Orbit Ephemeris Failure Detection in a GNSS Regional

Paper
Int’l J. of Aeronautical & Space Sci. 16(1), 89–101 (2015)
DOI: http://dx.doi.org/10.5139/IJASS.2015.16.1.89
Orbit Ephemeris Failure Detection in a GNSS Regional Application
Jongsun Ahn* and Young Jae Lee**
Konkuk University, 120 Neundong-ro, Gwangin-gu, Seoul 143-701, South Korea
Dae Hee Won***
University of Colorado at Boulder, 431 UCB, Boulder, CO 80309 USA
Hyang-Sig Jun**** and Chanhong Yeom*****
Korea Aerospace Research Institute, 169-84 Gwahang-no, Yuseong-gu, Daejeon 305-806, South Korea
Sangkyung Sung****** and Jeong-Oog Lee*******
Konkuk University, 120 Neundong-ro, Gwangin-gu, Seoul 143-701, South Korea
Abstract
To satisfy civil aviation requirements using the Global Navigation Satellite System (GNSS), it is important to guarantee system
integrity. In this work, we propose a fault detection algorithm for GNSS ephemeris anomalies. The basic principle concerns
baseline length estimation with GNSS measurements (pseudorange, broadcasted ephemerides). The estimated baseline length
is subtracted from the true baseline length, computed using the exact surveyed ground antenna positions. If this subtracted
value differs by more than a given threshold, this indicates that an ephemeris anomaly has been detected. This algorithm
is suitable for detecting Type A ephemeris failure, and more advantageous for use with multiple stations with various long
baseline vectors.
The principles of the algorithm, sensitivity analysis, minimum detectable error (MDE), and protection level derivation are
described and we verify the sensitivity analysis and algorithm availability based on real GPS data in Korea. Consequently, this
algorithm is appropriate for GNSS regional implementation.
Key words: GNSS, Integrity, Ephemeris Failure Detection, Baseline Length Estimation
1. Introduction
sensors, has been conducted to address integrity issues [1]
[2]. The main line of integrity issues includes threat definition,
development of detection algorithms within time to alert, and
the protection level (PL) for confidence in the user navigation
solution. One of the threats facing GNSS involves orbit
ephemerides, which are generated periodically at a ground
control facility (a GPS Operation Control Segment) and are
transmitted to the user by satellites [3]. The frequency of the
ephemeris anomaly tends to decrease, however. Because the
ramifications for user position error is critical, the system
must include monitoring and detection processing [4][5].
The Global Navigation Satellite System (GNSS) is used to
compute user navigation solutions with certain accuracy
at any time and location of interest. With increasing GNSS
applications, the civil aviation community has been trying to
implement primary navigation systems using GNSS. To ensure
aircraft safety in the civil aviation implementation of GNSS,
the implemented system must meet integrity requirements.
Research on various GNSS implemented systems and on the
use of ground facilities, additional satellites, and navigation
* Ph. D Candidate
Professor
**
*** Research Associate
**** Principal Researcher
***** Principal Researcher
******Professor
******* Professor, Corresponding Author: ljo7@konkuk.ac.kr
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/bync/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Received: November 5, 2014 Revised : February 26, 2015 Accepted: March 2, 2015
Copyright ⓒ The Korean Society for Aeronautical & Space Sciences
(89~101)14-083.indd 89
89
http://ijass.org pISSN: 2093-274x eISSN: 2093-2480
2015-03-30 오후 3:48:25
introduce
concept
of the
describe
introduce the concept of the algorithm,
describethetest
statistics
andalgorithm,
a threshold,
analyzetest statistics and a threshold, analy
introduce the concept of the algorithm, describe test statistics and a threshold, analyze
[14] between
test derive
statistics
ephemeris failure, and derive the epheme
sensitivity [14] between test statisticssensitivity
and ephemeris
failure, and
theand
ephemeris
sensitivity [14] between test statistics and ephemeris failure, and derive the ephemeris
introduce
the MDE
concept
offor
thealgorithm
algorithm,availability.
describe test
level
(EPL) and
[15]
In statistics
section 3,and
weafoth
protection level (EPL) and MDE [15]protection
for algorithm
availability.
In section
3, we
focus on
protection level (EPL) and MDE [15] for algorithm availability. In section 3, we focus on
sensitivity [14]
between
testusing
statistics
failure,
and deri
evaluating
above
real and
GPSephemeris
data. So, we
evaluated
sen
evaluating the algorithm described above
using the
realalgorithm
GPS data.described
So, we evaluated
sensitivity
evaluating the algorithm described above using real GPS data. So, we evaluated sensitivity
protection performance
level (EPL) and
MDE [15]
forephemeris
algorithmprotection
availability.
In se
Int’l J. of Aeronautical & Space Sci. 16(1), 89–101
(2015)
and availability
and
level
fo
analysis
and availability performanceanalysis
using MDE
and ephemeris
protectionusing
levelMDE
for landing
analysis and availability performance using MDE and ephemeris protection level for landing
evaluating
the of
algorithm
described
above using real GPS data. So, we
aircraft
usingstations
real
GPS
data
multiple
reference
aircraft using real GPS data of multiple
reference
(RSs)
in Korea.
Finally,
we stations (RSs) in Korea. Finally, w
stations
(RSs)
in
Korea.
Finally,
we
describe
the
conclusions
Ephemeris anomalies are conventionally
divided
into
aircraft using real GPS data of multiple reference stations (RSs) in Korea. Finally, we
analysis andand
availability
performance
using MDE and ephemeris prot
describe
the
future work
in section 4.
describe
theoccurrence
conclusions and future
work
in section
4.
and future
work
inconclusions
section
4.
two groups (Types A and B) with respect
to the
describefailure
the conclusions
and future work in section 4.
of satellite maneuvers. Type A is an ephemeris
event
aircraft using real GPS data of multiple reference stations (RSs) in Ko
during the process of a satellite maneuver, whereas Type B
describe
the conclusions
and future work in section 4.
2. Algorithm
Baseline
Length
Estimation
Algorithm
can be issued in ephemeris generation
transmission
2. Baseline
Length
Estimation
Algorithm
2. or
Baseline
Length but
Estimation
2.
Baseline
Length
Estimation
Algorithm
no satellite maneuver is involved [6].
This
introduces
concept
of the
proposed
algorithm (test statistics and
This
introduces the concept
of
thesection
proposed
algorithm
(test
statistics
and
threshold),
This section
introduces
thetheconcept
of
the
proposed
Various methods have been proposed
forsection
the detection
This section introduces the concept of the proposed algorithm (test statistics and threshold),
2. satellite
Baseline
Length
Estimation
Algorithm
(testthestatistics
and
analyses
the
of both Type A and Type B ephemeris
anomalies.
Type of testalgorithm
analyses
sensitivity
ofposition
testthreshold),
statistics
according
to satellite
position error due to e
analyses
the sensitivity
statistics
according
to
error
due
to ephemeris
sensitivity
of
test
statistics
according
to
satellite
position
B events can be detected by examining
the
consistency
analyses the sensitivity of test statistics according to satellite position error due to ephemeris
Thisthe
section
introduces
the concept of the
proposed algorithm (tes
failure,
and derives
MDE,
which
the
and derives
the MDE,error
whichdue
is the
performance
parameter
faultisdetection,
the parameter of fault detection
to ephemeris
failure,
andof
derives
theperformance
MDE,and
which
between the broadcast ephemeris failure,
and prior
validated
failure,
and derives
the MDE,
which
is the performance
parameter
of fault
detection,
andthe
the
is the
performance
parameter
of fault
detection,
and
ephemeris. The magnitude of detectable
satellite
position
analyses
the sensitivity
of test statistics
according to satellite position
ephemeris protection
level.
ephemeris protection level.
error due to a Type B anomaly depends on
the validated
time level. ephemeris protection level.
ephemeris
protection
failure,
and derives the MDE, which is the performance parameter of
of a prior ephemeris. Representative2.1algorithms
include
2.1 Principle of
the Algorithm
Principle of the Algorithm
of the Algorithm
the ephemeris-ephemeris test, YE-TE 2.1
test,Principle
and almanac2.1 Principle of the
Algorithm
ephemeris
protection
level. the baseline length between reference st
Thethedetection
estimates
The
detection
methodology estimates
baselinemethodology
length between
reference
station (RS)
ephemeris test. They compute satellite position and monitor
premeasured
premeasured
antenna
antenna
locations
locations
of
of
RSs
RSs
and
and
the
the
broadcast
broadcast
ephemeris
ephemeris
of
of
thethe
satellite
satellite
[14].
[14].
The detection
methodology
estimates
the
baseline
length
between
reference
station
(RS)
The
detection
methodology
estimates
the
baseline
2.1
Principle
of the Algorithm
consistency of results within a given threshold
However,
using
range
measurements
of be
RSvalidated.
and the broadcast
ephemeris to be vali
antennas [7].
using
range measurements antennas
of RS and
the broadcast
ephemeris to
The
length
between
reference station (RS) antennas using range
i range
i T T measurements
i i T of
T RS and the broadcast ephemeris to be validated. The
these algorithms have the limitation of requiring
validated
antennas ausing
i T Tdetection methodology
i T T
betwee
ABe AB i length
e AeAeABe AB i measurements
e BeBeerror-free
i 
i  to
RS
 iantenna
i The
i be
ephemeris
 isestimates
of
RS
and
the
broadcast
to
betheRSbaseline
is eassumed
because
antennalength
location
is
baseline
because
location
computed
A
iAephemeris
iA  iis i assumed
 BiBto ibei baseline
 A

x Type
x AB length
eerror-free
prior ephemeris, and hence cannot detect
(1)(1)
AB
AB
AB
A eAeAeAB
B
B eBeBeAB
T T
T T








baseline
length
is
assumed
to
be
error-free
because
RS
antenna
location
is
computed
validated.
The
baseline
length
is
assumed
to
be
error-free
e
e
e
e
e
e
e
e
A
A
AB
AB
B
B
AB
AB
failure. On the other hand, Type A failure is relatively more
antennas
using
range
measurements
of RS and
the broadcast ephem
accurately
precisely.
Incomputed
this section,
wefordescribe
accurately and precisely. In because
this section,
we and
describe
the is
estimation
method
single
RS antenna
location
accurately
and the estimation method
difficult to detect than Type B because there
is
no
validated
accurately and precisely. In this section, we describe the estimation method for single
baseline
length
assumed
to be
error-free
because
RSone
antenn
In thislength
section,
we on
describe
the
estimation
method
based
two is
RSs.
According
to the
First Cosine
Law,
side
prior ephemeris to compare with the ephemeris
of a satellite
baseline length
based on twoprecisely.
RSs. baseline
According
to the
First
Cosine
Law,
one side length
of
for
single
baseline
length
based
on
two
RSs.
According
to
the
baseline
length
based
on
two
RSs.
According
to
the
First
Cosine
Law,
one
side
length
of
just after an orbit maneuver. One method of detecting Type
and precisely.
In
this section,
we describe the estima
triangle
can
beaccurately
computed
the other
two
length
triangle can be computed using
theCosine
other
two
sides’
length
andusing
their
induced
angle.
In Fig.
(1), and their induced angle.
First
Law,
one
side length
of triangle
can
be sides’
computed
A failure is to monitor the range measurement correction
triangle can be computed using
thethe
other
twotwo
sides’
length
andand
theirtheir
induced
angle.angle.
In Fig.
using
other
sides’
length
induced
In(1), to the First Cosine
baseline
length
basedimplementation.
on
two RSs.
According
(pseudorange correction, PRC) derived
this law can be
applied
thisfrom
law the
can broadcast
be applied in GNSS implementation.
One
sideinofGNSS
the triangle
represents One
the side of the triangle rep
(1), this
law can be applied
in GNSS
implementation.
Onethe
ephemeris and the location of the ground
this station
law canantenna
be applied inFig.
GNSS
implementation.
One side
of the
triangle represents
triangle
can
be
computed
using
the
other
two sides’ length and their i


side of
the) triangle
represents
length
the
displacement
ofthe
theother
displacement
vector
( of
) determined
and
the other
sides’ lengths can be determ
xˆ AB
of theerror
displacement
vector
( xˆlength
and
sides’the
lengths
can
be
with
[7]. Other methods estimate satellitelength
position
with
AB
ˆ
vector
and
the
other
sides’
lengths
can
be
determined
length
of
the
displacement
vector
(
)
and
the
other
sides’
lengths
can
be
determined
with
x
AB
range measurements [8][9] or estimate the differential
this law can be applied in GNSS implementation. One side of th
range
measurements
received
RSs.
range
measurements
received
by two
RSs. The
The
 Ai ,  Bi ) angle
measurements
(  Ai ,  Bi )with
received
by
two RSs. The (induced
is then
computed
by induced angle is then co
range of ground stations with short range
baseline
vectors and
i
i
ˆ
)
received
by
two
RSs.
The
induced
angle
is
then
computed
bythe other sides’ lengths
range measurements (  A , induced
angle
is
then
computed
by
the
unit
vector
of
length of the displacement vector ( x AB ) and
B
range measurements [10][13]. However, these algorithms
 
ˆ
i i
ˆ
the
unit
vector
(
)
of
the
baseline
and
unit
vectors
( eAi , eBi ) correspond
x AB ( eA , eB ) corresponding
thebaseline
baselineand
andunit
unit vectors
vectors
thedirection
unit vector
( x AB ) of the
corresponding to
to the
the
have some weaknesses. First, when the
of satellite
i i
ˆ
i
i
the unit vector ( x ) of premeasured
the baseline and
unit
vectors
( eof
corresponding
tobythe
eB )( 
two RSs. The induced an
range
measurements
antenna
locations
and
broadcast
A , RSs
B ) received
A , the
position error is orthogonal to the line-of-sight, ephemerisAB
ephemeris
of
the
satellite
[14].
premeasured
antenna
locations
of
RSs
and
the
broadcast
ephemeris
of
the
satellite
[14].
failure on test statistics is small. Next, there is the limitation

4baseline and unit vectors ( ei , ei
4 the(A,unit
vector
(
of
the
xˆ AB ) i.
Figure
Figure
1.
1.
Geometry
Geometry
condition
condition
of
of
two
two
RS
RS
antennas
antennas
(A,
B)
B)
and
and
a
satellite
a
satellite
i.
B
A
i
T
i
T
of baseline length (short baseline, 100-400 m), and using
i
T
i
T
e A  e AB 4
e B  e AB
x AB  iA  i T  iB  i T  iA   e A  e AB   iB   e B  e AB 
(1) (1)
carrier phase measurement that must resolve integer




e A e AB
e B e AB
ambiguity.
4
premeasured antenna locations of RSs and the broadcast ephemeris of the satellite [14].
To supplement these algorithms, we propose a detection
The test statistic (TS) is defined by the following equation
The
test
test
statistic
statistic
(TS)
(TS)
is is
defined
defined
byby
thethe
following
following
equation
equation
(2),
(2),
computed
computed
byby
subtracting
subtracting
thethe
algorithm using non-limited baselineThe
length
and
code
Tsubtracting
i
T
(2), computedeiAby
estimated
baseline length
i
T
i
T
 e AB
e B  ethe
AB
i
i
i
i


















x
e
e
e
e
measurement for Type A failures. This methodology, based
(1)
A
B
i
T
i
T
 A   A AB  B  B AB 
  AB
e Apremeasured
epremeasured
e B one
e AB
AB
the
one
). ). ).
estimated
baseline
baseline
length
length xˆ ABxˆ AB from
from
thethe
premeasured
one
( x( ABx AB
on the law of cosines in trigonometry,estimated
estimates
the
baseline
lengths of multiple ground antennas.
  inˆ the

i i detail
We describe the methodology in more
TS

TS

xAB xABxˆAB
(2)(2)
AB
AB xAB
following sections. In section 2, we introduce the concept
of the algorithm, describe test statistics
and
a threshold,
AsAs
shown
shown
in in
equation
equation
(1),
(1),
thethe
test
test
statistic’s
statistic’s
variation
variation
depends
depends
onon
thethe
range
range
measurements
measurements
analyze sensitivity [14] between test statistics and ephemeris
Figure 1. Geometry condition of two RS antennas (A, B) and a satellite i.
and
and
thethe
induced
induced
angle.
angle.
The
range
range
measurements
measurements
areare
assumed
assumed
to to
bebe
fault-free
fault-free
and
and
areare
notnot
failure, and derive the ephemeris protection
level
(EPL)
and The
MDE [15] for algorithm availability. In section 3, we focus
treated
treated
further
further
in in
this
this
work.
work.
WeWe
focus
focus
onon
thethe
test
test
statistic’s
statistic’s
variation
variation
due
due
to to
induced
induced
angle
angle
on evaluating the algorithm described
above
using
real
GPS
The test statistic (TS) is defined by the following equation (2), computed by subtracting the
data. So, we evaluated sensitivity analysis and availability
error
error
resulting
resulting
from
from
broadcast
broadcast
ephemeris
ephemeris
failure.
failure.


performance using MDE and ephemeris protection level
estimated baseline length xˆ AB
from the premeasured one ( x AB ).
Figure
1.
Geometry
condition
of
two
RS
antennas
(A,
B)
and
a satellite
i. be
for landing aircraft using real GPS data of
multiple
reference
Fig. 1. Gerrors
eometry
condition
of two
RS
antennas
(A,conditions
B)conditions
and a satellite
i.be
However,
However,
range
range
measurement
measurement
errors
in in
the
the
ephemeris
ephemeris
under
under
normal
normal
must
must
i
TS

AB


xAB  xˆAB
(2)
accounted
accounted
forfor
in in
thethe
test
test
statistic’s
statistic’s
accuracy.
accuracy.
The
The
range
range
measurement
measurement
errors
errors
areare
defined
defined
asas
As shown in equation (1), the test statistic’s variation depends on the range measurements
i
i i test statistic (TS) is defined by the following equation (2), computed by subtracting the
(3)(3) to be fault-free and are not
 uu R
 uRui Ii Ii TiTi ccbu bu90
bi b 
 u induced angle. The range measurements are assumed
The
uthe
and
DOI: http://dx.doi.org/10.5139/IJASS.2015.16.1.89
i i
i


estimated baseline length xˆ AB
from the premeasured one ( x
).
treated furtheri ini this work. We focus
on the test statistic’s variation dueABto induced angle
: True
range
range
(m)
(m)
RuR: uTrue
 from ˆbroadcast ephemeris failure.
error resulting
i
TS

AB
xAB  xAB
(2)
However, range measurement errors in the ephemeris under normal conditions must be
(89~101)14-083.indd 90
5 5
2015-03-30
오후 3:48:26
As shown in equation (1), the test statistic’s variation depends on the
range measurements
Figure
1. 1.
Geometry
condition
of of
two
RSRS
antennas
(A,(A,
B)B)
and
a satellite
i. i.
Figure
Geometry
condition
two
antennas
and
a satellite
. Geometry condition of two RS antennas (A, B) and a satellite i.
The
testtest
statistic
(TS)
is defined
byby
thethe
following
equation
(2),(2),
computed
byby
subtracting
thethe
The
statistic
(TS)
is defined
following
equation
computed
subtracting
est statistic (TS) is defined by the following equation (2), computed
by subtracting
the Ahn Orbit
Jongsun
 Ephemeris Failure Detection in a GNSS Regional Application

thethe
premeasured
one
( x(ABxAB ). ).
estimated
baseline
length
estimated
baseline
length xˆ ABxˆ AB from
from
premeasured
one


d baseline length xˆ AB
from the premeasured one ( x AB ).


methods posed a contradiction in this algorithm because
i i
(2) (2)(2)
TS

xABxAB xˆABxˆAB
TS

AB
AB

ˆ
these methods use the broadcast
ephemeris
(to be
Klobuchar
model
xAB  xAB
(2)
Ionospheric delay
Klobuchar
model Message Iono. Parameter
model
validated
bydelay
test
statistics). The test Klobuchar
statistics
and
validated
(with
GPS
Navigation
Ionospheric
delay
Ionospheric
AsAs
shown
equation
(1),
the
test
statistic’s
variation
As
shown
ininin
equation
(1),(1),
thethe
testtest
statistic’s
variation
depends
onon
thethe
range
measurements
(with
Navigation
Message
Iono.
Parame
(with
GPSGPS
Navigation
Message
Iono.
Parameter
shown
equation
statistic’s
variation
depends
range
measurements
Klobuchar
model
Klobuchar
Klobuchar
model
model
Troposheric
delay
Sasstamonien
model
ephemeris
must
be
independent Klobuchar
for
reliablemodel
ephemeris
Ionospheric
delay
Ionospheric
Ionospheric
delay
delay
Ionospheric
delay
depends
onvariation
the range
measurements
and
the induced
angle.
own in equation (1), the test
statistic’s
depends
on the range
measurements
Troposheric
delay
Sasstamonien
model
Troposheric
delay
Sasstamonien
model
(with
GPS
Navigation
Message
Iono.
Parameters)
(with
(with
GPS
GPS
Navigation
Navigation
Message
Message
(with
Iono.
Iono.
GPS
Parameters)
Parameters)
Navigation
Message Iono. Parameter
Klobuchar
model
Satellite
/ Receiver
clock
Broadcast
ephemeris
fault
detection.
Innot
addition, the required
range
is not /aWeighted least square
Ionospheric
delay
and
the
induced
angle.
The
range
measurements
are
assumed
beTroposheric
fault-free
and
are
and
the
induced
angle.
The
range
measurements
are
assumed
be
fault-free
and
are
not
The
range
measurements
are
assumed
to be fault-free
and to to
(with
GPS
Navigation
Iono.
Parameters)
Satellite
/delay
Receiver
clock
Broadcast
ephemeris
/ Weighted
square
Satellite
/Sasstamonien
Receiver
clock
Broadcast
ephemeris
leastleast
square
Troposheric
delay
Sasstamonien
model
Troposheric
Troposheric
delay
delay
Sasstamonien
model
modelMessageSasstamonien
model / Weighted
i
I
:
Ionosphere
delay
error
(m)
relative value, but an absolute measurement. The methods
nduced angle. The range measurements are assumed to be fault-free and are not
are not treated further in this work.
We focus
on theclock
test
Troposheric
Sasstamonien
model / /Weighted
Satellite
/ /Receiver
Broadcast
ephemeris
least
square
Satellite
Satellite
Receiver
/delay
Receiver
clock Satellite
Broadcast
/Broadcast
Receiver
ephemeris
clock
ephemeris
Weighted
/ Weighted
Broadcast
least
least
square
square
ephemeris / Weighted least square
i clock
treated
further
in in
this
work.
WeWe
focus
onon
thethe
testtest
statistic’s
variation
to to
induced
angle
T : Troposphere
delay
error
(m)
fordue
mitigating
range
measurement errors are summarized
treated
further
this
work.
focus
statistic’s
induced
angle
statistic’s
variation
due
to
induced
angle
error
resulting
from variation due
/ Receiver
Broadcast
ephemeris
/ Weighted least square
urther in this work. We focus
on the test
statistic’s
variation
dueSatellite
to induced
angle bclock
clock
error
Table
1. (s)
u : Receiver in
broadcast
ephemeris
failure.
The threshold for ephemeris failure detection is determined by the probability prop
error
resulting
from
broadcast
ephemeris
failure.
error
resulting
from
broadcast
ephemeris
failure. bi : Satellite clock
TheThe
threshold
for ephemeris
ephemeris
failure
detection
is by the
threshold
failure
detection
is determined
by the
probability
The
threshold
ephemeris
failure
detection
is determined
probability
propp
error
(s) for for
ulting from broadcast ephemeris
failure.range measurement errors in the ephemeris
However,
i
:
Noise,
multipath,
etc.
inufor
determined
by must
the
property
of
the test
statistics
The
threshold
ephemeris
failure
detection
isisprobability
determined
by
the
probability
property
ofofof
The
threshold
threshold
for
ephemeris
ephemeris
failure
The
failure
threshold
detection
detection
for
ephemeris
isbe
determined
determined
failure
byby
the
the
detection
probability
probability
is determined
property
property
by the
prop
the
test
statistics
and
system
continuity
requirements.
We assume
thatprobability
test statistics
fo
However,
range
measurement
inThe
the
ephemeris
under
normal
conditions
under
normal
conditions
musterrors
be
accounted
forfor
the
However,
range
measurement
errors
in
the
ephemeris
under
normal
conditions
must
be
test
statistics
and
system
continuity
requirements.
We
statistic
the
test
statistics
and
system
continuity
requirements.
Wetest
assume
thatthat
testtest
statistics
fo
ver, range measurement errors in
the ephemeris
under
normal conditions
must
beforc :ephemeris
The threshold
failure
detection
is determined
by
the probability
ofassume
andthe
system
continuity
requirements.
We
assumeproperty
that
Speed
of
light
(m/s)
test statistic’s accuracy. The range measurement errors are
the
test
statistics
and
system
continuity
requirements.
We
assume
that
test
statistics
aa a thatrequirement
the
the
test
test
statistics
statistics
and
and
system
system
continuity
the
continuity
test statistics
requirements.
requirements.
and
system
We
We
assume
continuity
assume
that
that
requirements.
test
test
statistics
statistics
follow
follow
assume
test statistics
fo
Gaussian
distribution
[11]
with
a zero
mean,
and
thatfollow
the
continuity
(LA
statistics
follow
a Gaussian
distribution
[11]
with
aWe
zero
accounted
for range
in in
thethe
testtest
statistic’s
accuracy.
The
range
measurement
errors
are
defined
as
accounted
statistic’s
accuracy.
The
measurement
errors
are
defined
as[11]
ed for in the test statistic’s accuracy.
measurement
errors
are defined
asrange
defined
asThefor
Gaussian
distribution
with
a zero
mean,
that
continuity
requirement
Gaussian
distribution
[11]
with
a zero
mean,
andand
thatfollow
the the
continuity
requirement
(LA(
the test
statistics
and system continuity
requirements.
We
assume
that
test
statistics
a
mean, and that the continuity requirement (LAAS CAT-I,
distribution
[11]
with
aazero
mean,
and
that
the
continuity
requirement
(LAAS
Gaussian
Gaussian
distribution
distribution
[11]
[11]
with
with
Gaussian
azero
zero
mean,
distribution
mean,
and
and
that
that
[11]
the
the
with
continuity
continuity
aofzero
mean,
requirement
requirement
that
(LAAS
(LAAS
the
continuity
requirement
(LA
i i
i i i i i iSeveral methods
i i
i i Gaussian
CAT-I,
which
the
probability
false
alarm
in case
of the
first rising
of the day)
is
are
introduced
to mitigate
part
of
the
error
inis
range
measurement.
First,
a and
i
i
i
i
(3)


R

I

T

c
b

b




(3)
(3)


R

I

T

c
b

b


which
is the
probability
of false
alarm
inalarm
case
of
the
first
u u u u
u u
 u (3)uGaussian distribution [11] withCAT-I,
 I  T  c  bu  b    u
CAT-I,
which
isthat
the
probability
ofrequirement
false
in
case
of the
rising
of the
which
is
the
probability
of false
alarm
in case
of the
firstfirst
rising
of the
day)day)
is
a zero
mean,
and
the
continuity
(LAAS
-4
rising
of
the
day)
iscase
1.9×10
(K
=3.73)
[12].
The
threshold
4 alarm
CAT-I,
which
the
probability
ofofof
false
ofofof
the
first
rising
of
the
day)
isis
FFA
CAT-I,
CAT-I,
which
which
isthe
the
probability
probability
CAT-I,
false
false
which
alarm
alarm
is in
the
incase
in
probability
case
the
the
first
of
first
false
rising
rising
alarm
ofof
the
in
the
day)
case
day)
ofisthe first rising of the day) is
model-based mitigation
method
isisis
proposed
for ionosphere
delay
and
troposphere
delay
error.
(4K
1.9
10
FFA  3.73 ) [12]. The threshold (TH) is also designed with respect to sat
4 
i i
i
KFFA
 3.73
3.73
( in
) [12].
(TH)
is also
respect
to
( Kdesigned
) [12].
TheThe
threshold
isisalso
designed
withwith
respect
to sat
1.9
10is 10
range
Ru R: uTrue
True range
range(m)
(m)
(TH)
also
with
respect
tothreshold
satellite
elevation
for designed
CAT-I,
which
isrange
the(m)
probability
of 1.9
false
alarm
case
of
the
first
rising
of the(TH)
day)
Ru : True
: True
(m)
FFA
4444
4
K

3.73
(
)
[12].
The
threshold
(TH)
is
also
designed
with
respect
to
satellite
K
K


3.73
3.73
K

3.73
1.9

10
(
(
)
[12].
)
[12].
The
The
threshold
threshold
(
(TH)
(TH)
is
is
)
also
[12].
also
designed
designed
The
threshold
with
with
respect
respect
(TH)
is
to
to
also
satellite
satellite
designed
with
respect
to
sat
1.9
1.9

10

10
1.9

10
IIii :: Ionosphere
delay
error
(m)
The
range
differential
method
then
estimates
range
errors
by
subtracting
the
true
range
from
navigation
integrity
and
continuity,
according
to
equation
FFA
FFA
FFA
FFA
FFA
Ionosphere delay error (m)
elevation for navigation
integrity and continuity, according to equation (4). Because
4
i
integrity
and
continuity,
according
to equation
Becaut
elevation
for for
navigation
integrity
and
continuity,
according
(4).(4).
Because
threshold
(TH)
is also
designed
with
respect
to
satellite
T
delayerror
error(m)
(m)1.9  10 ( KFFA  3.73 ) [12]. The
(4).elevation
Because
testnavigation
statistics
are affected
by
corrected
rangeto equation
T :: Troposphere
Troposphere
delay
5
5 5 pseudorange correction). The receiver clock error is also
the
range
measurements
(also
called
elevation
for
navigation
integrity
and
continuity,
according
to
equation
(4).
Because
test
elevation
elevation
for
for
navigation
navigation
integrity
integrity
elevation
and
and
continuity,
for
continuity,
navigation
according
according
integrity
to
to
equation
and
equation
continuity,
(4).
(4).
Because
Because
according
test
test
to
equation
(4).
Because
buu:: Receiver
Receiverclock
clockerror
error(s)
(s)
statistics
are affected
by corrected
measurements,
measurements,
the threshold
hasrange
a stringent
boundarythe
asthreshold has a stringentt
statistics
affected
by corrected
range
measurements,
threshold
a string
statistics
are are
affected
by corrected
range
measurements,
the the
threshold
has has
a stringent
i
elevation
for
navigation
integrity
and
continuity,
according
to
equation
(4).
Because
test
i
bb :: Satellite
Satellite clock
clock error
error(s)
(s)
satellite elevation increases relative to the case of a satellite
reduced
by
subtraction
of
range
measurements
between
satellites
on
common
RS
(called
the
statistics
are
affected
by
corrected
range
measurements,
the
threshold
aastringent
statistics
statistics
are
are
affected
affected
byby
corrected
corrected
statistics
range
range
are
measurements,
affected
measurements,
by corrected
the
the
threshold
threshold
rangehas
measurements,
has
has
astringent
stringent
the threshold
hasata stringent
boundary
as
satellite
elevation
increases
relative
to the case
of a satellite
low eleva
Noise, multipath,
multipath,etc.
etc.
ui :: Noise,
at low
elevation.
boundary
as satellite
elevation
increases
to the
a satellite
at low
el
boundary
as satellite
elevation
increases
relative
to the
casecase
of aofsatellite
at low
eleva
statistics are affected by corrected
range measurements,
the threshold
has
a relative
stringent
(m/s)
c : Speed
Speed of
of light
light
(m/s)
double
difference between
satellites).
Aselevation
described
above,
the
combination
of
subtraction
of
boundary
asasas
satellite
increases
relative
to
the
case
of
a
satellite
at
low
elevation.
boundary
boundary
satellite
satellite
elevation
elevation
boundary
increases
increases
as
relative
satellite
relative
to
elevation
to
the
the
case
case
increases
of
of
a
a
satellite
satellite
relative
at
at
low
low
to
elevation.
the
elevation.
case
of
a
satellite
at
low
eleva
(4)
THiTSi i   iTSi i   K FFA  iTSi i 
(4)
 

 

boundary as satellite elevation increases
THTSrelative
  at low elevation.
TH TS
KFFAKofFFAasatellite
  TS toTSthe
case
(4) (4)
TS  TS 
receivers and satellites gives
from
perspective
of relative
range
i ii i the best
i ii performance
i iii iithe
i
i
i i : standard deviation of test statistics (m)
:
test
statistics
mean
(m),


Several methods
toTH
mitigate
part
of
the





K


TS
TSi(4)
i
i
i
TH
TH










K
K

TH









K





















(4)
(4)
(4) (m)(m)
al methods are introduced to mitigate
part of the are
errorintroduced
in range measurement.
First,
a
TSTS
TSTS
FFA
TSTS
TSTS
TSTS
FFA
FFA
FFA 
TSTS
TS
TS
FFA
: test
statistics
mean
deviation
of test
statistics
: test
Test
statistics
mean
(m)(m),
statistics
mean
(m),
deviation
of test
statistics
TSTS : standard
TS
TS : standard
i
i
i TS :
TH


K


K

error in range measurement.
First,
a
model-based
mitigation


elevation
angle
(degrees),
:
sigma
multiplier
based
on
the
continuity req
(4)
i TS
ii:ii
ii i   
i
i FFA
TS
FFA
TS
measurement accuracy.TSTS
Standard
deviation
of
test
: :test
statistics
mean
(m),
::TS:standard
deviation
of
test
test
: test
statistics
statistics
mean
mean
(m),
(m),
test
:elevation
standard
statistics
deviation
deviation
mean
(m),
ofof
test
test
statistics
statistics
:Kstandard
(m)
(m)
deviation
of
test
statistics
(m)

statistics
KTS
TSTS:TSTSelevation
:standard
angle
(degrees),
:(m)
sigma
multiplier
based
on the
continuity
angle
(degrees),
:FFA
sigma
multiplier
based
on
the
continuity
requ
TSTS
FFA
ased mitigation method is method
proposedisforproposed
ionosphere
and troposphere
delay
error.
i
i
fordelay
ionosphere
delay
and
troposphere
:
test
statistics
mean
(m),
:
standard
deviation
of
test
statistics
(m)
TS

:
Elevation
angle
(degrees)
TSK
angle
(degrees),
:
sigma
multiplier
based
on
the
continuity
requirement
K
K
::elevation
K
elevation
: elevation
angle
angle
(degrees),
(degrees),
:FFA
elevation
:
sigma
:
sigma
angle
multiplier
multiplier
(degrees),
based
based
on
on
the
:
the
sigma
continuity
continuity
multiplier
requirement
requirement
based
on
the
continuity
requ
FFA
FFA
FFA
However,
in this work,
a model-based
mitigation method
is used to mitigate range FFA
delay error. The range
differential
method
then estimates
: Sigma multiplier
continuity
requirement
ge differential method then estimates range errors by subtracting the
range from
 true
: elevation
angle (degrees),2.2KSensitivity
multiplierbased
basedononthe
the
continuity
requirement
FFA sigmaAnalysis
range errors by subtracting the true range from the range
Sensitivity
Analysis
2.2 2.2
Sensitivity
Analysis
measurement
errors
arising
from
thealso
ionosphere
and troposphere. The receiver clock bias is
e measurements (also called
pseudorange correction).
Thepseudorange
receiver
clock
error is
measurements
(also called
correction).
The
2.2
Sensitivity
Analysis
2.2
2.2
Sensitivity
Sensitivity
Analysis
Analysis
2.2
Sensitivity
Analysis
2.2
Sensitivity
Analysis
Various
geometry
conditions can be formed between ground baseline vectors and s
Various
geometry
conditions
be formed
between
ground
baseline
vectors
Various
geometry
conditions
cancan
be formed
between
ground
baseline
vectors
andan
s
receiver clock error is also reduced
bySensitivity
subtraction
of range
2.2
Analysis
estimated
by
computation
of
the
user’s
position
in
a
process
using
the
weighted
least squares
by subtraction of range measurements between satellites on common RS (called the
Various
geometry
conditions
can
be
formed
between
Various
geometry
can
be
formed
between
ground
baseline
vectors
and
satellites.
Various
Various
geometry
geometry
conditions
conditions
Various
can
can
be
be
formed
geometry
formed
between
between
conditions
ground
ground
can
baseline
baseline
be
formed
vectors
vectors
between
and
and
satellites.
ground
satellites.
baseline
vectors
and
measurements between satellites on common
RS
(calledconditions
the
Consequently, the detection performance of this algorithm depends on the geometrys
Consequently,
the
detection
of
this
depends
on the
geometc
Consequently,
the
detection
performance
of this
algorithm
depends
on the
geometry
Various
geometry
conditions
can
bemeasurements,
formed
between
baseline
andalgorithm
satellites.
baseline
vectors
andperformance
satellites.
Consequently,
double
difference
between
satellites).
described
method.
the adverse
effect on
accuracy ofground
range
weground
recognized
thatvectors
difference between satellites).
As described
above,
theDespite
combination
ofAssubtraction
ofabove,
Consequently,
the
detection
performance
of
this
algorithm
depends
on
the
geometry
condition.
Consequently,
Consequently,
the
the
detection
detection
performance
Consequently,
performance
of
of
the
this
this
detection
algorithm
algorithm
performance
depends
depends
on
on
of
the
the
this
geometry
geometry
algorithm
condition.
condition.
depends
on
the
geometry
In
this
section,
we
derive
the
relationship
between
satellite
position
error
and
the
testc
the detection performance of this algorithm depends on
the combination of subtraction of receivers and satellites
In this
section,
we
derive
the
relationship
between
satellite
position
error
In this
section,
wealgorithm
derive
thedepends
relationship
between
satellite
position
error
andand
the the
test
Consequently,
the detection
performance
of
this
on
the
geometry
condition.
s and satellites gives the best
performance
from
the
perspective
of
relative
range
combining
single
and
double
difference
methods
posed
a
contradiction
in
this
algorithm
the geometry condition. In this section, we derive the
gives the best performance from the perspective of relative
InInIn
this
section,
we
derive
the
relationship
between
satellite
error
and
the
test
statistics
this
this
section,
section,
we
we
derive
derive
the
the
In
relationship
relationship
this
section,
between
between
we
derive
satellite
satellite
the
relationship
position
position
error
error
between
and
and
the
the
satellite
test
test
statistics
position error
and
analyze
the
properties
ofposition
this
algorithm
with
regard
tostatistics
geometry
[14].and the test
relationship
between
satellite
error
and
the
test
range
measurement
accuracy.
and
analyze
the
properties
ofposition
this
algorithm
with
regard
to geometry
[14].
and
analyze
the
properties
ofposition
this
algorithm
with
regard
to
geometry
[14].
In
this
section,
we
derive
the
relationship
between
satellite
error
and
the
test
statistics
ment accuracy.
because these methods use the broadcast ephemeris (to be validated by test statistics). The
[14].
 i ,regard
statistics
and
analyze
thetoof
properties
of
this
algorithm
with
However, in this work, a model-based
mitigation
method
i , f with
f
and
analyze
the
properties
ofofof
this
algorithm
with
regard
geometry
[14].
and
and
analyze
analyze
the
the
properties
properties
and
this
this
analyze
algorithm
algorithm
the
with
properties
with
regard
regard
to
to
this
geometry
geometry
algorithm
[14].
to
geometry
[14].
In equation (5) and Fig. 2, we define eiA, f ,i , feiB, f toi , f be unit line-of-sight (LOS) vect
Intoequation
and
Fig.
2, we
define
to unit
be unit
line-of-sight
(LOS)
v
In
equation
(5) (5)
and
Fig.to
2,geometry
we
define
line-of-sight
(LOS)
vecto
eA e, A eB, etoB be
regard
geometry
[14].
ver, in this work, a model-based
method
is measurement
used
to validated
mitigate
range
and analyze
the
properties
this
algorithm
with
regard
[14].
is usedmitigation
to mitigate
range
errors
arising
frombeofindependent
test
statistics
and
ephemeris
must
for
ephemeris
fault
 i, ifi,,reliable
i, f i, f
ffi , f  i, ifi
,, ffi , f
InInIn
equation
(5)
and
Fig.
2,2,we
define
be
unit
line-of-sight
from
ee e,(5)
e, eand
equation
equation
(5)
(5)
and
and
Fig.
Fig.
2,we
we
In
define
equation
define
,(5)
Fig.
toFig.
bebe
2,
unit
unit
we
line-of-sight
line-of-sight
define
(LOS)
, (LOS)
to
vectors
vectors
be
from
line-of-sight
from
(LOS)
vecto
eto
eB vectors
In
equation
2,
we
define eA (LOS)
to
be unit
unit

i
the ionosphere and troposphere. The
receiver
clock
bias
BBand
B to
 ,Af AA A  iB,(A,
RS antennas
B) to the satellite i when
ephemeris failure occurs. eiAi and
eiB ,
f
ment errors arising from the ionosphere and troposphere. The receiver
clock bias
isand Fig. 2, wetwo
i
equation
(5)range
define
, but
tovectors
beto
unit
line-of-sight
(LOS)
vectors
eAiantennas
eB(A,
In addition,
required
value,
an
absolute
measurement.
two
(A,
B)
to
the
satellite
i when
ephemeris
occurs.
and
line-of-sight
(LOS)
from
two
RS
antennas
(A, B)from
to failure
two
RS RS
antennas
B)
the
satellite
i when
ephemeris
failure
occurs.
eA eand
e
is estimated bydetection.
computation
of theInthe
user’s
position
inisanot a relative
A
B,
 i ii i
 i ii i i ii i
i
i


the
satellite
i
when
ephemeris
failure
occurs.
and
,
two
RS
antennas
(A,
B)
to
the
satellite
i
when
ephemeris
failure
occurs.
and
,
is


e
e
two
two
RS
RS
antennas
antennas
(A,
(A,
B)
B)
to
to
the
the
two
satellite
satellite
RS
antennas
i
when
i
when
(A,
ephemeris
ephemeris
B)
to
the
failure
failure
satellite
occurs.
occurs.
i
when
ephemeris
failure
occurs.
and
and
,
,
is
is
and
e
e
e
e
e
e
process
using
weighted
squares least
method.
Despite
d by computation of the user’s
position
in the
a process
usingleast
the weighted
squares
B,
A
Bii i represents
 i A AAi A, and iBBBR
geometry
rangeephemeris
between
RS
and
the
satellite
the faulty
vecto
The methods for mitigating
range measurement
errors
are
summarized
in
Table
1.
iis

two
RS
antennas
(A,
B)
to
the
satellite
i
when
failure
occurs.
and
,
e
e
is geometry
range
between
RS and
and
thesatellite
satellite
and
the adverse effect on accuracy of range measurements, we
represents
faulty
ve
geometry
range
between
RS
i i,, and
represents
the the
faulty
vector
geometry
range
between
and
the the
satellite
B R  R
A i , and
RS
i ii i
i
Despite the adverse effectrecognized
on accuracy that
of range
measurements,
we
recognized
that
represents
the
faulty
vector
of
satellite
i
[13].
combining
singlegeometry
and
double
difference
range
between
RS
and
the
satellite
i
,
and

R
represents
the
faulty
vector
of


R
R
represents
represents
the
the
faulty
faulty
vector
vector

R
of
represents
of
the
faulty
vector
geometry
geometry
range
range
between
between
RS
RS
and
geometry
and
the
the
satellite
satellite
range
i
between
,
i
and
,
and
RS
and
the
satellite
i
,
and
satellite i [13].

satellite
i [13].
satellite
i [13].
geometry range between RS and
the
satellite
i , and  Ri represents the faulty vector of
ng single and double difference methods posed a contradiction in this algorithm
satellite
i i[13].
satellite
satellite
[13].
i [13].
satellite
1. Mitigation
methods
for
range measurement
error.i [13].
Table 1. MitigationTable
methods
for range measurement
error.
7
satellite
i
[13].
these methods use the broadcast ephemeris (to be validated by test statistics). The
 

 

Error components
Mitigation Methods
77 7
stics and validated ephemeris must be independent for reliable ephemeris fault
Klobuchar model
7
Ionospheric delay
(with GPS Navigation Message Iono. Parameters)
n. In addition, the required range is not a relative value, but an absolute measurement.
6
Troposheric delay
Sasstamonien
model
Satellite
/ Receiverinclock
Broadcast ephemeris / Weighted least square
hods for mitigating range measurement errors
are summarized
Table 1.
Mitigation methods for range measurement error.
The threshold for ephemeris failure detection is determined by the probability property of
omponents
91
7
7
7
http://ijass.org
Mitigation Methods
the test statistics and system continuity requirements. We assume that test statistics follow a
6
(89~101)14-083.indd 91
Gaussian distribution [11] with a zero mean, and that the continuity requirement (LAAS
CAT-I, which is the probability of false alarm in case of the first rising of the day) is
2015-03-30 오후 3:48:27
(7)

eBi  cos  B cos B iˆ1  cos  B sin B iˆ2  sin  B iˆ3
Figure 3. Local coordinates.


In local coordinates, eAi and eBi can be defined as in equation (7) with local elevation
Inangle
equation
(7), the local elevation angles (  A ,  B ) from reference stations (A, B) c
(  ) and local azimuth angle ( ).


eAi  cos  A cos Aiˆ1  cos  A sin Aiˆ2  sin  Aiˆ3
assumed
as similar (

B ), ˆbecause
 B i3
eBi  cos  B cos B iˆ1  cos  B sinA B iˆ2  sin
 
Int’l J. of Aeronautical & Space Sci. 16(1), 89–101 (2015)
igure
Figure2.2.Geometry
Geometryofofephemeris
ephemerisfault
faultcondition.
condition.
Figure 2. Geometry of ephemeris fault condition.
km) isIn equation
quite (7),
large
relative to the baseline length between two reference stations. If this
the local elevation angles (  ,  ) from reference stations (A, B) can be
A
 f fault


 f i

Figure 2. Geometry of ephemeris
eAi ,
eAi  condition.
eAi , eBi ,
eB   eBi
B
because
range from GNSS satellite and reference (over 20,000
(5)
range(reference
from GNSS satellite
(over 20,000can be assumed to be an
assumed as similar
(  A   B ), because
assumption
is adopted,
triangle
A, and
B, reference
and satellite)
km) is quite large relative to the baseline length between two
km) is quite large relative to the baseline length between two reference stations. If this
reference stations. If this assumption is adopted, triangle
isosceles
triangle. Using this geometry property the local elevation angles (  A ,  B ) ar
assumption is adopted, triangle (reference A, B, and satellite) can be assumed to be an
(reference
A, B, and satellite) can be assumed to be an isosceles
isosceles Using
triangle. Using
this geometry property
the local
angles (  A ,  Bangles
) are
triangle.
geometry
property
theelevation
local
elevation
regardless
of
reference
position.
expressed as this
 regardless ofas
(θAexpressed
reference
position. of reference position.
, θB) areas expressed
θ regardless
i
Using
equation
(7)ininequation
equation
(6), E AB
becomes
Using
(6),
becomes
Using equation
equation
(7) in(7)
equation
(6),
becomes
E
(6)
i
 i ,T  i
(5)
i  I  e  e   R
e 
i
Substituting
equation
(5)equation
into
equation
Substituting
Substitutingequation
equation(5)(5)into
into
equation
equation(2)
yields
equation
(6).
(6).(2) yields equation (6).
(2)yields
Substituting equation (5) into equation (2) yields equation (6).
Substituting equation (5) into equation (2) yields equation (6).

(7) GNSS satellite and reference (over
range from
 i ,T i ,T  i ,T i ,T i  i  i ,T i ,TTS
 i AB
i iT T Ri ,T  eAi ,T  eAi  eBi ,T  eBi   eAB
i i


TSTS

R

R
e
e
e
e
e
e
e
e










  A A A A BB BB T eABeAB
AB AB
(6)
i

T 
TS AB
   R i ,T  eAi ,T  eAi  eBi ,T  eBi   eAB
(6)(6)
i
  eAB
 Ri ,T   E AB

i ,T i ,T
i i T T 
(6)
i TeABeAB

R
 R  i,TEAB
E AB



  eAB  
 R   EAB

eAB  :Unit baseline vector ( xAB )
:
Unit
baseline
vector
(
)




eAB: Unit
eAB
: Unitbaseline
baselinevector
vector
(i x( xeABi ,T) e i  e i ,T  e i
E
eAB
:Unit
baseline vector
( xABABAB
) A A B B
i i
i ,
T i ,T i  i  i ,
T i ,T i  i




EE
E


e
e

e

e

e

e

e

e
i
i
,
T
i
i
,
T
i
AB AB  e
A A  eA 
A e B B e B B
AB
A
A
B
B
Figure 3. Local coordinates.
T
i
AB

cos   cos   cos  
cos   sin cos  sin cos  sin  cos   cos  cos  
(8)


2
2
2
2
 cos  cos
cos sin
 
cos  sin  cos
sin  
sin  cos
cos  
sin  cos(8)
 cos
 sinAsin
E  
  sin
  cos
 sin
 B  sin  cos   cos A  cos B  
A  cos  B  
A
B

  cos  sin   cos  cos 

cos  sin   sin  sin 
0

T

i
2
2
2
2 

cos

cos

sin

cos

sin

cos

sin

sin

sin  cos   sin A  sin B  
E




 AB  

A
A
B
B
A
B

cos  sin   cos A  cos B 
cos  sin   sin A  sin B 
0




2
T
i
AB
2
2
A
2
A
A
2
B
B
A
B
B
A
2
A
2
B
2
A
A
B
B
A
B
A
B
B
The baseline unit vector of equation (6) can be derived as
The baseline
baseline
unit vector
vector of
of9equation
equation (6)
(6) can
can be
be derived
derived as
as equation
equation (9)
(9) [10],
[10],
The
equation
(9) [10], unit
The baseline
unit vector of equation (6) can be derived as equation (9) [10],
Equation
(6)
can
be
simplified
in
local
coordinates
that
use
the
baseline
vector
as
the
x-axis.
baselineTTunit
vector of equation (6) can be derived as
(9)
Equation
(6)
can be simplified in local coordinates that
 The
The
as equation
equation
(9) [10],
[10],
Figure
Figure
3. 3.
Local
Local
coordinates.
coordinates.
(9)
 [1
[1baseline
0] unit vector of equation (6) can be derived
eeAB
00 0]
(9) (9)
AB 
Equation
Equation
(6)(6)
can
can
bebe
simplified
simplified
in
local
localcoordinates
coordinates
that
that
use
use
thebaseline
baseline
vector
vector
asx-axis.
asthethex-axis.
x-axis.
Equation
(6)
can
be
simplified
inin
local
coordinates
that
use
thethe
baseline
vector
as the

T
use
the
baseline
vector
as
the
x-axis.
Fig.
(3)
shows
the
local
Figure (3) shows the local coordinates
i
 iand the geometry of the satellite. eAB  [1 0 0]T
(9)
T
andtheesatellite.
In localand
coordinates,
eA of
(9)
[1
00 0]
9
B can be defined as in equatione
AB 
coordinates
the
geometry
(9)
e(7)
with
[1 local
0]elevation
ABand
Figure
(3)
shows
the
local
coordinates
and
the
geometry
of
the
satellite.
substituting
equation
(9) into
into equation
equation
(6) yields
yields
[18]
igure
Figure
(3)(3)
shows
shows
thethe
local
local
coordinates
coordinatesand
and
thethe
geometry
of
of
the
the
satellite.
satellite.
and
substituting
equation
(9)
(6)
[18]
geometry



and
substituting
equation
(9)
into
equation
(6)
yields
[18]
i i
i i
In
local
coordinates,
and
bebedefined
asas
inas
equation
andeB ecan
can
be
defined
defined
in in
equation
equation
(7)(7)
with
with
local
local
elevation
elevation
InIn
local
local
coordinates,
coordinates,
eA eAand
Bcan
and
substituting
equation (9) into equation (6) yields [18]
( 2.)Geometry
and
local angle
azimuth
( ).azimuth
and
substituting
(9)
Figure
of ephemeris
fault
condition.
iequation
and
substituting
equation
(9) into
into equation
equation (6)
(6) yields
yields [18]
[18]
(7)angle
with local
elevation
(θ) angle
and local
angle (ψ).
TT
i
ii
ii
(10) (10)

TSAB
EAB
(10)

TS
E
R
AB
AB T   R
E i TLL   Ri
i
angle
angle
(
) and
local
local
azimuth
azimuth
angle
angle
((
). ).
()and

TS AB
(10)

i
iii
eAi  cos  A cos Aiˆ1  cos  A sin Aiˆ2  sin  Aiˆ3

TS
(10)
 EEABABABii EEABiABiLTLL TTT RReeii ABAB

TS
(10)

EAB
(11)

E
(11)
AB


AB
(7)
(7)
AB

i
i 
i i i
LL
T e
(11)


ˆ
ˆ
ˆ

E
E
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ

(11)







cos
cos
sin
sin

i

i

i
i
i
cos
cos
cos
cos
sin
sin
sin
sin
eAeA cos
 ecos




i

i





i

i



i
i
AB
T AB
 EAB
B A A
B
BA2Af 2 B
TT
A 1fA 1 Bi 1 A Ai
 2 A 3A 3 B 3
i L
i 
cos

E
22e cos22   cos22 
(11)
 EAB

EAB
(11)
(7)(7)
eAi ,
eA   eA , eBi ,
eBi   eBi
T
L 
AB (5)
AB 
AB cos  AA  cos  BB 
i i
cos2eAB
L
2
2
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
Substituting
equation
(5)
into
equation
(2)
yields
equation
(6).



sin
 Bcos
 Bi1Bi1 cos
 Bisin
  i2 isin
 Bi3B i3 8
eB eB cos
 cos
 cos
 sin
 22 cos 2   cos 2 A  cos 2 B 
TT
TT
B cos
,B
T  i B i2B
ii
In equation i(7), Ithe
angles (θA, θB) from
 e local
 e  elevation
 R
2 
2

cos cos
cos
 cos
sin2 
AA  cos
cos
sin


 BB
 cos
 EEiABAB TLL cos
AA sin
cos
BBBBsin
2 cos   cos
A
e  8
T



E
cos

cos

sin

cos

sin
i




2
i
reference
stations
(A,
B)
can
be
assumed
as
similar
(θ
≈θ
),
AB
A
A
B
B  
from
reference
(A,cos
B)
In equation (7), the local
sin
cos
 AA cos
cos
BB 
B
 elevation angles (  A ,  BA) 
 can
sin
be


 cos
 cos
cos




 cos
 EEABABi stations
LTLL cos
A 
B sin

cos2cos
cos
 AA sin
sin

cos

 BB 
A 
B sin

sin
cos
cos
88


A
B
 B)B)

cos
be
) from
reference
reference
stations
stations
(A,(A,
can
can
be  cos
InIn
equation
equation
(7),
thethe
local
elevation
(7),
T angles
cos
 sin
sin
cos
A 
cos
 BB 
 cos
local
 elevation
  angles
 ( (A, A,B)Bfrom
i


assumed
asi ,Tsimilar
(over 20,000 A
  eAB range from GNSS satellite and reference
TS AB
 R
 eAi ,T (eAi A eBi,T B e),Bi because
(6) (10)
 ( ( i  T ),
Equation
(10)20,000
shows
the sensitivity
sensitivity relationship
relationship between
between test
test statistics
statistics and
and the
the fa
f
Equation
shows
 ), because
range
range
from
from
GNSS
GNSS
satellite
satellite
and
and
reference
reference
(over
(over
20,000the
assumed
assumed
asas
similar
 B eBABbecause
similar
Ri ,T large
  EA ABArelative
Equation
(10)If
shows
thesensitivity
sensitivityrelationship
relationshipbetween
between test statistics and the fa
Equation
(10)
shows
km) is
quite
to the baseline length between two reference
stations.
this the
Equation
(10)
shows
the sensitivity
relationship
between
test statistics
and
fa


Equation
(10)the
shows
sensitivity
relationship
between
statistics
and the
the is
fa
satellite
location
resulting
from
ephemeris
failure.
When
thetest
Schwarz
inequality
is
test
statistics
and
fault the
in from
satellite
location
resulting
fromthe
satellite
location
resulting
ephemeris
failure.
When
Schwarz
inequality
eAB
: Unit
baseline
vector
( xreference
)
ABreference
km)
km)
is assumption
is
quite
quite
large
large
relative
relative
to
to
the
the
baseline
baseline
length
length
between
between
two
two
stations.
stations.
If
If
this
this
satellite
location
resulting
from
ephemeris
failure.
When
the
Schwarz
inequality
is
is adopted, triangle
(reference
A,
B,
and
satellite)
can
be
assumed
to
be
an




i
ephemeris
failure. When
thefrom
Schwarz
inequality
is applied
satellite
location
resulting
ephemeris
failure.
When the
Schwarz
inequality
is
E AB
 eAi ,T  eAi  eBi ,T  eBi
satellite
from
Schwarz geometry
inequalityco
is
equationlocation
(10), we
weresulting
can derive
theephemeris
sensitivityfailure.
relationWhen
in the
thethe
worst-case
equation
(10),
relation
in
inassumed
equation
(10),
we can
can derive
derive the
the sensitivity
sensitivity relation
in
the worst-case geometry con
assumption
assumption
is is
adopted,
adopted,
triangle
triangle
(reference
(reference
A,A,
B,B,
and
and
satellite)
satellite)
can
can
bebe
assumed
to
to
be
be
an
an
equation
(10),
we
can
derive
the
sensitivity
relation
in
the
worst-case
geometry
con
 A , we
isosceles triangle. Using this geometry property the local elevation
angles ((10),
are derive the sensitivity relation in the worst-case geometry con
B ) can
equation
worst-case
geometry
condition.
equation
(10),
we
 derive the sensitivity relation in the worst-case geometry con
TT can
ii
ii
(12)
TS

E

Rii
(12)
TS

E
R



AB
AB


,

,

T
isosceles
isosceles
triangle.
triangle.
Using
Using
this
this
geometry
geometry
property
property
the
the
local
local
elevation
elevation
angles
angles
(
(
)
are
)
are
AB
AB
Equationas(6) can
be simplified
in localposition.
coordinates that use the baseline
i A A vector
i
i x-axis.
LLas the

regardless
of reference
expressed
(12)
TS AB
 BEBAB

R

i
i T
i
i
i LT  R i
(12)
TS
  E AB
(12)
(12)
TS AB
R
AB   E AB  L
L
Figure
(3)
shows
the
local
coordinates
and
the
geometry
of
the
satellite.


regardless
regardless
of
of
reference
reference
position.
position.
expressed
expressed
as
as
i
Using equation (7) in equation (6), E AB becomes
This shows
shows that
that the
the sensitivity
sensitivity increases
increases when
when the
the local
local elevation
elevation angle
angle and
and the
the
This
This
shows
that
the
sensitivity
increases
when
the
local elevation angle and the a
i i
This
shows
that
the
sensitivity
increases
when
the
local
Using
Using
equation
equation
(7)
(7)
in
in
equation
equation
(6),
(6),
becomes
becomes
E AB
E AB2
2
2 condition.
Fig. 2. Geometry
of 2ephemeris
fault
This
that
sensitivity
increases
when
elevation
angle and
 Figure
 the
fault
condition.
cos
 2.
cosGeometry
 A  cos
 Bof
cos   sin
 A cos
 A  sin B cos B  sin
 cos
  cosshows
 A  cosand
 ephemeris
B  the
This
shows
theand
sensitivity
increases
whentothe
the
local
elevation
and the
the
elevation
angle
azimuth
angle
are close
0°local
andgiven
angle
are
closethat
to
0°
and
90°, respectively.
respectively.
Accordingly,
given
these angle
properties,
thi
angle
are
close
to
90°,
Accordingly,
these
properties,
thi
(8)

0°
T
i
2
2
2
2
 cos   cos A sin A  cos B sin B 

cos

sin

sin

sin

cos

sin

sin






E AB  cos


angle
are
close
to
0°
and
90°,
respectively.
Accordingly,
given
these
properties,
thi
A
B
A
BAccordingly, given these properties, this
90°,
respectively.
2
2
2
2
2
2
2
2


cos
0° and 90°, respectively. Accordingly, given these properties, thi
  cos
  cos
 AAcos
 B B 
coscos
  sin
 
sinAcos
 AAsin
sinBcos
 B B sinsin
 cos
 cos
  cos
angle
 AAare
cos
close
 cos

 cos
 cos
 B  to
A cos
B cos
B
angle
are
90°, respectively.
Accordingly,
theseangle.
properties,
cos  sin   cos A  cos B 
cos  sin   sin A  sin B 
0 close
(8)
(8)and
 
for
 to

0°
is sufficient
sufficient
for
fault
detection
for rising
risingfor
satellites
of low
lowgiven
elevation
angle.
From athi
a lo
l
is
fault
detection
for
satellites
of
elevation
From
T T
algorithm
is
sufficient
fault detection
rising satellites
 for
2
2
2
2
 cos
B  
cos
 2 cos
  cos
 AsinA 
sinAAcos
 Bsin
sinB
coscos
 2 sin
  sin
 2AAsin
 B2  B 
sinsin
 cos
 cos
  sin
 
sinAAsin
sinB
 cos
 sin

 B

 E ABi EABi 
 
B
is sufficient for fault detection for rising satellites of low elevation angle. From a lo

of islow
angle.detection
From a local
azimuth
perspective,
for
satellites
of
angle.
aa lo
 sin
 sin
  cos
  cos
 AAcos
 B
coscos
 sin
 sin
  sin
 
sinAAsin
sinB
0 sufficient
0 elevation
 cos

 B
 B
  coscos
for
 fault
is
sufficient
for rising
rising
satellites
of low
low elevation
elevation
angle. From
From
lbb
azimuth
perspective,
various baseline
baseline
vectors
are required
required
using multiple
multiple
RS and
and
 
for
 fault detection
azimuth
perspective,
various
vectors
are
using
RS
Substituting equation (5) into equation (2) yields equation
(6). baseline
various
vectors
are
required
using
multiple
RS
and
azimuth perspective, various baseline vectors are required using multiple RS and b
azimuth
perspective,
various
baseline
vectors
are
required
using
multiple
RS
b
baseline
should
be
extended
to increase
the
azimuth
perspective,
various
vectors
aresatellite’s
required
using
multiple
RS and
and
length length
should
be extended
extended
to baseline
increase
the satellite’s
satellite’s
induced
angle.
This means
means
thab
length
should
be
to
increase
the
induced
angle.
This
tha
8
induced
angle. be
This
meanstothat
the algorithm
above
is angle. This means tha
length should
extended
increase
the satellite’s
induced
9
length
should
be
extended
to
the
induced
angle.
This
means
tha
 i ,T  i ,T  i  i ,T  i T 
length
should
be is
extended
to increase
increase
the satellite’s
satellite’s
induced
angle.
Thissuch
means
tha
algorithm
above
is
applicable
for
wide-area
implementations
of GNSS
GNSS
such
as Spa
Spa
applicable
for
wide-area
implementations
of
GNSS
such
i
algorithm
above
applicable
for
wide-area
implementations
of
as
TS AB  R  eA  eA  eB  eB   eAB
9 9
algorithm
above
is
applicable
for
wide-area
implementations
of
GNSS
such
as
Spa
as
Space-Based
Augmentation
System
(SBAS)
and
Ground
(6) is applicable for wide-area implementations of GNSS such as Spa
algorithm
above

T 
algorithm
above
is applicable
for
wide-area
implementations
of GNSS
such as
Spa
i
Augmentation
System
(SBAS)(GRAS)
and
Ground
Regional
Augmentation
System
(GRA
Augmentation
System
(SBAS)
and
Ground
Augmentation
System
(GRA
  eAB
 Ri ,T   E AB

Regional
Augmentation
System
ratherRegional
than narrowAugmentation
System
(SBAS)
and
Ground
Regional
Augmentation
System
(GRA

 Augmentation System (SBAS) and Ground Regional Augmentation System (GRA
implementations.
eAB : Unit baseline vector ( xABarea
)than
Augmentation
System
(SBAS) and Ground Regional Augmentation System (GRA
than
narrow-area
implementations.
narrow-area
implementations.




i
i ,T
i
i ,T
i
than narrow-area implementations.
E AB  eA  eA  eB  eB
than
narrow-area
implementations.
than
narrow-area
implementations.
2.3
Ephemeris
Protection
Level and MDE
Fig. 3. Local coordinates.
GNSS
implementations
conduct
a two-step process
2.3
Ephemeris
Protection
Level
and MDE
MDE
Equation
(6) can be simplified in local coordinates that use
the
baseline
vector
as theLevel
x-axis.
2.3
Ephemeris
Protection
and
Figure
3. Local
coordinates.
2.3 Ephemeris Protection Level and MDE
2.3 Ephemeris Protection Level and MDE
Ephemeris
Protection Level
andaaMDE
Figure (3) shows the local coordinates and the geometry of 2.3
theGNSS
satellite.
GNSS
implementations
conduct
two-step process
process to
to ensure
ensure navigation
navigation integrit
integri
implementations
conduct
two-step
i
i
GNSS
implementations
conduct
a two-step process to ensure navigation integrit
In
local
coordinates,
and
can
be
defined
as
in
equation
(7)
with
local
elevation
e
e
GNSS implementations conduct a two-step process to ensure navigation integrit
B
A
DOI: http://dx.doi.org/10.5139/IJASS.2015.16.1.89
92
GNSS implementations conduct a two-step process to ensure navigation integrit
angle (  ) and local azimuth angle ( ).
(89~101)14-083.indd

eAi  cos  A cos Aiˆ1  cos  A sin Aiˆ2  sin  Aiˆ3

e92Bi  cos  B cos B iˆ1  cos  B sin B iˆ2  sin  B iˆ3
10
10
10
10
10
(7)
2015-03-30 오후 3:48:28
When affection
of zero
nominal
of test statistics
is considered,
equation (13) is
distribution
mean
 and standard
  with
 condition
  deviation
   TSi . nthe
n
TSi i  
KK   
TS
i i 
i i
i i


others
SSvertvert

XX SS
x xthat
x x to Sa Svert
the test statisticsT correspond
Gaussian
vert
others
  EE

 i i EEi i T  
i i11


 A ABi ABiL L  
A



n









distribution
with
zero
standard deviation
 and
TS .
i
i
i
FFA
TS 
 Kmean
 TS
 x  S i  i
 X vert
 S vert
x  S vert

vert
others
 i Ei T 
  i Ei T 
i 1
  L   i A  AB L 
  A i AB
n
i i
i i
i i
FFA
FFA
TSTS
derived
as
equation
(15).
We assume
vert
vert
vert
vert
i i
i i T T
AA
ABAB L L i
 
 
K FFA   TS

(1(
(1
TS
i
i
i
 third
 x on
 right
 xside
side
Svert
Sof

 second
Xsecond
 S and
(15)
The
and
terms
on
the
hand
equation
(15)are
are
developedinto
int
The
terms
the
hand
of
(15)
developed

vertequation
others
  third

  right

i 1
E
E




Jongsun Ahn Orbit Ephemeris Failure
Detection
in
a
GNSS
Regional
Application



information of faulty satellites, identified using fault detection algorithms, is broadcast
to the
The
second and third terms on the right hand side of equation (15) are developed into
equation
(16).
equation
(16).
ormation of faulty satellites,
identified
usingsatellites,
fault detection
algorithms,
is
broadcast
toalgorithms,
theidentifiedisusing
information
of
faulty
identified
using
fault
detection
broadcast
the
information
of
faulty
satellites,
fault to
detection
algorithms,
is ofbroadcast
to the
user within a limited time; then, an airborne user computes the protectionThe
level
corresponding
second
terms on the
right
hand side(16).
equation (15) are developed into
equation
(15)
arethird
developed
equation
to information
ensure navigation
integrity.identified
First, the
information
equation
(16).
of faulty satellites,
using
fault detection algorithms,
is and
broadcast
to theinto
2 2
  (16).

i
r within a limited time;user
then,
an airborne
user
computes
protection
level
corresponding
a limited
time;
then, anthe
airborne
user
computes
thethen,
protection
level
corresponding
n n
of within
faulty
satellites,
identified
using
fault
detection
equation
user
within
a limited
an
airborne
user
computes
corresponding
2 2 level
2 2
TSiprotection
to the
available
satellites’
geometry
used
to compute
thetime;
navigation
solution.
2 2 protection
i i 2 2  The
i i
TS
   x2xthe
S

S







user
within
a
limited
time;
then,
an
airborne
user
computes
the
protection
level
corresponding
 Sn Sverti verti  others

vert
others
vert
algorithms, is broadcast to the user within a limited time;
 i 2i  EEi i Ti T   
i 1i 1
2
2

 position
LTSL the
he available satellites’to
geometry
used to
computegeometry
the navigation
The
2
theisavailable
satellites’
used
tosolution.
compute
theprotection
navigation
solution.
protection
A
A compute
ABAB
i
i
tohypotheses
the the
available
geometry
used
protection
 navigation
toThe
level
computed
baseduser
on various
andsatellites’
compared
with the
allowable
then,
an airborne
computes
protection
level
Svert
x   solution.


Sverti  The

others 
T


(16)
(16)
i
i

to the available satellites’ geometry used to compute the navigation solution.
protection
S    The
x

S






i
1

  AE  EAB  L 
corresponding
to the
available
satellites’
geometry
usedwith the allowable




el is computed based on
various
hypotheses
andon
compared
with
the
allowable
position
level
is
computed
based
various
hypotheses
and
compared
position
i
i
n nwith the allowable position (16)

level
is computed
based on
various ihypotheses
 and compared
(16)
2 2
error
bound, known
as the alertsolution.
limit
(AL),
to check GNSS
implementation
i   availability.
i i 2 2
i i
(16)
TSTS
x x
position
to level
compute
the navigation
The protection
level
SSvertvert
with
S Svert
others
is computed
based on various hypotheses
and compared
the allowable










vert
others
T iT 

n
i
i
i
i
 i  EE
2
2

i 1i 1
i
i
or bound, known as theerror
alert
limit (AL),
to as
check
GNSS
implementation
availability.
is computed
based
on
hypotheses
and
compared
bound,
known
the various
alert
limit
(AL),
to check
GNSS
implementation
availability.
   check
LTSL GNSS
AB
error
bound,
known
as the
alert limit(AL),
availability.
x  ximplementation
 S 
 S vert
  others

[12][16]
SSvertAAto iAB
T
i

E

i 1
errorthe
bound,
known position
as the alerterror
limit (AL),
to check
with
allowable
bound,
knownGNSS
as implementation
 E ABL 
  Aavailability.

][16]
[12][16]
[12][16]
thethis
alert
limit
tothe
check
GNSS
implementation
In
section
we (AL),
derived
vertical
PL (VPL
),
which
estimates
the
impact
of
one
He
[12][16]
i i
i i
availability. [12][16]
i is
i of
thestandard
standard
deviation
Equation
(16)isinto
issubstituted
substituted
intoeqe
where
others
where
deviation
. .Equation
(16)
others
where
isis
the
standard
deviation
of
. (16)
Equation
isthe
the
standard
deviation
of of
is substituted
equation into

others
others
n this section we derivedIn
thethis
vertical
PLwe
(VPL
estimates
the impact
of one
section
derived
the
vertical
PL section
(VPL
),
estimates
thewhere
impact
of
one
others
others . Equation
He), which
Hean
In facility)
this
wewhich
derived
the vertical
PL
(VPL
i MDE,
He), which estimates the impacti of one
satellite
failure
(undetected
at
a
ground
on
airborne
user’s
position,
and
where
isinto
the standard
deviation
ofapplying
. Equation
(16) is substituted into eq


In In
this
section
we
derived
the
vertical
PL
(VPL
),
which
others
others
He
(16)
is
substituted
equation
(15)
and
the
K
this section we derived the vertical PL (VPL ), which estimates the impact of one
MD
i
vert
i
vert
i
A
i
AB
T
i
AB
T
i
others
2
i
A
L
L
2
i
vert
i
vert
i
TS
2
i
A
i
TS
i
A
i
AB
n
2
i
AB
T
L
n
T
L
i
vert
2
2
i
others
i 1
i 1
i
vert
2
He
KK
(15)
and
applying
the
with with
the
probability
of missed detection,
thedetection,
protection
MD associated
estimates
the
impact
one
satellite
failure
(undetected
ellite failure (undetected
at
a ground
facility)
on of
anat
airborne
user’s
position,
MDE,
and
applying
the
associated
with
theprobability
probability
ofmissed
missed
detection,the
theprot
pro
and
the
the
satellite
(undetected
a ground
facility)
on
anand
airborne
position,
and
MDE,
associated
with
the
probability
of position,
missed
detection,
the
MDassociated
MD
satellite
failure
(undetected
atuser’s
a(15)
ground
facility)
on
an Kairborne
user’s
and
MDE, of
based
onfailure
the proposed
ephemeris
fault
detection
algorithm.
MDE
is(15)
used
toapplying
compute
the
K MD associated with the probability of missed detection, the prot
(15) position,
and applying
the
at satellite
a ground
facility)
on an airborne
user’s
position,
failure
(undetected
at a ground
facility)
on anand
airborne protection
user’s
and
MDE,
levelasisequation
derived
level is derived
(17)as equation (17)
ed on the proposed ephemeris
detection
algorithm.
MDE
isonto
used
toalgorithm.
compute
the
based
onfault
the proposed
ephemeris
fault
detection
MDE
islevel
used
to
compute
the
based
the
proposed
ephemeris
fault
detection
MDE is used to compute the
isderived
derived
equation
(17)
MDE,
based
on as
the
proposed
ephemeris
fault
detection
level
ispseudorange
asasalgorithm.
equation
(17)
P-Value,
broadcast
integrity
parameters
the
user
together
with
the
is derived
based on MDE
the proposed
ephemeris
algorithm. MDE level
is used
to compute
the i(17)
 as equation

algorithm.
is used to
computefault
the Pdetection
Value, broadcast
n
2
2
  K FFA  K MD    TS 
i
i
i
i
Value, broadcast as integrity
parameters
to as
theintegrity
user together
with the
pseudorange
P-Value,
broadcast
parameters
to
the
user
together
with
the
pseudorange
VPL
S

P-Value,
broadcast
as
integrity
parameters
to
the
user
together
with
(17)
 Svert
  others
 (17)

He
vert
correction
(PRC).
To derive ato
VPL
an estimated
position
error
bei T
 K MDpseudorange
  can
i xi the
He,user
as integrity
parameters
the
togetheruser’s
withconservative
the
i
i 1
n
n

E


P-Value, broadcast as integrity parameters to the user together with
the
pseudorange


K

A
AB L  
22 i i
22
K
K

K






i
FFA
MD
TS
i
FFA
MD
TS
i i
i i 
i
n
 K
2
2
K MD   TSxxKKMDMD 
pseudorange
correction
(PRC).
To derive
a user’s
VPL
, can
an beVPL
VPLHeposition
Svertverterror

Sestimated
SSvert

(17)
(17)
Heconservative
FFA
i
i
rection (PRC). To derive
a VPLHe(PRC).
, an estimated
user’s
conservative
position
error
correction
derive
VPL
beconservative

He
others
He, an estimated
i can
correction
(PRC).
To
derive
a VPL
, aniHe
user’s
can
beothers
HeVPL
S vert
x  K MDerror
Svert

shown
in equation To
(13)
usinga the
Cauchy-Schwarz
inequality.
[12]
(17)

i i
i i T T T    position

vert   others 

1

i
1

i

E

E
i
i
estimated
user’s
conservative
can be shown

correction
(PRC).
To derive aposition
VPLHe, error
an estimated
user’s conservative position
be
  error

A A can
ABAB
1

i
E
L algorithm,
A  LAB
 ofthe
The
the
proposed
equation
L  
TheMDE
MDE of
algorithm,
equation in
(18),
is defined in the first term on
 proposed
 in shown
wn in equation (13) using
inequality.
[12] in equation
shown
inCauchy-Schwarz
equation
(13)
using
theCauchy-Schwarz
Cauchy-Schwarz
inequality.
[12]
inthe
equation
(13)
inequality.
shown
(13) using
the Cauchy-Schwarz
inequality.
[12] shown
i using the
n



R
(18), is defined in the first term on the right hand side of
shown
in
equation
(13)
using
the
Cauchy-Schwarz
inequality.
[12]
i
i
i
i
[12]  S
the(13)
right hand side of equation (17). According to the integrity requirement, the MDE of the
 x   Svertothers
X
vert
vert 
i i
equation (17). According to the integrity requirement,
n
i
  Ri 




R
i n1


n
A


fault
detection
algorithm
is
a confidential
value
corresponding
to
probability
of
int

R
TheMDE
MDE
the
proposed
algorithm,
shown
equation
(18),
defined
inthe
thefirst
first
The
ofofthe
algorithm,
shown
ininin
equation
(18),
in
te
i
i
i
i
i i 
i
The
MDE
ofproposed
the
proposed
shown
equation
(18),isis
isdefined
defined
in
the
first
te
ii
i (13)
i fault
i detection
algorithm
is algorithm,
a confidential
value
corresponding
to
probability
of
int
 i  x X
X vert
Svert
 Svert
 i   Rxi   Svert
S
(13)
the
MDE
of
the
fault
detection
algorithm
is
a confidential
n 



X
x
S


vert 
vertothers
others S vert
(13)
fault
detection
algorithm
is
a
confidential
value
corresponding
to
probability
of
int
fault
detection
algorithm
is
a
confidential
value
corresponding
to
probability
of
integrity

vert
vert
others
i
i 
i
i
12
i 
1 Xi
  A 
i

1


Svert A  i  x   Svertothers   A  (13) i the
1 value corresponding
(13)
i
vert  
to probability
integrity
The
theright
right
hand
side
equation
(17).According
According
tothe
therisk.
integrity
requirement,
theMDE
MDE
hand
side
of
equation
(17).
tois
integrity
requirement,
the
Pof
The
P-Value
(side
),
derived
using
theofMDE,
one
of the
integrity
parameters
broad
the right
hand
Ai of equation (17). According to the integrity requirement, the MDE o
A 
i 1

P
The
P-Value
(
i ), derived using the MDE, is one of the integrity parameters broad
i


A
i
P
,
derived
using
the
MDE,
is
one
of
the
integrity
P-Value
The
P-Value
(
),
derived
using
the
MDE,
is
one
of
the
integrity
parameters
broad
P
),
MDE,
is
one
of
the
integrity
parameters
broadcast
to
The
P-Value
(
A
where Svert is the vertical component of the projection matrix of the range error
source
A
parameters
broadcast
to
the
user.
The
P-Value
represents
the
user.
The
P-Value
represents
the
decorrelation
due
to
ephemeris
error
between
the
i
i
irange
is
the
verticalcomponent
component
of
the
user.
The
P-Value
represents
the
decorrelation
due
to ephemeris error between the
where Svert
is the verticalwhere
component
ofthe
thevertical
projection
matrix
ofofthe
error
source
is
the
projection
matrixcomponent
of theuser.
range
source
Svert
12
where
is projection
the
vertical
of error
the
projection
matrix
oferror
the
range
error
source
Svert
1212
i
The
P-Value
the
decorrelation
due
to ephemeris
error between
user.
The(ionosphere,
P-Value
represents
the
decorrelation
due
to the
ephemeris
the RSthe
and
decorrelation
due torepresents
ephemeris
between
RS
anderror between
i error and 
is nominal
range
measurement
error
onto
the
position
others component
where
is
the
vertical
of
the
projection
matrix
of
the
range
error
source
matrix
of theSrange
error
source
onto
the
position
error
and
vert
aircraft
[19].
theaircraft
aircraft[19].
[19].
i
i
i error
is
nominal
range
measurement
error
(ionosphere,
o the position error andonto
is nominal
error
(ionosphere,
others
is nominal
range
measurement
(ionosphere,
the
position
errorrange
and measurement

aircraft
[19].
aircraft
[19].
the
position
error
and 
others onto
others is nominal range measurement error (ionosphere,
troposphere,
multipath,
receiver
noise,
etc.)
i
nominal range measurement errori (ionosphere,
onto the position
errorreceiver
and noise,
troposphere,
multipath,
 K FFA  K MD TSiii
others isetc.)
MDE
i
(18)
i   K FFA  K MD
T   TS
posphere, multipath, receiver
noise,
etc.)
(18)
K
troposphere,
multipath,
receiver noise,
etc.)
 K FFA
K FFA
i MD
etc.)
MDE
(18)
i 
TS
troposphere,
multipath,
receiver
noise,
i
MD 
TS T
isisthe
satellite
position
error
due
to
the
undetected
E AB
 but
i real
MDE i MDE
 failure,



R
(18)
the
satellite
position
error
due
to
the
undetected
ephemeris
in
a
(18)
T
T
L
EiAB 
troposphere, multipath, receiver noise, etc.)

i
L
 EAB LinEABa real
L
i to the
i failure,
 Ri is the satellite position
duesatellite
tobut
the in
undetected
ephemeris
inposition
aephemeris
real error
 Ri error
isfailure,
the
position
error
due
undetected
MDEtoiibut
isisfailure,
hard
to but
figure
ephemeris
a real case,

R
the
satellite
the
undetected
ephemeris failure, but in a real (19)
P
i  due
i
A
MDE
i 
PAi i MDE
(19)
i ephemeris
 Ri Risi hard
MDE
is the
position
due
to the undetected
failure,
in a real
TS i in
case,
to satellite
figure out
using error
equation
(2). So, we
assume
that
worst
case
(19)
iAAi but
(19) (19)
out using TSi in equation (2). So, we assume that worst case PA  PA   i

A
A
i
i
i i
i
e,  R is hard to figure
intoequation
(2).
So,
we
assume
that
worst
case
using
Ri isTS
TS
case,
hard
figure
out
using
in has
equation
(2).
So,
weusing
assumeTSthat
case (2). So, we assume that worst case
of out
ephemeris
failure
detection
processor
occurred,
R
case,
is
hard
to
figure
out
inworst
equation
i
i
of the
ephemeris
failure
detection
processor
has
occurred,
the
same
as
algorithm
threshold.
Then,

R
TS
case,
is
hard
to
figure
out
using
in
equation
(2).
So,
we
assume
that
worst
case
same as algorithm threshold. Then, equation (4) is
AsAs
shown
(18),
the MDE
MDE of
ofthe
theproposed
proposed
shownininequation
equation
(18), the
algorithm depends on the
As shown in equation (18), the MDE of the proposed algorithm depends on the
As
shown
in
equation
(18),
the
MDE
of
the
proposed
algorithm
depends
theo
ephemeris failure detection
processorfailure
has occurred,
same
as
algorithm
threshold.
Then,
substituted
into
the
worstthe
case
of
equation
(12),
and
the
As
shown
in
equation
(18),
the
MDE
of
the
proposed
algorithm
on the on
norm
of
ephemeris
detection
processor
has
occurred,
the
same
as
algorithm
threshold.
Then,
algorithm
depends
on
the
norm
of
the
worst-case
sensitivity
ephemeris
detection
processor
has occurred,
the same as algorithm threshold. Then, depends
equation (4) is substituted into the of
worst
case of failure
equation
(12), and
the satellite
position error
sensitivity
vector,
which
fluctuates with
the geometry condition. Howe
of ephemeris
processor
has (14).
occurred, the samevector,
asworst-case
algorithm
threshold.
Then,
which
fluctuates
with
the
geometry
condition.
satellite
positionfailure
error isdetection
derived with
equation
worst-case sensitivity
sensitivity vector,
vector, which
which fluctuates
fluctuates with
with the
the geometry
geometry condition.
condition. Howev
Howe
worst-case
ation (4) is substitutedequation
into the worst
case of equation
(12),
andcase
the(4)satellite
position
error
worst-case
sensitivity
vector,
which
fluctuates
withposition
the
geometry
However, the
(4)
is substituted
into the
worst
ofisequation
(12),
and
the
satellite
position
erroralgorithms
equation
substituted
into
the
worst
case
ofprevious
equation
(12),
and
the
satellite
However,
the
use
a fixed
MDE
aserror
the condition.
i
is
derived
with
equation
(14).
i
i
previous
algorithms
useerror
a fixed MDE as the P-Value. Equation (19) is substituted i
FFA   TS 
equation
the satellite
position

(4)
  TS  into the worst case of equation (12), and
 Ksubstituted
TS is
FFA
,
(14)
P-Value.
Equation
(19) is
substituted
into as
equation
(17); then
previous
algorithms
use
a fixed MDE
the P-Value.
Equation (19) is substituted
(14)
 equationT(14). ,
 R i with
T
previous
algorithms
use
a
fixed as
MDE
as the P-Value.
Equation
is substituted
erived
with equation (14).
previous
algorithms
use
a
fixed
MDE
the P-Value.
Equation
(19) is(19)
substituted
into i
is derived
(14)
is
derived
with
equation
(14).
i

the
EPL
of
the
proposed
algorithm
is
derived
in
equation
B L
 Eequation
AB  L
equation (17); then the EPL of the proposed algorithm is derived in equation (21).
is derived with
(14).
equation
(17); then
then the
the EPL
EPL of
of the
the proposed
proposed algorithm
algorithm is
is derived
derived in
in equation
equation (21).
(21).

   K  
(21).
[17] then
equation
(17);
equation
(17);
the EPL
of the proposed
algorithm is derived
in equation
(21). [17]
,
(14)
R 
E 
n
2
2
When affection of nominal condition 11
of test statistics
i
i
i
n
2

VPLiHe
Svert
PAii x  K
(20)
n  S vert
i
i
i  2  i

n of nominal condition of test When
statistics
is
considered,
the
equation
(13)
is
n MD

2
2  Si i 2 2 others
affection
of nominal
condition
of test
statistics
is considered,
(20)

VPLi Hethe iequation
Sivert PAi x(13)
 Kis
(20)
i

i
i

is considered,
the
equation
(13)
is
derived
as
equation
MD
vert
others
i 1 S

VPL
S
P
x

K


VPL
S
P
x

K
S

11
(20) (20)




11
 verti 1  vertothers  others
He
vert
A MD
MD

vert
A 11
When affection of nominal condition of test statistics is considered,
the equation (13) is He
1

i
i 1
(15).
assume
that the to
testGaussian
statistics correspond
to
on (15). We assume that the
testWe
statistics
correspond
11 statistics
derived
as equation
(15). We aassume
that the test
correspond to a Gaussian i
as equation (15). We assume that the test statistics correspond to a Gaussian
a derived
Gaussian
distribution with zero mean and standard
VPLA,eph  max VPLiA,eph 
(21)
VPLA,eph  max
(21)
(21)
i VPLi A , eph 
i
VPL

max
VPL
(21)
VPL

max
VPL
(21)
i


distribution 
with
zero
mean
and
standard
deviation
.

deviation
.
zero mean and standard deviation


A
,
eph
A
,
eph
A , eph
A , eph
TS with zero mean and standard deviation 
distribution
TS .
 

i
TS
FFA
T
i
AB L
i
TS
K FFA  
i
A
i
TS
i
    
i
TS
T
E 
i
AB
L


 x  Si 
vert

 i

 A


i
i
i
 X ivert
 S vert
i  Ki FFA in TST  x  Svert
TS

E


i
i
A
AB
 x i   KLSFFA
  TSi
 
i

TS i

  i E i T
 AB L
  A

n
 x  S i  i
vert
others


i 1
TS i
i 
The derived VPLHe applies the geometry condition and
The derived VPLHe applies the geometry condition and is expected to ensure int
The derived
VPLHeintegrity
applies the
geometry
condition
and is expected to ensure int
 (15) (15)
is derived
expected
to ensure
and
increase
availability.
n The
The
derived
the
geometry
condition
is expected
to ensure
int
VPL
applies
the geometry
condition
and is and
expected
to ensure
integrity
He applies
HeVPL
i
i

(15)

vert
others x  S

S

X

S
x
T
(15)
Further
and validation
are
described
in the following

i vert 
vert
vert
vert
othersdetails
increase
availability.
Further
details
and
validation
are
described
in
the
following
s
T
T
i 1 i E i

  i Ei

E AB  
The
increase availability. Further details and validation are described in the following s
i 1


L 
Athird
AB terms
A hand
AB  L side
increase
availability.
Further
details
and
validation
are
described
in
the
following
s
sections.
increase
availability.
Further
details
and
validation
are
described
in
the
following
section
second
and
on
the
right
of




L 



The second and third terms on the right hand side of equation (15) are developed into
equation (16).
3. Experiment
and
Results
d third terms on the right handThe
sidesecond
of equation
(15)terms
are developed
intohand side of equation (15)
and
third
on the right
are developed
into
3. Experiment
and
Results


http://ijass.org
93

3.
Experiment
and
Results
3.
Experiment
and
Results


 x     S   
S  

In this section, we evaluate the proposed algorithm using real GPS data for mult
 E  
  (16).
equation
In this section, we evaluate the proposed algorithm using real GPS data for mul
(16)
this section,
we evaluate
the proposed
algorithm
using
realdata
GPSfor
data
for mult
In this In
section,
we evaluate
the proposed
algorithm
using real
GPS
multiple





reference
stations.
First,
sensitivity
equation
(Section
2.1),
is
evaluated
with
respec
2
 S
x    S2   

reference stations. First, sensitivity equation (Section 2.1), is evaluated with respe
 E




n


reference
stations.
First,
sensitivity
equation
(Section
2.1),
is
evaluated
with
respec
reference
stations.
First,
sensitivity
equation
(Section
2.1),
is
evaluated
with
respect
to


i


2
2
n
  x 2   S i   i i 2 
 TS
  x  2   S i  2  i  2

geometry
conditions
(local
elevation
angle,
local
azimuth,
baselines
length),
and
th
vert
others
S
T



vert
vert
others

geometry conditions (local elevation angle, local azimuth,
baselines
length),
and th
i 1
  i Ei T 
(89~101)14-083.indd
93
2015-03-30
오후length),
3:48:29
i 1
AB  L 
geometry
conditions
(local elevation
local azimuth,
baselines
andwe
th
geometry
conditions
(local elevation
angle, angle,
local azimuth,
baselines
length),
and then
AB  deviation of

where
isAthestandard
. Equation (16) is substituted into equation
2
i
vert
i
vert
i
TS
2
i
A
2
i
AB
i
TS
i
A
i
AB
i
T
L
n
T
L
n
i
vert
2
2
i
others
i
others
i 1
i 1
i
vert
2
i
2
Int’l J. of Aeronautical & Space Sci. 16(1), 89–101 (2015)
3. Experiment and Results
The distinct property is that the effect of the local elevation
angle is different for the two algorithms. Both algorithms have
a similar tendency at the local azimuth angle. However, the
In this section, we evaluate the proposed algorithm using
sensitivity of the proposed algorithm increases as elevation
real GPS data for multiple reference stations. First, sensitivity
angle decreases, in contrast with the comparison algorithm’s
equation (Section 2.1), is evaluated with respect to geometry
tendency. As described, the purpose of these algorithms
conditions (local elevation
angle, Evaluation
local azimuth,
baselines
3.1 Sensitivity
Results
is ephemeris failure detection in the absence of a verified
length), and then we compute the threshold based on test
ephemeris.
Thisrespect
case isto
frequent
for the
first rising
evaluate
the sensitivity
the proposed algorithm
with
ephemeris
failure,
we satellite of
statistics. Finally, we To
check
the applicability
of of
algorithm
the
day.
Thus,
the
proposed
algorithm
is
better
because of its
using MDE and EPL in case of landing aircraft at Gimpo
compare the norm of the worst-case sensitivityhigh
vector
with
that
of
a
similar
algorithm
based
sensitivity for a rising satellite with a low elevation angle.
International Airport.
From a baseline length perspective, the proposed algorithm
on short baseline vectors and range measurements.
Using real GPS data, we present a
has high sensitivity with respect to large induced angle
3.1 Sensitivity Evaluation Results
resulting from baseline length. The comparison algorithm
simulation for verification of sensitivity properties.
To evaluate the sensitivity of the proposed algorithm with
has a similar tendency with baseline length; however, it has
respect to ephemeris failure,
we compare
thefor
norm
of the usesaalimit
baseline
of the basic assumption
The relevant
algorithm
comparison
shorton
baseline
andlength
carrierbecause
phase measurement.
worst-case sensitivity vector with that of a similar algorithm
that LOS vectors of two reference station about the target
The reason
its range
selection
is that its information
(range
baseline,
broadcast
based on short baseline
vectorsforand
measurements.
satellite
aremeasurement,
quasi-parallel.[8]
This means
that the proposed
Using real GPS data, we present a simulation for verification
algorithm does not have to consider the baseline length
ephemeris, etc.) for generation of test statistics is analogous to the proposed algorithm and its
of sensitivity properties.
limitation and is applicable to various GNSS augmentation
The relevant algorithm
comparison
uses aonshort
systems:
not only
the local
system
(GBAS)
detectionfor
properties
also depend
the geometry
condition
between
the area
baseline
vector
andbut also widebaseline and carrier phase measurement. The reason for
area systems (e.g., GRAS, SBAS). The various long baseline
its selection is thatthe
itssatellites.
information (range measurement,
vectors, which can be implemented with the local azimuth
baseline, broadcast ephemeris, etc.) for generation of test
and local elevation angle, are required to improve detection
Figure 4 shows the norm of the sensitivity vector of the proposed algorithm and the
statistics is analogous to the proposed algorithm and its
performance of the proposed algorithm. The geometry of
detection propertiescomparison
also dependalgorithm.
on the geometry
condition
various azimuth
angles canisbe
implemented
It can be see that the sensitivity
of both algorithms
affected
by the by multiple
between the baseline vector and the satellites.
RSs that are not on the same line. However, this is difficult
geometry
condition.
Figure 4 shows the
norm of
the sensitivity vector of the
with baseline vector geometry for various local elevations,
proposed algorithm and the comparison algorithm. It can be
especially when based on long baseline length. Fortunately,
see that the sensitivity of both algorithms is affected by the
satellites at low elevation angles are more frequent than
geometry condition.
those with high elevation angles (close to 90°) over a 24-h
Worst Case Sensitivity w.r.t Geometry (Proposed Algorithm)
-8
-3
x 10
Magnitude of Sensicivity Vector (Dimensionless)
EA =0 Deg
5
4
3
EA = 30 Deg
2
EA = 60 Deg
1
0
0
EA = 90 Deg
0.05
0.1
0.15
0.2
Induced Angle (Deg)
0.25
0.3
0
30
60
90
120
150
Local Azimuth Angle (Deg)
180
Magnitude of Sensitivity Vector (Dimensionless)
4.5
6
Worst Case Sensitivity w.r.t Geometry
x 10
4
3.5
EA
EA
EA
EA
3
2.5
=
=
=
=
0 degee
30 degee
60 degee
90 degee
2
1.5
1
0.5
0
0
20
40
60
80
100
120
Local Azimuth(Degree)
140
160
180
Fig. 4. Magnitude of the sensitivity vector with respect to the geometry condition
Figure 4. Magnitude of the sensitivity vector with respect to the geometry condition
(The proposed algorithm (left) and the comparison algorithm (right)).
(The proposed algorithm (left) and the comparison algorithm (right)).
2. Tendency
ofcorresponding
increasing sensitivity
corresponding
to geometry conditions.
Table 2. Tendency Table
of increasing
sensitivity
to geometry
conditions.
Geometry conditions
Local elevation angle
Local azimuth angle
Baseline length
Comparison algorithm
14
close to 90°
close to 90°
Increase (limited)
Proposed algorithm
close to 0°
close to 90°
Increase
The distinct property is that the effect of the local elevation angle is different for the two
94
algorithms. Both algorithms have a similar tendency at the local azimuth angle. However, the
DOI: http://dx.doi.org/10.5139/IJASS.2015.16.1.89
sensitivity of the proposed algorithm increases as elevation angle decreases, in contrast with
the comparison algorithm’s tendency. As described, the purpose of these algorithms is
(89~101)14-083.indd 94
2015-03-30 오후 3:48:32
Jongsun Ahn Orbit Ephemeris Failure Detection in a GNSS Regional Application
angle, can be an important factor for detection performance,
period. However, this is left to be considered in further work.
as shown in both Figs. 5 and 6.
Next, we tried to verify the sensitivity equation in equation
(12) for the proposed
algorithm.
As
shown
in
Table
3,
the
Next, we tried to verify the sensitivity equation in equation (12) for the proposed algorithm.
various baseline vectors are composed with Suwon (SUWN),
3.2 Algorithm Realization and Availability Test using
Tablefrom
3, theSeoul
various
baseline
composed
Nonsan (NONS), As
andshown
Jeju in
(JEJU)
(SOUL)
in vectors areGPS
Data with Suwon (SUWN),
Korea, and we use error-free range measurement to examine
Nonsan (NONS), and Jeju (JEJU) from Seoul (SOUL)
in Korea, and we use error-free range
In this section, we realized and evaluated the proposed
the influence of the geometry condition. A satellite position
algorithm
using real
GPS data
of multiple
fault (1 km, XYZ) ismeasurement
imposed on all visible satellites
for 24 h.
thesensitivity
influence
of the geometry
condition.
A satellite
position
fault RSs. The main
Next, we triedtotoexamine
verify the
equation
in equation
(12) for
the proposed
algorithm.
items were evaluation of the correction result of the range
Figure 5 shows the mean and standard deviation of test
error.Suwon
Then, (SUWN),
we determined the threshold,
statistics with local
elevation
angle.
As various
expected,
the test
km, XYZ)
is imposed
on all baseline
visible
satellites
for 24
h.
As(1shown
in Table
3, the
vectors measurement
are
composed
with
MDE, and conducted availability testing based on the EPL
statistics show a decreasing tendency with respect to high
at Gimpo
International
multiple baseline
in Korea,
and we use Airport
error-freewith
range
elevation angle. Nonsan (NONS), and Jeju (JEJU) from Seoul (SOUL)
conditions in Korea.
Figure 6 shows the mean and standard deviation of test
Table 3. Baseline
lengths
the simulated
measurement
examine
theofinfluence
thecondition.
geometry
condition.
A satellite
As shown
in Fig.
7, the position
multiplefault
baseline conditions (6
statistics with local
azimuth to
angle.
As expected,
the oftest
baselines) were deployed using four reference stations
statistics show an From
increasing
as close to 90° local
Seoultendency
(1 km, XYZ)
is imposedtoonSuwon
all visible
satellites for
24 h. (NONS)
(SUWN)
tooperated
Nonsan
to Jeju
(JEJU)Information Institute
by the National
Geographic
azimuth angle in similar
elevation angle.
(SOUL)
(NGII) in Korea.
Finally, long baseline length, which causes larger induced
Baseline length
39.37km
160.18km
459.26km
Table of3.the
Baseline
lengths
of the simulated condition.
Table 3. Baseline lengths
simulated
condition.
From
Seoul5 shows the mean and standard deviation of test statistics with local elevation angle.
Figure
to Suwon (SUWN)
to Nonsan (NONS)
to Jeju (JEJU)
(SOUL)
As expected, the test statistics show a decreasing tendency with respect to high elevation
Baseline length
39.37km
160.18km
459.26km
angle.
Mean of TS w.r.t Elevation Angle
Standard Deviation of TS w.r.t Elevation Angle
14
Figure
5 shows the mean and
standard deviation 15of test statistics with local
elevation angle.
SOUL-SUWN (39.37km)
SOUL-SUWN (39.37km)
SOUL-NONS (160.18km)
SOUL-JEJU (459.26km)
12
SOUL-NONS (160.18km)
SOUL-JEJU (459.26km)
As expected, the test statistics show a decreasing tendency with respect to high elevation
10
10
8
6
14
TS STD (m)
TS Mean (m)
angle.
Mean of TS w.r.t Elevation Angle
4
15
12
2
SOUL-SUWN (39.37km)
SOUL-NONS (160.18km)
SOUL-JEJU (459.26km)
10
10
20
8
30
40
50
60
Elevation Angle (Degree)
70
80
10
90
0
0
10
20
TS STD (m)
0
0
TS Mean (m)
Standard Deviation of TS w.r.t Elevation Angle
5
SOUL-SUWN (39.37km)
SOUL-NONS (160.18km)
SOUL-JEJU (459.26km)
30
40
50
60
Elevation Angle (Degree)
70
80
90
Figure 5. Variation of test statistics in the ephemeris failure condition with respect to local
6
Fig. 5. Variation of test statistics
in the ephemeris failure condition with respect to local elevation angle and baseline length: mean (left) and stanelevation angle and baseline length: mean (left) 5and standard deviation (right).
dard deviation (right).
4
2
30
30 of test statistics with local azimuth angle.
Figure
6 shows the mean and standard deviation
0
0
Test Statistics w.r.t Local Azimuth Angle (Elevation Angle 30~40 Deg)
0
10
20
Test Statistics w.r.t Local Azimuth Angle (Elevation Angle 70~80 Deg)
30
40
50
60
70
80
90
Elevation Angle (Degree)SOUL-JEJU (459.26km)
0
10
20
30
40
50
60
Elevation Angle (Degree)
70
80
90
As expected,
the test
statistics
show
an increasing
as closewith
to 90°
local azimuth
Figure
5. Variation
of test
statistics
in the
ephemeris tendency
failure condition
respect
SOUL-JEJU
(459.26 km) to local
20
20
elevation angle and baseline length: mean (left) and standard deviation (right).
angle in similar elevation angle.
15
SOUL-NONS (160.18km)
25
Test Statistics (m)
Test Statistics (m)
25
15
10
10 test statistics with local azimuth angle.
Figure
6 shows the mean and standard deviation of
SOUL-NONS (160.18km)
16tendency as close
SOUL-SUWN (39.37km)
SOUL-SUWN
As expected, the test
statistics show an increasing
to(39.37km)
90° local azimuth
5
0
5
0
20
40
60
80
100
120
Azimuth Angle (Degree)
angle in similar elevation
angle.
140
160
180
0
0
20
40
60
80
100
120
Azimuth Angle (Degree)
140
160
180
Fig. 6. Variation of test statistics in the ephemeris failure condition with respect to local azimuth angle and baseline length: in elevation angle 30Figure 6. Variation of test statistics in the ephemeris failure condition with respect to local
40° (left) and in elevation angle 70-80° (right).
azimuth angle and baseline length: in elevation angle 30-40° (left) and in elevation angle
16
70-80° (right).
http://ijass.org
95
Finally, long baseline length, which causes larger induced angle, can be an important factor
for detection performance, as shown in both Figs. 5 and 6.
(89~101)14-083.indd 95
2015-03-30 오후 3:48:35
are shown in Table 4.
Int’l J. of Aeronautical & Space Sci. 16(1), 89–101 (2015)
Prior to generating test statistics, we corrected the range
measurement error sources, which include the ionosphere,
troposphere, and satellite/receiver clock bias, as described
in Section 1. Fig. 8 shows the resulting range correction error
at four reference stations of all visible satellites based on
24-h data. It canare
be shown
seen that
the ramifications
of range error
in Table
4.
sources are reduced in all visible satellites. Statistical results
(mean, standard deviation, and histogram) are shown in
Table 4.
To ensure reliability, many data samples are required
are shown in Table 4.
for analysis in terms of statistics for determination of the
threshold. In the present work, the mean and standard
deviation of test statistics are computed using 1-week
data received from 2014.9.1 to 2014.9.7 (sampling period
30 s). The plot on the left hand side of Fig. 9 shows the test
statistics mean (red lines with diamond markers) and
standard deviation (blue lines with circular markers) with
local elevation angle. A standard deviation model of test
statistics is derived in equation (22). Shown in the plot on the
right hand side of Fig. 9, the threshold with respect to local
elevation angle is derived in equation (23), which uses the
multiplier
(KFFA) associated
with for
thethe
false
alarm rate of the
Figure 7. Baseline
construction
simulation.
CAT-I requirement.[12]
To ensure reliability, many data samples are required for analysis in terms of statistics for
determination of the threshold. In the present work, the mean and standard deviation of test
statistics are computed using 1-week data received from 2014.9.1 to 2014.9.7 (sampling
period 30 s). The plot on the left hand side of Fig. 9 shows the test statistics mean (red lines
with diamond markers) and standard deviation (blue lines with circular markers) with local
elevation
angle. Aconstruction
standard deviation
of test statistics is derived in equation (22).
Figure
7. Baseline
for the model
simulation.
Shown in the plot on the right hand side of Fig. 9, the threshold with respect to local elevation
angle is derived in equation (23), which uses the multiplier (KFFA) associated with the false
alarm rate
of the
CAT-I requirement.[12]
Fig. 7. Baseline construction
for the
simulation.
Fig. 8. Range measurement mitigation results (24 h).
Figure 7. Baseline construction for the simulation. Figure 8. Range measurement mitigation results (24 h).
Test Statistics w.r.t Elevation Angle
4.5
4
Table
4. Range mitigation error (mean and standard deviation).
8
3.5
3
Statistics
Mean (m)
Standard deviati
Before mitigation
10.24
4.78
After mitigation
-0.56
2.97
7
2.5
Threshold (m)
STD (m), Mean (m)
Threshold w.r.t Elevation Angle
9
Exponential Curve Fitting of STD
Mean
2
1.5
6
5
1
0.5
4
0
-0.5
0
10
20
30
40
50
Elev (Deg)
60
70
80
90
3
0
10
20
30
40
50
Elev (Deg)
60
70
80
90
Figure 8. Range measurement mitigation results (24 h).
Figure 9. Daily variation of test statistics and threshold with respect to local elevation angle.
Fig. 9. Daily variation of test statistics and threshold with respect to local elevation angle.
18
Table 4.error
Range
mitigation
error (mean and standard deviation).
Table 4. Range mitigation
(mean
and standard
i
0.01149deviation).

 TS
  2.431 e
(22)
i
Statistics
 Thresh

  K FFA   TSi   Mean (m)
(23)Standard deviation (m)
Before
 : mitigation
Elevation angle (degrees)10.24
4.78
After mitigation
2.97
Figure 8. Range measurement mitigation results (24 h).
-0.56
Table 4. Range mitigation error (mean and standard deviation).
5.Table
False
alarm
and
according
to the
CAT-I
requirement
False probability
alarm
probability
and multiplier
according
to the
CAT-I
requirement
Table 5. False alarmTable
probability
and5.multiplier
according
to
themultiplier
CAT-I
requirement
Statistics
Requirement
Requirement
CAT-I
BeforeCAT-I
mitigation
Mean
(m)Alarm
False Alarm
False
Rate Rate
18
-4
4
10.241.9  10 1.9 x 10
Standard
deviation
(m)
)
Multiplier
Multiplier
(KFFA)(KFFA
3.74
4.78
3.74
Table 6. Missed detection
probability and multiplier according
to the CAT-I requirement
After mitigation
-0.56
2.97
DOI: http://dx.doi.org/10.5139/IJASS.2015.16.1.89
Requirement
CAT-I
(89~101)14-083.indd 96
Probability of 96
missed detection
Multiplier (Kmd)
1.0 x 10-3
18 19
3.1
2015-03-30 오후 3:48:35
1
1.5
0.5
1
0
0.5
-0.5
Threshold
2
5
-0.5
6
5
4
4
0
0
6
Threshold (m)
STD (m), M
STD (m), Mean (
2.5
2
1.5
10
0
20
10
30
20
30
40
50
60
Elev (Deg)
40
50
Elev (Deg)
70
60
80
70
3
90
80
0
90
3
10
0
20
10
30
20
30
40
50
60
Elev (Deg)
40
50
Elev (Deg)
70
60
80
70
90
80
90
Jongsun
Ahnrespect
Orbit Ephemeris
Failure Detection
in a GNSS Regional Application
Figure 9. Daily variation of test statistics and threshold
with
to local elevation
angle.
Figure 9. Daily variation of test statistics and threshold with respect to local elevation angle.

 TSi 
i  2.431 e0.01149
0.01149



2.431

e


i
 Thresh

TS  K FFA   TSi i 
i
 Thresh 
 TS  
  K FFA (degrees)
 : Elevation angle
 : Elevation angle (degrees)
TS and Threshold (m)
(22) (22) MDE corresponding to six baselines. It is clear that the MDE
(22)
of the proposed algorithm fluctuates with respect to time
(23)
(23) (23)because of the variable worst-case sensitivity vector related
to geometry of the baseline and the satellite.
Consequently, for detection performance the appropriate
θ : Elevation angle (degrees)
baseline is selected based on the least MDE value monitored
Figure 10 shows an example of the test statistics and
among
the various RS geometries in real time. The right
threshold
on alarm
PRN 11
of the test
In according
the nominal
Table
5. False
probability
andperiod.
multiplier
to the CAT-I
requirement
Table 5. False
alarm
probability
multiplier
according
CAT-Iside
requirement
of Fig. 11 shows the least MDE value of all visible
condition,
we can
see that
the testand
statistic
does not
exceedto the hand
satellites(K
and
conditions for a 24-h period.
Requirement
Alarm Rate
Multiplier
)
the threshold, but does trigger aFalse
false alarm.
FFAbaseline
Requirement
False
Multiplier
(K
)
4 Alarm Rate
FFA
CAT-I
3.74 However, the figure showed unexpected MDE
1.9  10
3.74
CAT-I
1.9  10 4
divergence,
to be investigated further. The given satellites
3.3
MDE and Availability Testing
(PRN 1, PRN 11, PRN 26, and PRN 27) are close to highThe MDE of all visible satellites (24 h) is computed with
elevation-angle status. A satellite at a high elevation angle
equation (3-4). The missed detection probability and
degrades the sensitivity of the algorithm and leads to
multiplier associated with the CAT-I requirement, as shown
an increase in the MDE value. Once again, this satellite
19
in Table 6, are is used to compute the MDE [18].
condition is not frequent for a 1 day (24-h) period, and this
19
The left hand side of Fig. 11 shows an example of the PRN 5
algorithm focuses on rising satellites with low elevation
angles. However, to ensure system integrity, the GNSS
Test Statistics and Threshold (PRN 11, Normal Condition)
augmentation system, which implements this algorithm,
8
TS
should also use additional ephemeris fault detection
6
Threshold
algorithms.
4
Next, we present the availability result based on the
ephemeris protection level that uses the MDE value. The
2
basic simulation condition considers an aircraft trying
0
to land at Gimpo International Airport on runway 32R
-2
(RWY 32R), located at a CAT-I decision height (DH) of
200 ft. The GPS constellation is based on 31 satellites.
-4
The reason for availability analysis at the DH is that the
-6
aircraft
determines
the requirement
use
of baseline
the ground
because
of5.the
variable
vector
related
geometry
of the
and landing-aided
Table
False
alarmworst-case
probability sensitivity
and multiplier
according
totothe
CAT-I
-8
facility at the DH (final location) associated with system
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
Local Time (Hours)
the satellite.
The analysis
Requirement
False Alarmrequirements.
Rate
Multiplierperiod
(KFFA) is 24 h, considering
-4
constellation (approximately
3.74
Fig. 10. Test statistics of normal CAT-I
condition with threshold (PRN 11) 1.9 x 10the repeatability of the GPS
Figure 10. Test statistics
of normal condition
with threshold
(PRN
Consequently,
for detection
performance
the11)
appropriate baseline is selected based on the
6. Missed
detectionaccording
probability
andCAT-I
multiplier
according to the CAT-I requirement
Table 6. Missed detectionTable
probability
and multiplier
to the
requirement
least MDE value monitored among the various RS geometries in real time. The right hand
Figure 10 shows an example
of the test statistics
and threshold on PRN 11 of the test Multiplier (K )
Requirement
Probability of missed detection
md
side of Fig. 11 shows the least MDE value of all visible satellites and baseline conditions for
-3
period. In the nominal condition,
does not exceed the
3.1
CAT-Iwe can see that the test statistic
1.0 x 10
a 24-h period.
threshold, but does trigger a false alarm.
2000
Minimum Detectable Error (Single Baselines, PRN 5)
4500
1600
4000
3.3 MDE and Availability
Testing
1400
INCH-JEJU
Minimum Detectable Error (Multiple Baselines, 24h)
5000
1800
PRN 11
PRN 1
PUSN-JEJU
3500
3000
1200
MDE (m)
MDE (m)
INCH-KANR
The MDE of all visible
satellites
(24 h)KANR-PUSN
is computed with equation (3-4). The missed
1000
KANR-JEJU
800
PRN 27
2500
PRN 28
2000
detection probability and600multiplier associated with the CAT-I requirement, as shown in
1500
1000
400
Table 6, are is used to compute
the MDE [18].
200
0
10
500
INCH-PUSN
11
12
13
Local Time (hours)
14
15
0
0
4
8
12
Local Time (hours)
16
20
24
Figure 11. MDE according to baseline vectors (PRN 5) and minimum MDE result (24 h).
Fig. 11. MDE according to baseline vectors (PRN 5) and minimum MDE result (24 h).
Table
6. Missed detection probability and multiplier according to the CAT-I requirement
Requirement
CAT-I
of missed
However,Probability
the figure showed
unexpected MDE divergence, to be investigated further. The
Multiplier (Kmd)
detection
http://ijass.org
97
given satellites (PRN
3 1, PRN 11, PRN 26, and PRN 27) are close to high-elevation-angle
3.1
1.0  10
status. A satellite at a high elevation angle degrades the sensitivity of the algorithm and leads
increase
in the
value.
Once
again,
this corresponding
satellite condition
is not frequent for a
The left hand sidetoofanFig.
11 shows
anMDE
example
of the
PRN
5 MDE
to six
(89~101)14-083.indd 97
2015-03-30 오후 3:48:36
The estimated aircraft position error due to the P-value, which represents ephemer
The estimated
estimated
aircraft
position
error
to P-value,
the P-value,
represents
ephemeri
the DH is that the aircraft determines the use of the ground landing-aided
facility
at theposition
DH
aircraft
error
duedue
to the
whichwhich
represents
ephemeris
failur
The estimated
aircraftisposition
error
the P-value,
which
ephemer
detection
performance,
shown on
thedue
righttoside
of Fig. 12.
The represents
result shown
in Fig
detection
is is
shown
on on
thethe
right
sideside
of Fig.
12. The
shownshown
in Fig. in
12 Fig.
detection
performance,
shown
right
of Fig.
12.result
The result
(final location) associated with system requirements. The analysis
periodperformance,
is 24 h, considering
detection
shown on to
theP-Value
right side
of Fig.
The result shown
in Figt
shows thatperformance,
position errorisaccording
is less
than12.
centimeter-scale
because
shows
that
position
error
according
to P-Value
is less
than than
centimeter-scale
becausebecause
the effect
shows
that
position
error
according
to
P-Value
is
less
centimeter-scale
the
repeatability
of
the
GPS
constellation
(approximately
11
h
58
s)
and
the
rotation
of
the
Int’l J. of Aeronautical & Space Sci. 16(1), 89–101 (2015)
shows
that position
accordingistosmall
P-Value
is lessthethan
centimeter-scale
because
of ephemeris
failureerror
decorrelation
between
aircraft
and the RS in
the cast
of ephemeris failure decorrelation is small between the aircraft and the RS in the case of
ephemeris
failure decorrelation is small between the aircraft and the RS in the case
earth. This property is advantageous for shortening the analysisofperiod.
The availability
of
failure(752.5
decorrelation
smallif between
the aircraft
and thethan
RS in
in this
the cas
small
displacement
However,
the displacement
sim
in ephemeris
this
simulation
condition,
theisproposed
algorithm
is is larger
11 h 58 s) and the rotation of the earth. This property is
small
displacement
(752.5
m).m).
However,
if the displacement
is larger
than in this simulation
small
displacement
(752.5
m).
However,
if
the
displacement
is
larger
than
in
this
sim
analysis
with
EPL
is
conducted
in
comparison
with
the
same
algorithm
used
in
the
sensitivity
more useful than the comparison algorithm because of
advantageous for shortening the analysis period. The
small
displacement
(752.5
m). However,
ifuseful
the displacement
is larger
than
in thisbeca
sim
condition,
theproposed
proposed
algorithm
is more
thancomparison
the comparison
algorithm
condition,
the
is more
useful
algorithm
because
of
the increasing
of thealgorithm
decorrelation
effect,than
as the
shown in
availability analysis with EPL is conducted in comparison
condition,
theisproposed
analysis in Section 3.1 [9]. The P-value of the comparison algorithm,
which
used to algorithm is more useful than the comparison algorithm beca
equation
(24).
with the same algorithm used in the sensitivity analysis in
condition,
theofproposed
algorithm
is
more
the (24).
comparison
increasing
effect,
as shown
in than
equation
the
increasing
ofthe
thedecorrelation
decorrelation
effect,
asuseful
shown
in equation
(24). algorithm beca
The
nominal
error
models
due
to
airborne
antenna
Section
3.1
[9].
The
P-value
of
the
comparison
algorithm,
the
increasing
of
the
decorrelation
effect,
as
shown
in
equation
(24).
compute the EPL, is shown in equation (24) [12].
The
nominal
models
due
to airborne
antenna
multipath/noise
(equation
25), 25),
the
increasing
oferror
the
decorrelation
as shown
inmultipath/noise
equation
The
nominalerror
models
due
toeffect,
airborne
antenna
(equation
multipath/noise
(equation
25),
ionosphere
(equation
26), (24).
which is used to compute the EPL, is shown in equation
The
nominal
error
models
due
to
airborne
antenna
multipath/noise
(equation
25),
troposphere
(equation
27),
and RS(equation
receivers27),
(equation
28)
(24) [12].
ionosphere
(equation
troposphere
and RS
receivers
(equation
28) are25),
The nominal
error26),
models
due to airborne
antenna
multipath/noise
(equation
ionosphere
(equation
26),
troposphere
(equation
27),
and
RS
receivers
(equation
28)
are applied to compute the EPL together with ephemeris
ionosphere (equation 26), troposphere (equation 27), and RS receivers (equation 28)
P   K FFA  K MD     / b
(24) (24)
applied
to compute
the EPL
together with ephemeris failure-based position error.
failure-based
position
error.
ionosphere
(equation
troposphere
(equation
27), failure-based
and RS receivers
(equation
applied to compute
the26),
EPL
together with
ephemeris
position
error.28)
  : Standard deviation of the test statistics (double differential
carrier:
0.3
cm)
applied
to
compute
the
EPL
together
with
ephemeris
failure-based
position
error.


σϕ : Standard deviation of the test statistics (double

i
i
10 i together
applied
to compute
the
ephemeris
position
error.
 multipath

0.13  0.53
e EPL
(25)

a0,with
failure-based
 i (25)
 , 
b : Baseline length (200 m)
AAD  a1, AAD e
 noise i
c , AAD
i
10
differential carrier: 0.3 cm)
i
 imultipath
i 0.13  0.53e i  , 
(25)
a

a
e


noise
0, AAD
1, AAD

 iono

(
X

2



v
)
(26)
i  Fpp   vert _ iono _ gradient 10
i
air  a
i c , AAD
 multipath
, air
(25)

a0, AAD
b : Baseline length (200 m)
i  i 0.13  0.53e  i
noise
1, AAD e  
i  F 
 iono
(26)
, AAD
c(26)
2gradient
10  ( Xi air  2    vair )
vert
_0.53
iono
i _e


0.13

,
(25)


a

a
e
 ipp

imultipath
noise
0, AAD
1, AAD
 R cos( ) 
i 
Fiono
(26)
  noise
where
is thelevel
noiseoflevel
of the double
difference
 2   ( X air  2   vair )
vert _ iono _
 1Fppi ecorresponding
pp
where
σϕ is the
the double
difference
carriercarrier measurement,
i gradient
i
cos(
 _)gradient
 h_Iiono
 iono
 F RR vert
(26)
2  ( X air  2   vair )
i 
Fppi  1 pp R e cos(
measurement, corresponding to an integrated multipath

) 62
i
e
R
h


h
i  
e b I10
1value
Fi 
  Rofcos(

h 
to an integrated
limiting
usedoffor RS [8].
The
 
limiting
antenna multipath
(IMLA) used
for antenna
RS [8]. (IMLA)
The value
i
Fpp
e  h )
(27)
tropo
 N h0 R
e
I
 62 i 1  e  h
pp 1 
sin ( )

 hI 10
h0 
i
b represents the baseline length between multiple RS
  h Re0.002

6
1

e


(27)
h


0
tropo
N the
represents the baseline length between multiple RS antennas and is similar
to
RS of10
2
i
antennas and is similar to the RS of Gimpo International
sin
( i ) 1  e hh0 
6
 tropo
 N h0  0.00210

(27)
i
 tropo
 N h0  0.002  sin 2 ( i ) 1  e h0 

(27)
(27)
Airport.
Gimpo International Airport.

0.002  sin 2 ( i ) 
i
i

c , AAD
0
The P-value variation over 24 h is shown in Fig. 12. It can
23
The that
P-value
h is shown
in Fig. 12.
It can be seen that the P-value of the
be seen
the variation
P-value ofover
the24
proposed
algorithm
varies
2
 i  0 ,GAD
with time due to the geometry condition, whereas that of the
a
a
e
(28)
23 2
i
proposed algorithm varies with time due to the
geometry condition,
whereas
that0,GAD
of the 1,GAD
(28)
 Pr_gnd
  a2,GAD
comparison algorithm does not; additionally, the
P-value
23 
M
23
magnitude
is generally
foradditionally,
the proposed
comparison
algorithmsmaller
does not;
thealgorithm.
P-value magnitude is generally smaller for
As shown on the left side of Fig. 12, pronounced ramifications
The parameters of the given model are shown in Tables
due
high-elevation-angle
observed
7, The
8, and
9. The airborne
accuracy
and
parameters
of thedue
given
are shown (AAD)
in Tables
7, 8, and 9. The airborne a
thetoproposed
algorithm. As satellites
shown onare
thealso
left side
of Fig.in12, pronounced
ramifications
to modeldesignator
the P-value, as expected. Fortunately, it is still within the
ground accuracy designator (GAD) are related to antenna
designator
(AAD)
andforground
accuracystation
designator
high-elevation-angle
are also
observed
in the
as expected.
Fortunately,
it isairborne
allowable
value (CAT-I),satellites
established
in ICAO
Annex
10, P-value,
for
performance
index
the
and(GAD)
the RS,are related to antenna perf
landing aircraft. [19].
respectively. [19]
index10,
forfor
thelanding
airborneaircraft.
station and the RS, respectively. [19]
still within the allowable value (CAT-I), established in ICAO Annex
The estimated aircraft position error due to the
Figure 13 shows the EPL result according to both
P-value,
algorithms. The results show that EPL will meet the availability
[19]. which represents ephemeris failure detection
Table
7. Errorcalled
modelthe
parameters
the normal
condition.
performance, is shown on the right side of Fig. 12. The
requirement,
alert limitin(AL),
established
in CAT-I
result shown in Fig. 12 shows that position error according
(10 m) and CAT-II/III (4.4 m). The effect of high elevation
to P-Value is less than centimeter-scale because the effect
angle on theContents
proposed algorithm is expected to increaseValues
25.5×10-6
slant
( not
of ephemeris failure decorrelation is small between the
the EPL (redIonosphere
line), but it
does
threaten
vert _ iono
_ gradient )to exceed the
aircraft and the RS in the case of small displacement
AL (which would
Earthresult
radiusin( Rae loss
) of availability) because of6378.1368 km
(752.5 m). However, if the displacement is larger than
considerable satellite geometry.
350 km
Ionosphere thickness ( hI )
22

-4
P-Value (Gimpo International Airport, 24h)
x 10
0.16
0.14
Dimensionless
PL (m)
0.12
0.1
0.08
1
100 s
Vehicle velocity ( vair )
39.02 m/s
Troposphere scale height ( h0 )
25,500 m
Troposphere refractivity uncertainty (  N )
No. Reference station ( M )
255
Comparsion algorithm
Proposed algorithm
0.18
2
752.5 m
Slant Range ( X air )
Time constant (  )
Estimated Position Error due to MDE (GMP RWY 32R DH, 24h)
0.2
Comparison Method
Proposed Method

4
0.06
0.04
0
0.02
Table
8. Airborne accuracy designator.
0
5
10
15
Local Time (Hours)
20
25
0
0
5
10
15
Local Time (Hours)
20
a0, AAD
AAD to P-Value (right).
Figure 12. P-Value (left) and estimated position error with respect
AAD-B
0.11
Fig. 12. P-Value (left) and estimated position error with respect to P-Value (right).
98 Table 9. Ground accuracy designator.
The estimated aircraft position error due to the P-value, which represents ephemeris failure
GAD
a0,GAD
detection performance, is shown on the right side of Fig. 12. The result shown
in Fig. 12
0.15
 i ≥ 35
GAD-C
i
shows that position error according to P-Value is less than centimeter-scalebecause
effect
< 35 the0.24
a1, AAD
 C , AAD
0.13
4.0
DOI: http://dx.doi.org/10.5139/IJASS.2015.16.1.89
(89~101)14-083.indd 98
a1,GAD
a2,GAD
 0,GAD
0.84
0
0.04
0.04
15.5
-
2015-03-30 오후 3:48:37
Jongsun Ahn Orbit Ephemeris Failure Detection in a GNSS Regional Application
Despite the explicit distinctions between the two
4. Conclusions and Future Work
algorithms, appreciable differences in EPL are not observed
in Fig. 13. If the ephemeris failure decorrelation due to
GNSS orbit ephemeris is broadcast to users for computing
aircraft-RS displacement increases (as in a wide-area
the position of a navigation satellite. The orbit ephemeris is
implementation), then the EPL based on the proposed
estimated by the ground facility, which is the operational
algorithm is more relatively valuable for system availability
segment
(OCS)
and CAT-II/III (4.4 m). The effect of high elevation angle on thecontrol
proposed
algorithm
is of the GPS, and affects the accuracy
than that of the comparison algorithm. These findings
of the user’s navigation solution. Currently, estimated
suggest
that to
the
proposed
will be
to not
performance
tends
to improve when using the differential
expected
increase
the algorithm
EPL (red line),
butsuitable
it does not
threaten to
exceed the AL
(which
only GBAS, but also SBAS and GRAS, which are implemented
GNSS (DGNSS) implementation, so ramifications of
result
in a loss of availability) because of considerable satellite
geometry.
for would
wide-area
applications.
ephemeris
error to the user are very small in limiting cases.
However, the integrity issue has been magnified recently
along with user accuracy requirements. In particular,
Comparsion algorithm
Proposed algorithm
integrity can determine system availability and any other
4
2
i
requirements faced within the aviation community.
 
a0,GAD  a1,GAD e i 0 ,GAD 2
2
i
  0 ,GAD
3.5
(28)a threat to integrity of the GNSS; thus,
 Pr_gnd


a
a0,GAD  a1,GAD e  i 
 2,GAD  2 Ephemeris constitutes
2
i
0 ,GAD
we
propose
an
ephemeris
a0,GAD  a1,M
e
(28) failure detection algorithm.
 Pr_gnd


a


GAD
2, GAD 2
i
M
3
(28)
 Pr_gnd

  a2,GAD 
Basically,
the
proposed
algorithm uses trigonometry
M
(the First Cosine Law) to estimate the baseline length of
The parameters of the given model are shown in Tables 7, 8, and 9. The airborne accuracy
2.5
reference
antennas
andaccuracy
compute the residual with
The parameters of the given model are shown in Tables
7, 8,station
and 9. The
airborne
The parameters
of the
givenaccuracy
model aredesignator
shown in (GAD)
Tables
7,
and
9. to
The
airborne
accuracyIf this residual (from
true baseline
length
surveyed
precisely.
designator
(AAD) and
ground
are8,related
antenna
performance
2
designator (AAD) and ground accuracy designator (GAD)
are related
to antenna
performance
test statistics)
exceeds
a threshold
value, then ephemeris
designator
(AAD)
andstation
groundand
accuracy
designator
(GAD)
are
related to antenna performance
1.5
index
for
the
airborne
the
RS,
respectively.
[19]
failure
is
detected.
0
5
10
15
20
Time (Hours)
index for the Local
airborne
station and the RS, respectively. [19]
For detailed algorithm analysis, we introduce the
index
for the
station and the
RS, respectively. [19]
EPL
result
at airborne
Gimpo International
Fig.Figure
13. EPL 13.
result
at Gimpo
International
Airport (24 h).Airport (24 h).
generation
of test statistics and a threshold value. An
Table 7. Error model parameters in the normal condition.
VPL (m)
4.5
Ephemeris VPL (GMP RWY 32R DH, 24h)




Table 7. Error model parameters in the normal condition.
Table
7. Error
model
parameters
in the
condition.
Table 7.
Error model
parameters
in the normal
condition.
Contents
Values
Despite
the explicit
distinctions
between
the
twonormal
algorithms,
appreciable
differences in
Contents slant ( 
Values -6
25.5×10
Ionosphere
vert _ iono _ gradient )
Contents
Values
-6
EPL are not observed inIonosphere
Fig. 13. If the
ephemeris
failure
decorrelation
due to aircraft-RS
25.5×10
slant (  vert _ iono _ gradient )
6378.1368
R
Earth
radius
(
)
e
25.5×10-6 km
Ionosphere slant (  vert _ iono _ gradient )
6378.1368
km
Earth
( Re ) implementation),
displacement increases Ionosphere
(as
in aradius
wide-area
then the EPL
based
the
350
km onkm
( hI )
6378.1368
Earth radiusthickness
( Re )
350
km
h
Ionosphere
( I)
752.5
m of the
Slant
Rangethickness
(X
proposed algorithm is more
relatively
valuable
system availability than
that
air )
350 km
Ionosphere
thickness
( hfor
I )
752.5
X
Slant
Range
(
)
100
s m
Time constant ( air )
752.5
m
X
Slant
Range
(
)
air
comparison algorithm. These
findings suggest
that the proposed algorithm
100 will
s m/sbe suitable to
Time constant
(  v)
39.02
Vehicle
velocity
(
)
100 s
Time constant (  ) air
39.02 m/s
Vehicle
velocity
( vheight
) ( hare) implemented for25,500
airwhich
m
not only GBAS, but alsoTroposphere
SBAS
and
GRAS,
wide-area
scale
0
39.02
m/s
Vehicle velocity ( vair )
25,500
m
h
Troposphere
scale
height
(
)
0
255
Troposphere
( N )
25,500 m
Troposphere refractivity
scale heightuncertainty
( h0 )
applications.
255
Troposphere
( N )
No.
Referencerefractivity
station ( Muncertainty
)
4255
Troposphere refractivity uncertainty (  N )
No. Reference station ( M )
4
No. Reference station ( M )
4
4. Conclusions
Future Work
Table and
8. Airborne
accuracy designator.
Table
8. Airborne
accuracy designator.
Table 8. Airborne
accuracy
designator.
GNSS orbit
is broadcast
usersa for computing thea position of a navigation
Tableephemeris
8. Airborne
accuracy to
designator.
 C , AAD
AAD
0, AAD
1, AAD
a0, AAD
a1, AAD
 C , AAD
AAD
AAD-B
0.11 ground facility,
0.13
4.0
satellite. The orbit ephemeris
is the operational
a1,which
 C , AAD
AAD is estimated by athe
0, AAD
AAD
AAD-B
0.11
0.13
4.0
0.11
4.0solution.
control segment (OCS) AAD-B
of the GPS, and affects
the accuracy of 0.13
the user’s navigation
9. Ground
accuracy designator.
Table 9. GroundTable
accuracy
designator.
Currently, Table
estimated
performance
tends
to improve when using the differential GNSS
9. Ground
accuracy
designator.
Table 9. Ground
accuracy designator. a
GAD
a1,GAD
a2,GAD
 0,GAD
0,GAD
GAD
a
a
a
 0,GAD
i
0,GAD
1,GAD
2,GAD
0.84
0.04
15.5
 ≥ 35 25 0.15
GAD
a0,GAD
a1,GAD
a2,GAD
 0,GAD
i
GAD-C
0.15
0.04
≥ 35
35
0.24
00.84
0.04
-15.5
ii <
GAD-C
0.15
0.84
0.04
15.5
 i ≥ 35
0.24
0
0.04
 < 35
GAD-C
0.24
0
0.04
 i < 35
Figure 13 shows the EPL result according to both algorithms. The results show that EPL
Figure 13 shows the EPL result according to both
99 algorithms. The results show that EPL
the EPL
result according
results show
that EPL
willFigure
meet 13
theshows
availability
requirement,
called to
theboth
alertalgorithms.
limit (AL),The
established
in CAT-I
(10 m)
will meet the availability requirement, called the alert limit (AL), established in CAT-I (10 m)
will meet the availability requirement, called the alert limit (AL), established in CAT-I (10 m)
(89~101)14-083.indd 99
24
24
24
http://ijass.org
2015-03-30 오후 3:48:37
Int’l J. of Aeronautical & Space Sci. 16(1), 89–101 (2015)
Proceedings of the 23rd International Technical Meeting of
the Satellite Division of the Institute of Navigation (ION GNSS
2010), Portland, OR, Sep 2010, pp. 3115–3122
[6] Haochen Tang, Sam Pullen, Per Enge, Livio Gratton,
Boris Pervan, Mats Brenner, Joe Scheitlin, and Paul Kline,
“Ephemeris Type A Fault Analysis and Mitigation for LAAS”,
Proceedings of Position Location and Navigation Symposium
(PLANS), 2010 IEEE/ION, 4-6 May 2010, pp. 654-666
[7] Gang Xie, Optimal On-airport Monitoring of the
Integrity of GPS-Based Landing System, Ph.D Thesis, Stanford
University, March 2004, pp. 39-43.
[8] Boris Pervan, Livio Gratton, “Orbit Ephemeris Monitors
for Local Area Differential GPS”, Journal of IEEE Transactions
on Aerospace and Electronic Systems, Vol. 41, No. 2, April
2005.
[9] Jiyun Lee, Sam Pullen, Boris Pervan, and Livio Gratton,
“Monitoring Global Positioning Satellite Orbit Errors for
Aircraft Landing System”, Journal of Aircraft, Vol. 43, No. 3,
May-June 2006.
[10] Boris Pervan, and Fang Cheng Chan, “Detecting
Global Positioning System Orbit Errors Using Short-Baseline
Carrier-Phase Measurements“ Journal of Guidance, Control ,
and Dynamics, Vol. 26, No. 1, January-February 2003.
[11] Gang Xie, Optimal On-airport Monitoring of the
Integrity of GPS-Based Landing System, Ph.D Thesis, Stanford
University, March 2004, pp. 26-43.
[12] Pullen S., Lee J, Luo, M., Pervan, B, Chan, F-C, and
Gratton, L., “Ephemeris Protection Level Equation and
Monitor Algorithm for GBAS”, Proceedings of the 14th
International Technical Meeting of the Satellite Division of the
Institute of Navigation, Salt Lak City, UT, Sept, 11-14, 2001.
[13] Chan, F.C., Detection of Global Positioning
Satellite Orbit Errors Using Short-Baseline Carrier Phase
Measurements, M.S.Thesis, Dept. of Mechanical, Materials,
and Aerospace Engineering, Illinois Inst. of Technology,
Chicago, Aug. 2001, pp. 17-19.
[14] Jongsun Ahn, Hyang-Sig Jun, Chan-Hong Yeom,
Sangkyung Sung, and Young Jae Lee, “Sensitivity Analysis
of Ephemeris Fault Detection Algorithm based on Baseline
Length Estimation Method”, Proceeding of KGS 2014, Nov.
2014, pp. 693-696.
[15] Jongsun Ahn, Eunsung Lee, Moon Beom Heo,
Sangkyung Sung, and Young Jae Lee, “Research of MDE of for
Baseline-based GPS Ephemeris Fault Detection Algorithm”,
Proceeding of KSAS 2014, Nov. 2014, pp. 330-332.
[16] Mastumoto, S, Pullen, S, Rotkowitz, M., and Pervan,
B., “GPS Ephemeris Verification for Local Area Augmentation
System (LAAS) Ground Station,” Proceedings of the 12th
International Technical Meeting of the Satellite Division of the
Institute of Navigation, Inst., of Navigation, Alexandria, VA,
analysis is conducted on the test statistics’ sensitivity vector,
corresponding to ephemeris failure, and on an availability
test, based on the ephemeris protection level in Korea. The
important feature is that the performance index (sensitivity
and MDE) depends on geometric parameters including
local elevation angle, local azimuth angle, and baseline
length between the satellite and the baseline vectors.
Consequently, this algorithm is more efficient for use with
multiple reference stations that can generate various long
baseline vectors, such as GRAS and SBAS. The availability
test conducted for Gimpo International Airport in Korea
was based on the EPL of a landing aircraft at CAT-I decision
height. In some cases, concern is warranted by an increase
in EPL (which threatens availability) because of increased
MDE. However, this effect is still small enough to meet the
stringent requirements (CAT-I and CAT-II/III) for precision
approaches and landing.
In future work, we will address the algorithm’s weakness
at high elevation angles and validate it using numerous
international GNSS reference data.
Acknowledgement
This work was supported partially by the Ministry of Land,
Infrastructure and Transport (MOLIT) under Grant 10AVINAV01.
References
[1] B. Hoffmann-Wellenhof H. and Lichtenegger, E. Walse,
2008, GNSS; GPS, GLONASS, Galileo & more, springer Wien
New York, pp. 269~273
[2] Jeong Hwan Yang, Chang Ho Kang, Sun Young Kim,
and Chan Gook Park, “International GNSS Interference
Detection and Characterization Algorithm uisng AGC
and Adaptive IIR Notch Filter”, International Journal of
Aeronautical and Space Sciences, Vol. 13, No. 4, 2012. 12, pp.
491-498
[3] Bradford W. Parkinson, et al., Global Positioning System
: Theory and Applications Volume 1, AIAA, 1996, pp. 121-139
[4] Liang Heng, Grace Xingxin Gao, Todd Walter, and
Per Enge, “GPS Ephemeris Error Screening and Results for
2006-2009”, Proceedings of the 2010 International Technical
Meeting of the Institute of Navigation (ION ITM 2010), San
Diego, CA, Jan 2010, pp. 1014–1022
[5] Liang Heng, Grace Xingxin Gao, Todd Walter, and Per
Enge: “GPS signal-in-space anomalies in the last decade:
Data mining of 400,000,000 GPS navigation messages,”
DOI: http://dx.doi.org/10.5139/IJASS.2015.16.1.89
(89~101)14-083.indd 100
100
2015-03-30 오후 3:48:37
Jongsun Ahn Orbit Ephemeris Failure Detection in a GNSS Regional Application
ephemeris faults”, Proceedings of the 14th International
Technical Meeting of the Satellite Division of the Institute of
Navigation, Salt Lake City, UT, Sept. 11-14, 2001.
[19] Draft, ICAO Annex 10, GBAS CAT II/III Development
Baseline SARPs, 17-29, May 2010 meeting of the Navigation
System Panel (NSP) Working Group
1999. 691-703.
[17] Specification Category 1 Local Area Augmentation
System Non-Federal Ground Facility, U.S. Federal Aviation
Administration, FAA/AND 710-2937, Washington, DC, May,
2001
[18] Shively. C, “LAAS Integrity Risk due to satellite
101
(89~101)14-083.indd 101
http://ijass.org
2015-03-30 오후 3:48:37