ORIGINAL ARTICLE

Assisted adaptive extended Kalman filter for low-costsingle-frequency GPS/SBAS kinematic positioning

Shiou-Gwo Lin

Received: 4 March 2013 / Accepted: 23 April 2014

� Springer-Verlag Berlin Heidelberg 2014

Abstract Using a low-cost single-frequency global

positioning system (GPS) receiver for kinematic posi-

tioning, the ambiguity resolution requires longer data

accumulation time compared to survey-grade dual-fre-

quency receivers. A satellite-based augmentation system

(SBAS) can provide added measurement to help solve this

problem. However, the SBAS signal strength is weaker

and the satellite orbit and clock errors are greater than

those of GPS satellites. The difference in phase mea-

surement quality between GPS and SBAS satellites must

be considered to prevent an unstable positioning result or

diverging position solution. This study proposes using the

assisted adaptive extended Kalman filter (AAEKF)

method to address this problem. The concept of AAEKF

involves using measurements from the reference station to

estimate the errors of each satellite. This information is

then employed to dynamically adjust the corresponding

measurement model of the extended Kalman filter. The

proposed method was validated with 24 h of experiment

data from four different baselines obtained using a con-

sumer-grade L1 GPS receiver. The experimental results

show that AAEKF can be successfully employed for GPS/

SBAS kinematic positioning. The ambiguity resolution

success rates of 2, 5, and 10 min of measurements

improved by about 3.2, 2.4, and 1.6 times, respectively,

and the positioning accuracy of the north, east, and height

directions improved by 14–44, 17–56, and 9–53 %, when

adding the SBAS measurement.

Keywords GPS � SBAS � Adaptive extended Kalman

filter � Kinematic positioning

Introduction

Centimeter-level accuracy can be achieved by using real-

time kinematic (RTK) positioning with low-cost, single-

frequency global positioning system (GPS) receivers and

open source software. Using such a RTK system could

considerably reduce the total project costs of many appli-

cations (Takasu and Yasuda 2008). However, without

precision code (P code) and dual-frequency carrier phase,

the baseline length is limited typically to \10 km, and the

ambiguity resolution requires a longer data accumulation

time. The disadvantage of the limited baseline length can

be improved with multiple reference station technology

(Odijk et al. 2000; Fotopoulos and Cannon 2001; Zhang

and Lachapelle 2001). The objective of this study is to

improve the ambiguity resolution efficiency. The ambigu-

ity resolution efficiency is related to the satellite geometry

and measurement accuracy. Generally, for dual-frequency

receivers, the initialization period of ambiguity resolution

is about 1 min or less (Tiberius and de Jonge 1995), and the

initialization period for single-frequency instruments is

about 15–45 min (Takasu and Yasuda 2008).

Many studies have adopted assistance from satellite-

based augmentation system (SBAS), GLONASS, or iner-

tial navigation system (INS) to increase the number of

measurements and improve the ambiguity resolution effi-

ciency of single-frequency GNSS receivers (Kozlov and

Tkachenko 1998; Skaloud 1998; Wirola et al. 2006; Bo-

riskin et al. 2007). Of these methods, SBAS assistance is a

convenient solution, because many low-cost single-fre-

quency receivers support SBAS. SBAS and GPS satellites

S.-G. Lin (&)

Department of Communications, Navigation and Control

Engineering, National Taiwan Ocean University,

2 Pei-Ning Road, Keelung 20224, Taiwan

e-mail: sglin7222@gmail.com

123

GPS Solut

DOI 10.1007/s10291-014-0381-9

have the same L1 carrier frequency and the same family of

pseudorandom noise (PRN) code. This intentional design

means that GPS receivers can easily support SBAS sys-

tems, and the corresponding pseudorange and carrier phase

measurements equal GPS L1 measurements. The most

substantial difference between SBAS and GPS L1 is the

space segment. The SBAS signal is broadcast by multi-

functional satellites; for example, the European Geosta-

tionary Navigation Overlay Service and Wide Area Aug-

mentation System are primarily broadcast by multiple

geostationary communication and navigation satellites and

the Multi-functional Satellite Augmentation System is

broadcast by multiple geostationary weather and naviga-

tion satellites. These satellites were not specifically

designed for navigation, nor are navigation their primary

function. The space-borne clock errors and orbit errors are

greater than those of GPS satellites. Analyzing the received

measurements of SBAS satellites shows that the signal

strength of SBAS measurements is weaker compared with

GPS satellites.

The GPS provides global coverage with four to eight

simultaneously observable satellites at an elevation above

15� at any time of day. If the elevation mask is reduced to

10�, four to ten satellites are visible (Hofmann-Wellenhof

et al. 1994). Hence, it was possible that five or fewer sat-

ellites could be visible in certain periods in the survey

campaign, because of poor satellite geometry or because of

obstruction by buildings or trees. Most regions in the world

can receive at least two SBAS satellite signals. When the

GPS satellite geometry is poor, the SBAS satellite data can

provide a significant improvement, which can reduce the

required ambiguity resolution time. Compared to GPS

measurement, the signal-to-noise ratio of SBAS measure-

ment is relatively low, and the satellite orbital and clock

errors are larger and unstable. When using SBAS mea-

surements for relative positioning, SBAS satellite errors

must be appropriately considered to obtain a less biased

positioning solution.

The extended Kalman filter (EKF) is the nonlinear

version of Kalman filter (KF), and it has been widely used

for GNSS real-time kinematic positioning (Leick 2004).

However, one of the disadvantages of the EKF is its high

dependence on a priori knowledge of the potentially

unstable process and measurement noise statistics. Using

incomplete a priori statistics when designing a KF can lead

to significant estimation errors and even divergence prob-

lems. The purpose of an adaptive filter is to reduce or limit

these errors by modifying the KF to real data (Beutler et al.

1988).

The GNSS relative positioning technique requires

high-quality GNSS receivers at reference stations. The

reference station collects GNSS measurements for each

satellite in view. Measurements from the reference sta-

tion can be combined with measurements from the rover

station to form the double-difference (DD) measurements

used as the basis for relative positioning. The measure-

ments from the reference station can also be used to

estimate the error components of each satellite (Loomis

et al. 1989; Jin 1996; Farrell and Givargis 2000). The

value of error components and the distance between the

rover and reference stations can be used to estimate the

statistical information of DD measurement noise, which

is called ‘‘assisted statistical knowledge.’’ To prevent

divergence problems caused by errors from the SBAS

satellite when using the EKF approach, assisted statistical

knowledge can be applied to modified adaptive filter.

This method is called assisted adaptive extended Kalman

filter (AAEKF) and was used in this study for GPS/SBAS

kinematic positioning.

Theory background

This study modifies and uses an adaptive KF for GPS/

SBAS kinematic positioning. The GPS measurement

equations, the KF, the adaptive KF, and the proposed

AAEKF are briefly described below.

GPS measurement

The measurement equation of the L1 carrier phase and

pseudorange can be written as (Leick 2004)

ug1p ¼

f1

cRg

p þ Tgp � Ig

p þMPug

1p

� �� f1 dtp � dtg

� �� N

g1p

þ mug

1p

ð1Þ

qg1p ¼ Rg

p þ Tgp þ Ig

p þMPqg

1p� c � dtp þ c � dtg þ mqg

1pð2Þ

where ug1p is the phase measurement from the p receiver to

the g satellite, qg1p is the pseudorange, Rg

p is the geometric

range, Tgp is the tropospheric delay, dtp and dtg are the p

receiver and g satellite clock errors, Igp is the ionospheric

delay, Ng1p is the phase ambiguity, MPug

1pis the phase

multipath, MPqg

1pis the pseudorange multipath, mug

1pis the

phase noise, and mqg

1pis the pseudorange noise.

Linear DD phase combinations between two satellites,

reference receivers, and rover receivers are highly effective

for eliminating common-mode orbital, timing errors, and

atmospheric errors. A DD model was employed for this

study, and its measurement equation can be written as

(Leick 2004)

GPS Solut

123

ugh1pq ¼

f1

cRgh

pq þ Tghpq � Igh

pq þMPugh

1pq

� �� N

gh1pq þ mugh

1pqð3Þ

qgh1pq ¼ Rgh

pq þ Tghpq þ Igh

pq þMPqgh

1pq

þ mqgh

1pq

ð4Þ

The multipath is primarily caused by reflecting surfaces

near the receiver; this effect depends on the specific

receiver–satellite–reflector geometry (Hofmann-Wellenhof

et al. 1994). It does not cancel in the DD model.

Kalman filters

Kalman filters are widely adopted for GNSS kinematic

positioning; their standard formalism can be written as

follows (Gelb 1974):

xk ¼ Tk�1xk�1 þ wk�1;wk�1�Nð0;Qk�1Þ ð5Þzk ¼ Hkxk þ mk; mk �Nð0;RkÞ ð6Þ

Eq. (5) is a dynamic equation, and (6) is a measurement

equation, where k represents the epoch of time, x is the

system state vector, T is the parameter transformation

matrix, and wk�1 is the noise of the dynamic model

N 0;Qk�1ð Þ with Gaussian distribution (mean 0 and

covariance matrix Qk�1), Hk represents the design matrix,

and mk represents the noise vector of measurements

Nð0;Rk�1Þ with Gaussian distribution (mean 0 and

covariance matrix Rk�1). KF estimates are divided into the

following two steps: the time update and the measurement

update. The equation for the time update is as follows:

xk �ð Þ ¼ Tk�1xk�1ðþÞ ð7Þ

Pk �ð Þ ¼ Tk�1Pk�1 þð ÞTTk�1 þQk�1 ð8Þ

where Pk represents the covariance matrix of the system

state, (2) represents the predicted results, and (?) repre-

sents the updated results. Thus, the measurement update

equation is

xk þð Þ ¼ xk �ð Þ þKk½zk �Hkxk �ð Þ� ð9ÞPk þð Þ ¼ I�KkHk½ �Pkð�Þ ð10Þ

Kk ¼ Pk �ð ÞHTk ½HkPk �ð ÞHT

k þ Rk��1 ð11Þ

where Kk is the Kalman gain matrix. The KF algorithm

begins with the initial condition values x0ð�Þ and P0ð�Þ.When new measurements of zk become available, the time

update and measurement update steps follow.

Adaptive Kalman filters

Kalman filters require a priori knowledge of the process

and measurement noise statistics. Achieving full compli-

ance of this requirement under all circumstances is diffi-

cult. However, using incomplete information may result in

significant errors and even divergence problems. The

application of adaptive filters is one of the common strat-

egies for preventing this problem (Mehra 1972; Gelb

1974). The main function of adaptive filters is to estimate

the difference between the actual filter residuals and the

estimated noise statistics and to correct the estimated sta-

tistics using the filter residuals. Adaptive filters typically

employ the covariance matching technique. Innovation

sequences are also used by adaptive filters for estimating

the noise covariance. Innovation sequences represent the

differences between obtained measurements and the esti-

mated state, as expressed in the following formula (Mehra

1972):

gk ¼ zk � zkð�Þ ð12Þ

where zk �ð Þ ¼ Hkxkð�Þ. The innovation covariance is

Ck ¼ E gkgTk

� �¼ HkPk �ð ÞHT

K þ Rk

¼ Hk Tk�1Pk�1 þð ÞTTk�1 þQk�1

� �HT

k þ Rk

ð13Þ

where Ck is considered the predicted innovation covari-

ance. Innovation sequences are easily affected by actual

unaccounted errors, such as unmodeled dynamics,

unknown initial conditions, and increased measurement

covariance. The actual innovation covariance can be esti-

mated using (Mehra 1972)

�Ck ¼1

M � 1

Xk

i¼k�Mþ1

gigTi ð14Þ

where M is a window size, and the value is determined

empirically. A larger M value results in less noise after

averaging, and a smaller M provides a superior capability

of tracking dynamics. The actual innovation covariance

shows the effect of unaccounted errors. The basic concept

of the covariance matching technique is to achieve con-

sistency between the predicted innovation covariance and

the actual innovation covariance. For example, if the actual

innovation covariance is significantly larger than the pre-

dicted innovation covariance, the process noise covariance,

or the measurement noise covariance, should be increased.

This ensures that the predicted innovation covariance is

consistent with the actual innovation covariance. However,

using (13) and (14) does not provide a unique solution

when the true values of Q and R are unknown. For cases

where the Q is known but R is not known exactly, R is

estimated as

�Rk ¼ �Ck � �HkPk �ð ÞHTk ð15Þ

In this case, the measurement covariance is modified to

improve the consistency of the predicted and actual inno-

vation covariance. Subsequently, the second case is con-

sidered, in which the value of R is known but the value of

Q is not exactly known. Here, the Pkð�Þ is increased to

GPS Solut

123

compensate for the difference between the predicted and

actual innovation covariance.

Assisted adaptive extended Kalman filter (AAEKF)

During applications in surveying and geodesy, the GNSS

receivers are typically static or move smoothly. In such

situations, the process noise covariance Q can be applied

with a priori value, then Eq. (15) can be used to adap-

tively correct the measurement noise variance and

achieve the objective of the adaptive filter. Generally, the

adaptive covariance of measurement �Rk is calculated

using (12)–(15). Because the noise of phase measurement

is approximately 1–2 mm, it is less than the noise of the

predicted measurement zkð�Þ in (14). The use of (14)

produces a �Ck with substantial noise, and the obtained�Rk cannot effectively determine the measurement noise.

In this study, a new method for estimating �Rk is pro-

posed. The concept of this new method is to use the

reference station measurements to estimate the GNSS

measurement errors. These errors are then analyzed to

determine the noise of DD measurement �Rk. Thus, the

adaptive filtering technique can be employed to deter-

mine the solution.

GNSS errors, as discussed in (1)–(4), can be classified

into the following categories: (1) satellite errors, that is,

errors from the satellite ephemeris and clock; (2) atmo-

spheric effects, that is, errors from ionospheric and tro-

pospheric effects; and (3) receiver errors, that is, errors

from receivers and the reception environments, such as

the multipath effects, receiver clock errors, and thermal

noise. Satellite errors are strongly correlated with dis-

tance and vary slowly over time; atmospheric effects are

also correlated with distance and time under normal

weather conditions. Receiver errors are nearly completely

common to different satellites. The DD model is a

between-satellite and between-receiver process. The

between-satellite difference can eliminate receiver clock

errors. The between-receiver difference can eliminate

most satellite and atmospheric errors; the extent of this

elimination is reduced as the baseline length increases;

these errors are distance-correlated errors. Hence, using

the value of distance-correlated error and baseline length,

the remaining DD errors can be estimated. Many studies

have applied measurements taken at reference station to

estimate the GNSS measurement errors (Loomis et al.

1989; Jin 1996; Farrell and Givargis 2000), and these

estimation methods are widely employed in DGPS

applications. Following these studies, a KF was used to

estimate the errors of a reference station. The estimation

parameters included the tropospheric, ionospheric, and

satellite errors. With these errors, the remaining errors of

DD measurements can be estimated using the formula

below. The single-difference (SD) tropospheric errors

Tgpq, ionospheric errors Ig

pq, and satellite errors Sgpq are

defined as (Parkinson and Spilker 1996; Beutler et al.

1988):

Tgpq ffi Tg

p 0ð Þ 1

sin hgq

� 1

sin hgp

!ð16Þ

Igpq ffi Ig

p 0ð Þ 1

sin agq� 1

sin agp

� ð17Þ

Sgpq ffi

lpq � Sgp

Rgp

ð18Þ

where 1sin ag

q¼ 1

cos sin�1 0:94792 cos hgqð Þ½ � ffi 1þ 2

96�hgq

90

� �3

. The

symbol hgp denotes the elevation angle from reference sta-

tion p to satellite g, Tgp 0ð Þ and Ig

p 0ð Þ are the vertical tro-

pospheric and ionospheric errors, respectively, a is the

elevation angle at the ionospheric point (Klobuchar 1987),

R is the distance from reference station p to satellite g, I is

the baseline length from the reference station to the rover

station, Sgp is the satellite error from reference station p to

satellite g, the relationship between Sgp and Sg

pq in (18) is the

worst case situation. The overall estimated SD distance-

correlated error dE is

dEgpq ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS

g2

pq þ Tg2

pq þ Ig2

pq

qð19Þ

In addition to the distance-correlated errors, the uncorre-

lated errors should also be considered. Uncorrelated errors

include multipath effects and thermal noise. The multipath

effects are generally not estimated and allowed to impose

their signature on the residuals. Considering the remaining

uncorrelated error, receiver thermal noise, and overall

estimated distance-correlated error, the estimated SD

measurement error of satellite g is dEg2

pq þ rg2

pq

� �1=2

, where

rgpq ¼ rg2

pq þ rg2

pq

� �1=2

. Additionally, rgp and rg

q represent

the receiver thermal noise from satellite g to receivers

p and q; their values are a function of the carrier-to-noise

ratio. The relationship between the receiver thermal noise

and carrier-to-noise ratio is discussed in a later section.

The adaptive factor can be applied to KF to suppress

the contribution of measurements with comparatively

larger errors (Yang et al. 2001). The adaptive factor of

each measurement changes according to the standardized

estimated error. Thus, the influence of measurements

with larger errors can be reduced by increasing the

variance to avoid a significant rise in standard errors.

The adaptive factor can be obtained using

GPS Solut

123

ag ¼cg; cg\k

k � ec2g

k2�1; cg� k

(; g ¼ 1; 2; . . .;m ð20Þ

cg ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidE

g2

pq þ rg2

pq

q

rgpq

ð21Þ

where cg is the standardized estimated error and k is

determined according to the confidence level. The value for

k is set as 3.29 in this study (0.1 % confidence level).

Considering the adaptive factor, the SD covariance matrix

CSD ¼ diag ðr1pqÞ

2; ðr2pqÞ

2. . .; ðrmpqÞ

2� �

can be rewritten as

CSD ¼ diag ða1 � r1pqÞ

2; ða2 � r2pqÞ

2. . .; ðam � rmpqÞ

2� �

.

DD measurements are linear functions of the SD mea-

surements. By applying the law of variance–covariance

propagation and by taking the SD covariance matrix into

account, the DD covariance matrix is

CDD ¼ B � CSD � BT ð22Þ

where B is the coefficient matrix. If the first satellite is

selected as the reference satellite, then B is

B ¼

�1

�1

..

.

�1

1

0

..

.

0

0

1

..

.

0

� � �� � �. .

.

. . .

0

0

..

.

1

2664

3775 ð23Þ

After obtaining CDD, Eq. (11) can be rewritten as

Kk ¼ Pkð�ÞHTk ½HkPkð�ÞHT

k þ CDD;k��1 ð24Þ

The effects of increased measurement covariance can be

compensated by reducing the magnitude of the Kalman

gain.

Experimental results

The experiments consist of two parts. In the first part of the

experiment, the receiver thermal noise of code and phase

measurements were analyzed. The second part involved

analysis of the impact of L1 kinematic positioning when

adding the SBAS measurements. The GPS receiver used

for our experiments was a consumer-grade GPS receiver

with ublox-4t chipset, which costs approximately $100.

The cutoff angle is 15�, and the sampling rate of ranging

measurements is 1 Hz. Three SBAS satellites can be

observed in Taiwan, that is, PRN128, PRN129, and

PRN137.

Noise analysis of the code and phase measurements

This study used a 0.5 m baseline to analyze receiver

thermal noise on the code and phase measurements. The

experiment was performed on January 13, 2013. The 24-h

data were analyzed. To combine the two satellites and two

receivers for double differences, a carrier-to-noise ratio (C/

N0) was computed using the following equation (de Bakker

et al. 2008),

ðC=N0Þij ¼ �10:0 log1

210�

C=N0i10 þ 10�

C=N0j10

� � �ð25Þ

This equation follows the inverse relationship between the

variance and the carrier-to-noise ratio when expressed in

dB–Hz. A factor of one half is added to normalize the

C=N0 to the undifferenced levels. The residual of the short

baseline DD code and phase measurements is shown in

Figs. 1 and 2, in which the horizontal axis denotes the

carrier-to-noise ratio, which was calculated using (25). The

GPS and SBAS satellite signals were plotted separately.

The figures show that the GPS code and phase accuracy

were 0.11 m and 0.015 cycle when the C/N0 = 50 dB–Hz,

and the SBAS satellites code and phase accuracy were

0.46 m and 0.021 cycle when the C/N0 = 45 dB–Hz.

During the measurement campaign, the SNRs of GPS and

SBAS measurements are approximately 50 and 45 dB–Hz,

respectively. The standard error, or root mean squared error

(RMSE), of the regression for the GPS satellite code and

the phase measurement were 0.122 m and 0.015 cycles,

respectively. The RMSE for the SBAS satellite code and

the phase measurement were 0.282 m and 0.015 cycles,

respectively.

Fig. 1 Relationship between pseudorange errors and C/N0 for a short

baseline

Fig. 2 Relationship between phase errors and C/N0 for a short

baseline

GPS Solut

123

Half-integer ambiguity problem

The value of ambiguity is essentially an integer. But, there

may be half-integer cycle slips due to the navigation mes-

sages demodulated error (‘‘bit error’’) in low-cost L1 GPS

receivers. The navigation messages are modulated onto the

carrier using the binary-phase shift keying (BPSK) method.

When a navigation data bit transition occurs, the carrier is

instantaneously phase shifted by 180�. The ‘‘bit error’’

occurs when a bit transition is incorrectly detected yet the

tracking loop continues tracking. This will cause a 180�phase shift error (half-integer ambiguity problem). The half-

integer ambiguity problem is usually recovered in a few

seconds, when the ‘‘bit error’’ is found and repaired by parity

check and analyzing of the validity of the navigation mes-

sages. The validity items include the transmission time,

broadcast ephemeris, and almanac. The probability of error

Pe for a BPSK signal with Gaussian noise can be written as

(Tsui 2005):

Pe ¼ erfc

ffiffiffiffiffiffiffiffiffiffiffiffiffiC

N0

� t

r� ð26Þ

where the erfc is the complementary error function, t is the

length of predetection integration data, which is 20 ms for

GPS navigation message. For example, at C/N0 = 30 dB–

Hz, the probability of error for GPS navigation message is

about 2.59 9 10-10. The half-integer ambiguity problem

caused by ‘‘bit error’’ is temporary, and the probability of

error for the GPS navigation data bit is very low when the

SNR thread is set as C/N0 [ 35–40 dB–Hz. Hence, the

half-integer ambiguity problem of GPS measurements is

categorized as a temporary error which is excluded by

outlier detection method (Lin and Yu 2013). The signal

strength of SBAS satellite is weaker than GPS satellite and

the symbol rate of SBAS satellite is 10 times faster than the

rate of GPS satellite. According to (26), the probability of

error for the SBAS navigation data bit is much higher than

that for the GPS satellite. In the event of an error bit, the

ambiguity takes the form of a half-integer. For this reason,

the ambiguity of SBAS satellite is regarded as half-integer.

The success rate of ambiguity resolution with/without

(w/o) the SBAS satellite

In order to assess the feasibility of structural monitoring

using low-cost single-frequency receivers, four continu-

ously operating GPS (CGPS) stations located in New

Taipei City, Taiwan, were installed on flat roofs. In Tai-

wan, the ionospheric effect has large temporal and spatial

variations in the total electron content (TEC) and TEC

gradients. To eliminate most ionospheric effects in DD, a

conservative interstation distance is not greater than Ta

ble

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GPS Solut

123

5.0 km. The baseline lengths of CGPS stations were 0.5 m,

2.1 km, 3.3 km, and 4.9 km. Data were collected from Day

5 of 2013. For this experiment, every 10 min over 24 h,

one datum segment was obtained. The number of tested

segments was 144. Additionally, 2, 5, and 10 min data

were used to resolve the ambiguity in each segment. The

LAMBDA method (Teunissen 1995) was used for ambi-

guity resolution. A chi-square test (confidence interval

1 - a = 0.99) and contrast statistic test (contrast value is

[4.0) were used to judge the reliability of the solution

(Counselman and Abbot 1989; Frei and Beutler 1990). The

true value of ambiguity was obtained using the known

baseline vector. A successful solution must pass the chi-

square and contrast tests, demonstrating an identical value

to the truth.

Table 1 lists the success rates of GPS-only, GPS/SBAS

EKF, and GPS/SBAS AAEKF solutions for the three time

durations. The results in the table indicate that the GPS/

SBAS AAEKF solutions provide the optimum success

rates for the four baseline lengths and three time durations.

For the 3.3 and 4.9 km experimental results, the success

rates of the GPS/SBAS EKF solutions were lower than

those of the GPS-only solutions with longer time durations.

However, the success rates of the GPS/SBAS AAEKF

solutions were consistently superior to those of the GPS-

only solutions, and the rate of improvement was approxi-

mately 3.2, 2.4, and 1.6 times for the 2, 5, and 10 min data

collection durations, respectively. The improvement shows

that adding the SBAS data was very helpful, especially the

shorter the length of data. The ionospheric effects and

SBAS satellite errors have large values and variations, and

it is difficult to model these errors during EKF initializa-

tion. However, these errors are corrected, and their effect

on the DD model is directly proportional to the length of

baseline, increasing as the baseline length. Regarding the

EKF, the imperfect error model significantly decreases the

success rate when the baseline length increases. Compared

with EKF, AAEKF yields a more gentle decrease in suc-

cess rate, proving that AAEKF more successfully models

these errors than does EKF. It is worth mentioning that

during the 24 h experiment, for three periods (each period

lasted approximately 10–30 min), the signals from only

four or five GPS satellites could be received. During these

periods, the ambiguity resolution was very difficult using

only GPS data. Additionally, if any satellite was obscured,

the ambiguity almost cannot be resolved using GPS satel-

lite data along. However, the use of three SBAS satellites

significantly benefited the ambiguity resolution.

The accuracy of kinematic positioning w/o the SBAS

satellite

The purpose of this experiment was to analyze the effect that

supplementing SBAS satellite data had on the kinematic

positioning accuracy. The applied data were the same as the

data mentioned in the previous section. These data were

categorized into only GPS data and both GPS and SBAS

data. The GPS/SBAS data were solved with AAEKF. The

first 900 s data were used for ambiguity resolution, and the

remaining 85,500 s data were used for kinematic position-

ing. The positioning accuracy of the four baselines is listed in

Table 2. The positioning results and the positional dilution of

precision (PDOP) are shown in Fig. 3. The results of the four

baselines were similar, and only the 2.1 km baseline is

Table 2 Accuracy of kinematic

positioning w/o the SBAS

satellite

Units are meter

50 cm baseline 2.1 km baseline 3.3 km baseline 4.9 km baseline

GPS-

only

RMSE

GPS ? SBAS

RMSE

GPS-

only

RMSE

GPS ? SBAS

RMSE

GPS-

only

RMSE

GPS ? SBAS

RMSE

GPS-

only

RMSE

GPS ? SBAS

RMSE

E 0.0036 0.0020 0.0064 0.0040 0.0077 0.0066 0.0103 0.0087

N 0.0054 0.0024 0.0083 0.0049 0.0126 0.0091 0.0165 0.0137

H 0.0161 0.0076 0.0274 0.0132 0.0318 0.0252 0.0367 0.0334

Fig. 3 Kinematic positioning of the 2.1 km baseline vector; units are

in meter; the corresponding mean and RMS error of GPS-only data

for the east, north, and height coordinate components were 0.10 and

0.64 cm, -0.09 and 0.83 cm, and -0.09 and 2.74 cm, respectively;

the corresponding mean and RMS error of GPS/SBAS data for the

east, north, and height coordinate components were 0.12 and 0.40 cm,

-0.12 and 0.49 cm, and 0.03 and 1.32 cm, respectively

GPS Solut

123

plotted. Table 2 shows that the positioning accuracy of L1

kinematic positioning achieves 1–2 cm, and the positioning

accuracy of the used SBAS data was shown to have increased

by about 50.9, 43.4, 20.9, and 13.8 % for 50 cm, 2.1 km,

3.3 km, and 4.7 km baselines, respectively. When the PDOP

increased, the addition of SBAS satellite data significantly

enhanced the positioning accuracy.

Conclusion

Using SBAS can enhance the satellite geometry. In situa-

tions with fewer satellites, adding the SBAS measurements

is extremely beneficial. However, the SBAS satellite signal

strength is weak and the satellite clock and orbit errors are

larger and less stable compared with those of a GPS

satellite. Ignoring these factors may worsen the solution.

This phenomenon was demonstrated by the ambiguity

resolution success rate experiment. The 3.3- and 4.9-km

experimental results demonstrated that the success rate of

GPS/SBAS EKF was less satisfactory than that of GPS

EKF at longer time durations. This study proposes a

strategy to solve this problem, which can be successfully

employed for single-frequency GPS/SBAS positioning. In

this study, the addition of SBAS satellite data was shown to

enhance the ambiguity resolution success rate and posi-

tioning accuracy. The GPS/SBAS AAEKF solutions were

superior to those of GPS-only EKF solutions, the ambi-

guity resolution success rate for 2, 5, and 10 min mea-

surements were improved by about 3.2, 2.4, and 1.6 times,

respectively, and the positioning accuracy of the north,

east, and height directions were improved by 20–80,

20–125, and 20–110 %.

The virtual reference station (VRS) (Vollath et al.

2000) and multiple reference station techniques do not

currently include SBAS satellite data. Therefore, the

method proposed in this study is only applicable for short

baselines (typically \5 km). Further investigation

regarding the addition of SBAS functions to the VRS and

multiple reference station techniques is required. In recent

years, various developments of satellite navigation sys-

tems have been carried out; for example, the GLONASS

in Russia, the Compass from China, and the Galileo from

European Union. If these systems are supplemented,

problems such as varying measurement quality between

different system data can occur. However, the method

proposed in this study can be employed to resolve this

problem.

Acknowledgments The authors are grateful to a research Grant

NSC 101-2221-E-019-070-MY3 from the National Science Coun-

cil. The generous provision of GPS data for this study by the Institute

of Earth Sciences Academia Sinica is greatly appreciated.

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Shiou-Gwo Lin received the

B.S. degree in Physics from the

Soochow University, the M.S.

degree in Space Science, and

the Ph.D. degree in Atmo-

spheric Physics from the

National Central University. He

is currently an assistant profes-

sor of Department of Commu-

nications, Navigation and

Control Engineering at National

Taiwan Ocean University. His

research interests include GNSS

software receiver, GNSS rela-

tive positioning, and signal

denoising.

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