Elevation-dependent Weighting Research Articles

Improved PPP-RTK positioning performance by using the elevation-dependent weighting model for the atmospheric delay corrections

With satellite phase bias products, integer ambiguity resolution (AR) enabled precise point positioning (PPP) can be achieved, that is, PPP-real-time kinematic (RTK). However, PPP-AR still requires minutes to converge to centimeter-level accuracy, as atmospheric delay parameters need to be estimated in the user un-combined (UC) model. To shorten the convergence time, reference-network-derived atmospheric delay corrections are often directly corrected for raw code/phase observables, without considering their uncertainties, thereby causing model errors. In this study, we propose a new ionosphere-weighted model, in which an elevation-dependent weight is adopted. To evaluate the proposed model, on-board kinematic experiments were conducted based on GPS/Galileo/BDS2/ BDS3 dual-frequency measurements. Compared with the positioning results of the traditional determined variance weight, the proportion of horizontal errors less than 5 cm is increased from 70% to 80%, and horizontal accuracy is improved by 22% and 9.8% at the 68th and 95th percentile, respectively, indicating that the proposed model can improve the PPP-RTK performance.

An Improved Strategy for Real-Time Troposphere Estimation and Its Application in the Severe Weather Event Monitoring

The water vapor content in the atmosphere is highly correlated with rainfall events, which can be used as a data source for rainfall prediction or drought monitoring. The GNSS PPP (Precise Point Positioning) technique can be used to estimate the troposphere ZWD (Zenith Wet Delay) parameter which can be converted into precipitable water vapor (PWV). In this study, we first investigate the impacts of the weighting strategies, observation noise settings, and gradient estimation on the accuracy of ZWD and positions. A refined strategy is proposed for the troposphere estimation with uncombined raw PPP model, down-weighting of Galileo/GLONASS/BDS code observation by a factor of 3, using a sine2-type elevation-dependent weighting function and estimating the horizontal gradients. Based on the strategy, the correlation of the estimated tropospheric parameters with the rainfall is analyzed based on the data from the “7.20” rainstorm in Henan Province, China. The PWV is first calculated based on the hourly global pressure and temperature (HGPT) model and compared with the results from ERA5 products. Results show their good consistency during the rainfall period or the normal period with a standard deviation of 3 mm. Then, the high correlation between the PWV and the heavy rain rainfall event is validated. Results show that the accumulated PWV maintains a high level before the rainstorm if a sustainable water supply is available, while it decreased rapidly after the rainfall. In comparison, the horizontal gradients and the satellite residuals are less correlated with the water vapor content. However, the gradients can be used to indicate the horizontal asymmetry of the water vapor in the atmosphere.

Improved Single-Frequency Kinematic Orbit Determination Strategy of Small LEO Satellite with the Sun-Pointing Attitude Mode

Kinematic orbit determination (KOD) of low earth orbit (LEO) satellites only using single-frequency global navigation satellite system (GNSS) data is a suitable solution for space applications demanding meter-level orbit precision. For some small LEO satellites with the sun-pointing attitude mode, the rotation of the GNSS antenna radiation pattern changes the observation noise characteristics. Since the rotation angle information of the antenna plane may not be available for most low-cost missions, the true elevation cannot be computed and a general elevation-dependent weighting model remains invalid for the onboard GNSS observations. Furthermore, the low-stability GNSS receiver clock oscillator of the LEO satellite at high speeds makes single-frequency cycle slip detection ineffective and difficult since the clock steering events occur frequently. In this study, we investigated the improved KOD strategy to improve the performance of orbit solution using single-frequency GPS and BeiDou navigation satellite system (BDS) observations collected from the Luojia-1A satellite. The weighting model based on exponential function and signal strength is proposed according to the analysis of satellite attitude impact, and a joint single-frequency detection algorithm of receiver clock jump and cycle slip is investigated as well. Based on the GPS/BDS-combined KOD results, it is demonstrated that the clock jump and cycle slip can be properly detected and observations can be effectively utilized with the proposed weighting model considering satellite attitude, which significantly improves the availability and accuracy of orbit solution. The number of available epochs is increased by 12.9% benefitting from this strategy. The orbital root mean square (RMS) precision improvements in the radial, along-track, and cross-track directions are 22.1%, 16.4%, and 6.5%, respectively. Combining BDS observations also contributes to orbit precision improvement, which reaches up to 28.8%.

SBAS DFMC service for road transport: positioning and integrity monitoring with a new weighting model

In 2017, the new generation satellite-based augmentation system (SBAS) test-bed was initiated by Australia and New Zealand, which supports the dual-frequency multi-constellation (DFMC) positioning with both GPS and Galileo signals. This new SBAS DFMC service allows the elimination of the first-order term of the ionospheric delays, and extends the service area to the entire footprint of the geostationary satellite. In addition to the satellite clock and orbital corrections, the integrity information is also broadcast by the SBAS satellite to users, so that protection levels can be computed to bound the positioning errors with a pre-defined probability of hazardous misleading information. Different from the aeronautical applications, the ground-based applications for road transport may suffer from new problems in different measurement environments, e.g. complicated multipath behaviours and frequent filter re-initialisations during positioning in urban areas. A new weighting model allowing different impacts of the elevation angles, the signal-to-noise-ratios and the smoothing time after re-initialisations is proposed and compared with the traditional elevation-dependent weighting model. The model is applied to the carrier-smoothed code measurements in different environments, i.e., the open-sky scenario, the suburban scenario and the urban scenario. It is found that the new weighting model effectively de-weights the large residuals in the suburban and the urban scenarios, where the mean values and the standard deviations of the overbounding excess-mass cumulative density function can be significantly reduced for the combined weighted noise and multipath. Using 1 Hz GNSS observations measured in these three measurement environments, the horizontal positioning errors (HPEs) and the horizontal protection levels (HPLs) are computed for different filter smoothing windows. Applying the new weighting model, significant reduction can be observed in the mean HPLs in the suburban and urban scenarios. Among them, the reduction in the HPLs have reached about 35–40% in the suburban scenario. The mean absolute HPEs are also reduced by about 10% in the urban scenario. However, when under the open-sky scenario, the traditional elevation-dependent weighting model is sufficient for the positioning and integrity monitoring using the SBAS DFMC service.

Considering different recent advancements in GNSS on real-time zenith troposphere estimates

Global navigation satellite system (GNSS) remote sensing of the troposphere, called GNSS meteorology, is already a well-established tool in post-processing applications. Real-time GNSS meteorology has been possible since 2013, when the International GNSS Service (IGS) established its real-time service. The reported accuracy of the real-time zenith total delay (ZTD) has not improved significantly over time and usually remains at the level of 5–18 mm, depending on the station and test period studied. Millimeter-level improvements are noticed due to GPS ambiguity resolution, gradient estimation, or multi-GNSS processing. However, neither are these achievements combined in a single processing strategy, nor is the impact of other processing parameters on ZTD accuracy analyzed. Therefore, we discuss these shortcomings in detail and present a comprehensive analysis of the sensitivity of real-time ZTD on processing parameters. First, we identify a so-called common strategy, which combines processing parameters that are identified to be the most popular among published papers on the topic. We question the popular elevation-dependent weighting function and introduce an alternative one. We investigate the impact of selected processing parameters, i.e., PPP functional model, GNSS selection and combination, inter-system weighting, elevation-dependent weighting function, and gradient estimation. We define an advanced strategy dedicated to real-time GNSS meteorology, which is superior to the common one. The a posteriori error of estimated ZTD is reduced by 41%. The accuracy of ZTD estimates with the proposed strategy is improved by 17% with respect to the IGS final products and varies over stations from 5.4 to 10.1 mm. Finally, we confirm the latitude dependency of ZTD accuracy, but also detect its seasonality.

Impact of GPS Processing on the Estimation of Snow Water Equivalent Using Refracted GPS Signals

Global navigation satellite system (GNSS) antennas buried underneath a snowpack have a high potential for in situ snow water equivalent (SWE) estimation. Automated and continuous SWE quantification independent of weather conditions could enhance snow hydrological monitoring and modeling. Accurate and reliable in situ data are needed for the calibration and validation of remote sensing data and snowpack modeling. A relative bias of less than 5% is achieved using sub-snow global positioning system (GPS) antennas (GPS refractometry) during a three full seasons time period in the Swiss Alps. A systematic overview regarding the temporal reliability of the sub-snow GPS derived results is, however, missing for this emerging technique. Moreover, GPS processing impacts the results significantly. Different GPS processing parameters are therefore selected and their influence on the SWE estimation is investigated. The impact of elevation-dependent weighting, the elevation cutoff angles, and the time intervals for SWE estimation are systematically assessed. The best results are achieved using all observations with an elevation-dependent weighting scheme. Moreover, the SWE estimation performance is equally accurate for hourly SWE estimation as for lower temporal resolutions up to daily estimates. The impact of snow on the coordinate solution is furthermore evaluated. While the east and north components are not systematically influenced by the overlying snowpack, the vertical component exhibits a significant variation and strongly depends on the SWE. The biased vertical component therefore provides an additional possibility to estimate SWE.

Sensitivity of GNSS tropospheric gradients to processing options

Abstract. An analysis of processing settings impacts on estimated tropospheric gradients is presented. The study is based on the benchmark data set collected within the COST GNSS4SWEC action with observations from 430 Global Navigation Satellite Systems (GNSS) reference stations in central Europe for May and June 2013. Tropospheric gradients were estimated in eight different variants of GNSS data processing using precise point positioning (PPP) with the G-Nut/Tefnut software. The impacts of the gradient mapping function, elevation cut-off angle, GNSS constellation, observation elevation-dependent weighting and real-time versus post-processing mode were assessed by comparing the variants by each to other and by evaluating them with respect to tropospheric gradients derived from two numerical weather models (NWMs). Tropospheric gradients estimated in post-processing GNSS solutions using final products were in good agreement with NWM outputs. The quality of high-resolution gradients estimated in (near-)real-time PPP analysis still remains a challenging task due to the quality of the real-time orbit and clock corrections. Comparisons of GNSS and NWM gradients suggest the 3∘ elevation angle cut-off and GPS+GLONASS constellation for obtaining optimal gradient estimates provided precise models for antenna-phase centre offsets and variations, and tropospheric mapping functions are applied for low-elevation observations. Finally, systematic errors can affect the gradient components solely due to the use of different gradient mapping functions, and still depending on observation elevation-dependent weighting. A latitudinal tilting of the troposphere in a global scale causes a systematic difference of up to 0.3 mm in the north-gradient component, while large local gradients, usually pointing in a direction of increasing humidity, can cause differences of up to 1.0 mm (or even more in extreme cases) in any component depending on the actual direction of the gradient. Although the Bar-Sever gradient mapping function provided slightly better results in some aspects, it is not possible to give any strong recommendation on the gradient mapping function selection.

Determination of zero difference GPS differential code biases for satellites and prominent receiver types

The differential code bias (DCB) is the differential hardware (e.g., the satellite or receiver) delay that occurs between two different observations obtained at the same or two different frequencies. There are two approaches used to estimate DCBs for receivers and satellites: the relative and absolute methods. The relative method utilizes a GPS network, while the absolute method determines DCBs from a single station (zero difference). Three receiver types based on the pseudo-range observables were used here to collect the GPS data: Codeless Tracking, Cross Correlation, and Non-Cross Correlation styles. According to its types, GPS receivers have responded to restrictions on the GPS signal structure in different ways. The main goal of the current research is providing a method to determine the DCBs of GPS satellites and dual frequency receivers. The developed mathematical model was based on spherical harmonic function and geometry-free combination of pseudo-range observables (C/A or/and P-code) according to receiver type. A new elevation-dependent weighting function with respect to GPS satellites in our algorithm was applied. The applied weighting function was used to consider the quality variation of satellite DCBs, which is caused by pseudo-range measurement errors. The code of the proposed mathematical model was written using MATLAB and is called “zero difference differential code bias estimation (ZDDCBE)”. This code was tested and evaluated using data from IGS GNSS stations and different types of GPS stations out of IGS network installed in Egypt and Saudi Arabia. The estimated values from the ZDDCBE code show a good agreement with the IGS analysis centers with a mean error of estimation for the receiver DCB equal 5.94%. Therefore, the ZDDCBE code can be used to estimate the DCB for any type of receiver regardless if the receiver is from IGS network or not.

A priori fully populated covariance matrices in least-squares adjustment—case study: GPS relative positioning

In this contribution, using the example of the Matern covariance matrices, we study systematically the effect of apriori fully populated variance covariance matrices (VCM) in the Gauss–Markov model, by varying both the smoothness and the correlation length of the covariance function. Based on simulations where we consider a GPS relative positioning scenario with double differences, the true VCM is exactly known. Thus, an accurate study of parameters deviations with respect to the correlation structure is possible. By means of the mean-square error difference of the estimates obtained with the correct and the assumed VCM, the loss of efficiency when the correlation structure is missspecified is considered. The bias of the variance of unit weight is moreover analysed. By acting independently on the correlation length, the smoothness, the batch length, the noise level, or the design matrix, simulations allow to draw conclusions on the influence of these different factors on the least-squares results. Thanks to an adapted version of the Kermarrec–Schon model, fully populated VCM for GPS phase observations are computed where different correlation factors are resumed in a global covariance model with an elevation dependent weighting. Based on the data of the EPN network, two studies for different baseline lengths validate the conclusions of the simulations on the influence of the Matern covariance parameters. A precise insight into the impact of apriori correlation structures when the VCM is entirely unknown highlights that both the correlation length and the smoothness defined in the Matern model are important to get a lower loss of efficiency as well as a better estimation of the variance of unit weight. Consecutively, correlations, if present, should not be neglected for accurate test statistics. Therefore, a proposal is made to determine a mean value of the correlation structure based on a rough estimation of the Matern parameters via maximum likelihood estimation for some chosen time series of observations. Variations around these mean values show to have little impact on the least-squares results. At the estimates level, the effect of varying the parameters of the fully populated VCM around these approximated values was confirmed to be nearly negligible (i.e. a mm level for strong correlations and a submm level otherwise).

Monument-antenna effects on GPS coordinate time series with application to vertical rates in Antarctica

We examine the electromagnetic coupling of a GPS antenna–monument pair in terms of its simulated affect on long GPS coordinate time series. We focus on the Earth and Polar Observing System (POLENET) monument design widely deployed in Antarctica and Greenland in projects interested particularly in vertical velocities. We base our tests on an absolute robot calibration that included the top ~0.15 m of the monument and use simulations to assess its effect on site coordinate time series at eight representative POLENET sites in Antarctica over the period 2000.0–2011.0. We show that the neglect of this calibration would introduce mean coordinate bias, and most importantly for velocity estimation, coordinate noise which is highly sensitive to observation geometry and hence site location and observation period. Considering only sub-periods longer than 2.5 years, we show vertical site velocities may be biased by up to ±0.4 mm/year, and biases up to 0.2 mm/year may persist for observation spans of 8 years. Changing between uniform and elevation-dependent observation weighting alters the time series but does not remove the velocity biases, nor does ambiguity fixing. The effect on the horizontal coordinates is negligible. The ambiguities fixed series spectra show noise between flicker and random walk with near-white noise at the highest frequencies, with mean spectral indices (frequencies <20 cycles per year) of approximately −1.3 (uniform weighting) and −1.4 (elevation-dependent weighting). While the results are likely highly monument specific, they highlight the importance of accounting for monument effects when analysing vertical coordinate time series and velocities for the highest precision and accuracy geophysical studies.