Quantifying the Uncertainty Range of 30-Year Daily Precipitation Change due to Global Warming Using Regional Frequency Analysis

This study quantified the uncertainty range of future change in the 30-year return level of daily precipitation due to global warming for Kagoshima Prefecture (except for the Amami Islands), Japan, based on regional frequency analysis, following the results of Ishihara (2010b). The uncertainty due to resampling variability was quantified with a Monte Carlo simulation based on the regional quantile function, using projection results from MRI-RCM20. The 5%–95% uncertainty ranges of the regional 30-year quantile for the present and future climates were 1.941 ± 0.117 and 2.217 ± 0.160, respectively. Moreover, the 5%–95% range of the simulated future change ratio of the regional 30-year quantile was 1.142 ± 0.107. Based on the previous result that the annual maximum daily precipitation averaged over the region is projected to increase by 3.3%, the 5%–95% uncertainty range of the future change ratio of the regional 30-year return level of daily precipitation in the region was projected to be 1.180 ± 0.111. This result indicates that the regional 30-year return level of daily precipitation in the region is likely to increase by 6.9%–29.1% in about 100 years.


INTRODUCTION
It is important to accurately assess future change in hydrological risk due to global warming, for which regional projections have been made using high-resolution climate models (e.g., Sayama et al., 2008;Wada et al., 2008;Ishihara, 2010b;Kobayashi et al., 2010).However, the detailed integration periods for these climate models are usually only about 20 years for both the present and future climates under a greenhouse gas emission scenario, making it difficult to reliably assess the future change in precipitation with a given return period at each grid site.Moreover, the smallest scale of weather events able to be represented by climate models is generally several times the horizontal resolution of the model.Considering these limitations, regional assessments of future change might be more appropriate than "at-site" analyses (Ishihara, 2010b;Kitajima et al., 2010).
Regional frequency analysis (Hosking and Wallis, 1997), which is one of the most useful regional assessment methods, has been used for various hydrological assessments (e.g., Toyama and Mizuno, 2002;Fowler and Kilsby, 2003;Mishra et al., 2009aMishra et al., , 2009b;;Ishihara, 2010a).Furthermore, the regional approach has been applied to the projection results of high-resolution climate models to assess the future change in hydrological risk arising from global warming (e.g., Ekström et al., 2005;Fowler et al., 2007;Ishihara, 2010b).Ishihara (2010b) assessed the nature of future change in the 30-year return level of daily precipitation due to global warming for Kagoshima Prefecture (except for the Amami Islands), Japan, by applying regional frequency analysis to projection results of a 20-km-mesh regional climate model (MRI-RCM20; Kurihara et al., 2005) developed by the Meteorological Research Institute (MRI).The main results of this analysis were as follows: 1.The regional 30-year quantiles in Kagoshima Prefecture for the present and future climates were 1.956 and 2.239, respectively, representing an increase of 14.5%.

The annual maximum daily precipitation averaged over
Kagoshima Prefecture was projected to increase by 3.3%.3. Consequently, the regional 30-year return level of daily precipitation in Kagoshima Prefecture was projected to increase by 18.3% in about 100 years.
The projection results obtained from climate models are subject to several sources of uncertainty that can be divided into three categories: (i) uncertainty of greenhouse gas emission scenarios (emissions uncertainty); (ii) modeling uncertainty of GCM and RCM arising from model resolution, physical parameterizations, and the forcing boundary conditions; and (iii) natural climatic variability (e.g., Fowler et al., 2007;Shiogama et al., 2007).Quantifying these uncertainties in projections of future climate change is crucial for the development and implementation of climate policy.
To quantify the above uncertainties, previous studies have used statistical multi-model approaches such as the Bayesian approach (e.g., Tebaldi et al., 2005;Ishizaki et al., 2010).Fowler et al. (2007) assessed the future change of precipitation extremes for nine regions in the UK under the greenhouse gas emission scenario SRES A2 using six regional climate model integrations.In their analysis, the uncertainty (iii) of the projection result for each region by each climate model was quantified with a nonparametric bootstrap resampling method, and total uncertainty ((ii) and (iii)) was quantified based on all the results by assuming the models have equal performance of projection.In estimating the return values using regional frequency analysis, only the Generalized Extreme Value (GEV) distribution was used in their assessment.
For the quantification of total uncertainty, the results by a number of climate models under some greenhouse gas emission scenarios are absolutely necessary.However, the quantifications of uncertainty (i) and (ii) over Japan are difficult as it is now, because sufficient numbers of the detailed projection experiments haven't been conducted yet.Therefore, the present study sought to quantify the uncertainty (iii) of future change in the 30-year return level of daily precipitation for Kagoshima Prefecture using projection results from MRI-RCM20, following the results of Ishihara (2010b).In this study, contrary to Fowler et al. (2007), a Monte Carlo simulation with a regional common quantile function was applied in performing the quantification.The resampling by a Monte Carlo simulation is also used for both the identification of regional homogeneity and choice of frequency distribution in order to quantify the robustness (uncertainty) of the homogeneity.Therefore, by applying this resampling simulation, the uncertainty range of the 30-year quantile due to the uncertainty of regional homogeneity should be quantified.In addition, a bootstrap or a jackknife method wasn't applied for resampling in this study, because it is unclear that each simulated region is homogeneous and a common frequency distribution of the region is specified.

DATA
This study used the same annual maximum daily precipitation data as those analyzed by Ishihara (2010b), which were projected by MRI-RCM20.A historical experiment run was performed to reproduce the actual climate over Japan for 1981-2000 using initial and boundary conditions from a coupled atmosphere-ocean general circulation model (MRI-CGCM2; Yukimoto et al., 2001), as developed by MRI.A future experiment run was performed to project the future climate over Japan for 2081-2100 using initial and boundary conditions from MRI-CGCM2 under the greenhouse gas emission scenario SRES A2.
For both the present and future climates, 18 grid points from MRI-RCM20 were selected for Kagoshima Prefecture (see Figure 1 in Ishihara, 2010b).

METHODOLOGY AND APPLICATION RESULTS
Details regarding the methodology of regional frequency analysis can be found in Hosking and Wallis (1997), Toyama and Mizuno (2002), and Ishihara (2010aIshihara ( , 2010b)).This section explains the methodology used to quantify the uncertainty range of the future change of the 30-year return level of daily precipitation, and application results are shown along with the procedure and the results reported by Ishihara (2010b).

Calculation of sample L-moments and sample Lmoment ratios
In Ishihara (2010b), r-th sample L-moments l r , sample L-CV t, and r-th sample L-moment ratios t r were calculated at each point for both the present and future climates.These values were used as unbiased r-th L-moments λ r , L-CV τ, and r-th L-moment ratios τ r , which are respectively defined as shown in Equation (1): where X k:n denotes the k-th smallest value from a sample of size n.
Regionally averaged values of the sample L-CV t R and sample L-moment ratios t r R are weighted in proportion to the sample record length n i at site i (total number N), and are defined as , , where t (i) and t r (i) denote sample L-CV and sample L-moment ratios at site i, respectively.
Future change in t R was greater than that in t r R , indicating that future change in variability (corresponding to the second sample L-moment) is projected to exceed the average (corresponding to the first sample L-moment) (see Table I in Ishihara, 2010b).

Identification of homogeneous region and choice of frequency distribution
Next, Ishihara (2010b) estimated the degree of heterogeneity in the region to assess whether the sites might be treated as a homogeneous region, and the optimal frequency distribution was selected from the generalized logistic distribution (GLO), generalized extreme-value distribution (GEV), lognormal distribution (LN3), Pearson type III distribution (PE3), and generalized Pareto distribution (GPA).
For these assessments, a Monte Carlo simulation with a kappa distribution fitted to the regional average sample Lmoment ratios is conducted for each climate (the regional average value of the first sample L-moment λ 1 R is set to 1).In each simulation, a region with sites having a record length the same as that of the observed data is generated by using random non-exceedance probabilities.The simulated region is homogeneous and has no cross-correlation or serial correlation.
According to Hosking and Wallis (1997), a kappa distribution has four parameters and includes as special cases the generalized logistic, generalized extreme-value, and generalized Pareto distributions; consequently, it is capable of representing many of the distributions found in environmental sciences.Moreover, the use of a kappa distribution is intended to avoid a prematurely early commitment to a particular distribution as the parent of the observed data.
In Ishihara (2010b), these simulations were repeated 1000 times for both the present and future climates.As a result, the homogeneities of the region were confirmed, and LN3 was selected as the optimal regional distribution for both climates, based on the simulation results.Figure 1   λ shows regional quantile functions based on LN3 for both the present and future climates.The quantile function for the future climate showed a wide range compared with that for the present climate, indicating higher variance and that larger or smaller quantiles (compared with the present climate) are likely to appear in the future.

Quantifying the uncertainty range of the 30-year return level
The 30-year return level Q i (0.967) at site i is obtained by where l 1 (i) is the first sample L-moment at the site and q(0.967) is the regional 30-year quantile based on LN3.
Here, in the case of a homogeneous region, a quantile function at each station in the region is expected to be the same as the regional function.In the present study, to assess the uncertainty range of the results presented above, a Monte Carlo simulation with regional quantile function based on LN3 was conducted for each climate, as noted above.
The procedure employed here was the same as that in the above simulation with a kappa distribution, and the detail algorithm of each simulation is as follows: 1.A region with sites having a record length the same as that of the original data is generated by applying random non-exceedance probabilities to the regional common LN3. 2. Region average sample L-moments and sample Lmoment ratios are estimated by equation (2). 3. The regional 30-year quantile is estimated by fitting LN3 to those region average sample L-moments and sample L-moment ratios (the regional average value of the first sample L-moment λ 1 R is set to 1).This procedure was repeated 2000 times, giving 2000 quantile values for both climates.
Figure 2 shows histograms and approximated normal distributions of quantiles for the present and future climates.Each of the distributions can be approximated by a normal distribution, based on the Kolmogorov-Smirnov test for normality.Having confirmed the normality of the distributions, the 5%-95% uncertainty range was calculated to be 1.645 times the standard deviation.For the present climate, the average and standard deviation were 1.941 and 0.071, respectively, corresponding to a 5%-95% uncertainty range for the simulated regional 30-year quantile of 1.941 ± 0.117.Similarly, for the future climate, the average and standard deviation were 2.217 and 0.097, respectively, indicating an uncertainty range of 2.217 ± 0.160.Both of the average values are slightly lower than the results reported by Ishihara (2010b).The standard deviation for the future climate is slightly larger than that for the present climate, reflecting the fact that the quantile function for the future climate shows a wider range than that for the present.
Based on these simulation results, the uncertainty range of the future change ratio of the regional quantile is estimated.Here, the ratio of random values selected from simulated quantiles for both climates was calculated, following Fowler et al. (2007); this procedure was repeated 2000 times.
Figure 3 shows a histogram and approximated normal distribution of the future change ratio of the 30-year quantile.This distribution also follows a normal distribution according to the Kolmogorov-Smirnov test for normality.The average and standard deviation are 1.142 and 0.065, respectively, indicating that the 5%-95% uncertainty range of the simulated future change ratio of the regional quantiles is 1.142 ± 0.107.Consequently, given that the annual maximum daily precipitation averaged over Kagoshima Prefecture was projected to increase by 3.3% (Ishihara, 2010b), according to Eq. ( 2), the 5%-95% uncertainty range of the future  change ratio of the regional 30-year return level of daily precipitation in Kagoshima Prefecture (except for the Amami Islands) is projected to be 1.180 ± 0.111.This result indicates that the regional 30-year return level of daily precipitation in the region is highly likely to increase by 6.9%-29.1% in about 100 years.

DISCUSSION AND CONCLUSIONS
This study quantified the uncertainty range of future change of the 30-year return level of daily precipitation, resulting from global warming, for Kagoshima Prefecture (except for the Amami Islands) using regional frequency analysis, following the results of Ishihara (2010b).The uncertainty due to resampling variability was quantified with a Monte Carlo simulation based on the regional quantile function LN3.
The 5%-95% uncertainty ranges of the regional 30-year quantile for the present and future climates were 1.941 ± 0.117 and 2.217 ± 0.160, respectively.Moreover, the 5%-95% range of the simulated future change ratio of the regional 30-year quantile was estimated to be 1.142 ± 0.107.Consequently, using the previous result that the annual maximum daily precipitation averaged over the region is projected to increase by 3.3%, the 5%-95% uncertainty range of the future change ratio of the regional 30-year return level of daily precipitation in Kagoshima Prefecture (except for the Amami Islands) was projected to be 1.180 ± 0.111.This result indicates that the regional 30-year return level of daily precipitation in the region is likely to increase by 6.9%-29.1% in about 100 years.
In this study, only one projection result by MRI-RCM20 was used, as noted above, since sufficient numbers of the detailed projection experiments haven't been conducted yet.However, a super-high-resolution atmospheric general circulation model with a horizontal grid size of about 20 km (MRI-GCM20) has already been developed by MRI as part of the KAKUSHIN Program, and the projection results have been used in various assessments (e.g., Mizuta et al., 2006;Kitajima et al., 2010;Kitoh et al., 2009).Furthermore, research groups have been developing high-resolution climate models able to reproduce the present-day local climate in areas of Japan characterized by complex mountainous terrain (e.g., Ishizaki and Takayabu, 2009).Therefore, the quantification of total uncertainty with these results is an important issue in the future.
In the development and use of these climate models, it is necessary to develop appropriate metrics with which to assess their reproducibility.These metrics are also essential when using the correctly weighted multi-model ensemble approach.Using the results obtained from multi-models and well-established metrics, it is important to assess the nature of future change in hydrological risk (and the accompanying uncertainty range) for all regions in Japan.Such information would be important in developing and implementing climate policy in Japan.

Figure 1 .
Figure 1.Regional quantile functions based on LN3 for the present climate and future climate.Quantile values are ranged within certain values in response to parameters estimated with region average L-moment ratios for both cases.

Figure 2 .
Figure 2. Histograms (bars) and approximated normal distributions (lines) of the 30-year quantiles for the present climate (blue) and future climate (red).Both the total areas of all bars and the areas under the curves are normalized to 1.

Figure 3 .
Figure 3. Histogram (bars) and approximated normal distribution (line) of future change ratio in regional 30-year quantile of daily precipitation.Both the total area of all bars and the area under the curve are normalized to 1.