## 1. Introduction

[2] Understanding climate persistence is one of the principal goals of statistical climatology. This study aims to better characterize climate persistence by comparing two parsimonious statistical representations of the temporal spectral density of climatic time series: the short-memory AR1 representation and the long-memory power law representation. (The term*statistical representation* is here used instead of *statistical model* to avoid confusion with numerical climate models.) This paper also places these two representations into a broader statistical context.

[3] The well known AR1 representation [e.g., *Hasselmann*, 1976] can be written as a stochastic process *X*_{t} = *ϕX*_{t−1} + *ε*_{t}, where *ϕ*∈ (0, 1) is the lag-one autocorrelation and*ε*_{t} represents white noise innovations. The AR1 process is *short-memory* with an autocovariance function that decays exponentially with lag *τ*, *γ*_{AR1}(*τ*) ∼ *ϕ*^{|τ|}, and a spectral density as a function of frequency *f*

that saturates to a constant near the origin (i.e. for |*f*| ≪ 1/2) (see Appendix A and, e.g., *Brockwell and Davis* [1998]). The *power law* representation [e.g., *Beran*, 1994] can be defined with reference to a fractionally integrated autoregressive process FAR(0, *d*), (1 − *B*)^{d} *X*_{t} = *ε*_{t} where *B* is the backshift operator satisfying *BM*_{t} = *M*_{t−1} for random variable *M*_{t}, and *d* is the order of fractional integration [e.g., *Beran*, 1994]. It is conventional to write *H* = *d* + 1/2, where *H*is theso-called “Hurst exponent” [*Hurst*, 1951]; it is assumed that *H* ∈ (1/2, 1). This process is *long-memory* with an algebraically decaying autocovariance function *γ*_{PL}(*τ*) ∼ |*τ*|^{2H−2} and a spectral density that increases asymptotically with decreasing frequency according to a power law [e.g., *Taqqu*, 2002]

[4] In this article, we adopt the viewpoint that estimates of *ϕ* and *H* provide related measures of persistence. For AR1, the fraction of predictable variance (FPV) of *X*_{t} given past history, i.e. *X*_{t−1}, is *ϕ*^{2}. For FAR(0, *d*) the FPV is 1 − Γ^{2}(3/2 − *H*)/Γ(2 − 2*H*), which is monotonically increasing in *H* ∈ (1/2, 1) [e.g., *Beran*, 1994]. Thus, greater values of *ϕ* and *H*correspond to greater FPV in each representation. These two measures of persistence can be combined via the so-called FAR(1,*d*) stochastic process (1 − *ϕ*_{1}*B*)(1 − *B*)^{d} *X*_{t} = *ε*_{t} that generalizes the AR1 and power law representations [e.g., *Stephenson et al.*, 2000]. The FAR(1, *d*) reverts to AR1 with lag-one autocorrelation*ϕ* = *ϕ*_{1} when *d* → 0 and to the power law process FAR(0, *d*) with slope −2*d* = 1 − 2 *H* when *ϕ*_{1} → 0. The FPV for FAR(1, *d*) is

which can be shown to revert to the FPV of the AR1 process when *d* → 0 (i.e. when *H* → 1/2) and to the FPV of the power law process FAR(0, *d*) when *ϕ*_{1} → 0.

[5] The AR1 and power law stochastic processes provide distinct limiting cases of persistence and the behavior of the spectral density for low frequencies |*f*| ≪ 1/2, which is controlled in practice by the length of a given time series. The AR1 representation captures the tendency for the variability of some climatic time series to saturate in the transition from weekly, to intraseasonal, annual, and decadal variability [e.g., *Frankignoul and Hasselmann*, 1977; *Feldstein*, 2000]. But the power law representation may be more appropriate for long instrumental and paleoclimate records, for which variability tends to build in the transition to decadal, centennial, and millennial timescales [e.g., *Bloomfield*, 1992; *Pelletier*, 1997; *Tsonis et al.*, 1999; *Caballero et al.*, 2002; *Eichner et al.*, 2003; *Fraedrich and Blender*, 2003; *Vyushin et al.*, 2004; *Huybers and Curry*, 2006]. This study explores the use of these two representations in the intermediate interannual-to-multidecadal range captured by the instrumental observational record and current climate simulations.

[6] While other higher order statistical representations [including FAR(1, *d*)] are available, this study focuses on these two distinct, parsimonious, and commonly used representations. It evaluates their validity, and examines the ability of climate models to reproduce observed patterns of their parameters, and explain selected aspects of their behavior. In particular, some understanding of *internal* climate persistence, i.e. the persistence that occurs in the absence of anthropogenic and natural radiative forcing, can be obtained by comparing simulations with and without external radiative forcings. Only a few studies have sought to compare these two representations on an even footing [e.g., *Stephenson et al.*, 2000; *Percival et al.*, 2001; *Caballero et al.*, 2002; *Vyushin et al.*, 2007; *Vyushin and Kushner*, 2009; *Franzke*, 2012] and none have systematically compared them in climate simulations and observations or linked them in the way presented below.

[7] Observational data and simulations used and statistical methods employed are described in Section 2. Section 3 evaluates the ability of models to capture the observed spatial distribution of *ϕ* and *H* and the dependence of these parameters on analysis timescale. Section 4 relates the two measures of persistence to each other and evaluates their validity. The study concludes with a discussion of implications of this analysis in Section 5. Appendix A outlines mathematical concepts in the study, and Appendix Bprovides details of a goodness-of-fit test employed inSection 4.