AMTAtmospheric Measurement TechniquesAMTAtmos. Meas. Tech.1867-8548Copernicus PublicationsGöttingen, Germany10.5194/amt-9-1303-2016Statistical framework for estimating GNSS biasVierinenJuhax@mit.eduCosterAnthea J.https://orcid.org/0000-0001-8980-6550RideoutWilliam C.EricksonPhilip J.NorbergJohannesHaystack Observatory, Massachusetts Institute of Technology, Route 40 Westford, 01469 MA, USAFinnish Meteorological Institute, P.O. Box 503, 00101 Helsinki, FinlandJuha Vierinen (x@mit.edu)30March2016931303131226July201511September201519January20161March2016This 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/9/1303/2016/amt-9-1303-2016.htmlThe full text article is available as a PDF file from https://amt.copernicus.org/articles/9/1303/2016/amt-9-1303-2016.pdf
We present a statistical framework for estimating global navigation satellite
system (GNSS) non-ionospheric differential time delay bias. The biases are
estimated by examining differences of measured line-integrated electron
densities (total electron content: TEC) that are scaled to equivalent vertical integrated densities.
The spatiotemporal variability, instrumentation-dependent errors, and errors
due to inaccurate ionospheric altitude profile assumptions are modeled as
structure functions. These structure functions determine how the TEC
differences are weighted in the linear least-squares minimization procedure,
which is used to produce the bias estimates. A method for automatic detection
and removal of outlier measurements that do not fit into a model of receiver
bias is also described. The same statistical framework can be used for a
single receiver station, but it also scales to a large global network of
receivers. In addition to the Global Positioning System (GPS), the method is
also applicable to other dual-frequency GNSS systems, such as GLONASS
(Globalnaya Navigazionnaya Sputnikovaya Sistema). The use of the framework is
demonstrated in practice through several examples. A specific implementation
of the methods presented here is used to compute GPS receiver biases for
measurements in the MIT Haystack Madrigal distributed database system.
Results of the new algorithm are compared with the current MIT Haystack
Observatory MAPGPS (MIT Automated Processing of GPS) bias determination algorithm. The new method is found to
produce estimates of receiver bias that have reduced day-to-day variability
and more consistent coincident vertical TEC values.
Introduction
A dual-frequency global navigation satellite system (GNSS) receiver can measure the line-integrated ionospheric
electron density between the receiver and the GNSS satellite by observing the
transionospheric propagation time difference between two different radio
frequencies. Ignoring instrumental effects, this propagation delay difference
is directly proportional to the line integral of electron density
. This is derived, e.g., by .
Received GNSS signals are noisy and contain systematic instrumental effects,
which result in errors when determining the relative time delay between the
two frequencies. The main instrumental effects are frequency-dependent delays
that occur in the GNSS transmitter and receiver, arising from dispersive
hardware components such as filters, amplifiers, and antennas. Loss of
satellite signal can also cause unwanted jumps in the measured relative time
delay and cause unwanted nonzero mean errors in the relative time delay
measurement. Because line-integrated electron density is determined from this
relative time delay, it is important to be able to characterize and estimate
these non-ionospheric sources of relative time delay.
The non-ionospheric relative time delay due to hardware is commonly referred
to as bias in the literature. For the specific case of GPS
measurements, the bias is often separated into two parts ordered by the
source of delay: satellite bias and receiver bias.
A GNSS measurement of relative propagation time delay difference including
the line-integrated electron density effect can be written as
m=b+c+∫SNe(s)ds+ξ,
where m is the measurement, b is the receiver bias, c is the satellite
bias, S is the path between the receiver and the satellite,
Ne(s) is the ionospheric electron density at position s, and
ξ is the measurement noise. The measurement is scaled to total electron content (TEC) units, i.e.,
1016m-2,
and therefore bias terms also have units of TEC. See and
references therein for a more detailed discussion.
For ionospheric research with GNSS receivers that perform measurements of the
form shown in Eq. (), the quantity of interest is usually the
three-dimensional electron density function Ne(s). However, this
quantity is challenging to derive from just GNSS measurements alone, as we
only observe one-dimensional line integrals through the ionosphere. The
problem is an ill-posed inverse problem called the limited-angle tomography
problem . The difficulty arises from the fact that line integrals
are measured only at a small number of selected viewing angles, and this
information is not sufficient to fully determine the unknown electron density
distribution without making further assumptions about the unknown measurable
Ne(s). These assumptions often impose horizontal and vertical
smoothness, as well as temporal continuity.
A considerable number of prior studies have attempted to solve this
tomographic inversion problem in three dimensions for beacon satellites as
well as for GPS satellites (see, e.g., and references therein).
Because of the large computational costs and complexities associated with
full tomographic solvers, much of the practical research is done using a
reduced quantity called the vertical total electron content (VTEC). As we
will describe in more detail below, VTEC in essence results from a reduced
parameterization of the ionosphere that is used to simplify the tomography
problem and make it more well-posed. VTEC processing is only concerned with
the integrated column density, and therefore the measurements are reported in
TEC units.
The fundamental assumption for vertical TEC processing is that a slanted line
integral measurement of electron density can be converted into an equivalent
vertical line integral measurement with a parameterized scaling factor
v(α):
∫VNe(s)ds≈v(α)∫SNe(s)ds,
where “V” is a vertical path, “S” is the associated slanted path,
α is the elevation angle, and v(α) is the scaling factor that
relates a slanted integral to a vertical line integral.
There are several ways that v(α) can be derived without resorting to
full tomographic reconstruction of the altitude profile shape. Typically, the
ionosphere above a certain geographic point is assumed to be described with
some vertical shape profile p(h) multiplied by a scalar Ne(h)=Np(h). One example of an often-used shape profile is the Chapman profile:
p(h)=exp1-z(h)-e-z(h),
where z(h)=(h-hm)/H, hm is the peak altitude of the ionosphere, and H
is the scale height . Another example is a slab with
exponential top and bottom side ramps as described by and
. Figure depicts the geometry and profile shape
assumptions in vertical TEC processing.
A scaled altitude profile model of the ionosphere assumes that the
ionosphere locally has a fixed horizontally stratified altitude profile shape
multiplied by a scalar. This makes it possible to relate slanted line
integrals to equivalent vertical line integrals using an elevation-dependent
scaling factor called the mapping function. The pierce point is located where
the ray pierces the peak of the electron density profile.
In more advanced models, the mapping function can be parameterized not only
by elevation angle but also by factors such as time of day, geographic
location, solar activity, and the azimuth of the observation ray. In
practice, this can be done by using a first-principles ionospheric model to
derive a more physically motivated mapping function.
Although the vertical TEC assumptions described above are not as flexible as
a full tomographic model that attempts to determine the altitude profile,
they provide model-to-data fits that are to first order good enough to
produce measurements that are useful for studies of the ionosphere. The
utility of this simplified model derives from the fact that it results in an
overdetermined, well-posed problem that can be inverted with relatively
stable results. The main practical difficulties in data reduction using the
simplified model are estimating the receiver and satellite biases b and
c, as well as handling possible model errors.
In this paper, a novel statistical framework for deriving these GNSS
measurement biases is described. The method is based on examining large
numbers of differences between slanted TEC measurements that are scaled with
the mapping function v(α). The differences between pairs of
measurements are assumed to be Gaussian normal random variables with a
variance that is determined by the properties of the two measurements, i.e.,
spacing in time, geographic distance, and elevation angle. While the Gaussian
assumption results in a numerically efficient system of equations, this
assumption is also supported by numerical evidence, which suggests that the
distribution of the differences of vertical TEC measurements is close to a zero
mean normal random distribution, with the standard deviation increasing when
the geographic distance between pierce points is increased, the
temporal distance is increased, or the elevation angle of the measurement
is decreased.
We will show how this general statistical framework can be used to estimate
biases in multiple special cases and finally compare the newly presented
method with an existing bias determination scheme within the MIT Haystack
MAPGPS (MIT Automated Processing of GPS) algorithm . We will refer to this new method for
bias determination as weighted linear least squares of independent
differences (WLLSID).
Receiver bias estimation
Let us denote Eq. () in a more compact form, but now with indexing
i to denote the index of a measurement, j to denote receiver, and k to
denote the satellite:
mi=bj(i)+ck(i)+ni+ξi.
Here ni is the line integral of electron density through the ionosphere
for measurement i. The receiver and satellite index associated with
measurement i is given by j(i) and k(i). Receiver noise is represented
with ξi.
Now consider subtracting slanted TEC measurements i and i′, which
are scaled with corresponding mapping function values vi and
vi′, which convert slanted TEC to equivalent vertical TEC. In this
analysis, it does not matter if these measurements are associated with the
same receiver or the same satellite, or even if they occur at the same time.
vimi-vi′mi′=vini-vi′ni′+vibj(i)-vi′bj(i′)+vick(i)-vi′ck(i′)+viξi-vi′ξi′
This type of a difference equation has several benefits. If measurements i
and i′ are performed at a time close to each other ti≈ti′ and have closely located pierce points xi≈xi′,
then we can make the assumption that vini≈vi′ni′, i.e., that the vertical TEC is similar.
We can statistically model this similarity by assuming that the difference of
equivalent vertical line-integrated electron content between two measurements
is a normally distributed random variable with variance
vini-vi′ni′=ξ̃i,i′∼N0,Si,i′,
where Si,i′ is the structure function that indicates what we
assume to be the variance of the difference of the two measurements i and
i′. This structure function would be our best guess of how different
we expect these two measurements to be.
We assume the structure function depends on the following factors: (1)
geographic distance between pierce points di,i′=|xi-xi′|, (2) difference in time between when the measurements were made
τi,i′=|ti-ti′|, (3) receiver noise of both
measurements ξi+ξi′, and (4) modeling errors that are
dependent on elevation angles αi and αi′ of the
measurements. The modeling errors in (4) are caused by inaccuracies in the
assumption that we can scale a slanted measurement into an equivalent
vertical measurement.
The following subsections describe the structure function behaviors for each dependent variable.
Geographic distance
In order to model the variability of electron density as a function of
geographic location, we assume the difference between two measurements to be
a random variable:
vini-vi′ni′∼N0,D(di,i′),
where in this work we use D(d)=0.5d in units of (TECu/100km). This implies that we assume the standard deviation of
difference of two vertical TEC measurements to grow at a rate of 0.5 TEC
units per 100 km of spacing between pierce points.
For the results in this paper, we use the functional form above, but this can
be improved in future work by a more complicated spatial structure function
D(xi,xi′,ti,ti′), which is a function of pierce point
locations xi and xi′, as well as the time of the measurements
ti and ti′. This function could for example be derived
experimentally from vertical TEC measurements themselves. This would allow
more accurate modeling of sunrise and sunset phenomena, as well as meridional
and zonal gradients.
Temporal distance
Two measurements do not necessarily have to occur at the same time, but one
would expect the two measurements to differ more if they have been taken
further apart from one another. This difference can also be modeled as a
normal random variable:
vini-vi′ni′∼N0,T(τi,i′),
where T(τi,i′) is a structure function that statistically describes
the difference in vertical TEC from one measurement to the other when the
time difference between the two measurements is τi,i′=|ti-ti′|.
In this work, we use T(τ)=20τ in units of
TECu/hour. This makes the assumption that the standard
deviation of the difference of two vertical TEC measurements grows at the
rate of 20 TEC units for each hour.
Again, an improved version of this time structure function could also be
obtained by estimating it from data, but this is the subject of a future
study.
Model and receiver errors
There are modeling errors that are caused by our assumption that we can scale
a slanted line integral to a vertical line integral as shown in Eq. (). First of all, this assumption does not correctly take into
account that the slanted path cuts through different latitudes and longitudes
and thus averages vertical TEC over a geographic area. In addition to this,
our mapping function assumes an altitude profile for the ionosphere that is
hopefully close to reality, but never perfect. The ionosphere can have
several local electron density maxima and can have horizontal structure in
the form of, e.g., traveling ionospheric disturbances, or typical ionospheric
phenomena such as the Appleton anomaly at the Equator or the ionospheric
trough at high latitudes.
In addition to this, GNSS receivers often have difficulty with low-elevation
measurements arising from near-field multi-path propagation, which is
different for both frequencies. These errors can in some cases severely
affect vertical TEC estimation and thus also bias estimation.
To first order, the errors caused by the inadequacies of the model
assumptions or anomalous near-field propagation increase proportionally to
the zenith angle. It is useful to include this modeling error in the
equations as yet another random variable. We have done this by assuming the
elevation-angle-dependent errors to be a random variable of the following
form:
vini-vi′ni′∼N0,E(αi)+E(αi′).
Here E(αi) is the structure function that indicates the modeling
error variance as a function of elevation angle. In this work, we use a
structure function where the variance grows rapidly as the elevation angle
approaches the horizon, expressed as E(αi)=20(cosαi)4. This form penalizes lower elevations more heavily.
The structure function that takes into account vertical TEC scaling errors
and receiver issues at low elevations can also be determined from vertical
TEC estimates, e.g., by doing a histogram of coincident measurements of
vertical TEC:
E(α)≈〈|〈vini〉-vi′ni′|2〉
for all i,i′, where |xi-xi′|<ϵd and
|αi′-α|<ϵα. Here ϵd
determines the threshold for distance between pierce points that we consider
to be coincidental, and ϵα determines the resolution of the
histogram on the α axis. Here the angle brackets 〈⋅〉 denote a sample average operator.
Generalized linear least-squares solution
If we assume that all random variables in the structure functions of the
previous section are independent random variables, we can simply add them
together to obtain the full structure function
Si,i′=D(di,i′)+T(τi,i′)+E(αi)+E(αi′).
The differences in Eq. () can be expressed in matrix form as
m=Ax+ξ,
where
,
with the measurement vector containing differences between vertically scaled measurements
m=⋯,vimi-vi′mi′,⋯T,
and the unknown vector x contains the receiver and satellite biases
x=b0,⋯,bN,c0,⋯,cMT.
For x, N indicates the number of receivers and M indicates the
number of satellites.
The random variable vector ξ∼N(0,Σ)
has a diagonal covariance matrix defined by the structure function of each
measurement pair used to form differences:
Σ=diagSi,i′,⋯.
The theory matrix A forms the forward model for the measurements
as a linear function of the receiver biases.
This type of a measurement is known as a linear statistical inverse problem
, and it has a closed-form solution for the maximum-likelihood
estimator for the unknown x, which in this case is a vector of
receiver and satellite biases:
x^=(ATΣ-1A)-1ATΣ-1m.
This matrix equation is often not practical to compute directly due to the
typically large number of rows in A. However, because the matrix
A is very sparse, the solution can be obtained using sparse linear
least-squares solvers. In this work, we use the LSQR package
for minimizing |Ãx-m̃|2, where
à and m̃ are scaled versions of the matrix
A and vector m. Each row of A and m are
scaled with the square root of the variance of the associated measurement
Si,i′ in order to whiten the noise. In practice, this
performs a linear transformation with matrix P that projects the
linear system into a space where the covariance matrix is an identity
matrix PTΣP=I.
Outlier removal and bad receiver detection
When a maximum-likelihood solution has been obtained, a useful diagnostic
examines the residuals r=|Ãx^-m̃|. If the residuals are larger than a certain threshold, they
can be determined to be measurements that do not consistently fit the model,
i.e., outliers.
Outliers can be caused by several different mechanisms. They can be of
ionospheric origin, where vertical TEC gradients are sharper than our
structure function expects them to be. They can also be simply caused by a
loss of lock in the receiver, which can result in a large erroneous jump in
slanted TEC.
These outlying measurements can be detected and removed by a statistical
test, for example |Ãx^-m̃|>4σ, where σ is the standard deviation of the residuals estimated
with σ=median|Ãx^-m̃|. After the removal of problematic measurements,
another improved maximum-likelihood solution, one not contaminated by
outliers, can be obtained. The procedure for outlier removal can be repeated
over several iterations to ensure that no problematic data are used for bias
estimation.
Bias estimation using time differences of measurements obtained with
a single receiver. Top panel shows the residuals of the maximum likelihood
fit to the data. The points shown with red are automatically determined as
outliers and not used for determining the receiver bias. These mostly occur
during daytime at low elevations. The center panel shows vertical TEC
estimated with the original MAPGPS receiver bias determination algorithm,
while the bottom panel shows vertical TEC measurements obtained using only
time differences using the new method described in this paper, assuming
constant receiver bias and known satellite bias. The VTEC results do not
differ significantly.
Special cases
The previous section described the general method for estimating bias by
using differences of slanted TEC measurements scaled by the mapping function.
However, in practice this general form rarely needs to be used. In the
following sections we describe several important and practical special cases,
including known satellite bias, single receiver bias estimation, and
multiple biases for each receiver.
Known satellite bias
If satellite bias is known a priori to a good accuracy, then it can be subtracted from the measurements and the difference equation. This reduces Eq. () to
vimi-vi′mi′=vini-vi′ni′+vibj(i)-vi′bj(i′)+viξi-vi′ξi′.
This form results in the same linear measurement equations, except that the
satellite biases are not unknown parameters. In this case, the theory matrix
will only have at most two nonzero elements for each row.
For GPS receivers, satellite biases are known to a good accuracy using a
separate and comprehensive analysis technique , and
therefore this special case is appropriate for bias determination for GPS
receivers.
Single receiver and known satellite bias
For the case that the satellite bias is known a priori and there is
furthermore only one receiver, then the matrix only has one column with the
unknown bias for the receiver.
This still results in an overdetermined problem that can be solved. The
solution of this special case mathematically resembles a known analysis
procedure that is often referred to as “scalloping” (P. Doherty, personal
communication, 2003; ). This latter technique depends on the
assumption that the concave or convex shape of all zenith TEC estimates
collected by a single receiver observed over a 24 h period should be
minimized. This same goal is obtained when time differences are minimized.
The main difference in this work is that the statistical framework uses a
structure function that weights differences of measurements based on time
between the measurements, the elevation angle, and the pierce point distance.
Figure shows an example receiver bias that is determined
using only data from a single receiver. In this case, time differences with
τi,i′ less than 2 h were used, in order to keep the
number of measurements manageable. We also used differences of measurements
between different satellites. A comparison of results with measurements
obtained with the standard MAPGPS algorithm shows quite similar results
between the two techniques.
Multiple biases
There are several reasons for considering the use of multiple biases for the
same satellite and receiver. This special case can also be handled by the
same framework.
If there is a loss of phase lock on a receiver, this might result in a
discontinuity in the relative time-of-flight measurement, which appears as a
discrete jump in the slanted TEC curve. Rather than attempting to realign the
curve by assuming continuity, it is possible, using our framework, to
simply assign an independent bias parameter to each continuous part of a TEC
curve. As long as there are enough overlapping measurements, the biases can
be estimated.
For GNSS implementations other than GPS, it is possible that satellite biases
are not known or cannot be treated as a single satellite bias. For example,
the GLONASS (Globalnaya Navigazionnaya Sputnikovaya Sistema) network uses a different frequency for each satellite, which
means that any relative time delays between frequencies caused by the
receiver or transmitter hardware will most likely be different for each
satellite–receiver pair. Because of this, it is natural to combine the
satellite bias and receiver bias into a combined bias, which is unique for
each satellite–receiver combination.
Receiver biases are also known to depend on temperature ,
because dispersive properties of the different parts of the receiver can
change as a function of temperature. If an independent bias term is assigned
to, e.g., each satellite pass, this also allows temperature-dependent effects
to be accounted for, as a single satellite pass lasts only part of the day.
Multiple bias terms can be added in a straightforward manner to the model
using Eq. (). This is the same equation that is used for the
known satellite bias special case. Here, bj(i) can be interpreted as an
unknown relative bias term that can vary from one continuous slanted TEC curve
to another. The meaning of j(i) in this case is different. It is a function
that assigns bias terms to measurements i. Each receiver does not
necessarily need to have one unknown bias parameter; it can have many.
An example of a measurement where the same satellite is observed using a
single receiver is shown in Fig. . In this case, the
satellite is measured in the morning first, and during the pass there is a
discontinuity in the TEC curve, most likely due to loss of lock. We give the
measurements before ∼ 05:00 UTC and from ∼ 05:00 to 06:00 UTC an independent bias
term b0 and b1. The same satellite is seen again in the evening at
19:00 UTC, and we again assign a new bias term to it: b2.
An example of a measurement of a single satellite collected by a
single receiver. A loss of phase lock occurs during the first pass of the
satellite, resulting in two receiver biases for that pass (b0: blue curve;
b1: green curve). During the next pass, a drift in the receiver bias could
have occurred, so another receiver bias (b2: red curve) is determined when
the satellite is measured during the end of the day.
Another multiple-bias example is shown in Fig. , which
displays measurements from 19 neighboring receivers in China. A few of these
receivers have discrete jumps in the slanted TEC curves that make it
impossible to assume a constant receiver bias during the course of the entire
day. This can be seen as a poor fit using the standard MIT Haystack MAPGPS
algorithm. When multiple bias terms are introduced (in the same way as
depicted in Fig. ), the measurements from these stations can
be recovered.
Vertical TEC with satellite bias estimated using the current version
of the MIT Haystack Observatory MAPGPS algorithm (Rideout and Coster, 2006)
shown above. Multiple receivers have problems with receiver stability, which
makes the assumption of unchanging receiver bias problematic and causes the
receiver bias determination to fail. Vertical TEC with receiver biases
obtained using the multiple-biases assumption is shown below. The new method
produces a more consistent baseline. The red dots show stations that are
plotted. The algorithm uses all of the data from the 19 stations marked with
orange and red dots. The stations marked with orange are used to assist in
reconstruction by using a larger geographic area.
Comparison
In order to test the framework in practice for a large network of GPS
receivers, we implemented the framework described in this paper as a new bias
determination algorithm for the MIT Haystack MAPGPS software, which analyzes
data from over 5000 receivers on a daily basis. We used the MAPGPS program to
obtain slanted TEC estimates. Then, instead of using the MAPGPS routines for
determining receiver biases, we used the new methods described in this paper.
We label results obtained using the new bias determination algorithm with
WLLSID.
When fitting for receiver bias, we assumed a fixed receiver bias for each
station over 24 h. We also assumed a known satellite bias, which was
removed from the slanted measurement. To keep the size of the matrix
manageable, we selected sets of 11 neighboring receiver stations and
considered each combination of measurements across receiver and satellites
occurring within 5 min of each other as differences that went into the
linear least-squares solution. For this comparison, we did not use time
differences.
Probability density function and cumulative density functions for
192 360 coincidences where vertical TEC was measurement within the same 30 s time interval and have pierce points less than 50 km apart from one
another. The new method (labeled as WLLSID) has significantly more <1 TEC
unit differences than the old method.
Global TEC map produced using two different methods for the St.
Patrick's Day storm on 17 March 2015. Top: a map produced with the
MAPGPS method. Bottom: a map produced with the new WLLSID bias
determination method.
To estimate the goodness of the new receiver bias determination, we compared
the method with the existing MAPGPS algorithm for determining receiver bias,
which utilizes a combination of scalloping, zero-TEC, and differential linear
least-squares methods . At latitudes higher
than 70∘, the zero TEC method is used. This method finds the value of
bias in such a way that the minimum value of TEC is 0. At low and
midlatitudes scalloping is first used. Scalloping finds the bias by finding
the optimally flat vertical TEC ±2 h around local noon. After
finding the bias with either zero TEC or scalloping, the values are refined
by using the differential TEC method described by .
Self-consistency comparison
As a measurement of goodness, we used the absolute difference between two
simultaneous geographically coincident measurements of vertical TEC |vini-vi′ni′|. The two measurements were considered coincident
if the distance between the pierce points was less than 50 km and the
measurements occurred within 30 s of each other. We also required that
the two measurements were not obtained using the same receiver. As a figure
of merit, we used the mean value of the absolute differences:
F=1N∑i≠i′|vini-vi′ni′|.
This figure of merit measures the self-consistency of the measurements, i.e.,
how well the vertical TEC measurements obtained with different receivers
agree with one another. The smaller the value, the more consistent the
vertical TEC measurements are.
All in all, we found 192 360 such coincidences for the 5220 GPS receivers in
the database over a 24 h period starting at midnight 15 March 2015. Biases for
the measurements were obtained both with the new and existing MAPGPS bias
determination methods (MAPGPS and WLLSID). The figure of merit for the
existing MAPGPS method was 2.25 TEC units, and the WLLSID method has a figure
of merit of 1.62 TEC units, which is about 30 % better.
The probability density function and cumulative density function estimates
for the coincident vertical TEC differences are shown in Fig. . The new method results in
significantly more <1 TEC unit
differences than the old method. It is evident from the cumulative
distribution function that both methods also result in some coincidences that
are in large disagreement with each other. The result occurs at least in part
due to our inclusion of elevations down to 10∘ in the comparison, and
it is therefore expected that some low-elevation measurements will be
significantly different from one another.
Receiver bias day-to-day change
We also investigated receiver bias variation from day to day. We arbitrarily
selected two consecutive quiet days: days 140 and 141 of 2015. We calculated
the sample mean day-to-day change in receiver bias across all receivers:
δb=1N∑i=0Nbi,140-bi,141,
where N is the number of receiver. In addition to this, we calculated the standard deviation using sample variance:
σb=1N-1∑i=0N(bi,140-bi,141)-δb2.
For the MAPGPS method, we found overall that δb=-0.2±0.05(2σ) TEC units and σb=1.6 TEC units. With the new WLLSID
method, we found that δb=0.02±0.05(2σ) TEC units and
σb=1.3 TEC units. This indicates not only that the day-to-day
variability is slightly smaller with the new method but also that the old
method has a statistically significant nonzero mean day-to-day change in
receiver bias, which is not seen with the new method. When the data are broken
down into high and equatorial latitudes, the result is similar.
Qualitative comparison
In order to qualitatively compare the MAPGPS bias determination method with
the WLLSID method, we produced a global TEC map with the WLLSID method and
the existing MAPGPS bias determination method. The processing involved with
making these TEC maps is described by Rideout and Coster
().
To highlight the differences between the two methods, we chose a geomagnetic
storm day (17 March 2015), where we would expect large gradients and more issues
with data quality. Because of this, the bias determination problem is more
challenging than on a geomagnetically quiet day.
The two maps are shown in Fig. . The two TEC maps show no
major differences in broad general features, which is to be expected. The
main differences between the two images is that there are visibly fewer
outliers produced by the new method. For example, the Asian and European
sectors are significantly smoother with the new method. The ionospheric
trough associated with the sub-auroral ionization drift (SAID) stretching
from Asia to northern Europe is much more clearly seen with the new method.
Probably due to the strong gradients associated with the storm, the old
method fails to derive good bias values for a large number of receivers in
China, resulting in negative TEC values, which are not plotted. The new
method finds these values of bias more reliably.
The polar regions have slightly more TEC when using WLLSID. This is because
the MAPGPS uses the zero-TEC method for receiver bias determination at high
latitudes, whereas the WLLSID method is applied in the same way everywhere.
Conclusions
In this paper, we describe a statistical framework for estimating bias of
GNSS receivers by examining differences between measurements. We show that
the framework results in a linear model, which can be solved using linear
least squares. We describe a way that the method can be efficiently
implemented using a sparse matrix solver with very low memory footprint,
which is necessary when estimating receiver biases for extremely large
networks of GNSS receivers.
We compare our method for bias determination with the existing MIT Haystack
MAPGPS method and find the new method results in smaller day-to-day
variability in receiver bias, as well as a more self-consistent vertical TEC
map. Qualitatively, the new method reproduces the same general features as
the existing MAPGPS method that we compared with, but it is generally less noisy
and contains fewer outliers.
The weighting of the measurement differences is done using a structure
function. We outline a few ways to do this, but these are not guaranteed to
be the best ones. Future improvements to the method can be obtained by coming
up with a better structure function, which can possibly be determined from
the data themselves, e.g., using histograms, empirical orthogonal function
analysis, or similar methods.
While we describe how differences result in a linear model, we do not explore
to a large extent in this work the possible ways in which differences can be
formed between measurements. Because of the large number of measurements,
obviously all the possible differences cannot be included in the model. In
this study, we only explored two types of differences: (1) differences between
geographically separated, temporally simultaneous measurements obtained with
tens of receivers located near each other and (2) differences in time less than
2 h performed with a single receiver. There are countless other
possibilities, and it is a topic of future work to explore what differences
to include to obtain better results.
We describe several important special cases of the method: known satellite
bias, single receiver and known satellite bias, and the case of multiple bias
terms per receiver. The first two are applicable for GPS receivers, and the
last one is applicable to GLONASS measurements, as well as measurements where
a loss satellite signal has caused a step-like error in the TEC curve.
Acknowledgements
GPS TEC analysis and the Madrigal distributed database system are supported
at MIT Haystack Observatory by the activities of the Atmospheric Sciences
Group, including National Science Foundation grants AGS-1242204 and
AGS-1025467 to the Massachusetts Institute of Technology. Vertical TEC
measurements using the standard MAPGPS algorithm are provided free of charge
to the scientific community through the Madrigal system at http://madrigal.haystack.mit.edu.
Edited by: M. Portabella
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