Ps LAMBDA Manual

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Ps-LAMBDA software package
Matlab implementation, Version 1.0

Sandra Verhagen and Bofeng Li

Mathematical Geodesy and Positioning, Delft University of Technology

GNSS Research Centre, Curtin University
1

Contents
1 Introduction

4

1.1

Application of Ps-LAMBDA software . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.2

Disclaimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.3

Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2 Integer estimation methods and their pull-in regions

6

2.1

Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.2

Admissible integer estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.2.1

Z-transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.2.2

Integer rounding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.2.3

Integer bootstrapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.2.4

Integer least squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

3 Success rate of integer estimation

16

3.1

Definition of success rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

3.2

Monte Carlo simulation based approximate success rate . . . . . . . . . . . . . . . . .

17

3.3

Success rate and its bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

3.3.1

IR success rate and its bounds . . . . . . . . . . . . . . . . . . . . . . . . . .

19

3.3.2

IB success rate and its bounds . . . . . . . . . . . . . . . . . . . . . . . . . .

19

3.3.3

ILS success rate and its bounds . . . . . . . . . . . . . . . . . . . . . . . . . .

20

4 Routines and their usage

24

4.1

Overview of Ps-LAMBDA software . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

4.2

The main Ps-LAMBDA routine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

4.2.1

Input arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

4.2.2

ILS success rate (SRILS.m) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

2

4.3

4.2.3

IB success rate (SRBoot.m) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

4.2.4

IR success rate (SRRound.m) . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

Routines used by Ps-LAMBDA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

5 Getting started and performance aspects

27

5.1

Getting started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

5.2

Performance aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

5.2.1

IR success rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

5.2.2

IB success rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

5.2.3

ILS success rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

5.2.4

Examples with other models . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

5.2.5

Which bounds or approximations to use? . . . . . . . . . . . . . . . . . . . . .

32

6 Availability, Liability and Updates

34

6.1

Availability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

6.2

Liability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

6.3

Updates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

Bibliography

34

3

Chapter 1

Introduction
Integer ambiguity resolution is the process of estimating the unknown ambiguities of carrier-phase
observables as integers. It applies to a wide range of interferometric applications, of which Global
Navigation Satellite System (GNSS) precise positioning is a prominent example. GNSS precise positioning can be accomplished anytime and anywhere on Earth, provided that the integer ambiguities
are successfully resolved. In the past two decades, the LAMBDA method has been popularly applied
in variety of GNSS applications, stemming from not only its computation efficiency but also its maximizing the probability of correct integer estimation. LAMBDA stands for “Least-squares AMBiguity
Decorrelation Adjustment”, which was invented by Teunissen (1993b) and developed at the Delft University of Technology in the 1990s. Recently, the new version of LAMBDA software (version 3.0) is
released, which provides more options of integer estimation methods and more efficient search strategy
(Verhagen and Li 2012).
Unsuccessful ambiguity resolution, when passed unnoticed, will too often lead to unacceptable errors in
the results. Therefore, it is crucial that one is able to evaluate the integer ambiguity estimation. Evaluation of the integer solution is based on the success rate defined as the probability of correct integer
estimation. This ambiguity success rate depends on the underlying mathematical model as well as on
the integer estimation method used, which is generally difficult to be exactly evaluated. It is therefore
necessary to find some easy-to-use approximations of the success rate. So far, a variety of success
rate approximations and bounds have been developed for integer least squares (ILS), integer bootstrapping (IB) and integer rounding (IR) (Hassibi and Boyd 1998; Teunissen 1998e; Teunissen 2000;
Teunissen 2001b; Verhagen 2005), no standard software however exists to evaluate them.
The Ps-LAMBDA software, as the first implementation in Matlab for evaluation of the ambiguity
success rates, was developed in Curtin University of Technology and Delft University of Technology. In
the Ps-LAMBDA software, besides the Monte-Carlo simulation based success rate approximation for
all integer estimation methods, all the other available approximations and lower and upper bounds of
success rate can be easily assessed for each integer estimation method.
4

1.1

Application of Ps-LAMBDA software

Since the success rate can be computed once the float ambiguity variance-covariance (VC)-matrix Qââ
is known, it can be computed without the need for actual data. As such, the success rate can be used
as a very important performance measure for:

• planning purposes (design computations): what is the performance to be expected given a certain
measurement set-up at a given time and location;

• deciding whether or not to fix the ambiguities to the integer estimates during the actual data
processing (in real-time or post-processing mode);

• research purposes, e.g. to study the impact of receiver noise characteristics, availability of more
signals / satellites, baseline length, etcetera.

1.2

Disclaimer

This Matlab implementation is intended for educational / research purposes. Readability of the code
has therefore been considered more important than computational efficiency. Hence, the code could
still be optimized.

1.3

Outline

In the next chapter, a brief review of the integer ambiguity resolution theory will be given first, followed
by the three integer estimators together with their pull-in regions in the order of complexity. The
third Chapter gives the definition of success rate and its evaluations for ILS, IB and IR methods. The
routines and their usage are introduced in Chapter 4. Some examples are given in Chapter 5 to show the
performance of success rate bounds in Ps-LAMBDA. The final chapter is about the software availability
and liability.

5

Chapter 2

Integer estimation methods and their
pull-in regions
2.1

Parameter estimation

The mixed integer GNSS linear(ized) model can be defined as:
y ∼ N(Aa + Bb, Qyy ), a ∈ Zn , b ∈ Rp

(2.1)

The notation ”∼” is used to describe ”distributed as”. The m-vector y contains the pseudorange
and carrier-phase observables, the n-vector a contains the DD integer ambiguities, b is the real-valued
parameter vector of length p, including baseline or position components and possibly tropospheric and
ionospheric delay parameters. The coefficient matrices are A ∈ Rm×n and B ∈ Rm×p , with [A B]
of full column rank. The VC-matrix Qyy is an (m × m) positive definite matrix. In most GNSS
applications, the underlying distribution is assumed to be the multivariate normal distribution (denoted
by “N”).
In general, a three-step procedure is employed to solve model (2.1) based on the least squares criterion.
In practice, a user may want to include a validation step after step 1 and step 2.

Step 1: Float solution
In the first step, the integer property of the ambiguities a is disregarded and the so-called float estimates
together with their VC-matrix are computed
#!
" # "
" #
a
Qââ Qâb̂
â
∼N
,
b̂
b
Qb̂â Qb̂b̂

(2.2)

with Qââ and Qb̂b̂ are the VC-matrices of the float ambiguity and baseline estimators, respectively and
Qâb̂ = QTb̂â their covariance matrix.
6

Step 2: Integer estimation
In the second step, the float ambiguity estimate â is used to compute the corresponding integer
ambiguity estimate, denoted as
ǎ = I(â)
(2.3)
with I : Rn 7−→ Zn the integer mapping from the n-dimensional space of reals to the n-dimensional
space of integers. In this step, there are different choices of mapping function I possible, which
correspond to the different integer estimation methods. Popular choices are ILS, IB and IR.
ILS is optimal, as it can be shown to have the largest success rate of all integer estimators (Teunissen 1999b).
IR and IB, however, can also perform quite well, in particular after the LAMBDA decorrelation has been
applied. Their advantage over ILS is that no integer search is required. Each of the methods will be
discussed in more detail in the following sections.

Step 3: Fixed solution
In the third step, the float solution of the remaining real-valued parameters solved in the first step are
updated using the fixed integer parameters,
b̌ = b̂ − Qb̂â Q−1
ââ (â − ǎ)

(2.4)

Its VC-matrix is obtained by application of the error propagation law:

1

0.01

0.5

0.005
dU [m]

dU [m]

Qb̌b̌ = Qb̂b̂ − Qb̂â Q−1
ââ Qâb̂

0

0

−0.5

−0.005

−1

−0.01

0.01
0.005

1
0.5

1
0

0.5
0

−0.5
dN [m]

−1

dN [m]

−0.5
−1

(2.5)

dE [m]

0.01
0
−0.005
−0.01

0.005
0
−0.005
−0.01

dE [m]

Figure 2.1: Position errors in East (dE), North (dN) and Up (dU) direction in meters for ambiguity
float solutions (left panel), ambiguity fixed solutions (right panel) in a short baseline. Note the different
scales in the left and right panels.
7

Figure 2.2: Scatterplot of horizontal position errors in meters for float solution (grey dots) and corresponding fixed solution. In this case, 93% of the solutions were correctly fixed (green dots), and 7%
was wrongly fixed (red dots).

Note the derivation of VC-matrix based on error propagation law is based on the assumption that the
integer ambiguity solution can be assumed to be deterministic. It holds true only when the success
rate of ambiguity solution is sufficiently close to 1. In such case, in general Qb̌b̌ ≪ Qb̂b̂ , since after
successful ambiguity fixing the carrier-phase measurements start to act as very precise pseudorange
measurements. Figure 2.1 shows a scatterplot of the float and fixed position errors based on 10,000
solutions with single epoch, dual-frequency GPS for a short baseline; the success rate is equal to 1. It
can be observed that the precision is improved with a factor 100, in agreement with the difference in
code and carrier-phase measurement noise.

However, incorrect integer ambiguity estimation may result in the opposite effect in terms of positioning
accuracy: rather than a dramatic precision improvement, a wrong ambiguity solution can cause very
large position errors, exceeding those of the float solution. This is illustrated in Figure 2.2, which shows
a scatterplot of horizontal float position errors for a case where the ambiguities are fixed correctly in only
93% of the cases. The corresponding fixed solutions are shown as either red or green dots: red if the
ambiguities are fixed incorrectly, green if they are fixed correctly. It can be seen that in all cases where
the ambiguities were fixed correctly, the position errors are very small. However, in case of unsuccessful
integer estimation the corresponding position errors tend to be of the same size or even much larger
than the corresponding float position errors. The figure shows only the horizontal positioning results,
for the vertical component the errors can be as large as 8 meters in this example. This clearly shows
that the fixed solution should only be used if the success rate is very high. It is therefore very important
to evaluate the success rate of integer solution before use it.
8

2.2

Admissible integer estimation

As previously mentioned there are many ways of computing an integer ambiguity vector ǎ from its
real-valued counterpart â. To each such method belongs a different mapping I : Rn 7→ Zn . Due to
the discrete nature of Zn , the map I will not be one-to-one, but instead a many-to-one map. This
implies that different real-valued ambiguity vectors will be mapped to the same integer vector. One
can therefore assign a subset Pz ⊂ Rn to each integer vector z ∈ Zn :
Pz = {x ∈ Rn | z = I(x)}, z ∈ Zn

(2.6)

The subset Pz contains all real-valued ambiguity vectors that will be mapped by I to the same integer
vector z. This subset is referred to as the pull-in region of z. It is the region in which all ambiguity
float solutions are pulled to the same fixed ambiguity vector z.
Using the pull-in regions, one can give an explicit expression for the corresponding integer ambiguity
estimator. It reads

1 if â ∈ P
X
z
(2.7)
ǎ =
z Pz (â), Pz (â) =
0 otherwise
z∈Zn

Since the pull-in regions define the integer estimator completely, one can define classes of integer estimators by imposing various conditions on the pull-in regions. One such class is referred to as the class
of admissible integer estimators. This class was introduced in Teunissen (1999a) and it is defined as
follows.
Definition
P
The integer estimator ǎ = z∈Zn z Pz (â) is said to be admissible if
(i)

S

z∈Zn

(ii)

Pz = Rn

Int(Pz 1 )

T

Int(Pz 2 ) = ∅, ∀z 1 , z 2 ∈ Z n , z 1 6= z 2

(iii) Pz = z + P0 , ∀z ∈ Zn
This definition is motivated as follows. The first condition states that the pull-in regions should not
leave any gaps and the second that they should not overlap. The absence of gaps is needed in order to
be able to map any float solution â ∈ Rn to Zn , while the absence of overlaps is needed to guarantee
that the float solution is mapped to just one integer vector. The last condition of the definition follows
from the requirement that I(x + z) = I(x) + z, ∀x ∈ Rn , z ∈ Zn . It states that when the float
solution is perturbed by z, the corresponding integer solution is perturbed by the same amount. This
property allows one to apply the integer remove-restore technique: I(â − z) + z = I(â).
The three popularly used integer estimation methods, IR, IB and ILS, are all examples of admissible
integer estimation methods.
9

1.5

1

0.5

0

−0.5

−1

−1.5
−1.5

−1

−0.5

0

0.5

1

1.5

Figure 2.3: 2D IR pull-in regions: unit squares.

2.2.1

Z-transformation

It will be explained later that it may be useful to apply a so-called Z-transformation to the ambiguity
parameters. A matrix is called a Z-transformation if it is one-to-one (i.e. invertible) and integer
(Teunissen 1995a). Such transformations leave the integer nature of the parameters in tact. If a
certain integer estimator is Z-invariant it means that if the float solution is Z-transformed, the integer
solution transforms accordingly. Hence:
ž = Z T ǎ if

ẑ = Z T â

(2.8)

A very useful Z-transformation is the decorrelating Z-transformation (Teunissen 1993a; Teunissen 1994;
Teunissen 1995a; Teunissen 1995b). It results in a more diagonal VC-matrix:
Qẑ ẑ = Z T Qââ Z

(2.9)

Once the transformed ambiguities are fixed ž, one can also apply the back-transformation to recover
the integer solution of original ambiguities, ǎ = Z −T ž.

2.2.2

Integer rounding

The simplest way to obtain an integer vector from the real-valued float solution is to round each of the
entries of â to its nearest integer. The corresponding integer estimator reads
ǎIR = ([â1 ], · · · , [ân ])T
where [·] stands for rounding to the nearest integer.
10

(2.10)

8
6
1

4
2
0

0

−2
−4

−1

−6
−8
−8 −6 −4 −2

0

2

4

6

8

−1

0

1

Figure 2.4: 2D IR pull-in regions and 50,000 float solutions. Left: original ambiguities â [cycles]; Right:
Z-decorrelated ambiguities ẑ [cycles].
The pull-in regions for rounding are n-dimensional unit cubes centred at the integer grid points:
Pz,IR = {x ∈ Rn | cTi (x − z) ≤

1
, i = 1, . . . , n}, z ∈ Zn
2

(2.11)

with ci the unit vector have a 1 as its ith entry and 0’s otherwise. Figure 2.3 shows the 2D pull-in regions
for rounding: all float solutions residing in a specific pull-in region will be fixed to the corresponding
integer grid point in the centre of the pull-in region, i.e. all are pulled to the same integer vector.
As an example, one float solution is shown with the red dot, and the corresponding integer solution is
depicted with the blue circle.
In general, the rounding estimator is not Z-invariant, i.e. ž IR 6= Z T ǎIR . Only if Z is a permutation
matrix, and thus the transformation is a simple reordering of the ambiguities, the estimator is Zinvariant. Note that the IR pull-in regions remain unaffected by the Z-transformation. Figure 2.4
shows an example for a 2-dimensional (2D) ambiguity vector. 50,000 samples of float ambiguities for a
given VC-matrix Qââ were simulated; these are shown as the red and green dots. The left panel shows
the original float samples (before Z-decorrelation), and the pull-in region P0,IR , in which all the green
samples reside. Hence, for all those samples the 0-vector is obtained after rounding. The right panel
shows the corresponding Z-decorrelated float ambiguity samples, as well as the surrounding pull-in
regions. In this case, many more float samples reside in P0,IR : 95% versus 23% before Z-decorrelation.
This shows that the choice for the parameterization of the float ambiguity vector is very important in
case of IR.

2.2.3

Integer bootstrapping

The IB estimator still makes use of IR, but it takes some of the correlation between the ambiguities into
account. The IB estimator follows from a sequential least squares adjustment and it is computed as
follows. If n ambiguities are available, one starts with the most precise ambiguity. Let the nth ambiguity
11

be the most precise one, hence we start with rounding ân to the nearest integer. The remaining float
ambiguities are corrected by virtue of their correlation with the last ambiguity. Then the last-but-one,
but now corrected, real-valued ambiguity estimate is rounded to its nearest integer and all remaining
(n − 2) ambiguities are then again corrected, but now by virtue of their correlation with this ambiguity.
This process is continued until all ambiguities are considered. The components of the bootstrapped
estimator ǎIB are given as
ǎn;IB = [ân ]
ǎj;IB = [âj|J ] = [âj −

n
X

σâj âi|I σâ−2
(âi|I − ǎi;IB )], ∀j = 1, . . . , n − 1
i|I
|
{z
}
i=j+1

(2.12)

li,j

The short-hand notation âi|I stands for the ith ambiguity obtained through a conditioning on the
previous I = {i + 1, . . . , n} sequentially rounded ambiguities. The real-valued sequential least squares
solution can be obtained by means of the triangular decomposition of the VC-matrix of the ambiguities:
Qââ = LT DL, where L denotes a unit lower triangular matrix with entries li,j (see Eq.(2.12)) and
D a diagonal matrix with the conditional variances σâ2i|I as its diagonal elements.
The IB pull-in regions are given as:
1
Pz,IB = {x ∈ Rn | cTi L−T (x − z) ≤ , i = 1, . . . , n}, z ∈ Zn
2

(2.13)

with ci the unit vector have a 1 as its ith entry and 0’s otherwise. Figure 2.5 shows an example of the
IB pull-in regions in the 2D case, which are parallelograms. The float solution is depicted with a red
point and its bootstrapped solution is shown with the blue circle.
Like IR, IB suffers as well from a lack of Z-invariance, i.e. ž IB 6= Z T ǎIB if ẑ = Z T â. From Eq.(2.12)

1.5

1

0.5

0

−0.5

−1

−1.5
−1.5

−1

−0.5

0

0.5

1

1.5

Figure 2.5: 2D IB pull-in regions: parallelograms.
12

8
6
1

4
2
0

0

−2
−4

−1

−6
−8
−8 −6 −4 −2

0

2

4

6

8

−1

0

1

Figure 2.6: 2D IB pull-in regions and 50,000 float solutions. Left: original ambiguities â [cycles]; Right:
Z-decorrelated ambiguities ẑ [cycles].
can be seen that changing the order will already result in a different outcome with bootstrapping. It
can be clearly seen from Figure 2.6 how IB is affected by the decorrelating Z-transformation. Here
96% of the Z-decorrelated float samples resides in P0,IB versus 29% of the original ambiguity samples.

2.2.4

Integer least squares

When solving the GNSS model of Eq.(2.1) in a least squares sense, but now with the additional
constraint that the ambiguity parameters should be integer-valued, the integer estimator of the second
step in the procedure becomes:
(2.14)
ǎILS = arg minn kâ − zk2Qââ
z∈Z

with k · k2Q = (·)T Q−1 (·). This estimator is known to be optimal, cf. (Teunissen 1999b), which means
that the success rate of integer estimation is maximized.
The ILS pull-in region is defined by (Teunissen 1999b):
Pz,ILS = {x ∈ Rn | |wz (x)| ≤
with
wz (x) =

1
kukQââ , ∀u ∈ Zn }
2

uT Q−1
ââ (x − z)
kukQââ

(2.15)

(2.16)

the orthogonal projection of (x − z) onto the direction vector u. Hence, Pz,ILS is the intersection of
banded subsets centered at z and having width kukQââ .
Figure 2.7 shows an example of the 2D ILS pull-in regions. As an example, one float solution is shown
with the red dot, and the corresponding integer solution is depicted with the blue circle. For this
particular float solution, the same solution as with bootstrapping, see Fig.2.5, is obtained.
13

1.5

1

0.5

0

−0.5

−1

−1.5
−1.5

−1

−0.5

0

0.5

1

1.5

Figure 2.7: 2D ILS Pull-in regions: hexagons.

8
6
1

4
2
0

0

−2
−4

−1

−6
−8
−8 −6 −4 −2

0

2

4

6

8

−1

0

1

Figure 2.8: 2D ILS pull-in regions and 50,000 float solutions. Left: original ambiguities â [cylces];
Right: Z-decorrelated ambiguities ẑ [cylces].
In contrast to IR and IB, the ILS estimator is Z-invariant: ž ILS = Z T ǎILS if ẑ = Z T â. Figure 2.8
shows an example how the decorrelation affects the ILS estimator. For the original VC-matrix Qââ (left
panel) the ILS pull-in region follows the distribution of the float samples much better than in case of IR
and IB, compare with the corresponding Figures 2.4 and 2.6. Due to the Z-invariance the percentage
of float samples in P0,ILS (the green dots) is 97% both for the original and Z-decorrelated ambiguities.
The percentages for all three integer estimators are summarized in Table 2.1.
An integer search is needed to determine ǎILS . The ILS procedure is efficiently mechanized in the
LAMBDA method. A key element of the LAMBDA method is the decorrelating Z-transformation, see
Section 2.2.1, which results in largely reduced search times. For more information on the LAMBDA
14

Table 2.1: Percentage of float solutions that is correctly fixed for the
(corresponding to Figures 2.4 , 2.6 and 2.8).
IR IB
Original ambiguities â
23 29
Z-decorrelated ambiguities ẑ 95 96

three integer estimation methods
ILS
97
97

method and its wide-spread applications see e.g. (Teunissen 1993a; Teunissen 1995b; Li and Teunissen 2011;
Chang et al. 2005; De Jonge and Tiberius 1996a; Hofmann-Wellenhof et al. 2001; Teunissen and Kleusberg 1998;
Leick 2004; Strang and Borre 1997; Misra and Enge 2001). It is pointed out that the new version
LAMBDA software (version 3.0) has been released recently with more efficient search strategy and
more options of integer estimation methods (Verhagen and Li 2012).

15

Chapter 3

Success rate of integer estimation
3.1

Definition of success rate

According to the definition of admissible integer estimation in section 2.2, the float ambiguity can be
correctly fixed to its integer if and only if it resides its corresponding pull-in region, i.e.,
ǎ = a ⇔ â ∈ Pa

(3.1)

In other words, the probability of correct ambiguity estimation, i.e., success rate Ps , is equal to the
probability that â resides in the pull-in region Pa with a the true but unknown ambiguity vector:
Z
Ps = P (ǎ = a) = P (â ∈ Pa ) = fâ (x|a)dx
(3.2)
Pa

The probability density function (PDF) of the float ambiguities, fâ (x|a), is assumed to be the normal
PDF with mean a:
fâ (x|a) = p

1
1
exp{− (x − a)T Q−1
ââ (x − a)}
2
det(2πQââ )

(3.3)

As the pull-in regions of the integer estimators are integer-translation invariant, the success rate can
also be evaluated as:
Z
Ps = fâ (x|0)dx
(3.4)
P0

An illustration is given in Figure 3.1 for the ILS estimator: in the left panel the PDF of a 2D float
ambiguity vector is shown, with the corresponding ILS pull-in regions underneath. The right panel
shows the probability masses for each integer grid point, equal to the integral of the PDF over the
corresponding pull-in regions. In this case, the success rate is equal to the probability mass at [0 0]T .
The integration over the pull-in region is very complex in case of ILS and IR and hence it is difficult
to evaluate (3.4) exactly. Therefore we need to develop some easy-to-use probabilistic bounds or
approximation to the exact success rate. The lower bound can then be used to infer whether ambiguity
16

2

0.9

0.7

probability mass

probability density

0.8
1.5

1

0.5

0.6
0.5
0.4
0.3
0.2
0.1

0

0

1

1

0

0
1

−1

−1

0

1
0

−1

Figure 3.1: Left: PDF and 2D pull-in regions for ILS. Right: corresponding probability mass function.
resolution can expected to be successful, while the upper bound will show when one can expect successful
ambiguity resolution to fail. Thus there is enough confidence that ambiguity resolution will be successful
when the lower bound is sufficiently close to one, while no such confidence exists when the upper bound
turns out to be too small. These probabilistic bounds have the different approximation degrees to the
actual success rate, but all of them usually possess of efficient numerical computation which is the
right motivation of probabilistic bound. In the next section, the available approximations and bounds
are desbribed for IR, IB and ILS methods in order of complexity.

3.2

Monte Carlo simulation based approximate success rate

The success rate of integer estimation can be approximated by means of Monte Carlo simulation. The
procedure is as follows. It is assumed that the float solution is normally distributed â ∼ N(a, Qââ ),
and thus the distribution is symmetric about the mean a. Hence, we may shift the distribution over a
and draw samples from the distribution N(0, Qââ ).
The first step is to generate independent random samples with normal distribution N(0, Qââ ). One
can generate the random sample â directly from the Matlab built-in function mvnrnd(0, Qââ ). Alternatively, one can first generate n independent samples from the univariate standard normal distribution
N(0, 1), and then collect these in a vector s. This vector is transformed by means of â = Gs, with G
equal to the Cholesky factor of Qââ = GGT . The result is a sample â from N (0, Qââ ). The sample
â is used as input for integer estimation. If the output of this estimator equals the null vector, then it
is correct, otherwise it is incorrect. This process can be repeated an N number of times, and one can
count how many times the null vector is obtained as a solution, say Ns times. The approximation of
the success rate follows then as:
Ns
Ps ≈
(3.5)
N
In the simulation, the number of sample plays an important role for the approximation precision. In order
to get good approximations, the number of samples N must be sufficiently large (Teunissen 1998c).
The disadvantage is that it may be very time-consuming to evaluate Eq.(3.5), especially in case of ILS,
since for each sample an integer search is required.
Figure 3.2 shows for four GNSS models how the approximation performs depending on the number of
17

0.977
0.976

0.9998

0.975

simulation−based approximation

simulation−based approximation

1
0.9999

0.9997
0.9996
0.9995
0.9994
0.9993
0.9992

4

10

5

10
number of samples

0.971
0.97
0.969

4

10

5

10
number of samples

6

10

0.615

simulation−based approximation

0.91
simulation−based approximation

0.972

0.967
3
10

6

10

0.912

0.908
0.906
0.904
0.902

0.61

0.605

0.6

0.595

0.59

0.9
0.898
3
10

0.973

0.968

0.9991
0.999
3
10

0.974

4

10

5

10
number of samples

0.585
3
10

6

10

4

10

5

10
number of samples

6

10

Figure 3.2: Examples of simulation-based success rate as function of number of samples. Each panel
shows the results for a different GNSS model.
samples used (similar results were obtained for many other GNSS positioning models). It follows that
at least 105 samples should be used to get a good approximation. At the same time it can be seen that
using more samples generally only has a small effect, in the order of 10−3 , especially in cases where
the success rate is close to 1. With 106 samples the approximation will be very close to the true value.
In the Ps-LAMBDA, it allows to evaluate the simulation based success rates for IR and ILS, where the
user may specify the number of samples to be used.

3.3

Success rate and its bounds

The probabilistic bounds are developed typically from the following three aspects:
• Simplify the PDF of the float solution by partially capturing its associated VC-matrix. For
example, the product of all individual success rate of each scalar integer in a integer vector is
used as lower bound of IR success rate of this integer vector.
• Simplify the geometric complexity of pull-in region. The simplified pull-in region has the simpler
geometry such that the computation of the PDF integral over this simplified pull-in region is
feasible.
• Use the success rate relations between different integer estimation methods. As mentioned, the
success rates also depend on the selected integer estimation method, since the pull-in region is
different for IR, IB and ILS. In Teunissen (1999b), it was proven that:
P (ǎIR = a) ≤ P (ǎIB = a) ≤ P (ǎILS = a)
18

(3.6)

The ordering is thus the same as the ordering in terms of complexity, since IR is the simplest and
ILS the most complex method. This means that if IR or IB provides a very sharp lower bound, a
user could decide to use the simpler integer estimation method if their success rate is close to 1
and still obtain (close to) optimal performance.

3.3.1

IR success rate and its bounds

Lower bound based on the diagonal VC-matrix
The n-fold integral over the IR pull-in region defined in (2.11) is difficult to evaluate. Only if the
VC-matrix Qââ is diagonal will the success rate become equal to the n-fold product of the univariate
success rates. In Teunissen (1998e) it was shown that this also provides a lower bound in case Qââ is
not diagonal:

n 
Y
1
)−1
(3.7)
Ps,IR = P (ǎIR = a) ≥
2Φ(
2σâi
i=1

with Φ(x) the cumulative normal distribution function:
1
Φ(x) = √
2π

Z

x

1
exp{− t2 }dt
2
−∞

Note when the float solution â is a scalar case, this lower bound is the exact success rate.
IR success rate improved by decorrelation
In Section 2.2.2 it was mentioned that IR is not Z-invariant. This holds for the IR success rates as well,
since the pull-in regions are unaffected by a Z-transformation, while the distribution of the transformed
ambiguities is changed to ẑ ∼ N (Z T a, Qẑ ẑ ). If IR is applied to the Z-decorrelated ambiguities, the
success rate will increase due to the improved precision of the decorrelated ambiguities, i.e.
P (ž IR = z) ≥ P (ǎIR = a)

3.3.2

(3.8)

IB success rate and its bounds

IB success rate
In case of bootstrapping the success rate can be evaluated exactly using (Teunissen 1998e):
Ps,IB = P (ǎIB = a) =

n 
Y
i=1

2Φ(


1
)−1
2σâi|I

(3.9)

where σâ2i|I with I = {i + 1, · · · , n} is the conditional variance of the ith float ambiguity âi on the float
ambiguities from (i + 1) to n, which is the ith diagonal element of D computed from the Cholesky
decomposition on the VC-matrix Qââ = LT DL. To our best knowledge, the IB success rate is one
of the most useful success rate evaluations from both computation efficiency and performance.
19

Upper bound based on ADOP
In Teunissen (2000) it was proven that such an upper bound is given by:

Ps,IB ≤ 2Φ

n


1
− 1 = PADOP
2ADOP

(3.10)

with ADOP being the Ambiguity Dilution of Precision given by:
ADOP =

p

1

det(Qââ ) n

(3.11)

in units of cycles. The ADOP is a diagnostic that captures the main characteristics of the ambiguity
precision. It was introduced in Teunissen (1997), described and analyzed in (Teunissen and Odijk 1997;
Odijk and Teunissen 2008) and is widely used, see the introduction of Odijk and Teunissen (2008).
There is an important property for ADOP that it is Z-invariant for the class of admissible ambiguity
transformation, i.e. det(Qââ ) = det(Qẑ ẑ ). This merit property makes this upper bound useful,
since one does not need apply the transformation and directly compute this upper bound. If it is too
small, it can be immediately concluded that IB will be not successful for any parameterization of the
ambiguities. When the ambiguities are completely decorrelated, the ADOP equals the geometric mean
of the standard deviations of the ambiguities, hence it can be considered as a measure of the average
ambiguity precision.

IB success rate improved by decorrelation
The IB success rate is not Z-invariant. IB may perform close to optimal if applied to the decorrelated
ambiguities ẑ (Teunissen 1998e; Verhagen 2005) and we have:
P (ž IB = z) ≥ P (ǎIB = a)

3.3.3

(3.12)

ILS success rate and its bounds

In general it is more difficult to evaluate the ILS success rate than IR and IB since its pull-in region
has a rather complicated geometry. So far, a variety of ILS success rate bounds have been developed,
of which some are based on bounding the complicated pull-in region with a simple region, while some
based on bounding the VC-matrix with a simple one.

Approximation based on ADOP
It was also proved in Teunissen (1999c) that PADOP , as upper bound (3.10) of IB success rate, can be
used as an approximation to the ILS success rate, i.e.,

Ps,ILS ≈ PADOP = 2Φ
20

n
1 

−1
2ADOP

(3.13)

Upper bound based on ADOP
Besides the ADOP based ILS success rate approximation, an upper bound for the ILS success rate
based on the ADOP can be given as:

Ps,ILS ≤ P χ2 (n, 0) ≤
with
cn =

(3.14)


2

n
n
2 Γ( 2 )

π

cn 
ADOP2
n

and Γ(•) gamma function. This bound was introduced in Hassibi and Boyd (1998), while the proof
was given in Teunissen (2000).

Upper and lower bounds based on bounding the pull-in region
In Teunissen (1998c) lower and upper bounds for the ILS success rate were obtained by bounding the
integration region (pull-in region) Pa,ILS . Obviously, a lower bound is obtained if the integration region
is chosen such that it is completely contained by the pull-in region, and an upper bound is obtained if
the integration region is chosen such that it completely contains the pull-in region. Let subsets La and
Ua are the integration regions for lower and upper bounds, respectively, it follows that La ⊂ Pa,ILS and
Ua ⊃ Pa,ILS . In that case the success rate lies in the interval
P (â ∈ La ) ≤ P (â ∈ Pa,ILS ) ≤ P (â ∈ Ua )

(3.15)

Both regions of integration should of course be chosen such that the corresponding probabilities are
easily evaluated in practice.
Given the definition of the ILS pull-in region Pa,ILS in Eq.(2.15), it follows that any finite intersection
of p < n banded subsets defined by w of Eq.(2.16) will enclose Pa,ILS .
1
Ua = {x ∈ Rn | |wi (x)| ≤ kci kQââ , i = 1, · · · , p} ⊃ Pa,ILS
2

(3.16)

The idea is illustrated in Figure 3.3 for the 2D case where Ua is chosen as the intersection of two
banded subsets. The probability P (â ∈ Ua ), however, cannot be evaluated exactly either, but can be
bounded from above to obtain Teunissen (1998c)
Ps,ILS ≤ P (â ∈ Ua ) ≤

p 
Y

2Φ(

i=1


1
) − 1 = PUB,region
2σvi|I

(3.17)

with the conditional standard deviation σvi|I of vector v. These are equal to the square root of the
diagonal elements of D from the LT DL-decomposition of Qvv with its elements given by:
σvi vj =

uTi Q−1
ââ uj
, ui , uj ∈ Z n
kui kQââ kuj kQââ
21

1.5

1

0.5

0

−0.5

−1

−1.5
−1.5

−1

−0.5

0

0.5

1

1.5

Figure 3.3: Integration region (red) containing P0,ILS and defined by the intersection of two banded
subsets.
where the ui , i = 1, . . . , p need to be linearly independent to guarantee the full-rank of VC-matrix Qvv .
The procedure for computation of this upper bound is as follows. LAMBDA is used to find the q ≫ n
closest integers ui ∈ Zn \{0} for â = 0. These q integer vectors are ordered by increasing distance to
the zero vector, measured in the metric Qââ . Start with U = u1 so that rank(U ) = 1. Then find
the first candidate uj . (j = 2, · · · , q) for which rank([u1 uj ]) = 2. Continue with U = [u1 uj ], and
find the next candidate that results in an increase in rank. Continue this process until rank(U ) = n.
It cannot guarantee that we can find n independent integer vectors from q integer vectors. In PsLAMBDA software, we initially set q = 100n and if we obtain that rank(U ) < n from these q integer
vectors, we augment U ultil its rank equal to n by trying ci , i = 1, · · · n with all elements of 0s except
1 at ith slot. For more information, one refers to Teunissen (1998c) and Verhagen (2005).
Note that in the higher dimensional case many subsets are necessary to obtain a tight upper bound,
and selection of the subset is rather complicated. In addition, it is computationally demanding, since
the determination of the subset involves the evaluation of many integer candidates to be obtained with
LAMBDA.
For the lower bound, one can find an ellipsoid La to expand the pull-in region Pa,ILS from inside
Teunissen (1998c)
n
o
1
La = x ∈ Rn | | kx − ak2Qââ ≤
min kuk2Qââ
4 u∈Zn \{0}
The concept is illustrated in Figure 3.4 for two different pull-in regions, corresponding to different
VC-matrices Qââ . Therefore lower bound P (â ∈ La ) can be evaluated based on the χ2 -distribution:


1
Ps,ILS ≥ P (â ∈ La ) = P χ2 (n, 0) ≤
min kuk2Qââ = PLB,region
4 u∈Zn \{0}
22

(3.18)

0.6

0.6

0.4

0.4

0.2

0.2

0

0

−0.2

−0.2

−0.4

−0.4

−0.6

−0.6

−0.6

−0.4

−0.2

0

0.2

0.4

−0.6

0.6

−0.4

−0.2

0

0.2

0.4

0.6

Figure 3.4: Two examples of the ellipsoidal region (green) contained by the pull-in region P0,ILS
(different shape of pull-in regions is due to different VC-matrices Qââ ).
Upper and lower bounds based on bounding VC-matrix
It is also possible to obtain a lower and an upper bound by bounding the actual VC-matrix from
above and below by diagonal matrices, and then to computing the ILS success rate belonging to these
diagonal matrices becomes straightforward (Teunissen 1998c). The simplest way of bounding the actual
ambiguity VC from above and below, is to make use of unit matrix scaled by its maximum and minimum
eigenvalues. This gives
λmin I n ≤ Qẑ ẑ ≤ λmax I n
(3.19)
where λmin and λmax are the minimum and maximum eigenvalues of Qẑ ẑ , and I n is an identity matrix
of order n. The ILS success rate bounds follow as:
n
n


1
1
PLB, eigen = 2Φ( √
) − 1 ≤ Ps,ILS ≤ 2Φ( √
) − 1 = PUB, eigen
2 λmax
2 λmin

(3.20)

Note that the two bounds coincide when the two extreme eigenvalues coincide. This is the case
when the ambiguity VC-matrix itself is a scaled unit matrix. In the real GNSS applications, these two
extreme eigenvalues will differ considerably when the VC-matrix of the original DD ambiguities is used.
In that case the above two bounds would become too loose to be useful. When using the decorrelated
ambiguities as produced by the LAMBDA method, the elongation of the ambiguity search space is
considerably reduced and the ratio of the two extreme eigenvalues is pushed towards its minimum of
one. Hence, the above bounds are much sharper using the eigenvalues of the transformed ambiguity VCmatrix than using the eigenvalues of the original DD ambiguity VC-matrix. In Ps-LAMBDA software,
the decorrelated ambiguity VC-matrix is employed.

23

Chapter 4

Routines and their usage
4.1

Overview of Ps-LAMBDA software

Figure 4.1 gives an overview of the structure of Ps-LAMBDA software. The options of approximations
and bounds of success-rate are shown for ILS, IB and IR methods.
Method

Option
1

3 (*)

6

AP : simulation

LB : IB exact

UB : ADOP

SR_ILS_ap_sim

SR_B_ex

SR_ILS_ub_adop

1

2

4

7

ILS

AP : ADOP

LB : region

UB : region

SR_ILS_ap_adop

SR_ILS_lb_region

SR_ILS_ub_region

5

8

LB : VC-matrix

UB : VC-matrix

SR_ILS_lb_vc

SR_ILS_ub_vc

SRILS

2

IB
SRBoot
3

IR
SRRound

1 (*)

2

EXACT

UB : ADOP

SR_B_ex

SR_ILS_ap_adop

1

2

3 (*)

AP : simulation

LB : VC-matrix

UB : IB exact

SR_R_ap_sim

SR_R_lb

SR_B_ex

Figure 4.1: Ps-LAMBDA: overview of available methods and options in routine SuccessRate. Default
option is indicated with (*). Names of underlying routines are shown as well. AP=approximation
(blue), LB=lower bound (green), UB=upper bound (red).

24

4.2

The main Ps-LAMBDA routine

The main routine is
Ps = SuccessRate(Qa, method, opt, decor, nsamp)
The VC-matrix Qa should be square (n × n), symmetric and positive-definite. The input and output
arguments will be described first, followed by some examples that which subroutine can be used if the
user is interested in the different methods.

4.2.1

Input arguments

The following input arguments are needed, depending on the method of choice:
Qa : VC-matrix of ambiguities
method : 1 - ILS [DEFAULT] (SRILS.m)
2 - IB
(SRBoot.m)
3 - IR
(SRRound.m)
opt : Approximation / bound to compute, depending on method
decor : 1 - decorrelation [DEFAULT]
1 - no decorrelation
nsamp : number of samples only used for simulation-based approximation.
The choice for “decor” is only relevant for IR and IB, since these estimators are not Z-invariant.
Decorrelation is always applied for ILS to ensure computation efficiency.
The output “Ps” is the computed success rate.

4.2.2

ILS success rate (SRILS.m)

The routine for computing the approximation / bound of ILS success rate is
Ps = SRILS(Qa,opt,nsamp);
Here there are 8 options for approximation / bounds of ILS success rate:
opt : 1
2
3
4
5
6
7
8
9

-

Monte-Carlo simulation based approximation
ADOP based approximation
Lower bound of IB success rate [DEFAULT]
Lower bound by bounding pull-in region
Lower bound by bounding covariance matrix
Upper bound based on ADOP
Upper bound by bounding pull-in region
Upper bound by bounding covariance matrix
All
25

(SRsm_ILS.m)
(SRa_adop.m)
(SRe_boot.m)
(SRlb_region.m)
(SRlb_vcv.m)
(SRub_adop.m)
(SRub_region.m)
(SRub_vcv.m)

One can invoke this subroutine by specifying method=1 in the main routine. Alternatively, one can also
directly invoke this subroutine. But now one needs to first apply the decorrelation transformation on
Qââ to ensure the computation efficiency.

4.2.3

IB success rate (SRBoot.m)

The routine for computing the approximation / bound of IB success rate is
Ps = SRBoot(Qa,opt,decor);
Here there are 2 options of success rate approximation / bounds:
opt : 1 - Exact success-rate [DEFAULT] (SRe_boot.m)
2 - ADOP-based upper bound
(SRa_adop.m)
3 - All
For subroutine “SRa adop”, it is independent on the ambiguity transformation, referring to subsection
3.3.2, while “SRe boot” depends on the decorrelation. If the input “decor” is true, Qââ will be
transformed firstly and then used for “SRe boot”.

4.2.4

IR success rate (SRRound.m)

The routine for computing the approximation / bound of IR success rate is
Ps = SRRound(Qa,opt,decor,nsamp);
It has 4 options for success rate approximations / bounds:
opt : 1
2
3
4

-

Monte-Carlo simulation based approximation
Lower bound based on diagonal VC-matrix
Upper bound based on IB success-rate
All

(SRsm_round.m)
(SRlb_round.m)
(SRe_boot.m)

All of these subroutines are dependent on the decorrelation. If “decor” is true, all subroutines will be
based on the decorrelated VC-matrix.

4.3

Routines used by Ps-LAMBDA

Two subroutines, decorrel.m and ssearch.m, used for ambiguity resolution in LAMBDA software
are used in Ps-LAMBDA software. Subroutine decorrel.m is used to conduct the decorrelating Ztransformation; while ssearch.m is used to implement the integer search with “search-and-shrink”
technique. For more information, one can refer to the manual of new version LAMBDA software
(Verhagen and Li 2012).
26

Chapter 5

Getting started and performance aspects
5.1

Getting started

It is most convenient to include the folder with the Ps-LAMBDA routines in your Matlab path. The
folder contains a demonstration routine SR GUI with examples of how the program can be used. Open
the routine in your editor; the comments will guide you through the different options.

Figure 5.1: Graphical User Interface of Ps-LAMBDA software.
The toolbox also includes a Graphical User Interface (see Figure 5.1) which allows the user to select an
input file which contains the VC-matrix Qââ and to compute all the desired bounds and approximations
27

for different integer estimation methods simultaneously.

5.2

Performance aspects

In this section the bounds and approximations for ILS, IB and IR methods, will be briefly assessed for
different GNSS models, where the different factors affecting the float ambiguity precision are varied as
shown in Table 5.1. An exponential elevation-dependent weighting is applied (more noise is assumed
for observations from low-elevation satellites) to the standard deviations of the observations and of the
ionosphere corrections. The scale factors applied to the VC-matrix Qââ can either be interpreted as
representing a different number of epochs, or a different measurement precision due to different receiver
quality. In following, the Monte-Carlo simulation based success rate is referred as actual success rate
with number of samples 106 .
Table 5.1: Measurement scenarios (standard deviations (STD) apply to zenith direction).
system
GPS - combined GPS+Galileo
times
49 different epochs
frequencies
L5 - L1+L5 - L1+L5+L2/E5b
STD of undifferenced observations code: 15 cm; phase: 1 mm
VC-matrix scale factors
0.25 - 0.5 - 1 - 2 - 4
STD of ionosphere corrections
5 - 15 mm

5.2.1

IR success rate
1

0.9

0.8

0.7

bounds

0.6

0.5

0.4

0.3

0.2

0.1

0

0

0.1

0.2

0.3

0.4
0.5
0.6
IR success rate

0.7

0.8

0.9

1

Figure 5.2: IR success rates: upper bound based on IB (red; Eq 3.9) and lower bound based on diagonal
VC-matrix (green; Eq 3.7) versus the actual IR success rate for the models from Table 5.1.
28

Figure 5.2 shows the lower bound and upper bound versus the actual IR success rates (all for the
decorrelated ambiguities). It can be seen that the lower bound is very tight, whereas the upper bound
based on the IB success rate is not as tight, thus indicating that IB may still significantly outperform
IR.

5.2.2

IB success rate
1

0.9

0.8

ADOP−based upper bound

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0

0.1

0.2

0.3

0.4
0.5
0.6
IB success rate

0.7

0.8

0.9

1

Figure 5.3: IB success rates: ADOP-based upper bound (Eq 3.10) versus the exact IB success rate (Eq
3.9) for the models from Table 5.1.
Figure 5.3 shows that the ADOP-based upper bound is in these cases often significantly higher than
the exact IB success rate P (ž IB = z). Better bounding performance is obtained for lower dimensions
n, which is due to the replacement of the n conditional standard deviations in Eq.(3.9) by a single
value equal to ADOP.

5.2.3

ILS success rate

Figure 5.4 (Left) shows how the IB success rate performs as a lower bound for ILS. In practice, the IB
success rate is commonly used as the best known lower bound, and these results confirm that especially
if the success rate is high, this is indeed the case. At the same time, it can be seen how ILS may
still significantly outperform IB for lower success rates. For these cases the ADOP-based upper bound
often gives a too optimistic value compared to the actual success rate. As is shown later, however,
the bounding performance improves for lower dimensions (cf. Figure 5.7). A similar conclusion can be
given for the ADOP-based approximation of the ILS success rate as shown in Figure 5.4 (Right). Only
in some of these cases can it be used as a coarse approximation. The approximation improves in case
of lower dimensions (cf. Figure 5.7).
29

1

0.9

0.9

0.8

0.8

0.7

0.7
ADOP−based approximation

1

bounds

0.6

0.5

0.4

0.6

0.5

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

0.1

0.2

0.3

0.4
0.5
0.6
ILS success rate

0.7

0.8

0.9

0

1

0

0.1

0.2

0.3

0.4
0.5
0.6
ILS success rate

0.7

0.8

0.9

1

Figure 5.4: ILS success rates: lower bound based on IB (green; Eq 3.9) and upper bound based on
ADOP (red; Eq 3.14) versus the actual ILS success rate (Left); ADOP-based approximation (Eq 3.10)
versus the actual ILS success rate (Right) for the models from Table 5.1.
1

0.9

0.8

0.7

bounds

0.6

0.5

0.4

0.3

0.2

0.1

0

0

0.1

0.2

0.3

0.4
0.5
0.6
ILS success rate

0.7

0.8

0.9

1

Figure 5.5: ILS success rates: lower (Eq 3.17) and upper bounds (Eq 3.17) based on bounding the
pull-in region versus the actual ILS success rate for the models from Table 5.1.

Figure 5.5 shows the lower and upper bound of the ILS success rate based on bounding the pull-in
region. It can be seen that the upper bound performs reasonably well, whereas the lower bound is
generally not tight at all - it will be close to zero unless the success rate is very close to 1. The bad
performance can be explained based on the 2D example on the right-hand side of Figure 3.4: the
ellipsoidal region may leave a large part of the ILS pull-in region uncovered. This will be the case when
there is a large variation in the variances σẑi ẑi (making the ellipsoidal region elongated).
30

1

0.9

0.8

0.7

bounds

0.6

0.5

0.4

0.3

0.2

0.1

0

0

0.1

0.2

0.3

0.4
0.5
0.6
ILS success rate

0.7

0.8

0.9

1

Figure 5.6: ILS success rates: lower and upper bounds (Eq 3.20) based on bounding the VC-matrix
versus the actual ILS success rate for the models from Table 5.1.
Figure 5.6 shows the lower and upper bound of the ILS success rate based on bounding the VC-matrix.
It can be seen that both bounds perform poorly. Similarly as with the ADOP-based approximation
of the ILS success rate, this is especially true for large n due to the replacement of the n conditional
standard deviations in Eq.(3.9) by the square root of the minimum or maximum eigenvalue, respectively.

5.2.4

Examples with other models

So far, the performance of the success rate bounds and approximations was analyzed based on the
linearized DD GNSS model parameterized in terms of the baseline unknowns. However, the geometryfree model is used for example for integrity monitoring or as a first step in the data processing. Here, we
will show an example based on a dual-frequency GPS model for one satellite-receiver pair (i.e. one DD
code and phase observation per frequency). The undifferenced code and phase standard deviations were
set to 15 cm and 1.5 mm, respectively. The float ambiguity VC-matrix (units are cycles2 ) obtained in
this way is:
"
#
1.2429 0.9683
Qââ =
(5.1)
0.9683 0.7547
In addition, a scaling is applied to analyze the performance for different precisions:
Qââ,f = f × Qââ

(5.2)

The ILS success rate approximations and bounds are shown in Figure 5.7 as a function of the scale
factor f . The lower bound based on the exact IB success rate is very sharp. Interestingly, this also holds
for the ADOP-based upper bound and approximation (the orange line is hardly visible, as it is plotted
below the graph of the simulation-based success rate). In this case the bounds based on bounding the
31

1

1

0.99

0.8

success rate

success rate

0.9

0.7

0.98

UB: VC−matrix
UB: region
UB: ADOP
LB: VC−matrix
LB: region
LB: IB exact
AP: ADOP
AP: simulation

0.97

0.6
0.96

0.5
0.4

1

2

3

4

5

6

7

8

9

0.95

10

1

1.2

1.4

1.6

1.8

2

f

f

Figure 5.7: ILS success rate bounds for 2-frequency geometry-free model with 2 ambiguities, f is the
scale factor applied to the VC-matrix (right panel shows same results, but only for smaller f ).
1
UB: VC−matrix
LB: VC−matrix
AP: simulation

0.95
0.9

success rate

0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5

1

2

3

4

5

6

7

8

9

10

f

Figure 5.8: ILS success rate bounds based on bounding the (2×2) scaled VC-matrix with both variances
equal to 0.02, and covariance equal to 0.0005. The scale factor is equal to f .
integration region are quite sharp if the success rate is high, but become less tight as the scale factor
increases, and consequently the success rate decreases.
In all results shown so far, the bounds based on bounding the VC-matrix Qââ are generally not tight
at all. An example where also these bounds will work well is when all variances are equal to a certain
value v and all the covariances equal to a value c, with v ≫ c:
σâ2i âi = v, σâi âj = c,

∀i, j = 1, . . . , n; i 6= j

(5.3)

Figure 5.8 shows the bounds for an example with n = 2, v = 0.02 and c = 0.0005. Again the scaling
according to Eq.(5.2) is applied.

5.2.5

Which bounds or approximations to use?

The results in this section show that the success rate bounds and approximations differ in their performance. The simulation-based approximations of the IR and ILS success rates work well if enough
samples are used. However, they may not be suitable for real-time applications as their computation
time may be long. Computation time will also be an issue for real-time applications if the upper bound
32

of the ILS success rate based on bounding the pull-in region is considered. For design and research
purposes, as well as for post-processing, computation time will not be an issue. All other bounds and
approximations can be used in real-time.
For the IR success rate, the lower bound was shown to perform well. For the ILS success rate, the lower
bound based on the exact IB success rate, and the upper bound based on bounding the pull-in region
generally perform very well for the GNSS models considered here. Furthermore, it was shown that the
other bounds and approximations may work well for certain applications where the dimension is lower
or the structure of the VC-matrix Qââ is different, see for example Figures 5.7 and 5.8.

33

Chapter 6

Availability, Liability and Updates
6.1

Availability

The Matlab implementation of the Ps-LAMBDA software Version 1.0 is available on request.

6.2

Liability

Use of the accompanying Ps-LAMBDA software is allowed, but no liability for the use of the software
will be accepted by the authors or their employer. Giving proper credits to the authors is the only
condition posed upon the use of the Ps-LAMBDA software. We ask you to refrain from passing the
software to third parties. Instead you are asked to pass our (e-mail / website) address to them, so we
can send the software upon their request or they can download freely from website. The reason is that
in this way we have a complete overview of the users of the software, enabling us to keep everyone
informed of further developments.

6.3

Updates

We welcome any suggestion for improvement of the code, the in-source documentation and the description in the report. We also would like to encourage you to communicate to us about results obtained
with the Ps-LAMBDA method, and comparisons made with other methods. We would also be much
obliged if you inform us in case you decide to use the method commercially. As said before, there
are no restrictions on that, other than properly acknowledging the designers of the method and their
employer. If you are planning to make a version in another language and would like to make it public,
we would like you to contact us, in order to coordinate the efforts.

34

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37



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