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.

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

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

A GNSS measurement of relative propagation time delay difference including
the line-integrated electron density effect can be written as

For ionospheric research with GNSS receivers that perform measurements of the
form shown in Eq. (

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.,

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

There are several ways that

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

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

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

Let us denote Eq. (

Now consider subtracting slanted TEC measurements

This type of a difference equation has several benefits. If measurements

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

We assume the structure function depends on the following factors: (1)
geographic distance between pierce points

The following subsections describe the structure function behaviors for each dependent variable.

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:

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

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:

In this work, we use

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.

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. (

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:

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:

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

The differences in Eq. (

The random variable vector

The theory matrix

This type of a measurement is known as a linear statistical inverse problem

When a maximum-likelihood solution has been obtained, a useful diagnostic
examines the residuals

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

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.

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.

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. (

For GPS receivers, satellite biases are known to a good accuracy using a
separate and comprehensive analysis technique

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;

Figure

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

Multiple bias terms can be added in a straightforward manner to the model
using Eq. (

An example of a measurement where the same satellite is observed using a
single receiver is shown in Fig.

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 (

Another multiple-bias example is shown in Fig.

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.

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

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

As a measurement of goodness, we used the absolute difference between two
simultaneous geographically coincident measurements of vertical TEC

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.

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:

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 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.

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.

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