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.

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 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

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

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

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

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

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

In order to extract ionospheric information from dual-frequency GNSS
measurements, the geometry-free linear combination can be used

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

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

Global VTEC representation in a solar geomagnetic coordinate system
using different resolution levels;

The selection of appropriate resolution levels

Reconstructed VTEC maps:

The KF (

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

For the sake of clarity, the GNSS STEC measurement

The two observation vectors

Although the satellite systems GPS and GLONASS refer to the same space
geodetic technique, they are operated by different agencies with a different
design, constellation and signal structure, which can lead to different
sensitivities within the parameter estimation. To account for this fact,
instead of assigning one variance factor, an individual variance factor for
each observation group is introduced. Furthermore, it is assumed that the
vectors

The modeling approach includes constraints defined for the different groups
of unknown parameters. One group of constraints preserves the spherical
geometry

Many time-varying models for representing the ionospheric dynamics are based
on a KF approach, for
instance, the physics-based model developed by

The selection of a proper coordinate system is probably one of the most
important issues in monitoring the temporal variations of the ionosphere. The
KF problem can be solved both in a Sun-fixed

The covariance matrix

The solution of the estimation
problem as defined in Eqs. (

Once the prediction step is performed, the corrected state vector and its
covariance matrix are computed by incorporating the new allocated ionospheric
measurements as

The covariance matrix computed by
Eq. (

Two types of equality constraints were defined by
Eqs. (

The composition of the ionospheric state vector

The filter post-processing step refers to secondary tasks that are not directly related to the filter but to the generation of products. For instance, the estimated ionospheric parameters and their covariance matrix in combination with other relevant data that would be informative for diagnosing and monitoring the filter results are stored in a database.

Overall scheme for the global VTEC estimation and the product generation.

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

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.

The geographical locations and the identifiers of the receiver sites used in the dSTEC analysis.

Results of the statistical evaluations presenting the differences between the observed and computed dSTEC values at the station PIMO. The investigation covers the days DOY 224 to DOY 238, 2016: (upper panel) mean of deviations, (middle panel) standard deviations and (bottom panel) RMS values of errors in terms of TECU.

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,

SILSO data, Royal Observatory of
Belgium, Brussels,

Results of the statistical evaluations presenting the differences between the observed and computed dSTEC values, which cover the days between DOY 224 and DOY 238, 2016: (upper panel) mean of deviations, (middle panel) standard deviations and (bottom panel) RMS values of deviation in terms of TECU.

The derivation of very accurate absolute STEC values from GNSS measurements
may be a challenging procedure since the observations include the DCBs of the
receivers and the transmitting satellites. Several research groups have
provided GNSS-based solutions regarding the TEC modeling with appropriate
approaches for quality assessment; for example, see

Ground tracks of the Jason-2 altimetry mission for 16 August 2016 (DOY 229). The colors show the magnitude of VTEC as acquired from the satellite measurement system.

Ground track of the Jason-2 altimetry satellite between 00:00 and 01:00 UTC on DOY 229, 2016. The colors show the magnitude of VTEC in TECU acquired from the satellite.

In Fig.

The dual-frequency altimeter onboard the Jason-2 satellite can directly
measure in the nadir direction using two different frequencies, which allow
the
extraction of VTEC data with less effort and without applying any mapping between
STEC and VTEC

VTEC values from the six analysis centers and Jason-2 between 00:00 and 01:00 UTC on DOY 229, 2016 in TECU.

Before using the altimeter data for the comparisons, a median filter with a
window size of 20 s was applied to smooth the data. It is worth mentioning
that Jason-2 radar altimetry provides data with a higher spatial and temporal
resolution compared to the VTEC maps. Therefore, a linear interpolation in
the spatial and time domains was applied to obtain VTEC values for the
corresponding time and location of the altimetry observations. The
interpolation is performed in the Sun-fixed coordinate system between
consecutive epochs. Figure

Comparison of VTEC values acquired from the analysis centers and the DGFI-TUM solutions with Jason-2 altimetry VTEC data between DOY 224 and DOY 238, 2016; (upper panel) mean deviations, (middle panel) standard deviations and (bottom panel) RMS values of deviations in terms of TECU.

For the sake of clarity, a selected data set between 00:00 and 01:00 UTC
from Fig.

For the entire test period, the results of the comparisons in terms of daily
mean, standard deviation and RMS values as well as their overall averaged
values for the entire test period are illustrated in
Fig.

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

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:

The authors declare that they have no conflict of interest.

The authors would like to thank the following services and institutions for providing the input data: IGS and its data centers, the Center for Orbit Determination in Europe (CODE, University of Berne, Switzerland), the Jet Propulsion Laboratory (JPL, Pasadena,California, USA), the European Space Operations Centre of European Space Agency (ESOC, Darmstadt, Germany) and the Universitat Politècnica de Catalunya/IonSAT (UPC, Barcelona, Spain).

This work was supported by the German Research Foundation (DFG) and the Technical University of Munich (TUM) in the framework of the Open Access Publishing Program. Moreover, the presented models were developed in the frame of the project “Development of a novel adaptive model to represent global ionosphere information from combining space geodetic measurement systems” (ADAPIO) (German title: “Entwicklung eines neuartigen adaptiven Modells zur Darstellung von globalen Ionosphäreninformationen aus der Kombination geodätischer Raumverfahren”), which was funded by the German Federal Ministry for Economic Affairs and Energy via the German Aerospace Center (DLR, Bonn, Germany). The topical editor, K. Hosokawa, thanks the two anonymous referees for help in evaluating this paper.