Preliminary experimental results of determining the geopotential difference between two synchronized portable hydrogen clocks at different locations
Wen-Bin Shen, Kuangchao Wu, Xiao Sun, Chenghui Cai, Ziyu Shen
PPreliminary experimental results of determining thegeopotential difference between two synchronized portablehydrogen clocks at different locations
Wen-Bin Shen , , ∗ Kuangchao Wu Xiao Sun Chenghui Cai Ziyu Shen Department of Geophysics, School of Geodesy and Geomatics/Key Laboratory of Geospace Environmentand Geodesy of Ministry of Education, Wuhan University, Wuhan 430079, China State Key Laboratory of Information Engineering in Surveying, Mapping and Remote Sensing, WuhanUniversity, Wuhan 430079, China School of Resource and Environment, Hubei University of Science and Technology, Xianning, Hubei,China ∗ Corresponding author: [email protected]
Date: August 17, 2020
Abstract
According to general relativity theory (GRT), by comparing time elapses between two preciseclocks located at two different stations, the gravity potential (geopotential) difference betweenthem can be determined, due to the fact that precise clocks at positions with different geopoten-tials run at different rates. Here, we provide preliminary experimental results of the geopotentialdetermination based on time elapse comparisons between two remote atomic clocks located atBeijing and Wuhan, respectively. After synchronizing two hydrogen atomic clocks at Beijing 203Institute Laboratory (BIL) for 20 days as zero-baseline calibration, namely synchronization, wetransport one clock to Luojiashan Time-Frequency Station (LTS), Wuhan, without stopping itsrunning. Continuous comparisons between the two remote clocks were conducted for 65 daysbased on the Common View Satellite Time Transfer (CVSTT) technique. The ensemble empir-ical mode decomposition (EEMD) technique is applied to removing the uninteresting periodicsignals contaminated in the original CVSTT observations to obtain the residual clocks-offsetsseries, from which the time elapse between the two remote clocks was determined. Based onthe accumulated time elapse between these two clocks the geopotential difference betweenthese two stations was determined. Given the orthometric height (OH) of BIL, the OH of theLTS was determined based on the determined geopotential difference. Comparisons show thatthe OH of the LTS determined by time elapse comparisons deviates from that determined byEarth gravity model EGM2008 by about 98 m. The results are consistent with the frequencystabilities of the hydrogen atomic clocks (at the level of − /day) applied in our experiments. a r X i v : . [ phy s i c s . g e o - ph ] A ug ere the EEMD technique was first introduced and applied in the subjects related to this study,especially for the purpose of removing the periodic signals from the original CVSTT observa-tions to obtain the residual clock offsets signals, from which one can more effectively extractthe geopotential-related signals. In addition, we used 85-days original observations to deter-mine the geopotential difference between two remote stations based on the CVSTT technique.Using more precise atomic or optical clocks, the CVSTT method for geopotential determinationcould be applied effectively and extensively in geodesy in the future. Keywords: relativistic geodesy, atomic clock, CVSTT technique, EEMD technique
Precise determination of the Earth’s geopotential field and orthometric height (OH) is a main taskin geodesy (Mazurova et al., 2017, 2012). With rapid development of time and frequency science, high-precision atomic clock manufacturing technology (Bloom et al., 2014; Hinkley et al., 2013; McGrew et al.,2018) provides an alternative way to precisely determine the geopotentials and OHs, which has been exten-sively discussed in recent years (Bondarescu et al., 2012; Lion et al., 2017; Shen et al., 2016, 2018), andopening a new era of time-frequency geoscience (Kopeikin et al., 2018; Shen et al., 2019).General relativity theory (GRT) states that a precise clock runs at different rates at different positionswith different geopotentials (Einstein, 1915). Based on GRT, Bjerhammar (1985) proposed an idea to de-termine the geopotentials via clock transportation. Later, Shen et al. (1993) put forward an approach todetermine geopotentials via gravity frequency shift (GFS) that is also based on GRT. Consequently, thegeopotential difference between two arbitrary points can be determined by comparing the running rates oftwo precise atomic clocks (Bjerhammar, 1985; Mai and Müller, 2013) or by comparing their frequencies(Shen, 1998; Shen et al., 1993, 2011, 2009b). In order to determine geopotentials with an accuracy of 0.1m s − (equivalent to 1 cm in OH), the atomic clocks with frequency stabilities of × − are required.High performance clocks and time-frequency transfer techniques have been intensively developingover the past 60 years (Akatsuka et al., 2008; Bondarescu et al., 2015; Mehlstäubler et al., 2018). Recently,optical-atomic clocks (OACs) with stabilities around − level have been successively generated (Camp-bell et al., 2017; McGrew et al., 2018; Nicholson et al., 2015), and in the near future mobile high-precisionsatellite-borne optical clocks will be in practical usage (Poli et al., 2014; Riehle, 2017; Singh et al., 2015).This provides a good opportunity to precisely determine the geopotentials via precise clocks. Therefore,high-precision clocks enable 1 cm level geopotential determination and the realization of world height sys-tem (WHS) unification (Koller et al., 2016; Müller et al., 2018; Shen et al., 2016).As mentioned before, there are two kinds of methods for determining the geopotentials via clocks.One method is to compare the time elapses of two clocks located at two stations using various techniques(Bjerhammar, 1985; Kopeikin et al., 2016; Shen and Shen, 2015). The other method is to compare frequency2ifference of the two clocks (Deschenes et al., 2015; Pavlis and Weiss, 2017; Shen et al., 2018, 2017). Itis true that there exist some successful transportation experiments for determining geopotential via high-precise fiber links (Grotti et al., 2018; Lisdat et al., 2016; McGrew et al., 2018; Takano et al., 2016), andthe corresponding results are quite successful. For example, Grotti et al. (2018) provided the transportationexperiments by using optical atomic clocks for their unprecedented stability, and the discrepancy betweenrelativistic method and conventional method variates 2 ∼ m /s in geopotential units. However, there areseldom transportation experiments using the satellite signals for comparing clock offsets in the relativisticgeodesy. In fact, satellite time and frequency signal transfer approach is most prospective in the future due tothe fact that it is not constrained by geographic conditions, for instance connecting two continents separatedby oceans. Kopeikin et al. (2016) discussed and provided experimental results of transportation experimentsfor geopotential difference determination via Common-View Satellite Time Transfer (CVSTT) technique,with the help of two hydrogen atomic clocks. The results are consistent with the clocks’ stabilities in theirexperiments with a discrepancy of hundreds of meters. However, in their experiments, the time comparisonperiod is quite short. For instance, the geopotential difference measurement lasted about 63900 s (17.7 hr),and the zero-baseline calibration lasted only 23790 s (6.6 hr).In this study, we focus on the clocks’ time elapse comparisons via the CVSTT technique, and thedetermination of geopotential difference based on clock transportation experiments. Here, the transportationexperiments last for a quite long time, with the zero-baseline measurement lasting for 20 days, and thegeopotential difference measurement lasting for 65 days with plenty of data sets. In addition, we first usethe EEMD technique to remove the uninterested periodic signals from the original CVSTT observations toobtain the residual clock offsets signals, and then extract the geopotential-related signals from the residualclock offsets signals. In our knowledge, this is a first application in CVSTT data processing for the purposeof determining the geopotential difference in geodetic community. In this study, the stability of H-masersused in the experiments is at − /day level, which means that the determination of geopotential differenceis limited to tens of meters in equivalent height. We admit that the performance of the H-masers cannotbe compared to the optical atomic clocks with unprecedented stability, and it cannot obtain perfect resultslike the optical atomic clocks via optical fiber. However, the H-masers have their special advantages, forexample, they can keep operation with a relative stable stability for a long period, which could be difficultfor the optical atomic clocks at the current stage. In the future, the optical atomic clock might be introducedin the similar experiments when the experimental conditions are mature.This paper is organized as follows. In Section 2, we briefly describe how to determine geopotential aswell as OH using two remote clocks. In Section 3, we focus on describing how to achieve the comparisonsof time elapses between two remote clocks based on the CVSTT technique, and how to cancel or greatlyreduce various error sources. In Section 4, we introduce ensemble empirical mode decomposition (EEMD)technique, and verify its effectiveness for extracting the linear signals of interest from the original CVSTT3bservations based on simulation experiments. In Section 5, we describe our experimental setup, records,and data procession. We provide experimental results and relevant discussions in Section 6, and drawconclusions in Section 7. Based on GRT, considering the proper time durations ∆ t A and ∆ t B recorded by two precise clocks C A and C B located at two stations A and B , respectively (see Fig. 1). Accurate to /c , the following equationholds (Bjerhammar, 1985; Mehlstäubler et al., 2018; Shen et al., 2009a; Weinberg, 1972): Figure 1:
Schematic representation of the clock measurement. W is geopotential on the geoid (solid curve), ∆ t i , W i and H i denote proper time duration, geopotential, and orthometric height (OH) at position i ( i = A, B ), respectively.The OH H i (dashed red curve) is the distance between i (cid:48) and i along the curved plumb line. A (cid:48) ( B (cid:48) ) is the intersectionpoint between the plumb line passing through A(B) and the geoid, B (cid:48)(cid:48) is the intersection point between the plumb line B (cid:48) B and the equi-geopotential surface W A . ∆ t A / ∆ t ∆ t B / ∆ t = 1 − c − W A + O ( c − )1 − c − W B + O ( c − ) (1)where ∆ t denotes a standardized time duration (measured by a standard clock at infinity or at the center ofthe Earth); ∆ t i / ∆ t denotes the elapsed time of clock C i ( i = A, B ); W i denotes geopotential at station i ; c = 299792458 m/s is the speed of light in vacuum.Here, we point out that, the deformation of the Earth surface caused by the tides are neglected, due tothe fact that the tidal influences are at tens of centimeters level. Considering the stability of the hydrogenatomic clocks used in the experiments, say at × − /day level, which is equivalent to tens of metersin OH, the effect caused by the tidal influences is much smaller (two orders of magnitude lower). In thefuture study, however, when the clock’s stability reaches − level, the tidal effects should be consideredcarefully.From formula (1), the clock-comparison-determined geopotential difference between stations A and B,4 W ( T ) AB can be expressed as follows (accurate to /c ): ∆ W ( T ) AB ≡ W ( T ) B − W ( T ) A = c (1 − ∆ t B ∆ t A ) (2)The OH can be determined based on the following expression (Heiskanen and Moritz, 1967; Jekeli,2000; Molodenskii, 1962): H i = W − W i ¯ g i (3)where W is the geopotential on the geoid, with W = ± s − (SÃ ˛anchez and Sideris,2017; SÃ ˛anchez et al., 2016), W i is the geopotential at location i , ¯ g i is the ’mean gravity value’ at a particularpoint on the segment of the plumb line between point i and i (cid:48) .Suppose the geopotential W A at station A is a priori given, by combining formulas (2) and (3), theclock-comparison-determined OH at staion B, H ( T ) B can be determined as follows: H ( T ) B = W − ( W A + ∆ W ( T ) AB )¯ g B = W − W A ¯ g B + c ¯ g B ( ∆ t B ∆ t A − (4)Finally, by combining formulas (3) ∼ (4) we can compare the discrepancy between clock-comparison-determined results ( H ( T ) B ) with the corresponding results determined by conventional approach ( H B ): D = H ( T ) B − H B = (cid:18) W − W A ¯ g B + c ¯ g B ( ∆ t B ∆ t A − (cid:19) − W − W B ¯ g B = W B − W A ¯ g B + c ¯ g B ( ∆ t B ∆ t A − (5)where W B − W A can be determined by levelling and gravimetry or directly determined by a given Earthgravity model. In this study we use EGM2008 to obtain the geopotentials W A and W B .The determination of ∆ W ( T ) AB as well as H ( T ) B requires a precise measurement of time comparison.Hence, one practical solution is to set clock C A as the standard, after a standard time elapses ∆ T A = ∆ T ,the clock C B elapses ∆ T B = ∆ T (cid:48) , correspondingly. Therefore, the time elapse difference between clocks C A and C B , α can be finally determined as follows (Shen and Shen, 2015): α = ∆ t B − ∆ t A ∆ t A ≡ ∆ T B − ∆ T A ∆ T A = ∆ T (cid:48) − ∆ T ∆ T (6)To realize a precise time comparison between two remote clocks, we need not only atomic clocks withhigh stability for maintaining the standard time elapse, but also a reliable time transfer technique that canprecisely measure the time elapses. 5 Common-View Satellite Time Transfer Technique
As early as in 1980, it was proposed that the CVSTT technique could be used for comparing clockoffsets between two remote clocks (Allan et al., 1985; Allan and Weiss, 1980). This technique was adoptedby the Bureau International des Poids et Mesures (BIPM) as one of the main methods for transferring in-ternational atomic time (TAI) signals (Allan and Thomas, 1994; Imae et al., 2004). Using this method,the uncertainty of comparing remote time may achieve several nanoseconds (Lewandowski et al., 1999;Lewandowski and Thomas, 1991; Ray and Senior, 2003; Rose et al., 2016; Sun et al., 2010; Yang et al.,2012). The main advantages of CVSTT technique lie in that the satellite clock errors are cancelled, and var-ious other errors, especially ionospheric and tropospheric errors, are largely reduced due to the simultaneoustwo-way observations (Allan and Weiss, 1980; Defraigne and Baire, 2011; Yang et al., 2012).In this study, we use the CVSTT technique for comparing the clock offsets. The reasons are stated asfollows. Firstly, the CVSTT technique is a stable and accurate method for comparing clock offsets betweentwo remote clocks, and this technique has been adopted by the Bureau International des Poids et Mesures(BIPM) as one of the main methods for transferring international atomic time (TAI) signals. Therefore, theresults of CVSTT technique are reliable and stable. Secondly, using CVSTT technique is relative cheap andconvenient in practice. Though the experimental results using optical atomic clocks via fiber links are quiteaccurate and stable indeed, the fiber-link is limited to a relative short distance and fixed stations with fiberlinks, and for a long distance even intercontinental comparison, the fiber-link is extremely challengeableand costly. In addition, the Two-way satellite time and frequency transfer (TWSTFT) technique is moreaccurate and stable than CWSTFT technique, but it is very expensive in economy, which could be applied inthe future when the condition is available. Lastly, though the carrier phase measurements in GNSS are moreprecise than the corresponding precise code measurements in time transfer, the absolute clock offsets cannotbe determined from carrier phase measurements due to the ambiguities issues (Defraigne and Bruyninx,2007). Therefore, the code measurements are necessary for determining the absolute clock offsets betweenremote time and frequency standards for a long time. Hence, at present experiment condition, the CVSTTtechnique and precise code measurements are adopted here.To better understand the text, here we briefly introduce the principle of the CVSTT technique. Supposethere are two ground stations ( A , B ) and a common-view satellite S . The coordinates of the two groundstations are denoted as ( x i , y i , z i ) ( i = A, B ), and the satellite coordinates are denoted as ( x S , y S , z S ). Atan appointed time, the i -th staion’s clock and satellite’s clock emits 1 purse per second (1 PPS) signals,simultaneously. The 1 PPS signal of the i -th station (denoted as 1PPS-local signal) arrives to the GNSStime transfer receiver (GNSS-TTR) via cables, and opens the door of the time interval counter (TIC) forstarting timing. At the same time, the 1PPS signal of the satellite S (denoted as 1PPS-S signal) is received6 igure 2: The schematic diagram of the CVSTT technique. The hydrogen atomic clock at one station (A or B)generates 1 pulse per second (1PPS) signals (denoted as 1PPS-local) and 10 MHz sinusoidal signals. When the 1PPS-local signals arrive at time interval counter (TIC), the TIC starts timing. The satellite’s clock also generates 1PPSsignals (denoted as 1PPS-S), simultaneously. When 1PPS-S signals arrive at TIC, the TIC stops timing. The clockoffsets between the i -th station ( i = A or B) and satellite S can be determined by counting the number of sinusoidalwaves between 1PPS-local and 1PPS-S. After data exchange, the clock offsets between the two ground stations A andB can be finally determined by taking common-view satellite S as common reference. The TIC is built-in GNSS TimeTransfer Receiver (GNSS-TTR). by the GNSS antenna of the ground station. After signals demodulation the 1PPS-S signal arrives to theGNSS-TTR via cables. And the 1PPS-S signal is used for closing the TIC’s door for stopping timing. The i -th staion’s clock also generates the 10 MHz sinusoidal signals with extremely high accuracy. Therefore,for the i -th station and at a certain timepoint, by counting the number of the sinusoidal waves between thestarting timepoint (the time of 1PPS-local signals arrive at TIC) and the stopping timepoint (the time of1PPS-S signals arrive at TIC), the time interval between the two 1PPS signals can be determined as follows(Allan and Weiss, 1980; Lee et al., 1990; Lewandowski and Thomas, 1991): T i = t i − t s + ρ is /c + δρ is /c + t r i (7)where T i is time interval measurement of the two 1 PPS signals at station i ; t i and t s are the emission7imepoints of 1PPS signals of the atomic clocks from i -th station and satellite S , respectively; δρ is is theerror source (e.g. ionospheric delay, tropospheric delay, and Sagnac effect); t r i is the receiver delay whichis usually calibrated by the manufacturing company using simulated signals (Lewandowski and Thomas,1991); and ρ is is the geometric distance between i -th station and satellite S , expressed as: ρ is = (cid:113) ( x S − x i ) + ( y S − y i ) + ( z S − z i ) (8)By applying equation (7), the clock offsets between the i -th station and satellite S can be expressed asfollows: ∆ t si = t i − t s = T i − ρ is /c − δρ is /c − t r i (9)Finally, the clock offsets between two ground stations A and B can be determined by taking the common-view satellite’s clock as common reference: ∆ t AB = t B − t A = ( t S + ∆ t SB ) − ( t S + ∆ t SA )= ( T B − ρ BS c − δρ BS c − t r B ) − ( T A − ρ AS c − δρ AS c − t r A )= ( T B − T A ) − ( ρ BS − ρ AS ) c − ( δρ BS − δρ AS ) c − ( t r B − t r A ) (10)We note that the cable delays from station’s atomic clock to TTR are measured beforehand with thevalue of 50.2 ns, as well as the cable delays from GNSS antenna to TTR with the value of 204.5 ns. And thecable delays are input into the TTR as a parameter. In fact, the cable delays are not significant in our studydue to the fact that we compare the time elapse between two remote clocks. In this study, the transportation experiments is conducted at Beijing 203 Institute Laboratory (BIL) andLuojiashan Time-Frequency Station (LTS). In the sequel, we analyze various error sources.The accuracy of the satellite positions depends on the ephemeris. The broadcast ephemeris is adoptedin conventional CVSTT technique. Suppose the satellite position error is ( δx s , δy s , δz s ), when ignoringthe position error of ground stations, the relation between satellite position error and time transfer error inCVSTT technique can be determined by using the first-order difference (Imae et al., 2004; Li et al., 2008;Sun et al., 2010): ∆ τ AB = ( l AS − l BS ) δx s c + ( m AS − m BS ) δy s c + ( n AS − n BS ) δz s c (11)8here l is = x s − x i ρ is m is = y s − y i ρ is n is = z s − z i ρ is (12)Based on formulas (11) and (12) and applying the error propagation law, the following expression canbe obtained: m (∆ τ AB ) = ( l AS − l BS c ) m ( δx s ) + ( m AS − m BS c ) m ( δy s ) + ( n AS − n BS c ) m ( δz s ) (13)For instance, when accuracy of broadcast ephemeris in individual axes is 2 m (Gao, 2004; Liu et al.,2017), the accuracy of time transfer error caused by satellite position error did not exceed 0.16 ns in theexperiments. The influences of the ground stations’ positions errors on the time transfer in CVSTT techniquecan be evaluated, similarly. Here the ground stations’ coordinates are determined based on the precise pointpositioning (PPP) method in advance, with accuracy level of better than 3 cm (Dawidowicz and Krzan,2014), and the corresponding time transfer error caused by the coordinates errors will not exceed 0.15 ns(see Table 1). The ionospheric effect on GNSS signal means a signal delay which could achieve several tens ofnanoseconds, due to the electron content of the atmospheric layer. And the effect becomes more dramaticduring an ionospheric storm. The zenith group delays are about 1 ∼
30 m, 0 ∼ ∼ f and f )are often available. Therefore, an ionosphere-free observation P can be constructed so that the first-orderterm of ionospheric effect can be removed completely, due to the fact that the ionospheric effect on a signaldepends on its frequency. The P is constructed as follows (Peng and Liao, 2009; Petit and Arias, 2009): P = 1 f − f ( f P − f P ) (14)where P and P are the precise-code observations with frequencies of f = 1575 . MHz and f =1227 . MHz, respectively.However, formula (14) only removing the first-order ionospheric effect. There are second- and third-order ionospheric effects (also denoted as higher-order ionospheric effects). The residual higher-order iono-9pheric effects, I r , can be expressed as follows (Morton et al., 2009): I r = qf f ( f + f ) + tf f q = 2 . × (cid:90) N e B cosθ B dst = 2437 (cid:90) N e ds + 4 . × (cid:90) N e B (1 + cos θ B ) ds (15)where N e is the number of electrons in a unit volume( /m ), B is the magnitude of the plasma magneticfield(T), θ B is the angle between the wave propagation direction and the local magnetic field direction, and s is the integral path. More details can be seen in Morton et al. (2009) and Zhang et al. (2013).By using CVSTT technique, the residual errors of ionospheric effects between ground stations A andB, I r ( AB ) , can be determined as follows: I r ( AB ) = q A − q B f f ( f + f ) + t A − t B f f (16)By using the constructed ionosphere-free observation P , we remove the first-order ionospheric effect,which contributes to more than 99% of the entire ionospheric effects (Defraigne and Baire, 2010; Harmeg-nies et al., 2013; Huang and Defraigne, 2016), and the residual errors of higher-order terms are about 2 ∼ P and P are m and m , respectively. By applying the error propagation law, the code noise of ionosphere-free observation, m , can be expressed as follows: m = ( f f − f ) m + ( f f − f ) m (17)Here, suppose the code noise m ≈ m , and take the values of f = 1575 . MHz and f = 1227 . MHz into formula (17), the relations of m ≈ . m can be obtained. The accuracy of P code measure-ment is about 0.29m, which is equivalent to 0.97 ns in time. Therefore, the accuracy 2.89 ns of the P codemeasurement is determined, and the corresponding time transfer accuracy of the time interval measurementin CVSTT technique, T B − T A , will not exceed 4.09 ns. Combining expressions (16) and (17), after im-plementation of strategy of ionosphere-free combination, the residual ionospheric errors I r ( AB ) in CVSTTtechnique will not exceed 0.8 ns (see Table 1). The troposphere is the lower layer of atmosphere that extends from ground to base of the ionosphere.The signal transmission delay caused by troposphere is about 2 ∼
20 m from zenith to horizontal direction(Chen et al., 2012; Xu, 2007). The tropospheric delay depends on temperature, pressure, humidity, and aswell as location of the GPS antenna (Feng et al., 2006). The total tropospheric delay can be divided intohydrostatic part and wet part, which can be expressed as follows (Boehm and Schuh, 2004; Kouba, 2008): ∆ L = ∆ L zh · mf h ( E, a h , b h , c h ) + ∆ L zw · mf w ( E, a w , b w , c w ) (18)10here ∆ L is the total tropospheric delay, ∆ L zh and ∆ L zw are the hydrostatic part and wet part in zenithdelays, respectively. mf h and mf w are the corresponding mapping functions (MF), respectively. MF is afunction of E , a h ( w ) , b h ( w ) , and c h ( w ) , E is the elevation angle in radian. a h ( w ) , b h ( w ) , and c h ( w ) are MFcoefficients.Boehm and Schuh (2004) introduced a rigorous approach for utilizing the Numerical Weather Models(MWM) for MF determinations. The first and most significant MF coefficients, a h and a w , are fitted withthe NWM from the European Centre for Medium-Range Weather Forecasts (ECMWF). The coefficient b h = 0 . , and coefficient c h is expressed as follows: c h = c + (cid:20)(cid:18) cos (cid:18) doy − · π + Ψ (cid:19) + 1 (cid:19) · c c (cid:21) · (1 − cosϕ ) (19)where c = 0 . is the day of the year; ϕ is the latitude; Ψ , c , and c specifies the Northern or SouthernHemisphere, respectively. For the wet part, the coefficients b w and c w are constants with b w = 0 . and c w = 0 . (Boehm et al., 2006).Then, the gridded VMF1 data is generated from the ECMWF NWM using four global grid files( . ◦ × . ◦ ) of a h , a w , ∆ L zh and ∆ L zw . Using VMF1, one can determine the tropospheric delay at any locationafter 1994 (Kouba, 2008). The tar-compressed files of all VMF1 epoch grid files are available at the VMF1website ( http://ggosatm.hg.tuwien.ac.at/DELAY/GRID/VMFG/ ). More details are availablein Boehm and Schuh (2004) and Kouba (2008).To achieve near real-time data processing and improve efficiency, a simple empirical model is em-ployed, which is expressed as follows (Parkingson et al., 1996): ∆ L trop = 2 . / ( sinE + 0 . (20)In order to verify the accuracy of the empirical model, we take the two stations (BIL and LTS) fromMJD 58151 to 58160 into calculation. We use VMF1 as a comparison, and compare the correction mag-nitude of the two models in CVSTT technique. The results are shown in Fig. 3, the elevation set-off inour experiment is ◦ . From Fig. 3, we conclude that the correction magnitude is tens of nanoseconds forindividual stations, and about 2 ns for the two stations via CVSTT technique. And the difference betweenempirical model and VMF1 is quite small for CVSTT technique (less than 0.025 ns). Therefore, the empir-ical model is precise enough in our study. In our experiments, after using empirical model for troposphericcorrection, the accuracy of the correction achieves 0.52 ns (see Table 1). The Sagnac effect is caused by the rotation of the Earth. During the signal propagation from satellite S to station i , the Sagnac effect (correction) is expressed as (Ashby, 2004; Tseng et al., 2011; Yang et al.,2012): S i = − ω e x S y i − x i y S c (21)11 igure 3: The tropospheric corrections between stations BIL and LTS, which last from MJD 58151 to MJD 58160.The tropospheric corrections for stations LTS and BIL are shown in (a) and (b), respectively. (c) is troposphericcorrections for the two stations via CVSTT technique. And (d) is the difference between empirical model and VMFImodel. It should be noted that the blue curve denotes tropospheric corrections by the empirical model, the green curvedenotes tropospheric corrections by the VMF1 model, and the red curve denotes the difference of the two models. where ω e is the Earth rotation rate, with a relative uncertainty of δω e = 1 . × − (Groten, 2000).Taking station A and station B into considertion, the corrections of signal delay of the total Sagnaceffect via CVSTT technique can be expressed as follows: ∆ S AB = S B − S A = [ y S ( x B − x A ) − x S ( y B − y A )] ω e /c (22)And the accuracy of Sagnac corrections via CVSTT technique can be determined through error propa-gation law, which is expressed as: m Sac = [( x B − x A ) m ( δy S ) + ( y B − y A ) m ( δx S )+ y S m ( δx A ) + y S m ( δx B ) + x S m ( δy A ) + x S m ( δy B )]( ω e /c ) + [ y S ( x B − x A ) − x S ( y B − y A )] m ( δω e ) /c (23)Taking the coordinates of BIL and LTS into consideration, results show that m Sac will not exceed 0.00312 able 1:
The accuracy of CVSTT technique due to different error sources and the corresponding accuracy of OHdetermiation in the transportation experiments.Error type Sources errors CVSTT accuracy(ns) OH accuracy(m)code noise 0.97 ns 4.09 43.42Broadcast ephemeris 2 m 0.16 1.70Ground station 3 cm 0.15 1.60Ionosphere - 0.80 8.50Troposphere - 0.52 5.52Sagnac . × − ns. Previous studies (Shen and Ding, 2014) demonstrates that the ensemble empirical mode decompo-sition (EEMD) (Wu and Huang, 2009) is an effective technique for isolating target signals from envi-ronmental noises. The EEMD technique is developed from the empirical mode decomposition (EMD)(Huang et al., 1998) ( https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.1998.0193 ). The signals series are decomposed into a series of Intrinsic Mode Functions (IMFs)and a residual trend r after EEMD decomposition. These IMFs series that are sifted stage by stagereflect local characteristics of the signals, while the residual trend r series reflects slow change of thesignals (Zhu et al., 2013). The specific steps of EEMD are stated as follows (Wu and Huang, 2009)( ):1) The overall signal series X ( t ) is determined after adding Gaussian white noise ω ( t ) into originalsignal series x ( t ) : X ( t ) = x ( t ) + ω ( t ) (24)2) The EMD method is applied to implement decomposition for X ( t ) subsequently, these IMFs aredetermined as: X ( t ) = n (cid:88) j =1 c j + r n (25)where c j denotes IMF j and r n denotes the residual trend.13) Adding different Gaussian white noise ω i ( t ) into x ( t ) and repeating step 1) and step 2), a series of c ij and r in are determined: X i ( t ) = n (cid:88) j =1 c ij + r in (26)4) According to zero mean principle of Gaussian white noise, the interference cuased by Gaussianwhite noise can be cancelled after averaging.Therefore, the IMF series c n ( t ) can be determined as: c n ( t ) = 1 N n (cid:88) i =1 c i,n ( t ) (27)Finally, the original signal series x ( t ) is decomposed into a series of IMFs series c n ( t ) and a residualtrend series r ( t ) by EEMD technique: x ( t ) = m (cid:88) n =1 c n ( t ) + r ( t ) (28)After EEMD decomposition of the signal series x ( t ) , the index of the orthogonality (IO) is calculatedto check the completeness of the decomposition, which is defined as follows (Huang et al., 1998): IO = T (cid:88) t =0 ( n +1 (cid:88) j =1 n +1 (cid:88) k =1 c j ( t ) c k ( t ) /X ( t )) (29)where X ( t ) is defined as: X ( t ) = n +1 (cid:88) j =1 c i ( t ) + 2 n +1 (cid:88) j =1 n +1 (cid:88) k =1 c j ( t ) c k ( t ) (30)If the decomposition is completely orthogonal, the cross terms in the right-hand side of expression (29)should be zero, namely IO = 0 ; and for the worst case IO = 1 . Here, using a simulation experiments we explain the advantages of the EEMD technique. Supposewe have a synthetic series S base ( t ) that consists of 3 periodic signals series, S i ( t ) = A i · sin(2 πf i t ) · exp( − − t ) (units:ns), where A = 10 , A = A = 5 , f = 2 / Hz, f = 1 / Hz, f = 0 . / Hz, respectively, and a linear signal series, S ( t ) = 3 · − t − − , with a data length of10 days and sampling interval 960 seconds. The S base ( t ) can be expressed as: S base ( t ) = S ( t ) + S ( t ) + S ( t ) + S ( t )= 10 · sin(2 π · · t ) · exp( − − t ) + 5 · sin(2 π · · t ) · exp( − − t )+ 5 · sin(2 π · . · t ) · exp( − − t ) + (3 · − · t − − ) (31)The results of the constructed series are shown in Fig. 4. We added noise signals N ( t ) into S base ( t ) with 2 % (Case 1), 5 % (Case 2) and 10 % (Case 3) of the standard deviation (STD) of S base ( t ) , and then14 igure 4: The waveforms of the synthetic series S base ( t ) (top slot) and different components S i ( t ) ( i =1,2,3,4) in thesubsequent slots, respectively. construct a signal series S ( t ) which contains noise. The S ( t ) can be expressed as: S ( t ) = S base ( t ) + N ( t ) (32)where the N ( t ) is added noise, which consists of five types of noises, expressed as (Ashby, 2015; Li andSun, 2011; Zhai et al., 2012): N ( t ) = N W − P M ( t ) + N F − P M ( t ) + N W − F M ( t ) + N F − F M ( t ) + N RW − F M ( t ) (33)where N W − P M ( t ) is the white noise phase modulation (W-PM), N F − P M ( t ) is the flicker noise phasemodulation (F-PM), N W − F M ( t ) is the white noise frequency modulation (W-FM), N F − F M ( t ) is the flickernoise frequency modulation (F-FM), N RW − F M ( t ) is the random walk noise frequency modulation (RW-FM).The constructed signal series S ( t ) with different noise magnitudes are shown in Fig. 5. For our presentpurpose, the linear signal series S ( t ) is the target signal which need to be extracted from the syntheticsignals series S ( t ) .The EEMD technique is applied to identify the periodic signals S ( t ) , S ( t ) and S ( t ) from S ( t ) . AfterEEMD decomposition, we use the index IO to check completeness of the decomposition. In our simulationexperiments, the values of IO are 0.0015 (Case 1), 0.0017 (Case 2) and 0.0017 (Case 3), respectively,suggesting that signal series S ( t ) are effectively decomposed in three different cases. The decomposedIMFs are shown in Figs. 6-8, and we found that the periodic signals S ( t ) , S ( t ) and S ( t ) can be clearlyidentified, respectively. The red dotted curves in Figs. 6-8 denote the original signals S i ( t ) ( i = 1, 2, 3) for15 igure 5: The waveforms of the constructed signal series S ( t ) and different magnitudes of the noise series N ( t ) .(a), (c) and (e) are the constructed signal series S ( t ) which are the sum of S base ( t ) and corresponding N ( t ) withmagnitude of 2 % , 5 % and 10 % of S base ( t ) , respectively; (b), (d) and (f) are the noise series N ( t ) with magnitudes of2 % , 5 % and 10 % of S base ( t ) , respectively. comparison purpose.Further, the Hilbert transform (HT) is executed to display the variety of instantaneous frequencies foreach IMF component. The corresponding Hilbert spectra and marginal spectra for IMFs (expect for r) inthree cases are shown in Figs. 9-11, where each of the subfigures (a)s shows the frequency variations ofeach IMF, and each of the subfigures (b)s shows measures of the total amplitude (or energy) contributionfrom each frequency value, representing the cumulated amplitude over the entire data span in a probabilisticsense (Huang et al., 1998).From Figs. 9-11, we see that the mode-mixing problem exists in EEMD decomposition, and it becomesmore obvious as noise increases from 2 % to 10 % . However, the three periodic signals S ( t ) , S ( t ) , and S ( t ) can be identified clearly. The set frequencies of S ( t ) , S ( t ) , and S ( t ) are 2 circle per day (cpd), 1cpd and 0.5 cpd, respectively, denoted as f − set = 2 cpd, f − set = 2 cpd, and f − set = 0 . cpd. AfterEEMD decomposition, the marginal spectra show that the corresponding values are f − case = 2 . cpd, f − case = 1 . cpd, f − case = 0 . cpd (Case 1); f − case = 2 . cpd, f − case = 1 . cpd, f − case = 0 . cpd (Case 2); f − case = 2 . cpd, f − case = 1 . cpd, f − case = 0 . cpd (Case3), respectively. And the detected signals corresponding to the original set signals S ( t ) , S ( t ) and S ( t ) are shown by the peaks denoted in green circles. In addition, there appear not-real signals with frequenciesaround 0.2 cpd after EEMD decomposition, which are denoted as F ( t ) and shown by the peaks denoted inred rectangle. The F ( t ) might be meaningless in physical explaining, and it will be an interference if wefocus on periodic signals. In this study, however, the target is to detect and identify the linear signal S ( t ) ,16hich means that all periodic signals are useless and should be removed. After all these periodic signals areremoved, we reconstruct a new signal series by summing the residual IMFs and the residual trend r. Thereconstructed signal series is denoted as S (cid:48) ( t ) .We take the signal series S ( t ) as real signal series, and try to recovery it by two different methods.The first method is to perform a least squares linear fitting on the signal series S ( t ) directly; and the secondmethod is to perform EEMD decomposition on S ( t ) and then reconstruct the new signal series S (cid:48) ( t ) byremoving periodic series IMFs, and the least squares linear fitting is performed on S (cid:48) ( t ) . The results areshown in Fig. 12.Here we use the standard deviation (STD) of the difference series between the real signal series S ( t ) and that determined by Method 1 or 2 to evaluate the reliability of the two methods. Using Method 1,namely the direct least squared linear fitting of the series S ( t ) , the STDs of the results are 1.31 ns (Case 1),1.63 ns (Case 2), and 2.16 ns (Case 3), respectively. Using Method 2, namely the least squared linear fittingof the reconstructed series S (cid:48) ( t ) , which is obtained after removing the periodic series via EEMD technique,the STDs of the results are 0.82 ns (Case 1), 1.16 ns (Case 2), and 1.87 ns (Case 3), respectively. Thecomparative results clearly suggest that the EEMD technique is effective for extracting the linear signals ofinterest by a priori removing the contaminated periodic signals from the original observations. In our experiments, we used two hydrogen time and frequency standards, with one is a fixed referenceclock C A (iMaser3000) and the other is a portable clock C B (BM2101-02). The experiments are conductedat Beijing 203 Institute Laboratory (BIL) and Luojiashan Time-Frequency Station (LTS), and the locations ofthe two stations are shown in Fig. 13 and Table 2. And we used the Modified Allan deviation (MDEV)(Allanet al., 1991) to evaluate the frequency stability of the two hydrogen atomic clocks, which is estimated froma set of frequency measurements for time averaging. The MDEV is expressed as follows (Allan et al., 1991;Bregni and Tavella, 1997; Lesage and Ayi, 2007): δ M ( τ ) = (cid:115) (cid:80) N − n +1 j =1 [ (cid:80) n + j − i = j ( x i +2 n − x i + n + x i )] τ n ( N − n + 1) (34)where N denotes the total number of the time series, τ denotes the basic time interval, and τ = nτ .The frequency stabilities of the two clocks on the different average time intervals are given in Table 3.The MDEV with shorter interval represents random noise of the clock, while with longer interval representssecular frequency drift (Kopeikin et al., 2016).The experiments are divided into two Periods (shown in Fig. 14). In Period 1 that spans from January09, 2018 to January 29, 2018, the zero-baseline measurement is implenented at BIL with clocks C A and C B able 2: The detailed information of the two ground stations (BIL and LTS) in the transportation experiments. Thecoordinates ( ϕ, λ, h ) denote the locations of GNSS antennas (under the frame of WGS 84); H ant and H clock denotethe OHs of GNSS antenna and corresponding location of hydrogen atomic clock C i , respectively. It should be notedthat the OH of GNSS antenna, H ant , is determined by EGM2008 gravity field model. And the OH of clock C i , H clock ,is determined by measuring the OH difference via a ruler.Stations ϕ ( ◦ ) λ ( ◦ ) h ( m ) H ant ( m ) H clock ( m ) BIL 116.26 39.91 .
51 79 .
09 51 . LTS 114.36 30.53 .
00 41 .
76 35 . Table 3:
The Modified Allan Deviation (MDEV) of the two hydrogen time and frequency standards (the fixed one:iMaser3000( C A ) and the portable one: BM2101-02( C B )).Time interval 1 s 10 s 100 s 1000 s 10000 s C A . × − . × − . × − . × − . × − C B . × − . × − . × − . × − . × − are placed in the same shielding room with the same geopotentials. According to the International GNSSTracking Schedule (Allan and Thomas, 1994; Defraigne and Petit, 2015), the clock comparisons between C A and C B , ∆ C AB , are conducted via CVSTT technique, and a series of clock offsets ∆ C AB ( t ) can beobtained, where ∆ C AB = C B − C A , and ∆ C AB ( t ) = C B ( t ) − C A ( t ) .When zero-baseline measurement is finished, the portable clock C B is transported to LTS from BILwhile keeps clock C A unmoved at the BIL, and during transportation the portable clock C B continuesoperating. After installation and adjustment of C B at the LTS, the clock comparisons between C A and C B are conducted in geopotential difference measurement, similarly. And this period is denoted as Period 2,which spans from Febrary 01, 2018 to April 07, 2018. It should be noted that there is no difference exceptfor the location diversity of clock C B , when comparing the setup difference of Period1 and Period 2. Inaddition, there are two days of missing data (from January 30, 2018 to January 31, 2018) due to the factthat during transportation (around 8 h) there are no observations and the observations from the day after there-observation are not stable and consequently not used.Due to the fact that temperature is a major factor which may disturb the performance of atomic clocksas well as the cable delays (Lisowiec and Nafalski, 2004; Weiss et al., 1999),we controll the environmentwith a relatively constant temperature in whole experiments. The temperature is held nearly constant ( ± . ◦ C ) for the fixed clock C A in Period 1 and Period 2 at BIL. For the portable clock C B , it is in thesame laboratory with clock C A during Period 1, and the temperature is locked at ± . ◦ C in Period 2 at18TS. Based on the observations via CVSTT technique, the clock offsets series ∆ C AB ( t ) in each Period aredetermined. Here, the clock offsets series ∆ C AB ( t ) of the observations for all common-view satellites dur-ing Period 1 and Period 2 are denoted by the black dotted curve (simply denoted as raw data) in Fig. 15. Andtake the satellite elevation angles into consideration, the initial CVSTT observations (initial observations)are obtained, which are denoted by the yellow curve in Fig. 15. After that, we performe data preprocessingon the initial observations by removing gross error, correcting clock jumps and inserting missing data, andthe preprocessed CVSTT data (preprocessed data) are finally obtained which denoted by the blue curve inFig. 15. After data preprocessing, the clock offsets series ∆ C AB ( t ) become more stable. In the followingdata processing, all operations are based on the preprocessed data sets.At the same time, it should be noted that, however, the observations from MJD 58137 to MJD 57141 arenot used due to a systematic malfunction, and the observations from MJD 58142 to MJD 57147 are also notused due to random movement of the GNSS antenna (the two clocks are not transported, but one antenna isreplaced). Therefore, the valid observations in Period 1 spans from MJD 58127 to MJD 57136 (seen in Fig.15(a)). In Period 2, since the stability of clock C B could be significantly influenced by various environmentfactors during its transportation to LTS from BIL, the initial observations at LTS contains large fluctuationsand noises (from MJD 58150 to 58161, seen in Fig. 15(b) and Fig. 20(b)), which will contaminate finaldetermination of the geopotentials. Hence, as for preprocessed data in Period 2, we adopt two differentpreprocessed data sets by whether contains the observations of the first 12 days, and the details are shownlater.Then, the EEMD technique is applied to the preprocessed data sets for removing the uninterestingperiodic components and extracting the geopotential-related signals. The preprocessed data sets in Period1 and Period 2 are decomposed into a series of IMFs (with frequencies from higher to lower) and a longtrend component r after EEMD decomposition, as shown in Figs. 16 and 17. Generally, the componentsof the EEMD convey a physical meaning due to the fact that the characteristic scales are physical (Huanget al., 1998). However, the first several high frequency components might be fictitious due to the fact thatthe sampling interval (960s) is too large to capture the high-frequency variations. As a result, the data arejagged at these frequencies (e.g. c in Fig. 16; c , c , c in Fig. 17).After using EEMD technique for preprocessed data sets in Period 1 and Period 2, we examine com-pleteness and orthogonality of the EEMD decomposition. The IO values corresponding to Period 1 andPeriod 2 are determined with values of − . and − . , respectively, demonstrating that the signalsare effectively decomposed. It should be noted that the IO value during Period 2 is worse than that in Period1. This might be due to the environmental difference of C B and C A during Period 2, or due to the difference19f the signal propagation paths during Period 2.The corresponding Hilbert spectra in Period 1 and Period 2 are given in Fig. 18. In order to measurethe energy contributions from each frequency value, the marginal spectra of all IMFs are given in Fig. 19,which represent the cumulated amplitude over the entire data span in a probabilistic sense. From Figs. 18and 19, the periodic signals with frequencies of around 0.2 cpd, 0.5 cpd, 1 cpd and 3.3 cpd of zero-baselinemeasurement are detected; and the periodic signals with frequencies of around 0.05 cpd, 0.36 cpd, 1 cpdand 2 cpd of geopotential difference measurement are detected. Finally, these periodic signals included inthe original CVSTT observation series are removed, and we reconstructed the residual clock offsets series ∆ C AB − re ( t ) by summing the residual components. The reconstructed clock offsets series ∆ C AB − re ( t ) areregarded as the geopotential-related signals, based on which we can determine the geopotential differencebetween C B (LTS) and C A (BIL). The reconstructed clock offsets series ∆ C AB − re ( t ) are denoted as EEMDresults, as described in section 4.2. After data processing, the corresponding EEMD results of clock offsets series ∆ C AB − re ( t ) in Pe-riod 1 and Period 2 are determined, as shown in Fig. 20, which provides the clock offsets series beforeand after EEMD technique via the CVSTT technique. We can find that the clock offsets series becomemore stable and smooth after removing periodic signals included in the preprocessed data sets. Concerningzero-baseline measurement, after EEMD decomposition and removing periodic IMFs, the EEMD resultsof the reconstructed clock offsets ∆ C AB − re ( t ) are determined directly, and the corresponding time elapsedifference, α zero , is then determined based on formula (6). We take this result as a system constant shift.For the geopotential difference measurement, however, the first 12-day observations contain large fluc-tuations and noises (seen in Fig. 20b) when compared to the later observations. Therefore, the preprocesseddata set in Period 2 are analyzed twice. For the first analysis we used the entire data span to implementEEMD decomposition (the red curve in Fig. 21), and for the second analysis we removed the first 12-dayobservations before EEMD decomposition for avoding data pollution (the green curve in Fig. 21). In addi-tion, for the geopotential difference measurement, we segment the corresponding ∆ C AB − re ( t ) into 10-dayunits, not only for matching the zero-baseline duration, but also for improving data utilization. Furthermore,by this strategy, the clock drift could be limited to a relative short time duration (10 days). And we useMDEV of each unit as the weighting factor for determining the final experimental results, which meansthat for a certain time duration, the higher the stability the greater the weight. By the above strategy, theinterference of clock drift could be limited to a relative low level. And based on each unit, a series of timeelapse differences α geo ( t ) are determined.According to equations (2) ∼ (6), by taking the results of α zero in Period 1 as system constant shift,the variation of the time elapse difference caused by the geopotentials, α ( t ) is finally determined with20 ( t ) = α geo ( t ) − α zero . The α ( t ) is caused by the geopotential difference between the two stations LT S and
BIL . Therefore, the clock-comparison-determined results of geopotential difference ∆ W ( T ) AB as well asthe OH of H ( T ) LTS is obtained. We note that to determine the OH of the station LTS, the OH of the station BILis a priori given. The results of α ( t ) and D ( t ) are shown in Figs. 21 and 22, respectively. It is noted thatthe D ( t ) = H ( T ) LTS ( t ) − H LTS is the discrepancy between the clock-comparison-determined results H ( T ) LTS ( t ) and the corresponding EGM2008 results H LTS . From Figs. 21 and 22, we can find that after removing theseperiodic signals the results become more stable besides smaller errorbars, which explain the validity of theEEMD technique in extracting the linear signals of interest. In addition, comparing the EEMD results incase 1 (EEMD Results 1) and case 2 (EEMD Results 2), the interference caused by transportation from BILto LTS is reduced by removing the first 12-days observations in Period 2.The results of the clock-comparison-determined geopotential difference ∆ W ( T ) AB and OH of clock C B atLTS H LTS are given in Table 4. The results denoted as “preprocessed” and “EEMD” are obtained by takingall measurements in Period 2 as one unit. The results denoted as “M-preprocessed” and “M-EEMD” areobtained by implementing the grouping strategy for the measurements in Period 2 with 10-days measure-ment as different units, and then we take the mean average of all units as the corresponding final results.The results denoted as “W-preprocessed” and “W-EEMD” are obtained by also implementing the groupingstrategy for the measurements in Period 2 with 10-days measurement as different units, and we take theweighted average of all units as the corresponding results. Here we assign weights based on MDEV of eachunit, which means that the higher the stability the greater the weight. Due to the fact that the frequencystability of the two hydrogen clocks used in experiments are at the level of − /day, the accuracy of thedetermined geopotential is limited to tens of meters in equivalent height.Compared to the corresponding EGM2008 model results which suggests that H LT S = 35 . m,the clock-comparison-determined results H ( T ) L T S = 133 . ± . m. The deviation between the clock-comparison-determined results and the model value is 97.7 m. The results are consistent with the frequencystability of the hydrogen atomic clocks used in our experiments. In this study, the clock comparison based on CVSTT technique for determining the geopotential dif-ference is investigated. Based on this approach, one can directly determine the geopotential differencebetween two ground points. The experimental results provided in Table 4 show that the discrepancy be-tween the clock-comparison-determined result and the corresponding model value is . ± . m. This isconsistent with the frequency stability of the transportable H-master clocks (iMaser3000 and BM2102-02)used in our experiments.Here we first used the EEMD technique to remove the periodic signals from the original CVSTTobservations to more effectively extract the geopotential-related signals from clock offsets signals. We21 able 4: The clock-comparison-determined geopotential difference ∆ W ( T ) AB and OH of clock C B at LTS. D = H ( T ) LTS − H LTS is the difference between H ( T ) LTS and the EGM2008 results H LTS ( H LTS = 35 . m).Time span Strategy ∆ W ( T ) AB ( m /s ) H ( T ) LTS (m) D(m)MJD(58150-58215) Preprocessed . ± . . ± . . ± . EEMD . ± . . ± . . ± . M-preprocessed . ± . . ± . . ± . M-EEMD . ± . . ± . . ± . W-preprocessed . ± . . ± . . ± . W-EEMD . ± . . ± . . ± . MJD(58162-58215) Preprocessed . ± . . ± . . ± . EEMD . ± . . ± . . ± . M-preprocessed . ± . . ± . . ± . M-EEMD . ± . . ± . . ± . W-preprocessed . ± . . ± . . ± . W-EEMD . ± . . ± . . ± . made comparisons between the results using EEMD and the results without using the EEMD technique, andour study shows that the results using EEMD is better, providing the deviation between the measured OH ofthe station LTS and the EGM2008 model one with D = 97 . ± . m. We stress that the EEMD techniqueis applied only to the preprocessing of the observations. If we don’t use it, we still obtain comparable resultswith D = 138 . ± . m, but a little worse. In addition, as preliminary experiments, we used about85-days observations to determine the geopotential difference between two stations based on the CVSTTtechnique, which might be applied extensively in geodesy in the future.In our present experiments, the residual errors caused by the ionosphere, troposphere, and Sagnaceffects can be neglected, for their effects are far below the accuracy level of the hydrogen clocks. How-ever, to achieve centimeter-level measurement accuracy, those influences should be taken into consider-ation. Namely, more precise correction models need to be considered. In addition, we should use thetime-comparison equation accurate to /c level based on GRT (Kopeikin et al., 2011).With rapid development of time and frequency science and technology, the approach discussed inthis study for determining the geopotential difference as well as the OH is prospective and thus, couldbe implemented as an alternative technique for establishing height datum networks, and has potentialapplications for unifying the WHS, if GRT holds. Our experimental results are preliminary, and furtherexperiments and investigations are needed to improve the results with high accuracy.22 cknowledgement . We would like to express our sincere thanks to three anonymous reviewers, theresponsible editor, and the Editor in Chief of the Journal of Geodesy, for their valuable comments andsuggestions, which greatly improved the manuscript. This study was supported by the NSFC (grant Nos.41721003, 41804012, 41631072, 41874023, 41429401, 41574007) and Natural Science Foundation ofHubei Province of China (grant No. 2019CFB611). References
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Proceedings of the Csee , 33(7):92–98. igure 6: The results of IMFs from the constructed signal series S ( t ) in Case 1 (with a 2 % of the magnitude of thenoise). (a) The constructed signal series S ( t ) and the components IMF1 ∼ IMF4 (black curves), (b) the componentsIMF5 ∼ IMF9 (black curves), (c) the residual trend r. Dotted red curves are original signals. igure 7: The resulting IMFs from the constructed signal series S ( t ) in Case 2 (with a 5 % of the magnitude of thenoise). (a) The constructed signal series S ( t ) and the components IMF1 ∼ IMF4 (black curves), (b) the componentsIMF5 ∼ IMF9 (black curves), (c) the residual trend r. Dotted red curves are original signals. igure 8: The resulting IMFs from the constructed signal series S ( t ) in Case 3 (with a 10 % of the magnitude of thenoise). (a) The constructed signal series S ( t ) and the components IMF1 ∼ IMF4 (black curves), (b) the componentsIMF5 ∼ IMF9 (black curves), (c) the residual trend r. Dotted red curves are original signals. a) (b) Figure 9:
The frequency variations of each IMF (expect for trend r) decomposed from S(t) in Case 1 (with 2 % of themagnitude of the noise). (a) The Hilbert spectrum of IMFs, where the color variations occurred in each skeleton curverepresent the corresponding energy variations of each IMF, and (b) the corresponding marginal spectrum of IMFs. (a) (b) Figure 10:
The frequency variations of each IMF (expect for trend r) decomposed from S(t) in Case 2 (with 5 % of themagnitude of the noise). (a) The Hilbert spectrum of IMFs, where the color variations occurred in each skeleton curverepresent the corresponding energy variations of each IMF, and (b) the corresponding marginal spectrum of IMFs. a) (b) Figure 11:
The frequency variations of each IMF (expect for trend r) decomposed from S(t) in Case 3 (with 10 % of themagnitude of the noise). (a) The Hilbert spectrum of IMFs, where the color variations occurred in each skeleton curverepresent the corresponding energy variations of each IMF, and (b) the corresponding marginal spectrum of IMFs. a)(b)(c) Figure 12:
The comparison of the linear fitting for the signal series S ( t ) (Method1) and the linear fitting for thereconstructed signal series S (cid:48) ( t ) (Method2). S ( t ) is a given real signal series. (a), (b) and (c) denote the cases ofnoise magnitudes of 2 % , 5 % and 10 % of S base ( t ) , respectively. igure 13: The distribution of the two ground stations of BIL and LTS in the experiments. The two stations are around1000 km away; the OH difference of the antennas and clocks ( H clock A − H clock B ) between BIL and LTS are 37.3 mand 16 m, respectively. igure 14: The schematic diagram of the transportation experiments. In Period 1, the zero-baseline measurement isimplemented with both C A and C B located at the BIL, and this period lasts from January 09, 2018 to January 29, 2018.After that the clock C B is transported to LTS from BIL, and the geopotential difference measurement is implementedin Period 2 with C A and C B located at BIL and LTS, respectively.The Period 2 lasts from Febrary 01, 2018 to April07, 2018. Figure 15:
The data preprocessing for clock offsets series ∆ C AB ( t ) via the CVSTT technique. (a) The data prepro-cessing of the zero-baseline measurement, (b) the data preprocessing of the geopotential difference measurement. Theraw data (black dotted curve) denotes the results of ∆ C AB ( t ) from all common-view satellites; the initial observa-tions (yellow curve) denotes the results of ∆ C AB ( t ) which is the weighted average based on satellite elevations; thepreprocessed data (blue curve) denotes the results of ∆ C AB ( t ) after data preprocessing on initial observations (aftergross error removing, clock jump correction, and data insertion). igure 16: The resulting EEMD components of clock offsets series ∆ C AB ( t ) in zero-baseline measurement (Period1) with both C A and C B located at BIL, which lasts from MJD 58127 to MJD 58136. (a) The preprocessed data andthe components IMF1 ∼ IMF4, (b) the components IMF5 ∼ IMF8 and the residual trend r . igure 17: The resulting EEMD components of clock offsets series ∆ C AB ( t ) in geopotential difference measurement(Period 2) with C A located at BIL and C B located at LTS, which lasts from MJD 58150 to MJD 58215. (a) Thepreprocessed data and the components IMF1 ∼ IMF4, (b) the components IMF5 ∼ IMF9, (c) the components IMF10 ∼ IMF13 and the residual trend r . igure 18: The corresponding Hilbert spectra of EEMD decomposition in zero-baseline measurement (a) and that ofin geopotential difference measurement (b).
Figure 19:
The corresponding marginal spectra of EEMD decomposition in zero-baseline measurement (a) and thatof in geopotential difference measurement (b). igure 20: The experimental results of clock offsets series via CVSTT technique. (a) The clock offsets series ofzero-baseline measurement (Period 1), (b) the clock offsets series of geopotential difference measurement (Period 2).It should be noted that the blue curve is the preprocessed data sets ∆ C AB ( t ) ; the green curve is the EEMD results ∆ C AB − re ( t ) , which is determined after EEMD technique; the red curve is a linear fitting for the ∆ C AB − re ( t ) . Figure 21: the variation of the time elapse difference α ( t ) ( α ( t ) = α geo ( t ) − α zero ). The α ( t ) is caused by geopo-tential difference between the two staions ( LT S and
BIL ). The preprocessed results (blue curve) are determinedbased on the preprocessed data set; the EEMD results1 (red curve) are determined based on EEMD results whichcontains whole observations in Period 2; the EEMD results2 (green curve) are determined based on EEMD resultswhich removes the first 12-day observations before EEMD technique. igure 22: The discrepancy between the clock-comparison-determined results and the corresponding EGM2008 re-sults D ( t ) ( D ( t ) = H ( T ) LTS ( t ) − H LTS ). The preprocessed results (blue curve) are determined based on the preprocesseddata set; the EEMD results1 (red curve) are determined based on EEMD results which contains whole observationsin Period 2; the EEMD results2 (green curve) are determined based on EEMD results which removes the first 12-dayobservations before EEMD technique.). The preprocessed results (blue curve) are determined based on the preprocesseddata set; the EEMD results1 (red curve) are determined based on EEMD results which contains whole observationsin Period 2; the EEMD results2 (green curve) are determined based on EEMD results which removes the first 12-dayobservations before EEMD technique.