Non-stationary Flood Frequency Analysis Using Cubic B-spline-based GAMLSS Model

Under changing environment,the most widely used non-stationary flood frequency analysis method is the Generalized Additive Models for Location,Scale and Shape(GAMLSS)model.However,the model structure of the GAMLSS model is relatively complex due to the large number of statistical parameters,and the relationship between statistical parameters and covariates is assumed to be unchanged in future,which may be unreasonable.In recent years,non-parametric methods have received increasing attention in the field of non-stationary flood frequency analysis.Among them,the linear quantile regression model(QR-L)and the non-linear quantile regression model of cubic B-spline model(QR-CB)have been introduced into the non-stationary flood frequency analysis studies because they do not need to determine statistical parameters and consider the relationship between statistical parameters and covariates.However,quantile regression models have difficulties in estimating non-stationary design flood,since the trend of the established model must be extrapolated infinitely to estimate design flood.Besides,the number of available observations becomes scarcer when the quantile regression model estimating design flood corresponding to higher return periods,leading to unreasonable design flood.Therefore,this paper proposes the cubic B-spline-based GAMLSS model(GAMLSS-CB)for non-stationary flood frequency analysis.In the GAMLSS-CB model,the relationship between statistical parameters and covariates is fitted by the cubic Bspline under the GAMLSS model framework.The GAMLSS-CB model combines the advantages of the GAMLSS model and cubic B-spline.The GAMLSS-CB model has a good performance and considers the complex relationship between statistical parameters and covariates.In addition,based on the GAMLSS-CB model,this paper adopts the Average Design Life Level(ADLL)method to estimate the design flood value.The annual maximum flood series of five stations in the Weihe River basin and the Pearl River basin are taken as examples.The main research contents and results of the paper are as follows:(1)Non-stationary analysis of the annual maximum flood series of five stations in the Weihe River Basin and the Pearl River Basin.In this paper,simple linear regression trend test,simple moving average trend test,Sen's slope estimation trend test,and MannKendall trend test are used to analyze the trend of annual maximum flood series of five stations.The Pettitt variation point test was used to analyze the variation points of the flood series,and the rationality of the variation positions of the flood series obtained by the Pettitt variation point test was verified by using the traditional T-test and the traditional F-test according to the mean value and variance respectively.This article assumes 95% confidence interval,according to the trend test analysis,it was found that the annual maximum flood series of Huaxian station,Xianyang station and Zhangjiashan station are significantly decreased,the annual maximum flood series of Gaodao station is significantly increased,and the annual maximum flood series of Dahuangjiangkou station shows an insignificant upward trend.Pettitt variation point test analysis shows that the variation point of Huaxian station occurred in 1985,the variation point of Gaodao station occurred in 1992,the variation point of Zhangjiashan station occurred in 1998,the variation point of Xianyang station occurred in 1981,and the variation point of Dahuangjiangkou station occurred in 1991.The traditional T-test and the traditional Ftest found that the variation position of flood series obtained by Pettitt variation point test is reasonable and reliable.(2)Construct non-stationary probability distribution models based on linearity and non-stationary probability distribution models based on nonlinearity: the linear-based GAMLSS model,the QR-L model,the QR-CB model and the GAMLSS-CB model.(3)The optimal model based on the linear/non-linear GAMLSS model is determined using AIC value,standard normal residual QQ graph,residual Worm graph and residual rank graph.This paper chooses Pearson ? distribution,Gamma distribution,Lognormal distribution,Weibull distribution,Gumbel distribution,Normal distribution and the generalized extreme value distribution as the candidate parameter distribution type of GAMLSS model.Comparing the performance of models based on different parameter distribution types,it is found that the linear-based GAMLSS model performs better with the Gamma distribution,Weibull distribution,and Lognormal distribution among the seven selected distribution forms.And the GAMLSS-CB model performs better with the Gamma distribution,Lognormal distribution,and Normal distribution among the seven selected distribution forms.Among the seven distributions selected by the two GAMLSS models,the Gumbel distribution is slightly worse.When the Gumbel distribution is selected as the parameter distribution,the AIC values is larger than the AIC values of other distributions.(4)Select the model probability coverage rate and Filliben correlation coefficient as the index items of the goodness of fit of model,comprehensively qualitatively and quantitatively compare the differences of the goodness of fit of models of the above four non-stationary probability distribution models.Qualitative analysis of model probability coverage rate shows that the goodness of fit of the QR-L model is the best model among the four non-stationary probability distribution models,followed by the GAMLSS-CB model,and the goodness of fit of the QR-CB model is relatively poor.Quantitative analysis of Filliben correlation coefficient shows that the order of the goodness of fit is: the GAMLSS-CB model> the linear-based GAMLSS model> the QR-L model> the QRCB model.Comprehensive qualitative and quantitative analysis,the GAMLSS-CB model is the optimal model among the four non-stationary probability distribution models,and the order of the goodness of fit is: the GAMLSS-CB model> the linear-based GAMLSS model> the QR-L model> the QR-CB model.(5)Comprehensive quantitative and qualitative analysis found that the GAMLSSCB model is the optimal model among the four non-stationary probability distribution models.Comparing the difference of the design flood estimated by the four non-stationary probability distribution models,it is found that the design flood estimated by the GAMLSS-CB model using the ADLL method is feasible,which can provide data basis for the construction of water conservancy projects.Comprehensive analysis of model goodness of fit and design flood value differences,this paper recommends using the GAMLSS-CB model when conducting non-stationary flood frequency research.