We validate two-dimensional ionospheric tomography reconstructions against EISCAT incoherent scatter radar measurements. Our tomography method is based on Bayesian statistical inversion with prior distribution given by its mean and covariance. We employ ionosonde measurements for the choice of the prior mean and covariance parameters and use the Gaussian Markov random fields as a sparse matrix approximation for the numerical computations. This results in a computationally efficient tomographic inversion algorithm with clear probabilistic interpretation.

We demonstrate how this method works with simultaneous beacon satellite and ionosonde measurements obtained in northern Scandinavia. The performance is compared with results obtained with a zero-mean prior and with the prior mean taken from the International Reference Ionosphere 2007 model. In validating the results, we use EISCAT ultra-high-frequency incoherent scatter radar measurements as the ground truth for the ionization profile shape.

We find that in comparison to the alternative prior information sources, ionosonde measurements improve the reconstruction by adding accurate information about the absolute value and the altitude distribution of electron density. With an ionosonde at continuous disposal, the presented method enhances stand-alone near-real-time ionospheric tomography for the given conditions significantly.

In ionospheric satellite tomography the electron density distribution
of the ionosphere is reconstructed from ground-based measurements of
satellite-transmitted radio signals. The use of tomographic methods
for ionospheric research was first suggested by

Mathematically ionospheric tomography is an ill-posed inverse problem
and cannot be solved without some additional stabilization or
regularization information. In ionospheric tomography the additional
information is often incorporated with the use of iterative
reconstruction algorithms such as algebraic reconstruction technique
with a strong initial model for the ionosphere

Regardless of the tomographic algorithm in use, the information provided by satellite to ground measurements is poor in the vertical direction. This is due to the limited measurement geometry, namely the lack of horizontal signal paths. Consequently, the peak altitude and the vertical gradient of the reconstructed ionosphere will be determined mostly by the regularizing prior assumptions that are built in to the employed tomography algorithm. In this study we employ the ionosonde measurements to give these assumptions for the vertical profile.

An ionosonde is a radar used to investigate the ionosphere.
An ionosonde transmits electromagnetic frequency pulses, sweeping through
the high-frequency (HF) range, and receives the signals reflected from
an altitude where the radar frequency matches a critical
frequency

Inclusion of ionosonde measurements in ionospheric tomography has been
studied by

In this article we continue the work presented in

The approach is applied to Scandinavian sector with tomographic
measurements from the TomoScand receiver chain

The dual-frequency signal transmitted from low Earth orbit (LEO)
satellites consists of frequencies of 150 and 400

For practical computations, we discretize Eq. (

Let us denote by

As the ionospheric tomography is an ill-posed problem, the maximum
likelihood estimate for Eq. (

From the posterior distribution we can then derive the most probable
value for the unknown parameters based on the prior distribution and
observed measurements, namely, the maximum a posteriori
estimator (MAP)

As we assume that the unknown parameters follow multivariate normal
distribution, the prior density function

It is very natural to represent the prior information as a probability
distribution. However, for the MAP estimator Eq. (

Unfortunately, the linear system allows also negative values in the
solution.
A large proportion of negative values would suggest
that the prior distribution differs drastically from the actual
ionospheric conditions and needs to be reconsidered.
Then again, small areas of negative values indicate that the model accuracy is less
than the corresponding absolute values.
Here, if some negative values are found, we add them as new
measurements into the linear system. We then set these new
measurements to zero with a small variance (

Two EISCAT UHF incoherent scatter radar measurement campaigns were
performed in November 2014 and March 2015. Three daytime and one nighttime COSMOS satellite overflights, suitable for two-dimensional
tomography, were measured with TomoScand receivers starting
approximately on 20 November 2014 at 12:50, 3 November 2015 at 13:50,
14 March 2015 at 13:20 and 21 November 2014 at 02:50 UTC.
The magnetic local time is approximately UTC

The IRI-2007 electron density profiles were taken
for the reconstruction times with longitude parameter
26

To validate the resulting tomographic reconstructions, for each
satellite overflight, the EISCAT UHF was set to perform a scan of four
measurements along the corresponding satellite track.
The altitude resolution used for EISCAT data analysis was 10

In the following three subsections we compare the EISCAT UHF
measurements to corresponding electron density profiles from the
obtained tomographic reconstructions. With the Overflight I the
reconstruction was made multiple times to choose the measurement
domain and prior parameters other than the prior mean. Based on these
trials the measurements used for the tomography were limited between
the latitudes of 55 and

The prior standard deviation (SD) is given as
a Chapman function for the vertical profile, with approximately the
same peak altitude as the prior mean, and the maximum electron
density approximately 40 % of the corresponding NeXtYZ maximum.
The Chapman profile was modified to have different scale heights for
above and below the maximum. The chosen values used here are 200 and
60

For numerical reasons, the prior distribution is built to have periodic boundary
conditions

After calibrating the parameters with the Overflight I, for the Overflights II and III the parameter values are adjusted only according to corresponding ionosonde measurements without additional tuning. For the Overflight IV the ionosonde profiles differ significantly from the previous ones. Hence also the prior standard deviation shape is adjusted to correspond to these conditions.

TomoScand receiver network and the satellite overflight ground track with four EISCAT UHF scan paths.

Reconstruction, phase curves and profile comparisons for Overflight I starting on 20 November 2014 at 12:50 UTC.

TomoScand receiver network and the satellite overflight ground track with four EISCAT UHF scan paths.

Reconstruction, phase curves and profile comparisons for Overflight II starting on 3 November 2015 at 13:50 UTC.

TomoScand receiver network and the satellite overflight ground track with four EISCAT UHF scan paths.

Reconstruction, phase curves and profile comparisons for Overflight III starting on 14 November 2015 at 13:20 UTC.

Reconstruction, phase curves and profile comparisons for Overflight IV starting on 20 November 2014 at 02:50 UTC.

In each of the following cases we first visualize the general
measurement setup on a map in Figs.

The COSMOS 2463 overflight (Fig.

The profile comparisons 1–4 in Fig.

The COSMOS 2407 overflight starts approximately on 3 November 2015
at 13:50 UTC (Fig.

The IRI-based profiles have very good agreement with the maximum densities of EISCAT scans. However the peak altitude is underestimated. The profiles taken from the reconstruction with zero-mean prior clearly disagree with the UHF measurement, in terms of both profile shape and peak electron density.

With the ionosonde-based prior, in Profile comparison 1 the prior mean and the closest UHF measurement are very similar and also the tomographic reconstruction is almost unchanged from the prior profile. Again, the electron density slightly increases southwards, which is well captured in the reconstruction. Both the peak density and altitude are very close to each other between the reconstruction and UHF profiles.

The COSMOS 2407 overflight starts on 14 March 2015 at 13:20 UTC
(Fig.

With IRI prior the maximum densities are slightly pronounced and the peak altitude remains below the UHF peak. With the zero-mean prior both the profile shapes and peak densities clearly disagree with the UHF, again. For the ionosonde case the best agreement in general profile shape is again visible, even though the errors in peak altitudes and densities are in the same level with the IRI-based reconstructions.

The COSMOS 2407 overflight starts on 21 November 2014 at 02:50 UTC
(Fig.

With IRI prior an F region is visible, although at the wrong altitude, but
the E-region peak is completely missing. The zero-mean prior spreads
electron density also to lower altitudes, but it cannot distinguish the
two-peak structure. With ionosonde the shape of the reconstruction
seems to be strongly dictated by the prior. Horizontal gradients in
F-region peak density are rather well reproduced in the
reconstruction, whereas the reconstructed E-region peak is almost
unchanged in the profile comparisons, although the UHF radar shows
significantly different peak density at each pointing direction. In
the reconstruction in the upper left panel of Fig.

The presented method for ionospheric tomography includes several prior parameters, and the selection of the corresponding values might seem arbitrary. The objective of this article is not to optimize all of the prior parameters, but to concentrate on the altitude profiles of the prior mean and the standard deviation. Based on trials with the algorithm and different data, the information on the vertical structures has the most crucial effect on the reconstruction quality. This is also evident in the presented results. When zero-mean prior is used, the peak altitude can be found relatively well, but the measurements do not contain enough information to produce steep enough vertical gradients. Then again, when a vertical profile is given within the prior, the reconstruction of peak electron density is improved significantly, but the peak altitude becomes less sensitive to measurements.

In horizontal direction, the gradients can be reconstructed rather well regardless of the prior mean in use. Hence, information on horizontal electron density structures (IRI model) is less important if the trade-off is the accuracy on the vertical structure.

When accurate vertical electron density profile is provided within the prior, the selection for the values of the other prior parameters is less critical. For all prior parameters the stabilizing effect is also rather intuitive. Decreasing the correlation lengths allows more small-scale variation in the reconstructions; however, getting close to the corresponding discretization can result in artifacts. The increment of correlation lengths smoothens the reconstruction, but very long correlation lengths can again produce unexpected behavior.

Errors of tomographic profiles compared with EISCAT UHF scans.

With all cases in the previous section, the use of horizontal correlation
length values between 1 and 10

As mentioned in Sect. 3, the standard deviation profile is parameterized as a Chapman function. Hence, the ionosonde profile cannot be used explicitly, but the choice of the parameter values can be done viably based on the ionosonde measurements. For the first three overflights only the peak standard deviation altitude and density were set according the corresponding ionosonde measurements. With Overflight IV, the ionosonde profiles are significantly different; thus also the scale heights of the prior standard deviation were changed. Altogether, the results for the overflights II, III and IV could be enhanced by optimizing the parameters through trial and error individually for each case, but the results show that already intuitive realistic choices of these parameters are enough to give reasonable solutions.

As the ionosonde measurements provide relatively accurate measurements of the ionospheric electron density, it would be straightforward to use them also as direct measurements above the instrument location. However, the satellite overflight hits rarely at the zenith of the ionosonde site, and the electron densities measured by ionosonde and tomographic receiver can vary largely. When 2-D assumption (i.e., small gradients in longitude) is used, the ionosonde measurement error should reflect this discrepancy. Hence the information for the projected ionosonde measurement points cannot be modeled as accurately as they are in their actual location, and the prior distribution provides substantially the same information. In the 3-D case the situation will be different as all of the measurements will be modeled in their actual locations.

Electron density profiles measured with the EISCAT UHF are routinely calibrated by means of comparing F-region peak electron density estimates from the UHF and the dynasonde. Thus, when the ionosonde-based prior is used, F-region peak densities above the Tromsø site are taken from the same instrument in both the tomography prior and the UHF results. Our tomography measurements and the ground truth UHF measurements are thus not completely independent. However, we anticipate that this is not a very serious problem, as the calibration data were not used for the validation. Furthermore, calibration does not affect the UHF density profile shape, but only its absolute values, and calibration is not performed for individual profiles, but the same scaling is used for a longer period of time. Especially, the actual validation measurements with beam steered far away from zenith are never used for calibration.

In this study the use of Bayesian statistical inversion with known prior distributions and with the inclusion of simultaneous ionosonde measurements for ionospheric tomography is validated. Most importantly we show that the prior distribution can be constructed based on the ionosonde measurements, which helps in constraining the otherwise poorly defined altitude profile shape of the tomographic reconstruction.

We demonstrate the applicability of the method with four satellite overflights measured with the TomoScand receiver network, and with EISCAT dynasonde measurements from the EISCAT Tromsø site. In comparisons we used International Reference Ionosphere 2007 and zero mean in building of the prior. The validation is made against simultaneous EISCAT UHF incoherent scatter radar measurements.

The biggest issue with IRI-2007 consists in the problems with the peak altitude. With zero mean it is the significant underestimation of the electron density. From both of the reference schemes it can be seen that the measurements cannot provide enough information on the vertical gradients of the ionosphere. The use of ionosonde in the building of the prior distribution outperforms the compared alternatives. The results show better agreement between the incoherent scatter radar measurements and the corresponding electron density profiles taken from the reconstruction. The reconstructions seem reasonable even further away from the ionosonde location. However, the electron density height profiles are dictated by the prior model and could be biased further away from the ionosonde. Use of multiple ionosondes and altering the prior profile in horizontal direction would be straightforward within the method and highly recommended.

The results also indicate that when reliable prior information is provided, the required prior parameters can be predetermined and the method used without additional tuning. This makes the operational stand-alone use feasible, at least for typical ionospheric conditions. With the lattice sizes in the reported scale and with a modern PC the required computations can be made in real time.

As in the Bayesian inference we are presenting the information as probability distributions, we also have direct access to the credible intervals. If the prior is truly realistic, the posteriori credible interval can be highly informative. However, it is important to note that when interpreting the posterior distribution and credible intervals derived from it, they are highly dependent on the given prior distribution. Posterior credible intervals should thus be used with caution.

The data for analyzed EISCAT
dynasonde results from Tromsø are available from the EISCAT Dynasonde Database
(

Ionospheric tomography measurements and analyzed data products used in this paper are freely available upon request from the Finnish Meteorological Institute.

The work of J. Norberg has been funded by Academy of Finland (decision no. 287679) and European Regional Development Fund (Regional Council of Lapland, decision no. A70179). The work of I. I. Virtanen has been funded by Academy of Finland (decision no. 285474). The work of L. Roininen, M. Orispää and M. Lehtinen has been funded by European Research Council (ERC Advanced Grant 267700 – InvProb) and Academy of Finland (Finnish Centre of Excellence in Inverse Problems Research 2012–2017, decision no. 250215).

Authors thank EISCAT staff, especially Jussi Markkanen, for kindly assisting in the EISCAT UHF radar experiments, and Yoshimasa Tanaka and Yasunobu Ogawa of NIPR for executing the EISCAT UHF radar experiments in March 2015. We also would like to thank EISCAT for providing the dynasonde data. EISCAT is an international association supported by research organizations in China (CRIRP), Finland (SA), Japan (NIPR and STEL), Norway (NFR), Sweden (VR) and the UK (NERC). Edited by: M. Nicolls