## Spectral Analysis

Spectral analysis investigates the noise component in the climate equation (Eq. 1.2). A Fourier transformation into the frequency domain makes it possible to separate short-term from long-term variations and to distinguish between cyclical forcing mechanisms of the climate system and broad-band resonances. Spectral analysis allows to learn about the climate physics.

The task is to estimate the spectral density function, and to test for harmonic (cyclical) signals. This poses more difficulties than, for example, linear regression because now we estimate a function and not just two parameters. Spectral smoothing becomes therefore necessary, and this brings a trade-off between estimation variance and frequency resolution.

The multitaper smoothing method achieves the optimal trade-off for evenly spaced time series. The method of choice for unevenly spaced records is Lomb-Scargle, which estimates in the time domain and avoids distortions caused by interpolation.

Bootstrap resampling enhances multitaper and Lomb-Scargle methods by providing a bias correction and CIs. It supplies also a detection test for a spectral peak against realistic noise alternatives in form of an AR(1) process ("red noise"). Section 5.2.8 introduces bootstrap adaptions to take into account the effects of timescale uncertainties on detectability and frequency resolution.

5.1 Spectrum

Let us assume in this chapter that the climate process in continuous time, X(T), has no trend and no outlier components and a constant

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Oceanographic Sciences Library 42, DOI 10.1007/978-90-481-9482-7_5, © Springer Science+Business Media B.V. 2010

variability, S,

Such a process could be derived from a "real" climate process, that is, with trend and so forth, by subtracting the trend and outlier components and normalizing (standard deviation). Techniques for quantifying trend and variability and detecting outliers are presented in Chapter 4.

It is then straightforward (Priestley 1981) to define a truncated process,

I 0 elsewhere, and express it as a Fourier integral,

where

This introduces the frequency, f. (The symbol i in the exponent denotes \/—1.) This is a useful quantity for describing phenomena that exhibit a periodic behaviour in time. The period (time units) is given by Tperiod = 1/f. If one associates X(T) with movement and kinetic energy, then 2n|GT/(f )2|df can be seen as the energy contribution of components with frequencies within the (arbitrarily small) interval [f ; f + df]. Regarding the truncation, because with T' ^ to also the energy goes to infinity, one defines the power, n|GT/(f)2|/T'. Because the previous formulas in this section apply to a time series rather than a stochastic process, one uses the expectation operator to define h(f)= lim {£ [2n|GT(f)2|/T']} . (5.5)

The function h(f) is called one-sided non-normalized power spectral density function of the process X(T), often denoted just as (non-normalized) spectrum. It is the average (over all realizations) of the contribution to the total power from components in X(T) with frequencies within the interval [f; f + df]. h(f) is defined for f > 0 and integrates to S2. A closely related function is g(f)= h(f) /S2, (5.6)

the one-sided normalized power spectral density function, which integrates to unity. A two-sided version of the spectrum, symmetric about f = 0, is also used (Bendat and Piersol 1986).

The functions h(f) and g(f) are the Fourier transforms of the auto-covariance and autocorrelation functions, r(t) and p(t), respectively, provided they exist (Priestley 1981: Section 4.8 therein):

and the symbol t is used to denote a lag in continuous time. The caveat refers to the fact that not all processes X(T) have a spectral representation; however, the existence of the Fourier transform of the autocovari-ance function r(t) of X(T) is a sufficient condition.

Turning to the discrete-time version of the climate process, X(i), we assume also here absent trend, absent outliers and constant variability and find

The spectral theory is in this case similar to the continuous-time case (Priestley 1981: Section 4.8.3 therein), except that the frequency range is now restricted in both directions and the discrete Fourier transform is invoked to calculate the power spectral density functions. For example, with even time spacing, d(i) = d > 0, o g(f) = (d/n) £ p(1)e-2nif1 dl, 0 < f < 1/(2d). (5.12)

Herein, l denotes a lag in discrete time. The frequency /Ny = (2d)-1 is denoted as Nyquist frequency; it sets the upper frequency bound.

5.1.1 Example: AR(1) process, discrete time

Consider the discrete-time AR(1) process (Section 2.1.1) with an autocorrelation parameter a on an evenly spaced timescale, d(i) = d > 0, with n = to points. Then (Priestley 1981: Section 4.10 therein), g(/) = 2d(1 - a2) /[1 - 2acos(2n/d) + a2] , 0 < / < 1/(2d).

Plots of the AR(1) spectrum (Fig. 5.1) show higher power at lower frequencies for a > 0; such a spectrum is, hence, called "red."

Spectrum,