## 1. Introduction

The city of Olympia, which is the capital of Washington and is located at the southern end of Puget Sound about 100 km south of Seattle, is facing an increasing threat of sea level rise. There is disagreement about how quickly it will happen and how drastic it will be. Planning for a 15-cm rise is very different from planning for an 80-cm rise. It may take years to get approval for any adaptation effort, such as building a sea wall, and thus planning must start early (Craig 1993). We aim to provide tools to aid planning efforts by recognizing that point estimates of future sea level rise do not capture the full range of possible outcomes. As we will discuss, under different climate scenarios there are ranges of both timing and estimated sea level that will occur in Olympia with given confidence. Although the data used in our analysis are focused to a particular region of the United States, the same underlying strategy to identify confidence ranges around expected time frames and their associated sea level remains transferable as a sound tool for decision making. The financial impact of different scenarios of sea level rise can be assessed using the methods of Hallegatte et al. (2013), who estimate the vulnerability of a city by comparing its annual average loss from flooding with its annual gross domestic product. Using our approach, one can translate the distribution of projected sea level increase into a distribution of annual financial loss and hence of vulnerability, rather than focusing on averages. This approach can, in turn, be used to assess the likely effect of and need for investments in adaptation measures.

Systematic quantification of uncertainty gives policymakers a better idea of what to prepare for in the future (Reilly et al. 2001; Katz 2002; Stephenson et al. 2012). Instead of point estimates (often without uncertainty measures; e.g., Mote et al. 2008) for each scenario, we calculate simultaneous confidence sets of sea level projections. These are realistic benchmarks for when preventive measures need to be taken against the threat of sea level rise and how long one can afford to wait before taking action. For example, in an Olympia city report (Simpson 2011), various types of sea walls were proposed in response to different rises in mean sea level. To plan adequately for building these walls, our projections give policymakers a range of years for when they can expect certain levels of sea level rise. Building the sea walls in the earlier years of this range would be advisable, since the projections are intended to help the city avoid excessive flooding. Such flooding typically occurs at spring tide. In 1978 the city encountered a high tide of over 5.5 m (the mean sea level is 2.5 m), and the range of tides (from lowest low to highest high) can reach 6.8 m (see online at http://tidesandcurrents.noaa.gov/benchmarks/9446969.html). Thus, even a small increase in mean sea level is likely to change the characteristics of flooding, such as the design-life level (Rootzén and Katz 2013) of a sea wall.

## 2. Data

In our analysis we use 18 climate models from phase 5 of the Coupled Model Intercomparison Project (CMIP5; Taylor et al. 2012) that project global mean temperatures for each of the four representative concentration pathways (RCPs; Moss et al. 2010) used in the Fifth Assessment Report of the Intergovernmental Panel on Climate Change (IPCC; Stocker et al. 2013) (at the time of writing, information on the climate models used could be found online at http://courses.washington.edu/statclim/GCMs_used.html). The projected global mean temperatures for each model in this ensemble are then used to project global sea level using a relationship [Eq. (1), below], fitted to historical data: the Goddard Institute for Space Studies global annual mean temperature series (Hansen et al. 2001) and global sea level data from the Commonwealth Scientific and Industrial Research Organisation (Church and White 2011). The former is computed from archives of land and sea air temperature measurements, and the latter is based on tide gauge readings using a spatial structure determined by satellite data. Using a long time series of annual tide gauge data for Seattle from the National Oceanic and Atmospheric Administration (Zervas 2009), we relate the global sea level to Seattle sea level, and, because Olympia does not have a long record of reliable sea level data, we use tide gauge calibration data (http://tidesandcurrents.noaa.gov/inventory.html?id=9446807) for Budd Inlet (the southern part of Puget Sound where Olympia is located) to relate Olympia to Seattle. [All data used are available (as links in the R software code file) at http://www.statmos.washington.edu/datacode.html.]

## 3. Methods

### a. Predicting global mean sea level from global mean temperature

*dH*

_{t}/

*dt*to temperature

*T*

_{t}using the equationwhere

*a*and

*T*

_{0}are parameters and

*ε*

_{t}is a noise series. We assume that the noise series is a moving-average process of order 2 (Box and Jenkins 1970), as determined by the R function “auto.arima” in the “forecast” package (Hyndman et al. 2013). Different from Rahmstorf’s statistical approaches (Rahmstorf 2007; Vermeer and Rahmstorf 2009), we do not smooth the series (because smoothing just adds complexity to the time series structure), do not add a term corresponding to

*dT*

_{t}/

*dt*(because it is not statistically significant), and do not make a reservoir correction. The details of these choices are given in Guttorp et al. (2014). Figure 1a shows the sea level data (Church and White 2011) and the fitted model, with the corresponding residual plot given in Fig. 1b. The estimated values are

^{−1}(standard error of 0.05) and

### b. Predicting Seattle sea level from global sea level

*Y*

_{t}to global sea level data

*H*

_{t}, namely,where

*ζ*

_{t}is an error term with temporal structure a moving average of order 1, again determined using the R “forecast” package (Hyndman et al. 2013). The estimated parameters are

### c. Predicting Olympia sea level from Seattle sea level

*Z*

_{t}at Budd Inlet (Olympia) to the Seattle sea level data, and Fig. 1f depicts the residuals from the model given in Eq. (3). According to the U.S. Geological Survey calibration sheets, the difference in mean sea level between Seattle and Budd Inlet is 51.3 cm. A shift model relating the two series iswhere

*η*

_{t}is an autoregressive moving-average (ARMA) (2, 2) error term, again determined using the “forecast” package in R. The estimated parameter is

### d. Propagation of variability

To project future sea levels in southern Puget Sound, we employ an ensemble of climate models from CMIP5 (Taylor et al. 2012). The climate model projections are used to get an idea of the variability due to model uncertainty. We then apply the fitted regression Eq. (1) to the temperature projections. Last, we use the joint distribution of the two downscaling estimates (to Seattle and to Olympia) and apply the downscaling Eqs. (2) and (3) to the estimated global sea level projections. When estimating confidence range, it is important to take all of the known uncertainties into account. Each prediction step above increases the uncertainty and widens the range. The final step is to use the work of Bolin and Lindgren (2014) to widen the pointwise intervals so as to make them simultaneous. This step enables us to produce valid projection intervals for the timing of a particular amount of sea level rise as well as estimating the probability of not exceeding a particular level before a given year. We use 90% confidence level to conform to the usage in the latest IPCC report (Stocker et al. 2013).

### e. Simultaneous confidence regions

**Z**= {

*Z*

_{2000}, …,

*Z*

_{2100}} in the period 2000–2100, conditionally on the climate model outputs, the past sea levels, and the estimated model parameters, is a mixture of

*K*= 18 Gaussian distributions (Bolin 2012), corresponding to each of the climate models used:Here,

*μ*_{k}is determined by climate model

*k*and the common precision matrix

*q*

_{0.05}(

*t*),

*q*

_{0.95}(

*t*)], for each time

*t*, where

*q*

_{α}(

*t*) denotes the

*α*quantile of the marginal distribution

*π*(

*Z*

_{t}).

*α*, the process stays inside the band at all times

*t*∈ [2000, 2100], can be constructed by considering the joint distribution for

**Z**. We construct this simultaneous band by finding the value of

*ρ*such thatFinding

*ρ*requires that we can calculate probabilities

*P*(

**a**<

**Z**<

**b**) efficiently. This is done using the sequential integration method by Bolin and Lindgren (2014), implemented in the R package “excursions.” Finding the probability of staying below a given level

*u*until a particular year

*T*requires calculating probabilities

*P*(

*Z*

_{t}<

*u*, 2000 <

*t*<

*T*), which also is done using the excursions package. More details about the procedure can be found in Guttorp et al. (2014).

## 4. Results

Figure 2 shows a simultaneous 90% confidence region for the sea level projections and the projections based on each of the 18 climate models. There is considerable uncertainty (due to local and global statistical model fitting) beyond the ensemble variability. The pointwise confidence intervals (dashed purple lines) are only correct when looking at sea level at a given individual year, whereas the simultaneous confidence intervals (black lines) are valid for all years at the same time, giving policymakers the tools to make informed decisions on phenomena that may span multiple years without having to perform any additional calculations.

Looking across the confidence set vertically at a given year (here 2075), we can visualize the additional contribution of each part of the chain of models to the overall uncertainty (Fig. 3). There is very little additional uncertainty added in going from Seattle to Olympia, and therefore the blue and purple lines are almost on top of each other.

Looking across the bands in the horizontal direction we obtain a 90% confidence interval for the time when Olympia may see a given level of sea level rise over the 1970–99 average sea level. As an example, Table 1 shows high and low estimates together with the mean first-occurrence year for each of the RCPs and two different levels. For example, a 50-cm rise is reached earlier under RCP 8.5 than for the other scenarios.

Projected 90% confidence intervals for reaching particular mean sea level rise in comparison with 1970–99 averages.

Figure 4 shows the probability of staying below a given level until a particular year. The probability of not exceeding 25 cm is hardly affected by the RCP used, whereas there are substantial differences between RCPs for staying below 50-cm sea level rise. The city planners in Olympia have agreed to plan for the high end (Mulkern 2013), which in our analysis would correspond to RCP 8.5. The price of building a sea wall needs to be weighed against the potential price of the cleanups due to the flooding that a 50-cm mean sea level rise (in conjunction with a storm surge, spring tide, and low pressure) could cause. Similar computations can easily be done for any sea level rise of interest.

## 5. Discussion

The estimation of variability in the model projections using an ensemble may not be all that accurate, since climate models, particularly from the same modeling group, tend to be statistically dependent (Jun et al. 2008; Knutti et al. 2010). Furthermore, the selection of models submitted to CMIP5 is perhaps not representative of all modeling efforts. Both of these effects may lead to a biased estimate of the variability. Because there is no obvious approach that enables more accurate estimation of the model variability, we will just argue conditionally upon the projections.

In this work we use frequentist methods, but similar calculations can be done using Bayesian analyses. In either case, the approach uses a hierarchical model, which is the predominant statistical approach to analyze complicated systems (Katz et al. 2013). One could extend the uncertainty analysis by including historical climate simulations in the modeling of global sea level from global mean temperature as done in Bhat et al. (2011). To apply the method to other cities, one just needs to develop a model that relates global sea level to the particular local sea level (Tebaldi et al. 2012). This is easiest when tide data are available at the particular location but may need a spatiotemporal prediction model at locations at which no such data have been collected.

In this paper, we have used CMIP5 temperature projections together with a semiempirical model relating historical global mean sea level to global mean temperature. Our method can be applied to any projection of global mean sea level, such as a combination of steric sea level rise from climate models and glacial models driven by climate model temperature projections as used in Stocker et al. (2013); see also Moore et al. (2013) and Orlić and Pasarić (2013) for comparisons between semiempirical and process-based projections and Church et al. (2013) for an evaluation of process-based projections. The uncertainty analysis in the process-based case would require, at a minimum, a sensitivity analysis of the glacial models. The resolution of global models is insufficient to project local sea level rise, and we are not aware of any dynamic downscaling approaches for local sea levels. Hence, regardless of how global sea level is projected, a statistical approach to downscaling appears to be necessary.

## Acknowledgments

We thank Claudia Tebaldi for proposing this project and helping us to get started. We acknowledge the World Climate Research Programme’s Working Group on Coupled Modelling for CMIP5 data. The project had partial support from the NSF Research Network on Statistics in the Atmosphere and Ocean Sciences (STATMOS) through Grant DMS-1106862 and by the Knut and Alice Wallenberg Foundation. The research was part of a class in statistical climatology at the University of Washington. Further material can be found at http://courses.washington.edu/statclim.

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