## 1.4.1.1 Wide Lane

In equation 1.16 for i = 1 and j = 1 the wide-lane (LWL) observable is derived, with a wavelength of 86.2 cm. Given that, the wide lane ambiguity N5 equals N1-N2. The almost five times greater wavelength than L1 enhances the ambiguity resolution and is particularly useful for solutions where double differencing is used. The wide lane fixed solution is superior to L1 or L2 ambiguity float solutions, but inferior to their corresponding fixed ones in the accuracy of the baseline components. Errors and noise effects are magnified when expressed in metres by approximately 6 times (see Table 1.2).

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Related Content

1. Derivation of equations for Linear Combinations between frequencies

2. Narrow Lane

3. Ionospheric Free Linear Combination

4. Geometry Free Linear Combination

5. Linear Combinations between receivers and satellites

6. Single Difference Observable

7. Double Difference Observable

8. Triple Difference Observable

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## 1.4.1 Linear Combinations of simultaneous observations between frequencies

In the following linear combinations subscripts 1 and 2 refer to the f1 and f2 GPS frequencies respectively. Frequency combinations are formed by linearly combining measurements from the two frequencies. It is also assumed that for the following linear combination derivations the initial phases of the transmitted and the replica generated receiver signals are zero, thus the ambiguity parameter N1a corresponds to the integer number of cycles from the receiver to the satellite.

For the carrier phase functional model of equation 1.1, the general linear combination model between frequencies f1 and f2 is given below in metres:

Equation1.16

For a linear combination in cycles the coefficients and  are introduced.

Equation1.17

From the relationship between phase and frequency of an electromagnetic wave the frequency fLC of the new inter frequency combination is:

Equation1.18

Thus the new wavelength λLC is:

Equation1.19

Dividing equation 1.16 with λLC the following relationships are derived for the multiplying factors of the ambiguities  and  of frequencies f1 and f2 respectively:

Equation1.20

Equation1.21

The ambiguity NLC of the resulting observable following from equations 1.20 and 1.21 can be written in cycles as:

Equation1.22

The standard deviation of the original phase observation can be propagated into the standard deviation of the phase combination expressed in metres as:

Equation1.23

The corresponding pseudorange linear combinations are formed using equation 1.2 and parameters α and β. The following table summarises the most common between frequencies linear combinations, indicating the values used for either i and j or a and b combinations, the wavelength λLC and noise σLC of the resulting observables. The wide lane LWL narrow lane LNL ionospheric free LIF and geometry free LGF linear combinations are summarized. The standard deviation σ of the L1 and L2 measurement noise is expressed in metres and is assumed to be the same for both. It follows from the table that the noise of the resulting observable is increased for all linear combinations. The highest noise increment is seen for the wide lane linear combination while the lowest is observed for the narrow lane one.

Table 1.2: Common Linear Combinations between frequencies

 L α β i j λLC[cm] σLC[m] LWL 4.53 -3.53 1 -1 86.19 5.76 LNL 0.56 0.44 1 1 10.70 0.71 LIF 2.545 -1.545 77 -60 0.6 2.97 LGF 1 -1 60 -77 ∞ 1.41

In the case of relative positioning and ambiguity resolution the selection of the appropriate linear combination follows from the baseline distance and the required accuracy specifications. If the user is very close to the reference station then the narrow lane combination can be used, since the ionospheric and tropospheric bias will be low. For longer baselines the wide lane combination can be constructed since it has an effective wavelength of about 86 cm that is much longer than the respective wavelength of either L1 or L2. The ionospheric free linear combination can alternatively be used. The special characteristics and uses of the linear combinations are discussed in the following sections.

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Related Content

1. Narrow Lane

2. Wide Lane

2. Ionospheric Free Linear Combination

3. Geometry Free Linear Combination

4. Linear Combinations between receivers and satellites

5. Single Difference Observable

6. Double Difference Observable

7. Triple Difference Observable

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