A quick look back at “A 20th century acceleration in global sea-level rise” (2006)

Let’s take a few minutes to look at the sea level rise projections of Church and White (Church, J. A., and N. J. White (2006), A 20th century acceleration in global sea-level rise, Geophys. Res. Lett., 33).  This dovetails with my recent posts about Vermeer’s and Rahmsorf’s 2009 attempt to predict 21^st century sea level rise.  Vermeer and Rahmstorf use Church’s and White’s sea level data, with an “artificial reservoir” correction by Chao, to construct their model relating sea level rise rate to temperature.  So it worth a quick look back Church’s and White’s paper and data before considering Chao’s reservoir correction.

Church’s and White’s paper is entirely centered around the idea of fitting sea level vs. time data form 1870 to 2001 to a quadratic.  This is important to note from the start because their fit coefficients are just plain wrong.

Their abstact …

Multi-century sea-level records and climate models indicate an acceleration of sea-level rise, but no 20th century acceleration has previously been detected. A reconstruction of global sea level using tide-gauge data from 1950 to 2000 indicates a larger rate of rise after 1993 and other periods of rapid sea-level rise but no significant acceleration over this period. Here, we extend the reconstruction of global mean sea level back to 1870 and find a sea-level rise from January 1870 to December 2004 of 195 mm, a 20th century rate of sea-level rise of 1.7 ±0.3 mm yr^-1 and a significant acceleration of sea-level rise of 0.013 ± 0.006 mm yr^-2. This acceleration is an important confirmation of climate change simulations which show an acceleration not previously observed. If this acceleration remained constant then the 1990 to 2100 rise would range from 280 to 340 mm, consistent with projections in the IPCC TAR.

I am going to talk about fitting the sea level vs. time to a quadratic. It is highly unlikely that nature follows such a simple formula, but I hope you can excuse my discourse by understanding that Church and White thought it was a good idea.  Here is their graphical presentation of the sea level data…

The middle plot shows the 1870 to 2001 sea level data fit to a quadratic.  That is, if the sea level is H, then…

Church’s and White’s coefficients

Strangely, Church and White do not explicitly give the coefficients a₂, a₁, and a₀.  However, they say two things about the formula: First, “Fitting a quadratic to the GMSL [global mean sea level] time series gives an acceleration (twice the quadratic coefficient) of 0.013” and second,  “The quadratic implies that the rate of rise was zero in about 1820.” Since the sea level is a relative measurement, we can set a₀ to anything we like.  So, from their two statements we have…

and

So, a₁ = -23.66, and therefore according to Church and White…

They say “The differences between the quadratic and the GMSL time series have an rms value of only 7.5 mm,” implying a good fit which yields a 28 to 34 cm sea level rise in the 21^st century

The correct coefficients

I hate to nit-pic, but when I downloaded the data that Church and White used from the Permanent Service for Mean Sea Level (PMSL) and fit it to a quadratic, I found the best fit actually gives an acceleration of 0.016 mm yr^-2 with an rms of 8.2 mm. (I have checked the best fit quadratic with two software packages, Excel and LabView, and have used a variety of different fit algorithms, including SVD, Givens, Householder and Cholesky.   Each method gives the same result:

I double checked this point by digitizing the sea level from the image from their paper (shown above), and fitting it to a quadratic.  Again, the same result: an acceleration of about 0.016 mm yr^-2 (about 2 x 0.0079).

Also, Church’s and White’s coefficients (a₂ = 0.0065 and a₁ = -23.66) result in an rms closer to 9.2 than their claimed 7.5.  This confirms that my coefficients (a₂ = 0.0079 and a₁ = -29.18) give a better fit to the data (rms = 8.2 mm) than theirs.

So What?

What difference does it make?  The difference in the sea level rise during the 21st century that results from choosing their coefficients vs. mine are negligible.   But, if they fumbled this central (to their paper), but simple step of getting these coefficients right, why would you put much faith in the rest of the paper.

But more fundamentally, fitting this data to a quadratic, and using that fit to extrapolate another 100 years is preposterous.  Why not fit to a third order polynomial, or a fourth order, or an exponential?

Let’s consider some other data fits

Look at the image from Church’s and White’s paper, above.  Do you notice a change in the character of the sea level rise somewhere between 1920 and 1930?  Church and White acknowledged this “clear change of slope at ~1930.”  They suggest…

“Another approach…is to do linear regressions on the two halves (1870–1935 and 1936–2001) of the record. The slopes are 0.71 ± 0.40 and 1.84 ± 0.19 mm yr_-1 respectively, implying an acceleration of 0.017 ± 0.007 mm yr_-2 (95%).”

Why would they choose 1935 as their break point?  Look at their data again – isn’t 1925 a better break point?  They seem to like using a quadratic for the 1870 to 2001 data .  Why not a quadratic for the 1925 to 2001 data?  Here’s what happens when you fit their sea level data from 1925 to 2001 to a quadratic…

Using the logic of extrapolating a quadratic fit out 1oo years (the same thing Church and White do when they fit to the 1870 to 2001 data), then the total sea level rise in the 21^st century will be a mere 13.3 cm.  Do I endorse such an estimate? No, I do not.  I am only using it to point out the danger of fitting to a quadratic and extrapolating far into the future.

Church’s and White’s 2009 update

In 2009 Church and White provided the PMSL with sea level data updated to include 2007.  The big change was not the added six years of data though.  The real difference is that the data is very different prior to about 1925…

Just for the fun of it, here are quadratic fits for the updated data from 1870 to 2007 and 1925 to 2007.

The 1870 to 2007 data give a positive acceleration that is considerably smaller than the acceleration reported in their paper,  resulting in 29 cm of sea level rise in the 21^st century.  The 1925 to 2007 data gives a negative acceleration and a mere 6 cm sea level rise in the 21^st century.  Again, I do not endorse either one of these extrapolations.  I am just pointing out the danger of using them.

Conclusion

Church’s and White’s effort at extracting a global sea level from multiple records was a useful endeavor, and I have no reason to argue with it (so far), but trying to extrapolate a fit to that data a century into the future is mathematically very hazardous.

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