Near real-time estimation of ionosphere vertical total electron content from GNSS satellites using B-splines in a Kalman filter

Abstract. Although the number of terrestrial global navigation satellite system (GNSS) receivers supported by the International GNSS Service (IGS) is rapidly growing, the worldwide rather inhomogeneously distributed observation sites do not allow the generation of high-resolution global ionosphere products. Conversely, with the regionally enormous increase in highly precise GNSS data, the demands on (near) real-time ionosphere products, necessary in many applications such as navigation, are growing very fast. Consequently, many analysis centers accepted the responsibility of generating such products. In this regard, the primary objective of our work is to develop a near real-time processing framework for the estimation of the vertical total electron content (VTEC) of the ionosphere using proper models that are capable of a global representation adapted to the real data distribution. The global VTEC representation developed in this work is based on a series expansion in terms of compactly supported B-spline functions, which allow for an appropriate handling of the heterogeneous data distribution, including data gaps. The corresponding series coefficients and additional parameters such as differential code biases of the GNSS satellites and receivers constitute the set of unknown parameters. The Kalman filter (KF), as a popular recursive estimator, allows processing of the data immediately after acquisition and paves the way of sequential (near) real-time estimation of the unknown parameters. To exploit the advantages of the chosen data representation and the estimation procedure, the B-spline model is incorporated into the KF under the consideration of necessary constraints. Based on a preprocessing strategy, the developed approach utilizes hourly batches of GPS and GLONASS observations provided by the IGS data centers with a latency of 1 h in its current realization. Two methods for validation of the results are performed, namely the self consistency analysis and a comparison with Jason-2 altimetry data. The highly promising validation results allow the conclusion that under the investigated conditions our derived near real-time product is of the same accuracy level as the so-called final post-processed products provided by the IGS with a latency of several days or even weeks.

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Real-time regional VTEC modeling based on B-splines using real-time GPS and GLONASS observations

10.5194/egusphere-egu21-9291 ◽

2021 ◽

Author(s):

Eren Erdogan ◽

Andreas Goss ◽

Michael Schmidt ◽

Denise Dettmering ◽

Florian Seitz ◽

...

Keyword(s):

Kalman Filter ◽

Real Time ◽

Carrier Phase ◽

Background Information ◽

Electron Content ◽

Vertical Total Electron Content ◽

Total Electron ◽

B Spline ◽

B Splines ◽

Cycle Slips

The project OPTIMAP is at the current stage a joint initiative of BGIC, GSSAC and DGFI-TUM. The development of an operational tool for ionospheric mapping and prediction is the main goal of the project.The ionosphere is a dispersive medium. Therefore, GNSS signals are refracted while they pass through the ionosphere. The magnitude of the refraction rate depends on the frequencies of the transmitted GNSS signals. The ionospheric disturbance on the GNSS signals paves the way of extracting Vertical Total Electron Content (VTEC) information of the ionosphere.In OPTIMAP, the representation of the global and regional VTEC signal is based on localizing B-spline basis functions. For global VTEC modeling, polynomial B-splines are employed to represent the latitudinal variations, whereas trigonometric B-splines are used for the longitudinal variations. The regional modeling in OPTIMAP relies on a polynomial B-spline representation for both latitude and longitude.The VTEC modeling in this study relies on both a global and a regional sequential estimator (Kalman filter) running in a parallel mode. The global VTEC estimator produces VTEC maps based on data from GNSS receiver stations which are mainly part of the global real-time IGS network. The global estimator relies on additional VTEC information obtained from a forecast procedure using ultra-rapid VTEC products. The regional estimator makes use of the VTEC product of the real-time global estimator as background information and generates high-resolution VTEC maps using real-time data from the EUREF Permanent GNSS Network. EUREF provides a network of very dense GNSS receivers distributed alongside Europe.Carrier phase observations acquired from GPS and GLONASS, which are transmitted in accordance with RTCM standard, are used for real-time regional VTEC modeling. After the acquisition of GNSS data, cycle slips for each satellite-receiver pair are detected, and ionosphere observations are constructed via the linear combination of carrier-phase observations in the data pre-processing step. The unknown B-spline coefficients, as well as the biases for each phase-continuous arc belonging to each receiver-satellite pair, are simultaneously estimated in the Kalman filter.Within this study, we compare the performance of regional and global VTEC products generated in real-time using the well-known dSTEC analysis.

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Real-time regional VTEC modeling based on B-splines using real-time GPS, GLONASS and GALILEO observations

10.5194/egusphere-egu2020-15908 ◽

2020 ◽

Author(s):

Eren Erdogan ◽

Andreas Goss ◽

Michael Schmidt ◽

Denise Dettmering ◽

Florian Seitz ◽

...

Keyword(s):

Kalman Filter ◽

Real Time ◽

Carrier Phase ◽

Background Information ◽

Electron Content ◽

Vertical Total Electron Content ◽

Total Electron ◽

B Spline ◽

B Splines ◽

Cycle Slips

The project OPTIMAP is at the current stage a joint initiative of BGIC, GSSAC and DGFI-TUM. The development of an operational tool for ionospheric mapping and prediction is the main goal of the project.The ionosphere is a dispersive medium. Therefore, GNSS signals are refracted while they pass through the ionosphere. The magnitude of the refraction rate depends on the frequencies of the transmitted GNSS signals. The ionospheric disturbance on the GNSS signals paves the way of extracting Vertical Total Electron Content (VTEC) information of the ionosphere.In OPTIMAP, the representation of the global and regional VTEC signal is based on localizing B-spline basis functions. For global VTEC modeling, polynomial B-splines are employed to represent the latitudinal variations, whereas trigonometric B-splines are used for the longitudinal variations. The regional modeling in OPTIMAP relies on a polynomial B-spline representation for both latitude and longitude.The VTEC modeling in this study relies on both a global and a regional sequential estimator (Kalman filter) running in a parallel mode. The global VTEC estimator produces VTEC maps based on data from GNSS receiver stations which are mainly part of the global real-time IGS network. The global estimator relies on additional VTEC information obtained from a forecast procedure using ultra-rapid VTEC products. The regional estimator makes use of the VTEC product of the real-time global estimator as background information and generates high-resolution VTEC maps using real-time data from the EUREF Permanent GNSS Network. EUREF provides a network of very dense GNSS receivers distributed alongside Europe.Carrier phase observations acquired from GPS, GLONASS and GALILEO constellations, which are transmitted in accordance with RTCM standard, are used for real-time regional VTEC modeling. After the acquisition of GNSS data, cycle slips for each satellite-receiver pair are detected, and ionosphere observations are constructed via the linear combination of carrier-phase observations in the data pre-processing step. The unknown B-spline coefficients, as well as the biases for each phase-continuous arc belonging to each receiver-satellite pair, are simultaneously estimated in the Kalman filter.Within this study, we compare the performance of regional and global VTEC products generated in real-time using the well-known dSTEC analysis.

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Real-Time Estimation of Discharge Current Oscillations Using an Iterated Extended Kalman Filter

10.1109/icops36761.2021.9588439 ◽

2021 ◽

Author(s):

Christine M. Greve ◽

Kentaro Hara

Keyword(s):

Kalman Filter ◽

Real Time ◽

Extended Kalman Filter ◽

Discharge Current ◽

Time Estimation ◽

Current Oscillations ◽

Real Time Estimation ◽

Iterated Extended Kalman Filter

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Real-time Estimation of Plasma Dynamics using an Extended Kalman Filter

AIAA Propulsion and Energy 2021 Forum ◽

10.2514/6.2021-3425 ◽

2021 ◽

Author(s):

Christine Greve ◽

Ken Hara

Keyword(s):

Kalman Filter ◽

Real Time ◽

Extended Kalman Filter ◽

Time Estimation ◽

Plasma Dynamics ◽

Real Time Estimation

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Adaptive Modeling of the Global Ionosphere Vertical Total Electron Content

Remote Sensing ◽

10.3390/rs12111822 ◽

2020 ◽

Vol 12 (11) ◽

pp. 1822

Author(s):

Eren Erdogan ◽

Michael Schmidt ◽

Andreas Goss ◽

Barbara Görres ◽

Florian Seitz

Keyword(s):

Total Electron Content ◽

Temporal Variations ◽

Model Parameters ◽

Electron Content ◽

Vertical Total Electron Content ◽

Model Uncertainties ◽

Data Set ◽

Total Electron ◽

B Spline ◽

Run Time

The Kalman filter (KF) is widely applied in (ultra) rapid and (near) real-time ionosphere modeling to meet the demand on ionosphere products required in many applications extending from navigation and positioning to monitoring space weather events and naturals disasters. The requirement of a prior definition of the stochastic models attached to the measurements and the dynamic models of the KF is a drawback associated with its standard implementation since model uncertainties can exhibit temporal variations or the time span of a given test data set would not be large enough. Adaptive methods can mitigate these problems by tuning the stochastic model parameters during the filter run-time. Accordingly, one of the primary objectives of our study is to apply an adaptive KF based on variance component estimation to compute the global Vertical Total Electron Content (VTEC) of the ionosphere by assimilating different ionospheric GNSS measurements. Secondly, the derived VTEC representation is based on a series expansion in terms of compactly supported B-spline functions. We highlight the morphological similarity of the spatial distributions and the magnitudes between VTEC values and the corresponding estimated B-spline coefficients. This similarity allows for deducing physical interpretations from the coefficients. In this context, an empirical adaptive model to account for the dynamic model uncertainties, representing the temporal variations of VTEC errors, is developed in this work according to the structure of B-spline coefficients. For the validation, the differential slant total electron content (dSTEC) analysis and a comparison with Jason-2/3 altimetry data are performed. Assessments show that the quality of the VTEC products derived by the presented algorithm is in good agreement, or even more accurate, with the products provided by IGS ionosphere analysis centers within the selected periods in 2015 and 2017. Furthermore, we show that the presented approach can be applied to different ionosphere conditions ranging from very high to low solar activity without concerning time-variable model uncertainties, including measurement error and process noise of the KF because the associated covariance matrices are computed in a self-adaptive manner during run-time.

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Real-Time Global Ionospheric Map and Its Application in Single-Frequency Positioning

Sensors ◽

10.3390/s19051138 ◽

2019 ◽

Vol 19 (5) ◽

pp. 1138 ◽

Cited By ~ 7

Author(s):

Liang Zhang ◽

Yibin Yao ◽

Wenjie Peng ◽

Lulu Shan ◽

Yulin He ◽

...

Keyword(s):

Real Time ◽

Total Electron Content ◽

Precise Point Positioning ◽

Single Frequency ◽

Electron Content ◽

Vertical Total Electron Content ◽

Total Electron ◽

Time Service ◽

Point Positioning ◽

Raw Observations

The prevalence of real-time, low-cost, single-frequency, decimeter-level positioning has increased with the development of global navigation satellite systems (GNSSs). Ionospheric delay accounts for most errors in real-time single-frequency GNSS positioning. To eliminate ionospheric interference in real-time single-frequency precise point positioning (RT-SF-PPP), global ionospheric vertical total electron content (VTEC) product is designed in the next stage of the International GNSS Service (IGS) real-time service (RTS). In this study, real-time generation of a global ionospheric map (GIM) based on IGS RTS is proposed and assessed. There are three crucial steps in the process of generating a real-time global ionospheric map (RTGIM): estimating station differential code bias (DCB) using the precise point positioning (PPP) method, deriving slant total electron content (STEC) from PPP with raw observations, and modeling global vertical total electron content (VTEC). Experiments were carried out to validate the algorithm’s effectiveness. First, one month’s data from 16 globally distributed IGS stations were used to validate the performance of DCB estimation with the PPP method. Second, 30 IGS stations were used to verify the accuracy of static PPP with raw observations. Third, the modeling of residuals was assessed in high and quiet ionospheric activity periods. Afterwards, the quality of RTGIM products was assessed from two aspects: (1) comparison with the Center for Orbit Determination in Europe (CODE) global ionospheric map (GIM) products and (2) determination of the performance of RT-SF-PPP with the RTGIM. Experimental results show that DCB estimation using the PPP method can realize an average accuracy of 0.2 ns; static PPP with raw observations can achieve an accuracy of 0.7, 1.2, and 2.1 cm in the north, east, and up components, respectively. The average standard deviations (STDs) of the model residuals are 2.07 and 2.17 TEC units (TECU) for moderate and high ionospheric activity periods. Moreover, the average root-mean-square (RMS) error of RTGIM products is 2.4 TECU for the one-month moderate ionospheric period. Nevertheless, for the high ionospheric period, the RMS is greater than the RMS in the moderate period. A sub-meter-level horizontal accuracy and meter-level vertical accuracy can be achieved when the RTGIM is employed in RT-SF-PPP.

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Global and Regional High-Resolution VTEC Modelling Using a Two-Step B-Spline Approach

Remote Sensing ◽

10.3390/rs12071198 ◽

2020 ◽

Vol 12 (7) ◽

pp. 1198 ◽

Cited By ~ 3

Author(s):

Andreas Goss ◽

Michael Schmidt ◽

Eren Erdogan ◽

Florian Seitz

Keyword(s):

Real Time ◽

Spectral Resolution ◽

The State ◽

Single Frequency ◽

External Information ◽

Electron Content ◽

Intermediate Step ◽

Total Electron ◽

B Spline ◽

Hourly Data

The ionosphere is one of the largest error sources in GNSS (Global Navigation Satellite Systems) applications and can cause up to several meters of error in positioning. Especially for single-frequency users, who cannot correct the ionospheric delay, the information about the state of the ionosphere is mandatory. Dual- and multi-frequency GNSS users, on the other hand, can correct the ionospheric effect on their observations by linear combination. However, real-time applications such as autonomous driving or precision farming, require external high accuracy corrections for fast convergence. Mostly, this external information is given in terms of grids or coefficients of the vertical total electron content (VTEC). Globally distributed GNSS stations of different networks, such as the network of the International GNSS Services (IGS), provide a large number of multi-frequency observations which can be used to determine the state of the ionosphere. These data are used to generate Global Ionosphere Maps (GIM). Due to the inhomogeneous global distribution of GNSS real-time stations and especially due to the large data gaps over oceanic areas, the global VTEC models are usually limited in their spatial and spectral resolution. Most of the GIMs are mathematically based on globally defined radial basis functions, i.e., spherical harmonics (SH), with a maximum degree of 15 and provided with a spatial resolution of 2.5 ° × 5 ° in latitude and longitude, respectively. Regional GNSS networks, however, offer dense clusters of observations, which can be used to generate regional VTEC solutions with a higher spectral resolution. In this study, we introduce a two-step model (TSM) comprising a global model as the first step and a regional model as the second step. We apply polynomial and trigonometric B-spline functions to represent the global VTEC. Polynomial B-splines are used for modelling the finer structures of VTEC within selected regions, i.e., the densification areas. The TSM provides both, a global and a regional VTEC map at the same time. In order to study the performance, we apply the developed approach to hourly data of the global IGS network as well as the EUREF network of the European region for St. Patrick storm in March 2015. For the assessment of the generated maps, we use the dSTEC analysis and compare both maps with different global and regional products from the IGS Ionosphere Associated Analysis Centers, e.g., the global product from CODE (Berne, Switzerland) and from UPC (Barcelona, Spain), as well as the regional maps from ROB (Brussels, Belgium). The assessment shows a significant improvement of the regional VTEC representation in the form of the generated TSM maps. Among all other products used for comparison, the developed regional one is of the highest accuracy within the selected time span. Since the numerical tests are performed using hourly data with a latency of one to two hours, the presented procedure is seen as an intermediate step for the generation of high precision regional real-time corrections for modern applications.

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Efficient global ionospheric modeling based on multi-source and massive observation data

10.5194/egusphere-egu2020-16618 ◽

2020 ◽

Author(s):

Xulei Jin ◽

Shuli Song ◽

Wei Li ◽

Na Cheng

Keyword(s):

Real Time ◽

Rapid Development ◽

Electron Content ◽

Vertical Total Electron Content ◽

Observation Data ◽

Total Electron ◽

Ionospheric Modeling ◽

Virtual Observation ◽

Data Missing ◽

Modeling Strategy

Abstract Ionosphere is an important error source of satellite navigation and a key component of space weather. With the rapid development of multiple Global Navigation Satellite System (GNSS) and other ionospheric research technologies, and the high precision and near real-time requirements for ionospheric products, it is necessary to carry out a research on multi-source data fusion, massive data processing and near-real-time solution of global ionosphere model (GIM); therefore, we modified the traditional ionospheric modeling technology and generate the GIM products (GIM/SHA). In view of the defect of ground-based GNSS data missing in the ocean regions, the method of adding virtual observation stations to the data missing regions in a large range was adopted, which not only enhanced the accuracy of the GIM in the ocean regions, but also avoided the weight determination among different data sources. In terms of near-real-time modeling, the multi-threaded parallel modeling strategy was adopted. Four GNSS (GPS, GLONASS, BEIDOU, Galileo) observation data, eight virtual observation stations and a server with a CPU frequency of 2.1 GHz and 16 threads were utilized. It took less than 30 minutes to construct the GIM by using parallel modeling strategy, which was 10.3 times faster than serial modeling strategy. The accuracy of the GIM/SHA was verified by using the ionospheric products of International GNSS Service (IGS) Ionosphere Associate Analysis Centers (IAACs) in the period of day of year (DOY) 200-365, 2019. Compared with the ionospheric products of CODE, ESA/ESOC, JPL, UPC, EMR, CAS and WHU, the vertical total electron content (VTEC) root mean squares (RMSs) were 1.09 TEC units (TECu), 1.51TECu, 2.32TECu, 1.88TECu, 2.24TECu, 1.25TECu and 1.38TECu, respectively. The result shows that the GIM/SHA have comparable accuracy with IGS ionospheric products. Satellite altimetry data was exploited to verify the accuracy of GIM/SHA in ocean regions, and it can be concluded that the accuracy of the GIM in ocean regions can be significantly reinforced by adding virtual observation stations in ocean regions. Multi-system and multi-frequency differential code bias (DCB) products (DCB/SHA) were simultaneously generated. Compared with IGS DCB products, the satellite DCB RMSs of DCB/SHA were 0.16ns for GPS, 0.08ns for GLONASS, 0.17ns for BEIDOU and 0.14ns for Galileo; the GNSS receiver DCB RMSs of DCB/SHA were 0.69ns for GPS, 1.06ns for GLONASS, 0.75 for BEIDOU and 1.03ns for Galileo. It can be proved that the accuracy of DCB/SHA are comparable to IGS DCB products.Keywords Multi-GNSS; GIM; Virtual observation station; Near real-time; VTEC; DCB

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