1. Introduction
Statistical distributions have been used for representing the frequency of precipitation. However, precipitation frequency may not necessarily be fitted to a statistical distribution when large variability exists in precipitation data. For example, observed and simulated precipitation associated with tropical cyclones tends to exhibit large variability. This is not only because of its larger values (i.e., heavier precipitation) than those accompanying other weather phenomena, but also due to variability in the occurrence and tracking of a tropical cyclone having a significant effect on the distribution of precipitation. Potentially important sources of this variability are the genesis and movement of tropical cyclones. The genesis and movement are affected by the surrounding environmental conditions and vary considerably in time and space. In addition, the average annual number of tropical cyclones that make landfall in Japan is approximately three. This value is considered to be a “relatively low-frequency” event when compared with additional precipitation events related to other weather phenomena such as baiu fronts (e.g., Ninomiya and Shibagaki 2007) and summer monsoons. Events that are “relatively” rare can be a source of large variability in projected precipitation.
Several studies have focused on precipitation associated with tropical cyclones in Japan using data extraction procedures to separate precipitation data that were affected by tropical cyclones from all precipitation data. Nakano et al. (2010) validated the reproducibility of tropical cyclone precipitation using regional climate model (RCM) simulation results that were obtained by a reanalysis of regional data. They found that the distribution of precipitation directly affected by tropical cyclones, defined as precipitation within a 300-km radius from the tropical cyclone center, was accurately reproduced by the RCM. In addition, the frequency distribution of hourly simulated precipitation that was accumulated as a result of tropical cyclone interference was also in agreement with observations.
As for future climate projections, using the method proposed by Murata et al. (2019) and Watanabe et al. (2019), precipitation projected by RCMs can be classified into two categories: precipitation associated with tropical cyclones, and others. Since their primary interest was phenomena associated with tropical cyclones, the projected precipitation accompanying simulated tropical cyclones in Japan was examined. Their results demonstrated that the frequency of extreme precipitation associated with tropical cyclones increased in future climate projections under the representative concentration pathway (RCP) 8.5 scenario. This increase in extreme precipitation was attributed to the intensification of precipitation even though the number of tropical cyclones approaching Japan was reduced.
In terms of global precipitation, Kitoh and Endo (2019) examined future changes in extreme precipitation associated with tropical cyclones using large ensemble simulation results. They found that the average annual maximum daily precipitation increased in warmer climates, excluding the western North Pacific basin, where results showed little change in extreme precipitation. They concluded that this small change was attributed to a decrease in the overall frequency of tropical cyclones in the western North Pacific basin.
One method that can be used for reducing variability in projected precipitation is to exclude information exhibiting large variability from all data. Precipitation data associated with tropical cyclones can therefore be considered a type of data that should be excluded. Moreover, precipitation without the influence of tropical cyclones also brings many disasters associated with heavy precipitation. The present study therefore focused on precipitation without the contribution of tropical cyclones and the comparison of statistical distributions, such as probability density function (PDF) and cumulative distribution function (CDF), between precipitation with and without the effects of tropical cyclones. Specifically, CDF of precipitation using sample data was compared based on parametric models. The degree of difference between the two CDFs was examined to verify the similarities between them after the contribution of tropical cyclones to total precipitation was excluded. The degree of the similarity after the exclusion was expected to be higher than that obtained from the original data. Statistical distributions suitable for precipitation on different time scales, such PDF and CDF, have already been proposed. They include the Gaussian distribution (e.g., Hasan and Dunn 2011) and lognormal distribution (e.g., Biondini 1976; Swift and Schreuder 1981; Reddy 1997; Wilks 2011; Mahajan et al. 2012).
The gamma distribution is suitable for fitting probable precipitation on a daily time scale (Groisman et al. 1999; Semenov and Bengtsson 2002; Wilby and Wigley 2002; Lui et al. 2019). This is because its positively skewed distribution is favorable toward precipitation frequency distributions although there are a number of other statistical distributions exhibiting positively skewed distributions as well (Wilks 2011). Recently, Martinez-Villalobos and Neelin (2019) formulated a theory as to why daily precipitation follows the gamma distribution by separating the distribution into precipitating and nonprecipitating regimes. Many observational studies have used the gamma distribution to analyze data. For instance, Groisman et al. (1999) applied the gamma distribution to measure daily precipitation across eight countries in the summer over the twentieth century. The gamma distribution has also been utilized for modeled data. For example, Wilby and Wigley (2002) estimated two parameters of the gamma distribution for daily precipitation in North America simulated by GCMs under anthropogenic forcing. Changes in the two parameters, representing the shape and scale of the gamma distribution, were consistent with the observed trends. More recently, Lui et al. (2019) used the gamma distribution for characterizing daily precipitation during Asian summer monsoons on the basis of GCM data. Interestingly, they defined simulated precipitation into two categories: that associated with tropical cyclones and others. Overall, precipitation not associated with tropical cyclones contributed appreciably to future changes in extreme precipitation.
The gamma distribution has also been used to calculate probable precipitation on an hourly time scale (Cho et al. 2004; Baldwin et al. 2005; Guinard et al. 2015). For example, Cho et al. (2004) examined the spatial characteristics of hourly precipitation in wet regions and found that the frequency distributions of hourly rainfall were represented better by the gamma distribution than the lognormal distribution. Guinard et al. (2015) examined projected changes in the spatial structure of precipitation in fields over North America. They fitted the gamma distribution to the intensity of hourly precipitation simulated by an RCM. Two parameters, representing the shape and scale of the gamma distribution, were employed to obtain information on the spatial heterogeneity of the structure of precipitation.
In Japan, several studies have used gamma distributions for probable precipitation on shorter time scales (i.e., daily and hourly). For example, in Suda (1991), PDFs of accumulated precipitation at 1- and 24-h intervals followed a gamma distribution. In their study, probable precipitation was estimated by applying gamma distributions to the observational data in Japan. They found that geographical distributions of probable precipitation on daily and hourly time scales were associated with topography and latitude, respectively, indicating that the duration and intensity of precipitation depended on geographical factors.
In this study, we evaluate goodness of fit of precipitation data obtained from observations and RCM simulations to statistical distributions such as PDF and CDF. First, a difference between the statistical distribution calculated from sample data and that derived from a theoretical basis was established. Next, we compared the degree of differences in two precipitation datasets; one with the contribution of tropical cyclones, and the other without. We focused on precipitation in relation to daily time scales where accumulated precipitation data in a 24-h period was used. As a statistical distribution suitable for daily precipitation, the gamma distribution was used in this study. Moreover, future changes in the two parameters of gamma distribution were examined, and the regional dependence of these changes in Japan was analyzed.
This paper is organized as follows: section 2 describes the data used in this study and analysis methods, including a description of the gamma distribution and section 3 presents our results with respect to goodness of fit to the gamma distribution in terms of the observed and simulated precipitation on a daily time scale. In particular, this section made a comparison between the results obtained from precipitation data with and without the contribution of tropical cyclones. Section 4 addresses the range of two parameters of the gamma distribution for our results and presents changes in these parameters under a warmer climate, in addition to changes in no-rain days. Finally, a summary of the main results of this study is presented in section 5.
2. Data and methods
a. Observational data
Rain gauge observations were used to derive measurements for daily accumulated precipitation. Observational data were acquired from the Automated Meteorological Data Acquisition System (AMeDAS) developed by Japan Meteorological Agency (JMA). The AMeDAS consists of a network of stations throughout Japan. The mean distance between two stations is approximately 17 km. The number of AMeDAS stations where precipitation data are available is approximately 1200. Observational data accumulated over 20 years, from September 1980 to August 2000 corresponding to the integration period of the model simulation as will be described later, were used in this study. For each AMeDAS station, records that contained less than 50% missing data over 20 years were taken into consideration when determining information usability. When comparing simulated and observational precipitation, model data corresponding to the land grid closest to each AMeDAS station were extracted.
b. Model data
Gridded datasets for the present and future climates over Japan (Murata et al. 2015) were used. These simulations were conducted using the nonhydrostatic regional climate model (NHRCM; Sasaki et al. 2008) developed by the Meteorological Research Institute (MRI) of JMA, based on the JMA nonhydrostatic model (JMA-NHM; Saito et al. 2006, 2007). NHRCM has successfully simulated regional climates over Japan (Sasaki et al. 2011; Murata et al. 2017; Kawase et al. 2020; Nosaka et al. 2019). The model domain in this study covered almost all of Japan. The four corners of the model domain were 26°N, 113°E; 15°N, 132°E; 57°N, 140°E; and 38°N, 162°E. The simulation ran for 20 years for each experiment: from 1 September 1980 to 31 August 2000 for the present climate and from 1 September 2076 to 31 August 2096 for the future climate. For the future climate, the RCP8.5 scenario was used. Boundary conditions for the NHRCM simulations were provided using an AGCM with a 20-km horizontal resolution (MRI-AGCM3.2S, hereafter referred to as AGCM20; Mizuta et al. 2012). See Murata et al. (2015) for further details.
c. Precipitation with and without the influence of tropical cyclones
The best tracking data provided by Regional Specialized Meteorological Center (RSMC)-Tokyo were used to identify the positions of tropical cyclones using observational data. The scheme developed by Murata et al. (2019) was utilized for tropical cyclones in the model data. This scheme, proposed for detecting tropical cyclones in high-resolution climate models, uses the relationship between two quantities representing the radial gradient and the tangential asymmetry in terms of mid- to upper-level thickness around a simulated vortex. The ability to distinguish tropical cyclones from extratropical cyclones is crucial to the accurate detection of tropical cyclones, located at midlatitudes around Japan.
Precipitation influenced by a tropical cyclone was defined as precipitation located within 500 km of the center of a tropical cyclone. This is the same definition used by Watanabe et al. (2019) and is consistent with previous studies (e.g., Kamahori 2012; Dare et al. 2012; Kitoh and Endo 2019). Precipitation without the influence of tropical cyclones was defined as precipitation that does not meet this criterion.
d. Data analysis method
Next, the parameters for each station were estimated using a precipitation sample. The maximum likelihood estimation was applied in order to estimate these parameters, although there are several other methods for estimating the gamma distribution parameters as well. The algorithm for the maximum likelihood estimation proposed by Minka (2002) was used. Although approaches proposed by Thom (1958) and Greenwood and Durand (1960) were also used, but the results did not change. Previous research on the application of maximum likelihood estimation to precipitation data includes Wilks (1990) and Husak et al. (2007).
Finally, the CDF for each station was calculated using the obtained parameters, k and θ. This CDF, followed by the gamma distribution, was compared with the CDF derived directly from the frequency distribution of a sample. For comparison, the goodness of fit between the two types of CDF (the theoretical and sampled), was examined, where the theoretical CDF was deduced from the estimated parameters regarding gamma distribution, and the empirical CDF was derived directly from the frequency distribution of a sample. As a goodness of fit index, root-mean-square error (RMSE) was used, although there are also a variety of other methods for evaluating the goodness of fit. Specifically, the squared difference in cumulative probability between the theoretical and empirical CDFs was calculated for each bin of precipitation, and the root mean was derived from the obtained square differences.
3. Results
a. Examples of a CDF for observed daily accumulated precipitation
Using observational data, several examples of theoretical and empirical CDFs are shown. Figure 1 shows these CDFs for observed daily accumulated precipitation associated with tropical cyclones at an observation station called Okinoerabu, located about 70 km northeast of the main island of Okinawa, in August in all years. Although these curves represent exceedances (1 − CDF), they will be referred to as CDF for convenience. For this plot, data sampled in August are selected considering August is the most representative month in the tropical cyclone season around Japan. The CDF for total precipitation (i.e., precipitation associated and not associated with tropical cyclones) is also shown. At Okinoerabu (Fig. 1), tropical cyclone precipitation accounted for approximately 52% of the total amount of precipitation.
CDFs of daily precipitation at Okinoerabu, located about 70 km northeast of the main island of Okinawa, Japan. The probability of exceeding a given precipitation in August is shown. Curves with and without open circles represent those derived from actual data sampled from simulations and data obtained based on Eq. (1), respectively. Red curves represent precipitation not associated with tropical cyclones, and black curves represent the whole precipitation, respectively.
Citation: Journal of Hydrometeorology 21, 12; 10.1175/JHM-D-20-0068.1
The empirical curve of precipitation not associated with tropical cyclones more closely resembles the theoretical curve at Okinoerabu, when compared with the relationship between the empirical and theoretical curves in total precipitation (Fig. 1). In other words, precipitation sampled without the effects of tropical cyclones has a better fit to the gamma distribution, when compared with total precipitation sampled. The values of RMSE is 0.009 when precipitation associated with tropical cyclones is excluded, whereas 0.018 when not excluded. This result indicates that the gamma distribution provides a better fit when data associated with tropical cyclones are excluded.
b. RMSE for all stations
In general, there is a tendency toward increasing goodness of fit when the contribution of tropical cyclone precipitation to the total precipitation is larger. Figure 2a shows the scatterplot of rRMSE versus rTCP in August in all years. It is found that rRMSE is negative and its magnitude is larger across a range of a higher rTCP, indicating that goodness of fit in precipitation increased by excluding precipitation data associated with tropical cyclones. The result qualitatively holds true for other time periods during which tropical cyclones approach Japan, such as July, September, and October (not shown). In these months, the values of rRMSE are more negative as rTCP is larger.
Scatter diagram of rTCP in Eq. (3) and rRMSE in Eq. (2) in August for the entire region of Japan derived from the AMeDAS observational data using (a) the maximum likelihood estimation and (b) the moment method.
Citation: Journal of Hydrometeorology 21, 12; 10.1175/JHM-D-20-0068.1
Sensitivity of these results to the choice of methods for estimating the parameters of the gamma distribution was examined. Figure 2b shows the same scatterplot as Fig. 2a but using the moment method instead of the maximum likelihood estimation. It is found that rRMSE is negative and its magnitude is larger across a range of a higher rTCP, similar to the case of the maximum likelihood estimation. Our conclusion that the goodness of fit improves with the rate of tropical cyclone precipitation is insensitive to the choice of methods to calculate parameters of a gamma distribution.
Sensitivity of the results to the choice of methods for measuring a goodness of fit was also investigated. An analysis using Kolmogorov–Smirnov distance or correlation coefficient was conducted. Our conclusion that the goodness of fit improves with the rate of tropical cyclone precipitation is insensitive to the choice of methods for measuring a goodness of fit (see appendix).
c. Results from the present climate simulation
Goodness of fit in precipitation data obtained from the NHRCM simulations was investigated in the same manner as the observational data. The limited number of the NHRCM data was selected in order to compare the results obtained from the observational data. The selected NHRCM data consisted of locations that corresponded to grid points closest to the AMeDAS stations.
Similar to the observational data results, when using the model data, there was a tendency toward higher goodness of fit in precipitation when there was a large contribution of tropical cyclone precipitation to the total precipitation. Figure 3a shows the same scatterplots as Fig. 2a, but used the model data obtained from the present climate simulation instead of the observational data. The values of rRMSE are negative and its magnitude is larger with a higher range rTCP. The result qualitatively holds true for other time periods during which tropical cyclones approach Japan, such as July, September, and October (not shown).
Displays results similar to Fig. 2a, but derived from the model data for (a) the present climate and (b) the future climate (C0).
Citation: Journal of Hydrometeorology 21, 12; 10.1175/JHM-D-20-0068.1
d. Results from the future climate simulations
Results similar to the present climate simulation were also obtained in future climate simulations. Figure 3b shows the same scatterplots as Fig. 3a, but used the data obtained from future climate simulations, instead of present climate simulations. The result from one simulation (C0) in August is shown in Fig. 3b.
Similar to the present climate simulation data, there was a tendency toward enhanced goodness of fit in precipitation with a large range of contributions of tropical cyclone precipitation to total precipitation using future climate data. The result qualitatively holds true for other experiments, such as C1, C2, and C3 (not shown).
4. Discussion
a. The range of parameters in the gamma distribution
The parameters of the gamma distribution, k and θ in Eq. (1), are useful to diagnose the forms of PDF and CDF. For instance, how probable is extreme precipitation can be estimated by examining the ranges of these parameters. In fact, k and θ are called the shape and scale parameters, as previously mentioned. A scatter diagram can be plotted in order to present the relationship between k and θ. For this plot, data sampled over the entire region (i.e., Japan as a whole) in August were selected.
It is found, from the observational data, that the ranges of the two parameters, k and θ, are limited (Fig. 4a). That is, the parameter k ranges from 0.5 to 1.0 and the parameter θ from 5 to 50 with few exceptions. There is a relationship between these parameters; θ decreases with an increase in k. Each parameter pair determines the form of a CDF. Several CDFs corresponding to selected parameter pairs are displayed in Fig. 5. The selected values of (k, θ) are (0.6, 50), (0.7, 30), (0.8, 20), and (0.9, 15). A comparison between the CDF forms was derived from (0.6, 50; blue line) and (0.9, 15; black line) showing that the former CDF (blue line) has larger values at a range of larger amount of precipitation when compared with the latter CDF (black line). This result indicates that the rate of heavier precipitation for the parameter pair of (0.6, 50) is higher than that for (0.9, 15). For instance, the value of CDF at 80 mm day−1 is about 0.1 when (k, θ) is (0.6, 50) and approximately 0.0 when (k, θ) is (0.9, 15).
Scatter diagram of k and θ in August for the entire region of Japan derived from (a) the AMeDAS observational data, (b) the model data for the present climate, and (c) the model data for the future climate (C0).
Citation: Journal of Hydrometeorology 21, 12; 10.1175/JHM-D-20-0068.1
CDFs of daily precipitation based on Eq. (1). The probability of exceeding a given precipitation is shown. Each curve is derived from a pair of typical values of k and θ.
Citation: Journal of Hydrometeorology 21, 12; 10.1175/JHM-D-20-0068.1
The distributions of the parameters, k and θ, for the present and future climate simulations are similar to that for the observational data. Figures 4b and 4c show the same scatterplots as Fig. 4a, but for experiments of the present climate and one of future climate, C0. Each scatter diagram for the present and future climate exhibits a decrease in θ with an increase in k, denoting the same tendency shown in the case of observational data. The future climate appears to have wider parameter space compared with that in the present climate, particularly at a range of θ more than 60. To clarify this point, changes in these parameters at each location were examined.
b. Changes in the gamma distribution parameters
Changes in the gamma distribution parameters of precipitation without the effects of tropical cyclones are examined (Figs. 6 and 7). Nontropical cyclone related precipitation is also important because Japan has experienced many disasters associated with such precipitation (e.g., precipitation related to baiu front in the rainy season). Before calculating changes in the parameters, daily precipitation data over the entire region of Japan are aggregated. Confidence intervals are derived from the 2.5th and 97.5th percentiles of 10 000 bootstrap samples. Each sample consists of 20 years of data, which are randomly selected from the original 20 years, where duplicated years are allowed.
Scatter diagram of future changes in k and θ derived from the model data over the entire region of Japan. The future change is defined as the difference between the present and future climates (future minus present). For each month, data for four members, C0, C1, C2, and C3, are plotted. Error bars for each data point denote two-sided 95% confidence interval of changes in k and θ, respectively.
Citation: Journal of Hydrometeorology 21, 12; 10.1175/JHM-D-20-0068.1
As in Fig. 6, but for each region of Japan: (a) the Sea of Japan side located in northern Japan, (b) the Pacific side of northern Japan, (c) the Sea of Japan side in eastern Japan, (d) the Pacific side of eastern Japan, (e) the Sea of Japan side in western Japan, (f) the Pacific side of western Japan, and (g) the Nansei Islands.
Citation: Journal of Hydrometeorology 21, 12; 10.1175/JHM-D-20-0068.1
Changes in θ, averaged over all locations, were in positive and changes in k were negative for all experiments (Fig. 6). That is, the averaged θ(k) in the future climate was larger (smaller) than that in the present climate. This means that the probability of heavy precipitation for the future climate is higher compared with that for the present climate considering there is a constraint condition between these two parameters (i.e., data are in a limited region of the parameter space, such as in Fig. 4). This result indicates more frequent occurrences of heavy precipitation in warmer climates. The result is evidence for the constraint condition between the two parameters. This constraint condition, indicating that extremes increase faster than the mean, has been pointed out in previous studies (e.g., Allen and Ingram 2002; Chou and Neelin 2004; Held and Soden 2006; Pendergrass and Hartmann 2014). These studies have demonstrated the magnitude of increases in the mean precipitation, about 1%–3% K−1, is less than Clausius–Clapeyron scaling, about 6%–7% K−1. Our results are consistent with these studies on more frequent occurrence of heavy precipitation when compared with the mean precipitation.
The relationship between changes in the two parameters k and θ, averaged over all locations, depends on local regions in Japan. Figure 7 shows the same scatterplots as in Fig. 6, but used data in each local region of Japan. The entire region (i.e., Japan as a whole) was divided into seven local climate regions (Fig. 2 in Murata et al. 2015). It was found that the magnitude of θ is approximately proportional to that of k change for all local regions. However, the magnitude of the constant of proportionality depends on local regions. The magnitude of the constants is relatively low on the Sea of Japan side of northern local region of Japan (Fig. 7a), whereas it is relatively high on the Pacific side of western local region of Japan (Fig. 7f). On the Sea of Japan side of the northern local region (Fig. 7a), data points are located on a relatively lower left part of the second quadrant when compared with data from other local regions (i.e., the magnitude of k change is relatively high and the magnitude of θ change is relatively low). In contrast, on the Pacific side of western local region of Japan (Fig. 7f), data are located on the upper-right part of the second quadrant when compared with data for other local regions (i.e., the magnitude of k change is relatively low and the magnitude of θ change is relatively high).
The relationship dependence between parameters k and θ in local regions indicates a difference in future changes in the statistical distribution of precipitation in these regions. The dominance of changes in parameter k, called the shape parameter, on the Sea of Japan side of northern local region (Fig. 7a) suggests that the shape of CDF will also change in the future climate. In fact, a reduction in the magnitude of k alternates the shape of the CDF when there is little change in θ. For instance, a change in k from 1.299 to 1.065 with a slight change in θ from 7.43 to 10.30 for C0 in October leads to an increase in precipitation ranging from 20 to 50 mm (Fig. 8a). It should be noted that heavy precipitation greater than 100 mm also increases although these heavy precipitation events do not occur frequently (Fig. 9a). In contrast, the dominant parameter for the change is θ, the scale parameter, on the Pacific side of western local region (Fig. 7g). This result suggests that the scale of CDF also changes in the future climate. In fact, an increase in θ shows a scale transition in CDF when k shows little change. For instance, a change in θ from 29.85 to 40.80 with a slight change in k from 0.703 to 0.611 for C0 in July leads to an increase in a wider range of precipitation, including amounts more than 100 mm (Figs. 8b and 9b).
CDFs of daily precipitation based on Eq. (1) for (a) the Sea of Japan side of northern local region for C0 in October, and (b) the Pacific side of western local region for C0 in July. The probability of exceeding a given precipitation is shown. A blue (red) curve for each panel represents the present (future) climate with two-sided 95% confidence interval in light blue (orange) lines.
Citation: Journal of Hydrometeorology 21, 12; 10.1175/JHM-D-20-0068.1
As in Fig. 8, but both the x and y axes are logarithmic.
Citation: Journal of Hydrometeorology 21, 12; 10.1175/JHM-D-20-0068.1
Previous studies for other regions of the world have shown that the scale parameter of the gamma distribution for precipitation increases with temperature (e.g., Groisman et al. 1999; Wilby and Wigley 2002; Watterson and Dix 2003; Martinez-Villalobos and Neelin 2018, 2019). Based on these findings, the studies have demonstrated that frequencies of heavy and extreme precipitation increase greatly when compared with those of light and moderate precipitation. These results are partly consistent with our results on the dominance of changes in the scale parameter on the Pacific side of western local region. However, not all local regions of Japan show the dominance of changes in the scale parameter. Changes in the shape parameter are dominant on the Sea of Japan side of northern local region. This is a finding different from the previous studies.
c. Sensitivity of difference in the number of no-rain days between observational and model data for the present climate
A sensitivity analysis of difference in the number of no-rain days between the observational and model data, for the present climate, was conducted. To do so, the number of rain data in the observational or model data, whichever is smaller, is scaled up to that of the other group (i.e., observational or model data) by replacing no-rain data to rain data whose daily precipitation is assumed to be 1 mm.
Each scatter diagram for the observational (Fig. 10a) and model data (Fig. 10b) exhibits a decrease in θ with an increase in k, denoting the same tendency shown in the case of the original data (Figs. 4a,b). The relationship between the two parameters (Fig. 10) is similar to that in the original one (Fig. 4) for both observational and model data, indicating that our results are robust to the difference in the number of no-rain days between the two datasets.
(a) As in Fig. 4a, but the number of rain data is scaled up to that of the model data (present climate) by replacing no rain with 1 mm when the number of rain data is smaller than that of the model data. (b) As in Fig. 4b, but the number of rain data is scaled up to that of the observational data by replacing no rain with 1 mm when the number of rain data is smaller than that of the observational data.
Citation: Journal of Hydrometeorology 21, 12; 10.1175/JHM-D-20-0068.1
d. Changes in the number of no-rain days
Changes in the number of no-rain days without the effects of tropical cyclones are examined (Figs. 11 and 12). The same method as section 4b is used to derive confidence intervals, based on the 2.5th and 97.5th percentiles of 10 000 bootstrap samples. Each sample, consisting of 20 years of data, are randomly selected from the original 20 years, where duplicated years are allowed. Changes in the number of no-rain days, averaged over all locations, were in positive for all experiments except for September (Fig. 11). In September, changes in three experiments out of four were positive. That is, the averaged no-rain days in the future climate was larger than that in the present climate overall.
Diagram of future changes in no-rain days per month derived from the model data over the entire region of Japan. The future change is defined as the difference between the present and future climates (future minus present). For each month, data for four members, C0, C1, C2, and C3, are plotted. Error bars for each data point denote the two-sided 95% confidence interval.
Citation: Journal of Hydrometeorology 21, 12; 10.1175/JHM-D-20-0068.1
As in Fig. 11, but for each region of Japan: (a) the Sea of Japan side located in northern Japan, (b) the Pacific side of northern Japan, (c) the Sea of Japan side in eastern Japan, (d) the Pacific side of eastern Japan, (e) the Sea of Japan side in western Japan, (f) the Pacific side of western Japan, and (g) the Nansei Islands.
Citation: Journal of Hydrometeorology 21, 12; 10.1175/JHM-D-20-0068.1
The magnitude of increases in no-rain days depends on local regions in Japan. Figure 12 shows the same diagram as in Fig. 11, but used data in each local region of Japan. In the Pacific Ocean side of eastern and western local regions, increases in no-rain days are noticeable in July and August (Figs. 12d,f). In contrast, remarkable increases in no-rain days occur in September and October over the northern local regions (Figs. 12a,b). These months of each local region correspond to those of local rainy season, suggesting that the noticeable increases in no-rain days are associated with strength or transition of summer and autumn rain fronts.
5. Summary and concluding remarks
The statistical distribution of daily accumulated precipitation with and without the influence of tropical cyclones were examined using data from rain gauge observations and regional climate model simulations. Both data projected for a warmer climate based on the RCP8.5 scenario and simulated data representing the present climate were used as model simulation data.
Effects of the shortage of sampled tropical cyclones on statistical distributions of precipitation were evaluated. To do so, a CDF created from a precipitation data sample for each observation station was compared using a theoretical CDF calculated from two parameters of the gamma distribution. These parameters were estimated based on sampled precipitation data using the maximum likelihood estimation. The goodness of fit between the empirical and theoretical CDFs was examined. As a goodness of fit index, RMSE was used.
In general, there was a tendency toward obtaining a minimal RMSE when precipitation data associated with tropical cyclones was removed. That is, when the tropical cyclone influence was excluded, the empirical gamma precipitation curve more closely resembled the theoretical curve, when compared with the relationship between the empirical and theoretical curves including tropical cyclone influence. The reduction of RMSE was enhanced when there was a larger contribution of tropical cyclones to the total precipitation.
The ranges of the two parameters of the gamma distribution (k and θ) were limited for both observations and model simulations: θ decreased as k increased. The limited ranges of the two parameters and the specific relationship between them determined the shape and scale of the CDF of the gamma distribution.
Changes in the two parameters of the gamma distribution revealed that a CDF for future climate indicated more frequent occurrences of heavy precipitation in warmer climates. In general, there was a relationship between the changes in the two parameters, when the θ increased, the k decreased. The strength of this correlation depended on local regions in Japan. On the Sea of Japan side of northern local region, changes in parameter k were more dominant, suggesting that the shape of CDF changed in the future climate. However, on the Pacific side of western local region, the dominant parameter exhibiting change was θ, also suggesting that the scale of CDF changed in the future climate. It should be noted that this approach is useful for estimating precipitation changes in a simplified manner because only the two parameters are necessary for the estimation.
The number of no-rain days increased overall. However, the magnitude of increases depended on local regions of Japan. Increases in no-rain days were noticeable in earlier months (i.e., July and August) in the Pacific Ocean side of eastern and western local regions, whereas in later months (i.e., September and October) over the northern local regions.
This approach used in this study has a negative effect on estimated changes in precipitation: precipitation associated with tropical cyclones are not included in statistical distributions of projected precipitation. A possible solution to this issue is to obtain reliable future projections of tropical cyclone tracks from a large dataset, such as d4PDF (Mizuta et al. 2017), and to utilize them to estimate statistical distributions of precipitation associated with tropical cyclones. The horizontal resolution of these large ensemble datasets is relatively coarse and thereby not sufficient to resolve precipitation in the mountainous regions of Japan; therefore, only information pertaining to tropical cyclone tracks should be derived from these. Further study is required to develop a new strategy to include the effects of precipitation associated with tropical cyclones in statistical distributions of precipitation. When precipitation associated with tropical cyclone is included, caution regarding the tail of a statistical distribution should be needed. Previous studies have shown heavier tails, compared with gamma distributions, for heavy precipitation (e.g., Cavanaugh and Gershunov 2015; Cavanaugh et al. 2015).
Another issue that needs to be addressed is the relationship between moisture and a gamma distribution of precipitation. Recently, Martinez-Villalobos and Neelin (2019) has shown a connection between moisture, expected to increase under global warming, and the scale parameter of the gamma distribution. Increases in moisture with a warmer climate leads to increases in the scale parameter, and hence increases in precipitation extremes. Further study is required to examine the relationship between moisture and the parameters of the gamma distribution for precipitation in Japan.
Future work also includes assessments of uncertainties in climate change projections of precipitation. According to Hawkins and Sutton (2009, 2011), sources of uncertainty can be partitioned into three parts: internal variability of climate systems, model uncertainty, and scenario uncertainty. Our method of partitioning precipitation data into the two categories may be applicable to assessing precipitation uncertainties in regional climate model projections.
Acknowledgments
This research was supported by JSPS KAKENHI under Grant JP16K00526. Part of the data used was supplied by SOUSEI and TOUGOU programs under Grant JPMXD0717935561 of Ministry of Education, Culture, Sports, Science, and Technology of Japan. A part of the dataset used for this study was provided from the Japanese 25-year Reanalysis (JRA-25), the cooperative research project carried out by the Japan Meteorological Agency (JMA) and the Central Research Institute of Electric Power Industry (CRIEPI). The NUMPAC computer library programs of Nagoya University and a code produced by Dr. John Burkardt of the University of South Carolina were used for calculating functions related to gamma distributions.
Data availability statement
Gridded datasets for the present and future climates over Japan produced by NHRCM can be accessed online (https://www.diasjp.net/en/).
APPENDIX
Sensitivity of Goodness of Fit to Calculation Methods
Both methods indicate tendencies toward improving the goodness of fit when the contribution of tropical cyclone precipitation to the total precipitation is larger (Fig. A1). It is found that rKSD (rCR) is negative (positive) and its magnitude is larger across a range of a higher rTCP, indicating that the gamma distribution provides a better fit when data associated with tropical cyclones are excluded.
As in Fig. 2a, but using (a) Kolmogorov–Smirnov distance and (b) correlation coefficient, instead of RMSE.
Citation: Journal of Hydrometeorology 21, 12; 10.1175/JHM-D-20-0068.1
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