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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.

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Notes

  1. With the scaling we’ve chosen a periodogram value is actually Chi-square per degree of freedom.

  2. The lowess smooth uses a fixed number of nearby data points rather than a fixed time width to define its window, but for data which are evenly sampled in time the two strategies are equivalent.

  3. This is not a violation of our precaution against analyzing smoothed data, the velocity estimate is the linear coefficient of the local polynomial regression used to create the smooth.

References

  • Bindoff NL, Willebrand J, Artale V, Cazenave A, Gregory JM, Gulev S, Hanawa K, Le Quere C, Levitus S, Nojiri Y, Shum CK, Talley LD, Unnikrishnan AS (2007) Observations: oceanic climate change and sea level, climate change 2007: the physical science basis. In: Solomon D, Qin D, Manning M, Marquis M, Averyt KB, Tignor M, Miller HL, Chen Z (eds) Contribution of working group 1 to the fourth assessment report of the intergovernmental panel on climate change, Cambridge University Press, Cambridge, UK and NY, USA

  • Boon JD (2012) Evidence of sea level acceleration at U.S. and Canadian Tide Stations, Atlantic Coast, North America. J Coast Res 28(6):1437–1445. doi:10.2112/JCOASTRES-D-12-00102.1

  • Boretti A (2012) Is there any support in the long term tide gauge data to the claims that parts of Sydney will be swamped by rising sea levels? Coast Eng 64:161–167. doi:10.1016/j.coastaleng.2012.01.006

  • Boretti A, Watson T (2012) The inconvenient truth: ocean level not rising in Australia. Energy Environ 23(5):801–818

  • Burton DA (2012) Comments on Assessing future risk: quantifying the effects of sea level rise on storm surge risk for the southern shores of Long Island, New York, by Christine C. Shepard, Vera N. Agostini, Ben Gilmer, Tashya Allen, Jeff Stone, William Brooks and Michael W. Beck. Nat Hazards 63(2):1219–1221

    Article  Google Scholar 

  • Calafat FM, Chambers DP (2013) Quantifying recent acceleration in sea level unrelated to internal climate variability. Geophys Res Lett 40:3661–3666. doi:10.1002/grl.50731

    Article  Google Scholar 

  • Chambers DP, Merrifield MA, Nerem RS (2012) Is there a 60-year oscillation in global mean sea level? Geophys Res Lett 39:L18607. doi:10.1029/2012GL052885

    Google Scholar 

  • Cleveland WS (1979) Robust locally weighted regression and smoothing scatterplots. J Am Stat Assoc 74(368):829–836

    Article  Google Scholar 

  • Church JA, White NJ (2006) A 20th century acceleration in global sea-level rise. Geophys Res Lett 33:L01602. doi:10.1029/2005GL024826

    Google Scholar 

  • Church J, White N (2011) Sea-level rise from the late 19th to the early 21st century. Surv Geophys 32(4–5):585–602

    Article  Google Scholar 

  • Church JA, Clark PU, Cazenave A, Gregory JM, Jevrejeva S, Levermann A, Merrifield MA, Milne GA, Nerem RS, Nunn PD, Payne AJ, Pfeffer WT, Stammer D, Unnikrishnan AS (2013) Sea level change. In: Stocker TF, Qin D, Plattner G-K, Tignor M, Allen SK, Boschung J, Nauels A, Xia Y, Bex V, Midgley PM (eds) Climate change 2013: the physical science basis. Contribution of working group I to the fifth assessment report of the intergovernmental panel on climate change. Cambridge University Press, Cambridge, UK and NewYork, USA

  • Dean RG, Houston JR (2013) Recent sea level trends and accelerations: comparison of tide gauge and satellite results. Coast Eng 75:4–9. doi:10.1016/j.coastaleng.2013.01.001

  • Douglas BC (1992) Global sea level acceleration. J Geophys Res 97(C8):12699–12706. doi:10.1029/92JC01133

    Article  Google Scholar 

  • Ekman M, Stigebrandt A (1990) Secular change of the seasonal variation in sea level and of the pole tide in the Baltic Sea. J Geophys Res 95(C4):5379–5383. doi:10.1029/JC095iC04p05379

    Article  Google Scholar 

  • Foster G, Rahmstorf S (2011) Global temperature evolution 1979–2010. Environ Res Lett 6(4):044022

    Article  Google Scholar 

  • Gross RS (2000) The excitation of the Chandler Wobble. J Geophys Res 27(15):2329–2332. doi:10.1029/2000GL011450

    Google Scholar 

  • Haubrick R Jr, Munk W (1959) The pole tide. J Geophys Res 64(12):2373–2388. doi:10.1029/JZ064i012p02373

    Article  Google Scholar 

  • Holgate SJ, Woodworth PL (2004) Evidence for enhanced coastal sea level rise during the 1990s. Geophys Res Lett 31:L07305. doi:10.1029/2004GL019626

    Google Scholar 

  • Houston JR, Dean RG (2011) Sea-level acceleration based on U.S. tide gauges and extensions of previous global-gauge analyses. J Coast Res 27(3):409–417

    Article  Google Scholar 

  • Hunter JR, Brown MJI (2013) Discussion of Boretti, A., ‘Is there any support in the long term tide gauge data to the claims that parts of Sydney will be swamped by rising sea levels?’, Coast Eng 64:161–167, June 2012. Coast Eng 75:1–3. doi:10.1016/j.coastaleng.2012.12.003

  • IPCC (2013) Climate change 2013: the physical science basis. Contribution of working group I to the fifth assessment report of the intergovernmental panel on climate change. Cambridge University Press, Cambridge

  • Jevrejeva S, Grinsted A, Moore JC, Holgate S (2006) Nonlinear trends and multiyear cycles in sea level records. J Geophys Res 111:C09012. doi:10.1029/2005JC003229

    Google Scholar 

  • Jevrejeva S, Moore JC, Grinsted A, Woodworth PL (2008) Recent global sea level acceleration started over 200 years ago? Geophys Res Lett 35:L08715. doi:10.1029/2008GL033611

    Article  Google Scholar 

  • Lee J, Lund R (2004) Revisiting simple linear regression with autocorrelated errors. Biometrika 91(1):240–245

    Article  Google Scholar 

  • Moore JC, Grinsted A, Jevrejeva S (2005) New tools for analyzing time series relationships and trends. Eos Trans AGU 86(24):226–232. doi:10.1029/2005EO240003

    Article  Google Scholar 

  • Parker A (2013a) Sea level trends at locations of the United States with more than 100 years of recording. Nat Hazards 65(1):1011–1021. doi:10.1007/s11069-012-0400-5

    Article  Google Scholar 

  • Parker A (2013b) Natural oscillations and trends in long-term tide gauge records from the Pacic. Pattern Recogn Phys 1:11–23. doi:10.5194/prp-1-11-2013

    Article  Google Scholar 

  • Percival DB, Walden AT (1993) Spectral analysis for physical applications. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • PSMSL (2013) Permanent service for mean sea level, data. http://www.psmsl.org/data/obtaining/

  • Rahmstorf S (2007) A semi-empirical approach to projecting future sea-level rise. Science 315(5810):368–370. doi:10.1126/science.1135456

    Article  Google Scholar 

  • Rahmstorf S, Perrette M, Vermeer M (2012) Testing the robustness of semi-empirical sea level projections. Clim Dyn 39:861–875. doi:10.1007/s00382-011-1226-7

    Article  Google Scholar 

  • Sallenger AH, Doran KS, Howd PA (2012) Hotspot of accelerated sea-level rise on the Atlantic coast of North America. Nat Clim Change 2:884–888. doi:10.1038/nclimate1597

    Article  Google Scholar 

  • Scafetta N (2013) Multi-scale dynamical analysis (MSDA) of sea level records versus PDO, AMO, and NAO indexes. Clim Dyn. doi:10.1007/s00382-013-1771-3

  • Schulz M, Mudelsee M (2002) REDFIT: estimating red-noise spectra directly from unevenly spaced paleoclimatic time series. Comput Geosci 28(3):421–426

    Article  Google Scholar 

  • Trupin A, Wahr J (1990) Spectroscopic analysis of global tide gauge sea level data. Geophys J Int 100:441–453. doi:10.1111/j.1365-246X.1990.tb00697.x

    Article  Google Scholar 

  • Woodworth PL (1990) A search for accelerations in records of European mean sea level. Int J Climatol 10:129–143. doi:10.1002/joc.3370100203

    Article  Google Scholar 

  • Woodworth PL, White NJ, Jevrejeva S, Holgate SJ, Church JA, Gehrels WR (2009) Evidence for the accelerations of sea level on multi-decade and century timescales. Int J Climatol 29:777–789. doi:10.1002/joc.1771

    Article  Google Scholar 

  • Woodworth PL, Player R (2003) The permanent service for mean sea level: an update to the 21st century. J Coast Res 19:287–295

    Google Scholar 

  • Zwiers FW, von Storch H (1995) Taking serial correlation into account in test of the mean. J Clim 8:336–351

    Article  Google Scholar 

Download references

Acknowledgments

We would like to acknowledge Dr. Alexander Glass for initiating the discussion that lead to this manuscript.

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Correspondence to Grant Foster.

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

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.

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Foster, G., Brown, P.T. Time and tide: analysis of sea level time series. Clim Dyn 45, 291–308 (2015). https://doi.org/10.1007/s00382-014-2224-3

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Keywords

  • Time series
  • Sea level
  • Climate oscillations