Helmert Transformation Parameters Research Articles

Оценка согласованности орбитальных реализаций систем координат, распространяемых глобальными навигационными спутниковыми системами, с Международной земной системой координат ITRF за 2020–2023 гг.

The compatibility assessment of the terrestrial reference frames orbital realizations broadcasted by the global navigation satellite systems (GPS, GLONASS, Galileo and BeiDou) with the International terrestrial reference frame ITRF during 2020–2023 was performed. The assessment is based on the analysis of daily Helmert transformation parameters determined in the Information and Analysis Center for Positioning, Navigation and Timing of the Central Research Institute for Machine Building within calculating a posteriori ephemeris and time information for each constellation. Based on the daily Helmert transformation parameters, annual reference frames coincidence parameters RSS7 and Λ and their standard deviations StDRSS7 and StDΛ on the monthly, 10-day and daily averaging intervals were computed, characterizing the stability of terrestrial reference frames orbital realizations. The obtained results show the following level of coincidence between the terrestrial reference frames broadcasted by global navigation satellite systems and ITRF

SOME ISSUES OF THE CALCULATION PROCESS OPTIMIZATION IN TERRESTRIAL LASER SCANNERS CALIBRATION

A conventional and widely used mathematical model of laser scanners calibration has such an imperfection as correlation between some of parameters during the adjustment process and quite complex numerical solution. A method of avoiding the correlation between scanner and polygon transformation parameters and calibration parameters as well as calibration numerical solution simplification is considered. It is proposed to implement a separate transformation parameters computation by using Kabsch-Umeyama algorithm that can compute Helmert transformation parameters even in case of gross errors present in data set. But direct transformation before adjustment leads to parametric equations and following numerical solution complication. Therefore it is also proposed to conduct a reverse transformation from polygon coordinate system into scanner coordinate system as it simplifies further equations linearization and allows simplifying the numerical solution in general. As it is more than one station to be scanned during the calibration process, one cannot simply swap coordinate systems. The polygon coordinate system remains as original one but reverse Helmert transformation is conducted. To prove this approach is correct, three datasets using different laser scanners and high-precise total station were obtained. Direct and reverse transformation between scanners and total station coordinate systems was made for all datasets and MSE of transformation was calculated for each case. Thus, the possibility of using the reverse transformation to optimize the process of the calibration parameters determining has been experimentally proven.

Study on systematic errors of BDS-3 broadcast ephemeris and their effects with Helmert transformation

Previous studies have not evaluated the systematic errors implied in the third generation of BeiDou-3 Navigation Satellite System (BDS-3) broadcast ephemeris. In this paper we evaluate the systematic pattern described by the Helmert transformation parameters, including translations, rotations, and scale. BDS-3 broadcast and precise ephemerides from December 2019 to 2022 are collected, and the characteristics of the transformation parameters as well as their effects on the signal in space error are analysed. The annual variation in the z-translation is obtained, and the similar amplitudes of 5.5 cm and phases of approximate 300 days are obtained for different years. When the rotation parameters are considered in the orbit comparison, the Root Mean Square (RMS) errors of the along- and cross-track orbital differences decrease from 29.1 to 12.5 cm and from 30.6 to 9.2 cm, respectively, because the three rotation parameters compensate for the majority of the errors in the BDS-3 broadcast ephemeris. Moreover, the high correlations in the obtained rotation parameters among the three orbital planes suggest that the orientation of the BDS-3 broadcast ephemeris is influenced by common model errors, i.e., uncertainty of Earth Rotation Parameters (ERPs). Further research is required because an offset of 1.5 × 10–9 for the scale parameter is observed. A degraded User Range Error (URE) for epochs of up to 84% is attained when the systematic pattern is considered, though the impact of the systematic pattern indicated by the z-translation and rotation parameters on the URE is less than 5.0 cm. With the refinement of the ERPs implemented in the new generation of broadcast ephemeris, we anticipate that the broadcast ephemeris performance of BDS-3 will be improved.

Impact analysis of arc length in multi-GNSS ultra-rapid orbit determination based on the one-step method

With the development of near real-time and real-time high-precision global navigation satellite system (GNSS) applications, users’ requirements for the accuracy and stability of ultra-rapid orbits have increased, especially for multi-constellation orbits. Generally, several factors affect the accuracy and stability of multi-GNSS ultra-rapid orbit determination; however, this study is focused only on analyzing the orbit arc length of ultra-rapid orbits of the GPS, GLONASS, BeiDou-2, and Galileo satellites based on the one-step method. To realize nearly ideal conditions in the process of analysis, one full year’s rapid orbit products during 2018 provided by the GeoForschungsZentrum multi-GNSS experiment analysis center were used as fitted observed orbits and also as a reference orbit for comparison to the predicted orbit. In terms of residual analysis of orbit difference and stability analysis of Helmert transformation parameters, the numerical results showed that the optimal orbit arc lengths of quad-constellation medium earth orbit (MEO) satellites for 6 h and 3 h predicted ultra-rapid orbits based on the one-step method were in the range of 44–45 h and 41–44 h, respectively. Overall, for both 6 h and 3 h predictions, when only considering the impact of the orbit arc length on the accuracy and stability of the orbit itself, 44 h can be considered as the overlapping optimal orbit arc length of quad-constellation MEO satellitesʼ ultra-rapid orbits generated by the one-step method. In addition, these ranges of orbit arc length can contribute to the multi-GNSS ultra-rapid orbit determination.

A sequential estimation approach to terrestrial reference frame determination

We review the main concepts underlying the determination of terrestrial reference frames (TRFs) through a recursive algorithm based on Kalman Filtering and Rauch-Tung-Striebel (RTS) smoothing which is currently adopted at Jet Propulsion Laboratory (JPL) to compute sub-secular frame products (JTRFs). We contextualize the TRF determination in the state-space framework and we emphasize connections between frame state, its observability through space-geodetic frame inputs and the similarity transformation which is central to frame definition. We elaborate on the notion of sub-secular frame, enabled by our approach, in constrast to standard TRF products which, secular by construction, are designed to represent the long-term mean physical properties of the frame. Comparisons of JTRF solutions to standard products such as the International Terrestrial Reference Frame (ITRF) suggest high-level consistency in a long-term sense with time derivatives of the Helmert transformation parameters connecting the two TRFs below 0.18 mm/yr. We discuss advantages and limitations of JPL approach to TRF determination and outline lines of inquiries that are currently being researched as part of JTRF development plan.

Impacts on Noise Analyses of GNSS Position Time Series Caused by Seasonal Signal, Weight Matrix, Offset, and Helmert Transformation Parameters

The noise characteristics of the Global Navigation Satellite System (GNSS) position time series can be biased by many factors, which in turn affect the estimates of parameters in the deterministic model using a least squares method. The authors assess the effects of seasonal signals, weight matrix, intermittent offsets, and Helmert transformation parameters on the noise analyses. Different solutions are obtained using the simulated and real position time series of 647 global stations and power law noise derived from the residuals of stacking solutions are compared. Since the true noise in the position time series is not available except for the simulated data, the authors paid most attention to the noise difference caused by the variable factors. First, parameterization of seasonal signals in the time series can reduce the colored noise and cause the spectral indexes to be closer to zero (much “whiter”). Meanwhile, the additional offset parameters can also change the colored noise to be much “whiter” and more offsets parameters in the deterministic model leading to spectral indexes closer to zero. Second, the weight matrices derived from the covariance information can induce more colored noise than the unit weight matrix for both real and simulated data, and larger biases of annual amplitude of simulated data are attributed to the covariance information. Third, the Helmert transformation parameters (three translation, three rotation, and one scale) considered in the model show the largest impacts on the power law noise (medians of 0.4 mm−k/4 and 0.06 for the amplitude and spectral index, respectively). Finally, the transformation parameters and full-weight matrix used together in the stacking model can induce different patterns for the horizontal and vertical components, respectively, which are related to different dominant factors.

Weekly inter-technique combination of SLR, VLBI, GPS and DORIS at the solution level

Constructing and maintaining a stable terrestrial reference frame (TRF) is one of the key objectives of fundamental astronomy and geodesy. The datumrealization for all the global TRF versions, such as ITRF2014 and its predecessor ITRF2008, assumes linear time evolution for transformation parameters and then imposes some conditions on these Helmert transformation parameters. In this paper, we investigate a new approach, which is based on weekly estimation of station positions and Helmert transformation parameters from a combination of the solutions of four space-geodetic techniques, i.e., Satellite Laser Ranging (SLR), Very Long Baseline Interferometry (VLBI), Global Positioning System (GPS) and Doppler Orbitography and Radiopositioning Integrated by Satellite (DORIS). For this study, an interval of one week is chosen because the arc length of the SLR solutions is seven days. The major advantage of this weekly estimated reference frame is that both the non-linear station motions and the non-linear origin motion are implicitly taken into account. In order to study the non-linear behavior of station motions and physical parameters, ITRF2008 is used as a reference. As for datum definition of weekly reference frame, on one hand SLR is the unique technique to realize the origin and determine the scale together with VLBI, and on the other hand the orientation is realized via no net rotation with respect to ITRF2005 on a subset of core stations. Given the fact that without enough collocations an inter-technique combined TRF could not exist, the selection and relative weight of local ties surveyed at co-location sites are critical issues. To get stable results, we first assume that, if there were no events such as equipment changes between the measurement epoch of the local tie and that of the space-geodetic solution, the relative position between the two co-located stations should be invariant and this local tie could be used for computing the inter-technique combined reference frame in those weeks during the stable period of this tie. The resulting time series of both station positions and transformation parameters are studied in detail and are compared with ITRF2008. The residual station positions in the weekly combined reference frame are usually in the range of two millimeters without any periodic characteristic, but the residual station positions, when subtracting the regularized station position in ITRF2008, may reach a magnitude of a few centimeters and seem to have a significant annual signal. The physical parameter series between the weekly reference frame and ITRF2008 also show the obvious existence of an annual signal and reach a magnitude of one centimeter for origin motion and two parts per billion (ppb) for scale.

TRANSFORMATION PARAMETERS BETWEEN UCS-2000 AND WGS-84

According to the Order of the Ministry of Agrarian Policy of Ukraine, published in 2016, about the procedure for using the national coordinate system UCS-2000, this was the first time officially presented parameters of the Helmert transformation from the UCS-2000 system to the ITRF-2000 system. However, all software products are used for communication between various coordinate systems as the main coordinate system, WGS-84. The Helmert transformation parameters between the UCS-2000 and WGS-84 systems are found for the new realization of WGS-84 (G1762) based on GPS data and the old realization of the WGS-84, based on the US Satellite Navigation System, known as DOPPLER Transit. It is shown that the use of the transformation parameters of the old realization WGS-84 for the processing of present GPS measurements will result in a systematic error of the order of 0.6 m. Obtained transformation parameters can be used as the first approximation to obtain accurate Helmert transformations based on GPS measurements at points with known coordinates in the UCS-2000 system. The described procedure for determining the parameters will be especially useful in the case when a more accurate connection will be established between the systems ITRFyy and WGS-84, than the current one.

Effect of Helmert Transformation Parameters and Weight Matrix on Seasonal Signals in GNSS Coordinate Time Series.

Seasonal signals caused by the Earth’s surface mass redistribution can be detected by Global Navigation Satellite Systems (GNSS). The authors analyze the effect of Helmert transformation parameters and weight matrices, as well as the additional draconic signals on seasonal signals, in the GNSS coordinate time series. Moreover, the contribution of environmental loading models to the GNSS position series is assessed. Position time series of 647 global stations, with spans of 2–21 years are collected to generate six cumulative solutions using different parameters estimated in a deterministic model, as well as weight matrices. Comparison among the different solutions indicates that Helmert transformation parameters and weight matrices can result in a root mean square of 0.1 mm and 0.3 mm for seasonal signals, respectively. Compared to the displacements obtained from environmental loading models, seasonal signals estimated with the Helmert parameters and full weight matrices considered seems to have the best agreement with the results of the loading model. Meanwhile, the additional draconic signals are not effective to be parameterized in the deterministic model with an observation time span less than 15 years, marginally. There are 62%, 72% and 90% of 647 stations with weight root mean squares (WRMS) reduced by removing the loading-model-induced changes from the GNSS residual series for the east, north and vertical components, respectively. Finally, to obtain a velocity estimation with a bias of less than 0.1 mm/yr induced by seasonal signals, the position series with a time span greater than seven years is suggested.

On the long-term stability of terrestrial reference frame solutions based on Kalman filtering

The Global Geodetic Observing System requirement for the long-term stability of the International Terrestrial Reference Frame is 0.1 mm/year, motivated by rigorous sea level studies. Furthermore, high-quality station velocities are of great importance for the prediction of future station coordinates, which are fundamental for several geodetic applications. In this study, we investigate the performance of predictions from very long baseline interferometry (VLBI) terrestrial reference frames (TRFs) based on Kalman filtering. The predictions are computed by extrapolating the deterministic part of the coordinate model. As observational data, we used over 4000 VLBI sessions between 1980 and the middle of 2016. In order to study the predictions, we computed VLBI TRF solutions only from the data until the end of 2013. The period of 2014 until 2016.5 was used to validate the predictions of the TRF solutions against the measured VLBI station coordinates. To assess the quality, we computed average WRMS values from the coordinate differences as well as from estimated Helmert transformation parameters, in particular, the scale. We found that the results significantly depend on the level of process noise used in the filter. While larger values of process noise allow the TRF station coordinates to more closely follow the input data (decrease in WRMS of about 45%), the TRF predictions exhibit larger deviations from the VLBI station coordinates after 2014 (WRMS increase of about 15%). On the other hand, lower levels of process noise improve the predictions, making them more similar to those of solutions without process noise. Furthermore, our investigations show that additionally estimating annual signals in the coordinates does not significantly impact the results. Finally, we computed TRF solutions mimicking a potential real-time TRF and found significant improvements over the other investigated solutions, all of which rely on extrapolating the coordinate model for their predictions, with WRMS reductions of almost 50%.

Comparison of Ultra-Rapid Orbit Prediction Strategies for GPS, GLONASS, Galileo and BeiDou.

Currently, ultra-rapid orbits play an important role in the high-speed development of global navigation satellite system (GNSS) real-time applications. This contribution focuses on the impact of the fitting arc length of observed orbits and solar radiation pressure (SRP) on the orbit prediction performance for GPS, GLONASS, Galileo and BeiDou. One full year’s precise ephemerides during 2015 were used as fitted observed orbits and then as references to be compared with predicted orbits, together with known earth rotation parameters. The full nine-parameter Empirical Center for Orbit Determination in Europe (CODE) Orbit Model (ECOM) and its reduced version were chosen in our study. The arc lengths of observed fitted orbits that showed the smallest weighted root mean squares (WRMSs) and medians of the orbit differences after a Helmert transformation fell between 40 and 45 h for GPS and GLONASS and between 42 and 48 h for Galileo, while the WRMS values and medians become flat after a 42 h arc length for BeiDou. The stability of the Helmert transformation and SRP parameters also confirmed the similar optimal arc lengths. The range around 42–45 h is suggested to be the optimal arc length interval of the fitted observed orbits for the multi-GNSS joint solution of ultra-rapid orbits.

Helmert transformation with mixed geodetic and Cartesian coordinates

It is common in geodetic practice that station geodetic coordinates in an old frame are measured/estimated by adjusting conventional leveling and vertical networks (together with a vertical datum), whereas their Cartesian coordinates in a new frame are measured/estimated by adjusting modern 3D Global Navigation Satellite System (GNSS) networks. To estimate Helmert transformation parameters between the two frames, one can simply convert the geodetic coordinates to their Cartesian counterparts in the same frame, and then perform a least-squares transformation with Cartesian coordinates in both frames. This stepwise approach is not optimal because of nonlinearity in the first step. In this work, a direct transformation is conducted. A functional model with mixed geodetic and Cartesian measurements is followed, and a realistic stochastic model considering measurement errors in both frames is adopted. Correlations between common and non-common stations are also taken into account. Weighted least-squares estimates of the transformation parameters and transformed/adjusted coordinates in the new frame are derived in detail, the latter representing the enlarged network of the new frame. Simulations are conducted, and the results validate the superiority of the proposed method compared with the stepwise method.

JTRF2014, the JPL Kalman filter and smoother realization of the International Terrestrial Reference System

AbstractWe present and discuss JTRF2014, the Terrestrial Reference Frame (TRF) the Jet Propulsion Laboratory constructed by combining space‐geodetic inputs from very long baseline interferometry (VLBI), satellite laser ranging (SLR), Global Navigation Satellite Systems (GNSS), and Doppler orbitography and radiopositioning integrated by satellite submitted for the realization of ITRF2014. Determined through a Kalman filter and Rauch‐Tung‐Striebel smoother assimilating position observations, Earth orientation parameters, and local ties, JTRF2014 is a subsecular, time series‐based TRF whose origin is at the quasi‐instantaneous center of mass (CM) as sensed by SLR and whose scale is determined by the quasi‐instantaneous VLBI and SLR scales. The dynamical evolution of the positions accounts for a secular motion term, annual, and semiannual periodic modes. Site‐dependent variances based on the analysis of loading displacements induced by mass redistributions of terrestrial fluids have been used to control the extent of random walk adopted in the combination. With differences in the amplitude of the annual signal within the range 0.5–0.8 mm, JTRF2014‐derived center of network‐to‐center of mass (CM‐CN) is in remarkable agreement with the geocenter motion obtained via spectral inversion of GNSS, Gravity Recovery and Climate Experiment (GRACE) observations and modeled ocean bottom pressure from Estimating the Circulation and Climate of the Ocean (ECCO). Comparisons of JTRF2014 to ITRF2014 suggest high‐level consistency with time derivatives of the Helmert transformation parameters connecting the two frames below 0.18 mm/yr and weighted root‐mean‐square differences of the polar motion (polar motion rate) in the order of 30 μas (17 μas/d).

DORIS data processing in the INASAN Analysis Center and the contribution to ITRF2014

We present results of the DORIS data processing for the ITRF2014 carried out in the Institute of Astronomy, Russian Academy of Sciences (short INA) for the time period 1993.0–2015.0. The evaluated SINEX free-network (inawd08) weekly solutions for station positions and polar motion have been submitted to the IDS Combination Center. We applied the upgraded models of the gravity field, troposphere, and used corrected data for SPOT-5 satellite which permitted us to improve the result precision for Helmert transformation parameters and EOP. The agreement with the IERS C04 solution turns out better than 0.1 mas with a dispersion of 1 mas. The evaluated amplitudes of annual geocenter variations derived from inawd08 weekly solutions are 3.4±0.5mm, 4.4±0.5mm, and 3.3±1.0mm for X, Y, and Z components, respectively. The recovered amplitudes and phases are in a good agreement with the geophysical models. The results of two single satellite campaigns related to the scale increase in 2012 and 1994 for IDS combination are studied. Unlike other Analysis Centers, we did not apply a phase center law for ground antennas. It was shown that the correction for the phase center variation (PCV) of ground DORIS antennas results in the shift of the mean scale factor by 1.24ppb (i.e., ∼8mm) with respect to the case without correction. The PCV model and the satellites HY-2A and SARAL are included into the INA latest solutions (inawd10) after the submission for the ITRF2014. The future work is discussed.

The development of the GSFC DORIS contribution to ITRF2014

The NASA GSFC DORIS analysis center has processed data from January 1993 to December 2014 and provided 1141 weekly solutions in the form of normal equations for incorporation into the DORIS solution for ITRF2014. The solution time series, designated as gscwd26, were based on tracking data to eleven DORIS satellites divided generally into seven-day arcs. With respect to the ITRF2008 submission (Le Bail et al., 2010), the measurement model was updated to model the beacon frequency variations at certain DORIS sites, to apply the DORIS antenna phase law for the Starec and Alcatel antennae, and to apply the antenna offset corrections in the NASA GSFC orbit determination software rather than using the data-supplied corrections. We show that computing the antenna offset corrections in the orbit determination software is superior to using the offset corrections that are supplied with the DORIS data, and that this improves the RMS of fit for SPOT-2, Envisat, SPOT-4, and SPOT-5. The updates for the force model included: (1) the development of improved nonconservative force modeling for SPOT-2, SPOT-3, SPOT-5, Envisat, and HY-2A, and (2) the application of an updated static gravity model based on GRACE and GOCE data, and weekly models of the variations in the low degree gravity field deduced independently from tracking by Satellite Laser Ranging (SLR) and DORIS. The post-ITRF2008 DORIS coordinate WRMS after the launch of Envisat and SPOT-5 is improved from 11.20 to 12.45mm with ITRF2008 (Le Bail et al., 2010), to between 8.50 and 9.99mm with the gscwd26 SINEX solution. The application of the DORIS antenna phase laws shifts the DORIS scale wrt DPOD2008 by +6.0mm from 1993/01/03 to 2002/06/06, and by +11.4mm from 2002/06/13 to 2011/10/30. The application of more detailed models of time-variable gravity reduces the slopes in the Helmert transformation parameters Tx, and Ty (w.r.t. DPOD2008) after 2005. The annual amplitude in these parameters is reduced from 3.2mm (for Tx), 4.1mm (for Ty), to 1.7mm (for Tx) and 2.8mm (for Ty).

Variance–Covariance Matrix of Transformed GPS Positions: Case Study for the NAD 83 Geodetic Datum

AbstractThis investigation introduces a rigorous mathematical transformation of variance–covariance matrices between a global geocentric frame and a plate-fixed geodetic frame. A practical example between the geocentric frame of International GNSS Service 2008 (IGS08) epoch 2005.00 and the geodetic frame North American Datum of 1983 (NAD 83) (2011) epoch 2010.00 was implemented. Although the theory is general, the transformation used here is controlled by the assumptions implicit in the definition of NAD 83. However, the same approach could be extended to future definitions of fixed-plate datums used by geodetic organizations for charting and mapping applications. Consequently, because the transformation between these two specific frames is assumed by definition to be a one-to-one errorless transformation, the uncertainties for the 14 Helmert transformation parameters between the two frames are assumed to be zero. Nevertheless, the formulation is complete and applicable to other specific datum-transformat...

Improved PPP performance in regional networks

It has been demonstrated that precise point positioning (PPP) is a powerful tool in geodetic and geodynamic applications. As is known, it provides solutions in the reference system of the satellite orbits. We focuses on the strategy to transform PPP solutions into the International Terrestrial Reference System (ITRS) by applying a set of local Helmert transformation parameters obtained from a regional network rather than using global parameters. In order to carry out this test, a regional network composed of 14 stations was analyzed using GIPSY-OASIS II software, over a period of 6 years. Two solutions differently aligned to the ITRS were compared in terms of accuracy, scattering, frequency content and local movements. One solution is aligned to IGb08 through the X-files provided by JPL, while the other is aligned to the European reference frame densification of IGb08 using customized regional X-files. Therefore, both are updated realizations of the ITRS. The test shows that a regional, instead of a global, alignment to the ITRS can significantly improve the repeatability of the solutions. A small improvement can also be found in terms of agreement with the regional densification of IGb08. The analysis of the signal content in the differently aligned time series allowed some differences to be found, in terms of both frequency and magnitude. These differences are mainly due to an evident common signal that is defined for the whole area and which is removed when using regional alignment. Finally, residual scattering was calculated after removing the modeled signals from each time series, which results in a scatter being significantly smaller for the regional solution than for the global solution. In order to obtain these results, the choice of the reference stations is a major question and therefore discussed in detail.

Configuration of the reference stations as the element of national reference frame reliability

According to the newest Polish regulations multifunctional system of precise positioning ASGEUPOS serves as the fundamental network and constitutes reference frame as well. The permanent GNSS observations gathered within the network are processed and analysed by the Centre of Applied Geomatics of Military University of Technology, which is one of the EPN Local Analysis Centres. IGS and EUREF recommendations are included. Weekly solutions are being stacked and, as the result, cumulative solution (coordinates expressed at the specified epoch and velocities of all stations) by using the CATREF software and minimal constraints approach is determined. Its reliability increases with the number of reference (sometimes described as fiducial) stations. According to the EUREF guidelines, reference stations have to belong to the EPN A class, but their geographical layout leaves much to be desired. The authors made several numerous tests using different configuration of reference stations to transfer ETRF2000 frame into Poland area, and to choose the best strategy of ASG-EUPOS alignment to the EPN. Eight different variants of ASG-EUPOS processing were investigated and each time the different set of coordinates was obtained. The differences reached several centimetres. The spatial distribution of the results on the area of Poland was shown and the optimum solution was pointed out. Furthermore, the analysis on the Helmert transformation parameters from individual TRF to the reference network were performed. These parameters do not have any geophysical interpretation, but provide very useful information about solution’s consistency and compliance. ARTICLE INFO

Closed-form and iterative weighted least squares solutions of Helmert transformation parameters

The Helmert transformation is the most common transformation between different geodetic systems. In 2-D, in contrast to higher dimensions, it is a well-known procedure how to determine the 4 transformation parameters in a closed form. Here we derive the closed-form weighted least squares solution in m-dimensional space for an arbitrary number (≥ m) of coordinate set-ups in two related systems. The solution employs singular value decomposition (SVD) for the rotation matrix, while the translation vector and scale parameters are obtained in simpler ways. To avoid the SVD routine, we also present an iterative approach to solve for the rotation matrix. The paper is completed with a test procedure for detecting outlying coordinate pairs.

Use of Starlette and LAGEOS-1&-2 laser measurements for determination and analysis of stations coordinates and EOP time series

The purpose of this article is to study the contribution of a LEO satellite (Starlette) laser measurements in the estimation of the geodetic products, such as station coordinates, Earth Orientation Parameters (EOP), and Geocenter component variations, over a 14 year period (1993–2007). Three data combinations are considered in the processing, namely LAGEOS-1 (LA-1), LAGEOS-1&-2 (LA-1&LA-2) and LAGEOS-1&Starlette (LA-1&STAR). The orbit computation of the different satellites is performed with GINS software and the laser data processing is carried out by MATLO software, with consideration of a recent GRACE gravity model (Eigen_Grace03s) in the Starlette orbit computation. The time series of results are projected according to ITRF2005, by TRANSFOR software, where the Helmert transformation parameters are obtained. A comparison of the different combinations is effectuated in terms of quality, periodic signals and noises of the weekly stations positions, EOP and Geocenter variations. The results revealed a degradation of positioning accuracy of about 3 to 5mm when using Starlette data according to LA-1&STAR solution, but also a better capability to determine the annual and semi-annual variations of the UP coordinates and Geocenter components.