Complex demodulation in VLBI estimation of high frequency Earth rotation components

<p>Variations in the Earth's rotation can be examined in the low-frequency and high-frequency temporal scales. The low-frequency variations are dominated by the annual and Chandler wobbles, while the high-frequency variations are primarily caused by tidal effects and mass redistributions within the system Earth. Depending on the purpose, the Earth Rotation Parameters (ERPs) can be estimated in different time resolutions using space-geodetic techniques, especially using GNSS. However, the residual signals between different space geodetic techniques or satellite constellations indicate system-specific differences, which have to be correctly identified.</p><p>This research provides the daily, and sub-daily series of Earth Rotation Parameters (ERPs) estimated using GPS, GLONASS, and Galileo observations. We test different sampling intervals of estimated ERPs from 1h to 24h. The GNSS-based sub-daily estimates have been compared with the external models of variations in ERPs induced by the ocean tides from the IERS 2010 Conventions, a new model by Desai-Sibois, and the VLBI-based model by Gipson.</p><p>Any system-specific ERPs are affected by the orbital and draconitic signals. The orbital signals are visible in all system-specific ERPs at the periods that arise from the resonance between the Earth's rotation and the satellite revolution period, e.g., 8.87h, 34.22h, 3.4 days, 10 days for Galileo; 7.66h, 21.29h, 3.9 days, 7.9 days for GLONASS; 7.98h (S3 tidal term), 11.97h (S2 tidal term), 23.93h (S1 tidal term) for GPS. In the Galileo and GLONASS solutions, the artificial non-tidal signals' amplitudes can reach up to 30 µas. The GPS-derived sub-daily ERPs suffer from the overlapping periods of the diurnal and semidiurnal tidal terms and the harmonics of the GPS revolution period. After recovery of 38 sub-daily tidal terms, the Galileo-based model is more consistent with the external models than the GPS-based model, especially in the prograde diurnal band. The results confirmed that the Desai–Sibois model is more consistent with GNSS observations than the currently recommended model by the IERS 2010 Conventions. Moreover, GPS-based length-of-day (LoD) is systematically biased with respect to the IERS-C04-14 values with a mean offset of −22.4 µs/day, because of the deep resonance 2:1 between the satellite revolution period and the Earth rotation. The Galileo-based and GLONASS-based solutions are almost entirely free of this issue. Against the individual system-specific solutions, the multi-GNSS solution is not affected by most of the system-specific artifacts. Thus, multi-GNSS solutions are clearly beneficial for the estimation of both daily and sub-daily ERPs.</p>

We derive an empirical model of the sub-daily polar motion (PM) based on the multi-GNSS processing incorporating GPS, GLONASS, and Galileo observations. The sub-daily PM model is based on 3-year multi-GNSS solutions with a 2 h temporal resolution. Firstly, we discuss differences in sub-daily PM estimates delivered from individual GNSS constellations, including GPS, GLONASS, Galileo, and the combined multi-GNSS solutions. Secondly, we evaluate the consistency between the GNSS-based estimates of the sub-daily PM with three independent models, i.e., the model recommended in the International Earth Rotation and Reference Systems Service (IERS) 2010 Conventions, the Desai–Sibois model, and the Gipson model. The sub-daily PM estimates, which are derived from system-specific solutions, are inherently affected by artificial non-tidal signals. These signals arise mainly from the resonance between the Earth rotation period and the satellite revolution period. We found strong spurious signals in GLONASS-based and Galileo-based results with amplitudes up to 30 µas. The combined multi-GNSS solution delivers the best estimates and the best consistency of the sub-daily PM with external geophysical and empirical models. Moreover, the impact of the non-tidal spurious signals in the frequency domain diminishes in the multi-GNSS combination. After the recovery of the tidal coefficients for 38 tides, we infer better consistency of the GNSS-based empirical models with the new Desai–Sibois model than the model recommended in the IERS 2010 Conventions. The consistency with the Desai–Sibois model, in terms of the inter-quartile ranges of tidal amplitude differences, reaches the level of 1.6, 5.7, 6.3, 2.2 µas for the prograde diurnal tidal terms and 1.2/2.1, 2.3/6.0, 2.6/5.5, 2.1/5.1 µas for prograde/retrograde semi-diurnal tidal terms, for the combined multi-GNSS, GPS, GLONASS, and Galileo solutions, respectively.

This paper studies the connection between the subdaily model for polar motion used in the processing of very long baseline interferometry (VLBI) observations and the estimated nutation offsets. By convention accepted by the International Earth Rotation Service, the subdaily model for polar motion recommended for routine processing of geodetic observations does not contain any daily retrograde terms due to their one-to-one correlation with the nutation. Nevertheless, for a 24-h VLBI solution a part of the signal contained in the polar motion given by the used subdaily model is numerically mistaken for a retrograde daily sidereal signal. This fictitious retrograde daily signal contributes to the estimated nutation, leading to systematic differences between the nutation offsets from VLBI solutions computed with different subdaily polar motion models. We demonstrate this effect using solutions for all suitable 24-h VLBI sessions over a time span of 11 years (2000--2011). By changing the amplitudes of one tidal term in the underlying subdaily model for polar motion and comparing the estimated parameters to the solutions computed with the unchanged subdaily model, the paper shows and explains theoretically the effects produced by the individual subdaily terms on the VLBI nutation estimates.

Until today the products of the International Earth Rotation and Reference Systems Service (IERS), like International Terrestrial Reference Frame (ITRF), International Celestial Reference Frame (ICRF) and Earth Orientation Parameters (EOP), are combined independently, neither intra-technique nor inter-technique combinations, including the full variance-covariance information are performed. To overcome this deficiencies in the present IERS product generation the IERS implemented a new structure in January 2001. This includes the new IERS Combination Research Centres (CRC) and the IERS Analysis Coordinator (AC). He is responsible for the long-term and internal consistency of the IERS products. To achieve the highest accuracy and consistency, it is crucial to proceed towards a fully rigorous combination of all the parameters common to more than one space geodetic technique. Since 2001 the IERS AC initiated and coordinated many different projects and campaigns towards this overall goal. Now the results of the last three years build the theoretical and practical base for the latest project, the Combination Pilot Project (CPP). This project will prepare the generation of a combined IERS product on a routine basis.

Recent efforts of tracking low Earth orbit and medium Earth orbit (MEO) satellites using geodetic very long baseline interferometry (VLBI) raise questions on the potential of this novel observation concept for space geodesy. Therefore, we carry out extensive Monte Carlo simulations in order to investigate the feasibility of geodetic VLBI for precise orbit determination (POD) of MEO satellites and assess the impact of quality and quantity of satellite observations on the derived geodetic parameters. The MEO satellites are represented in our study by LAGEOS-1/-2 and a set of Galileo satellites. The concept is studied on the basis of 3-day solutions in which satellite observations are included into real schedules of the continuous geodetic VLBI campaign 2017 (CONT17) as well as simulated schedules concerning the next-generation VLBI system, known as the VLBI Global Observing System (VGOS). Our results indicate that geodetic VLBI can perform on a comparable level as other space-geodetic techniques concerning POD of MEO satellites. For an assumed satellite observation precision better than 14.1 mm (47 ps), an average 3D orbit precision of 2.0 cm and 6.3 cm is found for schedules including LAGEOS-1/-2 and Galileo satellites, respectively. Moreover, geocenter offsets, which were so far out of scope for the geodetic VLBI analysis, are close to the detection limit for the simulations concerning VGOS observations of Galileo satellites, with the potential to further enhance the results. Concerning the estimated satellite orbits, VGOS leads to an average precision improvement of 80% with respect to legacy VLBI. In absolute terms and for satellite observation precision of 14.1 mm (47 ps), this corresponds to an average value of 17 mm and 7 mm concerning the 3D orbit scatter and precision of geocenter components, respectively. As shown in this study, a poor satellite geometry can degrade the derived Earth rotation parameters and VLBI station positions, compared to the quasar-only reference schedules. Therefore, careful scheduling of both quasar and satellite observations should be performed in order to fully benefit from this novel observation concept.

Additional results are presented concerning a study that consider improvements over present Earth Rotation Parameter (ERP) determination methods by directly combining observations from various space geodetic systems in one adjustment. Earlier results are extended, showing that in addition to slight improvements in accuracy substantial (a factor of three or more) improvements in precision and significant reductions in correlations between various parameters can be obtained (by combining Lunar Laser Ranging (LLR), Satellite Laser Ranging (SLR) to Lageos, and Very Long Baseline Interferometry (VLBI) data in one adjustment) as compared to results from individual systems. Smaller improvements are also seen over the weighted means of the individual system results. Although data transmission would not be significantly reduced, negligible additional computer time would be required if (standardized) normal equations were available from individual solutions. Suggestions for future work and implications for the new International Earth Rotation Service (IERS) are also presented.

VLBI: A fascinating technique for geodesy and astrometry

Amplitude-frequency analysis of the Earth's rotaryoscillatory motion due to lunisolar gravita� tional and tidal perturbing torques is performed using methods of classical mechanics. Dynamic processes that occur in Earth's rotation parameters (ERP) and Earth's potential due to tidal deformations lead to distortions in Earth's shape and fluctuations in the gravitational field (1, 2). The results of the numerical modeling of oscillatory processes in the Earth's pole motion and variations in the second zonal harmonic δc20 of the geopotential are discussed. The amplitude and phase of oscillations of the Earth's pole are determined using the dynamic Euler-Liou� ville equations. The power spectral densities of the time series of Earth's polar coordinates are compared with those of δc20 variations in the geopotential. This harmonic is found to be a function of the amplitude and phase of the oscillatory process of the Earth's pole. 1. The study of temporal variations in the geopo� tential (2, 3) due to Earth's rotaryoscillatory motion is of scientific and practical interest. The observed ERP variations and variations in the Earth's gravita� tional field are highly interrelated; dynamic processes that lead to appreciable changes in both ERPs and geophysical phenomena have acquired the most detailed reflection in diurnal oscillations (4). The refinement of the Earth's gravitational field involves replacing the statistical geoid by a geoid that corresponds to the deformable shape of the Earth (due to oceanic and rigidbody tides and other factors). Due to increasing demands on the accuracy of the coordinatetime maintenance of navigation systems, investigations related to the refinement of the gravita� tional field and the shape of the Earth have developed intensively (2). Note that the modeling of tidal deformations on a rotating geoid can be used to reach high characteristics of atomic clock synchronization (the stability of atomic frequency is 10 -14 ) for objects that are located on different continents of the globe. Differential equations of the Earth's rotaryoscilla� tory motions with allowance for the harmonic struc� ture of the tidal coefficients were derived from the dynamic Euler-Liouville equations in the problem considering "a deformable Earth-Moon" system in the solar gravitational field (4):

Missing or incorrect consideration of azimuthal asymmetry of troposphere delays is a considerable error source in space geodetic techniques such as Global Navigation Satellite Systems (GNSS) or Very Long Baseline Interferometry (VLBI). So-called horizontal troposphere gradients are generally utilized for modeling such azimuthal variations and are particularly required for observations at low elevation angles. Apart from estimating the gradients within the data analysis, which has become common practice in space geodetic techniques, there is also the possibility to determine the gradients beforehand from different data sources than the actual observations. Using ray-tracing through Numerical Weather Models (NWMs), we determined discrete gradient values referred to as GRAD for VLBI observations, based on the standard gradient model by Chen and Herring (J Geophys Res 102(B9):20489–20502, 1997. https://doi.org/10.1029/97JB01739) and also for new, higher-order gradient models. These gradients are produced on the same data basis as the Vienna Mapping Functions 3 (VMF3) (Landskron and Böhm in J Geod, 2017. https://doi.org/10.1007/s00190-017-1066-2), so they can also be regarded as the VMF3 gradients as they are fully consistent with each other. From VLBI analyses of the Vienna VLBI and Satellite Software (VieVS), it becomes evident that baseline length repeatabilities (BLRs) are improved on average by 5% when using a priori gradients GRAD instead of estimating the gradients. The reason for this improvement is that the gradient estimation yields poor results for VLBI sessions with a small number of observations, while the GRAD a priori gradients are unaffected from this. We also developed a new empirical gradient model applicable for any time and location on Earth, which is included in the Global Pressure and Temperature 3 (GPT3) model. Although being able to describe only the systematic component of azimuthal asymmetry and no short-term variations at all, even these empirical a priori gradients slightly reduce (improve) the BLRs with respect to the estimation of gradients. In general, this paper addresses that a priori horizontal gradients are actually more important for VLBI analysis than previously assumed, as particularly the discrete model GRAD as well as the empirical model GPT3 are indeed able to refine and improve the results.

Precise orbit determination of the satellite “Starlette” with laser ranging data

The noise floor of empirical models of the prograde diurnal Earth rotation could reach as low as 1 µas as shown by several recent studies. In another aspect, the differences between these empirical models with the theoretical model predictions given by International Earth Rotation and Reference Systems Service (IERS) Conventions (2010) for certain diurnal frequencies are more than 10 µas (e.g., K1). The triaxiality of the core is ignored in the theoretical model given by IERS Conventions (2010) because it is highly uncertain. To explain the differences between the empirical models and the theoretical model, we consider the possible influence of a triaxial core. We use the differences between the empirical models and the theoretical model predictions given by IERS Conventions (2010) as inputs to invert the triaxiality parameter of the core. In the inversion, we assume the ocean tide response is subjected to the admittance theory. So extra six admittance parameters are introduced to model the differences between the smooth responses inferred from the empirical models and that given by theoretical model predictions from IERS Conventions (2010). The results show that adding core triaxiality into the theoretical model could narrow the differences between the empirical models and the theoretical model at the prograde diurnal band. As for the dynamical equatorial flattening of the core, the estimates inverted based on the different empirical models are consistent within standard deviation. The results also suggest that the principal axes of the triaxial core do not coincide with the principal axes of the whole Earth.

Ocean tidal effects on universal time and polar motion (UTPM) are investigated at four nearly diurnal (K1, P1, O1, and Q1) and four nearly semidiurnal (K2, S2, M2, and N2) frequencies by analyzing very long baseline interferometry (VLBI) data extending from 1978 to 1992. We discuss limitations of comparisons between experiment and theory for the retrograde nearly diurnal polar motion components due to their degeneracy with prograde components of the nutation model. Estimating amplitudes of contributions to the modeled VLBI observables at these eight frequencies produces a statistically highly significant improvement of 7 mm to the residuals of a fit to the observed delays. Use of such an improved UTPM model also reduces the 14–30 mm scatter of baseline lengths about a time‐linear model of tectonic motion by 3–14 mm, also with high significance levels. A total of 28 UTPM ocean tidal amplitudes can be unambiguously estimated from the data, with resulting UT1 and PM magnitudes as large as 21 μs and 270 microarc seconds (μas) and formal uncertainties of the order of 0.3 μs and 5 μas for UT1 and PM, respectively. Empirically determined UTPM amplitudes and phases are compared to values calculated theoretically by Gross from Seller's global ocean tide model. The discrepancy between theory and experiment is larger by a factor of 3 for UT1 amplitudes (9 μs) than for prograde PM amplitudes (42 μas). The 14‐year VLBI data span strongly attenuates the influence of mismodeled effects on estimated UTPM amplitudes and phases that are not coherent with the eight frequencies of interest. Magnitudes of coherent and quasi‐coherent systematic errors are quantified by means of internal consistency tests. We conclude that coherent systematic effects are many times larger than the formal uncertainties and can be as large as 4 μs for UT1 and 60 μas for polar motion. On the basis of such realistic error estimates, 22 of the 31 fitted UTPM ocean tidal amplitudes differ from zero by more than 2σ.

The paper deals with development of the theory of perturbed rotational motion of a celestial body with variable geometry of the masses. Its main task is to study the impact of annual and semi-annual variations of the Earth's mass geometry (a component of its inertia tensor), as well as a component of its relative angular momentum, on the movement of the Earth's poles and its axial rotation. The body is considered to be a free (isolated), and the problem formulation corresponds to the classical Liouville problem on rotation of a variable body. Euler conical movement of the axially symmetric body with an arbitrary constant half-angle is assumed as the unperturbed motion. In the classical theory of the Earth's rotation this angle is usually assumed to be zero.In the last 20 years, accuracy to determine the Earth rotation parameters owing to using methods of space geodesy and method of Very Long Baseline Interferometry (VLBI) has increased by about three orders of magnitude and has made about i.e., in angle measure it is about 10 - 20 arc-microseconds. According to experts, the theory of the Earth's rotation with such precision is not created yet. The paper is focused just on the new dynamic studies of the Earth rotation at a higher level of accuracy than has been done in previous studies, using a new approach to the problem, based on the new forms of the equations of motion (in the Andoyer variables) and the analytical methods of perturbation theory (small parameter method).The problem of perturbed rotational motion with variable geometry and variable mass relative angular momentum in the first approximation is solved in Andoyer variables and projections of the angular velocity of the planet rotation. The analytical solution allows us to run applications to study dynamic effects from above factors for various bodies in the solar system, including the Earth. The solution allowed us to obtain the following parameters of the fundamental effects in the Earth's rotation: the annual and semi-annual fluctuations of the axis pole of Earth's rotation, annual and semiannual variations in the axial Earth's rotation. The theoretical values of the parameters are in good agreement with modern observation data of the Earth's rotation and the cyclical variations in the geo-potential coefficients.

Summary Until now, the polar motion resonance (PMR) complex frequency has been determined in the seasonal and retrograde diurnal band of the polar motion. In this study, this resonance is studied in the prograde diurnal band, where polar motion is mainly composed of periodic terms caused by the diurnal oceanic tide. The resonance parameters (period and quality factor) are encompassed in the frequency transfer function between generating tidal potential and polar motion, and can be estimated accordingly. To this aim, we gather three published sets of prograde diurnal terms determined from Global Navigation Satellite System (GNSS) and Very Long Baseline Interferometry (VLBI), to which we append our own estimates based upon a processing of the VLBI delays over the period 1990–2020. Then, by fitting the PMR parameters so that the prograde diurnal terms match the corresponding components of the tide generating potential, we obtained a resonance period of about 401 d and an equivalent quality factor of −22, differing from those reigning in the seasonal band (PPMR ≈ 431 d; QPMR ≈ 56–255) and in the retrograde diurnal band (PPMR ≈ 380 d; QPMR ≈ −10). Our estimates confirm strikingly the theoretical prediction derived from the tidal ocean angular momentum derived from the FES 2014 ocean tide model.

We present BKG´s current activities in the area of combined data processing of different space-geodetic techniques. The primary goal of the combined analyses is the improvement of the consistency between the techniques through common parameters, mainly Earth Rotation Parameters (ERP), and thereby to improve also the resulting ERP. In previous studies, we have investigated different combination approaches using VLBI and GNSS data and generated ERP time series with latencies of about 1-2 or 14 days, depending on the input data we used. In this way, we achieved a significant improvement in accuracy, especially for the dUT1 series, compared to the individual technique-specific solutions. The processing is based on homogenized datum-free normal equations (provided via SINEX files), which allow a rigorous combination on the normal equation level instead of the observation level.Our main objective is to generate an ERP product that is characterized by a continuous, daily and regular resolution and the shortest possible latency, especially for the highly variable dUT1. The mandatory requirement for achieving these characteristics is the rapid availability of the input data on the daily basis, especially of the VLBI Intensive sessions. The time series of daily SINEX files of the legacy (S/X) VLBI Intensive sessions show some gaps in the past. The reasons for this are manifold and can be found throughout the entire VLBI processing chain, i.e., from the observation, i.e. station ability to participate, to the analysis, i.e. the data quality revision. However, in the last two years, an increasing number of VGOS Intensive campaigns has been conducted in addition to the legacy Intensives. As a result, the Intensive series is nowadays almost without gaps and there are even more than one Intensive sessions (up to 6) available per day. In our recent studies, we compare the VGOS and legacy Intensives data in terms of their latency and the quality of the resulting dUT1 estimates and integrate them into our combination process. We highlight also the challenges of extending the combination with the new VGOS data. Its incorporation will eventually pave the way for establishing an operational ERP product at BKG.