https://orcid.org/0009-0004-7159-6515
Figures
Abstract
Climate scientists employ various techniques to study the sea level rise (SLR), one of which is semi-empirical approach where the historical relationship between the SLR and global temperature is extracted from the data and parameterized for future SLR projections. It has been documented that semi-empirical models tend to have large variations in the projections depending on the data and methodologies. This study examines the statistical properties of the data used to construct the semi-empirical models and propose a new specification as a regime switch error correction model. We show that the proposed model has sound statistical foundation and good performance. The out-of-sample model projection of cumulative SLR from 2001–2020 is within 10% of the actual SLR. The model projects that in 21st century, the average and the 5%-95% range (in parenthesis) cumulative sea level rise will be 0.28m (0.20–0.36m), 0.41m (0.33–0.48m), or 0.68m (0.60–0.76m), respectively, under the SSP1-2.6/2-4.5/5-8.5 scenarios. These projections are aligned with IPCC AR5 while lower than IPCC AR6. They are also within the range of the projections in recent studies.
Citation: Fu R, Fu K (2024) A regime switch error correction model to project sea level rise. PLOS Clim 3(6): e0000369. https://doi.org/10.1371/journal.pclm.0000369
Editor: Lin Liu, First Institute of Oceanography Ministry of Natural Resources, CHINA
Received: January 9, 2024; Accepted: April 28, 2024; Published: June 10, 2024
Copyright: © 2024 Fu, Fu. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: The Global Mean Temperature (GMT) is the Global Land-Ocean Temperature Index from GISS Surface Temperature Analysis from National Aeronautics and Space Administration (NASA), downloadable from NASA website: https://data.giss.nasa.gov/gistemp/tabledata_v4/GLB.Ts+dSST.txt The Global Mean Sea Level (GMSL) data can be downloaded from the National Oceanic and Atmospheric Administration (NOAA) website: climate.gov/sites/default/files/Climate_dot_gov_dashboard_SeaLevel_Jan2021update.txt.
Funding: The authors received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist. This paper represents the views of the authors and does not necessarily reflect those of either Sophyics Technology or Wells Fargo Bank, N.A., its parent company, affiliates, and subsidiaries.
Introduction
Among many acute and chronic impacts of climate change, sea level rise (SLR) is a major hazard that is focused by the researchers because of its widespread and devastating impact to the coastal regions. Although the risk of rising sea level is qualitatively known, the magnitude of such risk is projected with large variations. Techniques to project sea level risk evolves over time. Historically, there are two broad categories of models used to project future sea level rise: Process-based models, where the projections are based on simulating physical processes, such as ocean thermal expansion, glacier melting and groundwater storage, and semi-empirical models, which project sea level rise based on historical observed relationship between sea level and temperature. The process-based model is inherently complex due to the large number of physical processes and their interactions included in the model. Semi-empirical model is much simpler to execute; however, the model projections are known to vary widely depending on the model specifications and the data used. In addition, similar to any empirical study, the parameters in semi-empirical models can be unstable because of structural breaks in the historical periods. In the past 10 years, new studies employ probabilistic methods and expert judgements to explore the range of likely projections, providing decision-makers additional insight on the tail distribution of the adverse outcomes.
Current study is within the school of semi-empirical models. We make three contributions to this class of technique for SLR projection: 1. We demonstrate that testing the statistical properties of the data is important for establishing the validity of the model specifications; 2. We statistically test structural breaks in the data and propose a regime-switch model as a result of our testing; 3. We validated our model by examining the 20-years’ out-of-sample model projections with the actual SLR. An uncertainty band was constructed by analyzing the properties of the model residuals and further validate the uncertainty measurement through examining the variations within the projection errors in the out-of-sample testing.
We hereby briefly describe two representative semi-empirical models. Rahmstorf [1] assumes a linear relationship between Global Mean Sea Level (GMSL) and Global Mean Temperature (GMT):
where H is GMSL and T is GMT, and T0 is an equilibrium temperature. Using data from 1880 to 2001, Rahmstorf estimates that the impact of global mean temperature on GMSL is 3.4mm/year/°C. This specification is later expanded in Vermeer and Rahmstorf [2] by adding a contemporaneous “rapid-response” term dT/dt:
This enhanced specification projects ~50% higher sea level rise, as compared to Rahmstorf [1].
Grinsted et al. [3] proposes a functional form where the equilibrium sea level is a linear function of temperature, instead of a constant as in Rahmstorf [1]:
where Seq is the equilibrium sea level and Շ governs the speed of current sea level S converging to equilibrium Seq. In addition to the historical data from 1850, the authors use two reconstructed temperature data from 0 AD and 200 AD and sea level data from 1700. The paper estimates the impact of temperature on GMSL ranging from 3.0 to 8.2mm°C-1 per year, depending on the data used. The speed of convergent to a new equilibrium (Շ) is estimated to range from 208 to 1193 years, again depending on the data.
Church et al. [4] summarizes the comparison of projected sea level rise by various models up to that point in the IPCC “Climate Change 2013” report (Table 1). The first three models are IPCC process-based models, and the rest are semi-empirical models with variations of Rahmstorf [1] and Grinsted et al. [3]. As shown in the table, the projections from the semi-empirical models have wide variations and are all significantly higher than the projections from the IPCC process-based models.
Numerous studies on SLR were published since 2013. Many of these recent studies emphasize on the probabilistic distribution of the SLR projections. The tail risk quantified by the probabilistic projections provides policy makers the magnitude of the potential high-risk events. Gardner et al. [13] has an excellent summary of the SLR projections over time. Table 2 are mostly adapted from Gardner et al. [13], by selecting the studies published after 2013 with clear labeling of RCP scenarios. As shown in the table, these studies have large variations in the SLR projections. Coupling the lowest 5% band with the highest 95% band among these studies (highlighted in shade in the table), the range of the SLR projections are 0.17–0.98m, 0.22–1.58m, and 0.37–2.43m, for RCP 2.6/4.5/8.5, respectively.
Materials and methods
Modeling framework
Our model framework is that of the Error Correction models widely used in Econometrics. Economists employ empirical techniques to study the relationship among economic variables. One of the most well-known approaches is the Error Correction model [31]:
(1)
(2)
The Eq (1) assumes a linear long-run equilibrium relationship between y and x. The Eq (2) describes a short-run dynamic where the change of y at time t is driven by the narrowing of the deviation μt-1 from the long-run equilibrium in the last period, and a contemporaneous reaction to the change in x. β3 is a constant drift term and can be set to zero if no drift is assumed. The model can be estimated in two-step as above, or in one-step with this transformation in Eq (3):
(3)
where
There are several extensions of this setup, such as assuming more complex short-term dynamics of y and x with multiple lags, or assuming nonlinear long term equilibrium relationships.
Data and testing
Similar to Rahmstorf [1] and Grinsted et al. [3], we use Global Mean Temperature (GMT) and Global Mean Sea Level (GMSL) yearly data spanning from 1880 to 2020 for our study. Global Mean Temperature (GMT) time series is the “Global Land-Ocean Temperature Index” from GISS Surface Temperature Analysis from National Aeronautics and Space Administration (NASA). We recognize that there are other potential explanatory variables that semi-empirical models could have considered, such as measurements of ocean thermal expansion or measurements of glacier/land ice melting, both of which contribute to the sea level rise. The main difficulties of using these alternative explanatory variables are the availability and quality of such measurements. Furthermore, to the extent that these measurements are correlated with or are direct results of the GMT change, the explanatory power of these alternative measurements in addition to GMT may be limited. Nevertheless, exploring additional or alternative explanatory variables can be a direction for future research. For GMSL, we use the data series from the National Oceanic and Atmospheric Administration (NOAA), produced by University of Hawaii Sea Level Center. NOAA describes this GMSL data as: “The early part of the time series comes from the sea level group of CSIRO (Commonwealth Scientific and Industrial Research Organisation), Australia’s national science agency,… The more recent part of the time series is from the University of Hawaii Sea Level Center… It is based on a weighted average of 373 global tide gauge records collected by the U.S. National Ocean Service, UHSLC, and partner agencies worldwide.” Since our model has annual observation units, we compute simple average of 4 quarterly observations from this data. Both data series are publicly available through the original sources (downloadable links in the “Data Availability” section).
For time series models, it’s critical to test the stationarity of the variables and to employ techniques guarding against spurious correlation among nonstationary variables. Dickey-fuller unit root test is commonly used for stationarity testing. Both GMSL and GMT variables fail to reject the existence of unit root at 10% significance level (critical value for t-stat at 10% significant level is -3.15), as shown in Table 3. Therefore, we do not have evidence to conclude that either series is stationary.
We then transform the GMSL and GMT to their first differences and test the unit root of the first differences. The test result (Table 4) rejects the null hypothesis of unit root. Therefore, we conclude that the first differences of both series are likely stationary.
In light of these test results, we find that the specifications in Rahmstorf [1] and Vermeer & Rahmstorf [2], where a stationary first difference in GMSL is modeled as a function of nonstationary GMT, are questionable on statistical grounds.
Grinsted et al. [3] specifies a long run equilibrium between GMSL and GMT, similar to that of Error Correction Model (ECM). However, since both GMSL and GMT are nonstationary, testing the co-integration is a necessary step to establish the validity of such equilibrium.
To test the co-integration relationship between GMSL and GMT, we first run this long-run equilibrium regression (Eq 4 and Table 5):
(4)
We then carry out unit root test on the residual μt. The unit root test shows the test statistics of -4.58, which is lower than the -4.04 critical value at 1% significance level. Therefore, we confirm that GMSL and GMT are co-integrated, which confirms the validity of the long run equilibrium relationship between these two variables for both Grinsted et al. [3] and Error Correction Model specification.
We further examine the residuals from Eq (4). The residual plot from the Eq (4) (Fig 1) shows a systematic overprediction in the first 20 years. Traditionally, 1850–1900 is considered as “pre-industrial” era. This suggests that there could be a structural break with two regimes of pre-1900 and post-1900.
To test the validity of a regime switch specification, we carry out Chow’s F-test, which is calculated by comparing sum of squared residuals between a single regression using combined data from 1880–2020 and two separate regressions with sub-periods breaking at year 1900. The sum of squared residual from regression (4) over the whole sampling period is 109,337, while the sum of squared residual for subperiod 1880–1889 and 1900–2020 are 2,206 and 56,739, respectively. The Chow’s F-test is then calculated as
Where 141 is the number of the observations and 2 is the number of parameters for the F-test. The 58.6 test statistics is greater than the critical value of 4.78 at 1% significance level for F (2, 137). Therefore, the statistical evidence confirms there is a regime switch at year 1900.
The Chow’s test suggests a regime switch Error Correction Model is appropriate for this data. Regime switch ECM is within the family of Threshold ECM, traditionally used for nonlinear co-integrated series (for example, Hansen and Seo 2002 [32]). We re-specify the long-run equilibrium with two indicator variables (year<1900) and (year > = 1900) for a regime break at year 1900 (Eq 5 and Table 6):
(5)
Plotting the residuals from Eq 5 to confirm that the systematic overprediction before 1900 is corrected in the regime switch specification (Fig 2).
Model specification, estimation and testing
After the data exploration and testing, we finalize our specification as a one-step regime switch error correction model, in Eq 6.
(6)
We estimate Eq 6 using data from 1880 to 2000, while reserving data from 2001–2020 for out-of-sample testing. The parameter estimates are shown in Table 7.
We notice that the two coefficients α4 and α5 governing the contemporaneous effects are not significant. We drop these two variables and re-run the regression (Eq 7 and Table 8).
We notice that one of the parameters, α1, slightly breaches the 10% significance level. We understand that limited data (141 yearly observations) and more complex non-linear specification reduce the power of the statistical testing. With solid theory and strong out-of-sample performance (shown later), we accept this model specification despite the marginal significance for this α1 parameter.
Re-arranging the terms in Eq (7), we recover the original ECM setup with transformation as shown in Eq (3) (constant drift term β3 is set to zero):
(8)
which is
(9)
Eq 9 implies that for every 1-degree Celsius increase in Global Mean Temperature, the long-run equilibrium sea level will increase by 213 mm. The coefficient α1 governs how quickly the convergency to the new equilibrium will take place. It takes 17 years for the sea level to reach half of the new equilibrium after a change in the Global Mean Temperature.
Residual plot of Eq (7) in Fig 3 below shows a pattern of white noise. In the out-of-sample performance validation in Fig 4 from year 2001 to 2020, the actual annual sea level rise is more volatile than the model projection, which is expected since there are other idiosyncratic factors outside the model. Overall, the model performs well with cumulative error around 10% in these 20 years’ out-of-sample period. In each of these 20 years, the actual SLR is within the “likely” range projected by the model, defined as 1 standard deviation around the mean model projections.
The two dotted lines show the “likely range”, from 16th to 84th percentile of the projections.
Constructing uncertainty measurements
To construct an uncertainty band around the mean prediction, we use the root mean squared error method by computing the standard deviation of the residuals from the model estimation. Since the model is specified with a regime break in 1900, we select the model residuals post 1990. Table 9 show the summary statistics of the residuals.
The median is close to 0, the skewness is close to 0 and Kurtosis is close to 3, suggesting that the residuals are approximately normally distributed. Standard deviation of the residual is 5.2mm, suggesting that 1-year ahead projection will have a +/- 5.2mm “likely” range for the 16th to 84th percentile around the mean projection.
For a n-year ahead projection, the variance of the sum of the n-year cumulative errors follow this general formula, where xi is the model error from the ith year projection:
Assuming the yearly errors are independent across projection horizon, variance of n-year ahead projection will be simply n times yearly variance. Therefore, one standard deviation of the n-year ahead projection is 5.2(n)1/2 mm. A more extreme 5th to 95th percentiles will equal to the mean projection +/- 1.645x(n)1/2x5.2mm, or 8.57(n)1/2mm.
One potential concern in this uncertainty calculation is that if the errors were positively correlated, instead of assumed independent, the uncertainty band would be understated. To address this concern, we compute the correlation between residuals and 1-year lagged residuals to be -0.34, and the correlation between residuals and 2-year lagged residuals to be 0.04, suggesting that by assuming independent, the uncertainty band of the mean estimates is likely overstated by about 14% (= 1-(1–0.34+0.04)1/2) in our current estimation.
Another sanity check on this uncertainty measurement is through examining the model errors in the out-of-sample estimates. A quick visual inspect in Fig 3 confirms that the errors in the out-of-sample period does not exhibit a wider dispersion than the in-sample residuals. We then compute the standard deviation of the residuals for the out-of-sample period from 2001–2020 to be 4.21mm, which is smaller than the 5.2mm in-sample standard deviation of the residuals, further supporting that using the standard deviations of in-sample residuals as a measurement of projection uncertainty is appropriate.
Results
We started by testing the statistical properties, namely stationarity and co-integration of the two time series: Global Mean Sea Level (GMSL) and the Global Mean Temperature (GMT). Our test shows that neither the Global Mean Sea Level (GMSL) nor the Global Mean Temperature (GMT) is stationarity, suggesting that the specifications in Rahmstorf [1] as well as derivatives of this model specification are not ideal on statistical ground. Our testing confirms that GMT and GMSL are co-integrated, which confirms a long-run equilibrium relationship between these two variables, as specified in Grinsted et al. [3]. We also detect that there is a statistically significant regime break in the relationship of GMSL and GMT at pre- and post- industrial eras (year 1900). GMT during the pre-industrial eras were relatively stable, fluctuating around a relatively stable mean. On the contrast, postindustrial GMT exhibited a rapid increase, particularly since the 2nd half of the 20th century. The regime break we observed in our model reflects the difference in the SLR responses to GMT between an increasing GMT environment (postindustrial) and a stable GMT environment (preindustrial). This difference fundamentally comes from the nonlinearity in the physical processes relating the temperature to the causes of SLR such as thermal expansion, ice melting, groundwater storage.
With these test results, we propose and estimate a nonlinear regime switch error correction model, with 2 regimes breaking at pre- and post- industrial era (Eq 9). The model fits the historical data well. In the out-of-sample testing, the model projected cumulative SLR from 2001–2020 is within 10% of the actual SLR, and each of the yearly actual SLR is within 1 standard deviation of the mean model projections.
The proposed model suggests that for every 1-degree Celsius increase in GMT, the long-run equilibrium sea level will increase by 213 mm. It takes 17 years for the sea level to reach half of the new equilibrium. Fig 5 display the dynamic response of SLR to 1°C increase in GMT over the course of 100 years. The solid line shows the mean projected path, while two dashed lines illustrate the 5th and 95th percentile bands. The uncertainties increase as the model projects further in the future, as one would have expected.
The solid line is the mean path, and the 2 dashed lines represent 5th and 95th percentile ranges.
The model projects that in 21st century, the average and the 5%-95% range (in parenthesis) cumulative sea level rise will be 0.28m (0.20–0.36m), 0.41m (0.33–0.48m), or 0.68m (0.60–0.76m), respectively, under the SSP1-2.6/2-4.5/5-8.5 (Fig 6 and Table 10). The sea level is projected to substantially level out at the turn of this century under SSP1-2.6 as we reach net zero but will continue the significant rise into the next century under the other two warmer scenarios. Compared with the recent studies in Table 2, where the 5–95 percentile bands are 0.17–0.98m, 0.22–1.58m, and 0.37–2.43m, for RCP 2.6/4.5/8.5, our projections are within the range of the uncertainties among recent studies but are on the lower end.
This figure shows the mean projections. The uncertainty ranges are given below in Table 10.
To have an apple-to-apple comparison of our model with the models in Church et al. [4] in Table 1, we also project the sea level rise from year 1990 (Fig 7) under SRES A1B scenario. The model projects that under SRES A1B scenario, by year 2100, the sea level will rise 0.57m from 1990 level, with 5th and 95th percentile band of o.48m and 0.66m, respectively. The magnitude of our projection is similar to those from IPCC AR5 models (Table 1), which projects 0.6m mean SLR with likely range from 0.42m to 0.8m.
The solid orange line is the actual SLR up to year 2020. The dotted black line is the mean projections. The dotted orange and blue lines are 5th and 95th percentile projections, respectively.
Discussion and conclusions
Time series variables are prone to have spurious relationships because these variables frequently share a common time trend. Therefore, it’s critical to establish the validity of the specifications for any empirical model by examining the statistical properties of these variables. In this study, we carry out commonly used statistical tests for time series data and find that the specifications for some of the semi-empirical model specifications in the past studies are questionable.
Stability of the model parameters is another common concern among empirical studies. This concern is particularly acute when there is a structural break in the data. Through testing, we find that the data shows two-regimes with pre- and post-industrial era break that can be built into the model specification.
Our proposed regime switch error correction model fits the historical data well. The out-of-sample testing on the recent 20-years’ actual SLR experience provides validation on both the accuracy of the model projections and the uncertainty measurements derived from the model residuals.
Comparing projections from the proposed model with the existing models, we find that the projections from our proposed model are close to the projections from the IPCC AR5 while lower than those from IPCC AR6. Our model projections are within the range of uncertainties among recent studies, although on the lower end of these ranges.
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