Introduction
The ionosphere is the upper part of the atmosphere where sufficient free
electrons exist to affect the propagation of radio waves, and its morphology
is mainly driven by solar radiation, particle precipitation and charge
exchange.
Over the last decade, global navigation satellite system (GNSS)
measurements have become one of the major tools for ionospheric sounding,
enabling the derivation of the total electron content (TEC) along a satellite-to-receiver ray path. There are several activities in the ionosphere
community aiming to estimate or model the ionospheric electron density based
on GNSS data and other ionospheric measurements.
The International Reference Ionosphere model (IRI; see ) is an empirical model based on historical ground- and
space-based data. It describes monthly averages of electron densities and
temperatures in an altitude range of about 50–1500 km in the non-auroral
ionosphere. Another empirical model is NeQuick (see ). It is
mainly driven by the monthly average solar flux F10.7 and ionospheric F2 peak
parameters computed by the International Telecommunication Union (ITU) foF2
and M(3000)F2 models; see . However, those models represent
median ionospheric behavior. Consequently the inclusion of actual ionospheric
measurements is essential to update the model and hence to improve the
electron density characterization.
Through the years different approaches have been developed and tested for
ionospheric imaging, combining actual direct or indirect measurements with
empirical or physical background models. We can identify
methods modifying the coefficients of an empirical model (see
), methods updating the model towards the
measurements without modification of its coefficients (see
), methods combining both (see
) and approaches using physical background models and
including the estimation of ionospheric drivers, such as neutral
winds, in the state vector (see ).
Since methods for data assimilation/ionospheric imaging were first developed, iterative methods have been used as computer resource-saving
approaches to assimilate data into background models, e.g., derivatives of the algebraic reconstruction technique
and the successive correction method (see ). However, such techniques have the disadvantage that the
incorporation of additional information (e.g., background and measurement
error covariances), which is extremely helpful for regularization of the
ill-posed inverse problem behind the ionosphere imaging, is hardly foreseen.
Thus, techniques which take advantage of spatial and temporal covariance
information of the ionosphere, such as optimal interpolation (OI), the 3-D and
4-D variational technique, Kalman-filter-based approaches, and geostatistical
approaches such as kriging, have been applied. In general, these methods
provide a best linear unbiased estimator/predictor but differ in their
mathematical frameworks and thus in their practical implementation; see e.g.,
.
As an example, introduced the Electron Density
Assimilative Model (EDAM) that incorporates different measurements into an
empirical background model by means of a Kalman filter. The majority of the
input data is GPS TEC derived from the ground-based GNSS stations of the
International GNSS Service (IGS). However, EDAM also deals with ionospheric
radio occultation (IRO), ionosonde data and in situ electron density
measurements; see . developed a
similar approach, the Ionospheric Data Assimilation Three-Dimensional
(IDA3-D) technique
based on 3-D variational data assimilation. Both EDAM and IDA3-D apply an
exponential time covariance model to forecast the electron density state
vector and its covariance matrix from one time step to the next.
The right choice of the
covariance matrix of the state vector (i.e., in this case the background
covariance), the determination of the time forecast model and the appropriate
choice of its parameters (for instance the correlation time) are critical to these kinds of approaches. However, until now
there have only been limited publications which explicitly cover these topics.
Flow chart of the 3-D simple kriging approach.
Variograms originating from geostatistics and describing the variation between
measurements depending on the distance separation are a popular tool to investigate spatial covariance; see,
e.g.,
. For the provision of vertical TEC (VTEC)
and its integrity/error bounds, this method is successfully applied within
the Wide Area Augmentation System (WAAS) and for the generation of global
ionospheric maps (GIMs); see and ,
respectively. In particular, both applications detrend the VTEC measurements
using a background model to derive the spatial covariance of the measurements
or, more specifically, the error covariance of the background. Afterwards this
information is used to estimate VTEC at ionospheric grid points using ordinary
kriging.
However, since electron density measurements are rarely available, especially at
altitudes above the F2 layer, it is difficult to obtain the electron
density's spatial covariance with variograms. In this paper, we develop an
approach enabling the estimation of the electron density's spatial covariance
model by means of direct and indirect ionospheric measurements. Based
on this information, the electron density for arbitrary points/grids is
calculated using 3-D simple kriging.
Methodology
The work flow of the approach is outlined in Fig. .
Following the general knowledge about the ionospheric behavior, we set up a
parametric spatial covariance model of the 3-D electron density. Based on the
ground-based slant TEC (STEC) measurements and the NeQuick model, the unknown
parameters of the spatial covariance model are derived using maximum
likelihood estimation (MLE). Afterwards the electron densities of a given
grid are calculated by 3-D simple kriging of linear functionals, i.e.,
integrals, incorporating the obtained covariance model, the NeQuick model and
the STEC measurements. The subsequent sections describe each step in more
detail.
Spatial covariance model of electron density
In order to establish a spatial covariance model of electron density,
information about the behavior of the ionosphere is necessary.
suggested the separation of the spatial covariance model
into horizontal and vertical components to take the geometric
anisotropy of the ionosphere, i.e., directionally dependent correlation
lengths, into account. and confirmed this approach with the
analysis of GPS and incoherent scatter radar observations revealing different
correlation lengths in latitude, longitude and height direction.
Furthermore, the investigations of and show that the
exponential covariance model might be appropriate to describe spatial dependencies of the ionosphere.
Moreover, the non-stationarity of the ionosphere should be considered within
the spatial covariance model. In other words, if we assume the electron
densities Ne(xi) at arbitrary coordinates xi as a Gaussian
random field (Ne(x1),...,Ne(xn)), then the corresponding
cumulative distribution functions are
N(μ1,σ12),…,N(μn,σn2) described by the
expectation values μ1,…,μn and variances σ12,…,σn2,
vary in space.Based on this information, we set up the following spatial covariance model
of the electron density with the unknown parameter vector θ=(θ1,…,θ4):
covθ(Ne(xi),Ne(xj)):=μ(Ne(xi))⋅μ(Ne(xj))⋅θ1⋅ch(hh;θ2,θ3)⋅cv(hv;θ4),
where xi,xj represent Earth-centered, Earth-fixed (ECEF)
coordinates of the WGS84 reference ellipsoid, μ(Ne(xi)),μ(Ne(xj)) are the expected electron densities at the coordinates
xi,xj, θ1 is the sill parameter and
ch(hh;θ2,θ3) and cv(hv;θ4) are the horizontal and
vertical spatial covariance models, respectively. The quantities ch and
cv are respectively driven by their correspondent model parameters
θ2,…,θ4 and the horizontal or
vertical distance, hh and hv, between two coordinates, xi and xj.The horizontal covariance model is defined as
ch(hh;θ2,θ3):=e-3hh,withhh:=θ2000θ2000θ3xi|xi|-xj|xj|.
By means of the normalization, the ECEF coordinates xi and
xj are projected to the unit sphere, and the influence of the height
component becomes negligible. Furthermore, the assumed anisotropic correlation
lengths in latitude and longitude direction are modeled by a diagonal matrix
containing the parameters θ2 and θ3
(see p. 98).Furthermore, the vertical covariance model is chosen as to be
cv(hv;θ4):=e-3hvθ4withhv:=|hgti-hgtj|,
where hgti and hgtj are the corresponding heights of xi and xj
over the WGS84 reference ellipsoid.
Considering Eqs. ()–(), it becomes clear that ch and cv describe
the anisotropy of the ionosphere and the expectation values
μ(Ne(xi)) and μ(Ne(xj)) are incorporated to take into
account the ionosphere's non-stationarity. For instance, let us assume
Ne(x1) and Ne(x2) to be around the ionospheric F2 peak height of 300 km, and Ne(x3) and Ne(x4) at a height
of about 2000 km, where the horizontal distances hh(x1,x2) and
hh(x3,x4) are equal. Then with Eq. () it follows
that covθ(Ne(x1),Ne(x2)) is usually higher than
covθ(Ne(x3),Ne(x4)).
Estimation of the spatial covariance model parameters using STEC
The spatial covariance model in Sect.
depends on the parameters θ=(θ1,…,θ4). In order to
estimate them, a background model and STEC measurements are used. Subsequently
we briefly describe the calculation of ionospheric STEC measurements as well
as the background model. Therefore, we particularly derive the relationship
between θ and the STEC measurements and outline the MLE of
θ.
Background model
As background model an arbitrary electron density model can be considered,
e.g., the NeQuick or the IRI model. Within this paper we apply the three-dimensional NeQuick model version 2.0.2 released in November, 2010 (B. Nava, personal communication, 15 January 2013). It serves
as a non-stationary trend model providing the expected electron density
μ(Ne(xi)) at a coordinate xi and the STEC along a ray
path s. Additionally, the NeQuick electron density background is
used within the 3-D simple kriging to stabilize the tomography of the
ionospheric electron density, which presents an ill-posed and strongly
underdetermined inverse problem.
The NeQuick model is currently being developed at the International Centre for Theoretical
Physics (ICTP) in Trieste, Italy, and at the University of Graz, Austria (see
). It is widely used in
ionospheric delay and TEC estimation for trans-ionospheric ray paths (see,
e.g.,
). The vertical electron density profiles of the NeQuick
model are modeled by summing up five semi-Epstein layers whose shape
parameters, such as peak ionization, peak height and semi-thickness, are
deduced from the ITU-R (ITU Radio-Communication Sector) foF2 and M(3000)F2
models (see ). Therefore, the modeled electron density
distribution inherits the spatial variances provided in the ITU-R maps via
the peak ionization and peak height information. Additionally, the impact of
the geomagnetic field on the ionospheric plasma density distribution is
determined using a specific geomagnetic parameter called modip which is
calculated from the Earth's magnetic field. The NeQuick model is driven by
the solar activity level, either by the Zurich sunspot number or by the solar
radio flux at 10.7 cm wave length (F10.7 index). In the present work, we used
the daily F10.7 index to drive the NeQuick model.
STEC measurements
GNSS STEC measurements represent integral measurements of the electron density
along a ray path s extending from a satellite position to a receiver position. By the combination of
GPS dual-frequency carrier-phase (L1, L2) and code pseudorange (P1, P2) measurements, we derive the low-noise
carrier-phase-relative STEC and the code-relative STEC. Subsequently, the code relative STEC is
smoothed by the carrier-phase relative STEC to obtain unambiguous relative STEC measurements with a
low noise level. However, the relative STEC measurements are impacted by the receiver and satellite inter-frequency biases. We use a model-assisted technique to separate the ionospheric delay (i.e., absolute STEC) and the receiver and satellite inter-frequency biases. For this, the two-dimensional Neustrelitz TEC Model (NTCM) is applied together with a mapping function based on a thin-shell ionosphere at 400 km height. For details about the absolute STEC estimation and the separation of inter-frequency satellite and receiver
biases we refer the reader to . Based on this approach, we estimate STEC for all
receiver–satellite link geometries having elevation angles equal to or greater than
10∘.
Relationship between electron density covariance and STEC measurements
We assume zero-mean Gaussian distributed and uncorrelated STEC measurement
errors εs ∼ N(0,σs2) and
state the STEC measurement model as follows:
STECs=∫sNe(s)ds+εs,
where Ne(s) values are the electron densities along the satellite–receiver ray path
s. Since the calibration of the STEC measurements is done accordingly to
, the assumption of uncorrelated measurement errors is tricky.
However for the purpose of the work, possible cross covariance errors are not considered.
investigated the calibration errors on experimental STEC
measurements determined by GPS. He found out that the leveling of the
carrier to the code measurements is mainly affected by the code multipath.
Consequently, a common choice of the measurement error variance
σs2 might be defined as dependent on the elevation angle of
the satellite-to-receiver configuration assuming an increasing error budget with
decreasing elevation angle. In this study, we set the minimum STEC error to 1 TECU.
Considering Eqs. () and (), the relationship between the spatial covariance model
of the electron density and the covariance of the STEC measurements results
in
covθ(STECs,STECr)=covθ∫sNe(s)ds+εs,∫rNe(r)dr+εr=∫s∫rcovθ(Ne(s),Ne(r))drds+cov(εs,εr)withcov(εs,εr):=σs2 or σr2,if s=r0otherwise.
Assuming that the STEC measurements form a Gaussian random field
STEC→=(STECs1,…, STECsn)T
with the expectation values μ=(μ(STECs1),…,μ(STECsn))T
and the corresponding covariance matrix
(Σθ)ij:=covθ(STECsi,STECsj) with i,j∈{1,…,n}, the multivariate Gaussian probability density function (pdf) of the STEC measurements fθSTEC→
depends on θ=(θ1,…,θ4)∈R+⋅R+⋅R+⋅R+ and is defined as
fθSTEC→=1(2π)n⋅|Σθ|e-12STEC→-μTΣθ-1STEC→-μ,
where |Σθ| is the determinant of the covariance matrix
and the expectation values μ of the STEC measurements are derived
from the NeQuick model.
Thus, the aim is the estimation of the parameters θ maximizing
the Gaussian pdf of the STEC measurements. This maximum likelihood approach
is an optimization problem and can be stated as follows:
argmaxθLSTEC→(θ)=argmaxθlnfθSTEC→=argmaxθln((2π)n|Σθ|)-12-12STEC→-μTΣθ-1STEC→-μ=argmaxθ-ln(|Σθ|)-STEC→-μTΣθ-1STEC→-μ.
The maximization problem can be transformed into a minimization problem, for
which different software package solutions exist. Within this paper, we
used the Python-based software SciPy to solve the problem formulated in Eq. (). In particular, the algorithm of Powell is
applied, which works iteratively and performs sequential one-dimensional
minimization along each variable θ1,…,θ4 without calculating
derivatives of the objective function. For more details we refer the reader to
and . The initial guess for the parameter
vector θ is made empirically. We assume an electron density
standard deviation of about 12 % resulting into θ1≈0.016.
Furthermore, we briefly examine the maximum horizontal distance hh between two
electron densities along ray paths with ionospheric piercing points in the
considered reconstruction area; see Sect. . At
this maximum distance, the correlation is assumed to be zero for the initial
guess. Based on the investigations of , we choose to set the
initial guess for the parameters θ2 and θ3 to about 1.5. The
parameter θ4 controls the vertical correlation length and is set to
about 300 km in agreement with the analyses of .
Correlation coefficients between selected coordinates (black dots) and
their adjacent coordinates at a height of 300 km for DOY 22 (2011) at 10:00, 12:00 and 14:00 UTC.
Figure illustrates the estimated electron density
covariance models for the three different latitudes 45∘ N,
50∘ N and 55∘ N at the Greenwich meridian at a height of
300 km. The black marked circle represents the corresponding coordinate. The correlation coefficient with its surrounding points is calculated at local
times 10:00, 12:00 and 14:00, and color-coded from blue (no correlation) to
red (fully correlated).
The estimated parameters of the electron density covariance confirm the
assumption of anisotropic correlation lengths in latitude and longitude. Thus, its principal behavior agrees with the TEC correlation
analyses of . Furthermore, we observe the temporal evolution of
the horizontal covariance reaching its peak at 12:00 on day of year (DOY)
22 (2011) coinciding with the local time variations of the TEC correlation
distances described in .
3-D simple kriging of the electron density
Once the parameters θ of the spatial covariance model of the
electron density are derived, the electron density at a WGS84 coordinate
x can be estimated using simple kriging of linear functionals (i.e.,
integrals; see ) as
Ne^(x)=μ[Ne(x)]+covθNe(x),STECs1⋮covθNe(x),STECsnT︸=:Σx⋅Σθ-1⋅STEC→-μ,wherecovθNe(x),STECsi=covθNe(x),∫siNe(si)dsi=∫sicovθ(Ne(x),Ne(si))dsi.
Consequently in order to estimate the electron density at an arbitrary WGS84
coordinate x, the product Σx⋅Σθ-1∈R1⋅n forms the weights
λ=(λ1,…,λn)T, which are used to add the
difference between the GNSS-based STEC measurements and the expected STEC, in
an optimal way, to the expected/modeled electron density μ[Ne(x)].
Moreover once the weights are calculated, the simple kriging estimation error
σSK2(x) at a point x is derived as (see p.
153)
σSK2(x)=covθNe(x),Ne(x)-λT⋅ΣxT.
For computational efficiency, Eq. () is extended to the dual
kriging equations, enabling the estimation of Ne at several WGS84 locations
x1,…,xm simultaneously:
Ne^(x1,…,xm)=μ[Ne(x1),…,Ne(xm)]T+covθNe(x1),STECs1…covθNe(x1),STECsn⋮covθNe(xm),STECs1…covθNe(xm),STECsn⋅Σθ-1⋅STEC→-μ,wherecovθNe(xk),STECsi=covθNe(xk),∫siNe(si)dsi.
Typical measurement geometry over a part of Europe for DOY 22 (2011)
using the IGS ground-station GNSS Network: IGS ground-stations (black
triangles), ionospheric piercing points of the STEC measurements (blue
circles). The ionosondes RO041 and DB049 (red triangles) are used for
validation.
Regional application
Validation scenario
In this study we apply the outlined method to a part of Europe
at 40–60∘ N and 30∘ W–30∘ E for DOY 22 (2011). We chose this region mainly
for two reasons. Firstly the availability of STEC measurements is relatively
good, and secondly within this region we expect better performance from the
NeQuick model, which represents the background of the method, and hence an
important input for the covariance and electron density estimation.
DOY 22 (2011) is within the current maximum of solar cycle 24 but reveals
quiet ionospheric conditions with a F10.7 of 84 flux units and an average
geomagnetic Kp index about 1. The STEC measurements are derived from the
1 Hz GPS L1 and L2 measurements of the International GNSS Service (IGS)
ground-station network. The measurements whose corresponding ionospheric piercing
points at a shell height of 400 km are within the described area are used for processing. On average, about 50 IGS stations with about 300 STEC
measurements are available for a 1 s epoch; see Fig. . Consequently, the tomography of the ionosphere is a
strongly underdetermined inverse problem with extremely limited angle
geometry. Since especially the height resolution is complicated (see
) we decide to make use of STEC measurements with an
elevation angle down to 10∘.
For validation, we chose the ionosondes DB049 in Dourbes, Belgium, at
50.1∘ N, 4.6∘ E and RO041 in Rome, Italy, at 41.9∘ N,
12.5∘ E; see Fig. . At these coordinates the
height profiles of the ionospheric electron density are reconstructed and
compared with the available ionosonde profiles downloaded from the Space
Physics Interactive Data Resource (SPIDR). The reconstructed F2 layer characteristics in particular, in terms of NmF2 and hmF2, are validated against the
measurements. For this purpose electron density profiles with a
1 km height resolution are estimated.
Example of the NeQuick modeled electron densities (left) and the
reconstructed electron densities (right) on DOY 22 (2011) at 12:00 UTC over 40–60∘ N and 30∘ W–30∘ E. The heights 200–350 km within the selected European region are shown.
Comparison of the NeQuick electron density profile (blue), the
reconstructed electron density profile (green) and the ionosonde profile
(red) of the ionosonde station DB049 on DOY 22 (2011) at 02:00, 12:00, 13:30 and
16:00 UTC.
Preliminary results
As an example for the application of the developed ionospheric tomography,
Fig. illustrates the electron density layers of DOY
22 (2011), 12:00 UTC, at altitudes between 200 and 350 km. On the
left-hand side, the background electron densities derived using the NeQuick model
are displayed, whereas the right-hand panel depicts the electron densities
calculated by the 3-D kriging. The reconstructed electron density layers
reveal lower electron densities than the one provided by the NeQuick model.
This indicates GPS STEC measurements are less than the one expected by the
NeQuick model.
Figures and show the
electron density profiles at Dourbes, Belgium, on DOY 22 (2011) at 02:00,
12:00, 13:30 and 16:00 UTC and at Rome, Italy, on DOY 22 (2011) at
02:00, 11:45, 13:30 and 16:00 UTC. For the presented profiles
an improvement of the NmF2 parameter is notable but simultaneously the peak
height hmF2 is apparently not correctly reconstructed.
Comparison of the NeQuick electron density profile (blue), the
reconstructed electron density profile (green) and the ionosonde profile
(red) of the ionosonde station RO041 on DOY 22 (2011) at 02:00, 11:45, 13:30 and
16:00 UTC.
Comparison of the NeQuick model (blue), the reconstructed (green)
and the ionosonde (red) F2 layer peak characteristics at the ionosonde
station locations of DB049 (left column) and RO041 (right column) on
DOY 22 (2011).
Comparison of the relative absolute errors of the NeQuick
model and the 3-D kriging at the ionosonde station locations of DB049 and
RO041 for DOY 22 (2011), all values are given in percent.
RO041
NmF2
RO041
hmF2
DB049
NmF2
DB049
hmF2
mean
RMS
mean
RMS
mean
RMS
mean
RMS
NeQuick
38.1
51.5
10.8
12.9
32.6
42.7
9.4
11.7
3-D kriging
25.0
35.5
11.3
13.5
25.6
39.3
10.0
12.5
In order to obtain a first assessment of the 3-D kriging, we derive the F2 layer peak density and height with the 3-D kriging for DOY 22 (2011). In Fig. and Table the
reconstructed F2 layer characteristics are validated against the
characteristics of DB049 and RO041 measured at 15 min cadence. It is
clear that the 3-D kriging can provide enhanced NmF2 estimations with
respect to the background model, especially at the ionosonde station RO041.
Additionally, the limitation regarding the hmF2 estimation becomes more evident.
For the selected DOY almost no difference between the hmF2 values given by
the NeQuick model and the 3-D kriging is found. These results are underpinned
by the mean and the root-mean-square relative absolute errors of the
NeQuick model and the 3-D kriging in Table . The
relative absolute error |ϵrel| is calculated as |ϵrel|=|y-y^|/y⋅ 100, where y is the measured value of the ionosonde
and y^ the F2 layer characteristic given by the NeQuick model and the
3-D kriging, respectively. For both ionosonde locations, the mean and the RMS
error of NmF2 are decreased whereas no reduction is obtained for the hmF2
errors. Similar results are obtained by .
compare foF2 and the maximum usable frequency factor M(3000)F2 estimated by
the Utah State University Global Assimilation of Ionospheric Measurements
(GAIM) model with Australian ionosonde station data. They observed that GAIM
reproduces foF2 better than M(3000)F2, which is related to hmF2. For a better
M(3000)F2/hmF2 reconstruction, the integration of additional ionosonde
profiles and the smart handling of these data within a TEC-rich environment
are noted to be crucial.
The developed 3-D kriging will be validated in more detail in future work and, in particular, the issue of the hmF2 reconstruction will be addressed.
Our goal is to enhance initialization of the background model by using the ionosonde F2 layer
measurements, as well as the assimilation of ionosonde electron density
profiles; see, e.g., .
Conclusions
The presented 3-D simple kriging of the ionospheric electron
density is a novel tool for ionospheric tomography and its development is
still in progress. This approach is based on the estimation of the electron
density's spatial covariance, which is one of the most crucial inputs for kriging and also for different data assimilation methods. We use the
relationship of this covariance to the covariance of the STEC measurements and
outline the possible estimation of its parameters using the STEC measurements.
Compared to the ionosonde electron density profiles, for the considered DOY
22 (2011) and locations the calculated electron density profiles show a
promising gain with respect to the background model, in particular for the
estimated NmF2.
In this study solely ground-based STEC measurements are incorporated.
Nevertheless the approach is extendable to various ionospheric measurements
such as ionosonde profiles and peak density measurements, radio occultation
measurements, space-based STEC measurements and in situ measurements. In the next stage of our research we will examine this topic in order to improve the
estimation of spatial covariance and electron densities. Our
first effort will be the integration of the ionosonde electron density
profiles, since ionosonde measurements are assumed to be the most reliable
and available data type, which can provide vertical information around and
below the F2 layer peak. showed that in specific cases
the assimilation of ionosonde data alone can yield even more accurate foF2
results than those obtained with the incorporation of GNSS TEC data in addition
to the ionosonde data.
Furthermore, focus will be directed towards the inclusion of temporal information, which
could be done, for instance, by developing a spatial–temporal
covariance function or embedding the approach into a Kalman filter
environment. Subsequently, the detailed validation will be one of the most
challenging tasks. Therefore we plan a study similar to
, investigating the capability of the 3-D
kriging to reconstruct the ionospheric characteristics, e.g., foF2, F2 layer
thickness and M(3000)F2. Based on these results, we will refine the approach
for the provision of ionospheric corrections for satellite-based radar
missions in regions with dense GNSS networks.