Keywords

1 Introduction

The Earth Orientation Parameters (EOP) describe the rotational part of the transformation between the Celestial Reference Frame (CRF) and the Terrestrial Reference Frame (TRF). They are represented by five components: the pole coordinates \(x_p\) and \(y_p\), the celestial pole offsets \(\delta X\) and \(\delta Y\), and the difference dUT1 between Universal Time UT1 and Coordinated Universal Time UTC (Thaller 2008; Bloßfeld 2015). The knowledge of accurate EOP plays an important role for various applications. This includes, for example, precise positioning and satellite navigation, precise orbit determination and Earth system monitoring, e.g. climate change studies (Gambis and Luzum 2011). The time component dUT1 is the most variable component among the EOP. It is dominated by significant and unpredictable variations which can only be measured with the quasi-space-fixed space-geodetic technique Very Long Baselines Inteferometry (VLBI) (Dermanis and Mueller 1978; Artz et al 2011; Thaller 2008). Since the VLBI observation and correlation process is not fully automated, continuous 24-hour observations are not yet possible (Nothnagel et al 2017). Therefore, the International VLBI Service for Geodesy and Astrometry (IVS) organizes two different geodetic VLBI session types with a limited observation time and a subset of radio telescopes. The 24-hour IVS-Rapid (RAP) observation campaigns are conducted every Monday (R1) and Thursday (R4) with an observation period of 24 hours. With a global network of up to ten antennas, these sessions are suitable for determining all five EOP components, but they have a rather long latency of up to two weeks until the final products in form of SINEX (Solution INdependent EXchange Format) files are available (Nothnagel et al 2017). The IVS additionally organizes daily single- (or triple-) baseline sessions with a large east-west extension and one hour duration. The so-called Intensives (INT) sessions are suitable only for the daily monitoring of the highly variable dUT1 (Robertson et al 1985; Leek 2015). The latency of the dUT1 estimates based on the VLBI INT sessions is about two days or less (Nothnagel et al 2017). Figure 1a illustrates the weekly session distribution of both VLBI observation campaigns.

Fig. 1
figure 1

Weekly distribution of the regular VLBI legacy sessions used for combination and their EOP parameterization: (a) initial offset/drift parameterization of input NEQs, (b) transformed piece-wise linear offset parameterization with two offsets per day, (c) seven-day piece-wise linear offset parameterization after combination

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At this stage, the two VLBI observation campaigns are analyzed separately and two independent EOP time series are estimated and published as official IVS EOP productsFootnote 1:

  • The Session EOP product (EOP-S) is a series of EOP results, one for each 24-hour geodetic session. The estimates are characterized by a high degree of accuracy. However, the temporal resolution is not daily and irregular.

  • The Intensive EOP product (EOP-I) is characterized by a daily but not regular temporal resolution, since it is referred to the mid-session epochs which vary from one Intensive to the next one. The high accuracy is limited to the observation interval of one hour per day.

The objective of our study is the development of a method in which data from both VLBI campaigns are combined in a common adjustment. In this way, we aim to generate a dUT1 time series characterized by a daily, continuous, and temporally regular resolution, with estimates e.g. at 12:00 UTC. By combining the VLBI data of the last seven consecutive days and estimating EOP as continuous piece-wise linear polynomials, we use the past days’ information to stabilize the estimated parameters and minimize random deviations. We expect a higher level of accuracy of the dUT1 time series, which becomes less dependent on the duration and timing of the observations. The use of a regular parameter epoch facilitates the comparison of the dUT1 time series of other techniques with the VLBI series. Furthermore, the EOP series can be used as input for EOP prediction algorithms, which usually require a time series with a daily and regular resolution.

For this purpose, we combine the data of the VLBI legacy S/X campaigns available within approx. two weeks (i.e. INT 1/2/3 and R1/R4) on the normal equation (NEQ) level. The combination at the NEQ level represents the second most rigorous combination method. It can be considered as a good approximation of the combination at the observation level, being the most rigorous combination approach, if the modeling and parameterization of common parameter have been handled in the same way for each input NEQ system (Thaller 2008; Seitz 2009; Schmid 2009; Rothacher et al 2011; Bloßfeld 2015). The solution is more consistent than the typical parameter-level combination approach typically used to generate the official IERS EOP products (Luzum et al 2001; Bizouard et al 2019). The advantages of combining at NEQ level rather than at the parameter level are, for example, that the correlations between parameters are taken into account and it is ensured that the same underlying reference frame is used.

2 Data Input and Combination Methodology

For the combination, we used NEQs in the period between 2014 and 2022 of the BKG IVS Analysis Center (AC), which were provided as SINEX files at the BKG IVS Data Center (DC) (Engelhardt et al 2021). The VLBI RAP NEQs contained all components of the EOP and their temporal derivatives (except for the drift parameter of the celestial pole), station coordinates, and radio source coordinates. The EOP were parameterized at the mid-session epochs. The parameterization of the VLBI INT NEQs was the same, except for the pre-eliminated celestial pole offsets and radio source coordinates. The weekly distribution of the respective sessions and their initial EOP parameterization is shown in Fig. 1a. In preparation for the combination, the NEQs first underwent a parameter transformation step, as depicted in Fig. 1b. The parameterization of the EOP was transformed from the offset/drift representation at mid-session epochs to a piece-wise linear representation consisting of two offsets at 00:00 and 24:00 UTC. A linear epoch transformation was used for the polar motion components. For the time component dUT1 and its negative temporal derivative the Length-of-Day (LOD), it should be noted that the short-time tidal variations were first subtracted according to the IERS Conventions before the reduced parameters \(\textsc {UT1}_R\) and \(\textsc {LOD}_R\) can finally be linearly transformed (Petit and Luzum 2010; Bloßfeld 2015). Since the 24-hour RAP campaigns usually start on Mondays and Thursdays at 17:00 and 18:30 UTC, respectively, the observation window of these sessions contains two consecutive days. The default parameterization of the official IVS products does not allow to separate the observations of both days in post-processing. Therefore, the second day with the majority of observations was chosen for converting the EOP from mid-session epochs to two offsets at 00:00 and 24:00 UTC (See Fig. 1b). In addition, the a priori values of the parameters were transformed to consistent reference series. For this purpose, we used the official products of the IERS, i.e. the IERS14C04 for the EOP (Bizouard et al 2019) and the ITRF2014 time series for the station coordinates (Altamimi et al 2016) (See Table 1). If no ITRF coordinates were available, we use the coordinates listed in the SINEX files as parameter a priori values.

Table 1 Summary of the a priori values and the respective constraints used for the combined solutions
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In the next step, the homogenized constraint-free NEQs of seven consecutive days were stacked to one NEQ system. As shown in Fig. 1c, the EOP were finally parameterized as continuous 7-day piece-wise linear polynomials with offsets every 24 hours at the day boundaries.

In this study, we compared two different combination approaches:

  • 7d-INT: The combination of the VLBI INT data of seven consecutive days with a latency of about two days.

  • 7d-INT+RAP: The combination of VLBI INT and VLBI R1/R4 data of seven consecutive days with a latency of about two weeks.

The analysis of both session types was performed with the same software using identical parameterization and background models. Therefore, no rescaling of NEQs was required before combination. We generated an EOP time series by using a sliding window approach. The window was shifted by one day over the daily NEQs which were combined into a 7-day NEQ system. This procedure was iterated over the entire series of daily NEQs.

For solving the datum-free session-wise and multi-day INT-only NEQs (1d-INT, 7d-INT) all EOP parameters except dUT1 had to be fixed to their a priori values (See Table 1). However, the advantage of the multi-day solution 7d-INT in comparison to the sessions-wise solution 1d-INT is that no constraining on the LOD parameter is required, since relative drift information is obtained by stacking consecutive sessions. Due to the sparse network all station coordinates were fixed to their a priori values (ITRF14). The radio source coordinates were pre-eliminated before generating the corresponding SINEX file and fixed to the a priori values (ICRF3) (Charlot et al 2020). The 24-hour VLBI sessions are appropriate for the determination of all five EOP, so no additional conditions were required. For the combined intra-technique solution 7d-INT+RAP, the pole coordinate information of the 24-hour sessions within the continuous polynomial is not sufficient to estimate high-quality pole coordinates over the entire seven-day period. At this point, it was necessary to apply supporting loose constraints with a threshold of 0.1 mas. This constraints stabilizes the pole coordinate estimates based on INT data only. For the session-wise solution 1d-RAP as well as for the intra-technique combined multi-day solution 7d-INT+RAP no-net-rotation (NNR) and no-net-translation (NNT) conditions were applied on the a priori coordinates of a subset of well-defined and stable VLBI stations. The radio source coordinates are fixed to their a priori values (ICRF3). After applying the constraints, summarized in Table 1, all these datum-free NEQ systems could be solved for the parameters to be estimated.

For the combination processing we use the Combination and Solution package of the DGFI Orbit and Geodetic parameter estimation Software (DOGS-CS), developed and maintained at DGFI-TUM (Deutsches Geodätisches Forschungsinstitut, Technische Universität München) (Gerstl et al 2004).

3 Resulting UT1-UTC Series

Dealing with multi-day time series allows to compare estimated values from the center of a multi-day session window with EOP values from the boundaries. Therefore, for each 7-day solution, we generated seven subseries by extracting the estimates at 00:00 and 24:00 UTC of the same day d from each multi-day solution. The analysis day d ranges from 0 to \(-6\) and represents the analyzed day within the polynomial, where day \(d\,{=}\,0\) is the rightmost and day \(d\,{=}\,{-}6\) the leftmost day on the time axis. For validation purposes, the dUT1 estimates at 00:00 and 24:00 UTC were interpolated to noon epochs, i.e., 12:00 UTC. We analyzed the Weighted Root Mean Square (WRMS) of the residuals of the estimated dUT1 values w.r.t. the IERSBulletinA series, interpolated at the same validation epochs (Luzum and Gambis 2014). The weighting factors for the WRMS were the reciprocal values of the individual dUT1 variance.

The WRMS values of the dUT1 differences at 12:00 UTC of both combination approaches 7d-INT and 7d-INT+RAP are summarized in Table 2. For validation purposes, we had additionally listed the WRMS level of the session-wise VLBI-INT solution 1d-INT as well. In the following, we will not go into more detail about the results of the session-wise and multi-day INT-only solutions (1d-INT, 7d-INT), as they are covered in an other publication. These can be found in Lengert et al (2022).

Table 2 Comparison of the different dUT1 solution types w.r.t. IERS BulletinA. The WRMS of the differences computed at 12:00 UTC epochs in \(\upmu \)s
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In summary, improvement in all WRMS values can be achieved when INT data and 24-hour VLBI data are combined with continuously parameterized EOP over 7-days. (See Table 2). The largest reduction in WRMS values of more than 8 \(\upmu \)s compared to the session-wise INT-only (1d-INT) solution is obtained for the three middle days of the polynomial \(d\,{=}\,{-}4\) to \(d\,{=}\,{-}2\). As expected, the addition of the 24-hour VLBI data stabilized the estimates over the entire polynomial and significantly flattened the symmetric, parabolic behavior of the WRMS of the 7d-INT solution with minima on the middle day. The WRMS values of the 7d-INT+RAP solution ranged at nearly equal level (from 15.0 \(\upmu \)s to 15.8 \(\upmu \)s), with the lowest values for the middle days. This corresponds to an improvement in WRMS values compared to seven-day INT-only (7d-INT) solution of 3.8 \(\upmu \)s and 5.1 \(\upmu \)s for the boundary days \(d\,{=}\,0\) and \(d\,{=}\,{-}6\), respectively. By using a continuous EOP parameterization, the accuracy was almost constant and less dependent on the irregularity of the VLBI observation periods.

4 Conclusion and Outlook

This paper presents the combined processing of VLBI INT and RAP data in a joint adjustment. The aim is to combine the strengths of both session types to estimate a dUT1 time series characterized by a daily, continuous and temporally regular resolution. We achieved a significant improvement in accuracy as evidenced by 35% lower WRMS values of the dUT1 residuals compared to the regular session-wise INT solution. By using a continuous EOP parameterization, the accuracy was almost at a constant level and less dependent on the irregularity of the VLBI observation period. The combination processing was based on homogenized, datum-free NEQs provided via SINEX files from the BKG IVS-AC, which allowed a combination on the NEQ level. We used the DOGS-CS software, developed and maintained at DGFI-TUM. Based on the improved combination method, we intend to set up a new operational VLBI EOP product at BKG, whose characteristics facilitates the comparability of different dUT1 time series with the VLBI series. In addition, the new VLBI EOP series is suitable as input data for EOP prediction algorithms.

There are still some challenges for the future work. The current datum definition of a combined 7-day solution assumes that the antennas of the INT session network are also included in the station network of the 24-hour sessions. This is usually the case, but there are some exceptions. If one or even two INT stations are not included in the 24-hour session, the short observation time of usually one hour may not be sufficient to estimate stable station coordinates. This has a direct effect on the accuracy and stability of the estimated EOP. In the next step we will investigate systematically the affected sessions and improve our datum definition.

In order to estimate a daily, continuous and regular dUT1 series, the daily and rapid availability of input data, especially of VLBI INT sessions, is a mandatory requirement. The series of the daily SINEX files of the legacy (S/X) VLBI INT campaigns has some gaps in the past. The reasons are manifold and can be found throughout the entire VLBI processing chain, i.e., from observation to analysis. However, in the last two years, an increasing number of VGOS INT campaigns has been conducted in addition to the legacy (S/X) INT sessions. As a result, the INT series is nowadays almost without gaps and there are even more than one INT sessions available per day. In the near future, we plan to extend the VLBI intra-technique combination by adding the new VGOS INT data.