## 1. Introduction

[2] Nonstationary oscillations (NSOs), also referred to as nonstationary sinusoids [*Kuznetsova and Tsirulnik*, 2004] or quasi periodicity [*Meyers and Pagani*, 2006], have been observed in climatological data such as global surface temperature anomalies (GSTA), the North Atlantic Oscillation (NAO) and the Pacific Decadal Oscillation (PDO) index values. For example, global surface temperature shows long-term (or low-frequency) nonstationary processes such as decadal [*Ghil and Vautard*, 1991] and multidecadal oscillations [*Schlesinger and Ramankutty*, 1994a]. However, these oscillations are often so irregular that they cannot be represented with a simple sine or cosine wave. In other words, the phase and modulus are changing with time, which implies nonstationarity.

[3] A stochastic model that reproduces an NSO is useful to predict the variations of climatic processes and study their impacts on other variables such as hydrologic regimes. It is, however, a difficult task to model a NSO process. For example, a simple linear and Autoregressive Moving Average (ARMA) model [*Salas et al.*, 1980] can be applicable for the stochastic modeling of climatic variables. However, the model assumes the applied time series is stationary and the NSO process is not adoptable. Another alternative is to employ data-adaptive simulation techniques such as the index sequential method [*Ouarda et al.*, 1997], block bootstrapping [*Efron and Tibshirani*, 1993; *Vogel and Shallcross*, 1996], and *k*-nearest neighbor resampling (KNNR) [*Lall and Sharma*, 1996]. However, these methods are also not able to capture a long-term NSO process.

[4] One applicable model for NSO processes is the shifting mean level (SML) model developed by *Salas and Boes* [1980] and *Sveinsson et al.* [2003] in which the long-term oscillation pattern is modeled with the shifting mean process. However, the correlation structure of the model decreases exponentially, implying that the oscillation is not properly conveyed. The best way of preserving a long-term NSO process in a stochastic model still remains in question.

[5] Another alternative might be (1) to extract the long-term NSOs from observed data into different frequency components and (2) to build a time series model for individual NSO components. Therefore, we need an algorithm that properly separates the long-term NSOs embedded in observed data into few components to be manageable. However, in general, extraction algorithms for frequency decomposition require preinformation about the observed signals, and their performance is downgraded when an overall trend exists [*Elsner and Tsonis*, 1994; *Schlesinger and Ramankutty*, 1994a; *Elsner and Tsonis*, 1996]. Other algorithms such as wavelet analysis [*Torrence and Compo*, 1998] and the multitaper method [*Thomson*, 2001] provide too many components to manipulate.

[6] Meanwhile, *Huang et al.* [1998] proposed a decomposition technique to disclose a hidden intrinsic NSO structure in a time series, named the empirical mode decomposition (EMD). From this decomposition, oscillation structures embedded in a time series with different frequency levels are expressed as intrinsic mode functions (IMFs). It has been proven that this data-adaptive decomposition method, EMD, extracts well the NSOs into a finite number of IMFs even if combined with a long-term trend. Furthermore, [*Wu and Huang* [2004] experimented with a Monte Carlo simulation to investigate how EMD performs on white noise. They derived a way to test the significance of IMFs from this experiment. EMD analysis has been applied in climate research for instance by *Xie et al.* [2002], *Li and Davis* [2006], *Pegram et al.* [2008], *McMahon et al.* [2008], and *Lee and Ouarda* [2010].

[7] In the present paper we propose an approach to model NSO processes. EMD is employed to capture the long-term NSO in the data. The decomposed components from EMD, called IMFs, are tested to determine whether an extracted component is induced from a white noise or from a physical force [*Wu and Huang*, 2004]. These IMFs are categorized into three types (overall trend, oscillatory, and residuals) and modeled according to their characteristics.

[8] For the overall trend, the change rate of the trend is fitted with a polynomial regression model if the trend component is significant from the test [*Wu and Huang*, 2004]. For the oscillatory component, a particular data-adaptive algorithm is proposed in this paper in order to extend the future evolution of the NSO process. To the authors' knowledge, no stochastic model exists for the extension of the NSO process when its time series varies smoothly with time and its frequencies and phases are not stationary. Finally, for the residuals, the sum of the insignificant components is treated either as random white noise or autocorrelated red noise. It is modeled accordingly with a short memory time series model. Parametric (e.g., normal random noise or lag-1 Autoregressive) or nonparametric approaches (e.g., bootstrapping or KNNR) can be adopted for this purpose. Note that even if a climatic system cannot be precisely predictable in a short-term range, the overall long-term change can be predictable. In this paper, we focus on the long-term oscillatory process instead of the short-term process.

[9] To validate the model performance, the model was tested with the synthetic data from a nonlinear chaotic system, the Rössler attractor [*Rössler*, 1976, 1995]. As a case study, the proposed model was applied to GSTA data. The GSTA observations of the last 30 years were truncated and compared to the data generated from the model. Finally, the next 50 years of data were generated to predict the evolution of GSTA data into the future.

[10] The paper is organized as follows. In section 2, we describe two fundamental models employed in the proposed approach, the KNNR and EMD. The procedures of the proposed model for the selected oscillation components, the parameter estimation approach, and the modeling of the other components are presented in section 3. The proposed model is applied to a synthetic nonlinear oscillation time series in section 4. The application of the procedure to the GSTA data is presented in section 5. Finally the summary and conclusions are presented in section 6.