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

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.

The properties of the atmosphere can be described by means of different variables, e.g., the temperature or the charge state. In the case of temperature, we distinguish between the troposphere (up to a height of about 15 km), the stratosphere (about 15 to 50 km), the mesosphere (about 50 to 90 km), the thermosphere (about 90 to 800 km) and the exosphere (above 800 km) using increasing height above the Earth's surface. In the case of the charge state, the atmosphere is split into the neutral atmosphere (up to a height of 80 km), the ionosphere (about 80 to 1000 km) and the plasmasphere (above 1000 km)

The ionosphere is mostly driven by the Sun; extreme UV (EUV), X-ray and solar particle radiation cause ionization processes. In geodesy, the main ionospheric impact is the influence of free electrons on radio wave propagation. This effect mainly depends on the signal frequency, i.e., the ionosphere is a dispersive medium

Observations of space geodetic techniques, such as the global navigation satellite systems (GNSS) and the Doppler Orbitography and Radiopositioning Integrated by Satellite (DORIS) tracking system as well as satellite altimetry and ionospheric radio occultation (IRO) are based on electromagnetic signal propagation; thus, they are disturbed by the ionosphere. Most of the techniques are not directly sensitive to the electron density, but to the integrated effect along the ray path. In Eq. (

The vertical total electron content (VTEC),

Equations (

Combining Eqs. (

if VTEC is given from an ionospheric model, the delay

or if the delay

There are several modeling strategies for generating GIMs; the most prominent approach is based on spherical harmonics (SH) and was introduced by

Generally, we distinguish between GIMs provided as “final”, “rapid”, “near real-time” (NRT) or “real-time” (RT) products. This classification is based on the latency of the underlying input data. For final products, for instance, only post-processed observations and orbits are used, whereas NRT products are based on rapid orbits and observations with a latency of some minutes up to a few hours.
GIMs are typically provided with a temporal resolution of 1 or 2 h and with a spatial resolution of

VTEC variations basically follow annual, seasonal, diurnal and semidiurnal periods. Earthquakes or incidental natural hazards can also cause small but visible signatures

As already confirmed by

To the knowledge of the authors, the spatial resolution of GIMs has not been investigated in detail to date. Most of the GIMs are based on series expansions in terms of SHs with a maximum degree of

Global distribution of the IPPs from GPS (red dots) and GLONASS (blue stars) measurements for 6 September 2017 collected within a 10 min interval between 12:55 and 13:05 UT. The regional maps at the top are “zoom-ins” of Europe and Indonesia.

By increasing the temporal resolution of the GIMs, the number of observations supporting the individual maps decreases. The two “zoom-in” maps at the top of Fig.

It is a well-known fact that SHs as global basis functions are not suitable for representing unevenly, globally distributed data. Consequently, in such cases, a series expansion in terms of localizing basis functions is more appropriate. In the following, we apply tensor products of polynomial and trigonometric B-splines as localizing 2-D basis functions. Besides the localizing features, B-splines additionally generate a multi-scale representation (MSR), also known as multi-resolution representation (MRR). The basic feature of a MSR is to split a target function into a smoothed, i.e., low-pass-filtered version, and a number of detail signals, i.e., band-pass-filtered versions via successive low-pass filtering

In this study, we compare global VTEC maps based on series expansions in terms of both globally defined SHs and localizing B-spline functions, including the MSR with respect to the spectral content. For this purpose, we use the SH degree as the common measure for the spectral content of a spherical signal. In detail, we study the interrelations between the SH degree, the spatial sampling intervals of the input data and the resolution levels of B-spline expansions. In addition, we discuss the influence of different temporal resolutions of the GIMs. For the estimation of the unknown series coefficients of the B-spline expansion, we use a Kalman filter (KF) procedure as explained by

The paper is outlined as follows: in Sect.

The 3-D signal VTEC

Note that we do not distinguish between geographical and geomagnetic spherical coordinates for latitude

In the following (Sect.

In the SH approach, the observation equation, Eq. (

As the VTEC observations

At DGFI-TUM we rely on B-splines as basis functions for ionosphere modeling, as they are (1) characterized by their localizing feature and (2) they can be used to generate a MSR. For VTEC modeling we rewrite Eq. (

To decompose VTEC into its spectral components via the MSR in Sect.

In the following, we apply polynomial quadratic B-splines

As can be seen from Fig.

Polynomial B-splines of level

For modeling the longitudinal variations of VTEC trigonometric B-splines

Following

The second option was introduced by

In Sect.

Given the numerical values 1 to 6 for the B-spline levels

Numerical values for the B-spline levels

From the spectral point of view the six inequalities from Eq. (

If the global sampling intervals

With a specified numerical value for

If the processing time of VTEC maps has to be considered, the level values

As already mentioned in the introduction, most of the GIMs produced by the IAACs are based on series expansions in SHs up to a maximum degree of

The VTEC GIMs of the IAACs are usually provided with a spatial resolution of

In order to calculate a VTEC value VTEC

Schematic representation of the four-point spatial interpolation to calculate the VTEC value at

Note, by applying the interpolation formula (

the chosen model approach, e.g., the SH or the B-spline expansion can be used directly to calculate VTEC values at any arbitrary point

the resolution intervals

For the calculation of a VTEC value VTEC

The previously described interpolation methods allow for the calculation of VTEC values VTEC

The B-spline functions as introduced in Sect.

Neglecting the time dependency, the B-spline approach Eq. (

In the context of the MSR the vectors

With

Following the argumentation of

The numerical entries of the

In Eq. (

A 2-D MSR of the signal

The previously described MSR refers to a successive low-pass filtering of the target function

A 1-D MSR of the signal

Besides the representation of a signal, e.g., VTEC, by means of approximations on different resolution levels with respect to latitude and longitude, the MSR also allows for the utilization of a powerful “data compression procedure”, as the numerical value of a large number of wavelet coefficients is generally close to zero depending on the signal structure

To estimate the elements of the unknown

In the linear formulation the Kalman filter consists (1) of the state equation

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

Using the estimations

The previously explained procedure allows for the dissemination of two products to the users:

The two products, i.e., the set of coefficients or the VTEC grid values reflect the two strategies of dissemination. In case of a SH expansion for RT applications as introduced in Sect.

For Product 2, the VTEC grid values Eq. (

In the following, the described modeling approach developed at DGFI-TUM is applied to real data. To be more specific, we use GPS and GLONASS NRT data in hourly blocks and apply ultra-rapid orbits. A detailed explanation of the data preprocessing and the setup of the full observation equations is presented by

List of GIM products used in this paper. Information on names, types and latencies are taken from the following references: (1)

For the evaluation of the data we have to define an appropriate coordinate system. Here we follow the standard procedure and use a sun-fixed geomagnetic coordinate system. To be more specific, we identify the coordinate system

While the scaling coefficients Eq. (

For validation purposes we rely on the dSTEC analysis which is currently regarded as the standard method for the quality assessment of VTEC models

This analysis method is based on the calculation of the difference between STEC observations STEC

Figure

Global distribution of the IPPs from GPS (red dots) and GLONASS (blue stars) measurements for 6 September 2017, at 13:00 UT.

The covariance matrices

Figure 8a shows with

While Fig. 8a and b show the results of Product 1 in the GSM system, Fig.

Estimated scaling coefficients

Note that for the visualization of VTEC and their standard deviations in Fig.

From the comparison of Fig.

Figure 8a, i.e., the plot of the set of scaling coefficients

We denote the two Multi-Scale Products 2 as “ophg” and “oplg”', where the first symbols refer to the OPTIMAP processing software, which was developed within a third-party funded project (see Acknowledgements). The “

As mentioned in the context of Table

Figure

VTEC maps “codg”

To numerically assess the comparability we apply the dSTEC analysis described in Sect.

Distribution of the 10 IGS receiver stations used for the dSTEC analysis.

The chosen set should not be used within the computation of the VTEC maps. Fulfilling both requirements at the same time is difficult and, thus, the set of stations shown in Fig.

RMS values for the “codg” (green) and “o1lg” (blue) products computed at the 10 receiver stations shown in Fig.

As can be seen, the RMS values vary between 0.3 and 1.6 TECU. By comparing the RMS values of “o1lg” with a mean RMS value of 0.80 TECU and “codg” with a mean RMS of 0.77 TECU we can state that the quality of these two products is very close to each other.

The results indicate that the overall quality of the NRT product “o1lg” is comparable with that of the final product “codg” including the developed and implemented preprocessing strategies and steps of the GNSS data (see Table

The two multi-scale VTEC products “ophg” and “oplg” have been introduced in the two equation blocks Eqs. (

Multi-scale VTEC products for solar storm events: high-resolution VTEC map “ophg” for 8 September 2017

As already mentioned in the context of Eq. (

Next, we focus on the solar storm during September 2017 and study the temporal sampling intervals of different GIMs. In summary, we distinguish between six products of different spectral content and different sampling intervals.

RMS values for the “o2hg”, “o1hg”, “othg”, “o2lg”, “o1lg” and “otlg” products computed at the 10 receiver stations shown in Fig.

Figure

The differences in the RMS values of the first three products, “o2lg”, “o1lg” and “otlg”, are caused by their different sampling intervals. Comparing the mean RMS values of 0.92 and 0.80 TECU for “o2lg” and “o1lg”, respectively, we find a relative improvement of approximately 13.0 %. By decreasing the sampling from

Relative improvements (in percentage) for a downsizing of the sampling interval of the “o2lg”, “o1lg”, “otlg”, “o2hg”, “o1hg” and “otlg” products.

Comparing the RMS values 0.90, 0.72 and 0.68 TECU of the “o2hg”, “o1hg” and “othg” products, respectively, we find relative improvements of 20 % and 24.4 % by downsizing the sampling interval from 2 to 1 h and finally to 10 min. A summary of the relative improvements is given in Table

In the next step, we compare the quality of the multi-scale products “ophg” and “oplg” directly. First, we compare “o2lg” with “o2hg” and obtain an improvement of approximately 2.2 % . In the same manner, we compare “o1lg” with “o1hg” and “otlg” with “othg” and find that improvements of 10.0 % and 11.7 % can be achieved, respectively. Table

Results (in percentage) of the comparisons of the high-resolution products “ophg” with the low-resolution products “oplg”. Positive (bold) numbers mean an improvement, and negative (italic) values represent a reduction in the quality.

From the investigations in Sect.

As the “othg” product outperforms all other products used in the previous sections we now compare it with UPC's “uqrg” product

RMS values for the “uqrg” and “othg” products computed at 9 IGS receiver stations during September 2017. The values in parentheses in the legend are the average RMS values over all 9 receiver stations for the entire test period between 1 and 30 September 2017.

As can be seen, the RMS values vary between 0.5 and 1.8 TECU but are mostly below 1.0 TECU. The dominant RMS value of “uqrg” at the “CHPI” station reduces its quality significantly. If “CHPI” is neglected, the mean RMS value of “uqrg” decrease to 0.59 TECU. Summarizing these investigations, we can state that the overall quality of the two products is very similar. Considering the fact that “othg” is a NRT product with a latency of less than 3 h, it also outperforms “uqrg” which is a rapid product with a latency of around 1 d (see Table

This paper presents an approach to model VTEC solely from NRT GNSS observations by generating a MSR based on B-splines; the unknown model parameters are estimated by means of an KF. Based on this approach, a number of products have been created which differ both in their spectral content and in their temporal resolution. From our investigations we state that the MSR provides B-spline models comparable to the standard GIMs of the IAACs, mostly based on SH expansions up to degree

Besides the facts, that our models can handle data gaps due to the utilization of localizing basis functions, the application of a KF to include a dynamic prediction procedure and the use of the MSR to create products of different spectral content at the same time, it should be mentioned that DGFI-TUM's products

are based on NRT GNSS observations only, i.e., are using input data with a latency of less then 3 h (in contrast, “codg” relies on post-processed data with a latency of larger than 3 weeks, and “uqrg” relies on rapid data with a latency of at least 1 d; cf. Table

rely on specially developed software modules (cf. Fig.

and can be disseminated to users with a delay of 2–3 h.

DGFI-TUM's processing modules, including (blue boxes) the download and preprocessing module for GNSS observations, the modeling module by means of B-splines, MSR and Kalman filtering (orange boxes) with possible output as Product 1 and Product 2 (yellow boxes) and the validation module.

In general, the dissemination of these products to users can be undertaken in two different ways: based on estimated scaling coefficients (Product 1) or by calculated VTEC grid values (Product 2). For RT applications, however, the dissemination in terms of Product 1 is preferred, in particular the usage of the RTCM format. In the scope of the developments in the recent years, RT applications have become more important, e.g., in unmanned or autonomous vehicle development; thus, the restriction of the RTCM message to allow only for SH coefficients needs to urgently be discussed. Particularly from the point of view that there are also other modeling methods, a modification of the RTCM format would be appropriate. The MSR allows for significant data compression to be obtained due the step-wise downsampling of the scaling coefficients according to the pyramid algorithm. Details represented in the signal

The results presented encourage the further development of high accuracy VTEC maps. By extending the models by a fourth dimension, i.e., modeling of the electron density directly, inaccuracies due to the mapping function can be avoided. To model the vertical structure of the electron density, additional observations have to be incorporated, e.g., from DORIS, satellite altimetry and ionospheric radio occultations. This would mitigate the inhomogeneity of the data distribution and, in turn, even higher levels of the B-spline expansion can be chosen.

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 concept for the paper was proposed by AG and discussed with all co-authors. AG compiled the figures and wrote the paper with assistance from MS. The paper and figures were reviewed by all co-authors.

The authors declare that they have no conflict of interest.

We are grateful to the Bundeswehr GeoInformation Centre (BGIC) and the German Space Situational Awareness Centre (GSSAC) for funding the “Operational tool for ionosphere mapping and prediction” (OPTIMAP) project. The approach presented was developed within this framework.

The authors express their thanks to the following services and institutions for providing the input data: IGS and its data centers, the Center for Orbit Determination in Europe (CODE, Berne, Switzerland) and the Universitat Politècnica de Catalunya/IonSAT (UPC, Barcelona, Spain). Furthermore, the authors acknowledge the developers of the Generic Mapping Tools (GMT) which was primarily used for generating the figures in this work.

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.

This paper was edited by Dalia Buresova and reviewed by Ilya Edemskiy and one anonymous referee.