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

April 24, 2010Let’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 a_{2}, a_{1}, and a_{0}. 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 a_{0} to anything we like. So, from their two statements we have…

and

So, a_{1} = -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 (a_{2} = 0.0065 and a_{1} = -23.66) result in an rms closer to 9.2 than their claimed 7.5. This confirms that my coefficients (a_{2} = 0.0079 and a_{1} = -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.

Tom, Great set of articles. I found a number of other errors in these SL papers and have written a couple of drafts, eg.

“That is, the higher levels of SL rise are supported by errors of logic; even the detrended series show a strong and significant correlation of temperature with r = 0.7, and no correlation of rate of rise with acceleration. In addition we find a number of inconsistencies in \cite{Jevrejeva:2007}. The acceleration coefficient is quoted as 0.01mm/year, but the coefficient is actually 0.004612 or 0.009224 (when doubled) using their data. Verification is complicated by lack of quoting of measures of uncertainties, coefficients for their equations, or anything. Its not clear why they rounded up when by rights, when it is conventional to round down to 0.009, and this would have biased their results upwards. We find another inconsistency in the estimates of sea level rise from the quadratic equation of 26.8cm (19cm for linear), with the stated results in Jevrejava of 34cm of sea level rise during the 21st century, an exaggeration of +26\%.”

I really doubt any of what Ive done would get published though, as statistical extrapolation is sufficient justification for projections of high SL rise, but errors and statistical extrapolation are not enough to refute them it seems.

by David Stockwell April 24, 2010 at 2:04 pmHi Tom,

Thanks for bringing sanity to the sea level “crisis.”

BTW, which software do you use to digitize data from an image?

“I double checked this point by digitizing the sea level from the image from their paper (shown above)”

by C3 Editor April 25, 2010 at 7:02 amI use code that I wrote in LabView version 7.1. Here is a screen shot…

Best Regards,

by tommoriarty April 25, 2010 at 3:38 pmTom

A few months ago I was trying to get NASA to clean up some bogus graphs at climate.nasa.gov/keyindicators and graphed the Church and White (original) data. They showed data from 1870 to 2000, so that was the period I used. A couple of months ago, NASA had the trend as 1.7mm from 1870-2000 — obviously bogus.

The best fit was a two line segment with a breakpoint at 1930. Since the trendlines are not right on top of the data near the ends, the transition point between the two trendlines isn’t the same as the end point of the regression period.

All of the rms errors are for annual data — I also plotted and calculated some monthly data and as expected it had slightly higher RMS errors.

The rms error of the trendlines are as follows:

1.7mm/yr) actually 1.67mm regression) ……….. 9.19mm rms error 1901-2000 (the regression period)

1.7mm/yr (actually 1.67mm regression) ……….. 17.7 mm rms error 1870-2000 (the period of the chart)

1.4mm/yr (actually 1.405mm/yr regression) ……. 12.4 mm rms error 1870-2000 (the period of the chart)

0.75mm/yr ….. 5.8mm rms error, 1870-1929

1.9mm/yr …….. 6.0 mm rms error, 1930-2000

2 line segment approximation, 0.75mm/yr & 1.9mm/yr, breakpoint at 1930 …5.8mm rms from 1870-2000

As can be easily seen in either the graphs or the above numerical results on quality of fit, the 1.7mm line is a poor fit to the data.

by Charlie A April 25, 2010 at 3:24 pmhttp://www.webcitation.org/5nmqTukFs is what NASA had for a sea level graph in December 2009. It was a correction to an even more bogus graph they had up when I first looked at it in August 2009. (In the upper right corner of the webcite page linked above, you can see a few other variants of the webpage that NASA has generated).

Nowdays, NASA has a graph of the updated Church and White data, with a link to the older dataset, and a note that the 1.7mm trendline is the estimate for the 20th century.

I don’t think it is some sort of AGW conspiracy. The NASA / Jet Propulsion Labs groups that writes the webpages just seems to be incompetent.

On a related page, http://climate.nasa.gov/evidence/ NASA says “Global sea level rose about 17 centimeters (6.7 inches) in the last century. The rate in the last decade, however, is nearly double that of the last century.4” — strangely, the footnote refers back to Church and White 2006, which made no such statement.

The older version of the NASA evidence page referred back to AR4 summary, which explicitly said that it was unknown whether the recent increase in GMSL rate-of-rise was evidence of warming or simply natural decadal variation. I guess NASA way of correcting that error is to simply point change the footnote to something unrelated to the statement. 🙂

by Charlie A April 25, 2010 at 3:38 pmI just stumbled across this page; sorry I’m almost a year late jumping in here!

I also looked at Church & White’s new and old data, as you did, and the coefficients which I found were also slightly different from theirs. But I asked them about it, and Dr. Church replied with some clarifications.

One clarification was that their quadratic & linear regression analyses fit minimum-variance unbiased estimators, which had the effect of weighting the older data a less since it had larger error bars. I did that with their 2009 data, and fit a quadratic starting with 1900; this was the result:

http://www.burtonsys.com/climate/c-w_1900-2007_weighted2.bmp

Note the negative quadratic term, indicating a slight deceleration in rate of sea level rise over the last 107 years (since 1900).

Because their newer data lowers 19th & early 20th century sea levels, the linear fit with their newer data yields a slightly higher rate of sea level rise: 1.92 mm/year.

However, I don’t think Church & White’s linear rates of sea level rise are meaningful, because in their 2006 paper they admitted adding an adjustment (I call it a fudge factor!) to increase the reported rate of global mean sea level rise! Here’s the remarkable admission quoted from their paper:

http://awurl.com/TCRPnD0dR#first_awesome_highlight

“An additional spatially uniform field is included in the reconstruction to represent changes in GMSL. Omitting this field results in a much smaller rate of GMSL rise…”That, along with GIA (which probably added 0.3 – 0.5 mm/yr to the rate of sea level rise), is apparently why their reported rates of sea level rise are so high.

A simple average of the sea level trends measured by the 159 GLOSS-LTT tide gauges around the world (which is the very best data we have on coastal sea levels) yields an average rate of sea level rise of only about 0.6 mm/year. (Note: 1/4 of the GLOSS-LTT coastal tide gauges show sea levels falling, rather than rising!) More sophisticated averaging, which takes into account the uneven geographical distribution of the tide gauges, yields a global average mean sea level rise of just over 1.1 mm/year:

http://www.burtonsys.com/climate/global_msl_trend_analysis.html#bydistance

But I’m not even sure Church & White’s data is meaningful for acceleration/decceleration calculations! Read that quote again:

“An additional spatially uniform field is included in the reconstruction to represent changes in GMSL. Omitting this field results in a much smaller rate of GMSL rise…”Does the word “spatially” seem odd there? It did to me, so I asked Church & White why they used the adjective “spatially.”

Surely, I assumed, since they were reporting acceleration trends, the “additional field” must at least have been temporally uniform. Wrong! I’ve yet to figure out what that “field” is, but Dr. Church told me that it was NOT temporally uniform!

by Dave Burton April 5, 2011 at 11:55 pmDave,

thanks for the very interesting comments. I will take a close look at your links when time allows.

Best Regards,

by tommoriarty April 6, 2011 at 10:42 amTom Moriarty

ClimateSanity

See also: http://tinyurl.com/nhazburt1

There, in addition to the paper, you’ll also find a download link for all the code that I used to do my calculations.

Dave

by daveburton February 18, 2013 at 4:21 pmhttp://www.sealevel.info

[…] There is only one reason to fit sea level vs. time data to a quadratic: to highlight an acceleration trend. It only makes sense to do so if you think that the trend is more or less uniform over time. I have warned against reading too much into a quadratic fit, and especially against using a quadratic fit to imply a future trend in sea level. […]

by Rahmstorf (2011): Robust or Just Busted (Part 2) – Quadratic Fits of Laughter « Climate Sanity July 6, 2012 at 8:34 pm