Mapping GPS positional errors with spatial linear mixed
models
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Citation
Militino, A. F., M. D. Ugarte, J. Iribas, and E. Lizarraga-Garcia.
“Mapping GPS Positional Errors Using Spatial Linear Mixed
Models.” Journal of Geodesy 87, no. 7 (July 2013): 675–685.
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http://dx.doi.org/10.1007/s00190-013-0638-z
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Springer-Verlag
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Wed May 25 20:57:06 EDT 2016
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Journal of Geodesy manuscript No.
(will be inserted by the editor)
Mapping GPS Positional Errors with Spatial Linear Mixed
Models
A. F. Militino, M. D. Ugarte, J. Iribas, and E. Lizarraga-Garcia
Received: date / Accepted: date
Abstract Nowadays, GPS receivers are very reliable
because of their good accuracy and precision; however,
uncertainty is also inherent in geospatial data. Quality of GPS measurements can be influenced by atmospheric disturbances, multipathing, synchronization of
clocks, satellite geometry, geographical features of the
observed region, low broadcasting coverage, inadequate
transmitting formats, or human or instrumental unknown errors. Assuming that the scenario and technical
conditions that can influence the quality of GPS measurements are optimal, that functional and stochastic
models that process the signals to a geodetic measurement are correct, and that all the GPS observables are
taken in the same conditions, it is still possible to estimate the positional errors as the difference between the
real coordinates and those measured by the GPS. In this
paper, three spatial linear mixed models, one for each
axis, are used for modelling real-time kinematic (RTK)
GPS accuracy and precision, of a multiple-referencestation network in dual-frequency with carrier phase
measurements. We interpret “accuracy” as unbias and
“precision” as a variance, but not for a particular signal
or a set of signals when they are processed to become
Militino, A. F. and Ugarte, M. D.
Department of Statistics and Operations Research
Public University of Navarre
Campus de Arrosadia, 31006 Pamplona, Spain
Tel.: +34-948-169206
Fax: +34-948.169204
E-mail: militino@unavarra.es
Iribas, J.
Department of Civil Engineering
Government of Navarre, Spain
Lizarraga-Garcia, E.
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, USA
an estimated coordinate. Along the paper, the proposed
models provide an estimate of the “accuracy” in terms
of bias defined as the difference between real coordinates and measured coordinates after being processed.
Moreover, our proposed models also provide an estimate of the “precision” through the standard errors of
the estimated differences. This is done using ten different transmitting formats. Mapping and quantifying
these differences can be interesting for users and GPS
professionals. The performance of these models is illustrated by mapping positional error estimates within the
whole region of Navarre, Spain. Sampled data have been
taken in 54 out of the 211 geodetic vertex points of this
region. Maps show interesting error patterns depending on transmitting formats, the different axes, and the
geographical characteristics of the region. Higher differences are found in regions with bad broadcasting coverage, due to the presence of mountains and high degree
of humidity.
Keywords Statistical estimation · Kriging · Geophysical measurements · Statistics · Spatial correlation
1 Introduction
The Global Positioning System (GPS) determines positions by measuring distances to Earth-orbiting satellites. Hofmann-Wellenhoff and Lichtenegger (2001) provide a comprehensive study on how GPS works. However, the measurement of a particular location is not exact. For example, Teunissen (1998) shows closed-form
expressions for the minimal detectable biases of single and dual-frequency pseudo-range and carrier-phase
data with three different single-baseline models. There
are still some relevant factors that can reduce the quality of the measurements, mainly due to: a) Atmospheric
2
disturbances, b) Timing of clocks, c) Multipathing, d)
Satellite ephemerides and e) Satellite geometry. a) Atmospheric disturbances consist mainly of the slowdown
that the signal experiences while passing through the
ionosphere, which delays the signal’s arrival at the receiver, thereby affecting the distance calculation. Lejeune et al. (2012) develop software to compute the positioning error due to the ionosphere for all baselines of
the Belgian GPS network, the so-called Active Geodetic Network (AGN). Hernández-Pajares et al. (2011)
provide a summary of the ionosphere as related to
space geodetic techniques. The amount of water vapor in the atmosphere can also affect the time delay.
Lenderink and van Meijgaard (2001) study structures
in predicted Integrated Water Vapor model for three cyclones (Kerstin, Liane, Monica) in 1995. Two versions of
the model with differences in the physics package were
compared: standard ECHAM4, and ECHAM4 with revised cloud and turbulence physics. The delay due to
water vapor, the wet delay, is difficult to model using
ground surface data. Thus, it is often estimated from
the GPS data. In order to obtain the most accurate
results from the GPS processing, a model of the horizontal distribution of the wet delay may be necessary.
Emardson and Jarlemark (1999) evaluate these models
through simulations. Banville and Langley (2013) provide a geometry-based approach with rigorous handling
of the ionosphere. Bock et al. (2010) show that the use
of modified GPS antennas can improve the propagation delays in the neutral atmosphere (troposphere). b)
An additional source of uncertainty is timing because
the clocks in the artificial satellites are very near perfect
(Han et al. 2001), but the clocks in the receivers are not
as good (Huang and Zhang 2012, Kenneth et al. 2008).
At this regard, Liu et al. (2004) model the between
observation-type delays for the purpose of precise positioning, under practical circumstances. c) Multipathing
is the reflection of the satellite signal off a building or
some other large reflective object before reaching the
receiver, which again delays its arrival as indicated by
Mekik and Can (2010). Under the assumption that the
surrounding environment remains unchanged, Phan et
al. (2012) show that multipath contamination of GPS
measurements can be formulated as a function of the
sidereal repeatable geometry of the satellite with respect to the fixed receiver. If the direct signal is also acquired, the software of most modern receivers can reject
the indirect signals. However, multipathing can confuse
the calculations for position. Iwase et al. (2013) detect
the multipath errors in the satellite signals and exclude
these signals to improve the positioning accuracy of
GNSS, in particular in an urban canyon environment.
d) Satellite ephemerides can also reduce the quality of
A. F. Militino, M. D. Ugarte, J. Iribas, and E. Lizarraga-Garcia
Fig. 1 Lakora, one of the 211 vertex points in Navarre; and
map of Navarre
measurements because the gravitational pull of the Sun
and Moon on a satellite can distort its orbit. Desmars
et al. (2009) introduce a new method to estimate the
accuracy of predicted positions at any time, in particular outside the observation period. e) The relative position of the satellites, known as the satellite geometry,
can also affect the accuracy. Cai et al. (2012) study the
position accuracy depending on the satellite geometry,
and satellite elevation. According to the existing correlation between day-to-day residual estimates, Wu and
Hsieh (2010) prove that a multipath mitigating algorithm improves the accuracy of the GPS height determination by at least 40%. Teunissen (1997) consider
the time-averaged GPS single-baseline model. This author studies its relation with the geometry-free model
and the geometry-based model in a qualitative sense.
The least-squares estimators of the model are also derived and their properties discussed. Finally, Luo et al.
(2011) illustrate the discrepancies between the normal
distribution assumption and reality, based on a large
and representative data set of GPS phase measurements
covering a range of factors, including multipath impact,
baseline length, and atmospheric conditions.
Estimation of GPS positional reliability has been extensively studied in the literature from different points
of view. For example, Hasewaga and Yoshimura (2007)
estimate the GPS positional accuracy under different
forest conditions. Rodriguez-Solano et al. (2012) study
the impact of Earth radiation pressure on the positional accuracy. Cai and Gao (2012) combine GPS and
GLONASS observations to improve accuracy. Odolinski (2012) takes into account the temporal correlation
for improving accuracy in real-time kinematic (RTK)
network data. However, the study of the GPS positional accuracy after processing data is scarce from the
stochastic perspective. Cressie and Kornak (2003) review a method for adjusting spatial inference in the
presence of data location error, particularly for data
that have a continuous spatial index. Specifically, their
paper derives a kriging methodology in the presence
Mapping GPS Positional Errors with Spatial Linear Mixed Models
Table 1 Abbreviations of network and the closest formats
abbreviation
format
N1
N2
N3
N4
N5
N6
NET i-max CMR
NET i-max CMR+
NET i-max Leica
NET i-max RTCM 2.3
NET i-max RTCM 3.1
NET max RTCM 3.1
C1
C2
C3
C4
CLOSEST CMR
CLOSEST Leica
CLOSEST RTCM 2.3
CLOSEST RTCM 3.1
of location error. The authors study two location error
models known as the coordinate-positioning model and
the feature-positioning model. In Cressie’s work, the
aim is to assess the effect on spatial prediction without making inferences about true feature locations. In
Barber et al. (2006), map positional errors are defined
as differences between locations represented in a spatial database and the corresponding unobservable true
locations. The authors focus on GIS positional errors,
explaining how estimates of positional error can be obtained, and providing estimates of the true location.
Three bivariate variables are used: a feature location
on the map, an associated GPS location, and a true
location for that feature. The authors cast the map positional error problem as one registering GPS locations
against the given map followed by inferring the true
location based upon their knowledge of GPS accuracy.
Wang et al. (2002a) study the dependence of the GPS
positioning precision on the station location. They use
a weighting function to account for an averaged number of satellites at different latitudes and demonstrate
that GPS positioning precision reduces when the latitude increases. They also quantify the positioning precision in the north-south direction, showing that it is
worse than in the east-west direction at all latitudes.
The same conclusion is given by Meng et al. (2004) and
Bosser et al. (2007), who study the impact of satellite
geometry. Lack of temporal dependence specification
between carrier phase measurements and the effect on
precise positioning is studied by Wang et al. (2002b).
The step of processing raw data to transform them
into geodetic measurements requires the specification of
a mathematical model consisting of a functional model
and a stochastic model (Tiberius, 2000). This process
involves computing unknown parameters from a set of
measurements or observations, preferably incorporating
some measures that express the quality of the estimated
parameter values (Tiberius, 1999). Least squares estimation constitutes the most popular technique to obtain the desired estimates. Then, it is clear that stochas-
3
tic models are a compulsory requirement for obtaining
final GPS measurements (Wang et al. 1998, Satirapod
et al. 2002, Wang et al. 2005, Amiri-Simkooei et al.
2009). However, this kind of models is not the objective
of our paper. Assuming that all these stochastic models
are already specified and correctly estimated, we consider that the differences between real and final measured coordinates in GPS are originated by unknown
reasons: if they were known, they could be corrected.
That is why three spatial linear mixed models are proposed to estimate the unknown sources of uncertainty.
Our goal is not to improve estimations by modifying
any parameter of these stochastic models. On the contrary, we are going to assume that all these stochastic
models are correct and perfectly estimated. Nonetheless, there may exist some misspecification or any other
kind of uncertainty that we cannot change due to atmospheric disturbances, satellite geometry, satellite elevation, multipahing or any other already cited. In that
case, we only assume that we can estimate the positional errors as the differences between the GPS measurements and the real ones. A map of these differences
in X, Y , and Z axes will provide an additional and
useful information to the user.
Then, this work addresses the estimation of GPS
positional errors. The observations have been taken
with a multiple-reference-station network in a realtime-kinematic system of dual-frequency using carrierphase measurements. The application is done providing
prediction of positional errors and their standard errors
in the whole region of Navarre, Spain.
The paper is organized as follows. Section 2 explains
the application and the terminology. Section 3 reviews
the theory of the spatial linear mixed-effects model and
defines the specific models for the three different axes
X, Y , and Z. It also explains how these models allow for
the estimation of positional errors. Section 4 provides
the results and, finally, the Summary and Conclusions
are presented.
2 Application
The application illustrates the performance of GPS
techniques based on the positioning in real time services
provided by Navarre active geodetic network infrastructure, called RGAN (see RGAN 2009). This network
consists of 14 beacon receiver stations strategically located along the region. The static network consists of
211 vertex points whose locations are fixed and spread
in the 10,000 km2 of the region. The RGAN network is
based on Leica Geosystems technology and uses Spider
as the maintenance’s software from the same company.
The equipment used in this work consists of a Leica
4
ATX 1230 and a GNSS Smart Rover made up of a Leica
1230 Sensor, a GNSS antenna and a RX 1250 XC Controller. As Brown et al. (2006) point out, combining the
Leica GPS Spider RTK network software with the Leica
System 1200 GPS receivers clearly improve the measurements, even when using network correction data at
a sampling rate of only 5s. The geodetic data are given
with the European Terrestrial Reference System 1989
(ETRS89) (see Bruyninx et al. 2001), which is the reference system used by RGAN and validated by the Spanish National Geographic Institute (IGN). A preliminary
work (Lizarraga-Garcia 2009) accomplished a simulation study where 54 sampled vertex points could be chosen to minimize the mean squared prediction error for
the whole province. Following that study, the Department of Civil Engineering of the Navarre Government
continued taking field measurements during 2011 and
2012. The sampled vertex points are regularly spread
along the whole territory with a mean distance of 40km
between them. They correspond to round and blue dots,
while square and red dots are the beacon stations in
Figures 2, 3, and 4. The sampled vertex points are
also located at different heights, varying between 207m
and 1844m with a mean height of 840m. Moreover,
the corresponding ground-truth data is also known for
these sampled vertex points. Ground-truth data have
been obtained in the same conditions as the sampled
measurements. That is, measurements for elaborating
ground-truth data are taken at the same time and with
the same rover as the sampled measurements in order
to avoid influence of different satellite geometry, atmospheric variability, broadcast connections or device handling. However, the way of processing the information is
different: while the sampled measurements are a limited
number of real-time measurements using the real-time
kinematics technique, RTK, the ground-truth data are
the result of static postprocessing measurements during
at least one hour. The difference between sample and
ground-truth coordinates, in X, Y , and Z will provide
the response variables Dif f X, Dif f Y , and Dif f Z of
the three stochastic positioning error models.
In this paper, real time kinematic (RTK) technique
with phase, dual-frequency and a multiple-referencestation network is used. Thus, GPS positioning is corrected by adjusting measurements taken by two or more
receivers. Positioning measurements of sampled vertex
points have been taken using 10 different transmitting
formats and they are based on both the closest reference
station (“CLOSEST” type) and the network of closer
stations (“NET” type). The names of the formats used
and their abbreviations are summarized in Table 1.
Radio technical position commission for maritime
services (RTCM) is a widely used protocol for com-
A. F. Militino, M. D. Ugarte, J. Iribas, and E. Lizarraga-Garcia
munication between the reference stations and the mobile receivers at the location being determined. The use
of RTCM standards allows for interoperation between
equipment from different manufacturers. Edition 3.1 includes an interoperable definition for RTK Network operation, which supports centimeter-level accuracy positioning service over large regions. Leica SpiderNet was
used to calculate single sites; however, MAX and iMAX
corrections are used in RTCM 2.3 and 3.1.
The RTCM “Version 3” message format is optimized for RTK operation, which requires more data
than earlier differential correction techniques, such as
in “Version 2” (see Brown et al., 2006). Among the
different formats, MAX format is very competitive because it selects automatically the optimum number of
sites for the cell to generate master-auxiliary corrections for each rover. The auxiliaries are chosen from the
surrounding stations to provide the best possible set of
corrections for the rover’s position. For older rovers, Leica GPS Spider provides an individualized version of the
master-auxiliary corrections, known as iMAX, that may
be transmitted using older versions of RTCM. It has the
advantage of using a lower bandwidth single site RTCM
2.3 or 3.0 format that can also be interpreted by older
receivers that do not support the new network messages. The use of the closest reference station provides
very accurate precision in each of the response variables
and even better precision than some iMAX formats.
MAX and iMAX show very similar values and a better
performance than other RTK network formats. “NET imax CMR” and “NET i-max CMR+” are Trimble formats and “NET i-max Leica” is a very specific format
used by Leica.
3 Spatial Linear Mixed-Effects Model
Traditionally, kriging has been largely used in spatial
modelling, as a minimum squared prediction method of
spatial variables where spatially correlated observations
come from a single realization of a spatial stochastic
process. In this work, repeated measurements in each
location have been taken in several transmitting RTK
GPS formats, yet the number of transmitting formats
for each location can differ. In order to analyze positional errors, spatial linear mixed-effects models are
proposed as an alternative to kriging. The models contain both fixed and random effects. Fixed effects allow
for modelling positional errors, and random effects allow for modelling their variance-covariance structure.
The inclusion of random effects reduces the number
of parameters in the specification of the variances and
gives more flexible models (see McCullogh and Searle,
Mapping GPS Positional Errors with Spatial Linear Mixed Models
2001). In particular, the same random effects associated with the repeated measurements in each location
express a kind of correlation within this group of data.
Moreover, spatially dependent data may share the same
covariance matrix of the error term as is done in kriging.
In this case, spatial linear mixed models can incorporate
the spatial dependence into the covariance matrix of the
error term in a similar way as kriging does, but they can
also incorporate the correlation between samples taken
under the same transmitting format through a common
random effect. Thus, spatial linear mixed models are an
alternative for modelling spatial linear models with repeated measurements (Militino et al. 2008). The general
spatial linear mixed model is written as
Y = Xβ + Zu + ,
(1)
where X is the design matrix (n × (p + 1)) of the unknown fixed effects, β = (β0 , β1 , . . . , βp )t is the (p + 1)
vector of fixed parameters and u = (u1 , . . . , uq )t is the
q-vector of random effects with an associated (n × q)
design matrix Z. This matrix consists of a collection
of zeros and ones that indicate the association between
random effects and observations. The random variables
u and are assumed to have mean equal to 0 and covariance matrix
t
cov[u, ] = σ
2
G 0
0 R
5
The vector of variance components θ may be estimated by using the fitting of constants method or
likelihood-based methods, such as the maximum likelihood (ML) or the restricted maximum likelihood
(REML) method. In this paper, the REML method is
used, and the restricted log likelihood function of Model
(1) is given by
1
1
lR (θ|Y) = − log|Vθ | − (Y − Xβ̂ θ )t Vθ−1 (Y − Xβ̂ θ )
2
2
n−p
1
log(2π).
(4)
− log|Xt Vθ−1 X| −
2
2
Here, β̂ θ is the estimate of β given by expression (2) and
both β̂ θ and Vθ depend on the vector of parameters θ.
The function given in expression (4) does not involve
β, and therefore the resulting variance parameter estimators are unbiased. For this and other advantages,
REML is usually recommended.
The asymptotic covariance matrix of the parameter
estimators can be obtained using the standard theory
of maximum likelihood, that is, calculating the inverse
of the observed information matrix evaluated at the optimum. Empirically, the approximate covariance matrix
of β̂ and û is calculated as the covariance of the prediction error, because neither of the predictors û nor
β̂ account for the uncertainty in the prediction of the
other. This covariance matrix is given by
.
"
D=E
Hence, the mean and variance of Y are given by E[Y] =
Xβ and V = ZGZt + R respectively.
Historically, one of the first solutions to the estimation process of fixed and random effects was proposed
by Henderson (1953). In principle, normality for the
random effects u does not need to be assumed; however,
it is a common assumption for inferential purposes. Assuming that R and G are known, Henderson obtained
the best linear unbiased estimator (BLUE) of β and the
best linear unbiased predictor (BLUP) of u, solving the
well-known mixed model equations which provide the
solution
β̂ = (Xt V−1 X)−1 Xt V−1 Y = AXt V−1 Y,
(2)
û = GZt V−1 (Y − Xβ̂),
(3)
where A = (Xt V−1 X)−1 . But G and R have to be estimated, and then β̂ and û are obtained substituting V
and G by Vθ̂ and Gθ̂ , respectively, where θ is the vector of variance components and θ̂ is its estimator. The
estimates for β and u are then called empirical BLUE
(EBLUE) and empirical BLUP (EBLUP), respectively.
β̂ − β
û − u
β̂ − β
û − u
t #
=
D11 Dt21
D21 D22
,
where
D11 = var(β̂ − β) = var(β̂) = (Xt V−1 X)−1 = A,
D21 = cov(β̂ − β, (û − u)t ) = −GZt V−1 XD11 ,
D22 = var(û − u) = (Zt R−1 Z + G−1 )−1
− D21 Xt V−1 ZG.
In practice, it is estimated by substituting G and V by
Gθ̂ and Vθ̂ , respectively.
Let xi,j , yi,j , and zi,j be the average of the three
measurements taken in each sampled vertex point,
where i = 1, . . . , nj indicates the observation within
each format and nj is the total number of observations of the jth-format (j = 1, . . . , 10). Coordinates
xi,j and yi,j correspond to the UTM coordinates, and
zi,j to the orthometric elevation. Let xi,true , yi,true ,
and zi,true be the corresponding ground-truth coordinates. Three types of measurement errors are considered here: differences in the X axis calculated as
Dif f Xi,j = xi,true − xi,j ; differences in the Y axis,
Dif f Yi,j = yi,true − yi,j ; and finally differences in the
Z axis (orthometric elevation) given by Dif f Zi,j =
6
A. F. Militino, M. D. Ugarte, J. Iribas, and E. Lizarraga-Garcia
zi,true − zi,j . Besides the measured coordinates, two
additional covariates are used as fixed effects in every
model: qalt and qposic corresponding to the standard
deviations of the observable height and position, respectively.
The three different response variables are fitted with
spatial linear mixed models, using libraries “nlme” for
parameter estimation (Pinheiro and Bates, 2000) and
“AICcmodavg” (Mazerolle, 2006) for standard errors of
the predictions. Both libraries can be obtained from the
free statistical package R (R Development Core Team
2012). See also Ugarte et al. (2008) for linear statistical
modelling in R. The stochastic models are given by
Dif f Xi,j = β0,x + β1,x xi,j + β2,x yi,j + β3,x zi,j
Table 2 Range of the variogram, power of the heteroscedastic model variance, and residual standard error for the three
response variables in Km
Variable
Range (r)
var power (δ)
residual s.e. (σ̂w )
DiffX
DiffY
DiffZ
6.15
1.70
2.59
0.99
1.32
1.34
0.27
0.15
0.23
Table 3 Summary of the predictions and their standard errors in mm by network and the closest formats for DiffX
format
min
+ β4,x qalti,j,x + β5,x qposici,j,x + uj,x
+ i,j,x ,
Dif f Yi,j = β0,y + β1,y xi,j + β2,y yi,j + β3,y zi,j
+ β4,y qalti,j,y + β5,y qposici,j,y + uj,y
+ i,j,y ,
Dif f Zi,j = β0,z + β1,z xi,j + β2,z yi,j + β3,z zi,j
error prediction
max range mean
prediction s.e.
min
max
N1
N2
N3
N4
N5
N6
-3.23
-3.87
-3.62
-3.52
-1.68
-2.47
8.52
7.70
5.46
4.80
7.35
6.50
11.75
11.57
9.08
8.32
9.03
8.97
1.93
1.85
1.20
1.47
1.53
1.47
0.34
0.35
0.36
0.33
0.34
0.34
2.76
2.00
1.44
1.50
1.76
1.61
C1
C2
C3
C4
-2.27
-3.35
-2.03
-3.12
7.32
5.19
6.10
6.86
9.59
8.54
8.13
9.98
2.58
1.71
2.22
1.83
0.35
0.33
0.37
0.37
1.89
1.49
2.15
1.85
+ β4,z qalti,j,z + β5,z qposici,j,z + uj,z
+ i,j,z .
Let us denote by w the directions x, y or z. Then,
uj,w is the jth random effect common to the observations taken at the same transmitting format j,
such that the overall vector uw = (u1,w , . . . , u10,w ) ∼
2
Gw ); and i,j,w , where i = 1, . . . , nj , is the
N (0, σw
i − jth random error, such that (1,1,w , . . . , 54,10,w ) ∼
2
N (0, σw
Rw ). The heteroscedasticity present in all the
three models has been corrected expressing model variances as a power function of the covariates, in this case
|qalti,j,w + qposici,j,w |2δw . Heteroscedasticity has been
tested by the likelihood ratio test in all the models. Estimation of covariance parameters has been obtained by
restricted maximum likelihood, and the exponential covariance function has been used. At lag h it corresponds
to
2
C(h) = c0 + σw
(exp(− k h k /rw )),
where c0 is the nugget effect and rw is the range.
4 Results
Measurement units are important when working with
statistical models as kriging, because the kriging procedure could be sensitive to these units. However, from
a statistical point of view, the projected coordinates
are preferable to angular coordinates such as longitude
and latitude because of two reasons. Firstly, singularities in matrix inversion processes are avoided and, secondly, they provide a constant distance relationship
anywhere on the map. On the other hand, UTM coordinates are easy to interpret for users that are nonexperts in Geodesy. Conversion between coordinates
can be done directly through mathematical expressions.
Anyway, similar results could be drawn when using alternative coordinates.
Estimates of the covariance range (rw ), variance
power (δw ), and residual standard error (σw ) are given
in Table 2. Covariance ranges of Dif f X, Dif f Y , and
Dif f Z vary between 6.15 and 1.7mm. As expected, the
spatial dependence shows that similar positional errors
are only found at very short distances, behavior particularly pronounced in the X direction. In the Y and
Z direction spatial dependence is smaller. The lowest
variance power (δw ) of the heteroscedastic variance corresponds to the X direction and the biggest one to the
Z direction. Thus, larger heteroscedasticity is expected
in the Z or Y direction with regard to X. Residual
standard errors (0.27, 0.15, and 0.23) are roughly similar, showing that the covariates explain accurately the
variability of the response variables.
Mapping GPS Positional Errors with Spatial Linear Mixed Models
7
(mm)
(mm)
6
1.6
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150
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Fig. 2 Prediction map of the positional errors (left) and their standard errors (right) in X for NET i-max Leica format
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Fig. 3 Prediction map of the positional errors (left) and their standard errors (right) in Y for NET i-max Leica format
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Fig. 4 Prediction map of the positional errors (left) and their standard errors (right) in Z for CLOSEST Leica format
8
A. F. Militino, M. D. Ugarte, J. Iribas, and E. Lizarraga-Garcia
Table 4 Summary of the predictions and their standard errors in mm by network and the closest formats for DiffY
format
min
error prediction
max range mean
prediction s.e.
min
max
N1
N2
N3
N4
N5
N6
-2.89
-3.87
-3.53
-3.41
-2.53
-2.56
15.56
6.49
2.78
4.76
6.69
6.38
18.45
10.36
6.31
8.16
9.23
8.95
1.10
0.89
-0.57
0.01
0.41
0.09
0.41
0.44
0.43
0.38
0.40
0.40
4.35
3.17
2.06
2.29
2.79
2.33
C1
C2
C3
C4
-2.29
-4.20
-2.30
-2.56
8.08
3.89
9.51
6.48
10.37
8.09
11.81
9.04
2.15
-0.08
0.89
0.84
0.43
0.38
0.46
0.44
2.87
2.08
3.45
2.81
Table 5 Summary of the predictions and their standard errors in mm by network and the closest formats for DiffZ
format
min
error prediction
max range
mean
prediction s.e.
min
max
N1
N2
N3
N4
N5
N6
-38.34
-26.16
-6.48
-24.29
-9.78
-10.11
-11.77
-6.35
12.48
-5.95
13.85
16.50
26.57
19.81
18.95
18.34
23.63
26.61
-23.87
-16.69
5.48
-14.04
4.90
6.98
3.69
3.70
3.71
3.69
3.69
3.70
8.50
6.48
5.01
5.30
5.95
5.35
C1
C2
C3
C4
-13.19
-5.98
-9.11
-11.83
12.52
12.51
18.59
9.64
25.71
18.50
27.70
21.48
1.42
6.19
6.13
2.91
3.70
3.69
3.71
3.71
6.12
5.02
6.85
6.01
Whereas the UTM coordinates X and Y and the
orthometric height Z are known, qalt and qposic covariates are not. As a result, qalt and qposic sampled
values of the closest sampled location are used for every
location of the prediction grid. That is why the maps
of the predictions have a polygonal shape that is defined by the closest locations to the sampled one (see
Figures 2, 3, and 4). Tables 3, 4, and 5 show the transmitting format, the minimum, maximum, range, and
mean of the prediction measurement errors, and their
minimum and maximum standard errors, for the whole
Navarre region. A summary of these numerical results
is plotted in Figures 5 (for network solutions) and 6
(for the closest reference station), where blue bars indicate the range of the predicted positional errors and
red bars depict the range of their standard error. The
prediction range and the standard errors of Dif f X and
Dif f Y are quite similar, yet in Dif f Z standard errors
are a bit bigger. Looking jointly at the range and mean
of the positional error prediction, and taking into ac-
count their standard errors, it can be concluded that
“NET i-MAX Leica” (N3 ) is the most accurate format
in the three axes. In the X direction, network formats
are more accurate than the closest formats, but in the
Y or Z direction the closest formats are very competitive, specifically “CLOSEST Leica” (C2 ) format. In X
and Z a positive bias always exists because prediction
means are always positive, but in the Y direction some
formats have a negative bias. Also, it is observed that
N3 format can be even more accurate in Y than in X,
although Z is always less accurate than planar coordinates. In N1 , N5 , C1 , and C3 transmitting formats, the
West-East or X direction has less range error than Y ,
which agrees with the result obtained by Wang et al.
(2002a), Meng et al. (2004), and Bosser et al. (2007).
However, this behavior does not happen in the rest of
the formats. Moreover, X error mean is always bigger
than Y error mean, yet range error in X is not always
bigger than in Y . Overall, network formats are not necessarily more precise than the closest formats in all the
axes. In the X direction, the prediction error range for
all formats is between -4mm and 9mm, in Y between -4
and 16mm, and in Z between -38mm and 19mm. In the
best format (N3 ), Dif f X can vary between -4mm and
6mm, and Dif f Y between -4mm and 3mm, and in the
C2 format Dif f Z varies between -6mm and 12.5mm.
In this work, we illustrate several maps where the X, Y,
and Z axes are expressed in UTM coordinates that were
scaled to Km and moved to a new origin defined by the
minimum of X and Y. With this new coordinates origin,
we could easily interpret distances to check spatial dependence and compare small geographical regions with
similar errors. The positional errors are mapped in a
grid of 1km × 1km of the whole region of Navarre. Figures 2 and 3 show predicted positional errors of Dif f X
and Dif f Y on the left, and their standard errors on
the right, for the whole Navarre in “NET i-MAX Leica”
format. Figure 4 shows the predictions of Dif f Z and its
standard errors in “CLOSEST Leica” format. The geographical pattern of these maps is similar in all directions. North and North-East directions present larger
positional and standard errors. They correspond to the
Pyrennees region, where humidity and mountains are
frequent and, perhaps, broadcast connections are also
weak. Finally, some spots in the center West direction
with larger positional errors than on the rest are also
seen. Perhaps a bad broadcast connection is also present
here.
Summary and Conclusions
The primary aim of this work is to analyze the quality of
the GPS measurements taken with a multiple-reference
Mapping GPS Positional Errors with Spatial Linear Mixed Models
CLOSEST_CMR
CLOSEST_Leica
35
25
20
5
5
10
15
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25
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diffZ
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pred
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se
pred
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pred
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15
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se
pred
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20
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pred
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diffX
diffX
NET_i−max_RTCM_3.1
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35
NET_i−max_RTCM_2.3
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diffZ
35
diffY
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5
5
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pred
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35
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pred
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NET_i−max_Leica
35
NET_i−max_CMR+
35
35
NET_i−max_CMR
9
diffX
diffX
diffY
diffY
diffZ
diffX
diffY
diffZ
diffZ
Fig. 5 Range summary of predictions and standard errors
for network formats
Fig. 6 Range summary of predictions and standard errors
for the closest formats
station network on RTK system with dual-frequency
and carrier-phase measurements. We assuming that all
uncontrollable sources of uncertainty are random and
they can be explained using spatial linear mixed-effects
models. These models provide a good methodology for
jointly predict the positional errors in different formats
when spatial dependence is assumed. The scenario for
illustrating the measurements is based on a high quality
device and a geodetic infrastructure of 14 beacon reference stations over a region of 10,000 km2 . RTK has been
the differential process used for correcting position measurements in real time. Also, 10 different formats have
been used to transmit data, allowing for a connection
to the closest or to a network of closer reference stations. The variables Dif f X, Dif f Y , and Dif f Z, calculated as differences among ground-truth coordinates
and measured GPS coordinates, have shown different
accuracy and precision performances between them and
within the formats, yet all of them provide centimeteraccuracy positions, less than 1cm for X or Y , and less
than 1.5cm for Z. Higher errors for Z have also been
reported in the literature (Meng et al. 2004).
The main contribution of our paper is the methodology used for providing positional error maps when using
a GPS receiver. Thus, this methodology could be easily
extended to other regions, although our study is limited to Spain. We have focused on a multiple-referencestation RTK network in dual-frequency with carrierphase measurements because a high performance is attained. However, other alternatives could also be used.
In addition, the statistical methodology proposed in
this paper is very useful to check the reliability of a
beacon station network in a certain region or to ascertain the optimal location of a new beacon station.
Consequently, although GPS commercial devices
present a high level of accuracy and precision, this can
be increased using appropriate transmitting formats. In
particular, network formats are not always better than
the closest formats. Here, the closest formats are very
competitive when measuring on the Y or Z axes, and
network formats are better for the X coordinate.
The transmitting formats are not a critical issue, but
it can influence the quality of the measurements. Thus,
we consider different models depending on the formats,
and deduce that there are slight differences between
them. This issue allows us to check if the location-based
error distributions are in fact independent or not of
these transmitting formats.
10
In addition, precision in Y is not always less than
in X for all the transmitting formats, contradicting the
theory. In specific formats, the precision of Y can be
even better than the prediction of the X coordinate.
On average, Y direction is more precise than X direction (because its means are lower than X means), but it
shows greater variability (because Y ranges are greater
than X ranges). Unfortunately, Z is always less accurate than planar coordinates and here the closest format
is very competitive with regard to network formats.
Final findings show that positional errors are not
completely independent of the locations. Valley or
mountain regions with high degrees of humidity present
both greater positional and standard errors, possibly
due to a weak broadcasting coverage.
To summarize, it has been shown that positional
error maps derived from spatial linear mixed models
can help the user to evaluate the quality of positional
measurements.
Acknowledgements The authors would like to thank the
comments made by the associate editor and the three referees.
They have contributed to improve this manuscript.
The authors would also like to thank the Department of
Civil Engineering of the Navarre Government for providing
the data, and more specifically to Miguel Ángel Jiménez de
Cisneros. This research has been supported by the Spanish
Ministry of Science and Education (project MTM2011-22664
jointly sponsored with Feder grants).
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