A semi-empirical approach to sea level rise

Published in Science, this Rahmstorf 2007 article provides a high-end estimate of sea level rise of over a meter by the end of the century (rate of 10mm/yr). Linear extrapolation puts the rate of increase at only 1.4mm and 1.7mm per year depending on start date (1860 or 1950).

The paper was followed by two critical comments, both bashing the statistics, and these are attached to the link above. Rahmstorf replied to those comments. The issues raised are familiar to readers of this, CA, Lucia, and other statistical blogs: significance, autocorrelation, etc. and worth a read.

Worthwhile as the comments are, they do not look into the problem of the end-treatment used by Rahmstorf, and I look at that here.

All of the papers projecting these high end rates, and they all depend on the assumption of recent ‘acceleration’ in sea levels. That is, seem to depend on the rate of increase getting faster and faster.

Rahmstorf 2007 paper uses the smoothing method most recently savaged at CA here, where it was shown despite all the high-falutin’ language to be equivalent to a simple triangular filter of length 2M, padded with M points of slope equal to the last M points. My main concern is that at this crucial end-section, the data has been duplicated by the padding, effectively increasing the number of data points of very high slope.

The figure below shows a replication of the Rahmstorf smoothing with and without padding (moved down for clarity) (code below). Two sea level data sets are shown, one by Church “A 20th century acceleration in global sea level rise” (used in Rahmstorf, data available from CSIRO here) another by Jevrejeva “Recent global sea level acceleration started over 200 years ago?” (data here)

It should be noted this data ends in 2001-2, a truncation bound to maximize recent temperature increases.

fig1

You can already see how much the padding adds to the impression of acceleration at the end. Also note that the simple triangular smooth is a very good replication to the Rahmstorf combobulation (lower fig 3 below).

rahm1

Having modeled the sea level rise, the next step in the replication is to calculate the rate of sea level increase, from the smoothed sea-level data (upper figure 3 above). The figure below shows my replication. The black line is a close as I can get, though it seems like Rahmstorf’s rate line rises even higher at the end. The red line shows that removing the padding effectively removes part of the increasing sea-level rise at the end.

fig2

The green and blue lines illustrate an undescribed part of the method. Figure 2 in Rahmstorf 2007 says that the rate curves are “time derivative of this sea-level curve”. In fact, it takes the calculation of a moving 15 year regression line to give a smooth rate curve. The green line is a 2 year regression, the time derivative, and it is still very bumpy.

rahm2

Which made me think, why not just apply a moving 30 year trend line to the raw sea level data without any other smoothing? The result is the blue line, which as you can see, is not too dissimilar in shape to the smoothed lines, although there is no uptick at the end.

He then goes on the bin the data into 5 year intervals to get the result in Fig2 – another ad-hoc smoothing technique. The Rahmstorf patented smoothing method so far has used singular spectrum analysis (SSA), convolution (filtering), and now linear regression and binning. Is there any form of smoothing this guy cannot use!

fig3

Above is my replication of Rahmstorf’s figure 2, with and without padding. The corrrelation between temperature (on the x axis) and rate of sea level rise (on the y axis) is the crux of the paper. The padded version is a fairly close replication. The unpadded version of course sees a number of points truncated at the end where the rate of sea level increase is highest.

I get a padded slope of 2.9 mm rise per degree C, and unpadded of 4.06 mm rise per degree C, both significant. Rahmstorf got 3.4 mm rise per degree C. He also claims that this figure is robust to changes in the smoothing from M=2 to M=17, and this appears to be the case from the plot below at M=2.

fig4

Looking at this data, when you take out the extension due to the padding, I see a cluster of points at about a rate of 2.0 mm of sea level rise, with a smaller cluster at a lower rate. The conclusion that the rate of sea level rise will increase as temperature increases (aka acceleration) depends very much on the position of the LOWER cluster. Relying on the middle cluster alone, there would be no correlation between rate of increase and temperature.

Based on these results, I would say that the padding is not contributing much of an artifact to the results. Need to look elsewhere for problems, and I have a few ideas.

Look here for non-turnkey code (but not hard to work out).

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0 thoughts on “A semi-empirical approach to sea level rise

  1. This stuff is a mess to read. Very poor explication. If you can’t even make your points clearly, how will you ever beat the bad guys?

  2. This stuff is a mess to read. Very poor explication. If you can't even make your points clearly, how will you ever beat the bad guys?

  3. David,
    Just a comment on your Fig 2. You’ve moved the Npad curves down for “clarity”. But it makes them look like very bad fits. Or more to the point, one can’t tell how well they fit. I presume they would actually coincide with the Pad curves without the shift.

    It isn’t clear what the PAD curves are. I presume that up to the end of the data shown, they are MRC or equivalent. But then they go beyond. What does this mean? Are they the padding points? If so, why not 15 years of them?

    But if you stop the PAD curves at the end of data, which is the fair thing to do, then they actually look like quite good smoothers.

    • Nick, I put in a 1.5M length pad, thats why they go beyond. They are good fits, just like the figure in the paper.

  4. David, Just a comment on your Fig 2. You've moved the Npad curves down for “clarity”. But it makes them look like very bad fits. Or more to the point, one can't tell how well they fit. I presume they would actually coincide with the Pad curves without the shift.It isn't clear what the PAD curves are. I presume that up to the end of the data shown, they are MRC or equivalent. But then they go beyond. What does this mean? Are they the padding points? If so, why not 15 years of them?But if you stop the PAD curves at the end of data, which is the fair thing to do, then they actually look like quite good smoothers.

  5. Nick, I put in a 1.5M length pad, thats why they go beyond. They are good fits, just like the figure in the paper.

  6. David–
    As a favor, could you do a “tweak” on R’s figure 2 for me?

    Instead of fitting
    dH/dt (t’)=aT(t’)

    Plot
    Fit
    dH/dt (t’)= a*T(t’) + b*H (t’) for each value of t’.

    Then show the graph using the best fit value of b. I guess I could down load the data, but I did a bit of scrawl, and I want to see what values you get. If the data aren’t too noisy, the ratio (-b/a) might turn out to be 3 C/30 mm (or not.)

    Thanks

    • I grabbed some of the data. (I read page 368 wrong. The ratio (-b/a should be about 3 C/30 meters. Heh. Which, for my first fit to post 1950 data, happens to work. Though I bet this is not “robust”.)

  7. David–As a favor, could you do a “tweak” on R's figure 2 for me? Instead of fitting dH/dt (t')=aT(t')Plot Fit dH/dt (t')= a*T(t') + b*H (t') for each value of t'. Then show the graph using the best fit value of b. I guess I could down load the data, but I did a bit of scrawl, and I want to see what values you get. If the data aren't too noisy, the ratio (-b/a) might turn out to be 3 C/30 mm (or not.)Thanks

  8. I grabbed some of the data. (I read page 368 wrong. The ratio (-b/a should be about 3 C/30 meters. Heh. Which, for my first fit to post 1950 data, happens to work. Though I bet this is not “robust”.)

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  10. It isn’t clear what the PAD curves are. I presume that up to the end of
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