The projected impacts of climate change involve large uncertainties. Considering these large uncertainties is critical to establish robust policies. For example, a water management plan which does not consider uncertainty causes large losses if extreme events occur that are not predicted by a projection of streamflow. Thus, quantifying uncertainties of a climateimpact projections has received significant attention.
A climateimpact projection consists of multiple stages. For example, a streamflow projection includes four stages: emission scenarios, global circulation models (GCMs), biascorrection techniques and hydrological models. Emission scenarios describe how future greenhouse gases emissions could evolve based on the set of assumptions about future energy use, technological change and population levels. GCMs are computational models simulating temperature and precipitation under the selected emission scenario. Bias correction techniques remove the systematic biases of the GCM outputs with respect to the observed data. Hydrological models produce projected values of the streamflow by using projected temperature and precipitation as an input. The projection of the streamflow in past can also be produced but in this case the projection is based on observed greenhouse gas emissions instead of the emission scenario. We use the terms
A climateimpact projection usually gathers multiple projection results based on several scenarios/models/techniques in each stage. Figure 1 is an example of a streamflow projection generated by two emission scenarios, five GCMs, three bias correction techniques and four hydrological models, which results in 120 (= 2 × 5 × 3 × 4) streamflow projection results.
The total of the uncertainties, is a mixture of uncertainties produced in each stage. However, quantifying the uncertainties of stages is not straightforward since we have only projected values obtained after the last stage. Various methods have been proposed in order to quantify uncertainties of each stage (e.g., Wilby and Harris, 2006; Bastola
To address these issues, there is a need to quantify the uncertainty of each source such that it is possible to evaluate the relative contribution to the total uncertainty of each uncertainty source. We call such analysis
Even though several studies on uncertainty decomposition have been proposed, there are still many areas to be improved. The first problem of the existing methods is that they do not fully utilize the information in projected values. Note that usually projected values are given at every month, but they use the average of all projected values instead of individual values. For example, the previous methods cannot be incorporated interannual variations. This is because the methods quantify uncertainty based on the averages of projected values over the future period (e.g., 50 years from 2030).
The second problem is that there is no room to use observed values in the analysis. Climate projection typically consists of two periods  control and scenario periods. In the control periods, we have projected values as well as observed values of a meteorological/hydrological variable of interest, while we have only projected values in the scenario period. The existing methods totally ignore the projected and observed values in the control period that may not yield a fully efficient result.
This study provides a Bayesian model for decomposing the total uncertainty into uncertainties of multiple sources. A Bayesian approach to analyze multimodel ensemble data has recently attracted much attention. Bayesian methods can express the complex relationships within the data, and between the data and prior knowledge in a transparent and clear structure. In addition, Bayesian methods provide a probabilistic representation of uncertainty. Tebaldi
Our proposed Bayesian model improves existing methods of decomposing the total uncertainty. First, it can model changes of mean as well as interannual variations, which makes it possible to incorporate the uncertainty of natural variability. As an illustration, we use a model in which the mean of projected values can change. In addition, the Bayesian model can utilize the projected and observed values in the observational period scientifically, which we believe leads to a better conclusion.
This article is organized as follows. Section 2 summarizes existing uncertainty decomposition methods. Section 3 provides a proposed Bayesian uncertainty decomposition method. Section 4 describes the study area and data involved and compares the proposed method with other uncertainty decomposition methods by analyzing the data. Finally, Section 5 provides the conclusion.
Throughout this article, we refer to both emission scenarios, GCMs, bias correction techniques and hydrological models as “simulators”. There are two reasons: 1) to prevent confusion with statistical “models” for the simulator outputs and the observations; 2) to simplify sentences, e.g., we write “simulator combinations” instead of “combinations of model/scenario/technique”.
In this section, we review existing methods for uncertainty decomposition. Suppose there are
First we note that the total uncertainty can be defined as
where
The ANOVA approach (Yip
which is equivalent to the sum of squares due to stage
where
The disadvantage of the ANOVA approach is that it ignores relationships or interactions between stages. To clarify the problem, consider a simple example where there are two stages having two simulators each. Suppose that four projection values are given as
To overcome this problem, Lee
For the previously described example, the uncertainty of the first stage is 50 when the variance measure is used. In this case, Lee’s approach quantifies the uncertainties of individual stages more reasonably than the ANOVA approach. But Lee’s approach does not decompose uncertainty properly because
Kim
The cumulative uncertainty can be viewed as the average of conditional uncertainties of the stages up to
with CU_{0} = 0. Kim
For the control period, we let
For the observed values in the control period, our Bayesian model assumes
where
For the description of the distributions of the projected values in the control period, we let Δ
We also centered the time variable
We impose a certain structural assumption on Δ
where
Note that we assume that {
Our Bayesian model is similar to the model of Buser
The total uncertainty can be understood as the variance of all detrended projected values
We define the total uncertainty
where
Under the assumption in Subsection 3.1,
where
where
which satisfies a natural requirement that
Lee
The above definition of the total uncertainty, uncertainties of each stage and each simulator are given as conditional on the parameters. Our Bayesian method estimates the posterior distribution of the parameters as well as the uncertainties based on the data. Considering the posterior distribution, we report the posterior expectations of the uncertainties as our estimated uncertainties. Let
respectively.
We let the prior of our model be as
We take a hierarchical prior for the simulator biases to prevent a nonidentifiable problem. Note that if any constant
The hyperparameters
The joint distribution of the data and parameters are proportional to
The conditional distributions of the parameters (except the interannual variation parameters) for the Gibbs sampling algorithm are given as:
where
The conditional distributions of the interannual variation parameters are awkward forms; therefore, we use an auxiliary variable (Damlen
where
Note that if a multivariate normal random vector
and variance
where
and calculate
Similarly we generate
where we define
and
where
Then we can sample the interannual variation parameters from the following conditional distributions
where
Lastly, the MetropolisHasting algorithm is used for the updating of
Then set
One iteration of the Gibbs sampler is summarized in Algorithm 1.
The target watershed is Yongdam Dam located at the upper basin of the Geum River in South Korea (Figure 2). The Yongdam Dam basin is located at latitude 35°35′–36°00′ and longitude 127°20′−127°45′, and it lies on the borders of Chungcheongnamdo, Jeollabukdo and Gyeongsangnamdo. The basin area is 930km^{2} accounting for about 9.5% of the Geum River basin area. The climate of the Geum River region dominated by the East Asian monsoon, which is divided into a warm and wet summer monsoon and a cold and dry winter monsoon. The monsoon drives an intense precipitation during the summer season, but sometimes it leads to drought. Therefore the projections for precipitation and streamflow are associated with a large uncertainty. Various waterrelated issues such as frequent extreme droughts and increased water demands have recently occurred in the Geum River region; therefore, uncertainty analysis is becoming increasingly important for water resource planning.
One iteration of the Gibbs sampler for Bayesian uncertainty decomposition
Sample 

Sample 

Sample 

Sample 

Sample ( 

Sample 

Sample 

Sample 

Sample Δ 

Sample Δ 

Sample 

Daily streamflow data for the Yongdam Dam basin were obtained from the Water Resources Management Information System webpage (http://www.wamis.go.kr). For the projection data, we used the emission scenarios, GCMs, bias correction techniques, and hydrological models presented in Table 1 or emission scenarios. We also adopted two representative concentration pathways (RCPs) scenarios, RCP4.5 and RCP8.5. Out of 27 GCMs contained in the Coupled Model Intercomparison Project Phase 5 (CMIP5), we selected the following four GCMs: CESM1BGC (Moore
Daily projection and observation data are aggregated to the annual data for two seasons, DJF (December, January and February) and JJA (June, July and August). Our model was applied independently for both seasons. The control period is from 1976 to 2005 and the scenario period is from 2016 to 2045.
Figures 3 and 4 draw the averaged projection values over the scenario period according to the simulators in each stage. For example, the dots above ‘GR4J’ in the right lower panel of Figure 3 are 16 projection values where hydrological model GR4J was used and the diamond shape dot is the average of the 16 projection values. The dots connected by lines are projection values using the same combination of simulators except the simulator at the xaxis. For the JJA season the hydrological model stage seems to be the most influential to the projection. The two hydrological models yield fairly different projection values. In contrast, the choice of bias correction technique does not appear to have a significant impact on projection values. The two projection values from the same combination of models, except the bias correction technique, are almost similar. For the DJF season, emission scenario and GCM seem to be more influential to the projection than the other stages.
We ran a single Markov chain with length 60,000. We saved every 50
Figures 5 and 6 present the estimated posterior densities of some easily interpretable parameters for the JJA and DJF seasons, respectively. For the JJA season, there is an apparent mean shift between the control and scenario periods but not for the DJF season. For both seasons, the dispersion of
There is a reduction in the linear trend for the JJA season. It shows that the degree of increase in streamflow gradually decreases over time. For the DJF season, the linear trend is negative in the control period, but positive in the scenario period.
Tables 2 and 3 present the posterior expectations of bias, interannual variation and uncertainty of each simulator for the JJA and DJF seasons, respectively. For both seasons, the interannual variation of RCP4.5 is greater than that of RCP8.5 and the interannual variation of Tank hydrological model is greater than that of GR4J hydrological model. This implies that RCP4.5 and Tank lead to the larger interannual variations. On the other hand, the bias of Tank hydrological model is higher than the bias of the GR4J hydrological model for the JJA season, but the relationship is reverse for the DJF season. It implies that Tank hydrological model projects more extreme streamflow compared to the GR4J hydrological model.
We compared the proposed Bayesian uncertainty decomposition method with the AVOVA approach (Yip
Our Bayesian model can be applied when there is only data in the scenario period. In this case, we exclude the unnecessary parameters
Tables 4 and 5 summarize the results of the uncertainty decomposition using the ANOVA method, the cumulative uncertainty method and the proposed Bayesian method using whole data and the Bayesian method using only the data in the scenario period for the JJA and DJF seasons. The percentage in the bracket indicates the proportion of the uncertainty of each stage that contributed to the total uncertainty. For the uncertainty measure in the cumulative uncertainty approach, we used the variance to quantify uncertainties for fair comparison. For the Bayesian model, the presented uncertainties are posterior expectation values and the proportions are computed as the ratio of each posterior expectation to the posterior expectation of the total uncertainty.
For the JJA season, the bias correction technique is the smallest contributor to the total uncertainty and the hydrological model is the largest contributor for all four methods. The uncertainty of GCMs is larger than that of emission scenarios for the three other methods except for the Bayesian method using whole data. For the DJF season, the hydrological model is the largest contributor among the four simulator stages for the two Bayesian methods but the emission scenario is the largest contributor for the other two methods. This is due to the large interannual variation of the Tank simulator, which is quantified by Bayesian methods.
As mentioned in Section 1, unlike the other methods, the proposed Bayesian method can incorporate the uncertainty of natural variability. When we use the whole data in the control and scenario periods, the estimated proportion of uncertainty of natural variability is the fourth largest for the JJA season and the largest for the DJF season. However, the uncertainty of natural variability when using only the data in the scenario period is similar to the JJA season, but different for the DJF season. The inconsistency of the results is due to the absence of historical observations.
The uncertainty decomposition results between the Bayesian model using whole data and the model using only the data in the scenario period, are quite different. This implies that the information from the data in the control period depends significantly on the uncertainty decomposition results.
We proposed a Bayesian method for decomposing the total uncertainty into uncertainty of each source with the following advantages: (1) it can borrow the information from projected and observed data in the control period; and (2) it can quantify the uncertainty of natural variability. We also provide a simple and efficient Gibbs sampling algorithm using auxiliary variables.
Future research will apply the proposed Bayesian method to meteorological/hydrological variables that are not expected to follow a normal distribution, e.g., maximum precipitation which may be assumed to follow a generalized extreme value distribution as in Jo
This work was supported by Korea Environmental Industry & Technology Institute (KEITI) through Advanced Water Management Research Program, funded by Korea Ministry of Environment (Grant. 83082)
Emission scenarios (ESs), global circulation models (GCMs), bias correction techniques (BCs), and hydrological models (HMs) used for streamflow projections
Stage  Simulator 

ESs  RCP 4.5, RCP 8.5 
GCMs  CESM1BGC, HadGEM2ES, MIROCESMCHEM, MRICGCM3 
BCs  BCCA, BCSD 
HMs  GR4J, Tank 
Posterior expectations of bias, variance, and uncertainty of each simulator for the JJA season
Stage  Simulator  Δ 


ES  RCP 4.5  0.0358  5.5000  2.8207 
RCP 8.5  −0.1292  2.7917  1.4372  
GCM  CESM1BGC  0.6574  1.1452  0.3080 
HadGEM2ES  0.7098  6.6361  1.7193  
MIROCESMCHEM  0.6176  4.0201  1.0402  
MRICGCM3  −0.0782  0.6669  0.2542  
BC  BCCA  0.9462  1.2556  0.6480 
BCSD  0.9441  2.1061  1.0825  
HM  GR4J  −0.5993  0.8461  1.5742 
Tank  2.4136  14.5822  8.5870 
ES = emission scenarios; GCM = global circulation model; BC = bias correction technique; HM = hydrological model.
Posterior expectations of bias, variance, and uncertainty of each simulator for the DJF season
Stage  Simulator  Δ 


ES  RCP 4.5  0.0697  0.0889  0.0463 
RCP 8.5  −0.0644  0.0049  0.0048  
GCM  CESM1BGC  0.0591  0.0132  0.0037 
HadGEM2ES  0.1579  0.0477  0.0129  
MIROCESMCHEM  0.1054  0.0360  0.0090  
MRICGCM3  0.0619  0.0107  0.0031  
BC  BCCA  0.1701  0.0072  0.0036 
BCSD  0.1679  0.0167  0.0084  
HM  GR4J  0.2019  0.0111  0.0058 
Tank  0.1599  0.1726  0.0857 
ES = emission scenarios; GCM = global circulation model; BC = bias correction technique; HM = hydrological model.
Results of decomposition of the total uncertainty into the sources including NV, ESs, GCMs, BCs, and HMs for the JJA season
Source  ANOVA  Cumulative  Bayesian (whole)  Bayesian (scenario) 

NV      2.5331 (11.51%)  2.2324 (9.63%) 
ES  0.0150 (0.43%)  0.3027 (8.63%)  4.2579 (19.35%)  1.1619 (5.01%) 
GCM  0.4751 (13.55%)  0.5276 (15.05%)  3.3217 (15.1%)  6.8689 (29.63%) 
BC  0.0010 (0.03%)  0.0011 (0.03%)  1.7305 (7.86%)  1.2469 (5.38%) 
HM  2.6745 (76.29%)  2.6745 (76.29%)  10.1611 (46.18%)  11.6702 (50.34%) 
RSS  0.3403 (9.71%)       
ANOVA denotes the ANOVA approach and Cumulative denotes the cumulative uncertainty approach, and Bayesian (whole) and Bayesian (scenario) denote the proposed Bayesian methods using the whole data and using only the data in the scenario period, respectively. RSS stands for residual sum of squares for the ANOVA approach. NV = natural variability; ES = emission scenarios; GCM = global circulation model; BC = bias correction technique; HM = hydrological model.
Results of decomposition of the total uncertainty into the sources including NV, ESs, GCMs, BCs, and HMs for the DJF season
Source  ANOVA  Cumulative  Bayesian (whole)  Bayesian (scenario) 

NV      0.1314 (41.75%)  0.0310 (9.61%) 
ES  0.0066 (28.26%)  0.0158 (67.47%)  0.0511 (16.24%)  0.0859 (26.61%) 
GCM  0.0058 (24.93%)  0.0073 (30.99%)  0.0287 (9.13%)  0.0528 (16.36%) 
BC  4 × 10^{−6} (0.02%)  1.48 × 10^{−4} (0.63%)  0.0120 (3.81%)  0.0349 (10.82%) 
HM  2.12 × 10^{−4} (0.91%)  2.12 × 10^{−4} (0.91%)  0.0915 (29.07%)  0.1181 (36.60%) 
RSS  0.0108 (45.88%)       
ANOVA denotes the ANOVA approach and Cumulative denotes the cumulative uncertainty approach, and Bayesian (whole) and Bayesian (scenario) denote the proposed Bayesian methods using whole data and only the data in the scenario period. RSS stands for residual sum of squares for the ANOVA approach. NV = natural variability; ES = emission scenarios; GCM = global circulation model; BC = bias correction technique; HM = hydrological model.