NKG2020 transformation: An updated transformation between dynamic and static reference frames in the Nordic and Baltic countries

2.1 NKG transformation

The NKG transformation methodology with associated parameters was published and presented by Häkli et al. (2016). Here, we followed the NKG transformation methodology but updated all associated data for determining a new updated transformation. The new version was labeled as the NKG2020 transformation. The main principle of the NKG2020 transformation remains the same as in the NKG2008 transformation, but the new data resulted in some changes. For determining the NKG2020 transformation parameters, we took the input coordinates from a multiyear GNSS (Global Navigation Satellite System) reference frame solution instead of a GPS (Global Positioning System) campaign that was used in the NKG2008 transformation. The multiyear NKG Repro1 solution (Lahtinen et al. 2019) was processed, and the coordinates and velocities were given in ITRF2014 (Altamimi et al. 2016). Associated intraplate velocities were interpolated from the NKG_RF17vel model, which is aligned to ETRF2014 (European Terrestrial Reference Frame 2014) (Altamimi 2018). According to NKG transformation methodology, this yields that also the transformation hub is aligned to ETRF2014. With this choice, we also avoided unnecessary extra transformations and utilized the latest ETRS89 realization. In the NKG2008 transformation, the intraplate velocities and the transformation hub were aligned to ETRF2000. The origin of the ETRF2014 coincides with that of the ITRF2014 and, therefore, the transformation hub suits better, e.g., for scientific applications like fitting gravimetric geoid models. For the same reason, ETRF2014 differs approximately 5–10 cm from the Nordic–Baltic ETRS89 realizations (which are pre-ETRF2014 realizations) but this still fits perfectly for the purpose. Moreover, all national coordinates were revised and, in many countries, also updated to reflect updates in the national reference frames.

All the new data induced updated national transformations as well. Previously, all national transformations were determined with the seven-parameter Helmert transformation, but as a new method in the NKG2020 transformation the Norwegian transformation is based on a correction grid.

The NKG2020 transformation method is summarized in Figure 1. It is important to note that even if the NKG2020 transformation was determined with the ITRF2014 coordinates, the method supports all ITRF realizations. The first step of the transformation equals the IERS transformation parameters between different ITRFyy realizations, and this enables input coordinates in any of the ITRFyy realizations (IERS 2023 [equation (1)]). This step is not needed if the input coordinates are already in ITRF2014. The second step (equation (2)) transforms the ITRF2014 coordinates to ETRF2014 with the formulae and parameters by EUREF (Altamimi 2018). This step transforms ITRF2014 coordinates at any epoch t to ETRF2014. According to the ETRS89 definition, the coordinates are moved back to epoch 1989.0 but the transformation considers only rigid plate motions, and any intra- or inter-plate deformations are not corrected. These two steps correspond to the EUREF recommendation (Altamimi 2018). The notations in equations (1) and (2) follow that of the IERS conventions (Petit and Luzum 2010) and EUREF Technical Note 1 (Altamimi 2018): T ₁, T ₂ and T ₃ correspond to the translation vectors (origin shifts) along the X-, Y- and Z-axes, R ₁, R ₂ and R ₃ rotations, and R ̇ 1 , R ̇ 2 and R ̇ 3 rotation rates around the X-, Y- and Z-axes, and D is the scale factor. The rotation signs in all equations follow the IERS conventions.

Figure 1

NKG2020 transformation method and the associated steps. The EUREF transformation forms the basis to which additional steps were added to account for the intraplate deformations as well as national reference frames. In the figure, t refers to any coordinate epoch, P _IERS refers to the ITRF transformation parameters estimated by IERS, P _EUREF to the ETRS89 transformation parameters estimated by EUREF, and V _{NKG_RF17vel} to the intraplate velocities from the NKG_RF17vel model.

The following steps are an add-on to the EUREF recommendation and necessary to account for the intraplate deformations due to the land uplift and possible differences between national and pan-European ETRS89 realizations. The third step accounts for the intraplate deformations with the velocities V from the NKG_RF17vel model and brings ETRF2014 coordinates to the common land uplift epoch 2000.0 (equation (3)). The first three steps are common to all Nordic and Baltic countries, and the resulting ETRF2014 coordinates at epoch 2000.0 constitute a transformation hub that was labeled as NKG_ETRF14. The remaining steps are country specific. The fourth step includes national transformations using either Helmert transformation, equation (4), or interpolation from a correction grid. These transformations correct for the pan-European coordinates to the national reference frames. The final step, equation (5), is yet needed to account for the intraplate deformations between the common epoch 2000.0 and the reference epoch (t _r) of the national ETRS89 realizations.

Steps 1–3 constitute the dynamic part of the NKG2020 transformation whereas steps 4–5 remain static (unless possible updates). More details about the background of the transformation steps are discussed by Häkli et al. (2016).

2.2 National transformations: Helmert transformation or least-squares collocation

Typically, three-dimensional transformations have been determined with the seven-parameter (or 14 parameters if rates of parameters are considered) Helmert transformation. The method has also been used for the national transformations in the NKG transformation. The NKG2020 transformation corrects for intraplate deformations with the NKG_RF17vel model but it considers the land uplift (GIA) effects and not deformations or uncertainties in the national reference frames. Subsequent Helmert transformation is a similarity transformation that considers only translation, rotation, and scale of the network. Following the IERS conventions and notations (Petit and Luzum 2010), the three-dimensional similarity transformation can be written with equation (6):

Rewriting equation (6), we get model of observation equations for the Helmert transformation in equation (7):

Observation equation for usual least-squares adjustment by parameters is given with equation (8):

where L is the observation vector, A is the design matrix of observations, x is the unknown Helmert parameters, and ε is the error (residual) of the least-squares adjustment. Omitting the derivation of the equations, we estimated the unknown parameters (or actually their increments) iteratively with equation (9) and finally the transformation errors (residuals) with equation (10) and standard error of unit weight m ₀ with equation (11).

In the equations, P is the weight matrix, and n is the number of points. We used equal weights for observations, P = I (identity matrix).

Helmert transformation is a similarity transformation that maintains the shape of the network from the source ITRF2014 solution that is corrected for the GIA effects with the deformation model. As a result, the transformation residuals describe the remaining deformations between the ETRF2014 and national ETRS89 coordinates at the transformation reference epoch 2000.0. Typically, the ITRF coordinates represent the most precise geometry from up-to-date GNSS observations whereas the geometry of national reference frames degrades over time. Consequently, the transformed coordinates inherit the geometry from a GNSS solution, but it may not coincide with the national coordinates good enough. In other words, the transformation does not provide accurate coordinates with respect to the other national geodata infrastructure, which may be a fundamental problem. In such cases, options are to upgrade either the transformation method or the national realization/coordinates. While a national reference frame is a basis for a national geospatial infrastructure, updating a national realization is a huge undertaking with wide consequences and typically not an option.

Determining the NKG2020 transformation for Norway revealed some inhomogeneities in the ETRS89 coordinates of the Norwegian CORS (Continuously Operating Reference Stations) network. The CORS network is the basis for thousands of lower-order passive benchmarks, and altogether, they are the basis for the geospatial data in Norway. Consequently, the residuals of the pure Helmert transformation were considered too large.

To make the residuals smaller, one can add the number of parameters and/or apply more local transformation. Several possible methods exist, for example, TIN (Triangulated Irregular Network)-based transformation used already in Norway and Finland for legacy transformations or transformations based on higher-order polynomials that are used in Denmark. Also, several gridding methods exist like least-squares collocation (LSC) or Kriging. Typically, the legacy methods are complex to implement in practice, but grids are rather easy to use in modern GIS (Geographic Information System) applications. Least-squares collocation is widely used in geodetic applications, and it provides continuous surface over the area of interest. Such a grid corrects for inconsistencies between the input coordinates up to the uncertainty level of the coordinates. Therefore, it can provide more accurate transformed coordinates with respect to the national geospatial infrastructure. Hence, LSC was chosen as the solution in Norway.

Observation equation for the least-squares collocation can be written with equation (12) (Moritz 1980):

The method of the LSC correction grid is a combination of a systematic part Ax, a stochastic part t, and errors n (typically called noise). The systematic part consists of parameters from the seven-parameter Helmert transformation and the stochastic part consists of signals from LSC. The sum of those parts is stored as corrections of a geocentric translation in a final three-dimensional grid. The Helmert parameters are estimated in the same way as in equation (9) but replacing P = C ⁻¹ where C is the variance-covariance matrix, leading to equation (13):

Then, the estimated signal at grid points can be estimated with equation (14):

where ŝ is the predicted signal, C _st is the covariance matrix between the signals t at observation points and signals s at grid points, and C is the total covariance matrix of the observations L. We estimate C with equation (15):

where C _tt is the covariance matrix of the signal t between observation points (here the fiducial points of the transformation) and C _nn is the covariance matrix of the noise n (here uncertainties of the fiducial points). Covariance matrices C _tt and C _st are estimated with a homogeneous and isotropic covariance function assuming that the mean of the random field (of observations) is constant, and the covariances depend only on the (spherical) distance between the points. For this, we used the first-order Gauss–Markov covariance function according to equation (16):

C ₀ is the observation variance at d = 0, r is the distance between the points in question, and D is the correlation length. For more details of the least-squares collocation, see e.g. Moritz (1980).

2.3 Uncertainty estimation

In Helmert transformations, residuals describe the consistency between the source and target coordinates and are typically used as an uncertainty estimate for the resulting coordinates and transformation (parameters). In the NKG2020 transformation, both input coordinates are already derived coordinates meaning that they include also uncertainties from the preceding coordinate operations and the original coordinates (Figure 1). For the used data, i.e., the associated coordinate solutions and time intervals, the residuals still describe the total uncertainty. But this is valid only for these specific data, and for other user cases, the data and time intervals are typically different; therefore, the residuals are only a part of the total uncertainty budget. We estimated also other sources of uncertainty to give a more realistic total uncertainty budget for the NKG2020 transformation.

The other uncertainties that should be considered include the source coordinates and the velocity model (for differing correction intervals). The NKG2020 transformation also includes the IERS and the EUREF transformation parameters, but their uncertainties, being a part of the ETRS89 definition, could be considered zero. The total uncertainty can be estimated in many ways, but one exact estimate is difficult to give. One important note is that the NKG2020 transformation is time-dependent and so is its uncertainty. Hence, the uncertainty can be divided into a constant and a time-dependent part. It is also important to remember that the NKG2020 transformation is meant to be an accurate link between the global ITRF and national ETRS89 coordinates for practical applications. Hence, instead of giving a theoretical total uncertainty budget, we decided to estimate the uncertainty empirically.

Our ITRF2014 coordinates for estimating the NKG2020 transformation parameters and the residuals were given at epoch t = 2015.0 (more details in Section 3.2). That can be considered as the epoch for the constant part of the uncertainty (residuals). We assessed the residuals as the uncertainty measure with redundant reference frame (GNSS) solutions and propagated their coordinates to epoch 2015.0. Keeping the epoch t = 2015.0 fixed, the velocity corrections, the national transformation parameters and the national coordinates remain constant, and the additional uncertainties (compared to the residuals) can be attributed to the input ITRF coordinates and, if applicable, to the first transformation step from other ITRF realizations to ITRF2014. Even if the ITRF coordinates typically have variance and covariance information available from GNSS solutions, they are usually quite optimistic, and empirical estimation gives the more realistic picture. We evaluated the constant part with three different reference frame solutions and give an estimate of the uncertainty by analyzing transformation accuracies with respect to the national reference coordinates (Section 5.1).

The time-variable part of the NKG2020 transformation is dominated by the uncertainty of the NKG_RF17vel velocity model. Such models are typically combined from several geodetic data techniques and/or other models, and consequently, also their uncertainty may be difficult to estimate reliably. We provide an estimate for the time-variable part of the uncertainty by transforming coordinates from different epochs to national coordinates. For this, we used ITRF2014 daily coordinate time series that were transformed with the NKG2020 transformation. Then, we analyzed the residual time series in the national ETRS89 realizations (Section 5.2).

We consider that with this methodology we can give more realistic and improved uncertainty estimates for the NKG2020 transformation.

3.1 National ETRS89 coordinates

Most of the Nordic–Baltic ETRS89 realizations were originally defined in the 1990s. The original realizations were typically defined or at least accessed mostly through passive benchmarks, whereas active CORS (Continuously Operating Reference Stations) networks are dominating today. Due to this and other technological developments, the national ETRS89 realizations have gotten some updates, but normally changes have been small or minimized in the coordinate domain. Some updates have taken place also since the NKG2008 transformation was published in 2016, and this along with some other improvements means that the NKG2020 transformation is a major update and, in most countries, also deprecates the NKG2008 transformation. The national ETRS89 coordinates were evaluated carefully with each Nordic and Baltic country, and they represent the official coordinates of the country for the NKG2020 transformation. Table 1 collects some key details of the Nordic and Baltic ETRS89 realizations as used in the NKG2020 transformation.

3.2 ITRF2014 coordinates

ITRF2014 coordinates for NKG2020 transformation were taken from the NKG Repro1 solution (Lahtinen et al. 2019). The solution includes approximately 280 CORS stations with the longest time series covering the period 1997.0–2017.1. However, many of the stations had much shorter time series but still usually more than three years. Due to the long time series, we were able to estimate the quality of the coordinates as well as the stations much better than in the NKG2008 transformation, where used ITRF2000 coordinates were based on a one-week campaign. We could identify and reject many young and unstable stations based on the time series analysis. Even though all stations are permanently installed, not all of them are founded on solid bedrock. Such stations are fit-for-purpose for positioning services, but some of them can be unsuitable for long-term reference frame maintenance or geodynamic studies.

Many stations have also undergone some unavoidable instrument replacements, e.g., due to failures. Especially, antenna replacements typically cause jumps in the time series that need to be considered. In such an event, the station velocity is typically constrained to remain the same, but the coordinates are affected by, e.g., uncertainties or differences in antenna calibration values. These different coordinates for different time spans are typically referred to as solution numbers. Optimally, solution numbers for the ITRF2014 coordinates should agree with the national coordinates; otherwise, there may remain discrepancies due to different antennas. Additionally, the epoch for ITRF2014 coordinates is optimally chosen to agree with the time interval of the selected solution number but it should preferably also be quite recent. All these required careful checks of solution numbers and communication with the station operators and/or country representatives.

Based on the evaluations, we chose to use epoch t = 2015.0 for the ITRF2014 coordinates. The NKG Repro1 solution is aligned to IGb14 (Rebischung 2020) and coordinates expressed at the frame reference epoch, 2010.0, from which they were propagated to the chosen epoch 2015.0 with the station velocities. We chose the coordinate solutions from the solution number at the same epoch 2015.0, which had the overall best agreement with the national coordinates. The solution number is relevant mostly for the recently updated national realizations whereas it is less relevant for the old realizations from the 90 s due to many other differences like the use of relative antenna type calibrations, fundamental development in GNSS signals and constellations, and various models used in processing. Some stations had to be rejected due to incompatible solution numbers (offsets), but every rejection was carefully analyzed and based on qualitative measures.

3.3 Velocity model NKG_RF17vel

In the NKG2020 transformation, we use the latest NKG land uplift (or deformation) model NKG_RF17vel (Figure 2). It is a 2D + 1D model where the vertical and horizontal velocities were estimated separately. Vertical velocities equal to those of the NKG2016LU_abs model described in Vestøl et al. (2019). Horizontal velocities were developed later with a similar methodology, and the final 2D + 1D model was released at the beginning of 2020.

Figure 2

NKG_RF17vel model intraplate velocities. Black vectors denote horizontal velocities in ETRF2014. Vertical velocities shown with the colormap. Unit: mm/year.

The NKG2016LU model is based on data from repeated precise leveling, GNSS time series (Kierulf et al. 2021), and GIA model that were combined with least-squares collocation (LSC). The horizontal velocities are based on the GNSS time series (Lahtinen et al. 2019; Kierulf et al. 2021), and the GIA model (Häkli et al. 2019) that were combined with a similar LSC approach. Minor, but unfortunate effect of the separate modeling is that the vertical velocities are aligned to IGb08 (Rebischung 2012) whereas the horizontal velocities are aligned to ETRF2014 (via IGb14). The systematic difference between vertical IGb08 and IGb14 velocities was estimated to be approximately 0.1 mm/year which was considered insignificant compared to the uncertainties. This, along with the fact that the NKG2016LU model had been released and was already in use, led to the conclusion not to alter the model due to this insignificant difference. This would have possibly caused more confusion. Furthermore, Häkli et al. (2016) discussed, thoroughly, why IGb08 (ITRF2008) is not suitable for the NKG transformation, and therefore, the horizontal velocities could not be transformed either. Hence, we consider the vertical velocities to be aligned to ITRF2014 within their uncertainties. Since the velocity difference is systematic, it did not affect the transformation residuals (they are adjusted to agree with the NKG_RF17vel model), but it may become visible with the future data sets (see also uncertainty discussion in Section 5).

The NKG_RF17vel model is a major improvement over the NKG_RF03vel model (Nørbech et al. 2006; Lidberg et al. 2007) that was used for the NKG2008 transformation. The biggest differences are the underlying station velocities and their uncertainties (the GNSS time series were extended from 1996–2004 to 1997–2017), improved GIA models, and enhanced combination strategy for horizontal velocities (from Helmert fit to LSC). Altogether, these improve the quality of the model but also the quantity of stations enabled a better selection of GNSS stations. The overall accuracy of the final NKG_RF17vel velocities was estimated as 0.1, 0.1, and 0.4 mm/year for North, East, and Up components, respectively. The final documentation of the NKG_RF17vel model is yet pending.

4.1 National transformation parameters and residuals

We derived the first coordinate set for estimating the national transformation parameters by applying equations (2) and (3) for the ITRF2014 coordinates described in Section 3.2. This yielded coordinates in ETRF2014 at epoch 2000.0, which is the common transformation hub for all countries and was labeled as NKG_ETRF14 (Section 4.3). We derived the second coordinate set from the national ETRS89 coordinates with reversed equation (5). From these coordinate sets, we estimated the national parameters with the seven-parameter Helmert transformation using equation (9), except for Norway where we ended up using a correction grid instead of the Helmert transformation (see next section). We estimated the national transformation residuals with equation (10), and they are plotted in (Figure 3). The residuals describe how well the two coordinate sets agree with each other at the transformation reference epoch 2000.0 and give an estimate of the expected transformation accuracy.

Figure 3

National residuals in 3D at the transformation reference epoch (2000.0). For the Norwegian case, values refer to coordinate differences between transformed and official national coordinates.

In most cases, the residuals are at a few millimeter levels (Table 2) proving that the Helmert transformation is still adequate for most countries. Larger residuals in Latvia can be explained by the early ETRS89 realization based on the EUREF‐BAL’92 campaign in 1992 with an estimated accuracy of a couple of centimeters. Latvia is underway to update the old LKS-92 realization with a new LKS-20 realization, and the parameters will be updated accordingly. Lithuanian residuals are larger than in the other countries. The horizontal residuals are rather good but in vertical some unexplained discrepancies remained. These will be investigated in the future. The final transformation parameters are given Table 3.

4.2 Correction grid for Norway

Initially, we estimated the Helmert parameters with a subset of Norwegian CORS stations similar to the other countries (NKG Repro1 solution includes only the most stable stations). Already the residuals for the subset revealed some inconsistencies. To evaluate the performance for the rest of the CORS network, we used another extended data set with ITRF2014 coordinates at epoch 2020.0 for 189 CORS stations and transformed the coordinates with the defined parameters. The accuracy (rms) was 2.5, 7.7, and 6.7 mm for North, East, and up components, respectively, and extreme values being more than 2 cm (Table 4). These were considered too large. We estimated also modified Helmert parameters with all 189 stations, but they gave only slightly better results.

Therefore, a method that corrects for weaknesses in the coordinates of the CORS network was needed. We used the gridding (LSC) method that was described in Section 2.2, and we estimated the final corrections as geocentric translations in a final three-dimensional grid. We used all 189 CORS stations with known ITRF2014 and national ETRS89 coordinates to develop the correction grid. We derived the input coordinates for LSC in the same way as described in the previous section yielding coordinates aligned to ETRF2014 at epoch 2000.0 and ETRF93 at epoch 2000.0. We applied LSC to the differences between these coordinates. We chose the LSC parameters based on tuning the signal and the noise. For the final grid, we used the observation variance C ₀ = 0.0008 m², the observation noise n = 0.025 m, and the correlation length D = 150 km.

The accuracy (rms) with the grid improved to 0.7, 0.7, and 2.7 mm for North, East, and Up components, respectively, and the extreme values remained below 1 cm (Table 4). The first version of the LSC correction grid was released in PROJ version 8.0.1 on May 1, 2021 (Figure 4). The grid was updated with the second version of the grid in PROJ 9.0.1 on April 1, 2022. The correction grid was updated due to coordinate changes in some CORS stations in Northern Norway.

Figure 4

Norwegian correction grids in geocentric Cartesian X, Y, and Z coordinates.

The correction grid fulfills the preset expectations and has been easy to implement to the PROJ transformation library that enables easy access for the users.

4.3 Transformation hub

As a result of the NKG2020 transformation, we created a common reference frame aligned to ETRF2014. We labeled this reference frame as NKG_ETRF14. The main purpose of NKG_ETRF14 is to act as a transformation hub but the solution can also be used for pan-Nordic–Baltic or other applications that extend over two or more countries. One important example is the fitting of Nordic–Baltic gravimetric geoid models to GNSS-levelling data sets that should be in consistent reference frames. Therefore, we evaluated its consistency with the official ETRF2014 solutions provided by EUREF. EUREF releases reference frame solutions in ETRF2014 and updates them regularly with cumulative solutions (Legrand 2022b).

We analyzed NKG_ETRF14 with the latest (at the time of writing) solution called EPN_ETRF2014_C2220 covering GNSS data up to GPS week 2220 (Legrand 2022a). We propagated the EPN ETRF2014 coordinates from the frame reference epoch to that of the NKG_ETRF14, i.e., 2000.0 using the published station velocities in ETRF2014. The differences between NKG_ETRF14 and EPN_C2220 coordinates for common stations are shown in Figure 5 and summarized in Table 5.

Figure 5

NKG_ETRF14 minus EPNC2220 (ETRF2014@2000.0).

The NKG_ETRF14 reference frame should not be used outside Nordic and Baltic countries. The reason is that the NKG_RF17vel model was aligned to ETRF2014 mostly with the data from the Nordic and Baltic countries and outside the coverage of geodetic data is worse, and the model is more dependent on the underlying GIA models. Therefore, we have excluded stations outside Nordic and Baltic countries from the Figure and Table. The results look mostly good but there are some larger differences that were not captured by the outlier analysis. They are likely to be caused by mismatching solution numbers and partly also by deviating station and model velocities. The overall agreement is 1.7, 1.5, and 7.4 mm for North, East, and Up components, respectively. Even without further outlier analysis, this proves that the NKG_ETRF14 is in good agreement with the official solutions.

4.4 Differences to the NKG2008 transformation

The previous version of the NKG transformation, the NKG2008 transformation, has been used in various applications, and therefore, it was important to evaluate the influences of updating to the new NKG2020 transformation. The differences in the transformation residuals at the transformation reference epoch 2000.0 are mostly small. These describe the consistency between new and previous data sets used for determining the parameters. The NKG2020 transformation is a major update over NKG2008. Considering the major improvements in the NKG2020, the differences in the residuals are surprisingly small.

However, the used intraplate velocity models differ significantly in some regions (Figure 6). In the Nordic and Baltic countries, the velocity differences remain at a few tenths of millimeter-per-year level for horizontal and up to a millimeter-per-year level for vertical velocities. These lead to differences in the transformed coordinates, and these differences are time-dependent. We took the NKG Repro1 solution and propagated the IGb14 coordinates to different epochs with the associated station velocities and then transformed these coordinates with both transformations. The results, indeed, show that the differences between the transformed coordinates are growing with time (two different epochs are shown in Figure 7). At the transformation reference epoch 2000.0, the differences are small unless the national coordinates have been updated between the NKG2008 and the NKG2020 transformations (which affect the transformation parameters). The difference does not explicitly tell which one has better accuracy, but in the next section, we show that the difference should be understood as an improvement.

Figure 6

NKG_RF17vel minus NKG_RF03vel_ETRF2000 (used in the NKG2008 transformation). NKG_RF03vel_ETRF2000 model was transformed to ETRF2014 before comparison to account for reference frame differences. Horizontal differences shown with grey vector and vertical differences with color map.

Figure 7

NKG2020 minus NKG2008 at transformation reference epoch 2000.0 (left) and at epoch 2020.0 (right). Horizontal differences shown with grey vector and vertical differences with color map.

5.1 Constant part

For the constant part, we extended the uncertainties from the residuals by using three independent data sets having ITRF2014 and ITRF2020 coordinates at the same epoch from which the NKG2020 transformation was determined, 2015.0. The redundant data sets give a more realistic picture of the NKG2020 transformation accuracy.

As the main solution, we used the NKG_Repro1_upd2020 solution (Lahtinen et al. 2022), which we considered the most dense, homogeneous, and up-to-date solution. It also consists of the stations that were regarded to be the most stable ones and suitable for geodynamical studies. As additional data sets, we used the EPN cumulative solution C2220 (Legrand 2022a) and the ITRF2020 solution by IERS (IGN 2023). The NKG and EPN solutions are aligned to IGb14 and the IERS solution to ITRF2020. We propagated all coordinates to epoch 2015.0 with the station velocities, transformed them with the NKG2020 transformation, and compared the resulting coordinates to the official national ETRS89 coordinates (Table 6).

Especially, the ITRF2020 solution has much fewer stations than the NKG solution in the Nordic and Baltic region; yet the overall coordinate differences (rms) between different solutions agree with each other horizontally within a few tenths of millimeters and vertically within approximately one millimeter. On one hand, the good agreement proves that the NKG2020 transformation performs well with different data sets. On the other hand, the results suggest that also ITRF2020 coordinates can also be transformed with the same accuracy-level as ITRF2014 coordinates. The sample was rather small but still the results indicate that the transformation works as planned and that it can be used for ITRF2020 coordinates without further modifications. We tested the recently released IGS20 solution (Villiger 2022) with the NKG2020 transformation as well, but the solution includes even fewer common stations for comparison. Hence, the results are not shown here but also they indicate similar accuracy as with the ITRF2020 solution.

The coordinate differences in Table 6 are the same within a few tenths of a millimeter compared to the residuals of the NKG2020 transformation (cf. Table 2). Hence, the additional component for the total uncertainty budget is negligible for the used GNSS solutions. However, this additional uncertainty component is directly proportional to the used input data and our values reflect the high quality of the GNSS solutions.

The initial results indicated that the differences include most likely some outliers due to, e.g., mismatches in solution numbers between the ITRF and the ETRS89 coordinates (offsets due to instrument changes; Section 5.2). Here, the results refer to the latest solution number in the ITRF solutions, but most likely not all national coordinates agree with this selection. However, checking all solution numbers would have required a thorough and laborious analysis and was disregarded. Instead, we applied a simple outlier detection with three-sigma rejection criteria to the results and considered this to fit the purpose. Differences from the NKG_Repro1_upd2020 solution are shown in Figure 8.

Figure 8

NKG_Repro1_upd2020@2015.0 transformed to the national ETRS89 realizations and compared to the official national ETRS89 coordinates. 3D coordinate differences shown in the plot.

5.2 Time-dependent part

We used daily coordinate time series from the cumulative NKG Repro1 upd2020 solution to evaluate the time-dependent part of the NKG2020 transformation. Cumulative solutions are typically available with regular intervals and can be used to account for, e.g., instrument changes or tectonic events in contrast to infrequently updated reference frame solutions. Typically, station coordinates and velocities with their uncertainties are provided as the result of a reference frame solution, and daily coordinate time series preceding the time series analysis may not be available. In our case, we had the NKG Repro1 upd2020 daily coordinate time series at our disposal. The advantage of daily coordinate time series is that all available data can be used for analysis, e.g. regarding the length of time series and the number of stations. Some stations may have been rejected from secular reference frame solutions if regarded as too unstable. Moreover, daily coordinate time series gives a more detailed picture of the station behavior (e.g., seasonal signals) than just a linear velocity. A disadvantage is that one has to perform the time series analysis oneself to extract, e.g., linear or residual velocities and their uncertainties.

We transformed the daily IGb14 coordinates with the NKG2020 transformation and compared them to the official ETRS89 national coordinates. We analyzed the residual time series in the national ETRS89 realizations with the Hector software (Bos et al. 2013). The data set was the same as for the NKG Repro1 upd2020 reference frame solution release and was therefore already cleaned from outliers and had pre-defined epochs for offsets too. Time series span from 3.3 to 23.5 years, average being almost 13 years. The number of solutions varies from 1–6 (offsets from zero to five), the average being 2. While being a tectonically rather silent region, except for the land uplift, we estimated the magnitudes of the offsets and one linear trend for the whole data period (secular velocity) as well as annual signals. For the estimation of velocity uncertainties, we used a combination of power-law noise plus white noise models.

The NKG2020 transformation accounts for the rigid plate motions and land uplift effects, and therefore, the residual velocities are optimally close to zero. The remaining non-zero velocities indicate either uncertainties in the NKG_RF17vel deformation model or uncertainties or some local effects in the station velocities. The residual velocities with the estimated uncertainties are shown in Figure 9.

Figure 9

Residual velocities in the national ETRS89 realizations (mm/year). Residual velocities in horizontal shown with black vectors and their uncertainties with error ellipses. Vertical residual velocities shown with colored circles and their uncertainties with vertical bars.

Most of the residual velocities are close to zero but quite many larger velocities exist. To analyze possible reasons, we show the length of the time series as well as a number of solutions (offsets) in the time series in Figure 10. Already this explains that most of the larger residual velocities are attributed to shorter time series (smallest circles in the figure). However, some larger vertical velocities are related to long time series as well. Many of these cases are related to discrete time series. Therefore, increasing number of offsets has a clear effect on the residual velocities. The NKG_RF17vel model is based on least-squares collocation that smooths the station velocity field considering the velocity uncertainties, and the aim is to describe the small-scale land uplift phenomenon. Therefore, non-zero residual velocities are most likely caused by the station time series that are either inaccurate or describe some local phenomena. For further analysis, we plotted the ratio of observations and solution numbers in Figure 11. This reflects the length of the uninterrupted time series. Now, most of the larger residual velocities are clearly attributed to the short uninterrupted time series. A couple of exceptions of long uninterrupted time series (large circles) with slightly larger vertical residual velocities pop up: KUUS and ARJ0. Both stations are affected by accumulating snow on the antenna radome every winter. The effect is treated slightly differently in the NKG Repro1 upd2020 (Lahtinen et al. 2022) and BIFROST2015 solutions (Kierulf et al. 2021). Latter was used to develop the NKG2016LU model (the vertical part of the NKG_RF17vel model) and thus included in the NKG2020 transformation. Also, the time spans and the processing setups are different, and altogether, these may explain 0.3–0.4 mm/year vertical residual velocities we observe. On the other hand, the figure shows that the discrete time series do not always result in bad velocities.

Figure 10

Same as Figure 9 but the circle sizes showing the lengths of time series (larger circle means longer time series). Time series vary between 3.3 and 23.5 years. Number of offsets (by means of solution numbers) in the time series shown as a number beside the circle for stations with more than two solution numbers.

Figure 11

Same as Figure 9 but the circle sizes show the ratio of observations and offsets (number of observations divided by the number of solutions). Results before outlier analysis on the left-hand side and after on the right-hand side. This describes the length of the uninterrupted time series. Increasing number of offsets affects the quality of time series.

Considering these ambiguities in the residual velocity analysis, it is difficult to give one definite answer to the transformation accuracy over time. However, the longest time series and/or longest uninterrupted time series suggest that the transformation is almost time-invariant see example from SUN0 station in Figure 12. At the other end, the residual velocities are dominated by inaccuracies and local effects in station velocities, see the examples from KRTN station in Figure 13 and OULU/OUL2 twin station in Figure 14. Somewhere between these ends, the uncertainties of the model and station velocities are converging to describe the actual time-variable accuracy for our purpose.

Figure 12

Example of the residual time series and velocities for SUN0 station in Sweden.

Figure 13

Example of the residual time series and velocities for KRTN station in Lithuania. Several offsets in the time series cause jumps in the coordinates but there are also significant remaining trends in the time series. The station was used for determining the NKG2020 transformation parameters but was found too unstable to be included in the NKG_RF17vel modeling and was captured also in our outlier analysis. Therefore, the residual velocities likely describe local effects.

Figure 14

Example of twin stations OULU and OUL2. The stations are located only 25 meters from each other, so in principle, the residual velocities should converge. The results show large differences in the residual velocities that are likely to be caused by, e.g., deviating time spans in the time series. OUL2 was rejected in the outlier analysis.

Table 7 summarizes the results for 148 time series. To derive statistics, we performed a simple outlier analysis (three-sigma criteria; Figure 11). Based on the analysis, we could reject most of the deviating stations (e.g., KRTN and OUL2 were captured in the outlier analysis). On average, the residual velocities are very close to zero suggesting well-aligned model velocities. We consider the rms in Table 7 as the best estimate for the time-dependent part of the NKG2020 transformation uncertainty and conclude that it is 0.1, 0.1, and 0.3 mm/year in North, East, and Up components, respectively. The extreme residual velocities are in the order of one millimeter per year.

Combining the constant and time-dependent parts from the NKG_Repro1_upd2020 solution, we conclude that the empirically estimated overall accuracy of the NKG2020 transformation is 1.7 mm ± 0.1 mm/year, 1.8 mm ± 0.1 mm/year, and 3.6 mm ± 0.3 mm/year in North, East, and Up components, respectively (1σ). The constant part is given at epoch 2015.0. These values suggest a few millimeter of stability over 10 years that can be considered a very good result.

5.3 Accuracy today (@2023.0)

Even though we derived the overall accuracy (covering constant and time-dependent parts), we evaluated the accuracy also empirically at a recent epoch 2023.0 from a multiyear secular reference frame solution. This is a common way to process ITRF coordinates, and therefore, we get another estimate for the actual accuracy. Similarly to Section 5.1, we used the NKG Repro1 upd2020 solution, propagated the coordinates to epoch 2023.0, transformed them with NKG2020 transformation, and compared the transformed coordinates to the official ETRS89 coordinates. The results are shown in Figure 15 and Table 8. The results show that the estimated total uncertainty with constant and time-dependent parts is quite conservative and that the reference frame solution seems to absorb a part of the uncertainties. Some differences can be caused by different analysis methods like data rejections, noise models, etc. In the end, this again is one more proof that the NKG2020 transformation performs well.

Figure 15

Accuracy today (simple outlier detection applied). IGb14@2023.0 transformed to the national ETRS89 realizations and compared to the official national ETRS89 coordinates.

6.1 Technical solution

We showed that the accuracy of the NKG2020 transformation is at the level of a few millimeters at the transformation reference epoch (residuals) in most cases. We extended the uncertainty estimate from residuals by using three independent data sets and found similar accuracies to the residuals. The improved deformation model NKG_RF17vel, as a part of the NKG2020 transformation, was shown to be able to produce almost static coordinates over time for the most stable reference stations. Overall, we found the empirical accuracy (uncertainty at epoch 2015.0): 1.7 mm ± 0.1 mm/year, 1.8 mm ± 0.1 mm/year, and 3.6 mm ± 0.3 mm/year in North, East, and Up components, respectively (1σ).

As a result, the accuracy degrades only a few millimeters in 10 years. It was also shown to operate equally with the recently released ITRF2020. Therefore, the NKG2020 transformation can be used as a long-term solution without immediate needs for updates. However, such needs may arise from updates in the national reference frames.

The NKG2020 transformation supersedes the previous version NKG2008 due to updated national realizations in many countries. Several improvements over the NKG2008 make NKG2020 more accurate, especially regarding the time-dependency. In many countries, the difference between the NKG2020 and NKG2008 transformations is almost zero at the transformation reference epoch 2000.0 but the differences are growing in time due to different deformation models.

As a new method, we presented a correction grid for the national transformations as a part of the NKG2020 transformation. Such a grid was developed and applied to Norwegian transformation and was shown to correct some deficiencies in regular Helmert transformation, thus improving the accuracy of the national transformation. Correction grid method gained interest also from other countries, but for the moment all countries except Norway still rely on national Helmert transformations.

6.2 Role of the NKG2020 transformation

The NKG2020 transformation is intended as a technical solution to harmonize the transformations from global ITRF coordinates to the national ETRS89 realizations in the Nordic and Baltic countries. As a collaborative work, it already has an official status in many countries, but this should always be confirmed by the appropriate national authority if needed. As a technical solution, the NKG2020 transformation offers an accurate link between global and national coordinates. As such it also provides an option for maintaining static national reference frames under the influence of crustal motions.

Nordic and Baltic reference frames, by means of static coordinates, get deformed over time due to the land uplift. Locally, in (traditional) relative surveying the deformations are normally insignificant but more and more geospatial solutions are relying on nationwide or even global positioning services. Nationwide networks may not be fixed (accurately) directly to the national ETRS89 realizations due to too large deformations. On the other hand, society and GIS systems need static coordinates that do not vary in time. We are also in the advent of global positioning services becoming in wider use. Such services typically provide coordinates in global dynamic reference frames.

The NKG2020 transformation provides a tool to keep the geospatial registries in the national ETRS89 realization and perform positioning in an accurate global dynamic reference frame. This is referred to as a two-frame approach or sometimes to as a semi-dynamic datum (or preferably reference frame). As an example, Finland has deployed the NKG2020 transformation for determining the highest order coordinates (“realization”) of the national EUREF-FIN reference frame. The NKG2020 transformation is used to define coordinates for lower-order active stations as well and for monitoring (maintenance) of the EUREF-FIN.

6.3 Access and standardization of reference frames and transformations

Globalization in the geospatial data domain means the analysis of data from different sources. At the same time, demands for accuracy are increasing. In many places, technical solutions exist (like the NKG2020 transformation) but the standardization is still lagging or is under development. Therefore, it is crucial to develop standardization for reference frames and transformations to ensure seamless and errorless data analysis without misinterpretations.

Several international geodetic registries have been developed to improve standardization regarding reference frames and transformations. Registries store the data including the methods and parameters defining geodetic reference systems and adopted transformations between them. This common information is available for various applications in the geosciences, such as GIS software decreasing the need for customized solutions thus ensuring more homogeneous results. One of the most popular geodetic registries is the EPSG registry (IOGP 2023). NKG joins the standardization work with plans to register the NKG2020 transformation to the EPSG registry. EPSG codes facilitate the use of coordinate reference systems and transformations and decrease risks for mistakes, e.g., in reference frame or transformation-related information. Also, submission to the ISO geodetic registry (ISO 2023) has been in consideration.

In addition to geodetic registries, some open geodetic code libraries exist. One widely used library for coordinate transformations is the PROJ software package for coordinate transformations (Evenden et al. 2022). PROJ is used by several GIS software and implementations, thus serving as a standardization for coordinate transformations. A reference implementation of the NKG2020 transformations was provided in the PROJ software package from version 8.1.0 and onwards. The implementation is compatible with the ISO 19111:2019 standard (ISO 2019) and the EPSG registry. At the time of writing, the transformations have not yet been submitted to the IOGP for inclusion in the EPSG registry but with the PROJ implementation in place, the necessary data modeling is in place.

A prerequisite for such libraries and software is proper data licenses. The NKG2020 transformation and associated data (e.g., NKG_RF17vel model) are attributed to the CC-BY4.0 license (CC 2023) to give wide permissions of use.

The initial goal of the NKG transformations was to harmonize the data and transformations. The PROJ implementation and open data licenses make the NKG2020 transformation and associated data widely available to users.

6.4 Future work

In the short term, we are planning to register NKG2020 in the EPSG registry. This facilitates the use of the transformations and makes them even more accessible. Latvia is about to update its national ETRS89 realization, and associated parameters will be calculated and released as soon as the realization will come into force. Other updates will be made when necessary.

In the longer term, NKG has plans for updating several fundamental data sets related to the NKG transformation. These include reprocessing of an extended (both spatially and temporally) GNSS solution aligned to the latest ITRF2020 reference frame. The velocity field with realistic uncertainty estimates together with an improved GIA model will serve as the basis for an updated deformation model. With the new deformation model also a new version of the NKG transformation becomes rational (NKG 2022).