Rahmstorf (2009): Off the mark again (Part 12). A mathematical comedy

Here is one more post about the laughably bad PNAS “Global Sea level linked to global temperature” by Vermeer and Rahmstorf.  Will this  fount of absurdity never run dry?

Much has been said about Rahmstorf’s data smoothing techniques.  But the little gem you are about read may make your head spin.

Remember the Chao reservoir correction?  This was the correction that VR2009 applied to the Church and White sea level data to compensate for water that has been impounded in man-made reservoirs.  Never mind the fact that VR2009 paid lip service to, but did not include, a counter-correction for water that has been pumped from the aquifers and has artificially added to the sea level.  Let’s look at some details of how VR2009 handled this correction.

Here is something amazing…

VR2009 had the 2006 Church and White sea level data, which is rather noisy.  They also had the Chao reservoir correction data, which is also noisy.  They correctly saw the need to smooth the noisy data.  It seems that they could have done it one of  two ways: smooth each set separately, then  add the smoothed Chao data to the smoothed Church and White data, or add the unsmoothed Chao data to the unsmoothed Church and White data and then smooth the result.

When I reproduced VR2009’s basic algorithm, I choose the first method.  But VR2009 doubled up on smoothing the Choa reservoir correction.  They smoothed the Chao data, added it to the unsmoothed Church and White data, then smoothed the sum again.  So, the Chao data was effectively smoothed twice.

But here is the really amazing thing:  Look at the overlay of Chao’s data, VR2009’s smooth for the Chao data, and my smooth for the Chao data…

Figure 1. Chao correction to sea level rise rate with VR2009 smooth and Moriarty smooth

Wow! All I can say is “Wow!”  Can you believe how terrible the VR2009 fit for the additional sea level rise rate is?  It’s just amazingly bad! 

How did VR2009 come up with this bizarre data smooth?

In the Matlab program file that VR2009 uses to find the relationship between sea level and temperature (sealevel2.m, get copy here) they first import the unsmoothed Church and White data (church_13221.txt, get copy here) with the following code…

% load the church & white sea level data
load church_13221.txt;
seayear = church_13221(:,1);
sealevel = church_13221(:,2)/10;

Two arrays are created, one with the year, one with the sea level.  The “/10” in the last line of code converts the sea level data from mm to cm.

Then they apply their Chao reservoir correction.  Instead of importing a time series with the Chao data, they apply a function…

% Apply Chao et al (2008) reservoir correction:
if chao == ‘y’
     sealevel = sealevel + 1.65 + (3.7/3.1415)*atan2(seayear-1978,13);
end

So, VR2009 claims the term “1.65 + (3.7/3.1415)*atan2(seayear-1978,13)” is a representation of the Chao reservoir correction.  Figure 1, above shows the derivative of the Chao reservoir correction (which you can see as figure 3 in Chao’s Science paper).  So the derivative of VR2009’s Chao correction term should at least be close to the derivative provided in Chao’s paper.  Alas, instead it looks like the blue peak in figure 1, above. 

How did VR2009 come up with this strange correction that “fits” the Chao reservoir correction to an inverse tangent (atan2) function?  VR2009 claims to use sophisticated single spectrum analysis (SSA) to smooth its sea level and temperature data.  But their SSA code yields a numerical result, not an analytic one (that is, a time series of numbers, not a formula).  So SSA was NOT used to generate VR2009’s Chao correction term.

If you use my smooth of the Chao data as a baseline, then the VR2009 fit is about 0.2 mm too low around 1960 and about 0.3 mm too high by 1980.  By using their fit to the Chao reservoir sea level rise rate correction, they have effectively increased the sea level rise rate from 1960 to 1980 by an additional 0.5 mm per year.  They have pushed the Chao sea leve rise rate correction to later in the century which, of course, fits their general theme.

The following plot shows the 2006 Church and White sea level data with the questionable VR2009 version of the  Chao reservoir correction data and my version of the Chao reservoir correction.  At first they do not look much different.  But consider this: The VR2009 version causes the average sea level rise rate from 1950 to 1970 to be 1.66 mm/year, and for 1970 to 1990 to be  1.99 mm/year.  That’s a 16% increase.  If my version is used there is an average DECREASE in sea level rise rate, from 1.87 mm/year to 1.78 mm/year.  That is a 5% drop.  Look at figure 1, above, and ask yourself “Whose smooth of the Chao data is better?”

I will not attempt to assign motivation for this laughably bad smooth of the Chao reservoir correction data.  Suffice it to say that it is just one more in long series of blunders and bizarre consequences for VR2009.

Read more about the comedy known as the PNAS “Global Sea level linked to global temperature” by Vermeer and Rahmstorf.

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