Subtraction of Correlated Noise in Global Networks of Gravitational-Wave Interferometers

Subtraction of correlated noise in global networks of gravitational-wave interferometers

Michael W. Coughlin,1 Nelson L. Christensen,2 Rosario De Rosa,3, 4 Irene Fiori,5 Mark Golkowski, 6 Melissa Guidry,7 Jan Harms,8 Jerzy Kubisz,9 Andrzej Kulak,10 Janusz Mlynarczyk,10 Federico Paoletti,11, 5 and Eric Thrane12 1Department of Physics, Harvard University, Cambridge, MA 02138, USA 2Physics and Astronomy, Carleton College, Northfield, MN 55057, USA 3INFN, Sezione di Napoli, Complesso Universitario di Monte S.Angelo, I-80126 Napoli, Italy 4Universitàdi Napoli ’Federico II’, Complesso Universitario di Monte S.Angelo, I-80126 Napoli, Italy 5European Gravitational Observatory (EGO), I-56021 Cascina, Pisa, Italy 6Department of Electrical Engineering, University of Colorado Denver, Denver, CO 80204, USA 7Department of Physics, College of William and Mary, Williamsburg, VA 23185, USA 8INFN, Sezione di Firenze, Sesto Fiorentino, 50019, Italy 9Astronomical Observatory, Jagiellonian University, Krakow, Poland 10AGH University of Science and Technology, Department of Electronics, Krakow, Poland 11INFN, Sezione di Pisa, I-56127 Pisa, Italy 12School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia The recent discovery of merging black holes suggests that a stochastic gravitational-wave background is within reach of the advanced detector network operating at design sensitivity. However, correlated magnetic noise from Schumann resonances threatens to contaminate observation of a stochastic background. In this paper, we report on the first effort to eliminate intercontinental correlated noise from Schumann resonances using Wiener filtering. Using magnetometers as proxies for gravitational-wave detectors, we demonstrate as much as a factor of two reduction in the coherence between magnetometers on different continents. While much work remains to be done, our results constitute a proof-of-principle and motivate follow-up studies with a dedicated array of magnetometers.

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Introduction. A stochastic gravitational-wave back- generation detectors, additional magnetic environmental ground (SGWB) is a potential signal source for ground- correlations have also been identified [8, 9]. These cor- based, second-generation interferometric gravitational- relations would contaminate the gravitational-wave data wave detectors such as Advanced LIGO [1] and Advanced streams and thus inhibit the detection of the SGWB. Virgo [2]. An astrophysical SGWB could be produced Correlated noise produces a systematic error that cannot by objects such as compact binary coalescences, pulsars, be reduced by integration over time and therefore is a magnetars, or core-collapse supernovae. A cosmological fundamental limit for SGWB searches. background could be generated by various physical pro- Global electromagnetic fields such as the Schumann cesses in the early universe [3, 4]. Previous analyses have resonances are an example of environmental correlations achieved interesting constraints on these processes [3–5]. between interferometers. By inducing forces on magnets In particular, with the recent discovery of a binary black- or magnetically susceptible materials in the test-mass hole merger [6], there is a chance of observing a SGWB suspension system, these fields are predicted to induce from these systems [7]. correlated noise in the spatially separated gravitational- Typical searches for a SGWB cross-correlate data from wave detectors. arXiv:1606.01011v1 [gr-qc] 3 Jun 2016 two spatially-separated interferometers, where the de- Schumann resonances are due to the very small at- tector noise is assumed to be Gaussian, stationary, and tenuation of extremely low frequency (ELF) electromag- uncorrelated between the two interferometers and much netic waves in the Earth-ionosphere waveguide, which is larger than the signal. In the case where the noise is formed by the highly conducting Earth and the lower uncorrelated, the sensitivity of the search for the SGWB ionosphere. The ELF waves are reflected by the lower increases with time, tobs, and with signal-to-noise ratio ionospheric layers at altitudes smaller than the half wave- 1/2 (SNR) proportional to tobs . Even though the interfer- length, enabling propagation of transverse electromag- ometers are spatially separated, with Advanced LIGO netic waves similar to a low-loss transmission line. The consisting of detectors in Livingston, Louisiana and Han- attenuation increases with frequency reaching the max- ford, Washington, and Advanced Virgo in Cascina, Italy, imum at the waveguide cut-off frequency of about 1500 correlated noise between the detectors has been identi- Hz. In the lower part of the ELF range, the attenua- fied [8, 9]. Stationary noise lines, such as those from the tion rate is particularly small: at 10 Hz it is roughly 0.25 60 Hz power line and 1 Hz timing GPS noise, present at dB/1000 km [12] . This enables observation of strong both LIGO sites, were notched in previous data analyzed ELF electromagnetic field pulses (ELF transients) prop- [4, 5, 10, 11]. Due to the increased sensitivity of second agating around the world several times. 2

The main source of ELF waves in the Earth-ionosphere 0.015pT/Hz1/2 at 14 Hz [18]. These magnetometers can waveguide are negative cloud-to-ground (-CG) atmo- potentially be used to subtract correlated magnetic noise spheric discharges, in which the vertical component of in gravitational-wave detectors. In previous work [8, 9], the dipole moment dominates and effectively generates Thrane et al. estimated the effect of the correlated strain the electromagnetic waves. An individual -CG discharge the Schumann resonances would generate for second- is an impulse of current associated with the charge trans- generation gravitational-wave detectors and in particular fer of about 2.5 C in the plasma channel that has a their effect on SGWB searches. In addition to explor- length of 2 to 3 km and lasts for about 75µs. A typ- ing simple Wiener filter schemes using toy models for the ical dipole moment (charge moment) of a discharge is gravitational-wave detector strain and magnetometer sig- about 6 C km [13]. The spectrum of an impulse gen- nals, they show how to optimally detect a SGWB in the erated by -CG is practically flat up to the cut-off fre- presence of unmitigated correlated noise. In this paper, quency of the waveguide. On Earth, mainly in the trop- we carry out a demonstration of Wiener filtering with a ics, many thunderstorm cells are always active and pro- goal of reducing the coherence between widely separated duce about 50 -CG discharges per second. Since the magnetometers (serving as proxies for gravitational-wave vertical atmospheric discharges radiate electromagnetic detectors). For our study, we use data from ultra-high- waves in all directions, it leads to interference of waves sensitivity magnetometers, which have been deployed for propagating around the world. As a consequence, the geophysical analyses. Using these previously deployed spectrum of atmospheric noise exhibit resonances. The magnetometers, allows us access to instruments with su- solutions to the resonance field in a lossless spherical cav- perb sensitivity, located in very magnetically quiet loca- ity were obtained for the first time by W.O. Schumann tions. As shown in [9], this is a requirement for successful [14]. The predicted eigenfrequencies (10.6, 18.4, 26, 33.5 subtraction. Unfortunately, the available magnetometers Hz) turned out to be much higher than the observed fre- are not situated optimally to reproduce the subtraction quencies, which are close to 8, 14, 20, 26 Hz [15], because scheme we envision for LIGO/Virgo. Ideally, one would of the dispersive character of the attenuation introduced want at least one pair of perpendicular magnetometers by the ionosphere, which detunes the Earth-ionosphere for each gravitational-wave detector. This witness pair cavity [16]. The peaks of the Schumann resonances are should be far enough from the detector to avoid the lo- relatively wide. Their quality factors for the first three cal magnetic noise, but close enough that it measures a Schumann resonance modes are about 4, 5, and 6, re- similar Schumann field as would be present at the detec- spectively. Due to the attenuation the coherence time of tor. For this study, the witness sensors are very far away the field in the Earth-ionosphere cavity does not exceed from the proxy gravitational-wave sensors. Despite these 1 second. limitations, the work that follows is an important first The Schumann resonance background is a global field. step to realistic subtraction of correlated magnetic noise The amplitude distribution of the following resonance in gravitational-wave detectors. modes depends on the time of day and year [17]. The In the most-ideal scenario, magnetometers would be spectral density of the first resonance mode is about stationed near the gravitational-wave detectors, not di- 1.0 pT/Hz1/2 and is different for different observers be- rectly on-site so as to be directly affected by the local cause of their location relative to the world thunderstorm varying magnetic fields, but not so far away as to reduce centers. The daily and yearly changes in the ampli- the coherence of the magnetometers. This is likely to be tude are of several tens of percent and are related to within a few hundreds of meters of the gravitational-wave the changes of the distance from the active thunderstorm detectors. We can test aspects of this by using existing centers (source-observer effect) and the intensity of dis- magnetometer infrastructure, although the distances be- charges. These factors have influence on the correlation tween these sites are significantly larger. Another caveat factor between fields measured in different locations on is that among the existing magnetometers, the ones near Earth and at different moments in time. to one another are orthogonal to one another, and so the Strong atmospheric discharges, which are much less efficacy of noise subtraction is limited. For these reasons, frequent, such as positive cloud-to-ground (+CG) dis- the work that follows functions as a first step to realistic charges and cloud-to-ionosphere discharges associated subtraction of correlated magnetic noise in gravitational- with Sprites and Gigantic Jets, have a smaller contri- wave detectors. bution to the Schumann resonance background. Their In this work, we will use ELF magnetometers from two influence on the fundamental limit for SGWB searches stations of the WERA project [28]. The Hylaty ELF sta- is negligible. However, these discharges generate ELF tion is located in the Bieszczady Mountains in Poland, impulses that have very high amplitudes, so a different at coordinates 49.2◦ N, 22.5◦ E, in an electromagnetic en- approach is required to analyze their influence on detec- vironment with a very low level of anthropogenic mag- tion of gravitational waves. netic field activity [19]. The Hugo Station is located in The Schumann resonances are detected at high SNR the Hugo Wildlife Area in Colorado (USA) at 38.9◦ N, with a worldwide array of extremely low frequency 103.4◦ W. Both stations include two ferrite core active (ELF) magnetometers, which generally have the fre- magnetic field antennas, one oriented to observe magnetic quency bandwidth of 3-300 Hz, with a sensitivity of ≈ fields along the North-South direction, the other oriented 3 to observe magnetic fields along the East-West direction. writing These instruments are sensitive to the Schumann reso- ˜ nances as well as transient signals from individual high s˜1(f) = h1(f) +n ˜1(f) + k1(f)m ˜ (f) (3) peak current lightning discharges. Such large discharges s˜ (f) = h˜ (f) +n ˜ (f) + k (f)m ˜ (f) are often associated with so called transient luminous 2 2 2 2 events that occur at stratospheric and mesospheric al- where h˜ (f),n ˜ (f), and k (f) are the gravitational-wave titudes [20]. The magnetometers are also sensitive to i i i strain, the independent instrumental noise, and the mag- atmospheric discharges, even when they have a very long netic coupling transfer function respectively.m ˜ (f) is continuing current phase. They have a lower cut-off fre- the correlated magnetic spectrum, which is the same at quency of 0.03 Hz with the overall shape of the spectrum both detectors. The effect of the local varying magnetic dominated by 1/f noise. field on the interferometers is contained within then ˜i(f) In addition to the ELF stations, we will also use mag- terms, and in principle, on-site magnetometers can be netic antennas at and nearby to the Virgo site. A tempo- used to monitor and subtract their effect from the data. rary station was created at Villa Cristina between June One metric for measuring this correlation is the coher- 22-25, 2015 and June 29 - July 3rd, 2015. This location ence c(f) is 12.72 km southwest from Virgo, and the magnetometer was placed on the ground floor of an uninhabited house ∗ s˜1(f)s ˜2(f) and oriented North-South. The house was not running c(f) = (4) electricity, and the nearest location served by electricity |s˜1(f)||s˜2(f)| is a small service building 500 m away. The house is sur- wheres ˜1(f) ands ˜2(f) are the Fourier transforms of the rounded by about a 2 km radius of woods, and there was two channels. some excess magnetic noise induced by nearby truck tran- A proposed method for mitigating magnetic noise in sits working on logging. In addition, there are 6 sensitive gravitational-wave detectors is to coherently cancel the magnetic antennas located inside of Virgo experimental noise by monitoring the magnetic environment. There halls, which are “Broadband Induction Coil Magnetome- are two potential ways this could be done. First, one ters”, model MFS-06 by Metronix. In the following anal- could directly correlate magnetometer and gravitational- ysis, we will take a single magnetic antenna from on-site wave strain data to calculate a Wiener filter. This has to represent Virgo’s magnetic environment, located in the the benefit of not relying on any magnetic coupling mod- North End Building along the West detector arm. The els, but the downside is that the Wiener filter will require Virgo detector North arm is rotated about 20 degrees very long correlation times to potentially disentangle lo- clockwise from geographic North. The Hylaty and Hugo cal magnetic foreground from Schumann resonances and stations are 8900 km away from one another. Hylaty and due to the small coherence between magnetometers and Villa Cristina and Hugo and Villa Cristina are 1200 km strain data. and 8700 km away respectively. This can be compared to The other option is to use magnetometers as witness the 3000 km separation between the two LIGO sites. sensors for magnetic noise produced at each test mass, Formalism. Typical searches for a SGWB use a apply measured magnetic coupling functions to strain cross-correlation method optimized for detecting an data, and subtract these channels from the strain data. isotropic SGWB using pairs of detectors [21]. This This has the benefit of not relying on weakly correlated method defines a cross-correlation estimator: measurements as in the case of the Wiener filter, but the downside is that there is no direct feedback on errors in Z ∞ Z ∞ the coupling model and cancellation performance. This ˆ 0 0 ∗ 0 ˜ 0 transfer function model, which will depend on the prop- Y = df df δT (f − f )˜s1(f)˜s2(f )Q(f ) (1) −∞ −∞ agation direction of the electromagnetic waves, will be difficult to compute precisely in practice also because of and its variance: the correlation of the Schumann resonances between test masses, and magnetic coupling can potentially also vary Z ∞ 2 T ˜ 2 with time. If, for example, the Schumann resonances σY ≈ dfP1(f)P2(f)|Q(f)| , (2) 2 0 produce very similar noise at two test masses, very pre- cise estimates of the coupling are required to perform 0 where δT (f − f ) is the finite-time approximation to the accurate subtraction (although this also implies that the Dirac delta function,s ˜1 ands ˜2 are Fourier transforms of Schumann resonances will produce smaller strain noise). time-series strain data from two interferometers, T is the Otherwise, measurement errors in the coupling function coincident observation time, and P1 and P2 are one-sided could dominate the error of the Schumann noise estimate strain power spectral densities from the two interferom- in the gravitational-wave strain channel. Both options eters. The SNR can be enhanced by filtering the data are similar in the sense that they both rely on the use of with an optimal filter spectrum Q˜(f) [21, 22]. Any cor- witness sensors, with transfer functions either provided related noise sources will appear in the inner product of by a Wiener filter in the first case or a magnetic coupling ∗ 0 the two strain channels,s ˜1(f)˜s2(f ). We can see this by model in the second case. 4

In this paper, we test noise subtraction between mag- 0 netometers as a first step towards subtraction of corre- 10 Poland lated noise from interferometer strain channels. Can- Colorado cellation filters applied to magnetometers can be imple- Virgo mented in the time-domain as vectors of real numbers Villa Cristina representing impulse responses and convolved with the −1 input signal to obtain the vector to subtract from the 10 gravitational-wave channel [23]. The idea is to predict Hz] √ the correlated noise seen in the target channel, which could be either a gravitational-wave strain channel, or, −2 as in the case presented in this study, a magnetometer, 10 using witness sensors, which are magnetometers. We are only able to test certain aspects of this calculation, as a gravitational-wave interferometer has more than one Magnetic Spectrum [nT/ test-mass with non-trivial couplings between them. The −3 10 witness sensors are used to predict and subtract the noise in the target channel. In the case where the witness sensors have infinite SNR, only the non-correlated residual remains. −4 To maximize the efficacy of the filter, it is necessary 10 5 10 15 20 25 30 to compute the filter during times of minimal local vary- Frequency [Hz] ing magnetic field activity, or equivalently, during times when the witness sensors are mostly detecting global electromagnetic noise. The calculation of the filter coeffi- FIG. 1: The median power spectral density of the North- cients depends on the autopower spectra of the input South Poland, North-South Colorado, Virgo, and Villa channels as well as the average correlation between chan- Cristina magnetic antennas. These are computed using 128 s nels. The noise cancellation algorithm can then be writ- segments. The Hylaty station in Poland, Hugo station in Col- ten symbolically as a convolution (symbol ∗) [23, 24]: orado, and Villa Cristina√ antennas all achieve peak sensitivities of less than 1 pT/ Hz, while the local varying magnetic M field at Virgo prevents those magnetic antennas from reaching X that sensitivity. r(f) = y(f) − (ai ∗ xi)(f) (5) m=1 where f is the frequency, r(f) is the residual (or the the magnetic antenna stationed inside of Virgo is dom- cleaned data channel), a is the correlation coefficient, inated by local varying magnetic fields as well as the i 50 Hz power-line and associated sidebands. For these y(f) is the target channel, and xi are the witness channels. The convolution is defined as local magnetic antennas, suppression of the power-line noise will be important to maximize the potential sub- N X traction of the underlying Schumann resonances. We can ai ∗ xi = a[n]x[i − n], (6) use this power-spectrum, in addition to the most recent i=−N magnetic coupling function published in [25], to determine the effect the global electromagnetic noise has on where N is the number of filter coefficients. In this anal- SGWB searches. We note here that the calculation of ysis, y(f) corresponds tos ˜i(f) from equation 3. One can common and differential mode coupling performed are predict the residual for a single witness sensor using the conservative and in the future, separate coupling func- equation [23] tions for each test mess would be ideal [9]. 1 We can relate the strain noise induced by the magnetic r(f) = . (7) fields, S (f), which is the magnetic power spectrum p1 − c(f)2 MAG multiplied by the magnetic coupling function, to ΩMAG by [4] where c(f) is the coherence calculated in equation 4. Data analysis. Figure 1 shows the power spectral den- 10π2 Ω = S (f)f 3 (8) sity (PSD) of magnetic antenna data for the sites of in- MAG 3H2 MAG terest in this analysis. The broad peaks at 8, 14, 21, 0 27, and 32 Hz in the spectra correspond to the Schu- where we assumed a value H0 = 67.8 km/s/Mpc for the mann resonances, while there are many sharp instrumen- Hubble constant [26]. The Schumann resonances induce −9 tal line features in the Virgo spectra and two in the Villa correlated noise such that ΩMAG = 1 × 10 , which is Cristina magnetic antenna as well. It is clear that the a potential limit for Advanced LIGO. Here we have in- ELF stations contain sufficient sensitivity to detect the tegrated over a year at design sensitivity and included Schumann resonances at high SNR. On the other hand, the Schumann frequency band. We note here that the 5

Schumann resonances are strongest in the fundamental strain channel data. In this case, both on-site and off- and first few harmonics and as well as the fact that the site magnetometers will be used in coordination to sub- magnetic coupling is the strongest at the lowest frequen- tract the effects of both local and global electromagnetic cies [25]. For these reasons, if instead of performing a fields from the strain data. If the local varying magnetic SGWB search from 10 Hz, one integrates from 25 Hz, the fields have a difference in phase and/or direction with effect from the Schumann resonances on SGWB searches the global magnetic fields, their use will likely inject ex- decreases by about an order of magnitude. tra noise from the Schumann resonances when subtrac- The left of figure 2 shows the coherence between the tion is performed. In our particular example, we take North-South Poland and North-South Colorado, Virgo, the North-South Poland magnetic antenna as our target and Villa Cristina magnetic antennas. The coherence be- channel and the East-West Poland magnetic antenna, the tween Poland and Colorado and Villa Cristina show clear two Colorado magnetic antennas, and the Villa Cristina peaks in the coherence spectrum, while the coherence be- magnetic antenna as the witness channels. Using both tween the Virgo and Poland stations are less pronounced of the magnetometers allows for coverage of both the due to very high local magnetic noise level at the Virgo North-South and East-West magnetic fields. The idea site. The right of figure 2 shows the variation in the co- is that using closely spaced, orthogonal magnetometers herence between the Hylaty station in Poland and Villa are likely to contain complementary information, even Cristina stations. These coherent peaks are potentially if local varying magnetic field disturbances will be corre- problematic for the SGWB searches. The most impor- lated between them. On the other hand, using co-located tant peaks for the gravitational-wave detectors are the magnetometers measuring magnetic fields in the same di- secondary and tertiary harmonics at 14 Hz and 20 Hz re- rection, assuming the local varying magnetic fields dom- spectively, as the primary is below the seismic wall of the inate, provides higher SNR for the Wiener filter if and gravitational-wave detectors at 10 Hz. In the future, it only if the intrinsic noise (not due to the global electro- will be beneficial to measure the coherence between quiet magnetic noise) in each magnetometer is independent. magnetometers stationed near LIGO and Virgo. As we This example shows the subtraction that can maximally show below, high coherence indicates the potential for be expected given the currently deployed sensors in this significant signal subtraction. study and properties of the local varying magnetic fields. Figure 3 shows the time-frequency coherence between It is also similar to the gravitational-wave detector case the North-South Poland and North-South Colorado sta- in that the magnetometers used to perform coherent sub- tions on the left and the coherence time-frequency map traction will be a combination of local magnetometers to between two co-located but orthogonal Hylaty station in subtract the local varying magnetic fields and magne- Poland magnetic antennas on the right. These plots in- tometers installed outside of the gravitational-wave de- dicate that the Schumann resonances are only present, tector beam arms to maximize coherence with the Schu- at least at detectable levels, at certain times and not mann resonances; this has the benefit of limiting the ef- others. The coherence between the two perpendicular fects from the local varying magnetic fields on site, which Hylaty station in Poland magnetic antennas show rela- can be quite strong [27]. In this case, using two witness tively weak coherence due to the fact that they overlook sensors subject to the same local varying magnetic fields different regions of the thunderstorm activity on Earth. is suboptimal, as any noise in the witness sensors limits In addition, their correlation is dominated by collocated the efficacy of the subtraction. In the case where the wit- magnetic fields, predominantly due to nearby thunder- ness sensors have the same limiting noise source, which is storms, which can lead to ADC saturations. generally a sum of instrumental and environmental/local Typically, Wiener filters are used to make a channel noise sources, the effective coherence will be dominated less noisy, e.g., [23]. The success of the filtering proce- by the local varying magnetic fields. This is true regard- dure is determined based on the observed reduction of the less of the target sensor, be it another magnetometer or channel’s auto power spectral density. However, when a gravitational-wave detector strain channel. This noise the goal is to minimize correlated noise in two channels, floor will set the possible SNR for Wiener filtered subtrac- we must use a different metric to determine the efficacy. tion, which must be high enough to provide the required In the case of Schumann resonances, the noise in ques- subtraction. tion contributes a very small amount to the auto power On the left of figure 4, we show the subtraction using spectral density in each gravitational-wave strain chan- a frequency domain Wiener filter between 1-32 Hz where nel, perhaps 0.1%. The key metric for our study is the we use 30 minutes of data to generate the filter. In this reduction in coherence between the two channels. analysis, we use a time with minimal local disturbances As described above, Wiener filtering allows for the sub- to estimate the Wiener filter and apply it across the en- traction of noise from a particular target channel using a tire run. This makes an implicit assumption that the set of witness channels. It is instructive to think of the correlation between the target and witness sensors is not target magnetometer as a proxy for a gravitational-wave changing significantly with time. We have checked that interferometer strain channel. The above analysis is sim- updating the Wiener filter every five minutes gives simi- ilar to the potential scenario in subtraction of the Schu- lar results, indicating that the time-scale does not make mann resonances from gravitational-wave interferometer a significant difference in this case for the magnetic field 6

1 Poland/Colorado Poland/Villa Cristina Poland/Virgo 0.9 0 Noise 10 0.8

0.7

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0.5 Coherence Coherence 0.4

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0 5 10 15 20 25 30 5 10 15 20 25 30 Frequency [Hz] Frequency [Hz]

FIG. 2: The plot on the left is the coherence between the North-South Poland, North-South Colorado, Virgo and Villa Cristina magnetic antennas over 4 days of coincident data divided into 128 s segments. In addition, we plot the expected correlation given Gaussian noise. The coherence between the Virgo and Villa Cristina antennas is similar to that of Virgo and North-South Poland. The plot on the right is the variation in the coherence between the Hylaty station in Poland and Villa Cristina. The colors represent the percentage of the segments which have any given coherence value. The white lines represent the 10th, 50th, and 90th coherence percentiles. subtraction. Magnetometers dominated by local varying imately a factor of 2 in coherence near the peak of the magnetic fields may require regular updates if the local dominant harmonic. varying magnetic field is changing often such that the Using the results of Thrane et al. [9], we can place sensitivity between the gravitational-wave detectors and these results in context for Advanced LIGO. The au- magnetometers to the Schumann resonances changes. In thors of that work showed that the integrated SNR from the case where the target sensor is a gravitational-wave correlated magnetic noise in one year of coincident data strain channel, it will likely be useful to regularly up- from the LIGO Hanford and Livingston detectors operat- date the filter due to possible changes in the magnetic ing at design sensitivity is between 24-470, depending on coupling function of the detector with time. The time magnetic field coupling assumptions, significantly limit- between updates will be affected by interferometer com- ing a potential measurement of ΩGW. With the help missioning activities interspersed with data acquisition. of recent commissioning activities to improve the mag- The coherence results discussed above give us an expecta- netic coupling functions, these numbers are likely to be tion of the amount of subtraction we can expect between a worst-case-scenario. The idea is that correlated noise magnetometers using equation 7. This is consistent with can only be safely ignored in SGWB searches if the SNR the result of the Wiener filter implementation. contribution from correlated noise is much less than 1. ∗ ∗ We now turn our attention to the metric most appro- As SNR ∝ r˜1r˜2m˜ 1m˜ 2, any reduction made in the power ∗ priate for searches for stochastic gravitational-wave back- spectrum of the magnetic noisem ˜ 1m˜ 2 will reduce the grounds. We can use the available magnetic antennas as SNR by that same factor. One possibility to improve a proxy for a 2-detector for a gravitational-wave inter- upon this subtraction would be to use multiple sensors ferometer network. We assign the North-South Colorado to improve the effective SNR of the witness sensors. An- and Villa Cristina magnetic antennas to be gravitational- other (perhaps more promising) possibility would be to wave strain channels and use the North-South Poland use magnetometers located closer to the gravitational- magnetic antenna to subtract the coherent Schumann wave interferometer site. resonances. On the right of figure 4, we measure the Conclusion. In summary, the magnetic fields associ- coherence between the North-South Colorado and Villa ated with Schumann resonances are a possible source of Cristina magnetic antennas before and after the subtrac- correlated noise between advanced gravitational-wave de- tion, to measure the effect that the Wiener filtering has tectors. The optimal method for subtracting correlated had on the correlations. We find a reduction of approx- noise in these detectors is Wiener filtering. In this pa- 7

1 1

180 180 0.9 0.9

160 160 0.8 0.8

140 140 0.7 0.7

120 0.6 120 0.6

100 0.5 100 0.5 Coherence Coherence Time [Days] 80 0.4 Time [Days] 80 0.4

60 0.3 60 0.3

40 0.2 40 0.2

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0 0 0 0 5 10 15 20 25 30 5 10 15 20 25 30 Frequency [Hz] Frequency [Hz]

FIG. 3: The plot on the left is a time-frequency map of the coherence between the North-South Poland and North-South Colorado magnetic antennas. The plot on the right is the same between the North-South and East-West Hylaty station in Poland magnetic antennas.

1 0.5 Pre-Subtraction Post-Subtraction 0.4 Noise 0.9

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0.7 0.1 Theoretical Wiener Filter 0.6 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Residual Spectrum / Original Spectrum Original / Spectrum Residual Frequency [Hz] Frequency [Hz]

FIG. 4: Success of the Wiener filtering procedure under two metrics: reduction in auto power spectral density and reduction in coherence. On the left is the ratio of the auto power spectral density before and after Wiener filter subtraction using the North- South Poland antenna as the target sensor and East-West Poland magnetic antennas, the two Colorado magnetic antennas, and the Villa Cristina magnetic antenna as witness sensors. The “theoretical” line corresponds to the expected subtraction from the maximum coherence between the target and witness sensors (in any given frequency bin) given by equation 7, while the “subtraction” line corresponds to the subtraction achieved by the Wiener filter. The “subtraction” line exceeds the performance over the “theoretical” over part of the band because the theoretical line, computed from a single channel, ignores the benefit of having multiple witness sensors. On the right is the coherence between the North-South Colorado and the Villa Cristina magnetic antenna before and after Wiener filter subtraction using the North-South Poland antenna. There is approximately a factor of 2 reduction in coherence at the fundamental Schumann resonance. per, we have described how the global electromagnetic glecting common-mode rejection, is a potential limit for fields create a potential limit for SGWB searches with ad- Advanced LIGO, where we have integrated over a year at vanced gravitational-wave detectors. In particular, with- design sensitivity and included the Schumann frequency out subtraction, the Schumann resonances induce corre- band. We have also discussed the implications of the co- −9 lated noise such that ΩMAG = 1×10 , using the most re- herence achieved between extremely low frequency mag- cent magnetic coupling function published in [25] and ne- netometers in their use in stochastic searches. This coher- 8 ence is sensitive to the magnetometer SNR of the Schu- It will be a challenge to measure the Schumann reso- mann resonances to the fundamental instrument noise as nances at the sites due to the local magnetic foreground. well as the local varying magnetic fields. Both the LIGO It will be important to perform a similar measurement and Virgo sites will benefit from sensitive magnetome- where the magnetometers are aligned and separated by ters at magnetically quiet locations that are outside of a 3 to 4 km distance, which will more closely mimic the the buildings housing the gravitational-wave detectors. scenario when magnetometers are used as witness sen- We have also shown that careful treatment of the mag- sors for gravitational-wave detectors. Finally, it will be netometer data, which include significant sensitivity fluc- useful to have coupling measurements at all of the test tuations due to local varying magnetic fields, will be re- masses at the gravitational-wave interferometers, which quired for subtraction of the Schumann resonances. We will determine the level of the correlations of the Schu- show that magnetometer pairs thousands of kilometers mann resonances at multiple test masses. apart are capable of reducing magnetic correlations by MC is supported by the National Science Founda- about a factor of 2 at the fundamental peak. This gives tion Graduate Research Fellowship Program, under NSF hope that magnetometers near to the interferometers can grant number DGE 1144152. NC work is supported by effectively subtract magnetic noise with Wiener filtering. NSF grant PHY-1505373. ET is supported through ARC There is significant work to be done looking forward. FT150100281.

[1] J Aasi et al. Advanced ligo. Classical and Quantum nite ground conductivity. IEEE Trans. Antennas Propag, Gravity, 32(7):074001, 2015. 61(4):2269–2275, 2013. [2] F Acernese et al. Advanced virgo: a second-generation [13] Rakov V. A. and M. A. Uman. Lightning Physics and interferometric gravitational wave detector. Classical and Effects. Cambridge University Press, 2003. Quantum Gravity, 32(2):024001, 2015. [14] Schumann, W. Uber¨ die strahlungslosen eigenschwingun- [3] BP Abbott, R Abbott, F Acernese, R Adhikari, P Ajith, gen einer leitenden kugel, die von einer luftschicht und B Allen, G Allen, M Alshourbagy, RS Amin, SB An- einer ionosphärenhülle umgeben ist. Zeitschrift für derson, et al. An upper limit on the stochastic Naturforschung A, 7(2):149–154, 1952. gravitational-wave background of cosmological origin. [15] M. Balser and C. Wagner. Observations of earth- Nature, 460(7258):990–994, 2009. ionosphere cavity resonances. Nature, 188(4751):638– [4] J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott, 641, 1960. M. Abernathy, T. Accadia, F. Acernese, C. Adams, [16] D. D. Sentman. Schumann resonance spectra in a two- R. Adhikari, C. Affeldt, and others. Upper limits on scale-height earth-ionosphere cavity. J. Geophys. Res., a stochastic gravitational-wave background using LIGO 101(D5):9479–9487, 1996. and Virgo interferometers at 600–1000 Hz. Phys. Rev. D, [17] A. Kulak, J. Mlynarczyk, S. Zieba, S. Micek, and 85:122001, Jun 2012. Z. Nieckarz. Studies of ELF propagation in the spherical [5] Aasi, J. et al. Improved upper limits on the stochastic shell cavity using a field decomposition method based on gravitational-wave background from 2009˘2010 ligo and asymmetry of schumann resonance curves. J. Geophys. virgo data. Phys. Rev. Lett., 113:231101, Dec 2014. Res., 111(A10304), 2006. [6] Abbott, B. P. et al. Observation of gravitational waves [18] Yu.P. Galuk, A.P. Nickolaenko, and M. Hayakawa. Knee from a binary black hole merger. Phys. Rev. Lett., model: Comparison between heuristic and rigorous solu- 116:061102, Feb 2016. tions for the schumann resonance problem. Journal of [7] B. P. et al. Abbott. Gw150914: Implications for the Atmospheric and Solar-Terrestrial Physics, 135:85 – 91, stochastic gravitational-wave background from binary 2015. black holes. Phys. Rev. Lett., 116:131102, Mar 2016. [19] A. Kulak, J. Kubisz, S. Klucjasz, A. Michalec, J. Mlynar- [8] E Thrane, N Christensen, and RMS Schofield. Correlated czyk, Z. Nieckarz, M. Ostrowski, and S. Zieba. Extremely magnetic noise in global networks of gravitational-wave low frequency electromagnetic field measurements at the detectors: Observations and implications. Physical Re- Hylaty station and methodology of signal analysis. Radio view D, 87(12):123009, 2013. Science, 49(6):361–370, 2014. 2014RS005400. [9] E. Thrane, N. Christensen, R. M. S. Schofield, and A. Ef- [20] Victor P. Pasko. Recent advances in theory of transient fler. Correlated noise in networks of gravitational-wave luminous events. Journal of Geophysical Research: Space detectors: Subtraction and mitigation. Phys. Rev. D, Physics, 115(A6):n/a–n/a, 2010. A00E35. 90:023013, Jul 2014. [21] B. Allen and J. D. Romano. Detecting a stochastic [10] LIGO Scientific Collaboration. Upper limit map of background of gravitational radiation: Signal process- a background of gravitational waves. Phys. Rev. D, ing strategies and sensitivities. Phys. Rev. D, 59:102001, 76:082003, 2007. 1999. [11] Abbott, B. et al. First cross-correlation analysis of in- [22] Nelson Christensen. Measuring the stochastic terferometric and resonant-bar gravitational-wave data gravitational-radiation background with laser- for stochastic backgrounds. Phys. Rev. D, 76:022001, Jul interferometric antennas. Phys. Rev. D, 46:5250–5266, 2007. Dec 1992. [12] A. Kulak and J. Mlynarczyk. ELF propagation pa- [23] M Coughlin, J Harms, N Christensen, V Dergachev, rameters for the ground-ionosphere waveguide with fi- R DeSalvo, S Kandhasamy, and V Mandic. Wiener fil- 9

tering with a seismic underground array at the sanford C Armitage-Caplan, M Arnaud, M Ashdown, F Atrio- underground research facility. Classical and Quantum Barandela, J Aumont, C Baccigalupi, AJ Banday, et al. Gravity, 31(21):215003, 2014. Planck 2013 results. i. overview of products and scientiﬁc [24] Saeed V. Vaseghi. Wiener Filters, pages 178–204. John results. arXiv preprint arXiv:1303.5062, 2013. Wiley & Sons, Ltd, 2001. [27] J Aasi et al. The characterization of Virgo data and [25] B. P. et al. Abbott. Characterization of transient noise its impact on gravitational-wave searches. Classical and in advanced ligo relevant to gravitational wave signal Quantum Gravity, 29(15):155002, 2012. gw150914. Accepted to Classical and Quantum Gravity, [28] http://www.oa.uj.edu.pl/elf/index/projects3.htm 2016. [26] Planck Collaboration, PAR Ade, N Aghanim,