The main intention of this study is to evaluate the two chosen NTL data sets in terms of their applicability in DGFI-TUM’s realization of the ITRS 2020. Besides, it should provide guidance for the decision process of picking any NTL data set for the realization of a secular reference frame. Having described and analysed the data of GCTI20 and ESMGFZ, we will now explain our final choice for the DTRF2020.

### Application at the normal equation level

The application of NTAL and HYDL in our previous ITRS realization, DTRF2014, is explained by Seitz et al. (2021). For the DTRF2020, it will basically be the same. The original input data from the International VLBI Service for Geodesy and Astrometry (IVS; Nothnagel et al. 2017), the ILRS, the IGS, and the International DORIS Service (IDS; Willis et al. 2010) for the geodetic space techniques have not been corrected for NTL at the observation level. Hence, we can only correct the input data afterwards at the NEQ level of the weighted least-squares estimation process that we use for the DTRF. (For a detailed description and comparison of the application levels for any site displacements see Glomsda et al. 2021.) This is not an issue, as we are combining the geodetic observations at the NEQ level, anyway. Furthermore, we ensure consistency by applying the same model for all techniques.

To ease the explanation, we provide a few formulas. The DTRF2020 will be a secular reference frame, which means that we are estimating linear motions. For each station (reference point) \(i\), these are represented by an offset vector \(\varvec{p}_i\) at some reference epoch \(t_0\) and a constant velocity vector \(\varvec{v}_i\), which provide a station position \(\varvec{s}_i\) at epoch \(t\) by:

$$\begin{aligned} \varvec{s}_i(t) \quad = \quad \varvec{s}_i(t_0) \, + \, (t - t_0) \, \dot{\varvec{s}}_i \quad =: \quad \varvec{p}_i \, + \, (t - t_0) \, \varvec{v}_i. \end{aligned}$$

(5)

The correction vectors \(\varvec{\Delta p}_i\), \(\varvec{\Delta v}_i\) to some a priori vectors \(\varvec{p}^0_i\), \(\varvec{v}^0_i\) for all stations \(i\) are gathered in the vector \(\varvec{\Delta x}\) (containing also corrections to other geodetic and auxiliary parameters) and obtained by solving the normal equation system (Koch 1999)

$$\begin{aligned} (\varvec{N} + \varvec{N}_D) \,\, \varvec{\Delta x} \quad = \quad \varvec{y} \, + \, \varvec{y}_D \quad = \quad \varvec{y}, \end{aligned}$$

(6)

with normal matrix \(\varvec{N}\), normal matrix of datum-conditions \(\varvec{N}_D\), and right-hand-side \(\varvec{y}\). The right-hand-side of datum-conditions \(\varvec{y}_D\) is equal to \(\varvec{0}\), since we are using minimum conditions. For the DTRF2020, the final normal matrix (right-hand-side) will contain the sum of many single normal matrices \(\varvec{N}_j\) (right-hand-sides \(\varvec{y}_j\)) which refer to individual VLBI, SLR, GNSS, and DORIS observation intervals \(j\).

Correcting for NTL at the NEQ level means applying the corresponding (xyz-coordinate) site displacements

$$\begin{aligned} \varvec{\delta }_j \quad = \quad (\delta _{j1_x}, \delta _{j1_y}, \delta _{j1_z}, \ldots , \delta _{jn_x}, \delta _{jn_y}, \delta _{jn_z},0,\ldots ,0) \end{aligned}$$

(7)

for all stations \(i = 1,\ldots ,n\) in the following way:

$$\begin{aligned} \varvec{y}^{NTL}_j \quad = \quad \varvec{y}_j \, - \, \varvec{N}_j \, \varvec{\delta }_j. \end{aligned}$$

(8)

Hence, the right-hand-side for each observation interval \(j\) is reduced by the product of the interval’s normal matrix and the vector of site displacements for this interval. \(\varvec{\delta }_j\) contains three non-zero values for each station, i.e., one site displacement for each coordinate of their instantaneous positions, which must be a single representative value of the displacements given for the respective observation interval. For VLBI, SLR, GNSS, and DORIS, the relevant observation intervals are (mostly) 24 hours, 1 week or 15 days, 1 day, and 1 week, respectively, and for the DTRF2020 we will apply average values of the corresponding site displacements. The original resolution of the NTL data by GCTI20 is 1 hour, of those by ESMGFZ it is 3 hours (NTAL, NTOL) or 1 day (HYDL, SLEL). As a consequence, both sets can provide at least one value for each technique’s observation intervals, and no bridging of gaps is necessary in either case.

### Centre of mass vs. centre of figure

The choice of (NTL) site displacements in the CM- or the CF-frame depends on the scope of application. If station positions are given in a CF-frame, and NTL is corrected at the solution level, the site displacements must be taken from the CF-frame as well. If NTL is corrected at the observation level or the NEQ level in VLBI analysis, the frame is irrelevant. This is because the site displacements at the two stations forming a baseline are subtracted from each other, and the geocentre motion is cancelled from the CM-frame displacements, leaving the same difference as for the CF-frame displacements (e.g., Eriksson and MacMillan 2014; Glomsda et al. 2021). Regarding GNSS, Männel et al. (2019) use CF-frame displacements for precise point positioning (PPP) solutions with fixed orbits, and CM-frame displacements for network solutions where the orbits are estimated.

CM is the dynamical origin of satellite orbits, hence the satellite techniques SLR, GNSS, and DORIS are basically able to realize this geocentre. The dedicated SLR satellites are spherical and best suited for determining CM: their cross-section is not attitude dependent, and so they are less affected by non-gravitational forces. The non-spherical GNSS and DORIS satellites, on the other hand, are more sensitive to their actual cross-sections and the non-gravitational forces. Due to aliasing effects, the latter distort the geocentre estimates of GNSS and DORIS solutions (e.g., Bloßfeld et al. 2016). For this reason, Helmert parameters are introduced to restore the degrees of freedom w.r.t. the origin for GNSS and DORIS. When applying NTL in a secular reference frame, the choice of CM- or CF-frame displacements is thus irrelevant for GNSS, DORIS, and VLBI (compare above). For SLR, however, the CM-frame displacements are the only option. Furthermore, CM is the origin of the ITRS, and it is just realized by assuming zero translation w.r.t. the origin of an SLR solution in the ITRS realizations of both IGN (Altamimi et al. 2016) and DGFI-TUM (Angermann et al. 2004; Seitz et al. 2021). As a consequence, and for consistency, we will use CM-frame displacements for all four techniques in the computation of the DTRF2020. Both GCTI20 and ESMGFZ provide these displacements, so either choice of data set is still possible.

### Model uncertainties

From the comparison of the NTL data sets, we learned that the agreement of (in particular) the atmospheric and (to a certain extent) the oceanic components is quite good. However, there are significant discrepancies for the hydrological component, and the total NTL displacements do not perfectly match the GNSS position residuals for neither of the two sets. Hence, we must accept for the time being that there is a model uncertainty with respect to the application of NTL effects, especially since there are many other geophysical models and providers of NTL data. A measure of this uncertainty could be the RMS values of the differences between the corresponding site displacements as given in Figs. 3 and 4. Keeping this in mind for the computation of the DTRF2020, we are convinced that the correction for NTL with either of our two data sets will still be beneficial.

### Trends in the displacement series

While the NTL data of GCTI20 and ESMGFZ have been deemed to be equivalent for the application in a secular reference frame up to here, we will now explain the reason to favour one over the other.

The modification of the right-hand-sides when applying NTL at the NEQ level, Eq. (8), is derived from the following approximation of the vector \(\varvec{f}^{NTL}_j\) of theoretical geodetic observations including the effect of NTL (Glomsda et al. 2021):

$$\begin{aligned} \varvec{f}^{NTL}_j \quad \approx \quad \varvec{f}_j \,\, + \,\, \varvec{A}_j \, \varvec{\delta }_j. \end{aligned}$$

(9)

\(\varvec{f}_j\) is the vector of theoretical observations in interval \(j\) without considering NTL, and \(\varvec{A}_j\) is the matrix of partial derivatives of the functional model \(f\) w.r.t. the estimated parameters in \(\varvec{\Delta x}\). The site displacements in \(\varvec{\delta }_j\) are hence implicitly added to the a priori station positions in interval \(j\), and the corresponding impact of NTL on the theoretical observations in \(\varvec{f}_j\) is approximated by the product \(\varvec{A}_j \, \varvec{\delta }_j\). In a secular reference frame, the instantaneous station positions from the observation intervals \(j\) are turned into long-term linear motions as given in Eq. (5). The application of \(\varvec{\delta }_j\) to the a priori values according to Eqs. (8) and (9) changes the instantaneous position estimates. In particular, offsets and drifts in the time-series of site displacements are transferred to the instantaneous positions and will ultimately affect the estimated station offsets \(\varvec{p}_i\) and velocities \(\varvec{v}_i\).

Trends in the displacement series are either geophysically driven or artefacts which can be attributed to (updates in) the background models. As long as these trends are stable over the entire observation period of each station, both cases can be handled well when computing secular reference frames: the individual offsets and drifts are removed from each displacement time-series, and the correction for NTL is performed with the detrended series. If the trends are real geophysical phenomena confirmed by the geodetic observations, the reduced offsets and drifts will be reflected in the estimated station positions and velocities of the secular frame. Thus, all linear motions are finally contained in the latter and not hidden in the NTL corrections. If the trends are artefacts only, which are not supported by the observations of the geodetic techniques, their reduction from the NTL time-series probably results in unaffected estimated positions and velocities, however.

In contrast, if the trend in a displacement series is not constant over time but changes repeatedly during the observation period of the respective station, the single estimated position and velocity of that station will be significantly distorted. To cope with this situation, there are, in our view, two possibilities. First, one could introduce new station position and velocity parameters whenever the trend in the displacement series changes significantly. This option would contradict the nature of a secular reference frame but facilitates both cases, geophysical and artefact trends, if the corresponding offsets and drifts are again removed from the displacement series between each two discontinuities. The alternative, assuming that trend changes are geophysically caused, is to apply the original displacement series (including all trend changes over the entire observation period of a station) as a correction. However, the estimated station positions and velocities would then only reflect one part of the linear movement, namely the joint long-term one, while all trend variations are included in the NTL corrections. This means that a user would have to re-add the site displacements to the station coordinates to get the actual station motion.

Revisiting the available NTL data in this respect, the current ESMGFZ data are not suitable for the application in DTRF2020. As we have seen in our analyses, their displacement time-series contain various changes in offset and drift over the period from 1976 to 2021 (in particular) in the CM-framework. If these were driven by actual geophysical effects, we would have to decide between introducing station position discontinuities or leaving the trends in the NTL corrections. However, these changes simply are the result of transitions between the various underlying atmospheric forcing models. Hence, applying the original ESMGFZ displacements as a whole is not an option, since this would distort the estimated station positions and velocities. On the other hand, the introduction of additional position and velocity parameters per station for such non-geophysical effects seems to be unjustified and possibly harmful in the context of a secular reference frame.

GCTI20, on the contrary, was processed from consistent underlying models between 1980 and 2021. The corresponding time-series of site displacements do not show any significant intermediate changes in their trends, suggesting that there are not even geophysically induced ones. It follows that single offsets and drifts can be removed from the displacement series which are—after the application of the detrended NTL corrections—properly reflected in the estimated long-term linear motion of each station in the DTRF2020. The purpose of a secular TRF thus has been satisfactorily realized.

### Geocentre motion

The behaviour of the geocentre motion contributions shown in Fig. 9 is intrinsically tied to that of the individual displacement time-series. For basically the same reason as given in the previous subsection, the GCTI20 data also has to be preferred over the ESMGFZ data when considering the realization of the DTRF2020 origin with NTL: there are changes in the trend of the geocentre motion contribution for ESMGFZ that are neither geophysically justified nor compatible with the contribution as inferred from the ILRS solution. Instead, they are introduced by the transitions in the underlying atmospheric forcing models and would likely distort the geocentre motion estimated in the DTRF2020. In contrast to that, the trends in geocentre motion as implied by the GCTI20 data are quite stable over the complete time interval (compare Table 2).

### Processing of NTL corrections

Following the above assessment, we will apply the CM-frame site displacements of GCTI20 in the DTRF2020. The displacements will be handled in the following way: for each station, the time-series for each NTL component is cut down to the corresponding observation interval. Each such truncated series is detrended, and the respective offsets and drifts are stored for the final DTRF2020 release. The residual time-series are used to compute 15-daily, weekly, or daily averages for each NTL component, which will be part of the release as well. Finally, the sum of all components’ averages is computed for each relevant observation interval, and each corresponding input normal equation is corrected for NTL by this sum. As a consequence, the trends in the original displacement series will be reflected by the estimated station positions and velocities of the DTRF2020. After all, this is the ultimate purpose of a secular reference frame: the total linear station motions have to be represented by the (estimated) station velocities.

Since the GCTI20 data will be prolongated every few months, the station positions of DTRF2020 can be extrapolated to future epochs with the aforementioned approach on a regular basis. Together with the separation of trends and the inclusion of NTOL, this is the main difference compared to the preceding DTRF2014 (w.r.t. NTL).