ANGEOAnnales GeophysicaeANGEOAnn. Geophys.1432-0576Copernicus PublicationsGöttingen, Germany10.5194/angeo-35-263-2017Near real-time estimation of ionosphere vertical total electron content from GNSS satellites using B-splines in a Kalman filterErdoganEreneren.erdogan@tum.dehttps://orcid.org/0000-0001-7468-0617SchmidtMichaelSeitzFlorianhttps://orcid.org/0000-0002-0718-6069DurmazMuratDeutsches Geodätisches Forschungsinstitut der Technischen Universität München (DGFI-TUM), Arcisstraße 21, 80333 München, GermanyGeomatics Engineering Division, Civil Engineering Department, Middle East Technical University (METU), 06800 Ankara, TurkeyEren Erdogan (eren.erdogan@tum.de)27February20173522632773November201614January201719January2017This 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://angeo.copernicus.org/articles/35/263/2017/angeo-35-263-2017.htmlThe full text article is available as a PDF file from https://angeo.copernicus.org/articles/35/263/2017/angeo-35-263-2017.pdf
Although the number of terrestrial global navigation satellite
system (GNSS) receivers supported by the International GNSS Service (IGS) is
rapidly growing, the worldwide rather inhomogeneously distributed observation
sites do not allow the generation of high-resolution global ionosphere
products. Conversely, with the regionally enormous increase in highly precise
GNSS data, the demands on (near) real-time ionosphere products, necessary in
many applications such as navigation, are growing very fast. Consequently,
many analysis centers accepted the responsibility of generating such
products. In this regard, the primary objective of our work is to develop a
near real-time processing framework for the estimation of the vertical total
electron content (VTEC) of the ionosphere using proper models that are
capable of a global representation adapted to the real data distribution.
The global VTEC representation developed in this work is based on a series
expansion in terms of compactly supported B-spline functions, which allow for
an appropriate handling of the heterogeneous data distribution, including data
gaps. The corresponding series coefficients and additional parameters such as
differential code biases of the GNSS satellites and receivers constitute the
set of unknown parameters. The Kalman filter (KF), as a popular recursive
estimator, allows processing of the data immediately after acquisition and paves
the way of sequential (near) real-time estimation of the unknown parameters. To exploit the advantages of the chosen data representation and
the estimation procedure, the B-spline model is incorporated into the KF
under the consideration of necessary constraints. Based on a preprocessing strategy, the developed approach
utilizes
hourly batches of GPS and GLONASS observations provided by the IGS data
centers with a latency of 1 h in its current realization.
Two methods for validation of the results are performed, namely the self
consistency analysis and a comparison with Jason-2 altimetry data. The highly
promising validation results allow the conclusion that under the investigated
conditions our derived near real-time product is of the same accuracy level
as the so-called final post-processed products provided by the IGS with a
latency of several days or even weeks.
Ionosphere (ionospheric disturbances; modeling and forecasting; instruments and techniques)Introduction
The ionosphere constitutes the upper part of the atmosphere, extending from
approximately 60 to 1500 km above the Earth's surface, enriched with free
electrons and ions . The knowledge of the structure and
dynamics of the ionospheric plasma has great importance for various
scientific applications and services, such as telecommunication through radio
signals, point positioning based on global navigation satellite systems
(GNSSs) ,
the monitoring of space weather events such as solar flares and coronal mass
ejections , the investigations of
ionospheric anomalies preceding or following a natural hazard such as
earthquakes , and investigation of the ionospheric effects on
thermospheric mass density, wind, and temperature .
The ionospheric plasma density varies with time and location and exhibits a
coupled system with its environment: the Sun, the Earth's lower atmosphere,
the thermosphere and the magnetosphere . Interactions
between
the thermospheric neutral winds and ionized plasma drive the ionospheric
charged particles in motion and lead to separation of charges, resulting in
the creation of a polarized electrical field E. During daytime, the
combination of this eastward-oriented electrical field E with the
Earth's horizontally northward-oriented magnetic field B causes an
upward E×B drift of the ionized plasma. Following the
upward drift, the plasma diffuses through the magnetic field lines and
creates two crests with high ionization at both sides of the magnetic
equator,
which is also known as fountain effect . In addition, at
high latitudes, the interaction of the Earth's magnetosphere with the
interplanetary magnetic field attached to the solar winds as well as to the
space weather events increases the complexity of the system. Today, the
advances in space-geodetic techniques, such as terrestrial GNSS, spaceborne
radio occultations to low-Earth-orbiting (LEO) satellites as well as
satellite altimetry, facilitates the monitoring of the structure of the ionosphere
with an improved spatial and temporal resolution. Particularly, GNSS offers
an attractive alternative to traditional methods, such as ionosondes, for
monitoring the electron content within the ionosphere in terms of volume and
global data distribution.
The International GNSS Service (IGS) delivers large volumes of GNSS data with
different latencies (e.g., real time, hourly) acquired from continuously
operating terrestrial GNSS receivers distributed worldwide. The four IGS
Ionosphere Associate Analysis Centers (IAACs), namely the Jet Propulsion
Laboratory (JPL), the Center for Orbit Determination in Europe (CODE), the
European Space Operations Center of the European Space Agency (ESOC) and the
Universitat Politècnica de Catalunya (UPC), monitor the ionosphere and
evaluate relevant parameters using dual frequency GNSS receivers. Several
modeling approaches for ionospheric parameters have been proposed. A common
approach, generally denoted as single-layer model (SLM), is based on the
assumption that the electrons in the ionosphere are concentrated within a
thin shell at a fixed altitude above the Earth
. The spatial variations of electron content
in this single layer are represented by a proper mathematical model such as
spherical harmonics , B-splines
, spatially defined total electron content (TEC) grids
, polynomials , wavelets
, or the MARS and BMARS approaches
. Furthermore, three-dimensional models that consider the
variation with altitude were also studied
. The IGS delivers post-processed global VTEC products by
combining VTEC maps from the different analysis centers mentioned before
. The combination of these products computed
with independent algorithms performed by the analysis centers contributes to
the accuracy, the integrity and the availability of the global VTEC.
In this context, one of the main goals of this study is to develop an
alternative approach that contributes to these modeling efforts by generating
near real-time products. To be more specific, series expansions in terms of
tensor products of compactly supported B-spline functions are used to
represent the spatial variation of the global VTEC; introductory studies on
this topic were published by , , and
. The advantages of B-spline representations for VTEC
modeling in comparison to spherical harmonics were also revealed by
. The results of this study show that the B-spline series
approach features in considerable improvements if the input data is
distributed heterogeneously.
Considering the increasing demands on high-precision (near) real-time global
ionosphere products, including global VTEC maps, an estimation strategy
becomes obviously important to appropriately handle the large amount of
ionosphere data as well as processing the data once it is available. The
Kalman filter (KF) is a popular filtering technique and does not require
storing measurements of the past since it is in the batch filtering for estimating
the current state of a system and data can be processed immediately after
acquisition . The KF has been successfully applied
in (near) real-time ionosphere modeling by many authors; see
, and . Here, apart from the
other studies, we focus on the implementation of a B-spline representation
into the KF for ionospheric parameter estimation.
In summary, the goal of the present study is to develop a near real-time
processing framework to globally monitor the spatial and temporal variations
within the ionosphere by exploiting the advantages of the B-spline series
expansion and the recursive filtering using GPS and GLONASS measurements.
Finally, it shall be discussed how the quality of this near real-time product
is in comparison to the so-called final, i.e., the post-processed
high-quality VTEC products of the
IGS and its IAACs provided with a latency of days or even weeks.
The paper is outlined as follows. Section explains
the theoretical foundations for obtaining ionosphere observations from raw hourly
GNSS data. In Sect. the B-spline representation
for global VTEC modeling is explained.
Section gives the background for the
sequential estimation algorithm using the KF. The section comprises subparts,
introducing the measurement model of the filter, the definition of model
constraints, the prediction model of the filter, the foundation of the Kalman
filtering, and the handling of the model constraints in the filter and data editing
concepts. Section summarizes
the entire near real-time estimation procedure from an application point of
view. Note that the exemplified generation of VTEC maps in this paper is
based just on GNSS data. Validation concepts – partly based on satellite
altimetry data – and the results are given in Sect. .
Finally, Sect. provides the conclusion and future work.
Ionosphere TEC measurements from GNSS
The free electrons within the ionosphere affect the propagation of
electromagnetic waves, i.e., they cause a frequency-dependent delay in the
transmitted radio signals. Although the ionospheric effect on GNSS signals
is not desirable in positioning and navigation, it provides valuable
information for the investigation of the electron content of the ionosphere.
The magnitude of the delay depends on the electron density Ne along
the ray path between the satellite s and the receiver r and is proportional
to the slant total electron content (STEC) defined as
STEC=∫rsNedl.Ne is measured in electrons per cubic meter. Following
Eq. (), the vertical total electron content
(VTEC) is defined as the integration of the electron density along the
vertical, i.e., in height direction.
In order to extract ionospheric information from dual-frequency GNSS
measurements, the geometry-free linear combination can be used
. Firstly, the equations leading to the ionospheric
observables Pr,Is and Lr,Is derived from
combinations of the pseudo-range (code) and carrier-phase measurements,
respectively, are defined as
Pr,Is=Pr,f2s-Pr,f1s=α⋅STEC+br+bs+ϵPLr,Is=Φr,f1s-Φr,f2s=α⋅STEC+Br+Bs+Carc,rs+ϵL,
where Pr,fis and Φr,fis
with i∈{1,2} stand for the pseudo-range and the carrier-phase
measurements observed simultaneously by the receiver r from satellite s. The
subscripts 1 and 2 denote the two carrier frequencies f1 and f2
with the corresponding wavelengths λ1 and λ2.
Br and Bs are the receiver and satellite
inter-frequency biases
(IFBs) for the carrier phase observations; similarly, br and
bs stand for the differential code biases (DCBs) of the receiver
and the satellite. α is a frequency-dependent constant factor.
Furthermore, Carc,rs is the ambiguity bias of the
carrier-phase, and ϵP and ϵL account for the
measurement errors.
The pseudo-range measurements are rather noisy but unambiguous, while the
carrier-phase data are significantly more precise but biased. To exploit the
precision of the phase measurements, an offset
CPBrs, including the terms IFB, DCB and
Carc,rs, is computed by averaging the differences between
Lr,Is and Pr,Is for every
continuous arc that shares a common phase bias
according to
CPBrs≈1N∑j=1NLr,Is-Pr,Isj,
where N is the number of observations continuously measured along the arc.
An elevation-dependent threshold is established to select the more precise
observations with higher elevation angle for the computation of
CPBrs. Then, a leveled geometry-free phase
observation Lr,4s̃ for a continuous arc
becomes
Lr,4s̃=Lr,Is-CPBrs=α⋅STEC+br+bs+ϵL4.
The leveling technique is applied to the hourly data sets of GPS and GLONASS
observations obtained from the IGS data servers.
Global VTEC representation with B-splines
As already mentioned in the introduction, appropriate approaches for
representing VTEC are two-dimensional series expansions in terms of spherical
harmonics or B-spline functions. In the latter case the basis functions can
be set up by tensor products of polynomial and trigonometric B-splines. To be
more specific, the B-spline representation of the global VTEC reads
VTEC(φ,λ)=∑k1=0KJ1-1∑k2=0KJ2-1dk1,k2J1,J2NJ1,k12(φ)TJ2,k22(λ), where the quantities dk1,k2J1,J2 mean the
unknown series coefficients, NJ1,k12(φ) are the end-point
interpolating polynomial B-spline functions of order three depending on the
latitude φ, and TJ2,k22(λ) are the trigonometric
B-splines of order three depending on the longitude λ. The values
J1 and J2 are called levels, the numbers k1 and k2 define
the geometrical positions of the two-dimensional basis functions
NJ1,k12(φ)⋅TJ2,k22(λ) on the sphere.
The value KJ1 stands for the number of polynomial B-spline functions
according to the associated level J1; its numerical value is given by
KJ1=2J1+2. Similarly, the number of trigonometric B-spline
functions KJ2 for the level J2 is defined by KJ2=3⋅2J2+2. The polynomial B-spline functions
NJ1,k12(φ) and the trigonometric B-spline functions
TJ2,k22(λ) are compactly supported, which means the
functions are different from zero only within a small subinterval (see
, and the references therein).
Global VTEC representation in a solar geomagnetic coordinate system
using different resolution levels; (a)J1=3,J2=1;
(b)J1=4,J2=2; (c)J1=4,J2=3;
(d)J1=5,J2=3. The circles and triangles represent the data
locations for GLONASS and GPS, respectively, and the colors indicate the
corresponding VTEC magnitudes related to the ionospheric pierce points (IPP)
obtained from a reference map. In all the four panels at the bottom and on
the left side the KJ2 trigonometric B-spline functions
TJ2,k22(λ) with k2=0,1,…,KJ2-1 and the
KJ1 polynomial B-spline functions NJ1,k12(φ) with
k1=0,1,…,KJ1-1 are visualized. The straight black lines on
the maps show the corresponding knot point locations. The red line shows the
prime meridian at Greenwich.
The selection of appropriate resolution levels J1 and J2 requires the
consideration of different criteria, namely the distribution of the input
data, the computational burden and the desired level of smoothness.
Figure clearly shows that the higher the level
chosen, the larger the number of B-spline basis functions, i.e., the higher
the resolution of the VTEC representation. Appropriate levels of the
B-splines can be obtained from the distribution of the input data
. This procedure has already been applied
successfully for ionosphere modeling from GPS occultation data
. In case of near real-time applications, the level
values may have to fulfill computation time constraints. A trade-off between
the resolution level and the computational burden becomes another necessary
issue since the higher the resolution, the larger the number of unknown
coefficients and the higher the time consumption the parameter estimation
procedure based on Kalman filtering requires. The GNSS data distribution is
generally rather inhomogeneous, with large data gaps, especially over the
oceans. B-spline representations with high level values can result in basis
functions without any data support. For instance, all the basis functions in
Fig. a are supported by GNSS observations. However,
this is not the case for the different levels of representations shown in the
other three panels of Fig. . Conversely, a
representation with level values chosen too low may not describe the spatial
VTEC variations sufficiently and may cause an undesired smoothing, i.e., a loss of
information. For example, a reference VTEC map used as input observation is
illustrated in Fig. a.
Figure b depicts the reconstructed map using the
B-spline levels J1=3 and J2=1: it smooths the details too much. An
increase in the level values to J1=4 and J2=3 results in a better
reconstruction of the reference map, as is shown in
Fig. c. Consequently, in order to achieve a
representation quality comparable to IGS products while considering the
aforementioned criteria, the B-spline levels are set to J1=4 and J2=3 in this study.
Reconstructed VTEC maps: (a) a reference VTEC map obtained
from IGS; (b) reconstruction following Eq. ()
with B-spline levels J1=3 and J2=1; (c) same
as (b) but with B-spline levels J1=4 and J2=3.
Sequential estimation of global VTEC
The KF () was used in this work as a
sequential estimator to compute the ionospheric parameters in near real time.
Since the KF is of recursive nature, measurements from the past do not need to
be stored . The current state is updated as soon as new
observations are available. This means a crucial advantage for (near)
real-time applications because this way the filter allows assimilation of
observations as soon as possible without waiting for another group of
observations.
Here, for the estimation of the ionospheric target parameters, the system
equations, including the measurement model and the prediction model, are linear
and the observations are assumed to have a Gaussian distribution. Then the
KF provides an optimal recursive estimator in terms of minimum variance
estimation (see ). The linear system
of equations in a discrete form is defined as
βk=Fkβk-1+wk-1yk=Xkβk+ek,
where k is the time stamp, Fk is the transition matrix,
βk is the vector of the unknown parameters, yk is
the vector of the measurements and Xk is the corresponding
design matrix. The measurement error vector ek and the vector
wk of the process noise are assumed to be white noise vectors with
the expectation values E(ek)=0 and
E(wk)=0; furthermore, the covariance matrices Σy and Σw, respectively,
fulfill the assumptions
E[wkwlT]=Σwδk,landE[ekelT]=Σyδk,l
and
E[wkelT]=0,
where δk,l is the delta symbol with δk,l=1 if k=l
and δk,l=0 for k≠l.
Equation () means that the vectors
wk and el are assumed to be mutually independent.
Measurement model
For the sake of clarity, the GNSS STEC measurement
Lr,4s̃ from Eq. () is
redefined to express both GPS and GLONASS measurement models separately as
yGPS+eGPS=αGPS⋅m(z)⋅VTEC+br,GPS+bGPSsyGLO+eGLO=αGLO⋅m(z)⋅VTEC+br,GLO+bGLOs,
i.e., Lr,4s̃∈{yGPS,yGLO}. The DCB values br,GPS and br,GLO
stand for the GPS and GLONASS receiver biases.
Furthermore,
bGPSs and bGLOs represent the
unknown DCB values of the respective satellites. The mapping function
m(z) depending on the zenith angle z projects STEC into the vertical by
VTEC=1m(z)STEC.
A widely accepted mapping function, namely the modified single-layer mapping (MSLM) function is defined as m(z)=1cosz′=11-sinz′2=1-ReRe+Hsinαmz2-1/2,
with αm=0.9782, the single-layer height H=506.7km and
the mean Earth radius Re=6371km, which are taken from
. The intersection of a signal path connecting satellite and
receiver (approximated as the line of sight) with the single layer is denoted
as the ionospheric pierce point (IPP). The angles z and z′ are the zenith
angles of the satellite at the receiver position and the IPP.
The two observation vectors yGPS and yGLO for
the GPS and the GLONASS measurements, respectively, build the measurement
vector y as given in
Eq. (). The state vector
β consists of the sub-vector d=(dk1,k2J1,J2) of the unknown B-spline coefficients
dk1,k2J1,J2 as defined in Eq. (), and the
sub-vectors br,GPS, br,GLO,
bGPSs and bGLOs of the
receiver and satellite DCBs. Consequently, the vectors β
and y read
β=dbr,GPSbGPSsbr,GLObGLOs,y=yGPSyGLO.
Although the satellite systems GPS and GLONASS refer to the same space
geodetic technique, they are operated by different agencies with a different
design, constellation and signal structure, which can lead to different
sensitivities within the parameter estimation. To account for this fact,
instead of assigning one variance factor, an individual variance factor for
each observation group is introduced. Furthermore, it is assumed that the
vectors yGPS and yGLO are uncorrelated.
Therefore, the measurement covariance matrix Σy reads
Σy=σGPS2PGPS-100σGLO2PGLO-1;
herein σGPS2 and σGLO2 are the two
unknown variance factors mentioned before, PGPS and
PGLO are given positive definite weight matrices. Since
the precision of the GNSS measurement depends on the elevation angle, a
common strategy is to adapt an elevation-dependent weighting scheme for each
individual observation. Therefore, a diagonal element pii of the weight
matrix P=(pij) is defined by (adapted from )
pii=1/(1+sin2(zi)),
where zi is the zenith angle of the ith measurement at a given GNSS
receiver location. The non-diagonal elements pij with i≠j are
usually set to zero.
Model constraints
The modeling approach includes constraints defined for the different groups
of unknown parameters. One group of constraints preserves the spherical
geometry since the two-dimensional B-spline model as introduced in
Eq. () is applied to the representation of a function
defined on a sphere. The constraints to be applied can be summarized as pole
constraints, pole continuity constraints and longitude periodicity
constraints. The first group of constraints forces the VTEC values to be the
same at the poles. The second group ensures the equality of tangent planes of
VTEC values at the poles, and the last group is to preserve continuity of the
VTEC values at the longitudinal boundaries. Considering these requirements
the constraint equation reads
Hdd=0,
with the matrix Hd of given coefficients .
To handle the rank deficiency problem related to the satellite and receiver
DCBs, a zero mean condition is usually applied by the IAACs within the
adjustment procedure. In the following, we also rely on this assumption and
introduce the constraint equations
HGPSbGPS=0HGLObGLO=0. Herein HGPS and
HGLO are matrices of given coefficients, respectively.
Prediction model
Many time-varying models for representing the ionospheric dynamics are based
on a KF approach, for
instance, the physics-based model developed by for
electron density modeling. The dynamic system within a KF is often realized
by empirical models such as the random walk or the
Gauss–Markov process .
The selection of a proper coordinate system is probably one of the most
important issues in monitoring the temporal variations of the ionosphere. The
KF problem can be solved both in a Sun-fixed or in an
Earth-fixed coordinate system . Since the effect of the
Earth's diurnal motion is mitigated, the ionosphere varies much slower in a
Sun-fixed system and could be assumed as static for a certain time interval.
Thus, we follow this argumentation and handle the global VTEC representation
according to Eq. () in a Sun-fixed coordinate system
spanned by the solar geomagnetic longitude, the geomagnetic latitude and the
radial distance from the origin, i.e., the geocenter. In this coordinate
system not only VTEC itself but also the B-spline coefficients vary rather
slowly. Hence, a random walk
approach could be performed for the prediction of the
coefficients from one measurement epoch to the next. Moreover, the satellite
and receiver DCBs are quite stable over a long period, even over a few days
. Consequently, they can also be modeled with a random walk
approach. In summary, all parameters of the ionospheric state vector are
treated as random walk processes in the time domain, which leads to a
transition matrix Fk as introduced in
Eq. () equal to the identity matrix
I.
The covariance matrix Σw of the process noise stands
for uncertainties introduced by deficiencies in the model. Here,
Σw is defined as a diagonal matrix with a variable
but unknown precision factor for each component of the ionospheric state
vector such that
Σw=σd2In00000σbr,GPS2Im00000σbGPSs2Ik00000σbr,GLO2Il00000σbGLOs2Ih,
where In,Im,Ik,Il and
Ih are identity matrices of different sizes. The quantities
σd2, σbr,GPS2,
σbr,GLO2, σbGPSs2
and σbGLOs2 are the unknown variance factors of
the process noise covariance matrix referring to the corresponding B-spline
coefficients as well as to the receiver and satellite DCBs for GPS and
GLONASS, respectively. If model uncertainties cannot be described precisely,
a common practice is manually conducting tests via multiple runs of the
filter . However, alternatively, adaptive methods can be
applied to compute the process noise covariance matrix as well as the
measurement covariance matrix in Eq. ()
during the running time (see ). In this research, the
process noise covariance matrix was constructed from an extensive data
analysis.
Kalman filtering
The solution of the estimation
problem as defined in Eqs. () and
() by using a KF consists generally
of the sequential application of a prediction step – also known as time
update – and a correction step – known as the measurement update. In the
prediction step, the current ionospheric state vector
β^k-1 and its covariance matrix
Σ^β,k-1 are propagated from the time
epoch k-1 to the next time epoch k using a proper prediction model in
order to obtain a predicted state vector βk- and the
related predicted covariance matrix Σβ,k- of
the next step k as
βk-=Fkβ^k-1,Σβ,k-=FkΣ^β,k-1FkT+Σw,
where the symbol “-” indicates predicted values. In our application, the
sampling interval, i.e., the step size from one epoch k-1 to the next epoch
k is set to 5 min.
Once the prediction step is performed, the corrected state vector and its
covariance matrix are computed by incorporating the new allocated ionospheric
measurements as
β^k=βk-+Kk(yk-Xkβk-)Σ^β,k=(I-KkXk)Σβ,k-,
where β^k and
Σ^β,k are the updated (or corrected)
state vector and the covariance matrix. The so-called gain matrix
behaves like a weighting factor between the new measurements and the
predicted state and is defined as
Kk=Σβ,k-,XkT(XkΣβ,k-XkT+Σy)-1.
The covariance matrix computed by
Eq. () does not guarantee symmetry and positive
definiteness due to computer rounding errors or an ill-posed character of the
problem . In order to preserve the symmetry and the
positive definiteness, an alternative form of
Eq. () can be obtained as
Σ^β,k=(I-KkXk)Σβ,k-(I-KkXk)T+KkΣyKkT,
where both terms are symmetric; the first term in
Eq. () is additionally positive
definite, and the second one is positive semi-definitive.
Kalman filtering with constraints
Two types of equality constraints were defined by
Eqs. () and (). In
order to simplify the notation we define the more general equation
Hβ=q
of constraints – matrix H and vector q are given – which
has to be incorporated into the KF. To solve such a problem, numerous
approaches have been proposed, for instance, a model reduction or the Kalman
gain restriction method (see , and references therein). In
our study, the so-called estimate projection method is applied and realized
as an additional step following the measurement update of the KF. The idea
behind this method is to project the estimated state vector
β^k onto the constraint surface. In other
words, it essentially solves the constrained minimization problem
minβ̃(β̃k-β^k)TPk(β̃k-β^k)suchthatHβ̃k=q, where Pk is a symmetric positive definitive
weighting matrix and β̃k means the projected
state vector. The former can be selected as Pk=Σβ,k-1 or Pk=I. The
solution of the minimization problem (Eq. ) is
given as
β̃k=β^k+Kc,k(q-Hβ^k)Σ̃β,k=(I-Kc,kH)Pk-1,
wherein the gain matrix reads
Kc,k=Pk-1HT(HPk-1HT)-1.
Data editing: pre and post-processing of the filter
The composition of the ionospheric state vector β as
defined in Eq. () can change in time. For
example, the current observation vector y at epoch k can include
observations from a new GNSS receiver not included in the previous time epoch
k-1. Thus, an additional unknown DCB has to be considered in the state
vector βk, and an additional preprocessing step prior to
the measurement update step has to be carried out for diagnosing the state
vector. To be more specific, the composition of the state vector has to be
checked at every cycle of the filter to detect a new allocation or a loss of
a GNSS receiver as well as the status of the GNSS satellites. If a DCB of the
predicted state vector has to be deleted since a receiver is lost, the
corresponding row and column also have to be removed from the predicted
covariance matrix. Conversely, if a new DCB has to be added, the predicted
state vector and the covariance matrix are extended and filled up with
predefined values.
The filter post-processing step refers to secondary tasks that are not
directly related to the filter but to the generation of products. For
instance, the estimated ionospheric parameters and their covariance matrix in
combination with other relevant data that would be informative for diagnosing
and monitoring the filter results are stored in a database.
Overall scheme for the
global VTEC estimation and the product generation.
Near real-time estimation and product generation
Although the presented filter is capable of running in real time,
measurements can only be assimilated with a time delay due to the latency
arising from the availability of hourly GNSS observations that have been
provided by the IGS data servers with at least a 1 h delay. Furthermore,
downloading and processing of the raw GNSS data as well as filtering
introduce additional delays. Therefore, the presented approach is called near
real time and is capable of generating global VTEC products usually with less
than a 1.5 h delay.
Figure shows the overall steps of the presented
approach: the hourly data processing, the filtering of the hourly data and
the product generation. The routines for downloading raw GNSS data, the
computation of the ionospheric observable from the raw data and storing of
the relevant data into a database are accomplished within the hourly data
processing step. A data preprocessing module, including the following steps,
runs sequentially at the beginning of every hour within the implemented
software. After the acquisition of the observations, a cutoff elevation angle
of 10∘ is applied to eliminate the very noisy measurements. Data arcs
containing cycle slips are detected and then split into parts. The arcs with
a number of observations less then a given threshold are rejected since the
leveling accuracy depends on the arc length .
Furthermore, an algorithm to detect and remove degraded observations is
performed using a so-called “3σ” outlier test, which is carried out
by screening the differences of the leveled carrier phase and the
pseudo-range measurements for each arc separately, i.e.,
drs=Pr,Is-Lr,4s. Thus, the observations exceeding the threshold
of 3σ are rejected from the data set.
The next step, the filtering of hourly data, includes the parameter
estimation procedures driven by the implemented KF. Finally, the estimated
ionospheric parameters are stored and utilized to generate the ionospheric
products, for instance, IONEX-formatted files including global VTEC maps.
Results and validation
To asses the quality of the VTEC products generated by the modeling approach
described before, two different evaluation methods are considered. The first
validation method, called self-consistency analysis, performs a very precise
sensitivity analysis using the differences in STEC observations to locally detect
temporal and spatial variations around a given GNSS receiver. The
second validation procedure shows the estimation quality of the VTEC maps on
water surfaces; the results are compared with altimeter data acquired from
the Jason-2 mission .
For the validation of our results we use the final, i.e., the post-processed,
products of the IGS and its IAACs because they are widely accepted as
well-established standards. Note that these final products are available with a
latency of days or even weeks, whereas our results are evaluated using
preprocessing strategies in near real time. We use statistical metrics,
namely the RMS value, the error mean value and the
standard deviation, to evaluate variations of the VTEC products with respect
to reference values derived from the self-consistency analysis and the
Jason-2 altimetry.
The geographical locations and the identifiers of the receiver sites
used in the dSTEC analysis.
Results of the statistical evaluations presenting the differences
between the observed and computed dSTEC values at the station PIMO. The
investigation covers the days DOY 224 to DOY 238, 2016: (upper panel) mean of
deviations, (middle panel) standard deviations and (bottom panel) RMS values
of errors in terms of TECU.
In this analysis, the VTEC products are labeled by the following standard
convention for the IONEX files, including VTEC maps, as “igsg”, “codg”,
“jplg”, “esag” and “upcg”, which are provided by the IGS and its IAACs,
namely CODE, JPL, ESOC and UPC. In this sense, the label “dfrg” throughout
this work refers to the estimated VTEC maps of the German Geodetic Research
Institute – Technical University of Munich (DGFI-TUM) with a temporal
resolution of the KF step
size set to 5 min, whereas “d1rg” is generated from “dfrg” and comprises
VTEC maps with a temporal resolution of 1 h.
A time interval between 11 August 2016 (day of year, DOY 224) and
25 August 2016 (DOY 238) covering 2 weeks of data was considered with the
following geomagnetic and ionospheric conditions: the 3 h Kp index
data
Kp index data, GFZ Potsdam, Germany,
ftp://ftp.gfz-potsdam.de/pub/home/obs/kp-ap.
, which quantifies the
intensity of the planetary geomagnetic activity, shows low variability with
Kp ≤ 5. The sunspot number
SILSO data, Royal Observatory of
Belgium, Brussels, http://www.sidc.be/silso/datafiles.
, which is a good
indicator of solar activity, shows a considerably high peak value of 80
for 16 August and the lowest value of 14 for 21 August. The characteristics
of these data sets are – as already mentioned – downloaded and
preprocessed in near real time. The following subsections are dedicated to
the results of these assessments.
Results of the statistical evaluations presenting the differences
between the observed and computed dSTEC values, which cover the days
between DOY 224 and DOY 238, 2016: (upper panel) mean of deviations, (middle
panel) standard deviations and (bottom panel) RMS values of deviation in
terms of TECU.
Self-consistency analysis
The derivation of very accurate absolute STEC values from GNSS measurements
may be a challenging procedure since the observations include the DCBs of the
receivers and the transmitting satellites. Several research groups have
provided GNSS-based solutions regarding the TEC modeling with appropriate
approaches for quality assessment; for example, see
, , and
. In the context of quality assessment, through a GPS
phase-continuous arc, differential STEC, i.e., dSTEC values, can be obtained
with an accuracy of less than 0.1 TECU . Note that
1 TECU is equivalent to 1016 electrons m-2. A test value
dSTECk for assessing the quality of the products at an epoch
k can be obtained by
dSTECk=dSTECobs,k-dSTECmap,k, where dSTECobs,k is the difference
of the GPS geometry-free linear combination at the epoch k with another
linear combination computed on the same continuous arc at a reference epoch
characterized by the highest elevation angle. The computed dSTEC values from
the VTEC maps denoted as
dSTECmap,k at the same epoch
k and the reference epoch are obtained by multiplying the
VTEC values with the
elevation-dependent mapping function (Eq. ). The
geographical locations and the names of the receiver stations selected for
the dSTEC evaluation are depicted in Fig. . The test receivers
are chosen globally and located at low and high latitudes, which can reveal
the VTEC model accuracy at the regions characterized by varying VTEC
activity. The receivers at the sites MKEA, ASPA, BOGT, CHPI, YKRO, DGAR and
PIMO are located at middle and low latitudes, whereas DUBO, PENC, URUM, SYOG
and MAC1 are established at higher latitudes. As an exemplified analysis, the
mean, the standard deviation and the RMS values of daily dSTEC variations are
computed using the data from the observation site PIMO as presented in
Fig. . The numbers within the parentheses shown on the
legends give the average values of the corresponding statistical measures for
the entire test period, which are also summarized in
Fig. . The biases of the VTEC maps at the PIMO
station show day-to-day variations from 0.1 to 1.2 TECU. The average RMS
errors of the dfrg and the d1rg solutions are 2.34 and 2.39 TECU,
respectively, which are in close agreement with the RMS values of the
analysis centers ranging from 1.99 to 2.72 TECU. Taking into account the
daily RMS variations plotted in Fig. , the VTEC
solutions dfrg and d1rg show larger values only at DOY 230 but smaller
deviations for the rest of the test period.
Ground tracks of the Jason-2 altimetry mission for 16 August 2016
(DOY 229). The colors show the magnitude of VTEC as acquired from the
satellite measurement system.
Ground track of the Jason-2 altimetry satellite between 00:00 and
01:00 UTC on DOY 229, 2016. The colors show the magnitude of VTEC in TECU
acquired from the satellite.
In Fig. , as a summary of statistical measures,
the average mean values and standard deviations as well as the average RMS
errors for the entire test period are visualized for each of the 12 receiver
sites and the seven VTEC models. As is expected, the dSTEC error for the sites
at low latitudes (ASPA, BOGT, YKRO, DGAR, PIMO) are general higher than
those obtained at high latitudes. Especially, the ASPA site has rather high
errors for each of the VTEC maps in terms of RMS error. The site is very
close to the geomagnetic equator, and it can suffer from poor
nearby data coverage due to its geographic location on an oceanic island. The
result of our solution dfrg has an average bias of 0.04 TECU. The biases
with respect to the other maps vary between -0.02 and 0.36 TECU. The
average standard deviation for dfrg is 1.78 TECU, which shows a moderate
accuracy compared to those of the other maps, which range from 1.66 to
1.96 TECU. The RMS error of dfrg has 1.81 TECU, whereas the RMS errors of
the other VTEC solutions vary between 1.70 and 2.00 TECU. A closer look
into the average RMS errors for each of the sites reveals that although the
presented approach shows a slightly higher deviation only for the ASPA
station, the accuracy at the remaining sites is compatible with that of the
analysis centers. Moreover, concerning the temporal resolution, the RMS error
of dfrg amounts to 1.81 TECU and is thus slightly better than d1rg
with 1.85 TECU. However, during high solar activity, a considerable
difference might be expected.
Validation by the Jason-2 altimetry
The dual-frequency altimeter onboard the Jason-2 satellite can directly
measure in the nadir direction using two different frequencies, which allow
the
extraction of VTEC data with less effort and without applying any mapping between
STEC and VTEC according to
Eq. (). The VTECs provided by Jason-2 are taken into
account as reference since they allow the evaluation of the errors of the global
VTEC maps over the oceans as well as over regions that exhibit poor
estimation quality due to the availability of only a few GNSS measurements.
It should be noted that although the Jason-2 satellite provides accurate,
direct and independent VTEC data, several studies reported that the
measurements are contaminated by an offset of around 3 TECU compared to the
GNSS-derived VTEC products .
VTEC values from the six analysis centers and Jason-2 between 00:00
and 01:00 UTC on DOY 229, 2016 in TECU.
Before using the altimeter data for the comparisons, a median filter with a
window size of 20 s was applied to smooth the data. It is worth mentioning
that Jason-2 radar altimetry provides data with a higher spatial and temporal
resolution compared to the VTEC maps. Therefore, a linear interpolation in
the spatial and time domains was applied to obtain VTEC values for the
corresponding time and location of the altimetry observations. The
interpolation is performed in the Sun-fixed coordinate system between
consecutive epochs. Figure shows
exemplified ground tracks of Jason-2 with the associated VTEC values for the
entire day of 16 August 2016 (DOY 229). High VTEC variations around the
equatorial regions can be clearly seen. This day has the highest sun spot
number during the test period.
Comparison of VTEC values acquired from the analysis centers and the
DGFI-TUM solutions with Jason-2 altimetry VTEC data between DOY 224 and
DOY 238, 2016; (upper panel) mean deviations, (middle panel) standard
deviations and (bottom panel) RMS values of deviations in terms of TECU.
For the sake of clarity, a selected data set between 00:00 and 01:00 UTC
from Fig. is depicted in
Fig. . The equatorial ionization
anomaly (EIA), which is characterized by two crests at both sides of the
geomagnetic equator and a trough around the equator , can
be seen in Figs. and
by carefully interpreting the colors
along the ground tracks. The VTEC values obtained from Jason-2 and the
seven VTEC maps, using the aforementioned interpolation method, are visualized
in Fig. for the ground track displayed
in Fig. . The camelback-shaped EIA in
terms of VTEC is illustrated by two regions, including the peak VTEC values
between 0.0 and 0.3 h of day (hod). The morphology of the EIA defined by the
Jason-2 VTEC is clearly represented by the VTEC maps. The DGFI-TUM solutions
are in good agreement with all the other VTEC values. For the whole day, a
mean bias of -1.23 TECU and a standard deviation of 3.43 TECU is computed
for the dfrg VTEC solution. The results are compatible with the solution
of the other VTEC maps showing mean biases ranging from -1.77 to 1.23 TECU
and standard deviations from 2.06 to 4.74 TECU.
For the entire test period, the results of the comparisons in terms of daily
mean, standard deviation and RMS values as well as their overall averaged
values for the entire test period are illustrated in
Fig. . The average relative biases of the VTEC
maps show variations between 0.5 and -2.0 TECU; our two solutions labeled
as dfrg and d1rg both have an average relative bias of -1.6 TECU.
The daily standard deviations vary between 3.2 and 5.0 TECU; their averages
deviate around 4.0 TECU and are also in accordance with those derived from a
recent comparison study by stating that the
daily deviations for the results of the IAACs can vary from a few TECU to
10 TECU. Our solutions have an average RMS error of 4.7 TECU, which does not
exceed that computed from the other analysis centers ranging between 4.0 and
4.7 TECU. A detailed look into the daily RMS values indicates that the
estimated VTEC values of our solutions are generally in good agreement with
the results of the analysis centers.
Conclusions and future improvements
A near real-time processing framework that is capable of automated data
downloading, data preprocessing, Kalman filtering and formatted product
generation is presented to provide VTEC maps as well as satellite and
receiver DCBs of GPS and GLONASS in near real time. The B-spline
representation of global VTEC is incorporated into the Kalman filter
procedure. The filter was also extended to integrate the equality
constraint equations comprising the spherical and DCB-related restrictions.
Coefficients of the B-spline model and the DCBs, which constitute the unknown
parameters, are recursively estimated by exploiting hourly GNSS observations
acquired from the IGS data centers with 1 h latency. The ionosphere
observable is derived from raw GNSS code and phase measurements using the
geometry-free linear combinations.
The validation of the proposed approach is carried out using GNSS data
downloaded in near real time covering a time span of 2 weeks. To summarize,
according to the self-consistency analysis, an RMS value of 1.81 TECU was
found. The four IAACs, CODE, JPL, ESOC and UPC, as well as the IGS combination
product exhibit comparable RMS errors between 1.70 and 2.00 TECU. Moreover,
the Jason-2 validation shows that the RMS error achieved by the proposed
method fits well with the results of the IAACs. Considering the comparisons,
specific to the test period it might be concluded that the estimated VTEC
products using the presented near real-time strategy shows promising initial
results in terms of accuracy and overall agreement with the post-processed
final products of IGS and its analysis centers, which are publicly available
with several days of latency. Furthermore, the results encourage further research
to improve the presented model as mentioned below.
One drawback associated with the KF is the requirement of the complete
knowledge of the prior information, i.e., the process noise covariance matrix
Σw, and the covariance matrix
Σy
of the measurement errors have to be given. The common practice of selecting
these matrices manually is conducted by doing tests through multiple runs of the
filter for different values of these parameters . However,
the test data may not be adequate for properly defining these matrices
beforehand or they can vary unpredictably throughout the time period
. If the prior information is not appropriate, the filter is
no longer optimal and can result in an estimation of poor quality, or even
worse, the filter may diverge. To cope with such situations in ionosphere
modeling, the implemented approach will be extended by adaptive methods to
estimate the covariance matrices in run time as a further improvement.
Moreover, the evaluations presented here show that the developed approach
exhibits promising results during a period of quiet solar activity, but
further tests have to be conducted using GNSS data sets covering a long time
span and downloaded in near real time during time periods of high solar
activity and solar events, e.g., solar flares and coronal mass ejections.
Data availability
The global VTEC maps in IONEX format used in the comparisons were acquired
from the Crustal Dynamics Data Information System (CDDIS) data center by the
following FTP server: ftp://cddis.gsfc.nasa.gov/gnss/products/ionex/.
The data of the Jason-2 altimetry mission are available via the FTP server:
ftp://data.nodc.noaa.gov/pub/data.nodc/jason2/ogdr/ogdr/. The hourly
available GNSS data from IGS sites were operationally downloaded in real time
through mirroring to the different IGS data centers, i.e., the CDDIS
(ftp://cddis.gsfc.nasa.gov/pub/gps/data/hourly/), the Bundesamt für
Kartographie und Geodäsie (BKG) (ftp://igs.bkg.bund.de/IGS/nrt/),
the Institut Geographique National (IGN)
(ftp://igs.ensg.ign.fr/pub/igs/data/hourly) and the Korean Astronomy
and Space Science Institute (KASI) (ftp://nfs.kasi.re.kr/).
Furthermore, ultrarapid orbits of GPS and GLONASS satellites utilized in the
data preprocessing step can be accessed through FTP servers, namely for GPS
via ftp://cddis.gsfc.nasa.gov/pub/gps/products and for GLONASS via
ftp://ftp.glonass-iac.ru/MCC/PRODUCTS/.
The authors declare that they have no conflict of
interest.
Acknowledgements
The authors would like to thank the following services and institutions for
providing the input data: IGS and its data centers, the Center for Orbit
Determination in Europe (CODE, University of Berne, Switzerland), the Jet
Propulsion Laboratory (JPL, Pasadena,California, USA), the European Space
Operations Centre of European Space Agency (ESOC, Darmstadt, Germany) and
the Universitat Politècnica de Catalunya/IonSAT (UPC, Barcelona, Spain).
This work was supported by the German Research Foundation (DFG) and the
Technical University of Munich (TUM) in the framework of the Open Access
Publishing Program. Moreover, the presented models were developed in the
frame of the project “Development of a novel adaptive model to represent
global ionosphere information from combining space geodetic measurement
systems” (ADAPIO) (German title: “Entwicklung eines neuartigen adaptiven
Modells zur Darstellung von globalen Ionosphäreninformationen aus der
Kombination geodätischer Raumverfahren”), which was funded by the German
Federal Ministry for Economic Affairs and Energy via the German Aerospace
Center (DLR, Bonn, Germany). The topical
editor, K. Hosokawa, thanks the two anonymous referees for help in evaluating
this paper.
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