On GPS positioning and integrity monitoring

Abstract

The paper reinvestigates the measurement model associated with
Global Positioning System (GPS) signal processing. It is argued that the
GPS positioning model is better formulated as a linear equation with
errors in both the data matrix and measurement variables. Furthermore,
depending on the nature of the measurement errors, the model is
categorized into unstructured and structured perturbation cases. In the
former, the total least squares method is proposed for position fixing
and clock bias determination. In the latter, an iteration method is
developed to search for the optimal solution. In addition to the
position update, both the total least squares and optimization methods
provide estimates of the model mismatch which leads to a measure of GPS
receiver autonomous integrity monitoring. Two new GPS fault detection
metrics are then proposed and discussed. The first integrity test
statistic is the norm of the residual vector in the total least squares
estimate. Statistical properties of this test statistic are obtained for
integrity monitoring. The second metric is a two-dimensional vector that
characterizes the norm of the residual vector and mismatch matrix, both
are outcomes of the total least squares method or the optimization
method. The positioning and integrity monitoring schemes are verified
using simulated examples

Full text

[11] Kralj, D., Hsu, T-T., and Carin, L. (1995)
Short-pulse propagation in a hollow waveguide: Analysis,
optoelectric measurement, and signal processing.
IEEE Transactions on Theory Technology, 43 (1995), 2144.
[12] Erd
´
elyi, A., Magnus, W., Oberhettinger, F., and Tricomi,
F. G. (1954)
Tables of Integral Transforms, Vol. II.
New York: McGraw-Hill, 1954.
[13] Borden, B. (1997)
An observation about radar imaging of re-entrant
structures with implications for automatic target
recognition.
Inverse Problems, 13 (1997), 1441.
On GPS Positioning and Integrity Monitoring
The paper reinvestigates the measurement model associated
with Global Positioning System (GPS) signal processing. It is
argued that the GPS positioning model is better formulated
as a linear equation with errors in both the data matrix and
measurement variables. Furthermore, depending on the nature
of the measurement errors, the model is categorized into
unstructured and structured perturbation cases. In the former,
the total least squares method is proposed for position fixing and
clock bias determination. In the latter, an iteration method is
developed to search for the optimal solution. In addition to the
position up date, both the total least squares and optimization
methods provide estimates of the model mismatch which leads to
a measure of GPS receiver autonomous integrity monitoring. Two
new GPS fault detection metrics are then proposed and discussed.
The first integrity test statistic is the norm of the residual vector
in the total least squares estimate. Statistical properties of this
test statistic are obtained for integrity monitoring. The second
metric is a two-dimensional vector that characterizes the norm
of the residual vector and mismatch matrix, both are outcomes of
the total least squares method or the optimization method. The
positioning and integrity monitoring schemes are verified using
simulated examples.
I. INTRODUCTION
The Global Positioning System (GPS) provides
three-dimensional position fix and time transfer based
upon the principle of time-of-arrival ranging. In
Manuscript received November 14, 1996; revised September 14 and
December 10, 1999; released for publication December 14, 1999.
Refereeing of this contribution was handled by X. R. Li.
IEEE Log No. T-AES/36/1/02613.
This work was supported by the National Science Council of the
Republic of China under Grant NSC 88-2623-D-006-015.
0018-9251/00/$10.00
c
° 2000 IEEE
solving the position and clock bias update, a linear
model is generally employed and the least squares
solution is obtained for correction; see [1, 7, 10, 14].
Indeed, it is known that the GPS measurements are
subject to errors due to clock bias, SA degradation,
satellite ephemeris error, atmospheric delay, multipath
offset, and receiver noise. Lumping these errors as a
user equivalent range error, a linear matrix equation
Hp = q + e canthenbeformulatedinwhichq is the
measurement residue, e models the measurement error,
p contains the position/time bias correction terms, and
H is the observation matrix which is composed of a
set of unit vectors from the computed user position to
the broadcast satellite positions.
The linear model associated with GPS
measurements is reinvestigated. Typically in GPS
signal processing, the observation matrix H is
assumed to be known. In practice, the observation
matrix consists of a set of unit vectors from the actual
user position to the actual GPS satellite positions.
Hence, the matrix H is subject to perturbations
associated with ephemeris errors and user’s position
errors. This discrepancy may be significant in the
initialization process and in the presence of GPS
satellite failure or signal interferences. Mathematically,
the resulting linearized model contains errors in
its variables (of the observation matrix). It is thus
appropriate to use the error-in-variables model to
describe the positioning operation. The errors which
act as perturbations on the matrix H can indeed be
further classified into two cases. In the unstructured
perturbation case, all entries of H are subject to
errors. In the structured perturbation case, each
ephemeris error is d ecomposed into two orthogonal
components, one along the observation vector and
another perpendicular to the vector. There then exists
a pattern on the perturbation of H.
With respect to these two perturbed models, two
different algorithms are developed. The total least
squares method is used to solve the unstructured
perturbation problem. The method of total least
squares has been widely in vestigated in statistical
literature and operational research. Applications of
the method to numerical linear algebra [6], modal
analysis, and system identification have been reported.
In comparison with the least squares method, the
total least squares method enjoys the properties of
data consistency at the expense of more numerical
computations [9]. On the other hand, the structured
perturbation problem can be tackled using the
solution of the (total) least squares as the starting
point for further refinement. The optimizatio n
principle is applied to the problem to seek for a
convergent solution for position update and clock bias
determination.
In both total least squares and optimization
methods, an estimate of the model mismatch is
CORRESPONDENCE 327

derived. This mismatch can thus serve as an indication
of GPS fault detection. Existing GPS integrity
monitoring schemes rely either on a wide-area
network for integrity broadcast [5, 11, 15] or receiver
autonomous integrity monitoring (RAIM) [3]. In the
latter, the signal processor within the GPS receiver
computes the measurement residue and compares it
against a prespecified threshold for false declaration
[2, 3, 12]. Since the residue is a combined effect
of various error sources, a tradeoff between false
alarm and missed detection is difficult to achieve.
The error-in-variables model, either the structured
or the unstructured form, essentially separates the
equivalent pseudo-range error and ephemeris error.
This enables a more balanced design in separating
error sources and monitoring GPS integrity. Two types
of GPS integrity monitoring metrics are proposed
and discussed based upon the total least squares or
optimization solutions.
In Section II, both the unstructured and structured
error-in-variables models for GPS measurement
processing are obtained. The total least squares
method is applied to the unstructured model and an
optimization method is developed for the structured
model. In both cases, the position/clock update and
the model mismatch are determined. The total least
squares method is compared with the conventional
least squares method. In Section III, the mismatch in
the total least squares method or the optimization is
used as an indication of GPS health monitoring. In
particular, the statistical properties of the range residue
in the total least squares method are investigated. In
Section IV, the simulation results are presented. In
Section V, the conclusion is given.
For a vector e, e
T
stands for the transpose of e.
Likewise, the transpose of the matrix H is denoted as
H
T
. I
n
stands for the n £n identity matrix and 0
m£n
is the m £n zero matrix. kek and kHk
F
stand for,
respectively, the vector norm of e and the Frobenius
norm of the matrix H. The trace and determinant of H
is denoted by traceH and det H, respectively.
II. GPS POSITIONING
A. GPS Measurement Model
Let u be the user’s position and s
i
be the broadcast
position of the ith GPS satellite. The pseudo-range
measurement with respect to the ith GPS satellite ½
i
is
given by
½
i
= ku ¡s
i
k+ b + e
i
(1)
where b represents the error associated with the GPS
receiver and e
i
stands for the errors due to the ith
satellite. Typically in GPS positioning, b is assumed
to be an offset term and e
i
is treated as a zero mean
noise. In this case, a standard approach for resolving
the user’s position u as well as the (clock) offset
b is to linearize the measurement equation (1) and
compute the least squares solution for position and
clock bias update. More precisely, let the coordinates
of u and s
i
in the same Cartesian coordinate system
be, respecti vely,
u =
2
6
4
x
y
z
3
7
5
and s
i
=
2
6
4
x
i
y
i
z
i
3
7
5
:
Suppose that the linearization point is at
u = u
0
=
2
6
4
x
0
y
0
z
0
3
7
5
and b = b
0
,
then the estimate of the pseudo-range measurement is
given by
½
i
0
= ku
0
¡s
i
k+ b
0
=
q
(x
i
¡x
0
)
2
+(y
i
¡y
0
)
2
+(z
i
¡z
0
)
2
+ b
0
:
Define
r =
2
6
4
x ¡x
0
y ¡y
0
z ¡z
0
3
7
5
, ± = ¡b + b
0
,andp =
·
r
±
¸
the linearized equation of (1) becomes
½
i
0
¡½
i
+ e
i
=[
h
i1
h
i2
h
i3
1
]p = h
T
i
r + ±
where
h
i1
=
x
i
¡x
0
p
(x
i
¡x
0
)
2
+(y
i
¡y
0
)
2
+(z
i
¡z
0
)
2
h
i2
=
y
i
¡y
0
p
(x
i
¡x
0
)
2
+(y
i
¡y
0
)
2
+(z
i
¡z
0
)
2
h
i3
=
z
i
¡z
0
p
(x
i
¡x
0
)
2
+(y
i
¡y
0
)
2
+(z
i
¡z
0
)
2
:
It is noted that the vector
h
i
=
2
6
4
h
i1
h
i2
h
i3
3
7
5
is the unit vector from the linearization point u
0
to the
broadcast position of the ith satellite s
i
. When there
are n GPS satellites in view, the linearized equations
can be assembled as a linear matrix equation
Hp = q + e (2)
328 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 36, NO. 1 JANUARY 2000

where
H =
2
6
6
6
6
4
h
11
h
12
h
13
1
h
21
h
22
h
23
1
.
.
.
.
.
.
.
.
.
.
.
.
h
n1
h
n2
h
n3
1
3
7
7
7
7
5
,
q =
2
6
6
6
6
4
½
1
0
¡½
1
½
2
0
¡½
2
.
.
.
½
n
0
¡½
n
3
7
7
7
7
5
,and
e =
2
6
6
6
6
4
e
1
e
2
.
.
.
e
n
3
7
7
7
7
5
:
In the GPS positioning equation (1), both the
pseudo-range ½
i
and the broadcast position s
i
are
subject to errors due to ephemeris errors, SA effects,
en vironmental effects, satellite failure, interferences,
noises, and so on. Thus, a more accurate measurement
model is
½
i
= ku ¡s
i
¡¢s
i
k+ b + e
i
(3)
where ¢s
i
is the difference between the broadcast
position and true position of the ith GPS satellite and
e
i
accounts for the other errors. Let
¢s
i
=
2
6
4
¢x
i
¢y
i
¢z
i
3
7
5
:
The linearized model of (3), taking u
0
and b
0
as the
nominal point, can be derived as
½
i
0
¡½
i
+ e
i
=[
h
0
i1
h
0
i2
h
0
i3
1
]p
where
h
0
i1
=
x
i
+ ¢x
i
¡x
0
p
(x
i
+ ¢x
i
¡x
0
)
2
+(y
i
+ ¢y
i
¡y
0
)
2
+(z
i
+ ¢z
i
¡z
0
)
2
h
0
i2
=
y
i
+ ¢y
i
¡y
0
p
(x
i
+ ¢x
i
¡x
0
)
2
+(y
i
+ ¢y
i
¡y
0
)
2
+(z
i
+ ¢z
i
¡z
0
)
2
h
0
i3
=
z
i
+ ¢z
i
¡z
0
p
(x
i
+ ¢x
i
¡x
0
)
2
+(y
i
+ ¢y
i
¡y
0
)
2
+(z
i
+ ¢z
i
¡z
0
)
2
:
In terms of matrix equations with respect to n
observable satellites, we have
H
0
p = q + e
where
H
0
=
2
6
6
6
6
4
h
0
11
h
0
12
h
0
13
1
h
0
21
h
0
22
h
0
23
1
.
.
.
.
.
.
.
.
.
.
.
.
h
0
n1
h
0
n2
h
0
n3
1
3
7
7
7
7
5
:
Clearly, the observation matrix H
0
is subject to
uncertainty mainly due to ephemeris error in this more
realistic model. A positioning method that accounts
for the error associated with the observation matrix is
worth investigating.
Let H
0
= H + G where G is the perturbation matrix
that is a function of ¢s
i
s. The resulting linear matrix
equation becomes
(H + G)p = q + e: (4)
This model is termed as the unstructured error-in-
variables model as the matrix G represents the errors
associated with the observation matrix and no a
priori structure on G is being imposed. The model
in (4) can indeed be more applicable than merely
addressing ephemeris error. Indeed, the matrix H
is composed of the unit vector from the linearized
point to the GPS satellite position. The linearization
point may be susceptible to error, especially in the
initialization period. Such an error can be captured
in G. The position of a GPS satellite is obtained
by decoding the navigation data. In the presence of
satellite failure, interferences, and corrupted navigation
data, the position may be erroneous. Again, the
matrix G can be used to model this error. Moreover,
the matrix G can also be used to account for the
difference in the synchronization between clocks in
GPS satellites.
In addition to the unstructured model, certain
structure can sometimes be imposed on G. It is noted
that the ephemeris error ¢s
i
can be decomposed into
two components: one is along the direction of h
i
and the other is orthogonal to h
i
. By absorbing the
along-component in e
i
, the perturbation matrix G in
(4) has the property that the ith row of G admits
[
g
i1
g
i2
g
i3
0
](5)
with the property that
g
i
=
2
6
4
g
i1
g
i2
g
i3
3
7
5
is orthogonal to h
i
,i.e.,
g
T
i
h
i
=0: (6)
The extent of the radial component of the ephemeris
error is characterized by the norm of g
i
.Thisgivesa
structured error-in-variables GPS measurement model.
In the following subsections, methods are devised
to solve p in (4) under structured or unstructured
perturbations.
B. Total Least Squares Method
In this subsection, the positioning problem of (4)
with respect to an unstructured G is discussed. Recall
CORRESPONDENCE 329

that in the least squares problem in solving (2), the
objective function kek
2
is minimized under the linear
system (2) constraint. The resulting least squares
solution is
p
ls
=(H
T
H)
¡1
H
T
q:
In the problem of (4) in finding the best p, both G
and e are subject to errors. This motivates the use of
the Frobenius norm of the augmented matrix [G e]
as the objective function to be minimized. In other
words,theideaistofindavectorp that best fits the
model in (4) in the sense that the model mismatch as
characterized by G and e is as small as possible. More
precisely, the problem can be stated as
min
p
kC[G e]Dk
F
(7)
subject to (4). In the above, C and D are compatible,
nonsingular, diagonal weighting matrices. Clearly,
the entries in C are used to weight the significance
of different GPS measurements (in terms of
signal-to-noise ratio, for example), while the entries
in D are related to the relative importance of satellite
position, user clock, and pseudo-range errors. The
weighting D offers some flexibility that cannot be
achieved in the weighted least squares problem.
Indeed, the error-in-variables model enables the
user to emphasize on the horizontal positioning
error, vertical positioning error, or clock bias by
adjusting D.
The total least squares method can be used
to solve the above problem (7). The procedure is
summarized as follows [6].
1) Perform a singular value decomposition of the
matrix C[H q]D such that
C[H q]D = U§V
T
:
In the above, U is an n £5 orthogonal matrix,
§ =
2
6
6
6
6
6
6
4
¾
1
0000
0 ¾
2
000
00¾
3
00
000¾
4
0
0000¾
5
3
7
7
7
7
7
7
5
,
and V is a 5 £5 orthogonal matrix. The singular
values of C[H q]D, ¾
i
’s, are in descending order
with the assumption that ¾
4
>¾
5
.
2) Partition U, V, §,andD such that
U =
2
6
4
U
1
|{z}
n£4
U
5
|{z}
n£1
3
7
5
V =
2
6
6
6
4
V
11
|{z}
4£4
V
15
|{z}
4£1
V
51
|{z}
1£4
V
55
|{z}
1£1
3
7
7
7
5
§ =
2
4
§
1
|{z}
4£4
0
4£1
0
1£4
¾
5
3
5
,and
D =
2
4
D
1
|{z}
4£4
0
4£1
0
1£4
d
5
3
5
:
The total least squares solution of p is given by
p
tls
= ¡
1
V
55
d
5
D
1
V
15
(8)
and the mismatches G
tls
and e
tls
are given by
[G
tls
e
tls
]=¡¾
5
C
¡1
U
5
[V
T
15
V
T
55
]D
¡1
: (9)
The main computation requirement in the
total least squares solution is the singular value
decomposition. The decomposition yields not only
the best solution p
tls
but also the mismatches G
tls
and
e
tls
. The vector p
tls
can then be used for position/clock
bias update. The mismatches between the model and
data can be used for integrity monitoring. It is also
noted that the resulting error equals to
minkC[G e]Dk
F
= kC[G
tls
e
tls
]Dk
F
= ¾
5
:
The least singular value ¾
5
thus represents the
amount of model mismatch that characterizes the
compatibility between the observation matrix H and
the measurement vector q.Further,thetotalleast
squares estimate p
tls
is related to the mismatches G
tls
and e
tls
by
G
tls
+ e
tls
p
T
tls
=0: (10)
In the above it is assumed that ¾
4
>¾
5
for simplicity
and clarity. In case that ¾
4
= ¾
5
, the total least squares
solution may no longer be unique [6]. However, all
the interpretations remain similar except that certain
manipulations become more involved.
When the weighting matrices are the identity
matrices, the total least squares estimate can be
expressed explicitly as
p
tls
=(H
T
H ¡¾
2
5
I
4
)
¡1
H
T
q (11)
provided that the inverse exists. The least squares
estimate and the total least squares estimate are indeed
related; see [8]. In particular,
p
tls
=[I
4
¡¾
2
5
(H
T
H)
¡1
]
¡1
p
ls
=[I
4
+ ¾
2
5
(H
T
H ¡¾
2
5
I
4
)
¡1
]p
ls
: (12)
330 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 36, NO. 1 JANUARY 2000

This implies that the least squares estimate is equal to
the total least squares estimate when and only when
¾
5
= 0. In this case, the vector q is in the range space
of H. The augmented matrix [H q] would then have
a rank of 4. This is also equivalent to the condition
that the mismatches e
tls
=0
n£1
and G
tls
=0
n£5
. Thus,
in the ideal case, both the total least squares estimate
and the least squares estimate are the same.
In GPS positioning, the navigation accuracy is
typically characterized as the product of the user
equivalent pseudo-range error and the dilution of
precision (DOP) factor. The former is related to the
uncertainties of each contributing error source and
the latter is attributed to the geometric distribution of
observable satellites. Suppose that the components
of the pseudo-range residual vector are independent,
identically distributed, zero mean Gaussian processes
with standard deviations ¾
0
. From (11), the covariance
of the positioning vector p
tls
can be found to be
cov(p
tls
)=¾
2
0
(H
T
H ¡¾
2
5
I
4
)
¡1
H
T
H(H
T
H ¡¾
2
5
I
4
)
¡1
:
In contrast, the covariance of the least squares
estimate is known to be cov(p
ls
)=¾
2
0
(H
T
H)
¡1
.Itcan
be shown that
det(H
T
H ¡¾
2
5
I
4
)
¡1
H
T
H(H
T
H ¡¾
2
5
I
4
)
¡1
¸ det(H
T
H)
¡1
:
Thus, the error ellipse associated with the total least
squares estimate is larger than that associated with
the least squares estimate. This result is not surprising
since in the total least squares method, errors resulting
from the observation matrix are explicitly accounted
for. Since there are potentially more error sources,
the distribution of the solution spreads out. To be fair,
in assessing the navigation accuracy u sing the total
least squares method, the user equivalent pseudo-range
error is smaller than that of the least squares method
since components due to ephemeris error and even
satellite clock errors in the total least squares method
are estimated and can thus be removed in forming ¾
0
.
One can further define a modified geometric DOP as
GDOP
0
=
q
tr(H
T
H ¡¾
2
5
I
4
)
¡1
H
T
H(H
T
H ¡¾
2
5
I
4
)
¡1
=
q
trH(H
T
H ¡¾
2
5
I
4
)
¡2
H
T
to better interpret the navigation error using the total
least squares method. Naturally, similar notions can
be carried out to define modified position DOP,
horizontal DOP, time DOP, and so forth.
C. Optimization Method
In this subsection, the structured model, i.e., when
the ith ro w of the matrix G admits the structure in (5),
is investigated. An optimization approach is used to
solve for the optimal
p =
·
r
±
¸
:
Note that the ith row of (4) can be rewritten as
(h
T
i
+ g
T
i
)r + ± = q
i
+ e
i
: (13)
A best match of the position update r and clock bias ±
can be accomplished through the minimization of the
following objective function:
J =
1
2
n
X
i=1
e
2
i
+
1
2
n
X
i=1
g
T
i
g
i
subject to (13) and the orthogonal condition (6).
The selection of the objective function bears similar
motivations as in the least squares and total least
squares cases. Introducing the Lagrange multipliers
¸
i
sand¹s, the objective function becomes
J
0
=
1
2
n
X
i=1
e
2
i
+
1
2
n
X
i=1
g
T
i
g
i
+
n
X
i=1
¸
i
h
T
i
g
i
+
n
X
i=1
¹
i
[(h
T
i
+ g
T
i
)r + ± ¡q
i
¡e
i
]:
The necessary conditions for optimum are
@J
0
@e
i
= e
i
¡¹
i
=0
@J
0
@g
i
= g
i
+ ¸
i
h
i
+ ¹
i
r = 0 (14)
@J
0
@r
=
n
X
i=1
¹
i
(h
i
+ g
i
) = 0 (15)
@J
0
@±
=
n
X
i=1
¹
i
= 0 (16)
@J
0
@¸
i
= h
T
i
g
i
=0
@J
0
@¹
i
=(h
T
i
+ g
T
i
)r + ± ¡q
i
¡e
i
=0: (17)
Some manipulations then lead to
¹
i
= e
i
¸
i
= ¡e
i
h
T
i
r
and
g
i
= ¡e
i
(I
3
¡h
i
h
T
i
)r: (18)
The latter two equalities are obtained through (14)
using the properties that h
i
is a unit vector and is
orthogonal to g
i
. Substituting (18) into (17) results
CORRESPONDENCE 331

in
e
i
=
h
T
i
r + ± ¡q
i
1+r
T
(I
3
¡h
i
h
T
i
)r
:
This, together with (16), gives the estimate of clock
bias as
± = ¡
P
n
i=1
h
T
i
r ¡q
i
1+r
T
(I
3
¡h
i
h
T
i
)r
P
n
i=1
1
1+r
T
(I
3
¡h
i
h
T
i
)r
: (19)
Hence,
e
i
=
h
T
i
r ¡q
i
1+r
T
(I
3
¡h
i
h
T
i
)r
¡
1
1+r
T
(I
3
¡h
i
h
T
i
)r
£
P
n
i=1
h
T
i
r ¡q
i
1+r
T
(I
3
¡h
i
h
T
i
)r
P
n
i=1
1
1+r
T
(I
3
¡h
i
h
T
i
)r
: (20)
Finally, combining (15) and (18) yields a
characterization of the vector r
r =
"
n
X
i=1
e
2
i
(I
3
¡h
i
h
T
i
)
#
¡1
"
n
X
i=1
e
i
h
i
#
: (21)
Only two unknowns e
i
and r are coupled in (20)
and (21). Thus, one can easily devise an iterative
scheme to determine these two variables. Normally,
given a good initial estimate, say, the (total) least
squares solution, the iterations will converge to the
optimal solution rather rapidly. The other variables can
then be determined using (18) and (19).
III. GPS INTEGRITY MONITORING
The use of error-in-variables model (4) is more
significant in dealing with anomalous GPS positioning
scenarios as described in this section. Existing
GPS integrity monitoring methods [16] such as
least squares residuals method [13], parity equation
method [17], approximate radial-error protected
method [4] rely on the use of residue comparison for
failure detection. Once the residue is greater than the
threshold, a fault is announced. The residue itself can
be in the domain of (pseudo-range) measurement or
(position) solution. In this section, the range residual
vector resulting from the total least squares solution
is investigated and compared with its least squares
counterpart. The statistical properties of a modified
sum of squared errors (SSE) which is the square of
the norm of the range residual vector is developed
to enhance the selection of the threshold for fault
declaration. One drawback of using a scalar for
fault detection lies in the selection of an appropriate
threshold. Indeed, the residue is a combined effect
of various error, noise, and failure sources, making
it difficult to tradeoff between an acceptable missed
detection and false alarm rate. Hence, another figure
of merit is proposed which is a test vector composed
of the norms of the model mismatches kGk
F
and kek.
The test vector is a direct outcome of the total least
squares or optimization method. A fault detection
plane is used to illustrate the significance of the
test vector . The test vector is further related to the
modified SSE.
With respect to the model in (2), the least squares
solution p
ls
can be used to provide a residue for
integrity monitoring. The range residual vector is
defined as
w
ls
= q ¡Hp
ls
=[I
n
¡H(H
T
H)
¡1
H
T
]q
= ¡[I
n
¡H(H
T
H)
¡1
H
T
]e: (22)
Once there is a failure on one of the pseudo-ranges,
the norm of the vector e increases and, consequently,
the norm of w
ls
increases. Hence, the norm of w
ls
or
the SSE serves as a measure for fault detection. Here,
the sum of squared errors is defined as SSE = w
T
ls
w
ls
.
Suppose that the pseudo-range measurement errors
are Gaussian distributed with the same variances,
then the sum of squared errors is known to possess
a chi-squared distribution with n ¡4 degrees of
freedom [13]. Indeed, after accounting for the number
of observable satellites and normalizing the error
statistics, a metric
r
SSE
n ¡4
has been used extensively for GPS fault detection
[2, 13]. This test statistic is known to be proportional
to the position error as well as the magnitude of the
faulty component. It can then serve as an intermediate
variable to protect the positioning accuracy in
the presence of anomalies. In a typical integrity
monitoring procedure, given a desired false alarm rate,
the corresponding threshold can be determined using
the statistical distribution. The task of RAIM is then
to check the test statistic against the threshold for the
declaration of fault.
In a similar fashion, the endeavor can be extended
to the error-in-variables model (4) and the total least
squares method. In this case, the range residual vector
becomes
w
tls
= q ¡Hp
tls
(23)
and it is appropriate to define the modified SSE as
SSE
0
= w
T
tls
w
tls
:
Since (H + G
tls
)p
tls
= q + e
tls
, the range residual vector
w
tls
can be rewritten as
w
tls
= G
tls
p
tls
¡e
tls
:
Together with (8) and (9), the range residual vector
can further be show n to equal
w
tls
= ¾
5
C
¡1
U
5
d
5
V
55
= ¡
1
d
5
V
2
55
e
tls
:
332 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 36, NO. 1 JANUARY 2000

Hence, the range residual vector is along the same
direction of e
tls
. Assume that C and D are the identity
matrices, using the fact that U
5
is a unit vector, the
modified SSE
0
can then be evaluated as
SSE
0
= w
T
tls
w
tls
=
¾
2
5
V
2
55
:
Recall that ¾
5
is the smallest singular value of the
augmented matrix [H q]. When q is in the range
space of H, the rank of the augmented matrix is 4.
In the presence of noises and failures, the rank of
the augmented matrix becomes 5. The singular value
¾
5
indeed characterizes how close the rank of the
augmented matrix is to 4 or how compatible the linear
matrix equation is. Naturally, when ¾
5
is large, the
linear equations associated with the model (4) become
less compatible and a fault is likely to occur. Thus, the
test statistic is proportional to ¾
5
.
When the weightings are identities, the range
residual vector (23) can be rewritten as
w
tls
= q ¡H(H
T
H ¡¾
2
5
I
4
)
¡1
H
T
q =(I
n
¡¾
¡2
5
HH
T
)
¡1
q:
It is noted that both w
tls
and p
tls
are nonlinear
functions of q. Yet it can be shown that w
tls
and p
tls
are linearly related. Indeed, we have
p
tls
= ¡¾
¡2
5
H
T
w
tls
:
In other words, the positioning error as revealed in
p
tls
depends linearly on the test statistic obtained
from the norm of the residual vector w
tls
. This linear
dependency is significant in integrity monitoring.
Consequently, many RAIM schemes based on the
least squares residuals can be appropriately modified
in the total least squares setting. The modified SSE
is equal to q
T
(I
n
¡¾
¡2
5
HH
T
)
¡2
q. Suppose that the
components of the pseudo-range vector have the same
independent zero-mean Gaussian distributions, then
it can be shown that the statistical distribution of the
modified SSE
0
is the summation of an n ¡4degrees
of freedom chi-square distribution and four Gamma
distributions with different p arameters; see Appendix
A for the details. Consequently, given a prescribed
false alarm rate, one can compute the threshold
using the statistical distribution and come up with
a RAIM scheme. The resulting probability density
function of SSE
0
also has a humped form. Yet it has
a smaller hump and longer tail in comparison with the
probability density function of SSE. Hence, given the
same false alarm rate, number of observable satellites,
and pseudo-range error, the threshold determined from
SSE
0
is greater than that of the SSE. This is essentially
consistent with the result derived earlier in comparing
the covariance of the p
tls
with that of p
ls
.
Another metric that can be devised for RAIM is
to assess the mismatch vectors. Since G
tls
and e
tls
represent the mismatches between the observation
model and measurement data, one can indeed
Fig. 1. Fault detection plane.
compare these two mismatches and assess the extents
of ephemeris error and pseudo-range error. The
2-dimensional kGk
F
versus kek plane is called the
fault detection plane (Fig. 1). The point (ke
tls
k,kG
tls
k
F
)
is located at (¾
5
jV
55
j,¾
5
q
1 ¡V
2
55
) in the plane when
the weightings C and D are identities. The distance
from the point to the origin, which is ¾
5
, thus serves
as a measure for fault detection. Furthermore, the
angle formed by the point and the kek-axis
µ =tan
¡1
q
1 ¡V
2
55
jV
55
j
is independent of ¾
5
and can be used to characterize
the relative significance between the measurement
vector error kek and the observation matrix error kGk.
Furthermore, the magnitude of
p
SSE
0
can be shown
to be
p
SSE
0
=
¾
5
jV
55
j
=
¾
5
cos µ
:
Thus the square root of the modified SSE is a product
of ¾
5
and 1= cos µ. The factor 1=cosµ can be regarded
as the contribution of the mismatch in the observation
matrix on the test statistic SSE
0
. When the mismatch
in the observation matrix is more significant, the
angle µ increases and the modified SSE increases
accordingly. On the other hand, when the model
mismatch due to the observation matrix is small, the
norm ke
tls
k is close to the square root of the modified
sum of squared errors. The two-dimensional metric,
either the point (ke
tls
k,kG
tls
k
F
) in the fault detection
plane or the measure (¾
5
,µ),canthenbeusedforGPS
fault detection and health monitoring.
IV. SIMULATION RESULTS
In this section, the simulation as well as test
results of the above positioning and integrity
monitoring algorithms are discussed. To assess the
two positioning algorithms discussed in Section II,
a record of GPS measurements is obtained. In this
25-epoch record, eight GPS satellites are observed
whose PRN number are, respectively, 2, 4, 7, 14, 16,
18, 24, and 29. Figs. 2—4, respectively depict the x,
y, and z axis positioning results using the total least
squares, optimization, and least squares methods.
Since there are no failures in this set of record, the
CORRESPONDENCE 333

Fig. 2. Comparison of total least squares, optimization, and least
squares methods in GPS positioning (X direction).
Fig. 3. Comparison of total least squares, optimization, and least
squares methods in GPS positioning (Y direction).
three methods essentially provide the same position
and clock bias estimates.
Using the afore-mentioned measurement data,
a simulated pseudo-range error is injected into one
of the pseudo-range measurement at the 15th epoch
(approximately 140 s). The least squares solution and
total least squares solution are then evaluated. Using
the position solution based upon healthy measurement
record as the nominal solution, the position errors of
the least squares and total least squares solutions are
plotted in Fig. 5. It is found that the total least squares
solution is less susceptible to faulty measurement. The
resulting time plot of the singular values is presented
in Fig. 6. Note that before the failure, the minimal
singular value ¾
5
is close to zero, meaning that the
total least squares and least squares solutions are
approximately the same. Once the failure occurs, there
is a jump on the minimal singular value, implying
Fig. 4. Comparison of total least squares, optimization, and least
squares methods in GPS positioning (Z direction).
Fig. 5. Position error using total least squares and least squares
methods.
Fig. 6. Singular values in total least squares positioning.
334 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 36, NO. 1 JANUARY 2000

Fig. 7. GPS fault detection using test statistic.
Fig. 8. GPS fault detection in the detection plane.
that the set of equations are no longer consistent and
hence a fault is likely to occur. The false can also be
detected in the test statistic (
p
SSE
0
)plotinFig.7.
Clearly, in the presence of faults, this test statistic
is able to react quickly and issue an alarm. In the
two-dimensional fault detection kek¡kGk
F
plane of
Fig. 8, the points (ke
tls
k, kG
tls
k
F
) before and after the
failure fall into two clusters. A decision boundary
based either on statistical reasoning or simulations
can be devised for receiver autonomous integrity
monitoring.
V. C O N C LU S I O N S
In this paper, an error-in-variable model of GPS
measurement is obtained. In contrast to standard
GPS measurement models, the observation matrix
in this model is subject to perturbations. The
perturbation can further be classified as unstructured
or structured cases. Wi th respect to both perturbed
models, algorithms are developed to determine the
position/clock bias update as well as estimates of the
mismatches. In the unstructured case, the total least
squares approach is adopted. In the structured case,
an optimization method is explored. Comparisons
between the total least squares method and the least
squares method are made. The issue of receiver
autonomous integrity monitoring is addressed using
the proposed model and solution. It is found that
the SSE is readily available in the total least squares
solution. Statistical properties of the modified SSE
are derived to enhance the selection of the threshold
in integrity monitoring. Furthermore, a test vector
which is the combination of the level of mismatch
in the observation matrix and pseudo-range vector
is proposed as an alternative in integrity monitoring.
The algorithms are verified using measurement and
simulated data.
APPENDIX. DISTRIBUTION OF MODIFIED SUM OF
SQUARED ERRORS
The n £4matrixH is of rank 4. Let H admit
the singular value decomposition H = U
T
h
§
h
V
h
in
which both U
h
and V
h
are orthogonal and §
h
is an
n £4 matrix with non-zero singular values on its
diagonal. The singular values of H are denoted by
¹
i
for i = 1,2,3,4. The matrix I
n
¡¾
¡2
5
HH
T
can
then be shown to admit the spectral decomposition
I
n
¡¾
¡2
5
HH
T
= U
T
h
(I ¡¾
¡2
5
§
h
§
T
h
)U
h
= U
T
h
D
h
U
h
in
which D
h
is a diagonal matrix whose ith diagonal
element ¸
i
is
¸
i
=
8
<
:
1 ¡
¹
2
i
¾
2
5
i = 1,2,3,4
1 i =5,6,:::,n
:
Let u = U
h
q, then the modified SSE can be written as
SSE
0
= q
T
(I ¡¾
¡2
5
HH
T
)
¡2
q = u
T
D
¡2
h
u =
n
X
i=1
¸
¡2
i
u
2
i
where u
i
is the i th component of u. Suppose that the
components of the pseudo-range residual vector are
identically distributed and independent, zero-mean
Gaussian processes w ith variances ¾
2
0
. Then the
components of the vector u are also Gaussian
processes with the same variances. The SSE
0
is a
weighted sum of n (one-dimensional) chi-squared
distributions. Since there are n ¡4 identical weights,
i.e., ¸
i
=1for i =5,:::,n, the statistical distribution
of the SSE
0
becomes a summation of four Gamma
distributions with different parameters (due to
different weights) and an n ¡4 degrees of freedom
chi-squared distribution. More precisely, the modified
SSE can be represented as SSE
0
= y = y
1
+ y
2
+ y
3
+
CORRESPONDENCE 335

y
4
+ y
5
. The joint density function of y
1
, y
2
, y
3
, y
4
,
and y
5
is
p(y
1
,y
2
,y
3
,y
4
,y
5
)
=
0
B
B
B
B
@
4
Y
i=1
µ
¸
2
i
2¾
2
0
¶
1=2
y
¡1=2
i
exp
µ
¡
¸
2
i
2¾
2
0
y
i
¶
¡
µ
1
2
¶
1
C
C
C
C
A
£
µ
1
2¾
2
0
¶
1=2
y
(n¡4=2)¡1
5
exp
µ
¡
1
2¾
2
0
y
5
¶
¡
µ
n ¡4
2
¶
where ¡ (¢) is the Gamma function. Given n, ¾
0
,
and ¸
i
for i = 1,2, 3, 4, the distribution p(y)canbe
evaluated using convolutions. Consequently, given
the desired false alarm rate, the threshold can be
determined.
JYH-CHING JUANG
Department of Electrical Engineering
National Cheng Kung University
Tainan , Tai wan
E-mail: (juang@mail.ncku.edu.tw)
REFERENCES
[1] Ackroyd, N., and Lorimer, R. (1990)
Global Navigation, A GPS User’s Guide.
London: Lloyd’s of London Press Ltd, 1990.
[2] Brown, R. G. (1992)
A baseline RAIM scheme and a note on the equivalence
of three RAIM methods.
In Proceedings of the National Technical Meeting, 1992.
[3] Brown, R. G. (1996)
Receiver autonomous integrity monitoring.
In B. W. Parkinson and J. J. Spilker Jr. (Eds.), Global
Positioning System: Theory and Applications,NewYork:
AIAA, 1996.
[4] Chin, G. Y., and Kraemer, J. H. (1992)
GPS RAIM: Screening out bad geometries under
worst-case bias condition.
Journal of Navigation, 1992—1993.
[5] Cole, R. W. (1992)
GPS integrity working group status.
In Proceedings of ION-GPS 92, 1992, 895—901.
[6] Golub, G. H., and Van Loan, C. F. (1983)
Matrix Computations.
Baltimore, MD: The John Hopkins University Press,
1983.
[7] Hofmann-Wellenhof, B., Lichtenegger, H., and Collins, J.
(1992)
Global Positioning System, Theory and Practice.
New York: Springer-Verlag, 1992.
[8] Van Huffel, S., and Vandewalle, J. (1989)
Algebraic connections between the least squares and total
least squares problems.
Numerische Mathematik (1989).
[9] Van Huffel, S., and Vandewalle, J. (1989)
On the accuracy of total least squares and least squares
techniques in the presence of errors on all data.
Automatica (1989).
[10] Kaplan, E. D. (Ed.) (1996)
Understanding GPS Principles and Applications.
Boston, MA: Artech House, 1996.
[11] Kee, C., and Parkinson, B. W. (1996)
Wide area differential GPS (WADGPS): Future
navigation system.
IEEE Transactions on Aerospace and Electronics Systems,
32, 2 (Apr. 1996), 795—808.
[12] Lee, Y. C. (1992)
Receiver autonomous integrity monitoring (RAIM)
capability for sole-means GPS navigation in the oceanic
phase of flight.
In IEEE PLANS, 1992.
[13] Parkinson, B. W., and Axelrad, P. (1988)
Autonomous GPS integrity monitoring using the
pseudorange residual.
Navigation (1988).
[14] Parkinson, B. W., and Spilker, J. J., Jr. (Eds.). (1996)
Global Positioning System: Theory and Applications.
New York: AIAA, 1996.
[15] Loh, R., and Fernow, J. (1994)
Integrity monitoring requirements for FAA’s GPS
wide-area augmentation system (WAAS).
In IEEE PLANS, 1994.
[16] RTCA (1991)
Minimum Operational Performance Standards for
Airborne Supplemental Navigation Equipment Using
Global Positioning System (GPS) DO-208.
Radio Technical Commission for Aeronautics, 1991.
[17] Sturza, M. A. (1988)
Navigation system integrity monitoring using redundant
measurements.
Navigation (1988—1989).
ProofofCFARbytheUseoftheInvariantTest
Many adaptive detectors regulate the detection threshold
adaptively through an estimate of the clutter power level, which is
formed by reference cell samples. This paper proves by the use of
the invariant test that a large class of adaptive detectors possesses
the constant false-alarm rate (CFAR) property when they satisfy
certain weak conditions. The proof reveals the mechanism of this
class of CFAR detectors and provides a general method of proving
the CFAR property.
I. INTRODUCTION
The typical radar signal detection problem is
to determine the existence of a target over certain
Manuscript received February 20, 1999, revised July 13, 1999;
released for publication December 14, 1999.
IEEE Log No. T-AES/36/1/02614.
Refereeing of this contribution was handled by L. M. Kaplan.
0018-9251/00/$10.00
c
° 2000 IEEE
336 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 36, NO. 1 JANUARY 2000