Journal of Global Positioning Systems (2002)
Vol. 1, No. 2: 96-105
Joint Treatment of Random Variability and Imprecision in GPS Data
Analysis
Hansjörg Kutterer
DGFI, Marstallplatz 8, D-80539 Munich, Germany
e-mail: [e-mail]; Tel: [phone]; Fax: [phone]
Received: 27 November 2002 / Accepted: 17 December 2002
Abstract. In the geodetic applications of the Global
Positioning System (GPS) various types of data
uncertainty are relevant. The most prominent ones are
random variability (stochasticity) and imprecision.
Stochasticity is caused by uncontrollable effects during
the observation process. Imprecision is due to remaining
systematic deviations between data and model due to
imperfect knowledge or just for practical reasons.
Depending on the particular application either
stochasticity or imprecision may dominate the uncertainty
budget. For the joint treatment of stochasticity and
imprecision two main problems have to be solved. First,
the imprecision of the original data has to be modelled in
an adequate way. Then this imprecision has to be
transferred to the quantities of interest. Fuzzy data
analysis offers a proper mathematical theory to handle
both problems. The main outcome is confidence regions
for estimated parameters which are superposed by the
effects of data imprecision. In the paper two applications
are considered in a general way: the resolution of the
phase ambiguity parameters and the estimation of point
positions. The paper concludes with numerical examples
for ambiguity resolution.
Key words: Fuzzy data analysis, imprecision, fuzzy
confidence regions, GPS, ambiguity resolution
1 Introduction
Today, the Global Positioning System (GPS) is
intensively used in geodetic applications as it is efficient
and easy to access. The GPS consists of nominally 24
satellites on six orbital planes. It supplies the broadcast
transmission of one-way microwave signals on two
frequencies from the satellites to the individual ground
stations. The 3D position of the GPS ground antenna and
the receiver clock offset can be determined by
simultaneously observing the signals of at least four
satellites. This yields the satellite-receiver distances either
directly using the code observations or indirectly via the
(carrier wave) phase observations. For the second type of
observations, the ambiguity parameters have to be
determined. For further reading on GPS and on ambiguity
resolution techniques see, e.g., Hofmann-Wellenhof et al.
(1997), Parkinson and Spilker (1996), Teunissen and
Kleusberg (1998).
GPS observations are biased by a variety of physical
effects which have to be considered and handled in data
processing. There are mainly three groups of causes. The
most important one is due to the propagation of the
signals. As the path of the GPS signals leads through the
complete atmosphere, ionospherically and
tropospherically caused travel-time delays have to be
taken into account. They are superposed by multipath
effects due to signal reflections in the vicinity of the
tracking GPS antenna. The second group comprises all
satellite effects like, e.g., signal transmission delays,
satellite clock errors, satellite orbit errors, and satellite
antenna offsets. Station and receiver effects like, e.g.,
signal reception delays and receiver clock errors belong
to the third group. In addition, the GPS data processing
results show characteristics due to the software and the
operator.
Several techniques can be applied to reduce or eliminate
most of the systematic effects such as the use of
correction models with fixed or free parameters or of
linear combinations of the GPS observations such as
double differencing. Longer-term periodic signals such as
diurnal ones can be weakened if the observation time is
sufficiently long. However, such effects can not be
eliminated completely due to the imperfect knowledge

Kutterer: Joint Treatment of Random Variability and Imprecision 97
and the approximate character of the models in use,
respectively. Hence, the uncertainty due to remaining
systematic effects (imprecision) must be taken into
account in addition to the random variability
(stochasticity) of the observations. Kutterer (2001a, 2002)
gives a general discussion of uncertainty in geodetic data
analysis. Imprecision is particularly relevant in case of
long distances between the GPS sites or very short
observation intervals as in neither case it is possible to
completely describe and remove systematic effects. This
paper can be seen as an extension and generalization of
the results given by Kutterer (2001b).
Fuzzy data analysis (Bandemer and Näther, 1992; Viertl,
1996) has proven to be an adequate mathematical tool to
handle imprecision. Moreover, the combination of
methods from stochastic and fuzzy theory allows the
extension of classical geodetic data analysis to account
for the effects due to superposed imprecision. In the
following, the basics of fuzzy data analysis are presented.
Two alternatives for the definition of fuzzy vectors are
discussed. If the classical formulas of statistics are
fuzzified by means of Zadeh’s extension principle,
stochasticity and imprecision can be treated
simultaneously. Thus, imprecise confidence regions for
the ambiguity parameters and for the point positions can
be defined and discussed. At the end of the paper the
results of simulation studies are given. They illustrate the
applicability of the theory and quantify the impact of
imprecision.
2 Basics of fuzzy data analysis
Fuzzy-theory was initiated by Zadeh (1965) in order to
extend classical set theory by describing the degree (of
membership) that a certain element belongs to a set. In
classical set theory the membership degrees are either 1
(is element) or 0 (is not element). In fuzzy set theory the
range of membership degree is [0,1]. Thus, a fuzzy set is
defined as
{}
[
~
~~ A (x , m(x)) x X , m : X 0 , 1
AA
=∈→
]
. (1)
The degree of membership is given by the membership
function which is denoted by. X is a classical set
such as the set R of the real numbers. Important notions
are the support of a fuzzy set (classical set with positive
degrees of membership), the height of a fuzzy set
(maximum membership degree), the core of a fuzzy set
(the classical set with membership degree equal to 1), and
the
α
-cut of a fuzzy set (classical set with membership
degree greater equal α ∈ [0,1]). For further reading see
standard references on fuzzy data analysis such as
Bandemer and Näther (1992) or Viertl (1996).
m(x)
A
~
The most important operation in fuzzy-theory is the
intersection of fuzzy sets. It is defined through the
resulting membership function
(
)
m min m, m
ABA B
~~~ ~
∩
= (2)
This definition is mostly used. Other consistent
extensions of the classical intersection operator are
available. See, e.g., Dubois and Prade (1980).
Fuzzy numbers can be defined based on fuzzy sets. A
fuzzy number is a fuzzy set with a single element core and
compact α-cuts. The L-fuzzy numbers defined by Dubois
and Prade (1980) are widely used. They are exclusively
considered in this paper. Their membership function is
given by a strictly decreasing non-negative reference
function L with [0,1] as the range of values.
mx
Lxx
xxxx
Lxx
xxxx
else
x
m
s
lm
m
s
u
~ ()
,
,
,
=
−




≤<
−




≥≥











0
m
L
(3)
Due to the single element core, L(0) = 1. For a graphical
sketch of a L-fuzzy number with a linear reference
function see Fig. 1. Formally, it can be represented by
. The mean point is denoted by x
~
X (
x, x)
ms
=
m. The
spread x
s serves as a scale factor. In practice, a typical
membership function vanishes outside the interval given
by the lower bound xl and the upper bound xu.
Fig. 1 L-fuzzy number with linear membership function (triangular
fuzzy number)
The extension principle (Zadeh, 1965)

98 Journal of Global Positioning Systems
(
)
~~~~
~~
B g (A,,A) :
m(y) = supmin (m(x),,m(x)) yY
1n
B
(x1,,x
n) X1Xn
g(x1,,x
n) y
A11Ann
=⇔
∀∈
∈××
=
K
K
KK
K
~
(4)
allows the generalization of functions with real arguments
to functions with fuzzy arguments. For L-fuzzy numbers,
the extended arithmetic rules are, e.g.,
~~
~~
~
X + Y
X - Y
=+
=− +
=
(x+y, xy)
(xy, xy)
aX (a x,a x)
mms sL
mmssL
msL
Addition
Subtraction
Multiplication by a real number
(5a, b, c)
The type of the reference function is preserved. The
arithmetic operations can be carried out simply based on
the mean points and the spreads. Please note that
subtraction is not the inverse of addition. In fuzzy data
analysis the spreads are regarded as measures of
fuzziness or imprecision, respectively. Obviously, they
are just added (linear propagation) in contrast to the
addition of variances (quadratic propagation of the
standard deviations) according to the Gaussian law of
error propagation.
Fig. 2 Two-dimensional fuzzy vectors by the minimum rule
There are several possibilities to combine fuzzy numbers
to a vector; see, e.g. Viertl (1996), Kutterer (2002). The
mostly used way is to build a fuzzy vector by the
minimum rule, i.e. using the minimum operator according
to Eq. (2). In the 2D case this reads as
(
)
m (m(x),m(y))
XY
~~
Z
z = min
~
(6)
For a graphical representation (linear reference function L
as in Fig. 1) see Fig. 2. Such fuzzy vectors are called non-
interactive (independent components).
A linear mapping of a fuzzy vector by the minimum rule
can be approximated by the tightest inclusion
m() ( , )
L
FZ ms
F zFz
~
z
=
(7)
The operation . yields the matrix of the absolute values
of the matrix components.
Interactive fuzzy vectors can be defined through
(
)
(
)
(
)
(
)
mh
T1
~
Zm
zzzUz
=− −
−
m
z
(8)
Fig. 3 Two-dimensional fuzzy vector of elliptic type
The function h is monotonously decreasing and non-
negative with h(0)=1. The spreads and the interaction of
the components are quantified in the positive definite
uncertainty matrix U. Interaction is principally present
due to the quadratic form which is the argument of h.
Fuzzy vectors according to Eq. (8) are called fuzzy
vectors of elliptic type. See Fig. 3 for a graphical
representation; the function h of non-negative real
arguments p and the matrix U are chosen as
(
)
( )
hp maxpx
y
s
s
=− =








10 0
0
12
2
2
,, U
Linear mappings of fuzzy vectors of elliptic type are
given in closed form by
(
)
(
)
(
)
(
)
m=
TT1
~~
Y =
−
−−





FZ mm
yyyFUFyy h (9)
The simplicity and closeness of Eqs. (8) and (9) is in
contrast to the problems of motivating and formulating
interactive (i.e. fuzzy-theoretically dependent)
components. The equivalence with the Gaussian error
propagation (variance propagation law) is obvious. But it
has to be kept in mind that the interpretation is different
since the membership functions must not be confused
with the independently defined density functions of
probability theory. Nevertheless, a quadratic propagation

Kutterer: Joint Treatment of Random Variability and Imprecision 99
of spreads is available by means of fuzzy vectors of
elliptic type; see Eq. (9).
3 Modeling and propagation of data imprecision
Fuzziness and imprecision are considered as being
identical in the following as it is common practice in
fuzzy data analysis. Hence, fuzzy data analysis can be
applied to handle the impact of observation imprecision
on the parameters of interest. As already motivated in
Section 1, there are several sources of imprecision in GPS
data acquisition and analysis. Hence, both stochasticity
and imprecision have to be considered in a general
combined approach. Stochasticity is assumed to be
superposed by imprecision. This is the basic condition of
the extension principle according to Eq. (4).
There are three steps to derive the imprecision of the
quantities of interest. First, the imprecision of a single
observation has to be described by means of a fuzzy (or
imprecise) number. This can be based on a questionnaire
to be completed by experts in order to assess the
particular application; see, e.g., Kutterer (2002) for
details. Second, the fuzzy numbers representing the
imprecise observations have to be combined to a fuzzy
(or imprecise) vector. This can be based on the two types
of fuzzy vectors given in Section 2; see Eq. (6) for the
definition of a fuzzy vector by the minimum rule and Eq.
(8) for a fuzzy vector of elliptic type. Third, the extension
principle according to Eq. (4) has to be applied to the
real-valued functional expressions. Here, the least-
squares estimator (LSE)
(
)
$
β = XWX XWy
TT
-1
(10)
m
of the (deterministic) parameters β in a Gauss-
Markoff model is considered first. Its variance-
covariance matrix (vcm) reads as
(
)
Σ
ββ
$$
=
−
σ
0
21
XWX
T
(11)
The column-regular [n×u]-dimensional configura-
tion matrix is denoted by X and the [n×n]-
dimensional regular weight matrix of the
observations by W. The vector of the observations
is represented by y. The a priori variance factor is
given by σ .
0
2
The second quantity of interest is the (1-γ)-
confidence region für the expected value µ of
β
which is given by
$
(
)
(
)
(
)
K
T
u11
2
−−
=≤






γγ
χ
$$$
$$
,
βµµ−βΣµ−β
ββ
−1
(12)
with the (1-γ)-fractile value of the
χ
χ
γ n,1
2
−
2
∈
−distribution with u degrees of freedom.
3.1 Extended least-squares estimator
In the first case (LSE according to Eq. (10)) the extension
principle reads as
(
)
()
(
)
mm
TT
u
$
~
$
sup ~
$
$
β
ββ
β
=∀
∈
=
y
XWX XWy
yy
R
R
n
-1
(13)
with the membership function of the vector of
the n imprecise observations and the
membership function of the vector of the u imprecise
estimated parameters. The use of fuzzy vectors by the
minimum rule yields
( )
~
yy
(
)
m
$
$
β
∼
β
(
)
( ) ( )
m
TT
m
TT
s
L
$
$
,
β
∼
β =




XWX XWyXWXXWy
-1 -1
(14)
according to Eq. (7). The use of fuzzy vectors of elliptic type yields
(
)
(
)
(
)
(
)
(
)
m
TT T
$
$$$ $$
β
∼
βββ ββ h =
T-1
−



−





m
-1 -1
m
UXWX XWWXXWX
according to Eq. (9) with
( )
$
β
m
TT
m
= XWX XWy
-1
This can be rewritten as
(
)
(
)
(
)
m
$$$
$$$ $$
βββ
−1
∼
βββ ββ h =
T
−−





m
U
m
(15a)
with the imprecision matrix
(
)
(
)
UXWX XWWXXWX
$$
ββ
=
TT T
-1 -1
U
(15b)

100 Journal of Global Positioning Systems
respectively; see, e.g., Kutterer (2002). Please note that
for both the fuzzy vectors by the minimum rule and the
fuzzy vectors of elliptic type the mean point vectors
β
are identical with the classical (precise) least-squares
estimators. Thus, both presented fuzzy extensions are
consistent with the real-valued case.
$
m
3.2 Extended confidence regions
In case of the confidence regions, see Eq. (12), the
extension principle reads as
(
)
()
( )
mm
K
K
~
$$
$
sup
$
1
1
−
=∀
∈−
γ
γ
βµ
µβ
β
∼
R
u
∈ β
$
β
m
$
m
(16)
Eq. (16) represents a constrained optimization problem.
The imprecise confidence region is the solution of this
problem. In order to obtain a closed-form expression for
the results, fuzzy vectors of elliptic type
( ) ( )
mh
T1
$$$
~
()
$
βββ
µµβµ =− −





−
m
U (17)
as given in Eqs. (15a, b) are solely considered in the
following. The function h is strictly decreasing for non-
negative arguments. Hence, the supremum or maximum
functional value, respectively, is obtained with the
minimum argument value (quadratic Euclidean distance
with respect to ) U
$$
ββ
(
)
(
)
(
)
d
2
2
,
$$ $$
,
$$
Um m
U
ββ ββ
µβµβ µβ =− −
−
T1
under the side condition
(
)
(
)
(
)
µβµµ−βΣµ−β
ββ
−1
∈= ≤






−−
K
T
u11
2
γγ
χ
$$$
$$
,
An equivalent side condition is
(
)
(
)
[]
µ−βΣµ−β
ββ
−1
$$
,,
$$
,
T
u
with
=∈
−
κκχ
γ
22
1
2
0
Thus, the objective function to be minimized with respect
to (w.r.t.) µ reads as
( )
( ) ( ) ( ) ( ) [
Φ µµ βµβµ−βΣµ−β
ββ ββ
−1
=−− −−




∈
−−
$$$$
,,
$$ $$
, mm
U
T1
λκκ
γ
T
u
with
22
1
2
0
]
χ
$
$
(18)
and with the Lagrangian multiplier λ.
Two special cases can be distinguished: Obviously, as
long as , there is always a
µβ
with and hence .
Consequently, the resulting value of the membership
function is equal to one. Hence, the obtained classical set
corresponds to the confidence region given by Eq. (12).
(
)
$
ββ ∈
−
K
m1
γ
µβ =
$
m
(
)
∈
−
K
1
γ
$
)
0,
$
Um
µβ =
(
d
2
2
,
$$
ββ
In all other cases, i.e., , there is
. Then the problem can be
(
)
$
ββ ∉
−
K
m1 γ
κχ
γ
2
1
2
==
− u,
constant
understood in a geometrical way as the determination of
the distance between the point and the hyperellipsoid
$
β
m
(
)
(
)
(
)
K
T
u11
2
−−
==






γγ
χ
$$$
$$
,
βµµ−βΣµ−β
ββ
−1
. As
now the distance is in any case positive, the
corresponding values of the membership function are less
than one. Hence, the confidence region according to Eq.
(12) is the core (see Section 2) of the extended
confidence region.
The objective function given in Eq. (18) now reads as
( )
(
)
(
)
(
)
(
)
Φ µµ βµβµ−βΣµ−β
ββ ββ
−1
=−− −−




=
−−
$$$$
,
$$ $$
, mm
U
T1
λκ
γ
T
u
with
22
1
2
κχ (19)
The determination of the stationary point which refers to
the minimum requires the differentiation of Φ w.r.t.
µ which yields
( )
(
)
(
)
∂
∂λ
Φ µ
µµβµ−β Σ
ββ ββ
−1
=− −=
−
22
$$
$$ $$
m
U
T1 T
0
and
( ) ( )
1
U
m
$$ $$
$
ββ ββ
−1
µβΣ µ−β
−
−− =λ 0
$
$
. (20)
Hence,
( ) ( )
11
µ−βΣβ−β
ββ ββ
−1
ββ
$$
$$$$$$
=−
−−−
UU λ
1
m (21)
Differentiation of Φ w.r.t. λ yields
( ) ( )
µ−β Σµ−β
ββ
−1
$$
.
$$
T
−=κ
2
0

Kutterer: Joint Treatment of Random Variability and Imprecision 101
Insertion of Eq. (21) into the last one finally yields the single equation
(
)
(
)
(
)
(
)
$$ $$
$$$$$$ $$$$
β−β ΣΣ Σβ−β
ββ ββββββ ββ
m
T
m
−−
−−
λλ UU
11
2
=κ
$$
$
which is nonlinear w.r.t. the single unknown λ. λ can be
determined numerically by a common root-finding
method. When its actual value is known, Eqs. (20) and
(21) yield
(
)
( )
(
)
11
µβΣ µ−βΣΣβ−β
ββ ββ
−1
ββ ββ
−1
ββ ββ
−1
ββ
−== −
−−−
$$
$$ $$$$ $$$$$$$$
m
UUUU λλλ
1
m (22)
Finally the membership function of the imprecise confidence region is obtained regarding Eq. (16) as
(
)
(
)
(
)
(
)
m
h
K1- T1
~
$$
(
$
)
,
$$
$$
,
$
γ
γ
γ
β
ββ
µβµβ ββ
ββ
=
∈
−−




∉





−
−−
1
1
1
K
K
m
m mm
U
. (23)
with
(
as given in Eq. (22).
)
µ−β
$
m
Please note that the second derivative of Φ w.r.t. µ
( )
( )
∂
∂λ
2
2
2
Φ µ
µΣ
ββ ββ
−1
=−
−
U
$$ $$
1
has to be positive definite to assure a minimum what can
easily be checked.
Eq. (23) describes the extension of classical confidence
regions to confidence regions where the originally
random-type quantities of interest are superposed by
imprecision. It is a significant generalization of the
corresponding Eq. (20) in Kutterer (2001b) where U
and had to be proportional. Like in the analysis of
GPS observations both the phase ambiguity search spaces
and the precision of point positions are represented by
confidence regions, Eq. (23) plays the key role in any
case when imprecision has to be taken into account.
$$
ββ
Σ
ββ
$$
Fig. 4 shows exemplarily for the 2D case the
superposition of a classical (1-γ) confidence ellipse and
an imprecise 2D vector of elliptic type. It is obvious that
the superposition of the two quantities does not yield an
elliptic quantity. The maximum membership degree is
obtained for the mean point of the imprecise vector and
the corresponding confidence ellipse. This is only valid
for the classical confidence region.
Fig. 4 Imprecise confidence ellipse as resulting from the superposition of a classical (precise) confidence ellipse and an imprecise vector. The
quantities are placed separately for the sake of better representation. The light-gray lines indicate isolines of the membership values.
The qualititative difference of the presented results
from the common unterstanding of accuracy is
obvious from the extended LSE and the extended
confidence regions. Actually, the introduced
combined measures of stochasticity and imprecision
are closer to the idea of accuracy in practical
applications. The classical statistical point of view
implies reduction of uncertainty just by repetition of
observations. If fuzzy-theory is used to model and
handle imprecision this is not possible. The amount
of imprecision is kept when observations are
repeated. Imprecision can only be reduced outside
the particular observation scenario as it is according
to common sense.

102 Journal of Global Positioning Systems
4 Extended GPS phase ambiguity search spaces
The linearized functional model of GPS code and phase
observations reads as
(
)
E yX A Z ==+ βξζ
(24)
Σ
with the expectation E(.), the real-valued parameters
ξ such as coordinates and the integer ambiguity
parameters ζ. The matrices A and Z denote the two
corresponding components of the configuration matrix X.
Please note that the vector y comprises both code and
phase observations or differences, respectively. For the
following there is no need to distinguish between
undifferenced and double-differenced observations. The
only impact is then on the adequate parametrization. The
rows of matrix Z which correspond with the code
observations are naturally equal to zero. The vcm or
dispersion matrix of y given by
(
)
D y
yy
= Σ (25)
A real-valued approximation of the integer ambiguity
parameters is obtained by a least-squares estimation
weighted by as
yy
−
1
$
ζ = Fy (26a)
with
(
)
(
)
FZAAAAZ
ZAAAA
yy yyyyyy
yy yyyyyy
=−










×−










−−
−−−
−−
−−
TTT
TTT
ΣΣ ΣΣ
ΣΣ ΣΣ
11
11
1
11
11
(26b)
what leads to
( )
ΣΣΣΣ ΣΣ
ζζ
$$
== −










−− −
−−−
FF ZAAAAZ
yyyyyyyyyy
11 1
11
1
TTTT (27)
for the vcm of the real-valued estimates of the integer
ambiguity parameters. Consequently, the corresponding
(1-γ)- confidence hyperellipsoid regarding Eq. (12) reads
as
( ) ( ) ( )
K
T
f1
2
−
=






γ
χ
$$$
$$
,
ζµ µ−ζΣµ−ζ
ζζ
−1
1 −
≤
γ
(28)
with f denoting the number of ambiguity parameters; see
Kutterer (2001b). The confidence region given in Eq.
(28) can be set up based on code observations only. It
serves as a search space for the integer ambiguity
parameters.
The methods proposed in literature for ambiguity
resolution differ mainly in the strategy how to identify the
„correct“ ambiguity parameter. In any case they depend
and rely on the adequateness of the models given in Eqs.
(24) and (25). In particular, the functional model has to
be accurate in the meaning that the existing errors are
only assignable to the observations and that they are all
and exclusively random. However, this does not hold in
general. This assumption is certainly not suitable for
short observation times or real-time applications and for
long baselines. Hence, the imprecision of the
observations has to be assessed and modelled as
mentioned above. Then it has to be superposed to the
search space by applying the procedure shown in Section
3, in particular by using Eq. (23).
As the extended search space is obviously enlarged, more
candidate vectors have to be taken into account for
ambiguity resolution. If a rounding procedure is applied
such as the LAMBDA method (Teunissen and Kleusberg,
1998), there is no change for the integer-estimated
ambiguity vector. However, due to the increased number
of candidates the separability of the best and the second-
best solution may be reduced which leads to more reliable
results. In all other methods like, e.g., On-The-Fly
algorithms (Abidin, 1993; Leinen, 2001), the degree of
imprecision given by the membership function of the
extended search space offers additional information for
the validity of the solution.
5 Extended error measures for GPS site positions
Imprecise (1-γ)-confidence regions (ellipses and
ellipsoids, respectively) for the 3D positions of GPS sites
can be given in analogy to the ambiguity resolution
presented in Section 4. As soon as the ambiguity
parameters are known (and fixed, respectively), the phase
observations can be used as highly precise distance
observations. The functional model according to Eq. (23)
simplifies to
(
)
E with yA yyZ == ξ , − ζ
, (29)
but the stochastic model represented by the vcm
( )
D y
yy yy
== ΣΣ
(30)
is unchanged because the introduced ambiguities are
considered as exact. Least-squares estimation of the
remaining real-valued unknown parameters like, e.g.,
position coordinates or tropospheric parameters, yields
(
)
$
ξΣ Σ =
−−−
AAA
yy yy
TT111
y (31)

Kutterer: Joint Treatment of Random Variability and Imprecision 103
with the corresponding vcm
( )
ΣΣ
ξξ
$$
=
−−
AA
yy
T1 1
(32)
The submatrix for a set of parameters such as the
coordinates of a particular point is obtained by means of a
selection matrix like, e.g.,
()( )
[
S0 I0
ii
=
×− ×− 33 1333i
]
u
(33)
Hence, for the position of the ith point it is
(
)
$
ξΣ Σ
ii
TT
=
−−−
SAA Ay
yy yy
111
(34)
and
( )
ΣΣ
ξξ,
$$
ii
T
i
T
=
−−
SAA S
yy
11
(35)
Its classical (1-γ)-confidence ellipsoid reads as
( ) ( ) ( )
K
ii
T
ii1
2
−
=






γ
χ
$$$
$$
ξµ µ−ξΣµ−ξ
ξξ,
−1
3,1 −
≤
γ
The corresponding imprecise confidence ellipsoid is
obtained by means of the procedure given in Section 3,
mainly using Eq. (23).
6. Examples
In the following, the impact of the proposed
superposition of stochasticity and imprecision on the size
of the search space is shown exemplarily. The main idea
is to extend the classical search space by scaling the
semi-axes of the respective confidence hyperellipsoid so
that the result is the tightest inclusion of the support of
the imprecise confidence region. From a practical point of
view this is an important first step to consider imprecise
observations. Below, several GPS real-time scenarios are
simulated and discussed. This section is organized as
follows. First, the general configuration and the
estimation procedure are given. Second, the modeling of
the imprecise observations and the derivation of the
imprecise vector of the ambiguity parameters are
described. Third, the resulting scaling factors for the
classical search spaces are compiled and discussed. The
results of the simulation runs were derived by means of
the procedure which was described in Section 3 and
which led to Eq. (23).
The scenarios are based on the nominal GPS
configuration with 24 satellites which was simulated
according to orbital elements published by Parkinson and
Spilker (1996). The respective solutions are based on
single epoch observations to all visible satellites. The
number of satellites was controlled by means of an
elevation mask: If n satellites were visible and m < n
satellites were considered, those n-m satellites with
lowest elevation were dropped. The (1-γ)-confidence
hyperellipsoids are based on a code-only approximate
position. The standard deviation of the code observations
was chosen as 0.3 m in order to obtain realistic
magnitudes. The integer ambiguity parameters were
approximated by means of a least-squares adjustment
(float solution) as it is common practice in GPS data
analysis. From Fig. 4 it is already clear that the resulting
imprecise confidence regions are no hyperellipsoids but
more complex quantities. They are fuzzy supersets of the
classical precise hyperellipsoids.
The actual ratio of stochasticity and imprecision of the
observations depends on the respective configuration. It
is part of the complete uncertainty budget. The relevant
types of uncertainty can be described and quantified by
experts in several ways using a detailed questionnaire: A
sensitivity analysis of the applied correction models for
example gives insight in critical parameters, site-
dependent effects such as multipath can be assessed by
studying the local situation, and extensive controlled
variations of the observation configuration indicate the
magnitude of external effects.
In the following examples some illustrative values were
chosen for the amount of imprecision. The imprecise
vector of the real-valued approximation of the ambiguity
parameters was represented as an imprecise vector of
elliptic type. It was deduced from its range of values
(convex polyhedron) which is directly computable from
the imprecision of the observations in the considered
cases because of the relatively low dimension of the
parameter space. This polyhedron was then enclosed by
the tightest possible hyperellipsoid in order to define the
support of an imprecise vector of elliptic type. A linear
reference function was chosen as in Figures 1 and 3 and
Eq. (3), respectively.
For the first simulation runs the imprecision of all code
observations was introduced as 0.03 m (10% of the value
of the standard deviation) what is a very restrained
assumption. Table 1 shows the scaling factors of the
semi-axes of the classical search space which were
obtained for GPS observation sites in three different
latitudes: equatorial region (Latitude = 0°), mean latitudes
(45°) and the poles (90°). It is obvious that the scaling
factor depends only slightly on the configuration - less on
to the latitude and more on the number of satellites. The
latter is due to the fact that the amount of imprecision
increases by the number of observations. There is no
significant dependence of the results on the time of
observation. By taking an average value of 1.25 one can
state that the assumed imprecision of 10% requires an
extension of the search space by 25%.

104 Journal of Global Positioning Systems
Tab. 1 Maximum scaling factor for the semi-axes of the classical
(precise) ambiguity search space in the real-time case (single epoch
observations) to take imprecision into account. The standard deviations
of the code observations equal 0.3 m, the imprecision equals 0.03 m.
Latitude →
# of sat. ↓ 0° 45° 90°
4 1.20 1.21 1.20
5 1.23 1.23 1.23
6 1.25 1.25 1.25
7 1.25 1.26 1.27
8 1.27 1.27 1.27
9 1.29
 1.28
In further simulations runs the ratio of imprecision and
stochasticity (in terms of standard deviations) was varied.
Table 2 shows the results which were derived for the GPS
observation site with latitude = 45° for different ratios. A
linear dependence of the scaling factor on the chosen
ratio can be found. In case of identical magnitudes of
stochasticity and imprecision (ratio=1) the semi-axes of
the search spaces need to be increased by a factor
significantly larger than 3.
Tab. 2 Maximum scaling factor for the semi-axes of the classical
(precise) ambiguity search space in the real-time case (single epoch
observations) to take imprecision into account. The standard deviations
of the code observations equals 0.3 m, the imprecision is varied.
% of
std.dev. →
# of sats. ↓
0.1⋅σcode 0.2⋅σcode 0.5⋅σcode 1.0⋅σcode 2.0⋅σcode
4 1.21 1.41 2.03 3.06 5.12
5 1.23 1.47 2.17 3.33 5.65
6 1.25 1.51 2.26 3.52 6.05
7 1.26 1.52 2.31 3.61 6.23
8 1.27 1.54 2.35 3.71 6.41
The quality of the approximation of the actual imprecise
search space by scaling the classical (precise) one
becomes poorer with increasing importance of
imprecision. In such cases the individual scaling factors
for the respective semi-axes can differ significantly. Fig.
4 illustrates this: The imprecise confidence ellipse is
principally obtained by superposing two ellipses;
however, the resulting quantity is not elliptic and cannot
be represented uniquely by an ellipse. Nevertheless, if the
maximum values for the scaling factors are taken the
inclusion property is kept in any case.
The examples indicate that the ratio of stochasticity and
imprecision plays a leading role in the extension of the
classical search space in order to take imprecision into
account. Hence, the traditional procedure of ambiguity
resolution is inadequate when imprecision dominates the
uncertainty budget. A rule-of-thumb for practitioners
reads as follows: The search space needs to be extended
even in the case of low imprecision. When the
observations are likely to be imprecise at least in the
same magnitude as stochasticity the semi-axes of the
search space should be lengthened by a factor of at least
3. In this way the quality of the validation of the resolved
ambiguity vector can be improved. Some remarks on this
topic were also made at the end of Section 4.
7 Conclusions and outlook
As imprecision has to be considered in a variety of
geodetic applications of the GPS the joint treatment of
stochasticity and imprecision in GPS data analysis is
important. Imprecision is an independent type of
uncertainty and in general it can not be reduced or
transformed to stochasticity. Hence, the common
modeling as given in Eqs. (24) and (25) is incomplete
because it does not take imprecision into account. Fuzzy-
theory allows to distinguish strictly between these two
types of uncertainty. For this reason it is suitable to
handle both stochasticity and imprecision. Moreover, it
allows to control the type of propagation of imprecision
from the observations to the parameters; see Eq. (7) for
linear propagation and Eq. (9) for quadratic propagation,
respectively. In both cases the classical least-squares
estimator is kept as mean point of the resulting fuzzy set.
The benefit of the joint treatment of stochasticity and
imprecision for the GPS community is two-fold. On the
one hand the resolution of the phase ambiguity
parameters can be improved by extending the classical
search space by simple scaling. This is a first step to more
reliable results in real-time GPS. On the other hand the
quality of the point positions determined by means of
GPS can be described more thoroughly. It is well known
that their formal precision is too optimistic. Imprecise
confidence regions can objectivy the common measures
of precision and accuracy since imprecision cannot be
reduced by repeated observations.
There are some issues which are worthwhile for further
studies. First, the uncertainty budget of GPS observation
configurations has to be evaluated thoroughly for the
practical application of the presented approach. Thus, a
look-up table for observation imprecision could be
worked out for typical configurations. Second, the notion
of imprecision was based here on L-fuzzy numbers which
imply identical left and right spreads. In a more general
formulation LL-fuzzy numbers can be used which have
identical left and right reference functions but different
spreads; see standard references on fuzzy-theory. In this
way the knowledge of possible asymmetries in remaining
systematic effects could be modeled what would lead
directly to biases in least-squares estimation which have
to be taken into account. Third, it is up to now not
sufficiently understood how imprecision propagates in
practice from the observations to the parameters of
interest. There could be more possibilities than the linear
and the quadratic propagation which were considered in
this paper.

Kutterer: Joint Treatment of Random Variability and Imprecision 105
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