CN109408989B - A Calculation Method for Designing Flood Hydrographs - Google Patents

Abstract

The invention provides a calculation method for designing a flood hydrograph. The method includes the following steps: adopting the annual maximum value method to select flood peaks and flood volume sequences of multiple time periods; using Mann-Kendall method to perform trend analysis on flood peaks and flood volume sequences of multiple time periods , and according to the trend analysis results, the GAMLSS model is used for fitting; the time-varying Kendall coefficients of flood peaks and floods in each period are calculated, the ARIMA model is used to fit the Kendall coefficient sequence, and the Kendall coefficients are converted into Copula parameters according to the correlation coefficient method; Calculate the conditional distribution of each time period based on the univariate return period of flood peaks; zoom in on typical flood hydrographs. The method of the invention fully considers the time-varying of the flood extreme value sequence itself and the time-varying correlation between extreme value sequences, which is more in line with the actual situation and has higher reliability.

Description

A Calculation Method for Designing Flood Hydrographs

technical field

The invention relates to the field of engineering hydrology, in particular to a deduction method for designing a flood hydrograph.

Background technique

The premise of the current design flood hydrograph calculation method is that the flood extreme value series satisfies the assumption of stationarity. The process includes: the first step is to calculate the flood design value and fit the Pearson distribution function under the condition of consistency; the second step is to amplify the flood hydrograph, using the traditional same-frequency amplification method to treat the flood peak and flood volume as two independent variables. Amplify separately, regardless of the correlation between the two. In fact, due to human activities and climate change, the inconsistency of hydrological sequences has become increasingly prominent. The traditional Pearson distribution function cannot describe this change, and considering the inherent laws of hydrology, there is a certain correlation between flood peaks and flood volume sequences. Such correlations are also time-varying due to climate change, and flood peaks and flood volumes do not always appear at the same frequency in general. Therefore, if the traditional method is still adopted, it is easy to ignore the above two kinds of time variability, and the reliability of the results will be questioned, which often leads to overestimation or underestimation of flood risk.

At present, some scholars have considered the impact of environmental changes on flood extreme events, and have conducted research on the inconsistency of the distribution function of extreme value series, but have failed to consider that the correlation between extreme value series also changes with the climate environment. change occurs.

SUMMARY OF THE INVENTION

In order to solve the above-mentioned defects and deficiencies in the prior art, the present invention provides a calculation method for designing a flood hydrograph, which fully considers the time-varying of the flood extreme value sequence itself and the time-varying correlation between extreme value sequences. .

In order to solve the above-mentioned technical problems, the present invention adopts the following technical solutions:

An inference method for designing a flood hydrograph, including the following steps:

According to the existing flood sequences, select flood peaks and flood volume sequences of multiple periods;

Perform trend analysis on the selected flood peaks and flood volume sequences in multiple time periods, and according to the trend analysis results, use the GAMLSS model to fit the flood peaks with trend changes and flood volume sequences in multiple time periods;

Calculate the Kendall correlation coefficient between the flood peak and the flood volume in each period, and fit the Kendall correlation coefficient;

According to the fitting results, the Kendall correlation coefficient between the flood peak in a certain year and the flood volume in each period is predicted, and the Copula function and its parameters between the two variables are obtained according to the correlation coefficient method;

According to the GAMLSS model and the Copula function and its parameters, a joint distribution is established, and the univariate design value of the flood peak is used as the control condition to calculate the most probable design value of the flood volume in each time period based on the conditional distribution of the flood peak frequency;

A typical flood hydrograph is scaled up based on the obtained design values.

Further, when using the GAMLSS model to fit, the annual maximum rainfall is used as a covariate to fit, and an appropriate extreme value distribution function is selected to estimate the time-varying parameters of the model.

Further, the parameters in the GAMLSS model are fitted by the maximum likelihood method, the best is selected according to the AIC minimum criterion, and the fitting evaluation is performed by the Worm diagram and the QQ diagram.

Further, the Kendall correlation coefficient is calculated by the sliding window method, and the two variables of flood peak and flood volume in a certain control period are respectively X=(x ₁ , x ₂ . . . x _N ) and _Y =(y ₁ , y ₂ . ), where N represents the total length of the flood peak or period flood sequence, and the calculation formula is as follows:

In the formula, τ _t is the t-th Kendall correlation coefficient, 1≤t≤Nn, n is the length of the sliding window, x _i and y _i respectively represent the i-th value of the two random variables, t≤i≤t+n -1, x _j , y _j respectively represent the j-th value of the two random variables, t+1≤j≤t+n; sgn is the indicator function:

Think of the Kendall correlation coefficient as a time series.

Further, the ARIMA(p,d,q) autoregressive model is used to fit the Kendall correlation coefficient, where p is the autoregressive term, q is the number of moving average terms, d is the number of differences made when the time series becomes stationary, Include the following steps:

Step 1: Carry out the stationarity test on the Kendall correlation coefficient sequence, if it passes, go to the next step; if not, continue to differentiate the sequence until the differenced sequence satisfies the stationarity test, and use the unit root ADF test;

Step 2: Obtain the difference times d of the model through step 1; take the AIC information criterion as the criterion, combine the ACF and PACF diagrams to limit the range of the autoregressive term p and the number of moving average terms q, and traverse the (p, q) combination to find out the (p,q) combinatorial model for the smallest AIC value.

Further, the Copula function and its parameters between the two variables are obtained according to the correlation coefficient method, including: only considering the variable parameter Copula without considering the variable structure Copula, using the Gumbel-Houggard Copula in the hydrological sequence analysis for fitting, that is, the corresponding Copula parameters The calculation formula is:

In the formula, θ _t is the parameter of Copula at time t;

The corresponding Copula function expression is:

In the formula, u=F _X (x) represents the marginal distribution of flood peaks, and v=F _Y (y) represents the marginal distribution of floods in the time period.

Further, on the condition that the flood peak occurs once in a century, the most probable design value of the flood volume in each period under the conditional distribution is calculated.

Further, according to the calculated design value, it is divided into three sections to enlarge the typical flood hydrograph.

Further, the method further includes: smoothing the typical flood hydrograph.

Compared with the prior art, the present invention fully considers the time-varying time-varying of the flood peak and the flood volume of each period and the time-varying correlation of the two, and provides a more scientific and reasonable calculation that is more in line with the actual situation for the calculation of the design flood hydrograph. method, using this method to predict flood risk with higher reliability.

Description of drawings

Fig. 1 is the flow chart of the method of the embodiment of the present invention;

Figure 2 shows the fitted Kendall coefficients and predictions and confidence intervals over 10 years;

Among them, the solid line is the measured value of the Kendall coefficient, the dotted line is the model fitting value, the dotted line is the predicted value within 10 years, and the gray filled part is the 95% confidence interval;

Fig. 3 shows the comparison chart of the 100-year flood hydrograph in 2016 calculated by the method of the present invention;

Figure 4 shows a comparison chart of the 100-year flood hydrograph in 2017 calculated by the method of the present invention.

Detailed ways

The present invention will be further described below in conjunction with specific embodiments. The following examples are only used to illustrate the technical solutions of the present invention more clearly, and cannot be used to limit the protection scope of the present invention.

As shown in Figure 1, it is a flowchart of a method for estimating a design flood hydrograph provided by an embodiment of the present invention, taking the flood sequence of Xixian Station from 1951 to 2015 for a total of 65 years as an example:

Step (1): The annual maximum flood extreme value sample sequence (including flood peak, maximum 30-day flood volume, and maximum 60-day flood volume) is selected by the annual maximum value method.

Step (2): The Mann-Kendall method is used to analyze the trend of each extreme value sequence in step (1). The results show that each extreme value sequence presents different degrees of inconsistency.

Step (3): According to the analysis result of step (2), use GAMLSS model to fit the extreme value sequence with trend change:

g _k (θ _k )=X _k β _k

Among them, θ _k represents k distribution statistical parameters, and this embodiment mainly considers two parameters, the mean θ ₁ and the variance θ ₂ , that is, θ _k =(θ ₁ ,θ ₂ ), and β _k is a regression coefficient vector with a length of J _k (J The value of _k is equal to the number of additive items in the covariate function), X _k is an explanatory variable matrix of n×J _k , the explanatory variable selected in this embodiment is the maximum rainfall, and a linear equation is used as the parameter function form ( The cubic linear equation is selected here), so the explanatory variable matrix and regression coefficient vector are respectively:

Among them, P represents the maximum rainfall, and the relationship between the parameter θ _k in the distribution function and the covariate factor (the maximum annual rainfall P of Xinyang Station is selected) is established. The parameters in the model are fitted by the maximum likelihood method, and the model is based on the minimum AIC. Criterion is the best, and Worm diagram and QQ diagram are used for fitting evaluation. The results show that the best distribution functions for both are Gamma distribution, and the function form of parameters is linear function, and a cubic linear equation (that is, J _k is 3) is selected.

Step (4): Use the Kendall correlation coefficient to describe the correlation between the flood peak and the flood volume in each period, and use the sliding window method to calculate the Kendall coefficient over the past 65 years. Set the flood peak and the maximum 30-day flood volume (or the maximum 60-day flood volume) as two variables. are respectively X _{= ( x 1} , _{x 2} . In the example, N is 65, the sliding window length is 35 years, and the sliding step is 1 year, that is, τ ₁ represents the correlation coefficient from 1951 to 1985, τ ₂ represents the correlation coefficient from 1952 to 1986, and so on for a total of 30 Kendall coefficients, each Kendall coefficient is calculated by the following formula:

In the formula, τ _t is the t-th (1≤t≤Nn) Kendall coefficient, n is the sample length (that is, the sliding window length), in this embodiment, n is 35, when studying the flood peak and the maximum 30-day flood, x _i , y _i represents the i-th value (t≤i≤t+n-1) taken from the flood peak and the maximum 30-day flood, respectively, x _j and y _j represent the j (t+1≤ j≤t+n) values, also when studying the flood peak and the maximum 60-day flood. sgn is the indicator function, and the corresponding form is:

Step (5): Consider the Kendall coefficient as a time series, and use the ARIMA(p,d,q) autoregressive model to fit the Kendall coefficient, where p is the autoregressive item, q is the number of moving average items, and d The number of differences made for the time series to become stationary, the steps are as follows:

Step 1: Check the stationarity of the hydrological time series. If it passes, go to the next step. If it fails, continue to differentiate the series until the differenced series satisfies the stationarity check, and use the unit root ADF check;

Step 2: Obtain the number of differences d of the model through step 1; take the AIC information criterion as the criterion, combine the ACF and PACF diagrams to limit the range of the autoregressive term p and the number of moving average terms q, and traverse the (p, q) combination to find the (p,q) combinatorial model for the smallest AIC value. The fitting results are shown in Figure 2, in which Figure 2 (left) represents the correlation coefficient between the flood peak and the maximum 30d flood volume, and the ARIMA (7,3,0) model is used for better fitting, and Figure 2 (right) represents the flood peak and The correlation coefficient of the maximum 60d flood was better fitted by the ARIMA(3,2,0) model. The Kendall coefficient within 10 years was predicted according to the ARIMA model, and the 95% confidence interval was given.

Step (6): Calculate the parameter θ _t according to the conversion relationship between the Kendall coefficient and the Copula parameter (ie, the correlation coefficient method). In this embodiment, only the variable parameter Copula is considered and the variable structure Copula is not considered, and the Gumbel-Houggard Copula commonly used in hydrological sequence analysis is used. Fitting is performed, that is, the corresponding parameters are calculated as:

In the formula, θ _t is the parameter of Copula at time t,

The corresponding Copula expression is:

In the formula, u=F _X (x) represents the marginal distribution of the flood peak, and v=F _Y (y) represents the marginal distribution of the maximum 30-day flood (or the maximum 60-day flood).

相应最可能组合值为：Step (7): According to the calculated Copula and GAMLSS models in 2016 and 2017, a joint distribution is established, and the conditional distribution is obtained, and the maximum value of the conditional probability density is calculated.

The corresponding most likely combination is:

分别表示条件概率密度值取最大时对应的最大30天设计洪量、最大60天设计洪量，W₃₀、W₆₀分别表示最大30天洪量及最大60天洪量两变量；

表示设计洪峰，Q_1％表示百年一遇的设计洪峰。In the formula, f ₁ and f ₂ respectively represent the maximum conditional probability corresponding to the maximum 30-day flood and the maximum conditional probability corresponding to the maximum 60-day flood,

respectively represent the maximum 30-day design flood and the maximum 60-day design flood corresponding to the maximum conditional probability density value, W ₃₀ and W ₆₀ respectively represent the two variables of the maximum 30-day flood and the maximum 60-day flood;

Represents a design flood, and Q _1% represents a design flood that occurs once in a century.

与

综上可得到洪峰与各时段的设计值，采用计算所得的设计值分成三段放大1954年的还原洪水过程的典型洪水过程线：Find the most probable values of the corresponding maximum 30 days and maximum 60 days according to the numerical solution

and

In summary, the design values of the flood peak and each period can be obtained, and the calculated design values are divided into three sections to enlarge the typical flood hydrograph of the restoration flood process in 1954:

Among them, _Q , w ₃₀ and w ₆₀ represent the flood peak in 1954, the flood volume in ₃₀ days and the flood volume in 60 days respectively. The magnification ratio of removing the maximum 30-day part is K ₃ , and then the hydrograph at the junction of magnification ratios is appropriately corrected, so that the design flood process of 2016 (Fig. 3) and 2017 (Fig. 4) once in a century can be obtained. From the results, since the flood peak univariate return period is used as the condition, the design value of the flood peak is consistent, but the flood volume in each period is significantly different, reflecting the impact of environmental changes on the design flood hydrograph. Therefore, the annual design flood hydrograph can be reviewed through a time-varying model in a changing environment combined with rainfall data.

The present invention has been disclosed above with preferred embodiments, but it is not intended to limit the present invention, and all technical solutions obtained by adopting equivalent replacement or equivalent transformation schemes all fall within the protection scope of the present invention.

Claims (9)

1. A method for calculating a flood process line, comprising the steps of:

selecting a flood peak and a plurality of period flood volume sequences according to the existing flood sequence;

performing trend analysis on the selected flood peak and the flood quantity sequences in multiple time periods, and fitting the flood peak with trend change and the flood quantity sequences in multiple time periods by adopting a GALSS model according to the trend analysis result;

calculating Kendall correlation coefficients between the flood peak and the flood volume in each period, and fitting the Kendall correlation coefficients;

predicting Kendall correlation coefficients between a flood peak of a certain year and the flood volume of each period according to the fitting result, and solving a Copula function and parameters between two variables according to a correlation coefficient method;

establishing joint distribution according to the GALSS model, the Copula function and parameters thereof, and calculating the most possible design value of the flood volume in each time period under the condition distribution based on the flood peak frequency by adopting a flood peak single variable design value as a control condition;

amplifying a typical flood process line according to the obtained design value;

wherein the GAMLSS model is:

g _k (θ _k )＝X _k β _k

wherein, theta _k Representing k statistical parameters of the distribution, taking into account the mean value theta ₁ And variance θ ₂ Two parameters, theta _k ＝(θ ₁ ,θ ₂ )；β _k Is of length J _k A regression coefficient vector; j. the design is a square _k The value of (a) is equal to the number of additive terms in the covariate function; x _k Is n × J _k The interpretation variable matrix of (2); n is the sample length;

selecting an explanation variable as the maximum rainfall, adopting a linear equation as a parameter function form, and respectively expressing an explanation variable matrix and a regression coefficient vector as follows:

where P represents the maximum rainfall.

2. The method of claim 1, wherein the fitting is performed using a GALSS model fitting, wherein a maximum rainfall per year is used as a covariate, and an extreme distribution function is selected and model time-varying parameters are estimated.

3. The method of claim 2, wherein the parameters in the GALSS model are fit using maximum likelihood, are preferred according to the minimum AIC criteria, and are evaluated by fitting using a Worm graph and a QQ graph.

4. The method of claim 1, wherein the Kendall correlation coefficient is calculated by a sliding window method, and the two variables of a flood peak and a flood volume in a certain control period are X ═ X (X), respectively ₁ ，x ₂ …x _N )、Y＝(y ₁ ，y ₂ …y _N ) Wherein N represents the total length of the sequence of flood peaks or period floods, and the calculation formula is as follows:

in the formula, τ _t Is the t-th Kendall correlation coefficient, t is more than or equal to 1 and less than or equal to N-N, N is the length of the sliding window, x _i 、y _i Respectively representing the ith values of two random variables, i is more than or equal to t and is less than or equal to t + n-1, x _j 、y _j J is more than or equal to t +1 and less than or equal to t + n; sgn is the indicator function:

the Kendall correlation coefficients are considered as a time series.

5. The method of claim 1, wherein the method of estimating a flood process line comprises fitting Kendall correlation coefficients using an ARIMA (p, d, q) autoregressive model, wherein p is an autoregressive term, q is a number of moving average terms, and d is a difference degree when a time series becomes stationary, and comprises the steps of:

the method comprises the following steps: carrying out stability inspection on the Kendall correlation coefficient sequence, and entering the next step if the Kendall correlation coefficient sequence passes the stability inspection; if not, continuously differentiating the sequence until the differentiated sequence meets the stationarity test and using the unit root ADF for testing;

step two: obtaining difference times d of the model through the step I; and (p, q) traversing the combination by taking the AIC information criterion as a criterion and combining the ACF and the PACF graph to define the range of the autoregressive term p and the moving average term number q, and finding out the (p, q) combination model with the minimum AIC value.

6. The method of claim 4, wherein the determining the Copula function and its parameters between two variables according to the correlation coefficient method comprises: and (3) fitting by using Gumbel-Houggard Copula in hydrologic sequence analysis only by considering the variable parameters Copula and not considering the variable structure Copula, namely the corresponding Copula parameter calculation formula is as follows:

in the formula, theta _t Is a parameter of Copula at time t;

the corresponding Copula function is expressed in the form:

wherein u is F _X (x) Denotes the edge distribution of the flood peak, v ═ F _Y (y) represents the edge distribution of the period flood.

7. The method of claim 1, wherein the most probable design value of flood volume in each period under the condition of flood peak one hundred years is calculated.

8. The method of claim 1, wherein the typical flood process line is amplified in three stages according to the calculated design value.

9. The method of claim 1, wherein the method further comprises: and (5) smoothing a typical flood process line.

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