ALGORITHMS FOR DESIGN COMPUTATIONS FOR INTEGRATED GPS – GALILEO

Sandra Verhagen and Peter Joosten

Mathematical Geodesy and Positioning Delft University of Technology Thijsseweg 11, 2629 JA Delft

The Netherlands S.Verhagen@geo.tudelft.nl

1 Abstract Future triple-frequency GNSSs like modernized GPS and Galileo require extension or modification of the observation models. These modifications involve the functional model as well as the stochastic model. Since the structure of the models will not change, this will generally not be difficult as long as we have information on the new signal structures and the expected precisions. However, one of the major benefits of the two future systems may lie in the possibility to use GPS and Galileo as one integrated system. The set up of the observation model of such an integrated system may be somewhat more complicated, and the structure will be different. This is especially important if we look at the algorithms needed for design computations. This contribution will give the general observation models for GPS, Galileo, and the integrated system that may be of interest in the future. We will focus on short to medium baseline lengths. The observation models will then serve as a basis for the derivation of expressions that allow for the computation of design parameters that provide a priori information on the expected performance. The focus will be on the implementation aspects. Eventually, this makes it possible to compare the performance of different systems and when using different observation models and ambiguity resolution methods. In order to illustrate how the design computations may be used, the performance of modernized GPS, Galileo, and integrated GPS-Galileo will be analyzed for some realistic scenarios. It will be shown that the use of an integrated system will be mainly interesting in situations where the individual systems cannot guarantee the required availability, for example in environments with a lot of constructions.

2 Introduction Future Global Navigation Satellite Systems (GNSS) are intended to provide higher precision, availability and integrity. With respect to GPS, these enhancements will be due to an additional carrier on L5 and an additional civil code on the L2 signal. Galileo will be a brand new system with at least three carriers and codes available for every user, plus integrity information for commercial users. So, the two systems on their own already will to some extend contribute to the improved performance. However, the most obvious improvement with respect to the current GPS may be expected from the

integrated GPS – Galileo system. This system will not just provide more signals, but also more satellites. In this contribution, we will therefore focus on the Integrated GPS – Galileo (IGG) system. At this moment, it is very interesting to know how the system might perform in terms of the precision and reliability that will be obtained. For that purpose, algorithms are set up that can be used for design computations. These algorithms should be usable for any scenario and allow easy comparison of the performance of different systems under different circumstances. Starting point will be the definition of the observation models that will be of interest. These models depend on the system design of the future GNSSs, so a brief overview of the current plans will also be given. The observation models will then serve as the basis for the derivation of expressions that allow for the computation of the design parameters. Various parameters will be considered. A user could then put conditions on the values that the parameters should take in order to meet his/her requirements. In this way, it is possible to make inferences on the availability of the system. Finally, design computations are carried out for some realistic scenarios in order to compare the performance of the different systems.

3 Observation models A wide variety of GPS observation models is available and extensively used for all kind of high-precision applications, ranging from geodesy and navigation to geophysics. The general form of all GNSS models for a single epoch can be written as:

( ) yQyDa

bNMyE == }{;}{ (1)

Where E{.} and D{.} are the expectation and dispersion operators. The double difference (DD) code and carrier phase observations are collected in is the (m-1)-vector y. The p-vector b contains all real-valued unknown parameters, including the baseline unknowns and for example atmospheric delays. In the sequel it will be referred to as the vector of baseline unknowns. The unknown integer ambiguities are denoted with the vector a, which is of order n. The functional relationship between the observations and the unknown parameters is captured by the matrices M and N. Finally, the precision of the observations is described by the variance-covariance matrix Qy. In this contribution we will focus on the roving receiver model for short to medium baseline lengths and for short time spans. This implies that the baseline unknowns will change every epoch, however, it is assumed that the satellite geometry does not change (in reality it changes slowly). For k epochs the model becomes:

( ) ykk

k

kkk QIyDa

bNeMIyE ⊗=⊗⊗= }{;}{ (2)

where we used the Kronecker product. As can be seen in eq.(2) the ambiguities will not change in time, and it is assumed that the observations are not correlated in time.

In case of medium baseline lengths, i.e. longer than 10 km, it will be necessary to account for the ionospheric delays. This can be done by introducing pseudo-observations that come from an ionospheric model. These pseudo-observations are subtracted from the actual observations, which implies that the vc-matrix of the observations has to be modified as to account for the uncertainty in the pseudo-observations, cf. (Odijk, 2002). This is referred to as the ionosphere weighted model. The general model as described above can be used for any GNSS. The structure is independent of the number frequencies. Note that in case of the ionosphere weighted model, correlation between the different observation types on the different frequencies is introduced, i.e. Qy is not a block-diagonal matrix. For IGG the double differences have to be formed for both systems separately even if the frequencies are the same, because of different signal characteristics. Model (1) for IGG becomes therefore:

==2

1

2

1

22

11

2

1;}{

Q

QQ

a

a

b

NM

NM

y

yE y (3)

The sub-indices refer to the two systems: 1 to GPS, 2 to Galileo. It is assumed that future IGG receivers will be able to collect the observations of both systems independently, i.e. without correlation. Table 1 gives an overview of the system parameters of modernized GPS and Galileo. It is assumed that for GPS the current 24-satellite constellation will be maintained, with 3 to 4 extra satellites. For Galileo the constellation as described in Salgado et al. (2001) is chosen. For IGG different frequency combinations are possible. For a dual-frequency receiver it seems most likely that a combination of GPS L1 and L5 with Galileo L1 and E5a would be chosen, although alternatively E5b or E5a+E5b could be chosen. IGG receivers need a front-end for each different frequency, but in principle it should be possible to use the three GPS frequencies and three of the Galileo frequencies (Eissfeller et al., 2002).

Table 1. GPS and Galileo system overview

system signal frequency [Mhz]

σp

[m]

open service

number of satellites

number of planes

L1 1575.42 0.3 yes

L2 1227.60 0.3 yes

GPS

L5 1176.45 0.1 yes

24 + ?

6

L1 1575.42 0.2 yes

E5a 1176.45 0.1 yes

E5b 1207.14 0.1 yes

E5a+E5b 1191.80 0.1 yes

Galileo

E6 1278.75 0.1 no

27 + 3

3

4 Design computations Design computations provide a priori information on the expected performance of a GNSS for a specific measurement scenario. For that purpose, no actual observations are needed. One could simply look at the standard deviations of the parameters that will be estimated, but other examples of design parameters are the internal and external reliability, which give information on the detectability and the impact of unmodeled biases respectively. Another example is the probability of correct integer estimation (the so-called success rate). This probability depends on the strength of the observation model, and also on the ambiguity resolution method that is used. Traditionally, design computations are considered especially interesting for setting up a measurement campaign. Users can choose the best measurement scenario with respect to e.g. time, duration, and observation types to be used. In the future there will be another important reason to carry out a design study: the user use it to decide which system to use: GPS, Galileo or IGG. This section will give an overview of interesting design parameters and the algorithms that are required in order to compute them. This is especially interesting for the new IGG model, since the structure of this model will be different compared to the standard GPS-only models (or Galileo-only).

4.1 Ambiguity resolution and success rates

For high-precision applications it is necessary to include the carrier phase measurements in the observation model. Once the integer cycle ambiguities are resolved, these carrier phase observations will start to act as very precise pseudorange measurements. An important prerequisite is thus that it will indeed be possible to resolve the integer ambiguities correctly. The probability that this will be the case is referred to as the success rate. It can be computed once the variance-covariance matrix of the unknown parameters is known. So, no actual observations are needed and the success rate can be considered as a very useful design parameter. In the following it will be explained how the success rates can be computed. The first step in GNSS data processing is to determine the so-called float solution of the unknown parameters, using a standard least-squares adjustment. After this step, real-valued estimates of the parameters are available together with their associated vc-matrix. Let this float solution be given as:

aba

abb

QQ

QQ

a

b

ˆˆˆ

ˆˆˆ;

ˆ

ˆ (4)

The next step is then to resolve the integer ambiguities, meaning that the float ambiguities are all mapped to integer values. The optimal solution will be obtained with integer least-squares as is used in the LAMBDA method, cf. (Teunissen, 1993), (Teunissen, 1999). The LAMBDA method actually consists of two steps. In the first step the float ambiguities are decorrelated in order to eliminate the discontinuity in the spectrum of the conditional variances. This discontinuity causes the search space to be very

elongated, so that the search for the correct integer values will be very inefficient. Decorrelation is attained by the transformation matrix ZT:

ZQZQaZz aT

zT

ˆˆ;ˆˆ == (5)

The elements of this matrix are all integers and in order to be volume preserving its determinant must be ±1. After decorrelation, a search space is defined and the search can be carried out. The integer solution that results in the minimum squared norm of the ambiguity residuals will be selected as the fixed solution ž. This means that there is a region Sz centered at each integer z; if the float solution falls in this region, it will be pulled to this integer, hence the name pull-in region. An example of two-dimensional pull-in regions is shown in figure 1. The integer solution needs to be transformed back to the integer values corresponding to the original float ambiguities by:

zZa T−= (6)

Finally, the float baseline solution is adjusted for the difference between the float and the integer ambiguities. This will result in a more precise solution, the so-called fixed solution. The fixed baseline solution is obtained as follows:

)ˆ(ˆ 1

ˆˆˆ aaQQbb aab−−= − (7)

If we look at the different steps described above, it becomes clear that integral ambiguity resolution for GPS and Galileo will not be a problem. The float ambiguity vector â will consist of two parts â1 and â2 now, corresponding to the GPS and Galileo phase observations respectively. The same applies for the vc-matrix Qâ which will have a form like:

=212

211

ˆˆˆ

ˆˆˆ

ˆ

aaa

aaa

a QQ

QQQ (8)

It is important to note that the float ambiguities of the two systems are correlated, so it is not correct to resolve the ambiguities for both systems separately, and thus independently. Fortunately, the input for LAMBDA simply consists of the float ambiguity vector together with the associated vc-matrix, so that we don't need to care about this.

−2 −1 0 1 2−2

−1

0

1

2

Figure 1. Two-dimensional integer least-squares pull-in regions.

−1

0

1−1

0

1

0

0.25

0.5

0.75

0.9

Figure 2. Probability mass function. The success rate equals the probability mass

located at the correct integer, in this example (0,0). The goal of integer ambiguity estimation is of course to improve the precision of the baseline estimates. This will certainly be the case if it can be assumed that the fixed ambiguities are non-stochastic, since then application of the propagation law of variances to eq.(7) will result in:

bbaaabbb

QQQQQQ ˆˆˆ1

ˆˆˆˆ <<−= − (9)

In reality, the fixed ambiguities are stochastic and their distribution function is a probability mass function. For an example see figure 2. Only if the probability mass located at the correct integer is very close to one, the fixed ambiguities may indeed be considered non-stochastic. This probability is the same as the success rate mentioned at the beginning of this section. It is important to know whether or not the success rate is sufficiently close to one. If it is not, there is a high probability that the ambiguities will be fixed incorrectly, resulting in a fixed baseline solution that is wrong. The success rate is given by:

==

aS

a dxxfaaP )()( ˆ (10)

where a is the correct integer value, Sa the corresponding pull-in region, and fâ(x) the probability density function of the float ambiguities, cf. (Teunissen, 1998a). It is assumed that these are normally distributed. Computation of the success rate, however, is difficult and therefore we use a lower bound and upper bound in practice. We will only consider the lower bound, which is given by the success rate of integer bootstrapping:

∏=

−Φ≥=n

i z Ii

aaP1 ˆ

12

12)(

|σ

(11)

where n is the number of ambiguities and

dyyx

x

}exp{2

1)( 2

21−=Φ

∞−π

(12)

The standard deviation Iiz |ˆσ is the square root of the conditional variance of the i-th

ambiguity, conditioned on the previous I=1,...,i-1 ambiguities. It can be obtained from the diagonal matrix D after an LDL

T-decomposition of the decorrelated vc-matrix of the ambiguities. This matrix can be computed without the need for actual observations, which shows that the success rate can indeed be considered as a design parameter. As input, we need Qâ, which follows from the observation model (2) as:

( ) 111ˆ

−⊥−= NPQNQ My

T

ka (13)

For the IGG model in (3) the sub-matrices are given by:

( )

( ) 2

1

22

1

2

1

221

1

111

1

111

ˆˆˆˆ

11

2

1

221

1

11

111ˆ

1221

2 ,1)(

NQMMQMMQMMQNQQ

iNQMMQMMQMMQNNQNQ

TTTT

k

T

aaaa

ii

T

i

TT

ii

T

iii

T

ikai

−−−−−

−−−−−−

+⋅−==

=+−= (14)

It would of course be interesting to know whether or not the success rates for IGG will be higher than for the individual systems. It can be expected that this will be the case, since there is more redundancy in model (3) and the model will be stronger because of a better geometry due to the larger number of visible satellites. On the other hand, there is a larger number of ambiguities to be resolved. It is possible that the lower bound of the success rate for IGG will be lower than for e.g. GPS-only, since by its definition in eq.(11) it can be seen that it is computed as a product of probabilities all smaller than or equal to one. So, if the number of ambiguities n becomes very large, the lower bound will become lower. It is thus possible that the decrease in the success rate due to the larger number of ambiguities will be larger than the gain due to improved geometry.

4.2 Internal and external reliability

The internal and external reliability of a system describe the detectability of model errors and the impact of unmodeled errors on the unknown parameters respectively. The internal reliability could thus be defined as the minimal detectable bias (MDB) that can

be detected using the appropriate test statistics (Baarda, 1967 and 1968). The MDB, ∇, can be computed with:

cPQc Ay

T ⊥−=∇

1

0λ (15)

where c specifies the model error that is considered - for GNSS for example a code

outlier or carrier slip; λ0 is the non-centrality parameter that depends on chosen values

of the confidence level and detection power; 111 )( −−−⊥ −=−= yT

yT

AA QAAQAAIPIP with PA the

orthogonal projector on the range space of the design matrix A.

In Teunissen (1998b) expressions were derived for the MDB of a code outlier and carrier slip for the observation model as given in eq.(2):

−−−⋅

+−−−

=∇

−−−

−−−⊥

slipcarrier )1(

1

outlier code)()1(

111

0

11111

0

dPQddPQddQdv

dPPQddPQddQd

Ny

T

kv

My

T

kv

y

T

MPNy

T

kMy

T

ky

T

N

λ

λ

(16)

The vector c in eq.(15) is now given by c=ci ⊗ d. The error is assumed to occur (code outlier) or start (carrier slip) in epoch i. The k-vector ci contains a one as the i-th entry and zeros otherwise in case of a code outlier; it contains ones as the last v entries in case of a carrier slip. The vector d selects the observation in which the error occurs. It is possible to elaborate further upon the expressions in (16), see e.g. (Verhagen, 2002). In case of IGG some of the terms in eq.(16) will change. Assume for example that we want to consider an error in a GPS observation. The equations below show what the terms in eq.(14) become for GPS-only (left) and for IGG (right):

11

111

21

11

1111

11

11

1111

11

11

111

21

2211

1111

1111

11

11

1111

11

11222211111111

11

1

)(

)(

dQPPQPPQPPQddPQd

dPQddPQd

dQMMQMMQMMQddPQd

dQddQd

MPMPMPMPMPMP

T

MP

T

NT

NT

TTTTM

T

TT

NNNNNNN

−−−−−−

−−

−−−−−−

−−

⊥⊥⊥⊥⊥⊥⊥ +↔

↔

+↔

↔

(17)

This indicates that the second and fourth term become smaller, and thus that the MDB will be also be smaller (because of the minus signs in the denominator of eq. (16)). So, errors in GPS observations will always be easier to detect in case of IGG than with GPS-only. On the other hand, the overall internal reliability of IGG needs not to be better compared to GPS-only and Galileo-only. It might be for example that the MDBs for Galileo-only are already much lower, so that the MDBs for errors in GPS observations are larger in case of IGG than the MDBs for Galileo-only. So far, only the internal reliability was considered. However, a user will generally be more interested in the external reliability, since it gives the impact of unmodeled errors on the unknown parameters. The external reliability could for example be computed as

the impact of a specific bias in the observations, c∇, on each of the unknown parameters. This is referred to as the minimal detectable effect (MDE), and it follows as:

∇⋅⋅

⊗=∇⋅⊗

⊗

⊗=

∇

∇−

−

−

−

dQNv

QMc

QQ

QQdc

QNe

QMI

QQ

QQ

a

b

y

T

y

T

i

aba

abb

i

y

T

k

y

T

k

aba

abb

1

1

ˆˆˆ

ˆˆˆ

1

1

ˆˆˆ

ˆˆˆ)(

ˆ

ˆ (18)

where ∇ b̂ is the bias in the float baseline solution and ∇â the bias in the float ambiguity

solution. Note that v=1 for a code outlier. For ∇ the value of the MDB can be used, since that is the largest bias due to a code outlier or carrier slip that will not be detected.

From eq.(18) it follows that the external reliability as represented by the MDE is given by a vector, whereas it would be more practical to have a scalar quantity as output. Thereby comes that a user is mainly interested in the baseline unknowns. Therefore, it might be more useful to compute the distance to the ‘true’ position, which we will call the minimal detectable position effect (MDPE), which follows directly from eq.(18) as:

2

3

2

2

2

1ˆˆˆ bbbMDPE ∇+∇+∇= (19)

where b̂∇ i , i=1,2,3 are the entries of the vector b̂∇ that refer to the position unknowns. If

this vector is given as Northing, Easting and Up (in meters), one could also consider the minimal detectable horizontal effect (MDHE) and the minimal detectable vertical effect (MDVE). A disadvantage of the parameters as described above is that they only refer to the float solution; the impact on the fixed solution is not yet considered. The MDE of the fixed baseline solution would be defined as:

)ˆ(ˆ 1

ˆˆˆ aaQQbb aab∇−∇−∇=∇ − (20)

However, this vector cannot be computed since the bias in the fixed ambiguities

depends on the float ambiguities themselves, i.e. this bias does not follow from ∇â directly. Still, it is possible to make inferences on the possible impact on the fixed solution by computing the so-called biased success rate (Joosten and Teunissen, 2002), that is the probability that the ambiguities will be fixed correctly in case the float

ambiguities are biased. It can be computed once the size of the bias, i.e. ∇â, is known:

aZzzLzL

aaP Tn

i z

i

z

i

IiIi

ˆˆwith 12

ˆ21

2

ˆ21)(

1 ˆ

1

ˆ

1

||

∇=∇−∇+

Φ+∇−

Φ== ∏=

−−

∇σσ

(21)

where L is the lower triangular matrix from the LDL

T-decomposition of the decorrelated vc-matrix of the ambiguities. Note that this probability refers to integer bootstrapping, not to integer least-squares.

4.3 Overview of design parameters

The following design parameters were described in this section:

• success rate: probability that ambiguities will be fixed correctly;

• MDB: minimal detectable bias in observations;

• MDE: minimal detectable effect of a bias on unknown parameters;

• MDPE: minimal detectable position effect;

• MDHE, MDVE: minimal detectable horizontal and vertical effect respectively;

• biased success rate: success rate if float solution is biased. The external reliability can now be described by the MDPE (or MDHE and MDVE) together with the biased success rate. The first parameter gives an indication of the possible impact of an unmodeled bias on the estimated float baseline parameters, the second gives the possible impact on the fixed solution in terms of the probability that the

ambiguities will (still) be fixed correctly in the presence of an unmodeled bias that is not detected. The goal of the computation of design parameters is to decide whether or not a positioning system can meet the user-specified requirements. Basically, a user will only be interested in a binary outcome of the design computations: yes, requirements are met; or no, requirements are not met. This is possible by defining upper bounds and/or lower bounds for some of the above-mentioned parameters, for example:

• success rate > 0.99; MDHE < 1.0 meter; MDVE < 1.0 meter. If the computed parameters fulfill these requirements, the system can be used. This approach also makes it possible to carry out availability studies: one could determine what percentage of the time a system would be available at a certain location. The next section shows some examples for GPS, Galileo and IGG.

5 Performance analysis As mentioned before, future GNSS users will be interested in the performance characteristics of the three available systems: GPS, Galileo and IGG. Maybe modernized GPS or Galileo on itself will provide the required precision, availability, integrity and reliability, otherwise IGG might be the solution. Therefore, we will give different examples of performance analyses for modernized GPS, Galileo and IGG. System specification were as follows:

• GPS: - 28 satellites (current status) - dual-frequency (L1 and L5) or triple-frequency (L1, L2 and L5)

• Galileo: - 30 satellites - dual-frequency (L1 and E5a) or triple-frequency (L1, E5a and E5b)

• IGG: - 28 + 30 satellites - dual-frequency (L1, L5 + L1, E5a)

A medium-length baseline was considered; an ionospheric standard deviation of 10 cm was used.

5.1 Graz

As a first example the following performance parameters were computed for a complete day, location Graz:

• success rates;

• MDHE and MDVE;

• biased success rates. The results are shown in figures 3 and 4. The MDEs are given with respect to a code outlier in the observation that results in the largest MDE. It turned out that the MDB’s for L1 outliers are larger than for L5 outliers, whereas the MDEs for L5 / E5a outliers are larger. That is because of the higher precision (and thus larger weight) that is assigned to these observations: a bias in an L5 / E5a code outlier has a larger impact on the final solution than a bias of the same size in an L1 observation. This indicates that it is difficult to make inferences on the performance based on only the internal reliability.

0 6 12 18 243

5

7

9

11

number of satellites (green)

0 6 12 18 240

1.5

3

4.5

6

MDE [m]

time

2−GPS

0 6 12 18 243

5

7

9

11

number of satellites (green)

0 6 12 18 240

1.5

3

4.5

6

MDE [m]

time

3−GPS

0 6 12 18 244

6

8

10

12

number of satellites (green)

0 6 12 18 240

1.5

3

4.5

6

MDE [m]

time

2−GAL

0 6 12 18 244

6

8

10

12

number of satellites (green)

0 6 12 18 240

1.5

3

4.5

6

MDE [m]

time

3−GAL

0 6 12 18 249

12

15

18

21

number of satellites (green)

0 6 12 18 240

1.5

3

4.5

6

MDE [m]

time

2−IGG

Figure 3. Minimal detectable horizontal effects (red) and vertical effects (blue) for GPS,

Galileo and integrated GPS-Galileo. Left: dual-frequency, right: triple-frequency. The number of satellites is shown in green (scaling on right side of the axes).

The biased success rates in figure 4 (blue) are shown for the outlier that results in the lowest value, i.e. they might refer to different outliers as the MDEs in figure 3. Hence, again only the worst case scenario is considered. It can be seen that GPS on itself performs badly, especially as compared to Galileo and IGG. This is due to the higher standard deviations assigned to the GPS L1 and L2 code observations (see table 1), and the lower number of satellites. The performance of IGG is best if we look at all design parameters, although there is not much difference with triple-frequency Galileo. Furthermore, it follows that the bias robustness of ambiguity resolution is much better for triple-frequency GPS and Galileo than for the dual-frequency equivalents.

0 6 12 18 243

5

7

9

11

number of satellites (green)

0 6 12 18 240

0.25

0.5

0.75

1

success rate

time

2−GPS

0 6 12 18 243

5

7

9

11

number of satellites (green)

0 6 12 18 240

0.25

0.5

0.75

1

success rate

time

3−GPS

0 6 12 18 244

6

8

10

12

number of satellites (green)

0 6 12 18 240

0.25

0.5

0.75

1

success rate

time

2−GAL

0 6 12 18 244

5

6

7

8

9

10

11

12

number of satellites (green)

0 6 12 18 240

0.25

0.5

0.75

1

success rate

time

3−GAL

0 6 12 18 249

12

15

18

21

number of satellites (green)

0 6 12 18 240

0.25

0.5

0.75

1

success rate

time

2−IGG

Figure 4. Success rates (red) and biased success rates (blue) for GPS, Galileo and integrated GPS-Galileo. Left: dual-frequency, right: triple-frequency. The number of

satellites is shown in green (scaling on right side of the axes).

5.2 World

The same settings as in the previous subsection were used in order to determine the availability of GPS, Galileo and IGG during the day for the entire world. The conditions as in section 4.3 were used, i.e. success rates should be larger than 0.99, and the MDHE and MDVE should be smaller than 1 meter. If these requirements are met, it is assumed that the system is available. Figure 6 shows the percentage of time that this is the case for the three systems. Obviously, IGG is the only system that is available at almost any time at any place.

2−GPS

−150 −100 −50 0 50 100 150

−80

−60

−40

−20

0

20

40

60

80

2−GAL

−150 −100 −50 0 50 100 150

−80

−60

−40

−20

0

20

40

60

80

2−IGG

−150 −100 −50 0 50 100 150

−80

−60

−40

−20

0

20

40

60

80

0 0.2 0.4 0.6 0.8 1 Figure 5. Availability of GPS, Galileo and integrated GPS-Galileo for one day.

So, for this example, IGG is to be preferred, but of course for other measurements scenarios and signal specifications it might well be that also GPS or Galileo provide the required performance. The results for triple-frequency GPS and Galileo are not shown here, but as expected the performance is much better than for the dual-frequency systems. Still, IGG will give the best availability statistics.

6 Concluding remarks It has been shown how design computations can be carried out for any of the future GNSSs: GPS, Galileo and integrated GPS-Galileo. Traditional design parameters are considered, as well as some new ones that are based on those well-known parameters.

The necessary algorithms are described and examples are shown in order to illustrate how they can be used. It has been shown that the proposed design parameters are easy to interpret and also allow for easy comparison of the performance for different scenarios or systems. The examples show that the performance of IGG in terms of precision and reliability will be better than for the individual systems, although triple-frequency GPS and Galileo might also provide the required performance. The triple-frequency systems will perform much better than the dual-frequency equivalents with respect to the bias robustness. The differences in the results for GPS and Galileo are due to the lower number of GPS satellites and the higher standard deviations assigned to the GPS code observations. So, it might well be that the results for GPS are too pessimistic.

References

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