OTHER DUAL FREQUENCY PHASE COMBINATIONS |
There are other linear combinations of L1 and L2 observables that could
be constructed, leading to a variety of observables with different wavelengths,
ionospheric amplification factors and data noise characteristics (see, for
example, ABIDIN, 1993). Desirable
features of the artificial observables that can be constructed from the
L1 and L2 observations, for data processing purposes, are:
The linear combinations of the L1 and L2 phases which preserve the integer
nature of the cycle ambiguity, can be expressed as (in units of cycles):
 i,jm = i.(L1) + j.(L2) |
(6.4-22a) |
where i and j are integer constants. The following are some properties of the linear combination observables (ABIDIN, 1993).
The cycle ambiguity:
| ni,j = i.n1 + j.n2 | (6.4-22b) |
The frequency:
| fi,j = i.f1 + j.f2 | (6.4-22c) |
The effective wavelength:
 i,j = c / fi,j |
(6.4-22d) |
The ionospheric effect:
 |
(6.4-22e) |
The ionospheric scale factor isf:
| dion(i,j) = isf.dion(L1) |
where
 |
(6.4-22f) |
The noise scale factor nsf:
 (i,j)
= nsf.(1) |
where
 |
(6.4-22g) |
and (1) is the standard deviation of the L1 phase observation
( in metres, and assuming equal noise on L1 and L2 when expressed in metric
units).
The parameters in the columns in Table below have been computed using the
relations in eqn (6.4-22).
| Phase Combination |
i,j (m) |
Noise nsf x L1 |
Ion. delay isf x dion(L1) |
Ambiguity |
|---|---|---|---|---|
| L1 | 0.190 | 1.0 | 1.0 | n1 |
| L2 | 0.244 | 1.28 | 1.65 | n2 |
| L3 | 0.190 0.244 |
3.2 | 0.0 (or) |
 1n1
+ 2n2  1n1
+ 2n2 |
| L4 | 1.63 | -0.65 | 1n1
+ 2n2 | |
| L5 | 0.862 | 6.4 | -1.28 | n1 - n2 |
| L6 | 0.107 | 0.8 | 1.28 | n1 + n2 |
| nsf | is the noise scale factor to relate the data "noise" of the linear combination to that of the L1 observation |
| isf | is the magnitude of the ionospheric scale factor needed to relate the ionospheric delay on the linear combination to the ionospheric delay on the L1 observation. |
Some well known linear combinations are:
| > wide-lane: | i = 1, | j = -1 | (L5) |
| > narrow-lane: | i = 1, | j = 1 | (L6) |
| > first GPS signal: | i = 1, | j = 0 | (L1) |
| > second GPS signal: | i = 0, | j = 1 | (L2) |
| > double wide-lane: | i = -3, | j = 4 | |
| > half wide-lane: | i = 2, | j = -2 | |
| > semi wide-lane: | i = -1, | j = 2 | |
| > ionosphere-free: | i =77, | j =-60 | |
| > "monster" wide-lane: | i = -7, | j = 9 |
Two interesting combinations are worth closer inspection. The "monster
wide-lane" combination has the largest of all the wavelengths, being
of the order of 14.65m! However, it suffers in two respects: (a) the noise
scale factor is about 350 (if the noise on L1 is 1mm, the noise on -7,9
is 0.35m!), and (b) the ionospheric bias is approximately 870 times the
delay on L1! An analysis eqn (6.4-22e) results in another ionosphere-free
combination that can be constructed with integer coefficients i = 77, j
= -60. The effective wavelength of the 77,-60 combination
is very small, 77,-60 
0.006m.
A number of linear combinations of L1 and L2 observations derived from
certain codeless GPS receivers, for which the effective wavelength of the
L2 measurement is half that of the L2 signal, are useful. For example, the
"half wide-lane" is the linear combination that is analogous to
the standard L5 combination, constructed using the full wavelength L1 and
the half wavelength L2 measurement. The effective wavelength of this combination
is 43cm, however the ionospheric scale factor and the noise scale
factor are the same as for the L5 combination.
The more "exotic" dual-frequency combinations will not be discussed in these notes. Some useful combinations of dual-frequency phase and dual-frequency pseudo-range data will be introduced in the next section.
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© Chris Rizos, SNAP-UNSW, 1999