Theor Appl Climatol DOI 10.1007/s00704-011-0497-6
ORIGINAL PAPER
Design rainfall depth estimation through two regional frequency analysis methods in Hanjiang River Basin, China
Yue-Ping Xu & Chaofeng Yu & Xujie Zhang & Qingqing Zhang & Xiao Xu
Received: 19 March 2011 /Accepted: 4 August 2011 # Springer-Verlag 2011
Abstract Hydrological predictions in ungauged basins are 1 Introduction of significant importance for water resources management. In hydrological frequency analysis, regional methods are Under enhanced greenhouse conditions, the increases in regarded as useful tools in estimating design rainfall/flood the frequency and magnitude of hydro-meteorological for areas with only little data available. The purpose of this events like storm, flood, typhoon, and droughts are very paper is to investigate the performance of two regional likely (Fowler and Ekström 2009). This consequently methods, namely the Hosking’s approach and the cokriging increased public awareness of extreme events, particularly approach, in hydrological frequency analysis. These two for regions which are very sensitive to climate change. methods are employed to estimate 24-h design rainfall Determination of design storm/flood with a specified depths in Hanjiang River Basin, one of the largest return period therefore becomes important for flood risk tributaries of Yangtze River, China. Validation is made assessment, flood management and infrastructure design through comparing the results to those calculated from the under anticipated climate change. Climate change greatly provincial handbook approach which uses hundreds of increases the uncertainty already existed in design storm/ rainfall gauge stations. Also for validation purpose, five flood obtained by hydrological frequency analysis. In hypothetically ungauged sites from the middle basin are order to reduce the uncertainty in design storm/flood, chosen. The final results show that compared to the several possible solutions are available, including consid- provincial handbook approach, the Hosking’s approach eration of climate change (Fowler et al. 2007)in often overestimated the 24-h design rainfall depths while hydrological frequency analysis, appropriate choice of the cokriging approach most of the time underestimated. probability distributions and parameter estimation meth- Overall, the Hosking’ approach produced more accurate ods (Xu and Booij 2007), incorporating physical mecha- results than the cokriging approach. nism in frequency analysis (Pandey et al. 1998;Xuand Tung 2009), and use of “regional” or “prehistoric” informa- tion (Frances and Salas 1994; Benito and Thorndycraft 2005; Neppel et al. 2010). : : : Under climate change, it is increasingly important to Y.-P. Xu (*) X. Zhang Q. Zhang X. Xu investigate the impact of climate change on the frequency Institute of Hydrology and Water Resources, Civil Engineering, and magnitude of extreme hydro-meteorological events and Zhejiang University, Hangzhou, China associated uncertainties. For example, Jones and Reid e-mail: [email protected] (2001) studied possible changes in extreme precipitation intensities through frequency analysis using regional C. Yu climate model (with greenhouse gas changes following Shanghai Water Conservancy – Engineering Design & Research Institute, the IS92a scenario for 2080 2100) over Britain. Results Shanghai, China indicated dramatic increases in the heaviest precipitation Y.-P. Xu et al. events over Britain. Prudhomme et al. (2003) proposed the approach to analyze the 24- and 2-h design precipitation for implementation of appropriate methods that incorporate Washington State. Yang et al. (2010) investigated the spatial climate change uncertainty in flood risk assessment. Fowler patterns of the flood risks in the Pearl River Delta region and Ekström (2009)used13regionalclimatemodel using Hosking’sapproach. integrations from the Prediction of Regional Scenarios and The use of geostatistical methods is another prevailing Uncertainties for Defining European Climate change risks alternative to estimate design storm/flood for ungauged areas and Effects (PRUDENCE) ensemble together with extreme in recent years. For example, Daviau et al. (2000)used value analysis to assess changes to seasonal precipitation geostatistical methods to obtain map sets of L-mean and L- extremes in nine UK rainfall regions by 2070–2100 under CV for flood timing and magnitude. Nezhad et al. (2010) the SRES A2 emissions scenario. They found out that the employed residual kriging in physiographical space to handle multimodel ensembles project increases across the UK in effectively any possible spatial trends within the hydrological winter, spring, and autumn extreme precipitation; although variables for regional frequency analysis. Castiglioni et al. there is uncertainty in the absolute magnitude of increases, (2010) applied Physiographic-Space Based Interpolation and these range from 5% to 30% depending upon region and Top-Kriging for the regionalization of Q355 (i.e., the stream- season. In general, these researches show that the impact of flow that is equaled or exceeded 355 days in a year, on climate change can be significant in hydrological frequency average) over a broad geographical region in central Italy. analysis. These researches demonstrated the usefulness of geostatis- The second topic of interest in hydrological frequency tical methods in regional frequency analysis. analysis is the use of “regional” or “prehistoric/historic” The purpose of this paper is twofold: (1) to data. The data from a single site for frequency analysis are investigate the spatial patterns of design storm/rainfall usually insufficient to obtain accurate estimates of quan- depth for different return periods for Hanjiang River tiles. Therefore, regional and historical information can be Basin, one of the largest tributaries of Yangtze River, often incorporated into hydrological frequency analyses to China; (2) to compare the performance of two regional reduce the uncertainty in the design storm/flood estimation. frequency analysis methods, i.e., geostatistical method The use of prehistoric or historic data can be found in, e.g., and Hosking’s approach in design rainfall depth estima- Fenske (2003) and Reis and Stedinger (2005). Fenske tion in ungauged areas. The structure of this paper is as (2003) investigated the use of paleoflood data for risk follows. First, study area and data used in this study analysis of US Army Corps of Engineers projects and will be described. Second, L-moment approach used in found out that the usefulness of paleoflood data in flood regional frequency analysis is briefly described. Third, frequency analysis depends on their accuracy and uncer- the two regional frequency analysis methods will be tainty. Reis and Stedinger (2005) have employed Bayesian introduced. Afterwards an exploratory data analysis is methods to incorporate historic flood information in flood made followed by results. Finally discussion and frequency analysis. Similarly, the use of regional informa- conclusion are given. tion is also very useful for hydrological frequency analysis, particularly in ungauged basins (Hosking and Wallis 1997). It may reduce the sampling uncertainty and result in more 2 Study area and data robust estimates by introducing more data as long as homogeneous regions can be established. Frequency anal- The data used in this study is from Hanjiang River, one of ysis using regional information is widely found in the the largest tributaries of Yangtze River, China. This river is literature. Yu and Chen (1998) developed regional rainfall the source of the Middle Route of well-known South-to- intensity-duration-frequency (IDF) relationship by general- North Water Transfer Project. It stretches 1′577 km from izing the parameters of dimensionless frequency curves Shannxi to Hubei Provinces, returning Yangtze River in using regression equations. Yu et al. (2004) developed IDF Wuhan city. The river passes through Gansu, Sichuan, scaling formulas based on simple scaling concept combined Henan, and Chongqin Provinces. The Hanjiang River is with Gumbel distribution and parameters of IDF scaling located at 106°15′∼114°20′ east longitude, 30°10′∼34°20′ formulas were then regionalized based on the relationship north latitude, with its basin area of 159,000 km2.Itis between the scaling exponent and the average of annual 1- 820 km long from northwest to southeast, 320 km at the day maximum rainfall. Norbiato et al. (2007) characterized widest, and 180 km at the narrowest. The basin has a sub- the severity of a flash flood generating storm occurred on tropical monsoon climate and the average annual precipi- August 29, 2003 on the upper Tagliamento river basin using tation is about 700∼1,200 mm. More precipitation can be Hosking’s approach (Hosking and Wallis 1997). Wallis et al. observed in the south than the north, in the west than the (2007) combined the PRISM mapping system with Hosking’s east in the upper basin. In the middle and lower basin, the Design rainfall depth estimation in Hanjiang River Basin, China precipitation increases. It is noted that precipitation at combination of order statistics and known to be robust to mountainous region is much more than that at hillock areas outliers and virtually unbiased for small samples. Details and the plain. about the L-moment approach can be found in Hosking and Daily rainfall data of 20 gauge stations are obtained from Wallis (1997). The L-moment is given by: China Meteorological Administration (CMA) and Bureau of Hydrology (BOH), Yangtze Commission. New time Z1 Xr r k series of annual maximum daily data are then abstracted » ðÞ 1 ðÞr þ k ! l þ ¼ xðFÞP ðFÞdF ¼ b ; ð1Þ from these daily data. Figure 1 shows the locations of r 1 r ðÞ2ðÞ k k¼0 k! r k ! Hanjiang River Basin and 20 rainfall gauge stations, with 0 basic information shown in Table 1. r ¼ 0; 1; 2;
3 L-moment approach Pr r k R1 P»ðFÞ¼ ð 1Þ ðrþkÞ!Fk b ¼ xðFÞFrdF With r 2 and r ðk!Þ ðr kÞ! 3.1 L-moment theory k¼0 0 Where F(x) is a cumulative distribution function and Both regional frequency analysis approaches above are x(F) is the quantile function. based on the L-moment theory. L-moments are linear The first four L-moments are therefore:
l ¼ ½ l ¼ 1 1 EX ; 2 2 EXðÞ1=2 XðÞ2=2 ; ð Þ l ¼ 1 þ l ¼ 1 þ 2 3 3 EXðÞ1=3 2XðÞ2=3 XðÞ3=3 ; 4 4 EXðÞ1=4 3XðÞ2=4 3XðÞ3=4 XðÞ4=4
Fig. 1 Locations of Hanjiang River Basin and 20 gauge stations Y.-P. Xu et al.
λ Where X(k/n) is the kth order statistic of a random order L-moments by the scale measure 2. L-moment ratios sample of size n from the distribution F.TheL-CVis are defined as: defined as: τ = λ /λ . 2 1 t ¼ l =l ; r ¼ 3; 4; ... ð6Þ In practice, the L-moments can be obtained from a finite r r 2 sample. The unbiased sample L-moments are given by: Therefore, τ3 is called the L-skewness and τ4 is the L-kurtosis. Xr ðÞ 1 r k ðÞr þ k ! l þ ¼ b ; r ¼ 0; 1; ; n 1 ð3Þ A selection of different probability distributions can be r 1 ðÞ2ðÞ k k¼0 k! r k ! made through the so-called L-moment ratio diagram (Hosking Pn and Wallis 1997), whose axes are L-skewness and L-kurtosis, b ¼ n 1 ði 1Þði 2Þ...ðr kÞ x ; k ¼ ; ; ; n Where k ðn 1Þðn 2Þ...ðn kÞ i:n 0 1 1, respectively. Figure 2 presents the L-moment ratios diagram i¼Kþ1 covering the useful probability distributions. It is convenient which is an unbiased estimator of βk. The first four unbiased estimators are: for plotting sample at-site or regional average L-moment ratios for comparison with the population values of some nP1 ðÞ ¼ ¼ i 1 XðÞi:n commonly used frequency analysis. b0 X ; b1 nnðÞ 1 ; i¼2 ð4Þ nP2 ðÞ ðÞ Pn ðÞ ðÞ ðÞ ¼ j 1 j 2 XðÞi:n ¼ j 1 j 2 j 3 XðÞi:n b2 nnðÞ 1 ðÞn 2 ; b3 nnðÞ 1 ðÞn 2 ðÞn 3 j¼3 i¼4 4 Two regional rainfall frequency analysis methods
The sample moments can be obtained as: 4.1 Method 1: regional frequency analysis using cokriging approach l1 ¼ b0; l2 ¼ 2b1 b0; l3 ¼ 6b2 6b1 þ b0; ð5Þ The regional frequency analysis based on geostatistical l ¼ 20b 30b þ 12b b 4 3 2 1 0 methods is realized through the interpolation of at-site first four L-moments, which is more reasonable than 3.2 L-moment ratio diagram interpolation of probability distribution parameters. The procedure follows the five steps listed below: (1) Sometimes, it is convenient to define dimensionless version Screening of the data and quality checking; (2) At-site of L-moments, which is achieved by dividing the higher- probability distribution selection by the goodness-of-fit
Table 1 Basic information of rainfall gauge stations
Station No. Station name Period Longitude (°E) Latitude (°N) Height (m) Source
1 Shangzhou 1953∼2008 109.97 33.87 742.2 CMA 2 Xixia 1957∼2008 111.5 33.3 742.2 CMA 3 Nanyang 1952∼2008 112.58 33.03 129.2 CMA 4 Yunxie 1952∼1991, 2007∼2008 110.82 32.85 201.9 CMA 5 Fangxie 1958∼2008 110.73 32.05 434.4 CMA 6 Laohekou 1951∼2008 111.67 32.38 90 CMA 7 Zaoyang 1957∼2008 112.75 32.15 125.5 CMA 8 Zhongxiang 1952∼2008 112.57 31.17 65.8 CMA 9 Tianmen 1954∼2008 113.17 30.67 34.1 CMA 10 Wuhan 1951∼2008 114.13 30.62 23.3 CMA 11 Hanzhong 1951∼2008 107.03 33.07 508.4 CMA 12 Foping 1957∼2008 107.98 33.53 1087.7 CMA 13 Shiquan 1959∼2008 108.27 33.05 484.9 CMA 14 Ankong 1952∼2008 109.03 32.72 290.8 CMA 15 Guotan 1956∼2008 112.6 32.52 89 BOH 16 Xindianpu 1953∼2008 112.3 32.42 88 BOH 17 Yicheng 1930∼1938, 1951∼1989, 2003∼2008 112.58 31.75 60 BOH 18 Gucheng 1929∼1938,1947,1950∼2008 111.57 32.25 91 BOH 19 Huangjiagang 1955∼2008 111.53 32.53 115 BOH 20 Xiangyang 1931∼1938, 1947∼1990, 1992,2003∼2008 112.17 32.03 70 CMA Design rainfall depth estimation in Hanjiang River Basin, China techniques; (3) Interpolation of at-site L-moments; (4) identical apart from a site-specific scaling factor. The index- Extraction of L-moments for ungauged sites; (5) flood method (Dalrymple 1960) is a convenient way of Estimation of design rainfall depth. pooling summary statistics from various data samples.
As with any statistical analysis, the first stage of Suppose that observed data Qi,j are available at N sites, hydrological frequency analysis is screening of the data where i is the ith site and j indicates the sample size of to be used. Errors in measurements must be checked observed data at site i. The at-site flood can be estimated carefully. Afterwards, assumptions behind frequency through: analysis should be checked, i.e., observations are independent, identically distributed, and stationary. ð Þ¼m ð Þ; ¼ ; ; ð Þ Qi F iq F i 1 ... N 7 After checking the data quality, appropriate at-site probability distributions are selected based on the observations through goodness-of-fit techniques and Where Qi(F) is the at-site flood, μi is the index flood and parameters are estimated. During this step, L-moment q(F) is the regional growth curve which is a dimensionless approach and ratio diagram are used to identify the quantile function common to every site. distribution and parameters. The at-site L-moments are Regional frequency analysis by means of the index then interpolated using one of the geostatistical meth- flood procedure involves four steps: (1) screening of the ods, cokriging. Cokriging is an interpolation technique data and quality checking; (2) forming of homogeneous that supports the analysis of a primary and secondary sub-regions; (3) choice of a distribution using the dataset, mapped across the same region, using a goodness-of-fit techniques; (4) estimation of design straightforward extension of standard Kriging techni- rainfall depth/flood. ques (Kitanidis 1997). Finally, design rainfall depth can Besides checking errors and assumptions, screening of be estimated. the data and quality checking also includes the procedure of assessing if the data used are appropriate for regional 4.2 Method 2: regional frequency analysis frequency analysis. This is often done through computing through Hosking’s approach discordance measure which is measured in terms of the L- moments of the observation data (Hosking and Wallis The method proposed by Hosking and Wallis (1993, 1997) 1997). The idea for the analysis of the discordance measure is based on the assumption that data from sites within a is that sites with gross errors in the data will stand out from homogeneous region can be pooled to reduce the uncer- the other sites and regarded as discordant. tainty in single site estimates. A homogeneous region Following the data screening, the identification of means that the frequency distributions of the N sites are homogeneous regions is the next step, which is the most
Fig. 2 L-moment ratio diagram Y.-P. Xu et al.
Table 2 Z statistics and inclination rate for 20 rainfall gauge stations whether a region is homogeneous or not (Hosking and Wallis 1993; Fill and Stedinger 1995). Here, the heteroge- No. Stations Zs β neity measure proposed by Hosking and Wallis (1997)is 1 Shangzhou −0.6643 0.0447 used and described. Hosking and Wallis (1997) argued that 2 Xixia −0.5208 −0.1717 if the observed sites form a homogeneous region, this 3 Nanyang 0.1514 −0.1469 region’s population L-moment ratios are likely to be close 4 Yunxie 0.1084 −0.1840 to the average of the sample L-moment ratios of the data. 5 Fangxie −0.6173 0.0367 The heterogeneity measure can be therefore calculated 6 Laohekou 0.2281 0.1500 through: 7 Zaoyang 0.5208 0.1634 8 Zhongxiang 0.3304 0.0290 V1 m ¼ V1 ð Þ 9 Tianmen 0.0290 0.1405 H1 s 8 10 Wuhan −0.3756 −0.0571 V1 11 Hanzhong 0.3086 −0.2040 12 Foping −0.4577 −0.1864 Where V1 is the weighted standard deviation of the at- site sample L-CVs; mV and sV are mean and standard 13 Shiquan 0.4015 0.0880 1 1 14 Ankong 0.3580 0.3073 deviation of the simulated V1 from Kappa distribution, “ 15 Guotan −0.0153 −0.4660 respectively. The regions can be regarded as acceptably ” ≤ “ ” 16 Xindianpu 0.3675 −0.2352 homogeneous if H1 1, possibly heterogeneous if H < “ ” ≥ 17 Yicheng −0.0767 0.0283 1 1 2,and definitely heterogeneous if H1 2. Hosking and Wallis (1997) also proposed alternative ways 18 Gucheng 0.2598 0.0211 to construct a heterogeneity measure H where V is a 19 Huangjiagang −0.2089 −0.3044 2 2 measure based on L-CVand L-skewness, and H where V 20 Xiangyang 0.0131 0.1533 3 3 are is a measure based on L-skewness and L-kurtosis. After the homogenous regions are formed, appropriate difficult and subjective part of Hosking’s approach. The probability distributions are selected based on L-moment sites’ frequency distributions are assumed identical except a approach and design rainfall depth/flood can be estimated site-specific factor. Different tests can be used to check through the index-flood method.
Table 3 At-site 24-h design rainfall depths for different re- Stations 5 years 10 years 20 years 50 years 100 years 200 years 500 years turn periods and stations Shangzhou 68.674 79.101 88.585 100.250 108.632 116.744 127.153 Xixia 101.747 120.616 138.984 162.920 180.924 198.915 222.747 Nanyang 122.758 146.430 168.709 196.934 217.703 238.140 264.808 Yunxie 90.954 109.670 127.693 150.972 168.361 185.656 208.458 Fangxie 77.750 91.506 104.784 121.969 134.826 147.626 164.520 Laohekou 93.982 116.878 140.883 174.080 200.184 227.070 263.728 Zaoyang 132.354 162.219 190.463 226.390 252.910 279.068 313.276 Zhongxiang 132.258 157.609 181.300 211.133 232.981 254.407 282.274 Tianmen 120.929 144.502 168.193 199.882 224.194 248.822 281.874 Wuhan 160.318 199.540 238.259 289.300 328.031 366.971 418.856 Hanzhong 86.464 98.606 109.473 122.648 132.006 140.985 152.410 Foping 92.196 112.630 133.276 161.007 182.350 204.017 233.155 Shiquan 97.628 111.865 124.544 139.849 150.681 161.047 174.202 Ankong 87.242 104.676 121.190 142.225 157.770 173.113 193.194 Guotan 129.776 158.641 185.817 220.255 245.601 270.547 303.104 Xindianpu 117.140 150.474 183.517 227.224 260.476 293.967 338.666 Yicheng 106.127 128.230 149.569 177.186 197.850 218.424 245.579 Gucheng 96.876 115.437 133.369 156.591 173.973 191.285 214.143 Huangjiagang 91.111 111.745 132.586 160.572 182.108 203.967 233.361 Xiangyang 98.765 117.759 136.089 159.803 177.541 195.199 218.501 Design rainfall depth estimation in Hanjiang River Basin, China
Fig. 3 Interpolation maps of l1, l2, l3, and l4 for the Hanjiang (a) l1 River Basin
(b) l2
(c) l3
(d) l4 Y.-P. Xu et al.
Fig. 4 Interpolation maps of (a) 5y 24-h design rainfall depths (mm) for different return periods for the Hanjiang River Basin
(b) 10y
(c) 20y
(d) 50y Design rainfall depth estimation in Hanjiang River Basin, China
Fig. 4 (continued) (e) 100y
(f) 200y
(g) 500y
5 Exploratory data analysis most important methods to examine the stationarity of hydrological series. Here, the Mann–Kendall method is Before applying regional frequency analysis methods, it used for its nonparametric characteristics (Mann 1945; is important to check if there is any trend in original Yu e e t a l . 2002). A brief introduction of this method is annual maximum daily data. Trend test is one of the given below.
Table 4 Critical discordance measure coefficients given by Hosking and Wallis (1997)
No. of stations 5 6 7891011121314≥15
Critical value 1.333 1.648 1.917 2.140 2.329 2.491 2.632 2.757 2.869 2.971 3.000 Y.-P. Xu et al.
Table 5 Sample L-moments and discordance measure Station name Period l1/mm tt3 t4 Di coefficients Shangzhou 56 54.2786 0.1861 0.1848 0.1323 0.6671 Xixia 52 81.8769 0.1904 0.2349 0.2290 0.5879 Nanyang 57 94.1053 0.2180 0.2626 0.1537 0.6275 Yunxie 42 70.3286 0.2211 0.2597 0.1298 1.0135 Fangxie 51 62.7412 0.1843 0.2293 0.2369 0.8307 Laohekou 58 76.9707 0.2173 0.2874 0.2697 0.9500 Zaoyang 52 96.8942 0.2687 0.2335 0.1845 2.1563 Zhongxiang 57 100.7053 0.2272 0.2115 0.1349 0.3303 Tianmen 55 99.4036 0.1880 0.2202 0.2325 0.7222 Wuhan 58 121.4672 0.2454 0.3400 0.2470 1.4290 Hanzhong 58 68.6793 0.1797 0.1571 0.1023 1.2657 Foping 52 73.9865 0.2106 0.2804 0.2280 0.5722 Shiquan 50 76.4100 0.1919 0.1520 0.1163 0.9817 Ankong 57 66.6772 0.2246 0.2549 0.1717 0.1132 Guotan 53 94.8849 0.2697 0.2167 0.1227 2.1559 Xindianpu 56 84.7304 0.2926 0.3610 0.1877 2.8095 Yicheng 53 82.0189 0.2267 0.2273 0.1942 0.2705 Gucheng 67 76.6896 0.2043 0.2181 0.2391 0.8683 Huangjiagang 54 72.6963 0.2233 0.2471 0.2314 0.5390 Xiangyang 59 78.0119 0.2096 0.1770 0.1501 0.5981
In MK test, the test statistic is calculated as: The value of inclination rate β is calculated through: Xn 1 Xn xi xj ¼ ð Þ b ¼ Median ; 8j < i ð11Þ S sgn xj xk 9 i j k¼1 j¼kþ1 8 < 1 if xj xk > 0 Where j
Table 6 Heterogeneity measures
Homogenous regions Sites H1 H2 H3
Entire HRB 7.8993 −0.7115 0.3821 Upper HRB Shangzhou, Xixia, Yunxian, Hanzhong, Foping, Shiquan, Ankang −0.2330 −0.6589 −0.2257 Lower HRB Nanyang, Fangxie, Laohekou, Zaoyang, Guotan, Xindianpu, Yicheng, 0.9168 −1.2853 −1.4012 Gucheng, Huangjiagang, Xiangyang, Zhongxiang, Tianmen, Wuhan Design rainfall depth estimation in Hanjiang River Basin, China
Fig. 5 L-moment ratio diagram for upper and lower HRB
Table 3 shows the estimated at-site 24-h design rainfall observed at the far west of the basin. For high return depth for different return periods and stations. periods, low value of design rainfall depths can be also In the present study, assume that spatial L-moments are found in the upper stream of the basin. An increasing realizations of a stationary stochastic process. The four L- pattern to the west and east can be noted as well, except in moments are then interpolated by the cokriging approach, the far upstream, a decrease trend can be found. namely l1, l2, l3, and l4 shown in Eq. 5. Figure 3 shows the interpolated four moments for the Hanjiang River Basin. 6.2 Results from Hosking’s approach
The value of l1 is the lowest in the middle part of the basin (however, upper basin). Then it increases to the 6.2.1 Screening data through the discordance measure west and east. The highest value can be observed in the downstream of the Hanjiang River Basin. Similar spatial Screening the data is of importance to check if the data are patterns can be observed for l2 except that lowest value appropriate for regional frequency analysis. This is done by of l1 can be found at the north part of the river basin. The calculating discordance measures. Table 4 shows the critical spatial pattern of l3 is rather simple compared to the first discordance measure coefficient for different number of two moments. The value of l3 increases all the way from stations. the west to the east of the basin, i.e., from the upstream to Table 5 shows the discordance measure coefficients for downstream of the basin. The value of the fourth moment the 20 stations in the Hanjiang River Basin, together with increases along the main riveringeneralexceptthatin corresponding sample L-moments. As shown in Table 4, the the north part of the basin very low value can be critical coefficient is 3.00. Therefore, the results in Table 5 observed. show that the largest Di are 2.80 and all 20 stations pass the Figure 4a–g shows the spatial patterns of 24-h design discordance measure test, indicating no discordant site. rainfall depths for different return periods. For small return periods, low value of design rainfall depths can be found in 6.2.2 Test for regional homogeneity the middle part of the basin (mostly in the upper stream though). The value of design rainfall depths increases to the The hypothesis of homogeneity is that the single-site frequency west and east, respectively, with the highest value found in distributions are the same for the selected region except for a the far downstream. Meanwhile, slightly low value can be site-specific scale factor. Due to the limited number of gauge
Table 7 Index flood for the upper HRB 5 years 10 years 20 years 50 years 100 years 200 years 500 years
1.2862 1.5438 1.7981 2.1381 2.4018 2.6728 3.0444 Y.-P. Xu et al.
Table 8 Index flood for the lower HRB 5 years 10 years 20 years 50 years 100 years 200 years 500 years
1.2818 1.5467 1.8121 2.1717 2.454 2.7469 3.1523 stations, the whole Hanjiang River Basin (HRB) is divided into respectively. It can be observed that the index floods for two parts: Upper HRB and Lower HRB. The heterogeneity the lower HRB are slightly higher than those for the upper measures described before are calculated for different parts of HRB. the basin and shown in Table 6. According to Hosking and Tables 9 and 10 show the estimated 24-h design rainfall
Wallis (1997), H statistics based on V2 and V3 (Eq. 8) lack depths for the upper HRB and the lower HRB, respectively. power to discriminate between homogeneous and heteroge- Compared to the results from at-site frequency analysis neous regions. Therefore, here only H1 is used for the test. In (Table 3), it can be observed that the 24-h design rainfall the first glance, the entire basin fails to form a homogeneous depths estimated through the Hosking’s approach are region with H1=7.8993, significantly higher than 2. However, slightly higher for small return periods while for high the Upper HRB and the Lower HRB satisfy the hypothesis of return periods, sometimes more than 30% increase can be homogeneity with a value of H1=−0.2330 and 0.9168, observed. Particularly, large deviations (>20%) from at-site respectively. Therefore, these two regions are used for design results can be observed at Shangzhou, Hanzhong, and rainfall depth estimation. Shiquan in the upper HRB, Nanyang, Fangxie, Zhong- xiang, Xiangyang, and Gucheng in the lower HRB for 500- 6.2.3 Choice of a frequency distribution through L-moment year return period. ratio diagram 6.3 Validation through provincial handbook approach After homogeneous regions are formed, the next step is to for ungauged stations choose a frequency distribution for the regions. L-moment ratio diagram is used here to choose the appropriate 6.3.1 Location of ungauged stations distribution due to its simplicity. Figure 5 shows the corresponding L-moment ratio diagram for upper HRB In the following, the performance of two regional frequency and lower HRB respectively. The x coordinate is L- analysis methods is compared through selecting five skewness and the y coordinate is L-kurtosis. It can be hypothetical ungauged stations from the middle basin (part easily observed that the lognormal distribution is the most of lower HRB). The validation approach is taken from appropriate for the upper HRB while the generalized BOHHB (2008), referred to as provincial handbook Extreme-value distribution is the best option for the lower approach in this paper. Such approach is often available HRB. Formations of these two distributions can be found in for each province in China and uses hundreds of rainfall Hosking and Wallis (1997). gauge stations for regional frequency analysis. The proba- bility distribution used in the approach is again Pearson 6.2.4 Design rainfall depth estimation through index-flood Type III, recommended by MWR (2006). Isolines of method parameters of Pearson Type III are provided by such approach. Based on the parameters, 24-h design rainfall Design rainfall depth can then be estimated through the depths can be computed. The provincial handbook ap- index flood method introduced before. Tables 7 and 8 show proach is often used in hydrological analysis for ungauged the index flood used for the upper and lower HRB, regions. The main reason of using this approach for
Table 9 24-h design rainfall depth for the upper HRB (mm)
Stations l1 5 years 10 years 20 years 50 years 100 years 200 years 500 years
Shangzhou 54.279 69.811 83.794 97.597 116.055 130.366 145.078 165.244 Xixia 81.877 105.306 126.399 147.221 175.063 196.651 218.844 249.264 Yunxie 70.329 90.453 108.571 126.457 150.372 168.915 187.977 214.107 Hanzhong 68.679 88.332 106.025 123.491 146.845 164.953 183.569 209.085 Foping 73.987 95.158 114.218 133.034 158.193 177.700 197.754 225.243 Shiquan 76.410 98.275 117.959 137.391 163.374 183.521 204.232 232.621 Ankang 66.677 85.757 102.934 119.891 142.564 160.145 178.217 202.990 Design rainfall depth estimation in Hanjiang River Basin, China
Table 10 24-h design rainfall depth for the lower HRB (mm)
Stations l1 5 years 10 years 20 years 50 years 100 years 200 years 500 years
Nanyang 94.105 119.673 145.420 172.383 210.907 242.719 277.164 327.215 Fangxie 62.741 79.788 96.954 114.930 140.614 161.824 184.789 218.158 Laohekou 76.971 97.883 118.942 140.996 172.505 198.525 226.699 267.636 Zaoyang 96.894 123.220 149.730 177.492 217.158 249.912 285.378 336.912 Guotan 94.885 120.665 146.625 173.811 212.654 244.729 279.461 329.925 Xindianpu 84.730 107.751 130.934 155.210 189.896 218.539 249.553 294.617 Yicheng 82.019 104.303 126.743 150.243 183.819 211.545 241.567 285.189 Gucheng 76.690 97.526 118.508 140.481 171.875 197.800 225.871 266.658 Huangjiagang 72.696 92.448 112.337 133.166 162.926 187.500 214.109 252.773 Xiangyang 78.012 99.207 120.551 142.903 174.839 201.210 229.765 271.256 Zhongxiang 100.705 128.066 155.619 184.473 225.699 259.742 296.603 350.163 Tianmen 99.404 126.411 153.608 182.088 222.782 256.384 292.769 345.637 Wuhan 121.467 154.469 187.703 222.505 272.230 313.291 357.752 422.355
validation in this study is due to its numerous stations and 6.3.2 Validation results ease of use. Figure 6 shows the locations of five hypothetical Figure 7a–e shows the estimated 24-h design rainfall ungauged stations with their geographic information shown depths calculated by three different methods for five in Table 11. For the cokriging approach, the parameters of hypothetical ungauged locations selected. Hereafter, the Pearson Type III distributions for five stations are interpo- results from the provincial handbook approach are used lated from maps shown in Fig. 3. as a reference.
Fig. 6 Location of five hypothetic stations Y.-P. Xu et al.
Ta b l e 11 Longitude and latitude of five hypothetical ungauged accurate results than the cokriging approach. In the stations meanwhile, it can be observed that the deviations from Stations Longitude (°E) Latitude (°N) results of the provincial handbook approach for both methods increase with return periods. Yangriwan 110.8167 31.7333 Shahe 112.9500 32.2833 Sandaohe 111.8333 31.7833 7 Discussion and conclusions Zhangjiaji 112.7833 31.5167 Songjiaji 112.6667 31.7333 This paper investigated the performance of two regional frequency analysis methods in Hanjiang River Basin through five hypothetically ungauged stations from the middle basin From the figures it can be noted that, in most stations, of the river. Although there are different performances in the Hosking’s approach overestimated the 24-h design various sites for both methods, it can be concluded that the rainfall depths (Yangriwan, Shahe, Zhangjiaji, and Song- Hosking’ approach produced more accurate results compared jiaji) while the cokriging approach often underestimated with the cokriging approach. However, it must be kept in mind (Shahe, Sandaohe, Zhangjiangji, and Songjiaji). For Yang- that geostatistical approach considers the spatial correlation riwan Station, both regional frequency analysis methods explicitly and can present the spatial pattern of design rainfall overestimated the 24-h design rainfall depths while for depths of different return periods nicely, as shown in this Sandaohe, both methods underestimated. In general, it can paper. This is why it is still one of the prevailing approaches in be observed that the Hosking’ approach produced more hydrological field.
(a) Yangriwan (b) Shahe 400 400 350 350 300 300 250 250 200 200 150 150 100 100 50 50 Design rainfall (mm) 0 Design rainfall (mm) 0 5y 10y 20y 50y 100y 200y 500y 5y 10y 20y 50y 100y 200y 500y Return period(year) Return periods(year) Cokriging approach Hosking's approach Cokriging approach Hosking's approach Provincial handbook approach Provincial hangbook approach
(c) Sandaohe (d) Zhangjiaji 400 400 350 350 300 300 250 250 200 200 150 150 100 100 50 50 Design rainfall(mm) 0 Design rainfall(mm) 0 5y 10y 20y 50y 100y 200y 500y 5y 10y 20y 50y 100y 200y 500y Return periods(year) Return periods(year)
Cokriging approach Hosking's approach Cokriging approach Hosking's approach Provincial hangbook approach Provincial hangbook approach
(e) Songjiaji 400 350 300 250 200 150 100
Design rainfall(mm) 50 0 5y 10y 20y 50y 100y 200y 500y Return periods(year) Cokriging approach Hosking's approach Provincial handbook approach
Fig. 7 24-h design rainfall depths based on two regional frequency analysis methods and validation approach Design rainfall depth estimation in Hanjiang River Basin, China
One interesting topic in hydrological analysis is to indices: PSBI and Top-Kriging. Hydrol Earth Syst Sci Discuss – use the scale-invariant concept originated from the 7:7231 7261. doi:10.5194/hessd-7-7231-2010 Dalrymple T (1960) Flood frequency methods. US Geo Survey, finding of existence of multi-fractal nature in rainfall Washington, Water Supply Paper 1543 A process. Applications of the scale-invariant concept in Daviau JL, Adamowski K, Patry GG (2000) Regional flood frequency hydrological frequency analysis are recent. For example, analysis using GIS, L-moment and geostatistical methods. – Yu et al. (2004) have investigated regional rainfall IDF Hydrol Process 14:2731 2753. doi:10.1002/1099-1085 (20001030)14:15<2731::AID-HYP89>3.0.CO;2-U formulations for ungauged sites based on the scaling Fenske J (2003) Application of Paleohydrology to Crops flood property of rainfalls. In their study, the piecewise simple frequency analysis. Research report. 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Cambridge University Press, able to provide recommendations for infrastructure design Cambridge and water management. Jones PD, Reid PA (2001) Assessing future changes in extreme Finally, the impact of climate change on hydrological precipitation over Britain using Regional Climate Model integra- tions. Int J Climatol 21:1337–1356. doi:10.1002/joc.677 frequency analysis can be significant as described in the Kitanidis PK (1997) Introduction to geostatistics: applications to introduction. The next step of this research is to investigate hydrogeology. Cambridge University Press, Cambridge the impact of climate change on design rainfall depth Mann HB (1945) Nonparametric tests against trend. Econometrica – estimation for the Hanjiang River Basin. 13:245 259 MWR (The Ministry of Water Resources of the People’s Republic of China) (2006) Regulation for calculating design flood of water resources and hydropower projects. China Water Power Press, Acknowledgements This research is financially supported by Beijing, pp 44–2006 National Natural Science Foundation of China (Project No. Neppel L, Renard B, Lang M, Ayral PA, Coeur D, Gaume E, Jacob N, 50809058), International Science & Technology Cooperation Program Payrastre O, Pobanz K, Vinet F (2010) Flood frequency analysis of China (2010DFA24320) and Doctorial Foundation of Ministry of using historical data: accounting for random and systematic Education (Project NO. 200803351029). The authors also would like errors. Hydrol Sci J 55(2):192–208. doi:10.1080/ to thank China Meteorological Administration and Bureau of 02626660903546092 Hydrology, Yangtze Commission for providing precipitation data of Nezhad MK, Chokmani K, Ouarda BMJ, Barbet M, Bruneau P (2010) the Hanjiang River Basin. 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