CONSIDERING VARIANCES OF QUASI-RANDOM EFFECTS IN RELATIVE GPS POSITIONING PERFORMED DURING DAYTIME AND NIGHTTIME PERIODS – A NOVEL TWO-STAGE APPROACH

In this paper, a new two-stage approach, involving an integral treatement of all quasi-random effects limiting the accuracy of relative GPS positioning and the method of moments to obtain final variance components regarding the effects of short-term (“far-field”) multipath (factor b), joint action of long-term (“near-field”) multipath and receiver antenna phase center offset and variations (factor 1 a ), as well as joint action of tropospheric and ionospheric refraction (factor 2 a ), is presented. In the study, GPS data collected on five baselines were used. Variance components of the quasirandom effects were obtained for the three relative GPS coordinates (e, n and u) using individually monthly datsets including daytimeand those including nighttime-wise ambiguity-fixed baseline solutions. The related results show that statistically significant inequality exists when comparing corresponding variances obtained for daytime and nighttime periods. It turned out that the following standard deviation estimates intervals are present (by the coordinates e, n and u, respectively): (a) daytime period: 3.3–6.9, 4.6–9.0 and 9.1–20.3 mm (factor b); 1.5–4.7, 1.9–7.0 and 3.4–21.9 mm (factor 1 a ); 0.0116– 0.3282, 0.0103–0.2365 and 0.1222–0.7818  mm/km (factor 2 a ); (b) nighttime period: 3.2–4.9, 4.7–7.3 and 8.4–15.4  mm (factor b); 0.8–3.8, 2.1–5.0 and 3.1–15.8 mm (factor 1 a ); 0.0118–0.2734, 0.0097–0.2289 and 0.0752–0.6315 mm/km (factor 2 a ).

Herein, however, the author uses a linear model involving effects limiting relative GPS positioning accuarcy at once. Namely, the effects, such as short-("far-field" multipath -FF MP) and long-term ("near-field" multipath -NF MP, receiver antenna phase center offset and variations -RAPCOV, tropospheric and ionospheric refraction -TI) unmodelled effects, are the only ones that must not be neglected when making a choice of baselines as in this study (each of them is of less than 300 km in potential reflectors of different characteristics, some of which were in the near vicinity while others were far away from an antenna. More detailed information related to the two MontePos stations can be found at the following link: http://www.nekretnine.co.me/me/Djelatnosti5.asp. On the other side, information regarding eight EPN stations are provided at the following address: http://epncb.oma. be/_networkdata/stationmaps.php.
It should be noted that no paper dealing with the previously mentioned effects in the way presented in this study has been published so far. That is where the motivation for writing this paper came from.

Input data
For the purpose of the study, a total number of 38548930 true errors (corresponding to the relative coordinates e, n and u) remained after outliers removal, were used. These solutions were obtained by using Trimble Total Control software in processing 0.033-Hz GPS data, collected, as previously said, at ending stations of five baselines. The baselines are located in Italy (TORI-IENG and IGMI-PRAT, being 5.6 and 13.6 km long, respectively), in Montenegro (BAR-PODG, 40 km long), in Poland (BOR1-WROC, 129.5 km long) and in Romania (BACA-BAIA, of 281.9 km in length).
In data processing, the processing interval of 30 s, precise orbits, elevation cutoff of 10°, ionosphere-free linear combination, default calibration model, Saastamoinen tropospheric model, MSIS meteorological model and OTF processing mode (as an auxiliary one) were used. Such settings allowed all unmodelled effects to be expressed in each ambiguity-fixed solution.

The two-stage method used in the study
In the first stage of the study, the two-way nested ANOVA was applied individually on monthly datasets of daytimeand nighttime-wise true errors for all of the three relative coordinates (e, n and u), whereby variance components , were calculated. Herein, the following denotations were introduced: a -the nesting factor (related to the unmodelled joint long-term effect, including NF MP, RAPCOV, TI); b -the nested factor, nested within a (related to the unmodelled short-term effect, i.e. FF MP); ε -the purely random error, nested within b. It should also be mentioned that the 3-minute constancy of the short-term effect (i.e. FF MP -the nested factor b) and the 90-minute constancy of all long-term effects (i.e. NF MP, RAPCOV, TI -the nesting factor a) were assumed in the calculations.
The concept of the first-stage-related method is shown in Figure 1.
Namely, here we introduce the true error model equation of an individual coordinate c as follows: whereby the accompanying stochastic model is based on the assumptions below: On the basis of (1) and (2a-e), one writes: whereby 2 ,c a σ , 2 ,c b σ and 2 ,c ε σ represent the variance components that are to be estimated in the ANOVA estimation procedure. With of an aim to avoid unnecessary transcription of the corresponding formulas, that procedure is not shown herein, so, for details, a reader is reffered to Anđić (2016).
In the second stage, with the aim of calculation of final variance components, an iterative method of moments was used. Namelly, as assumptions regarding unmodelled and purely random effects are where , ; 1; ,c ε σ , regarding, respectively, the joint effect of NF MP and RAPCOV, the joint effect TI, the effect of FF MP and the purely random error, are estimated through the following iterative procedure: Step 1. Adopting initial weight matrices: Step 2. Calculating initial estimates of variance components: Step 3. Calculating weight matrices using variance components estimates obtained in Step 2: Step 4. Calculating new (final) estimates of variance components: Step 5. Repeating steps 3 and 4 until the following conditions are met (t denotes an iteration number):

Results
In Tables 1 to 6, the results of ANOVA estimation performed for the daytime and nighttime within the fouryear period considered (2008)(2009)(2010)(2011), are given. The results are square roots of epoch-wise variance components estimates (i.e. standard deviations estimates) obtained using individually monthly datasets including daytime-and those including nighttime-wise true errors. In doing so, a linear model based on two-way nested classification with random effects with no interactions was used (see Anđić, 2016Anđić, , 2019a.
Square roots of the final epoch-wise variance components estimates, by months within the period 2008-2011, are presented graphically in Figures 2 to 5.