## 1. Introduction

Sea level rise is one of the important societal aspects of climate change but unfortunately there is large uncertainty about expected changes. The most recent assessment report from the Intergovernmental Panel on Climate Change (IPCC; Solomon et al. 2007) specifically mentions that the published projections cannot be taken as the upper limit because the full effect of Greenland ice streams and other effects are only partially included owing to a lack of understanding of the relevant processes.

As an alternative to process-based, physical models, Rahmstorf (2007a) suggested applying semiempirical models and he formulated a statistical relationship between global averages of surface temperature and the rate of sea level rise. The work caused some debate (Holgate et al. 2007; Schmith et al. 2007; von Storch et al. 2008; Rahmstorf 2007b), which led to a modification of the model (Vermeer and Rahmstorf 2009) debated in Taboada and Anadón (2010) and Vermeer and Rahmstorf (2010). Parallel to this, Grinsted et al. (2010) and Jevrejeva et al. (2009, 2010) formulated similar semiempirical models. Common to these semiempirical methods is that they yield larger projected global sea level changes by the end of this century than stated in Solomon et al. (2007) and therefore they have caused concern.

Our reservation about the statistical approach applied in the above work is that it does not take into account the possible nonstationarity of the time series. Such nonstationarity can be due to deterministic trends or stochastic trends. We consider that linear deterministic trends and any stochastic trend to be a consequence of the statistical model that we assume is used to generate the data. A typical stochastic trend is a realization of a so-called integrated process, consisting of a random walk generated by the accumulation of independent errors plus a stationary process. Neglecting such trends implies a risk of misleading and biased determination and estimation of statistical models (Yule 1926; Granger and Newbold 1974).

The presence of stochastic trends in climate series is well-documented (e.g., Kaufmann and Stern 1997, 2002; Kaufmann et al. 2006a,b; Richards 1993, 1998; Stern and Kaufmann 1999, 2000, and references therein). Within the field of econometrics there exists an extensive body of literature on analyses of time series containing a combination of stochastic and deterministic trends. We find it peculiar that these statistical tools, which are applied routinely in economical analyses, are rarely applied in the analysis of climate data, with Kaufmann and Stern (1997, 2002), Kaufmann et al. (2006a,b), Liu and Rodriguez (2005), Mills (2009), Stephenson et al. (2000), Stern (2006), and Stern and Kaufmann (1999, 2000) as notable exceptions. A central concept in these works is cointegration, which is also the backbone of the method we will apply. The concept of cointegration and common stochastic trends is thanks to Clive W. J. Granger, who in 2003 was awarded the Nobel Prize in economics for it. A popular explanation of cointegration can be found in Murray (1994).

## 2. Data

We use annual averages of the combined land–ocean surface temperature data described in Hansen et al. (2010) and shown in Fig. 1a. As the primary global sea level dataset we use that of Church and White (2006), constructed by combining satellite altimeter data from the beginning of the 1990s with conventional sea level records to obtain a record of global average sea level back to 1880. Additionally, we use the dataset of Jevrejeva et al. (2009), who use a “virtual stacking method” based solely on conventional sea level records (shown in Fig. 1b). Radiative forcing data on an annual basis from natural and anthropogenic sources, separated into volcanic, solar, greenhouse gases, and man-made tropospheric aerosols, were taken from Crowley (2000). As an alternative forcing dataset we used Myhre et al. (2001), with in total nine anthropogenic and natural forcing components. For each of these datasets, we calculate the total radiative forcing by simple addition of the individual components (shown in Fig. 1c). These data are analyzed in the overlapping period of 1880–1998 in what follows.

Over this period, the global sea level (Fig. 1b) has been increasing almost steadily, while the temperature (Fig. 1a) has a more complicated development with an increase until around 1940, then a slight decrease until around 1980, followed by an increase. In particular, we note that the increase in sea level begins before the warming (and the increase in forcing). The forcing has also been steadily increasing throughout the period with intermittent negative spikes caused by major volcanic eruptions. None of these negative forcing spikes occur in the 1930s, 1940s, or 1950s.

The reasons for this complicated temperature behavior are still unclear: it could be due to variations in the external forcing, mainly anthropogenic and/or volcanic aerosols, and/or it could be due to internal variability in the climate system. Studies using coupled atmosphere–ocean GCMs arrive at the conclusion that the full range of anthropogenic and natural external forcings can account for the gross features in the observed global temperature changes over the twentieth century while natural external forcings alone cannot (e.g., Broccoli et al. 2003; Knutson et al. 2006; Stott et al. 2006). Other studies claim that a residual remains, which cannot be accounted for by external forcing (Andronova and Schlesinger 2000; Kravtsov and Spannagle 2008). It is our aim that the present statistical analysis will shed light on this problem.

The series described here possibly exhibit stochastic trends. This means that the series of time differences play a central role in this type of analysis and it is therefore illustrative to plot these two derived series (Fig. 2). They appear stationary and much more like “ordinary” time series when compared to the original series shown in Figs. 1a,b. The fact that the time difference series appear stationary is a sign that the original series are integrated. We also performed an augmented Dickey–Fuller test. This is a univariate test with the null hypothesis that the series is integrated. We specify two lags of the variable and constant term included and for the temperature series we get a *p* value of 0.83, while we get 0.99 for the sea level series. This means that the null hypothesis cannot be rejected for any of the series.

## 3. Method

Since the method we intend to apply is nonstandard within the climate community, we will explain it in some detail. More comprehensive descriptions can be found in, for example, Hendry and Juselius (1999, 2000) and Juselius (2006).

*T*,

_{t}*h*) of the annual global anomalies of surface temperature and sea level. We model it statistically as a vector-autoregressive (VAR) model, which in its simplest form (first order) is a dynamic linear stochastic model:where

_{t}_{1}−

The rank of

*T*and

_{t}*h*themselves are both given as a random walk plus a stationary process, a so-called I(1) process.

_{t}The model (4)–(5) is called the error correction form of the VAR (VECM), which we recall is only possible if the two series are cointegrated. By the transformation to the VECM description, the two coupled equations for the time differences in the VAR description separate into two equations relating the time differences of the original variables, *T _{t}* and

*h*, to the disequilibrium error

_{t}*z*and random and uncorrelated disturbances.

_{t}Here,

*T*and

_{t}*h*into account.

_{t}In applications of VAR model fitting, the first step is to determine the order of the model. This is a balance between wanting a parsimonious model on one hand and retaining a sufficient number of lags to ensure normally distributed and uncorrelated residuals on the other hand. This is because having residuals with these properties ensures valid statistical inference.

The technique to ensure this is to begin by fitting a high-order model. The residuals

Next we determine the rank of the impact matrix

If cointegration is found, our VAR model is reformulated as a VECM as described above and can be estimated using the maximum-likelihood procedure described in Johansen and Juselius (1990). This includes the estimation of all parameters in the model and their confidence intervals and significance test statistics.

Engle and Granger (1987) pointed out that once the parameters *z _{t}* by (5) or (9) and use its lagged value as a regressor variable in the estimation of (7) or (8) by ordinary least squares regression. Here, ordinary test statistics, like a

*t*test, can be used and make it easy to test the significance of each independent variable in (7) or (8) and to re-estimate restricted models with insignificant variables omitted. A backward elimination procedure is applied where in each iteration, we perform doubled-sided

*t*-ratio tests on all coefficients and successively omit the variable with the highest

*p*value. We stop the procedure when all coefficients have

*p*values less than 0.05.

## 4. Application of the method

We apply the method described in the previous section to the global temperature series and the sea level series by Church and White (2006). We expect both series to exhibit an upward deterministic trend arising from the increase through time of the external radiative forcing and therefore we assume the unrestricted-constant case [see (7)] in the following analysis.

By inspecting the residuals from fitted VAR models of successively lower order, as described in the previous section, we find that VAR model of order two is a satisfactory description of the data. When applying the cointegration test we conclude that the impact matrix *p* value around 0.005 for the “rank-0” hypothesis, which we therefore reject, while we obtain a *p* value of 0.25 for the “rank-1” hypothesis, which we therefore cannot reject. Based on this, we regard the two series as cointegrated and describe their mutual time development as a VECM of order two.

*t*ratio in parentheses below an estimated parameter.

*H*, which is in direct thermal contact with the lower atmosphere on annual time scales, is involved in the thermal expansion. In that case, we have approximately

This value is obviously an order of magnitude larger than the generally agreed values for the thickness of the ocean mixed layer. Therefore, we can conclude that the thermal expansion is not confined to the mixed layer, but rather, the calculation indicates that the deeper-water masses are also heating. This is in accordance with Barnett et al. (2001).

As mentioned in section 2, the disequilibrium error should be a stationary series (i.e., without any random-walk character). This is confirmed by plotting *z _{t}*, given by (10), against time (Fig. 4), although it admittedly contains some decadal-term deviations (e.g., around 1940). This means that the series cannot be described as a simple autoregressive-type relaxation. We will discuss the temporal evolution of the disequilibrium error later in the paper.

*t*test, differs among the parameters. Retaining only significant terms, identified by the backward elimination procedure described above, the equations reduce toand

In (12), the estimated significantly negative value for

The different properties of the adjustment coefficients in the two Eqs. (12) and (13) can be illustrated by the partial residual scatterplot of

In summary, our statistical analysis shows that the points (*T _{t}*,

*h*) tend to be clustered along the cointegration line in the (

_{t}*T*,

_{t}*h*) plane. The two variables, however, behave very differently. The time differences in the global sea level exhibit a deterministic positive trend, representing the influence of the radiative forcing, while the time differences in the global temperature have the role of keeping (

_{t}*T*,

_{t}*h*) near the cointegration line.

_{t}We can understand this result from a physical point of view, as follows: because of the much larger heat capacity of the ocean compared to that of the atmosphere, any change of the upper-ocean temperature, being due to external forcing or to internal climate variability, will spread to the atmospheric surface layer. On the other hand, if the surface temperature of the atmosphere changes, the change will vanish as soon as the associated heat anomaly spreads to the upper ocean. We can say that the surface air temperature *adjusts* to the average temperature of the upper ocean, which, as remarked earlier, is strongly related to the sea level because of thermal expansion (Domingues et al. (2008). The result is therefore in opposition to the physical thinking behind the model proposed by Rahmstorf (2007a), where adjustment of the sea level to temperature is assumed.

To investigate whether the reason for the above was due to the specific nature of the sea level data used, we redid the whole analysis using the sea level series by Jevrejeva et al. (2009), but this left all of the above results virtually unchanged.

## 5. Cointegration analysis conditioned on the external radiative forcing

In the analysis presented until now, the forcing of the sea level rise comes about through a deterministic trend, determined by the drift parameter

*F*and lagged values of

_{t}*F*,

_{t}*T*, and

_{t}*h*. This model is fitted and cointegration is tested for, based on the rank of the impact matrix

_{t}*F*as a weakly exogeneous variable and can formulate a second-order VAR model aswhich in the case of cointegration can be transformed toAs in (7), model (15) relates the time differences to a combination of adjustment terms, in which the disequilibrium error now includes the external forcing and is given byshort-run dynamics determined by the 2 × 1 matrix

_{t}We conduct an analysis with the total external forcing calculated from Crowley (2000) specified as a weakly exogenous variable. We thus assume no feedback from changes in global temperature forcing variables like the atmospheric carbon dioxide concentration.

As previously explained, our motivation for introducing the total radiative forcing as a weakly exogenous variable in the model was to replace the deterministic trend of sea level as a driver of the model. As a consequence of this, we expect no deterministic trends in our VAR model (14) and should test and estimate it under the restricted-constant assumption. However, this estimating yields residuals

We conduct the cointegration test and find that the impact matrix *p* value of <0.03 for the rank-0 hypothesis, which we therefore reject, while we obtain a *p* value of 0.91 for the rank-1 hypothesis, which we therefore cannot reject. Thus, a reasonable description of the data is the cointegrated model (15)–(16).

*h*has been reduced from −3.3 K m

_{t}^{−1}in the unconditional case to −1.4 K m

^{−1}in the conditional case.

We plot the disequilibrium error for both the unconditional and conditional model (Fig. 6) and see that this variable is almost unchanged, which tells us that the effect of introducing the external radiative forcing as an exogenous variable is small. There is still a large positive anomaly around the 1940s, which means that this anomaly cannot be explained by the external forcing.

For the time differences in temperature, the first term in (18) is an adjustment to the equilibrium cointegrating relation, while the second term represents a short-run response in temperature to an increase in total forcing. This means that time differences in temperature are explained by either disequilibrium from the cointegration line or by immediate time differences in the external forcing.

For the time differences in sea level, we see from (19) that the sea level in this model also contains no term representing adjustment to the cointegration line; in fact, (19) is identical to (13). This unexpected result will be discussed in the next section.

We repeated the whole analysis using the total forcing calculated from Myhre et al. (2001) instead. This caused only minor changes to the estimation of (17) and the form of (18) and (19) was virtually unchanged as well as the estimation of the parameters. In particular, we note that here also the time differences of the sea level do not depend on the disequilibrium error.

## 6. Discussion

### a. Robustness of the analysis

Whereas there is more or less agreement between different estimates of global surface temperature, this is not the case for estimates of global average mean sea level. This is because the basic data, which are historical sea level records, are sparsely distributed and mainly along the coastline, and the interior ocean completely lacks information on sea level prior to the satellite era. This leaves room for variations due to the methodology applied in calculating the global average (see Christiansen et al. 2010).

There is also much uncertainty about the historical radiative forcing. This is because both the series of historical concentrations of greenhouse gases, aerosols, and other radiative constituents as well as the model to transform these concentrations to net forcing differ.

We carried out the above analysis with two different global sea level time series and two different time series of external forcing and saw that the main conclusions remained unchanged, which gives confidence to our results.

### b. Causes for the warming in the 1940s and the present warming

We have seen the disequilibrium error exhibit decadal-scale undulations from its average. The most prominent one is the period of above 0.1-K and up to 0.2-K positive deviations around 1940. In other words, in this period surface temperatures are relatively high or, equivalently, sea level is relatively low. This period, known as the early twentieth-century warming (E20CW), is known as a period with relatively high surface temperatures compared to surrounding decades, both averaged over the globe and in particular over the North Atlantic and the Arctic (e.g., Johannessen et al. 2004; Polyakov et al. 2003; Wood and Overland 2010). The 1960s and 1970s are characterized by moderately negative disequilibrium anomalies, while the period from 1980 onward has smaller positive values of the disequilibrium error. This last period is known as the present warming (PW). This intriguing difference in the disequilibrium error between the two warming periods of the twentieth century may tell us something about the different nature of these warming periods.

Whereas the PW is to a large extent ascribed to increased external forcing from greenhouse gases, it is still unclear whether the E20CW is entirely caused by changes in the external forcing due to changes in solar or volcanic activity or the emission of man-made aerosols (Stott et al. 2000) or can in part be attributed to a multidecadal variability mode in the thermohaline overturning circulation of the World Ocean (Delworth and Knutson 2000).

Polyakov et al. (2010) used hydrographic observations to identify such a variability mode in the North Atlantic with maximum amplitude around 1940 and with an out-of-phase relation between the temperature in the surface of the ocean and at depth. We would expect a quite modest signal in the sea surface height from such a mode, since expansion at the top would partly be compensated for by contraction at depth and vice versa. Therefore, we would see a high disequilibrium error in such a situation and thus our analysis supports the view that the E20CW is at least in parts attributable to an anomaly in the thermohaline overturning circulation.

If the PW is mainly caused by radiative anomalies and only to a minor degree by anomalies in the overturning circulation, we would not have the expansion/contraction described above and therefore we would see modest disequilibrium errors. This is indeed what we have found from our analysis, from which we conclude that there is a minor contribution from overturning anomalies but the main contribution is from radiative forcing.

### c. Investigating the reasons behind the undetected adjustment of sea level: Monte Carlo experiments

In our last analysis with forcing included we would a priori expect that sea level changes were driven by the total external radiative forcing. Therefore, it is surprising that sea level does not react to the disequilibrium error given by (17).

Could this result be an effect of having too short a period of data at hand? To investigate this we performed Monte Carlo experiments based on the model described in Eq. (15), as follows: a greater than 1100-yr-long time series of external forcing was constructed from the forcing series by Crowley (2000) by repeating the series a sufficient number of times and in each repetition adding an offset to the series. The offset has a magnitude so that the final series is steadily increasing. We then use this long forcing series to construct, in a Monte Carlo simulation, artificial series of

Looking at these histograms, shown in Fig. 7, we note that the adjustment coefficient ^{−1} but with a large spread, so that the distribution is not clearly separated from zero for the short case. For the long case, the distribution is almost separable from zero. From this we conclude that we need on the order of 1000 yr of data to detect any reaction of the sea level to disequilibrium.

## 7. Summary and concluding remarks

We have introduced a statistical tool, rather unknown within geophysics but widely used within the field of econometrics, and applied it to the problem of analyzing the relationship between global land–ocean surface temperature and sea level. We have found that the time series for global surface temperature and sea level both contain stochastic trends and therefore the risk of spurious regression is present, which in turn justifies the use of these methods. We found that the two time series shared a common stochastic trend, that is, they cointegrate. This is in accordance with our physical expectations, since we expect both series to be driven by the external radiative forcing in the long run. Accordingly, the common time development can be described by a VECM introduced in the beginning of the paper.

Three main conclusions can be drawn by fitting the VECM to data. First, we show that temperature adjusts to the cointegration equilibrium, while there is no detectable adjustment of sea level. We can understand this in physical terms. The ocean represents a much larger heat capacity than the atmosphere and therefore the surface air temperature adjusts to the average temperature of the upper ocean, which again is proportional to the sea level anomaly. This mechanism has also been shown in GCM modeling experiments (Hoerling et al. 2008; Dommenget 2009).

Second, by studying the disequilibrium error we have learned that there is an important difference between the two warming periods during the twentieth century in that the E20CW has a positive disequilibrium error, while the PW has close to zero disequilibrium error. We have suggested that the reason for this could be that while the PW is mainly caused by changes in the external radiative forcing, the E20CW seems to be at least partly connected to multidecadal variations internal to the climate system and to the thermohaline overturning circulation of the ocean.

In the second part of our analysis we introduced the external radiative forcing as an external, explanatory variable. Contrary to our expectations, this did not extensively change our overall conclusions. We expected the forcing to determine the development of sea level, but could not find any sign of this in our data material. A Monte Carlo experiment revealed that we need on the order of 1000 yr of data to detect this effect. This is in contrast to results from global climate model experiments, where a clear signal from expansion of the seawater is seen in twentieth-century integrations (e.g., Gregory et al. 2001) as a result of the increased forcing. At present, we can only suggest explanations for this discrepancy. As seen in Fig. 1b, the sea level has also been rising well before the forcing (Fig. 1c) increases in the mid-twentieth century. This secular sea level rise seems to have persisted for at least some centuries (Kearney 2001) and may dominate even today, which may be the reason why we cannot detect any influence from the external radiative forcing on sea level.

Another matter is data quality. There is an ongoing discussion on the discrepancy between the observed sea level rise and the contributions from the different components, when seen over the whole twentieth century (Munk 2002; Mitrovica et al. 2006; Woodworth 2006). One explanation for this “enigma” could be the poorer quality of the estimated historical sea level back in time, which would probably also affect our analysis. Besides that, the confidence we can have in the historical forcings is unknown. One meaningful next step would therefore be to do a similar analysis on a transient coupled GCM experiment, to see whether we arrive at similar conclusions when using such consistent data. This work will be described in a subsequent paper.

This work was supported by the European Science Foundation under Contract EW06-047 (LESC). The Gnu Regression, Econometrics and Time-series Library was widely used in the analyses.

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