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The 5

th

IVS General Meeting Proceedings, 2008, p. 178­182

VLBI Data Structure, Analysis Strategies and Software

Influence of the Elevation-Dep endent Weighting on Parameter Estimates: A Case of CONT05
Zinovy Malkin
Central Astronomical Observatory at Pulkovo of Russian Academy of Sciences e-mail: malkin@gao.spb.ru

Abstract. In this paper, results are presented on studies which have been performed to investigate the impact of the cut-off elevation angle and elevation-dependent weighting on the EOP estimates and baseline length repeatability.

1. Intro duction
The precision and accuracy of observations made through the Earth's atmosphere grow with decreasing of the elevation due to larger air mass and difficulties in modelling refraction effects at low elevation. From this point of view observations should be made in the near-zenith zone when possible. On the other hand, inclusion in processing of observations made at low elevations is important when highly correlated parameters are estimated simultaneously. In such a case using observations made at in a widest range of elevation allows one to mitigate the correlations between unknowns. To meet these mutually exclusive requirements, elevation-dependent weighting (EDW) is used. In a special case of step-like weighting function, one rejects the observations made at the elevation less than the given limit, so-called cut-off elevation angle (CEA). It is well known that EDW has a significant impact on the results of processing of the space geodesy observations. In particular, several studies on the sub ject were made in the framework of the IVS VLBI2010 Committee activity (href="http://ivscc.gsfc.nasa.gov/pipermail/ivs-v2c/). Those results were based on simulation. Gipson in [1] applied elevation dependent additive noise to the measurement error instead of using a weighting factor as it is usually being made. He tested his method with the actual CONT05 observations. Results of both mentioned and other results are sometimes contradictory. This gave an impulse to the present work where some results are presented of investigation of the impact of the CEA and EDW on the baseline length repeatability (BLR) and Earth rotation parameters (EOP) estimates. 178


2. Test Description and Results
For this test, CONT05 observations were processed with the OCCAM software with different EDW functions including CEA, keeping all other options as used during routine processing: Kalman filter mode (KF), random walk model for clocks (PSD=1.5 ps2 /s) and ZTD (PSD=0.25 ps2 /s), one NS and EW troposphere gradient estimate for each 24h session. For the continuous EDW mode (continuous weighting function), the measurement error is multiplied by a factor We = (sin e0 / sin e)p , (1)

where e0 and p are EDW parameters, e is the source elevation. Such a weighting function provides a smooth stepless change in weight for any e0 . One can see that a case of e0 = 90 , p = 1 gives merely We = 1/ sin e, which is close to actual mapping function used in the last works by MacMillan and Gipson (private communications). EDW mode with e0 = 10, p = 2 was implemented in the OCCAM/GROSS software [2] for routine data processing. It will be referred hereafter as "normal mode". In a case of CEA we have We = 1, 10
3

if e >= e0 otherwise

(2)

The latter line corresponds to the KF realization used in OCCAM. For VLBI delay, measurement error coming from correlator is multiplied by two We values computed for both the stations. In our test, e0 = 3(2)25 were used for CEA test, and e0 = 10, 25, 45, 90 , p = 1, 2 were used for continuous EDW mode. BLR obtained with different CEA are shown in Figure 1. The case of e0 = 3 includes all the observations without weighting, since no CONT05 observations were made at the elevation less than 4 . Table 1 shows EDW test results. Different EDW modes are denoted as `e0 p' correspondingly to Eq.(1). Test results are given for quadratic approximation in percent with respect to the case of CEA with e0 = 3 . One can see that several EDW modes show about the same improvement in the BLR.
Table 1. Baseline length repeatability for different EDW models

EDW mode 10 1 10 2 25 1 25 2

Baseline length, 103 km 3 6 9 12 98.0 97.2 97.3 97.6 95.3 95.0 95.5 96.1 102.6 92.8 87.2 83.8 95.7 91.8 91.4 91.9

EDW mode 45 1 45 2 90 1 90 2

Baseline length, 103 km 3 6 9 12 101.7 90.0 84.6 81.9 104.0 97.2 101.8 107.6 101.3 90.5 85.4 82.8 109.6 127.3 161.1 191.1

For investigation of change in the baseline length obtained with different CEA and EDW models we compared them (mean of 15 daily estimates) to 179


Baseline length repeatability, mm

0 0 2 4 6 8 10 12 14 Baseline length, thousand km

Baseline length repeatability, mm

60 e0 e0 50 e0 e0 40 e0 e0 e0 30 e0 e0 20 e0 e0 e0 10

= = = = = = = = = = = =

3 5 7 9 11 13 15 17 19 21 23 25

deg deg deg deg deg deg deg deg deg deg deg deg

20 18 Normal mode e0 = 3 deg 16 e0 = 5 deg e0 = 7 deg 14 e0 = 9 deg 12 e0 = 11 deg 10 8 6 4 2 0 0 2 4

6

8

10

12

14

Baseline length, thousand km

Figure 1. Baseline length repeatability for different CEA (quadratic regression).

the length value obtained with e0 = 3 (all the observations included). The comparison shows that change in the baseline length essentially depends on stations involved in the baseline under consideration. For stations located among others, for which observation sky coverage is close to uniform, change in baseline length may reach several millimeters with increasing of CEA. For outside stations, HARTRAO and especially TIGOCONC for CONT05, one can see very asymmetric sky coverage, and change in baseline length may reach 3 cm as CEA grows from 3 to 25 . EDW mode shows much less change in the baseline length, and in this case increasing e0 leads to greater change in the baseline length too up to 1 cm for TIGOCONC and HARTRAO. Figure 2 shows EOP statistics for different CEA. Notation used is the following: Xp, Yp ­ terrestrial pole coordinates, Xc, Yc ­ celestial pole coordinates. Results of comparison of CONT05 EOP with the IGS time series are presented in Figure 3. Xp and Yp wrms with respect to IGS EOP series are computed after removing the bias. Table 2 shows the main EDW test results. Notation of the EDW modes is the same as used in Table 1. The column labelled "All" corresponds to inclusion of all observations without weighting. One can see again that several EDW modes show about the same EOP precision and accuracy.

3. Conclusions
The preliminary conclusions from this test are the following. The BLR steadily grows with the CEA increasing, remaining practically the same up to 9 . The same can be said for the bias in the baseline length. Maximal impact of the CEA on the baseline length bias is seen for the outside network stations with the most asymmetric observations sky coverage. The Xp, Yp and UT1 uncertainties grow with the increasing CEA after about 10 . Most probably, this reflects the fact that only about 6% of the total number of CONT05 observations were made at the elevations below 10 . The Xc and Yc uncertainties and scatter depend on the CEA much less. Xp bias w.r.t. IGS slightly depends on the CEA, except the maximum tested CEA values, evidently unrealistic. In contrast, Yp bias substantially changes 180


70 Uncertainty, microarcseconds 60 50 40 30 20

Uncertainty, microseconds

Xp Yp mean

2 1.8 1.6 1.4 1.2 1

UT1

3

5

7

9

11 13 15 17 19 21 23 25

3

5

7

9 11 13 15 17 19 21 23 25

Cut-off elevation angle, deg 23 Uncertainty, microarcseconds 22 21 20 19 18 17 3 5 7 9 11 13 15 17 19 21 23 25 Cut-off elevation angle, deg 0.4 0.3 Correlations 0.2 0.1 0 -0.1 -0.2 -0.3 3 5

Cut-off elevation angle, deg

Xc Yc mean

Xp/Yp Xp/UT1 Yp/UT1 Xc/Yc

7

9 11 13 15 17 19 21 23 25

Cut-off elevation angle, deg

Figure 2. Statistics of EOP obtained from CONT05.
100 Bias, microarcseconds 50 0 -50 -100 -150 3 5 7 9 11 13 15 17 19 21 23 25 Cut-off elevation angle, deg Xp+100 Yp-100 Xp Yp mean

160 WRMS, microarcseconds 140 120 100 80 60 40

3

5

7

9 11 13 15 17 19 21 23 25

Cut-off elevation angle, deg

Figure 3. Comparison of EOP obtained from CONT05 with the IGS EOP series.

with increasing CEA. Most probably, this can be explained by the CONT05 network orientation, for which the longitude of the central meridian 0 = 265 just corresponds to the Yp direction. Some statistics, such as the uncertainty and the scatter of the Xc and Yc, as well as the WRMS of Xp and Yp w.r.t. IGS have the minimum at the CEA around 15 , which is interesting and deserves a supplement investigation. As one can expect, the correlations between EOP comprising Xp and Yp grow with increasing CEA, but remain small due to good CONT05 network geometry. The same can be expected for the IVS2010 network. The correlation between Xc and Yc remain practically the same for

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Table 2. EOP statistics for different EDW models. See explanation in text

Statistics Xp uncertainty Yp uncertainty UT1 uncertainty Xc uncertainty Yc uncertainty Xp bias w.r.t. IGS Yp bias w.r.t. IGS Xp wrms w.r.t. IGS Yp wrms w.r.t. IGS

Unit All µas 26 µas 25 µs 1.1 µas 21 µas 19 µas ­79 µas +135 µas 77 µas 73 10 1 26 24 1.1 21 18 ­75 +135 72 74 10 2 25 24 1.1 20 18 ­79 +128 76 72

EDW 25 1 25 2 26 28 24 26 1.1 1.1 20 19 18 18 ­77 ­73 +132 +129 69 65 69 61

mode 45 1 27 25 1.0 20 18 ­80 +134 68 68

45 2 37 32 1.3 22 21 ­86 +127 69 63

90 1 30 26 1.1 22 20 ­81 +129 64 71

90 2 51 43 1.6 27 27 ­123 +92 86 83

all tested CEA, except the maximum tested CEA value, evidently unrealistic. For most of the tests performed, the best result was obtained with EDW applied to the low-elevation observations. However, the test results are not always unambiguous. Further adjustment of the weighting method may be fruitful. Finally, we can conclude that inclusion of the low-elevation observations, properly weighted, improves the baseline length and EOP estimates. On the contrary, filtering the observations using the CEA method may lead to degradation of geodetic results. However, this should be mentioned that the conclusions drawn from the result obtained in this paper has been proven with standard geodetic VLBI observations, where rather few observations were made at low elevations, as mentioned above. Perhaps, special R&D sessions with more uniform distribution of observations over the sky, including observations at very low elevations, may be useful for more detailed study on optimal scheduling and processing of the space geodesy observations. It ought be mentioned that all the EDW modes considered in this paper in fact modify only diagonal elements of the corresponding covariance matrix. According to Gipson's work [1] best result can be achieved in a case of account also for correlations between observations. It seems to be interesting to investigate how this approach will work for the KF estimator.

References
[1] Gipson, J. Incorporating Correlated Station Dependent Noise Improves VLBI Estimates. In: J. Boehm, A. Pany, H. Schuh (Eds.), Proc. 18th European VLBI for Geodesy and Astrometry Working Meeting, Vienna, Austria, 12-13 Apr 2007, 129­134. [2] Malkin, Z., Skurikhina, E. OCCAM/GROSS Software Used at the IAA EOP Service for processing of VLBI Observations. Trans. IAA, 2005, v. 12, 54-67. (in Russian)

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