# Don’t estimate acceleration by fitting a quadratic…

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.

Conclusion

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.

Matlab code:

% script to produce idealised sea level curves

x=[-50:50];

a = 1.42e-5; b = -0.0159; c=2;

rate= a* x.^3 + b*x + c;

plot([0:100],rate,’r’);

slr = cumsum(rate); % integrate the rate to get sea level

plot([0:100],slr/10,’r’);

p = polyfit(x,slr,2);

acceleration = 2 * p(1)

References

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.

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