8.4.4 Some Dual-Frequency Procedures

USING THE L5 WIDE-LANE OBSERVABLE



The L5 observable approach is only appropriate for double-difference solutions. First an ambiguity-free solution is obtained using L5 observables. Because the L5 observable has a relatively long wavelength ( 0.86 metres), the L5 ambiguities can be more easily resolved than either the L1 or L2 ambiguities for short and medium length baselines, even in the presence of the ionospheric biases (they are still present in the L5 observable -- Table in section 6.4.6). An ambiguity-fixed solution is then obtained where perhaps one would not have been possible if only single-frequency data had been processed.

How good is an L5 solution? An L5 ambiguity-free solution is inferior to an L1 (or L2) ambiguity-free solution. If the L5 ambiguities can be resolved then the ambiguity-fixed solution is superior to an L1 or L2 ambiguity-free solution. Some commercial GPS software make use of the L5 observable. There are however a number of more sophisticated procedures that have been developed for high precision applications (in which ambiguity resolution is desired for baselines in excess of 100km in length), and which now use the L5 observable to aid ambiguity resolution from very short observation sessions ("rapid static" techniques, or "on-the-fly" ambiguity resolution -- section 5.5.2), or to assist in cycle slip detection and repair.

One approach is to form double-differences of the geometry-free L4 observable from the L1 and L2 phase observations and to decouple the L1 and L2 terms using eqn (6.4-27):

(8.4-2a)


Several comments can be made with respect to this relation:

The table below lists combinations of c4 and c5 for small values of c1 and c2. For example, a 1 cycle slip on L1 and L2 will cause only a 0.283 cycles (L1 wavelength size) jump in L4, making that combination difficult to spot in the L4 data series.


Another approach for cycle slip detection/repair, or ambiguity resolution, is to use L3 in combination with L5, and to decouple the L1 and L2 ambiguities using the relation:

(8.4-3)


The values of n5 and n3 can be assumed to have been derived from two separate Least Squares ambiguity-free solutions. Alternatively, n5 can be obtained from eqn (6.4-26a). This expression is ionosphere-free, and hence the combination of L3 and L5 processing can be used for ambiguity resolution and cycle slip detection/repair for very long baselines (as in the case of GPS crustal motion surveys) -- see discussion below following eqn (8.4-6). However, dual-frequency pseudo-range data would be required in order to determine the P(L6) quantity.

On the other hand, a similar expression to eqn (8.4-3) can be written for cycle slip detection and repair, by substituting c5 for n5, c3 for n3, etc. In this case the L3 and L5 data series are screened and when a "jump" is detected, the L1 and L2 cycle slips can be decoupled. Table below shows the expected jumps in L3 and L5 for small jumps in L1 and L2. Note that for certain combinations of L1 and L2 slips (for example, 4 cycles on L1 and 5 cycles on L2), the L3 signature is very small (being only 0.264 cycles of L1 size). An ideal cycle slip detection algorithm should therefore screen all data series: L1, L2, L3, L4 and L5.


L3 and L4 signatures (in units of L1 cycles) for small L1 and L2 cycle slips.

L5 = c1 - c2 L3 = 2.546c1 - 1.984c2 L4 = c1 - 1.283c2

l c1 - c2 l c1 c2 L3 L4

0

± 1

± 1

0.562

0.283

0

± 2

± 2

1.124

0.567

0.5

± 0

± 0.5

0.992

0.642

0.5

± 1

± 0.5

1.554

0.358

1

± 1

± 2

1.422

1.567

1

± 2

± 3

0.860

1.850

1

± 3

± 4

0.298

2.133

1

± 4

± 5

0.264

2.417

1

± 5

± 4

4.794

0.132

1

± 5

± 6

0.827

2.700

1

± 6

± 7

1.389

2.983

2

± 5

± 7

1.157

3.983

2

± 6

± 8

0.595

4.267

2

± 7

± 9

0.033

4.550

2

± 8

± 10

0.529

4.833


Another procedure for resolving L1 and L2 integer ambiguities from the resolved wide-lane ambiguities n5 is to attempt to resolve the narrow-lane ambiguities n6, and then to use the relations:

(8.4-4)

This is does not appear to be an easy task because of the short wavelength of the L6 observables ( 0.11 metres). However, there are two positive aspects to using L5 and L6 combinations in this way:

If n5 is even --> n6 must be even
If n5 is odd --> n6 must be odd

In this way the effective wavelength of the L6 ambiguity is doubled, and is 0.21 metres, hence making the resolution of the n6 much easier.


Use can also be made of the L4 geometry-free / ionosphere-free observable (eqn (6.4-27)) and the ionosphere-free n3 ambiguity values, and then decouple them using the relation:

(8.4-5)

If dual-frequency pseudo-range data is not available, eqn (8.4-5) can still be used but only after making the assumption that P(L4) =0. This technique is often used for baselines up to 100km, if observed using C/A code dual-frequency receivers (no P(L2) observations). (Note, that L4 in eqn (8.4-2) is equivalent to (L4) + P(L4) in eqn (8.4-5).)

If the narrow-lane pseudo-range combination is available then, through the combination of eqns (6.4-17b) and (6.4-21):

(8.4-6)

the wide-lane ambiguities are obtained directly, without the need for a Least Squares search procedure. Furthermore, due to the peculiar influence of the ionosphere on phase and pseudo-range data (section 6.2.7), the combination above has the unique property of being free of any ionospheric delay. It is advisable however to average the data combination in eqn (8.4-6) in order to beat down the multipath errors in the pseudo-range data. Once the wide-lane ambiguities have been determined, then one of several methods described earlier can be used to decouple the L1 and L2 ambiguities (using either the L4, L3 or L6 observables).

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© Chris Rizos, SNAP-UNSW, 1999