Keywords

1 Introduction

The International Height Reference System, IHRS, was defined in 2015 by the International Association of Geodesy, IAG; see IAG Resolution No. 1 in Drewes et al. (2016). A common global vertical reference is needed for many applications, for instance to investigate and monitor climate related changes in the Earth system (Ihde et al. 2017). The vertical coordinates in IHRS are given by the geopotential numbers, CP, which are defined as the difference between the conventional value W0 = 62,636,853.4 m2s−2 (Sánchez et al. 2016) and the geopotential value at point P, WP. CP can be converted to different types of physical heights, but the preferred type is not specified in the IAG resolution. Ihde et al. (2017) point out that the computation of orthometric heights introduce discrepancies caused by dissimilarities in the hypotheses and recommend the use of normal heights.

The specification and establishment of the first IHRS realisation, the International Height Reference Frame (IHRF), is now one of the highest priorities for the international geodetic community. The global IHRF reference network will realise the IHRS at the highest level. One possibility to determine geopotential values referring to the IHRF is the combination of a gravity field model and ellipsoidal heights determined by a space geodetic technique, most often GNSS (Global Navigation Satellite System). The underlying gravimetric model is crucial in the realisation process (Tocho et al. 2022). Regional high resolution gravity field modelling will be used when available. Otherwise, a suitable combined global Earth Gravitational Model (EGM) of high resolution will be used instead.

The global realisation is to be supplemented by regional and national realisations to provide local access to IHRF and to enable the best possible unification of height datums. In the strategy paper of Sánchez et al. (2021b), it is outlined how the IHRS may be realised on the regional/national level. It is specified that the pointwise realisation may be densified by precise levelling to provide local accessibility to the frame with low uncertainty at short distances up to about 100 km.

1.1 Purpose and Delimitations

The main purpose of the paper is to investigate a selection of methods (Table 1) to make use of a precise levelling network when computing a regional or national IHRS realisation. The paper presents a case study for Sweden using the best GNSS dataset, gravimetric quasigeoid model and precise levelling network currently available. The pointwise IHRS realisation is made following the guidelines of Sánchez et al. (2021a, b) based on the latest Nordic/Baltic gravimetric quasigeoid model NKG2015 (Ågren et al. 2016), the third precise levelling of Sweden (Ågren and Svensson 2011) and the NKG2016LU postglacial land uplift model (Vestøl et al. 2019). Only levelling assisted methods that use potential numbers of the pointwise realisation as fixed in the adjustment are investigated. In a forthcoming study, the plan is to find out how the pointwise geopotential numbers and levelling network should be properly weighted relative to each other.

Table 1 Investigated IHRS realisations/solutions
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The study is a part of a larger project aiming for the best possible realisation of IHRS for Sweden including the transformation to the national height frame RH 2000. Later, the project can hopefully be extended to the Nordic/Baltic level within the Nordic Geodetic Commission (NKG).

2 Method

For a levelling assisted realisation of IHRS, a pointwise realisation is needed to provide fixed (or weighted) potential numbers for the height network adjustments. Sections 2.1 and 2.2 presents the input data and conversions made prior to the levelling network adjustments. After that, in Sect. 2.3, we briefly describe the different height network adjustments. Finally, we outline how the levelling assisted IHRS realisations were compared to the pointwise realisation (Sect. 2.4).

2.1 Gravimetric Geoid Model and Ellipsoidal GNSS Heights Used for the Pointwise Realisation

As mentioned in the introduction, the study is limited to using the current official Nordic gravimetric NKG2015 quasigeoid model (Ågren et al. 2016) for the pointwise IHRS realisation. The model was computed using the Least Squares Modification of Stokes’ formula with Additive corrections (LSMSA) method, also named the KTH method (Sjöberg 1991, 2003). The global satellite-only geopotential model GO_CONS_GCF_2_DIR_R5 (Bruinsma et al. 2013) with maximum degree 300 and regional gravity data from the NKG gravity database were used. The NKG2015 version we use here utilises the W0 value of IHRS, the zero permanent tide concept, and the land uplift epoch 2000.0. It should be mentioned that the officially released version of NKG2015 includes a correction for the permanent tide and a zero-level shift to approximately adapt the model to the Nordic/Baltic height systems, but the pure gravimetric model specified above is used in this paper.

The pointwise solution is based on a dataset of 187 evenly distributed high-quality GNSS stations over Sweden that includes the Swedish global IHRF station ONSA0. The dataset has one station every 35–50 km and the coordinates are given in the official Swedish ETRS89 realisation SWEREF 99 (Jivall et al. 2022). The location of the GNSS stations can be seen in the figures in the result chapter. At least 48 hours of GNSS observations with Dorne Margolin antennas and processing with the Bernese software (Dach et al. 2015) have been used to determine the coordinates of the stations. A list of the used versions of the Bernese software can be found in Jivall et al. (2022). The dataset is also well connected to the precise levelling network; see Sect. 2.3. Like in the ITRF2014, spatial positions in SWEREF 99 are given in the tide-free concept.

2.2 Transformations and Epoch Unification

The postglacial land uplift in the Nordic area makes it crucial to be consistent regarding reference epochs for any kind of geodetic data, models or reference systems (Ekman 1996). All computations and comparisons were thus made in the reference epoch 2021.04 as this epoch was agreed for the first IHRF computation (Sánchez et al. 2021a). All input data were thus converted to this epoch prior to the computations.

The quasigeoid model was converted from the reference epoch 2000.0 to 2021.04 using the geoid change model of NKG2016LU (Vestøl et al. 2019). The GNSS dataset was converted from SWEREF 99 to ITRF2014 epoch 2021.04 applying the NKG transformation method according to Häkli et al. (2016). This method contains a seven parameter Helmert transformation together with an epoch conversion based on the velocity field model NKG_RF17vel (Lantmäteriet 2021).

Corrections to align the permanent tide from the tide-free and zero tide concept in the input data sources to the mean tide concept in the IHRF was applied as specified in Mäkinen (2021).

2.3 Height Network Adjustments

The Swedish precise levelling network is part of the Baltic Levelling Ring (BLR) and is the basis for the national Swedish realisation of the European Vertical Reference System, RH 2000 (Ågren and Svensson 2011). The Swedish levelling observations were measured during approximately 30 years, between 1975 and 2003. The adjustment of RH 2000 was made in Nordic cooperation and was finalised in 2005. In total, the Swedish part of the network consists of around 50,000 height benchmarks, of which 5108 are classified as nodal benchmarks. In the current study, the Swedish precise levelling network was extended by selected lines from other countries in BLR, see Fig. 1, and reduced to include only the measured height differences between nodal benchmarks. The resulting network includes 3380 nodal benchmarks, of which 187 are common to the pointwise IHRS realisation.

Fig. 1
figure 1

The Swedish precise levelling network (dark blue) extended with selected parts of the Baltic Levelling Ring (light blue). Green markers represent the fixed pointwise IHRF stations in the adjustments

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The precise levelling observations, which are the geopotential differences between nodal benchmarks, were converted to the epoch 2021.04 using the postglacial land uplift model NKG2016LU prior to the adjustment. The least squares adjustment was performed using a standard Gauss-Markoff model (Koch 1999), which is also referred to as adjustment by elements in geodesy (Fan 1997). The levelling observations were weighted using the standard model for levelling assuming weights proportional to the inverse of the length of the levelling lines. Besides this, the variance components for data from different countries presented in Mäkinen et al. (2006) were introduced to change the relative weighting between the countries.

This study includes three different levelling assisted realisations (solutions 2 to 4 in Table 1). Solution 2 was made using a minimum constraint adjustment with one station fixed, namely the Swedish station in the global IHRF network, ONSA0, which is marked by a star in Fig. 1. For reference, 187 similar adjustments were made with respect to each of the 187 stations of the pointwise realisation, one at a time. Solution 3 is the mean of all these solutions.

The relative a posteriori standard uncertainties of the adjusted heights of the Swedish precise levelling network are less than about 8–10 mm over 200 km, i.e. relative to a fixed station 200 km away (Ågren and Svensson 2011). According to Sánchez et al. (2021b), high-quality precise levelling can be used in combination with pointwise IHRF stations up to about 100 km to acquire higher resolution and high accuracy locally. Solution 4 is the result of a constrained adjustment with a random selection of fixed stations from the pointwise realisation under the condition to get as closely as possible to 200 km between the fixed stations, cf. the black dots in Fig. 4. The 200 km distance was chosen as it corresponds to a relative standard uncertainty of 10 mm in the levelling (cf. the beginning of this paragraph) and for all levelling stations to be closer than 100 km to the nearest selected pointwise IHRF station anywhere in the network (cf. Sánchez et al. 2021b).

2.4 Comparisons

The four solutions from Table 1 were finally compared with each other. The adjusted geopotential numbers for the height network (solutions 2 to 4) were compared with the pointwise IHRS realisation at the 187 IHRF stations. Statistics of the differences between the solutions were computed as minimum, maximum, mean, and standard deviation. The constrained adjustment solution 4 was compared with both the pointwise realisation and the mean of the minimum constraint adjustments, solution 3.

3 Results

The geopotential numbers at the IHRF stations (same as GNSS stations) were compared according to Sect. 2.4. Statistics for the differences between the solutions are presented in Table 2 and illustrated in the corresponding Figs. 2, 3, 4 and 5.

Table 2 Statistics for the differences between the four solutions in the study. Unit: gpu
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Fig. 2
figure 2

Differences between the minimum constraint adjustment with respect to the global Swedish IHRF station (ONSA0 in green) and the pointwise IHRS realisation. Unit: gpu

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Fig. 3
figure 3

Differences between the mean of the minimum constraint adjustments solution and the pointwise IHRS realisation. Unit: gpu

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Fig. 4
figure 4

Differences between the solution with a selection of fixed IHRF stations at an internal distance of 200 km (solution 4) and the pointwise IHRS realisation (black dots). Unit: gpu

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Fig. 5
figure 5

Differences between the solution with a selection of fixed IHRF stations (solution 4) and the mean of the minimum constraint adjustments (solution 3). Note the different scale compared to Figs. 2, 3 and 4. Unit: gpu

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4 Discussion

With the minimum constraint adjustment with respect to one station, the absolute reference level of the network is relying on one single fixed station. Using the Swedish global IHRF station, ONSA0, as fixed (solution 2), the mean difference and standard deviation compared to the pure pointwise solution (solution 1) are −0.039 gpu and 0.016 gpu, respectively, see Fig. 2. To use only the global IHRF station as fixed is clearly not very representative for the whole of Sweden. Using another station from the pointwise IHRS realisation as fixed, the mean difference will be in the interval from −0.040 gpu to 0.062 gpu. Discrepancies of solutions with one fixed station are closely related to the quality of the pointwise IHRS realisation of the fixed station as the levelling observations remain the same in all compared adjustments. Uncertainties in the gravimetric model and GNSS heights are important factors for the quality of the pointwise realisation, but the uncertainty of other geodynamic modelling required in the realisation process is also crucial. In this case study, the post glacial land uplift was handled by the NKG2016LU model (Vestøl et al. 2019).

The mean of the minimum constraint adjustments, solution 3, is basically a free adjustment solution fitted to the pointwise IHRS realisation with a one-dimensional shift. The minimum and maximum differences compared to solution 1 are −0.062 gpu and 0.040 gpu, respectively, and the standard deviation is 0.016 gpu, see Fig. 3. The shape of the solution relies on the levelling observations only. In one way, solution 3 is not a realistic way to realise IHRS as it demands a pointwise realisation to compute the mean difference. However, assuming that suitable pointwise realisation is available, this kind of solution might be a good option in case one considers the relative uncertainty of the precise levelling to be significantly lower than the relative uncertainty of the pointwise IHRF solution over the whole target area.

For the constrained adjustment with fixed stations every 200 km, solution 4, the minimum and maximum differences to the pointwise realisation are in the same range as solution 3. The standard deviation is slightly lower, 0.015 gpu, and the mean difference is 0.015 gpu, see Fig. 4. The selection of 200 km distance between the fixed stations are based on the motivation in Sect. 2.3. With the constrained adjustment with fixed IHRF stations every 200 km, the shape of solution 4 mainly follows the NKG2015 model over longer distances than 200 km and the levelling over shorter distances. This is considered as a good solution since the accumulated relative standard uncertainty for the levelling network over 200 km (8–10 mm; see Sect. 2.3) is of about the same magnitude as the relative standard uncertainty of the NKG2015 model, which has a very small distance dependence. This means that levelling is better than NKG2015 for shorter distances and NKG2015 is better over longer distances.

The long wavelength systematic pattern in the difference between solutions 4 and 3, see Fig. 5, is either caused by accumulated errors in the levelling network at longer distances or uncertainties in the lower degrees of the gravimetric model. Accumulated long wavelength errors in the levelling network will result in this kind of pattern with a different shape for solution 3 (mean of the minimum constraint adjustments) compared to solution 4. On the other hand, uncertainties in the lower degrees of the gravimetric model will affect the pointwise realisation over longer distances instead, but for very long distances it is well known that gravity field modelling is much better than precise levelling. The errors of the ellipsoidal GNSS heights are considered to be almost uncorrelated and will not produce this kind of long wavelength systematic pattern. The mean difference between solutions 4 and 3 is 0.002 gpu, the minimum and maximum differences are −0.015 gpu and 0.028 gpu, respectively, and the standard deviation is 0.009 gpu. The deviation represents mainly the difference in long wavelength shape between the levelling network and the gravimetric model.

5 Conclusions

The result of this paper shows that a minimum constraint adjustment with respect to one station in the height network is not optimum for a levelling assisted realisation of IHRS in the whole of Sweden.

It can be concluded that a constrained adjustment of the Swedish height network can be used to densify a sparse pointwise realisation. It is shown that a constrained adjustment with 200 km between the fixed stations performs about as well as the mean of the minimum constrained adjustments of the levelling network. It should be noted that a careful consideration of the uncertainty of the levelling network and the pointwise realisation should form the basis of the choice of distance between fixed stations.

A densified IHRS realisation based on the adjustment of a levelling network will provide IHRF potential numbers for a large number of height benchmarks. The Swedish network consists of 3380 nodal benchmarks and about 50,000 benchmarks in total, which will provide the basis for the work on height datum unification and for computing transformation surfaces between the national height reference frame and the IHRF.