A General Criterion of Integer Ambiguity Search (GPS) Part 2

An Equivalent Ambiguity Search Criterion and its Properties

Suppose undifferenced GPS observation equation and related LS normal equation are

tmpD-797_thumb[2]

Where all symbols have the same meanings as that of Eqs. 7.117 and 7.118. Equation 8.39 can be diagonalised as (cf. Sect. 7.6.1)

tmpD-798_thumb[2]

The related equivalent observation equation of the diagonal normal Eq. 8.41 can be written (cf. Sect. 7.6.1)

tmpD-799_thumb[2]


where all symbols have the same meanings as that of Eqs. 7.140 and 7.142.

Suppose GPS observation equation is Eq. 8.38 and the related least squares normal equation is Eq. 8.39, where X2 = N (Nis the ambiguity sub-vector) and X1= Y (Y is the other unknown sub-vector). The general criterion is (cf. Eq. 8.35)

tmpD-800_thumb[2]

wheretmpD-801_thumb[2]and index 0 denotes the float solution.

The search process in the ambiguity domain is a process to find out a solution X (which includes Nin the searching area and the computed Y) so that the value of 5(dX) reaches the minimum. The optimality property of this criterion is obvious.

For the equivalent observation Eq. 8.43, the related least squares normal equation is Eq. 8.41. The related equivalent general criterion is then (putting the diagonal cofactor of Eq. 8.41 into Eq. 8.44 and taking Eqs. 8.40 and 8.42 into account)

tmpD-803_thumb[2]

where index 1 is used to distinguish criterion of Eq. 8.45 from Eq. 8.44. The observation equations 8.38 and 8.43 are equivalent, and the related normal Eqs. 8.39 and 8.41 are also equivalent. Therefore, the Criterion 8.45 is called an equivalent criterion of the general Criterion 8.44.

Furthermore, Y and N shall be consistent to each other because they are presented in the same normal Eqs. 8.39 and 8.41. Using condition W = N and notation of Eq. 8.42, one has from Eqs. 8.26 and 8.24

tmpD-804_thumb[2]

Putting Eq. 8.46 into Eq. 8.45, one has

tmpD-805_thumb[2]

It is notable that the second term 5(dN) of the equivalent criterion Eq. 8.45 is exactly the same as the commonly used least squares ambiguity search (LSAS) criterion of Eq. 8.30 (cf., e.g., Teunissen 1995; Leick 1995; Hofmann-Wellenhof et al. 1997; Euler and Landau 1992; Han and Rizos 1997). Through Eq. 8.47 one may clearly see the differences between the criteria of Eqs. 8.30 and 8.45. When the results searched using Eq. 8.30 are different from that of using Eq. 8.45, the results from the search using Eq. 8.30 shall be only sub-optimal ones due to the optimality and uniqueness property of Eq. 8.45. The first term on the right side of Eq. 8.45 signifies an enlarging of the residuals due to the coordinate change caused by ambiguity fixing (cf. Sect. 8.3.3). The second term on the right side of Eq. 8.45 signifies an enlarging of the residuals due to the ambiguity change caused by ambiguity fixing (cf. Sect. 8.3.4). Equation 8.45 takes both effects into account.

1. Optimality and Uniqueness Properties of the Equivalent Criterion

The float solution X0 is the optimal and unique solution of Eq. 7.117 under the least squares principle. Criterion Eq. 8.45 is equivalent to criterion Eq. 8.44. A X leads to the minimum oftmpD-806_thumb[2]in Eq. 8.45, which will lead to the minimum oftmpD-807_thumb[2]in Eq. 8.44 and consequentially the minimum oftmpD-808_thumb[2]in Eq. 8.36; therefore using criterion of Eq. 8.45, analogously, the searched vector X is the optimal solution of Eq. 8.38 under the least squares principle and integer ambiguity properties. The uniqueness property is obvious. If one hasthen by using Eq. 8.45, one may assume that X1 must be equal to X2.

It is notable that Eqs. 8.44 and 8.45 are equivalent for use in searching; however, they are neither the same nor equal. For computing the precision,tmpD-810_thumb[2]in Eq. 8.36 has to be computed using Eq. 8.44.

Numerical Examples of the Equivalent Criterion

Several numerical examples are given here to illustrate the behaviour of the two terms of the criterion. The first and second terms on the right-hand side of Eq. 8.45 are denoted astmpD-811_thumb[2]is the equivalent criterion of the general criterion and is denoted as 8(total). The termtmpD-812_thumb[2]is the LSAS criterion. Of course, the search is made in the ambiguity domain. The search area is determined by the precision vector of the float solution. All possible candidates are tested one by one, and the relatedtmpD-813_thumb[2]are compared with each other to find out the minimum.

In the first example, precise orbits and dual frequency GPS data of 15 April 1999 at station Brst (N 48.3805°, E 355.5034°) and Hers (N 50.8673°, E 0.3363°) are used. The session length is 4 hours. The total search candidate number is 1 020. Results of the two delta components are illustrated as 2-D graphics with the 1st axis of search number and the 2nd axis of delta in Fig. 8.1. The red and blue lines representtmpD-814_thumb[2]respectively.tmpD-815_thumb[2]reaches the minimum at the search No. 237, andtmpD-816_thumb[2]at 769. 8(total) is plotted in Fig. 8.2, and it shows that the general criterion reaches the minimum at the search No. 493. For more detail, a part of the results are listed in Table 8.1.

tmpD-817_thumb[2]reaches the second minimum at search No. 771. This example shows that the minimum oftmpD-818_thumb[2]may not lead to the minimum of total delta, because the related tmpD-819_thumb[2]is large. If the delta ratio criterion is used in this case, the LSAS method will reject the found minimum and explain that no significant ambiguity fixing can be made. However, because of the uniqueness principle of the general criterion, the search reaches the total minimum uniquely.

Two components of the equivalent ambiguity search criterion

Fig. 8.1. Two components of the equivalent ambiguity search criterion

Equivalent ambiguity search criterion

Fig. 8.2. Equivalent ambiguity search criterion

The second example is very similar to the first one. The delta values of the search process are plotted in Fig. 8.3, wheretmpD-836_thumb[2]is much smaller thantmpD-837_thumb[2]reaches the minimum at the search No. 5 andtmpD-838_thumb[2]at 171. 8(total) reaches the minimum at the search No. 129. The total 11 ambiguity parameters are fixed and listed in Table 8.2. Two ambiguity fixings have just one cycle difference at the 6th ambiguity parameter. The related coordinate solutions after the ambiguity fixings are listed in Table 8.3. The coordinate differences at component x and z are about 5 mm. Even the results are very similar; however, two criteria do give different results.

In the third example, real GPS data of 3 October 1997 at station Faim (N 38.5295°, E 331.3711°) and Flor (N 39.4493°, E 328.8715°) are used. The delta values of the search process are listed in Table 8.4. BothtmpD-839_thumb[2]reach the minimum at the search No. 5. This indicates that the LSAS criterion may sometimes reach the same result as that of the equivalent criterion being used.

Table 8.1. Delta values of searching process

Search No.

tmpD-844 tmpD-845 tmpD-846

237

183.0937

97.8046

280.8984

493

181.7359

97.9494

279.6853

769

93.3593

315.2760

408.6353

771

96.0678

343.5736

439.6414

Table 8.2. Two kinds of ambiguity fixing due to two criteria

Ambiguity No.

1

2

3

4

5

6

7

8

9

10

11

LSAS fixing

0

0

1

0

0

0

-1

0

0

-1

-1

General fixing

0

0

1

0

0

-1

-1

0

0

-1

-1

Table 8.3.

Ambiguity fixed coordinate solutions (in meters)

Coordinates

X

y

z

LSAS fixing

0.2140

-0.0449

0.1078

General fixing

0.2213

-0.0465

0.1127

Table 8.4.

Deltas of the ambiguity search process

Search No.

tmpD-847 tmpD-848 tmpD-849

1

248.5681

129.0555

377.6236

2

702.6925

58.9271

761.6195

3

889.5496

107.9330

997.4825

4

452.1952

42.3226

494.5178

5

186.7937

112.3030

299.0967

6

739.0487

55.9744

795.0231

7

931.4125

89.9074

1021.3199

8

592.1887

38.0969

630.2856

 

Example of equivalent ambiguity search criterion 8.3.9  

Fig. 8.3. Example of equivalent ambiguity search criterion 8.3.9

Conclusions and Comments

Conclusions

A general criterion and its equivalent criterion of integer ambiguity searching are proposed in this section. Using these two criteria, the searched result is optimal and unique under the least squares minimum principle and under the condition of integer ambiguities. The general criterion has a clear geometrical explanation. The theoretical relationship between the equivalent criterion and the commonly used least squares ambiguity search (LSAS) criterion is obvious. It shows that the LSAS criterion is just one of the terms of the equivalent criterion of the general criterion (this does not take into account the residual enlarging effect caused by coordinate change due to ambiguity fixing). Numerical examples show that a minimum 5(dN) may have a relatively large 5(dY), and therefore a minimum 5(dN) may not guarantee a minimum 5(total). For an optimal search, the equivalent criterion or the general criterion shall be used.

Comments

The float solution is the optimal solution of the GPS problem under the least squares minimum principle. Using the equivalent general criterion, the searched solution is the optimal solution under the least squares minimum principle and under the condition of integer ambiguities. However, the ambiguity-searching criterion is just a statistic criterion. Statistic correctness does not guarantee correctness in all applications. Ambiguity fixing only makes sense when the GPS observables are good enough and the data processing models are accurate enough.

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