QUT Digital Repository: http://eprints.qut.edu.au/

Wang, Jun, Feng, Yanming, & Wang, Charles (2010) A Modified inverse integer Cholesky decorrelation method and the performance on ambiguity resolution. In: Proceedings of CPGPS 2010 Navigation and Location Services: Emerging Industry and International Exchanges, August, 2010, Shanghai, China.

© Copyright 2010 Scientific Research Publishing

A Modified Inverse Integer Cholesky Decorrelation Method and the Performance on Ambiguity Resolution

Jun Wang，Yanming Feng，Charles Wang

Queensland University of Technology, Brisbane, Australia e-mail: jun.wang@student.qut.edu.au

Abstract: One of the research focus in the integer least squares problem is the decorrelation technique to reduce the number of integer parameter search candidates and improve the efficiency of the integer parameter search method. It remains as a challenging issue for determining carrier phase ambiguities and plays a critical role in the future of GNSS high precise positioning area. Currently, there are three main decorrelation techniques being employed: the integer Gaussian decorrelation, the Lenstra–Lenstra–Lovász (LLL) algorithm and the inverse integer Cholesky decorrelation (IICD) method. Although the performance of these three state-of-the-art methods have been proved and demonstrated, there is still a potential for further improvements. To measure the performance of decorrelation techniques, the condition number is usually used as the criterion. Additionally, the number of grid points in the search space can be directly utilized as a performance measure as it denotes the size of search space. However, a smaller initial volume of the search ellipsoid does not always represent a smaller number of candidates.

This research has proposed a modified inverse integer Cholesky decorrelation (MIICD) method which improves the decorrelation performance over the other three techniques. The decorrelation performance of these methods was evaluated based on the condition number of the decorrelation matrix, the number of search candidates and the initial volume of search space. Additionally, the success rate of decorrelated ambiguities was calculated for all different methods to investigate the performance of ambiguity validation.

The performance of different decorrelation methods was tested and compared using both simulation and real data. The simulation experiment scenarios employ the isotropic probabilistic model using a predetermined eigenvalue and without any geometry or weighting system constraints. MIICD method outperformed other three methods with conditioning improvements over LAMBDA method by 78.33% and 81.67% without and with eigenvalue constraint respectively. The real data experiment scenarios involve both the single constellation system case and dual constellations system case. Experimental results demonstrate that by comparing with LAMBDA, MIICD method can significantly improve the efficiency of reducing the condition number by 78.65% and 97.78% in the case of single constellation and dual constellations respectively. It also shows improvements in the number of search candidate points by 98.92% and 100% in single constellation case and dual constellations case.

Keywords:

Modified Inverse Integer Cholesky Decorrelation, LLL, Condition Number, Ambiguity Resolution

1. Introduction

Integer ambiguity resolution is the key to use carrier phase measurements from Global Navigation Satellite System (GNSS) for high precise positioning. Given the GNSS linear observation equations

L=Aδx+BN+e (1)

and the criterion:

{ }21

x,Nmin , ,n p

LR Z

δ−− − ∈ ∈

QL Aδx BN δx N (2)

where L is m-vector of ‘observed minus computed’ double-difference (DD) observations, A is the m n× design matrix for the vector of real-valued unknowns δx , B is the m p× design matrix for the vector of integer DD ambiguities N, LQ is the variance matrix of

observables and e is the vector of unmodelled effects and measurement noise.

Solving the above mixed integer least-squares (MILS) problem has been proved equivalent to solve the integer least-squares (ILS) problem:

2

1N ˆ

ˆmin , p

N

Z−

⎧ ⎫⎪ ⎪− ∈⎨ ⎬⎪ ⎪⎩ ⎭Q

N N N           (3)

where is a floating ambiguity vector, having the corresponding variance-covariance matrix . For more details on the procedure of solving MILS or ILS, see (Teunissen, 1995; Hassibi and Boyd, 1998; Grafarend, 2000; Chang & Zhou, 2007).

The integer ambiguity search space is defined as

1 2ˆ

ˆ ˆ( ) ( )TN χ−− − ≤N N Q N N (4)

It is actually a hyper-ellipsoid centered at , its shape and orientation are governed by and its size can be controlled by 2χ . In general, has high correlation since the DD operation and correlation between measurements errors. Hence, the integer ambiguities search space can be very large. In order to make the search process easier and more efficient, different decorrelation techniques have been developed. The essence of decorrelation is to apply an admissible integer unimodular matrix Z to eliminate the off-diagonal elements of or reduce the size of the correlation coefficients. This can be expressed as

ˆ ˆˆ ˆ , ,= = = T

dec dec N NdecN ZN N ZN Q ZQ Z     (5)

Therefore the search space (4) can be transformed as

1 2ˆ

ˆ ˆ( ) ( )TN χ−− − ≤dec dec dec decdec

N N Q N N (6)

The condition number is usually used to show the performance of decorrelation methods. For instance, the well-known least-squares ambiguity decorrelation adjustment (LAMBDA) method based on integer Gaussian decorrelation was proposed by Teunissen, see (Teunissen et al., 1995, 1996, 1998). A detailed description and implementation of this method can be found in (De Jonge & Tiberius, 1996). Another algorithm named Lenstra–Lenstra–Lovász (LLL) was originally developed for lattice basis reduction, which can also be used to reduce the condition number of matrix(Lenstra, Lenstra, & Lovász, 1982). This algorithm was suggested for the decorrelation of the integer ambiguities (Hassibi and Boyd, 1998; Grafarend, 2000). Based on a modified LLL algorithm, Chang and Zhou (2007) developed a free Matlab package for solving MILES problems and demonstrated a higher computation efficiency than LAMBDA. Xu (2001) developed a random simulation approach to compare the performance of different decorrelation method, but the simulation seems more general, without referring to any particular satellite-receiver geometry, observation span and weighting system. The non-informativeness guarantees the statistical fairness of comparing different methods numerically because these three factors may favor a particular method. Xu also proposed an inverse integer Cholesky decorrelation method and demonstrated this

method outperformed LAMBDA and LLL method. However, the performance of these decorrelation methods in dealing with practical high dimension cases is still question mark (Xu, 2001; Svendsen, 2006). In the near future, more frequency signals, e.g. L1, L2 and L5 and more navigation satellites systems, e.g. GPS and Galileo could be used. Introducing more observations from three frequency signals and dual constellations changes the condition number of . Figure 1 shows the condition number of and the corresponding decorrelated matrix both in dual and triple frequencies cases. It is easy to find that the condition numbers of are larger than the original one. Figure 2 compares the condition numbers of and from single constellation and dual constellation cases; the later simulates dual-GNSS constellations. It is seen that the condition numbers of

are larger than these for the single constellation. In the present paper, a modified inverse integer Cholesky decorrelation method is proposed to further decorrelate .

Figure 1. Condition numbers of and in L1L2 and L1L2L5 cases. Left plot: the floating ambiguity variance-covariance matrix ; Right plot: the decorrelated ambiguity

variance-covariance matrix

On the other hand, the condition properties of decorrelation methods might not directly link to the ambiguity searching efficiency. A simple way is to count the grid points within search space. Although the volume of search space has been demonstrated to be a fair approximation of the number of grid points on the average(P. Teunissen, De Jonge, & Tiberius, 1996), the relation between the grid point number and condition

number have not been specified. In the later context of this work, we will also compare the number of search grid points of different decorrelation method and showing the dependence between them.. In addition, ambiguity reliability is a crucial issue, but the impact of different decorrelation methods on ambiguity reliability have not been well discussed (Peter J. G. Teunissen, Joosten, & Odijk, 1999; Verhagen, 2005).

 Figure 2. Condition numbers of and in GPS and DCS cases. Left plot: the floating ambiguity variance-covariance matrix ; Right plot: the decorrelated ambiguity variance-

covariance matrix .

Since the success rate of bootstrapping could be a very good approximation of ILS (P. J. G. Teunissen, 1998; Verhagen, 2003), in this contribution, we will compute the success rate of ambiguity bootstrapping with different decorrelation methods. We also notice the work by Henkel (2007 and 2009) investigating the impact of decorrelation transformation in integrity analysis since the decorrelation implementation can inflate biases.

The rest of paper is organized as follows. Section 2 briefly introduces different decorrelation techniques and proposes a modified inverse integer Cholesky decorrelation method. Section 3 presents the random simulation method, the concept of generating dual constellations and the criterion used to compare the performance of different decorrelation methods. Section 4 discusses the experimental results from four computation scenarios. The main findings of the paper are summarized in the final section. 

2. Decorrelation techniques

The variance-covariance matrix of DD float ambiguities has high correlations with off-diagonal elements. The target of decorrelation is to find a unimodular matrix Z to reduce the off-diagonal elements or reduce the size of the correlation coefficients. Since the matrix Z should be admissible and integer, the absolute decorrelation is impossible in most cases. The LAMBDA method based on integer Gaussian decorrelation has been proved to be highly efficient for ambiguity resolution in most situations(P. J. G. Teunissen, De Jonge, & Tiberius, 1995; P. J. G. Teunissen, de Jonge, & Tiberius, 1997). Although the LLL algorithm is developed for lattice basis reduction, the method can also be used to reduce the correlation of

in ILS (Sanzheng, 2008). The inverse integer Cholesky decorrelation (IICD) method uses the Cholesky decomposition instead of Gaussian decomposition for GPS decorrelation(Xu, 2001). Based on IICD method, a modified inverse integer Cholesky decorrelation (MIICD) is proposed.

2.1 Integer Gaussian decorrelation

Integer Gaussian decorrelation is actually performed as a sequence of integer Gaussian eliminations and permutations. Assuming there exist three elements iiq ,

jjq and  ijq of that satisfy / min( , ) 1/ 2ij ii jjq q q > , then the unimodular matrix can be constructed as

1

1

/ 1

1

ij iiq q

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎡ ⎤− ⎣ ⎦⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

1Z

O

M O

L

O

      (7)

if ii jjq q≤ ,or

jj

1

1 /

1

1

ijq q

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎡ ⎤− ⎣ ⎦⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

1Z

O

L

O M

O

     (8) 

if  ii jjq q> . Here operator [ ] denotes rounding to the nearest integer. Repeating the above steps, the final Z transformation matrix can be expressed as

Z=ZnLZ2Z1 (9)

Thus, the decorrelated matrix can be obtained by equation (5).

2.2 Lenstra–Lenstra–Lovász algorithm

The LLL algorithm was first introduced for decorrelation in GPS ambiguity resolution by Hassibi and Boyd (1998), followed with the contributions by Grafarend (2000), Xu (2001) and Chang (2007). The original LLL algorithm use the integer Gram-Schmidt orthogonalization process, however the Givens reflection based LLL algorithm can be numerically more robust than the original one (F. T. Luk & Qiao, 2007; Franklin T. Luk & Tracy, 2008). Since the matrix is positive and definite, it can always be factorized as . Compute the reduced or almost orthogonal basis V0, so that V0=V , where Z is unimodular. After these steps, we have

(10)

Due to V0 is almost orthogonal, the target of decorrelation can be achieved with .

2.3 Inverse integer Cholesky decorrelation (IICD) method

The inverse integer Cholesky decorrelation (IICD) method applies the LDLT factorization as follows:

(11)

where L is unit lower triangular matrix, and D is a diagonal matrix with positive elements. Although L cannot be directly used for ambiguity decorrelation due to the real-valued elements, [L] is obviously unimoudular as well as [L-1]. Thus, we let Z1= [L-1] and can compute the decorrelated matrix as

H1=Z1 (12)

Since in most cases Z1is not equivalent to L, H1is no longer diagonal. Repeating the process like

Hn=Zn (13)

until the condition number of Hn reach the predetermine vale, the final decorrelation can be express as(Xu, 2001)

L L (14) To obtain larger off-diagonal elements of L, we may rearrange the diagonal elements of and Hi (i=1,…,n-1) in ascending order. Before finishing this section, we would like to make some arguments on this method. Firstly, how big the predetermined condition number of the decorrelated matrix should be? The answer is not easy to say, because it relates to the dimension and formation of the original matrix. Secondly, since this method involves iteration process, sorting and stopping criteria would be very important for the IICD method (private communication with Dr. Xu on 25 June 2010). Simply comparing the condition number of Hn and Hn-1 may lead to wrong decision, because it is very likely to happen that cond(Hn)<cond(Hn-1) and cond(Hn)<cond(Hn-2) with cond(Hn-1)>cond(Hn-2), where cond() is the condition number operator. To overcome this shortcoming of IICD, we will propose a new method in the next section.

2.4 Modified inverse integer Cholesky decorrelation (MIICD) method

Instead of using the predetermined condition number as the iteration stopping criteria, we consider applying whether the abs(Zn) is an identity matrix to stop the process of inverse integer Cholesky decorrelation, where abs(.) is the absolute value operator. In addition, we may also rearrange the diagonal elements of H in descending order after stopping iteration and repeat the decorrelation process. From our experience, the condition numbers of Hi usually decrease with fluctuation, so we will record the condition number of Hi and transformation matrix Zi each time while conducting the procedure of decorrelation. This function let us be able to find the smallest condition number by searching all the value cond(Hi). Another iteration stopping criteria in this method is the predetermined iteration number. Figure 3 depicts this modified inverse integer Cholesky decorrelation method.

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δ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎢ ⎥⎣ ⎦

XL A B 0 eN

eL A 0 BN

(19)

It is noted that in (19), two data sets have the same coordinates systems, but different sets of ambiguity parameters.

 

 

 

Figure 4. The eigenvalues distribution of GPS DD floating

ambiguities. Left plot: the three largest eigenvalues; Right plot:

the other eigenvalues

3.3 Measuring performance

Conditioning properties are often used to compare the performance of different decorrelation techniques; however condition numbers only reveal the ratio of the square of semi-major axis and semi-minor axis of search ellipsoid. It can only partially reflect the ILS search efficiency; thus, the search numbers of grid points and the initial search space size are directly used to compare the impact of different decorrelation methods on search efficiency with the same search method. The details on how to compute the search numbers of candidates and initial search space size can see the instruction of LAMBDA or MILES (De Jonge & Tiberius, 1996; Chang & Zhou, 2007).

Furthermore, considering ambiguity validation requirements, the value of ratio-test and the success rate are computed. In general, the larger ratio value, the more confidence level of correct integer ambiguity results (Verhagen, 2004; P. J. G. Teunissen & Verhagen, 2008). The actual success rate of ILS is difficult to calculate; nevertheless the success rate of

bootstrapping is a lower bound and a very good approximation of ILS (P. J. G. Teunissen, 1998;  Feng & Wang, 2010), which can be computed as

0

t

21/ 21 0 |0 |

ˆ ˆP( | ) P( | )

(1 exp[ ]2(2 )

ILS Boostrapping

i

i i IR i I

z dQQ σπ σ=

≥

−= −∏ ∫

2

N N

x ) x   (20) 

where the diagonal elements of |i IQ can be calculated by factorization based on (11).

4. Experiments

The purpose of this section is to demonstrate the decorrelation performance of LAMBDA, LLL, IICD and MIICD methods in different situations. Four computation scenarios are set up follows:

• Scenario 1: Performing LAMBDA, LLL, IICD and MIICD decorrelation with randomly simulated definite-positive covariance matrices;

• Scenario 2: Performing LAMBDA, LLL, IICD and MIICD decorrelation with randomly simulated definite-positive covariance matrices where eigenvalues are constrained to certain values as discussed in Section 3;

• Scenario 3: Performing LAMBDA and MIICD decorrelation in ILS processing of a real GPS data set for a 21 km baseline ;

• Scenario 4: Performing LAMBDA and MIICD decorrelation in ILS processing of the same data set as Scenario 3, but added with virtual GNSS data.

For simulation experiments, 300 Q matrix samples were randomly generated for performance evaluation in Scenario 1 and Scenario 2. We then set the condition number of original positive definite matrix Q based on the sample dimension size (as shown in Figure 5). The condition number is set as 1×104, if 4 dim 20, or 1×105, if 20 dim 24, where dim () is the matrix dimension operator.

Figure 6 shows the condition numbers of the original simulation matrix Q and the results for four decorrelation methods for Scenario 1. It is obvious that all decorrelation methods can significantly decrease the condition number of Q. Particularly, the condition

numbers resulted from LLL and MIICD decorrelation are smaller than 200 and in most cases are smaller than 100. Meanwhile, the condition numbers of other two methods, LAMBDA and IICD are slightly higher, mostly below 200 with occasional peaks to 400.

 

Figure 5. Demensions of the 300 random simulation examples

Figure 6 Condition Numbers of simulation Q examples and results by LAMBDA, LLL, IICD and MIICD in Scenario 1

For Scenario 2, constraint eigenvalues of Q were

generated for performance evaluation as discussed in Section 3.1. The condition numbers of the original Q matrix and four decorrelation methods are shown in Figure 7. In this scenario, though most samples were

successfully decorrelated by these methods, results indicate that the decorrelation may not occur at some epochs based on given stopping criteria on condition number comparison. For instance, at epoch 206, the decorrelation did not happen with IICD method.

Figure 7 Condition Numbers of simulation Q examples and results by LAMBDA, LLL, IICD and MIICD in Scenario 2

Table 1 summarizes the improvement in term of condition numbers with respect to the LAMBDA results in Table 1. It is clear that MIICD has the best performance in terms of conditioning property in these two cases. Therefore, for simplicity, only MIICD and LAMBDA method are chosen for evaluation in the practical scenarios. Table 1 Statistics of the results by LLL, IICD and MIICD with the condition numbers of the simulated examples by comparison of LAMBDA

Scenario 1 Scenario 2

Method Number of examples whose conditionings smaller than LAMBDA

LLL 161 (53.67%) 213 (71.00%)

IICD 178 (59.33%) 225 (75.00%)

MIICD 235 (78.33%) 245 (81.67%)

For Scenario 3 and Scenario 4 experiments, data

were collected for 24 hours duration with 15 seconds of

0 50 100 150 200 250 3004

6

8

10

12

14

16

18

20

22

24

Dim

ensi

ons

Epochs

0 50 100 150 200 250 3000

1

2x 105 Original

0 50 100 150 200 250 3000

200

400LAMBDA

0 50 100 150 200 250 3000

100

200LLL

0 50 100 150 200 250 3000

200

400IICD

0 50 100 150 200 250 3000

100

200

Con

ditio

n N

umbe

r

Epochs

MIICD

0 50 100 150 200 250 300012

x 105 Original

0 50 100 150 200 250 30005

10x 104 LAMBDA

0 50 100 150 200 250 30005

10x 104 LLL

0 50 100 150 200 250 30005

10x 104

Con

ditio

n N

umbe

r

Epochs

MIICD

0 50 100 150 200 250 300012

x 105 IICD

sampling rate and 15 degrees of mask angle. The virtual Galileo constellation (VGC) used for Scenario 4 is generated from the collected dataset with time latency of 2 hours.

Figure 8 and Figure 9 show the initial search space sizes of two methods in the cases of Scenario 3 and Scenario 4 respectively. In order to avoiding the effects of different search algorithm, we use the same search method from MILES software package (Chang & Zhou, 2007). It can be seen that there is no distinct trend of these two methods in the initial search space size computation. Additionally, Scenario 4 is shown to have larger initial search space sizes which are due to the increased ambiguity candidates in Dual Constellation Systems (DCS).

 

Figure 8. Initial search space sizes of the24 hours results by

LAMBDA and MIICD for Scenario 3

 

Figure 9. Search space sizes of the24 hours results by LAMBDA

and MIICD for Scenario 4

The condition numbers of LAMBDA and MIICD methods for Scenario 3 and Scenario 4 have been computed and shown in Figure 10 and Figure 11 respectively. It can be clearly seen that the condition number results of LAMBDA and MIICD have similar trends and fluctuations and the latter method has better performance in most cases. In particular, the MIICD method has significant performance improvement in the DCS case where the peak condition number of LAMBDA is larger than 8000 while that of MICCD is smaller than 2000.

 

Figure 10. Condition numbers of the 24 hours results by

LAMBDA and MIICD for Scenario 3

 

Figure 11. Condition numbers of the 24 hours results by

LAMBDA and MIICD for Scenario 4

The search candidate number of LAMBDA and

MIICD methods for Scenario 3 and Scenario 4 is shown in Figure 12 and Figure 13 respectively. It can be clearly observed that the search candidate numbers of LAMBDA are generally larger than MIICD method. Similarly to the condition number results, the

improvement in the search candidate numbers of MIICD method is more significant in the DCS case (Scenario 4). For instance in Scenario 4, the search numbers of LAMBDA are around 1×105 between epoch 2400-2800, while the MIICD search numbers are mostly less than 200 with the peak of 4000 during the time.

 

Figure 12 Search numbers of grid points of the24 hours results

by LAMBDA and MIICD for Scenario 3

 

Figure 13 Search Candidates numbers of the24 hours results by

LAMBDA and MIICD for Scenario 4

The success rates of LAMBDA and MIICD method

have been computed based on (20) for Scenario 3 Scenario 4 and shown in Figure 14 and Figure 15 respectively. It is seen that in most cases the success rate of LAMBDA are higher than MIICD, in particularly for the MSC case as shown in Figure 15. It makes no sense to relate the search number of grid

points to the success rate, because it may decrease the search number with different search method.

  

Figure 14 Success rates of the24 hours results by LAMBDA and

MIICD for Scenario 3   

 

Figure 15 Success rates of the24 hours results by LAMBDA and MIICD for Scenario 4 

Figure 16 shows the ratio-test values of MIICD both in the case of Scenario 3and Scenario 4. Since the ratio-test values of LAMBDA are equal to those of MIICD, thus it is not presented. This interesting fact implies the performance of different decorrelation methods is the same in terms of ambiguity validation even the success rates may be different. Results from Figure 16 have shown that there is no big difference in the ratio-test values between the single constellation and the dual constellations scenarios. 

0 500 1000 1500 2000 2500 30000

0.2

0.4

0.6

0.8

1

Suc

cess

Rat

e

Epochs

LAMBDAMIICD

0 500 1000 1500 2000 2500 30000.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Suc

cess

Rat

e

Epochs

LAMBDAMIICD

Comment [w1]: need a better statement. 

 

Figure 16 Ratios of the24 hours results by MIICD for Scenario 3

and Scenario 4

Figure 17 shows the position errors in the case of DCS, indicating that the errors are bounded in [-4cm, 4cm].

 

Figure 17 Position results of MIICD for Scenario 4

 

To investigate relations between condition numbers and search candidate numbers, initial search space sizes and search candidate numbers, we can either draw the scatter plots or calculate the correlation coefficients. Figure 18 shows the scatter plots of these three parameters. A linear dependence is clearly shown between the condition number and the search candidate number (upper plot), but not distinct relation is shown for the search space sizes and the search candidate number (lower plot).  

   

 Figure 18 Scatter Plot of the search candidates number with the condition number and search space size respectively. Upper plot: the search candidates numbers and the condition numbers scatter plot; Lower plot: the search candidates numbers and the search space size scatter plot.

Table 2 presents the correlation coefficients which also verify that the condition number is highly related to the candidate search number. On the other hand, the correlation coefficient 0.8050 also reveals that the condition numbers cannot totally be represented by the search candidate numbers. Table 2. The correlation coefficients between search candidates numbers and condition numbers, search candidates numbers and search space sizes.

The correlation coefficients Search Candidates Numbers

Condition numbers 0.8050

Search space sizes 0.0329

Table 3 summarizes the statistics of LAMBDA and MIICD methods in the cases of Scenario 3 and Scenario 4. There are 2880 samples for 24 hours data both in Scenario 3 and Scenario 4. From the above figures and tables, we can obtain the following useful observations: • MIICD has better performance than LAMBDA by comparing the condition numbers and the search grid point numbers, especially in the high

0 1000 2000 3000 4000 50000

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ize

Search Candidates Number

dimension case and the dual constellation system case. For instance, 97.78% of the condition numbers and 100% of the search grid point numbers of MIICD method are smaller than those of LAMBDA method. In terms of the condition numbers, the improvement percentage (78.65%) of Scenario 3 is very close to the simulated case (78.33%) of Scenario 1; • In general, the initial search space size of MIICD are larger than that of LAMBDA both in single GPS system and dual constellations system, however it doesn’t mean that more search candidate steps are required. The search space size and the search candidate number for ambiguity resolution are uncorrelated as was pointed out by Teunissen (1996). A practical example is presented in this paper to highlighted the fact; • Although the success rates of MIICD are generally lower than those of LAMBDA, the AR results of MIICD is not necessary less reliable than those results of LAMBDA as the same ratio-test values were obtained for both methods. Therefore, further investigation is needed before we can relate the success rate of bootstrapping ambiguity to its reliability (Verhagen, 2005).

Table 3  Statistics of MIICD with the condition numbers, search numbers of candidates, search efficiency, search space sizes and success rates of the practical examples by comparison of LAMBDA.

MIICD <LAMBDA Scenario3 Scenario 4

Condition numbers 2265 (78.65%)

2816 (97.78%)

Search numbers 2849 (98.92%)

2880 (100%)

Search space sizes 1283 (44.55%)

1072 (37.22%)

Success rates 2878 (99.93%)

2880 (100%)

Ratio-test values equal equal

 

5. Conclusions

With the target of speeding up the ambiguity resolution process, different decorrelation techniques have been developed (P. J. G. Teunissen, 1995; Hassibi & Boyd, 1998; Xu, 2001). Some properties with mainly focus on the condition number of different decorrelation

techniques have been investigated (Svendsen, 2006). In spite of demonstrating that the invest integer Cholesky decorrelation (IICD) method outperforms the LAMBDA method and LLL algorithm in random simulation cases by Xu (2001), the practical examples have not been given.

In this contribution, we have proposed a modified inverse integer Cholesky decorrelation (MIICD) and the whole procedure. Four different experiments results including simulation data and real data have demonstrated that further improvement is achieved by MIICD. Even though the performance of MIICD in success rate computation is slightly deteriorated, the improvement of condition number and search step number is more distinct. However as a result of the random simulation Scenario 2, MIICD cannot guarantee the outstanding performance in all the situations, even the dimension of matrix is less than 25. Other techniques to data cannot be either. In addition, it was point out there is no evidence that the initial search space size relates to the number of search candidates. The condition number has high linear dependence with the number of search candidates, but it is still impossible to totally reflect on each other. Finally, it is shown the fact that we can get the same ratio-test value even with different success rate. This may deserve attentions and interests of people in the ambiguity resolution reliability investigation.

Acknowledgments 

This work was partially supported by CRC for Spatial Information project 1.04 "Delivering Precise Positioning Services in Regional Areas" 2007-2010. The first author acknowledges the financial support by China Scholarship Council (Grant No: 2008466001). Last but not least, the author would like to thank Dr. Xu Peiliang from Kyoto University for his constructive suggestions and comments.

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