Assessing the uncertainty in projecting local mean
sea level from global temperature
Peter Guttorp1,2
David Bolin3
Alex Januzzi1
Daniel Jones1
Marie Novak1
Harry Podschwit1
Lee Richardson1
Aila Särkkä4
Colin D. Sowder1
Aaron Zimmerman1
September 21, 2013
1
University of Washington, Seattle, USA
Norwegian Computing Center, Oslo, Norway
3
Umeå University, Umeå, Sweden
4
Chalmers Technical University and Gothenburg University, Gothenburg, Sweden
2
Abstract
The process of moving from an ensemble of global climate model temperature projections to local sea level projections requires several steps. We estimate
sea level in Olympia, Washington (a city quite concerned with sea level rise[1]
since parts of downtown are barely above highest high tide) by relating global
mean temperature to global sea level, global sea level to sea levels at Seattle,
Washington, and finally relating Seattle to Olympia. There has long been a realisation that accurate assessment of the precision of projections is needed for
science-based policy decisions[2, 3, 4]. When a string of statistical and/or deterministic models is connected, the uncertainty of each individual model needs to
be accounted for. Here we quantify uncertainty for each model in our system, and
assess the total uncertainty in a cascading effect throughout the system. We then
visualise the projected sea level rise over time with its total estimated uncertainty
simultaneously for the years 2000-2100, the increased uncertainty due to each of
the component models at a particular projection year, and estimates of the time
a certain sea level rise will first be reached.
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Using Uncertainty to Make Decisions
Olympia is facing an increasing threat of sea level rise, but there is disagreement about
how quickly and drastically it will happen. 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 mitigation
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effort, such as the building of a sea wall, and thus planning must start early. We
provide tools to help in this planning. The financial impact of different sea level rise
can be assessed using the methods of Hallegatte et al.[5], and using our approach one
can translate this into a distribution of impacts.
Quantification of the uncertainty throughout a system gives policymakers a more
accurate idea of what they can expect in the future. Instead of point estimates (often
without uncertainty measures[6]) for each scenario, the simultaneous 90% confidence
set of projections in Figure 1 gives 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[7], various types of
sea walls were proposed in response to different rises in mean sea level. In order to plan
adequately for building these, 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 meters (the mean sea level is 2.5m), and the range
of tides (from lowest low to highest high) can reach 6.8 m 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[8] of a sea
wall.
In our analysis we use 18 climate models from the Coupled Model Intercomparison Project Phase 5 (CMIP5[9]) that project global mean temperatures for each of
the four Representative Concentration Pathways (RCPs)[10] used in the Fifth Assessment Report of the Intergovernmental Panel on Climate Change. For the climate
models used, see 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 equation (1), fitted to the GISS global annual mean temperature series[11] and the global sea level data from CSIRO[12]. Using tide gauge data
from Seattle, we relate the global sea level to Seattle sea level, and since Olympia does
not have a long record of reliable sea level data, we use USGS calibration data for Budd
Inlet (the part of Puget Sound where Olympia is located) to relate Olympia to Seattle.
Figure 1 shows a simultaneous 90% confidence region for the sea level projections,
and the projections based on each of the 18 climate models. There is seen to be
considerable uncertainty (due to local and global statistical model fitting) beyond the
ensemble variability. Also, it would be very misleading to interpret the pointwise bands
(green lines) as simultaneous (black lines). Looking across the confidence set vertically
at a given year (here 2075) we can visualise the additional contribution of each part of
the chain of models to the overall uncertainty (Figure 2). 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 relative to the 1970-1999 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.
Figure 3 shows the probability of staying below a given level until a particular year.
2
2000
2050
80
60
40
2100
1950
2000
2050
Year
RCP 6.0
RCP 8.5
60
40
0
20
Anomaly (cm)
60
40
20
0
2100
80
Year
80
1950
Anomaly (cm)
20
0
20
40
60
Anomaly (cm)
80
RCP 4.5
0
Anomaly (cm)
RCP 2.6
1950
2000
2050
2100
1950
Year
2000
2050
2100
Year
Figure 1: 90% Olympia sea level projection simultaneous confidence set (black lines)
for the years 2000-2100 for four climate scenarios. The sea level data are shown in blue
and end in 2009. The red lines are the projections without uncertainty based on each
of the climate models. The green lines connect pointwise confidence intervals for each
year.
3
30
40
50
60
70
20
30
40
50
60
Anomaly (cm)
RCP 6.0
RCP 8.5
70
0.08
0.04
0.00
0.00
0.04
Density
0.08
0.12
Anomaly (cm)
0.12
20
Density
0.08
0.00
0.04
Density
0.08
0.00
0.04
Density
0.12
RCP 4.5
0.12
RCP 2.6
20
30
40
50
60
70
20
Anomaly (cm)
30
40
50
60
70
Anomaly (cm)
Figure 2: 2075 Olympia sea level projections with uncertainty due to different sources
for four climate scenarios . The black line is the median projection (with no uncertainty), while the grey histogram corresponds to the spread of the climate models, the
red curve adds the uncertainty due to the relation between global temperature and
global sea level, the blue line that due to downscaling global sea level to Seattle, and
the green that due to the projection of Budd Inlet (Olympia) sea level.
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Level (cm)
25
50
Level (cm)
25
50
Level (cm)
25
50
Level (cm)
25
50
RCP
Earliest
year
2029
2068
RCP
Earliest
year
2029
2065
RCP
Earliest
year
2029
2062
RCP
Earliest
year
2029
2062
2.6
Mean
year
2051
2099
4.5
Mean
year
2050
2088
6.0
Mean
year
2051
2088
8.5
Mean
year
2048
2078
Latest
year
2083
After 2100
Latest
year
2074
After 2100
Latest
year
2075
After 2100
Latest
year
2067
2097
Table 1: Projected 90% confidence intervals for reaching particular mean sea level rise
compared to 1970–1999 averages.
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1.0
P(Y(s)<50,s<t)
1.0
P(Y(s)<25,s<t)
0.8
0.6
0.4
Probability
0.6
RCP 2.6
RCP 4.5
RCP 6.0
RCP 8.5
0.0
0.2
0.4
0.0
0.2
Probability
0.8
RCP 2.6
RCP 4.5
RCP 6.0
RCP 8.5
2000
2020
2040
2060
2080
2100
2000
Year
2020
2040
2060
2080
2100
Year
Figure 3: Projected probability of Olympia sea level rise staying below a given level (25
cm in left panel, 50 cm in right panel) for four climate scenarios.
One can see that the probability of not exceeding 25 cm hardly is affected by the RCP
used, while 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[13], 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 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 interesting sea level rise.
The estimation of variability in the model projections using an ensemble may not
be all that accurate, since climate models, particularly from the same modelling group,
tend to be statistically dependent[14, 15]. Furthermore, the selection of models submitted to CMIP5 is perhaps not representative of all modelling efforts. Both these effects
may lead to a biased estimate of the variability. As there is no obvious approach that enables more accurate estimation of the model variability, we will just argue conditionally
upon the projections.
While in this work we use frequentist methods, similar calculations can be done
using Bayesian analyses. In either case the approach uses a hierarchical model, which
is the predominant statistical approach to analyse complicated systems[16]. In order
to apply the methodology to other cities, one just needs to develop a model relating
global sea level to the particular local sea level[17].
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1880
1920
1960
Year
2000
(c)
−10
−5
0
5
Global sea level anomaly (cm)
260
250
240
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Budd Inlet sea level (cm)
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−10
−5
0
5
Settle sea level anomaly (cm)
(b)
−15
Sea level relative to 1970−99 average
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Seattle sea level (cm)
Figure 4: Observed and fitted global sea levels (panel a); Seattle sea level plotted
against global sea level for 1899–2009 with fitted line (b); Budd Inlet sea level plotted
against Seattle sea level for calibration data in 1996 with fitted line (c).
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2.1
Methods
Predicting global mean sea level from global mean temperature.
Rahmstorf[18] proposed to relate global sea level change dHt /dt to temperature Tt using
the equation
dHt /dt = a(Tt − T0 ) + t ,
(1)
where a and T0 are parameters and t a noise series. We assume[19] that the noise
series is a moving average process of order two[20], and as opposed to Rahmstorf’s
approaches[18, 21], and as explained in our paper[19], we see no reason to smooth the
series, add a term corresponding to dTt /dt, or make a reservoir correction. Figure 4(a)
shows the sea level data[12] and the fitted model. The estimated values are â = 0.18
(standard error 0.05) and T̂0 = -1.12 (0.26). All computations are made using the
statistical software R[22], and the code and data used are available at http://www.
statmos.washington.edu/datacode.html.
2.2
Predicting Seattle sea level from global sea level.
Figure 4(b) indicates the adequacy of a linear model, fitting Seattle sea level data Yt to
global sea level data Ht , namely
Yt = α + βHt + ζt
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(2)
where ζt is an error term with temporal structure a moving average of order 1. The
estimated parameters are α̂ = -0.96 (0.51) and β̂ = 1.23 (0.07).
2.3
Predicting Olympia sea level from Seattle sea level.
Figure 4(c) shows 9 months of daily calibration data relating daily mean sea level
Zt at Budd Inlet (Olympia) to the Seattle sea level data. According to http://
tidesandcurrent1s.noaa.gov/station_retrieve.shtml?type=Bench+Mark+Data+Sheets
the difference in mean sea level between Seattle and Budd Inlet is 51.3cm. A shift model
relating the two series is
Zt = Yt + γ + ηt
(3)
where ηt is an error term. The estimated parameter is γ̂ = 51.1 (0.19), in good agreement
with the benchmark value. Once we correct for the average sea level from 1970-1999,
the shift disappears and the uncertainty due to the shift is just the very small standard
error of γ̂.
2.4
Propagation of variability.
In order to project future sea levels in south Puget Sound, we use an ensemble of
climate models from CMIP5[9]. We will use the climate model projections to get an
idea of the variability due to model uncertainty. We will then sample from the joint
distribution of the regression estimates, and apply the fitted regression equation (1) to
the temperature projections. Finally we sample from the joint distribution of the two
downscaling estimates (to Seattle and to Olympia) and apply the downscaling equations
(2) and (3) to the estimated global sea level projections. When estimating confidence
range, it is important to take all 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[23, 24] to widen the pointwise intervals so as to make them
simultaneous. This enables us to produce valid 90% projection intervals for the time of
a particular amount of sea level rise as in Table 1, as well as estimating the probability
of not exceeding a.particular level before a given year (Figure 3).
2.5
Simultaneous confidence regions.
The joint distribution of the Olympia sea level Z = {Z2000 , . . . , Z2100 } in the period
2000 – 2100, conditionally on the climate model outputs, the past sea levels, and the
estimated model parameters, is a mixture of Gaussian distributions
K
X
1
1 |Q|
T
√ exp − (Z − µk ) Q(Z − µk ) .
π(Z) =
K 2π
2
k=1
(4)
Here, µk is determined by climate model k and the common precision matrix Q is determined by the stochastic MA models, the uncertainty in model (3), and the uncertainty
8
in the lower integration bound in the integrated version of equation (1). A pointwise
confidence band is constructed by finding the interval [q0.05 (t), q0.95 (t)], for each time t,
where qα (t) denotes the α quantile in the marginal distribution π(Zt ).
A simultaneous confidence band, such that the probability that the process stays
inside the band at all times t ∈ [2000, 2100] with probability 1 − α can be constructed
by considering the joint distribution for Z. We construct this simultaneous band by
finding the value of ρ such that
P(qρ (t) < Zt < q1−ρ (t), 2000 < t < 2100) = 1 − α.
(5)
Finding ρ requires that we can calculate probabilities P(a < Z < b) efficiently. This is
done using the sequential integration method by Bolin and Lindgren[24], implemented
in the R package excursions.
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Acknowledgements
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 global temperature product was obtained from the
Goddard Institute for Space Sciences web pages, and the global sea level product from
the CSIRO web pages. The Seattle sea level data as well as the Olympia/Budd Inlet calibration data were obtained from the US Geological Survey web pages. The project had
partial support from the NSF Research Network on Statistics in the Atmosphere and
Ocean Sciences (STATMOS) through grant DMS-1106862. This 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. Correspondence and requests
for materials should be addressed to Peter Guttorp (email: [email protected]).
11