The accuracy and availability of satellite-based applications, like Global Navigation Satellite System (GNSS) positioning and remote sensing, crucially depend on the knowledge of the ionospheric electron density distribution. The tomography of the ionosphere is one of the major tools for providing links to specific ionospheric corrections and studying and monitoring physical processes in the ionosphere and plasmasphere. In this work, we apply an ensemble Kalman filter (EnKF) approach for the 4D electron density reconstruction of the topside ionosphere and plasmasphere, with the focus on the investigation of different propagation models, and compare them with the iterative reconstruction technique of simultaneous multiplicative column normalized method plus (SMART

The ionosphere is the charged part of the upper atmosphere extending from about 50 to 1000 km and going over in the plasmasphere. The characteristic property of the ionosphere is that it contains sufficient free electrons to affect the propagation of trans-ionospheric radio signals from telecommunication, navigation or remote sensing satellites by refraction, diffraction and scattering. Therefore, knowledge of the 3D electron density distribution and its dynamics are of practical importance. Around 50 % of the signal delays or range errors of L-band signals used in the Global Navigation Satellite System (GNSS) originate from altitudes above the ionospheric F2 layer, consisting of topside ionosphere and plasmasphere (see Klimenko et al., 2015; Chen and Yao, 2015). So far, especially the topside ionosphere and plasmasphere, has not been well described.

The choice of the ionospheric correction model has an essential impact on the accuracy of the estimated ionospheric delay and its uncertainty. A widely used approach for ionospheric modelling is the single-layer model, whereby the ionosphere is projected onto a 2D spherical layer, typically located between 350 and 450 km. However, 2D models are usually not accurate enough to support high-accuracy navigation and positioning techniques in real time (see, for example, Odijk, 2002; Banville, 2014). More accurate and precise positioning is achievable by considering the ionosphere as a 3D medium. There are several activities in the ionosphere community aiming to describe the mean ionospheric behaviour by the development of 3D electron density models based on long-term historical data. Two widely used models are the International Reference Ionosphere (IRI) model (see Bilitza et al., 2011) and the NeQuick model (see Nava et al., 2008).

Since those models represent a mean behaviour, it is essential to update them by assimilating actual ionospheric measurements. There have been a variety of approaches developed and validated for ionospheric reconstruction by the combination of actual observations with an empirical or a physical background model. Hernandez-Pajares et al. (1999) present one of the first GNSS-based data-driven tomographic models, which considers the ionosphere as a grid of 3D voxels, and the electron density within each voxel is computed as a random walk time series. The voxel-based discretization of the ionosphere is further used, for instance, in Heise et al. (2002), Wen et al. (2007), Gerzen and Minkwitz (2016), Gerzen et al. (2017) and Wen et al. (2020). These authors reconstruct the 3D ionosphere by algebraic iterative methods. An alternative is to estimate the electron density as a linear combination of smooth and continuous basis functions, like, for example, spherical harmonics (SPHs; Schaer, 1999), B-splines (Schmidt et al., 2008; Zeilhofer, 2008; Zeilhofer et al., 2009; Olivares-Pulido et al., 2019), B-splines and trigonometric B-splines (Schmidt et al., 2015), B-splines and Chapman functions (Liang et al., 2015, 2016), and empirical orthogonal functions and SPHs (Howe et al., 1998).

Besides the algebraic methods, other techniques benefitting from information on spatial and temporal covariance information, such as optimal interpolation, the Kalman filter, 3D and 4D variational techniques and kriging, are applied to update the modelled electron density distributions (see Howe et al., 1998; Angling et al., 2008; Minkwitz et al., 2015 and 2016; Nikoukar et al., 2015; Olivares-Pulido et al., 2019).

Moreover, there are approaches based on physical models which combine the estimation of the electron density with physically related variables, such as neutral winds or the oxygen / nitrogen ratio (see Wang, et al., 2004; Scherliess et al., 2009; Lee et al., 2012; Lomidze et al., 2015; Schunk et al., 2004, 2016; Elvidge and Angling, 2019).

In general, the majority of data available for the reconstruction of the ionosphere and plasmasphere are slant total electron content (STEC) measurements, i.e. the integral of the electron density along the line of sight between the GNSS satellite and receiver. Often, STEC measurements provide limited vertical information, and hence, the modelling of the vertical the electron density distribution is hampered (see, for example, Dettmering, 2003). The estimation of the topside ionosphere and plasmasphere poses a particular difficulty since direct electron density measurements are rare, and low plasma densities at these high altitudes contribute only marginally to the STEC measurements. Ground-based STEC measurements are especially dominated by electron densities within and below the characteristic F2 layer peak. Consequently, information about the plasmasphere is difficult to extract from ground-based STEC measurements (see, for example, Spencer and Mitchell, 2011). Thus, in the presented work, we concentrate on the modelling of the topside part of the ionosphere and plasmasphere and utilize only the space-based STEC measurements.

In this paper, we introduce an ensemble Kalman filter (EnKF) to estimate the topside ionosphere and plasmasphere based on space-based STEC measurements. The propagation of the analysed state vector to the next time step within a Kalman filter is a key challenge. The majority of the approaches, working with EnKF variants, use physics-based models for the propagation step (see, for example, Elvidge and Angling, 2019; Codrescu et al., 2018; Lee et al., 2012). In our work, we investigate the question of how the propagation step can be realized if a physical model is not available or if the usage of a physical model is rejected as computationally time consuming. We discretize the ionosphere and the plasmasphere below the GNSS orbit height by 3D voxels, initialize them with electron densities calculated by the NeQuick model and update them with respect to the data. We present different methods for how to perform the propagation step and assess their suitability for the estimation of electron density. For this purpose, a case study of quiet and perturbed ionospheric conditions in 2015 is conducted to investigate the capability of the estimates to reproduce assimilated STEC and to reconstruct independent STEC and electron density measurements.

We have organized the paper as follows: Sect. 2 describes the EnKF with the different propagation methods and the generation of the initial ensembles by the NeQuick model. Section 3 outlines the validation scenario with the applied data sets. Section 4 presents the obtained results. Finally, we conclude our work in Sect. 5 and provide an overview of the next steps.

Information about the STEC along the satellite-to-receiver ray path

The discretization of the ionosphere by a 3D grid and the assumption of a
constant electron density function within a fixed voxel allows the
transformation of Eq. (1) into a linear system of equations as follows:

As a regularization of the inverse problem in Eq. (2), a background model often provides the initial guess of the state vector

We apply EnKF to solve the inverse problem defined in Sect. 2.1. Evensen (1994) introduces the EnKF as an alternative to the standard Kalman filter (KF) in order to cope with the non-linear propagation dynamics and the large dimension of the state vector and its covariance matrix. In an EnKF, a collection of realizations, called ensembles, represent the state vector

Let

We define the ensemble covariance matrix

In the analysis step of the EnKF, the a priori knowledge of the state vector

For the propagation of the analysed solution to the next time step, we test
different propagation models described in Sect. 2.4. In order to generate the initial ensembles

In this section, we introduce different models to propagate the analysed
solution to the next time step. With all of them, we propagate the ensembles
20 min in time. Generally, these propagation models can be described as

Note that, beyond the presented methods, we had additionally tested a propagation model based on persistence, i.e.

The method rotation assumes that, in geomagnetic coordinates, the ionosphere
remains invariant in space while the Earth rotates below it (see Angling and
Cannon, 2004). Thus, we propagate the analysed ensemble

To calculate rot

Here we assume the electron density differences between the voxels of the
analysis and the background model to be a first-order Gauss–Markov sequence.
These differences are propagated in time by an exponential decay function
(see Nikoukar et al., 2015; Bust and Mitchell, 2008; Gerzen et al., 2015).

Note that, similar to the method described here, we also tested the application of rot

For the third method, we define the propagation model as a combination of
the propagation models described in the previous subsections, in particular, as the following:

The systematic error

Thereby,

The matrix

In order to generate the ensembles, we vary the F10.7 input parameter of the
NeQuick model (see Sect. 2.2). First, we analysed the sensitivity of the NeQuick model to F10.7. Based on the results, we calculate a vector

An example of the variation of the generated ensembles is provided by
Fig. 1. Particularly, we show, in this figure, the distribution of the differences between the ensemble of electron densities

The distribution of the ensemble residuals for a chosen altitude
and selected universal times (UTs), for all latitudes and longitudes, with panel

In order to provide a benchmark for the described methods, we apply SMART

For SMART

Within this study, the EnKF with different propagation methods is
applied and validated for the tomography of the topside ionosphere and
plasmasphere. Two periods with quiet (DOY 041–059 in 2015) and perturbed (DOY 074–079 in 2015) ionospheric conditions are analysed. In this scope, we
investigate the ability to reproduce assimilated STEC and to estimate
independent STEC measurements and in situ electron density measurements of
the Swarm Langmuir probes (LPs). In addition, we apply the tomography
approach SMART

We estimate the electron density over the entire globe, with a spatial
resolution of 2.5

We use the solar radio flux of F10.7, the global planetary 3 h index Kp and the geomagnetic disturbance storm time (DST) index to characterize the
ionospheric conditions during the periods of DOY 041–059 and DOY 074–079 in
2015. In the February period (DOY 041–059 in 2015), the ionosphere is evaluated as being quiet, with F10.7 solar flux unit (sfu) range between 108 and 137s, a Kp index below 6 (on 2 d between 4 and 6; below 4 during the rest of the period) and DST values between 20 and

As input for the tomography approaches and for the validation, we use
space-based calibrated STEC measurements of the following low earth orbit (LEO) satellite missions: COSMIC, Swarm, TerraSAR-X, MetOpA and MetOpB and GRACE. Please note that, in 2015, the orbit height of the COSMIC and MetOp satellites was

COSMIC, available at:

Swarm, available at:

TerraSAR-X, available at:

MetOpA and MetOpB, available at:

GRACE, available at:

The STEC measurements of the Swarm satellites are available at

The LPs on board the Swarm satellites provide in situ electron density
measurements, with a time resolution of 2 Hz. For the present study, the LP
in situ data are acquired from

Lomidze et al. (2018) assess the accuracy and reliability of the LP data (December 2013 to June 2016) by nearly coincident measurements from low- and mid-latitude incoherent scatter radars, low-latitude ionosondes and COSMIC satellites, which cover all latitudes. The comparison results for each Swarm satellite are consistent across these different measurement techniques. The results show that the Swarm LPs underestimate the electron density systematically by about 10 %.

In this section, the different methods are presented with the following
colour code: blue for the method rotation, green for the method exponential
decay, light blue for the method rotation with exponential decay and magenta
for NeQuick and red for SMART

Rotation with exponential decay reconstructed electron density, represented by layers at different heights between 490 and 827 km

At the end of each EnKF analysis step, we have, for each of the considered
methods, 100 ensembles representing the electron density values within the
voxels. The EnKF-reconstructed electron densities are then calculated as the
ensemble mean. Figure 2a and b present the electron densities reconstructed by the method rotation with exponential decay, i.e.

Method rotation reconstructed electron density represented by
layers at different heights between 490 and 827 km

Figure 3 displays the electron density layers
reconstructed by the method rotation, i.e.

Reconstructed minus NeQuick-modelled electron density represented
by layers at different heights between 490 and 827 km.

In the following, we discuss Figs. 4–7 in order to understand the deviations between the reconstructions produced by the different methods. In
Fig. 4, the differences between the reconstructed and the modelled electron densities, i.e.

Differences between reconstructed and NeQuick-modelled electron
density in percent, represented by layers at different heights between 490
and 827 km.

The locations of the LEO satellites for the STEC measurements used in the reconstruction.

Taking into account the differences in Fig. 5, for instance around 120

In order to supplement the understanding of the differences between the
propagation methods, Fig. 7a, c and e present the differences

In conclusion, the different behaviour of the propagation methods, in combination with the sparse measurement geometry, might serve as an explanation for the substantial differences observed in the VTEC maps shown in Figs. 2 and 3.

In this section, we check the ability of the methods to reproduce the
assimilated STEC measurements. For that purpose, we calculate STEC along a
ray path,

Furthermore, we consider the RMS of the deviations, in detail, as follows:

To calculate

Plausibility check of the residuals calculated as measured STEC minus estimated STEC.

Figure 8a and b depict the distribution of the residuals for the quiet period and for the perturbed period respectively. The corresponding residual median, standard deviation (SD) and root mean square (RMS) values are also presented in the Fig. 8. It is worth mentioning here that, during the quiet period, the measured STEC is below 150 total electron content unit (TECU). For all DOYs of the perturbed period, except for DOY 076, the measured STEC is below

The NeQuick model seems to underestimate the measured topside ionosphere and
plasmasphere STEC during both periods. During both periods, SMART

Interestingly, the median values are higher during the quiet period, while the SD values are on the same level when compared between perturbed and quiet periods. The reason, therefore, is probably that the assimilated STEC values have, on average, lower magnitude during the days in the perturbed period compared to those during the quiet period (which explains the lower median), except for the storm on DOY 076, while on DOY 076 they are significantly higher (which explains the comparable SD).

Plausibility check for the quiet period.

Plausibility check for the perturbed period.

Figures 9 and 10 plot

Plausibility check for the quiet period. Distributions of the
delta STEC

Plausibility check for the perturbed period. Distributions of
the delta STEC

Figures 11 and 12 present the distribution of

In order to validate the methods with respect to their capability to estimate independent STEC, the LEO satellites of Swarm A and GRACE have been used. The STEC measurements of these satellites are not assimilated by the tested methods.

For each of the three LEO satellites, the residuals between

In general, for the quiet period, the STEC measurements of Swarm A vary below 105 TECU, and they are below 170 TECU for the second period. For the GRACE satellite, the STEC measurements are below 282 TECU for the quiet period and below 264 TECU for the second period.

Histograms of the STEC residuals

Histograms of the STEC residuals

Figures 13 and 14 display the histograms of the STEC residuals during the quiet period for Swarm A and GRACE respectively. Presented are the distributions of the residuals dTEC and the absolute residuals

Histograms of the STEC residuals

Histograms of the STEC residuals

Again, the NeQuick model seems to underestimate the measured STEC during
both periods for GRACE and Swarm A satellites. Compared to the NeQuick
model, during both periods the methods of SMART

Regarding the STEC of Swarm A, the lowest residuals and the most reduction,
in comparison to the NeQuick model, are achieved by SMART

Regarding the STEC of GRACE during the quiet period, the lowest residuals
and the most reduction in comparison to the background, are achieved by the
exponential decay, followed by SMART

Comparing the quiet and storm conditions, in general an increase in the RMS and SD of the Swarm A residuals is observable for the NeQuick model and all tomography methods regarding both satellites. This is not the case for the GRACE residuals.

In this section, we further extend our analyses to the validation of the
methods with independent LP in situ electron densities of the three Swarm
satellites. According to the locations of the LP measurements, the estimated
electron density values are interpolated (by a 3D interpolation, using the
MATLAB built-in function of scatteredInterpolant.m) from the 3D electron density reconstructions. For each satellite, the measured electron density

Figure 17 depicts the distribution of the residuals d

Validation with LP data. Distribution of Swarm A, B and C (separated) electron density residuals for the quiet period.

Validation with LP data. Distribution of the Swarm absolute and absolute relative electron density residuals for the quiet period.

The electron densities of the NeQuick model are, in median, slightly higher
than the LP in situ measurements for all three satellites during both
periods. The median and SD values for the

Validation with LP data. Distribution of the Swarm A, B and C (separated) electron density residuals for the perturbed period.

Validation with LP data. Distribution of the Swarm absolute and absolute relative electron density residuals for the perturbed period.

The methods of SMART

In this paper, we assess three different propagation methods for an ensemble
Kalman filter approach in the case that a physical propagation model is not
available or is discarded due to the computational burden. We validate these
methods with independent STEC observations of the satellites of GRACE and Swarm A and with independent Langmuir probes data of the three Swarm satellites. The methods are compared to the algebraic reconstruction method SMART

Overlooking all the validation results, the methods of SMART

The plausibility check in Sect. 4.2 shows that all methods successfully reduce the STEC residuals and provide better results than the background model. SMART

Although the EnKF with the method rotation reproduces the assimilated STEC data well, less accurate estimates are obtained in the validation with independent data. We assume that this has two main reasons. First, as the only propagation method, rotation is not anchored by the background model. Second, the number of the assimilated measurements is low compared to the number of unknowns, and the available measurements are unevenly distributed and angle limited. Both together could lead to increased deviations in the estimates of the truth.

The methods SMART

Concerning the estimation of independent electron densities of the Langmuir
probes, SMART

Another approach for improving the reconstructions could be to precondition the background model, for example, in terms of F2 layer characteristics or the plasmapause location (see, for example, Bidaine and Warnant, 2010; Gerzen et al., 2017).

To obtain a comprehensive final impression of the performance of the investigated methods and to gain insight into the ability of the methods to correctly characterize the shapes of the electron density profiles, we intend to continue the validation of the methods with additional, independent measurements of the plasmasphere and topside ionosphere, for example, coherent scatter radar data.

The data that were used can be found in Sect. 3.3 or are available upon request.

TG contributed the main ideas to the methods presented in this article, implemented the EKF, carried out the validation and wrote most of the sections. DM helped to fine-tune the propagation methods, took care of the operation of the background model and contributed sections to the final article. All co-authors helped to interpret the results, read and comment on the article.

The authors declare that they have no conflict of interest.

We thank NOAA (

This study was performed as part of the MuSE project (grant no. 273481272) and funded by the DFG as a part of the Priority Programme of DynamicEarth (grant no. SPP-1788).

This paper was edited by Steve Milan and reviewed by two anonymous referees.