… if your data do not look like a quadratic!
This is a post about global sea-level rise, but I put that message up front so that you’ve got it even if you don’t read any further.
Fitting a quadratic to test for change in the rate of sea-level rise is a fool’s errand.
I’d like to explain why, with the help of a simple example. Imagine your rate of sea-level rise changes over 100 years in the following way:
Fig. 1. Rate of sea-level rise as it changes over 100 years. This is a fictitious example chosen for illustrative purposes. It’s simply a polynomial curve, see appended matlab script.
It starts at 1 mm/year, then in the middle of the century it hovers for a while around 2 mm/year (namely between 1.8 and 2.2), and in the end it climbs to 3 mm/year. There can be no question about the fact that the rate of sea-level rise increases overall during those 100 years. It increases by a factor of three, from one to three millimetres per year, although not at a steady rate. You could fit a linear trend to the above curve and also find an increase in the rate – although a linear trend would not be a great description of what is going on, because the increase in rate is clearly not linear. (Note that a linear increase in the rate corresponds to a quadratic sea-level curve.)
You can easily compute the sea-level curve that follows from the above rate by integrating it over time, and it looks like this:
Fig. 2. This sea level curve is the integral of the curve in Fig. 1 and thus contains the same information, but when viewed in this way it is hard to judge by eye whether sea-level rise has accelerated. The better way to answer this question is by looking at the rate curve, i.e. Fig. 1.
Now here it comes: if you fit a quadratic (by the standard least-squares method) to this sea-level curve, the quadratic term (i.e. the acceleration) is negative! So by this diagnostic, sea level rise supposedly has decelerated, i.e. the rate of rise has slowed down! This clearly is nonsense and misleading (we know the truth in this case, it is shown in Fig. 1), and this nonsense results from trying to fit a quadratic curve to data that do not resemble a quadratic. You can call it a misapplication of curve fitting, or the use of a bad model.
Now to real data
Is this just a bizarre, unrealistic example? No! Because the basic shape of this example resembles the observed global sea-level curve from about 1930 to 2000. The red curve below is the rate of rise as diagnosed from the Church&White (2006) global sea-level data set, as shown and described in more detail in Rahmstorf (Science 2007):
Fig. 3. Rate of global sea-level rise based on the data of Church & White (2006), and global mean temperature data of GISS, both smoothed. The satellite-derived rate of sea-level rise of 3.2 ± 0.5 mm/yr is also shown. The strong similarity of these two curves is at the core of the semi-empirical models of sea-level rise. Graph adapted from Rahmstorf (2007).
Why would it have such a funny shape, with the rate of rise hovering around 2 mm/year in mid-century before starting to increase again after 1980? I think the reason is physics: the warmer it gets, the faster sea-level rises, because for example land ice melts faster. The sea-level rate curve has an uncanny similarity to the GISS global temperature, shown here in blue. The rate of SLR may well have stagnated in mid-century because global temperature also did not rise between about 1940 and 1980, and in the northern hemisphere even dropped.
Houston & Dean
The “sea-level sceptics” paper by Houston and Dean in 2011 claimed that there is no acceleration of global sea-level rise, by doing two things: cherry-picking 1930 as start date and fitting a quadratic. The graphs above show how this could give them their result despite the clear threefold acceleration from 1 to 3 mm/yr during the 20th Century. In our rebuttal published soon after (Rahmstorf and Vermeer 2011), we explained this and concluded:
Houston and Dean’s method of fitting a quadratic and discussing just one number, the acceleration factor, is inadequate.
(There’s quite a few other things wrong with this paper and several responses have been published in the peer-reviewed literature (5? I lost track); we also discussed it at Realclimate.)
So the bottom line is: the quadratic acceleration term is a meaningless diagnostic for the real-life global sea-level curve. Instead, one needs to look at the time evolution of the rate of sea-level rise, as has been done in a number of peer-reviewed papers. For example, Rahmstorf et al. (2012) in their Fig. 6 show the rate curve for the Church&White 2006 and 2011 and the Jevrejeva et al. 2008 sea-level data sets, corrected for land-water storage in order to isolate the climate-driven sea-level rise. In all cases the rate of rise increases over time, albeit with some ups and downs, and recently reaches rates unprecedented in the 20th Century or (for the Jevrejeva data) even since 1700 AD. Similar results have been obtained for regional sea-level on the German North Sea coast (Wahl et al. 2011).
Pitfalls of rate curves
When looking at real data, one needs to be aware of one pitfall: unlike the ideal example shown at the outset, real tide gauge data contain spurious sampling noise due to inadequate spatial coverage, so it is not trivial to derive rates of rise. One needs to apply enough smoothing (as in Fig. 3 above) to remove this noise, otherwise the computed rates of rise are dominated by sampling noise and have little to do with real changes in the rate of global sea-level rise. Holgate (2007) showed decadal rates of sea-level rise (linear trends over 10 years), but as we have shown in Rahmstorf et al. (2012), those vary wildly over time simply as a result of sampling noise and are not consistent across different data sets (see Fig. 2 of our paper). Random noise in global sea level of just 5 mm standard deviation is enough to render decadal rates meaningless (see Fig. 3 of our paper)!
The quality of the data set is important – some global compilations contain more spurious sampling noise than others. Personally I think the approach taken by Church and White (2006, 2011) probably comes closest to the true global average sea level, due to the method they used to combine the tide gauge data.
And one needs to consider boundary effects at the beginning and end of the data series. Boundary effects at the start of the curve are not a big deal because the rate curve is rather flat before the 20th Century. And luckily, at the end this is also not a big problem since we have the satellite altimeter data starting from 1993 as an independent check on the most recent rate of sea-level rise, which confirm that it is now a bit over 3 mm/year, where also the smoothed rate curve in Fig. 3 ends. We now have almost 20 years of altimeter data that show a trend consistent with the tide gauges, but less noisy, since the satellite data have good global coverage.
So remember: don’t fit a quadratic to data that do not resemble a quadratic. Instead, look at the time evolution of the rate of sea level rise. And remember there is something called physics: this time evolution must be expected to have something to do with global temperature. And indeed it does.
A note for the technically minded:
The quadratic fit to the sea-level curve can be written as:
SL(t) = a t^2 + b t + c, where t= time and a, b and c are constants.
The rate of rise is the time derivative:
rate(t) = 2a t + b.
Often 2a is called the acceleration. That is because when we are talking about acceleration, it is the rate of rise that is of prime interest. The question is: how does this rate of sea-level rise change over time? And not: how quadratic does the sea-level curve look? Hence the second, rate equation is the relevant one, and we call 2a and not a the acceleration in the quadratic case. 2a is the slope of the rate(t) curve.
Now the interesting thing is that in the example given above, you get a negative a when you fit a quadratic to the sea-level data in Fig. 2, but you get a positive a when you make a linear fit to the rate curve in Fig. 1. You’d probably find the latter more informative since it has to do with how the rate of rise has changed, which is the question of prime interest. But as argued above, for such a time evolution it is neither a good idea to fit sea level with a quadratic nor to fit the rate curve with a straight line – it’s a bad model that gives inconsistent results.
% script to produce idealised sea level curves
a = 1.42e-5; b = -0.0159; c=2;
rate= a* x.^3 + b*x + c;
slr = cumsum(rate); % integrate the rate to get sea level
% compute quadratic fit
p = polyfit(x,slr,2);
acceleration = 2 * p(1)
Church, J. A., and N. J. White (2006), A 20th century acceleration in global sea-level rise, Geophys. Res. Let., 33(1), L01602.
Church, J. A., and N. J. White (2011), Sea level rise from the late 19th to the early 21st Century, Surveys in Geophys., 32, 585-602.
Holgate, S. (2007), On the decadal rates of sea level change during the twentieth century, Geophys. Res. Let., 34, L01602.
Houston, J., and R. Dean (2011), Sea-level acceleration based on US tide gauges and extensions of previous global-gauge analysis, J. Coast. Res., 27(3), 409-417.
Rahmstorf, S. (2007), A semi-empirical approach to projecting future sea-level rise, Science, 315(5810), 368-370.
Rahmstorf, S., and M. Vermeer (2011), Discussion of: Houston, J.R. and Dean, R.G., 2011. Sea-Level Acceleration Based on U.S. Tide Gauges and Extensions of Previous Global-Gauge Analyses., J. Coast. Res., 27, 784–787.
Rahmstorf, S., M. Perrette, and M. Vermeer (2012), Testing the Robustness of Semi-Empirical Sea Level Projections, Clim. Dyn., 39(3-4), 861-875.
Wahl, T., J. Jensen, T. Frank, and I. Haigh (2011), Improved estimates of mean sea level changes in the German Bight over the last 166 years, Ocean Dyn., 61, 701-715.