Page 1

Response to Comments on

“A Semi-Empirical Approach to

Projecting Future Sea-Level Rise”

Stefan Rahmstorf

Additional analysis performed in response to Holgate et al. and Schmith et al. shows that the semi-

empirical method for projecting future sea-level rise passes the test of predicting one half of the

data set based on the other half. It further shows that the conclusions are robust with respect to

choices of data binning, smoothing, and detrending.

T

ysis of the link between sea-level rise and global

warming,andtomakethecomputercodeusedin

theanalysisavailableforusebyotherresearchers

(see Supporting Online Material).

Holgate et al. raise two issues. The first,

shown in Fig. 1 in (1), concerns what they call a

“clustering” of data in the scatter plot of temper-

ature versus rate of sea-level rise. However, this

clustering is an artifact of the authors’ plotting

annual data points based on a 15-year smoothed

sea-level record, resulting in data points that are

not independent but highly autocorrelated. This

isthereasonIbinnedthedatapointsinmyscatter

plot [figure 2 in (3)]. This does not “further re-

duce the degrees of freedom,” as Holgate et al.

claim,butratherreflectsthefactthattheresimply

are not more degrees of freedom in these data

after the smoothing. In fact, as the comment by

Schmith et al. (2) correctly observes, it would

have been more consistent to use 15-year bins.

Using15-yearbins,r=0.9andP=0.002includ-

ing the trend, or r = 0.7 and P = 0.04 for a

detrended version of the analysis (see below).

Thus,thecorrelationisstillsignificantatthe99%

level with trend and at the 95% level even

without trend. Note that the binning affects only

the look of the graph, not the statistical fit (i.e.,

slope and base temperature), and the particular

smoothing procedure used has only a minor

impact.Thefuturesealevelprojectionspresented

in (3) are robust to changes in these technicalities

of the analysis.

The second issue that Holgate et al. raise is

whether the semi-empirical formula proposed in

(3) passes a simple test of predicting one half of

thedatasetbasedontheotherhalfofthedataset.

That this is indeed the case is demonstrated in

Figs. 1 and 2. Figure 1 shows the predicted rate

of sea-level rise, and of sea level itself, exactly as

in figure 3 in (3), but using only the first half of

the data set (1880 to 1940) for deriving the

he technical comments by Holgate et al.

(1) and Schmith et al. (2) provide a wel-

come opportunity to present further anal-

statistical fit. The slope found in this case is

0.42mm/yearper°C,andthebasetemperatureis

–0.42°C (relative to the period 1951 to 1980).

The result shows that sea level for the period

1940 to 2000 is predicted well (to within 2 cm of

observed sea level) by the semi-empirical

formula, based only on the sea-level data before

1940. Figure 2 shows the same for a hindcast of

sea level for the period 1880 to 1940, based only

on the data after 1940 (in this case, sea level is

integrated backward from the present). In each

case, the error margins (dashed lines) are small

enough to give useful predictions despite using

only 60 years of data, and the observed sea level

is well within those error margins of the method.

These error margins are computed the same way

as shown by the dashed gray lines in figure 4 in

(3). The semi-empirical method thus passes this

simple test very well, and its validity is thereby

confirmed. The algorithms used here are the

same as in (3). The fact that Holgate et al. show

different results in their figure 2 is due to their

using a different method, which involves de-

trending each half of the data separately (and

likely some other differences). Comparing the

graphs shows that the performance of their

method is not as good as that of the method used

in (3). The acceleration in sea-level rise between

the first period (1880 to 1940) and the second

period (1940 to 2000) due to global warming is

captured by my semi-empirical model but not

bythealternativeapproachproposedbyHolgate

et al.

The comment by Schmith et al. (2) further

raises the issue of the trend of the series being

included in the correlation. Whether an analysis

with trend or after removal of a linear (or higher

order) trend is more useful depends on what one

TECHNICALCOMMENT

Potsdam Institute for Climate Impact Research, 14473

Potsdam, Germany. E-mail: rahmstorf@ozean-klima.de

0

1

2

3

4

5

6

Rate of Change (mm/yr)

1880 1900 1920 1940 1960 1980 2000

Year

Fig. 1. (Top) Observed rate of sea-level rise (red)

and that forecast using the simple empirical

model (blue), trained using data for the period

1880 to 1940. (Bottom) Observed sea level (red)

and that predicted using the empirical model

(blue), by integrating the blue curve from the top

panel forward in time. Dashed lines show the

error estimate for the prediction, as in (3).

–20

–15

–10

–5

0

5

10

Sea Level (cm)

0

1

2

3

4

5

6

Rate of Change (mm/yr)

1880 1900 1920 1940 1960 1980 2000

Year

Fig. 2. (Top) Observed rate of sea-level rise (red)

andthatforecastusingthesimpleempiricalmodel

(blue), trained using data for the period 1940 to

2000. (Bottom) Observed sea level (red) and that

predicted using the empirical model (blue), by

integrating the blue curve from the top panel

backward in time. Dashed lines show the error

estimate for the prediction, as in (3).

–20

–15

–10

–5

0

5

10

Sea Level (cm)

1880 1900 1920 1940 1960 1980 2000

Year

Fig. 3. Fifteen-year averages of the global

mean temperature (blue, °C) and rate of sea

level rise (red, cm/year), both detrended.

–0.1

–0.05

0

0.05

0.1

°C and cm/year

28 SEPTEMBER 2007VOL 317

SCIENCE

www.sciencemag.org

1866d

on September 28, 2007

www.sciencemag.org

Downloaded from

Page 2

is interested in. In this case, the common trend of

globaltemperatureandtherateofsea-levelriseis

one of the most interesting aspects of the data. If

the rate of sea-level rise had not increased while

temperatures warmed, the basic idea behind my

analysis would have been falsified right away.

Nevertheless, even the detrended series show a

strong and significant correlation, with r = 0.7.

This is evident from Fig. 3, which shows the

temperature (blue) and the rate of sea-level rise

(red) in their detrended versions using 15-year

bins. Using the detrended data for the fit, the

agreement with past observed sea level is not

quite as good, the sea-level projections for the

year 2100 are raised by about one-third (e.g., to

93 cm instead of 69 cm for the B1 scenario), and

the statistical error estimate for these projections

is increased by up to a factor of three.

Schmith et al. also raise the possibility of

“nonsense correlations,” that is, real correlations

thatdonothaveacausalbasis.Thiscanofcourse

neverberuledout;datacanonlyfalsifybutnever

prove a hypothesis. However, the starting point

of my analysis and my paper was not a correla-

tion found in data but rather the physical rea-

soningthatachangeinglobaltemperatureshould

to first order be proportional to a change in the

rate of sea-level rise. The analysis shows that the

data of the past 120 years are indeed consistent

with this expectation, and the expected connec-

tion is statistically significant. The observational

data therefore strongly support the hypothesis I

put forward.

References

1. S. Holgate, S. Jevrejeva, P. Woodworth, S. Brewer,

Science 317, 1866 (2007); www.sciencemag.org/cgi/

content/full/317/5846/1866b.

2. T.Schmith,S.Johansen,P.Thejll, Science 317,1866(2007);

www.sciencemag.org/cgi/content/full/317/5846/1866c.

3. S. Rahmstorf, Science 315, 368 (2007).

Supporting Online Material

www.sciencemag.org/cgi/content/full/317/5846/1866d/DC1

Computer algorithm and data files

20 February 2007; accepted 4 September 2007

10.1126/science.1141283

www.sciencemag.org

SCIENCE

VOL 31728 SEPTEMBER 2007

1866d

TECHNICAL COMMENT

on September 28, 2007

www.sciencemag.org

Downloaded from