Normal Height Connection across Seas by the Geopotential-Difference Method: Case Study in Qiongzhou Strait, China

In geodesy, there are mainly two kinds of height systems: the geometric height based on reference ellipsoid, and physically meaningful heights (e.g., orthometric and normal heights). The latter heights are usually derived from a specific equipotential surface of the Earth’s gravity field, which is commonly called a vertical datum. Usually, the vertical height datum is defined as the equipotential surface that goes through the mean sea level (MSL) at the specific reference tide gauge [i.e., China’s vertical datum is defined by the MSL of the Yellow Sea at the Dagang tide gauge at Qingdao in terms of 1952–1979 tide gauge records (e.g., Zhai et al. 2011)]. However, as an effect of sea surface topography (SST), the regional vertical datums derived from different tide gauges in the various spots are not consistent with each other, where the associated offsets among different regions exceed several meters (e.g., Jiao et al. 2002; Ekman 1991). Even in the same country, the vertical datum offsets derived from different tide gauges or the sea level observations from different time spans in the same tide gauge reach several decimeters (e.g., Luz et al. 2008). As a basis of height, the establishment of a regional unified vertical datum is always one of the main targets in physical geodesy, which has been studied by many geodesists, such as Xu (1992), Ekman (1999), Lehmann (2000), Sansò and Venuti (2002), Jekeli and Dumrongchai (2003), Ihde and Sanchez (2005), Featherstone and Kuhn (2006), Ardalan and Safari (2005), and Zhang et al. (2009).

Countries, such as China, with extensive coastlines and hundreds of islands often suffer from the absence of a common and unified vertical datum. Generally, the hydrostatic leveling (Madsen and Tscherning 1990), oceanic dynamic leveling (Mather et al. 1976), and trigonometric leveling methods (Li and Jiang 2001) could be used for height-datum unification. However, the hydrostatic leveling method is time-consuming, costly, and the final accuracy would be largely doubtful, whereas oceanic dynamic leveling requires long-term tidal observations, which are not available in many cases. Moreover, due to the effect of atmospheric refraction, the accuracy of long-distance measurements derived from trigonometric leveling is relatively low. Thus, these approaches are seldom used in height-datum connection across seas for a distance over 10 km (Guo et al. 2005).

Recently used methodology, such as the global positioning system (GPS)/leveling approach, uses GPS/leveling data to fit a local geoid, which can be extrapolated to the unknown height reference surface. It is usually combined with the global geopotential model (GGM) based on remove-compute-restore methodology (Nahavandchi and Sjöberg 1998; Pan and Sjöberg 1998), which improves the quality of height connection in many cases. The GPS/leveling method is easily realized and has been extensively used in height-datum connection (Deng et al. 2013). However, the accuracy of this method largely depends on the distribution of GPS/leveling data, the precision of GGM, as well as the distance for the height-datum connection. The accuracy of height connection based on the GPS/leveling approach for a distance up to 20 km only reaches the decimeter level (Guo et al. 2005). Other methods compute the regional geoid/quasi-geoid over land and sea on the basis of geodetic boundary value theory (GBVP) (Kotsakis 2008; Ardalan et al. 2010). However, the geoid/quasi-geoid modeling requires high-accuracy and high-resolution heterogeneous gravity data over the whole regions, which are not available in many engineering applications.

The key issue of height-datum unification is to determine the geopotential difference among different height systems (Rummel and Teunissen 1988). GPS/leveling data and GGM could also be used to determine the geopotential number of a local vertical datum (Burša et al. 2004). However, because GGMs only accurately reflect the middle- and low-frequency components of the gravity field, the accuracy of this approach is still at the decimeter level. However, as the marine gravity field has been improved significantly by satellite altimetry data in terms of both spatial resolution and accuracy, it is possible to precisely determine the geopotential difference between different height reference surfaces. The rest of the paper is as follows. First, the geopotential-difference method is proposed for long-distance height connection across seas. In addition, the heterogeneous data sets used for height-datum unification are introduced. Then, a case study is investigated, where the Chinese height datum 1985 (CNHD85) is connected to the local vertical datum (LVD) over Hainan island. The last section contains the main summary and conclusion of this study.

According to the geopotential theory, geodetic leveling is a direct measurement of potential difference by combining spirit leveling and gravity observations along the traverse. With the development of the satellite altimetry technique, a similar method could be proposed to determine the potential difference between different height datums across seas.

The mean dynamic topography (MDT) is defined as the distance between the mean sea surface (MSS) and the geoid, which is also expressed as the normal height of the points on the MSS. Whereas the MSS and the geoid vary up to ±100 m relative to the reference ellipsoid on a global scale, the MDT only deviates within a few meters. Typically, the difference in MDT between two adjacent points at sea is similar to the difference in normal heights between two points acquired by spirit leveling on land. This is practicable because the effects introduced by various error sources on these two adjacent points are almost the same as if they were sufficiently close to each other. Assume there is a virtual leveling line that connects the benchmark point on the land to the one at the island. Along this line, a certain number of virtual stations (e.g., the tripods in Fig. 1) are chosen that just play the same role as the real stations on land for the measurement of potential difference (see Fig. 1). In such a way, the geopotential difference between two different height datums across seas is computed by cumulatively combining the difference in MDT between two stations together with the gravity observations along this profile.

As shown in Fig. 1, there are two different height datums: datum $M$ refers to the height system on the mainland, whereas datum $N$ is typically the LVD over the island, which should be connected to the height datum on the mainland. The benchmarks located close to the coastline should be chosen for height connection (see Fig. 1). Two points ($A$ and $B$, respectively) situated on the land and island are selected for height system unification, whereas $A\prime$ and $B\prime$ are the points on the MSS that are closest to $A$ and $B$, respectively. The geopotential difference between $A\prime$ and $B\prime$ is mathematically described aswhere $W_{A\prime }$ and $W_{B\prime }$ = gravity potential value of $A\prime$ and $B\prime$, respectively; $g$ = gravity value at the specific station; and $dh$ = difference in MDT between two neighboring stations.

In practical computation, $\Delta W_{A\text{'}B\text{'}}$ is approximately estimated aswhere $M$ = number of virtual stations; $g_i$ = gravity value at the ith station, which is computed by summing the altimetric gravity anomaly and normal gravity value; $g_m^i$ = mean gravity value between the $i$th and $i+1$-th station; $H_i^{\mathrm{MDT}}$ = MDT value; and ${\epsilon }_i$ = correction for nonparallel equipotential surfaces, which is computed as follows (Heiskanen and Moritz 1967):where ${\gamma }_0^i$ = normal gravity on the reference ellipsoid surface; and ${\gamma }_m^i$ = mean normal gravity from the MSS to the reference ellipsoid surface.

The MDT value for the $i$th station is calculated aswhere $H_i^{\mathrm{MSSH}}$ = MSS height; and $N_i^{\mathrm{Geoid}}$ = geoidal height, which is computed by using the GGM (Pavlis et al. 2012)where $GM$ = geocentric gravitational constant; $(r,\theta ,\lambda )$ = spherical polar coordinates of the computation point; $a$ = length of the semimajor axis of the geocentric reference ellipsoid; ${\overline{C}}_{nm}$ and ${\overline{S}}_{nm}$ = fully normalized geopotential coefficients of degree n and order m; ${\overline{P}}_{nm}(\text{cos}\hspace{0.17em}\theta )$ = fully normalized associated Legendre functions; and $\gamma$ = mean normal gravity.

Generally, the GPS/leveling benchmarks for the height-datum connection should be chosen as the data points near the coastline with low altitude. Moreover, because $A$ and $A\prime$ are close to each other, the difference in the geoidal heights between these two points is supposed to be negligible. Thus, the difference in ellipsoidal height between $A$ and $A\prime$ is expressed as the difference in normal height between themwhere $H_A^{\gamma }$ and $H_{A\prime }^{\gamma }$ = normal height of $A$ and $A\prime$, respectively; $H_{A\prime }^{\mathrm{MSSH}}$ = MSS height of $A\prime$; and $H_A^{\mathrm{GPS}}$ = ellipsoidal height of $A$.

The gravity potential of $B\prime$ is computed bywhere $W_{A\prime }$ is described aswhere $W_0$ = geopotential number of the height datum over the mainland; $H_{A\prime }^{\gamma }$ = normal height of $A\prime$; and ${\gamma }_m^{A\prime }$ = mean normal gravity from the reference ellipsoid surface to $A\prime$, which is computed as (Heiskanen and Moritz 1967)where ${\gamma }_0^{A\prime }$ = normal gravity of the projection of $A\prime$ along the normal gravity line on the reference ellipsoid surface. After spherical approximation, $\partial \gamma /\partial H$ is approximately computed as its mean value on the global scale (i.e., –0.3086 $\mathrm{mGal/m}$. Thus, Eq. (9) is rewritten as

Similarly, the normal height of $B\prime$ relative to the mainland’s height datum is computed bywhere ${\gamma }_{B\prime }^m$ = mean normal gravity from the reference ellipsoid surface to $B\prime$.

Neglecting the difference in the geoidal height between $B\prime$ and $B$, the normal height of $B\prime$ is derived bywhere $H_B^{\gamma }$ and $H_{B\prime }^{\gamma }$ = normal height of $B$ and $B\prime$ (both of them refer to the height datum over the mainland); $H_B^{\mathrm{GPS}}$ = ellipsoidal height of $B$; and $H_{B\prime }^{\mathrm{MSSH}}$ = MSS height of $B\prime$.

To summarize, the normal height of $B$ under the mainland’s height system is described as

Compared to the recently used GPS/leveling approach (e.g., Deng et al. 2013), the method proposed in this paper has two main advantages. First, the differences in the GGM-derived geoidal heights between two neighboring points other than the geoid heights themselves are used to compute the geopotential difference between two height datums. In such a way, the long-wavelength errors of the GGM could be reduced. In addition, more high-quality data sets are incorporated (e.g., satellite altimetry–derived gravity anomalies and MSS model), which play a complementary role to the GGMs in precisely determining the marine gravity field.

For a practical height-datum connection, because the geopotential difference is independent of the selection of spirit leveling line, and the leveling line that connects different height datums across seas could be chosen arbitrarily, the satellite altimetry–derived gravity anomalies and MSS heights on the virtual stations should be interpolated based on the gridded models [e.g., DTU10 (Andersen 2010)]. In this paper, the Shepard method is applied for interpolation, which can be expressed as (William and James 1978)where $f_i$ = given value at the sampling point; $n$ = number of sampling points; $(x,y)$ = interpolation point; $\mu$ = weighting exponent; ρ = weight function; and $r_i$ = radius between sampling and interpolation point, which can be computed aswhere (x_i, y_i) = sampling point. The weight function ρ(ri) is computed asand $R$ = search radius.

To validate the previously discussed method, the normal height connection between Guangdong and Hainan, which are separated by Qiongzhou Strait (Fig. 2), was selected as a case study. The Guangdong height system refers to CNHD85, which is derived from the mean sea level of the Yellow Sea determined by the tide gauge records at Qingdao over 26 years (Zhai et al. 2011), whereas the height system over Hainan is based on a LVD obtained from the tide gauge observations at Xiuying. These two regions are separated by Qiongzhou Strait, and the minimum distance across the strait is over 20 km, which makes some of the traditional methods unsatisfactory in many engineering applications. Based on the method mentioned earlier, the GPS/leveling, GGM, altimetry-derived MSS, and gravity data sets are combined to connect these two different height datums. The data sets used in this case study are described as follows.

The GPS/leveling control network is composed of 12 stations, six of which (i.e., QG01–QG06) are located in Guangdong, whereas the remaining six are situated in Hainan (i.e., QH01–QH06). Trimble (Sunnyvale, California) and Leica (Norcross, Georgia) GPS receivers were used to collect the data, and these observations were processed by GAMIT/GLOBK software, with the final precise ephemerides supplied by the International GPS Service for Geodynamics (IGS). Six GPS/leveling points in Guangdong were measured by second-order leveling under CNHD85, whereas the remaining six points in Hainan were also observed from second-order leveling under the local height datum. Fig. 2 shows the distribution of GPS/leveling points over Guangdong and Hainan.

MSS and marine gravity anomalies were derived from the DTU10 model, which makes use of multisatellite altimetry data sets [e.g., TOPEX/Poseidon, Jason-1, Jason-2, European Space Agency’s Remote Sensing Satellite (ERS-2)] over 17 years (Andersen 2010). The DTU10 model maps with a resolution of 1′ × 1′, the accuracy of the MSS derived from DTU10 (DTU10MSS) is better than 3 cm, and the precision of gravity anomalies obtained from DTU10 (DTU10GRAV) is approximately at 4 mGal (Andersen 2010). The DTU10MSS and DTU10GRAV models over Qiongzhou Strait are shown in Figs. 3 and 4, respectively.

To choose the optimal GGM for accurately computing the marine geoid, four recently published GGMs [i.e., Earth Gravity Model 2008 (EGM2008) with d/o 2190 (Pavlis et al. 2012), European Improved Gravity Model of the Earth by New Techniques 6C (EIGEN-6C) (d/o 1420), EIGEN-6C3STAT (d/o 1949), and EIGEN-6C4 (d/o 2190) (Förste et al. 2011, 2012, 2014)] were tested for their performance in the height-datum connection. These four GGMs are computed by combining satellite gravity-related observations, ground-based gravity data, and satellite altimetry data. The accuracy of these four GGMs is 0.239, 0.238, 0.236, and 0.235 m, respectively, when compared with the globally distributed GPS/leveling data, which are more accurate than other models (see http://icgem.gfz-potsdam.de/ICGEM/evaluation/evaluation.html).

The GPS/leveling points QG03, QG05, and QG06 in Guangdong under CNHD85 were chosen as the starting points, all of which are close to the coastline. Similarly, QH01 and QH06 in Hainan under the local vertical datum were selected as the unknown points, the normal height of which under CNHD85 should be computed. As mentioned earlier, various leveling lines could be chosen between two fixed GPS/leveling points across the ocean. Thus, it is necessary to quantify the effects on the height connection caused by the choice of leveling lines. EGM2008 serves as the reference model, and QG03 and QH01 are the original and unknown points, respectively. As shown in Fig. 5, four leveling lines were chosen to study the effects on the computation of the normal height of QH01, where the colored stars represent the virtual stations along these lines. Table 1 shows the normal heights of QH01 under CNHD85 derived from various leveling lines, which indicate that there are no substantial differences among the results derived from various leveling lines. This is reasonable because the geopotential difference between two fixed points is independent of the selection of leveling lines. Tiny differences show up among these solutions as the errors in the computational model; however, the maximum value is within 1 mm, which is negligible in this study. Based on these results, one representative leveling line could be selected to determine the geopotential difference between two GPS/leveling points across seas.

To connect two height datums across Qiongzhou Strait, QG03, QG05, and QG06 were chosen as the starting points, and QH01 and QH06 were selected as the unknown points, respectively. Figs. 6 and 7 show the configuration of the leveling lines. Table 2 shows the normal heights of the unknown points derived from different starting points based on four GGMs. It is noticeable that the results computed from QG06 show large difference when compared to the solutions derived from the other two points. This may be caused by the poor quality of the GPS/leveling data over QG06 as well as the relatively low accuracy of the satellite altimetry data sets around QG06. As QG06 does not locate in the open sea area, the quality of the neighboring satellite altimetric data is suspicious, which may affect the accuracy of height-datum connection. However, the real reason for the unreliable solutions obtained from QG06 needs further investigation, including rechecking the GPS/leveling benchmarks in QG06 in the future. For the time being, the authors conclude that QG06 cannot be chosen as the starting point for the height-datum connection. Thus, only QG03 and QG05 are applied for height-datum unification in the following part.

It is also worth mentioning that even the normal heights of the unknown points derived from QG03 and QG05 differ from each other in the magnitude of several centimeters. A single GPS/leveling point is not sufficient for representing the geopotential number of a specific height system, which may deviate from the true value. The incorporation of more high-quality GPS/leveling data sets may help to derive more reliable results. However, as the authors do not have enough high-quality GPS/leveling points over the area of interest, the final normal height of the unknown point under CNHD85 is estimated as the weighted average of the two normal heights derived from QG03 and QG05.

Moreover, there are various normal heights derived from different GGMs, as shown in Table 2, where the differences among these solutions reach several centimeters, which means the choice of GGM also affects the quality of normal height connection. Because of the lack of direct observations for CNHD85’s normal heights of unknown points in Hainan (e.g., derived from oceanic dynamic or trigonometric leveling measurement), it is difficult to evaluate the absolute accuracy of the height-datum connection based on this method. Thus, an indirect approach is introduced to evaluate the relative accuracy of the height-datum connection by the geopotential-difference method. This approach makes use of the measurements derived from the second-order leveling network, which have been implemented under the LVD in Hainan. Although the normal heights of GPS/leveling points in Hainan under CNHD85 are unknown, the height difference of two points can be precisely determined by using the second-order leveling results, which can be treated as observations. In another way, the height differences of these two points in Hainan could be computed by the method proposed in this research, which is regarded as the computed values. Thus, the differences between the observed and computed values are used to evaluate the relative accuracy of height connection across the strait, which is the so-called error of the closure of the height difference. Table 3 shows the closing errors of the height differences between QH01 and QH06 derived from various starting points, which shows the height differences between the unknown points are independent of the selection of the starting points. However, the choices of the GGM have nonnegligible effects on the normal height connection. The relative accuracies for normal height computation based on the EGM2008, EIGEN-6C, and EIGEN-6C4 models are 0.012, 0.0132, and 0.009 m, respectively, all of which reach the tolerance requirement of the closure of the height difference for the second-order spirit leveling, whereas the accuracy of the solutions based on the EIGEN-6C3STAT model is 0.022 m, which is relatively low and only meets the requirements for the third-order spirit leveling. These results show the error in GGMs is one of the main error sources that affect the quality of this method, which also indicates the GGMs used in the height-datum connection should be carefully chosen by trial and error. In addition, the specific GGM that derives the best result is supposed to be the optimal one in the local region. Based on these results, the EIGEN-6C4 model was finally incorporated for height-datum unification across Qiongzhou Strait, and the final weight-averaged normal height of QH01 and QH06 under CNHD85 is shown in Table 4. The corresponding results also indicate that the vertical datum difference between CNHD85 and the LVD over Hainan is approximately at the level of 0.186 m.

The authors propose the geopotential-difference method for long-distance normal height connection based on GPS/leveling, GGM, and satellite altimetry data sets. For a case study, the CNHD85 was connected to the LVD in Hainan across Qiongzhou Strait. The results show that this method is independent of the selection of the virtual leveling lines. However, the choice of GGM has nonnegligible effects on the normal height connection, where the differences among the solutions derived from various GGMs reach several centimeters. Furthermore, an indirect approach was used for evaluating the quality of the height connection across the strait based on the high-accuracy spirit leveling measurements in the LVD. The evaluation results show that the relative accuracies for normal height computation based on the EGM2008, EIGEN-6C, EIGEN-6C4, and EIGEN-6C3STAT models are 0.012, 0.0132, 0.009, and 0.022 m, respectively. The relative accuracy of the solutions based on the former three GGMs meets the tolerance requirement of the closure of the height difference for the second-order leveling, whereas the accuracy of the solutions based on the EIGEN-6C3STAT model is relatively low, which only meets the requirements for the third-order spirit leveling. These results indicate the error in GGMs is one of the main error sources that affect the quality of this method, and the GGMs used in the height-datum connection should be chosen carefully. The final CNHD85 normal heights of GPS/leveling benchmarks in Hainan are computed based on the EIGEN-6C4 model, and the corresponding results indicate that the vertical datum difference between CNHD85 and LVD over Hainan is approximately at the level of 0.186 m. It is also noticeable that the indirect approach used for evaluation purposes could not show the absolute accuracy for the height-datum connection based on the geopotential-difference method. The future investigation involves incorporating more independent data for cross validation (e.g., computing the high-quality and high-resolution geoid based on heterogeneous data sets or incorporating the direct observations derived from trigonometric leveling measurements). In addition, because the confidentially kept ground-based gravity data in China have not been incorporated for the computation of these GGMs, the quality of GGMs (e.g., EGM2008 or EIGEN-6C4) may be suspicious in a local region over China. Future work may involve applying the method in other interesting regions (e.g., Europe or Australia), which may provide users with more insight regarding the accuracy of the method. It is practicable that the different height datums could be connected in a more accurate way based on this method, with increasing accuracy of the heterogeneous gravity-related data sets.

The authors thank the two anonymous reviewers for their beneficial suggestions. This research was mainly supported by the National Natural Science Foundation of China (41374023, 41131067); the Key Laboratory of Geospace Environment and Geodesy, Ministry of Education, Wuhan University (15-02-08); and the State Scholarship Fund from the Chinese Scholarship Council (201306270014). Generic Mapping Tools (GMT) was used to draw the figures.