**7.4.3 Structure of the source data**

Thanks to astronomical observations data of the moon and planets motion, maintained from the middle of the seventeenth century, we now have an opportunity to reconstruct the picture of the Earth's rotation over the past 350 years at least approximately. The Figure 4 represents a diagram of the *UT*1 parameter variations starting from the 1656 to the present day in the Terrestrial Dynamic Time system *TT*. The diagram was drawn on the basis of the 1 *T UT TT* differences data, represented in (AE SSSR, 1970 and Konstantinov A. I, Fleer A.G., 1971).

Analyzing the diagram in the Figure 4 we can note that the Earth's rotation speed in past centuries varied slightly and rather evenly until the late nineteenth century, when a sharp slowdown occurred. During the past 100 years some periods of Earth's rotation deceleration and acceleration can be traced with specific intervals of 20-30 years. Nowadays the Earth's rotation slowdown is observed, which is likely to be continued during the next 15 to 20 years.

In the Figure 5 taken from (IERS) the dots indicate the Earth's pole wandering in 1996-2000., and the solid line indicate the trajectory of the mean pole from 1890 to 2000.

Fig. 5. Wandering of the Earth's North Pole

Analyzing the data, represented in the Figure 5, we can note that during the whole time of observations the mean pole was being displaced at a speed of about 10 cm/year along a complicated zigzag curve mainly in the direction of North America (meridian 290 ° EL).

#### **7.5 Overview of the proposed ERP prediction method**

The proposed method is based on approximation of periodic and quasi-periodic processes of the Earth's rotation using a polyharmonical mathematical model built on an ERP observations sample for at least 100 years. The calculation technique is based on the sequential adjustment of the original ERP time series by corrections introduction for tidal harmonics (for *dUT*1 series), calculated on the basis of known physical and mathematical theory of the sun and moon motions, as well as harmonics corrections, obtained with

Analyzing the data, represented in the Figure 5, we can note that during the whole time of observations the mean pole was being displaced at a speed of about 10 cm/year along a complicated zigzag curve mainly in the direction of North America (meridian 290 ° EL).

The proposed method is based on approximation of periodic and quasi-periodic processes of the Earth's rotation using a polyharmonical mathematical model built on an ERP observations sample for at least 100 years. The calculation technique is based on the sequential adjustment of the original ERP time series by corrections introduction for tidal harmonics (for *dUT*1 series), calculated on the basis of known physical and mathematical theory of the sun and moon motions, as well as harmonics corrections, obtained with

Fig. 4. Earth's rotation variations for the period from 1650 to 2010.

Fig. 5. Wandering of the Earth's North Pole

**7.5 Overview of the proposed ERP prediction method** 

statistical methods (for all ERP series) . Generalized formula for *dUT*1 parameter prediction can be written down as follows:

$$d\text{LIT1}\_{\text{prg},i} = d\text{LIT1}\_{\text{real},0} - d\text{LIT1}\_{\text{pu},0} + d\text{LIT}\_{\text{pu},i} + \Delta T\_{\text{pr},i} + \Delta T\_{\text{fr},i} + \Delta T\_{\text{sz},i} \tag{36}$$

where , 1*prg <sup>i</sup> dUT* is the prediction of 1 *dUT* for the *i* -day;

,0 1*real dUT* is the known value of 1 *dUT* for the reference day;

,0 1*pa dUT* is the prediction of 1 *dUT* for the reference day made by means of the autoregression;

, 1*pa i dUT* is the prediction of 1 *dUT* for the *i* -day made by means of the autoregression;

*Tpr i*, , *Ttr i*, , *Tsz i*, are the corrections for the lunar and solar tides in the oceans, and for trend and seasonal variations at the *i* -day.

Tidal harmonics corrections are being calculated in accordance with the method accepted in the IERS (McCarthy, 2003). Consideration of the trend influence corrections by means of the harmonic model composed of harmonics with periods from 1 year to 70 years. Calculation of these harmonics parameters is performed by means of a specially developed for this purpose method of step-by-step summation of intervals of the time series. (Tissen et al., 2009). In order to calculate corrections for seasonal variations of the dUT1 parameter the polyharmonic model shall be used as a sum of a year, half-year wave and an unlimited number of harmonics with periods less than a year. Calculation these harmonics parameters is performed at time intervals of about 15-25 years. The existing methods of dUT1 prediction approximate these variations by a sum of year and half-year harmonics, whose parameters are estimated within the period of 4-6 years. In order to consider short-term stochastic components of the ERP series variations a mathematical model's component in a form of the autoregression shall be used as a restriction to smoothness.

Predictions for the pole coordinates shall be made in a similar way, except that in the equation (36) the influence of the lunar-solar tides shouldn't be taken into account because of their smallness.

### **7.6 Main results**

Reports on the results of the ERP predictions obtained using the method developed in SNIIM from 2007 to 2011 were made many times at national conferences in Moscow and St. Petersburg, and at an international conference in Warsaw (Tissen & Tolstikov, 2011) in October 2009 (Tissen et al. , 2009).

Since October 2010 the method is being testing on the basis of the results of SNIIM/SSGA participation in the Earth Orientation Parameters Combination of Prediction Pilot Project (EOPCPPP). Participation in this project presupposes the daily data transfer into the main IERS centres: the Space Research Center of the Polish Academy of Science, Warsaw (eopcppp@cbk.waw.pl) and the United States Naval Observatory (USNO), Washington, District Columbia (eopcppp@maia.usno.navy.mil). Figures 5-7 show evaluations of the RMS predictions for universal time dUT1 and pole coordinates: *<sup>P</sup> x* , *<sup>P</sup> y* at intervals from 1 to 90 days by all EOPCPPP participants. The evaluations above, except for 3 participants, were received during the period of the project from September 2010 to November 2011

Fig. 6. Comparison of RMS for universal time predictions for 90 days made by EOPCPPP participants from September 2010 to November 2011.

Fig. 7. Comparison of RMS for pole coordinates predictions: *XP* and *YP* (top down, respectively) for 90 days made by EOPCPPP participants from September 2010 to November 2011.

Fig. 6. Comparison of RMS for universal time predictions for 90 days made by EOPCPPP

Fig. 7. Comparison of RMS for pole coordinates predictions: *XP* and *YP* (top down, respectively)

for 90 days made by EOPCPPP participants from September 2010 to November 2011.

participants from September 2010 to November 2011.

Fig. 8. Comparison of RMS for pole coordinates predictions: *XP* and *YP* (top down, respectively) for 90 days made by EOPCPPP participants from September 2010 to November 2011.

Graphic data in the Figures 6-8 demonstrates an advantage of the ERP prediction method of the SNIIM/SSGA in comparison to the methods used by the other EOPCPPP participants, including those with official global suppliers of the ERP predictions used by the IERS (USNO).

During the first 13 months of the pilot project EOPCPPP, which involves more than 10 different methods for ERP prediction, the method used by SNIIM-SSGA showed the best results. In particular, a prediction accuracy of the universal time parameter, which is considered as the most important and difficult to predict one, within the period of less than 30 days was 1.19 times higher by the SNIIM - SSGA in 350 realizations than by the IERS and 1.93 times higher than by SSTF of Russia. Similar results were also obtained in longer intervals of up to 90 days and more. On the IERS website «eopcppp@maia.usno.navy.mil» ERP predictions files from all the EOPCPPP project participants are published daily with their metrological characteristics according to the evaluations of RMS, absolute error (MAE) and standard deviation.
