Helmert Parameters Research Articles

Методика преобразования трехмерных положений пунктов между геоцентрическими и референцными системами координат для региональных территорий

A technique was developed for converting three-dimensional positions of points for regional territories between geocentric and reference coordinate systems (RCS) by matching the Helmert parameters with the geoid height model for the reference coordinate system and the three-dimensional positions of points in both SC. The technique is based on replacing the quasigeoid height model to the EGM2008 geoid height model transformed for reference coordinate systems. For regional territories, the accuracy of determining the Helmert parameters and the accuracy of transformations of coordinates and heights of points at the level of accuracy of the positions of points in the general and reference coordinate systems is ensured. Examples of calculations of the agreed Helmert parameters with the transformed elevation model of the EGM2008 geoid between the reference coordinate system SK-95 and WGS-84, SK-42, GSK-2011 for the territory of the Novosibirsk region are given.

Методика формирования цифровых моделей высот геоида для референцных систем координат

A technique was developed for generating digital models of geoid heights for reference coordinate systems using the Trimble Bussines Center software (TBC software). The source models can be the publicly available high-precision global geoid elevation models EGM2008, EIGEN6C4 and other models whose files with the *.ggf extension are compatible with the TBC software. The technique is based on the possibility of exporting orthometric heights from the TBC software in ASCII format to a user-specified area on the surface of the global WGS-84 ellipsoid for a uniform grid of nodes in lati-tude and longitude. The resulting data table is the starting point for calculating geoid heights for reference coordinate systems, which are determined by the parameters of the corresponding reference ellipsoids and Helmert parameters - spatial position, orientation and linear scale relative to the global coordinate system, according to Standard. The initial data are sufficient to calculate the digital model of the geoid heights for an arbitrary reference coordinate system. The expediency of the developed technique lies in the possibility of its use for a much more accurate calculation of the Helmert parameters for regional and other territories compared to the accepted standard values and, accordingly, a more accurate transformation of the three-dimensional positions of points between the general earth and reference coordinate systems (these issues will be discussed in a subsequent publication by the author). Approbation of the technique was carried out for several regional territories: Novosibirsk and Moscow regions, the territory of Western Siberia, the Republic of Belarus in the SK42 and SK95 coordinate systems, and the results are presented graphically in the form of geoid height maps.

Broadcast Ephemeris with Centimetric Accuracy: Test Results for GPS, Galileo, Beidou and Glonass

Here we test the capability of the Broadcast Ephemeris Message, in both its GPS-like (Keplerian ellipse with secular and periodic perturbations) and Glonass-like (numerical integration of a 9D state vector) formats, to reproduce a corresponding precise ephemeris. We start from a daily Rinex 3.04 navigation file for multiple GNSS and the corresponding SP3 precise orbits computed by CNES (Centre National d’Etudes Spatiales) for GPS, Glonass, Galileo and CODE (Center for Orbit Determination in Europe) for Beidou, and compute broadcast ECEF coordinates and clocks. The pre-fit discrepancies are converted by least squares to corrections to the broadcast ephemeris parameters in two-hour consecutive arcs (for GPS, Galileo and Beidou) and to a set of seven Helmert parameters for the entire day, to align in origin, orientation and scale to the common GNSS IGS14 Reference Frame. The test cases suggest that the Broadcast Ephemeris Message, complemented with Reference Frame information, can reproduce the precise ephemeris and clocks with centimetric accuracy for intervals at least equal to the respective validity times, typically 2 h. The broadcast ephemeris of Glonass consists of three initial positions and velocities at epoch, three constant Lunisolar accelerations for the satellite position, and of three polynomial coefficients for the satellite clock. The 9D vector of state is numerically integrated to generate position and velocity data within the validity time (0.5 h) of the message. To test the capability of this model to reproduce the corresponding values of a precise ephemeris, the 9D vector of state and clock polynomials are adjusted until the rms (root mean squared spread) of the post-fit residuals relative to a precise orbit (CNES’s in our case) is minimum. We show in one example (one satellite for one day) that the Glonass type of message can reproduce a precise ephemeris and clock with a rms spread of 0.025 m over one-hour arcs. Volume computations on one month of data with all available satellites confirm the test results. For GPS, Glonass, Galileo and Beidou, the best fitting clock values predicted by our second order polynomials, based on a 15 min sampling, are shown to fit the corresponding high rate clocks (30 s sampling) of MGEX with zero bias and a rms spread of 0.062 ns (GPS G01), 0.023 ns (Galileo E01), 0.43 ns (Glonass R01), 0.086 ns (Beidou C07) and 0.086 ns (Beidou C12). Modifications to the GPS-like message structure and Glonass algorithm are proposed to increase the validity time by including the effect of the 3rd zonal harmonic of the Earth’s gravity field. The potential of the RTCM messages for broadcasting the improved navigation message is reviewed.

The First Realization of BeiDou Coordinate System and Its Assessment

The BeiDou Navigation Satellite System (BDS) has used the BeiDou Coordinate System (BDCS) as its geodetic reference system since December 2017. After a brief description of the background, definitions of the origin, scale, and orientation of BDCS are given, which are consistent with China’s Geodetic Coordinate System 2000 (CGCS2000). Then the first realization of the BDCS aligning with the International Terrestrial Reference Frame 2014 (ITRF2014) is addressed. Coordinates at epoch 2018.0 and velocities at eight BDS monitoring stations were obtained achieved by processing four-phase GPS data for a global navigation satellite system (GNSS) network, including the monitoring stations, and by the frame alignment and linear regression fitting of station coordinate series. Assessment of the precision of the coordinates and Helmert parameters with respect to ITRF2014 reveals that the differences between the first frame of BDCS and ITRF2014 are less than 1 cm, which is compatible with WGS84 (G1762), PZ-90.11, and the latest Galileo terrestrial reference frame.

Multi-constellation GNSS orbit combination based on MGEX products

Abstract. The mission of the International GNSS Service (IGS) is to deliver highly accurate GNSS data and products to the scientific users and the community. Among these products, precise orbits, and clocks for GPS and GLONASS are available to the public. These products are system-wise combinations based on solutions provided by the Analysis Centers (AC). Over the past years, the IGS has been putting efforts in extending the service to other navigation satellite systems within the Multi-GNSS Experiment and Pilot Project (MGEX). Several ACs contribute by providing solutions containing not only GPS and GLONASS but also Galileo, BeiDou, and QZSS. However, there is no official MGEX combination so far. Therefore, we started to develop a new combination algorithm aiming at a fully consistent multi-constellation solution. We apply two different strategies focusing on the alignment of the orbits to the International Terrestrial Reference Frame (ITRF). In the first strategy, we use the Earth Rotation Parameters (ERP) to align the orbits, whereas in the second strategy Helmert parameters provided by the Terrestrial Frame Combination Center (TFCC) are applied. Based on the alignment we compare the GPS orbit products from both strategies with the official IGS orbits. These preliminary results show that the ERP strategy agrees with the official orbits around by 30 mm whereas, with the second strategy, the agreement is around 15 mm.

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.

Estimating network effect in geocenter motion: Applications

AbstractThe network effect is the error associated with the subsampling of the Earth surface by space geodetic networks. It is an obstacle toward the precise measurement of geocenter motion, that is, the relative motion between the center of mass of the Earth system and the center of figure of the Earth surface. In a complementary paper, we proposed a theoretical approach to estimate the magnitude of this effect from the displacement fields predicted by geophysical models. Here we evaluate the effectiveness of our estimate for two illustrative physical processes: coseismic displacements inducing instantaneous changes in the Helmert parameters and elastic deformation due to surface water movements causing secular drifts in those parameters. For the first, we consider simplified models of the 2004 Sumatra‐Andaman and the 2011 Tōhoku‐Oki earthquakes, and for the second, we use the observations of the Gravity Recovery and Climate Experiment, complemented by an ocean model. In both case studies, it is found that the magnitude of the network effect, even for a large global network, is often as large as the magnitude of the changes in the Helmert parameters themselves. However, we also show that our proposed modification to the definition of the center of network frame to include weights proportional to the area of the Earth surface that the stations represent can significantly reduce the network effect in most cases.

Estimating network effect in geocenter motion: Theory

AbstractGeophysical models and their interpretations of several processes of interest, such as sea level rise, postseismic relaxation, and glacial isostatic adjustment, are intertwined with the need to realize the International Terrestrial Reference Frame. However, this realization needs to take into account the geocenter motion, that is, the motion of the center of figure of the Earth surface, due to, for example, deformation of the surface by earthquakes or hydrological loading effects. Usually, there is also a discrepancy, known as the network effect, between the theoretically convenient center of figure and the physically accessible center of network frames, because of unavoidable factors such as uneven station distribution, lack of stations in the oceans, disparity in the coverage between the two hemispheres, and the existence of tectonically deforming zones. Here we develop a method to estimate the magnitude of the network effect, that is, the error introduced by the incomplete sampling of the Earth surface, in measuring the geocenter motion, for a network of space geodetic stations of a fixed size N. For this purpose, we use, as our proposed estimate, the standard deviations of the changes in Helmert parameters measured by a random network of the same size N. We show that our estimate scales as and give an explicit formula for it in terms of the vector spherical harmonics expansion of the displacement field. In a complementary paper we apply this formalism to coseismic displacements and elastic deformations due to surface water movements.

A Direct Georeferencing Method for Terrestrial Laser Scanning Using GNSS Data and the Vertical Deflection from Global Earth Gravity Models

Terrestrial laser scanning is an efficient technique in providing highly accurate point clouds for various geoscience applications. The point clouds have to be transformed to a well-defined reference frame, such as the global Geodetic Reference System 1980. The transformation to the geocentric coordinate frame is based on estimating seven Helmert parameters using several GNSS (Global Navigation Satellite System) referencing points. This paper proposes a method for direct point cloud georeferencing that provides coordinates in the geocentric frame. The proposed method employs the vertical deflection from an external global Earth gravity model and thus demands a minimum number of GNSS measurements. The proposed method can be helpful when the number of georeferencing GNSS points is limited, for instance in city corridors. It needs only two georeferencing points. The validation of the method in a field test reveals that the differences between the classical georefencing and the proposed method amount at maximum to 7 mm with the standard deviation of 8 mm for all of three coordinate components. The proposed method may serve as an alternative for the laser scanning data georeferencing, especially when the number of GNSS points is insufficient for classical methods.

Учет влияния глобальных геотектонических процессов на результаты точного местоопределения в государственной или местной системе отсчета по радионавигационным наблюдениям спутников GPS NAVSTAR

Abstract: The transformation of the surveyed points’ coordinates, which was performed through the satellite geodesy methods is discussed. The authors report about the coordinate points transformation from International Terrestrial Reference System to the state reference system, as well as in the local reference system. In the work the problem and reason of changes in the communication parameters of the international terrestrial and local reference systems in the local area, in course of time are described. A method of updating the previously found Helmert parameters for a limited area of geodetic works is proposed. To do this the authors propose formulae describing the motion of a random point of the Eurasian lithospheric plate in the Russian Federation boundaries.

Laborde projection in Madagascar cartography and its recovery in WGS84 datum

The International Ellipsoid 1924 was adopted in Madagascar in 1925 for the definition of the Tananarive 1925 datum, while Laborde projection have been used for Madagascar cartography in 1926. Direct and inverse Laborde formulas are reviewed to convert cartographic and geodetic coordinates. The Hotine approximation of the Laborde projection is also discussed. From the geodetic point of view, there is the need to recover the Madagascar cartographic heritage in the WGS84 datum; datum transformations in the Molodensky form are known by literature, with the three dimensional accuracy of some meters. A new set of Helmert parameters has been estimated taking in account geoid correction and giving metric accuracy.

Orbit fitting based on Helmert transformation

Orbit fitting is used in many GPS applications. For example, in Precise Point Positioning (PPP), GPS orbits (SP3 orbits) are normally retrieved either from IGS or from one of its Analysis Centers (ACs) with 15 minutes’ sampling, which is much bigger than the normal observation sampling. Therefore, algorithms should be derived to fit GPS orbits to the observation time. Many methods based on interpolation were developed. Using these methods the orbits fit well at the sampling points. However, these methods ignore the physical motion model of GPS satellites. Therefore, the trajectories may not fit the true orbits at the periods in between 2 sampling epochs. To solve this problem, we develop a dynamic approach, in which a model based on Helmert transformation is developed in GPS orbit fitting. In this orbit fitting approach, GPS orbits at sampling points are treated as pseudo-observations. Thereafter, Helmert transformation is built up between the pseudo-observations and dynamically integrated orbits at each epoch. A set of Helmert parameters together with corrections of GPS initial orbits are then modeled as unknown parameters. Results show that the final fit orbits have the same precision as the IGS final orbits.

A semi-analytical estimation of the effect of second-order ionospheric correction on the GPS positioning

SUMMARY We developed a semi-analytical method to evaluate the effect of the second-order ionospheric correction on GPS positioning. This method is based on the semi-analytical positioning error simulation method developed by Geiger and Santerre in which, assuming the continuous distribution of the satellites, a normal equation is formed to estimate the positioning error taking all the contributions of the ranging error by the visible satellites into account. Our method successfully reproduced the averaged time-series of three IGS sites which is comparable to the rigorous simulation. We then evaluated the effect of the ionospheric error on the determination of the reference frame. We evaluated the additional Helmert parameters that are required for the ionospheric effect. We found that the ionospheric effect can lead to annual scale changes of 0.1 ppb, with an offset of 1.8 mm and a semi-annual oscillation of 1 mm in the z-direction. However, these values are too small to explain the current deviations between the GPS-derived reference frame and the ITRF reference frame. Next, we estimated the apparent scale changes due to the ionospheric error in the GEONET coordinate time-series in Japan. We could qualitatively reproduce the observed semi-annual scale changes peaking at the equinoxes and having asymmetrical amplitudes between the vernal and autumnal equinoxes.