Time and Tide

A new paper (Foster and Brown 2014, Time and Tide: Analysis of Sea Level Time Series, Climate Dynamics, doi:10.1007/s00382-014-2224-3) looks at how some authors have analyzed sea level data, and how they’ve gone wrong.

There has been a spate of papers in the last several years analyzing sea level data (including tide gauge records) to look for acceleration, or its absence, or deceleration, or changes in acceleration … all in the vain hope that if one can show that sea level didn’t accelerate consistently throughout the 20th century then there’s no reason to believe it will accelerate in the 21st century (an idea which reveals complete ignorance of how qualified scientists estimate future sea level rise). Most such papers don’t say so explicitly, but the implication is dripping from their writings. And at least one of them did say so explicity — our old friend Dave Burton

“Since the rate of sea level rise has not increased significantly in response to the last 3/4 century of CO2 emissions, there is no reason to expect that it will do so in response to the next 3/4 century of CO2 emissions. The best prediction for sea level in the future is simply a linear projection of the history of sea level at the same location in the past, or about 7-8 inches by 2080, for Long Island.”

Besides seeking to dispute future sea level rise based on utter nonsense, there are other commonalities among these papers. When calculating regression on sea level data (to estimate trends, accelerations, and their uncertainties) almost all of them ignore the issue of autocorrelation. To make matters worse, most of them use monthly data rather than annual averages, but the autocorrelation in monthly data is far stronger than in annual data so that makes the problem worse. Perhaps silliest of all, most of them ignore the annual (seasonal) cycle in tide gauge data, which should be removed (or accounted for in some other way) for best results.

But there’s one such paper which did not ignore the issue of autocorrelation in tide gauge data, or the presence of an annual cycle. Nicola Scafetta actually did raise these issues. He just got it wrong.

Here’s what he had to say about the presence of autocorrelation in tide gauge data:

Records with a monthly resolution are adopted because the statistical error for the regression coeffcients is normally smaller than using the annual resolution records due to the fact that the monthly data are not Gaussian-distributed around an average trend but are autocorrelated because constrained by the annual seasonal cycle…”

In my opinion, this is absurd nonsense, wrong on just about every point.

First of all, the fact that the monthly data are not necessarily Guassian-distributed around an average trend, is pretty much irrelevant. At least, according to the Gauss-Markov theorem.

Second, the data are autocorrelated even if the annual cycle is removed. The seasonal cycle just adds to the autocorrelation that’s already there. Which is a lot.

But most important, autocorrelation does not make the statistical errors for regression smaller. It just looks like it because Scafetta failed to compensate for autocorrelation. Instead he takes his uncertainty estimates straight from standard statistical software, which computes them by assuming a white-noise model. Which ain’t right. For the data studied by Scafetta, it ain’t even close The real effect of autocorrelation is not to make the statistical error for the regression smaller, it’s to make Scafetta’s reported values, based on a white-noise model, just plain wrong.

Now for the real kicker: Scafetta even goes out of his way to compare the standard errors based on different treatments of the analysis (using monthly vs annual averages, including an annual cycle in the regression or not), then chooses the approach that gives the smallest standard errors. But those “smallest standard errors” are just plain wrong. The only reason they’re the most small is that they’re the most wrong. Yes, Nicola Scafetta went out of his way to choose the analysis approach which would give him the most wrong answers.

The data really do show strong autocorrelation apart from that imposed by the seasonal cycle. Consider e.g. the monthly tide gauge data from Fremantle, Australia. Using a quadratic polynomial of time to estimate the long-term trend and a 2nd-order Fourier series to model the seasonal cycle, the residuals are still strongly autocorrelated.


Using an AR(1) model for the noise suggests that the variance of a regression estimate is slightly more than twice the “naive” estimate (i.e. the white-noise estimate, which is what Scafetta used). Using a more realistic noise model like ARMA(1,1) suggests that the variance is slightly more than three times the naive estimate.

Considering the vast chasm between the actual uncertainties and those which Scafetta reports, it’s no surprise that he finds the overall results to be “highly ambiguous” — unless of course you explain things his way.

As for the annual cycle, Scafetta at least tries. When fitting straight lines (to estimate rates) or quadratic polynomials (to estimate acceleration) he simultaneously fits a sinusoid with period 1 year to account for the annual cycle. Too bad he didn’t actually study the annual cycle. Here it is for the tide gauge data from New York (a “folded plot” of the residuals after subtracting a lowess smooth, showing two full cycles just for extra context):


It’s visually evident that the annual cycle is not just a sinusoid. This is further confirmed by a Fourier power spectrum:


Clearly we need at least a 2nd-order Fourier series to capture the demonstrable signal power in the seasonal cycle.

In one sense, the annual-cycle issue isn’t that big a deal because a sinusoid does capture most of the signal power in the annual cycle. But in another sense, it is a big deal because it demonstrates that Scafetta didn’t actually study this, he just threw a sinusoid at it and figured that was good enough.

The autocorrelation issue is a big deal, any way you look at it. In fact it’s a huge deal. Many others might be faulted for ignoring it, but at least we recognize that a lot of people (including scientists) aren’t aware of it and don’t know how to compensate. But when you yourself raise the issue, then get it wrong, I start to think …

Incidentally, most of Scafetta’s paper is about his application of something he calls “multi-scale analysis” to demonstrate that there are oscillatory patterns in tide gauge records. Lots of them. At decadal time scales, at bi-decadal time scales, and a 60-70 year quasi-periodic oscillation which he seems to think is ubiquitous (and of course related to ocean oscillations). Well, I’d venture to say he got that stuff wrong too.

But that is a topic for another day.

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14 responses to “Time and Tide

  1. This is quite wonderful. I wonder if you would consider following it up with an analysis of _is_ likely to be future sea level rise? Or at least perhaps discuss the use of probability statement used by those who do make such projections?

  2. David B. Benson

    Good. Are you prepared to make a 10 year projection?

  3. Horatio Algeranon

    You will no longer be welcome at Harry Hoodwinki’s shows.

    “Harry Hoodwinki”
    — by Horatio Algeranon

    Harry Hoodwinki did a trick
    With a crooked magic stick
    When the secret was revealed
    Harry’s fate was henceforth sealed

  4. Nice, another example of Scafetta’s hammer. It gets a bit boring, no?

    Not that there aren’t any others who use a similar hammer – I am sure Tamino will enjoy Richard Telford’s blog entry on similar cyclical fetishism in the solar system:


  5. How did papers with these mistakes get through peer review? They aren’t denial journals like Energy & Environment.

    [Response: Peer review is necessary, but not sufficient, for quality science. Also, the journal in which Scafetta published his article wasn't the Journal of the Royal Statistical Society -- it was Climate Dynamics, and it's to their credit that they also published a paper highly critical of what they had published before.]

    • Why publish something mistaken?

      First, journals do that all the time. That is how “controversies” get worked out.

      Second, while there is really no algorithm for any regression analysis, time series analysis has far less of an algorithmic nature than OLS. The range for judgement variances and errors is far greater.

      Then there are reviewer problems.

      I have decades of regression stats experience, but time series I approach with a lot of caution.

  6. Horatio Algeranon

    — by Horatio Algeranon

    Makes some people blind.
    Yesterday’s ideology
    Begets tomorrow’s mind.

  7. Aaron Lewis

    El Nino can push California sea levels up (or down) by a foot – that is just pushing water around – not raising or dropping global sea level. When the water goes up (or down) in California, it is going to go down (or up) somewhere else.

    Thus, a hundred mm of “noise” is inserted into SL data on the 4 to 7 year El Nino period, and in super El Nino years, (1972, 1983, 1997) a couple of hundred mm of noise is inserted into tide gauge data. And, each of the super El Ninos resulted in larger California SL changes than the last.

    The increasing magnitude of the super El ninos suggests that the energy of the system is increasing and increasing the noise in tide gauge data.

    [Response: I haven't noticed an increase in the noise level of tide gauge data. That doesn't mean it isn't there -- but I haven't noticed it. As for super el Nino events, they're pretty rare so it's hard to define robust trends.]

  8. Isn’t part of the problem with scientists such as Scafetta is that they are looking for luni-solar influences at the multi-decadal level before looking at what evidence exists at much shorter time frames?

    In other words they want to solve the longer-range trends of global warming in terms of natural variability without going through a detailed analysis of what can happen at monthly and yearly time-frames.

  9. How do such errors pass peer review? Are there no reviewers who understand statistics?

    • Only about 104% of reviewers understand all 16% of statistics … give or take a towelled republican.

      [Response: LOL]

  10. Lol, Scafetta sounds like classic denier nonsense where they are seeking for multi-decadal trends that is supposedly going to explain global warming. Of course they invent these statistical “natural cycles” without stating anywhere that they don’t have a physical explanation for what makes the “cycle” tick. Similarly Scafetta doesn’t have any for his as well – and certainly not since a large part of sea level rise in the future will certainly come from melting land ice – which can be both observed, measured and modelled pretty well – although there are still questions about the rate of melt which if anything is severely underestimated in IPCC AR5. What I find odd is that many of these so called “scientists” like Scafetta often eludes discussing e.g. the rate of melting of land ice or take in the fact of thermal expansion from global warming which is established science today. A classic denier trait.

  11. The NS quote about standard errors should have been a red flag if his reviewers knew statistics. So I’m guessing they didn’t, but the journal brought in sharper folks to read yours?

    Nice work and confirmation again that this blog makes a unique, valuable contribution.

  12. Not squarely on topic, but relevant (and, I hope, amusing): a compendium of denialist argument (or at least rhetoric), versified and set in ‘classic blues rock’ style:

    It’s a basic demo; I hope to perform this at the Atlanta rally in support of the new EPA clean air regs.