Time and tide: analysis of sea level time series

Abstract

A number of recent papers have examined sea level data, both local tide gauge records and regional/global averages, to estimate not only how fast sea level is rising but how the rate has changed over time, i.e. its pattern of acceleration and deceleration. In addition, a number of claims of cyclic/quasi-periodic variations have been proposed. However, many of these papers contain technical problems which call their results into question. In particular, the issue of autocorrelation is often ignored, and even when it is addressed its impact has sometimes been misinterpreted. Autocorrelation does more than just affect the standard errors of regression analysis, it can also make the spectrum of a noise process distinctly “red” and therefore be highly suggestive of low-frequency periodic or pseudo-periodic behavior when none is present. If any analysis is applied which acts as a band-pass filter, it can further exaggerate the illusion of oscillatory behavior. These issues are highlighted in a small number of recent papers, in order to improve the quality of future work on this subject.

Appendix: Spectral response of ARMA(1,1) noise

Let $x_\alpha$ be the realization of an ARMA(1,1) process, so that

$$\langle x_{\alpha } x_{\beta } \rangle = \left\{ {\begin{array}{*{20}l} {\sigma ^{2} } \hfill & {\alpha = \beta } \hfill \\ {\sigma ^{2} \lambda \rho ^{{|\beta - \alpha |}} } \hfill & {\alpha \ne \beta } \hfill \\ \end{array} } \right.$$

(41)

In the case $\lambda =1$ we have an AR(1) process.

We define the discrete Fourier transform of $x_\alpha$ at frequency $f$ as

$$\begin{aligned} X(f) = N^{-1} \sum _{\alpha =1}^N x_\alpha e^{-i 2\pi f t_\alpha }, \end{aligned}$$

(42)

and we define the periodogram as

$$\begin{aligned} {\mathcal {X}}(f)&= {N \over \sigma ^2} | X(f) |^2\nonumber \\&= {1 \over N \sigma ^2 } \sum _{\alpha =1}^N \sum _{\beta =1}^N x_\alpha x_\beta e^{i 2\pi f (t_\beta - t_\alpha )}. \end{aligned}$$

(43)

When the time series is evenly sampled, $t_\beta - t_\alpha = (\beta - \alpha ) \tau$ where $\tau$ is the time spacing. Defining $\phi = 2\pi f \tau$ we have, we have in this case

$$\begin{aligned} {\mathcal {X}}(f) = {1 \over N \sigma ^2 } \sum _{\alpha =1}^N \sum _{\beta =1}^N x_\alpha x_\beta e^{i (\beta - \alpha ) \phi }. \end{aligned}$$

(44)

When we substitute Eq. (41) into Eq. (44) we get the expected value of the periodogram

$$\begin{aligned} \langle {\mathcal {X}} \rangle&= {1 \over N \sigma ^2} \sum _{\alpha =1}^N \sum _{\beta =1}^N \sigma ^2 \rho _{(\alpha -\beta )} e^{i (\beta -\alpha )\phi }\nonumber \\&= {1 \over N} \sum _{\alpha =1}^N \sum _{\beta =1}^N \rho _{(\alpha -\beta )} e^{i \phi (\beta -\alpha )}. \end{aligned}$$

(45)

We now separate the double sum into three parts: terms for which $\alpha = \beta$, those with $\alpha < \beta$, and those with $\alpha > \beta$. The terms with $\alpha = \beta$ all have $\rho _o = 1$, and there are $N$ such terms, so they contribute

$$\begin{aligned} \left( {1 \over N} \right) N = 1. \end{aligned}$$

(46)

We also note that $\rho _k = \lambda \rho ^k$, so we are motivated to define the complex number

$$\begin{aligned} z = \rho e^{i \phi } = \rho e^{i 2\pi f \tau }. \end{aligned}$$

(47)

so that the terms with $\alpha < \beta$ contribute

$$\begin{aligned} S = {\lambda \over N} \sum _{\alpha =1}^{N-1} \sum _{\beta =\alpha +1}^N z^{(\beta -\alpha )} = {\lambda \over N} \sum _{\alpha =1}^{N-1} \sum _{\beta =1}^{N-\alpha } z^\beta . \end{aligned}$$

(48)

Because $\rho _{-k} = \rho _k$ for all $k$, the terms with $\alpha < \beta$ are the complex conjugates of the terms with $\alpha > \beta$ and their sum is $\bar{S}$, the complex conjugate of $S$. Finally we can write the periodogram as

$$\begin{aligned} \langle {\mathcal {X}} \rangle = 1 + S + \bar{S} = 1 + {\lambda \over N} \sum _{\alpha =1}^{N-1} \sum _{\beta =1}^{N-\alpha } z^\beta + {\lambda \over N} \sum _{\alpha =1}^{N-1} \sum _{\beta =1}^{N-\alpha } \bar{z}^\beta , \end{aligned}$$

(49)

where a bar indicates the complex conjugate.

The sums in Eq. (49) are directly calculable, leading first to

$$\begin{aligned} S = {\lambda \over N} \sum _{\alpha =1}^{N-1} {z - z^{N-\alpha +1} \over 1-z} = {\lambda z \over N(1-z)} \sum _{\alpha =1}^{N-1} (1 - z^{N-\alpha }), \end{aligned}$$

(50)

and then directly to

$$\begin{aligned} S&= {\lambda z \over N(1-z)} \sum _{\alpha =1}^{N-1} (1 - z^\alpha ) = {\lambda z \over N(1-z)} \left[ N - 1 - {z - z^N \over 1-z} \right] \nonumber \\&= {\lambda z \over 1-z} \left[ 1 - {1 - z^N \over N(1-z)} \right] . \end{aligned}$$

(51)

We now note that as $N \rightarrow \infty$, $S \rightarrow \lambda z/(1-z)$ so the expected value of the periodogram goes to

$$\begin{aligned} \langle {\mathcal {X}} \rangle&= 1 + S + \bar{S} \rightarrow 1 + {\lambda z \over 1-z} + {\lambda \bar{z} \over 1 - \bar{z}}\nonumber \\&= 1 + \lambda \left( {z + \bar{z} - 2 z \bar{z} \over (1 - z) (1 - \bar{z})} \right) . \end{aligned}$$

(52)

Now we can substitute $z \bar{z} = \rho ^2$ and $z + \bar{z} = 2 \rho \cos \phi$ to get

$$\begin{aligned} \langle {\mathcal {X}} \rangle&= 1 + 2 \lambda \left( {\rho \cos \phi - \rho ^2 \over 1 + \rho ^2 - 2 \rho \cos \phi } \right) \nonumber \\&= 1 - \lambda + { \lambda (1 - \rho ^2) \over 1 + \rho ^2 - 2 \rho \cos \phi }. \end{aligned}$$

(53)

Finally, when $\lambda = 1$ we recover the result for an AR(1) process.