The accuracy and availability of satellite-based applications like GNSS positioning and remote sensing crucially depends on the knowledge of the ionospheric electron density distribution. The tomography of the ionosphere is one of the major tools to provide link specific ionospheric corrections as well as to study and monitor physical processes in the ionosphere.
In this paper, we introduce a simultaneous multiplicative column-normalized
method (SMART) for electron density reconstruction. Further, SMART
The comparative case study is implemented over Europe during two periods of
the year 2011 covering quiet to disturbed ionospheric conditions. In
particular, the performance of the methods is compared in terms of the
convergence behaviour and the capability to reproduce sTEC and electron
density profiles. For this purpose, independent sTEC data of four IGS
stations and electron density profiles of four ionosonde stations are taken
as reference. The results indicate that SMART significantly reduces the
number of iterations necessary to achieve a predefined accuracy level.
Further, SMART
The ionosphere is the upper part of the atmosphere extending from about 50 to 1000 km and going over into the plasmasphere. The characteristic property of the ionosphere is that it contains sufficient free electrons to affect radio wave propagation. The electron density distribution is driven mainly by solar radiation, particle precipitation and charge exchange; it varies widely in both space and time. Thus, real-time determination of the ionospheric electron density distribution becomes important from the satellite applications perspective as well as for understanding ionosphere dynamics. Global Navigation Satellite System (GNSS) observations, which provide the total electron content (TEC) along a receiver-to-satellite ray path, have become one of the major tools for ionospheric sounding.
The ionosphere community carries out several activities that are aimed at describing the ionospheric behaviour by developing electron density models, based on historical GNSS data and other ionospheric measurements. For instance, the International Reference Ionosphere model (IRI; see Bilitza, 2001; Bilitza and Reinisch, 2008) describes empirically monthly averages of the electron density and temperature, based on historical ground- and space-based data. NeQuick (see Nava et al., 2008) is also an empirical model driven mainly by solar activity level and ionospheric F2 layer parameters, which are computed based on historical vertical sounders data (see ITU-R 1995, Sect. 3.3).
Since those models represent a median ionospheric behaviour, ingestion of actual ionospheric measurements is essential to update them. Several approaches have been developed and validated for ionospheric reconstruction by a combination of actual observations with an empirical or a physical background model. Galkin et al. (2012) present a method to update the IRI coefficients, using vertical sounding observations of a 24 h sliding window. Bust et al. (2004) use a variational data assimilation technique to update the background, combining the observations and the associated data error covariances. Also, other techniques, taking advantage of spatial and temporal covariance information, such as optimal interpolation, Kalman filter and kriging, have been applied (see e.g. Angling and Cannon, 2004; Angling and Khattatov, 2006; Angling et al., 2008; Gerzen et al., 2015; Minkwitz et al., 2015; Pérez, 2005) to update the modelled electron density distributions. Moreover, there are approaches based on physical models that combine the estimation of electron density with physical related variables, such as neutral winds or oxygen/nitrogen ratio (see Schunk et al., 2004; Wang et al., 2004).
In the beginning and even now, when looking for computer resource-saving approaches, algebraic iterative methods have been used to ingest data into background models, e.g. derivatives of the Algebraic Reconstruction Technique (e.g. ART, MART), column-normalized methods (e.g. SART) and the successive correction method (SCM) (see Daley, 1991; Heise et al., 2002; Wen et al., 2007, 2008; Li et al., 2012; Pezzopane et al., 2013). Those methods are working without the modification of the model coefficients but by updating the background in the area surrounding the available measurements. Lorenc (1986) discusses the differences in mathematical framework and implementation for the majority of the above-mentioned methods.
In this paper, we introduce a multiplicative column-normalized method,
called SMART. Further to this, SMART
Information about the total electron content, along the
receiver-to-satellite ray path
By discretization of the ionosphere into a 3-D grid and assuming the electron
density function to be constant within a fixed voxel, we can transform Eq. (1) to a linear system of equations (LSE):
In the following chapters, algebraic iterative methods are presented to
solve this LSE. All the methods work with an initial guess
Originating with Kaczmarz (1937), the ART algorithm was suggested for medical
computerised tomography by Gordon et al. (1970). ART works iteratively,
starting with the initial guess
The relaxation parameter
When applied to a consistent LSE, ART was shown to converge to the minimum-norm least-squares solution for relaxation parameters equal to 1. The behaviour of this algorithm for inconsistent systems, when relaxation parameters are allowed, is studied by Censor et al. (1983) and Eggermont et al. (1981). Atkinson and Soria (2007) compare the ART method to other algebraic methods (e.g. with MART, AART, SIRT).
The simultaneous Algebraic Reconstruction Technique (SART) is a kind of
refinement of ART towards a column-normalized method. SART is successfully
used for tomographic problems (see Andersen and Kak, 1984; Kunitsyn and
Tereshchenko, 2003). The
The use of the weighted mean of the deviations is the major refinement of the column-normalized methods in comparison with the classical row action methods, such as ART, which innovate separately for each ray path.
In this chapter, the simultaneous multiplicative column-normalized method
SMART is introduced. The
Equation (6) can be interpreted as follows: for a
voxel
Until now we have studied the convergence behaviour of SMART empirically but have
not proved the convergence. An advantage of the multiplicative methods, such
as SMART, is that they automatically guarantee non-negative estimates of the
We developed SMART
Thereafter, assuming electron densities covariance between the ray path
intersected voxels and those not intersected by any TEC ray path, an
extrapolation is done from intersected to not intersected voxels. For this
purpose, exactly one iteration of a 3-D SCM (see Kalnay, 2011; Daley, 1991)
is applied:
The horizontal and especially the vertical correlation lengths of the
ionospheric electron densities have not been completely known until now. In the
algorithm developed here, these key parameters are chosen empirically. But,
we are currently working on methods that facilitate a better estimation of
the correlation lengths for the 3-D electron densities (see Minkwitz et al.,
2015). The ratio of error variances
The solar radio flux index F10.7 (left panel) and the global planetary 3 h index Kp (right panel) for the periods DOY 000–039/2011 (blue) and DOY 282–319/2011 (black). The investigated periods are highlighted in bold font.
The methods outlined here are developed and tested for the ionosphere tomography during two contrasting periods of the year 2011, one with quiet and the other with disturbed ionospheric conditions. In the following sections the chosen periods and reconstruction data base are described in detail.
Two periods of the year 2011 are selected for assimilation and case studies: DOYs 009–022 (January) and 294–298 (21–25 October). The geomagnetic and solar activities during these two periods are indicated in Fig. 1. The right-hand panel shows the global planetary 3 h index Kp as a measure of geomagnetic activity. The left-hand panel presents the variation of the solar radio flux at 10.7 cm wave length (F10.7 index), which serves as indicator of solar activity.
For a fuller understanding of the temporal evolution of the ionosphere, the
panels cover not just the targeted periods, but also a few more days
before and after these periods. The days investigated within this study are
displayed in bold font. The data have been acquired from the Space Physics
Interactive Data Resource of NOAA's National Geophysical Data Center (SPIDR)
and the World Data Center for Geomagnetism (WDC) Kyoto. According to Suard
et al. (2011), the ionospheric conditions can be assessed as quiet during
DOY 009–022 (blue line) and as disturbed during DOY 294–298 (black line)
with F10.7 between 130 and 170 and a severe geomagnetic storm on DOY 297–298
with a Kp index above 7. Also the geomagnetic index DST indicates a
geomagnetic storm during the night from DOY 297 to DOY 298, with DST values
below
In this study we apply the described methods to reconstruct the electron
density in the extended European region covering the geographic latitudes
To regularise the inverse problem in Eq. (2), the initial guess for the ionosphere tomography by algebraic methods is usually calculated by a background model. Therefor an arbitrary electron density model can be deployed. In this study we apply the three-dimensional NeQuick model version 2.0.2, released in November 2010.
The NeQuick model was developed at the International Centre for Theoretical Physics (ICTP) in Trieste/Italy and at the University of Graz/Austria (see Hochegger et al., 2000; Radicella and Leitinger, 2001; Nava et al., 2008). The vertical electron density profiles are modelled by parameters such as peak ionisation, peak height and semi-thickness, deduced from the ITU-R models (see ITU-R, 1995). We use the daily F10.7 index to drive the NeQuick model.
As mentioned in Sect. 2, we use the ground-based
absolute sTEC as input for the tomography approaches and also for the
validation. The unambiguous relative sTEC is derived by the combination of
GPS dual-frequency carrier-phase and code-pseudorange measurements. Then,
the absolute sTEC and the receiver and satellite inter-frequency biases are
separated by a model-assisted technique. The Neustrelitz TEC model
(Jakowski, et al., 2011a), together with a single-layer mapping function
(assuming shell height of 400 km), is applied for the calibration procedure.
For more details, we refer to Jakowski et al. (2011b). For this study, the
GNSS data of the global International GNSS Service (IGS) 1 s high rate
receiver network were acquired via
For one reconstruction epoch, the available sTEC data are collected within a 10 min interval and averaged regarding the ray path geometry. On average, around 80–90 stations and 600–700 averaged sTEC measurements become available in the reconstructed area. Comparing the measurement number with the number of unknowns in Eq. (2), we get a strongly underdetermined inverse problem with extremely limited angle geometry (see also Garcia and Crespon, 2008). Therefore, to regularise this inverse problem, we decided to use the corresponding vertical vTEC data, in addition to the sTEC measurements. Indeed, we validated the investigated tomography methods also without the additional use of vTEC values (i.e. assimilating only the ground-based sTEC) and detected a slight increase of the residuals statistics. This motivated us to concentrate on the results of assimilation, where both slant and vertical TEC is applied.
The independent IGS stations used for validation purposes.
Electron density layers calculated by the NeQuick model (top),
SMART (middle row) and SMART
This section is organised as follows: first, we present, by way of an
example, the reconstructed 3-D electron density. Subsequently, the
investigated tomography methods are validated for the two periods of the
year 2011 by comparing the following:
the convergence behaviour; the ability to reproduce the assimilated TEC; the reconstructed sTEC with independent ground-based sTEC data; the reconstructed electron densities with ionosonde electron density
profiles.
The results obtained by different methods are colour-coded as follows:
NeQuick model, orange; ART, light blue; SART, blue; SMART, red;
SMART
Figure 2 presents the 3-D electron densities at
different altitudes for DOY 009/2011 at 12:00 UTC. The top panel is
calculated using the pure NeQuick model; the middle panel depicts the SMART
reconstruction and the bottom panel the SMART
The figures deduced from SART and ART are similar to those deduced from
SMART and hence are not presented here. It is notable that the SMART result is
rather patchy, which is usual for locally working reconstruction methods
applied to sparse, unevenly distributed data. The application of 3-D SCM
within SMART
To compare the convergence behaviour of the investigated methods, we count
the number of iterations needed by the methods ART, SART and SMART to
achieve a predefined threshold of
Number of iterations needed to achieve
In the above equation,
Figure 3 shows the decrease in the mean deviation
The left-hand panel of Fig. 4 illustrates the
number of iterations
The right-hand panel of the same figure illustrates that, during the disturbed period, ART could not reach the threshold at any epoch. Using SART, the threshold is achieved only for approximately half of the processed epochs and even SMART has four short time intervals missing the threshold.
When applying the methods ART, SART and SMART to the LSE (2), the iteration process is stopped after
performing 100 iteration steps. To check how well the methods work, we
consider the mean deviation
Additionally, we look at the percentage reduction of the mean deviation
achieved by the tomography methods after 100 iteration steps, in comparison
to
Mean TEC deviation for NeQuick (orange diamonds), SMART
Left-hand panel:
The
As expected, the subsequent application (after 100 iterations with SMART) of the
3-D SCM method within SMART
At the beginning of the disturbed period high
Decrease of
Measured sTEC, validation station mas1 (27.76
Except for the two
Figure 8 displays the extent of
The reconstruction outcomes are highly dependent on the quality and
availability of data and on the accuracy of the background. Therefore, for
this first comparison, we concentrate on the European region covering the
geographic latitudes 20 to 60
For validating the outlined methods regarding their capability to estimate
independent sTEC, four IGS stations are chosen. They are listed in
Table 1. These stations are not used for
tomography. For each station, the measured sTEC (namely
For each IGS validation station, the residuals between the reconstructed
values and the measured TEC values are calculated as
Histograms of the absolute (left-hand panel) and relative (right-hand panel) sTEC residuals during the quiet period. For IGS validation stations from top to bottom: ffmj, pado, ajac, mas1 (north to south).
Histograms of the absolute (left-hand panel) sTEC residuals and sTEC residuals (right-hand panel) over all the four validation stations during the quiet period.
Figure 9 depicts the
Figure 10 displays the histograms of the sTEC
residuals during the quiet period for the four reference stations, from top
(north) to bottom (south): ffmj, pado, ajac, mas1. The distribution of the
relative residuals for the methods SART, SMART and SMART
The general behaviour of the residuals during the disturbed period is very similar to that during the quiet period. Thus, we just present the corresponding statistics of the absolute residuals in Table 3.
During both periods at all stations, the NeQuick model seems to overestimate
the sTEC values visible in the negative relative residuals. A similar
overestimation was observed by Nigussie et al. (2012). The authors
assimilated the GNSS ground-based sTEC data into the NeQuick model with an
alternative least square approach. Afterwards, the results obtained before
and after the assimilation were compared with GNSS sTEC of four independent
ground-based stations located in East Africa. They even detected a higher
level of overestimation by the pure NeQuick model. This higher level can be
explained probably due to the low-latitude locations of the therein chosen
validation stations and by the 10
The ionosonde stations used for validation purposes.
The statistics of the absolute sTEC residuals (all in TECU) for the ground-based validation stations during the disturbed period.
For the quiet period, the medians of the NeQuick relative residuals range
between
Histograms of the absolute (left-hand panel) sTEC residuals and sTEC residuals (right-hand panel) over all the four validation stations during the disturbed period.
During both periods, all the compared tomography methods could significantly
decrease all the statistics of the absolute residuals at each validation
station, as compared to the corresponding background values. Again, at each
station, the reduction achieved by the SMART
Figures 11 and 12
present the histograms and the statistics of the residuals and absolute
residuals over all the four validation stations for the quiet and the storm
period, respectively. Notably, the NeQuick model, once again overestimates
the sTEC values. The overall statistics confirm that the performance of
SMART
A comparison of the overall statistics of quiet and storm conditions shows
an increase of
In this section the investigated tomography methods are compared in terms of their capability to estimate the vertical electron density profiles. Therefore, the 3-D reconstructions are validated with vertical sounding data of four ionosonde stations, listed in Table 2. The ionosonde profiles of these ionosondes are downloaded from SPIDR.
Vertical
According to the ionosonde locations, the electron density profiles are
deduced from the 3-D electron density reconstructions. Since the
reconstructions are calculated with resolution of 30 km altitude (below
1000 km height), the derivation of the F2 layer characteristics,
Thus, instead of comparing the profiles in terms of
Figure 13 presents the profiles for the DOY 009/2011 at 12:00 UTC at the four ionosondes. The different methods are colour-coded as has been done in the figures of foregoing sections. For Fig. 13, the reconstructed profiles are interpolated by the piecewise cubic Hermite interpolation to the higher resolution of the ionosonde profiles (usually 10 km).
At JR055 and DB049, the electron density values of the NeQuick model, and
also of all the methods being compared, are smaller than the ionosonde
measurements for altitudes below 180 km. Above this altitude, the electron
density values of the NeQuick model, ART and SART at the DB049 station are
higher than the ionosonde values, whereas those of SMART and SMART
At EB040, the E and F2 layer peak heights given by the NeQuick model are completely different from the ionosonde values. As a result, the NeQuick modelled electron densities and consequently all the reconstructed electron densities are smaller than those of the ionosonde at all altitudes.
At GM037, the estimated E layer peak height and also the densities, estimated by the investigated methods below 200 km altitude, match the measured values. The ionosonde profile of this station is provided with 1 km altitude resolution, but seems unsmoothed above the 200 km altitude.
The median of the relative ionosonde DB049 residuals versus the corresponding altitude during the quiet (left) and disturbed (right) periods.
The
Figure 14 points out the results of the comparison between the profiles. The altitude-dependent median values of the relative residuals at DB049 station are shown. During both periods, the electron densities estimated by the NeQuick at the lower altitudes from 120 to 180 km, are significantly lower than those of the ionosonde station. This can be explained most probably due to the difference in the estimation of the ionospheric layers peak heights (especially the E-layer seems to be problematic), which causes different shapes of the model and ionosonde electron density profiles.
To elaborate this point further, attention is invited to
Fig. 15, which shows the
It is important to realise here that such huge deviations between ionosonde and modelled profiles could be induced, at least partly, by the inaccuracy of the ionosonde profiles themselves (see McNamara, 2006; Gerzen et al., 2015). McNamara (2006) addresses this topic in a comprehensive way, especially by pointing out the weakness in electron density estimation between E and F layers and in determination of layers height.
The 90 % values (left) and the medians (right) of the relative residuals versus the corresponding altitude during the quiet period. For the validation ionosonde stations from top to bottom: JR055, DB049, EB040, GM037 (north to south).
The 90 % values (left) and the medians (right) of the relative residuals versus the corresponding altitude during the disturbed period.
Because of these reasons, we restrict our further comparison to the area
that usually provides the most reliable ionosonde data for altitudes ranging
from 210 km to altitudes that are just above the corresponding ionosonde
For the quiet period, the low median at 210 km altitude for
JR055 is conspicuous. Probably, the 210 km altitude cut-off used is too low for this
station, and thus we observe a similar behaviour as at DB049 in
Fig. 14. Regarding the median values during the
quiet period, SMART and SMART
Regarding the 90 % bound, again during the quiet period, the results
provided by ART and SART methods are very similar to those provided by
NeQuick, except for the 210 km altitude. The bounds for SMART and SMART
For the disturbed period at DB040 and JR055, the behaviour of the median and
the 90 % bound for the tomography methods is similar to the NeQuick
values. Also at EB040, it is hard to name any method as the best-performing
one, because the methods perform differently at different altitudes.
The high negative median values obtained by NeQuick, SART
and ART at altitudes between 330 and 390 km are conspicuous. The sharp increase of the
90 % bound for the SART, SMART and SMART
From a comparison of the statistics of the quiet and the disturbed periods
(especially at the lower altitudes), a significant increase in the 90 %
bound becomes visible for the residuals of NeQuick and all tomography
methods at all stations, except GM037. At the low-latitude station GM037,
the behaviour of the 90 % bound differs for the quiet and disturbed periods
in dependence on the altitude: during the quiet period, very high values of
the 90 % bound are obtained at altitudes above 360 km. In contrast,
during the disturbed period, the highest bound values are obtained below 360 km, which is similar to the 90 % bound behaviour at the other ionosondes.
The increase in the ionosonde
In the present work, our main goal has been to introduce the algebraic
tomography methods SMART and SMART
The SMART method shows the best performance, in terms of convergence speed, especially visible during the storm period, followed by SART and ART. The reduction in the mean TEC deviation achieved by SMART, SART and ART, after 100 iterations, in comparison to the background (NeQuick model) initial mean deviation, is up to 90, 85 and 40 % respectively.
For the purpose of validation, we selected sTEC GNSS observations of four independent ground-based IGS stations and the vertical electron density profiles of four ionosonde stations in the European region. Two periods within the year 2011, one with quiet ionospheric conditions and the other with disturbed conditions, were investigated.
In summary, comparison of the sTEC results of this case study reveals that all the investigated tomography methods improve the background. During both periods and at each validation station, all the four methods could successfully reduce the median, RMS and SD values of the absolute sTEC residuals, in comparison to the background values.
SMART
The first validation with vertical sounding data reveals, on the one hand, the difficulties involved in correct characterisation of the electron density profile shapes, when only ground-based TEC is used for tomography. This is in agreement with the results deduced by similar studies (see e.g. Minkwitz et al. 2015; McNamara et al., 2008, 2011). On the other hand, the validation emphasises the need for careful treatment and filtering of ionosonde profile data. Here, bigger discrepancies between the background estimated and the true (or ionosonde) ionospheric layer heights cause significant differences in the electron density profile shape estimation and thus, huge differences between the modelled and true (or ionosonde) electron densities at the same altitudes. This problem seems to be difficult to solve by mere ingestion of ground-based data.
To get comprehensive 3-D reconstructions in the future, the step of assimilation of data providing more information about the vertical distribution, like ionosonde profiles and ionospheric radio occultation profiles, may prove important and promising (see McNamara et al., 2007; Angling 2008). Moreover, in order to improve the data coverage and measurement geometry we will assimilate space-based GNSS sTEC and further ground-based sTEC measurements available due to the recent development and modernisation of the different GNSS (e.g. BDS, Galileo and GLONASS – see e.g. Li et al., 2012, 2015). Further, adjustment of the background in terms of F2 layer characteristics (because the F2 layer dominates the shape of the whole profile) before starting the assimilation procedure seems to be helpful (see e.g. Bidaine and Warnant, 2010). In this context, because of the limitations of ionosonde profile estimation, filtering of data is a further important topic (see e.g. McNamara, 2006 and Gerzen et al., 2015).
Additionally, we are currently working on methods that enable better estimation
of the correlation lengths and error bounds for the 3-D electron densities
(see Minkwitz et al., 2015). This information can be used to improve upon
the SMART
We thank the IGS (