue
line in Figure 3.

Figure 5: The left panel shows the sky area enclosed within the $50\%$
credible interval (CI) in square degrees versus the network SNR of the
corresponding detector network. Similarly on the right panel, except for the
$90\%$ CI. Figure 6: Search area, $\mathcal{A}$ (Equation 28) versus network
SNR for the HL (blue dots) and HLV (orange stars) networks.

###  C Sky localisation

Expanding detector networks improves sky localisation of GW events, as has
been shown by various studies on coherent network detections e.g
2011CQGra..28l5023S  2018ApJ…854L..25P  and 2019PhRvD.100l4003P ; see Section
II. In this section, we compare the accuracy of BayesWave in locating the
source with the HL and HLV networks. We use two separate measures: (i) sky
area enclosed within the $50\%$ and $90\%$ credible intervals (CI) and (ii)
search area, $\mathcal{A}$.

For every injection, BayesWave produces posterior distributions for the sky
location (in the form of right ascension and declination) of the GW signal. We
first look at the sky area enclosed within $50\%$ and $90\%$ credible
intervals (CIs) of the posterior distribution of source location. In the left
panel of Figure 5, we show the plot for sky area enclosed within the $50\%$ CI
versus network SNR for each injection, and similarly for the the $90\%$ CI on
the right panel. For all three detector configurations, we note that the area
within the $50\%$ and $90\%$ CIs measured in square degrees (deg2)
fundamentally reduces with increasing network SNR due to improved accuracy in
arrival time differences 2018ApJ…854L..25P . However, both sky areas are
generally an order of magnitude smaller for the HLV network compared to the HL
network. Upon addition of the KAGRA detector, we observe further reduction in
the sky area, but not as drastic as that between the HL and HLV networks since
KAGRA is less sensitive than Virgo. The areas enclosed within both $50\%$ and
$90\%$ CIs reduces with increasing $\mathcal{I}$ due to the additional arrival
time differences which further constrain the location of each source. These
results reiterate that accuracy of sky localisation improves at fixed CI as
$\mathcal{I}$ increases.

We also compare the inferred sky location with the true injected location of
the source. We introduce another metric - the search area, $\mathcal{A}$, the
hypothetical sky area observed by a detector before it correctly points
towards the true location. To define this quantity mathematically, we first
denote the posterior probability density function (PDF) of sky location as
$p_{\text{sky}}(\phi,\theta)$. If the true location of the source is
$(\phi_{t},\theta_{t})$ and $p_{0}=p_{\text{sky}}(\phi_{t},\theta_{t})$, then
all points within $\mathcal{A}$ should have $p_{\text{sky}}\geq p_{0}$.
Mathematically, we write 2015ApJ…800…81E ; 2017ApJ…839…15B

$\mathcal{A}=\int H[p_{\text{sky}}(\phi,\theta)-p_{0}]\,d\Omega$ (28)

where $H$ is the Heaviside step function and $d\Omega$ is the surface area
element on the celestial sphere i.e. $d\Omega=\cos\delta d\theta d\phi$. In
Figure 6 we plot the search area, $\mathcal{A}$ against network SNR for both
the HL and HLV networks. The HLVK search area is slightly smaller than the HLV
search area, which in turn is significantly smaller than the HL search area,
consistent with Figure 5.

Overall, we see that sky localisation improves remarkably when a detector of
high-sensitivity is added to the network. If a less sensitive detector is
added, the improvements are small but not negligible.

##  VII Conclusion

The aim of this study is to compare the performance of BayesWave in recovering
GW waveforms from detector networks of different sizes. We derive an analytic
scaling for the Bayes factor between the signal and glitch models,
$\mathcal{B}_{\mathcal{S},\mathcal{G}}$. We then inject a set of simulated BBH
signals of fixed masses at different SNRs into simulated O4 detector data of
the HL, HLV and HLKV network. We quantify BayesWave’s performance in signal
identification with $\mathcal{B}_{\mathcal{S},\mathcal{G}}$ and the
performance in waveform reconstruction with overlap,
$\mathcal{O}_{\text{net}}$. We also compare the accuracy of sky localisation
between the two networks.

We find that events of similar injected SNR analysed using the HLV and HLKV
network have higher $\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}$ than those
using the HL network. This agrees with theoretical prediction of the Bayes
factor scaling:

$\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}\propto\mathcal{I}N\ln\text{SNR}_{\text{net}}.$
(29)

Previous work 2016PhRvD..94d4050L  demonstrated that BayesWave is unique
amongst GW umodelled burst searches in that the so-called “complexity” of the
signal in time-frequency plane plays a crucial role in the detection
statistic, rather than just the signal’s strength. This is understood through
the factor of $N$ in Equation 29: a signal with more complex structure needs
more wavelets to accurately reconstruct the waveform. In this work, we expose
another novel feature of the BayesWave algorithm: the detection statistic is
also influenced by the number of detectors i.e. the factor of $\mathcal{I}$ in
Equation 29. Events of similar injected SNR (SNR${}_{\text{net}}$) analysed
using larger detector networks have higher
$\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}$, indicating detection confidence
increases more than we would expect purely from the increase in
SNR${}_{\text{net}}$.

The network overlap, $\mathcal{O}_{\text{net}}$, between the injected and
recovered waveforms increases with SNR${}_{\text{net}}$. We also show that
$87\%$ of the HLKV network, $86\%$ of the HLV network and $81\%$ of the HL
network injections have $\mathcal{O}>0.8$. Since larger detector networks can
detect signals at higher SNR, they pick up more details of the true waveform.
Thus, BayesWave can reconstruct the waveforms more accurately.

Finally in Section C, we quantify accuracy of sky localisation with the sky
area enclosed within the $50\%$ and $90\%$ credible intervals (CI). We find
that both areas decrease with increasing SNR${}_{\text{net}}$ and are
generally an order of magnitude smaller for the HLV networks than the HL
network. The reduction of sky area is less significant upon the addition of
the KAGRA detector due to its low sensitivity compared to Virgo. The search
area, $\mathcal{A}$ also decreases with increasing SNR${}_{\text{net}}$ and
increasing number of detectors. The overall results suggest that increasing
the number of detectors at different geographical locations improves sky
localisation, consistent with previous analyses 2011CQGra..28l5023S ;
2018ApJ…854L..25P ; 2019PhRvD.100l4003P .

With the global detector network growing in size, the outlook for improving
detection confidence with unmodelled burst searches is promising. Prospective
work along the lines of the research presented in this paper may include
injecting different waveform morphologies to compare detection confidence
between detector networks of different sizes. We also recommend looking into
quantifying and comparing the outcomes of BayesWave in recovering simulated
signals from more realistic detector noise (i.e. in the presence of glitches)
between different detector configurations.

In summary, BayesWave shows significant improvements in terms of waveform
recovery and parameter estimation when working with a larger detector network.
This promising result suggests that with more detectors joining the global
network in the future, we will be able to reconstruct generic GW burst signals
more accurately using BayesWave making detections with higher Bayes factor and
hence with higher confidence.

## Acknowledgements

Parts of this research were conducted by the Australian Research Council
Centre of Excellence for Gravitational Wave Discovery (OzGrav), through
project number CE170100004. The authors are grateful for computational
resources provided by the LIGO Laboratory and supported by National Science
Foundation Grants PHY-0757058 and PHY-0823459. We thank Bence Bécsy for his
helpful comments.

##  Appendix A Fisher Information Matrix

Each wavelet has its Fisher Information Matrices (FIMs), $\Gamma$ written in
terms of its five intrinsic parameters $\\{t_{0},f_{0},Q,\ln A,\phi_{0}\\}$

$\Gamma=\text{SNR}^{2}\begin{pmatrix}\frac{4\pi^{2}f_{0}^{2}(1+Q^{2})}{Q^{2}}&0&0&0&-2\pi
f_{0}\\\
0&\frac{3+Q^{2}}{4f_{0}^{2}}&-\frac{3}{4Qf_{0}}&-\frac{1}{2f_{0}}&0\\\
0&-\frac{3}{4Qf_{0}}&\frac{3}{4Q^{2}}&\frac{1}{2Q}&0\\\
0&-\frac{1}{2f_{0}}&\frac{1}{2Q}&1&0\\\ -2\pi f_{0}&0&0&0&1\end{pmatrix}_{.}$
(30)

FIMs contain information on local curvature of the likelihood of wavelet
parameters which accelerates convergence by proposing jumps in the MCMC
algorithm towards regions of higher likelihood 2015CQGra..32m5012C . BayesWave
uses FIMs to update wavelet parameters by drawing proposals from a
multivariate Gaussian distribution

$q(\mathbf{y}|\mathbf{x})=\frac{\det\Gamma}{(2\pi)^{2}}\exp\left(-\frac{1}{2}\Gamma_{ij}\Delta
x^{i}\Delta x^{j}\right)$ (31)

where $\Delta x^{i}=x^{i}-y^{i}$ denotes the displacement in intrinsic
parameter $i$ before and after the update.

##  Appendix B Assumptions for Bayes Factor Scaling

Laplace approximations for the logarithmic signal ($\mathcal{S}$) and glitch
($\mathcal{G}$) model evidences are given by Equations 23 and 24 respectively.
In order to see how $\mathcal{B}_{\mathcal{S},\mathcal{G}}$ scales with the
waveform parameters, we make some assumptions to simplify the two logarithmic
evidences. In this work we use the same assumptions as in Ref.
2016PhRvD..94d4050L .

Loud signals typically have optimal extrinsic parameters across the detector
network, so the SNR in each detector will be approximately equal such that

$\text{SNR}_{i,n}\approx\frac{\text{SNR}_{\text{net},n}}{\sqrt{\mathcal{I}}}$
(32)

where $\text{SNR}_{i,n}$ is the SNR of the $n$-th wavelet in detector $i$. We
use a further simplifying assumption that the SNR of each wavelet is the same

$\text{SNR}_{\text{net},n}\approx\frac{\text{SNR}_{\text{net}}}{\sqrt{N}},$
(33)

which has been empirically validated. We assume that the glitch model in each
detector uses similar reconstruction parameters as the signal model, and as
such the quality factors of all wavelets are approximately equal:

$Q^{\mathcal{G}}_{i,n}\approx Q^{\mathcal{S}}_{n}\equiv Q$ (34)

and similarly,

$N^{\mathcal{G}}\approx N^{\mathcal{S}}\equiv N.$ (35)

Recall that $N^{\mathcal{G}}$ indicates the number of wavelets used in the
glitch model for a single detector, so for an $\mathcal{I}$-detector network,
the total number of wavelets used in glitch models across the entire network
is $\mathcal{I}N$.

Equations 23 and 24 can be simplified to

$\ln
p\left(d\,\middle|\,\mathcal{S}\right)\simeq\frac{\text{SNR}_{\text{net}}^{2}}{2}-\frac{5N}{2}-N\ln(V_{\lambda})+\sum_{n=1}^{N}\ln\left(\bar{Q}_{n}\right)-5N\ln(\frac{\text{SNR}_{\text{net}}}{\sqrt{N}})+\left(2+\ln\frac{\sqrt{\det
C_{\Omega}}}{V_{\Omega}}\right)$ (36)

$\ln
p\left(d\,\middle|\,\mathcal{G}\right)\simeq\frac{\text{SNR}_{\text{net}}^{2}}{2}-\mathcal{I}\left[\frac{5N}{2}+N\ln(V_{\lambda})-\sum_{n=1}^{N}\ln\left(\bar{Q}_{n}\right)+5N\ln(\frac{\text{SNR}_{\text{net}}}{\sqrt{N\mathcal{I}}})\right].$
(37)

##  Appendix C Scaling of Bayes factor with $\mathcal{I}$

Figure 7: Log signal-to-glitch Bayes factor,
$\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}$ of sine-Gaussian wavelet recoveries
versus network signal-to-noise ratio, SNR${}_{\text{net}}$. The solid lines
with colours corresponding to the data symbols are analytic predictions of
$\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}$ given by Equation 39.

Our results in Section A show that $\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}$
scales with SNR, $N$, and $\mathcal{I}$. As per Equation 6, $N$ itself depends
on both the SNR of the signal, and the waveform morphology. In order to
specifically test the scaling of $\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}$
with $\mathcal{I}$ alone, we inject a set of sine-Gaussian wavelets as
coherent signals into detector noise for the HL, HLV and HLKV network and then
recover them using BayesWave. Because sine-Gaussian wavelets are the basis of
reconstruction for BayesWave, the number of wavelets used is $N=1$, with no
dependence on SNR.

The dataset used this analysis is a set of 150 single sine-Gaussian wavelets.
The parameters of each wavelet are randomly drawn from the following
distributions: $t_{0}\in[1.5,2.5]$ s (where $t=1\ {\rm s}$ is the center of
the analysis window), $f_{0}\in[32,1000]$ Hz, $Q\in[0.1,40]$ and
$\phi_{0}\in[0,2\pi]$. The SNR of the signals are drawn randomly from a
uniform distribution and SNR $\in[10,50]$, and the amplitude is then found
viz.

$A=\text{SNR}\sqrt{\frac{2\sqrt{2\pi}f_{0}S_{n}(f_{0})}{Q}}$ (38)

(see 2018PhRvD..97j4057M  for details). As we are injecting a coherent signal,
we also require four extrinsic parameters as described in Section A. These
parameters are also drawn randomly from uniform distributions such that
$\alpha\in[0,2\pi]$,
$\delta\in[-\nicefrac{{\pi}}{{2}},\nicefrac{{\pi}}{{2}}]$, $\psi\in[0,2\pi]$
and $\epsilon\in[-0.99,0.99]$.

In Figure 7, we plot $\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}$ of each
injection against ${\rm SNR}_{\text{net}}$ for the HL, HLV and HLKV network
injections. We note that $\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}$ increases
with SNR${}_{\text{net}}$ and is generally higher for networks with greater
$\mathcal{I}$, as predicted from Equation 26. Since $N$ in this case does not
depend on SNR, we can be certain that the differences in
$\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}$ between the different detector
configurations at comparable SNR${}_{\text{net}}$ are entirely due to
$\mathcal{I}$.

In order to compare the analytic and empirical scaling of
$\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}$ with $\mathcal{I}$, we fit analytic
approximation of $\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}$ for each detector
network with a generalised expression

$\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}\approx(\mathcal{I}-1)[5\ln\text{SNR}_{\text{net}}+a]+\frac{5}{2}\mathcal{I}\ln\mathcal{I}+b.$
(39)

This expression is derived from Equation 25 with $N=1$. We define the
constants $a=\frac{5}{2}-\ln(V_{\lambda})+\ln(Q)$ and $b=2+\ln\frac{\sqrt{\det
C_{\Omega}}}{V_{\Omega}}$. The prior volumes, $V_{\lambda}$ and $V_{\Omega}$
are respectively the same for all detector configurations. We do not have an
analytic expression for $\det C_{\Omega}$ as the FIM approximations is
inadequate; the extrinsic parameter space contains degeneracy between
parameters, resulting in multimodal, non-Gaussian likelihood distribution
which spans the full extent of the prior range (see 2016PhRvD..94d4050L  for
details). We present the fits as three solid lines in Figure 7, where we have
determined by-eye $a=-10$ and $b=4$. The same values of $a$ and $b$ are used
for all three fits and they are broadly consistent with the empirical results.

We see general agreement between the empirical results and predicted scaling
for $\ln\mathcal{B}_{\mathcal{S},\mathcal{G}}$, which further confirms our
results in Section A that the Bayes factor does not only depend on
SNR${}_{\text{net}}$ but also on $\mathcal{I}$. We again note that these
scalings are estimations, and due to the different sensitivities of the
detectors we do not expect exact agreement between analytic prediction and
empirical results.

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