GENERAL FORM: PSEUDO-RANGE & PHASE DATA "IONOSPHERE-FREE" COMBINATIONS |
A general form of the linear combination of dual-frequency pseudo-range
and carrier phase data for the construction of "ionosphere-free"
/ "geometry-free" observables that can be used for ambiguity estimation
is given in HAN & RIZOS (1995a).
The integer ambiguities of any such four observable combination is given
by:
ni,j = i. (L1) + j.(L2)
+ a.P(L1)+ b.P(L2) |
(6.4-31a) |
where
| ni,j = i.n1 + j.n2 | (6.4-31b) |
 |
(6.4-31c) |
 |
(6.4-31d) |
with an effective wavelength of:
 i,j = c / (i.f1 + j.f2) |
(6.4-31e) |
The critical issue with regards to such ambiguity estimation is the standard deviation of such estimated ambiguities, taking into account the measurement noise of the observable types and how they propagate into the combinations. Obviously, if the standard deviation is greater than the magnitude of the effective wavelength, the estimated ambiguities are weakly determined and resolving them to their nearest likely integer values is not possible.
Applying the Law of Propagation of Variances to eqn (6.4-31a), the standard deviation of the estimated ambiguities is (in units of cycles of the effective wavelength):
 |
(6.4-32) |
where 
(1), (2), (P1)
and (P2) are the standard deviations of the L1, L2 phase
and L1, L2 pseudo-range measurements respectively.
An expression for the noise scale factor, relating the data "noise" of the linear combination to that of the L1 phase observation is:
| nsf = ( (i,j) . i,j ) / (1) |
(6.4-33) |
Table below summarises the characteristics of some useful dual-frequency
carrier phase / pseudo-range ionosphere-free combinations. Note that in
the case of some of the combinations, the standard deviation is dominated
by the uncertainty in the pseudo-ranges. In particular, note the range of
values of (i,j) in the second last column of the table.
These values have been computed assuming that (1) = (2)
= 0.01cyc, and (P1) = (P2) = 0.3m.
The "best" ionosphere-free / geometry-free combination is that
with the smallest standard deviation (i,j), which is the combination
i = 1, j =-1. This expression is the same as that which appears in eqn (6.4-26).
Hence it appears that the wide-lane ambiguity can be well determined using
this data combination technique, however it is difficult to compute any
other ambiguities, even the L1 and L2 ( n1, n2 ) integer
ambiguities. The reason is that the estimated ionospheric delay using the
pseudo-range data is not accurate enough to correct these other carrier
phase combinations (compare the last column of this Table with the corresponding
column in Table in section 6.4.6).
| Data combinationi,j |
i,j (m) |
a | b |  i,j |
nsf |
|---|---|---|---|---|---|
| 1,0
( (L1) ) |
0.190 | 21.501 | 16.246 | 8.085 | 808 |
| 0,1
( (L2) ) |
0.244 | 20.849 | 16.754 | 8.024 | 1028 |
| 77,-60 |
0.006 | 404.638 | 245.690 | 142.020 | 448 |
| 1,-1 |
0.862 | 0.652 | -0.508 | 0.248 | 112 |
| -1,2 |
0.341 | 20.197 | 17.262 | 7.971 | 1428 |
| 2,-2 |
0.431 | 1.304 | -1.016 | 0.497 | 113 |
| -3,4 |
1.628 | 18.892 | 18.278 | 7.886 | 6746 |
| -7,9 |
14.653 | 37.133 | 37.065 | 15.740 | 121197 |
| nsf | is the noise scale factor to relate the data noise of the linear combination to that of the L1 phase observation. |
HAN & RIZOS (1995a) present alternate formulae for eqns (6.4-31) and (6.4-32) which are based on single frequency phase and pseudo-range data. The problem with such linear combinations that give integer estimates of the ambiguity (or ambiguities) is that the ionospheric effect cannot be eliminated, hence the uncertainties associated with the ambiguity estimates are greater.
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© Chris Rizos, SNAP-UNSW, 1999