Unification of global height system at centimeter level using precise frequency signal links
UUnification of global height system at centimeter level usingprecise frequency signal links
Ziyu Shen Wen-Bin Shen , , ∗ Shuangxi Zhang , School of Resource, Environmental Science and Engineering, Hubei University of Science andTechnology, Xianning, Hubei, China Time and Frequency Geodesy Research Center, Department of Geophysics, School of Geodesy andGeomatics, Wuhan University, Wuhan, China Key Lab of Surveying Eng. and Remote Sensing, Wuhan University, Wuhan, China ∗ Corresponding author: [email protected]
Date: August 14, 2020
Abstract
The realization of International Height Reference System (IHRS) is one of the major tasks of the Inter-national Association of Geodesy (IAG). A main component of the IHRS realization is the global verticaldatum unification, which requires the connection of the existing local vertical height reference systems(VHS). However, it is difficult to estimate the offsets between two local height systems by conventional ap-proaches when they are far apart. In this paper, we formulate a framework for connecting two local VHSsusing ultra-precise frequency signal transmission links between satellites and ground stations, which isreferred to as satellite frequency signal transmission (SFST) approach. The SFST approach can directlydetermine the geopotential difference between two ground datum stations without location restrictions, andconsequently determine the height difference of the two VHSs. Simulation results show that the China’sVHS and the US’s VHS can be unified at the accuracy of several centimeters, provided that the stabilityof atomic clocks used on board the satellite and on ground datum stations reach the highest level of cur-rent technology, about . × − τ − / for an averaging time τ (in seconds). The SFST approach ispromising to unify the global vertical height datum in centimeter level in future, and it also provide a newway for the IHRS realization. Keywords: relativistic geodesy, satellite frequency signal transmission, vertical height reference system,global vertical height datum unification
Reference frames with long-term stability and homogeneous consistency worldwide plays a key rolein establishing various theoretical frameworks, e.g., gravity field, Earth rotation, geodynamics, as well as1 a r X i v : . [ phy s i c s . g e o - ph ] A ug xtensive applications, such as global navigation satellite system, precise positioning, precise observationsof any subject in space, etc. The International Terrestrial Reference System (ITRS) and its realization,International Terrestrial Reference Frame (ITRF, Petit and Luzum, 2010) provide a globally unified geometricreference frame with accuracy at millimeter level. However, currently an equivalent high-precise globalphysical reference frame that reflects the EarthâĂŹs gravity field is still missing (Ihde et al., 2017). In orderto establish a consistent and accurate physical reference frame, the International Association of Geodesy(IAG) released the IAG Resolution No.1. for the definition and realization of an International HeightReference System (IHRS) in 2015 (Sánchez et al., 2016), which design a physical world height system asthe basis for monitoring effects generated by gravity field variation. Similar to the geometric referencesystem and frame, the realization of IHRS is the establishment of the International Height Reference Frame(IHRF). The IHRS is defined by an equipotential surface of the Earth’s gravity field, where the geopotentialvalue at the surface is the conventional value W = 62636853 . m /s (zero-height level), and the verticalcoordinates are geopotential numbers C p = W − W P (Ihde et al., 2017; Sánchez et al., 2016). A keyconcept of realizing IHRS is the unification of local vertical height systems (VHSs, which refer to localisolated level) around the world, and connect them to the global one. Since local VHSs are usually based onmean sea level (MSL) determined by tide gauges, and the MSL is not an equi-qeopotential surface, differentlocal VHSs exhibit inconsistencies with respect to each other up to ∼ meters (Sideris, 2015). How tofind out the offset between arbitrary two different height systems’ origins (datums) is the main challenge forrealizing IHRS, and various approaches have been tested and discussed.Currently there are four approaches that are extensively discussed and practically applied for the heightsystem unification. They are briefly explained as follows, each of which presents advantages and drawbacks.(1) The conventional approach is leveling with gravity reductions. This is mainly used for the realizationof local VHS and the accuracy can reach sub-millimeter level between neighboring leveling point (Sánchezand Sideris, 2017). However, leveling is laborious and time-consuming while the errors accumulate overlong distances. In addition, the main drawback of leveling is that it cannot connect two continents separatedby the ocean, which makes it impractical for the realization of a global VHS (Ihde et al., 2017).(2) Oceanic leveling (Stöcker-Meier, 1990), in contrary, is suitable for connecting different local heightsystems separated by oceans. For example, ocean models can provide a mean dynamic topography correctionto height datums of countries with coastlines, thus realize the unification. Though the uncertainty ofoceanographic modeling method can be better than a decimeter (Woodworth et al., 2012), the ocean levelingis limited to height datums near a coastlines, and for high precision it requires years of continuous observationdata of tide gauges with adequate density distribution (Woodworth et al., 2012), which is unavailable in manyplaces such as Africa areas.(3) The third method is estimating the anomalous potential by solving the geodetic boundary valueproblem (GBVP) (Rummel and Teunissen, 1988). It can provide a global solution for height unification, and2he precision in well-surveyed regions reaches several centimeters (Amjadiparvar et al., 2016; Gerlach andRummel, 2013; Rangelova et al., 2016). But in sparsely surveyed regions the precision drops to decimeterlevel (Sánchez and Sideris, 2017). Another drawback of GBVP method lies in that it requires a priorinformation of potential or height values from various sources (global geopotential model, tide gauge data,gravity observation data, et al.); and the errors in these a prior information will influence the precision ofGBVP method, and the use of different kinds of a prior information in different regions makes it difficult tounify the height datums in these regions.(4) The fourth method is applying global gravity models (GGMs) with high precision. The EGM2008,for example, is complete with degree and order of spherical harmonics up to 2159 (Pavlis et al., 2012), andwe can directly compute the potential C ( P ) of any given point in the ITRF coordinates by introducing itinto the spherical harmonic expansion equation. However, at present the GGMs method meets the problemof precision and resolution trade-off. For instance, the GOCE series models (see e.g., Hirt and Kuhn (2012);also see the released products from ESA ( ) and the International Centre for Global EarthModels ( icgem.gfz-potsdam.de/ICGEM )) can reach the accuracy of 1 cm (even higher) but with poorresolution of ◦ × ◦ . In contrary, although the EGM2008 model has a relatively high resolution of (cid:48) × (cid:48) ,its average accuracy is only about 10 to 20 cm (Pavlis et al., 2012). Another drawbacks of the GGMs methodlies in that different models usually give rise to quite obvious discrepancies, because of different standardsand conventions used.Currently it is difficult to establish IHRF with high precision by any of the approaches describedabove. In order to get out of the difficulties, another method, relativistic geodetic method, has gained anincreasing number of attention and discussion. The relativistic geodetic method is based on the generaltheory of relativity (Einstein, 1915): precise cocks at positions at different geopotentials run at differentrates. Therefore geopotentials or geopotential difference between arbitrary two stations can be measured byprecise clocks, and the corresponding height propagation based on this method is referred to as “chronometricleveling” (Bjerhammar, 1985; Vermeer, 1983). Since relativistic geodetic method requires ultra-high preciseclocks (e.g., for the precision of 1 cm, the stability of clocks should reach × − ), it was not payedattention for the purpose in practical applications for a long time because of the limit of clock precision.However, with the fast development of high-precision clock manufacturing technology in recent years, theoptical-atomic clocks (OACs) with uncertainty and accuracy around × − and even higher level havebeen generated in various laboratories (Huang et al., 2019; McGrew et al., 2018; Mehlstäubler et al., 2018;Oelker et al., 2019). That guarantees the feasibility of actual applications of the relativistic geodetic methods.Consequently, more and more scientists pay great attention to various potential applications of the relativisticgeodetic methods (Flury, 2016; Kopeikin et al., 2011; Müller et al., 2008; Puetzfeld and Lämmerzahl, 2019).In order to compare clocks in different places, the most precise method is to connect them via opticalfibre link (OFL) (Riehle, 2017). Thereby an increasing number of discussions and experiments on clocks3onnected by OFL have been carried out and discussed (Lion et al., 2017; Lisdat et al., 2016; Shen et al., 2019;Takano et al., 2016). Recently the most precise measurement in OFL chronometric leveling is conducted byGrotti et al. (2018), who use transportable optical clocks with uncertainties around × − to determinethe geopotential difference between two points in a mountain area between France and Italy. Though theirexperiments show a height discrepancy of around 20 cm between the OFL observed result and that determinedby conventional approach (leveling and gravity measurement), the σ uncertainties is limited to around 17m (Grotti et al., 2018). In addition, Wu et al. (2019) proposed a method to unify several local height systemsby clock networks connected by OFLs. According to their simulation results, the height systems of WestEuropean region can be unified at a precision better than 1 cm, under the assumption that the clock frequencyuncertainty is × − . Although relativistic geodetic methods are now practical and can reach highprecision, the adoption of OFL limits its development. That is because the cost for optical fibers will increaserapidly as the distance between two clocks increases, or as the number of stations in a network increases.Alternatively, we can compare two clocks by microwave frequency links in space, and even if the twoclocks are not inter-visible, they can be abridged by a satellite, and the geopotential difference between themcan be measured (Shen et al., 1993). This method is regarded as satellite frequency signal transmission(SFST) approach, which is detailedly discussed in Shen et al. (2016, 2017). Given the assumption that thestability of OACs is × − within an hour, simulation experiments show that the precision of geopotentialdifference between two stations on ground can reach several centimeters in height (Shen et al., 2017).Although its precision is slightly lower than that of the OFL approach, the SFST approach is much moreconvenient and cost much less, and it is promising for the global VHS unification and the IHRS realization.In this paper, we propose an approach for unifying global VHS by providing example how to connectarbitrary two different local VHSs using SFST approach. In section 2 we briefly describe the concept of heightreference system and SFST method. In section 3 we conduct simulation experiments of VHS unification bya SFST network, and present our results. In the last section we provide discussion and conclusions aboutthis work and potential improvements and applications in future. The International Height Reference System (IHRS) is a geopotential reference system co-rotatingwith the Earth, as defined by the International Association of Geodesy (IAG) in 2015 (Drewes et al.,2016). According to IHRS, the geopotential on the geoid (simply geoidal potential) is a constant value W = 62636853 . / s − , and the vertical coordinates are defined as (Ihde et al., 2017) C P = − ∆ W P = W − W P , (1)4here C P is denoted as geopotential number, ∆ W P is the geopotential difference between the potential W P at the considered point P and the geoidal potential W .A vertical height reference system (VHS) is defined by geographic elevation or depth in relation to areference surface (which is usually the local sea level) (Ihde et al., 2008; Luz et al., 2002; Sanchez, 2007).It has close connection to the concept of IHRS because its reference surface could be the geoidal surfacewith W , and the value C P (given in m / s − ) can be converted to a physical height H P (given in m) by thefollowing equation (Hofmann-Wellenhof and Moritz, 2005; Torge and Müller, 2012) H P = C P ˆ g = W − W P ˆ g , (2)where H P can be orthometric height (OH), normal height or dynamic height, depending on the types of ˆ g applied.The OH is a geometric length measured along the plumb-line from the ground point i ( i = P, Q ) to itscorespondent point i (cid:48) on the geoid ( i (cid:48) = P (cid:48) , Q (cid:48) , see Fig. 1). For the OH case the ˆ g in Eq. (2) is expressed as ˆ g = ¯ g = 1 H P (cid:90) H P g ( h ) dH P , (3)where ¯ g is the "mean value" of gravity g ( h ) along the plumb-line. The normal height and dynamic heightare approximations of OH (Hofmann-Wellenhof and Moritz, 2005), and OH is for practical purposes theheight above sea level (in fact the height above the geoid). In this paper we will regard OH as the verticalcoordinates of VHS. Figure 1:
Red dashed red curve denotes the global geoid, the two solid blue curves denote the W = W P and W = W Q surfaces, respectively. Bold blue curve denotes the plumbline, along which the height integration is executed. Currently there are many different local VHSs which are difficult to be unified because the global meansea level is not an equi-geopotential surface. An important component of realization IHRS is to unify theglobal VHS, which requires defining a global reference surface that is assumed to be available all over theworld (Ihde et al., 2017).
According to the general theory of relativity, we have the following relationship between the geopo-tentials W P and W Q and the clock frequencies f P and f Q for two points P and Q (Bjerhammar, 1985;5einberg, 1972) f P f Q = 1 − W Q /c − W P /c , (4)where c is the speed of light in vacuum. Since the Earth’s gravity field is weak, we have the followingapproximation W P − W Q = f P − f Q f c + O ( c − ) , (5)where f = ( f P + f Q ) / , and O ( c − ) denote higher order terms which can be neglected if the stations P and Q are stationary near the Earth’s surface.Eq.(5) is sufficient for fibre link measurement between two clocks located on ground. However, whenwe use microwave links to compare two clocks located respectively on board a satellite and at a groundstation, the case is much more complex. For example, a satellite might move in high velocity which givesrise to big Doppler effects. In addition, the ionosphere and troposphere will influence the frequency ofmicrowaves propagating in space with medium, and the rotation and tidal effects of the Earth will changethe status and environments of ground station. In order to address these problems, recently we formulatedthe SFST approach for determining the geopotential difference between a satellite and a ground station orbetween two ground stations (Shen et al., 2016). The main idea and formulations are briefly introduced asfollows, details of which are referred to Shen et al. (2016, 2017).Referring to Figure 2, the SFST contains three microwave links. An emitter at a ground station E emitsa frequency signal f e at time t . When the signal is received by a satellite S at time t , it immediatelytransmits the received signal f (cid:48) e and emits a frequency signal f s simultaneously. These two signals that aresimultaneously transmitted and emitted from the satellite are received by a receiver at ground station E attime t , which are noted as f (cid:48)(cid:48) e and f (cid:48) s , respectively. During the period of the emitting and receiving, theposition of the ground station in space has been changed from E to E (cid:48) . The satellite transmits and emitssignals at the same instant as it receives signal, so its position in the signal links is supposed to be the pointS at time t . There might be a small amount of latency, due to the fact that during the transmitting, theposition of the satellite is slightly different at the time as it receives and emits the signals. However, theun-synchronization influence is very small, which can be neglected for the SFST as explained in Shen et al.(2017)). The output frequency shift ∆ f is expressed by a combination of three frequencies as (Vessot andLevine, 1979; Vessot et al., 1980) ∆ ff e = f (cid:48) s − f s f e − ( f (cid:48)(cid:48) e − f (cid:48) e ) + ( f (cid:48) e − f e )2 f e , (6)The beat frequency ∆ f as expressed by Eq.(6) has cancelled out the first-order Doppler effect due tothe relative motion between satellite and ground station. As for the second-order Doppler effect and Earth’srotation influence, it is expressed as (Vessot and Levine, 1979) ∆ ff e = φ s − φ e c − | (cid:126)v e − (cid:126)v s | c − (cid:126)r se · (cid:126)a e c + O ( c − ) , (7)6here φ s − φ e is the gravitational potential difference between the satellite and the ground station, (cid:126)v e and (cid:126)v s are velocities of ground station and satellite (spacecraft) respectively, (cid:126)r se is vector from satellite to groundstation, (cid:126)a e is centrifugal acceleration vector of ground station, and O ( c − ) denote higher order terms than c − . On the right hand side of Eq.(7), the second term denotes the second-order Doppler shift predictedby special relativity, and the third term represents the effect of EarthâĂŹs rotation during the signal’spropagation time.If the higher order terms O ( c − ) are omitted, Eq.(7) holds only at the accuracy level of − (Caccia-puoti and Salomon, 2011), and in this case it need not consider other influence factors such as the residualionospheric effects, tidal effect etc. To achieve one-centimeter level measurement in height, we consideredhigher order terms until O ( c − ) and various influence factors for satellite-ground microwave links, andderived a theoretical formula that holds at the accuracy level better than − , expressed as (Shen et al.,2017) ∆ φ es c ≡ φ s − φ e c = ∆ ff e − v s − v e c − (cid:88) i =1 q ( i ) + Λ f + δf + O ( c − ) , (8)where Λ f is the sum of all correction terms (it contains corrections of ionospheric and tropospheric effects,tidal effects and influence of celestial bodies), δf is the sum of all error terms, q ( i ) ( i = 1 , , , ) arequantities related to the positions and velocities of the ground station and satellite, second Newtonianpotential, vector potential, and higher-order post Newtonian terms. The order terms higher than O ( c − ) aresafely omitted. The detailed expressions of the relevant quantities are referred to Shen et al. (2017). Basedon Eq.(8), when the output frequency shift ∆ f is measured and relevant quantities (such as position, speed,and acceleration of ground station and satellite) are given, the gravitational potential difference φ es can beobtained. We also discussed how to determine the geopotential difference between two ground stations,which are connected to a same satellite simultaneously (Shen et al., 2017). Since the satellite can serveas a "bridge" to connect the two ground sites, the geopotential difference between these two sites can beobtained. Simulation experiments show that the precision of the geopotential difference between two groundsites determined by SFST method is about ∼ cm in height (Shen et al., 2017), under the assumption thatthe accuracy of OACs is × − , which has been achieved recently (Huang et al., 2019; McGrew et al.,2018; Oelker et al., 2019). Suppose we have two ground datum stations, Chinese height datum station at Qingdao and Americanheight datum station at San Francisco (which were assumed), denoted respectively as P and Q , located ontwo continents which are connected to the same satellite via SFST links simultaneously, cf. Fig. 3.Suppose the gravitational potential difference ∆ φ P Q between the datum points in China and US has beendetermined using the SFST approach as described in section 2.2, then we may determine the geopotential7ifference ∆ W P Q by the following equation ∆ W P Q = ( φ Q − φ P ) + ( Z Q − Z P ) , (9)where Z P and Z Q are centrifugal force potentials at P and Q , respectively; and Z is expressed as Z = 12 ω ( x + y ) , (10)where ω is the angular velocity of the Earth rotation, x and y are coordinates defined in the geocentricEarth-fixed Cartesian coordinate system o − xyz (e.g. ITRF2008, see Petit and Luzum (2010)).Suppose the height of point P (noted as H P ) is given, and the geopotential difference ∆ W P Q has beenmeasured by SFST method; then the height of point Q (noted as H Q ) can be determined based on Eqs. (2)and (3), expressed as H Q = W − W Q ¯ g Q = W − W P − ∆ W P Q ¯ g Q ,H P = W − W P ¯ g P , (11)where ¯ g P and ¯ g Q are the average gravity values along the plumb-lines P P (cid:48) and QQ (cid:48) , respectively (see Fig.1). It should be noted that ¯ g i ( i = P, Q ) can not be directly calculated by Eq. (3), because we do not knowexactly the density distribution as well as the gravity distribution g ( h ) inside the Earth.We can see that besides the influence of the given value of H P , the accuracy of the determined H Q depends on that of ∆ W P Q ; consequently it is related to the stabilities of the optical atomic clocks. Sincewe cannot precisely determine the “mean value” ¯ g ( i ) , in practical applications in plain region, ¯ g i is usuallyreplaced by the following formula (Hofmann-Wellenhof and Moritz, 2005) ¯ g i = g i + 4 . × − H i , (12)where g i , in gals (cm/s ), is the gravity at ground point i , which can be measured by absolute gravimeter,and H i , in meters, is the height difference between i and i (cid:48) ( i (cid:48) = P (cid:48) , Q (cid:48) ) (see Fig. 1). Therefore accordingto Eqs. (11) and (12), we obtain a practical formula for determining H Q , expressed as H Q = H P · ( g P + 4 . × − H P ) − ∆ W P Q g Q + 4 . × − H Q , (13)where ∆ W P Q applies the geopotential unit (g.p.u, 1 g.p.u=1,000 gal.m), and iteration procedure could beapplied if needed.For the purpose of connecting VHSs, since the heights H P and H Q of the two height datum stationsare relatively small (say less than 100 m), using Eq.(13) is sufficient. For instance, suppose H P = 0 , ∆ W P Q = − , gal.m (which is equivalent to 100 m near Earth’s surface), the maximum errorcaused by using Eq.(13) will not exceed . mm. The reason is stated as follows. In the mentionedcase, | H Q | = ∆ W P Q / ( g Q + 0 . . The error caused by the uncertainty δ ¯ g i of the chosen meangravity ¯ g i will not exceed . gal. Then, we have | δH Q | = (∆ W P Q /g Q ) δ ¯ g i = 100 mδ ¯ g i /g Q ≤ m × . gal/ gal = 0 . mm.Now the height of the site Q (in US) is determined under the same basis (geoid) as is that of the site8 (in China). Therefore, these two local VHSs are unified. Based on the same principle, the SFST methodcan also be applied in the establishment of regional height system, and the geopotential difference (or heightdifference) of arbitrary two points located in this region can be directly determined, solving the regionalheight system (regional geoid) tilt problem. In this section we conducted simulation experiments to verify the SFST method of connecting twoVHSs. The main idea of the experiment is to compare a set of true values to a set of simulated observationvalues, as depicted in Fig. 4, which is explained in the following subsections.
We chose two datum stations, Qingdao Datum Station (QDDS, which is located at Qingdao Guanxiang-shan mountain and served as a height reference datum of China’s VHS) and San Francisco Datum Station(SFDS, which is located at California Academy of Science and supposed to be the height datum station ofUS’s VHS), and connected them via a GNSS-type satellite, referring to Fig. 5. The experiment time spanis set for 1.5h, from 7:00 am to 8:30 am, March 30, 2019. The satellite should be inter-visible to both thetwo ground datum stations during the experiment time; thus we chose the GPS navigation satellite SVN-56which satisfies the requirement. The trace of SVN-56 during our experiment time is depicted in Fig. 5The orbit information of the GPS navigation satellite SVN-56 was obtained from the precise ephemerisprovided by IGS( ), which is regarded as true values. The given coordinates ofQDDS and SFDS (which are also regarded as true values) can be transferred from LLA to ECEF positionsfor later calculations. The frequency links are designed to be established for every 5 second, hence we geta set of observation values for every 5 second. Since the time interval between two data set in the preciseephemeris is 15 min, we use polynomial interpolation (Horemuž and Andersson, 2006) to acquire the dataset in 5-second interval (true values). Then we use EGM2008 model (Pavlis et al., 2012) to calculate thegravitational potential values of satellite and two ground sites corresponding to the “observation” time points.These gravitational potential values are regarded as true values, namely the errors caused by EGM2008 arenot considered (the accuracy of EGM2008 is about 10-20 cm at ground, and better than 1 cm at GNSSsatellite altitude). Then the true value of the geopotential difference between QDDS and SFDS, ∆ W QD − SF ,can be obtained.The microwave signals’ frequencies will be affected by ionosphere and troposphere medium. Hencewe use the International Reference Ionosphere Model (Bilitza et al., 2017; Rawer et al., 1978) to obtainthe electron density values, and Earth Global Reference Atmospheric Model (Leslie and Justus, 2011) toobtain the temperature and pressure values, which are used to estimate the ionospheric and tropospheric9 able 1: The input datas used in simulation experiments. The coordinates are based on ITRF14Entities Values of ParametersSatellite ID SVN-56 (GPS Navigation Sat.)Coord. from (-19167.235509, 3652.729794, 18038.749481)to (-26493.102586, 424.868409, 3830.004962)Qingdao DS LLA (36.06974 ◦ N, 120.32172 ◦ E, 77.472 m)ECEF (m) (-2605813.108, 4455436.499, 3734494.956)OH (m) 72.260San Francisco DS LLA (37.76985 ◦ N, 122.46616 ◦ W, 75.878 m)ECEF(m) (-2709867.959, -4259189.792, 3885328.909)OH (m) 109.126Gravity field model EGM2008Ionospheric model International Reference IonosphereTropospheric model Earth Global Reference Atmospheric ModelTide correction ETERNAObservation duration from 7:00 am to 8:30 am, March 30, 2019Mearsurement interval 5 sHeight systems diff. 1.000 m (China HS is higher than US HS) influences on the signals’ frequencies (Millman and Arabadjis, 1984; Namazov et al., 1975). The height andgeopotential of the two ground sites will be also influenced by periodical tidal effect, which is well modeled(Voigt et al., 2017) and can be removed by some mature softwares such as ETERNA (Wenzel, 1996) andTsoft (Van Camp and Vauterin, 2005). In our experiment we use ETERNA to generate and analyses tidesignals. We also considered the influences of other planets (such as Venus, Jupiter etc.) besides the Sun andthe Moon. The relevant planet correction models are referred to Shen et al. (2017).In our experiment, the two datum stations are connected to SVN-56 simultaneously via SFST links.Relevant input parameters are listed in Table 1. It should be noted that the OH of QDDS is released byChinese government as China’s height datum origin, but the US has no corresponding height datum origin.Therefore the OH of SFDS is deduced from EGM2008. We assume the height difference between China’sVHS and US’s VHS is . m (as assumed true value), and China’s VHS is higher than US’s VHS. According to Eqs. (8) and (9), the geopotential difference between QDDS and SFDS, ∆ ˆ W QD − SF ( t ) ,can be measured as time series ∆ ˆ W QD − SF ( t ) = ∆ ˆ φ QD − s ( t ) − ∆ ˆ φ SF − s ( t ) + ( Z SF − Z QD ) , (14)10here ∆ ˆ φ QD − s ( t ) and ∆ ˆ φ SF − s ( t ) are respectively the observed gravitational potential differences betweenQDDS and the satellite as well as SFDS and the satellite, at time t , Z QD and Z SF are centrifugal forcepotentials of QDDS and SFDS respectively.The observed values ∆ ˆ W QD − SF ( t ) are different from true geopotential difference value ∆ W QD − SF because they are influenced by various error sources. In this simulation experiment we have consideredclock error e clk , ionosphere residual error e ion , troposphere residual error e tro , satellite’s position andvelocity errors e pos and e vel , tidal correction residual error e tide , and asynchronism error e asy . We expectthat ∆ ˆ φ QD − s ( t ) and ∆ ˆ φ SF − s ( t ) are measured at the same time t , but in practice they might have slightdifference, which will introduce the asynchronism error. The above mentioned various errors are consideredas noises, which are added to the true values. The total errors e all are expressed in the following form e all = e clk + e ion + e tro + e pos + e vel + e tide + e asy , (15)The magnitude and behavior of each kind of error play important role in this experiment; thereby we needto investigate different error models based on different error sources to make the simulation case more closeto the real case.Currently the most precise OACs have demonstrated . × stability at 1 second, and . × in 1 hour for two clocks comparison (Oelker et al., 2019); therefore we set the error magnitude of e clk as . × . Although there are many kinds of random noises that affect OAC signals (Major, 2013), the mostprominent components are white frequency modulation and random walk frequency modulation (Galleaniet al., 2003). Correspondingly the behaviors of clock errors are modeled as the following equation e clk ( t ) = a clk + b clk · t + c clk · φ ( t ) + d clk · (cid:90) t ξ ( t ) dt, (16)where a clk , b clk , c clk and d clk are constant coefficients, φ ( t ) and ξ ( t ) are both standard white Gaussiannoises. Each term in the right side of Eq.(16) has clear physical meaning; specifically a clk denotes the initialfrequency difference, b clk · t is the drift term, c clk · φ ( t ) is the white noise component, and d clk · (cid:82) t ξ ( t ) dt represents the random walk effect. As we set proper values of constant coefficients in accordance with theperformance of OACs in Oelker et al. (2019), a series of frequency comparison data with errors embeddedcan be generated.For other error sources, their magnitudes are discussed in detail in Shen et al. (2017) and listed in Table2. It should be noted that though the residual influences of ionosphere and troposphere for SFST methodare at the centimeter level (corresponding to frequency shift of − level), we have established correctionmodels (Shen et al., 2017) to reduce their influences to the millimeter level. The Earth tide effects couldreach up to 60 cm at maximum (Poutanen et al., 1996), but the residual error in vertical direction aftercorrections can be limited to 2 mm for solid Earth tide (Li et al., 2018), and 8 mm for ocean tide loading(Penna et al., 2008).Since there are no mature mathematical models for above mentioned errors and their influences aremuch smaller than the clock errors (see Table. 2), we adopt a general error model which contains systematic11 able 2: Error magnitudes of different error sources in determining the gravitational potential difference between asatellite and a ground station. They are transformed to relative frequency (modified after Shen et al. (2017))Influence factor (Residual) Error magnitude in ∆ f /f e ionospheric correction residual δf ion ∼ . × − tropospheric correction residual δf tro ∼ . × − tidal correction residual δf tide ≤ − position & velocity δf vepo ∼ . × − (10 mm and 0.1mm/s b )asynchronism δf delay ∼ − c (below 1 ms)clock error δf osc ∼ . × − a After tri-frequency combination; b Satellite’s position errors are assumed as 10 mm (Kang et al., 2006), velocity errors are assumed as 0.1mm/s (Sharifiet al., 2013). (initial) offset, drift and white Gaussian noises for each of the error source, expressed as the followingequation e j ( t ) = a j + b j · t + c j · φ i ( t ) , ( j = ion, tro, pos, vel, tide, asy ) (17)where a j , b j and c j are constant coefficients, which are randomly set in accordance with the error magnitudeslisted in Table 2According to Eqs. (16) and (17), we can generate the noise signals based on the magnitudes and natureof the error sources at any time. Then these noises are added to relevant true values, and we get a set ofrelevant "Observed" values, based on which the geopotential difference ∆ ˆ W QD − SF ( t ) is determined usingEqs. (8) and (14). The next step is converting the geopotential difference to corresponding height difference.Without loss of generality, assuming the zero-height surface of China’s VHS is just coinciding with the W surface, based on China’s VHS, the height of SFDS can be calculated by Eq. (13), expressed as ˆ H SF ( t ) = H QD · ( g QD + 0 . H QD ) − ∆ ˆ W QD − SF ( t ) g SF + 0 . H SF ( t ) , (18)where H Q = 72 . m is the height of QDDS in China’s VHS. In this case, the observed VHS differencebetween China and US can be obtained as ∆ ˆ H V HS ( t ) = ˆ H SF ( t ) − H SF , (19)where H SF = 109 . m is the height of SFDS in US’s VHS, and the unification of the two VHSs isrealized. However, if the zero-height surface of China’s VHS does not coincide with the W surface (this is12n general the real case), Eq. (18) is not rigorous, and in this case Eq. (11) should be modified as H QD = W + δW China − W QD ¯ g QD ,H SF = W + δW US − W SF ¯ g SF , ˆ H SF = W + δW China − W QD − ∆ W QD − SF ¯ g SF , (20)where δW China is the geopotential difference between the zero-height surface of China’s VHS and the W surface, δW USA is the geopotential difference between the zero-height surface of US’s VHS and the W surface. If δW China is unknown, the derived height of SFDS ˆ H SF ( t ) can not be calculated based on theheight of QDDS H QD even though their geopotential difference is given; therefore the height differencebetween the two VHSs cannot be strictly determined. However, as the W surface and a VHS’s zero-heightsurface are close to the mean sea level, δW i ( i = China, U S ) are relatively small (usually less than 10 m / s − (Sideris, 2015)); the error introduced by Eqs. (18) and (19) can be neglected at current precisionlevel of centimeter. In addition, if the value of δW i can be obtained (which is very promising in future, seediscussions in Sec. 4), we can also unify the two VHSs based on rigorous equation. Therefore for brevityand without loss of generality, we just use Eqs. (18) and (19) for the height unification calculation. Bycomparing the observed height difference ∆ ˆ H V HS ( t ) and the true difference ∆ H V HS = 1 m, the reliabilityof SFST approach for height system unification can be verified.
Since the experiment time span in one times (from 7:00 to 8:30, on March 30, 2019) is 1.5 h andthe measurement interval is 5 s, there are 1080 observation values in total. The results in first experiment(namely from 7:00 to 8:30, on March 30, 2019) are shown in Fig. 6, with the mean offset 3.08 cm and theSTD value of 21.45 cm (see the first row of Table 3).We can see that although the STD is relatively large, at the decimeter level, the mean offset value issmall, at the centimeter level. That is because the main component of the clock errors is white noises, whilethe drift and random walk effects are not quite obvious in the results. Therefore, since the stability of theclock can be significantly improved after a period (say one hour) of integration, as demonstrated by Oelkeret al. (2019), the height unification accuracy could be improved after a period of integration.In order to improve the accuracy of the results, we may use multi-observations, namely we may useobservations in different time periods in one day or different days. To improve the precision, with differentrandomly chosen coefficients a i , b i , c i and d i in Eqs. (16) and (17) we run 10 times simulation experimentsin total. The behaviors of the offset signals are similar. Thus we only display the mean offset values andSTD values in Fig. 7. We can see that the mean offsets are limited to centimeter level, and the largest meanoffset is 4.77 cm in the 8th experiment. The final results are listed in Table 3.In practical applications, we may use several days’ data to estimate the height difference. For instance,13 able 3: The results of 10 simulation experiments. Relevant parameters are listed in Table 1Experiment Height diff. between China’s Offset to true STDNo. VHS and US’ VHS (m) value (1 m) (m)1 1.0308 0.0308 0.21452 1.0061 0.0061 0.21513 1.0257 0.0257 0.21304 0.9975 -0.0025 0.20565 1.0399 0.0399 0.21166 1.0277 0.0277 0.20947 0.9961 -0.0039 0.21128 1.0477 0.0477 0.20799 0.9781 -0.0219 0.211010 1.0180 0.0180 0.2108Average 1.0168 0.0168 0.2110 if 10 different experiments are conducted in different time periods (e.g. continuous 10 days, every day from7:00 to 8:30), the final results can be improved after taking average.
In this study we formulated an approach to unify different local vertical height systems in centimeterlevel via ultra-high precise frequency signal links between one satellite and two datum stations separatedby oceans, and performed simulation experiments addressing this issue by taking an example of connectingthe China’s VHS and the US’s VHS based on the SFST approach. Simulation experiment results show that,the deviation between the calculated result based on the “observations” and the true result is around 2 cm,with an accuracy level (STD) of 2 decimeters in 1.5 h, provided that the OACs’ stability achieves the level of . × − in one second. Results of simulation experiments confirmed the reliability of the SFST approachin unifying VHSs, and the precision could be greatly improved by more observations in more time periods.A prerequisite for the SFST approach is frequency synchronization before the measurement, whichmeans that the output frequency of OACs’ oscillators should be identical if their locations have the samegeopotential value. The error of a prior synchronization will also affect the precision of height unificationbased on relativistic geodetic methods. Two clocks can be easily synchronized in the same position; butwhen they are separated at large distance as in our case, it is very challenging to precisely synchronize them.However, it can be realized by a combined method of fiber connection and repeated clock transportation.How to precisely synchronize two separated clocks is a meticulous technique problem, which is not quiterelevant to the main topic of this study. Hence, we will address that problem in a separated paper.14ith quick development of time-frequency science, ultra-high precise OACs (say at × − level orbetter within one hour) have been developed and still under improvement, which make the SFST approachprospective for the unification of VHSs at centimeter level. The SFST approach is also prospective forrealizing the IHRS. As a preliminary study we only connect two stations in this work. However, if a globallycovered SFST network is established, the VHSs all over the world can be unified.Compared to conventional methods, the main merits of SFST method lie in that it can directly determinethe geopotential difference between two arbitrary stations in a relatively short period (say several hours or oneday), and it is not subjected to distance or obstacles such as mountains or oceans. However it also has somelimitations. The first problem is the requirement of ultra-high precise clocks and relevant equipments, whichmakes the measurement relatively difficult currently. Another problem is that its precision is slightly lowerthan the optical fibre links method. Therefore at present, the best practice is to adopt the SFST method as asupplement of conventional methods. For example, we can use SFST method to connect the benchmarks oftwo VHSs far apart, and use conventional methods and optical fibre links method for local VHS unification. Acknowledgements
We would like to express our sincere thanks to Jüergen Kusche, and three anonymous Reviewers fortheir valuable comments and suggestions, which greatly improved the manuscript. This study is supported byNSFC (grant Nos. 41804012, 41631072, 41721003, 41574007 and 41429401), Natural Science Foundationof Hubei Province of China (grant No. 2019CFB611), the Discipline Innovative Engineering Plan of ModernGeodesy and Geodynamics (grant No. B17033), DAAD Thematic Network Project (grant No. 57173947)and ISSI 2017 Supporting Project.
References
Amjadiparvar, B., Rangelova, E., and Sideris, M. G. (2016). The GBVP approach for vertical datum unification: recent results innorth america.
Journal of Geodesy , 90(1):45–63.Bilitza, D., Altadill, D., Truhlik, V., Shubin, V., Galkin, I., Reinisch, B., and Huang, X. (2017). International reference ionosphere2016: From ionospheric climate to real-time weather predictions.
Space Weather , 15(2):418–429.Bjerhammar, A. (1985). On a relativistic geodesy.
Bull. Am. Assoc. Hist. Nurs. , 59(3):207–220.Cacciapuoti, L. and Salomon, C. (2011). Atomic clock ensemble in space ( ACES ).
European Space Agency, (Special Publication)ESA SP , 327(385):295–297.Drewes, H., Kuglitsch, F., Adám, J., and Rózsa, S. (2016). The geodesist’s handbook 2016.
J. Geodesy , 90(10):907–1205.Einstein, A. (1915). Die feldgleichungen der gravitation.
Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften(Berlin). , 1:844–847.Flury, J. (2016). Relativistic geodesy.
J. Phys. Conf. Ser. , 723(1):012051.Galleani, L., Sacerdote, L., Tavella, P., and Zucca, C. (2003). A mathematical model for the atomic clock error.
Metrologia ,40(3):S257. erlach, C. and Rummel, R. (2013). Global height system unification with GOCE: a simulation study on the indirect bias term inthe GBVP approach. J Geod , 87(1):57–67.Grotti, J., Koller, S., Vogt, S., Häfner, S., Sterr, U., Lisdat, C., Denker, H., Voigt, C., Timmen, L., Rolland, A., Baynes, F. N.,Margolis, H. S., Zampaolo, M., Thoumany, P., Pizzocaro, M., Rauf, B., Bregolin, F., Tampellini, A., Barbieri, P., Zucco, M.,Costanzo, G. A., Clivati, C., Levi, F., and Calonico, D. (2018). Geodesy and metrology with a transportable optical clock.
Nat.Phys. , 14(5):437–441.Hirt, C. and Kuhn, M. (2012). Evaluation of high-degree series expansions of the topographic potential to higher-order powers:TOPOPOTENTIAL TO HIGHER-ORDER POWERS.
J. Geophys. Res. , 117(B12).Hofmann-Wellenhof, B. and Moritz, H. (2005).
Physical geodesy . Springer.Horemuž, M. and Andersson, J. V. (2006). Polynomial interpolation of GPS satellite coordinates.
GPS Solutions , 10(1):67–72.Huang, Y., Guan, H., Zeng, M., Tang, L., and Gao, K. (2019). Ca + ion optical clock with micromotion-induced shifts below10 − . Phys. Rev. A , 99(1):011401.Ihde, J., Mäkinen, J., and Sacher, M. (2008). Conventions for the definition and realization of a european vertical reference system(EVRS)–EVRS conventions 2007.
EVRS Conv , pages 1–10.Ihde, J., Sánchez, L., Barzaghi, R., Drewes, H., Foerste, C., Gruber, T., Liebsch, G., Marti, U., Pail, R., and Sideris, M. (2017).Definition and proposed realization of the international height reference system (IHRS).
Surv. Geophys. , 38(3):549–570.Kang, Z., Tapley, B., Bettadpur, S., Ries, J., Nagel, P., and Pastor, R. (2006). Precise orbit determination for the GRACE missionusing only GPS data.
J. Geodesy , 80(6):322–331.Kopeikin, S., Efroimsky, M., and Kaplan, G. (2011). Relativistic geodesy. In
Relativistic Celestial Mechanics of the Solar System ,volume 83 of
Bulletin of the American Astronomical Society , pages 671–714. Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim,Germany.Leslie, F. W. and Justus, C. G. (2011). The NASA marshall space flight center earth global reference atmospheric model-2010version. Technical report.Li, F., Lei, J., Zhang, S., Ma, C., Hao, W., E, D., and Zhang, Q. (2018). The impact of solid earth-tide model error on troposphericzenith delay estimates and GPS coordinate time series.
Survey Review , 50(361):355–363.Lion, G., Panet, I., Wolf, P., Guerlin, C., Bize, S., and Delva, P. (2017). Determination of a high spatial resolution geopotentialmodel using atomic clock comparisons.
J. Geodesy , 91(6):597–611.Lisdat, C., Grosche, G., Quintin, N., Shi, C., Raupach, S. M. F., Grebing, C., Nicolodi, D., Stefani, F., Al-Masoudi, A., Dörscher,S., Häfner, S., Robyr, J.-L., Chiodo, N., Bilicki, S., Bookjans, E., Koczwara, A., Koke, S., Kuhl, A., Wiotte, F., Meynadier, F.,Camisard, E., Abgrall, M., Lours, M., Legero, T., Schnatz, H., Sterr, U., Denker, H., Chardonnet, C., Le Coq, Y., Santarelli, G.,Amy-Klein, A., Le Targat, R., Lodewyck, J., Lopez, O., and Pottie, P.-E. (2016). A clock network for geodesy and fundamentalscience.
Nat. Commun. , 7:12443.Luz, R. T., Fortes, L. P. S., Hoyer, M., and Drewes, H. (2002). The vertical reference frame for the americas — the sirgas 2000 GPScampaign —. In
Vertical Reference Systems , pages 302–305. Springer Berlin Heidelberg.Major, F. G. (2013).
The Quantum Beat: The Physical Principles of Atomic Clocks . Springer Science & Business Media.McGrew, W. F., Zhang, X., Fasano, R. J., Schäffer, S. A., Beloy, K., Nicolodi, D., Brown, R. C., Hinkley, N., Milani, G., Schioppo,M., Yoon, T. H., and Ludlow, A. D. (2018). Atomic clock performance enabling geodesy below the centimetre level.
Nature ,564(7734):87–90.Mehlstäubler, T. E., Grosche, G., Lisdat, C., Schmidt, P. O., and Denker, H. (2018). Atomic clocks for geodesy.
Rep. Prog. Phys. ,81(6):064401.Millman, G. H. and Arabadjis, M. C. (1984). Tropospheric and ionospheric phase perturbations and doppler frequency shift effects.
Nasa Sti/recon Technical Report N , 85.Müller, J., Soffel, M., and Klioner, S. A. (2008). Geodesy and relativity.
J. Geodesy , 82(3):133–145.Namazov, S. A., Novikov, V. D., and Khmel’nitskii, I. A. (1975). Doppler frequency shift during ionospheric propagation of ecameter radio waves (review). Radiophys. Quantum Electron. , 18(4):345–364.Oelker, E., Hutson, R. B., Kennedy, C. J., Sonderhouse, L., Bothwell, T., Goban, A., Kedar, D., Sanner, C., Robinson, J. M., Marti,G. E., Matei, D. G., Legero, T., Giunta, M., Holzwarth, R., Riehle, F., Sterr, U., and Ye, J. (2019). Demonstration of 4.8 × stability at 1 s for two independent optical clocks. Nat. Photonics , 13(10):714–719.Pavlis, N. K., Holmes, S. A., Kenyon, S. C., and others (2012). The development and evaluation of the earth gravitational model2008 (EGM2008). research: solid earth , 117(B4):406.Penna, N. T., Bos, M. S., Baker, T. F., and Scherneck, H.-G. (2008). Assessing the accuracy of predicted ocean tide loadingdisplacement values.
J. Geodesy , 82(12):893–907.Petit, G. and Luzum, B. (2010). IERS conventions (2010).Poutanen, M., Vermeer, M., and Mäkinen, J. (1996). The permanent tide in GPS positioning.
J. Geodesy , 70(8):499–504.Puetzfeld, D. and Lämmerzahl, C., editors (2019).
Relativistic Geodesy: Foundations and Applications . Springer, Cham.Rangelova, E., Sideris, M. G., Amjadiparvar, B., and Hayden, T. (2016). Height datum unification by means of the GBVP approachusing tide gauges. In
VIII Hotine-Marussi Symposium on Mathematical Geodesy , pages 121–129. Springer InternationalPublishing.Rawer, K., Bilitza, D., and Ramakrishnan, S. (1978). Goals and status of the international reference ionosphere.
Rev. Geophys. ,16(2):177.Riehle, F. (2017). Optical clock networks.
Nat. Photonics , 11:25.Rummel, R. and Teunissen, P. (1988). Height datum definition, height datum connection and the role of the geodetic boundaryvalue problem.
Bull. Am. Assoc. Hist. Nurs. , 62(4):477–498.Sanchez, L. (2007). Definition and realisation of the SIRGAS vertical reference system within a globally unified height system.In Tregoning, P. and Rizos, C., editors,
Dynamic Planet: Monitoring and Understanding a Dynamic Planet with Geodetic andOceanographic Tools IAG Symposium Cairns, Australia 22–26 August, 2005 , pages 638–645. Springer Berlin Heidelberg, Berlin,Heidelberg.Sánchez, L., Čunderlík, R., Dayoub, N., Mikula, K., Minarechová, Z., Šíma, Z., Vatrt, V., and Vojtíšková, M. (2016). A conventionalvalue for the geoid reference potential 0W0.
J. Geodesy , 90(9):815–835.Sánchez, L. and Sideris, M. G. (2017). Vertical datum unification for the international height reference system (IHRS).
Geophys.J. Int. , 209(2):570–586.Sharifi, M. A., Seif, M. R., and Hadi, M. A. (2013). A comparison between numerical differentiation and kalman filtering for a leosatellite velocity determination.
Artificial Satellites , 48(3):103–110.Shen, W., Chao, D., and Jin, B. (1993). On relativistic geoid.
Bollettino di geodesia e scienze affini , 52(3):207–216.Shen, Z., Shen, W.-B., Peng, Z., Liu, T., Zhang, S., and Chao, D. (2019). Formulation of determining the gravity potential differenceusing Ultra-High precise clocks via optical fiber frequency transfer technique.
J. Earth Sci. , 30(2):422–428.Shen, Z., Shen, W.-B., and Zhang, S. (2016). Formulation of geopotential difference determination using optical-atomic clocksonboard satellites and on ground based on doppler cancellation system.
Geophys. J. Int. , 206(2):1162–1168.Shen, Z., Shen, W.-B., and Zhang, S. (2017). Determination of gravitational potential at ground using Optical-Atomic clocks onboard satellites and on ground stations and relevant simulation experiments.
Surv. Geophys. , 38(4):757–780.Sideris, M. (2015). Geodetic world height system unification. In Freeden, W., Zuhair Nashed, M., and Sonar, T., editors,
Handbookof Geomathematics , pages 3067–3085. Springer Berlin Heidelberg.Stöcker-Meier, E. (1990). Theory of oceanic levelling for improving the geoid from satellite altimetry.
Bull. Am. Assoc. Hist. Nurs. ,64(3):247–258.Takano, T., Takamoto, M., Ushijima, I., Ohmae, N., Akatsuka, T., Yamaguchi, A., Kuroishi, Y., Munekane, H., Miyahara, B., andKatori, H. (2016). Geopotential measurements with synchronously linked optical lattice clocks.
Nat. Photonics , 10(10):662.Torge, W. and Müller, J. (2012).
Geodesy . Walter de Gruyter.Van Camp, M. and Vauterin, P. (2005). Tsoft: graphical and interactive software for the analysis of time series and earth tides. omput. Geosci. , 31(5):631–640.Vermeer, M. (1983). Chronometric Levelling . Geodeettinen Laitos, Geodetiska Institutet.Vessot, R. F. C. and Levine, M. W. (1979). A test of the equivalence principle using a space-borne clock.
Gen. Relat. Grav. ,10(3):181–204.Vessot, R. F. C., Levine, M. W., Mattison, E. M., Blomberg, E. L., Hoffman, T. E., Nystrom, G. U., Farrel, B. F., Decher, R., Eby,P. B., Baugher, C. R., Watts, J. W., Teuber, D. L., and Wills, F. D. (1980). Test of relativistic gravitation with a Space-Bornehydrogen maser.
Phys. Rev. Lett. , 45(26):2081–2084.Voigt, C., Förste, C., Wziontek, H., Crossley, D., Meurers, B., Pálinkáš, V., Hinderer, J., Boy, J.-P., Barriot, J.-P., and Sun, H.(2017). The data base of the international geodynamics and earth tide service (IGETS). In
EGU General Assembly ConferenceAbstracts , volume 19, page 4947.Weinberg, S. (1972).
Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity . Wiley, NewYork.Wenzel, H.-G. (1996). The nanogal software: Earth tide data processing package ETERNA 3.30.
Bull. Inf. Marées Terrestres ,124:9425–9439.Woodworth, P. L., Hughes, C. W., Bingham, R. J., and Gruber, T. (2012). Towards worldwide height system unification using oceaninformation.
Journal of Geodetic Science , 2(4):302–318.Wu, H., Müller, J., and Lämmerzahl, C. (2019). Clock networks for height system unification: a simulation study.
Geophys. J. Int. ,216(3):1594–1607. igure 2: Ground station E emits a frequency signal f e at time t , denoted by uplink (blue line). Satellite S transmitsthe received signal f (cid:48) e (the downlink denoted by blue line) and emits a new frequency signal f s at time t (the downlinkdenoted by dark-blue line). The ground station receives signals f (cid:48)(cid:48) e and f (cid:48) s at time t at position E (cid:48) . φ is gravitationalpotential (GP), (cid:126)r is position vector. igure 3: Connection of China HS originated at Qingdao datum and USA HS originated at San Francisco datum viasatellite frequency signal transmission.
Figure 4:
The scheme of the simulation experiment. igure 5: Experiments are conducted at the time duration when the satellite SVN-56 moves from position S to position S (cid:48) (from 7:00 am to 8:30 am, March 30, 2019). igure 6: The offset between true values and estimated values of Height datum difference determined by SVN-56satellite.
Figure 7: