Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Supplementary Section Detailed Methodology The adopted GLS regression method is summarised below from Madsen et al. (2002) and Haddad et al. (2011). Consider ŷ i to be an estimate of the annual maximum series (AMS) parameter at station i. The following linear relationship is considered: p yˆ i 0 X k ik i i (A1) k 1 where X ik are predictor variables (climatic and physiographic characteristics), k are the regression coefficients, i is the random sampling error associated with ŷi , and i is the residual model error. To evaluate equation A1, the covariance structure of the sampling error must be known. The sampling error variances of the AMS parameters can be estimated by Monte Carlo simulations (Madsen et al., 2002). Estimates can be derived for the sampling error variances (diagonal of error covariance matrix) by substituting the population parameters by the sample estimates. It must be noted though, to solve the GLSR equations, the error covariance estimator should be independent, or nearly so, of the AMS parameter estimate ŷ i (Stedinger and Tasker, 1985). Following a similar approach as outlined by Madsen et al. (2002), estimates of the sampling error variance that is nearly independent of the three AMS parameters were obtained. For the estimation of inter-site correlation for the various parameters, we considered concurrent records of annual maximum rainfall series across all the sites within a selected region. The inter-site correlation between the sample mean values ij is equal to the correlation coefficient between concurrent rainfall events of sites i and j. The correlation between higher order sample moments depends on the order of the moment (Stedinger, 1983; Madsen et al., 2002). For the L-CV and L-SK estimates, the inter-site correlation coefficient is approximated by ij ij2 and ij ij3 2 3 respectively. The estimated cross correlation coefficients have reasonably large sampling uncertainties associated with them, especially if the concurrent record length is small. Relatively better estimates of cross correlation can be found when the sample 1 cross correlation coefficients are smoothed by relating them to the distance between sites. In this study, the following exponential correlation function was used: ij d ij d 1 ij (A2) where dij is the distance between sites i and j and and are parameters estimated from the data for each duration required. The Bayesian Information Criterion (BIC) and the Akaike Information Criterion (AIC) were used to evaluate the goodness-of-fit of the candidate distributions. The AIC has generally been used in hydrological applications to select the flood frequency model (Laio et al., 2009). However in this study the second order variant of AIC, called AICc was used (given by equation A3), where n is the sample size and P is the number of parameters of the desired probability distribution. AICc accounts for the biases in smaller sample size. As reported, AICc should be used when n/p < 40 to avoid bias (Calenda et al., 2009). AIC c - 2 (Y) 2 P n (n - P - 1) (A3) Where (Y) is the log-likelihood maximised function and P is the number of model parameters fitted to the available sample. In practice, after the computation of the AICc, for all of the operating models, one selects the model with the minimum AICc value. The BIC is very similar to the AIC, but is developed in a Bayesian framework: BIC - 2 (Y) ln( n) P (A4) The BIC penalizes more heavily for small sample size and for models with high values of P. Since (Y ) depends on the sample, the candidate models can be compared using AIC and BIC only if fitted on the same sample. In this study, the competing distributions were fitted to the same samples. It should be noted here that other goodness-of-fit tests (such as Chi-square and Kolmogorov-Simirnov tests) could have 2 been adopted (e.g. Rahman et al., 2013), but it was deemed to be adequate adopting the AICc and BIC tests in this study. A number of goodness-of-fit measures and statistical diagnostics were used to assess 2 the regression equations. A pseudo coefficient of determination ( RGLS ) (Reis et al. 2005) was used (defined in equation A5) as the traditional coefficient of determination made little sense with the GLSR as it neglects sampling variability portion of the total error. Outlier statistics and various diagnostics plots were used to identify outlier sites. 2 The RGLS is given by: 2 RGLS n[ˆ 2 (0) ˆ 2 (k )] ˆ 2 (k ) 1 nˆ 2 (0) ˆ 2 (0) (A5) where ˆ 2 (k ) and ˆ 2 (0) are the model error variances when k and no predictor 2 variables are used, respectively. In this case, RGLS measures the improvement of a GLSR model with k predictor variables against the estimated error variance for a model without predictor variables. The standardised residual (rsi) is the residual ri divided by the square root of its variance and was calculated as: rsi ri [i x i ( X T Λ 1 X) 1 xTi ]0.5 where λi is the diagonal of Λ, (A6) where is the error covariance matrix and xi is catchment characteristic at site i. To assess the adequacy of the estimated design rainfall the Z score was plotted as a quantile-quantile (QQ)-plot (See equation A7) (for easier interpretation) to assess if the underlying assumptions of normality of residuals were satisfied. 3 Z score log10 yi log10 yˆi (A7) 2 yi ˆ 2 yˆi Here the numerator is the difference between the at-site rainfall quantile and regional rainfall quantile (estimated from the developed regression equation) and the denominator is the square root of the sum of the variances of the at-site ( 2 yi ) and regional ( ˆ 2 ŷ ) rainfall quantiles in logarithm space. References Calenda G, Mancini CP, Volpi E. 2009. Selection of the probabilistic model of extreme floods: The case of the River Tiber in Rome. Journal of Hydrology 27: 1-11. Haddad K. Rahman A. Green J. 2011. Design Rainfall Estimation in Australia: A Case Study using L moments and Generalized Least Squares Regression. Stochastic Environmental Research & Risk Assessment 25(6): 815-825. Laio F, Di Baldassarre G, Montanari A. 2009. Model selection techniques for the frequency analysis of hydrological extremes. Water Resources Research 45: W07416.doi:10.1029/2007/WR006666. Madsen H, Mikkelsen PS, Rosbjerg D, Harremoes P. 2002. Regional estimation of rainfall intensity duration curves using generalised least squares regression of partial duration series statistics. Water Resources Research 38(11): 1-11. Rahman S.A, Rahman A, Zaman M, Haddad K, Ashan A, Imteaz MA. 2013. A Study on Selection of Probability Distributions for At-site Flood Frequency Analysis in Australia. Natural Hazards, 69: 1803-1813. Reis DS Jr, Stedinger, JR, Martins ES. 2005. Bayesian generalised least squares regression with application to log Pearson type 3 regional skew estimation. Water Resources Research 41: W10419 doi: 10.1029/2004WR003445. Stedinger JR. 1983. Estimating a regional flood frequency distribution. Water Resources Research 19(2): 503-510. Stedinger JR, Tasker GD. 1985. Regional hydrologic analysis, 1.Ordinary, weighted, and generalised least squares compared. Water Resources Research 22(9):1421-1432. 4