6. Observables
6.2 Data combinations
84 6. Observables occur at the satellite during signal emission). Multipath is interference be-tween the direct and the reflected signal and is largely random; however, it may also appear as a short term bias. Wells et al. (1987) report a similar effect called "imaging" where a reflecting obstacle generates an image of the real antenna which distorts the antenna pattern. Both effects, multi path and imaging, can be considerably reduced by selecting sites protected from reflections (buildings, vehicles, trees, etc.) and by an appropriate antenna design. It should be noted that multi path is frequency dependent. There-fore, carrier phases are less affected than code ranges where multipath can amount to the meter level as stated by Lachapelle (1990). More details on the multipath problems are given in Sect. 6.5.
The measurement noise, an estimation of the satellite biases, and the contributions from the wave propagation are combined in the User Equiv-alent Range Error (UERE). This UERE is transmitted via the navigation message. In combination with a DOP factor, explained in Sect. 9.5, UERE allows for an estimation of the achievable point positioning precision.
6.2 Data combinations Therefore,
is the frequency and
,\ =
~I
is the wavelength of the linear combination.
85
(6.13)
(6.14)
In the case of GPS, the linear combination of L1 and L2 carrier phases
q;Ll and q;L2 for the simplest nontrivial cases in Eq. (6.11) are nl = n2 = 1, yielding the sum
q; L1+L2 = q; Ll
+
q; L2 (6.15)and nl
=
1, n2=
-1, leading to the differenceq;L1-L2
=
q;L1 - q;L2' (6.16)The corresponding wavelengths according to Eq. (6.14) are
'\Ll+L2
=
10.7 cm(6.17)
'\L1-L2 = 86.2 cm
where the numerical values for the carrier frequencies
ILl
andh2'
see Table 5.2, p. 71, have been substituted. The combination q;L1+L2 is de-noted as narrow lane and q;LI-L2 as wide lane, cf. for example Beutler et al. (1988), Wiibbena (1988).A slightly more complicated linear combination results from the choice
nl
=
1 n2= -- hI h2
(6.18)which is frequently denoted as L3 signal, thus
q;L3
=
q;Ll --I h2
q;L2.L1 (6.19)
Now that the significant linear combinations have been defined, the ad-vantage of these combinations will be shown. The L3 combination, for exam-ple, is used to reduce ionospheric effects, cf. Sect. 6.3.2., and the ambiguity resolution combines wide and narrow lane signals, cf. Sect. 9.1.3.
Assuming a certain noise level for the phase measurements, it is seen that the noise level will increase for these linear combinations. Applying the error propagation law and assuming the same noise for both phases, the noise of the sum or the difference formed by q; L1 and q; L2 is higher by the factor
V2
than the noise of a single phase. Of course, to compute this factor correctly, one must take into account the different noise levels.
86 6. 0 bservables 6.2.2 Phase and code pseudorange combinations
The objective here is to show the principle of the smoothing of code pseu-doranges by means of phase pseupseu-doranges. A first extensive investigation of this subject was provided by Hatch (1982). Applications and improve-ments were proposed later by the same author and are given in Hatch and Larson (1985) and Hatch (1986). Other slight variations can be found in e.g. Lachapelle et al. (1986) or Meyerhoff and Evans (1986). Today, phase and code pseudorange combinations are an important part of real-time tra-jectory determination.
Assuming dual frequency measurements for epoch tt, the P-code pseudo-ranges RLl(tl), RL2(tl) and the carrier phase pseudopseudo-ranges «PLl(tt), «PL2(tl) are obtained. Assume also, the code pseudoranges are scaled to cycles (but still denoted as R) by dividing them by the corresponding carrier wavelength.
Using the two frequencies fLl,!L2, the combination R(tl) =
hl
RLl(tl) - h2 RL2(tt)hl+h2
is formed for the code pseudoranges and the wide lane signal
(6.20)
(6.21) for the carrier phase pseudoranges. From Eq. (6.20) one can see that the noise of the combined code pseudorange R(t!) is reduced by a factor of 0.7 compared to the noise of the single code measurement. The increase of the noise in the wide lane signal by a factor of
v'2
has no effect because the noise of the carrier phase pseudoranges is essentially lower than the noise of the code pseudoranges. It is worth noting that both signals R(td
and«p(tt) have the same frequency and thus the same wavelength as the reader may verify by applying Eq. (6.13). This is not the case with the approach given in Hatch (1986) where a plus sign is chosen for the code pseudorange combination (this can be regarded as a weighted mean).
Combinations of the form (6.20) and (6.21) are formed for each epoch.
Additionally, for all epochs ti after tl, extrapolated values of the code pseu-doranges R(ti)ex can be calculated from
(6.22) The smoothed value R(ti)sm is finally obtained by the arithmetic mean
(6.23)
6.2 Data combinations 87 Generalizing the above formulas for an arbitrary epoch ti (with the preceding epoch ti-l ), a recursive algorithm is given by
R(ti)
=
ILl RLl(ti) - IL2 RL2(ti) ILl+
IL2~(ti)
=
~Ll(ti) - ~L2(ti),R(ti)ex
=
R(ti-dsm+
(~(ti) - ~(ti-l))'R(ti)sm
= 2"
1 (R(ti)+
R(ti)ex)which works for all i
>
1 under the initial condition R(tt} = R(tt}ex = R(tl)sm.The above algorithm assumes the data is free of gross errors. However, carrier phase data are sensitive to changes in the integer ambiguity (Le., cycle slips). To circumvent this problem, a variation of the algorithm is given in Lachapelle et al. (1986). Using the same notations as before for an epoch ti, the smoothed code pseudorange is obtained by
with w as a time dependent weight factor. For the first epoch i
=
1, theweight is set w = 1, thus putting the full weight on the measured code pseudorange. For consecutive epochs, ;he weight is reduced continuously and thus emphasizes the influence of th.~ carrier phases. To get an idea for the reduction factor, Lachapelle et al. (1986) proposed a reduction of the weight by 0.01 from epoch to epoch for :a. kinematic experiment with a data sampling rate of 1.2 seconds. After two minutes, only the smoothed value of the previous epoch (augmented by the carrier phase difference) is taken into account. Again, in case of cycle slips, the algorithm would fail. A simple check of the carrier phase differ·ence for two consecutive epochs by the Doppler shift X time may detect data irregularities such as cycle slips.
After the occurrence of a cycle slip, the weight is reset to w
=
1 which fully eliminates the influence of the erroneous carrier phase data. The key of this approach is that cycle slips must be detected but do not have to be corrected, cf. Hein et al. (1988).Another smoothing algorithm for code pseudoranges is given by Meyer-hoff and Evans (1986). Here, the phase changes obtained for instance by the integrated Doppler shift between the epochs ti and tl are denoted by
~~(ti, tl) with tl being the starting epoch for the integration. Accordingly from each code pseudorange R(ti) at epoch ti, an estimate of the code pseu-dorange at epoch tl can be given by
(6.25)
88 6. Observables where the subscript i on the left-hand side indicates the epoch that the code pseudorange R(ft) is computed from. Obtaining consecutively for each epoch an estimate, the arithmetic mean R( tl)m of the code pseudorange for n epochs is calculated by
1 n
R(tt}m
= - L:
R(tt}in i=l (6.26)
and the smoothed code pseudorange for an arbitrary epoch results from (6.27) The advantage of this procedure lies in the reduction of the noise in the initial code pseudorange by averaging an arbitrary number n of measured code pseudoranges. Note from the three formulas (6.25) through (6.27) that the algorithm may also be applied successively epoch by epoch where the arithmetic mean must be updated from epoch to epoch. Using the above notations, formula (6.27) also works for epoch t}, where of course ~ ~(tl' tl) is zero and there is no smoothing effect.
All the smoothing algorithms are also applicable if only single frequency data are available. In this case R(ti), ~(td, and ~~(ti, tt) denote the single frequency code pseudorange, carrier phase pseudorange, and phase differ-ence, respectively.