High Resolution Vertical Total Electron Content Maps Based on Multi-Scale B-spline Representations

Abstract. For more than two decades the IGS (International GNSS Service) Ionosphere Associated Analysis Centers (IAAC) provide global maps of the vertical total electron content (VTEC). In general, the representation of a two- or three-dimensional function can be performed by means of a series expansion or by using a discretization technique. Whereas in the latter case for a spherical function such as VTEC usually pixels or voxels are chosen, in case of a series expansion mostly spherical harmonics (SH) are used as basis functions. The selection of the best suited approach for ionosphere modelling means a trade-off between the distribution of available data and their possibility to represent ionospheric variations with high resolution and high accuracy. Most of the IAACs generate Global Ionosphere Maps (GIMs) based on SH expansions up to the spectral degree n = 15 and provide them with a spatial resolution of 2.5° × 5° with respect to latitude and longitude direction, and a temporal sampling of two hours. In the recent years it was frequently claimed to improve the spatial sampling of the VTEC GIMs to a spatial resolution of 1° × 1° and to a temporal sampling of about 15 minutes. Enhancing the grid resolution means a interpolation of VTEC values for intermediate points but with no further information about variations in the signal. A degree 15 in the SH case for instance corresponds to a spatial sampling of 12° × 12°. Consequently, increasing the grid resolution requires at the same time an extension of the spectral content, i.e. to choose a higher SH degree value than 15. Unlike most of the IAACs, the VTEC modelling approach at DGFI-TUM is based on localizing basis functions, namely tensor products of polynomial and trigonometric B-splines. This way, not only data gaps can be handled appropriately and sparse normal equation systems are established for the parameters estimation procedure, also a multi-scale-representation (MSR) can be set up, to determine GIMs of different spectral content directly by applying the so-called pyramid algorithm and to perform highly effective data compression techniques. The estimation of the MSR model parameters is finally performed by a Kalman-Filter driven by near real-time (NRT) GNSS data. Within this paper we realize the MSR and create multi-scale products based on B-spline scaling and wavelet coefficients and VTEC grid values. We compare these products with different final and rapid products of the IAACs, e.g., the SH model from CODE (Berne) and the voxel solution from UPC (Barcelona). In opposite to that, DGFI-TUM's products are solely based on NRT GNSS observations and ultra-rapid orbits. Nevertheless, we can conclude that DGFI-TUMs high-resolution product (`othg') outperforms all products used within the selected time span of investigation, namely September 2017.

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High-resolution vertical total electron content maps based on multi-scale B-spline representations

Annales Geophysicae ◽

10.5194/angeo-37-699-2019 ◽

2019 ◽

Vol 37 (4) ◽

pp. 699-717 ◽

Cited By ~ 6

Author(s):

Andreas Goss ◽

Michael Schmidt ◽

Eren Erdogan ◽

Barbara Görres ◽

Florian Seitz

Keyword(s):

High Resolution ◽

Spatial Resolution ◽

Sampling Interval ◽

Basis Functions ◽

Electron Content ◽

Vertical Total Electron Content ◽

Spectral Content ◽

Total Electron ◽

Multi Scale ◽

Temporal Sampling

Abstract. For more than 2 decades the IGS (International GNSS Service) ionosphere associated analysis centers (IAACs) have provided global maps of the vertical total electron content (VTEC). In general, the representation of a 2-D or 3-D function can be performed by means of a series expansion or by using a discretization technique. While in the latter case, pixels or voxels are usually chosen for a spherical function such as VTEC, for a series expansion spherical harmonics (SH) are primarily used as basis functions. The selection of the best suited approach for ionosphere modeling means a trade-off between the distribution of available data and their possibility of representing ionospheric variations with high resolution and high accuracy. Most of the IAACs generate global ionosphere maps (GIMs) based on SH expansions up to the spectral degree n=15 and provide them with a spatial resolution of 2.5∘×5∘ with respect to the latitudinal and longitudinal directions, respectively, and a temporal sampling interval of 2 h. In recent years, it has frequently been claimed that the spatial resolution of the VTEC GIMs has to be increased to a spatial resolution of 1∘×1∘ and to a temporal sampling interval of about 15 min. Enhancing the grid resolution means an interpolation of VTEC values for intermediate points but with no further information about variations in the signal. n=15 in the SH case, for instance, corresponds to a spatial sampling of 12∘×12∘. Consequently, increasing the grid resolution concurrently requires an extension of the spectral content, i.e., to choose a higher SH degree value than 15. Unlike most of the IAACs, the VTEC modeling approach at Deutsches Geodätisches Forschungsinstitut der Technischen Universität München (DGFI-TUM) is based on localizing basis functions, namely tensor products of polynomial and trigonometric B-splines. In this way, not only can data gaps be handled appropriately and sparse normal equation systems be established for the parameter estimation procedure, a multi-scale representation (MSR) can also be set up to determine GIMs of different spectral content directly, by applying the so-called pyramid algorithm, and to perform highly effective data compression techniques. The estimation of the MSR model parameters is finally performed by a Kalman filter driven by near real-time (NRT) GNSS data. Within this paper, we realize the MSR and create multi-scale products based on B-spline scaling, wavelet coefficients and VTEC grid values. We compare these products with different final and rapid products from the IAACs, e.g., the SH model from CODE (Berne) and the voxel solution from UPC (Barcelona). In contrast to the abovementioned products, DGFI-TUM's products are based solely on NRT GNSS observations and ultra-rapid orbits. Nevertheless, we can conclude that the DGFI-TUM's high-resolution product (“othg”) outperforms all products used within the selected time span of investigation, namely September 2017.

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Dissemination of High-Resolution Ionosphere Information from VTEC B-spline Expansions for Single-Frequency Positioning

10.5194/egusphere-egu21-14179 ◽

2021 ◽

Author(s):

Andreas Goss ◽

Manuel Hernández-Pajares ◽

Michael Schmidt ◽

Eren Erdogan

Keyword(s):

High Resolution ◽

Series Expansion ◽

Single Frequency ◽

Space Representation ◽

Model Parameters ◽

Electron Content ◽

Vertical Total Electron Content ◽

Total Electron ◽

B Spline ◽

Voxel Representation

The ionospheric signal delay is one of the largest error sources in GNSS applications and may cause in case of a single-frequency receiver a positioning error of up to several meters. To avoid such an inaccuracy some of the Ionosphere Associated Analysis Centers (IAAC) of the International GNSS Service (IGS) provide the user the Vertical Total Electron Content (VTEC) as Real-Time Global Ionosphere Maps (RT-GIM) via streaming formats. Currently, the only data format used for the dissemination of these ionospheric corrections is based on the State Space Representation (SSR) message and the RTCM standards.Mathematically most of the RT-GIMs are based on modeling VTEC as series expansions in spherical harmonics (SH) up to a highest degree of n = 15 which corresponds to a spatial resolution of 12&#176; in latitude and longitude and is therefore, too low for modern GNSS applications such as autonomous driving. However, the SSR VTEC message allows the dissemination of SH coefficients only up to a maximum degree of n = 16.To avoid the drawbacks of expanding VTEC in SHs other approaches such as a voxel representation or a B-spline series expansion have been proven to be appropriate candidates for global and regional modelling with an enhanced resolution. In order to provide in these cases the significant model parameters to the user, the application of the SSR VTEC message requires a transformation of the model parameters into SH coefficients. In this contribution a methodology will be presented which describes a fast transformation of the B-spline approach into a SH representation with high accuracy by minimizing the information loss.To test the method, a high-resolution VTEC GIM modeled as a series expansion in B-splines is transformed into SH representations of different highest degree values; the results are validated via dSTEC analysis as well as via an example of single frequency positioning and show a significantly improved accuracy compared to the IGS GIMs.

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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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Analysis of a new regional ionospheric assimilated H2PT model for Europe

10.5194/egusphere-egu2020-1524 ◽

2020 ◽

Author(s):

Paulina Woźniak ◽

Anna Świątek ◽

Mariusz Pożoga ◽

Łukasz Tomasik

Keyword(s):

Spatial Resolution ◽

Negative Impact ◽

Satellite System ◽

Electron Content ◽

Maximum Height ◽

Vertical Total Electron Content ◽

Dual Frequency ◽

International Gnss Service ◽

Statistical Parameters ◽

Total Electron

The signal emitted by the GNSS (Global Navigation Satellite System) satellite, on the way to the receiver located on the Earth&#8217;s surface, encounters a heterogeneous layer of ionized gas and free electrons, in which the radio wave is dispersed. As the ionosphere is the source of the highest-value errors among the different factors that affect GNSS positioning accuracy, it is necessary to minimize its negative impact. Various methods are used to compensate for the ionospheric delay, one of which is the usage of models. The intensity of the processes occurring in the ionosphere is closely related to the Sun activity. As a consequence, with respect to a given location on the Earth's surface, the activity of the ionosphere changes throughout the year and day. Therefore, a model dedicated to a specific region is especially important in case of high-precision GNSS applications. The assimilated H2PT model was based on the dual-frequency observations from GNSS stations belonging to EPN (EUREF Permanent Network), as well as on ionosondes participating in the DIAS (European Digital Upper Atmosphere Server) project. The H2PT model covers the Europe area, data with a 15-minutes interval were placed in similar to IONEX (IONosphere Map EXchenge) files in two versions of spatial resolution: 1- and 5-degree. Data provided by the H2PT model are the VTEC (Vertical Total Electron Content) values and the hmF2 (maximum height of the F2 layer) parameters. The subject of this research is the comparison of the H2PT model with NeQuick-G model and IONEX data published by IGS (International GNSS Service) in the context of TEC values as well as determining differences between regional hmF2 data and its commonly used fixed value for the entire globe, amounting to 450 km. In order to perform the analysis, appropriate visualizations were made and statistical parameters determined. Additionally, data from selected periods of positive and negative disturbances were analysed in details based on the developed time series. The relatively high temporal and spatial resolution is undoubtedly an advantage of the H2PT model, because unlike global models, the regional one allows conscientious analysis of the ionosphere characteristics for the area of Europe. Importantly, solutions regarding hmF2 show significant deviations from the fixed value approximated for the whole Earth. Taking into account the parameter appropriate for a given location and time during GNSS data processing may improve the obtained positioning quality.&#160;

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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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Comparison of spherical harmonic and B spline models for the vertical total electron content

Radio Science ◽

10.1029/2010rs004609 ◽

2011 ◽

Vol 46 (6) ◽

Cited By ~ 30

Author(s):

Michael Schmidt ◽

Denise Dettmering ◽

Matthias Mößmer ◽

Yuanyuan Wang ◽

Jiantong Zhang

Keyword(s):

Total Electron Content ◽

Spherical Harmonic ◽

Electron Content ◽

Vertical Total Electron Content ◽

Total Electron ◽

B Spline ◽

Spline Models

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

Annales Geophysicae ◽

10.5194/angeo-35-263-2017 ◽

2017 ◽

Vol 35 (2) ◽

pp. 263-277 ◽

Cited By ~ 14

Author(s):

Eren Erdogan ◽

Michael Schmidt ◽

Florian Seitz ◽

Murat Durmaz

Keyword(s):

Kalman Filter ◽

Real Time ◽

Data Distribution ◽

Time Estimation ◽

Electron Content ◽

Vertical Total Electron Content ◽

Unknown Parameters ◽

Total Electron ◽

B Spline ◽

Real Time Estimation

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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A Bayesian Approach to Improve Regional and Global Ionospheric Maps using GNSS Observations

10.5194/egusphere-egu2020-282 ◽

2020 ◽

Author(s):

Saeed Farzaneh ◽

Ehsan Forootan

Keyword(s):

Total Electron Content ◽

Spatial Resolution ◽

Spherical Harmonics ◽

Bayesian Approach ◽

Electron Content ◽

Vertical Total Electron Content ◽

Numerical Application ◽

Total Electron ◽

Base Functions ◽

Gnss Measurements

The Global Ionosphere Maps (GIMs) are generated on a daily basis at the Center for Orbit Determination in Europe (CODE) using the observations from about 200 Global Positioning System (GPS)/GLONASS sites of the International GNSS Service (IGS) and other institutions. These maps contain Vertical Total Electron Content (VTEC) values, which are estimated in a solar-geomagnetic reference frame using a spherical harmonics expansion up to degree and order 15. Although these maps have wide applications, their relatively low spatial resolution limits the accuracy of many geodetic applications such as those related to Precise Point Positioning (PPP) and navigation. In this study, a novel Bayesian approach is proposed to improve the spatial resolution of VTEC estimations in regional and global scales. The proposed technique utilises GIMs as a prior information and updates the VTEC estimates using a new set of base-functions (with better resolution than that of spherical harmonics) and the GNSS measurements that are not included in the network of GIMs. To achieve the highest accuracy possible, our implementation is based on a transformation of spherical harmonics to the Slepian base-functions, where the latter is a set of bandlimited functions that reflect the majority of signal energy inside an arbitrarily defined region, yet they remain orthogonal within this region. The new GNSS measurements are considered in a Bayesian update estimation to modify those of GIMs. Numerical application of this study is demonstrated using the ground-based GPS data over South America. The results are also validated against the VTEC estimations derived from independent GPS stations.Key words: Spherical Slepian Base-Functions, Spherical Harmonics, Ionospheric modelling, Vertical Total Electron Content (VTEC)

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Characterization of multi-scale ionospheric irregularities using ground-based and space-based GNSS observations

Satellite Navigation ◽

10.1186/s43020-021-00047-x ◽

2021 ◽

Vol 2 (1) ◽

Author(s):

YuXiang Peng ◽

Wayne A Scales ◽

Michael D Hartinger ◽

Zhonghua Xu ◽

Shane Coyle

Keyword(s):

Formation Flying ◽

Satellite System ◽

Ionospheric Irregularities ◽

Mission Design ◽

Electron Content ◽

Total Electron ◽

Spatial And Temporal Scales ◽

Multi Scale ◽

Global Navigation Satellite ◽

Gnss Observations

AbstractIonospheric irregularities can adversely affect the performance of Global Navigation Satellite System (GNSS). However, this opens the possibility of using GNSS as an effective ionospheric remote sensing tool. Despite ionospheric monitoring has been undertaken for decades, these irregularities in multiple spatial and temporal scales are still not fully understood. This paper reviews Virginia Tech’s recent studies on multi-scale ionospheric irregularities using ground-based and space-based GNSS observations. First, the relevant background of ionospheric irregularities and their impact on GNSS signals is reviewed. Next, three topics of ground-based observations of ionospheric irregularities for which GNSS and other ground-based techniques are used simultaneously are reviewed. Both passive and active measurements in high-latitude regions are covered. Modelling and observations in mid-latitude regions are considered as well. Emphasis is placed on the increased capability of assessing the multi-scale nature of ionospheric irregularities using other traditional techniques (e.g., radar, magnetometer, high frequency receivers) as well as GNSS observations (e.g., Total-Electron-Content or TEC, scintillation). Besides ground-based observations, recent advances in GNSS space-based ionospheric measurements are briefly reviewed. Finally, a new space-based ionospheric observation technique using GNSS-based spacecraft formation flying and a differential TEC method is demonstrated using the newly developed Virginia Tech Formation Flying Testbed (VTFFTB). Based on multi-constellation multi-band GNSS, the VTFFTB has been developed into a hardware-in-the-loop simulation testbed with external high-fidelity global ionospheric model(s) for 3-satellite formation flying, which can potentially be used for new multi-scale ionospheric measurement mission design.

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