Calibration of a superconducting gravimeter with an absolute atom gravimeter
S. Merlet, P. Gillot, B. Cheng, R. Karcher, A. Imanaliev, L. Timmen, F. Pereira Dos Santos
CCalibration of a superconducting gravimeter withan absolute atom gravimeter
P. Gillot , B. Cheng ‡ , R. Karcher , A. Imanaliev §(cid:107) , L.Timmen ¶ , S. Merlet and F. Pereira Dos Santos ∗ LNE-SYRTE, Observatoire de Paris - Universit´e PSL, CNRS, Sorbonne Universit´e,61 avenue de l’Observatoire, 75014 Paris, France. Institute of Geodesy, Leibniz University Hannover, Scheinderberg 50, 30167Hannover, GermanyE-mail: [email protected]
Abstract.
We present in this article a 27 days long common view measurement of anabsolute cold atom gravimeter (CAG) and a relative iGrav superconducting gravimeter,which we use to calibrate the iGrav scale factor. We investigate the impact of theduration of the measurement on the uncertainty in the determination of the correlationfactor and show that it is limited to about 3 ‰ by the coloured noise of our cold atomgravimeter. A 3 days long measurement session with an additional FG5X absolutegravimeter allows us to directly compare the calibration results obtained with twodifferent absolute meters. Based on our analysis, we expect that with an improvementof its long term stability, the CAG will allow to calibrate the iGrav scale factor to theper mille level after only one day of concurrent measurements.
1. Introduction
Because of their high sensitivity, low drift and reasonable maintenance costs,superconducting gravimeters (SG) [1] are today the key instruments for the continuousmonitoring of gravity variations. Nevertheless, being relative meters, they need to becalibrated and their drift to be determined, the methods for this being summarisedfor instance in Ref. [2] and [3]. For their calibration, one can either use long tidalmeasurements [4], induce controlled gravity changes by displacing masses, or the SGitself [5, 6, 7, 8], perform co-located measurements with relative spring gravimeters [9, 10]or with absolute gravimeters (AG) [11, 12, 13], this last method being today the most ‡ Present address: Institute of Optics, The Zhejiang Provincial Key Laboratory of Quantum PrecisionMeasurement, College of Science, Zhejiang University of Technology, Hangzhou 310023, China § Present address: Laboratoire National de M´etrologie et Essais (LNE), 29 avenue Roger Hennequin,78197 Trappes cedex, France (cid:107)
Orcid: 0000-0002-8397-6927 ¶ Orcid: 0000-0003-2334-5282 + Orcid: 0000-0002-4746-2400 ∗ Orcid: 0000-0003-0659-5028 a r X i v : . [ phy s i c s . i n s - d e t ] J u l alibration of a superconducting gravimeter with an absolute atom gravimeter ‰ is obtained in less than a week [13, 2, 15, 16, 3, 17].For applications in geophysics [18], the accurate determination of the SG scalefactor is important, and a long term stability of the gravity measurements is desirable.This motivates the regular intercomparison of SGs with AGs in order to track SGsdrifts, potential changes in their scale factor, as well as offsets related to maintenanceoperations, or uncontrolled systematic effects.Atom gravimeters based on atom interferometry [19] offer new measurementcapabilities, by combining high sensitivities [20, 21, 22, 23] and accuracies at the bestlevel of a few tens of nm.s − [20, 24, 25], to the possibility to perform continuousmeasurements [20, 26, 27, 28, 29]. Being absolute meters, their scale factor is accuratelydetermined and do not need calibration. This is essential for applications in the frameof metrology, such as for the determination of the Planck constant with a Kibblebalance [30] and new realisation of the kilogram in the revised International Systemof Units [31]. The study of their long term stability, as for any type of AGs, requiresthe precise knowledge of temporal fluctuations of gravity, in order to be able to separatethem from fluctuations of systematic effects in the sensors. Even the best tide modelsare not enough for that purpose, as they do not account for all processes that do changethe local value of gravity. This prevented us for a long time to assess the long termstability of our Cold Atom Gravimeter (CAG), such as in Ref. [26] where one couldnot assess whether the long term stability of a 12 days continuous gravity measurementwas limited by the instrument or by the tidal model [26]. In 2013, the comparison ofour measurements with an improved tidal model allowed to demonstrate a stability of2 nm.s − at 1 000 s measurement time [22]. Nevertheless, direct comparisons betweendifferent sensors, and in particular with SGs, is preferable. In 2015, the GAIN gravimeterof Humboldt-Universit¨at zu Berlin reached a remarkable stability of 0.5 nm.s − whencompared to an Observatory SG (OSG) [23].Since the very beginning of 2013, we operate a superconducting gravimeter in ourgravity laboratory at LNE [32], were the CAG operates since the end of the CIPM KeyComparison CCM.G-K1 during ICAG’09 [33], except when taken out to participate tocomparisons in other laboratories [34, 35] or to demonstrate the capabilities of atominterferometers [36] in the LSBB underground facility for the MIGA project [37]. Wepresent in this paper a study of the calibration of the iGrav-005 [38] with the CAG,exploiting a one month-long common view g measurement campaign, and discuss theuncertainty of this process. alibration of a superconducting gravimeter with an absolute atom gravimeter Figure 1.
Picture of the LNE gravimetry laboratory. A 6 m ×
2. Continuous common view gravity measurement with atom andsuperconducting gravimeters
The LNE gravimetry laboratory is equipped with a pillar of 33 square meters, largeenough to accommodate several AGs at a time for intercomparisons, onto which wehave installed in 2013 an iGrav SG ( alibration of a superconducting gravimeter with an absolute atom gravimeter th to May the 4 th of 2015.Gaps in the data correspond to the removal of measurements perturbed either by anearthquake, that drastically increase the noise, or by a failure of the CAG, due to lasersout of lock. Figure 2.
Continuous gravity signals as measured by the CAG (in black) and theiGrav005 (in blue) from April the 7 th to May the 4 th of 2015. Data are averagedover the same duration of 177 s for both instruments. The difference between the twoinstrument measurements after the calibration of the iGrav scale factor is representedin grey on the bottom graph. A shorter sample spanning over a week-end is highlightedin black.
3. Instrumental delays
As for tidal analysis [39], an accurate timing of the data is required when comparing thegravity variations of the two instruments to calibrate the iGrav output signal [9, 40]. Theeffect of a lack of synchronisation on the calibration of the SG depends on the amplitudeof gravity variations and the duration of the common view measurement. For a 11 dayssession, which allows to observe tides with large amplitudes, a 1 s difference between thetwo instrument timings leads to an effect of the order of 0.5 ‰ on the CF determination.For a one day session, the effect varies from 0.2 ‰ to 1 ‰ depending on the magnitudeof the tides. The time stamping of CAG data is performed by the clock of its controlcomputer, which is locked on UTC via the NTP protocol, whereas the iGrav SG usesGPS time via a GPS receiver. In addition, one should also take into account delays dueto the time response of the sensors to gravity changes. While the CAG suffers negligibledelay when considering the time of the measurement at the middle of the interferometer,appreciable delays are present in the case of the SG, owing not only to their mechanicalresponse function but also on the use of additional filters. A precise determinationof the response function can in principle be performed via self calibration, but thisfunctionality is not available in our iGrav. While the theoretical transfer function given alibration of a superconducting gravimeter with an absolute atom gravimeter − with a delayof 11(1) s, in agreement with the theoretical estimation. Finally, this delay could alsobe extracted from the analysis of the time correlation between the signals of the twometers of figure 2, which allows for the determination of a delay of 10.3(3) s. Note thatthese last two methods determine the overall SG delay, including additional delays bythe data acquisition system [42].
4. Direct calibration of the superconducting gravimeter
Methods to calibrate SGs have already been investigated in detail in [13] and laterimproved in [40]. The relationship between the two signals (expressed in nm.s − forthe CAG and in Volts for the iGrav) can simply be determined via a simple linearregression, such as illustrated in figure 3 where we have considered two sets of data ofdifferent lengths. The first set displayed in grey corresponds to the whole measurementperiod of 27 days, while the second in black corresponds to a more quiet period of1.7 days, starting before midnight on a Friday and ending after midday on the nextSunday, the noise during week-ends being reduced by the absence of on site humanactivity. We obtain two calibration factors (CF) of respectively -898.25(20) nm.s − /Vand -899.00(50) nm.s − /V, in agreement within their uncertainties, which are givenhere by the errors of the fits, ie the standard errors of the regression slopes. The firstcalibration factor was then used to convert the SG voltage samples into gravity data,and the difference between the calibrated SG and CAG measurements was calculated.This difference is displayed in the bottom part of figure 2. A statistical analysis ofthis difference over the two sets of data was then performed by calculating their Allanstandard deviations, which are displayed on figure 4. For the selected 1.7 day period,the Allan standard deviation averages down to 0.5 - 0.6 nm.s − , with a τ − / slopecharacteristic of white noise, as already observed in [23]. As for the Allan standarddeviation of the 27 days common view measurement, it also decreases with the sameslope down to the 2 - 3 nm.s − level for about 3 000 - 4 000 s, but reaches some kind ofplateau for larger averaging times.
5. Segmented duration analysis
To investigate through a statistical analysis the uncertainty associated with thecalibration factor determination, several independent such determinations would berequired. We thus take advantage of the time-length of the measurement to carryout a segmented analysis and calculate an iGrav calibration factor CF for each day ofmeasurement. The 27 resulting one-day calibration factors CF are displayed in figure 5, alibration of a superconducting gravimeter with an absolute atom gravimeter - 1 . 5 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0- 1 0 0 0- 5 0 005 0 01 0 0 01 5 0 0 CAG (nm.s-2) i G r a v o u t p u t ( V )- 8 9 8 . 2 5 ( 2 0 ) n m . s - 2 / V- 8 9 9 . 0 0 ( 5 0 ) n m . s - 2 / V
Figure 3.
Calibration of the iGrav output signal for the whole 27 d measurementperiod (in grey) and for a selected shorter 1.7 d-long period (in black). (cid:1) g ( (cid:2) ) (nm.s-2) (cid:1) ( s ) Figure 4.
Allan Standard deviations of the difference between the iGrav and the CAGgravity signals, for the whole 27 d measurement period (in grey) and for a selectedshorter 1.7 day-long period (in black). with uncertainties given by the errors of the linear fits. Remarkably, the standarddeviation of the one-day CFs, which amounts to 2.77 nm.s − /V, is three times largerthan the mean value of the errors of the fits of 1.05 nm.s − /V. This tends to indicate thatthe errors of the fits underestimate the uncertainty in the CF determinations. The peakto peak variation of the one-day CFs is 10 nm.s − /V, twice smaller than in Ref. [2],where a similar analysis was carried out between a SG and a FG5 AG for a similar27 days long measurement.We stress here that the observed variations are not correlated with changes in theamplitude of the SG noise, which can in practice impact the CF as discussed in Ref. [40]. alibration of a superconducting gravimeter with an absolute atom gravimeter CF (nm.s-2/V)
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Figure 5. iGrav one-day calibration factors. The error bars are the errors of theindividual fits.
To better understand this behaviour and compare the results we obtain withsimulated data, we generated a synthetic AG signal, obtained out of the iGrav outputsignal converted into a gravity signal with the first CF obtained in figure 3, to whicha white noise of the same amplitude as the short term noise of the CAG was added.We then used this synthetic signal to calibrate the iGrav, repeating our segmentedanalysis, but for different measurement durations, spanning from 7 h to 200 h. As thetotal common view measurement is 27 d long, we obtained several distributions withnumbers of samples ranging from 91 to 3 respectively. We report on figure 6 the standarddeviations of these distributions as grey diamonds, as well as the corresponding meansof the errors of the fits as open blue dots, and we find a fair agreement between them.This shows that the mean errors of the fits are good estimates of the uncertainties in theCF determinations when the differential noise between the sensors is white. By contrast,the same analysis performed with the real CAG signal shows to a different behaviour.Indeed, the standard deviations, which are displayed as blue dots on figure 6, clearlyfeature a plateau, showing that measurements longer than about a day do not help toreduce the uncertainty on the CF determination. On the other hand, for durations ofup to a day or so, the errors of the CF fits could be taken as reasonable estimators of theuncertainty of the CFs, despite being about twice overoptimistic. With this analysis,we understand that the behaviours we observe with the real data are related to colourednoise. Though at this stage, the question remains in principle open whether the colourednoise arises from the CAG or from the iGrav, we attribute it to the CAG. Indeed, theAllan standard deviations of the residuals of the gravity data corrected from tides, with alibration of a superconducting gravimeter with an absolute atom gravimeter
8a tidal model obtained with a spring gravimeter [32], and from atmospheric effects,show for the CAG a behaviour similar to the grey curve of figure 4 whereas, for theiGrav005, it is about three times lower for a 10 000 s averaging time.
Statistics on CF (nm.s-2/V)
S a m p l e s i z e a n a l y s e ( d )
Figure 6.
Statistic analysis of iGrav CF determinations for different durations ofmeasurement segmentation. Standard deviations of the distribution of the CFs arerepresented in blue dots: full dots for the real CAG data, and open dots for thesynthetic AG signal. Grey diamonds display the means of the errors of the CF fits forthe synthetic signal.
6. Comparison of calibrations with different type absolute gravimeters
In Ref. [43], the authors calibrated a SG with four FG5 AGs during a single commonview measurement session. The different CFs they obtained agreed with each other,demonstrating the robustness of using any FG5 AG for such calibration. Yet, onecould not exclude a possible unaccounted-for bias because the experiment was limitedto only FG5 AGs, which motivates carrying similar studies with AGs relying on differenttechnologies.To do so, we took the opportunity of a measurement campaign organised in theframe of the ITOC project [44] to welcome again in the LNE gravimetry laboratory thefree fall corner cube gravimeter FG5-220 of Institute of Geodesy of Leibniz Universityof Hannover, in its improved version [45], namely the FG5X-220. As a remark,measurements at 2 host stations on the pillar were performed with the FG5X-220,in good agreement with the results of a previous measurement campaign performed in2009 [24] with the FG5-220. But, of particular interest for the present study, we tookadvantage of a week-end to perform a common view continuous measurement betweenthe three instruments (iGrav, CAG, FG5X) to repeat the analysis presented above insection 5. Note that for these measurements, the FG5X-220 performed one free fall alibration of a superconducting gravimeter with an absolute atom gravimeter ‰ level with an FG5, according to [13].Figure 7 presents the results of the statistical analysis of the iGrav CFs determinedfor segment sizes varying from 4 h to 62.5 h, corresponding respectively to number of15 to 1 samples. Surprisingly, the analysis does not lead to the same calibration factors.They differ by about 5 to 6 nm.s − /V. Note that we verified that the FG5X values werenot affected by aliasing effects due to the 30 s measurement period [46]. The means ofthe errors of the CF fit, which we take here as fair estimates of the uncertainties in theCF determinations, is three times better for the CAG determination due to its bettershort term sensitivity [22]. Nevertheless, in principle these uncertainty associated tothe FG5X could be reduced by increasing the repetition rate. Expressed as in manypapers on the determination of the CF of relative gravimeters [13, 2, 15, 16, 3, 10, 47], inless than a day, the CAG, respectively the FG5X-220, allows there for a determinationwith errors from the fits of 0.7 ‰ , respectively 2.3 ‰ which, for a free fall corner cubegravimeter, is consistent with the results of [13]. As shown by previous measurements,the CAG potentially allows for a precision on the CF of the iGrav around 1 ‰ after onlya day of measurement. Mean CF error (nm.s-2/V)
S a m p l e s i z e a n a l y s e d ( d )
CF (nm.s-2/V)
Figure 7.
Mean calibration factors of the iGrav005, and mean CF fit errors, obtainedwith the CAG (full dots and diamonds) and the FG5X-220 (opened squares anddiamonds) for different durations of segmentation of the measurements.
As the common view measurement with the FG5X was not exactly 3 day long, wethen split the measurement data into 3 slightly overlapping periods of 1 day length inorder to perform three 1-day analysis such as the one presented in figure 5. The overlapbetween two consecutive segments is close to 10%. Figure 8 displays the results of these1-day analysis for both instruments. As in the analysis of section 5, the CFs we obtain alibration of a superconducting gravimeter with an absolute atom gravimeter
CF (nm.s-2/V)
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Figure 8. iGrav calibration factors determined during three consecutive one-daycommon view measurements obtained with the CAG (blue dots) and FG5X-220(opened squares). The error bars are the individual fit errors.
7. Repeated calibrations over 7 years
To conclude with the iGrav005 CF, the figure 9 displays as black dots the results ofrepeated calibration campaigns realized with the CAG since 2013. The dispersion ofthe CFs is there comparable to the one of the CFs obtained in this paper for one-day calibrations (displayed on the same figure as blue dots), and comparable to the15 nm.s − /V peak-to-peak fluctuations obtained again with FG5 in other works (over 5months in [48] and over 10 years in [3]).Based on these results, we finally evaluate the mean calibration factor of theiGrav005 to be (897 . ± .
7) nm.s − /V.
8. Conclusion
We have performed the calibration of the relative SG iGrav005, using a 27 days longcommon view measurement with the SYRTE atomic absolute gravimeter CAG. Thisallowed to evaluate the long term stability of the residuals obtained by taking thedifference between gravity data of the CAG and the calibrated SG. The Allan standarddeviation of these residuals can reach 0.5 nm.s − after averaging over two days for a alibration of a superconducting gravimeter with an absolute atom gravimeter CF (nm.s-2/V)
M J D
Figure 9.
Calibration factors of the iGrav005 obtained with the CAG since 2013 (fulldots). The blue dots display the 1-day calibration CFs presented in figures 5 and 8.The opened squares display the CF obtained with the FG5X-220 in figure 8. selected quiet period, but tend to flicker at a level of about 2 - 3 nm.s − when averagingover the whole period. We attribute this behaviour to instabilities of CAG systematiceffects rather than instabilities of the SG. By carrying out a detailed statistical analysisand a comparison with simulated data, we show how this instability imposes a limit onthe uncertainty of the determination of the SG calibration factor, of about 3 ‰ . Thiscould be improved well below the ‰ level with an improved long term stability of theCAG, as good as 0.5 ‰ after 2 days as demonstrated with a selected quiet set of data.A comparison with the calibrations realized with a corner cube FG5X gravimeter hasalso been performed, which shows the better performance of the CAG. Moreover, theiGrav calibration factors determined by these two types of sensors, which differ by morethan 5 nm.s − /V, seem to exhibit correlated fluctuations, which could be related toinstabilities of the iGrav005. A longer common view measurement session would beuseful to confirm this hypothesis, which we plan to carry on in the future, after anupgrade of the CAG to improve its long term stability. Acknowledgments
This research is carried on within the kNOW and ITOC projects, which acknowledgesthe financial support of the EMRP. The EMRP was jointly funded by the EuropeanMetrology Research Programme (EMRP) participating countries within the EuropeanAssociation of National Metrology Institutes (EURAMET) and the European Union.B.C. thanks the Labex First-TF for financial support. This work has been supportedby the Paris ˆIle-de-France R´egion in the framework of DIM SIRTEQ. alibration of a superconducting gravimeter with an absolute atom gravimeter References [1] J.M. Goodkind, The superconducting gravimeter,
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