Future improvements in EOP prediction

Future improvements in EOP prediction W. Kosek Space Research Centre, Polish Academy of Sciences, Bartycka 18A, 00-716 Warsaw, Poland e-mail: [email protected]

Abstract. The Earth orientation parameters (EOP) are determined by space geodetic techniques with very high accuracy corresponding to a few millimeters on the Earths surface. However, the accuracy of their prediction, even for a few days in the future, is several times lower and still unsatisfactory for practical use. Wavelet decomposition of the EOP data and prediction of their different frequency components reveals that the increase of x, y pole coordinate and UT1-UTC data prediction errors up to about 100 days in the future are mostly caused by irregular short period oscillations with periods less than half a year. These irregular short period variations in x, y pole coordinates data are mostly excited by the equatorial components of joint atmospheric and ocean excitation functions while in UT1-UTC data are mostly excited by axial component of atmospheric excitation function. The main problem of each prediction technique is to predict simultaneously long and short period oscillations of the EOP data. The nature of short period oscillations in EOP data is mostly stochastic and longer period seasonal oscillations can be modeled using deterministic method. It has been shown that the combination of the prediction methods which are different for deterministic and stochastic part of the EOP can provide the best accuracy of prediction. Several prediction techniques, involving the least-squares extrapolation for prediction of the deterministic part and autoregressive method to predict short period stochastic part are good candidates for the prediction algorithms of the EOP data. The main problem of each prediction technique is to predict simultaneously long and short period oscillations of the EOP data and this problem can be solved by the combination of wavelet transform decomposition with the autocovariance prediction method. Keywords Prediction, Earth Orientation Parameters, wavelet transform.

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Introduction

Increase of accuracy of space geodetic techniques during the last decades caused increase of determination accuracy of Earth orientation parameters which are x, y pole coordinates, universal time UT1-UTC and celestial pole offsets. However, the prediction errors of x, y and UT1-UTC data are much greater than their determination errors. Now the determination errors of the EOP data correspond to about 3-4 mm on the earth’s surface. The ratio of the EOP prediction errors to their determination errors increases with the prediction length and even for a few days in the future the EOP prediction errors are hundred times greater than their determination errors. The future EOP data are needed to compute real-time transformation between the celestial and terrestrial reference frames. This transformation is for example important for the NASA Deep Space Network, which is an international network of antennas that supports: interplanetary spacecraft missions, radio and radar astronomy observations and selected Earth-orbiting missions. The EOP predictions are published by the International Earth Rotation and Reference Systems Service (IERS) Rapid Service/Prediction Centre (RS/PC). Pole coordinates data in the IERS RS/PC are predicted by the combination of the least-squares (LS) and autoregressive (AR) method (e.g. Kosek et al. 2004) and UT1-UTC data are predicted using the axial component of atmospheric angular momentum (e.g. Johnson et al. 2005). The importance of the EOP prediction lead to organize the Earth Orientation Parameters Prediction Comparison Campaign (EOP PCC) which started on October 2005 and ended in March 2008. The aim of this campaign was to join scientists who work on the EOP predictions and then compare results of different prediction techniques and algorithms provided by them during this campaign using equal and well-defined

rules (Kalarus et al. 2009). Some of the prediction algorithms including Kalman filter (Gross et al. 1998), fuzzy interface system (Akyilmaz and Kutterer 2004), autocovariance prediction (Kosek 2002), combination of the least-squares (LS) extrapolation and autoregressive (AR) prediction (Kosek et al. 2004, 2008) as well as combination of the discrete wavelet transform with autocovariance (DWT+AC) prediction (Kosek and Popi´ nski 2006) were involved in the EOPPCC. Recently, Niedzielski and Kosek (2008) applied a multivariate autoregressive model comprising length of day and axial component of atmospheric angular momentum time series and gained the improvement of UT1-UTC predictions especially during the ENSO events.

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Data

The following data sets were used in the analysis: 1) x, y pole coordinates data, universal time UT1-UTC and length of day ∆ data from the IERS: EOPC04 IAU2000.62-now from 1962.0 to 2009.6 with the sampling interval ∆t = 1 day, http://hpiers.obspm.fr/iers/eop/eopc04 05/, 2) Equatorial and axial components of atmospheric angular momentum from NCEP/NCAR, aam.ncep.reanalysis.* from 1948.0 to 2009.3 with the sampling interval ∆t = 0.25 day, ftp://ftp.aer.com/pub/anon collaborations/sba/. These data were interpolated with 1 day sampling interval using linear interpolation. 3) Equatorial components of ocean angular momentum: c20010701.oam in Jan. 1980 - Mar. 2002 with the sampling interval ∆t = 1 day and ECCO kf066b.oam from Jan. 1993 to Dec. 2008 with the sampling interval ∆t = 1 day, http://euler.jpl.nasa.gov/sbo/sbo data.html,

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Prediction algorithms

To compute the prediction of pole coordinates data the combination of LS extrapolation and AR prediction was be applied, denoted as LS+AR combination in this paper. In this LS+AR prediction algorithm first the LS model which consists of the Chandler circle, annual and semi-annual ellipses and linear trend is fit to the complex-valued pole coordinates data. The difference between pole coordinates data and its LS model is equal to the LS residuals. Prediction of pole coordinates data is the sum of the LS extrapolation model and the AR prediction of the LS residuals. Figure 1 shows the mean prediction

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Figure 1: Mean prediction errors of pole coordinates data computed by the LS (dots or circles) and LS+AR (thin or dashed line) methods in 1984-2009. To fit the LS model 10 years (circles or thin line), 6 years (dots) and 4 years (dashed line) of pole coordinates data were used. errors of x, y pole coordinates data computed by the LS method and the LS+AR combination. In the LS method different lengths of pole coordinates data were used to fit the LS model. The length of the AR model to fit the LS residuals was equal to 850 days. The smallest mean prediction errors of the LS+AR combination are obtained when the length of pole coordinates data to fit the LS model is equal to 10 years. To compute the prediction of UT1-UTC data the same LS+AR combination was applied to the length of day ∆ data from which the model of tidal oscillations δ∆ was removed (McCarthy and Petit 2004). To obtain such non-tidal ∆−δ∆ data leap seconds were removed from UT1-UTC data to get UT1-TAI data. First difference of UT1-TAI time series is equal to length of day ∆ data. In this prediction algorithm the LS model consists annual, semi-annual 18.6, 9.3 year oscillations and linear trend. Prediction of UT1-UTC data is computed by integrating the prediction of ∆ − δ∆ data and adding tide model and leap seconds. Figure 2 shows the mean prediction errors of UT1-UTC data computed by the LS+AR combination. In the LS method different lengths of ∆ − δ∆ data were used to fit the LS model. The length of the AR model to fit the LS residuals was equal to 1.5 years. The smallest mean prediction errors of the LS+AR combination are obtained when the length of day ∆ − δ∆ data to

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Figure 2: Mean prediction errors of UT1-UTC data computed by the LS+AR method in 19842009. To fit the LS model 5 years (dots), 10 years (thin line) or 15 years (triangles) of ∆ − δ∆ data were used. fit the LS model is equal to 10 years. To compute the prediction of UT1-UTC data by the DWT+AC combination (Kosek and Popi´ nski 2006) first we need to compute as in the previous method ∆ − δ∆ data. In this prediction method the ∆ − δ∆ data are decomposed into frequency components using the discrete wavelet transform band pass filter (DWT BPF) (see Figure 8). This filter enables decomposition of the signal into frequency components in such a way that their sum is exactly equal to the input time series. In the DWT+AC predicion method each frequency component is predicted separately by the autocovariace (AC) prediction (Kosek 2002) and the prediction of ∆−δ∆ is the sum of predictions of all the frequency components. Prediction of UT1-UTC data were computed from prediction of ∆ − δ∆ data. This DWT+AC method was used to copute weekly UT1-UTC prediction during the EOP PCC. Figure 3 shows comparison of three mean absolute errors of UT1-UTC data for 30 day in the future computed from DWT+AC, AR predictions provided by L. Zotov and Kalman filter predictions provided by R. Gross in the EOP PCC. The best prediction results of UT1-UTC data were obtained when the Kalman filter was applied (Gross et al. 1998), however other two methods give comparable results, especially for short term predictions.

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Geophysical causes of the EOP prediction errors

To examine the influence of fluid excitation functions on prediction errors of x, y pole coordinates data first the pole coordinates model data were computed from the equatorial components

Figure 3: The chosen mean absolute errors of UT1-UTC data computed by Kalman filter (thin line), DWT+AC (circles) and AR (dashed line) methods during the EOPPCC. of these excitation functions. The differential equation of polar motion is given by the following formula (Brzezi´ nski 1992): i · m(t)/σ ˙ ch + m(t) = χ(t)

(1)

in which m(t) = x(t)−i·y(t) are the pole coordinates data to be computed, χ(t) = χ1 (t) + iχ2 (t) are equatorial components corresponding to atmospheric and ocean excitation functions, σch = [1 + i/(2Q)]2π/Tch is the complex-valued Chandler frequency, Tch = 433 days and Q = 170 is the quality factor. Solution of this equation in discrete time moments can be obtained using the trapezoidal rule of numerical integration: m(t + ∆t) = m(t)exp(iσch ∆t)− iσch ∆t[χ(t + ∆t) + χ(t)exp(iσch ∆t)]/2,

(2)

where ∆t is the sampling interval of data. The maps in Figure 4 shows the time varying LS+AR prediction errors up to one year in the future of pole coordinates data or the pole coordinates model data computed from atmospheric, ocean and join atmospheric-ocean excitation function. It can be noticed that big prediction errors in 1980-82 can be explained by ocean excitation function and big prediction errors in 2006-07 can be explained by joint atmosphericocean excitation. Figure 5 shows the corresponding mean prediction errors of pole coordinates data and of the pole coordinates model data computed from atmospheric, ocean and joint atmospheric-ocean excitation functions. The contributions of atmospheric or ocean angular momentum excitation to the mean prediction errors of pole coordinates data from 1 to about 50-100 days in the future is of the same order and explain about 60% of the total prediction error. When the prediction length increases then the mean prediction errors caused by ocean excitation become

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Contribution of different frequency bands to the EOP prediction errors

In order to find a contribution of different frequency bands of the EOP data on their prediction errors the DWT BPF was applied. Figure 6 shows an example of the reconstruction of x pole coordinate data into frequency components using the DWT BPF with the Shannon wavelet function (Benedetto and Frazier 1994; Kosek et al. 2009). The sum of these frequency components is exactly equal to the x pole coordinate data. The frequency component with index j = 4 correspond to the sum of the Chandler and annual oscillations and the frequency component with index j = 5 corresponds to the semiannual oscillation. Components with frequency indices j < 4 and j > 4 correspond to longer and shorter period oscillations, respectively. The pole coordinates model data were computed by summing

Figure 5: The mean LS+AR prediction errors of IERS pole coordinates data (thin line) and of pole coordinates model data computed from AAM (circles), OAM (triangles) and AAM+OAM (dots) excitation functions.

the chosen frequency components but always including the component j = 4 corresponding to the sum of the Chandler and annual oscillations. Figure 7 shows the mean prediction errors of the IERS x, y pole coordinates data up to 30 days in the future, and of the pole coordinates model data computed by summing the chosen DWT BPF frequency components. If pole coordinates data model are composed of Chandler, annual and shorter period variations then the mean prediction errors are almost equal to the mean prediction error of the IERS pole coordinates data. If pole coordinates model data are composed of the Chandler, annual and longer period variations then the mean prediction error are very small. It means that the Chandler and annual oscillations with variable amplitudes and phases have meaningless influence on the short term prediction errors of pole coordinates data. The short term mean prediction errors for a few days in the future of the pole coordinates model data obtained after removal two frequency components j = 10 and j = 11 with the highest frequencies are several times smaller than the mean prediction errors of the IERS pole coordinates data. In order to find a contribution of different frequency bands of UT1-UTC data on their prediction errors the DWT BPF using the Meyer

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Figure 7: The mean LS+AR prediction errors of the IERS pole coordinates data (thin line) and the pole coordinates data model computed by summing the frequency components corresponding to the Chandler, annual and shorter period variations (triangles), frequency components from 0 to 9 (dashed line) and the Chandler, annual and longer period variations (circles).

Figure 8: Frequency components of the IERS ∆ − δ∆ data computed by the DWT BPF with the Meyer wavelet function.

wavelet function (Popi´ nski and Kosek 1995) was applied. Figure 8 shows an example of the reconstruction of ∆ − δ∆ data into the frequency components using the DWT BPF with the Meyer wavelet function. The frequency component with the frequency index j = 8 corresponds to the annual oscillation and with the frequency index j = 7 corresponds to the semi-annual oscillation. Frequency indices j > 8 and j < 8 correspond to shorter and longer period variations, respectively. The sum of these frequency components is exactly equal to the ∆ − δ∆ data. Figure 9 shows the mean prediction errors of UT1-UTC data and UT1-UTC model data up to 30 days in the future. If UT1-UTC data are composed of the annual, semiannual and shorter period oscillations then the mean prediction errors are almost equal to the mean prediction errors of the IERS UT1-UTC data. If the model UT1UTC data are composed of the annual, semiannual and longer period variations then the mean prediction errors are very small. It means that the annual and semi-annual oscillations with variable amplitudes and phases have meaningless influence on the short term prediction errors of UT1-UTC data. The short term mean prediction errors for a few days in the future of the model UT1-UTC data obtained after removal two highest frequency components with indices j = 1 and j = 2 are several times smaller than the mean prediction errors of the IERS UT1-UTC data.

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Figure 9: The mean LS+AR prediction errors of the IERS UT1-UTC data (thin line) and the model UT1-UTC data computed by summing the frequency components corresponding to the sum of annual, semi-annual and shorter period variations (triangles), frequency components from 3 to 13 (dashed line) and the annual, semi-annual and longer period variations (circles).

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Conclusions

Short term prediction errors of pole coordinates data are caused by wideband short period oscillations in joint atmospheric-ocean excitation functions. Some big prediction errors of pole coordinates data in 1981-1982 are caused by wideband oscillations in ocean excitation functions and in 2006-2007 are caused by wideband oscillations in joint atmospheric-ocean excitation functions. The recommended prediction method for pole coordinates data is the LS+AR combination and Kalman filter is recommended for prediction of UT1-UTC data. Short term prediction errors of the EOP data are not caused by variable amplitudes and phases of the most energetic oscillations in these data. Short term EOP prediction errors of the EOP data are much smaller when these data are smoothed by removing the frequency components computed by the DWT BPF. Since short term EOP prediction errors are mostly caused by atmospheric and ocean excitation functions these functions together with their predictions should be involved in the future EOP prediction algorithms using Kalman filter or multivariate autoregressive techniques.

Acknowledgements The research was supported by Polish Ministry of Science and Higher Education through the grant no. N N526 160136 under leadership of Dr Tomasz Niedzielski.

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