Positioning and Navigation Using the Russian Satellite System

wavelength is still in the order or below the noise level of carrier phase measurements and thus too short
for fixing the integer ambiguities. Frequency number 19 will only be used until 2005. see Section 3.4.
Beyond 2005, the largest common denominator will be 9 for the pair of frequency numbers -4 and 5.
This would correspond to a wavelength of approximately 0.6 mm. Generally, it can be stated that the
traditional range of GLONASS frequency numbers is better suited to form such pairs, due to the wider
range of frequencies.
Some frequency numbers, however, are not suited to form such pairs with common denominators,
e.g. numbers −5 and 3. Added to 2848, these numbers are either prime numbers themselves or all their
smallest prime factors is so large that the nearest multiple of that prime factor is outside the GLONASS
frequency range.
Given a mixed GPS/GLONASS double difference, the largest common denominator of any pair of
the coefficients 210056, 75 · (2848 + n) is 62 for GLONASS frequency number. This would increase
the wavelength from some 880 nm to around 55 µm. But here, again, for some of the pairs common
denominators do not exist.
On the other hand, with such small wavelengths, ambiguity fixing to integers will not be necessary,
but fixing to thousands (GLONASS/GLONASS) or even hundreds of thousands (GPS/GLONASS) of
cycles might be sufficient.
The noise σ
λ
of a carrier phase measurement can be written as σ
λ
= λ · σ
ϕ
with σ
ϕ
being the noise
of the phase measurement at cycle level. The noise of a linear combination can then be expressed as
σ
λ
kS ,kr
= λ
k
S
,k
r
k
S 2
+ k
r2
· σ
ϕ
Considering σ
ϕ
to be constant for all GPS and GLONASS carrier phase measurements, and with
λ
k
S
,k
r
= λ
∗
from above, the ratio of
σ
λ
kS ,kr
σ
λ
L
=
λ
k
S
,k
r
λ
L
k
S 2
+ k
r2

8.4 A Proposed Solution to the Frequency Problem
117
n
0
1
2
3
4
5
6
7
8
9
10
11
12
2848 + n
2848
2849
2850
2851
2852
2853
2854
2855
2856
2857
2858
2859
2860
0
2848
–
1
2
1
4
1
2
1
8
1
2
1
4
1
2849
1
–
1
1
1
1
1
1
7
1
1
1
11
2
2850
2
1
–
1
2
3
2
5
6
1
2
3
10
3
2851
1
1
1
–
1
1
1
1
1
1
1
1
1
4
2852
4
1
2
1
–
1
2
1
4
1
2
1
4
5
2853
1
1
3
1
1
–
1
1
3
1
1
3
1
6
2854
2
1
2
1
2
1
–
1
2
1
2
1
2
7
2855
1
1
5
1
1
1
1
–
1
1
1
1
5
8
2856
8
7
6
1
4
3
2
1
–
1
2
3
4
9
2857
1
1
1
1
1
1
1
1
1
–
1
1
1
10
2858
2
1
2
1
2
1
2
1
2
1
–
1
2
11
2859
1
1
3
1
1
3
1
1
3
1
1
–
1
12
2860
4
11
10
1
4
1
2
5
4
1
2
1
–
13
2861
1
1
1
1
1
1
1
1
1
1
1
1
1
14
2862
2
1
6
1
2
9
2
1
6
1
2
3
2
15
2863
1
7
1
1
1
1
1
1
7
1
1
1
1
16
2864
16
1
2
1
4
1
2
1
8
1
2
1
4
17
2865
1
1
15
1
1
3
1
5
3
1
1
3
5
18
2866
2
1
2
1
2
1
2
1
2
1
2
1
2
19
2867
1
1
1
1
1
1
1
1
1
1
1
1
1
20
2868
4
1
6
1
4
3
2
1
12
1
2
3
4
21
2869
1
1
19
1
1
1
1
1
1
1
1
1
1
22
2870
2
7
10
1
2
1
2
5
14
1
2
1
10
23
2871
1
11
3
1
1
9
1
1
3
1
1
3
11
24
2872
8
1
2
1
4
1
2
1
8
1
2
1
4
n
13
14
15
16
17
18
19
20
21
22
23
24
2848 + n
2861
2862
2863
2864
2865
2866
2867
2868
2869
2870
2871
2872
0
2848
1
2
1
16
1
2
1
4
1
2
1
8
1
2849
1
1
7
1
1
1
1
1
1
7
11
1
2
2850
1
6
1
2
15
2
1
6
19
10
3
2
3
2851
1
1
1
1
1
1
1
1
1
1
1
1
4
2852
1
2
1
4
1
2
1
4
1
2
1
4
5
2853
1
9
1
1
3
1
1
3
1
1
9
1
6
2854
1
2
1
2
1
2
1
2
1
2
1
2
7
2855
1
1
1
1
5
1
1
1
1
5
1
1
8
2856
1
6
7
8
3
2
1
12
1
14
3
8
9
2857
1
1
1
1
1
1
1
1
1
1
1
1
10
2858
1
2
1
2
1
2
1
2
1
2
1
2
11
2859
1
3
1
1
3
1
1
3
1
1
3
1
12
2860
1
2
1
4
5
2
1
4
1
10
11
4
13
2861
–
1
1
1
1
1
1
1
1
1
1
1
14
2862
1
–
1
2
3
2
1
6
1
2
9
2
15
2863
1
1
–
1
1
1
1
1
1
7
1
1
16
2864
1
2
1
–
1
1
1
4
1
2
1
8
17
2865
1
3
1
1
–
1
1
3
1
5
3
1
18
2866
1
2
1
2
1
–
1
2
1
2
1
2
19
2867
1
1
1
1
1
1
–
1
1
1
1
1
20
2868
1
6
1
4
3
2
1
–
1
1
3
4
21
2869
1
1
1
1
1
1
1
1
–
1
1
1
22
2870
1
2
7
2
5
2
1
2
1
–
1
2
23
2871
1
9
1
1
3
1
1
3
1
1
–
1
24
2872
1
2
1
8
1
2
1
4
1
2
1
–
Table 8.1: Largest common denominators of GLONASS/GLONASS double difference coefficients, fre-
quency range until 2005.

118
8 OBSERVATIONS AND POSITION DETERMINATION
n
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
2848 + n
2841
2842
2843
2844
2845
2846
2847
2848
2849
2850
2851
2852
2853
2854
-7
2841
–
1
1
3
1
1
3
1
1
3
1
1
3
1
-6
2842
1
–
1
2
1
2
1
2
7
2
1
2
1
2
-5
2843
1
1
–
1
1
1
1
1
1
1
1
1
1
1
-4
2844
3
2
1
–
1
2
3
4
1
6
1
4
9
2
-3
2845
1
1
1
1
–
1
1
1
1
5
1
1
1
1
-2
2846
1
2
1
2
1
–
1
2
1
2
1
2
1
2
-1
2847
3
1
1
3
1
1
–
1
1
3
1
1
3
1
0
2848
1
2
1
4
1
2
1
–
1
2
1
4
1
2
1
2849
1
7
1
1
1
1
1
1
–
1
1
1
1
1
2
2850
3
2
1
6
5
2
3
2
1
–
1
2
3
2
3
2851
1
1
1
1
1
1
1
1
1
1
–
1
1
1
4
2852
1
2
1
4
1
2
1
4
1
2
1
–
1
2
5
2853
3
1
1
9
1
1
3
1
1
3
1
1
–
1
6
2854
1
2
1
2
1
2
1
2
1
2
1
2
1
–
Table 8.2: Largest common denominators of GLONASS/GLONASS double difference coefficients, fre-
quency range beyond 2005.
will become about 1.4 for both L
1
(λ
L
≈ 19 cm) and L
2
(λ
L
≈ 24 cm), GPS/GLONASS and GLO-
NASS/GLONASS carrier phase measurements (L
2
GPS at full wavelength) and 2.2 for L
2
GPS/GLO-
NASS carrier phase measurements with GPS at half wavelength. This means, the noise of a double
difference carrier phase measurement formed in this way is about 1.4 (respectively 2.2 for L
2
with GPS
at half wavelength) times higher than the noise of the original measurements.
An example of positioning results using GPS and GLONASS double difference positioning with car-
rier phases is shown in Figure 8.10. Positions were computed from data logged by two 3S Navigation
R-100/R-101 receivers, which were set up at known locations at the Institute of Geodesy and Naviga-
tion. Pseudorange and carrier phase measurements were logged every second for approximately one hour
each, of which some forty minutes were common to both receivers. One of these receivers was used as
reference station, the other was treated as the user station. Its position was determined in this example.
Observation epochs at both receivers were not exactly synchronized. The data are the same as the data
already used for the pseudorange positioning example. The plot shows the deviation from the known
location of the antenna of the user station in the horizontal plane. GPS positions were computed from
L
1
carrier phase measurements and raw L
1
C/A-code pseudorange measurements. GLONASS positions
were computed from dual-frequency carrier phase measurements and raw dual-frequency P-code measure-
ments. Wherever possible, the ionospheric free linear combinations of the code observables were formed.
These observables used are not really identical for GPS and GLONASS, but with dual-frequency mea-
surements readily available on GLONASS, the best possible results for each system are determined. The
inter-channel biases
∆δt
Sr
U R,ICB
for the GLONASS satellites have been neglected. GLONASS satellite
positions were converted from PZ-90 to WGS84 using the transformation according to (Roßbach et al.,
1996). Carrier phase ambiguities have not been fixed. The pseudorange observations have been included
in the positioning to increase the number of observations, even though these observations are much noisier
than the carrier phase measurements and thus may adversely affect the accuracy of the results.
For the combined GPS/GLONASS processing, mixed GPS/GLONASS double differences – i.e. GPS
satellite and GLONASS reference satellite or vice versa – have been allowed. GLONASS/GLONASS and
GPS/GLONASS double differences were formed using the algorithm described above. Auxiliary wave-
lengths were left at their original values, i.e. they were not enlarged by applying common denominators
of the double difference coefficients.
Figure 8.10 shows the deviations from the known position in the horizontal plane. The converging of
the Kalman filter towards the true position is clearly visible. Figure 8.11 shows the time series of the 3D
deviation from the true position. The GPS only solution exhibits some amount of oscillations, probably
due to remaining S/A effects, because the observation epochs at reference and user stations were not
exactly synchronized. This solution converges towards the true position only very late. The GLONASS

8.4 A Proposed Solution to the Frequency Problem
119
Position Deviation [m] from Center E 11 37’ 43.783” N 48 04’ 39.911”
◦
GPS
×
GLONASS
GPS+GLONASS
East/West Deviation [m]
-4
-3.2
-2.4
-1.6
-0.8
0
0.8
1.6
2.4
3.2
4
North/South
Deviation
[m]
-4
-3.2
-2.4
-1.6
-0.8
0
0.8
1.6
2.4
3.2
4
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