A Matched Filter Technique For Slow Radio Transient Detection And First Demonstration With The Murchison Widefield Array

2.7 MB·

by  L. Feng

Draftversion January16,2017
PreprinttypesetusingLATEXstyleemulateapjv.01/23/15
A MATCHED FILTER TECHNIQUE FOR SLOW RADIO TRANSIENT DETECTION AND FIRST
DEMONSTRATION WITH THE MURCHISON WIDEFIELD ARRAY
L. Feng1, R. Vaulin1,2, J. N. Hewitt1, R. Remillard1, D. L. Kaplan3, Tara Murphy4,5, N. Kudryavtseva6,
P. Hancock7,5, G. Bernardi8, J. D. Bowman9, F. Briggs10, R. J. Cappallo11, A. A. Deshpande12,
B. M. Gaensler4,5,13, L. J. Greenhill14, B. J. Hazelton15, M. Johnston-Hollitt16, C. J. Lonsdale11,
S. R. McWhirter11, D. A. Mitchell17,5, M. F. Morales15, E. Morgan1, D. Oberoi18, S. M. Ord17,5, T. Prabu12,
N. Udaya Shankar12, K. S. Srivani12, R. Subrahmanyan12,5, S. J. Tingay7,5,19, R. B. Wayth7,5, R. L. Webster20,5,
7 A. Williams7 and C. L. Williams1
1 1 MIT Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology,
0 Cambridge, MA 02139, USA
2 2 Sqrrl Data Inc., 125 CambridgePark Drive, Suite 401, Cambridge, MA 02140, USA
3 Department of Physics, University of Wisconsin–Milwaukee, Milwaukee, WI 53201, USA
n
4 Sydney Institute for Astronomy, School of Physics, The University of Sydney, NSW 2006, Australia
a
5 ARC Centre of Excellence for All-sky Astrophysics (CAASTRO)
J
6 Institute of Cybernetics at Tallinn University of Technology, Akadeemia tee 21, 12618 Tallinn, Estonia
3 7 International Centre for Radio Astronomy Research, Curtin University, Bentley, WA 6102, Australia
1 8 Department of Physics and Electronics, Rhodes University, PO Box 94, Grahamstown 6140, South Africa
9 School of Earth and Space Exploration, Arizona State University, Tempe, AZ 85287, USA
] 10 Research School of Astronomy and Astrophysics, Australian National University, Canberra, ACT 2611,
E
Australia
H 11 MIT Haystack Observatory, Westford, MA 01886, USA
. 12 Raman Research Institute, Bangalore 560080, India
h 13 Dunlap Institute for Astronomy and Astrophysics, University of Toronto, 50 St. George Street,
p
Toronto, ON M5S 3H4, Canada
- 14 Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA
o
15 Department of Physics, University of Washington, Seattle, WA 98195, USA
r
t 16 School of Chemical & Physical Sciences, Victoria University of Wellington, Wellington 6140, New
s Zealand
a 17 CSIRO Astronomy and Space Science (CASS), PO Box 76, Epping, NSW 1710, Australia
[
18 National Centre for Radio Astrophysics, Tata Institute for Fundamental Research, Pune 411007, India

1 19 Osservatorio di Radio Astronomia, Istituto Nazionale di Astrofisica, Bologna, Italy, 40123
v 20 School of Physics, The University of Melbourne, Parkville, VIC 3010, Australia
7 Draft version January 16, 2017
5
5 ABSTRACT
3
Many astronomical sources produce transient phenomena at radio frequencies, but the transient
0
sky at low frequencies (<300MHz) remains relatively unexplored. Blind surveys with new widefield
.
1 radio instruments are setting increasingly stringent limits on the transient surface density on various
0 timescales. Although many of these instruments are limited by classical confusion noise from an en-
7 sembleoffaint,unresolvedsources,onecaninprincipledetecttransientsbelowtheclassicalconfusion
1 limittotheextentthattheclassicalconfusionnoiseisindependentoftime. Wedevelopatechniquefor
: detecting radio transients that is based on temporal matched filters applied directly to time series of
v
images rather than relying on source-finding algorithms applied to individual images. This technique
i
X has well-defined statistical properties and is applicable to variable and transient searches for both
r confusion-limitedandnon-confusion-limitedinstruments. Usingthe MurchisonWidefield Arrayasan
a example, we demonstrate that the technique works well on real data despite the presence of classical
confusion noise, sidelobe confusion noise, and other systematic errors. We searched for transients
lasting between 2 minutes and 3 months. We found no transients and set improved upper limits on
the transient surface density at 182MHz for flux densities between 20–200mJy, providing the best
∼
limits to date for hour- and month-long transients.
Subject headings: methods: data analysis – techniques: interferometric – radiation mechanisms: non-
thermal – radio continuum: general – stars: variables: general.
1. INTRODUCTION (Gaensler et al. 2005), to explosive events such as su-
pernovae (Soderberg et al. 2010) and gamma-ray bursts
Many astrophysical systems and physical processes
(Frail et al. 1997). In addition, there are tran-
give rise to radio variability and transient phenom-
sient sources of the unknown nature, such as the
ena. These range from propagation effects, e.g. ex-
Galactic Center radio transients (Hyman et al. 2005),
treme scattering events seen in active galactic nu-
fast radio bursts (Thornton et al. 2013), and sources
clei (Fiedler et al. 1987), to magnetic activity in ul-
with no known counterparts at other wavelengths
tracool dwarfs (Hallinan et al. 2007) and magnetars
(Thyagarajanet al. 2011); as well as hypothesized
2
sourcesofradioemission,suchasexoplanets(Lazio et al. longer integration times. This, in turn, limits the abil-
2004), orphan afterglows (Levinson et al. 2002), and ity of source-finding algorithms to identify faint sources
gravitationalwavecounterpartsfromcompactbinaryco- in an image. The theoretical estimate of the confu-
alescence (Nakar & Piran 2011). sion limit for the MWA is 6mJy at 150MHz and
′ ∼
Despite the numerous observed and predicted ra- 5 angular resolution (Tingay et al. 2013), while P(D)
dio transient sources, few transients have been de- analysis, using different source count estimates, sug-
tected at radio frequencies < 300MHz. However, re- gests that the classical confusion noise for the MWA at
cent blind surveys with new widefield radio interferom- 154MHz is 1.7mJybeam−1 at 2.3′ angular resolution
eters, such as the Murchison Widefield Array1 (MWA, (Franzen et∼al. 2016).
Lonsdale et al. 2009, Tingay et al. 2013) and the Low- The classical confusion noise is largely independent of
Frequency Array2 (LOFAR, van Haarlem et al. 2013), time, however, so it is, in principle, not a limit for de-
are setting increasingly stringent limits on the sur- tecting fainter but varying brightness (Condon 1974). A
face density of radio transients at different sensitivi- simple method for detecting brightness variations below
ties and over a wide range of timescales. In partic- the classical confusion noise is image subtraction. For
ular, Bell et al. (2014) reported a limit of < 7.5 example, one can subtract images taken at the same lo-
10−5deg−2 above 5.5Jy at 154MHz for minute-to-hou×r calsiderealtimetoremovebothclassicalconfusionnoise
transients; Carbone et al. (2016) reported a limit of < andsidelobeconfusionnoise,thelatterofwhichisdueto
1.0 10−3deg−2 above 500mJy at 152MHz for minute- residual synthesized beam sidelobes. For many surveys,
× however, the images are taken at different local sidereal
to-hour transients; Cendes et al. (2014) reported a limit
of <2.2 10−2deg−2 above500mJy at149MHz for 11- times, and sidelobe confusion noise can dominate. In
× this case, image subtraction can be prone to artifacts.
min transients; Rowlinson et al. (2016) reported a limit
of < 6.4 10−7deg−2 above 210mJy at 182MHz for Even without image subtraction, CLEAN artifacts im-
30-sectran×sients up to <6.6 10−3deg−2 for 1-yr tran- pact radio transient searches. For instance, Frail et al.
× (2012) found that many transient candidates reported
sients;Polisensky et al.(2016)reportedarangeoflimits
byBower et al.(2007)wereinfactartifactsorthe result
( 10−4deg−2 above 100mJy) for 10-min and 6-hour
∼ ofcalibrationerrors,andthosethatwerenotidentifiedas
transients at 340MHz; and Murphy et al. (submitted)
such were detections at lower significance. A technique
reported a limit of < 1.8 10−4deg−2 above 100mJy that couldmeasure or accountfor the distribution of ar-
×
between 150–200MHz for transients that lasted 1–3yr. tifacts would be able to identify astrophysical transient
Jaeger et al. (2012) reported a transient detection that
candidates more reliably.
corresponded to a surface density of 0.12deg−2 above As many radio transients are expected to be faint
2.1mJy over 12hours at 325MHz, while Stewart et al. (.mJy; seeMetzger et al.2015), we seekamethod that
(2016) reporteda transientdetection thatimplied a sur- not only takes advantage of the time-independent na-
facedensityof1.5 10−5deg−2above7.9Jyfor 10-min ture of the classical confusion noise to detect transients
× ∼
transients at 60MHz. without relying on source-finding algorithms that could
These surveys, searching for slow transients that are otherwiselimit the sensitivity ofthe search,butalsohas
on the timescale of a snapshot or longer (generally well-defined statistical properties that take into account
& min), are usually limited by the image root-mean- the distribution of artifacts.
square (RMS) noise and the detection threshold of a We adapted the well-known matched filter technique
source-finding algorithm. One component of the RMS (Helmstrom 1968) to detect radio transients in the pres-
noiseistheclassicalconfusionnoiseσc,whicharisesfrom ence of classical confusion noise by drawing on the ex-
abackgroundoffaint,unresolvedsources(Condon1974). perience of the LIGO community (Laser Interferome-
This is a spatial noise in the image that depends on the ter Gravitational-Wave Observatory), which developed
sourcedistributioninthe skyandthe instrumentresolu- techniques to detect gravitational wave signals in the
tion: presence of non-Gaussian noise (e.g. Biswas et al. 2012,
Sc dn Allen et al. 2012). This technique operates on the im-
σ2 =Ω S2 dS, (1)
c b dS age pixel level, has well-defined statistical properties,
Z0 and is applicable to variable and transient searches for
where Ωb is the solid angle of the synthesized beam, S bothconfusion-limitedandnon-confusion-limitedinstru-
is the source flux density, and dn/dS is the differen- ments. Weappliedthistechniquetosearchforslowtran-
tialnumberdensityofsources(Thyagarajanet al.2013). sients in the MWA data. In Section 2, we describe the
The upper limit of integration Sc is the flux density of mathematical framework for our radio transient detec-
a source detected at a particular signal-to-noise ratio tion technique and derive its statistical properties. In
q =Sc/σc; usually Sc, referred to as the confusion limit, Section 3, we describe the MWA data reduction proce-
is determined iteratively until q =5. dure. In Section 4, we discuss the performance of this
As the instrument resolution improves and more techniqueonrealMWAdatatodemonstrateitspotential
sources are resolved, the classical confusion noise de- forsensitivetransientsearches. InSection5,wedescribe
creases. For a modest angular resolution like that of thetransientsearchanalysisusingthistechnique,discuss
′
the MWA at 2, however,the classicalconfusion noise theresultsofoursearch,andsetanimprovedupperlimit
∼
canbecomealimitingfactorintheimageRMSnoisebe- on the transient surface density at 182MHz, the best to
causetheclassicalconfusionnoisedoesnotdecreasewith date for hour- and month-long transients. In Section 6,
we conclude that our technique is capable of detecting
1 http://mwatelescope.org/ fainttransientsanddiscussareasofimprovementaswell
2 http://www.lofar.org/ as future work.
Matched Filter and Radio Transient Detection 3
2. THEORY rors. The likelihood functions are thus the following:
Thepresenceofclassicalconfusionnoiselimitstheabil- N b2(x c)2
ity of source-finding algorithms to identify sources with p(x H )= exp i i− p(c H )dc,
flux densities near or below the confusion limit, but the | 0 Z N0 −i=1 2σi2 ! | 0
sourcepopulationthatcontributestotheclassicalconfu- X
sionnoiseis independentoftime unless they aregenuine (5)
transient or variable events. Thus a transient detection
technique that searches for brightness variations on top p(x H )= exp N b2i(xi−c−Afi)2
oaflgaorcitohnmstaisntneseigdneadltwoidthetoeucttrterlayninsigenotnsiagnsaolusracpe-pfirnodaicnhg- | 1 Z N1 −Xi=1 2σi2 !× (6)
p(c H )p(A H )dcdA.
ing the confusion limit. In this section we describe a 1 1
| |
technique that identifies transients in individual image
and are the normalization factors for a multi-
0 1
pixels despite the classical confusion noise. N N
variate normal distribution. b is the value of the ith
i
Adapted from matched filter techniques, which have
primary beam for the given pixel. p(c H ), p(c H ),
0 1
been used in engineering applications (Helmstrom 1968) | |
and p(A H ) are the probability distributions of c and
1
and gravitational wave astronomy (Allen et al. 2012, |
A given the respective hypotheses. We assume flat pri-
Biswas et al. 2012), this technique searches for bright-
ors; in other words, we assume that c and A are in-
ness variations on top of a constant signal in individual
dependent and uniformly distributed, and we estimate
pixels without using source-finding algorithms. We de-
thembyusingaleast-squaresapproach(Sivia & Skilling
rive a new transient detection statistic from this tech-
2006). To do that, we solve Equations 5 and 6 by ap-
nique, discuss its statistical properties, and relate it to
proximatingthe integralwiththevalue atitsextremum,
the sensitivity of a radio transient search. Although our i.e. p(x H ) const exp( χ2 /2) where χ2 is the
formalismis derived for transient detection in the image | j ≈ × − min min
solution to χ2 =0. Specifically, for Equation 5, where
domain,itissimilartotheformalismforsourcedetection ∇
in the visibility domain (Trott et al. 2011). N b2(x c)2
Todeterminewhetherornotthereisatransientsignal χ2 i i− , (7)
in a particular image pixel, we compare two hypothe- 0 ≡ i=1 σi2
ses: thetransientisabsent(the nullhypothesisH ),and X
0
its extremum is computed by solving
the transient is present (the alternative hypothesis H ).
1
Usually,H onlyincludesrandomnoise,butinthiscase,
we add a c0onstant background3 to H0 to represent the dχ20 = N −2b2i(xi−c) =0; (8)
time-independentbrightnesscontributionfromconfusion dc σ2
(cid:12)c=c0 i=1 i (cid:12)c=c0
sources(orothersteadysources)becauseweareonlyin- (cid:12) X (cid:12)
terested in the change in brightness over time: and for Equat(cid:12)(cid:12)ion 6, where (cid:12)(cid:12)
H0 :xi =c+σi, (2) χ2 N b2i(xi−c−Afi)2, (9)
H1 :xi =c+Afi+σi. (3) 1 ≡ i=1 σi2
X
Forafixedpixel, xi isthemeasuredbrightnessintheith its extrema are computed by solving the two equations
snapshot (i= 1,2,...,N), c is the constant background
RthMatSfonloloiswes(tthheerdmisatrl,ibsuidtieolnobcehacroancftuesriiozne,dabnydσoc,thσeirisrathne- ∂χ21 = N −2b2i(xi−c−Afi) =0 and
∂c σ2
tdroamnsieernrotrssi)gnmaela,swuhreedrefoAr eisacthhesnoavpesrhalolt,amanpdlitAufdieisfotrhae (cid:12)(cid:12)(cid:12)c=c1 Xi=1 i (cid:12)(cid:12)(cid:12)c=c1 (10)
light curve template f f ,f ,...,f . Giventhe two (cid:12) (cid:12)
hypotheses and the da≡ta{x1= 2x1,x2,N..}.,xN , we com- ∂χ21 = N −2fib2i(xi−c−Afi) =0.
pute the ratio of the likelihoo{d functions kno}wn as the ∂A σ2
(cid:12)A=A1 i=1 i (cid:12)A=A1
Bayesfactoror the likelihoodratio as partof hypothesis (cid:12) X (cid:12) (11)
(cid:12) (cid:12)
testing (Neyman & Pearson1933,Kass & Raftery 1995): (cid:12) (cid:12)
Wenotethatthechoiceofusingflatpriorsinthisderiva-
Λ(x)= p(x|H1), (4) tionallowsustoobtainanalyticalsolutionsandsimplifies
p(x H ) the computation process to demonstrate the technique.
0
|
The choiceofpriorswillaffectthe outcome atlowsignal
where p(x Hi) is the probability of observing x given amplitudes, but accounting for this effect is beyond the
|
that Hi is true for i=(0,1). scope of this work.
To derive an analytical result, we assume that the im- For H , the solution c = c is an estimate of the con-
0 0
agenoisefollowsa Gaussiandistributionwithµ=0and stant background level, including the contribution from
σim = σ1,σ2,...,σN ,but we showlater that ourtran- confusionsources,anditisgivenbytheweightedaverage
{ }
sient analysis does not rely on this assumption. For the of the data:
MWA data,this noise is a combinationofthermalnoise, b2x /σ2
(residual)sidelobeconfusionnoise,andotherrandomer- c0 =hxi≡ ib2/iσ2i . (12)
P i i
3 Pure random noise isa special case where the constant back- ForH1,thesolutionc=c1 isPanestimateoftheconstant
groundiszero. backgroundlevelin the presence ofa transientsignal; c
1
4
becomes c in the absence of the transient signal. The
0
othersolutionA=A ,whichisunitless,istheamplitude
1
of the transient signal given the predefined template f,
which has brightness units.
c = x A f , (13)
1 1
h i− h i
(x,f f )
A = −h i . (14)
1 (f f ,f f )
−h i −h i
Thenotation x representsthe weightedaverageofxas
in Equation 1h2,iand (x,f) b2(x f )/σ2 denotes the
“weighted” inner product b≡etweein xi aindif. Note that
an equivalent way of writing P(x,f f ) is (x x ,f),
−h i −h i
which we interpret as how well the light curve template
matches the data after the constant component is sub-
tracted. However, (f f ) is simpler and computation-
ally less intensive to−cahlcuilate since f is predefined and
identical for all pixels, so we write Equation 14 in its
given form.
Fig.1.— Theoretical distributions of the transient detection
Having determined the best fit values for c and A, we statisticρ: background(gray-filled)andsignal(red-line)witharbi-
substitute Equations 12, 13, and 14 into the approxima- traryimagenoiseσim andsignalamplitudeA1. Asσim decreases,
tions of Equations 5 and 6, which then lets us solve for bthuetwthidetmhseaonf tohfethdeistsrigibnuatliodnisstrinibcuretaiosne iancccroeradsiensgfatostσerρa∝cc1o/rdσiinmg,
nEoqrumatailoizned4.byAsaΛco(xns)taisnte,ssweentriaewllyritaerEaqtiuoaotifonex4poinnetnhtes troatµio1n∝be1t/wσei2emn,bsoactkhgerosuennsditaivnidtysiigmnparlo.vSeismaislatrhlye,rethiesbbreitgthertesrepthae-
form Λ(x) const exp ρ2/2σ2 and define the new signal(largerA1),thebetter thesensitivity.
≈ × ρ
quantities ρ to be the “detection statistic” and σ to be
the standard deviation of t(cid:2)he ρ dis(cid:3)tribution: ρ template that provided this value. Using the ratio ρ/σρ
ensuresthatallthepixelsaredrawnfromthesamestan-
N b2 dard normal distribution. Note that ρ˜ is the maximum
ρ=(x,f f )= i x (f f ), (15) of a Gaussian random variable, so its distribution is no
−h i σ2 i i−h i longer Gaussian. The trials factor arising from the bank
i=0 i
X oftemplatesisincludedintheρ˜distributionthroughthe
N b2 maximization procedure, unlike the ρ distribution for a
σρ = (f −hfi,f −hfi)=v σi2(fi−hfi)2. (16) single template where there is no need to correctfor the
ui=0 i trials factor. We handle the non-Gaussianity of the ρ˜
p uX
t distribution in Section 4.
ρ is a modified versionof the matched filter (Helmstrom There are two more steps before transient identifica-
1968) and determines the likelihood that x contains the tion: characterize the detection significance (reliability)
signal A f. In other words, it determines which hy-
1 and measure the detection efficiency (completeness). As
pothesis is favored (transient absent or present) and by we describe these steps in detail in Section 4, we sum-
how much. As ρ is a linear superposition of Gaussian marize them here. To characterize the detection signif-
random variables, its distribution is also Gaussian with
icance, one designates a small part of the image as the
width σρ. When the transientsignalis absent, the mean playgroundregionwherethereareassumedtobenotran-
of ρ is µ0 = 0. When the transient signal is present, sients, corrects the cumulative distribution of ρ˜ in the
the mean of ρ is shifted by the signal and becomes
playgroundregionbythetrialsfactor,i.e. thenumberof
µ1 = A1(f − hfi,f − hfi) = A1σρ2. This is illustrated synthesized beams in the search region compared to the
in Figure 1. number in the playground region, and extrapolates the
ρandσρ arealsorelatedtothe morefamiliarquantity tailofthis correcteddistributionto atolerableprobabil-
σ through the weighted inner product: ρ A /σ2 ity of false alarm. The extrapolated distribution deter-
im ∝ 1 im
and σ 1/σ . As σ decreases, both ρ (or µ ) and mines the significance ofanydetectionduring the actual
ρ im im 1
∝
σ increase,butρ(orµ )increasesfasterthanσ ,which search. To measure the efficiency, one applies the detec-
ρ 1 ρ
leads to a better separation of background and signal tion threshold to the injected transients and computes
(see Figure 1) and hence an improved sensitivity. Sim- the fraction that is recovered. This procedure handles
ilarly, the brighter the transient (larger A ), the better non-Gaussianity in the data, such as artifacts or side-
1
the sensitivity. lobe confusion noise as well as the effects of maximiza-
So far we considered ρ for a single template. How- tion, thus making it a powerful technique. The pipeline
ever, in a real search, we use a bank of templates and implementation is described in Appendix A.
maximize ρ over the various parameters that character-
izethesetemplates,suchasstarttimesanddurations,so 3. DATA REDUCTION
the practical statistic is
TheMWAisalow-frequencyradiointerferometercon-
sisting of 128 aperture arrays known as “tiles” located
ρ˜=max[(ρ/σ );(t ,t )], (17)
ρ 0 dur
at the Murchison Radio-astronomy Observatory in the
wherewecalculateρ/σ foreverytemplateandallpixels, Murchison Shire of Western Australia; for a complete
ρ
andthen,foreachpixel,savethemaximumvalueandthe description of the instrument, see Tingay et al. (2013).
Matched Filter and Radio Transient Detection 5
TABLE 1
Summaryof observationsused inour analysis.
Date TimeRange(UT) NumberofSnapshots
2013-09-02 16:08:08–17:39:36 46
2013-09-04 16:00:16–18:32:48 74
2013-09-06 15:52:22–18:24:54 72
2013-09-09 15:40:39–18:13:03 75
2013-09-11 15:32:47–18:05:11 75
2013-09-13 15:24:55–17:57:19 72
2013-09-17 15:09:11–17:41:35 76
2013-09-19 15:01:19–17:31:43 71
2013-09-30 14:17:59–16:50:31 76
2013-10-02 15:07:03–16:42:39 47
2013-10-04 14:02:15–16:34:47 76
2013-10-08 13:46:31–16:19:03 76
2013-10-10 13:39:43–16:12:15 76
2013-10-23 12:48:39–15:21:03 76
2013-10-25 12:40:47–15:13:11 76
2013-10-29 12:25:03–14:57:27 75
2013-11-18 11:18:31–13:38:55 70
2013-11-29 11:30:41–12:56:09 42
Note. —EachsnapshotiscenteredontheEOR0field,
or (RA,Dec) = (0◦, 27◦), and integrated over 112s at Fig.2.— An example snapshot of the EOR0 field. It was inte-
182.4MHz. − gratedover112sandcleanedintheXXpolarization,plottedwith
the J2000 coordinate grid and a squared color scale. The dashed
circlehasaradiusof10◦,whichisapproximatelytheouterbound-
The data for this paper were taken according to the aryofthefieldincludedinouranalysis.
commensal MWA observing proposals4 G0009 (“Epoch
package5 (v4.1.0) (McMullin et al. 2007) and the wide-
of Reionisation,” EOR) and G0005 (“Search for Vari-
field imager wsclean (Offringa et al. 2014).
able and Transient Sources in the EOR Fields with the
We built a point-source sky model for each snap-
MWA”) for Semester 2013-B. These observations were
shot observation, using the 11 brightest point sources
doneusingthe “point-and-drift”strategy,wherethe pri-
in the field after primary beam attenuation accord-
marybeampointing(beamformersetting)changedevery
ing to the MWA Commissioning Survey Catalog
20–30minutestotrackthefieldafteritdriftedacrossthe
(Hurley-Walker et al. 2014). We generated the model
field of view of the instrument.
visibilities and the calibration solutions in CASA, using
Weused1251snapshotobservationsoftheEOR0field,
◦ ◦ in particular the tools componentlist, ft, bandpass,
which wascentered on(RA,Dec)=(0 , 27 ), takenon
− and gencal. Then we performed one iteration of phase
18 nights between 2013 September 2 and November 30.
and amplitude self-calibration with wsclean.
We included only snapshots taken when the field cen-
◦ Finally, we averaged over many observations on the
ter was<20 fromthe zenith, correspondingto 2hof
∼ samenightthecalibrationsolutionsgeneratedinthepre-
observationeachnight,andexcludeddatafrom2013Oc-
vioussteptoproduceasinglecalibrationsolutionforthis
tober15becauseofknownionosphericactivity(Loi et al.
night. Thiswasdoneintwosteps: (1)weselectedalistof
2015). Each snapshot is a multi-frequency synthesis im-
observations for which the aegean source finder (v951)
ageintegratedover112swithabandwidthof30.72MHz
(Hancock et al. 2012) detected at least 1500 sources in
centered on 182.40MHz. Table 1 lists a summary of the
each112-ssnapshotaswefoundthistobeapracticalin-
observations.
dicator of image quality; (2) we averagedthe calibration
amplitudeandphasesolutionsforeachtile,polarization,
3.1. Preprocessing
andfrequencychannelfortheselectedobservationswhile
Raw interferometric data were converted into the ignoring the highest and lowest 10% of the data.
uvfits format (Greisen 2012) by the Cotter MWA Thisprocedureprovidedstableandsmoothcalibration
preprocessing pipeline (Offringa et al. 2015). During solutions, which were not expected a priori to vary sig-
this process, Cotter used AOFlagger (Offringa et al. nificantly over time and frequency. It also gave more
2010,Offringa et al. 2012) to flag radio-frequency inter- reliable estimates of the source flux densities.
ference,frequencychannelsaffectedbybandpassaliasing
(240kHz oneachedgeof a 1.28-MHzcoarsechannel), as 3.3. Imaging
well as known bad tiles, which might vary from night to
After we applied the average calibration solutions to
night depending on the state of the instrument. To de-
each snapshot, we generated multi-frequency synthesis
crease the file size, Cotter also averaged the data to 1-s
images over 30.72MHz bandwidth in the instrumental
time resolution and 80-kHz frequency resolution.
XX and YY polarizations, setting wsclean to use uni-
′
form weighting that gave a synthesized beam of 2, a
3.2. Calibration ′ ∼
pixel size of 0.5, and an image size of 4096 4096 pix-
We developed a data reduction pipeline based on els, which corresponded to a field of view of×34◦ 34◦.
the Common Astronomy Software Applications (CASA) Figure 2 shows an example snapshot of the EOR0×field.
4 http://mwatelescope.org/astronomers/ 5 http://casa.nrao.edu/
6
3.4. Primary Beam Correction
We adopted an empirical approach for primary beam
correction. While there are theoretical primary beam
models for the MWA (Sutinjo et al. 2015), they are less
accurate at frequencies & 180MHz. We found that the
source light curves extracted from images corrected by
the theoretical primary beam model at 182MHz showed
systematic trends of 4% flux change per hour that
∼
dependedonthe rightascensionofthe source. Toallevi-
ate this effect, we measuredthe empiricalprimary beam
value at eachsourcelocationby comparingthe observed
flux of the source to its catalog flux, which we assumed
to be true and static, and fitted a smoothing spline to Fig.3.—(a)Distributionofρ˜forallthepixelsintheplayground
the measured beam values. This procedure reduced the region, with the x-axis limited to 0 ρ˜ 9. (b) The same dis-
tribution of ρ˜but without the “bad”≤pixe≤ls, i.e. the pixels in the
systematictrendsofthelightcurvesto.2%fluxchange
region where the primary beam is poorly modeled and the pixels
per hour and removed most of the systematic variations thatcontain brightsourcesaffected bytheprimarybeam system-
in the light curves. A similar approach to empirical pri- atics. The tail in the background distribution, which extends to
mary beam correction was done by Thyagarajanet al. ρ˜ 140intheleftpanel,isremovedbymaskingthosebadpixels.
∼
(2011) for the Very Large Array; for a more detailed
illustrated in Figure 3b. The distribution after masking
discussion on the empirical primary beam we used, see
is what we work with and refer to as the “background
Appendix B.
distribution.”
The background distribution of ρ˜ lets us derive the
4. DEMONSTRATION
probability of false alarm P (equivalent to reliability)
FA
In this section we demonstrate how a transient search fordifferentvaluesofρ˜. HereP istheprobabilitythat
FA
with our matched filter technique proceeds and that it our experiment contains a false positive. Every experi-
performs well on real data. Before we identify possible menthasitsownobservationtimescaleandskycoverage,
transient signals according to the matched filter detec- which need to be factored accordingly. We start with a
tion statistic, we need to characterize the background quantity closely related to P : the number of back-
FA
distribution of ρ˜ max(ρ/σ ). Using ρ˜ instead of ρ ground detections N( ρ˜) above a particular threshold.
ρ
≡ ≥
ensures that all pixels are drawn from the same distri- We scale N( ρ˜) as derived from the playground re-
≥
bution. Characterizing the background determines the gion to the search region according to the number of
∗
significanceofany detectionandprovidesa thresholdρ˜ synthesized beams for the different sky areas, thus ac-
for us to compare the performance of different searches counting for the trials factor correctly. The trials fac-
and to measure the search efficiency. tor from the number of templates is already accounted
We used a small area ( 10%) labeled as the “play- for in the background distribution when we maximized
∼
groundregion”inthe imagesforthe backgroundcharac- over different templates. Then we fit a smooth func-
terization. Inthisregion,weassumedthattherewereno tionto the tailofN( ρ˜)to extrapolatethe background
≥
transientevents;existinglimitsontherateofradiotran- ratefor largervalues ofρ˜wherewe mightnothave mea-
sients at low frequencies suggest that these events are surements as the playground region contains much less
relatively rare (e.g. Jaeger et al. 2012,Rowlinson et al. data than the search region. Non-Gaussianity and non-
2016), so our assumption should be valid. Significant thermal components of the image noise, such as side-
transients would appear as a tail in the ρ˜ distribution lobe confusion and image artifacts, are modeled in this
that we would examine further. empirical fit. We determine the fit parameters by min-
To demonstrate our transient detection technique, we imizing the negative log-likelihood for a Poisson distri-
present the results for one light curve template: the bution, where N( ρ˜) is treated as the Poisson mean.
≥
top-hat with a duration of 15days and a brightness of As illustrated in Figure 4, an exponential function of
1Jybeam−1, searched over different start times t0. In the form fN(ρ˜) = Nˆexp( ρ˜/ρˆ), where Nˆ and ρˆare the
principle, we could have treated the observation time of fit parameters, fits the ta−il of N( ρ˜) well within the
every 112-s snapshot as a unique t0, but since computa- errorsσ √N. Theprobabilityf≥orN( ρ˜)to benon-
tion time scaled with the number of search parameters, N ≡ ≥
zero, assuming a Poisson distribution, is P(N > 0) =
dwaetash,iwfthedicht0inbyth∼is1c0a%seocfotrhreesdpuornadteiodntowhthereestthaertreowf eevre- 1−P(N =0)=1−e−N. WetakePFA =P(N >0),and
since we require P 1, P =P(N >0) N( ρ˜).
ery night of the observation. The distribution of ρ˜ for FA ≪ FA ≈ ≥
Havingdeterminedf (ρ˜),we chooseatolerablevalue of
the entire playground region is presented in Figure 3. N
P (for example P = 10−3) and solve for ρ˜∗ such
As evident in Figure 3a, there is a significant tail in FA FA
∗ ∗
that P = f (ρ˜ ) = N( ρ˜ ). For our exponential
the ρ˜distribution. This is due to residualprimarybeam FA N
≥
function, the threshold is of the form
effects as the pixels in the tail are located in the regions
where the primary beam is poorly modeled or that con- ρ˜∗ =ρˆ(logNˆ logP ). (18)
tain bright sources (> 100mJy). We masked the pixels − FA
◦ ∗
beyond 10 from the phase center and excluded the Thethresholdρ˜ dependsonhowmuchofthedistribu-
∼
source pixels by creating a 20 20-pixel square mask tiontailisusedinthefit. ForFigure4,wefittedthe tail
centered on the (RA, Dec) of ea×ch source. After mask- 500 points, which gave ρ˜∗ =7.98 at P =10−3. When
FA
∗
ing,wemanagedtoremovethetailinthedistributionas we fitted 100 points, we obtained ρ˜ = 7.37, whereas
Matched Filter and Radio Transient Detection 7
Fig. 4.— Cumulative background distribution of ρ˜(black solid Fig.5.— Flux density sensitivity comparison between the
line). Theshaded area isthe errorregionas computed from√N. matched filter transient detection technique (using a 15-day top-
Only the tail of the distribution (500 data points to the right of hat template) and a source detection technique for the same set
thevertical dotted line)wasusedforfitting. Thefit(bluedashed of 2-minsnapshots. Each black dot is A∗ =(ρ˜∗/σρ)=(7.98/σρ)
1li0n7e)exips(anρ˜e/x0p.3o3n)e.nTtihalefruendcutcieodn:χf2Nfo(ρr˜)th=eNfiˆteixsp0(.−25ρ˜./ρˆT)h=e fi2t.3w8a×s otTifohnaetpdhiaxrseehsl-hdaoonltdd)lieanatechaisbptlahureeticmcureoladsrsiarinasdfl6iuσuxismfdre(oanmnsitteyhxeasmeEnOpsliReti0vsiotfiuyerlc2de5c.d0eenmtteeJcry-.
tfdoheretnetchtueiosedn−detthatoirlescshoomofltdphuietsecρ˜a∗tlhc=eul7faa.t9liso8en..alaArtmPpFrAob=ab1il0it−y3,PtFhAe;trsaeenstieexntt afiqlulgtoeotrreidttehcimnhntfihoqeruettehaxecthsfioaevrmetesheasembteaotttfcehrimesdaengfieslistteirwvititteyhchotunhtiaqnutheae.sTonueheredcem-tfiaontcdphirneodg-
duce a deeper integration image. The sensitivity decreases away
from the phase center because of increased noise, but the signifi-
whenwefitted50points,weobtainedρ˜∗ =7.26. Thisde- canceremainsthesame. Thegapbetween2◦ and4◦ isbecauseof
pendenceimpliesthatasinglethresholdvalueisnotvery thediscontinuousplaygroundregionsampling;seetheinsetwhere
black rectangles markthe playground regions, each of which con-
reliable for identifying transient events near the thresh-
old, so in the actual search, we opted for the “loudest isnisdtespoefndseenvtelryalfosrmeaallcehrp8a6tc×h,86so-ptixheelsptraitpcehsesi.n Aσi∗mcoisrrceasplcounladtetdo
event statistic” (Brady et al. 2004,Biswas et al. 2009), thepatches wherethenoisepropertiesaredifferent.
which we describe in more detail in Section 5. Never-
theless, the threshold is useful for comparing how well the data.
different searches perform, for providing an estimate of It is ρ˜that matters in this technique. Since our tech-
the brightness sensitivity, and for verifying the recovery nique uses ρ˜and not flux density as a metric to identify
of injected transients. transientsources,itiscapableofdetectingsourcesfainter
The flux density sensitivity of the search can be cal- thanthe medianflux density sensitivity atthe same sig-
∗ ∗
culated according to A = ρ˜ /σρ (see Section 2; note nificance (reliability) but a lower efficiency (complete-
∗
ρ˜ already contains a factor of σρ unlike ρ). Strictly ness),makingitpotentiallymorepowerfulthanthetech-
∗
speaking, A is unitless, so one needs to multiply by the niques that apply a more stringent flux density thresh-
template f to convert it into brightness units, but we old. Forourcleanimagesanda15-daytop-hattemplate,
drop f for a simpler notation as we choose f to have P =10−3 correspondsto ρ˜∗ =7.98 anda median flux
units of 1mJybeam−1. Because σρ depends on the pri- deFnAsitysensitivityof25.0mJy;notethatfitting50points
mary beam, as presented in Equations 12 and 16, pixels instead of 500 gives ρ˜∗ = 7.26, which corresponds to a
closer to the edge of the primary beam have different median flux density sensitivity of 22.7mJy, a value not
values of σρ compared to the pixels closer to the center very different from 25.0mJy.
of the primary beam. This implies that we have non-
uniform brightness sensitivity across the image, where 5. ANALYSIS
we are more sensitive to fainter transientsourcestoward
We ran three separate blind transient searches on the
thecenteroftheprimarybeam. ThisisillustratedinFig-
MWA data. Each search used a different set of light
ure5,wherewealsoshowanexampledetectionthreshold
curve templates and thus was sensitive to transients on
6σ for a source finder.
im a different timescale.
Toquantifythefluxdensitysensitivityinasinglenum-
∗
ber, we computed A separately for each pixel, then we
∗ 5.1. Defining the Searches
computedthemedianvalueofA integratedover1beam.
∗
Since the distribution of σ and hence A is skewed, the Thetimescalesspannedbythedatarangedfrom2min
ρ
median is a better estimator of the flux density sensitiv- to 3 months, but we did not have uniform sensitivity
ity than the mean. However, we stress that this value over all possible timescales. Some timescales could not
only provides an estimate of how well the search might be probed because of gaps in the observations, while
performandshouldnotbeinterpretedasastrictthresh- other particular timescales suffered from systematic ef-
old because a single value for the flux density sensitivity fects, such as those due to the periodic change in the
is a convenience and cannot capture the complexity of primary beam pointing.
8
TABLE 2
Summaryof our searches.
Search Template Duration ρ˜∗
1: minute top-hat 4m 7.6
2: hour top-hat 1.5h 7.9
3: day-to-month top-hat 2d,4d,7d,9d 8.0
11d,15d,17d,28d
30d,32d,36d,38d
51d,53d,57d,77d,88d
Note. —ρ˜∗ isthethresholdwithPFA=10−3.
We characterized the background distribution of ρ˜by
running the pipeline on the playground region. This re-
gion was 10% of the image, which we assumed con-
∼ ◦
tained no transients. First we divided the inner 13
∼
of the image, or 3096 3096pixels, into 86 86-pixel
Fig.6.— Expected thresholds ρ˜∗ for transients on different squares. This was beca×use the pipeline ran on×one CPU
timescales,characterizedbythematchedfilterstatistic,whichde- core allocated 3GB of RAM at any given time and
pends on the image noise, the light curve template, and how the
observationsweretaken. Smallervaluesofρ˜∗ correspondtobetter wouldencountermemoryissuesifitprocessedmorethan
sensitivities. Theverticalbluedotted linemarksthetimescalefor 100 100 pixels each with 1251 brightness measure-
∼ ×
consecutivechangesintheprimarybeampointing;aswehadpoor ments. Because of primary beam systematics, we only
sensitivityon this timescale, weexcluded itfromour search. The searchedfortransientsintheinner 10◦,or2096 2096
vertical grey region marks the approximate timescales that cor- ∼ ×
pixels, of the image. Then we chose the playground re-
respond to the gaps between observations where we had no data.
Basedonthisplot,wedividedourtransientsearchintothreeparts: giontobe nine172 172-pixelpatchesdividedintorows
×
min,hour,andday-to-months. ofthreeacrosstheimage. Thischoicesampleduniformly
across the image to capture any spatial noise variation.
As mentioned in Section 4, we masked certain pixels
Inordertodeterminewhichsearchestorunandwhich toremovethetailinthe backgroundρ˜distribution. One
templates to use, we did a test run on the playground can consider the source mask as an “auxillary channel”
region to compare the expected transient thresholds for for the search, where events that occur within a certain
various timescales. Figure 6 shows the expected thresh- area of a (bright) source are vetoed. Brighter sources
olds from the test run; we sampled roughly logarithmi- have larger sidelobes and hence a larger “veto” area. To
callybetween2minand3monthsbutalsosampledevery definethevetoareaandrefinethebackgroundcharacter-
5min between 15 and 50 min. There is a drastic in- ization, we performed an empirical fit to determine the
∼
crease in threshold around the 30-min timescale, which size of the source mask as a function of source flux den-
corresponds to the time between consecutive changes in sity. We picked 6 sources with flux densities above 5Jy,
the primary beam pointing. Hence we excluded that measured their sidelobe contamination areas in the un-
timescalefromoursearchanddefinedthefollowingthree maskedtransientskymapforthehoursearch,andfitted
searches: minute, hour, and day-to-months. Despite the a straight line through the two points with the steepest
∗
variation in ρ˜ values for the day-to-month templates, slope to obtain the most conservative relationship be-
themedianfluxdensitysensitivitieswereaboutthesame, tween the size of the source mask and the source flux
so we groupedthem together. The specific templates we density:
usedforthesearchesarelistedinTable2. Becauseofthe ∆n =6.6S+0.5, (19)
pix
primary beam systematics, we avoided templates with
where ∆n is the number of pixels to mask on each
durationsbetween15–60min. The 4-mintemplate is ca- pix
side of the source and S is the value of the source flux
pable of recovering transients with durations < 15min
density. If the fit returned ∆n < 10, however, we set
as the difference between ρ computed from the 4-min pix
∆n =10togivea20 20-pixelmaskregion. Thisisa
template and ρ computed from the perfectly matched pix
×
conservative value to account for the size of the synthe-
template is <10%. As each night consisted of 2hours
∼ sizedbeam. We also manually flagged31 double sources
of observation, we chose the 1.5-hour template that is
thatweremisidentifiedassinglesourcesbyaegean. We
halfway between 1 and 2hours for the second search.
note that by masking known, bright sources > 100mJy,
Finally, for the last search, we considered all possible
we were no longer able to search for light curve varia-
durations sampled by the data, excluding the gaps, and
tions in these sources although it would not affect our
separated by at least 1-day.
search for fainter transients. This was not ideal for a
Each search consisted of three parts: (1) characteriz-
transient search, but it was necessary given the primary
ing the background to determine the significance of any
beamsystematicsin ourdata; with improvementonpri-
detection,(2)measuringtheefficiencyatwhichthetran-
mary beam modeling and calibration techniques, it is
sient events were detected, and (3) identifying the tran-
conceivablethatone does notneedto maskthose pixels.
sient candidates if there were any.
We thendividedthe playgroundregioninto twoparts,
A and B, by choosing alternating 86 86-pixel squares.
5.2. Characterizing the Background ×
Playground A was used to characterize the background
Matched Filter and Radio Transient Detection 9
TABLE 3
Summaryof our injection runs.
Injection Template Duration PeakFlux
1: minute top-hat 2–12m <1300mJy
2: hour top-hat 1–2h <450mJy
3: day-to-month top-hat 1–90d <150mJy
FRED τ1=1–2d,τ2=30–40d <160mJy
Note. — We sampled uniformly in durations and peak fluxes as
well as start times (not listed) that spanned the entire 3 months of
observation. For the FRED profile, τ1 is the characteristic rise time,
andτ2 isthecharacteristicdecaytime.
by running the same search on the injection region.
The injection region consisted of 104 random pixels
sampled over the entire image. For each pixel, we simu-
lated a transient light curve by drawing randomly from
uniform distributions of brightness, start times, dura-
Fig.7.—Cumulativebackground distributionofρ˜fortheplay-
ground region for the three searches. The blue dashed line is an tions, and other parameters that define the transient
exponential fittothetailoftheplaygroundAdistribution,where shape. The values for these parameters were identi-
the tail consists of 100 points to the right of the vertical dotted fied by their pixel number and recorded in an injection
line. The yellow shaded region is the √N error region for the fit, file independent of and unknown to the search pipeline.
showing that the distributions from playground A and B agree
Rather than simulating measurement uncertainties, we
withintheerrorbars. Allcurves arenormalizedtothenumber of
synthesized beams in playground B. The fit determines ρ˜∗ at a corrupted these simulated transient light curves by in-
predeterminedPFA=N( ρ˜)forN( ρ˜) 1whenitisnormal- jecting them into the actual data, i.e. added the bright-
izedtothenumberofsynt≥hesizedbea≥msin≪thesearchregion. See nessofthesimulatedtransienttotheobservedlightcurve
textforfurtherdiscussion.
in the corresponding pixel at the corresponding times.
This preserved the noise properties of the real data.
∗
distributionofρ˜andtosetathresholdρ˜ byextrapolat-
These light curves, with injected transient signals, were
ingthetailofthedistributiontoourchoiceoffalsealarm
then processed through the pipeline in the exact same
probability P (the reliability of detected candidates),
FA mannerasthetransientsearchitself. Inotherwords,the
whileplaygroundB wasusedtoverifythatextrapolation
pipeline did not know whether or not the light curves it
and the trials factor normalization. The extrapolation
wasprocessingcontainedinjectionsornot. Theinjection
wasdone byfitting anexponentialfunctionto the tailof
run remained blind in this manner and would not have
thecumulativedistributionofρ˜asdescribedinSection4.
biased the results. The pipeline returned a list of tran-
For all three of our searches, we chose P = 10−3 for
FA sientcandidates that passedthe same criteria appliedto
∗
ρ˜ , which meant that the probability of detecting a false
thesearchregion. Weidentifiedthesecandidatesbytheir
positive in each search (experiment) is 10−3. If the
pixel number and compared the transient parameters as
≤
distribution is Gaussian,which it is not, this probability
recovered by the pipeline, which had been corrupted by
corresponds to a significance of 3.3-σ. Figure 7 shows
noiseintherealdata,totheir“true”valuesstoredinthe
the extrapolationand verificationfor the three searches.
injection file. We note that gaps in the observed data
Whilethenormalizationwassettothenumberofsynthe-
made comparisonsofstart times and durations tricky as
sizedbeams(independentpixels)inplaygroundBforthe
aninjectedtransientmighthavea“true”starttimethat
verification process, the threshold was determined after
occurred during a gap in the observation. We corrected
normalizing the background distribution to the number
for this by identifying the time of the closest observed
of synthesized beams in the searchregion. The values of
snapshotbefore the “true” start (or end) time and com-
∗
ρ˜ for the three searches are listed in Table 2.
pared the “effective” start times and durations since a
We mention that another possible way to determine
real transient that starts before an observation is indis-
the background distribution is to shuffle the images in
tinguishablefromatransientthatstartsatthebeginning
time and then use the entire image instead of defining a
of an observation.
playground region. This avoids the need to extrapolate
For all three searches, we injected transients with the
thetailoftheρ˜distribution,butwecautionthatrandom
top-hat profile, sampling uniformly in transient dura-
shuffling could break any temporal correlations in the
tion, start time, and brightness (amplitude). For the
systematic errors and change the noise distribution. We
day-to-month search, we also injected transients with
did not do this in our analysis. Future workis necessary
the fast-rise-exponential-decay(FRED) profile to mimic
todeterminethetimescaleonwhichtoshuffletheimages
whatrealtransientsmightlooklike,e.g. radioflaresfrom
that would preserve the true noise distribution.
X-ray binaries (Lo 2014), sampling uniformly in charac-
teristic rise and decay times, start time, as well as peak
5.3. Measuring the Efficiency
flux. All injection parameters are listed in Table 3.
Theefficiency(completeness)ofeachsearchcharacter- After running the search on the injection region, we
∗
izes the fraction of real transients successfully recovered appliedacutonρ˜,choosingonlythe eventswithρ˜ ρ˜ .
≥
and plays a role in the upper limit calculation of the Theseweretherecoveredtransientevents. Bycomputing
transient surface density. We determined the efficiency the ratio of the number of recovered events to the total
10
Fig. 8.— Efficiency (completeness) of recovered transients for Fig.9.—Recoveryoftransientparameters. Thepipelineisable
the three searches. The efficiency increases at lower fluxdensities torecover injectedtransient parametersfairlyaccurately; seealso
asthesearcheddurationincreasesbecauselongerdurationsimply Figure10. Thereisaslightlybiggerspreadintherecovereddura-
longer integration times and lower image noise. Panel (d) shows tions,butthatisbecausethesearchusedalimitedsetofdurations
theefficiencyforwhenthetransientswereinjectedwiththeFRED comparedtotheuniformsamplingofinjecteddurations. Theflux
profile,whichisqualitativelydifferentfromthetop-hat templates densities recovered for the FRED profile injections are systemat-
usedinthesearch. Itisnotdrasticallydifferentfromtheefficiency ically lower than the injected peak fluxes, and that is a result of
forwhenthetransientswereinjectedwiththetop-hatprofileasin the qualitative difference between the injected and the searched
panel (c), demonstrating that the top-hat template is capable of light curve profiles; the FRED profile is sharper than the top-hat
recoveringtransientsthatarenottop-hatinshape. profile sothe recovered flux density is smeared out. Although we
onlyshow thisfortheday-to-month search, thepipelineperforms
similarlyfortheothertwosearches.
number of injected events, we determined the efficiency
as a function of flux density for each search, as shown
in Figure 8. Since sensitivity improves with lower image
noiseorlongerintegrationtime,theday-to-monthsearch
was able to recover fainter transient sources at a higher
efficiency than the minute or hour search.
Since we also knew the true injection parameters, we
checked the accuracy at which the pipeline recovered
these parameters, as shown in Figures 9 and 10. The
pipeline was able to recover injected parameters fairly
accurately (.10–30% 1-σ errors), even when the search
was run with top-hat templates on injected transients
with the FRED profile. This demonstrates that we are
capable of detecting real transients in the data, if they
are present, despite using simple top-hat templates for
the search. This is because a top-hat template will al-
wayshavesignificantoverlapwithatransientsignalthat
rises then falls on a similar timescale, regardless of the
exact shape of the transient signal (see Equation 15).
However,the significance of a detection from a template
that does not match exactly will be less than that from Fig.10.— Accuracy of recovered transient parameters, cor-
anexactmatch,asshownfromtheloweredefficiencyand responding to the panels in Figure 9. The accuracy of each
accuracyofrecoveringFREDprofilesusingtop-hattem- light curve parameter p is quantified by its fractional error:
plates. On the other hand, if there is a detection, one (izpeindjb−ypthreec)s/tpairntjteimxceepdtifffeorrentcheeresltaatritvetitmoet,hewhinicjehctiesdcdhuarraatcitoenr-:
can rerun the search with better-matched templates to (t0,inj−t0,rec)/(tdur,inj). The means of the distributions are (a)
measure the transient properties more accurately. 1.9%, (b) 10.2%, (c) 3.3%, (d) 26.3%; and the standard devia-
t−ionsare(a)7.8%,(b)35.5%,(c)11.6%,(d)14.3%.
5.4. Identifying the Candidates
Finally,weranthepipelineonthesearchregion,which we could produce a “transient” sky map, as shown in
contained 3 106 image pixels or 2 105 synthe- Figure 11, where each pixel contains the corresponding
∼ × ∼ ×
sizedbeams,whereonesynthesizedbeamcontained 16 ρ˜ instead of brightness. This map visualizes the vari-
∗ ∼
image pixels. We applied the cut ρ˜ ρ˜ on individual ability on a particular timescale across the image, and a
≥
pixels to identify transient candidates. If multiple pix- transient candidate would stand out as a source.
elspassedthecut,wegroupedtheadjacentonestogether We found no transient candidates. The properties of
andconsideredthemasonecandidatebecausetheimage the loudest events are listed in Table 4. The ρ˜ distri-
was oversampled. Since we computed ρ˜for every pixel, butions from the search region agree very well with the