1. Introduction
Climate change is becoming one of the most pressing issues around the world. It has already started to affect every continent, country, community, and individual. As the largest economy in Canada, the province of Ontario is now facing extraordinary challenges under a changing climate. In recent years, people are seeing more frequent and intense weather anomalies, shorter duration of ice cover on lakes, and fluctuating water levels in lakes, rivers, and streams (Ontario Ministry of the Environment 2011a). For example, local residents in southern Ontario suffered the hottest summer yet in 2010, which was deemed the hottest year on record by Environment Canada, leading to an increasing requirement for investments in energy systems, especially in the electricity system (Ontario Ministry of Energy 2010). In response to these challenges, the province of Ontario has been taking prudent steps to protect its health, economy, and communities from the harmful effects of climate change (Ontario Ministry of the Environment 2011b). The realization of such an adaptation initiative substantially depends upon how well we know and how confident we are about the negative impacts of a changing climate in the context of Ontario. It thus leads to the development of reliable climate projections at finer resolutions over the domain of Ontario, aiming at providing decision makers or policy makers with helpful information for assessing plausible future effects of climate change at regional scales.
Forecasts of future climate change with current state-of-the-art climate models can provide potential evidence for decision makers and policy makers to help determine how to adapt to or mitigate climate change. However, these forecasts are inevitably uncertain due to our incomplete understanding of the climate system in terms of its complicated physical processes and natural variability, as well as the response to rising levels of greenhouse gases (Allen et al. 2000; Murphy et al. 2004; Stainforth et al. 2005; Stott and Kettleborough 2002; Webster et al. 2003). This further results in considerable uncertainties about the rates of change that can be expected, such as changes in extremes of temperature and precipitation and sea level rise (Karl and Trenberth 2003). No single model can be powerful enough to tackle such uncertainties all at once, so it is necessary to utilize results from a range of coupled models (Houghton et al. 2001).
Previously, a number of climate research projects based on multimodel ensemble (MME) or perturbed physics ensemble (PPE) approaches have been carried out to explore techniques for quantifying uncertainties of future climate change (e.g., Barnett et al. 2006; Furrer et al. 2007; Giorgi and Francisco 2000; Giorgi and Mearns 2002, 2003; Greene et al. 2006; Harris et al. 2013; Murphy et al. 2007; Sexton et al. 2012; Tebaldi and Sansó 2009; Tebaldi et al. 2005; Watterson and Whetton 2011). The MME approach usually consists of various GCMs developed at different modeling centers around the world to sample both structural and parametric uncertainties to a limited degree (Meehl et al. 2007; Taylor et al. 2012), but it is unable to sample the consequences of either type of uncertainty in a systematic fashion since it is assembled on an opportunistic basis from current available models (Murphy et al. 2007; Sexton et al. 2012). The PPE usually consists of variants of a single base model with perturbed parameters limited in a space of possible model configurations (Collins et al. 2006; Murphy et al. 2004; Webb et al. 2006; Yokohata et al. 2010). The main advantage of the PPE approach is that it allows greater control over the design of experiments to sample parametric uncertainties within a single model framework. Both ensemble approaches can generate a large number of projections of future climate for various scenarios, but how to combine these multiple projections and interpret them into policy-relevant information has become a major challenge in recent years, due to the lack of verification of climate projections, the problems of model dependence, bias, and tuning, and the difficulty in making sense of an “ensemble of opportunity” (Knutti et al. 2010; Tebaldi and Knutti 2007).
There are obviously different ways to synthesize modeling results of MMEs or PPEs. A straightforward way is to calculate the multimodel averages for given diagnostics or variables where each model is weighted equally (Räisänen and Palmer 2001). In many cases, combining ensemble results through Bayesian methods or weighted averages, where weights are determined by comparing model predictions to observations, shows improved performance better than simple averages (e.g., Barnston et al. 2003; Krishnamurti et al. 1999, 2000; Robertson et al. 2004). The article by Tebaldi and Knutti (2007) presents a comprehensive literature review on the existing published methods to obtain best estimates of climate change based on multiple projections through ensemble approaches. It concludes that the combined estimates, which are generally expressed in a probabilistic way based on a variety of statistical methods (e.g., Giorgi and Mearns 2002, 2003; Tebaldi et al. 2005), may give more helpful information for impact studies and decision making given that none of the model projections for future climate can be validated at this stage. For example, as each estimate comes with a specific level of occurrence (i.e., probability), it thus allows for planning appropriate adaptation strategies in advance by balancing the tradeoff between the adapting costs and the potential damages of climate change at different probabilistic levels.
However, previous probabilistic studies have mainly focused on quantifying climate change uncertainties at global scales based on various GCM ensembles (e.g., Furrer et al. 2007; Greene et al. 2006; Tebaldi and Knutti 2007; Tebaldi and Sansó 2009). Additional downscaling work needs to be done before feeding the probabilistic projections into impact models (e.g., hydrological modes or crop models) because of the coarse resolution of GCM outputs, by either statistical approaches (e.g., Wang et al. 2013; Wilby et al. 2004) or regional climate models (RCMs) . A small number of papers published in recent years (e.g., Benestad 2004; Luo et al. 2005; Manning et al. 2009; New et al. 2007) have approached the problem of deriving regional probabilistic projections at a finer resolution to facilitate the assessment of regional and local climate change impacts, but they are characterized by a less general approach and are tailored to specific regions or impacts studies.
The objective of this research is to develop high-resolution probabilistic projections of temperature changes for the province of Ontario, Canada, where uncertainties in observational errors, model reliability, and temporal correlation of climate change signals will be reflected through a Bayesian hierarchical model. Specifically, a 25-km RCM ensemble is first developed and assembled by using a subset of 17-member PPE ensemble (McSweeney et al. 2012), which is developed based on the Hadley Centre Coupled Model, version 3 (HadCM3), as boundary conditions to drive the Providing Regional Climates for Impacts Studies (PRECIS) regional modeling system (Jones et al. 2004) from 1950 to 2099. A Bayesian statistical model is then advanced based on the method proposed by Tebaldi et al. (2005) for generating probabilistic projections of climate change at grid point scale, by treating the unknown quantities of interest as random variables to quantify their uncertainties in a statistical way. Observations for current climate and simulations of the PRECIS ensemble are fed into the Bayesian model to derive posterior distributions of all the uncertain quantities, which are next used to construct probabilistic projections of temperature changes over Ontario. This paper is organized as follows: section 2 describes the development of RCM ensemble using PRECIS; section 3 details the formulation of a Bayesian hierarchical model; section 4 presents the results including the main findings at 12 selected weather stations and maps of projected temperature changes at different probabilistic levels over the entire province; and the last section summarizes the study and states the main conclusions.
2. Development of RCM ensemble
a. Study area
As the second largest province in Canada, Ontario is located in the east-central area of Canada and covers more than 106 km2. As shown in Fig. 1, Ontario is bounded by Quebec to the east, Manitoba to the west, Hudson Bay and James Bay to the north, and the Great Lakes to the south. With a population of more than 13.5 million, Ontario is home to about 2 in 5 Canadians. More than 85% of the total population lives in urban centers, largely in cities on the shores of the Great Lakes. In summer, temperatures in Ontario can soar above 30°C, whereas in winter they can drop below −40°C. It has been reported by Ontario Ministry of the Environment (2011a) that the average temperature in Ontario has gone up by as much as 1.4°C since 1948. In southern Ontario, the number of days over 30°C will likely more than double by 2050. This will lead to hotter summers with more heat waves. The greatest change will likely take place in the far north in winter. Warmer winters will make it more difficult for wildlife and people in the far north to adapt. For example, some dramatic changes have already started near the coast of Hudson Bay and James Bay. Furthermore, warmer temperatures are causing the loss of permafrost, which may result in changes in the local landscape and the stability of houses and buried water pipes. The water levels in the Great Lakes have been found to be dropping with warmer winters, less ice cover, and hotter summers. This change will directly influence more than 70% of people in the province who depend on the Great Lakes for their drinking water.
In 2007, Ontario introduced its climate change action plan as a framework for action to reduce greenhouse gas (GHG) emissions (Ontario Ministry of the Environment 2011c). Following that, the government of Ontario (Ontario Ministry of the Environment 2011b) reported its adaptation strategy and action plan to create a vision and framework for collaboration across government, businesses, communities, institutes, and individuals, aiming at minimizing the negative impacts of a changing climate by taking prudent actions to adapt well to climate change. Such initiatives require a better understanding of how the climate of Ontario will change in both short and long terms, through a regional climate modeling approach.
b. Regional climate modeling using PRECIS
Regional climate modeling is an essential way for conducting thorough assessments of climate change impacts by providing impact researchers with regional detail of how future climate might change. An RCM is a powerful tool to add small-scale detailed information of future climate change to the large-scale projections of a GCM. RCMs are physics-based climate models and as such represent most or all of the processes, interactions, and feedbacks between the climate system components that are represented in GCMs (Jones et al. 2004). By taking coarse-resolution information from GCMs, RCMs can develop temporally and spatially finescale information using their higher-resolution representation of the climate system. In this study, we apply a widely used RCM developed at the Hadley Centre, PRECIS, to facilitate regional climate modeling over Ontario. PRECIS is a flexible, easy to use, and computationally inexpensive RCM designed to provide detailed climate scenarios (Wilson et al. 2011). It can be applied easily to any area of the globe to generate detailed climate change projections, with the provision of a simple user interface as well as a visualization and data-processing package. PRECIS is able to run at two different horizontal resolutions: 0.44° × 0.44° (approximately 50 km × 50 km) and 0.22° × 0.22° (approximately 25 km × 25 km), with a vertical resolution of 19 atmospheric levels using a hybrid coordinate system (a linear combination of a terrain-following 
Using a single GCM projection to drive RCMs can provide us with some information about the expected changes under a given emission scenario for the region of interest, but it does not provide us with more information about how confident we should be in those changes (Bellprat et al. 2012). The ensemble approach through either MMEs or PPEs is widely accepted as an effective way to explore the range or spread of projections from multiple members, which enables us to gain a better understanding of the uncertainties in climate modeling. The Hadley Centre has published 17 sets of boundary data from a perturbed physics ensemble (i.e., HadCM3Q0-Q16, known as QUMP), which is based on HadCM3 under the Special Report on Emissions Scenarios (SRES) A1B emission scenario, for use with PRECIS in order to allow users to generate an ensemble of high-resolution regional simulations (McSweeney and Jones 2010). Downscaling the 17-member PPE ensemble with PRECIS would require very large inputs of computing resources, data storage, and data analyses. To explore the range of uncertainties while minimizing these requirements, we select a subset of five members (i.e., HadCM3Q0, Q3, Q10, Q13, and Q15) from the QUMP ensemble according to the Hadley Centre’s recommendation (see http://www.metoffice.gov.uk/precis/qump). HadCM3Q0 is first selected as it is the standard, unperturbed model using the original parameter settings as applied in the atmospheric component of HadCM3. Selection of the remaining four members is based on 1) their performances in simulating key features of the climate over Ontario, and 2) their ability to sample the range of outcomes of future changes simulated by the full 17-member ensemble (Bellprat et al. 2012).
We run five PRECIS experiments driven by boundary conditions from the selected GCM members from 1950 to 2099 at its highest horizontal resolution (i.e., 25 km). This allows us to carry out undermentioned probabilistic analyses by providing full simulation coverage from the present day to the future. The PRECIS model outputs are extracted and divided into four 30-yr periods: one baseline period (1961–90), and three future periods (2020–49, 2040–69, and 2070–99), representing its simulations for the province of Ontario under current and future climate forcings.
c. Data
We focus on the temperature responses over the province of Ontario under the SRES A1B emission scenario (Nakicenovic and Swart 2000). Three temperature-related variables, daily maximum temperature (Tmax), daily mean temperature (Tmean), and daily minimum temperature (Tmin), are extracted from the outputs of five PRECIS runs. Projection on each variable is divided into four 30-yr periods: 1961–90, 2020–49, 2040–69, and 2070–99. We regard the period of 1961–90 as a baseline so that the model outputs on this period can be used to represent simulation on current climate, while projections of future climate are reflected in the remaining three periods and simulation on each period will be separately treated. Observed temperature means for the baseline period are derived from the raster-gridded climate dataset provided by the National Land and Water Information Service, Agriculture and Agri-Food Canada. The dataset contains 10-km gridded point estimates of daily maximum temperature and minimum temperature, which are interpolated from daily Environment Canada climate station observations using a thin plate smoothing spline surface fitting method implemented by the ANUSPLIN V4.3 software package (NLWIS 2007). Daily mean temperature on each grid is calculated using (Tmax + Tmin)/2. The observed temperature means derived from the 10-km dataset are then regridded to the 25-km grids of PRECIS model.
3. Bayesian hierarchical model
a. Likelihoods
b. Prior distributions
The statistical model described by Eqs. (2), (4), and (7) are formulated using a set of parameters: {
The true values of current and future climate means,
The regression coefficient (
- Based on the estimation of Giorgi and Mearns (2002) in terms of natural variability of observed temperature at different regions for winter and summer seasons, we assume gamma prior density for
where
- We assume gamma distributions for
Similarly, gamma distribution for
Here we set 
c. Posterior distributions
From Eq. (11) we can obtain full conditional distribution for each parameter by ignoring all other parameters that are constant with respect to the parameter of interest. In our case, full conditional distributions for all parameters are well-known distributions such as the gamma or Gaussian ones. We can therefore perform MCMC simulation through a Gibbs sampler. The formulation of the full conditional distributions for all parameters, the steps of Gibbs sampler as well as the implementation of MCMC algorithm are given in appendixes A, B, and C.
4. Results
a. Capability of hindcasting current climate
As mentioned above, a five-member subset was screened out from the QUMP ensemble as boundary conditions to drive five individual PRECIS runs. To validate the capability of these five runs in simulating key features of current climate over Ontario, we choose 12 weather stations (see Table 1) from the weather and climate observing networks owned by Environment Canada to assess their overall performance. As shown in Fig. 1, these 12 stations are spatially distributed across the landmass of Ontario, allowing for capturing the spatial characteristics of model performance. Monthly temperature means for the baseline period at each weather station are calculated separately based on the outputs of five PRECIS runs. We then draw boxplots for monthly means of Tmax, Tmean, and Tmin separately at the 12 stations. Observed means are plotted over the corresponding boxplots to help us visually check if the observed climate is well simulated by the five-member ensemble.
Twelve selected weather stations (UA denotes an upper air reporting station).
Figures 2–4 show the boxplots for Tmax, Tmean, and Tmin respectively. Observed monthly means of Tmax and Tmean at 12 weather stations are well captured by the aggregated simulations from the ensemble runs. However, the ensemble simulations slightly overestimate Tmin in some weather stations located in the middle and northern areas. In particular, the observed means of Tmin at stations SLA, TVA, and MUA for winter (December–February) and spring (March–May) are less than the lower bound of simulated results. Otherwise, the observations are still fully covered by the interval bounded by the maximum and minimum values of five simulated values. Overall, the climate means and patterns over Ontario can be smoothly reflected by the PRECIS ensemble.
Monthly means of Tmax at 12 weather stations.
Citation: Journal of Climate 27, 14; 10.1175/JCLI-D-13-00717.1
Monthly means of Tmean at 12 weather stations.
Citation: Journal of Climate 27, 14; 10.1175/JCLI-D-13-00717.1
Monthly means of Tmin at 12 weather stations.
Citation: Journal of Climate 27, 14; 10.1175/JCLI-D-13-00717.1
b. Posterior distributions at selected weather stations
By applying the Bayesian model, we derive posterior distributions for all random parameters based on the assumptions for the likelihoods and priors. Here we focus on analyzing the MCMC results of all parameters in terms of simulating three temperature variables (i.e., Tmax, Tmean, and Tmin) at the 12 weather stations under current and future climate forcings.
1) Temperature changes
Figure 5 shows posterior distributions of 
Posterior distributions of 
Citation: Journal of Climate 27, 14; 10.1175/JCLI-D-13-00717.1
To further analyze the temperature changes (i.e., 
Projected temperature changes (
2) Model reliability and inflation-deflation effects
We use 
Posterior distributions of 
Citation: Journal of Climate 27, 14; 10.1175/JCLI-D-13-00717.1
Posterior distributions of 
Citation: Journal of Climate 27, 14; 10.1175/JCLI-D-13-00717.1
Overall performance of five PRECIS runs. The model with a higher rank shows better performance than the one with a lower rank.
The inflation/deflation parameter (
3) Temporal correlation
By introducing parameter 
Posterior distribution of 
Citation: Journal of Climate 27, 14; 10.1175/JCLI-D-13-00717.1
4) Observational errors
Figure 9 shows the posterior densities of 
Posterior distribution of 
Citation: Journal of Climate 27, 14; 10.1175/JCLI-D-13-00717.1
c. Maps of projected temperature changes
The purpose of this section is to display a number of maps to help understand the geographical patterns of temperature changes across the domain of Ontario at different probability levels. We calculates the changes of three temperature variables at nine probability levels (i.e., 10%, 20%, …, 90%) offline (the results can be downloaded from http://env.uregina.ca/moe/pdownload/). We here show the changes at 10%, 50%, and 90% probability levels. The maps are at 25-km resolution covering the landmass and all water areas over Ontario.
Figure 10 shows the projected changes of Tmax at different probability levels. The central estimates of change are projected to increase significantly with time periods, and so are the changes at 10% and 90% probability levels. Specifically, in the period of 2020–49, the central estimates of change are projected to be generally between 2° and 3°C across most of the province, with slightly larger changes in the northeast next to the James Bay. The changes of Tmax at 50% probability level in 2040–69 would go up to [4, 5]°C in most of the domain. Furthermore, the projected central estimates of change in 2070–99 would be as high as [6, 8]°C in the northeast, with slightly smaller changes between 5° and 6°C in the southeast and the west of Ontario. It seems that the projected changes are very unlikely to be less than 2°C (in 2020–49) and very unlikely to be greater than 10°C (in 2070–99).
Tmax changes at 10%, 50%, and 90% probability levels.
Citation: Journal of Climate 27, 14; 10.1175/JCLI-D-13-00717.1
The changes of Tmin, as shown in Fig. 11, reveal similar patterns as those of Tmax, but with relatively smaller magnitudes compared to the changes of Tmax. The central estimates of changes in mean daily minimum temperature are projected to be [2, 3]°C in 2020–49, [4, 5]°C in 2040–69, and [5, 7]°C in 2070–99, respectively. The changes in Tmin are very unlikely to be less than 1°C (in 2020–49) and very unlikely to be greater than 10°C (in 2070–99).
Tmin changes at 10%, 50%, and 90% probability levels.
Citation: Journal of Climate 27, 14; 10.1175/JCLI-D-13-00717.1
Figure 12 shows that, as with Tmax and Tmin, there is a temporally gradient increase in terms of changing magnitudes of daily mean temperature from 2020–49 to 2070–99. In addition, each map shows a gradient decreasing pattern for Tmean changes from the northeast, where the central estimates of changes in Tmean can be as high as 8°C in 2070–99, to the southwest, where the projected changes in the same period at 50% probability level can be 6°C or less. Such a spatially distributed pattern may be somewhat related to the large water bodies (i.e., Hudson Bay and James Bay) surrounding most parts of northern Ontario.
Tmean changes at 10%, 50%, and 90% probability levels.
Citation: Journal of Climate 27, 14; 10.1175/JCLI-D-13-00717.1
5. Summary and conclusions
In this study, we developed a high-resolution RCM ensemble for the province of Ontario, Canada, by using a subset of 17-member PPE ensemble from HadCM3 as boundary conditions to drive the PRECIS regional modeling system from 1950 to 2099. An advancement to the method proposed by Tebaldi et al. (2005) was presented to facilitate the calculation of the probabilities of temperature changes at grid point scale. The advanced method treated the unknown quantities of interest, such as observational errors, model reliability, and temporal correlation, as well as true climate signals, as random variables with uninformative prior distributions. A Bayesian hierarchical model was employed to quantify the uncertainties of these quantities in a statistical framework based upon a limited number of explicit assumptions. By feeding the observations for current climate and simulations from the RCM ensemble into the Bayesian model, we obtained posterior distributions of all the uncertain quantities of interest through MCMC simulations. Following that, we computed the probabilities of changes in Tmax, Tmean, and Tmin at ~1800 25-km grids for three 30-yr periods: 2020–49, 2040–69, and 2070–99. Posterior distributions for all uncertain parameters at 12 selected weather stations were derived and analyzed in detail to investigate the practical significance of the proposed statistical model. We then presented maps of projected temperature changes at different probability levels to help understand the spatial patterns across the entire province.
The results show that there is likely to be a significant warming trend throughout the twenty-first century over Ontario. The central estimate of change in mean temperature in 2020–49 would be 2°–4°C, but this value is likely to increase dramatically in future time periods (3°–5°C in 2040–69, and 5°–8°C in 2070–99). The maps for Tmax and Tmin reveal the similar temporally increasing trends in terms of the magnitude of change from 2020–49 to 2070–99. In addition, there is obviously a spatially gradient decrease in the changes of three temperature variables from the northeast (with higher changes) to the southwest (with lower ones). The probabilistic projections suggest that people in Ontario are very unlikely to suffer a change less than 2°C to mean temperature in the forthcoming decades and very unlikely to suffer a change greater than 10°C to the end of this century.
The Bayesian hierarchical model presented here is not restricted to the high-resolution PRECIS ensemble developed for the province of Ontario. It can be applied directly to other GCM or RCM ensembles to construct more reliable projections for future climate change by quantifying related uncertainties in a statistical framework, and thus to provide useful information for assessing the risks and costs associated with climatic changes at global and regional scales.
Acknowledgments
This research was supported by the Major Project Program of the Natural Sciences Foundation (51190095), the Program for Innovative Research Team in University (IRT1127), Ontario Ministry of Environment and the Natural Science and Engineering Research Council of Canada.
APPENDIX A
Bayesian Model Formulation
- Likelihoods for existing data
Prior distributions for parameters
- Assume uniform prior densities on the real line for
- Posterior is given by, up to a normalizing constant,
Thus, we can obtain full conditional distributions for each parameter by ignoring all terms that are constant with respect to the parameter:
- Full conditional for
- Full conditional for
- Full conditional for
- Full conditional for
- Full conditional for
- Full conditional for
APPENDIX B
Gibbs Sampling Algorithm
Step 2. Update each parameter in turn:
Sample a value of
Sample a value of
Sample a value of
Sample a value of
Sample a value of
Sample a value of
Step 3. Repeat step 2 M − 1 times to produce a Markov chain of length M.
APPENDIX C
MCMC Simulation
We run the Gibbs sampling for a total of 260 000 iterations for all parameters at each 25-km grid specified by the PRECIS model. The first 10 000 iterations are treated as random drawings in the burn-in period during which the MCMC simulation forgets about the initial values for all parameters. After that, we save only one iteration result from every 50. Thus, we can get a total of 5000 values for each parameter, representing a sample from its posterior distribution. Estimated changes of Tmax, Tmean, and Tmin at nine probability levels (10%, 20%, …, 90%) for three 30-yr periods (2020–49, 2049–69, and 2070–99), calculated based on the CDFs of MCMC samples, are available online at http://env.uregina.ca/moe/pdownload.
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