3 Combination of Earth orientation parameters

The International Earth Rotation Service (IERS), founded in 1988 jointly by the IAU and IUGG, mainly in order to monitor the Earth orientation parameters (Universal Time, polar motion and celestial pole offsets), collects and analyzes the observations by several space techniques. They comprise already mentioned VLBI and GPS, but also satellite laser ranging (SLR), lunar laser ranging (LLR) and, most recently, Doppler orbit determination and radiopositioning integrated on satellite (DORIS). Now the discussions take place within IERS how to combine the results of all these techniques into a single representative solution. The proposed method of combined smoothing is a contribution to solving this problem.

The accuracy of these techniques sometimes strongly depend on the frequency of the observed phenomenon, as demonstrated e.g. by Vondrák & Gambis (2000). The most striking difference is between VLBI (that refers the observations to extragalactic objects) and satellite methods (that refer the observations to the orbits of the satellites). The motions of extragalactic objects with respect to inertial reference system are negligible, therefore the stability of the celestial frame is very high at any frequency. On the other hand, the motions of the satellite orbits with respect to inertial reference system depend not only on the gravitational field of the Earth and its time changes but also on numerous non-gravitational forces. Therefore these motions can be modeled with uncertainties that grow with period.

Figure 6: Transfer functions of the smoothing used to combine the data in Fig. 5, plotted against period P in days (full line corresponds to $\varepsilon \,=\,25 \,\,10^2$day-6, dashed line to $\bar\varepsilon\,=\,4 \,\,10^4$day-4)

The most important consequence is that only the short-periodic part of Universal Time can be measured by satellite methods what is in practice assured by determining the length-of-day changes instead of Universal Time. On the other hand, the satellite methods are capable of providing much more frequent measurements, monitoring thus shorter periodic motions of the Earth's orientation in space and their time derivatives.

In the following we demonstrate the capability of the method proposed above to combine Universal Time with length-of-day changes, and polar motion with its time derivatives.

Figure 7: Enlarged cutoff of Fig. 5 around a spurious peak caused by improper choice of coefficients of smoothing (UT1R in top, lodR in bottom)

Figure 8: Transfer functions of the smoothing used to combine the data in Figs. 9, 10 and 11, plotted against period P in days (full line corresponds to $\varepsilon$=100day-6, dashed line to $\bar{\varepsilon}$=20day-4)

3.1 Combining Universal Time and length-of-day changes

As already said in the Introduction, the proposed method is especially convenient to combine the observations of Universal Time, in the form of its difference from the uniform time scale given by atomic clocks (UT1-TAI, observed weekly by VLBI) and the length of day changes (l.o.d., observed daily by GPS). We chose the series covering four years (1995.0-1999.0) as determined at Shanghai Astronomical Observatory, China (VLBI), and at the Astronomical Institute of Berne University, Switzerland (GPS). The two observed quantities are tied by a simple relation:

$${\rm l.o.d.} = -{\rm d(UT1-TAI)/d}t,$$

in which l.o.d. is given in seconds of time if the time derivative on the r.h.s. is expressed in seconds of time per day. Before using both series in the combination, we removed from them the theoretical solid Earth tidal variations with periods shorter than 35 days after Yoder et al. (1981); these corrections lead to series denoted as UT1R-TAI and lodR. The a priori values of the average standard deviations of both series are $7.3~\mu {\rm s}$ for UT1 of VLBI and $1.5~\mu {\rm s}$ for l.o.d. of GPS.

Figure 9: Combined smoothing of UT1R-TAI measured by VLBI and lodR measured by GPS, using $\varepsilon$=100day-6, $\bar{\varepsilon}$=20day-4 (leading to standard deviations equal to $2.6~\mu {\rm s}$ and $24.8~\mu {\rm s}$, respectively). Smoothed curves are shown as full lines and residuals (enlarged scale on the right) as grey crosses. Reduced values of UT1R-TAI+27.5+0.00183(MJD-48987) are displayed in top, lodR in bottom.

Figure 10: Combined smoothing of pole coordinate x and its rate measured by GPS, using $\varepsilon$=100day-6, $\bar{\varepsilon}$=20day-4. Smoothed curves are shown as full lines and residuals (enlarged scale on the right) as grey crosses. Coordinate x is displayed in top, its rate in bottom

Figure 11: Combined smoothing of pole coordinate y and its rate measured by GPS, using $\varepsilon$=100day-6, $\bar{\varepsilon}$=20day-4. Smoothed curves are shown as full lines and residuals (enlarged scale on the right) as grey crosses. Coordinate y is displayed in top, its rate in bottom

Firstly, we applied a set of coefficients of smoothing that makes a posteriori standard deviations equal to their average a priori values, using iterations as outlined in Sect. 2.4 (namely $\varepsilon$ = 25 10² day^-6, $\bar{\varepsilon}$ = 4 10⁴ day^-4). The results are shown in Fig. 5, UT1R in top and lodR in bottom part of the figure. The values UT1R-TAI have a very large negative trend; therefore we plot their reduced values, with a constant and linear trend removed. Since the observed values are hardly distinguishable from the smoothed curve (black full line), the residuals in the sense "observed - smoothed'' (displayed as gray crosses) are plotted in the enlarged scale marked on the right. The full line in lower graph is negatively taken time derivative of the one in the upper part.

The combined smoothing, applied in this case, has transfer function plotted in Fig. 6. It is a very weak smoothing that passes completely all periods longer than approximately one day, too weak to remove the true noise of the observations as one can see from the lower curve of lodR that is rather ragged.

As the tests with simulated data revealed (see Sect. 2.5 above), the combined smoothing with this combination of $\varepsilon$, $\bar{\varepsilon}$ should not generally give satisfactory results. Really, a detailed inspection of the results discloses that the smoothed curve, although running almost perfectly through both series of observed values, sometimes forms sudden spurious peaks between two points with l.o.d. observations, at the epochs with only UT1 observed. A typical example of this effect is demonstrated in Fig. 7 that is a closeup of a part of Fig. 5. This effect is due to the very weak smoothing applied (almost interpolation in this case), and also because much weaker smoothing (large value of $\bar{\varepsilon}$) is used for observed first derivatives.

Therefore we made another try and used a stronger smoothing, following the rules given in Sect. 2.5. We assumed that the shortest period of the signal contained in the data is about one week; using P_0.5=3days to calculate the coefficients of smoothing (see Sect. 2.4) leads to $\varepsilon$=100day^-6, $\bar{\varepsilon}$=20day^-4. The solution yields aposteriori standard deviations equal to 2.6$\mu$s and 24.8$\mu$s, respectively, the values that are evidently much different from the average a priori values given by the analysis centers.

The result is displayed in Fig. 9, and the transfer functions are given in Fig. 8; they are shifted to the right with respect to the ones depicted in Fig. 6 and they are nearly identical. This combination gives much better results than the preceding one; the smoothing is still rather weak not to suppress real signal but sufficiently efficient to remove the observational noise. The residuals disclose that UT1 as observed by VLBI seems to be very accurate (maybe more than one would expect from their formal standard deviations). The l.o.d. values as given by GPS are obviously more noisy than their formal standard deviations hint - they rather represent the internal precision of the method (without taking into account the instabilities of the modeled satellite orbits with respect to inertial reference frame) than accuracy. The residuals of lodR thus mostly reflect the long-periodic deviations of GPS-determined lod that are not fully compatible with the first derivative of VLBI-based UT1. The behavior of the residuals e.g. clearly demonstrate that the l.o.d. as determined by GPS before and after 1996.7 systematically differ by about 40$\mu$s. It is necessary to mention in this respect that Berne University is probably the only GPS analysis center that provides the free solution of l.o.d., without frequent constraints to VLBI results, and that the date of systematic step in the results correspond to the change in the model used by this center.

3.2 Combining polar motion and its rate

Another example of using the new method is given by the observation of polar motion; Astronomical Institute of the University of Berne provides not only the instantaneous coordinates of the pole but also their rate; both series are mutually independent in spite of the fact that they are based on the observations by the same technique - GPS. The data used in this study, covering roughly the interval 1993.5-1999.5, were subject to combined smoothing. The average a priori standard deviations of the series are respectively 23.4 $\mu$ arcsec, 22.0 $\mu$ arcsec in x and y, and 18.6 $\mu$ arcsec/day, 18.2 $\mu$ arcsec/day in their daily rates.

Although we made many tests, using solutions with different combinations of coefficients of smoothing, we were never able to find $\varepsilon$, $\bar{\varepsilon}$ that would lead to a posteriori values equal to average a priori values given above. It obviously reflects the fact that the formal standard deviations as reported by the analysis centers are so much underestimated that the observed function values and their first derivatives are not mutually compatible at the given level of accuracy.

Therefore we finally decided to use the same coefficients of smoothing as used in the last example of combining UT1 and lod, i.e. $\varepsilon$=100day^-6, $\bar{\varepsilon}$=20day^-4 whose transfer functions are shown in Fig. 8. They lead to approximately the same a posteriori standard deviations in x, y (respectively 20.0 $\mu$ arcsec and 20.9 $\mu$ arcsec) but the standard deviations of their rates are significantly larger (respectively 96.1 $\mu$ arcsec per day and 99.1 $\mu$ arcsec per day). The results are depicted in Figs. 10 and 11.

It can be seen that the accuracy of GPS-determined polar motion and its rate substantially improved after 1995. The coefficients of smoothing applied seem to be well chosen to suppress the noise of the observations, without affecting the real signal in the data.The combination of both types of observables (although not fully compatible on the level of their formal standard deviations), helps improve the solution.