The ionosphere constitutes the upper part of the atmosphere, extending from approximately 60 to 1500 km above the Earth's surface, enriched with free electrons and ions . The knowledge of the structure and dynamics of the ionospheric plasma has great importance for various scientific applications and services, such as telecommunication through radio signals, point positioning based on global navigation satellite systems (GNSSs) , the monitoring of space weather events such as solar flares and coronal mass ejections , the investigations of ionospheric anomalies preceding or following a natural hazard such as earthquakes , and investigation of the ionospheric effects on thermospheric mass density, wind, and temperature .

The ionospheric plasma density varies with time and location and exhibits a coupled system with its environment: the Sun, the Earth's lower atmosphere, the thermosphere and the magnetosphere . Interactions between the thermospheric neutral winds and ionized plasma drive the ionospheric charged particles in motion and lead to separation of charges, resulting in the creation of a polarized electrical field E. During daytime, the combination of this eastward-oriented electrical field E with the Earth's horizontally northward-oriented magnetic field B causes an upward E×B drift of the ionized plasma. Following the upward drift, the plasma diffuses through the magnetic field lines and creates two crests with high ionization at both sides of the magnetic equator, which is also known as fountain effect . In addition, at high latitudes, the interaction of the Earth's magnetosphere with the interplanetary magnetic field attached to the solar winds as well as to the space weather events increases the complexity of the system. Today, the advances in space-geodetic techniques, such as terrestrial GNSS, spaceborne radio occultations to low-Earth-orbiting (LEO) satellites as well as satellite altimetry, facilitates the monitoring of the structure of the ionosphere with an improved spatial and temporal resolution. Particularly, GNSS offers an attractive alternative to traditional methods, such as ionosondes, for monitoring the electron content within the ionosphere in terms of volume and global data distribution.

The International GNSS Service (IGS) delivers large volumes of GNSS data with different latencies (e.g., real time, hourly) acquired from continuously operating terrestrial GNSS receivers distributed worldwide. The four IGS Ionosphere Associate Analysis Centers (IAACs), namely the Jet Propulsion Laboratory (JPL), the Center for Orbit Determination in Europe (CODE), the European Space Operations Center of the European Space Agency (ESOC) and the Universitat Politècnica de Catalunya (UPC), monitor the ionosphere and evaluate relevant parameters using dual frequency GNSS receivers. Several modeling approaches for ionospheric parameters have been proposed. A common approach, generally denoted as single-layer model (SLM), is based on the assumption that the electrons in the ionosphere are concentrated within a thin shell at a fixed altitude above the Earth . The spatial variations of electron content in this single layer are represented by a proper mathematical model such as spherical harmonics , B-splines , spatially defined total electron content (TEC) grids , polynomials , wavelets , or the MARS and BMARS approaches . Furthermore, three-dimensional models that consider the variation with altitude were also studied . The IGS delivers post-processed global VTEC products by combining VTEC maps from the different analysis centers mentioned before . The combination of these products computed with independent algorithms performed by the analysis centers contributes to the accuracy, the integrity and the availability of the global VTEC.

In this context, one of the main goals of this study is to develop an alternative approach that contributes to these modeling efforts by generating near real-time products. To be more specific, series expansions in terms of tensor products of compactly supported B-spline functions are used to represent the spatial variation of the global VTEC; introductory studies on this topic were published by , , and . The advantages of B-spline representations for VTEC modeling in comparison to spherical harmonics were also revealed by . The results of this study show that the B-spline series approach features in considerable improvements if the input data is distributed heterogeneously.

Considering the increasing demands on high-precision (near) real-time global ionosphere products, including global VTEC maps, an estimation strategy becomes obviously important to appropriately handle the large amount of ionosphere data as well as processing the data once it is available. The Kalman filter (KF) is a popular filtering technique and does not require storing measurements of the past since it is in the batch filtering for estimating the current state of a system and data can be processed immediately after acquisition . The KF has been successfully applied in (near) real-time ionosphere modeling by many authors; see , and . Here, apart from the other studies, we focus on the implementation of a B-spline representation into the KF for ionospheric parameter estimation.

In summary, the goal of the present study is to develop a near real-time processing framework to globally monitor the spatial and temporal variations within the ionosphere by exploiting the advantages of the B-spline series expansion and the recursive filtering using GPS and GLONASS measurements. Finally, it shall be discussed how the quality of this near real-time product is in comparison to the so-called final, i.e., the post-processed high-quality VTEC products of the IGS and its IAACs provided with a latency of days or even weeks.

The paper is outlined as follows. Section  explains the theoretical foundations for obtaining ionosphere observations from raw hourly GNSS data. In Sect.  the B-spline representation for global VTEC modeling is explained. Section  gives the background for the sequential estimation algorithm using the KF. The section comprises subparts, introducing the measurement model of the filter, the definition of model constraints, the prediction model of the filter, the foundation of the Kalman filtering, and the handling of the model constraints in the filter and data editing concepts. Section  summarizes the entire near real-time estimation procedure from an application point of view. Note that the exemplified generation of VTEC maps in this paper is based just on GNSS data. Validation concepts – partly based on satellite altimetry data – and the results are given in Sect. . Finally, Sect.  provides the conclusion and future work.

The free electrons within the ionosphere affect the propagation of electromagnetic waves, i.e., they cause a frequency-dependent delay in the transmitted radio signals. Although the ionospheric effect on GNSS signals is not desirable in positioning and navigation, it provides valuable information for the investigation of the electron content of the ionosphere. The magnitude of the delay depends on the electron density Ne along the ray path between the satellite s and the receiver r and is proportional to the slant total electron content (STEC) defined as STEC=∫rsNedl. Ne is measured in electrons per cubic meter. Following Eq. (), the vertical total electron content (VTEC) is defined as the integration of the electron density along the vertical, i.e., in height direction.

In order to extract ionospheric information from dual-frequency GNSS measurements, the geometry-free linear combination can be used . Firstly, the equations leading to the ionospheric observables Pr,Is and Lr,Is derived from combinations of the pseudo-range (code) and carrier-phase measurements, respectively, are defined as Pr,Is=Pr,f2s-Pr,f1s=α⋅STEC+br+bs+ϵPLr,Is=Φr,f1s-Φr,f2s=α⋅STEC+Br+Bs+Carc,rs+ϵL, where Pr,fis and Φr,fis with i∈{1,2} stand for the pseudo-range and the carrier-phase measurements observed simultaneously by the receiver r from satellite s. The subscripts 1 and 2 denote the two carrier frequencies f1 and f2 with the corresponding wavelengths λ1 and λ2. Br and Bs are the receiver and satellite inter-frequency biases (IFBs) for the carrier phase observations; similarly, br and bs stand for the differential code biases (DCBs) of the receiver and the satellite. α is a frequency-dependent constant factor. Furthermore, Carc,rs is the ambiguity bias of the carrier-phase, and ϵP and ϵL account for the measurement errors.

The pseudo-range measurements are rather noisy but unambiguous, while the carrier-phase data are significantly more precise but biased. To exploit the precision of the phase measurements, an offset CPBrs, including the terms IFB, DCB and Carc,rs, is computed by averaging the differences between Lr,Is and Pr,Is for every continuous arc that shares a common phase bias according to CPBrs≈1N∑j=1NLr,Is-Pr,Isj, where N is the number of observations continuously measured along the arc. An elevation-dependent threshold is established to select the more precise observations with higher elevation angle for the computation of CPBrs. Then, a leveled geometry-free phase observation Lr,4s̃ for a continuous arc becomes Lr,4s̃=Lr,Is-CPBrs=α⋅STEC+br+bs+ϵL4. The leveling technique is applied to the hourly data sets of GPS and GLONASS observations obtained from the IGS data servers.

As already mentioned in the introduction, appropriate approaches for representing VTEC are two-dimensional series expansions in terms of spherical harmonics or B-spline functions. In the latter case the basis functions can be set up by tensor products of polynomial and trigonometric B-splines. To be more specific, the B-spline representation of the global VTEC reads VTEC(φ,λ)=∑k1=0KJ1-1∑k2=0KJ2-1dk1,k2J1,J2NJ1,k12(φ)TJ2,k22(λ) , where the quantities dk1,k2J1,J2 mean the unknown series coefficients, NJ1,k12(φ) are the end-point interpolating polynomial B-spline functions of order three depending on the latitude φ, and TJ2,k22(λ) are the trigonometric B-splines of order three depending on the longitude λ. The values J1 and J2 are called levels, the numbers k1 and k2 define the geometrical positions of the two-dimensional basis functions NJ1,k12(φ)⋅TJ2,k22(λ) on the sphere. The value KJ1 stands for the number of polynomial B-spline functions according to the associated level J1; its numerical value is given by KJ1=2J1+2. Similarly, the number of trigonometric B-spline functions KJ2 for the level J2 is defined by KJ2=3⋅2J2+2. The polynomial B-spline functions NJ1,k12(φ) and the trigonometric B-spline functions TJ2,k22(λ) are compactly supported, which means the functions are different from zero only within a small subinterval (see , and the references therein).

The selection of appropriate resolution levels J1 and J2 requires the consideration of different criteria, namely the distribution of the input data, the computational burden and the desired level of smoothness. Figure  clearly shows that the higher the level chosen, the larger the number of B-spline basis functions, i.e., the higher the resolution of the VTEC representation. Appropriate levels of the B-splines can be obtained from the distribution of the input data . This procedure has already been applied successfully for ionosphere modeling from GPS occultation data . In case of near real-time applications, the level values may have to fulfill computation time constraints. A trade-off between the resolution level and the computational burden becomes another necessary issue since the higher the resolution, the larger the number of unknown coefficients and the higher the time consumption the parameter estimation procedure based on Kalman filtering requires. The GNSS data distribution is generally rather inhomogeneous, with large data gaps, especially over the oceans. B-spline representations with high level values can result in basis functions without any data support. For instance, all the basis functions in Fig. a are supported by GNSS observations. However, this is not the case for the different levels of representations shown in the other three panels of Fig. . Conversely, a representation with level values chosen too low may not describe the spatial VTEC variations sufficiently and may cause an undesired smoothing, i.e., a loss of information. For example, a reference VTEC map used as input observation is illustrated in Fig. a. Figure b depicts the reconstructed map using the B-spline levels J1=3 and J2=1: it smooths the details too much. An increase in the level values to J1=4 and J2=3 results in a better reconstruction of the reference map, as is shown in Fig. c. Consequently, in order to achieve a representation quality comparable to IGS products while considering the aforementioned criteria, the B-spline levels are set to J1=4 and J2=3 in this study.

The KF () was used in this work as a sequential estimator to compute the ionospheric parameters in near real time. Since the KF is of recursive nature, measurements from the past do not need to be stored . The current state is updated as soon as new observations are available. This means a crucial advantage for (near) real-time applications because this way the filter allows assimilation of observations as soon as possible without waiting for another group of observations.

Here, for the estimation of the ionospheric target parameters, the system equations, including the measurement model and the prediction model, are linear and the observations are assumed to have a Gaussian distribution. Then the KF provides an optimal recursive estimator in terms of minimum variance estimation (see ). The linear system of equations in a discrete form is defined as βk=Fkβk-1+wk-1yk=Xkβk+ek, where k is the time stamp, Fk is the transition matrix, βk is the vector of the unknown parameters, yk is the vector of the measurements and Xk is the corresponding design matrix. The measurement error vector ek and the vector wk of the process noise are assumed to be white noise vectors with the expectation values E(ek)=0 and E(wk)=0; furthermore, the covariance matrices Σy and Σw, respectively, fulfill the assumptions E[wkwlT]=Σwδk,landE[ekelT]=Σyδk,l and E[wkelT]=0, where δk,l is the delta symbol with δk,l=1 if k=l and δk,l=0 for k≠l. Equation () means that the vectors wk and el are assumed to be mutually independent.

Although the presented filter is capable of running in real time, measurements can only be assimilated with a time delay due to the latency arising from the availability of hourly GNSS observations that have been provided by the IGS data servers with at least a 1 h delay. Furthermore, downloading and processing of the raw GNSS data as well as filtering introduce additional delays. Therefore, the presented approach is called near real time and is capable of generating global VTEC products usually with less than a 1.5 h delay.

Figure shows the overall steps of the presented approach: the hourly data processing, the filtering of the hourly data and the product generation. The routines for downloading raw GNSS data, the computation of the ionospheric observable from the raw data and storing of the relevant data into a database are accomplished within the hourly data processing step. A data preprocessing module, including the following steps, runs sequentially at the beginning of every hour within the implemented software. After the acquisition of the observations, a cutoff elevation angle of 10∘ is applied to eliminate the very noisy measurements. Data arcs containing cycle slips are detected and then split into parts. The arcs with a number of observations less then a given threshold are rejected since the leveling accuracy depends on the arc length . Furthermore, an algorithm to detect and remove degraded observations is performed using a so-called “3σ” outlier test, which is carried out by screening the differences of the leveled carrier phase and the pseudo-range measurements for each arc separately, i.e., drs=Pr,Is-Lr,4s. Thus, the observations exceeding the threshold of 3σ are rejected from the data set.

The next step, the filtering of hourly data, includes the parameter estimation procedures driven by the implemented KF. Finally, the estimated ionospheric parameters are stored and utilized to generate the ionospheric products, for instance, IONEX-formatted files including global VTEC maps.

To asses the quality of the VTEC products generated by the modeling approach described before, two different evaluation methods are considered. The first validation method, called self-consistency analysis, performs a very precise sensitivity analysis using the differences in STEC observations to locally detect temporal and spatial variations around a given GNSS receiver. The second validation procedure shows the estimation quality of the VTEC maps on water surfaces; the results are compared with altimeter data acquired from the Jason-2 mission .

For the validation of our results we use the final, i.e., the post-processed, products of the IGS and its IAACs because they are widely accepted as well-established standards. Note that these final products are available with a latency of days or even weeks, whereas our results are evaluated using preprocessing strategies in near real time. We use statistical metrics, namely the RMS value, the error mean value and the standard deviation, to evaluate variations of the VTEC products with respect to reference values derived from the self-consistency analysis and the Jason-2 altimetry.

In this analysis, the VTEC products are labeled by the following standard convention for the IONEX files, including VTEC maps, as “igsg”, “codg”, “jplg”, “esag” and “upcg”, which are provided by the IGS and its IAACs, namely CODE, JPL, ESOC and UPC. In this sense, the label “dfrg” throughout this work refers to the estimated VTEC maps of the German Geodetic Research Institute – Technical University of Munich (DGFI-TUM) with a temporal resolution of the KF step size set to 5 min, whereas “d1rg” is generated from “dfrg” and comprises VTEC maps with a temporal resolution of 1 h.

A time interval between 11 August 2016 (day of year, DOY 224) and 25 August 2016 (DOY 238) covering 2 weeks of data was considered with the following geomagnetic and ionospheric conditions: the 3 h Kp index data

Kp index data, GFZ Potsdam, Germany, ftp://ftp.gfz-potsdam.de/pub/home/obs/kp-ap.

A near real-time processing framework that is capable of automated data downloading, data preprocessing, Kalman filtering and formatted product generation is presented to provide VTEC maps as well as satellite and receiver DCBs of GPS and GLONASS in near real time. The B-spline representation of global VTEC is incorporated into the Kalman filter procedure. The filter was also extended to integrate the equality constraint equations comprising the spherical and DCB-related restrictions. Coefficients of the B-spline model and the DCBs, which constitute the unknown parameters, are recursively estimated by exploiting hourly GNSS observations acquired from the IGS data centers with 1 h latency. The ionosphere observable is derived from raw GNSS code and phase measurements using the geometry-free linear combinations.

The validation of the proposed approach is carried out using GNSS data downloaded in near real time covering a time span of 2 weeks. To summarize, according to the self-consistency analysis, an RMS value of 1.81 TECU was found. The four IAACs, CODE, JPL, ESOC and UPC, as well as the IGS combination product exhibit comparable RMS errors between 1.70 and 2.00 TECU. Moreover, the Jason-2 validation shows that the RMS error achieved by the proposed method fits well with the results of the IAACs. Considering the comparisons, specific to the test period it might be concluded that the estimated VTEC products using the presented near real-time strategy shows promising initial results in terms of accuracy and overall agreement with the post-processed final products of IGS and its analysis centers, which are publicly available with several days of latency. Furthermore, the results encourage further research to improve the presented model as mentioned below.

One drawback associated with the KF is the requirement of the complete knowledge of the prior information, i.e., the process noise covariance matrix Σw, and the covariance matrix Σy of the measurement errors have to be given. The common practice of selecting these matrices manually is conducted by doing tests through multiple runs of the filter for different values of these parameters . However, the test data may not be adequate for properly defining these matrices beforehand or they can vary unpredictably throughout the time period . If the prior information is not appropriate, the filter is no longer optimal and can result in an estimation of poor quality, or even worse, the filter may diverge. To cope with such situations in ionosphere modeling, the implemented approach will be extended by adaptive methods to estimate the covariance matrices in run time as a further improvement. Moreover, the evaluations presented here show that the developed approach exhibits promising results during a period of quiet solar activity, but further tests have to be conducted using GNSS data sets covering a long time span and downloaded in near real time during time periods of high solar activity and solar events, e.g., solar flares and coronal mass ejections.

The global VTEC maps in IONEX format used in the comparisons were acquired from the Crustal Dynamics Data Information System (CDDIS) data center by the following FTP server: ftp://cddis.gsfc.nasa.gov/gnss/products/ionex/. The data of the Jason-2 altimetry mission are available via the FTP server: ftp://data.nodc.noaa.gov/pub/data.nodc/jason2/ogdr/ogdr/. The hourly available GNSS data from IGS sites were operationally downloaded in real time through mirroring to the different IGS data centers, i.e., the CDDIS (ftp://cddis.gsfc.nasa.gov/pub/gps/data/hourly/), the Bundesamt für Kartographie und Geodäsie (BKG) (ftp://igs.bkg.bund.de/IGS/nrt/), the Institut Geographique National (IGN) (ftp://igs.ensg.ign.fr/pub/igs/data/hourly) and the Korean Astronomy and Space Science Institute (KASI) (ftp://nfs.kasi.re.kr/). Furthermore, ultrarapid orbits of GPS and GLONASS satellites utilized in the data preprocessing step can be accessed through FTP servers, namely for GPS via ftp://cddis.gsfc.nasa.gov/pub/gps/products and for GLONASS via ftp://ftp.glonass-iac.ru/MCC/PRODUCTS/.