CN105808868B - A kind of hydrological models combination Uncertainty Analysis Method based on Copula function - Google Patents
A kind of hydrological models combination Uncertainty Analysis Method based on Copula function Download PDFInfo
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Abstract
The invention discloses a kind of hydrological models combination Uncertainty Analysis Methods based on Copula function, comprising the following steps: (1) collects the data information in basin；(2) hydrological model simulation basin Outlet Section discharge process is established；(3) the marginal probability distribution function of measured discharge and analogue flow rate is determined；(4) the joint probability distribution function of Copula function building measured discharge and analogue flow rate is utilized；(5) measured discharge conditional probability distribution function when solving given analogue flow rate；(6) the median and indeterminacy section of measured discharge are obtained.The present invention can consider the uncertainty of model parameter and model structure simultaneously more fully hereinafter, obtain the comprehensive uncertain of hydrological model.The present invention independently of certainty mathematics model, can with the certainty mathematics model cooperated integration of arbitrarily complicated degree, without adding any to model it is assumed that providing the theoretical frame of versatility to analyze the synthesis uncertainty of hydrological model.
Description
Technical field
The present invention relates to Watershed Hydrologic Models, specifically a kind of hydrological models combination based on Copula function is not true
Method for qualitative analysis.
Background technique
Watershed Hydrologic Models refer to the mathematical modulo for being built up complicated hydrology phenomenon and process through generalization with analogy method
Type.Currently, hydrological model be widely used in Flood Prevention drought resisting, water resource administrative system, water environment and the ecosystem protection,
Climate change and mankind's activity are to fields such as Southeast Tibetan Plateau analyses.In fact, Hydrology is complicated, nonlinear process,
The River Basin Hydrology process that true complexity is described using relatively simple mathematical formulae often will appear distortion, lead to hydrological model
It is inevitably present uncertainty.Therefore, the uncertainty for analysing in depth hydrological simulation result, can provide more for policymaker
Add sufficient risk information, there is important scientific meaning and application value.
The uncertainty of hydrological model generally includes two aspects of parameter uncertainty and structural uncertainty.For model
The uncertain problem of parameter, Britain hydrologist Beven and Binley were proposed in 1992 based on the general of bayesian theory
Suitable likelihood uncertainty estimation (Generalized Likelihood Uncertainty Estimation, GLUE) method,
Be widely used both at home and abroad (the .GLUE method such as Zhang Liru, Guan Yiqing, Wang Jun analyze Hydro-Model Parameter Calibration Technology it is probabilistic
Study [J] hydroelectric generation, 2010,36 (5): 14-16).Although GLUE method principle is simple, easily operated, exists and not pass through
The bayes method of allusion quotation, subjective judgement parameter feasible zone threshold value, the parameter Posterior probability distribution inquired into do not have apparent statistics
The problems such as learning meaning, so that the reliability and applicability of the method are queried.
For this purpose, have scholar propose based on Markov Chain Monte Carlo (Markov Chain Monte Carlo,
MCMC) bayes method of random sampling technology studies uncertainty (Liang Zhongmin, Li Binquan, the Yu Zhong of Hydro-Model Parameter Calibration Technology
The such as wave analyze [J] Hohai University journal (natural science edition) based on the TOPMODEL parameter uncertainty of bayesian theory,
2009,37(2):129-132).MCMC methodology avoids direct solution integral difficulty, can derive and anticipate with significant statistics
Justice each parameter Posterior distrbutionp of model, but by MCMC methodology be applied to practical main hydrologic problems when whether have enough computational efficiencies,
Can the problems such as actual requirement be met all have larger dispute, needs further to be studied.
Bayesian model weighted average (Bayesian Model Averaging, BMA) method is then most widely used at present
(Dong Leihua, Xiong Lihua, ten thousand people are based on Bayesian model weighted average side to general hydrological model structural uncertainty analysis method
Hydrological model analysis of uncertainty [J] Journal of Hydraulic Engineering of method, 2011,42 (9): 1065-1074).However, research shows that the side BMA
Method is actually still a kind of weighted average method, it will usually carry out smoothing techniques, and then reduction pair to analogue flow rate result
The simulation effect of peak flow.In addition, the expectation-maximization algorithm for solving BMA model needs hypothesized model forecasting runoff equal
Normal Distribution, and actual hydrologic process is often nonnormal, it is necessary to positive state space is transformed by normal state quantile
It is handled again, this certainly will will affect the precision and accuracy of model.In addition, current research only relates separately to hydrology mould mostly
The uncertain problem of shape parameter or model structure, it is less while considering that the synthesis of coupling model structure and model parameter is not true
It is qualitative.It would therefore be highly desirable to the effective hydrological models combination Uncertainty Analysis Method that studies science.
The comprehensive uncertain uncertainty for finally concentrating on model output result of Hydro-Model Parameter Calibration Technology and structure.
Thus, the synthesis analysis of uncertainty of hydrological model is substantially considered as being to solve for measured discharge when setting models output flow
Conditional probability density function and distribution function.Copula function theory can connect the edge distribution of multiple stochastic variables
Carry out tectonic syntaxis distribution, the analytical expression of solving condition distribution is widely used (Guo in hydrographic water resource field
It is raw to practice, application and progress [J] hydrology of the .Copula such as Yan Baowei, the Xiao Yi function in multivariable hydrological analysis calculating,
2008,28(3):1-7).It is Arbitrary distribution that it, which can permit edge distribution, can preferable simulation hydrologic process it is non-linear and non-
Normal state feature.Currently, Copula function is introduced into the synthesis analysis of uncertainty research of hydrological model without document.
Summary of the invention
The purpose of the present invention is overcoming the shortcomings of the prior art, a kind of hydrological model based on Copula function is provided
Comprehensive Uncertainty Analysis Method.
A kind of hydrological models combination Uncertainty Analysis Method based on Copula function of the present invention, comprising the following steps:
Step 1, the actual measurement rainfall, evaporation and data on flows data in basin are collected；
Step 2, selector interflow domain produces the hydrological model of afflux characteristic, according to rainfall, evaporation and the flow number in step 1
It is exported using established hydrological model simulation basin disconnected according to data using the parameter of optimization algorithm calibration hydrological model
Face discharge process；
Step 3, according in step 1 measured discharge and step 2 obtained in analogue flow rate data information, choose it is appropriate
Marginal probability distribution function line style, and estimate the parameter of marginal probability distribution function；
Step 4, using the joint probability distribution function of Copula construction of function measured discharge and analogue flow rate, and estimate
The parameter of Copula function；
Step 5, the joint probability distribution function that the marginal probability distribution function and step 4 estimated according to step 3 construct pushes away
The analytical expression of measured discharge conditional probability distribution function when seeking given analogue flow rate；
Step 6, according to the analytical expression of the resulting conditional probability distribution function of step 5, according to statistical principle, meter
The median for obtaining measured discharge is calculated as deterministic simulation as a result, obtaining the uncertain simulation under given confidence level simultaneously
Section.
In the step 2, user selects Watershed Hydrologic Models structure appropriate, is according to the actual conditions in specific basin
Conceptual hydrological model or hydrological distribution model, including but not limited to Xinanjiang model, Sacramento model,
TOPMODEL model, TANK model, VIC model or MIKE SHE model.
In the step 2, parameter automatic rating method is used when calibration Hydro-Model Parameter Calibration Technology, selected objective function is residual
Poor quadratic sum minimum criteria, used optimization algorithm are SCE-UA algorithm.
In the step 3, P-III type is distributed the marginal probability distribution function line as measured discharge and analogue flow rate
Type.
In the step 3, using the parameter of linear Moment method estimators marginal probability distribution function.
In the step 4, using the joint probability distribution of Frank Copula construction of function measured discharge and analogue flow rate
Function, using the parameter of Kendall rank correlation Y-factor method Y estimation Frank Copula function.
The present invention directly post-processes the output result of hydrological model, while considering hydrological model structure and parameter
Uncertainty.Using the joint probability distribution function of Copula function building measured discharge and analogue flow rate, given by solving
Measured discharge conditional probability distribution function when analogue flow rate is determined, to obtain median and the uncertainty area of measured discharge
Between, the comprehensive uncertain of hydrological model is analyzed accordingly.
Compared with prior art, the beneficial effects of the present invention are:
1, can only individually consider with conventional hydrological model Uncertainty Analysis Method model parameter or model structure not really
Qualitative difference, the present invention can consider the uncertainty of model parameter and model structure simultaneously more fully hereinafter, obtain hydrology mould
The comprehensive uncertainty of type.
2, the present invention can cooperate with the certainty mathematics model of arbitrarily complicated degree and collect independently of certainty mathematics model
At without adding any to model it is assumed that providing the theoretical frame of versatility to analyze the synthesis uncertainty of hydrological model
Frame.
3, the present invention allows measured discharge and analogue flow rate to have any type of marginal probability distribution function, can be accurate
Ground captures the non-linear and Singular variance correlation structure between measured discharge and analogue flow rate.
4, the present invention quantitatively describes the comprehensive uncertain of hydrological model in the form of probability distribution, and provides specified set
Uncertain section under letter is horizontal makes user's quantitative various uncertainties of consideration of energy in decision, estimates various decision wind
Danger and consequence realize the organic coupling of simulation and decision process.
Detailed description of the invention
Fig. 1 is the flow chart of the method for the present invention.
Fig. 2 is the schematic diagram of measured discharge and analogue flow rate scatter plot.
Fig. 3 is measured discharge theoretical margin probability-distribution function value (being calculated using the distribution of P-III type) and experience side
The schematic diagram of edge probability-distribution function value comparative situation.
Fig. 4 is analogue flow rate theoretical margin probability-distribution function value (being calculated using the distribution of P-III type) and experience side
The schematic diagram of edge probability-distribution function value comparative situation.
Fig. 5 is to be combined generally using the theoretical joint probability distribution function value that Frank Copula function is calculated with experience
The schematic diagram of rate distribution function value comparative situation.
The schematic diagram of the conditional probability distribution function curve of measured discharge when Fig. 6 is given analogue flow rate.
Fig. 7 is measured discharge, the median analog result being calculated according to the method for the present invention and 90% uncertain mould
The schematic diagram of quasi- section comparative situation.
Specific embodiment
The invention will be further described below by way of examples and with reference to the accompanying drawings.
As shown in Fig. 1-Fig. 7, a kind of hydrological models combination Uncertainty Analysis Method based on Copula function collects stream
Actual measurement rainfall, evaporation and the flow data in domain establish hydrological model simulation basin Outlet Section discharge process, are determining actual measurement stream
On the basis of the marginal probability distribution function of amount and analogue flow rate, measured discharge and analogue flow rate are constructed using Copula function
Joint probability distribution function, by solving measured discharge conditional probability distribution function when given analogue flow rate, to obtain reality
The median of measurement of discharge and uncertain section.Fig. 1 is the calculation flow chart of the present embodiment, is followed the steps below:
1. collecting actual measurement rainfall, evaporation and the data on flows data in basin.
The time scale that rainfall, evaporation and data on flows data are surveyed in this specific implementation is day.Rainfall data refers to
The face average rainfall for studying basin is calculated by representativeness rainfall websites multiple on basin using Thiessen polygon method.
Basin Evapotranspiration measurement Data can be obtained from the evaporating dish measured data of weather station.Flow data refers to the representativeness of basin Outlet Section
The measured discharge process at hydrometric station is obtained from the Water Year Book at hydrometric station.
2. establishing hydrological model simulation basin Outlet Section discharge process.
According to the weather in basin, geology and geomorphology actual conditions, select Xinanjiang model as basin water in this specific implementation
Literary model structure.
According to rainfall, evaporation and the data on flows data in step 1, SCE-UA optimization algorithm is used in this specific implementation certainly
The parameter of hydrological model selected by dynamic calibration.
The objective function of parameter rating of the model is residual sum of squares (RSS) minimum criteria in this specific implementation, it can be described as reality
Residual sum of squares (RSS) between measurement of discharge and analogue flow rate is minimum, is shown below:
Wherein, hjAnd sjRespectively measured discharge and analogue flow rate, number of segment when n is indicated.
As shown in Fig. 2, giving measured discharge and analogue flow rate scatter plot, wherein analogue flow rate be by rate set
Xinanjiang River hydrological model calculate and obtain.
3. determining the marginal probability distribution function of measured discharge and analogue flow rate.
According in step 1 measured discharge and step 2 obtained in analogue flow rate data information, it is general to choose edge appropriate
Rate distribution function line style, and estimate its parameter, this step includes two sub-steps:
3.1 selection marginal probability distribution function line styles
Due to the overall distribution frequency curves of measured discharge and analogue flow rate be it is unknown, usually select can good fit it is more
The line style of number hydrology sample data series.China compares by analysis for many years, and discovery P-III type distribution is for China major part river
Preferably, recommendation uses in engineering practice for the hydrological data fitting of stream.
The marginal probability distribution function line as measured discharge and analogue flow rate is distributed using P-III type in this specific implementation
Type.
The parameter of 3.2 estimation marginal probability distribution function line styles
After curve type of frequency distribution is selected, the parameter of estimation frequency distribution is next carried out.Currently used method
Mainly there are moments method, maximum-likelihood method, suitable collimation method, probability-weighted moment, weight-function method and linear moments method (L- moments method) etc..Wherein,
L- moments method is the actual parameter estimation method generally acknowledged both at home and abroad at present, and maximum feature is that do not have to the maximum and minimum of sequence
Conventional square is so sensitive, and the estimates of parameters acquired is more steady.
The parameter of L- Moment method estimators marginal probability distribution function line style is used in this specific implementation.
As shown in Figure 3 and Figure 4, the measured discharge being calculated using the distribution of P-III type is set forth, analogue flow rate is managed
By marginal probability distribution functional value and experience marginal probability distribution functional value comparison diagram.Wherein, experience marginal probability distribution function
Value is calculated using one-dimensional mathematic expectaion formula.
4. utilizing the joint probability distribution function of Copula function building measured discharge and analogue flow rate.
According to what is estimated in the measured discharge in step 1, analogue flow rate data information and step 3 obtained in step 2
Marginal probability distribution function chooses joint of the Copula function appropriate as contiguous function construction measured discharge and analogue flow rate
Probability-distribution function, and estimate its parameter, this step includes two sub-steps:
4.1 selection Copula functions
Assuming that H, S respectively indicate measured discharge and analogue flow rate, h, s are respectively corresponding implementation value.FH(h)、FS(s) it is
Marginal probability distribution function, corresponding probability density function are fH(h)、fS(s).By Sklar theorem it is found that the joint probability of H, S
Distribution function can be indicated with a dimensional Co pula function C:
FH,S(h, s)=Cθ(FH(h),FS(s))=Cθ(u,v) (2)
Wherein, θ is the parameter of Copula function；U=FH(h), v=FSIt (s) is marginal probability distribution function.
In this specific implementation, using the joint probability distribution of Frank Copula construction of function measured discharge and analogue flow rate
Function, expression formula are as follows:
The parameter of 4.2 estimation Copula functions
In this specific implementation, using the parameter of Kendall rank correlation Y-factor method Y estimation Frank Copula function.
The relationship of Kendall related coefficient τ and parameter θ are as follows:
Enable { (x1,y1),…,(xn,yn) indicate from the middle n observation extracted of continuous random variable (X, Y) with press proof
This, then have in the sampleThe different observation of kind combines (xi,yi) and (xj,yj).Sample Kendall rank correlation coefficient τ passes through
Following formula calculates
Wherein, sign () is sign function.
When practical calculating, the Kendall rank correlation coefficient τ of two variable samples is first calculated using formula (5), is recycled formula (4)
Inverse goes out parameter θ.Since analytic solutions are not present in formula (4), solve to obtain numerical solution using Newton iteration method in this specific implementation.
As shown in figure 5, giving the theoretical joint probability distribution function value being calculated using Frank Copula function
With the comparative situation of experience joint probability distribution function value.Wherein, experience marginal probability distribution functional value uses two-dimemsional number term
Formula is hoped to be calculated.
5. solving measured discharge conditional probability distribution function when given analogue flow rate.
When given analogue flow rate S value s, the value of corresponding measured discharge H is simultaneously not exclusive, but changeable, only
Be occur different values probability it is different, there is a conditional probability distribution functions
FH|S(h)=P (H≤h | S=s) (6)
By Copula function, conditional probability distribution function FH|S(h) it can indicate are as follows:
As shown in fig. 6, giving the conditional probability distribution function curve of measured discharge when given analogue flow rate.
6. obtaining median and the uncertainty section of measured discharge.
Conditional probability distribution function FH|S(h) corresponding probability density function is
fH|S(h)=cθ(u,v)fH(h) (8)
Wherein, cθ(u, v) is the density function of Copula function, analytical expression are as follows:
Obtain the conditional probability density function f of stochastic variable HH|S(h) it after, according to statistical principle, can be calculated
The median of measured discharge is as deterministic simulation as a result, obtaining the uncertain simulation section under given confidence level simultaneously.
The median h of measured dischargemIt is solved by following formula:
Certain confidence level (1- ξ) is selected, enabling measured discharge value appear in the probability at distribution both ends is ξ, so that it may
Define the interval estimation of measured discharge.The upper and lower limit of the confidence of stochastic variable H is provided by following two formula respectively:
In formula: ξ1+ ξ2=ξ is significance；ξ1And ξ2Can be arbitrarily selected according to practical problem, this specific implementation
In take ξ1=ξ2=ξ/2.
Therefore have
P(hl≤H≤hu)=1- ξ (13)
That is [hl,hu] be stochastic variable H confidence level (1- ξ) interval estimation, according to confidence interval can to actual measurement flow
It measures the uncertain of estimated value and carries out quantitative assessment.
In view of formula (10)-(12) can not obtain analytic solutions, solve to obtain using dichotomy tentative calculation in this specific implementation
Numerical solution.
As shown in fig. 7, the median analog result and 90% for giving measured discharge, being calculated according to the method for the present invention
Uncertainty simulation section comparative situation.Wherein, median analog result is the quantile of conditional probability distribution function 50%；It gives
Determine significance ξ=0.1, the quantile of conditional probability distribution function 5% and 95% is calculated, they are set forth
The confidence lower limit and upper limit value in 90% uncertain simulation section.
To sum up, the present invention is based on basin actual measurement rainfall, evaporation and flow data, it is disconnected to establish the outlet of hydrological model simulation basin
Face discharge process utilizes Copula function on the basis of determining the marginal probability distribution function of measured discharge and analogue flow rate
The joint probability distribution function for constructing measured discharge and analogue flow rate, by solving measured discharge condition when given analogue flow rate
Probability-distribution function, to obtain median and the uncertainty section of measured discharge.The present invention is retouched in the form of probability distribution
The comprehensive uncertain of hydrological model is stated, the uncertain section under specified confidence level is provided, makes user can be fixed in decision
The considerations of amount various uncertainties, realize the organic coupling of simulation and decision process.
Claims (6)
1. a kind of hydrological models combination Uncertainty Analysis Method based on Copula function, it is characterised in that including following step
It is rapid:
Step 1, the actual measurement rainfall, evaporation and data on flows data in basin are collected；
Step 2, selector interflow domain produces the hydrological model of afflux characteristic, is provided according to rainfall, evaporation and the data on flows in step 1
Material simulates basin Outlet Section stream using established hydrological model using the parameter of optimization algorithm calibration hydrological model
Amount process；
Step 3, according in step 1 measured discharge and step 2 obtained in analogue flow rate data information, choose edge appropriate
Probability-distribution function line style, and estimate the parameter of marginal probability distribution function；
Step 4, using the joint probability distribution function of Copula construction of function measured discharge and analogue flow rate, and estimate Copula
The parameter of function；
Step 5, according to step 3 estimate marginal probability distribution function and step 4 construct joint probability distribution function inquire into
Determine the analytical expression of measured discharge conditional probability distribution function when analogue flow rate；
Step 6, it is calculated according to the analytical expression of the resulting conditional probability distribution function of step 5 according to statistical principle
To measured discharge median as deterministic simulation as a result, obtaining the uncertain simulation region under given confidence level simultaneously
Between.
2. the method as described in claim 1, it is characterised in that: in the step 2, user is according to the practical feelings in specific basin
Condition selects Watershed Hydrologic Models structure appropriate, be conceptual hydrological model or hydrological distribution model, including but not limited to
Xinanjiang model, Sacramento model, TOPMODEL model, TANK model, VIC model or MIKE SHE model.
3. the method as described in claim 1, it is characterised in that: in the step 2, parameter is used when calibration Hydro-Model Parameter Calibration Technology
Automatic rating method, selected objective function areWherein, hjAnd sjRespectively measured discharge and analog stream
Amount, number of segment when n is indicated, the objective function are adopted using the residual sum of squares (RSS) minimum criteria of measured discharge and analogue flow rate as foundation
Optimization algorithm is SCE-UA algorithm.
4. the method as described in claim 1, it is characterised in that: in the step 3, regard the distribution of P-III type as measured discharge
With the marginal probability distribution function line style of analogue flow rate.
5. the method as described in claim 1, it is characterised in that: in the step 3, using linear Moment method estimators marginal probability point
The parameter of cloth function.
6. the method as described in claim 1, it is characterised in that: real using Frank Copula construction of function in the step 4
The joint probability distribution function of measurement of discharge and analogue flow rate estimates Frank Copula using Kendall rank correlation Y-factor method Y
The parameter of function.
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