Don’t estimate acceleration by fitting a quadratic…

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)!

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