AMTAtmospheric Measurement TechniquesAMTAtmos. Meas. Tech.1867-8548Copernicus PublicationsGöttingen, Germany10.5194/amt-9-1859-2016Bayesian statistical ionospheric tomography improved by incorporating ionosonde measurementsNorbergJohannesjohannes.norberg@fmi.fiVirtanenIlkka I.https://orcid.org/0000-0002-7111-8888RoininenLassihttps://orcid.org/0000-0002-7014-6684VierinenJuhaOrispääMikkoKauristieKirstiLehtinenMarkku S.Finnish Meteorological Institute, Helsinki, FinlandSodankylä Geophysical Observatory, University of Oulu, Sodankylä, FinlandIonospheric Physics, University of Oulu, Oulu, FinlandDepartment of Mathematics, Tallinn University of Technology, Tallinn, EstoniaHaystack Observatory, Massachusetts Institute of Technology, Westford, MA, USAJohannes Norberg (johannes.norberg@fmi.fi)28April2016941859186926August201521September201515March201617March2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://amt.copernicus.org/articles/9/1859/2016/amt-9-1859-2016.htmlThe full text article is available as a PDF file from https://amt.copernicus.org/articles/9/1859/2016/amt-9-1859-2016.pdf
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.
Introduction
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 .
provide a good overview on the development of the topic.
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 .
Bayesian statistical inversion was applied to ionospheric tomography
first by . The Bayesian approach provides an
interpretable approach for the stabilization as the additional
information is given as a prior probability distribution of unknown
parameters. However, in the work of , the prior
distribution is not defined by its covariance, but by an assumption of
smoothness resulting from the limiting of the differences of
neighboring pixels. This is an often valid assumption, but the
relation between the prior parameters and the physical quantities is
not clear. Recently, have described a method in
which the prior can be built in a computationally efficient way as
a probability distribution with a known covariance structure. The
prior is parameterized with physical units and can be understood as
a probability distribution for realizations of 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 . For ordinary mode polarization the critical frequency is
the plasma frequency of the local electron density. Because refractive
index along the signal path differs significantly from that of a vacuum,
conversion of signal travel time into reflection height is not
trivial, but the electron density profile along the path needs to be
taken into account. The reflections and the travel times at multiple
frequencies can be used to estimate an electron density profile of the
ionosphere. Because the ionosonde relies on reflection, it can
directly measure only the bottom side of the ionospheric altitude
profile up to the peak of the electron density profile. Also, it is
not very effective for observing local minima, e.g., the valley region
between the E and F regions of the ionosphere. Ionosonde
measurements provide recurrent and accurate but geographically
localized information of the ionospheric electron density profile. In
mesoscale tomographic analysis, it is often the best information
available, even if the analyzed region is somewhat displaced from the
ionosonde site.
Inclusion of ionosonde measurements in ionospheric tomography has been
studied by , where ionosonde measurements were used to
form the background profile for an iterative reconstruction algorithm.
The study had mixed results on the impact of ionosonde measurement
inclusion. They also observed up to 70 % differences between the
ionosonde and incoherent scatter radar-derived electron density
profiles. More recently used ionosonde measurements
to set vertical basis functions for the inversion, as well as using
them as local measurements of peak density and bottom-side profile
gradients. The inclusion improved the tomographic results
significantly, but the sensitivity to ionosonde measurement bias was
also underlined. also combined ionosonde data with GPS
measurements for ionospheric tomography. The presented method
concentrates on estimating parameterized local electron density profile
at the location of the ionosonde. For latitudinal and longitudinal changes,
only the first-order dependence of vertical total electron content was considered.
In this article we continue the work presented in and
include the ionosonde measurements in the Bayesian statistical
inversion approach for ionospheric tomography. For comparison, we
analyze the data also with the prior mean taken from the International
Reference Ionosphere (IRI) model, and with a zero-mean prior.
The IRI model is chosen as it is a well-known ionospheric model,
and unlike the ionosonde, it provides information also on horizontal
electron density gradients. The zero-mean prior is included to demonstrate
the performance with simpler and more general prior information. The zero-mean
prior carries essentially similar information to the prior model used in .
We construct the prior mean electron density profile for the entire
ionospheric tomography domain according to the chosen information
source. This assumption is then controlled with the prior covariance,
as it states how strictly the reconstruction should follow the prior
mean. As the prior distribution is parameterized with physical units,
the method provides clear understanding on information used for the
tomographic reconstruction. Hence the approach makes the inversion
possible with less ad hoc adjustment. This is a very important aspect
for achieving reliable operational near-real-time tomography results.
The approach is applied to Scandinavian sector with tomographic
measurements from the TomoScand receiver chain and
ionosonde data from the European Incoherent Scatter Scientific
Association (EISCAT) dynasonde in Tromsø, Norway. The IRI model used
for the comparison is the International Reference Ionosphere 2007
(IRI-2007) . We validate the results with EISCAT
ultra-high-frequency (UHF) incoherent scatter radar measurements
carried out on 20 and 21 November 2014 and 11 and 14 March 2015 in
Tromsø.
Methodology
The dual-frequency signal transmitted from low Earth orbit (LEO)
satellites consists of frequencies of 150 and 400 MHz. The
ionospheric refraction causes a phase shift to propagating
electromagnetic waves. This phase shift is proportional to density of
electrons along the signal path and can be modeled
as
mt,sat,rec=γsat,rec+∫Lt,sat,recNe(z)dl+εt,sat,rec,
where mt,sat,rec is the measured relative total electron content
at time t between the satellite sat and receiver rec, and εt,sat,rec the corresponding measurement
error. Ne(z) is the two-dimensional continuous field of
electron densities with coordinates z=(z1,z2)∈R2. The
integral is defined over the measurement signal path Lt,sat,rec. The
receiver–satellite specific constant γsat,rec is due to the unknown
amount of electron content when the satellite is first
observed.
For practical computations, we discretize Eq. () for all
measurements.
The discretized measurement
model for the ionospheric tomography is given as
M=AX+E.
The measurement vector is M∈Rnm.
Theory matrix A∈Rnm×nx
gives the measurement geometry between the satellite measurement
points and receiver locations. The vector of unknown parameters
X∈Rnx includes both electron densities and
the 2π-ambiguity constants γ. The measurement error vector
is E∈Rnm. The number of measurements
is given as nm and the number of unknown parameters as
nx.
Let us denote by x and m the realizations of the random variables
X and M, respectively. We can then write the
likelihood function for unknown parameters, given the measurements as
L(x|m)=DE(Ax-m),
where DE is the probability density function of measurement
errors. From here on we assume that E∼N(0,ΣE):
the measurement errors follow a multivariate normal distribution
with zero mean and covariance ΣE∈R(nm)×(nm).
As the ionospheric tomography is an ill-posed problem, the maximum
likelihood estimate for Eq. () cannot be solved without
including some additional information regarding the unknown
parameters. Here we use Bayesian statistical inversion
to give this information as a prior
distribution. We assume that the unknown X follows a multivariate
normal distribution X∼N(μ,Σpr), where vector μ∈Rnx is the mean value and the matrix
Σpr∈R(nx)×(nx) the
covariance. Again, the vector μ as well as the matrix Σpr
consists of parts for both the unknown electron densities and the unknown
γ parameters. We denote the prior probability density function with
Dpr(x). Following the Bayes' theorem, we then obtain the
posterior distribution for X as
Dpost(x|m)=DE(Ax-m)Dpr(x)∫RnxDE(Ax-m)Dpr(x)dx,
where the denominator is a normalization constant and we can write
Dpost(x|m)∝DE(Ax-m)Dpr(x).
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)
xMAP=ΣpostATΣE-1m+Σpr-1μ,
where
Σpost=ATΣE-1A+Σpr-1-1
is called posterior covariance.
As we assume that the unknown parameters follow multivariate normal
distribution, the prior density function Dpr(x) is defined
with its mean and covariance. In Bayesian statistical approach for
ionospheric tomography, the prior mean can be understood as the most
probable state of the ionosphere before the actual satellite
measurements. With the covariance we can express how reliable the
information of prior mean is and how correlated the ionospheric
electron densities are. Actual values of these parameters should be
based on all information we have at our disposal, i.e., on other
measurements, models, statistical data and the physics of ionosphere.
In the performed experiments, we use three different schemes to
compose the prior: IRI-2007 ionospheric model, zero mean and, most
importantly, the ionosonde measurements. The prior covariance is
given as a squared exponential, i.e., as a Gaussian-shaped function
that is defined with its amplitude (variance or standard deviation) and
correlation length. The correlation length is given separately for
horizontal and vertical directions and is defined here as the distance
where the covariance drops to 10 % of variance.
It is very natural to represent the prior information as a probability
distribution. However, for the MAP estimator Eq. () only the
precision matrix Σpr-1 (i.e., the inverse
of the prior covariance) is required besides the prior mean. In
it is shown how the precision matrix of a known
covariance can be constructed with a sparse matrix representation with
Gaussian Markov random fields. The approach provides us with the
interpretation of a probability distribution, yet it keeps the
approach computationally feasible, in comparison to operating with
full covariance matrices.
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 (10-20) and solve the
system again. We note that here this positivity constraint
is mostly a cosmetic ad hoc method which will be
reconsidered in future studies.
Experiments
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 + 2.5 h.
The altitude of COSMOS satellites is
approximately 1000 km and the duration of measurements from an
overflight is roughly 10 min. For the ionosonde prior mean the NeXtYZ
analyzed EISCAT dynasonde results from Tromsø (see Sect. Data availability) were
collected. The ionosonde electron density profiles that were measured
during each satellite overflight were averaged together to form
one
profile. We denote the resulting profile with
μNeXtYZ. The NeXtYZ provides also a modeled profile for
the top-side ionosphere, but to gain better control over the prior, we
here give the top side as an exponential profile. The complete
altitude profile for the prior mean based on ionosonde measurement can
be written as
μionosonde(z)=μNeXtYZ(zpeak)exp-z-zpeaks,zpeak<z≤zmaxμNeXtYZ(z),0≤z≤zpeak,
where z is the altitude with the maximum zmax=1000 (km) and
zpeak=argmaxz(μNeXtYZ(z)), i.e.,
the altitude of the maximum electron density. The parameter s
defines how rapidly the electron density decreases at the higher
altitudes.
The IRI-2007 electron density profiles were taken
for the reconstruction times with longitude parameter
26∘. With the IRI-2007 we obtain a two-dimensional profile
with latitudinal variation for the complete domain where the
ionospheric tomography takes place.
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 km.
The UHF data were calibrated against the EISCAT
Tromsø dynasonde. The calibration data were taken from periods when
the radar was not scanning and the ionosphere was reasonably
stable. Each few-hours-long continuous radar run was calibrated
separately.
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 75∘ and the elevation
angles over 20∘.
The chosen sampling rate of 0.5 Hz then produces between 100 and 200
suitable measurements from each receiver station.
The corresponding measurement errors are estimated from the original
20 Hz sampling rate data.
The measurement errors are assumed to be independent, resulting in a
diagonal measurement error covariance matrix.
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 km correspondingly. In vertical direction the prior
correlation length was chosen to be 200 km and in the
horizontal 2∘.
The s parameter for the upper profile of
the prior was chosen to be 140 km. This results as a slightly
steeper gradient for the top-side ionosphere than provided by NeXtYZ.
With the zero-mean prior we use the same prior standard deviation as
with the ionosonde case but, to allow larger changes in electron
density, the maximum is set to 80 % of the NeXtYZ maximum.
For all of the experiments, the prior mean values for the γ parameters are
set to zero, and the prior standard deviations as large as 1020 m-3,
nearing an uninformative prior.
The resolution for the domain is 200×200, resulting in pixel size
of 5 km× 0.1∘.
For numerical reasons, the prior distribution is built to have periodic boundary
conditions . Here, the given vertical prior profile constrains
the values of highest and lowest altitudes so strongly
that boundary effects in that direction are prevented.
To avoid boundary effects in horizontal direction, the correlation lengths at
the boundaries are decreased to 10 % of the initial values and the actual inversion
is carried out in a larger domain than is our actual interest.
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.
TomoScand receiver network and the satellite overflight ground track with four EISCAT UHF scan paths.
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. ,
, and . The
results are presented in
Figs. , ,
and , first as two-dimensional altitude–latitude
reconstructions of electron densities, i.e., the MAP estimates where
the ionosonde prior is used. On top of the reconstruction the EISCAT
UHF scans are shown with white paths. We then compare the prior and
posteriori distribution parameters to corresponding EISCAT UHF scan
locations by assuming a longitudinally uniform ionosphere. The ionosonde
prior means are plotted with solid green lines and the 95 %
prior credible intervals (Ionosonde prior mean
±2 × prior SD) with dashed
green lines. The profiles taken from the reconstruction with ionosonde
prior are plotted with solid black lines (MAP ionosonde) and the
corresponding 95 % posterior credible intervals with dashed
black lines (MAP ionosonde ±2 × posterior SD). The electron density profiles obtained with
EISCAT UHF scans are plotted with red. The blue dashed line is
a profile taken from the reconstruction where the prior is based on
IRI-2007 profile (MAP IRI) and the cyan dashed line from the
reconstruction with zero-mean prior (MAP ZERO). In
Table the relative mean errors for
profile peak electron densities and the mean errors for peak altitudes
are given. In addition to the profile comparisons, we show the
relative phase difference measurements used for the inversion for each
station, as well as the corresponding measurements predicted from the
reconstruction obtained with ionosonde prior.
Overflight I
The COSMOS 2463 overflight (Fig. ) starts on
20 November 2014 at 12:50 UTC. The direction of the satellite track is
from north to south. The relative phase difference curves in the top
right panel of Fig. indicate smooth ionosphere, with
some local structures visible in the Tromsø station curve. The
ionosonde measurements used for the prior are from 12:54, 12:56, 12:58
and 13:00 UTC.
The largest differences between the ionosonde profiles were at
330 km altitude, with standard deviation of 2.3×1011 m-3.
The peak altitudes range from 320 to 340 km.
In Fig. , the obtained tomographic
reconstruction is shown in the top left panel. On top of the
reconstruction are plotted the four EISCAT UHF measurements performed
at (1) 12:53:00–12:54:10, (2) 12:55:03–12:56:03,
(3) 12:56:20–12:57:20 and (4) 12:57:35–12:58:35 UTC. The
latitude–longitude directions of the measurement can be seen in
Fig. . Hourly averaged Kp and F10.7 indices at
13:00 UTC were 1.3 and 164.1, respectively.
The profile comparisons 1–4 in Fig. show that the
southward increment of electron density is captured by all three
reconstructions. In the profiles based on the IRI-2007 and zero-mean
prior reconstructions the maximum electron density is significantly
lower than in the EISCAT UHF profiles and shape of the profiles
clearly disagrees with the UHF measurements in comparisons 1 and 2.
With IRI-2007 the peak altitude is underestimated in all of the
profiles. The ionosonde prior shows a good agreement between shapes of
the corresponding profiles. Although the satellite rises almost to
zenith above Tromsø, F-region peak density estimates from the
ionosonde are about 30 % higher than the calibrated UHF
measurements. However, the prior standard deviation enables large
enough changes to capture the correct level in the MAP estimate. With
the ionosonde prior the most glaring difference between the UHF and
tomographic profiles is in the altitude of the peak electron
densities.
Overflight II
The COSMOS 2407 overflight starts approximately on 3 November 2015
at 13:50 UTC (Fig. ). The direction of the satellite
track is from north to south. The relative phase difference curves in
Fig. indicate a smooth ionosphere. Based on the
ionosonde measurements collected at 13:54, 13:56, 13:58 and 14:00 UTC
the electron density level is expected to be lower than in Overflight I.
The largest differences between the ionosonde profiles were at
260 km altitude, with standard deviation of 0.6×1011 m-3.
The peak altitudes range between 260 and 280 km.
The new prior profiles for this overflight are shown in
the lower four panels of Fig. . Besides the altitude
profiles for prior mean and standard deviation, the other parameters
remain unchanged. In the top left panel of Fig. the
reconstruction and the EISCAT UHF measurement projections from
(1) 13:54:28–13:55:28, (2) 13:55:50–13:56:50, (3) 13:57:11–13:58:11,
and (4) 13:58:26–13:59:30 UTC are shown. Hourly averaged Kp and
F10.7 indices at 14:00 UTC were 2.3 and 129.9, respectively.
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.
Overflight III
The COSMOS 2407 overflight starts on 14 March 2015 at 13:20 UTC
(Fig. ). The direction of the satellite track is from
north to south. The ionosonde measurements used for the prior were
collected at 13:26, 13:28, 13:30 and 13:32 UTC.
The largest differences between the ionosonde profiles were at
310 km altitude, with standard deviation of 0.5×1011 m-3.
The peak altitudes range from below 250 km to almost 320 km.
The reconstruction
and the EISCAT UHF measurement directions at (1) 13:27:45–13:28:45,
(2) 13:29:01–13:30:02, (3) 13:30:20–13:31:21 and (4)
13:31:35–13:32:35 UTC are shown in the top left panel of
Fig. . Hourly averaged Kp and F10.7 indices at
13:00 UTC were 1.7 and 114.3, respectively.
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.
Overflight IV
The COSMOS 2407 overflight starts on 21 November 2014 at 02:50 UTC
(Fig. ). Direction of the satellite track is from
north to south. The relative phase difference curves in
Fig. indicate more small-scale structures in ionosphere
than in the previous measurements. The ionosonde measurements were
collected at 02:56, 02:58, 03:02 and 03:04 UTC, as the data for 03:00 are missing.
The largest differences between the ionosonde profiles were at
180 km altitude, with standard deviation of 0.7×1011 m-3.
The measurements show a strong E region at 100 km
altitude. As the ionosonde measurements indicate that the electron
density is not concentrated on one altitude, the maximum of the prior
standard deviation is here set to the lower E-region peak of the
ionosonde profile and the upper-scale height is increased to
600 km to allow more variation also around the higher F-region
peak. Otherwise the prior profiles are formed similarly to previous
cases. The reconstruction and the EISCAT UHF measurement directions at
(1) 02:57:40–02:58:50, (2) 02:59:15–03:00:30, (3) 03:00:50–03:02:05
and (4) 03:02:25–03:03:35 UTC are shown in the top left panel of
Fig. . Hourly averaged Kp and F10.7 indices at
03:00 UTC were 3.3 and 158.6, respectively.
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.
a southward decrement in E-region density is visible between the
receivers, where the information provided by the measurements is
higher. Directly above the receivers information about the vertical
profile is very poor and the reconstruction relies on the prior
information. Hence the lower layer remains.
Discussion
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.
Relative mean error ofMean error ofpeak density (%)peak altitude (km)Overflight IIonosonde541IRI2755Zero5274Overflight IIIonosonde517IRI658Zero5415Overflight IIIIonosonde433IRI631Zero6033Overflight IVIonosonde540IRI1284Zero6150
With all cases in the previous section, the use of horizontal correlation
length values between 1 and 10∘ and vertical correlation
lengths between 20 and 500 km were carried out without
drastically unrealistic changes in reconstructions. The peak value of
standard deviation was also altered in a range from 20 to 100% with
anticipated results.
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.
Conclusions
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.
Data availability
The data for analyzed EISCAT
dynasonde results from Tromsø are available from the EISCAT Dynasonde Database
(http://dynserv.eiscat.uit.no/DD/Iono_form.php). The IRI-2007 electron density profiles are available from the
IRI-2007 website
(http://omniweb.gsfc.nasa.gov/vitmo/iri_vitmo.html).
Ionospheric tomography measurements and analyzed data products used in this paper are freely
available upon request from the Finnish Meteorological Institute.
Acknowledgements
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
References
Andreeva, E. S.:
Radio tomographic reconstruction of ionization dip in the plasma near the Earth,
J. Exp. Theor. Phys. Lett., 52, 142–148, 1990.
Austen, J. R., Franke, S. J., and Liu, C. H.:
Ionospheric imaging using computerized tomography,
Radio Sci., 3, 299–307, 1988.
Bilitza, D. and Reinisch, B. W.:
International Reference Ionosphere 2007: improvements and a new parameters,
Adv. Space Res., 42, 599–609, 2008.
Breit, G. and Tuve, M. A.:
A Test of the Existence of the Conducting Layer,
Phys. Rev., 28, 554–575, 1926.Bust, G. S. and Mitchell, C. N.:
History, current state, and future directions of ionospheric imaging,
Rev. Geophys., 46, RG1003,
10.1029/2006RG000212, 2008.
Chartier, A. T., Smith, N. D., Mitchell, C. N., Jackson, D. R., and Patilongo, P. J. C.:
The use of ionosondes in GPS ionospheric tomography at low latitudes,
J. Geophys. Res.,
117, A10326,
10.1029/2012JA018054, 2012.
Chiang, K. Q. Z., and Psiaki, M. L.:
GPS and Ionosonde Data Fusion for Ionospheric Tomography,
Proc. ION GNSS+ 2014,
9–12 September 2014, Tampa, FL, USA, 1163–1172, 2014.
Davies, K.:
Ionospheric Radio IEE Electromagn. Waves ser.,
IET, London, UK, 1990.EISCAT Dynasonde Database: available at:
http://dynserv.eiscat.uit.no/DD/Iono_form.php, last access: 27 April 2016.IRI-2007: International Reference Ionosphere – IRI-2007, available at:
http://omniweb.gsfc.nasa.gov/vitmo/iri_vitmo.html, last access: 27 April 2016.
Kaipio, J. and Somersalo, E.:
Statistical and Computational Inverse Problems, Applied Mathematical Sciences,
Springer, New York, USA, 2005.
Kersley, L., Heaton, J. A. T., Pryse, S. E., and Raymund, T. D.:
Experimental ionospheric tomography with ionosonde input and EISCAT verification,
Ann. Geophys.,
11, 1064–1074, 1993.
Markkanen, M., Lehtinen, M., Nygren, T., Pirttilä, J., Helenius, P., Vilenius, E., Tereshchenko, E. D., and Khudukon, B. Z.:
Bayesian approach to satellite radiotomography with applications in the Scandinavian sector,
Ann. Geophys.,
13, 1277–1287, 1995.Norberg, J., Roininen, L., Vierinen, J., Amm, O., McKay-Bukowski, D., and Lehtinen, M. S.:
Ionospheric tomography in Bayesian framework with Gaussian Markov random field priors,
Radio Sci.,
50, 138–152,
10.1002/2014RS005431, 2015.Vierinen, J., Norberg, J., Lehtinen, M. S., Amm, O., Roininen, L., Väänänen, A., and McKay-Bukowski, D.:
Software defined beacon satellite receiver software for ionospheric tomography,
Radio Sci.,
49, 1141–1152,
10.1002/2014RS005434, 2014.Zabotin, N. A., Wright, J. W., and Zhbankov, G. A.:
NeXtYZ: three-dimensional electron density inversion for dynasonde ionograms,
Radio Sci., 41, RS6S32, 10.1029/2005RS003352, 2006.