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Clock networks for height system unification: a simulation study

Clock networks for height system unification: a simulation study SUMMARY The unification of local height systems has been a classical geodetic problem for a long time, the main challenges of which are the estimation of offsets between different height systems and the correction of tilts along the levelling lines. It has been proposed to address these challenges with clock networks. The latest generation of optical clocks as well as the dedicated frequency links, for example optical fibres, are now approaching to deliver the comparison of frequencies at the level of 1.0 × 10−18. It corresponds to an accuracy of about 1.0 cm in height difference. Clock networks can thus serve as a powerful tool to connect local height systems. To verify the idea, we carried out simulations using the EUVN/2000 (European Unified Vertical Network) as apriori input. Four local height systems were simulated from the EUVN/2000 by introducing individual offsets and tilts, and were reunified by using measurements in clock networks. The results demonstrate the great potential of clock networks for height system unification. In case that the offsets between different height systems and tilts along national levelling lines in both longitudinal and latitudinal directions are considered, three or four clocks measurements for each local region are sufficient for the unification. These clocks are to be interconnected and should be properly arranged so that they can sense the levelling tilts where necessary. Our results also indicate that even clocks with one magnitude poorer accuracy than the desired ones can still unify the height systems to some extent, but it may cause a shift for the reunified system. Geodetic instrumentation, Geopotential theory, Reference systems, Satellite geodesy 1 INTRODUCTION Establishing a consistent and accurate global height system is one of the major tasks of the International Association of Geodesy (IAG) which released a resolution for the definition and realization of an International Height Reference System (IHRS) in 2015 (Drewes et al.2016; Ihde et al.2017). Such a height system is essential for engineering constructions, for example building a bridge or tunnel that connects different countries, and scientific research in geodynamics, geophysics and oceanography, as well as for environmental monitoring such as global sea level changes, natural disasters (e.g. floods and droughts) and surface deformations (e.g. crustal motion and subsidence), see Sideris (2014). To realize a global height system, a widely discussed strategy is to unify the local height systems worldwide. The datum of a local height system is usually tied to tide gauges which measure the mean sea level locally. However, the local mean sea level does not exactly coincide with the geoid, that is a selected equipotential surface of the Earth’s gravity field. The inconsistencies are up to ±1–2m in some regions (Torge & Müller 2012), mainly caused by the sea surface topography. Consequently, there exist offsets between different vertical datums, and the determination of these offsets is one of the main issues to be solved in height system unification. In addition to the datum, local height systems are composed of many levelling points. The height values of these points are related to the datum by spirit levelling. However, height errors accumulate during spirit levelling over long distances. As a result, tilts appear in national vertical networks, at the level of 1–3 cm/100 km (Gruber et al.2014). The estimation of those tilts is a further major task in height system unification. Three approaches have been extensively discussed in the literature for height system unification, namely geodetic levelling, oceanographic modelling and the Geodetic Boundary Value Problem (GBVP) approach (Colombo 1980; Pavlis 1991; Rummel 2001; Sánchez 2012). Each of them has advantages and drawbacks. The classical geodetic method which applies spirit levelling and terrestrial gravimetry is a time-consuming way to connect local height systems. In terms of accuracy, it can achieve submillimetre relative accuracy over short distances, while the absolute accuracy with respect to a global datum can reach up to ±2 m (Sánchez & Sideris 2017). Oceanographic modelling is a powerful approach to connect the local height systems that are separated by oceans. Unfortunately, it suffers from the poor availability of observations at high temporal and spatial resolutions and heavily relies on the assumption about hydrostatic and geostrophic equilibrium (Sánchez 2012; Woodworth et al.2012). Additionally, the GBVP approach has been modified and applied to estimate the offsets between different height systems (Rummel & Teunissen 1988; Gerlach & Rummel 2013; Sideris 2014; Amjadiparvar et al.2016; Sánchez & Sideris 2017). Its major drawbacks, however, are the use of different standards in different regions (e.g. gravity data: gravity anomaly or gravity disturbance; tide systems: mean or zero tide system for computing the potential difference; gravity field models: different models with different spherical harmonic degree and order for removing the long-wavelength contributions) and the restricted accessibility to terrestrial gravity data (Rummel 2001; Sideris 2014; Sánchez & Sideris 2017). In further works, some vertical datum unification experiments are based on the GNSS/geoid (Global Navigation Satellite System) method, using some GOCE (Gravity field and steady-state Ocean Circulation Explorer) geoid or GOCE-based combined geoid solutions (Gruber et al.2012, 2014). However, the gravity field model might show poor performance in sparsely surveyed regions. For extreme cases, the errors are up to ±1 m (Gruber et al.2014), which are even larger than some datum offsets. Alternatively, we investigate the application of relativistic geodesy and the use of clock networks for height system unification. Einstein’s theory of general relativity predicts that the gravity potential difference (closely related to the height difference) can be derived from the comparison of clocks’ frequencies. This concept is called relativistic geodesy and was described in detail by Bjerhammar (1985, 1986). Thanks to the rapid development of high-performance clocks and frequency transfer techniques, frequency comparison between two distant clocks can hopefully soon achieve the level of 1.0 × 10−18, equivalent to about 1.0 cm in terms of height difference. Hence, clocks appear as a promising option for connecting local height systems, especially where no spirit levelling surveys are possible. Using clocks for height system unification has been one of the geodetic research topics covered by geo-Q (Collaborative Research Centre 1128 ‘Relativity geodesy and gravimetry with quantum sensors’),1 funded by the Deutsche Forschungsgemeinschaft (DFG). Under the scope of geo-Q, we are evaluating different approaches for suitable application in practice before the real clock data are available. Through simulations, we address several research questions that are of interest within both the geodesy and physics communities: How well can clock networks unify local height systems? How many clocks are required for the unification? How can the clock network be optimised? And how does the clocks’ accuracy affect the unification? The knowledge gained from this study will benefit the practical application when sufficient data in clock networks become available in the future. The paper is organised as follows. The relevant background knowledge is reviewed in Section 2, such as the definition of a height system, the theory of relativistic geodesy and the development of clock networks. The simulation strategy and the input data are described in Section 3. Section 4 presents our results and discussions. Some concluding remarks and future work are addressed in the last section. 2 HEIGHT SYSTEM, RELATIVISTIC GEODESY AND OPTICAL CLOCKS 2.1 Height system By definition, height describes the vertical distance of any geospatial point to a reference surface. Nowadays, because of the widely applied GNSS positioning method, the ellipsoidal height (with the ellipsoid surface as reference) can be conveniently obtained. However, due to their pure geometric nature, the GNSS measurements lack the physical interpretation of heights. They cannot properly reflect, that is the direction of the water flow. To address the physical meaning, heights are differentially determined with reference to an equipotential surface such as the geoid. There are three types of physical height, namely orthometric, normal and dynamic height (Torge & Müller 2012). Each of them fundamentally refers to the geopotential number CP, that is the gravity potential difference between the geoid (with potential value W0) and the point P (with potential value WP) in question (Jekeli 2000; Torge & Müller 2012): \begin{eqnarray*} C_P = C(P) = W_0 - W_P . \end{eqnarray*} (1) The geopotential number can be appropriately scaled to a height value using \begin{eqnarray*} H_P = \frac{C_P}{\hat{g}_P} = \frac{W_0 - W_P}{\hat{g}_P} , \end{eqnarray*} (2) where |$\hat{g}_P$| is an appropriate gravity value, depending on the type of height. Specifically, the orthometric, normal and dynamic height are obtained when |$\hat{g}_P$| is chosen as mean gravity along the plumb line, mean normal gravity along the normal plumb line and normal gravity at mid-latitude. For more details, we refer to Jekeli (2000) and Torge & Müller (2012). While the height values at the same point P vary in accordance with the gravity values introduced in eq. (2), the geopotential numbers (eq. 1) are unique. To realize a physical height system, the reference equipotential surface has to be defined first. Typically, the mean sea level is taken as reference. It is observed and determined by tide gauges, and coincides with a local equipotential surface. A reference benchmark (or a datum point) is then defined and tied to the tide gauge, and its height above the mean sea level is measured and assigned. In addition to the datum point, some high-order control points realize the framework of such a practical height system. The heights of all control points are connected to the datum by spirit levelling and gravimetry. A major problem arises when the mean sea level is used as the height reference. In fact, the mean sea level determined at each tide gauge locally coincides with a different equipotential surface. These surfaces do not strictly coincide with the same global equipotential surface (i.e. the geoid, see Fig. 1). The difference between these surfaces reaches the same magnitude (up to ±2 m) as the so-called sea surface topography, which represents the discrepancy between the mean sea surface and the geoid (Torge & Müller 2012). Figure 1. View largeDownload slide Offsets between different height datums. The equipotential surfaces for the local height system j and j+1 exhibit discrepancies with respect to the global geoid. The potential offsets ΔW0, j and ΔW0, j+1 (resp. height offsets) have to be estimated for height system unification. Figure 1. View largeDownload slide Offsets between different height datums. The equipotential surfaces for the local height system j and j+1 exhibit discrepancies with respect to the global geoid. The potential offsets ΔW0, j and ΔW0, j+1 (resp. height offsets) have to be estimated for height system unification. In addition, there are tilts along the levelling lines in local or national height systems. Tilts along the longitudinal and latitudinal directions were detected when unifying national height systems in Europe (Rülke et al.2012; Gruber et al.2014). The tilts can reach about 1–3 cm/100 km and are mainly attributed to the accumulated errors of spirit levelling over long distances. Furthermore, different conventions for the national height systems and the heterogeneity of data sets (e.g. different epochs and network configurations) may cause tilts along levelling lines. As an example, Fig. 2 shows the tilts of the national height systems in France, Germany and Spain, estimated from GOCE gravity field models combined with GNSS/levelling. Figure 2. View largeDownload slide Tilts of national height systems. The estimated tilts along the longitudinal and latitudinal directions are (1.0, −2.7), (0.9, −1.4) and (0.8, −3.0) cm/100 km for France, Germany and Spain, respectively. The number and the figures are taken from Gruber et al. (2014). Figure 2. View largeDownload slide Tilts of national height systems. The estimated tilts along the longitudinal and latitudinal directions are (1.0, −2.7), (0.9, −1.4) and (0.8, −3.0) cm/100 km for France, Germany and Spain, respectively. The number and the figures are taken from Gruber et al. (2014). Over 100 local height systems are in use worldwide. Their unification is one major research topic of IAG. The main objective is the precise determination of vertical datum offsets and possible tilts in the levelling networks. As discussed in Section 1, various approaches have been considered to address this issue. However, each of these approaches has its limitations. Therefore, further effort is required in order to realize the precise unification of local height systems. 2.2 Relativistic geodesy Einstein’s theory of general relativity predicts that clocks tick at different rates if they are transported with different speeds or they are under the influence of different gravitational potential. Considering the case of two earthbound clocks at rest (on the Earth surface), the change of the clocks’ frequencies is proportional to the corresponding difference in the gravity potential (sum of the gravitational potential and the centrifugal potential) between the two clocks (Bjerhammar 1985, 1986). It reads \begin{eqnarray*} \frac{\Delta f_{21}}{f_1} = \frac{f_2 - f_1}{f_1} = \frac{W_2 - W_1}{c^2} + O(c^{-4}) , \end{eqnarray*} (3) where f1 and f2 are the proper frequencies at points of 1 and 2 at the Earth’s surface, while W1 and W2 are the corresponding gravity potential values, c is the speed of light. Higher order terms are neglected. Based on eqs (1) and (3), we obtain \begin{eqnarray*} \frac{\Delta f_{21}}{f_1} = \frac{(W_2 - W_0) + (W_0 - W_1)}{c^2} = - \frac{C_2 - C_1}{c^2} , \end{eqnarray*} (4) where C1 and C2 are the corresponding geopotential numbers. In geodesy, the differences between geopotential numbers can be straightforwardly used to derive physical height differences. For example, the difference between two orthometric heights |$\Delta H_{21}^{\ast }$| can be obtained as (Müller et al.2018): \begin{eqnarray*} \Delta H_{21}^{\ast } &=& H_{2}^{\ast } - H_{1}^{\ast } = H_{1}^{\ast } \frac{\Delta \bar{g}_{12}}{\bar{g}_2} + \frac{C_2 - C_1}{\bar{g}_2} \nonumber \\ &=& H_{1}^{\ast } \frac{\Delta \bar{g}_{12}}{\bar{g}_2} - \frac{c^2}{\bar{g}_2} \frac{\Delta f_{21}}{f_1} , \end{eqnarray*} (5) where |$\Delta \bar{g}_{12} = \bar{g}_1 - \bar{g}_2$| is the difference of the mean gravity values (i.e. the mean along the plumb line between the Earth’s surface and the geoid) at both locations. Note that eq. (5) can be formulated for normal heights accordingly. Then, the mean normal gravity has to be used, that is |$\bar{\gamma }$| instead of |$\bar{g}$|⁠. This method to obtain heights through the comparison of clocks is called chronometric levelling (Vermeer 1983). The scheme is shown in Fig. 3. For an approximate estimation, a relative frequency inaccuracy of 1.0 × 10−18 corresponds to an uncertainty of about 1.0 cm in height. Clocks have the advantage to connect distant areas, for example they deliver physical height differences for the observed points without being affected by accumulated levelling errors or by some smoothing effects when combined global gravity field models (determined from multisource gravimetry data) are used. Therefore, high-performance clocks, for example at the level of 1.0 × 10−18 will be a powerful tool for delivering physical heights and with them, a well-suitable approach for the connection of height systems. Figure 3. View largeDownload slide Scheme of chronometric levelling (adapted from Müller et al.2018). Figure 3. View largeDownload slide Scheme of chronometric levelling (adapted from Müller et al.2018). 2.3 Optical clock networks When relativistic geodesy was described by Bjerhammar in 1980s, the hydrogen maser was probably the most accurate clock, with a relative frequency uncertainty of 10−13. The hydrogen maser is a kind of atomic clock, which uses an electron transition frequency in the microwave band as frequency standard. Microwave clocks were improved by three orders of magnitude in the past decades, now approaching the level of 10−16 (Heavner et al.2014), see also Fig. 4. This is however about two orders of magnitude away from our desired level in geodesy. Another type of atomic clock using optical transitions, the so-called optical clock, was proposed and rapidly developed. Since 1990, the uncertainty of optical clocks was improved by about two orders of magnitude every ten years, see Fig. 4. Recently, two optical clocks in laboratories approached the 10−18 level (Bloom et al.2014; Huntemann et al.2016). Moreover, a transportable optical clock that is more likely to be used for a field measurement campaign in future was reported at the level of 10−17 (Koller et al.2017). Figure 4. View largeDownload slide Evolution of fractional frequency uncertainty of atomic clocks based on microwave (Cs clocks) and optical transitions. Figure 4. View largeDownload slide Evolution of fractional frequency uncertainty of atomic clocks based on microwave (Cs clocks) and optical transitions. To support the comparison of distant clocks, frequency transfer techniques, such as fibre link, free air link and radio frequency link (Riehle 2017), have been extensively experimented and developed. Among them, the frequency link via optical fibres shows the best performance, which reached an extreme accuracy of 2.5 × 10−19 over 1400  km, cf. Lisdat et al. (2016). This makes the comparison of accurate clocks over long distances possible and opens the door for novel geodetic applications. Considerations on the benefit of clocks in geodesy were addressed by Chou et al. (2010); Delva & Lodewyck (2013). The suitability of chronometric levelling using optical clocks was demonstrated through experiments. Two clocks connected by a 75-m-long fibre delivered a height difference of 37 ± 15 cm. This finding agreed quite well with the known value of 33 cm (Chou et al.2010). A master (reference) and slave (secondary) clock separated by 15 km delivered the height difference with an uncertainty of 5 cm for a height difference of 1516 m (Takano et al.2016). Very recently, a transportable clock was first used for a field measurement campaign in a mountain area between France and Italy (Jacopo et al.2018). The potential difference inferred from chronometric levelling is 10 034 m2 s−2, which is in a good agreement with the value of 10 032.1 m2 s−2 determined independently by geodetic methods. Vice versa, the relativistic redshift effect, that is the shift of clock’s frequency caused by the gravity field, has to be corrected for the frequency standards. This effect was estimated by geodetic methods, for example the GNSS/geoid approach, with one case study in the United States (Pavlis & Weiss 2003, 2017) and another one in Europe (Denker et al.2018). In addition to the corrections related to the static part of gravity field, temporal variations caused by tides and non-tidal mass redistribution were computed and discussed by Voigt et al. (2016), with the biggest contribution from solid Earth tides and ocean tides, being relevant at a 10−17 level. Clocks were also considered to realize a space–time reference which can provide precise time worldwide, a valuable reference for geodesy (Berceau et al.2016). The impact of highly precise clocks for the realization of geodetic reference frames, timescales and the determination of the Earth’s gravity field was thoroughly discussed by Müller et al. (2018). Furthermore, Philipp et al. (2017) defined the geoid as a level surface of a time-independent redshift potential under the formalism of general relativity without any approximation. A simulation study was ran by Lion et al. (2017) to evaluate the contribution of clock measurements for determining the high-resolution geopotential. Sustainable efforts are being made to improve the capability of optical clocks and to study the application of relativistic geodesy, for example in the research project geo-Q at Leibniz Universität Hannover. We are optimistic that the frequency comparison between clocks can be realized at the level of 10−18 in near future. This will definitely open a new window for geodesy and beyond. 3 THE SIMULATION FRAMEWORK 3.1 Description of the simulator A simulator has been designed to verify the idea of using clocks for height system unification. The scheme of the simulator is depicted in Fig. 5. It takes a known height system as input. All levelling points of this height system are divided into several groups and each group forms a local height system. The generated local height systems are then reunified using simulated potential differences that are assumed to be obtained from observed frequency differences between clocks. Figure 5. View largeDownload slide The scheme of the simulator. Figure 5. View largeDownload slide The scheme of the simulator. The known normal heights of the levelling points are taken as true signals, while errors are treated individually for each local height system and are simulated independently. Based on the previous discussion, random errors (white noise) and systematic errors (biases and tilts) are considered in the simulation. Random height errors are simplified as white noise. Individual biases (accounting for the offsets between height systems) and tilts (responsible for the slopes along levelling lines) are introduced so that realistic height values in each local height system are obtained. We assume that a certain number of clocks is used in each local height system and all the clocks to be interconnected (every two clocks are connected with each other). We can thus obtain |$\left({k \atop 2} \right)$| measurements, with k the total number of clocks. These measurements, that are the potential differences (resp. the frequency differences between clocks), are simulated from the true height differences of EUVN/2000, cf. eq. (2). Concerning the errors, clock-based measurements are idealized to be independent and consistent. Moreover, clocks are not affected by the offsets and tilts of different height systems. Therefore, white noise with a desired level of 1.0 × 10−18 in terms of relative frequency is added to the clock-based measurements. A joint adjustment of the normal heights and the potential differences is finally applied to reunify the local height systems. Through this procedure, heights in a local system are adjusted to a unified system where all levelling points are tied to a unique datum point. The performance of the unification is assessed by comparing the heights in the reunified system and the true heights in the a priori system. 3.2 Data preparation The EUVN/2000 (European Vertical Reference Network) is adopted as a priori system in our study. It is an integrated height reference frame for Europe, which was completed in 2000 and published by Ihde et al. (2000). Although the EUVN/2000 is not the latest solution, it can fully meet our purpose for the simulation, because heights of levelling points are only taken as signals, while noise is simulated independently. The EUVN/2000 consists of about 200 high quality levelling points covering very well the European continent, see Fig. 6. Now, we classify all the levelling points into four groups. The height datums of these groups are supposed to be tied to well-known tide gauges in: (1) Marseille in France, (2) Newlyn in United Kingdom, (3) NAP (Normaal Amsterdams Peil) in the Netherlands and (4) Genova in Italy. Basically, levelling points are assigned to one of these groups based on the spatial distance. However, the administrative attribution is also taken into consideration. For example, the height datum of some North-European countries are known as the NAP, so that the levelling points in these countries are naturally assigned to the group of NAP. Fig. 6 shows the final division and Table 1 lists the exact number of levelling points for each group. Figure 6. View largeDownload slide Geographic distribution of EUVN/2000 levelling points and the classification of these points into four groups. Figure 6. View largeDownload slide Geographic distribution of EUVN/2000 levelling points and the classification of these points into four groups. Table 1. Number of levelling points for each local group. Total # 29 16 73 84 202 Total # 29 16 73 84 202 View Large Table 1. Number of levelling points for each local group. Total # 29 16 73 84 202 Total # 29 16 73 84 202 View Large Each point in EUVN/2000 is available with its 3-D coordinates, including the plane coordinates (latitude and longitude) and the normal heights. The plane coordinates are kept as noise-free as they can be determined with very high accuracy in practice, for example mm level. And, they are only used to geo-locate the levelling points. For the heights, random and systematic errors (both offsets and tilts) are introduced. As random error, white noise with a magnitude of 1.0 cm is generated for all levelling heights. The offsets are treated as constant biases and added to the levelling heights for each local system. The tilts along the latitudinal and longitudinal directions are considered, with value of 1–3 cm/100 km. In the end, four local height systems with individual errors are obtained. 3.3 Reunification We can set up the functional model for the normal heights in a local system as \begin{eqnarray*} H_i^L = \frac{C_i^U}{\overline{\gamma }_i} + a^L \Delta X_i^L + b^L \Delta Y_i^L + c^L , \end{eqnarray*} (6) where |$H_i^L$| is the normal height of point i in local height system L, |$C_i^U$| is the geopotential number of the corresponding point in the unified system U, |$\overline{\gamma }_i$| is the mean normal gravity, aL,bL and cL are error parameters that represent the tilts and the offsets, |$\Delta X_i^L, \Delta Y_i^L$| are relative coordinates in latitudinal and longitudinal directions. The functional model for the clock-based data is \begin{eqnarray*} \Delta W_{ij} = W_i^U - W_j^U = - (C_i^U - C_j^U), \end{eqnarray*} (7) where |$\Delta W_{ij}$| are the gravity potential differences delivered by clocks. Considering the number of observations (including both |$H_i^L$| and |$\Delta W_{ij}$|⁠), we have \begin{eqnarray*} N = \sum _{L =1}^4 n_L + \binom{k}{2} , \end{eqnarray*} (8) where nL is the number of levelling points in the local height system L, k is the total number of clocks (⁠|$k = \sum _{L=1}^{4} k_L$|⁠, kL the number of clocks in the local height system). As mentioned previously, all clocks are interconnected so that |$\binom{k}{2}$| measurements are available. However, this might not be exactly true in practice as not all clocks are always linked with each other. The unknowns include the geopotential numbers of all levelling points in the unified system |$C_i^U$| and the error parameters aL, bL, cL. The total number of unknowns M is \begin{eqnarray*} M = \sum _{L=1}^4 (n_L + n_e) - 1, \end{eqnarray*} (9) where ne is the number of error parameters, which is 1, 2 or 3 depending on the error types (see more in Section 3.2). Since one datum is adopted as the unique one for the reunified system, we have one less bias to estimate. Supposing a sufficient number of clocks is available to fulfil the condition |$\binom{k}{2} \gt \sum _{L=1}^{4} n_e - 1$|⁠, the solution can be solved in a least-squares adjustment. Rewriting the function models in a linear matrix form, it reads \begin{eqnarray*} {\left[\begin{array}{c}\boldsymbol {H} \\ \boldsymbol {\Delta W} \end{array}\right]} = {\left[\begin{array}{cc}\boldsymbol {A}_1 & \quad \boldsymbol {A}_2 \\ \boldsymbol {B}_1 & \quad \boldsymbol {B}_2 \end{array}\right]} {\left[\begin{array}{c}\boldsymbol {x}_1 \\ \boldsymbol {x}_2 \end{array}\right]} , \end{eqnarray*} (10) where |$\boldsymbol {H}$| and |$\boldsymbol {\Delta W}$| are observation vectors for normal heights and potential differences, |$\boldsymbol {x}_1$| is the unknown geopotential numbers to be estimated \begin{eqnarray*} \boldsymbol {x}_1 = [ \underbrace{\ldots , C_i^U, \ldots }_{L_1}, \cdots , \underbrace{\ldots , C_j^U, \ldots }_{L_4} ]^T , \end{eqnarray*} (11)|$\boldsymbol {x}_2$| denotes the error parameters. For example, when offsets and tilts in both latitudinal and longitudinal directions are considered, |$\boldsymbol {x}_2$| is written as \begin{eqnarray*} \boldsymbol {x}_2 = [ \underbrace{a^L, b^L, c^L}_{L=1}, \cdots , \underbrace{a^L,b^L,c^L}_{L=4} ]^T. \end{eqnarray*} (12) Note that one of the offset cL can be removed from |$\boldsymbol {x}_2$| by adopting one datum as the final one for the reunified system. |$\boldsymbol {A}_1, \boldsymbol {A}_2, \boldsymbol {B}_1, \boldsymbol {B}_2$| are corresponding design matrices and their elements are provided by eqs (6) and (7). By a least-squares adjustment, the unknowns are solved (Koch 2013) as \begin{eqnarray*} \boldsymbol {x} = (\boldsymbol {A}^T \boldsymbol {P} \boldsymbol {A})^{-1} (\boldsymbol {A}^T \boldsymbol {P} \boldsymbol {l}), \end{eqnarray*} (13) where $\boldsymbol {x} = {\left[\begin{array}{c}\boldsymbol {x}_1 \\\boldsymbol {x}_2 \end{array}\right]}$ ⁠, $\boldsymbol {A} = {\left[\begin{array}{cc}\boldsymbol {A}_1 & \boldsymbol {A}_2 \\\boldsymbol {B}_1 & \boldsymbol {B}_2 \end{array}\right]}$ is the completed design matrix, $\boldsymbol {l} = {\left[\begin{array}{c}\boldsymbol {H} \\\boldsymbol {\Delta W} \end{array}\right]}$ is the observation vector, $\boldsymbol {P} = {\left[\begin{array}{cc}\boldsymbol {P}_1 & \boldsymbol {0} \\\boldsymbol {0} & \boldsymbol {P}_2 \end{array}\right]}$ ⁠, and |$\boldsymbol {P}_1, \boldsymbol {P}_2$| represent weighting matrices for |$\boldsymbol {H}$| and |$\boldsymbol {\Delta W}$|⁠. They are written as |$\boldsymbol {P}_1 = \boldsymbol {I}$| and |$\boldsymbol {P}_2 = \frac{\sigma _{0,\boldsymbol {H}}^2}{\sigma _{0,\boldsymbol {\Delta W}}^2}\boldsymbol {I}$|⁠, where |$\boldsymbol {I}$| represents an identity matrix as observations for each type are assumed to be independent and at the same accuracy level. Note, |$\sigma _{0,\boldsymbol {H}}^2$| and |$\sigma _{0,\boldsymbol {\Delta W}}^2$| should be given in the same physical unit. When the geopotential numbers in the reunified height system are estimated, they are converted to heights and compared to the a priori values. The differences are labelled as adjusted errors (ea). The differences between the heights in the local system and the a priori values are called true errors (et). \begin{eqnarray*} e_a = H_i^{Esti, U} - H_i^{True}, \end{eqnarray*} (14) \begin{eqnarray*} e_t = H_i^{Obs., L} - H_i^{True}, \end{eqnarray*} (15) where |$H_i^{Esti, U}$| are estimated heights in the reunified system, |$H_i^{True}$| are corresponding true heights in the a priori system, |$H_i^{Obs., L}$| are heights in the local system. The adjusted errors can be used to evaluate the performance of the unification, and their comparison to true errors helps to understand the procedure of unification. 4 RESULTS AND DISCUSSIONS Based on the framework described above, we ran three simulation cases, that is CASE A, B, C, with different types of biases and tilts. In these simulations, we investigated how well clock networks can unify local height systems. We also discuss some open questions about clock networks, that is the number of clocks, the optimisation of clock networks and the impact of clocks’ performance. We took only biases into account for CASE A. The biases for the four local systems were set as –18.0, 25.0, 0.0, 8.0 cm, which are at the reasonable range of offsets between national height systems in Europe. We also tested other choices of biases, but they did not affect our final conclusion. Note that the bias for the third group was 0.0 cm, since NAP was adopted as the final datum for the reunified system. In addition to biases, tilts along the latitudinal direction were included for CASE B. The tilts were specified as 3.0, –2.0, 1.5, –3.0 cm/100 km. For a more thorough case, that is CASE C, biases and tilts along both the latitudinal and longitudinal directions were considered, with the tilts along the longitudinal directions as 2.0, 3.0, –1.5 and -2.0 cm/100 km. 4.1 Results for the reunification In order to ensure the success of the unification, a sufficient number of clocks (more than the requirement) were assumed for each case. Considering the complexity of errors in the three cases, we applied two, three and four clocks for CASE A, B, C. This number of clocks was used in each of the four local systems. All clocks were assumed to be identical (the same type of clock at the same accuracy level) so that homogenous measurements can be obtained. The clock-based data is at the desired level of 1.0 × 10−18 in terms of relative frequency. The result of CASE A is shown in Table 2. The input parameters are compared with the recovered ones, and the RMS (Root Mean Square) of the true and adjusted errors are given. The true errors are dominated by systematic errors, that is the offsets. The RMSs of true errors for Group , and are 17.90, 25.06 and 8.09 cm, respectively. They are around the introduced offsets –18.0, 25.0 and 8.0 cm. Similarly, the results of CASE B and C are shown in Tables 3 and 4. The true errors for CASE B and C are slightly more complicated than those of CASE A (they show significant differences to the offsets), due to the influence of tilts. Despite the divergence of the true errors, the adjusted errors for all three cases were reduced to the same level, around 1.0 cm. This value is at the level of the random height errors and clock errors. Table 2. Results for CASE A. In this case, offsets between different local height systems were considered. The input and recovered parameters (cL) are compared. The RMSs of true errors et and adjusted errors ea are shown as well. Unit is cm. Note that the bias for the third group was not estimated. Input parameters Recovered parameters RMS of et RMS of ea Offset Offset [cm] [cm] [cm] [cm] −18.0 −17.36 17.90 1.14 25.0 25.23 25.06 0.91 0 − 1.01 1.01 8.0 8.32 8.09 1.00 Input parameters Recovered parameters RMS of et RMS of ea Offset Offset [cm] [cm] [cm] [cm] −18.0 −17.36 17.90 1.14 25.0 25.23 25.06 0.91 0 − 1.01 1.01 8.0 8.32 8.09 1.00 View Large Table 2. Results for CASE A. In this case, offsets between different local height systems were considered. The input and recovered parameters (cL) are compared. The RMSs of true errors et and adjusted errors ea are shown as well. Unit is cm. Note that the bias for the third group was not estimated. Input parameters Recovered parameters RMS of et RMS of ea Offset Offset [cm] [cm] [cm] [cm] −18.0 −17.36 17.90 1.14 25.0 25.23 25.06 0.91 0 − 1.01 1.01 8.0 8.32 8.09 1.00 Input parameters Recovered parameters RMS of et RMS of ea Offset Offset [cm] [cm] [cm] [cm] −18.0 −17.36 17.90 1.14 25.0 25.23 25.06 0.91 0 − 1.01 1.01 8.0 8.32 8.09 1.00 View Large Table 3. Results for CASE B. In this case, offsets between different local height systems and tilts over national levelling lines in the latitudinal direction were considered. The input and recovered parameters (aL, cL) are compared. Unit for aL is cm/100 km and unit for cL is cm. The RMSs of true errors et and adjusted errors ea are shown as well. Unit is cm. Note that the bias for the third group was not estimated. Input parameters Recovered parameters RMS of et RMS of ea Offset Lat. tilt Offset Lat. tilt [cm] [cm/100 km] [cm] [cm/100 km] [cm] [cm] −18.0 3.0 −18.87 3.12 11.09 1.21 25.0 −2.0 25.69 −2.05 16.16 0.95 0 1.5 − 1.45 18.43 1.12 8.0 −3.0 8.70 −3.09 31.79 1.24 Input parameters Recovered parameters RMS of et RMS of ea Offset Lat. tilt Offset Lat. tilt [cm] [cm/100 km] [cm] [cm/100 km] [cm] [cm] −18.0 3.0 −18.87 3.12 11.09 1.21 25.0 −2.0 25.69 −2.05 16.16 0.95 0 1.5 − 1.45 18.43 1.12 8.0 −3.0 8.70 −3.09 31.79 1.24 View Large Table 3. Results for CASE B. In this case, offsets between different local height systems and tilts over national levelling lines in the latitudinal direction were considered. The input and recovered parameters (aL, cL) are compared. Unit for aL is cm/100 km and unit for cL is cm. The RMSs of true errors et and adjusted errors ea are shown as well. Unit is cm. Note that the bias for the third group was not estimated. Input parameters Recovered parameters RMS of et RMS of ea Offset Lat. tilt Offset Lat. tilt [cm] [cm/100 km] [cm] [cm/100 km] [cm] [cm] −18.0 3.0 −18.87 3.12 11.09 1.21 25.0 −2.0 25.69 −2.05 16.16 0.95 0 1.5 − 1.45 18.43 1.12 8.0 −3.0 8.70 −3.09 31.79 1.24 Input parameters Recovered parameters RMS of et RMS of ea Offset Lat. tilt Offset Lat. tilt [cm] [cm/100 km] [cm] [cm/100 km] [cm] [cm] −18.0 3.0 −18.87 3.12 11.09 1.21 25.0 −2.0 25.69 −2.05 16.16 0.95 0 1.5 − 1.45 18.43 1.12 8.0 −3.0 8.70 −3.09 31.79 1.24 View Large Table 4. Results for CASE C. In this case, offsets between different local height systems and tilts over national levelling lines in both the latitudinal and longitudinal directions were considered. The input and recovered parameters (aL, bL, cL) are compared. Unit for aL, bL is cm/100 km and unit for cL is cm. The RMSs of true errors et and adjusted errors ea are shown as well. Unit is cm. Note that the bias for the third group was not estimated. Input parameters Recovered parameters RMS of et RMS of ea Offset Lat. tilt Lon. tilt Offset Lat. tilt Lon. tilt [cm] [cm/100 km] [cm] [cm/100 km] [cm] [cm] −18.0 3.0 2.0 −17.07 3.12 1.78 22.93 1.28 25.0 −2.0 3.0 22.60 −1.94 3.11 62.35 0.90 0 1.5 −1.5 − 1.39 −1.46 15.39 1.13 8.0 −3.0 −2.0 6.22 −2.95 −1.94 53.44 1.09 Input parameters Recovered parameters RMS of et RMS of ea Offset Lat. tilt Lon. tilt Offset Lat. tilt Lon. tilt [cm] [cm/100 km] [cm] [cm/100 km] [cm] [cm] −18.0 3.0 2.0 −17.07 3.12 1.78 22.93 1.28 25.0 −2.0 3.0 22.60 −1.94 3.11 62.35 0.90 0 1.5 −1.5 − 1.39 −1.46 15.39 1.13 8.0 −3.0 −2.0 6.22 −2.95 −1.94 53.44 1.09 View Large Table 4. Results for CASE C. In this case, offsets between different local height systems and tilts over national levelling lines in both the latitudinal and longitudinal directions were considered. The input and recovered parameters (aL, bL, cL) are compared. Unit for aL, bL is cm/100 km and unit for cL is cm. The RMSs of true errors et and adjusted errors ea are shown as well. Unit is cm. Note that the bias for the third group was not estimated. Input parameters Recovered parameters RMS of et RMS of ea Offset Lat. tilt Lon. tilt Offset Lat. tilt Lon. tilt [cm] [cm/100 km] [cm] [cm/100 km] [cm] [cm] −18.0 3.0 2.0 −17.07 3.12 1.78 22.93 1.28 25.0 −2.0 3.0 22.60 −1.94 3.11 62.35 0.90 0 1.5 −1.5 − 1.39 −1.46 15.39 1.13 8.0 −3.0 −2.0 6.22 −2.95 −1.94 53.44 1.09 Input parameters Recovered parameters RMS of et RMS of ea Offset Lat. tilt Lon. tilt Offset Lat. tilt Lon. tilt [cm] [cm/100 km] [cm] [cm/100 km] [cm] [cm] −18.0 3.0 2.0 −17.07 3.12 1.78 22.93 1.28 25.0 −2.0 3.0 22.60 −1.94 3.11 62.35 0.90 0 1.5 −1.5 − 1.39 −1.46 15.39 1.13 8.0 −3.0 −2.0 6.22 −2.95 −1.94 53.44 1.09 View Large The true errors and adjusted errors for all levelling points are depicted, cf. Fig. 7. The systematic errors for different height systems are clearly visible in the curves of true errors, where biases are distinguishable for CASE A and tilts show significant effects for CASE B and C. However, the adjusted errors for all three cases are more consistent as they behave like random errors and vary around zero. Figure 7. View largeDownload slide True and adjusted errors for CASEs A, B and C. True errors are differences between the heights in the local system and the a priori values, while adjusted errors are differences between the heights in the reunified system and the a priori values. Circles represent clocks. Figure 7. View largeDownload slide True and adjusted errors for CASEs A, B and C. True errors are differences between the heights in the local system and the a priori values, while adjusted errors are differences between the heights in the reunified system and the a priori values. Circles represent clocks. For a further understanding of the errors’ behaviour, we plotted the geographical distribution of the true and adjusted errors for all levelling points in Fig. 8. We can easily distinguish different local height systems in the geographical distribution of the true errors. The virtual boundaries caused by the biases are apparent for CASE A. In CASE B, the trend changes of the height errors along the latitudinal direction are visible for each local region, while additional changes along the longitudinal direction are visible for CASE C. The virtual boundaries as well as the trend changes disappeared in the plots of the adjusted errors, where the errors look consistent over the whole region of the unified system. Figure 8. View largeDownload slide Geographic distribution of the true errors (in the left-hand column) and adjusted errors (in the right-hand column) for CASEs A, B and C. The figures on the top row show the results for CASE A, while those on the middle and bottom rows show the results for CASE B and CASE C. The magenta symbols shown in the figures of adjusted errors represent clocks, where each of the four types of symbols (⁠|$\vartriangle ,\triangledown , \square$|⁠, ☆) indicate clocks used for one local height system. Figure 8. View largeDownload slide Geographic distribution of the true errors (in the left-hand column) and adjusted errors (in the right-hand column) for CASEs A, B and C. The figures on the top row show the results for CASE A, while those on the middle and bottom rows show the results for CASE B and CASE C. The magenta symbols shown in the figures of adjusted errors represent clocks, where each of the four types of symbols (⁠|$\vartriangle ,\triangledown , \square$|⁠, ☆) indicate clocks used for one local height system. We can conclude from the above that a proper clock network is well suited for the unification of local height systems. A clock network, composed of a sufficient number of high-performance clocks, can support the estimation of the offsets between different height systems and the tilts along levelling lines of each local height system with high accuracy. In this study, the accuracy of the heights in the reunified system is at the level of 1.0 cm, which corresponds to the introduced random errors of the heights and the clock-based measurements. In the above simulations, we just assigned some clocks for each case. But, the number and the spatial distribution of the clocks were not addressed. However, these factors affect the unification of the local height systems and the cost to install and operate such a clock network. These major issues are discussed in the following. 4.2 Number of clocks An optimal number of clocks should be determined, considering both the performance of the unification and the cost to operate such a clock network. As pointed out previously, the number of clocks should fulfil the overall condition |$\binom{k}{2} \gt \sum _{L=1}^4 n_e - 1$| so that an unambiguous solution can be computed (ne is the number of error parameters, which is 1, 2, 3 for CASE A to CASE C). Because at least one clock should be used for each local height system, k is larger than or equal to four in our simulation cases. From another perspective, it is costly to realize a clock network in practice, since clocks and the dedicated links between them are expensive. We thus should optimally determine the number of clocks to achieve the best cost-performance ratio. To study the optimised number of clocks, we ran further simulations based on CASE C, where four clocks for each local height system have shown a good performance for height unification. Now, we ran three new tests on the number of clocks in each of the four regions. The used numbers are given in brackets: (2, 2, 3, 3), (3, 3, 3, 3) and (5, 5, 5, 5). We see from Figs 9(a) and 10(a) that the adjusted errors are much larger for groups one and two than for the other two groups. We suppose this occurs because the tilts for groups one and two were not correctly estimated. Two clocks are not enough for this purpose, even if the total number of clocks per region could meet the overall requirements. When three clocks are used, the adjusted errors are significantly reduced to the level of the other two groups, see Figs 9(b) and 10(b). However, the performance of the unification was slightly worse compared to the case where four clocks were used. A further improvement was not evident when the number of clocks was increased to five. Figure 9. View largeDownload slide True and adjusted errors for different scenarios where different numbers of clocks were tested for CASE C. The used numbers are given in brackets. Figure 9. View largeDownload slide True and adjusted errors for different scenarios where different numbers of clocks were tested for CASE C. The used numbers are given in brackets. Figure 10. View largeDownload slide Geographic distribution of adjusted errors for different scenarios of CASE C, where different numbers of clocks were tested. Note, adjusted errors in the marked area of Fig. 10(b) are larger than in other areas due to the lack of clocks there. The magenta symbols represent the clocks, and each of the four types of symbols (⁠|$\vartriangle ,\triangledown , \square$|⁠, ☆) indicate clocks used for one local height system. Figure 10. View largeDownload slide Geographic distribution of adjusted errors for different scenarios of CASE C, where different numbers of clocks were tested. Note, adjusted errors in the marked area of Fig. 10(b) are larger than in other areas due to the lack of clocks there. The magenta symbols represent the clocks, and each of the four types of symbols (⁠|$\vartriangle ,\triangledown , \square$|⁠, ☆) indicate clocks used for one local height system. We also ran simulations to identify a suitable number of clocks for the other two cases (the results are not shown here). For CASE A, one clock for each region is enough to estimate the offsets between different height systems. For CASE B, one more clock is required to estimate the tilts appearing along the levelling lines. With slightly more clocks than the basic requirement, the height unification can be achieved with higher accuracy. We can conclude that a few clocks, that is one to three for each local region (with a proper spatial distribution, cf. Section 4.3), can meet the basic requirement for height unification. One clock for each region is sufficient for the estimation of offsets between different height systems. One more ‘domestic’ clock (in the same local region) is needed to correct the tilt over the latitudinal or longitudinal directions. For the case where offsets and tilts along both directions are taken into account (it is more likely to be the real case in practice), at least three clocks for each region are required for the unification. And depending on the availability of slightly more clocks, the performance of unification can be further improved. Furthermore, we found that the adjusted errors in some regions were larger than in others (see the marked regions in Fig. 10b) for the case where three clocks were used. There, no clocks were used in the marked areas. However, the errors can be reduced when one more clock is assigned, see the results for the case of four clocks. This gave us a hint that in addition to the number of clocks, their spatial distribution of the clocks also plays an important role. This is further discussed in the next section. 4.3 Spatial distribution of the clocks The previous discussion addressed the sensitivity of the vertical datum unification to the quantity of clocks available in each region. On one hand, a proper arrangement of clocks can detect the systematic errors of a specific height system. On the other hand, clocks should be arranged to cover as much as possible the research area. Considering both aspects, we assign the clocks to the levelling points which are geographically located as shown in Fig. 11. More specifically for the cases where one to five clocks are used, the clocks’ sites have been selected according to the following schemes. One clock: at the geographic centre of the research region, cf. Fig. 11(a). Two clocks: four variable choices (Figs 11 b–e), along the main- or off-diagonal directions, or along the latitudinal or longitudinal direction. Three clocks: at any three of the four corner sites (cf. the case of four clocks Fig. 11f). Four clocks: at the four corner sites (Fig. 11f). Five clocks: at the four corner sites and the geographic centre, cf. Fig. 11(g). Figure 11. View largeDownload slide The spatial distribution of clocks, for the number of clocks from one to five. Figure 11. View largeDownload slide The spatial distribution of clocks, for the number of clocks from one to five. We took CASE B in which offsets and tilts along the latitudinal directions were included to study the effect of the clocks’ distribution on the unification. The number of clocks were assumed as 2, 2, 3, 2 for each of the local systems. One more clock was assumed for the third region to make sure that the levelling points belonging to this group (the datum selected for the final vertical unification) will be accurately unified. In fact, we compared four different choices, where the spatial distributions of the used clocks were ‘diagonal’, ‘latitudinal’, ‘longitudinal’ and a commixture of them. The results are shown in Figs 12 and 13. The unification was well performed by setting clocks along the diagonal or latitudinal directions. However, when clocks were set along the longitudinal direction, the unification of the height systems one and four was not good. For the ‘commixture’ case, the result for height system one became good. There the clocks were placed along the diagonal direction. The result for height system four remained poor since the clocks were still put along the longitudinal direction. Figure 12. View largeDownload slide Geographic distribution of adjusted errors for different scenarios of CASE B, where spatial distributions of the clocks were tested. The clocks were arranged along the diagonal, latitudinal, longitudinal directions and a commixture of them. The magenta symbols represent clocks, and each of the four types of symbols (⁠|$\vartriangle ,\triangledown , \square$|⁠, ☆) indicate clocks used for one local height system. Figure 12. View largeDownload slide Geographic distribution of adjusted errors for different scenarios of CASE B, where spatial distributions of the clocks were tested. The clocks were arranged along the diagonal, latitudinal, longitudinal directions and a commixture of them. The magenta symbols represent clocks, and each of the four types of symbols (⁠|$\vartriangle ,\triangledown , \square$|⁠, ☆) indicate clocks used for one local height system. Figure 13. View largeDownload slide True errors and adjusted errors for different scenarios of CASE B, where spatial distributions of the clocks were tested. The clocks were arranged along the diagonal, latitudinal, longitudinal directions and a commixture of them. Figure 13. View largeDownload slide True errors and adjusted errors for different scenarios of CASE B, where spatial distributions of the clocks were tested. The clocks were arranged along the diagonal, latitudinal, longitudinal directions and a commixture of them. This analysis shows that clocks should be properly arranged so that they can sense the tilts where necessary. If one clock is available for each region, it can solve the unification of CASE A where only offsets between height systems were considered. In principle, this clock can be assigned on any levelling point. When two clocks are available, the tilt of vertical network might be estimated. The placement of clocks in the longitudinal direction can sense the north–south tilt, and the placement in the latitudinal direction can sense the east–west tilt. In case that the direction of tilt is unknown, it is recommended to place the clock along the (off-)diagonal direction. This placement can well sense the north-south or the east–west tilt. For the case of three clocks, they can be distributed at any three of the corner sites. The placement like a triangle can sense the north–south and the east–west tilts simultaneously. However, the clocks should be arranged to cover most levelling points in practice, because the performance of unification in the area without clocks may be poorer. For the cases of four and five clocks, only one choice of spatial distribution is considered, which can well meet the requirement for the unification. Furthermore, some other factors should be taken into account in practice, for example a frequency link should be available via fibres or satellites. The clocks’ sites should be finally decided on all of these factors, however, this discussion is beyond the scope of this study. 4.4 Clock accuracy So far, clock-based data with a relative frequency uncertainty of 1.0 × 10−18 has been assumed. However, the best clocks available nowadays can achieve one order of magnitude lower than the presented here. Consequently, we performed further experiments to assess the suitability of the existing clocks for the vertical datum unification. The CASE C was chosen for further simulations. Four identical clocks were applied for each local system, which has been shown to be adequate for the unification. We run two more simulations by using clocks with lower accuracy, that is with a relative frequency uncertainty of 5.0 × 10−18 and 20.0 × 10−18, which correspond to about 5.0 and 20.0 cm in height. The results are displayed in Figs 14 and 15. Clocks at the level of 5.0 × 10−18 show a good performance for the unification. The adjusted height errors of the levelling points for the four local systems were reduced to the level of about 1.7 cm and look consistent in the reunified system. When clocks with random errors at the magnitude of 20.0 × 10−18 were used, it seems that the unification can only be achieved to some extent. The tilts along the levelling lines were adjusted somehow, but the offsets were not exactly estimated. The normal heights for the reunified system exhibited a positive shift of about 4.5 cm. However, we should note that the random clock errors are even larger than some height offsets. Figure 14. View largeDownload slide True errors and adjusted errors for different scenarios of CASE C, where clocks with different levels of uncertainty were assumed. Figure 14. View largeDownload slide True errors and adjusted errors for different scenarios of CASE C, where clocks with different levels of uncertainty were assumed. Figure 15. View largeDownload slide Geographic distribution of adjusted errors for different scenarios of CASE C, where clocks with different levels of uncertainty were assumed. The magenta symbols represent clocks, and each of the four types of symbols (⁠|$\vartriangle ,\triangledown , \square$|⁠, ☆) indicate clocks used for one local height system. Figure 15. View largeDownload slide Geographic distribution of adjusted errors for different scenarios of CASE C, where clocks with different levels of uncertainty were assumed. The magenta symbols represent clocks, and each of the four types of symbols (⁠|$\vartriangle ,\triangledown , \square$|⁠, ☆) indicate clocks used for one local height system. The results show that clocks nowadays can unify height systems to some extent, but still lack of accuracy for achieving a highly accurate unified system, for example at the level of 1.0 cm. In this work, the height offsets were assumed only up to 25 cm, however, in practice, this value can reach to 1–2 m in some regions. In such cases even today, clocks may help to unify the height systems to a certain acceptable level, for example at the sub-dm level. Moreover, the accuracy of clock-based data can be improved through repeated measurements, which in principle increases the accuracy by |$\sqrt{n}$| when n times of observations are obtained. Last but not the least, more advanced clocks shall be developed at various research institutes. To make clock comparisons more operational, highly accurate space-based links were favourable, which shall be developed in the coming years. 5 CONCLUSIONS AND FUTURE WORK The rapid development of high-performance optical clocks opens the door to realize relativistic geodesy. In particular, it provides the possibility to estimate physical height differences between two distant points with high accuracies. Clocks are thus expected to be powerful observing tools in future, which may lead to revolutionary changes in geodesy. In this work, we considered to use clocks for height system unification which has been a classical geodetic problem for a long time. Through simulations, we demonstrated the great potential of clocks for this issue. On the assumption that offsets between height datums and tilts along levelling lines in both longitudinal and latitudinal directions were present, three or four clocks measurements for each local region are required to perform the unification. These clocks should be properly arranged so that they can cover most of the network area and sense the tilts where necessary. For example, if clocks are arranged in the longitudinal direction, they will not be sensible to the tilts along the north–south direction. In case that the direction of tilt (in the longitudinal or the latitudinal direction) is unknown, it is suggested to place the clocks along the (off-)diagonal direction. The results also reveal that the clocks nowadays still lack of accuracy to achieve a highly accurate unified height system, for example at the level of 1.0 cm. Clocks with large errors at the level of dm may result in an overall shift of the unified height system. However, this gap could be filled through repeated measurements or the further development of clocks. In reality, however, the character of the systematic errors of national vertical networks might be unknown or more complicated than our assumptions. In this case, the clock-based method can be combined with other geodetic methods, for example the GNSS/geoid approach. The clock part can contribute to remove offsets and tilts from the post-fit residuals, and thereby helps to identify other implicit systematic errors of vertical networks. Then, we can properly extend the functional model to adapt to the more complicated cases. On the other hand, complex errors can be reduced/simplified by subdividing the research region. In the sub regions, our assumption on the character of the systematic errors (being offsets and tilts) can hold true, and therefore our conclusions are correct as well. As a preliminary study, we have assumed some factors at ideal conditions, for example clocks with identical accuracies. This does not hold true in the actual situation. Different types of clocks or clocks developed by different countries have diverse performances. As a result, the frequency comparisons between clocks may have different precision and may be correlated. A refined stochastic model should be applied to assign proper weights to individual measurements and cope with the correlations between different measurements. We also idealised the clock-based data with only random errors. But more likely, clocks have their own biases or drifts or other instrument-related systematic errors. These factors should be well treated in practical applications. Revisiting the goal to establish an international height system, the resolution of IAG in 2015 suggested to adopt the gravity potential as the vertical coordinate. Clocks definitely have the unique advantage in realizing such a height system as the potential difference between different points can be directly obtained through the comparison of clocks. A hybrid clock network which probably includes Earth-bound clocks, space-based clocks and transportable clocks, might be designed to realize a consistent and accurate global height system. Moreover, clock measurements are considered to be combined with other kinds of gravimetric observations to potentially improve the Earth’s gravity field models. This might also be helpful to improve the performance of height system unification by using the GNSS/geoid method. ACKNOWLEDGEMENTS The authors thank the DFG Sonderforschungsbereich (SFB) 1128 ‘Relativistic Geodesy and Gravimetry with Quantum Sensors (geo-Q)’ for financial support. Hu Wu acknowledges the International Space Science Institute (ISSI) for providing the opportunity to present this work at the Workshop on “Spacetime Metrology, Clocks and Relativistic Geodesy”, March 19-23, 2018. We also gratefully thank the two anonymous reviewers whose comments helped to improve and clarify this manuscript. Footnotes 1 https://www.geoq.uni-hannover.de REFERENCES Amjadiparvar B. , Rangelova E. , Sideris M.G. , 2016 . The GBVP approach for vertical datum unification: recent results in North America , J. Geod. , 90 ( 1 ), 45 – 63 . https://doi.org/10.1007/s00190-015-0855-8 Google Scholar Crossref Search ADS Berceau P. , Taylor M. , Kahn J. , Hollberg L. , 2016 . Space-time reference with an optical link , Class. Quant. 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Definition and proposed realization of the International Height Reference System (IHRS) , Surv. Geophys. , 38 ( 3 ), 549 – 570 . https://doi.org/10.1007/s10712-017-9409-3 Google Scholar Crossref Search ADS Jacopo G. et al. , 2018 . Geodesy and metrology with a transportable optical clock , Nat. Phys. , 14 , 437 – 441 ., doi:10.1038/s41567-017-0042-3 . Google Scholar Crossref Search ADS Jekeli C. , 2000 . Heights, the geopotential, and vertical datums, Tech. rep., Department of Civil and Environmental Engineering and Geodetic Science, Ohio State University . Koch K.-R. , 2013 . Parameter Estimation and Hypothesis Testing in Linear Models , Springer Science & Business Media . Koller S.B. , Grotti J. , Vogt S. , Al-Masoudi A. , Dörscher S. , Häfner S. , Sterr U. , Lisdat C. , 2017 . Transportable optical lattice clock with 7 × 10−17 uncertainty , Phys. Rev. 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Sci. , 2 ( 4 ), 325 – 342 . https://doi.org/10.2478/v10156-012-0002-x Sánchez L. , Sideris M.G. , 2017 . Vertical datum unification for the International Height Reference System (IHRS) , Geophys. J. Int. , 209 ( 2 ), 570 . https://doi.org/10.1093/gji/ggx025 Takano T. et al. , 2016 . Geopotential measurements with synchronously linked optical lattice clocks , Nat. Photon. , 10 , 662 – 666 . https://doi.org/10.1038/nphoton.2016.159 Google Scholar Crossref Search ADS Torge W. , Müller J. , 2012 . Geodesy , De Gruyter . Vermeer M. , 1983 . Chronometric levelling, Tech. rep., Finnish Geodetic Institute . Voigt C. , Denker H. , Timmen L. , 2016 . Time-variable gravity potential components for optical clock comparisons and the definition of international time scales , Metrologia , 53 ( 6 ), 1365 . https://doi.org/10.1088/0026-1394/53/6/1365 Google Scholar Crossref Search ADS Woodworth P. , Hughes C. , Bingham R. , Gruber T. , 2012 . Towards worldwide height system unification using ocean information , J. Geod. Sci. , 2 ( 4 ), 302 – 318 . https://doi.org/10.2478/v10156-012-0004-8 © The Author(s) 2018. Published by Oxford University Press on behalf of The Royal Astronomical Society. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Geophysical Journal International Oxford University Press

Clock networks for height system unification: a simulation study

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Oxford University Press
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© The Author(s) 2018. Published by Oxford University Press on behalf of The Royal Astronomical Society.
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Abstract

SUMMARY The unification of local height systems has been a classical geodetic problem for a long time, the main challenges of which are the estimation of offsets between different height systems and the correction of tilts along the levelling lines. It has been proposed to address these challenges with clock networks. The latest generation of optical clocks as well as the dedicated frequency links, for example optical fibres, are now approaching to deliver the comparison of frequencies at the level of 1.0 × 10−18. It corresponds to an accuracy of about 1.0 cm in height difference. Clock networks can thus serve as a powerful tool to connect local height systems. To verify the idea, we carried out simulations using the EUVN/2000 (European Unified Vertical Network) as apriori input. Four local height systems were simulated from the EUVN/2000 by introducing individual offsets and tilts, and were reunified by using measurements in clock networks. The results demonstrate the great potential of clock networks for height system unification. In case that the offsets between different height systems and tilts along national levelling lines in both longitudinal and latitudinal directions are considered, three or four clocks measurements for each local region are sufficient for the unification. These clocks are to be interconnected and should be properly arranged so that they can sense the levelling tilts where necessary. Our results also indicate that even clocks with one magnitude poorer accuracy than the desired ones can still unify the height systems to some extent, but it may cause a shift for the reunified system. Geodetic instrumentation, Geopotential theory, Reference systems, Satellite geodesy 1 INTRODUCTION Establishing a consistent and accurate global height system is one of the major tasks of the International Association of Geodesy (IAG) which released a resolution for the definition and realization of an International Height Reference System (IHRS) in 2015 (Drewes et al.2016; Ihde et al.2017). Such a height system is essential for engineering constructions, for example building a bridge or tunnel that connects different countries, and scientific research in geodynamics, geophysics and oceanography, as well as for environmental monitoring such as global sea level changes, natural disasters (e.g. floods and droughts) and surface deformations (e.g. crustal motion and subsidence), see Sideris (2014). To realize a global height system, a widely discussed strategy is to unify the local height systems worldwide. The datum of a local height system is usually tied to tide gauges which measure the mean sea level locally. However, the local mean sea level does not exactly coincide with the geoid, that is a selected equipotential surface of the Earth’s gravity field. The inconsistencies are up to ±1–2m in some regions (Torge & Müller 2012), mainly caused by the sea surface topography. Consequently, there exist offsets between different vertical datums, and the determination of these offsets is one of the main issues to be solved in height system unification. In addition to the datum, local height systems are composed of many levelling points. The height values of these points are related to the datum by spirit levelling. However, height errors accumulate during spirit levelling over long distances. As a result, tilts appear in national vertical networks, at the level of 1–3 cm/100 km (Gruber et al.2014). The estimation of those tilts is a further major task in height system unification. Three approaches have been extensively discussed in the literature for height system unification, namely geodetic levelling, oceanographic modelling and the Geodetic Boundary Value Problem (GBVP) approach (Colombo 1980; Pavlis 1991; Rummel 2001; Sánchez 2012). Each of them has advantages and drawbacks. The classical geodetic method which applies spirit levelling and terrestrial gravimetry is a time-consuming way to connect local height systems. In terms of accuracy, it can achieve submillimetre relative accuracy over short distances, while the absolute accuracy with respect to a global datum can reach up to ±2 m (Sánchez & Sideris 2017). Oceanographic modelling is a powerful approach to connect the local height systems that are separated by oceans. Unfortunately, it suffers from the poor availability of observations at high temporal and spatial resolutions and heavily relies on the assumption about hydrostatic and geostrophic equilibrium (Sánchez 2012; Woodworth et al.2012). Additionally, the GBVP approach has been modified and applied to estimate the offsets between different height systems (Rummel & Teunissen 1988; Gerlach & Rummel 2013; Sideris 2014; Amjadiparvar et al.2016; Sánchez & Sideris 2017). Its major drawbacks, however, are the use of different standards in different regions (e.g. gravity data: gravity anomaly or gravity disturbance; tide systems: mean or zero tide system for computing the potential difference; gravity field models: different models with different spherical harmonic degree and order for removing the long-wavelength contributions) and the restricted accessibility to terrestrial gravity data (Rummel 2001; Sideris 2014; Sánchez & Sideris 2017). In further works, some vertical datum unification experiments are based on the GNSS/geoid (Global Navigation Satellite System) method, using some GOCE (Gravity field and steady-state Ocean Circulation Explorer) geoid or GOCE-based combined geoid solutions (Gruber et al.2012, 2014). However, the gravity field model might show poor performance in sparsely surveyed regions. For extreme cases, the errors are up to ±1 m (Gruber et al.2014), which are even larger than some datum offsets. Alternatively, we investigate the application of relativistic geodesy and the use of clock networks for height system unification. Einstein’s theory of general relativity predicts that the gravity potential difference (closely related to the height difference) can be derived from the comparison of clocks’ frequencies. This concept is called relativistic geodesy and was described in detail by Bjerhammar (1985, 1986). Thanks to the rapid development of high-performance clocks and frequency transfer techniques, frequency comparison between two distant clocks can hopefully soon achieve the level of 1.0 × 10−18, equivalent to about 1.0 cm in terms of height difference. Hence, clocks appear as a promising option for connecting local height systems, especially where no spirit levelling surveys are possible. Using clocks for height system unification has been one of the geodetic research topics covered by geo-Q (Collaborative Research Centre 1128 ‘Relativity geodesy and gravimetry with quantum sensors’),1 funded by the Deutsche Forschungsgemeinschaft (DFG). Under the scope of geo-Q, we are evaluating different approaches for suitable application in practice before the real clock data are available. Through simulations, we address several research questions that are of interest within both the geodesy and physics communities: How well can clock networks unify local height systems? How many clocks are required for the unification? How can the clock network be optimised? And how does the clocks’ accuracy affect the unification? The knowledge gained from this study will benefit the practical application when sufficient data in clock networks become available in the future. The paper is organised as follows. The relevant background knowledge is reviewed in Section 2, such as the definition of a height system, the theory of relativistic geodesy and the development of clock networks. The simulation strategy and the input data are described in Section 3. Section 4 presents our results and discussions. Some concluding remarks and future work are addressed in the last section. 2 HEIGHT SYSTEM, RELATIVISTIC GEODESY AND OPTICAL CLOCKS 2.1 Height system By definition, height describes the vertical distance of any geospatial point to a reference surface. Nowadays, because of the widely applied GNSS positioning method, the ellipsoidal height (with the ellipsoid surface as reference) can be conveniently obtained. However, due to their pure geometric nature, the GNSS measurements lack the physical interpretation of heights. They cannot properly reflect, that is the direction of the water flow. To address the physical meaning, heights are differentially determined with reference to an equipotential surface such as the geoid. There are three types of physical height, namely orthometric, normal and dynamic height (Torge & Müller 2012). Each of them fundamentally refers to the geopotential number CP, that is the gravity potential difference between the geoid (with potential value W0) and the point P (with potential value WP) in question (Jekeli 2000; Torge & Müller 2012): \begin{eqnarray*} C_P = C(P) = W_0 - W_P . \end{eqnarray*} (1) The geopotential number can be appropriately scaled to a height value using \begin{eqnarray*} H_P = \frac{C_P}{\hat{g}_P} = \frac{W_0 - W_P}{\hat{g}_P} , \end{eqnarray*} (2) where |$\hat{g}_P$| is an appropriate gravity value, depending on the type of height. Specifically, the orthometric, normal and dynamic height are obtained when |$\hat{g}_P$| is chosen as mean gravity along the plumb line, mean normal gravity along the normal plumb line and normal gravity at mid-latitude. For more details, we refer to Jekeli (2000) and Torge & Müller (2012). While the height values at the same point P vary in accordance with the gravity values introduced in eq. (2), the geopotential numbers (eq. 1) are unique. To realize a physical height system, the reference equipotential surface has to be defined first. Typically, the mean sea level is taken as reference. It is observed and determined by tide gauges, and coincides with a local equipotential surface. A reference benchmark (or a datum point) is then defined and tied to the tide gauge, and its height above the mean sea level is measured and assigned. In addition to the datum point, some high-order control points realize the framework of such a practical height system. The heights of all control points are connected to the datum by spirit levelling and gravimetry. A major problem arises when the mean sea level is used as the height reference. In fact, the mean sea level determined at each tide gauge locally coincides with a different equipotential surface. These surfaces do not strictly coincide with the same global equipotential surface (i.e. the geoid, see Fig. 1). The difference between these surfaces reaches the same magnitude (up to ±2 m) as the so-called sea surface topography, which represents the discrepancy between the mean sea surface and the geoid (Torge & Müller 2012). Figure 1. View largeDownload slide Offsets between different height datums. The equipotential surfaces for the local height system j and j+1 exhibit discrepancies with respect to the global geoid. The potential offsets ΔW0, j and ΔW0, j+1 (resp. height offsets) have to be estimated for height system unification. Figure 1. View largeDownload slide Offsets between different height datums. The equipotential surfaces for the local height system j and j+1 exhibit discrepancies with respect to the global geoid. The potential offsets ΔW0, j and ΔW0, j+1 (resp. height offsets) have to be estimated for height system unification. In addition, there are tilts along the levelling lines in local or national height systems. Tilts along the longitudinal and latitudinal directions were detected when unifying national height systems in Europe (Rülke et al.2012; Gruber et al.2014). The tilts can reach about 1–3 cm/100 km and are mainly attributed to the accumulated errors of spirit levelling over long distances. Furthermore, different conventions for the national height systems and the heterogeneity of data sets (e.g. different epochs and network configurations) may cause tilts along levelling lines. As an example, Fig. 2 shows the tilts of the national height systems in France, Germany and Spain, estimated from GOCE gravity field models combined with GNSS/levelling. Figure 2. View largeDownload slide Tilts of national height systems. The estimated tilts along the longitudinal and latitudinal directions are (1.0, −2.7), (0.9, −1.4) and (0.8, −3.0) cm/100 km for France, Germany and Spain, respectively. The number and the figures are taken from Gruber et al. (2014). Figure 2. View largeDownload slide Tilts of national height systems. The estimated tilts along the longitudinal and latitudinal directions are (1.0, −2.7), (0.9, −1.4) and (0.8, −3.0) cm/100 km for France, Germany and Spain, respectively. The number and the figures are taken from Gruber et al. (2014). Over 100 local height systems are in use worldwide. Their unification is one major research topic of IAG. The main objective is the precise determination of vertical datum offsets and possible tilts in the levelling networks. As discussed in Section 1, various approaches have been considered to address this issue. However, each of these approaches has its limitations. Therefore, further effort is required in order to realize the precise unification of local height systems. 2.2 Relativistic geodesy Einstein’s theory of general relativity predicts that clocks tick at different rates if they are transported with different speeds or they are under the influence of different gravitational potential. Considering the case of two earthbound clocks at rest (on the Earth surface), the change of the clocks’ frequencies is proportional to the corresponding difference in the gravity potential (sum of the gravitational potential and the centrifugal potential) between the two clocks (Bjerhammar 1985, 1986). It reads \begin{eqnarray*} \frac{\Delta f_{21}}{f_1} = \frac{f_2 - f_1}{f_1} = \frac{W_2 - W_1}{c^2} + O(c^{-4}) , \end{eqnarray*} (3) where f1 and f2 are the proper frequencies at points of 1 and 2 at the Earth’s surface, while W1 and W2 are the corresponding gravity potential values, c is the speed of light. Higher order terms are neglected. Based on eqs (1) and (3), we obtain \begin{eqnarray*} \frac{\Delta f_{21}}{f_1} = \frac{(W_2 - W_0) + (W_0 - W_1)}{c^2} = - \frac{C_2 - C_1}{c^2} , \end{eqnarray*} (4) where C1 and C2 are the corresponding geopotential numbers. In geodesy, the differences between geopotential numbers can be straightforwardly used to derive physical height differences. For example, the difference between two orthometric heights |$\Delta H_{21}^{\ast }$| can be obtained as (Müller et al.2018): \begin{eqnarray*} \Delta H_{21}^{\ast } &=& H_{2}^{\ast } - H_{1}^{\ast } = H_{1}^{\ast } \frac{\Delta \bar{g}_{12}}{\bar{g}_2} + \frac{C_2 - C_1}{\bar{g}_2} \nonumber \\ &=& H_{1}^{\ast } \frac{\Delta \bar{g}_{12}}{\bar{g}_2} - \frac{c^2}{\bar{g}_2} \frac{\Delta f_{21}}{f_1} , \end{eqnarray*} (5) where |$\Delta \bar{g}_{12} = \bar{g}_1 - \bar{g}_2$| is the difference of the mean gravity values (i.e. the mean along the plumb line between the Earth’s surface and the geoid) at both locations. Note that eq. (5) can be formulated for normal heights accordingly. Then, the mean normal gravity has to be used, that is |$\bar{\gamma }$| instead of |$\bar{g}$|⁠. This method to obtain heights through the comparison of clocks is called chronometric levelling (Vermeer 1983). The scheme is shown in Fig. 3. For an approximate estimation, a relative frequency inaccuracy of 1.0 × 10−18 corresponds to an uncertainty of about 1.0 cm in height. Clocks have the advantage to connect distant areas, for example they deliver physical height differences for the observed points without being affected by accumulated levelling errors or by some smoothing effects when combined global gravity field models (determined from multisource gravimetry data) are used. Therefore, high-performance clocks, for example at the level of 1.0 × 10−18 will be a powerful tool for delivering physical heights and with them, a well-suitable approach for the connection of height systems. Figure 3. View largeDownload slide Scheme of chronometric levelling (adapted from Müller et al.2018). Figure 3. View largeDownload slide Scheme of chronometric levelling (adapted from Müller et al.2018). 2.3 Optical clock networks When relativistic geodesy was described by Bjerhammar in 1980s, the hydrogen maser was probably the most accurate clock, with a relative frequency uncertainty of 10−13. The hydrogen maser is a kind of atomic clock, which uses an electron transition frequency in the microwave band as frequency standard. Microwave clocks were improved by three orders of magnitude in the past decades, now approaching the level of 10−16 (Heavner et al.2014), see also Fig. 4. This is however about two orders of magnitude away from our desired level in geodesy. Another type of atomic clock using optical transitions, the so-called optical clock, was proposed and rapidly developed. Since 1990, the uncertainty of optical clocks was improved by about two orders of magnitude every ten years, see Fig. 4. Recently, two optical clocks in laboratories approached the 10−18 level (Bloom et al.2014; Huntemann et al.2016). Moreover, a transportable optical clock that is more likely to be used for a field measurement campaign in future was reported at the level of 10−17 (Koller et al.2017). Figure 4. View largeDownload slide Evolution of fractional frequency uncertainty of atomic clocks based on microwave (Cs clocks) and optical transitions. Figure 4. View largeDownload slide Evolution of fractional frequency uncertainty of atomic clocks based on microwave (Cs clocks) and optical transitions. To support the comparison of distant clocks, frequency transfer techniques, such as fibre link, free air link and radio frequency link (Riehle 2017), have been extensively experimented and developed. Among them, the frequency link via optical fibres shows the best performance, which reached an extreme accuracy of 2.5 × 10−19 over 1400  km, cf. Lisdat et al. (2016). This makes the comparison of accurate clocks over long distances possible and opens the door for novel geodetic applications. Considerations on the benefit of clocks in geodesy were addressed by Chou et al. (2010); Delva & Lodewyck (2013). The suitability of chronometric levelling using optical clocks was demonstrated through experiments. Two clocks connected by a 75-m-long fibre delivered a height difference of 37 ± 15 cm. This finding agreed quite well with the known value of 33 cm (Chou et al.2010). A master (reference) and slave (secondary) clock separated by 15 km delivered the height difference with an uncertainty of 5 cm for a height difference of 1516 m (Takano et al.2016). Very recently, a transportable clock was first used for a field measurement campaign in a mountain area between France and Italy (Jacopo et al.2018). The potential difference inferred from chronometric levelling is 10 034 m2 s−2, which is in a good agreement with the value of 10 032.1 m2 s−2 determined independently by geodetic methods. Vice versa, the relativistic redshift effect, that is the shift of clock’s frequency caused by the gravity field, has to be corrected for the frequency standards. This effect was estimated by geodetic methods, for example the GNSS/geoid approach, with one case study in the United States (Pavlis & Weiss 2003, 2017) and another one in Europe (Denker et al.2018). In addition to the corrections related to the static part of gravity field, temporal variations caused by tides and non-tidal mass redistribution were computed and discussed by Voigt et al. (2016), with the biggest contribution from solid Earth tides and ocean tides, being relevant at a 10−17 level. Clocks were also considered to realize a space–time reference which can provide precise time worldwide, a valuable reference for geodesy (Berceau et al.2016). The impact of highly precise clocks for the realization of geodetic reference frames, timescales and the determination of the Earth’s gravity field was thoroughly discussed by Müller et al. (2018). Furthermore, Philipp et al. (2017) defined the geoid as a level surface of a time-independent redshift potential under the formalism of general relativity without any approximation. A simulation study was ran by Lion et al. (2017) to evaluate the contribution of clock measurements for determining the high-resolution geopotential. Sustainable efforts are being made to improve the capability of optical clocks and to study the application of relativistic geodesy, for example in the research project geo-Q at Leibniz Universität Hannover. We are optimistic that the frequency comparison between clocks can be realized at the level of 10−18 in near future. This will definitely open a new window for geodesy and beyond. 3 THE SIMULATION FRAMEWORK 3.1 Description of the simulator A simulator has been designed to verify the idea of using clocks for height system unification. The scheme of the simulator is depicted in Fig. 5. It takes a known height system as input. All levelling points of this height system are divided into several groups and each group forms a local height system. The generated local height systems are then reunified using simulated potential differences that are assumed to be obtained from observed frequency differences between clocks. Figure 5. View largeDownload slide The scheme of the simulator. Figure 5. View largeDownload slide The scheme of the simulator. The known normal heights of the levelling points are taken as true signals, while errors are treated individually for each local height system and are simulated independently. Based on the previous discussion, random errors (white noise) and systematic errors (biases and tilts) are considered in the simulation. Random height errors are simplified as white noise. Individual biases (accounting for the offsets between height systems) and tilts (responsible for the slopes along levelling lines) are introduced so that realistic height values in each local height system are obtained. We assume that a certain number of clocks is used in each local height system and all the clocks to be interconnected (every two clocks are connected with each other). We can thus obtain |$\left({k \atop 2} \right)$| measurements, with k the total number of clocks. These measurements, that are the potential differences (resp. the frequency differences between clocks), are simulated from the true height differences of EUVN/2000, cf. eq. (2). Concerning the errors, clock-based measurements are idealized to be independent and consistent. Moreover, clocks are not affected by the offsets and tilts of different height systems. Therefore, white noise with a desired level of 1.0 × 10−18 in terms of relative frequency is added to the clock-based measurements. A joint adjustment of the normal heights and the potential differences is finally applied to reunify the local height systems. Through this procedure, heights in a local system are adjusted to a unified system where all levelling points are tied to a unique datum point. The performance of the unification is assessed by comparing the heights in the reunified system and the true heights in the a priori system. 3.2 Data preparation The EUVN/2000 (European Vertical Reference Network) is adopted as a priori system in our study. It is an integrated height reference frame for Europe, which was completed in 2000 and published by Ihde et al. (2000). Although the EUVN/2000 is not the latest solution, it can fully meet our purpose for the simulation, because heights of levelling points are only taken as signals, while noise is simulated independently. The EUVN/2000 consists of about 200 high quality levelling points covering very well the European continent, see Fig. 6. Now, we classify all the levelling points into four groups. The height datums of these groups are supposed to be tied to well-known tide gauges in: (1) Marseille in France, (2) Newlyn in United Kingdom, (3) NAP (Normaal Amsterdams Peil) in the Netherlands and (4) Genova in Italy. Basically, levelling points are assigned to one of these groups based on the spatial distance. However, the administrative attribution is also taken into consideration. For example, the height datum of some North-European countries are known as the NAP, so that the levelling points in these countries are naturally assigned to the group of NAP. Fig. 6 shows the final division and Table 1 lists the exact number of levelling points for each group. Figure 6. View largeDownload slide Geographic distribution of EUVN/2000 levelling points and the classification of these points into four groups. Figure 6. View largeDownload slide Geographic distribution of EUVN/2000 levelling points and the classification of these points into four groups. Table 1. Number of levelling points for each local group. Total # 29 16 73 84 202 Total # 29 16 73 84 202 View Large Table 1. Number of levelling points for each local group. Total # 29 16 73 84 202 Total # 29 16 73 84 202 View Large Each point in EUVN/2000 is available with its 3-D coordinates, including the plane coordinates (latitude and longitude) and the normal heights. The plane coordinates are kept as noise-free as they can be determined with very high accuracy in practice, for example mm level. And, they are only used to geo-locate the levelling points. For the heights, random and systematic errors (both offsets and tilts) are introduced. As random error, white noise with a magnitude of 1.0 cm is generated for all levelling heights. The offsets are treated as constant biases and added to the levelling heights for each local system. The tilts along the latitudinal and longitudinal directions are considered, with value of 1–3 cm/100 km. In the end, four local height systems with individual errors are obtained. 3.3 Reunification We can set up the functional model for the normal heights in a local system as \begin{eqnarray*} H_i^L = \frac{C_i^U}{\overline{\gamma }_i} + a^L \Delta X_i^L + b^L \Delta Y_i^L + c^L , \end{eqnarray*} (6) where |$H_i^L$| is the normal height of point i in local height system L, |$C_i^U$| is the geopotential number of the corresponding point in the unified system U, |$\overline{\gamma }_i$| is the mean normal gravity, aL,bL and cL are error parameters that represent the tilts and the offsets, |$\Delta X_i^L, \Delta Y_i^L$| are relative coordinates in latitudinal and longitudinal directions. The functional model for the clock-based data is \begin{eqnarray*} \Delta W_{ij} = W_i^U - W_j^U = - (C_i^U - C_j^U), \end{eqnarray*} (7) where |$\Delta W_{ij}$| are the gravity potential differences delivered by clocks. Considering the number of observations (including both |$H_i^L$| and |$\Delta W_{ij}$|⁠), we have \begin{eqnarray*} N = \sum _{L =1}^4 n_L + \binom{k}{2} , \end{eqnarray*} (8) where nL is the number of levelling points in the local height system L, k is the total number of clocks (⁠|$k = \sum _{L=1}^{4} k_L$|⁠, kL the number of clocks in the local height system). As mentioned previously, all clocks are interconnected so that |$\binom{k}{2}$| measurements are available. However, this might not be exactly true in practice as not all clocks are always linked with each other. The unknowns include the geopotential numbers of all levelling points in the unified system |$C_i^U$| and the error parameters aL, bL, cL. The total number of unknowns M is \begin{eqnarray*} M = \sum _{L=1}^4 (n_L + n_e) - 1, \end{eqnarray*} (9) where ne is the number of error parameters, which is 1, 2 or 3 depending on the error types (see more in Section 3.2). Since one datum is adopted as the unique one for the reunified system, we have one less bias to estimate. Supposing a sufficient number of clocks is available to fulfil the condition |$\binom{k}{2} \gt \sum _{L=1}^{4} n_e - 1$|⁠, the solution can be solved in a least-squares adjustment. Rewriting the function models in a linear matrix form, it reads \begin{eqnarray*} {\left[\begin{array}{c}\boldsymbol {H} \\ \boldsymbol {\Delta W} \end{array}\right]} = {\left[\begin{array}{cc}\boldsymbol {A}_1 & \quad \boldsymbol {A}_2 \\ \boldsymbol {B}_1 & \quad \boldsymbol {B}_2 \end{array}\right]} {\left[\begin{array}{c}\boldsymbol {x}_1 \\ \boldsymbol {x}_2 \end{array}\right]} , \end{eqnarray*} (10) where |$\boldsymbol {H}$| and |$\boldsymbol {\Delta W}$| are observation vectors for normal heights and potential differences, |$\boldsymbol {x}_1$| is the unknown geopotential numbers to be estimated \begin{eqnarray*} \boldsymbol {x}_1 = [ \underbrace{\ldots , C_i^U, \ldots }_{L_1}, \cdots , \underbrace{\ldots , C_j^U, \ldots }_{L_4} ]^T , \end{eqnarray*} (11)|$\boldsymbol {x}_2$| denotes the error parameters. For example, when offsets and tilts in both latitudinal and longitudinal directions are considered, |$\boldsymbol {x}_2$| is written as \begin{eqnarray*} \boldsymbol {x}_2 = [ \underbrace{a^L, b^L, c^L}_{L=1}, \cdots , \underbrace{a^L,b^L,c^L}_{L=4} ]^T. \end{eqnarray*} (12) Note that one of the offset cL can be removed from |$\boldsymbol {x}_2$| by adopting one datum as the final one for the reunified system. |$\boldsymbol {A}_1, \boldsymbol {A}_2, \boldsymbol {B}_1, \boldsymbol {B}_2$| are corresponding design matrices and their elements are provided by eqs (6) and (7). By a least-squares adjustment, the unknowns are solved (Koch 2013) as \begin{eqnarray*} \boldsymbol {x} = (\boldsymbol {A}^T \boldsymbol {P} \boldsymbol {A})^{-1} (\boldsymbol {A}^T \boldsymbol {P} \boldsymbol {l}), \end{eqnarray*} (13) where $\boldsymbol {x} = {\left[\begin{array}{c}\boldsymbol {x}_1 \\\boldsymbol {x}_2 \end{array}\right]}$ ⁠, $\boldsymbol {A} = {\left[\begin{array}{cc}\boldsymbol {A}_1 & \boldsymbol {A}_2 \\\boldsymbol {B}_1 & \boldsymbol {B}_2 \end{array}\right]}$ is the completed design matrix, $\boldsymbol {l} = {\left[\begin{array}{c}\boldsymbol {H} \\\boldsymbol {\Delta W} \end{array}\right]}$ is the observation vector, $\boldsymbol {P} = {\left[\begin{array}{cc}\boldsymbol {P}_1 & \boldsymbol {0} \\\boldsymbol {0} & \boldsymbol {P}_2 \end{array}\right]}$ ⁠, and |$\boldsymbol {P}_1, \boldsymbol {P}_2$| represent weighting matrices for |$\boldsymbol {H}$| and |$\boldsymbol {\Delta W}$|⁠. They are written as |$\boldsymbol {P}_1 = \boldsymbol {I}$| and |$\boldsymbol {P}_2 = \frac{\sigma _{0,\boldsymbol {H}}^2}{\sigma _{0,\boldsymbol {\Delta W}}^2}\boldsymbol {I}$|⁠, where |$\boldsymbol {I}$| represents an identity matrix as observations for each type are assumed to be independent and at the same accuracy level. Note, |$\sigma _{0,\boldsymbol {H}}^2$| and |$\sigma _{0,\boldsymbol {\Delta W}}^2$| should be given in the same physical unit. When the geopotential numbers in the reunified height system are estimated, they are converted to heights and compared to the a priori values. The differences are labelled as adjusted errors (ea). The differences between the heights in the local system and the a priori values are called true errors (et). \begin{eqnarray*} e_a = H_i^{Esti, U} - H_i^{True}, \end{eqnarray*} (14) \begin{eqnarray*} e_t = H_i^{Obs., L} - H_i^{True}, \end{eqnarray*} (15) where |$H_i^{Esti, U}$| are estimated heights in the reunified system, |$H_i^{True}$| are corresponding true heights in the a priori system, |$H_i^{Obs., L}$| are heights in the local system. The adjusted errors can be used to evaluate the performance of the unification, and their comparison to true errors helps to understand the procedure of unification. 4 RESULTS AND DISCUSSIONS Based on the framework described above, we ran three simulation cases, that is CASE A, B, C, with different types of biases and tilts. In these simulations, we investigated how well clock networks can unify local height systems. We also discuss some open questions about clock networks, that is the number of clocks, the optimisation of clock networks and the impact of clocks’ performance. We took only biases into account for CASE A. The biases for the four local systems were set as –18.0, 25.0, 0.0, 8.0 cm, which are at the reasonable range of offsets between national height systems in Europe. We also tested other choices of biases, but they did not affect our final conclusion. Note that the bias for the third group was 0.0 cm, since NAP was adopted as the final datum for the reunified system. In addition to biases, tilts along the latitudinal direction were included for CASE B. The tilts were specified as 3.0, –2.0, 1.5, –3.0 cm/100 km. For a more thorough case, that is CASE C, biases and tilts along both the latitudinal and longitudinal directions were considered, with the tilts along the longitudinal directions as 2.0, 3.0, –1.5 and -2.0 cm/100 km. 4.1 Results for the reunification In order to ensure the success of the unification, a sufficient number of clocks (more than the requirement) were assumed for each case. Considering the complexity of errors in the three cases, we applied two, three and four clocks for CASE A, B, C. This number of clocks was used in each of the four local systems. All clocks were assumed to be identical (the same type of clock at the same accuracy level) so that homogenous measurements can be obtained. The clock-based data is at the desired level of 1.0 × 10−18 in terms of relative frequency. The result of CASE A is shown in Table 2. The input parameters are compared with the recovered ones, and the RMS (Root Mean Square) of the true and adjusted errors are given. The true errors are dominated by systematic errors, that is the offsets. The RMSs of true errors for Group , and are 17.90, 25.06 and 8.09 cm, respectively. They are around the introduced offsets –18.0, 25.0 and 8.0 cm. Similarly, the results of CASE B and C are shown in Tables 3 and 4. The true errors for CASE B and C are slightly more complicated than those of CASE A (they show significant differences to the offsets), due to the influence of tilts. Despite the divergence of the true errors, the adjusted errors for all three cases were reduced to the same level, around 1.0 cm. This value is at the level of the random height errors and clock errors. Table 2. Results for CASE A. In this case, offsets between different local height systems were considered. The input and recovered parameters (cL) are compared. The RMSs of true errors et and adjusted errors ea are shown as well. Unit is cm. Note that the bias for the third group was not estimated. Input parameters Recovered parameters RMS of et RMS of ea Offset Offset [cm] [cm] [cm] [cm] −18.0 −17.36 17.90 1.14 25.0 25.23 25.06 0.91 0 − 1.01 1.01 8.0 8.32 8.09 1.00 Input parameters Recovered parameters RMS of et RMS of ea Offset Offset [cm] [cm] [cm] [cm] −18.0 −17.36 17.90 1.14 25.0 25.23 25.06 0.91 0 − 1.01 1.01 8.0 8.32 8.09 1.00 View Large Table 2. Results for CASE A. In this case, offsets between different local height systems were considered. The input and recovered parameters (cL) are compared. The RMSs of true errors et and adjusted errors ea are shown as well. Unit is cm. Note that the bias for the third group was not estimated. Input parameters Recovered parameters RMS of et RMS of ea Offset Offset [cm] [cm] [cm] [cm] −18.0 −17.36 17.90 1.14 25.0 25.23 25.06 0.91 0 − 1.01 1.01 8.0 8.32 8.09 1.00 Input parameters Recovered parameters RMS of et RMS of ea Offset Offset [cm] [cm] [cm] [cm] −18.0 −17.36 17.90 1.14 25.0 25.23 25.06 0.91 0 − 1.01 1.01 8.0 8.32 8.09 1.00 View Large Table 3. Results for CASE B. In this case, offsets between different local height systems and tilts over national levelling lines in the latitudinal direction were considered. The input and recovered parameters (aL, cL) are compared. Unit for aL is cm/100 km and unit for cL is cm. The RMSs of true errors et and adjusted errors ea are shown as well. Unit is cm. Note that the bias for the third group was not estimated. Input parameters Recovered parameters RMS of et RMS of ea Offset Lat. tilt Offset Lat. tilt [cm] [cm/100 km] [cm] [cm/100 km] [cm] [cm] −18.0 3.0 −18.87 3.12 11.09 1.21 25.0 −2.0 25.69 −2.05 16.16 0.95 0 1.5 − 1.45 18.43 1.12 8.0 −3.0 8.70 −3.09 31.79 1.24 Input parameters Recovered parameters RMS of et RMS of ea Offset Lat. tilt Offset Lat. tilt [cm] [cm/100 km] [cm] [cm/100 km] [cm] [cm] −18.0 3.0 −18.87 3.12 11.09 1.21 25.0 −2.0 25.69 −2.05 16.16 0.95 0 1.5 − 1.45 18.43 1.12 8.0 −3.0 8.70 −3.09 31.79 1.24 View Large Table 3. Results for CASE B. In this case, offsets between different local height systems and tilts over national levelling lines in the latitudinal direction were considered. The input and recovered parameters (aL, cL) are compared. Unit for aL is cm/100 km and unit for cL is cm. The RMSs of true errors et and adjusted errors ea are shown as well. Unit is cm. Note that the bias for the third group was not estimated. Input parameters Recovered parameters RMS of et RMS of ea Offset Lat. tilt Offset Lat. tilt [cm] [cm/100 km] [cm] [cm/100 km] [cm] [cm] −18.0 3.0 −18.87 3.12 11.09 1.21 25.0 −2.0 25.69 −2.05 16.16 0.95 0 1.5 − 1.45 18.43 1.12 8.0 −3.0 8.70 −3.09 31.79 1.24 Input parameters Recovered parameters RMS of et RMS of ea Offset Lat. tilt Offset Lat. tilt [cm] [cm/100 km] [cm] [cm/100 km] [cm] [cm] −18.0 3.0 −18.87 3.12 11.09 1.21 25.0 −2.0 25.69 −2.05 16.16 0.95 0 1.5 − 1.45 18.43 1.12 8.0 −3.0 8.70 −3.09 31.79 1.24 View Large Table 4. Results for CASE C. In this case, offsets between different local height systems and tilts over national levelling lines in both the latitudinal and longitudinal directions were considered. The input and recovered parameters (aL, bL, cL) are compared. Unit for aL, bL is cm/100 km and unit for cL is cm. The RMSs of true errors et and adjusted errors ea are shown as well. Unit is cm. Note that the bias for the third group was not estimated. Input parameters Recovered parameters RMS of et RMS of ea Offset Lat. tilt Lon. tilt Offset Lat. tilt Lon. tilt [cm] [cm/100 km] [cm] [cm/100 km] [cm] [cm] −18.0 3.0 2.0 −17.07 3.12 1.78 22.93 1.28 25.0 −2.0 3.0 22.60 −1.94 3.11 62.35 0.90 0 1.5 −1.5 − 1.39 −1.46 15.39 1.13 8.0 −3.0 −2.0 6.22 −2.95 −1.94 53.44 1.09 Input parameters Recovered parameters RMS of et RMS of ea Offset Lat. tilt Lon. tilt Offset Lat. tilt Lon. tilt [cm] [cm/100 km] [cm] [cm/100 km] [cm] [cm] −18.0 3.0 2.0 −17.07 3.12 1.78 22.93 1.28 25.0 −2.0 3.0 22.60 −1.94 3.11 62.35 0.90 0 1.5 −1.5 − 1.39 −1.46 15.39 1.13 8.0 −3.0 −2.0 6.22 −2.95 −1.94 53.44 1.09 View Large Table 4. Results for CASE C. In this case, offsets between different local height systems and tilts over national levelling lines in both the latitudinal and longitudinal directions were considered. The input and recovered parameters (aL, bL, cL) are compared. Unit for aL, bL is cm/100 km and unit for cL is cm. The RMSs of true errors et and adjusted errors ea are shown as well. Unit is cm. Note that the bias for the third group was not estimated. Input parameters Recovered parameters RMS of et RMS of ea Offset Lat. tilt Lon. tilt Offset Lat. tilt Lon. tilt [cm] [cm/100 km] [cm] [cm/100 km] [cm] [cm] −18.0 3.0 2.0 −17.07 3.12 1.78 22.93 1.28 25.0 −2.0 3.0 22.60 −1.94 3.11 62.35 0.90 0 1.5 −1.5 − 1.39 −1.46 15.39 1.13 8.0 −3.0 −2.0 6.22 −2.95 −1.94 53.44 1.09 Input parameters Recovered parameters RMS of et RMS of ea Offset Lat. tilt Lon. tilt Offset Lat. tilt Lon. tilt [cm] [cm/100 km] [cm] [cm/100 km] [cm] [cm] −18.0 3.0 2.0 −17.07 3.12 1.78 22.93 1.28 25.0 −2.0 3.0 22.60 −1.94 3.11 62.35 0.90 0 1.5 −1.5 − 1.39 −1.46 15.39 1.13 8.0 −3.0 −2.0 6.22 −2.95 −1.94 53.44 1.09 View Large The true errors and adjusted errors for all levelling points are depicted, cf. Fig. 7. The systematic errors for different height systems are clearly visible in the curves of true errors, where biases are distinguishable for CASE A and tilts show significant effects for CASE B and C. However, the adjusted errors for all three cases are more consistent as they behave like random errors and vary around zero. Figure 7. View largeDownload slide True and adjusted errors for CASEs A, B and C. True errors are differences between the heights in the local system and the a priori values, while adjusted errors are differences between the heights in the reunified system and the a priori values. Circles represent clocks. Figure 7. View largeDownload slide True and adjusted errors for CASEs A, B and C. True errors are differences between the heights in the local system and the a priori values, while adjusted errors are differences between the heights in the reunified system and the a priori values. Circles represent clocks. For a further understanding of the errors’ behaviour, we plotted the geographical distribution of the true and adjusted errors for all levelling points in Fig. 8. We can easily distinguish different local height systems in the geographical distribution of the true errors. The virtual boundaries caused by the biases are apparent for CASE A. In CASE B, the trend changes of the height errors along the latitudinal direction are visible for each local region, while additional changes along the longitudinal direction are visible for CASE C. The virtual boundaries as well as the trend changes disappeared in the plots of the adjusted errors, where the errors look consistent over the whole region of the unified system. Figure 8. View largeDownload slide Geographic distribution of the true errors (in the left-hand column) and adjusted errors (in the right-hand column) for CASEs A, B and C. The figures on the top row show the results for CASE A, while those on the middle and bottom rows show the results for CASE B and CASE C. The magenta symbols shown in the figures of adjusted errors represent clocks, where each of the four types of symbols (⁠|$\vartriangle ,\triangledown , \square$|⁠, ☆) indicate clocks used for one local height system. Figure 8. View largeDownload slide Geographic distribution of the true errors (in the left-hand column) and adjusted errors (in the right-hand column) for CASEs A, B and C. The figures on the top row show the results for CASE A, while those on the middle and bottom rows show the results for CASE B and CASE C. The magenta symbols shown in the figures of adjusted errors represent clocks, where each of the four types of symbols (⁠|$\vartriangle ,\triangledown , \square$|⁠, ☆) indicate clocks used for one local height system. We can conclude from the above that a proper clock network is well suited for the unification of local height systems. A clock network, composed of a sufficient number of high-performance clocks, can support the estimation of the offsets between different height systems and the tilts along levelling lines of each local height system with high accuracy. In this study, the accuracy of the heights in the reunified system is at the level of 1.0 cm, which corresponds to the introduced random errors of the heights and the clock-based measurements. In the above simulations, we just assigned some clocks for each case. But, the number and the spatial distribution of the clocks were not addressed. However, these factors affect the unification of the local height systems and the cost to install and operate such a clock network. These major issues are discussed in the following. 4.2 Number of clocks An optimal number of clocks should be determined, considering both the performance of the unification and the cost to operate such a clock network. As pointed out previously, the number of clocks should fulfil the overall condition |$\binom{k}{2} \gt \sum _{L=1}^4 n_e - 1$| so that an unambiguous solution can be computed (ne is the number of error parameters, which is 1, 2, 3 for CASE A to CASE C). Because at least one clock should be used for each local height system, k is larger than or equal to four in our simulation cases. From another perspective, it is costly to realize a clock network in practice, since clocks and the dedicated links between them are expensive. We thus should optimally determine the number of clocks to achieve the best cost-performance ratio. To study the optimised number of clocks, we ran further simulations based on CASE C, where four clocks for each local height system have shown a good performance for height unification. Now, we ran three new tests on the number of clocks in each of the four regions. The used numbers are given in brackets: (2, 2, 3, 3), (3, 3, 3, 3) and (5, 5, 5, 5). We see from Figs 9(a) and 10(a) that the adjusted errors are much larger for groups one and two than for the other two groups. We suppose this occurs because the tilts for groups one and two were not correctly estimated. Two clocks are not enough for this purpose, even if the total number of clocks per region could meet the overall requirements. When three clocks are used, the adjusted errors are significantly reduced to the level of the other two groups, see Figs 9(b) and 10(b). However, the performance of the unification was slightly worse compared to the case where four clocks were used. A further improvement was not evident when the number of clocks was increased to five. Figure 9. View largeDownload slide True and adjusted errors for different scenarios where different numbers of clocks were tested for CASE C. The used numbers are given in brackets. Figure 9. View largeDownload slide True and adjusted errors for different scenarios where different numbers of clocks were tested for CASE C. The used numbers are given in brackets. Figure 10. View largeDownload slide Geographic distribution of adjusted errors for different scenarios of CASE C, where different numbers of clocks were tested. Note, adjusted errors in the marked area of Fig. 10(b) are larger than in other areas due to the lack of clocks there. The magenta symbols represent the clocks, and each of the four types of symbols (⁠|$\vartriangle ,\triangledown , \square$|⁠, ☆) indicate clocks used for one local height system. Figure 10. View largeDownload slide Geographic distribution of adjusted errors for different scenarios of CASE C, where different numbers of clocks were tested. Note, adjusted errors in the marked area of Fig. 10(b) are larger than in other areas due to the lack of clocks there. The magenta symbols represent the clocks, and each of the four types of symbols (⁠|$\vartriangle ,\triangledown , \square$|⁠, ☆) indicate clocks used for one local height system. We also ran simulations to identify a suitable number of clocks for the other two cases (the results are not shown here). For CASE A, one clock for each region is enough to estimate the offsets between different height systems. For CASE B, one more clock is required to estimate the tilts appearing along the levelling lines. With slightly more clocks than the basic requirement, the height unification can be achieved with higher accuracy. We can conclude that a few clocks, that is one to three for each local region (with a proper spatial distribution, cf. Section 4.3), can meet the basic requirement for height unification. One clock for each region is sufficient for the estimation of offsets between different height systems. One more ‘domestic’ clock (in the same local region) is needed to correct the tilt over the latitudinal or longitudinal directions. For the case where offsets and tilts along both directions are taken into account (it is more likely to be the real case in practice), at least three clocks for each region are required for the unification. And depending on the availability of slightly more clocks, the performance of unification can be further improved. Furthermore, we found that the adjusted errors in some regions were larger than in others (see the marked regions in Fig. 10b) for the case where three clocks were used. There, no clocks were used in the marked areas. However, the errors can be reduced when one more clock is assigned, see the results for the case of four clocks. This gave us a hint that in addition to the number of clocks, their spatial distribution of the clocks also plays an important role. This is further discussed in the next section. 4.3 Spatial distribution of the clocks The previous discussion addressed the sensitivity of the vertical datum unification to the quantity of clocks available in each region. On one hand, a proper arrangement of clocks can detect the systematic errors of a specific height system. On the other hand, clocks should be arranged to cover as much as possible the research area. Considering both aspects, we assign the clocks to the levelling points which are geographically located as shown in Fig. 11. More specifically for the cases where one to five clocks are used, the clocks’ sites have been selected according to the following schemes. One clock: at the geographic centre of the research region, cf. Fig. 11(a). Two clocks: four variable choices (Figs 11 b–e), along the main- or off-diagonal directions, or along the latitudinal or longitudinal direction. Three clocks: at any three of the four corner sites (cf. the case of four clocks Fig. 11f). Four clocks: at the four corner sites (Fig. 11f). Five clocks: at the four corner sites and the geographic centre, cf. Fig. 11(g). Figure 11. View largeDownload slide The spatial distribution of clocks, for the number of clocks from one to five. Figure 11. View largeDownload slide The spatial distribution of clocks, for the number of clocks from one to five. We took CASE B in which offsets and tilts along the latitudinal directions were included to study the effect of the clocks’ distribution on the unification. The number of clocks were assumed as 2, 2, 3, 2 for each of the local systems. One more clock was assumed for the third region to make sure that the levelling points belonging to this group (the datum selected for the final vertical unification) will be accurately unified. In fact, we compared four different choices, where the spatial distributions of the used clocks were ‘diagonal’, ‘latitudinal’, ‘longitudinal’ and a commixture of them. The results are shown in Figs 12 and 13. The unification was well performed by setting clocks along the diagonal or latitudinal directions. However, when clocks were set along the longitudinal direction, the unification of the height systems one and four was not good. For the ‘commixture’ case, the result for height system one became good. There the clocks were placed along the diagonal direction. The result for height system four remained poor since the clocks were still put along the longitudinal direction. Figure 12. View largeDownload slide Geographic distribution of adjusted errors for different scenarios of CASE B, where spatial distributions of the clocks were tested. The clocks were arranged along the diagonal, latitudinal, longitudinal directions and a commixture of them. The magenta symbols represent clocks, and each of the four types of symbols (⁠|$\vartriangle ,\triangledown , \square$|⁠, ☆) indicate clocks used for one local height system. Figure 12. View largeDownload slide Geographic distribution of adjusted errors for different scenarios of CASE B, where spatial distributions of the clocks were tested. The clocks were arranged along the diagonal, latitudinal, longitudinal directions and a commixture of them. The magenta symbols represent clocks, and each of the four types of symbols (⁠|$\vartriangle ,\triangledown , \square$|⁠, ☆) indicate clocks used for one local height system. Figure 13. View largeDownload slide True errors and adjusted errors for different scenarios of CASE B, where spatial distributions of the clocks were tested. The clocks were arranged along the diagonal, latitudinal, longitudinal directions and a commixture of them. Figure 13. View largeDownload slide True errors and adjusted errors for different scenarios of CASE B, where spatial distributions of the clocks were tested. The clocks were arranged along the diagonal, latitudinal, longitudinal directions and a commixture of them. This analysis shows that clocks should be properly arranged so that they can sense the tilts where necessary. If one clock is available for each region, it can solve the unification of CASE A where only offsets between height systems were considered. In principle, this clock can be assigned on any levelling point. When two clocks are available, the tilt of vertical network might be estimated. The placement of clocks in the longitudinal direction can sense the north–south tilt, and the placement in the latitudinal direction can sense the east–west tilt. In case that the direction of tilt is unknown, it is recommended to place the clock along the (off-)diagonal direction. This placement can well sense the north-south or the east–west tilt. For the case of three clocks, they can be distributed at any three of the corner sites. The placement like a triangle can sense the north–south and the east–west tilts simultaneously. However, the clocks should be arranged to cover most levelling points in practice, because the performance of unification in the area without clocks may be poorer. For the cases of four and five clocks, only one choice of spatial distribution is considered, which can well meet the requirement for the unification. Furthermore, some other factors should be taken into account in practice, for example a frequency link should be available via fibres or satellites. The clocks’ sites should be finally decided on all of these factors, however, this discussion is beyond the scope of this study. 4.4 Clock accuracy So far, clock-based data with a relative frequency uncertainty of 1.0 × 10−18 has been assumed. However, the best clocks available nowadays can achieve one order of magnitude lower than the presented here. Consequently, we performed further experiments to assess the suitability of the existing clocks for the vertical datum unification. The CASE C was chosen for further simulations. Four identical clocks were applied for each local system, which has been shown to be adequate for the unification. We run two more simulations by using clocks with lower accuracy, that is with a relative frequency uncertainty of 5.0 × 10−18 and 20.0 × 10−18, which correspond to about 5.0 and 20.0 cm in height. The results are displayed in Figs 14 and 15. Clocks at the level of 5.0 × 10−18 show a good performance for the unification. The adjusted height errors of the levelling points for the four local systems were reduced to the level of about 1.7 cm and look consistent in the reunified system. When clocks with random errors at the magnitude of 20.0 × 10−18 were used, it seems that the unification can only be achieved to some extent. The tilts along the levelling lines were adjusted somehow, but the offsets were not exactly estimated. The normal heights for the reunified system exhibited a positive shift of about 4.5 cm. However, we should note that the random clock errors are even larger than some height offsets. Figure 14. View largeDownload slide True errors and adjusted errors for different scenarios of CASE C, where clocks with different levels of uncertainty were assumed. Figure 14. View largeDownload slide True errors and adjusted errors for different scenarios of CASE C, where clocks with different levels of uncertainty were assumed. Figure 15. View largeDownload slide Geographic distribution of adjusted errors for different scenarios of CASE C, where clocks with different levels of uncertainty were assumed. The magenta symbols represent clocks, and each of the four types of symbols (⁠|$\vartriangle ,\triangledown , \square$|⁠, ☆) indicate clocks used for one local height system. Figure 15. View largeDownload slide Geographic distribution of adjusted errors for different scenarios of CASE C, where clocks with different levels of uncertainty were assumed. The magenta symbols represent clocks, and each of the four types of symbols (⁠|$\vartriangle ,\triangledown , \square$|⁠, ☆) indicate clocks used for one local height system. The results show that clocks nowadays can unify height systems to some extent, but still lack of accuracy for achieving a highly accurate unified system, for example at the level of 1.0 cm. In this work, the height offsets were assumed only up to 25 cm, however, in practice, this value can reach to 1–2 m in some regions. In such cases even today, clocks may help to unify the height systems to a certain acceptable level, for example at the sub-dm level. Moreover, the accuracy of clock-based data can be improved through repeated measurements, which in principle increases the accuracy by |$\sqrt{n}$| when n times of observations are obtained. Last but not the least, more advanced clocks shall be developed at various research institutes. To make clock comparisons more operational, highly accurate space-based links were favourable, which shall be developed in the coming years. 5 CONCLUSIONS AND FUTURE WORK The rapid development of high-performance optical clocks opens the door to realize relativistic geodesy. In particular, it provides the possibility to estimate physical height differences between two distant points with high accuracies. Clocks are thus expected to be powerful observing tools in future, which may lead to revolutionary changes in geodesy. In this work, we considered to use clocks for height system unification which has been a classical geodetic problem for a long time. Through simulations, we demonstrated the great potential of clocks for this issue. On the assumption that offsets between height datums and tilts along levelling lines in both longitudinal and latitudinal directions were present, three or four clocks measurements for each local region are required to perform the unification. These clocks should be properly arranged so that they can cover most of the network area and sense the tilts where necessary. For example, if clocks are arranged in the longitudinal direction, they will not be sensible to the tilts along the north–south direction. In case that the direction of tilt (in the longitudinal or the latitudinal direction) is unknown, it is suggested to place the clocks along the (off-)diagonal direction. The results also reveal that the clocks nowadays still lack of accuracy to achieve a highly accurate unified height system, for example at the level of 1.0 cm. Clocks with large errors at the level of dm may result in an overall shift of the unified height system. However, this gap could be filled through repeated measurements or the further development of clocks. In reality, however, the character of the systematic errors of national vertical networks might be unknown or more complicated than our assumptions. In this case, the clock-based method can be combined with other geodetic methods, for example the GNSS/geoid approach. The clock part can contribute to remove offsets and tilts from the post-fit residuals, and thereby helps to identify other implicit systematic errors of vertical networks. Then, we can properly extend the functional model to adapt to the more complicated cases. On the other hand, complex errors can be reduced/simplified by subdividing the research region. In the sub regions, our assumption on the character of the systematic errors (being offsets and tilts) can hold true, and therefore our conclusions are correct as well. As a preliminary study, we have assumed some factors at ideal conditions, for example clocks with identical accuracies. This does not hold true in the actual situation. Different types of clocks or clocks developed by different countries have diverse performances. As a result, the frequency comparisons between clocks may have different precision and may be correlated. A refined stochastic model should be applied to assign proper weights to individual measurements and cope with the correlations between different measurements. We also idealised the clock-based data with only random errors. But more likely, clocks have their own biases or drifts or other instrument-related systematic errors. These factors should be well treated in practical applications. Revisiting the goal to establish an international height system, the resolution of IAG in 2015 suggested to adopt the gravity potential as the vertical coordinate. Clocks definitely have the unique advantage in realizing such a height system as the potential difference between different points can be directly obtained through the comparison of clocks. A hybrid clock network which probably includes Earth-bound clocks, space-based clocks and transportable clocks, might be designed to realize a consistent and accurate global height system. Moreover, clock measurements are considered to be combined with other kinds of gravimetric observations to potentially improve the Earth’s gravity field models. This might also be helpful to improve the performance of height system unification by using the GNSS/geoid method. ACKNOWLEDGEMENTS The authors thank the DFG Sonderforschungsbereich (SFB) 1128 ‘Relativistic Geodesy and Gravimetry with Quantum Sensors (geo-Q)’ for financial support. Hu Wu acknowledges the International Space Science Institute (ISSI) for providing the opportunity to present this work at the Workshop on “Spacetime Metrology, Clocks and Relativistic Geodesy”, March 19-23, 2018. We also gratefully thank the two anonymous reviewers whose comments helped to improve and clarify this manuscript. Footnotes 1 https://www.geoq.uni-hannover.de REFERENCES Amjadiparvar B. , Rangelova E. , Sideris M.G. , 2016 . The GBVP approach for vertical datum unification: recent results in North America , J. Geod. , 90 ( 1 ), 45 – 63 . https://doi.org/10.1007/s00190-015-0855-8 Google Scholar Crossref Search ADS Berceau P. , Taylor M. , Kahn J. , Hollberg L. , 2016 . Space-time reference with an optical link , Class. Quant. 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Towards worldwide height system unification using ocean information , J. Geod. Sci. , 2 ( 4 ), 302 – 318 . https://doi.org/10.2478/v10156-012-0004-8 © The Author(s) 2018. Published by Oxford University Press on behalf of The Royal Astronomical Society. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)

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Geophysical Journal International – Oxford University Press

Published: Mar 1, 2019

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