Tropospheric Errors Research Articles

Improved GPS tropospheric path delay estimation using variable random walk process noise

Accurate positioning using the Global Positioning System relies on accurate modeling of tropospheric delay. Estimated tropospheric delay must vary sufficiently to capture true variations; otherwise, systematic errors propagate into estimated positions, particularly the vertical. However, if the allowed delay variation is too large, the propagation of data noise into all parameters is amplified, reducing precision. Here we investigate the optimal choice of tropospheric constraints applied in the GipsyX software, which are specified by values of random walk process noise. We use the variability of 5-min estimated positions as a proxy for tropospheric error. Given that weighted mean 5-min positions closely replicate 24-h solutions, our ultimate goal is to improve 24-h positions and other daily products, such as precise orbit parameters. The commonly adopted default constraint for the zenith wet delay (ZWD) is 3 mm/√(hr) for 5-min data intervals. Using this constraint, we observe spurious wave-like patterns of 5-min vertical displacement estimates with amplitudes ~ 100 mm coincident with Winter Storm Ezekiel of November 27, 2019, across the central/eastern USA. Loosening the constraint suppresses the spurious waves and reduces 5-min vertical displacement variability while improving water vapor estimates. Further improvement can be achieved when optimizing constraints regionally, or for each station. Globally, results are typically optimized in the range of 6–12 mm/√(hr). Generally, we at least recommend loosening the constraint from the current default of 3 mm/√(hr) to 6 mm/√(hr) for ZWD every 300 s. Constraint values must be scaled by √(x/300) for alternative data intervals of x seconds.

High-precision tropospheric correction method for NRTK regions with significant height differences

Abstract In response to the issue of poor network real-time kinematic (NRTK) service performance in regions with significant height differences, an improved tropospheric height correction (ITHC) method is proposed. Precise point positioning (PPP) is employed to compute the troposphere delay at base stations. Subsequently, a tropospheric vertical profile fitting model (TVPFM) is established for the vertical reduction of the troposphere in regions with significant height differences. In this case, the tropospheric errors introduced by the height differences between the base and rover stations can be calculated. Finally, the tropospheric error can be corrected during the generation of virtual observations, ensuring high-accuracy positioning of NRTK rovers. With the troposphere delay computed based on the PPP approach, datum errors introduced by inaccurate tropospheric correction methods are mitigated. To reduce the dependence of the troposphere delay on height, ECMWF reanalysis v5 (ERA5) data are employed to fit the TVPFM. Experimental analysis demonstrates that the troposphere exhibits distinct vertical variation characteristics, allowing for its segmentation into three layers. Consequently, a piecewise TVPFM is established. Observations obtained from the continuously operating reference stations network located in Yunnan, China, are utilized for validation. The selected stations exhibit a maximum height difference of approximately 2 km. The experimental results exhibit a notable enhancement in correction accuracy with the ITHC in comparison to conventional correction methodologies. Specifically, the ambiguity fixing rate demonstrates a noteworthy improvement of 13.3%, accompanied by a substantial increase in positioning accuracy by 51.4%.

Characterizing the spatial structure and aliasing effect of ocean tide loading on InSAR measurements

Ocean tide loading (OTL) displacements, shown as long-wavelength errors in Interferometric Synthetic Aperture Radar (InSAR), must be considered in large-scale applications. Despite efforts to explore the impacts of OTL on InSAR, most studies use individual interferograms and simple metrics, which fail to characterize the spatial structure of OTL. Moreover, the OTL contribution to InSAR time series remains relatively unexplored. The aliasing effect and related biases due to OTL, which are common to space-geodetic time series, are primarily theoretical with few practical observations for InSAR. This study comprehensively explores the statistical properties of OTL and their impacts on InSAR measurements, using the Southwest United Kingdom and Northwest France as study areas. Spatially, OTL artifacts on interferograms exhibit an escalating magnitude along the principal direction that aligns with the coastline's orientation. Temporally, the aliasing effect originating from OTL introduces periodic signals with prominent 15/64-day cycles into the Sentinel-1 InSAR time series, causing high velocity biases (up to ∼1 cm/yr) and uncertainties (up to ∼5 mm/yr) for short time spans. Applying OTL correction mitigates the noise level in the displacement time series, leading to a 16% improvement in accuracy, as validated against the Global Navigation Satellite System (GNSS). The study proposes the “overlapping effect” concept, which links InSAR tropospheric delay errors and OTL effects. It underscores the importance of accurate error assessment and removal. Neglecting this interaction may result in a 13% underestimation of the tropospheric error correction efficacy.

Risk overbounding for ionospheric gradient monitor using geometry-free double differenced carrier phase measurements

Ionosphere anomaly can cause large spatial gradients in the double-differenced carrier phase (DDCP) measurements. Taking advantage of this characteristic, the ionospheric gradient monitor (IGM) is designed with multiple ground reference stations to detect threatening ionospheric gradients for safety of life applications. An optimal IGM should be most sensitive to ionospheric gradients and least sensitive to other errors. However, current IGM suffers from the influence of tropospheric error, which can degrade monitor performance under extreme weather conditions by causing risks of false alarms and missed detections. To address this issue, an alternative IGM is proposed using the geometry-free combination of DDCP as the test statistic. Ambiguity resolution procedure is designed for estimating the ambiguity term in the test statistic. The risk induced by the wrong ambiguity fix and tropospheric error are analyzed and bounded with required averaging period and minimum baseline length. The results show that the proposed theoretical IGM is capable of achieving probability of false alarm of 10–8 and probability of missed detection of 10–6 with a filtering period of 612 s and baseline length of 371.7 m or a filtering period of 544 s and baseline length of 384.4 m. The experimental results using data collected from Hong Kong Satellite Positioning Reference Station Network demonstrate that the performance of the proposed theoretical IGM is comparable to that of the existing detection algorithm.

A Machine Learning-Based Tropospheric Prediction Approach for High-Precision Real-Time GNSS Positioning.

For high-precision positioning applications, various GNSS errors need to be mitigated, including the tropospheric error, which remains a significant error source as it can reach up to a few meters. Although some commercial GNSS correction data providers, such as the Quasi-Zenith Satellite System (QZSS) Centimeter Level Augmentation Service (CLAS), have developed real-time precise regional troposphere products, the service is available only in limited regional areas. The International GNSS Service (IGS) has provided precise troposphere correction data in TRO format post-mission, but its long latency of 1 to 2 weeks makes it unable to support real-time applications. In this work, a real-time troposphere prediction method based on the IGS post-processing products was developed using machine learning techniques to eliminate the long latency problem. The test results from tropospheric predictions over a year using the proposed method indicate that the new method can achieve a prediction accuracy (RMSE) of 2 cm, making it suitable for real-time applications.

A Review of Selected Applications of GNSS CORS and Related Experiences at the University of Palermo (Italy)

Services from the Continuously Operating Reference Stations (CORS) of the Global Navigation Satellite System (GNSS) provide data and insights to a range of research areas such as physical sciences, engineering, earth and planetary sciences, computer science, and environmental science. Even though these fields are varied, they are all linked through the GNSS operational application. GNSS CORS have historically been deployed for three-dimensional positioning but also for the establishment of local and global reference systems and the measurement of ionospheric and tropospheric errors. In addition to these studies, CORS is uncovering new, emerging scientific applications. These include real-time monitoring of land subsidence via network real-time kinematics (NRTK) or precise point positioning (PPP), structural health monitoring (SHM), earthquake and volcanology monitoring, GNSS reflectometry (GNSS-R) for mapping soil moisture content, precision farming with affordable receivers, and zenith total delay to aid hydrology and meteorology. The flexibility of CORS infrastructure and services has paved the way for new research areas. The aim of this study is to present a curated selection of scientific papers on prevalent topics such as network monitoring, reference frames, and structure monitoring (like dams), along with an evaluation of CORS performance. Concurrently, it reports on the scientific endeavours undertaken by the Geomatics Research Group at the University of Palermo in the realm of GNSS CORS over the past 15 years.

Geodetic evidence of land subsidence in Cirebon, Indonesia

The subsidence phenomena occur in numerous cities in Indonesia (Jakarta, Bandung, and Semarang) and have been observed over the last few decades using geodetic techniques such as interferometric synthetic aperture radar (InSAR) and the Global Navigation Satellite System (GNSS). Three variables are considered the leading causes of substantial subsidence in those cities: urban development, excessive groundwater removal, and geological context. Considering these considerations, we assess surface displacement in the surrounding area of Cirebon, Indonesia, which has similar background circumstances. We used 43 Sentinel-1A images covering from 2014 to 2021 and processed them using the small baseline subset (SBAS) method obtaining 78 interferograms for time series analysis. A weighted correction from two weather models was used to correct tropospheric errors. The results from the InSAR time series were compared with the continuous GNSS time series (2010–2021) processed using GAMIT/GLOBK GNSS processing softwares. Cirebon experienced a slow land subsidence rate with an average magnitude of -2.6 mm/yr. Furthermore, we underline that more significant subsidence rates were found in several locations. Conic-shaped subsidence with a diameter of around 1 km was observed in a highly inhabited area, with a maximum rate of up to -17 mm/yr near the cone’s center. We also found more prominent land subsidence along the coastline with magnitudes up to -25 mm/yr. Surface lowering with a magnitude of up to -32 mm/yr also affects power plant regions close to Cirebon city. Meanwhile, the fastest subsidence rate was observed over the salt evaporation field area of about -50 mm/yr. We attribute the natural and anthropogenic factors to causing the subsidence in this area.

Novel algorithms for pair and pixel selection and atmospheric error correction in multitemporal InSAR

Entering the SAR's golden era began with the launch of Sentinel-1A/B satellites in 2014 and 2016 with 6–12 day revisit time, much larger stacks of high-resolution SAR images are available over a given area to perform time series analysis. Algorithms that deal with large stack sizes face several challenges, including interferometric phase quality degradation due to signal decorrelations, phase closure error caused by applied multilooking, and tropospheric phase delay. Here, we present an improved SBAS-type algorithm suitable for processing a large stack of SAR images at an arbitrary resolution. We develop a new pair selection strategy that applies dyadic downsampling combined with widely used Delaunay Triangulation to identify an optimal set of interferometric pairs that minimize systematic errors due to short-lived signals and closure errors. We develop and apply a novel statistical framework that selects elite pixels accounting for distributed and permanent scatterers. Also, we implement a new tropospheric error correction that takes advantage of smooth 2D splines to identify and remove error components with fractal-like structures. We demonstrate the effectiveness of the algorithms by applying them to 3 large datasets of Sentinel-1 SAR images measuring non-linear surface deformation over various terrains. Compared with independent GNSS observations, we find that over the rural/natural terrains adjacent to San Andreas fault in southern California, our approach yields a standard deviation of 0.48 cm for time series differences in both ascending and descending tracks. While in urban areas, such as Los Angeles, standard deviation difference with GNSS time series is 0.30 cm.

Block Optimization and Outlier Detection in GEO SAR Turbulent Tropospheric Error Compensation

Geosynchronous Synthetic Aperture Radar (GEO SAR) has important potential applications in the fields of atmospheric water cycle and digital weather forecast, and establishing the connection between GEO SAR imaging and atmospheric water vapor inversion is a key step. The turbulent variation of atmospheric phase delay will cause the SAR image defocus, and turbulent tropospheric delay phase estimation based on block autofocus method in low-orbit high resolution space-borne SAR can be used for reference, but this method is very sensitive to the data block selection, modulation frequency (FM) outlier detection and correction. This Letter gives some considerations to these two problems of block selection and outlier correction. Based on the idea of Extreme Learning Machine (ELM), a mathematical model with nonlinearity as the core and linear model as easy to solve is established. When the parameters affecting the turbulent troposphere are known, the model can be directly used to obtain the block optimization selection, namely, the number and size of blocks. When the parameters are unknown, the model can be used to roughly inverse the parameters and provide initial values for iteration. For the second problem, using the prior information of the correlation of the FM estimated by the adjacent sub-blocks, we proposes a Local Outlier Factor (LOF) method to identify the error point by defining the distance threshold. The effectiveness of the proposed algorithm is verified by Sentinel-1 data injected with tropospheric error.

A Novel Tropospheric Error Formula for Ground-Based GNSS Interferometric Reflectometry

A Novel Tropospheric Error Formula for Ground-Based GNSS Interferometric Reflectometry

Gulf of Mexico Loop-Current Signature Observed From GNSS-R Phase Altimetry Based on Spire Global CubeSat Data

Global Navigation Satellite System (GNSS) coherent ocean reflections collected at grazing-angles by Spire Global’s CubeSats are explored to retrieve Sea Level Anomalies (SLA) over the Gulf of Mexico. Dual-frequency GPS carrier phase estimations are derived from the 50-Hz I and Q samples generated onboard the low Earth orbit CubeSats. First order ionospheric phase advancement, troposphere delay, and precise orbit solutions for GPS satellites and CubeSats mitigate the range measurement errors. Comparisons between collocated specular reflection tracks from various Spire CubeSats and GPS satellites configurations demonstrate the self-consistency of the resulting SLA solutions. The Spire retrievals are then validated against conventional altimetry SLA products. There is a Root Mean Square (RMS) difference of ~ 20 cm between the Spire and the gridded model mesoscale SLAs, unless the reflected elevations are very low. To better evaluate the performance of the Spire retrievals they are examined over sharp ocean topography. About 40% of coherent reflections with elevations above 12° detected the signature of the energetic Loop-Current during the 2020-2021 period. The Spire retrievals were also compared with collocated Sentinel-3 tracks. One track with elevations above 12° demonstrated high correlation and agreement with the Sentinel-3 retrievals. The other two low-elevation tracks showed large magnitude discrepancies, but similar trends once appropriate scaling factors are applied. The study demonstrates the potential for GNSS reflectometry’s application to SLA mapping while also highlighting where significant improvements in error mitigations are needed. A particularly large error source is the troposphere error for signals with lower elevations.

The Effects of Differential Tropospheric Error on the Measurement of Wide-Swath Interferometric Altimetry

The Effects of Differential Tropospheric Error on the Measurement of Wide-Swath Interferometric Altimetry

A New Approach for Improving GNSS Geodetic Position by Reducing Residual Tropospheric Error (RTE) Based on Surface Meteorological Data

Positioning error components related to tropospheric and ionospheric delays are caused by the atmosphere in positioning determined by global navigation satellite systems (GNSS). Depending on the user’s requirements, the position error caused by tropospheric influences, which is commonly referred to as zenith tropospheric delay (ZTD), must be estimated during position determination or determined later by external tropospheric corrections. In this study, a new approach was adopted based on the reduction of residual tropospheric error (RTE), i.e., the unmodeled part of the tropospheric error that remains included in the total geodetic position error, along with other unmodeled systematic and random errors. The study was performed based on Global Navigation Satellite System (GLONASS) positioning solutions and accompanying meteorological parameters in a defined and harmonized temporal-spatial frame of three locations in the Republic of Croatia. A multidisciplinary approach-based analysis from a navigational science aspect was applied. The residual amount of satellite positioning signal tropospheric delay was quantitatively reduced by employing statistical analysis methods. The result of statistical regression is a model which correlates surface meteorological parameters with RTE. Considering the input data, the model has a regional character, and it is based on the Saastamoinen model of zenith tropospheric delay. The verification results show that the model reduces the RTE and thus increases the geodetic accuracy of the observed GNSS stations (with horizontal components of position accuracy of up to 3.8% and vertical components of position of up to 4.37%, respectively). To obtain these results, the Root Mean Square Error (RMSE) was used as the fundamental parameter for position accuracy evaluation. Although developed based on GLONASS data, the proposed model also shows a considerable degree of success in the verification of geodetic positions based on Global Positioning System (GPS). The purpose of the research, and one of its scientific contributions, is that the proposed method can be used to quantitatively monitor the dynamics of changes in deviations of X, Y, and Z coordinate values along coordinate axes. The results show that there is a distinct interdependence of the dynamics of Y and Z coordinate changes (with almost mirror symmetry), which has not been investigated and published so far. The resultant position of the coordinates is created by deviations of the coordinates along the Y and Z axes—in the vertical plane of space, the deviations of the coordinate X (horizontal plane) are mostly uniform and independent of deviations along the Y and Z axes. The proposed model shows the realized state of the statistical position equilibrium of the selected GNSS stations which were observed using RTE values. Although of regional character, the model is suitable for application in larger areas with similar climatological profiles and for users who do not require a maximum level of geodetic accuracy achieved by using Satellite-Based Augmentation Systems (SBAS) or other more advanced, time-consuming, and equipment-consuming positioning techniques.

Estimation of Terrestrial Water Storage Variations in Sichuan-Yunnan Region from GPS Observations Using Independent Component Analysis

GPS can be used to measure land motions induced by mass loading variations on the Earth’s surface. This paper presents an independent component analysis (ICA)-based inversion method that uses vertical GPS coordinate time series to estimate the change of terrestrial water storage (TWS) in the Sichuan-Yunnan region in China. The ICA method was applied to extract the hydrological deformation signals from the vertical coordinate time series of GPS stations in the Sichuan-Yunnan region from the Crustal Movement Observation Network of China (CMONC). These vertical deformation signals were then inverted to TWS variations. Comparative experiments were conducted based on Gravity Recovery and Climate Experiment (GRACE) data and a hydrological model for validation. The results demonstrate that the TWS changes estimated from GPS(ICA) deformations are highly correlated with the water variations derived from the GRACE data and hydrological model in Sichuan-Yunnan region. The TWS variations are overestimated by the vertical GPS observations the northwestern Sichuan-Yunnan region. The anomalies are likely caused by inaccurate atmospheric loading correction models or residual tropospheric errors in the region with high topographic variability and can be reduced by ICA preprocessing.

Analysis on the performances of the GNSS tropospheric delay correction models

Tropospheric delay is one of the important factors affecting GNSS positioning accuracy, and there are different ways to deal with the multiple measurement situations. In short baseline measurements, the difference method is commonly used to eliminate tropospheric errors. However, it cannot be used in long baseline measurements or complex weather since it still has great influences on precision measurement after difference calculation. Therefore, modelling method is usually used to reduce tropospheric delay. As it is well known, there are three types of commonly used tropospheric delay correction models, which are suitable for different situations. When any model is used to solve the tropospheric delay in a large scale, there is always an error between the model value and the actual one. In order to investigate the applicability of the three models in different atmospheric conditions, we actually used the measured meteorological data provided by IGS (International GNSS Service) stations as a reference, and then calculated the ZTD (Zenith Tropospheric Delay) with the different models, including Hopfield model, Saastamoinen model and Black model. The calculation results indicate that Saastamoinen model is the most robust and practical model.

Single Epoch Ambiguity Resolution of Small-Scale CORS with Multi-Frequency GNSS

The network real-time kinematic (RTK) technique uses continuously operating reference stations (CORS) within a geographic area to model the distance dependent errors, allowing users in the area to solve ambiguities. A key step in network RTK is to fix ambiguities between multiple reference stations. When a new satellite rises or when maintenance happens, many unknown parameters are involved in the mathematical model, and traditional methods take some time to estimate the integer ambiguities reliably. The purpose of this study is the single-epoch ambiguity resolution on small-scale CORS network with inter-station distance of around 50 km. A new differencing scheme is developed to explore the full potential of multi-frequency Global Navigation Satellite System (GNSS). In this scheme, a differencing operation is formed between satellites with the closest mapping functions. With the new differencing scheme, tropospheric error can be mostly neglected after the correction, as well as the double-differencing operation. Numerical tests based on two baselines of 49 km and 35 km show that the success rate of ambiguity resolution can reach more than 90%. The single-epoch ambiguity resolution for reference stations brings many benefits to the network RTK service, for example, the instantaneous recovery after maintenance or when a new satellite rises.

Joint Correction of Tropospheric and Orbital Errors in SAR Differential Interferograms

Synthetic Aperture Radar (SAR) satellite imagery is a source of data widely employed in the quantification and analysis of an earthquake coseismic displacement. However, due to the signal path along the atmosphere and to other sources, the interferometric phase becomes compromised. In this work, a methodology for the correction of tropospheric and orbital errors in the differential interferogram is presented. This methodology was applied to a couple of Sentinel-1A data. The phenomenon studied was the 11th November 2018 Zeribet el Oued earthquake, Mw. 5.2 (The state of Biskra, South East of Algeria). It was possible to correct both tropospheric and orbital errors, where the dominant one was the tropospheric delay, a displacement error of 4 cm was added to the differential interferogram by this noise source. The correction of orbital error led to a better interpretation of the coseismic displacement.

Integrated High-Accuracy Correction Technology of Radio-Wave Refraction for Deep-Space (High-Orbit) Targets

The radio-wave refraction error caused by the troposphere and ionosphere badly affects accuracy in terms of the navigation, positioning, measurement, and control of a target; it is the main source of errors in high-accuracy measurement and control systems. The high-accuracy technology needed for radio-wave refraction error correction (mainly in the troposphere and ionosphere) has been the focus of research for a long time. At present, the correction methods used for radio-wave refraction errors have a low accuracy. For an S-band radio-wave signal, the accuracy of refraction error correction can generally only reach m-level (elevation angle of 15° and above), and thus has difficulty meeting the requirements of dm-level accuracy refraction error correction for deep-space and high-orbit targets. To improve the accuracy of radio-wave refraction error correction for deep-space and high-orbit targets, a novel correction method for tropospheric and ionospheric range error due to refraction is proposed in this study, on the basis of the measured data from a water vapor radiometer and dual-frequency Global Navigation Satellite System (GNSS). The comprehensive calibration test is conducted in combination with the Chinese Area Positioning System (CAPS) in Kunming. Results show that this method can effectively correct the range error due to refraction that is caused by the troposphere and ionosphere. For an S-band radio-wave signal, the accuracy of refraction error correction can reach dm-level accuracy (elevation angle of 15° and above), which is 50% higher than that achieved with traditional methods. This work provides an effective support system for major projects, such as lunar exploration and Mars exploration.

On the Assessment GPS-Based WRFDA for InSAR Atmospheric Correction: A Case Study in Pearl River Delta Region of China

The accuracy and applications of synthetic aperture radar interferometry (InSAR) are severely suppressed by tropospheric error. Numerical Weather Models (NWMs) and GPS-derived tropospheric delays have been widely used to correct the tropospheric error considering their complete spatial coverage or high accuracy. However, few studies focus on the fusion of both NWMs and GPS for the tropospheric error correction. In this study, we used the Weather Research and Forecasting (WRF) to obtain NWMs with a higher spatial-temporal resolution of 3 km and 20 s from both ERAI (79 km and 6 h) and ERA5 (0.25° and 1 h). After that, we utilized the WRF Data Assimilation (WRFDA) system to assimilate the GPS ZTD into these enhanced NWMs and generate merged NWMs products. The tropospheric correction effectiveness from different NWMs products was evaluated in a case in the Pearl River Delta region of China. The results showed that all the NWMs products could correct the stratified component in the interferogram but could not mitigate the turbulence well, even after improving the spatial-temporal resolution. As for the trend component, the merged NWMs products showed obvious superiority over other products. From the statistics perspective, the stdev of the interferogram decreased further over 20% by the merged NWMs products than other products when using both ERAI and ERA5, indicating the significant effectiveness of GPS ZTD assimilation.

Modified Interpolation Method of Multi-Reference Station Tropospheric Delay Considering the Influence of Height Difference

In network real-time kinematic (NRTK) positioning, atmospheric delay information is critical for generating virtual observations at a virtual reference station (VRS). The traditional linear interpolation method (LIM) is widely used to obtain the atmospheric delay correction. However, even though the conventional LIM is robust in the horizontal direction of the atmospheric error, it ignores the influence of the vertical direction, especially for the tropospheric error. If the height difference between the reference stations and the rover is large and, subsequently, tropospheric error and height are strongly correlated, the performance of the traditional method is degraded for tropospheric delay interpolation at the VRS. Therefore, considering the height difference between the reference stations and the rover, a modified linear interpolation method (MLIM) is proposed to be applied to a conventional single Delaunay triangulated network (DTN). The systematic error of the double-differenced (DD) tropospheric delay in the vertical direction is corrected first. The LIM method is then applied to interpolate the DD tropospheric delay at the VRS. In order to verify the performance of the proposed method, we used two datasets from the American NOAA continuously operating reference stations (CORS) network with significant height differences for experiments and analysis. Results show that the DD tropospheric delay interpolation accuracy obtained by the modified method is improved by 84.1% and 69.6% on average in the two experiments compared to the conventional method. This improvement is significant, especially for low elevation satellites. In rover positioning analysis, the traditional LIM has a noticeable systematic deviation in the up component. Compared to the conventional method, the positioning accuracy of the MLIM method is improved in the horizontal and vertical directions, especially in the up component. The accuracy of the up component is reduced from tens of centimeters to a few centimeters and demonstrates better positioning stability.