Polar motion prediction using the combination of SSA and Copulabased analysis
Abstract
Keywords
Copula SSA Polar motion EOP PredictionAbbreviations
 ANN
artificial neural network
 AR
autoregressive
 CDF
cumulative distribution function
 CLME
canonical maximum likelihood estimation
 DORIS
Doppler Orbitography and Radiopositioning Integrated by Satellite
 EOP
Earth orientation parameters
 EOP PCC
Earth orientation parameters prediction comparison campaign
 GFZ
German Research Centre for Geosciences
 GGOS
Global Geodetic Observing System
 GNSS
Global Navigation Satellite Systems
 IAG
International Association of Geodesy
 ICRF
international celestial reference frame
 IERS
International Earth Rotation and Reference Systems Service
 IFME
Inference for Margins Estimation
 LLR
Lunar Laser Ranging
 LS
least squares
 MAE
mean absolute error
 RMS
rootmeansquare
 PM
polar motion
 TRF
terrestrial reference frame
 VLBI
Verylongbaseline interferometry
 SLR
Satellite Laser Ranging
 SSA
singular spectrum analysis
Introduction
Polar motion (PM) describes the movement of the Earth’s rotation axis w.r.t the Earth surface. The study of PM provides valuable information for studying many geophysical and meteorological phenomena (Barnes et al. 1983; Wahr 1982, 1983; Mathews et al. 1991; Gross et al. 2003; Chen and Wilson 2005; Gross 2015; Seitz and Schuh 2010; Schuh and Böhm 2011).
Since the 1960s, highly accurate PM coordinates can be obtained by different space geodesy techniques. These techniques include: Satellite Laser Ranging (SLR) (Coulot et al. 2010), Lunar Laser Ranging (LLR) (Dickey et al. 1985), Doppler Orbitography and Radiopositioning Integrated by Satellite (DORIS) (Angermann et al. 2010), Global Navigation Satellite Systems (GNSS) (Dow et al. 2009; Byram and Hackman 2012), and verylongbaseline interferometry (VLBI) (Schuh and SchmitzHübsch 2000; Nilsson et al. 2010, 2011, 2014).
Methodology
In this study, we developed and explored the integration of Copulabased analysis and SSA for precisely predicting PM.
Singular spectrum analysis
To maximize the prediction performance, we need a mathematical tool to retrieve all timecorrelated information from the time series. As a matter of fact, the existence of excitations of PM can profoundly affect the forecasting procedure, particularly in longer intervals. Therefore, the exploitation of efficient techniques is crucial to minimize the risk of having gross errors.
SSA is a nonparametric spectral estimation method which can be used for decomposing a time series into the sum of interpretable components, e.g., trend, periodic components, and noise, without a priori assumption about the constituent components (Golyandina et al. 2001).
SSA is able to remove redundancies and groups uncorrelated information into informative empirical functions which can reveal main aspects of the time series. The mentioned functions are used as bases of a subspace in which the time series is a member of and can be exploited for modeling the time series in a desired level of details. Therefore, the model can simulate the future entries of the time series using these base functions.
The SSA method for trend extraction can be succinctly expressed as two stages:
Decomposition
Reconstruction
Copulabased analysis
Characteristic of Copulas
Empirical Copula

\((r_1), (r_2) \ldots , (r_n)\) denote the pairs of ranks of the variable \((x_1),(x_2), \ldots , (x_n)\),

\((s_1), (s_2) \ldots , (s_n)\) denote the pairs of ranks of the variable \((y_1),(y_2), \ldots , (y_n)\),

n is the length of the data vector,

1(...) is the indicator function. If the condition is true, the indicator function is equal to 1. Otherwise, the indicator function is equal to 0.
Archimedean Copula

for all \(u \in (0,1), \phi (u) < 0\), \(\phi\) is decreasing,

for all \(u \in (0,1), \phi (u) < 0\), \(\phi\) is convex,

\(\phi (1)=0\),
 (1)
Clayton Copulas
The generator of the Clayton Copula (see Fig. 1) is given byTherefore, the cumulative distribution function (CDF) for Clayton Copula is defined as (Clayton 1978):$$\begin{aligned} \phi ^{Cl}(x)=\frac{1}{\theta }(t^{\theta }1) \end{aligned}$$(16)where \(\theta\) is restricted on the interval \([1,\infty )\). If \(\theta = 0\), it shows the independence case and when \(\theta \rightarrow \infty\), indicate high dependency in the lower rank space.$$\begin{aligned} C_\theta (u,v)= \max [(u^{\theta }+v^{\theta }1),0]^{\frac{1}{\theta }} \end{aligned}$$(17)  (2)
Frank Copula
The generator of the Frank Copula (see Fig. 2) is given byThe parameter \(\theta\) is defined over \(\in (\infty , \infty )\lbrace 0 \rbrace\). The CDF for Frank Copula is given by (Joe 1997; Lee and Long 2009)$$\begin{aligned} \phi ^{Fr}(t)= \ln \left\{ \frac{\mathrm{e}^{\theta t}1}{\mathrm{e}^{\theta }1}\right\} \end{aligned}$$(18)Frank Copula allows to model data with positive and negative dependency. The large positive and negative \(\theta\) indicate high dependency, and \(\theta = 1\) implies total independence. The Frank Copula is a suitable method for two data sets with the same dynamic characteristics (Rodriguez 2007).$$\begin{aligned} C_\theta (u,v)= \frac{1}{\theta }\ln \left( 1+ \frac{(\mathrm{e}^{\theta u}1)(\mathrm{e}^{\theta v})}{\mathrm{e}^{\theta }1}\right) \end{aligned}$$(19)  (3)
Gumbel Copulas
Gumbel Copula (see Fig. 3) is famous for its ability to capture strong upper tail dependence and weak lower tail dependence. Gumbel Copula is used to model asymmetric relationship in the data (Jaworski et al. 2010). The Gumbel Copula generator is written as:The CDF for Gumbel Copula is given by (Nelsen 2007)$$\begin{aligned} \phi (t)^{\rm Gu}=(\ln t)^\theta \end{aligned}$$(20)The Copula parameter \(\theta\) is on the interval \([1, +\infty )\). If \(\theta\) is equal 1, Copula shows independence. When \(\theta \rightarrow \infty\), the Gumbel Copula indicates high dependence between the random variables.$$\begin{aligned} C_\theta (u,v)= \mathrm{e}^{((\ln (u)^\theta )+(\ln (v)^\theta ))^{\frac{1}{\theta }}} \end{aligned}$$(21)
Three ordinary families of Archimedean Copulas (Clayton, Frank, and Gumbel Copula) and their generator, parameter space, and their formula
Family  Generator  Parameter  Formula 

Clayton  \(\phi ^{Cl}(x)=\frac{1}{\theta }(t^{\theta }1)\)  \(\,1 \le \theta\)  \(C_\theta (u,v)= \max [(u^{\theta }+v^{\theta }1),0]^{\frac{1}{\theta }}\) 
Frank  \(\phi ^{Fr}(t)=\ln \left\{ \frac{\mathrm{e}^{\theta t}1}{\mathrm{e}^{\theta }1}\right\}\)  \(\,\infty< \theta <\infty\)  \(C_\theta (u,v)= \frac{1}{\theta }\ln (1+ \frac{(\mathrm{e}^{\theta u}1)(\mathrm{e}^{\theta v})}{\mathrm{e}^{\theta }1})\) 
Gumbel  \(\phi (t)=(\ln t)^\theta\)  \(1 \le \theta\)  \(C_\theta (u,v)= \mathrm{e}^{((\ln (u)^\theta )+(\ln (v)^\theta ))^{\frac{1}{\theta }}}\) 
Copula parameter estimation
The widely used estimation method for the Copula parameter is the maximum likelihood (ML) estimation methodology (Joe 1997). The Copula parameters in this study are derived from ML estimation. The canonical maximum likelihood estimation (CLME) and inference for margins estimation (IFME) are two methods for estimation of the Copula parameter (Joe and Xu 1996). For both methods, the first step is marginal distribution estimation. Then, a pseudosample of the transformed observation is used to estimate the Copula parameter. In the IFME method, the theoretical marginal distribution parameters are estimated, and in the CMLE the univariate marginals are the empirical distribution functions (Giacomini et al. 2009). It is assumed that the sample data \((X_1, X_2, X_3, \ldots , X_n)\) are n independent and identically distributed (iid) random variables. These data are transformed into uniform variates \((r_1, r_2, r_3, \ldots , r_n)\).
Computation of conditional CDF for Archimedean Copula
Simulating from Copulabased conditional random data
 (1)
Independent identical distribution (iid)transformation of input time series.
 (2)
Compute the marginal distribution \(F_X (x)\) and \(F_Y (y)\) of the input data x and y.
 (3)
Transform data to rank space using the estimated marginal distributions of data with \(u_i\) and \(v_i\) in rank space.
 (4)
Compute the empirical Copula to the dependence structure of random variables using the ranktransformed data.
 (5)
Fit a theoretical Copula function \(C_\theta (u,v)\).
 (6)
Compute the conditional Copula function.
 (7)
Sample random data from the conditional Copula CDF.
 (8)
Transfer the sample back to the data space using the inverse marginal.
Error analysis
Calculation and analysis
Data description
In this paper, the \({\hbox{PM}}_{x}\) and \({\hbox{PM}}_{y}\) time series (see Fig. 4) are from the International Earth Rotation and Reference Systems Service (IERS) combined earth orientation parameter (EOP) solutions 08 C04 (available at http://hpiers.obspm.fr/eoppc/analysis/excitactive.html). The EOP 08 C04 series is derived from different geodetic techniques, and it is consistent with ITRF 2008. The EOP 08 C04 time series cover the period 1962 to the present. The sampling interval is one day.
Data processing and analysis
In this study, we defined an algorithm for PM prediction which is shown in Fig. 5. The observed PM time series can be split up into two parts. The first part is dealing with periodic effects such as Chandler wobble, annual variation, and influences of solid Earth tides and ocean tides on PM. The SSA is used to model the periodic terms of the PM. Then, the difference between the observed PM and SSA estimated data is modeled by using the Copulabased analysis method. After that, the periodic terms of PM are extrapolated using the SSA a priori model. Also, the anomaly part is predicted using the Copulabased model. Finally, the anomaly solution is added to the SSAforecasted time series.
 (1)SSA Periodic Terms Estimation

Selecting window parameter (L) considering the dominant periods of the time series and the prediction interval,

Forming trajectory matrix (\(\mathbf{X}\)) using L,

Singular value decomposition of \(\mathbf{X}\),

Selecting a proper group of singular values and corresponding singular vectors,

Reconstruction of \(\mathbf{X}\),

Calculation of the trend by applying diagonal averaging to \(\mathbf{X}\).

 (2)Copula Anomaly Modeling

Subtract the observed PM time series by SSAreconstructed time series,

Forming the trajectory matrix of residual time series using window length L and time delay of 1 day,

Compute the marginal distribution of each column of the matrix,

Transform data to the rank space,

Compute the empirical Copula between the column i and i+1,

Fit the theoretical Copula model by applying appropriate goodnessoffit tests,

Compute the conditional Copula,

Sample random data from the conditional Copula CDF and transfer the sample back to the data space using the inverse marginal,

For each value of one input time series, one obtains an ensemble of possible values for other time series.

SSA periodic terms estimation
Window length selection is a crucial step in SSA which has a significant impact on the decomposition of the time series. The appropriate choice for L in a periodic time series with a period T is a window length proportional to the period, meaning that the L / T is an integer. Figure 6 depicts the main periods of PM time series (Golyandina and Zhigljavsky 2013). So, the Chandler period as the main period of both time series would be a reasonable choice. Making the closest choice to the half of the length of the time series (if possible, least common multiple of the Chandler and annual periods) is recommended by Golyandina and Zhigljavsky (2013), but is avoided due to the processing time.
After selection of the window length, the number of singular vectors or empirical functions for reconstruction of the time series should be determined. The goal of this procedure is to find and apply a proper set of constructive components. Most significant periodicities as well as excitation mechanisms are rather lowfrequency components and reveal their impact in the first few singular vectors while highfrequency components fall in later singular vectors. The singular value spectrum reflects the importance of each singular vector. Figure 7 suggests that in order to achieve an accuracy of about 1 mas in polar motion modeling, we need to utilize at least first 70 singular vectors which correspond to using all components with periods more than or equal to 14 days.
Having the window length and the number of singular values determined, we construct the trajectory matrix. As it can be seen in Fig. 8 the data between the year of 1997 and 2003 is used as the training period. The cyan curve is the SSAreconstructed \({\hbox{PM}}_{x}\) time series. Prediction of the future entries starts by adding initial guess of future entries to the end of the time series. Then, iteration of the SSA process is done until the result of two successive iterations has a difference less than a certain threshold. This will map the initial values to the estimated periodic terms of the time series. The residual part of the difference between original \({\hbox {PM}}_{x}\) time series and SSA estimated time series is named anomaly of \({\hbox {PM}}_{x}\) which has a stochastic behavior. Therefore, the anomaly part will be investigated by Copulabased technique.
Copula anomaly modeling
Marginal distributions
Distribution  Formula  Parameters 

Extreme value (Kotz and Nadarajah 2000)  \(f(x;\mu , \sigma )= \sigma ^{1}\exp (\frac{x\mu }{\sigma })\exp \left( \exp \left( \frac{x\mu }{\sigma }\right) \right)\)  Location \(\mu\) scale \(\sigma\) 
Generalized extreme value (Hosking et al. 1985)  \({\displaystyle f(x;\mu , \sigma ,\xi )={{\left\{ \begin{array}{ll}{\big (}1+\xi ({\tfrac{x\mu }{\sigma }}){\big )}^{1/\xi }&{}{\text {if}}\ \xi \ne 0\\ \mathrm{e}^{(x\mu )/\sigma }&{}{\text {if}}\ \xi =0\end{array}\right. }}}\)  Location \(\mu\) scale \(\sigma\) shape \(\xi\) 
Generalized Pareto (Hosking and Wallis 1987)  \(f(x;\sigma ,\xi )= f_{{(\xi ,\mu ,\sigma )}}(x)={\frac{1}{\sigma }}\left( 1+{\frac{\xi (x\mu )}{\sigma }}\right) ^{{\left( {\frac{1}{\xi }}1\right) }}\)  Location \(\mu\) scale \(\sigma\) shape \(\xi\) 
Goodnessoffit test for marginal distribution of \({\hbox {PM}}_{x}\)
Distributions  AIC  BIC  RMSE 

Extreme value  574.60  582.66  0.04 
Generalized extreme value  511.14  523.23  0.01 
Generalized Pareto  758.22  770.31  0.13 
Goodnessoffit test for marginal distribution of \({\hbox {PM}}_{y}\)
Distributions  AIC  BIC  RMSE 

Extreme value  310.99  319.05  0.03 
Generalized extreme value  261.25  273.34  0.01 
Generalized Pareto  523.99  536.09  0.14 
Estimating empirical Copula
Once the univariate marginal distribution is fitted, the dependence structure between the time series has to be investigated. The first step is to calculate the empirical Copula using Eq. (14). As it can be seen in Fig. 10, there is a scatter plot of two adjacent columns, and it shows a scatter linear dependency structure with the heavy tail. This kind of dependency structure can be correctly modeled using the Archimedean Copula.
Fitting a theoretical Copula function
Goodnessoffit test for Copula model
Copula name  Clayton  Frank  Gumbel 

Mean(\(S_{n}\))  43.57  12.13  17.58 
365dayahead prediction
We utilized 6 years of observed PM time series, from January 1997 to December 2002, for the 365dayahead prediction. To verify the reliability of this method, the results were compared with the IERS Bulletin A predictions (https://datacenter.iers.org/web/guest/bulletins//somos/5Rgv/version/6). The IERS Bulletin A contains the PM parameters and the predicted PM for one year into the future, and they are released every seven days by IERS Rapid Service/Prediction Center (RS/PC), hosted by the U.S. Naval Observatory (USNO) (Petit and Luzum 2010; Gambis and Luzum 2011). The predictions of PM from the IERS Bulletin A were produced by LS + AR method. In the current prediction method, the PM prediction was the sum of the LS extrapolation model (including the Chandler period, annual, semiannual, terannual, and quarter annual terms), and the AR predictions of the LS extrapolation residuals (Kosek et al. 2007).
Discussion of results
Success rate of \({\hbox {PM}}_{x}\) prediction [%]
Method\(\backslash\)year  2003  2004  2005  2006  2007  2008  2009  Average 

SSA  55.29  33.31  26.52  40.16  22.90  45.94  64.70  41.26 
SSA + Clayton  61.71  33.88  31.91  40.17  22.91  45.96  64.95  43.07 
SSA + Frank  58.31  34.31  33.61  42.50  22.91  45.97  64.99  43.22 
SSA + Gumbel  55.90  33.81  28.31  41.00  22.90  45.94  64.97  41.83 
Success rate of \({\hbox {PM}}_{y}\) prediction [%]
Method\(\backslash\)year  2003  2004  2005  2006  2007  2008  2009  Average 

SSA  35.95  44.99  25.43  45.28  39.50  29.21  39.50  37.12 
SSA + Clayton  35.99  44.84  24.93  41.27  39.57  29.14  39.65  36.48 
SSA + Frank  35.94  44.82  25.54  46.66  39.45  29.30  39.70  37.36 
SSA + Gumbel  38.45  44.60  26.66  44.46  39.36  29.55  39.74  37.54 
Conclusions
The improvement in the Earth rotation prediction is a relevant, timely problem, as confirmed by the fact that the International Astronomical Union (IAU) Commission A2, the International Association of Geodesy (IAG), and the IERS have at present two Joint Working Groups on Prediction (JWGP) and on Theory of Earth rotation and validation (JWGThER). According to the United Nations (UN) resolution in 2015, the primary objective of these JWGs is to assess and ensure the level of consistency of earth orientation parameter (EOP) predictions derived from theories with the corresponding EOP determined from analyses of the observational data provided by the various geodetic techniques. Therefore, accurate EOP predictions are essential to avoid any systematic drifts and/or biases between the international celestial and terrestrial reference frames (ICRF and ITRF). The results illustrate that the proposed method could efficiently and precisely predict the PM parameters. As clearly demonstrated, the SSA + Copula algorithm shows better performance for \({\hbox {PM}}_{x}\) prediction in comparison with the SSA prediction. The Copulabased analysis is fully successful in its aim to increase the accuracy of PM prediction by modeling the stochastic part of the PM and subtracting PM by SSAreconstructed time series. We suspect the main error contributions come from SSA extrapolation part. So, further investigations about the SSA training time will be required to clarify this issue. Also, SSA + Copula prediction method shows periodic errors, and these errors have a significant impact on the mean absolute error. Therefore, these occasional errors should be further investigated to have a noticeable progression in the PM prediction accuracy.
Notes
Authors' contributions
SM did most of the data analysis and writing of the manuscript. MH carried out the SSA studies and wrote a part of the manuscript. SB conceived and designed the study. RH, JMF, and HS participated in the design of the study and helped to improve the manuscript. All authors read and approved the final manuscript.
Acknowledgements
We are grateful to the International Earth Rotation and Reference Systems Service (IERS) for providing the Polar motion data. We like to thank the two anonymous reviewers for their comments which helped to improve the paper.
Competing interests
The authors declare that they have no competing interests.
Funding
The corresponding author is supported by an offer of financial assistance by GFZ German Research Centre for Geosciences, Potsdam, Germany. SB and JF works were partially supported by Projects AYA201679775P (AEI/FEDER, UE). Also, SB was supported by the European Research Council (ERC) under the ERC2017STG SENTIFLEX project (Grant Agreement 755617).
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
References
 Akulenko L, Kumakshev S, Markov YG, Rykhlova L (2002) Forecasting the polar motions of the deformable earth. Astron Rep 46(10):858–865CrossRefGoogle Scholar
 Akyilmaz O, Kutterer H (2004) Prediction of earth rotation parameters by fuzzy inference systems. J Geodesy 78(1–2):82–93Google Scholar
 Akyilmaz O, Kutterer H, Shum C, Ayan T (2011) Fuzzywavelet based prediction of earth rotation parameters. Appl Soft Comput 11(1):837–841CrossRefGoogle Scholar
 Angermann D, Seitz M, Drewes H (2010) Analysis of the DORIS contributions to IRTF2008. Adv Space Res 46(12):1633–1647CrossRefGoogle Scholar
 Bárdossy A, Li J (2008) Geostatistical interpolation using copulas. Water Resour Res 44(7):1–15CrossRefGoogle Scholar
 Bárdossy A, Pegram G (2009) Copula based multisite model for daily precipitation simulation. Hydrol Earth Syst Sci 13(12):22–99CrossRefGoogle Scholar
 Barnes R, White A, Wilson C (1983) Atmospheric angular momentum fluctuations, lengthofday changes and polar motion. Proc R Soc Lond A 387(1792):31–73CrossRefGoogle Scholar
 Bizouard C, Gambis D, (2009) The combined solution c04 for earth orientation parameters consistent with international terrestrial reference frame 2005. In: Geodetic reference frames, Springer, pp. 265–270Google Scholar
 Broomhead DS, King GP (1986) Extracting qualitative dynamics from experimental data. Phys D 20(2–3):217–236CrossRefGoogle Scholar
 Byram S, Hackman C (2012) Highprecision GNSS orbit, clock and EOP estimation at the United States Naval Observatory. In: Position location and navigation symposium (PLANS), 2012 IEEE/ION, IEEE, pp. 659–663Google Scholar
 Chen J, Wilson C (2005) Hydrological excitations of polar motion, 1993–2002. Geophys J Int 160(3):833–839CrossRefGoogle Scholar
 Chin TM, Gross RS, Dickey JO (2004) Modeling and forecast of the polar motion excitation functions for shortterm polar motion prediction. J Geodesy 78(6):343–353CrossRefGoogle Scholar
 Clayton DG (1978) A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65(1):141–151CrossRefGoogle Scholar
 Coulot D, Pollet A, Collilieux X, Berio P (2010) Global optimization of core station networks for space geodesy: application to the referencing of the STR EOP with respect to ITRF. J Geod 84(1):31CrossRefGoogle Scholar
 Dickey J, Newhall X, Williams J (1985) Earth orientation from Lunar Laser Ranging and an error analysis of polar motion services. J Geophys Res Solid Earth 90(B11):9353–9362CrossRefGoogle Scholar
 Dow JM, Neilan RE, Rizos C (2009) The international GNSS service in a changing landscape of Global Navigation Satellite Systems. J Geod 83(3–4):191–198CrossRefGoogle Scholar
 Embrechts P, McNeil A, Straumann D (2002) Correlation and dependence in risk management: properties and pitfalls. In: Dempster MAH (ed) Risk management: value at risk and beyond, vol 1. Cambridge University Press, Cambridge, pp 176–223CrossRefGoogle Scholar
 Escarela G, Carriere JF (2003) Fitting competing risks with an assumed Copula. Stat Methods Med Res 12(4):333–349CrossRefGoogle Scholar
 Freedman A, Steppe J, Dickey J, Eubanks T, Sung LY (1994) The shortterm prediction of universal time and length of day using atmospheric angular momentum. J Geophys Res Solid Earth 99(B4):6981–6996CrossRefGoogle Scholar
 Gambis D, Luzum B (2011) Earth rotation monitoring, UT1 determination and prediction. Metrologia 48(4):S165CrossRefGoogle Scholar
 Genest C, Favre AC (2007) Everything you always wanted to know about Copula modeling but were afraid to ask. J Hydrol Eng 12(4):347–368CrossRefGoogle Scholar
 Genest C, Rivest LP (1993) Statistical inference procedures for bivariate Archimedean Copulas. J Am Stat Assoc 88(423):1034–1043CrossRefGoogle Scholar
 Giacomini E, Härdle W, Spokoiny V (2009) Inhomogeneous dependence modeling with timevarying Copula. J Bus Econ Stat 27(2):224–234CrossRefGoogle Scholar
 Golyandina N, Zhigljavsky A (2013) Singular Spectrum Analysis for time series. SpringerVerlag, BerlinCrossRefGoogle Scholar
 Golyandina N, Nekrutkin V, Zhigljavsky AA (2001) Analysis of time series structure: SSA and related techniques. Chapman and Hall, New YorkCrossRefGoogle Scholar
 Gross RS (2015) Earth rotation variations—long period. Phys Geod 11:215–261Google Scholar
 Gross RS, Fukumori I, Menemenlis D (2003) Atmospheric and oceanic excitation of the earth’s wobbles during 1980–2000. J Geophys Res Solid Earth 108(B8):2370CrossRefGoogle Scholar
 Hosking JR, Wallis JR (1987) Parameter and quantile estimation for the generalized Pareto distribution. Technometrics 29(3):339–349CrossRefGoogle Scholar
 Hosking JR, Wallis JR, Wood EF (1985) Estimation of the generalized extremevalue distribution by the method of probabilityweighted moments. Technometrics 27(3):251–261CrossRefGoogle Scholar
 Jaworski P, Durante F, Härdle WK, Rychlik T (2010) Copula theory and its applications. In: proceedings of the workshop held in Warsaw, 25–26 September 2009, Vol. 198, Springer Science & Business MediaGoogle Scholar
 Joe H (1997) Multivariate models and multivariate dependence concepts. CRC Press, Boca RatonCrossRefGoogle Scholar
 Joe H, Xu JJ (1996) The estimation method of inference functions for margins for multivariate models, Technical report, Department of Statistics, University of British ColumbiaGoogle Scholar
 Kalarus M, Kosek W (2004) Prediction of earth orientation parameters by artificial neural networks. Artif Satell J Planet Geod 39(2):175–184Google Scholar
 Kalarus M, Schuh H, Kosek W, Akyilmaz O, Bizouard C, Gambis D, Gross R, Jovanović B, Kumakshev S, Kutterer H et al (2010) Achievements of the earth orientation parameters prediction comparison campaign. J Geod 84(10):587–596CrossRefGoogle Scholar
 Kosek W, McCarthy D, Luzum B (1998) Possible improvement of earth orientation forecast using autocovariance prediction procedures. J Geod 72(4):189–199CrossRefGoogle Scholar
 Kosek WI, Kalarus M, Niedzielski T, Capitaine N (2007) Forecasting of the Earth orientation parameters: comparison of different algorithms. Observatoire de Paris, ParisGoogle Scholar
 Kotz S, Nadarajah S (2000) Extreme value distributions: theory and applications. World Scientific, SingaporeCrossRefGoogle Scholar
 Laux P, Vogl S, Qiu W, Knoche H, Kunstmann H (2011) Copulabased statistical refinement of precipitation in RCM simulations over complex terrain. Hydrol Earth Syst Sci 15(7):2401–2419CrossRefGoogle Scholar
 Lee TH, Long X (2009) Copulabased multivariate GARCH model with uncorrelated dependent errors. J Econom 150(2):207–218CrossRefGoogle Scholar
 Malkin Z, Miller N (2010) Chandler wobble: two more large phase jumps revealed. Earth Planets Space 62(12):943–947CrossRefGoogle Scholar
 Mathews P, Buffett BA, Herring TA, Shapiro II (1991) Forced nutations of the earth: influence of inner core dynamics: 1. Theory. J Geophys Res Solid Earth 96(B5):8219–8242CrossRefGoogle Scholar
 Modiri S, Lorenz C, Sneeuw N, Kunstmann H, (2015) Copulabased estimation of largescale water storage changes: exploiting the dependence structure between hydrological and grace data. In: EGU general assembly conference abstracts, Vol. 17Google Scholar
 Nelsen RB (2007) An introduction to copulas. Springer Science & Business Media, BerlinGoogle Scholar
 Nilsson T, Böhm J, Schuh H (2010) Subdiurnal earth rotation variations observed by VLBI. Artif Satell 45(2):49–55CrossRefGoogle Scholar
 Nilsson T, Böhm J, Schuh H (2011) Universal time from VLBI singlebaseline observations during CONT08. J Geod 85(7):415–423CrossRefGoogle Scholar
 Nilsson T, Heinkelmann R, Karbon M, RaposoPulido V, Soja B, Schuh H (2014) Earth orientation parameters estimated from VLBI during the CONT11 campaign. J Geod 88(5):491–502CrossRefGoogle Scholar
 Patton AJ (2006) Modelling asymmetric exchange rate dependence. Int Econ Rev 47(2):527–556CrossRefGoogle Scholar
 Patton AJ (2009) Copulabased models for financial time series. In: Mikosch T, Kreiß JP, Davis RA, Andersen TG (eds) Handbook of financial time series. Springer, Berlin, pp 767–785CrossRefGoogle Scholar
 Petit G, Luzum B (2010) IERS conventions (2010), Technical report, Bureau International des poids et mesures sevres (France)Google Scholar
 Plag HP, Pearlman M (2009) Global geodetic observing system: meeting the requirements of a global society on a changing planet in 2020. Springer, BerlinCrossRefGoogle Scholar
 Rachev S, Mittnik S (2000) Stable Paretian models in finance. Willey, New YorkGoogle Scholar
 Rodriguez JC (2007) Measuring financial contagion: a copula approach. J Empir Finance 14(3):401–423CrossRefGoogle Scholar
 Schuh H, Behrend D (2012) VLBI: a fascinating technique for geodesy and astrometry. J Geodyn 61:68–80CrossRefGoogle Scholar
 Schuh H, Böhm S (2011) Earth rotation. In: Gupta HK (ed) Encyclopedia of solid earth geophysics. Springer, Dordrecht, pp 123–129CrossRefGoogle Scholar
 Schuh H, SchmitzHübsch H (2000) Short period variations in earth rotation as seen by VLBI. Surv Geophys 21(5–6):499–520CrossRefGoogle Scholar
 Schuh H, Ulrich M, Egger D, Müller J, Schwegmann W (2002) Prediction of earth orientation parameters by artificial neural networks. J Geod 76(5):247–258CrossRefGoogle Scholar
 Seitz F, Schuh H (2010) Earth rotation. In: Xu G (ed) Sciences of geodesyI. Springer, Berlin, pp 185–227CrossRefGoogle Scholar
 Sklar M (1959) Fonctions de repartition an dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8:229–231Google Scholar
 Stamatakos N (2017) IERS rapid service prediction center products and services: improvement, changes, and challenges, 2012 to 2017. In: Proceedings of the Journées Systèmes de référence spatiotemporels’Google Scholar
 Trivedi PK, Zimmer DM et al (2007) Copula modeling: an introduction for practitioners. FoundTrends® Economet 1(1):1–111Google Scholar
 Vautard R, Yiou P, Ghil M (1992) SingularSpectrum Analysis: a toolkit for short, noisy chaotic signals. Phys D 58(1–4):95–126CrossRefGoogle Scholar
 Verhoest NE, van den Berg MJ, Martens B, Lievens H, Wood EF, Pan M, Kerr YH, Al Bitar A, Tomer SK, Drusch M et al (2015) Copulabased downscaling of coarsescale soil moisture observations with implicit bias correction. IEEE Trans Geosci Remote Sens 53(6):3507–3521CrossRefGoogle Scholar
 Vogl S, Laux P, Qiu W, Mao G, Kunstmann H (2012) Copulabased assimilation of radar and gauge information to derive biascorrected precipitation fields. Hydrol Earth Syst Sci 16(7):2311–2328CrossRefGoogle Scholar
 Wahr JM (1982) The effects of the atmosphere and oceans on the earth’s wobble—I. Theory. Geophys J Int 70(2):349–372CrossRefGoogle Scholar
 Wahr JM (1983) The effects of the atmosphere and oceans on the earth’s wobble and on the seasonal variations in the length of day—II. results. Geophys J Int 74(2):451–487Google Scholar
 Wang W, Wells MT (2000) Model selection and semiparametric inference for bivariate failuretime data. J Am Stat Assoc 95(449):62–72CrossRefGoogle Scholar
 Willmott CJ, Matsuura K (2005) Advantages of the mean absolute error (MAE) over the root mean square error (RMSE) in assessing average model performance. Clim Res 30(1):79–82CrossRefGoogle Scholar
 Włodzimierz H (1990) Polar motion prediction by the leastsquares collocation method. In: Boucher C, Wilkins GA (eds) Earth rotation and coordinate reference frames. Springer, US, pp 50–57CrossRefGoogle Scholar
 Xu X, Zhou Y, Liao X (2012) Shortterm earth orientation parameters predictions by combination of the leastsquares, AR model and Kalman filter. J Geodyn 62:83–86CrossRefGoogle Scholar
 Yue S (1999) Applying bivariate normal distribution to flood frequency analysis. Water Int 24(3):248–254CrossRefGoogle Scholar
 Zhang L, Singh VP (2007) Bivariate rainfall frequency distributions using Archimedean Copulas. J Hydrol 332(1–2):93–109CrossRefGoogle Scholar
 Zotov L (2005) Regression methods of earth rotation prediction. Mosc Univ Phys Bull 5(2005):36–50Google Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.