Correlated magnetic noise across Virgoand spatially separated gravitational-wave detectors

Melissa A. Guidry∗

College of William and Mary, Williamsburg, Virginia 23187, USAEuropean Gravitational Observatory (EGO), I-56021 Cascina, Pisa, Italy

(Dated: August 12, 2015)

The stochastic gravitational-wave background may be detected within the near future using earth-based interferometric detectors. Untreated correlated noise across two or more detectors provides afundamental limit on the sensitivity of stochastic searches. One method of reduction is the use ofmagnetometers to subtract the correlated noise, where a Wiener filter scheme may be implemented.Effective subtraction requires precise measurement of global electromagnetic fields in areas with lowlocal electromagnetic contamination. We conduct such measurements near the Virgo detector, firstdeveloping an off-site quiet standard for comparison and then characterizing the on-site magneticenvironment. An on-site quiet location is successfully determined but its efficiency is limited bylocal magnetic noise.

I. INTRODUCTION

Gravitational-waves (GWs) have long been known toexist since their indirect detection by Hulse and Taylorin 1975 [1]. By accurately measuring the change in pe-riod of a pulsar in a binary system, it has been predictedthat the two objects will eventually in-spiral toward eachother partly due to the radiation of GWs. Direct obser-vation of GWs using earth-based interferometeric detec-tors would provide a new window for astrophysics andtest current theories of gravitation. The most sensitiveof these detectors include EGO’s Virgo (Pisa, Italy) andLIGO (Hanford, Washington and Livingston, Lousiana).The detectors are currently upgrading to ten-fold bettersensitivity (Advanced Virgo (AdV) and Advanced LIGO(aLIGO)), with the first joint data taking foreseen for2016 [2].

GW astronomy hopes to measure the stochasticgravitational-wave background (SGWB), which arisesfrom a large number of random, independent events com-bining to form an isotropic signal. In cosmological modelsas opposed to astrophysical, the SGWB may be createdfrom inflationary physics [3][4][5][6], cosmic strings [7][8],and pre-big-bang physics [9][10]. Its detection may allowa probe into the earliest moments after the big-bang [6].

The search strategy for the SGWB is to cross-correlatestrain data channels from at least two detectors. Byperforming a year-long integration of data, it is possibleto increase sensitivity to signals that would normally bemasked by strain noise. A key assumption in this tech-nique is that the noise in each detector is uncorrelated.Uncorrelated noise is reduced with continued integrationwhereas correlated noise is not, producing a systematicbias. A low level of correlated noise is best achieved inspatially separated detectors where local noise sourcesare not shared. However, global electromagnetic (EM)fields are a source of correlated noise for such detectors.

∗ maguidry@email.wm.edu

These fields may induce forces on magnets mounted di-rectly on the test masses for position control (such asin initial Virgo and AdV) or on magnets higher up thetest-mass suspension system (such as in aLIGO).

In our study, we focus on Schumann resonances: globalEM resonances in the cavity formed by the surface of theEarth and the ionosphere. The cavity is excited by ap-proximately 100 lightning strikes per second around theworld, producing fields on the Earth’s surface of 0.5−1.0pT Hz−1/2 [11]. Observed over time, 10 pT bursts ap-pear over a 1 pT background [12], at a rate of ≈ 0.5Hz. The primary resonance is at 7.8 Hz, with secondaryand tertiary resonances at 14.3 Hz and 21.8 Hz respec-tively. The amplitudes and peak frequencies vary slightlywith proximity to lightning strikes and exhibit a spectralwidth of ≈ 20% (where spectral width is the full widthat half maximum of the peak divided by the central fre-quency).

Previously, magnetic field data have been taken us-ing magnetometers housed in the Virgo and LIGO obser-vatories. Averaged over three months, coherence peakswere found at the known Schumann resonance frequen-cies on the order of 10−4 [13]. Using measurements of thecoupling between magnetic fields and initial LIGO testmass motion, the level of correlated strain noise presentin initial LIGO SGWB analyses has been inferred. Theamount of coupling at which correlated noise becomescomparable to expected SGWB sensitivity has been esti-mated, and we now strive to meet the magnetic isolationspecifications.

EM-induced correlated noise may be reduced by min-imizing magnetic coupling to the strain channel throughthe removal of magnetic components. For example,smaller sized magnetic actuators will be used in theAdV upgrade. However, it is unclear whether second-generation detectors will be capable of achieving suf-ficient magnetic isolation to avoid limiting our SGWBsearch capabilities. An alternative method is to use mag-netometers to subtract correlated noise, where a Wienerfilter scheme may be implemented. Effective subtractionrequires precise measurement of global EM fields near

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detectors of interest. This demands low local EM con-tamination around the measurement station.

In this paper we report on measurements of globalEM fields at electromagnetically quiet locations in andaround the Virgo detector site.

II. FORMALISM

A. Schumann formula

Our planet may be modeled as a solid conductingsphere surrounded by a thin dielectric atmosphere [14].The thickness of the air layer is small in comparison tothe Earth’s radius a ≈ 6400 km. The conductivity ofair increases by six orders of magnitude when ascendingtoward the ionosphere.

The lower parts of the atmosphere are like a thin di-electric layer located between two conductors, the plasmaof the ionosphere and the Earth’s surface. Thus, a cavitywith the properties of a spherical waveguide is formed.When the wavelength of EM waves in the cavity becomeson the order of the Earth’s circumference λ ≈ 2πa, aglobal resonance may occur [14].

Natural Schumann resonance excitation is mostlycaused by lightning discharges. Lightning dischargesradiate EM energy at frequencies below 100 kHz [15].These signals are very weak as the distance from the dis-charge grows, but the waveguide behaves like a resonatorat extremely low frequencies.

In an ideal cavity, the resonant frequency of the n-thmode fn is determined by the Schumann formula [16]:

fn =c

2πa

√n(n+ 1). (1)

For the Earth-ionosphere cavity particular resonancesare: f1 = 10.6, f2 = 18.3, and f3 = 25.9 Hz. The ob-served peak frequencies are lower than those predicted bythe Schumann formula. This is because the real Earth-ionosphere waveguide is not a perfect EM resonant cavity.The observed peaks are lower in frequency and wide witha quality factor Q ranging from 4 to 6, attributed to fi-nite conductivity of the lower ionosphere [14]. Theorieshave since been adapted to account for the observed peakqualities.

We may distinguish Schumann resonances by takingthe discrete time signal from a magnetometer and look-ing at how the power of the time series is distributedwith frequency. While the amplitudes and peak frequen-cies are known to vary slightly, distinct structures areobserved around the nominal peak frequencies in the fre-quency domain.

B. Converting ADC output

For each sensor, the data we want to analyze are mea-sured in Volts at the input of the analog-to-digital con-

verter (ADC) where they are converted into binary bits.The first scaling factor applied to the raw data is theconversion from bits back to Volts. If the data arriveas binary signed integers, they are converted to floating-point numbers by multiplication with the factor

VLSB =Vmax − Vmin

2n, (2)

where VLSB is the voltage corresponding to a least sig-nificant bit, Vmax is the ADC maximum input voltage,and n is the number of bits of the ADC. Generally,Vmin = −Vmax. We then scale our data by a sensor-specific calibration constant, from Volts to the corre-sponding physical unit. We thus have a time series ofequidistant samples to be processed.

Our sensors record data at a fixed sampling frequencyfs. If a discrete Fourier transform (DFT) is computed onan N -sample long signal, the width of a frequency bin,also called frequency resolution, is given by

fres =fsN. (3)

From the Nyquist theorem [17], it follows that the max-imum useful frequency is exactly half of the samplingfrequency. Thus we obtain N/2 frequency bins of thewidth fs/N in the output.

C. Spectral density and coherence

We use the sequence of time samples of the digitizedsignal to calculate the power spectral density (PSD). ThePSD describes how the squared amplitude of a signal isdistributed with frequency. We then take the square rootof the power spectral density to calculate the amplitudespectral density (ASD). The ASD is often used becauseit can be directly related to the parameters of the exper-iment [18].

We define our PSD for channel x to be

Px(f) =|sx|2

µ, (4)

and our cross-PSD for channels x and y to be

Pxy(f) =sxsy

∗

µ, (5)

where sx is the DFT for channel x and µ is a discreteFourier transform normalization constant (see AppendixA, Eq. A5). Finally, we define coherence between the twochannels to be

coh(f) =|Pxy(f)|2

Px(f) Py(f). (6)

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The overline denotes time-averaging over N segments.If sx and sy are independent, Gaussian, and stationaryrandom variables, then 〈coh(f)〉 = 1/N [13]. Deviationsfrom this value are evidence that one of these three as-sumptions is violated. This is useful for determining iftwo channels are correlated. For a full derivation of theseconcepts and their implementation, see Appendix A.

D. Wiener filtering

Wiener filtering is used to subtract noise from a mainchannel using a witness channel when the transfer func-tion from witness to main is not known a priori [19]. Inour case, the strain channel s acts as the main whereasa magnetometer channel w acts as the witness. Giventwo witness sensors w1 and w2 for detecting correlatedmagnetic noise m in two strain channels s1 and s2,

w1(t) = m(t) + η1(t),

w2(t) = m(t) + η2(t).(7)

The eta terms represent noise in each witness sensor, ei-ther electronic or from local magnetic fields. This noiseis uncorrelated between the two witness sensors. We maydefine the witness sensor noise auto-power as N (f).

The witness-to-main transfer functions are estimatedto be

r1(f) =s1(f)w∗1(f)

|w1(f)|2,

r2(f) =s2(f)w∗2(f)

|w2(f)|2.

(8)

The hats are used to denoted estimated quantities. Thus,the Wiener-subtracted strain data are given by

s′1(f) = s1(f)− r1(f)w1(f),

s′2(f) = s2(f)− r2(f)w2(f).(9)

Here s′1 and s′2 are referred to as the “cleaned” channels.In the limit where the estimated transfer functions equalthe real transfer functions, the correlated noise goes tozero and the subtraction is perfect [19].

In practice, this is not the case. As the witness sensornoise grows larger than the correlated noise, the estima-tion of the transfer function deviates more from reality.With larger deviation, Wiener filtering fails. Thus wecan determine the success of our estimation, and thusour filter, using the “witness signal-to-noise ratio”

ρ2w(f) =M(f)

N (f), (10)

where M is the cross-power of m1 and m2. For a wit-ness signal, ρw(f) must be sufficiently high in order to

provide successful Wiener filtering. This ratio may bequalitatively assessed as the resolution of the Schumannresonances in the frequency domain of the witness chan-nel. For more information, see Ref. [19].

III. PROCEDURE AND RESULTS

A. Experimental setup

The data acquisition system (DAQ) utilized a low-noise coil-based magnetometer and a three-axis seis-mometer. A digital recorder with time-stamp capabilitieswas used for long-term measurements and a spectrum an-alyzer was used for short-term measurements.

Metronix MFS-06 magnetometers were used for allmagnetic measurements. This sensor utilizes a sensorcoil consisting of a high permeable ferrite core and sev-eral thousand copper turns. The magnetometer outputsa voltage signal, propoertional to magnetic flux throughthe coil.

This magnetometer may function in two modes, “chop-per on” and “chopper off”. The “chopper on” mode ofoperation is used to reduce low frequency sensor noise dueto thermal drifts. The different modes allow frequencysensitivities of 0.00025 to 500 Hz and 10 Hz to 10 kHz,respectively. The output sensitivity varies with respectto the frequency. The sensitivity function was measuredand used in subsequent conversions from Volts to Tesla(see Appendix B, Fig. 13). The chopper was turned on tomaintain higher sensitivity near the first Schumann res-onance (7.8 Hz). The nominal noise floor from the spec-ifications was used for ASD comparison (see AppendixB, Fig. 14). In order to power the magnetometer with a±15 V source as specified, a custom DC-to-DC converterwas used with a 38 A, rechargeable 12 V battery.

A Nanometrics Trillium seismometer was used withthe magnetometer to investigate possible correlations be-tween seismic activity and magnetic noise. Together, thetwo sensors output signals to a Nanometrics Centaur dig-ital recorder during measurements spanning over multi-ple days. It consists of a 24-bit ADC, a GPS antennareceiver, and a 16 GB SD card for removable storage. ACrystal Instruments CoCo80 spectrum analyzer was usedfor short measurements. Unless otherwise specified, thesample rate was set to fs = 250 Hz with an input voltagerange of Vmax − Vmin = 4 V.

It was essential that we limit low frequency noise (1−20Hz) at the installation in order to resolve our resonances.A common source of this noise is seismic vibrations ofthe equipment. These vibrations modulate the Earth’sstatic magnetic field lines through the magnetometer aswell as generate a field simply from moving the metalliccomponents (e.g. wires) within the vicinity. This noisewas suppressed by either burying the sensors and theirwires several centimeters under the ground or placing themagnetometer inside a building. This setup also reducedthermal drifts within the seismometer.

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B. Quiet standard

In order to find a quiet location, “quietness” must firstbe defined. A location was deemed quiet if the Schu-mann resonances could be resolved by eye in the ASDplot without largely reducing the data set. The resolu-tion of Schumann resonances is indicative of a high ρw,whereas a location is more efficient if less data are dis-carded. A location with a duty cycle of & 25% is desired(N. Christensen, personal communication, June, 2015).

FIG. 1. Magnetic and seismic spectrograms at Villa Cristinabetween June 22-25 2015. The first four Schumann resonancesare observed in the magnetometer spectrogram. Glitcheswithin both spectrograms are identified.

The data are reduced using a post-processing selectionthat removes extraordinarily large increases in magneticnoise that are transient in nature, known as glitches. Adata set consists of a time-series of 10-second-long ASDsbetween 0 Hz and the Nyquist frequency. The filter usesa cutting threshold, typically taken as the average ASDvalue across the entire time period and across the 10-30Hz frequency range. Small average amplitudes in this fre-quency range proved to be most indicative of resolvableSchumann resonances. The ASDs where the average am-plitude value in the 10-30 Hz range is above the thresholdare then removed. The threshold may be lowered untilthe resonances are resolved or statistics become too low.

Before testing on-site locations, we measured a quietstandard which would act as a lower limit for future com-parisons. This was the average of 160 hours worth ofdata taken inside Villa Cristina, a scout house located 13km South-West of Virgo in the Livorno Hills (for a pic-ture of the surrounding environment, see Appendix C,Fig. 15). These data were collected June 22 to June 25,and June 29 to July 3 (all weekdays). The magnetome-ter and seismometer were placed inside the scout housewith the magnetometer length and seismometer y-axisoriented geodetic North-South.

FIG. 2. Villa Cristina: Five Schumann resonances (7.8, 14.3,20.8, 27.3, and 33.8 Hz) are clearly resolved after cutting only3% of 160-hours-worth of data from June and July 2015. Theblack line is the full set of averaged ASDs. The red line isthe average of the ASDs that are removed from the set. Theblue line is the average of the set after the less quiet ASDsare removed.

Spectrograms for 70 hours worth of Villa Cristina dataare shown in Fig. 1. The first four Schumann resonancesare observed as broad bands across the entire time win-dow. In the magnetometer spectrogram, a specific set ofglitches were attributed to a thunderstorm by looking atcoincidences with a Virgo-based lightning counter. Mag-netic glitches were more common during daylight hours,attributed to the movement of heavy metal machinerycutting down trees in the area during this time. Thissource is observed as periodic stretches of glitches in theseismometer spectrogram. Despite this noise, the firstfive Schumann resonances were clearly resolved after cut-ting only 3% of data using the aforementioned cuttingprocedure (see Fig. 2). The Villa Cristina site exhibitedhigh efficiency at a potentially high ρw value.

The transfer function between seismic activity and lo-cal magnetic noise was not assumed to be linear. How-ever, as a preliminary exploration of the coupling, the co-herence was calculated between the seismometer channeland the magnetometer channel. In Fig. 3, the squaredcoherence is calculated between the magnetometer andeach of the seismometer axes for the first 70 hours ofuncut data. In the uncut magnetometer average ASDplot, three peaks were evident at 26, 35, and 37.8 Hz. Inthe coherence plot, we see coherence peaks at these samefrequencies.

The largest coherence was found at 37.8 Hz for thex-axis of the seismometer. To better understand the cor-relation, the ASD values in the time domain were plottedfor this peak frequency in Fig. 4. We see a periodic in-crease in seismic and magnetic activity stretching from5:00 to 12:00 UTC each day. Here we attribute this to thetree removal in the area at these times. The largest cen-

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FIG. 3. Coherence between the magnetometer and threeaxes of the seismometer data over 70 hours at Villa Cristina.

tral peak in the magnetometer data (purple) occurredduring a thunderstorm, and thus does not seem to fitthe periodic pattern or appear as strongly in the seismicdata. We conclude that the three noise peaks in the un-cut 70-hour magnetic average ASD plot are either fully orpartially seismically induced. It is important to note thatthe ρw value for this location, as well as future stations,will vary depending on the level of seismic activity.

The coherence was also calculated between the VillaCristina data and data from magnetometers housed inthe LIGO Hanford Observatory (LHO). The LHO mag-netometers with the smallest noise were used for the full160-hour comparison. These were found to be the threeaxes of the “H1:PEM-CS” setup, near the center of themain LHO interferometer (see Appendix C, Fig. 16 for apicture of the surrounding environment). It is observedthat the LHO magnetometer noise is about one order ofmagnitude above the Schumann peaks.

All three magnetometers were coherent at two mainfrequencies. We see a broad peak characteristic of Schu-mann resonance coherence at 7.8 Hz, as well as a narrowpeak characteristic of electrical noise at exactly 60 Hz,the mains frequency of the United States (see Fig. 5).However, we saw very small coherence at 50 Hz, themains frequency of Europe. This is most likely due tospectral leakage from the much larger amplitude of the60 Hz component in the LHO magnetometer channels.Overlaying the coherence with the clean average ASDsbetween two of the magnetometers, the coherence peakexactly lines up with the primary resonance and there isno noise peak in the LHO ASD, providing good evidencethat this is indeed Schumann induced.

Previous Schumann resonance coherence analysis wasdone using magnetometers housed in the observatories atVirgo and the LHO [13]. These data were averaged overthree months and resulted in a coherence at 7.8 Hz on theorder of 10−4. Using a week of Villa Cristina data, thiscoherence improved by nearly two orders of magnitude.

FIG. 4. ASD values over time for 37.8 Hz. Top: The magne-tometer data. Bottom: The seisomometer x-, y-, and z- axesdata.

C. On-site location testing

Magnetic measurements were taken at different on-sitelocations (Fig. 6). These specific locations were chosento avoid being close to electrical powerlines running overthe detector. While electronic noise appears as a narrowpeak in an ASD plot, an increase in electronic noise oftenresults in larger noise across all frequencies. For example,a 50 Hz power source is modulated, with the potential toproduce noise across all frequencies. The sidebands of afrequency modulated signal extend out either side of themain carrier, and cause the bandwidth of the overall sig-nal to increase beyond that of the unmodulated carrier.As the 50 Hz component increases, so does this noise.

Short measurements were taken using either the digitalrecorder or the spectrum analyzer, with the magnetome-ter axis oriented both magnetic North-South and East-West. An average ASD plot was generated for each site.Using the digital recorder, hours of clean data were aver-aged (sites Central Building (CEB) 1, North End Build-ing (NEB) 1). Using the spectrum analyzer, minutes ofdata were averaged (sites West End Building (WEB) 1-4,A, B, C, D, E, NEB 2-5).

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FIG. 5. Coherence between Villa Cristina and the LHOH1:PEM-CS y-axis magnetometer (green line). The purpledashed line indicates the average coherence value expected foruncorrelated noise. These two lines are plotted with respectto the right vertical axis. The average clean ASDs (red andblue) are plotted with respect to the left vertical axis.

FIG. 6. Virgo site: 15 locations along the detector whereEM measurements were conducted. All measurements weredone between June and July 2015.

At each location, two numbers were chosen for com-parison: the ASD value at 50 Hz (Fig. 7, top) and theaverage ASD value between 10-30 Hz (Fig. 7, bottom).In Fig. 7, the size and color of the dot are both propor-tional to the ASD value (for exact values, see AppendixE, Table I). We found that that the 50 Hz component wasa decent predictor of how noisy a location may be, but itwas the 10 − 30 Hz average that was most indicative ofnoise that would obscure Schumann resonances. Site Bhad the smallest 10− 30 Hz average ASD value with theaxis oriented East-West.

FIG. 7. Top: 50 Hz ASD component for each location tested.Bottom: 10-30 Hz average ASD components.

D. On-site quiet installation

A 14-day long installation was undertaken at Site Bwith the magnetometer and seismometer buried. Initialdata analysis revealed three out of five main resonancesresolved when cutting 35% of the data (see Fig. 8), wellwithin the suggested duty cycle of & 25%. However,on the second day of data collection the measurementsbecame contaminated by local magnetic noise. This noiseappeared in the form of 0.5 Hz sidebands around the 50Hz component. When analyzing a time where sidebandswere always on, no matter how extensive the cutting theresonances were unresolvable (see Appendix D, Fig. 17).

A 230 kV powerline running over the West arm waspredicted as the source. Four electrical powerlines runover the Virgo site. Each one was investigated by mea-suring directly underneath with a small size tri-axial fluxgate FL3-100 magnetometer and the spectrum analyzer.A diagram of the powerlines (bottom) tested as well astheir respective magnetic ASD plots (top) is shown inFig. 9.

Two lines showed signs of sidebands: 300 m down theroad from Virgo (pink) and 1500 m down the West Arm(dark blue). However, it was the 1500 m line that seemed

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FIG. 8. Site B: Three out of four main Schumann resonanceswere resolved after cutting only 35% of data. 22 hours of datawere taken between July 16-17, 2015.

to display 0.5 Hz sidebands consistent with the Site Binstallation (black line). This is a 230 kV line which runsacross both interferometer arms.

Further measurements revealed that the sidebandswere only visible in the ASD plot when the measurementwas taken next to the door of the West tunnel, whichhappened to be our first and most convenient measuringlocation. Measurements along the outside of the Westtunnel revealed that the 0.5 Hz sideband noise was onlypresent when next to tunnel doors. Investigating insidethe tunnel, we found increased amplitudes of the side-bands next to various electrical wires. While we are stillhunting for the 0.5 Hz sideband source, this analysis pro-vides evidence that the source is internal to Virgo.

It was observed that the WEB magnetometers dis-played a sensitivity to the 0.5 Hz sidebands as well (red),with the vertical magnetometer being particularly sensi-tive. Coherence was calculated between the WEB mag-netometers and the Interruptible Power Supply (IPS)monitors. At sideband frequencies, no coherence wasfound.

A spectrogram was generated between 45 and 50 Hzspanning back to February using WEB vertical magne-tometer data (see Fig. 10) in order to see if the 0.5 Hzsideband noise had occurred previously. The dark blueblock represents a period when data collection was tem-porarily offline. The first sideband event was observedon July 7 2015 at about 16:19:12 UTC. Since then, thesidebands have turned on and off without a clear patternin time.

On June 26 2015, we noticed that our NEB magne-tometers were saturating nearly 50% of the time be-cause of a too intense 50 Hz magnetic signal. We relo-cated them in the building in order to reduce saturation.WEB magnetometers were not saturated by showed avery prominent noise bump at 14 Hz. We moved them in

FIG. 9. Top: Average ASDs for surveys done under powerlines, as well as for the Site B and WEB (vertical) magne-tometers. Bottom: A map relating different ASD plot lines(colors) to different powerline sites tested.

the building as well in an attempt to reduce this noise.Some magnetometers gained sensitivity during this time,which was right before we began to see the sideband noisein the WEB vertical magnetometer channel. It is un-clear whether this sideband noise is newly occurring or ifit was always present and our magnetometer simply wasnot sensitive enough to detect it.

Between July 16-30, the sideband noise was present45.1% of the time. This was determined by smoothingSite B ASD values at 45.5 Hz over the time window anddetermining if they went over a “sideband on” thresholddetermined through averaging. This frequency was cho-sen due to its relatively low noise levels during “sidebandoff” times and strong increase in power during “sidebandon” times. In Fig. 11, the binary on-off counts have beenoverlaid with the Site B magnetometer spectrogram todemonstrate the sensitivity of the method. This algo-rithm tends to work well when the sideband noise oc-curs for several hours and there are few glitches. Despitethe smoothing, glitches with enough power will send theASD value over the threshold, creating a false count. Thesmoothing also has the negative capability of ignoring a“sideband on” time that is very short (less then 3 hours)in duration. Most sideband noise occurences are severalhours in duration and high-power glitches are not fre-

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FIG. 10. A spectrogram using WEB vertical magnetometer data dating back several months to look for sideband emergence.We determine that the sideband noise is a relatively new feature.

FIG. 11. A magnetic spectrogram for Site B displaying thesideband noise turning on and off. A digital count is overlaidwith the plot (black line) which was used to determine thepercentage of time the noise was present.

quent. Hence a ±5% error on the occurence percentagemay be assumed as a conservative estimate.

Without the sideband noise, Site B demonstrated aduty cycle of ≈ 65%. With the sideband noise on nearlyhalf of the time, this duty cycle was significantly reduced.The Site B ASD data were averaged and cut for the fulltime window, which included both “on” and “off” times(see Fig. 12). Cutting was done using thresholds aver-aged over 7-30 Hz as opposed to 10-30 Hz in order tobetter resolve the primary resonance. We see Schumannresolution at a duty cycle of 26%. With optimized cut-ting, we predict that this value may be further increased.The lack of resolution of the primary resonance at 7.8 Hzcontinues to be investigated.

FIG. 12. Site B ASD plot between July 16-30, the fulltime window for the installation. Cutting was done usingthresholds averaged over 7 − 30 Hz as opposed to 10 − 30Hz in order to better resolve the primary resonance. We seeSchumann resolution at a duty cycle of 26%.

IV. IMPLICATIONS

For successful Wiener filtering, Schumann resolutionwith a duty cycle of & 25% is preferred. If the sidebandnoise persists, this duty cycle requirement is barely metat the Site B installation with rough cutting on the dataset. With optimized cutting on the data set, this installa-tion site may be well within the duty cycle requirement.If the sideband noise is understood and suppressed, thesite will be further improved for long-term Schumannresonance resolution. Due to the internal nature of thesource to the Virgo site, suppression is feasible.

The lack of resolution of the 7.8 Hz resonance isworth investigating but is not limiting for future filter-

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ing schemes. Our strain sensitivity is greater than 10 Hzand magnetic noise has been observed to couple linearlyinto the strain channel. The primary resonance will notbe directly used for subtraction. It is the secondary andtertiary resonances that are most important for corre-lated noise subtraction and these are regularly resolvedat Location B.

A permanent installation along the West Arm wouldrequire several modifications. A long-term power sourcewould need to be realized as opposed to a rechargeablebattery. This power could be drawn from a power linealong the arm, and be modified in such a way as not toproduce 50 Hz noise when used with the magnetometer.The data would need to be acquired continuously or atleast automatically, without the use of a memory card.This could be realized in the form of an ADC board whichstreams data to the main Virgo DAQ system (which uti-lizes the main GPS clock) using ethernet wires. Cableswould need to stretch across the road between the sensorsand the West arm, and this may require digging a smallditch under the road. Ideally, two more magnetometerscould be utilized in order to measure along all three axes.

Once a permanent quiet magnetic station is established

near Virgo, coherence may be measured with an upcom-ing quiet station near the LHO. Higher coherence over ashort time is an indication of a high ρw in the witnesssensors. Moving forward, the magnetic data will be usedto test the success of different subtraction methods todetermine which should be used for future science runs.

ACKNOWLEDGMENTS

I gratefully acknowledge the use of environmental mon-itoring data from the Virgo and LIGO experiments. Ithank the National Science Foundation for funding theIREU program and the University of Florida for its or-ganization. I thank my advisors Dr. Irene Fiori andFederico Paoletti for allowing me the incredible oppor-tunity to work in a fulfilling and encouraging researchenvironment. I thank all of the EGO physicists and tech-nicians who advised and aided in my project along theway. Lastly, I thank my advisor at my home institutionDr. Eugeniy Mikhailov for advising me to work in thehills of Tuscany instead of the desert.

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[19] E. Thrane, N. Christensen, R. Schofield, A. Effler, Phys.Rev. D 90, 023013 (2014).

10

Appendix A: Full PSD definitions

The estimation of spectral densities is done using thediscrete Fourier transform (DFT). The DFT takes avector of N complex numbers sk, k = 0...(N − 1),and transforms it into a vector of N complex numberssm, m = 0...(N − 1) [18]. We define our DFT as

sm =

N−1∑k=0

sk exp(−2πimk

N),

m = 0 ... N − 1.

(A1)

Here sk are defined as complex numbers, but the timeseries of the digitized input signal is always real.

The DFT assumes that the input is a finite data setwith an integer number of periods, where the two end-points of the time waveform are interpreted as thoughthey were connected together. In practice, the measuredsignal is rarely an integer number of periods. The finitetime signal then looks like a truncated waveform with dif-ferent characteristics from the original continuous-timesignal. This effect appears as sharp transition changesin the measured signal, which show up in the output asnon-physical high-frequency components aliased between0 and the Nyquist frequency. This is known as spectralleakage.

Spectral leakage may be minimized through window-ing, where the amplitude of the discontinuities at theboundaries of each finite sequence are manually reduced.A window function to be used with a DFT of length Nis defined by a vector of real coefficients wk. It is usedby multiplying the time series sk with the window beforeperforming the DFT. In our analysis, we use a Hanningwindow, defined as

wk =1

2[1− cos(2π · k

N)],

k = 0 ... N − 1.

(A2)

Finally, we may normalize our DFT results to generateour power spectrum (PS)

PSrms(fm) =2 · |sm|2

C21,

m = 0 ... N − 1,

(A3)

where fm = m · fres and C1 is the normalization con-stant

C1 =

N−1∑k=0

wk. (A4)

We may calculate our power spectral density (PSD,denoted Px for the xth signal) by dividing the PS by theeffective noise bandwidth

Px(fm) =2 · |sm|2

fs · C2,

m = 0... N − 1,

(A5)

where C2 is the normalization constant

C2 =

N−1∑k=0

wk2. (A6)

For an amplitude spectral density (ASD), we simplytake the square root of the corresponding PSD. How-ever, this simple procedure described above tends to benoisy because the standard deviation of the spectrum es-timate in a frequency bin is equal to the estimate. Thuswe take the average of M estimates and hence reducethe standard deviation of the averaged result by a fac-tor of 1/

√M . This method of averaging several spectra

while using a window function is also known as “Welch’smethod”. In this report, all PSDs are calculated in thisway.

If a time series is split into several non-overlappingsegments of length N to be processed using a windowfunction, we end up throwing out a large portion of thedata stream. Some data are saved by letting the seg-ments overlap. For the Hanning window, the recom-mended overlap is 50%, which is used throughout thisreport.

We must calculate the cross-power spectral density be-tween the two data channels to find the coherence be-tween them. For discrete signals sx and sy, we calculatethe DFTs sx and sy

∗, where the latter is the complexconjugate of the DFT. Thus,

Pxy(f) =2 · sxsy∗

fs · C2, (A7)

where the coherence is then

coh(f) =|Pxy(f)|2

Px(f) Py(f). (A8)

11

Appendix B: Magnetometer specifications

FIG. 13. The sensitivity of the magnetometer MFS-06, withthe chopper on (CHON) and off (CHOFF). The sensitivityfunction for the magnetometer was measured by applying aknown voltage and measuring the output voltage. This con-version was then applied to subsequent measurements for ac-curate ASD comparisons.

FIG. 14. The MFS-06 nominal noise floor with Schumannresonances from Villa Cristina overlaid to demonstrate rela-tive amplitude. It is well below all Schumann measurementsand was thus excluded in plots in this report where the mag-netometer was used.

Appendix C: Supplementary diagrams

FIG. 15. The location of Villa Cristina (43.539625,10.410031) in the Livorno hills. This site was specificallychosen due to its spatial separation from many sources ofanthropogenic noise. Created using Google maps.

FIG. 16. The location of the LHO magnetometers “H1:PEM-CS” used in our coherence analysis. The diagram wastaken from the LIGO web-based data viewer, ligoDV-web(https://ldvw.ligo.caltech.edu/).

12

Appendix D: Supplementary figures

FIG. 17. An ASD plot for Site B during a “sideband on” timeis shown. No matter how much data are cut, the resonancesare not resolvable.

Appendix E: Magnetic surveys

Loc./Or. ASD(50 Hz) ASD(10-30 Hz)

CEB 1 - EW 4082 6.247NEB 1 - EW 4026 1.465NEB 2 - EW 2099 1.027NEB 3 - EW 1740 0.915NEB 4 - EW 2047 1.884NEB 5 - EW 2000 1.153WEB 1 - EW 1391 1.345WEB 2 - EW 535.1 2.256WEB 3 - NS 918.4 1.68WEB 4 - EW 2763 1.169

A - NS 4897 0.665B - EW 2621 0.626C - NS 10447 1.177D - EW 6362 0.706E - EW 4535 0.808

TABLE I. A table with ASD values for different magneticsurvey locations with the magnetometer orientations (North-South, NS; East-West, EW). The units for the ASD values

are always pT/√Hz.