An ensemble approach to quantify global mean sea-level rise over the 20th century from tide gauge reconstructions

There are four steps in the construction of the ensemble estimate of GMSL change: (a) construct the central estimate timeseries; (b) construct the ensemble timeseries of internal uncertainty; (c) construct the ensemble timeseries of structural uncertainty; (d) combine the timeseries. Each of these steps is described below in sequence.

The choice of central estimate is made based on the application. For this study, the focus is on determining long-term changes in GMSL. Since Calafat et al (2014) found the RSOI approach for CW2011 to be reasonable for these purposes, we include this RSOI estimate in our ensemble. Of the two reconstructions that employ a KS method to inform the long-term GMSL changes, we use the more recent study of DA2019. Similarly, of the two VS-based reconstructions we use the more recent study of FR2020, which is more up-to-date and includes a more comprehensive treatment of uncertainties. The central estimate timeseries is constructed by simply taking the average of CW2011, DA2019 and FR2020, i.e. each of the three methods—RSOI, KS and VS (table 1)—is given equal weight in our ensemble central estimate.

The next step is to estimate the internal uncertainty. Internal uncertainty is the term we use to describe the uncertainty estimates that are associated with each individual GMSL reconstruction (figure 1(a)). Different estimation methods are used among the products and result in substantial differences in both the magnitude and time evolution of internal uncertainty. We take a conservative approach to constructing the ensemble internal uncertainty by selecting the largest value at any given time across all available products (including 'hybrid' products, figure 1(a), dashed black line). If there were good scientific justification to do so, the estimate could be based on a subset of reconstructions. However, our approach assumes that all internal uncertainty estimates are equally valid and hence all are included.

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As discussed by Thorne et al (2005), structural uncertainty arises from the choice of approach in the reconstruction of observational datasets. This is particularly important when the spatial sampling density is low (as is the case for tide gauge reconstructions of GMSL change), as there are different approaches to account for the missing information, such as RSOI or the VS method. We estimate the structural uncertainty through computing the standard deviation of the central estimates of all products relative to the ensemble average (figure 1(a), grey shading). To do this, we first express every timeseries to a reference period by subtracting the average value for 1991–2010. This is the latest 20 year period that is common to all products and offers improved coverage and more numerous gauge records compared to earlier periods (e.g. Calafat et al 2014). Note that the central estimate timeseries is also computed relative to the same 1991–2010 reference period.

Finally, we assume that the internal and structural uncertainty estimates are independent with a normal distribution and we therefore add the timeseries in quadrature to provide a timeseries of the total observational uncertainty. This timeseries is then combined with the ensemble central estimate to provide an ensemble timeseries of GMSL change with uncertainties (figure 1(b)).

To assess the robustness of our results, we carry out two sensitivity tests. In the first, we use a much longer reference period of 1902–2010 to explore the impact this has on our results. The choice of reference period has an impact on the estimate of structural uncertainty, as shown in figure S1 (available online at stacks.iop.org/ERL/16/044043/mmedia). For the second test, we eliminate hybrid products from our ensemble central estimate, by replacing the DA2019 timeseries with DA2017. This test affects the central estimate timeseries but not the ensemble-based uncertainties, as shown in figure S3. Overall, the impact of the sensitivity tests on our results are small, with the largest variations in central estimates and total uncertainties for GMSL change not exceeding a few mm for all periods considered in this study (figures S2 and S4). Further discussion is provided in section 4.3.

We compare our ensemble timeseries to individual products and past assessments of observed GMSL reported by the IPCC on the basis of two change metrics.

The first, which we refer to as a 'delta' or Δ, is simply the difference in GMSL between two years in the historical record. The uncertainty in each delta value is estimated by combining the total uncertainty for each of the two years in quadrature—under the assumption that the uncertainties are independent. While we consider this assumption to be reasonable for years that are temporally distant, it may tend to overestimate the uncertainty for shorter periods (e.g. interannual-to-decadal timescale). Deltas are perhaps the simplest approach to quantifying sea-level change but they can be sensitive to end-point effects in the presence of substantial year-to-year variability. Deltas represent a logical choice for budget studies and may have advantages in accommodating accelerations in rates of GMSL change, compared to linear-trend estimates.

The second metric of change we use is the slope of a linear fit to the GMSL central estimate applied over a specified period, which we refer to as a 'trend'. This approach has been widely used in the IPCC AR5 and SROCC reports to characterize the average rate of change of key indices, such as globally averaged surface temperature, globally integrated ocean heat content, and GMSL. While a trend over a longer period is effective for removing the effects of internal variability, the assumption of a linear trend may be less appropriate in case of substantial accelerations in the rate of GMSL change. For consistency with IPCC assessments, we compute the uncertainty in the trend based on the approach of Santer et al (2008), which accounts for serial autocorrelation in the timeseries. Note that this approach does not make use of the a priori uncertainty estimates presented in figure 1. The uncertainty arises purely from the variability and temporal autocorrelation in the central estimate timeseries (2008, 2012).

Estimates of internal uncertainty (expressed as the 90% confidence interval) across the GMSL reconstructions range from between about 30 and 40 mm at the start of 20th century to between about 5 and 25 mm at their lowest values in the early 21st century (figure 1(a)). While all estimates show a substantial decrease in the estimated internal uncertainty over the first half of the 21st century, the non-hybrid estimates tend to level off around 1960. In contrast, the two hybrid estimates (HA2015 and DA2019) continue their much smoother reduction of uncertainty into the 21st century. Our conservative ensemble estimate of internal uncertainty is based on the maximum value across all products for any given year over the period 1900–2010 (figure 1(a), dashed line). Using this criterion, the ensemble internal uncertainty estimate is based on the HA2015 and DA2017 estimates, with values ranging from >40 mm to about 25 mm.

The estimate of ensemble structural uncertainty is computed as the standard deviation of the ensemble relative to the ensemble mean, when all timeseries are expressed relative to their 1991–2010 mean value (figure 1(a), grey shading). The structural uncertainty (expressed as the 90% confidence interval) is largest prior to the mid-20th century, with an average value of around 20 mm between 1900 and 1950. Between about 1950 and 1970 there is a substantial reduction in structural uncertainty, which has an average value of around 5 mm between 1980 and 2020. The estimate of structural uncertainty has quite a lot of residual noise, with typical year-to-year variations of several mm. Overall, the ensemble structural uncertainty is about 50% smaller than the ensemble internal uncertainty, making a lesser but still important contribution to the total uncertainty.

The estimates of structural and internal uncertainty are combined in quadrature to estimate the ensemble total uncertainty (figure 1(b), light grey shading), under the assumption that they are essentially independent and have a normal distribution. Total uncertainty ranges from about 50 mm in the early 20th century to about 25 mm in the early 21st century. The ensemble central estimate is determined by taking the mean of the three non-hybrid products, i.e. CW2011, DA2017 and FR2020. The ensemble central estimate has a very similar timeseries to FR2020, with some reduction in the higher frequency variability as a result of averaging across several estimates. We note that the CW2011 product, which is based on an RSOI method, shows less variability than products based on a VS approach (DA2017 and FR2020).

In this section we compare changes in GMSL from our ensemble estimate with the individual central estimates of the products listed in table 1 and the assessment ranges presented in recent IPCC reports. We consider estimates of GMSL change based deltas and trends (see section 2) for the periods 1901–2010, 1901–1990, 1901–1993, 1993–2010 and 1971–2010. These periods are informed by the periods used in past IPCC assessments and the availability of satellite altimeter data from 1993.

For all periods considered, the 90% confidence interval of each delta based on our ensemble GMSL estimate encompasses the individual products (figure 2(a)). Our ensemble estimate is consistent with the assessment of AR5 (Church et al 2013a) for the period 1901–2010, with a slightly lower central estimate and substantially larger uncertainties (table 2). We note that reconstructions published since the AR5, tend to show lower rates of GMSL rise over the first half of the 20th century than earlier studies (figure 1(b); Hay et al 2015).

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The deltas presented in figure 2(a) can be simply converted to a mean rate of GMSL rise for each period by dividing by the period length in years (figure 2(b)). Unsurprisingly, the comparison to AR5 is very similar when expressed as an average rate. Our ensemble estimate for the period 1901–1990 shows remarkably good agreement with the assessment presented in the recent SROCC (Oppenheimer et al 2019). The ensemble estimate uncertainties for 1993–2010 and 1971–2010 seem large compared to the spread of individual GMSL reconstructions. Since structural uncertainty is relatively small during these periods, the ensemble total uncertainty is dominated by the internal uncertainty estimate (figure 1).

The comparison of average linear trends and uncertainties following Santer et al (2008) (figure 2(c)) presents a very different picture to the analysis based on deltas (figure 2(b)). While the ensemble central estimate is broadly similar to the rates based on deltas, there is a large reduction in the estimated uncertainties. As a result, for every period, one or more individual GMSL reconstructions exhibit a trend that lies outside the ensemble 90% confidence interval. The AR5 assessment for 1901–2010 has similar, albeit slightly larger, uncertainties than our ensemble. The AR5 assessment was based on the reconstruction of CW2011, using ordinary least squares linear fit with the uncertainty representing the 90% confidence interval (Rhein et al 2013; their Table 3.1) using a similar method to Santer et al (2008) to account for serial autocorrelation. We note that our ensemble approach (section 3) will tend to reduce variability in our central estimate timeseries and may explain why our uncertainties are smaller than those of AR5, based on the Santer et al method.

The SROCC (Oppenheimer et al 2019) used an ensemble estimate that augmented the GMSL reconstructions available at the time of AR5 with the additional estimates of Hay et al (2015) and Dangendorf et al (2017) to arrive at an assessed rate of GMSL rise of 0.8–2.0 mm yr⁻¹ for the period 1901–1990. We arrive at very similar 90% confidence interval for this period, i.e. 1.0–2.0 mm yr⁻¹, based on the standard deviation of trends computed using Church and White (2011), Ray and Douglas (2011), Jevrejeva et al (2014), Hay et al (2015) and Dangendorf et al (2017). The SROCC approach is similar to the ensemble estimate presented here and shows remarkably good agreement with our estimated uncertainties based on deltas. If we were to carry out our ensemble method using an identical ensemble to SROCC, we might expect to see larger overall uncertainties arising from the estimate of internal uncertainty in addition to structural uncertainty.

The ensemble estimates of GMSL change are highly consistent with values based on the satellite altimeter timeseries presented by Frederikse et al (2020) for the period 1993–2010 (figures 2(a)–(c); table 2). For the GMSL change based on deltas, as expected, the satellite altimeter timeseries have much smaller uncertainties than our tide gauge reconstruction ensemble (figures 2(a) and (b)), which nevertheless encompass our ensemble central estimate (table 2). A comparison of trends based on Santer et al (2008) yields very similar central estimates and uncertainty values (figure 2(c) and table 2).

The trend results presented here call into question the suitability of the Santer et al approach for estimating uncertainties in GMSL rise, which does not account for structural or internal uncertainty in the ensemble (figure 1), and appears to underestimate the overall trend uncertainty (c.f. figure 2(b) and (c)). We note that this method was devised primarily in the context of estimates of global mean surface temperature (GMST) change, which has very different observational properties to GMSL. Compared to GMSL, estimates of GMST are supported by more numerous and well-distributed observations and the methodological uncertainties are relatively small compared to the observed signals (e.g. Morice et al 2012). GMST is subject to large internannual-to-decadal variability associated with ENSO and other modes of climate variability and therefore Santer et al (2008) is a very useful approach for quantifying uncertainties in the underlying multi-decadal trends. For climate indices with lesser variability and larger observational uncertainties (e.g. GMSL, ocean heat content) our results suggest that it is important to incorporate a priori estimates of internal and structural uncertainty when assessing the observed changes. We would therefore caution against using the Santer et al (2008) method for assessing trend uncertainty in GMSL. Our results suggest that the spread in linear trends across the ensemble is a more reliable basis for estimating uncertainty in GMSL change.

As a fist sensitivity test, we recompute our GMSL ensemble using a longer reference period of 1902–2010 (figure S1), since the choice of reference period is somewhat subjective. This results in an estimate of structural uncertainty that is more time-constant and does not show the substantial reduction after 1950 see in figure 1(a). Relative to our original ensemble, the total uncertainty increases in the last few decades of the timeseries. The sensitivity to reference period arises due the different decadal rates of change seen across the reconstructions, particularly during the period 1940–1980 (figure 1(b)). While the change in reference period has no impact on the central estimates of GMSL change, it does increase the uncertainty of the delta in GMSL for the periods 1993–2010 and 1971–2010 by 6 mm and 4 mm, respectively (figure S2). The effect on the uncertainty for the longer periods does not exceed ~1 mm.

In a second sensitivity test, we eliminate hybrid estimates from our central estimate ensemble by replacing DA2019 with DA2017, while retaining the 1991–2010 reference period. While the multi-decadal characteristics of GMSL are largely unchanged, the new ensemble has substantially larger interannual and decadal variability (figure S3). However, the uncertainties in delta GMSL are unaffected by the change of central estimate, and differences in the central estimate does not exceed 3 mm for any period (figure S4).