CN108875130A - A kind of design flood peak amount condition most probable combined method based on Copula function - Google Patents
A kind of design flood peak amount condition most probable combined method based on Copula function Download PDFInfo
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Abstract
The design flood peak amount condition most probable combined method based on Copula function that the present invention provides a kind of.Pass through the joint distribution function of crest discharge and day part magnanimity in Copula construction of function observed flood event, multivariable condition most probable combinatorial formula based on derivation calculates the most probable combined value of flood peak and day part magnanimity for different flood control tasks, important reference frame is provided for drafting for design flood design standard, provides a new approach for the peak amount combinatory analysis of design flood.
Description
Technical field
The present invention relates to combined probability analysis field, in particular to a kind of design flood peak amount item based on Copula function
Part most probable combined method.
Background technique
Design flood refers to the flood with certain standard and a certain specific function, its function is：It is equal to frequency
Design standard carries out Flood Frequency Analysis calculating, corresponding design flood is obtained, then as the engineering according to planning and designing
The relative risk of flood control safety accident is exactly equal to the design standard [1] formulated.It is well known that building hydraulic engineering is resisting flood
In play an important role, be reduce flood the problems such as an effective non-engineering measure.And hydrologic(al) frequency analysis meter
It is an indispensable important link in Water Conservancy Works, i.e., carries out frequency analysis by flood data, push away
Ask the design flood of certain standard.The reasonability of design flood is directly related to the reliability and safety of engineering construction.
Existing hydrologic(al) frequency analysis mostly uses univariate analysis method, i.e., a certain characteristic value is only selected to carry out variable as frequency
Rate calculation basis.And flood event is made of many factors with correlation, traditional calculation method artificially will be present
The flood factor of correlation is separated, then hardness combination (most dangerous combination, same frequency combination etc.), can not comprehensively excavate flood
The syntagmatic of flood peak and magnanimity in water event can not be truly reflected flood characteristics, and then erroneous estimation hydrologic characteristic value, lead
Flood risk is caused, economic loss is caused.
Extensive use of the Copula function in hydrology field compensates for the deficiency of legacy frequencies calculation method, can flexible structure
The characteristic for making Joint Distribution between aleatory variable can preferably reflect the correlation of each factor in flood event.For this purpose, Xiao Yi etc. [2]
Based on the joint distribution function of Copula construction of function flood flood peak and magnitude, the determination side of designed flood hydrograph is analyzed
Method；Yan Baowei etc. [3] forms calculation method based on the flood that Copula function has derived each subregion flood relevance of consideration；Liu Zhang
Monarchs etc. [4] are formed based on the most probable area of Copula Functional Analysis design flood.
In the peak amount combined aspects of design flood, at present document be mostly with Copula construction of function crest discharge and when
The joint distribution function of Duan Hongliang；In rare literature research situation known to crest discharge (or period magnanimity) design value,
The possibility combined situation of his design value.There is presently no based on Copula function researching and designing flood crest discharge and day part flood
The research for measuring most probable combination value calculating method, lacks reliable hydrologic characteristic value (crest discharge, period magnanimity) combinatory analysis.
Bibliography of the present invention is as follows：
[1] Liang Zhongmin, clock safety, Hua Jiapeng, wait hydrology and water conservancy calculate [M] Chinese Water Conservancy water power publishing house, 2008.
[2] Xiao Yi, Guo Shenglian, Liu Pan wait designed flood hydrograph method [J] Wuhan University of the based on Copula function
Journal (engineering version), 2007 (04):13-17.
[3] Yan Baowei, Guo Shenglian, Guo Jing wait design flood area composition research [J] the waterpower of based on Copula function
Generate electricity journal, 2010 (06):60-65.
[4] Liu Zhangjun, Guo Shenglian, Li Tianyuan wait step reservoir design flood most probable area to form method computation formula
[J] hydroscience progress, 2014 (04):575-584.
Summary of the invention
The technical problem to be solved by the present invention is to：There is provided it is a kind of based on Copula function consider crest discharge (when
In the case where Duan Hongliang) giving, the condition most probable of corresponding day part magnanimity (crest discharge) forms calculated result, deeply excavates
The syntagmatic of crest discharge and period magnanimity in flood event, and in the method for this standard for formulating design flood
The Joint Distribution that the present invention passes through crest discharge and day part magnanimity in Copula construction of function observed flood event
Function, the multivariable condition most probable combinatorial formula based on derivation calculate the crest discharge and day part for being directed to different flood control tasks
The most probable combined value of magnanimity, thus using this combined value as the design standard of design flood.
The present invention uses following technical scheme to solve above-mentioned technical problem：
A kind of design flood peak amount condition most probable combined method based on Copula function, includes the following steps：
Step 1. collects and arranges the crest discharge and day part magnanimity data that extract year observed flood series；
Step 2., should using the crest discharge and day part magnanimity data extracted in P-III type distribution function fit procedure 1
Marginal distribution function in P-III type distribution function, that is, Copula function；And carry out the test of fitness of fot；
Step 3. measures the correlation between crest discharge and day part magnanimity, selects suitable Copula function, estimates
It is fitted the parameter of optimal Copula function, constructs multivariate joint probability distribution function；
Step 4. is calculated using the condition most probable combinatorial formula suitable for any dimension variable derived with crest discharge
Or period magnanimity is condition, the condition most probable combination of day part magnanimity or crest discharge；
Step 5. is using combined value the drafting as design flood of the crest discharge being calculated in step 4 and period magnanimity
Standard, and compared with traditional same frequency combination method, the reasonability of the method for inspection.
In the step 1, the flood data of collection include historical flood data, and are examined by three property.
In the step 2, frequency calculating is carried out to flood data using the method that is uniformly processed, line is fitted using computer optimization and estimates
Mean value E (X), coefficient variation Cv, the coefficient of skew Cs of P-III type curve are counted, and following P-III type is converted into according to formula (1)
Distribution density function fX(x) corresponding α, β, γ in expression formula (2)：
Wherein, Γ (α) is the gamma function of α, and e is the truth of a matter of exponential function.
Using χ2The method of inspection is fitted goodness inspection.
It is preferred using crosscheck and AIC criterion in symmetrical Archimedean Copula class function in the step 3
Multidimensional Copula function estimates Copula function parameter using maximum-likelihood method.
The derivation of the condition most probable combinatorial formula suitable for any dimension variable in the step 4 further comprises son
Step：
4.1 calculate stochastic variable X1Take design value x1When, surplus variable XiConditional probability distribution function, wherein i=2,
3 ..., n, expression formula are shown below：
Wherein, stochastic variable X1Represent crest discharge or its corresponding a certain period magnanimity, xiIt is stochastic variable XiTake
Value, C (u1,u2,u3,…,un-1,un) it is Copula function,For crest discharge and its corresponding day part flood
The marginal distribution function of amount；
4.2 calculate the corresponding conditional joint probability density function of (3) formula, and expression formula is shown below：
In formula：
For Copula letter
Several density functions, is indicated in following formula with c；For XiProbability density function, wherein i=2,3 ..., n；
4.3f(x2,x3,x4,…,xn|x1) corresponding x when being maximizedMiValue, wherein i=2,3 ..., n, form combinationAs condition most probable combines, and is maximum point using the point that partial derivative is zero
Property obtains expression formula and is shown below：
(5) formula is equivalent to expression formula as follows according to the property of Copula function and distribution function by 4.4：
In formula,
4.5 inquire into XiRelationship between the density function and its derived function of the P-III distribution of obedience, is shown below：
WhereinFor XiThe parameter of corresponding P-III distribution density function；
4.6 (7) formula is brought into (6) formula obtain condition most probable combination general formula expression formula, be shown below：
The invention adopts the above technical scheme compared with prior art, has the following technical effects：
1, single argument conventional method is overcome to the limitation inquiring into flood event crest discharge and combining with day part magnanimity, benefit
With Copula construction of function joint distribution function, better describe between the inherent law of flood event and each characteristic attribute
Correlation.
2, the multivariable condition most probable combination general formula using Copula function has innovatively been derived, has been calculated instead of the hydrology
In common full specific combination content to retain sovereignty over a part of the country analysis method, obtain more objective multivariable combinatorial analysis result, more
Close to engineering reality, the reference frame of important strong operability is provided for drafting for design flood design standard.
Detailed description of the invention
Fig. 1 is the flow chart of the method for the present invention；
Fig. 2 is the Q- of yearly maximum wind velocity flow, three days magnanimity, 15 days magnanimity experience Joint Distributions and theoretical Joint Distribution
Q figure；
When Fig. 3 is using crest discharge as condition (design value for taking P=10%), conditional joint probability density function is three-dimensional
Figure；
Fig. 4 is using crest discharge as the condition most probable combinational estimation result of three days magnanimity of condition and crest discharge and three
The schematic diagram of its magnanimity common frequence method calculated result comparative situation；
Fig. 5 be using crest discharge as the condition most probable combinational estimation result of 15 days magnanimity of condition and crest discharge with
The schematic diagram of 15 days magnanimity common frequence method calculated result comparative situations.
Fig. 6 is using three days magnanimity as the condition most probable combinational estimation result of the crest discharge of condition and three days magnanimity and flood
The schematic diagram of peak flow common frequence method calculated result comparative situation；
Fig. 7 be using three days magnanimity as the condition most probable combinational estimation result of 15 days magnanimity of condition and three days magnanimity with
The schematic diagram of 15 days magnanimity common frequence method calculated result comparative situations.
Fig. 8 is using 15 days magnanimity as the condition most probable combinational estimation result of the crest discharge of condition and 15 days magnanimity
With the schematic diagram of crest discharge common frequence method calculated result comparative situation；
Fig. 9 is using 15 days magnanimity as the condition most probable combinational estimation result of three days magnanimity of condition and 15 days magnanimity
With the schematic diagram of three days magnanimity common frequence method calculated result comparative situations.
Specific embodiment
Present invention is further described in detail with reference to the accompanying drawings and detailed description, the example of the embodiment
It is shown in the accompanying drawings, is exemplary below with reference to the embodiment of attached drawing description, for explaining only the invention, and cannot
It is construed to limitation of the present invention.
Detailed process of the invention is shown in Fig. 1, includes the following steps：
Step 1：It collects and arranges the yearly maximum wind velocity flow day part magnanimity corresponding with its for extracting observed flood series
Data；
Compiled altogether in this specific example 116 years yearly maximum wind velocity flow and the secondary flood corresponding three hours
Period magnanimity data, and joined 8 historic flood data.
Step 2：Using the crest discharge and day part magnanimity data extracted in P-III type distribution function fit procedure 1, i.e.,
It determines the marginal distribution function in Copula function, and carries out the test of fitness of fot；
Frequency calculating is carried out to the flood data comprising historical flood using the method that is uniformly processed in this specific example, using meter
Calculation machine optimizes the parameter of suitable line estimation P-III type distribution function, crest discharge Q and corresponding three days magnanimity W3, 15 days magnanimity W15
Distribution function expression formula distribution be denoted asIts corresponding density function is denoted as respectively And pass through χ2It examines.
Step 3：The correlation between crest discharge and day part magnanimity is measured, suitable Copula function is selected, is estimated
It is fitted the parameter of optimal Copula function, constructs multivariate joint probability distribution function；
The excellent of symmetrical Archimedean Copula function is carried out using crosscheck and AIC criterion in this specific example
Choosing, final choice Gumbel Copula Function Fitting crest discharge Q and corresponding three days magnanimity W3, 15 days magnanimity W15Joint
Distribution function estimates Copula function parameter using maximum-likelihood method.Wherein joint distribution function is denoted as Cθ(u1,u2,u3).Its
Middle θ is the parameter of Copula function,
See Fig. 2, the Q-Q for giving selected Copula function experience Joint Distribution and theoretical Joint Distribution schemes.
Step 4：It is calculated using the condition most probable combinatorial formula suitable for any dimension variable of derivation with the flood peak (period
Magnanimity) it is condition, the condition most probable of day part magnanimity (flood peak) combines.
The step of derivation condition most probable combinatorial formula includes six sub-steps, is illustrated in conjunction with this example：
4.1 calculating take design value x in crest discharge Q1pWhen, corresponding three days magnanimity W3, 15 days magnanimity W15Conditional probability
Distribution function.Its expression formula is shown below：
Similarly, it calculates separately in three days magnanimity W3, 15 days magnanimity W15Take design value x2p、x3pWhen, corresponding surplus variable
Three-dimensional condition probability-distribution function.
Its expression formula is shown below respectively：
4.2 calculate when crest discharge Q takes design value x1pWhen, corresponding three days magnanimity W3, 15 days magnanimity W15Conditional probability
Density function.
See Fig. 3, when giving using crest discharge as condition (design value for taking P=10%), conditional joint probability density letter
Number three-dimensional graph.
Similarly, it calculates in three days magnanimity W3, 15 days magnanimity W15Design value x is taken respectively2p、x3pWhen, corresponding surplus variable
Three-dimensional condition probability density function.
Its expression formula is shown below respectively：
4.3f(x2,x3|x1) corresponding x when being maximized2M、x3MValue forms combination (x1p,x2M,x3M)、f(x1,x3|x2)
Corresponding x when being maximized1M、x3MValue forms combination (x1M,x2p,x3M)、f(x1,x2|x3) corresponding x when being maximized2M、x3M
Value forms combination (x1M,x2M,x3p) it is the combination of condition most probable.
Be zero point using partial derivative it is the property of maximum point, expression formula can be respectively obtained and be shown below：
(15) formula, (16) formula, (17) formula can be equivalent to following institute according to the property of Copula function and distribution function by 4.4
The expression formula shown：
4.5 inquire into XiRelationship between the density function and its derived function of the P-III distribution of obedience, is shown below：
WhereinFor XiThe parameter of corresponding P-III distribution density function；
4.6 bring (21) formula into (18) formula, (19) formula, respectively obtain with crest discharge Q, with three days magnanimity in (20) formula
W3, with 15 days magnanimity W15The expression formula that general formula is combined for the most probable of condition, respectively such as (22) formula, (23) formula, (24) formula institute
Show：
In this example, using fast convergence rate, the strong quasi-Newton method of stability solves (22) formula, (23) formula, (24) respectively
Three Nonlinear System of Equations of formula.
Step 5：Using combined value the drafting as design flood of the crest discharge being calculated in step 4 and period magnanimity
Standard, and compared with traditional same frequency combination method, the reasonability of the method for inspection.
Wherein crest discharge Q and three days magnanimity W3, 15 days magnanimity W15Choose the design value under multiple design frequencies.With frequency
Rate combined method is to choose crest discharge Q and three days magnanimity W3, 15 days magnanimity W15Design value combination under same frequency, respectively
It combines obtained design value and draws curve.
See Fig. 4, gives three days magnanimity W using crest discharge Q as condition3Condition most probable combinational estimation result and flood
Peak flow Q and three days magnanimity W3Common frequence method calculated result comparative situation.Wherein observation is crest discharge Q and three days magnanimity W3
Eyeball evidence.
See Fig. 5, gives three days magnanimity W using crest discharge Q as condition15Condition most probable combinational estimation result and flood
Peak flow Q and three days magnanimity W15Common frequence method calculated result comparative situation.Wherein observation is crest discharge Q and 15 days magnanimity
W15Eyeball evidence.
Three days magnanimity W that condition most probable combination calculates3, 15 days magnanimity W15Design value closer to eyeball
According to three days magnanimity W that same frequency combination calculates3, 15 days magnanimity W15Design value be slightly above condition most probable combination side
Three days magnanimity W that formula calculates3, 15 days magnanimity W15Design value.
The analytic process of Fig. 6 to Fig. 9 is similar with Fig. 4, Fig. 5, condition most probable combined method proposed by the present invention and same frequency
Method, which is compared, some superiority.
To sum up, the present invention overcomes single argument conventional methods to the office inquiring into flood event flood peak and combining with day part magnanimity
Limit, using the tectonic syntaxis distribution function of Copula function, better describe flood event inherent law and each feature category
Correlation between property.And innovative derivation using the multivariable condition most probable of Copula function combines general formula, generation
For the analysis method of common specific combination (same frequency combination) in hydrology calculating, more objective multivariable combinatorial analysis is obtained
Calculated result provides a new approach more close to engineering reality for the peak amount combinatory analysis calculating of design flood, to set
Drafting for design flood standard of meter provides important reference frame.
Those skilled in the art of the present technique are appreciated that unless otherwise defined, all terms used herein (including technology art
Language and scientific term) there is meaning identical with the general understanding of those of ordinary skill in fields of the present invention.Should also
Understand, those terms such as defined in the general dictionary, which should be understood that, to be had and the meaning in the context of the prior art
The consistent meaning of justice, and unless defined as here, it will not be explained in an idealized or overly formal meaning.
The above examples only illustrate the technical idea of the present invention, and this does not limit the scope of protection of the present invention, all
According to the technical idea provided by the invention, any changes made on the basis of the technical scheme each falls within the scope of the present invention
Within.
Claims (7)
1. a kind of design flood peak amount condition most probable combined method based on Copula function, which is characterized in that including following
Step：
Step 1., which is collected and arranged, extracts year observed flood data, including crest discharge and day part magnanimity data；
Step 2. utilizes the crest discharge and day part magnanimity data extracted in P-III type distribution function fit procedure 1, the P-
Marginal distribution function in type III distribution function, that is, Copula function；And carry out the test of fitness of fot；
Step 3. measures the correlation between crest discharge and day part magnanimity, selects suitable Copula function, evaluation fitting
The parameter of optimal Copula function constructs multivariate joint probability distribution function；
Step 4. using derive be suitable for any dimension variable condition most probable combinatorial formula calculating with crest discharge or when
Duan Hongliang is condition, the condition most probable combination of day part magnanimity or crest discharge；
The combined value of the crest discharge being calculated in step 4 and period magnanimity is drafted mark as design flood by step 5.
It is quasi-.
2. the design flood peak amount condition most probable combined method according to claim 1 based on Copula function, special
Sign is, in the step 1, the flood data of extraction include the historical flood data examined by three property.
3. the design flood peak amount condition most probable combined method according to claim 1 based on Copula function, special
Sign is, in the step 2, carries out frequency calculating to flood data using the method that is uniformly processed, fits line using computer optimization
Estimate mean value E (X), coefficient variation Cv, the coefficient of skew Cs of P-III type curve, and following P-III is converted into according to formula (1)
Type distribution density function fX(x) corresponding α, β, γ in expression formula (2)：
Wherein, Γ (α) is the gamma function of α, and e is the truth of a matter of exponential function.
4. the design flood peak amount condition most probable combined method according to claim 1 based on Copula function, special
Sign is, in the step 2, using χ2The method of inspection is fitted goodness inspection.
5. the design flood peak amount condition most probable combined method according to claim 1 based on Copula function, special
Sign is, in the step 3, selects suitable Copula function including the use of crosscheck and AIC criterion symmetrical
Preferred multidimensional Copula function in Archimedean Copula class function；The parameter packet of the optimal Copula function of evaluation fitting
It includes and Copula function parameter is estimated using maximum-likelihood method.
6. the design flood peak amount condition most probable combined method according to claim 1 based on Copula function, special
Sign is that the derivation of the condition most probable combinatorial formula suitable for any dimension variable in the step 4 further comprises son
Step：
4.1 calculate stochastic variable X1Take design value x1When, surplus variable XiConditional probability distribution function, wherein i=2,3 ...,
N, expression formula are shown below：
Wherein, stochastic variable X1Represent crest discharge or its corresponding a certain period magnanimity, xiIt is stochastic variable XiValue, C
(u1,u2,u3,…,un-1,un) it is Copula function,For crest discharge and its corresponding day part magnanimity
Marginal distribution function；
4.2 calculate the corresponding conditional joint probability density function of (3) formula, and expression formula is shown below：
In formula：
For the density of Copula function
Function is indicated in following formula with c；For XiProbability density function, wherein i=2,3 ..., n；
4.3 f(x2,x3,x4,…,xn|x1) corresponding x when being maximizedMiValue, wherein i=2,3 ..., n, form combinationAs condition most probable combines, and is maximum point using the point that partial derivative is zero
Property obtains expression formula and is shown below：
(5) formula is equivalent to expression formula as follows by 4.4：
In formula,
4.5 inquire into XiRelationship between the density function and its derived function of the P-III distribution of obedience, is shown below：
WhereinFor XiThe parameter of corresponding P-III distribution density function；
4.6 (7) formula is brought into (6) formula obtain condition most probable combination general formula expression formula, be shown below：
7. the design flood peak amount condition most probable combined method according to claim 1 based on Copula function, special
Sign is, is calculated by quasi-Newton method using crest discharge or period magnanimity as condition in the step 4, day part magnanimity or flood
The condition most probable of peak flow combines.
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Cited By (4)
| Publication number | Priority date | Publication date | Assignee | Title |
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| CN110377989A (en) * | 2019-07-08 | 2019-10-25 | 武汉大学 | Two Variational Design flood calculation method of nonuniformity based on hydrothermal reaction coupling balance |
| CN110598315A (en) * | 2019-09-10 | 2019-12-20 | 太原理工大学 | Uncertainty analysis method for basin non-uniformity design flood under variable conditions |
| CN111079086A (en) * | 2019-12-06 | 2020-04-28 | 华中科技大学 | Multi-element joint distribution-based multiple risk assessment method for water resource system |
| CN111984693A (en) * | 2020-07-17 | 2020-11-24 | 河海大学 | Method for calculating design flood for large-scale river and lake communication based on Copula function |
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2018
- 2018-05-07 CN CN201810424328.4A patent/CN108875130A/en not_active Withdrawn
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| Publication number | Priority date | Publication date | Assignee | Title |
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| CN110377989A (en) * | 2019-07-08 | 2019-10-25 | 武汉大学 | Two Variational Design flood calculation method of nonuniformity based on hydrothermal reaction coupling balance |
| CN110377989B (en) * | 2019-07-08 | 2022-08-05 | 武汉大学 | Non-uniformity two-variable design flood calculation method based on hydrothermal coupling balance |
| CN110598315A (en) * | 2019-09-10 | 2019-12-20 | 太原理工大学 | Uncertainty analysis method for basin non-uniformity design flood under variable conditions |
| CN110598315B (en) * | 2019-09-10 | 2022-11-18 | 太原理工大学 | Uncertainty analysis method for basin non-uniformity design flood under variable conditions |
| CN111079086A (en) * | 2019-12-06 | 2020-04-28 | 华中科技大学 | Multi-element joint distribution-based multiple risk assessment method for water resource system |
| CN111079086B (en) * | 2019-12-06 | 2021-11-02 | 华中科技大学 | Multi-element joint distribution-based multiple risk assessment method for water resource system |
| CN111984693A (en) * | 2020-07-17 | 2020-11-24 | 河海大学 | Method for calculating design flood for large-scale river and lake communication based on Copula function |
| CN111984693B (en) * | 2020-07-17 | 2023-12-08 | 河海大学 | Method for pushing out design flood for large-scale river-dredging continuous lake based on Copula function |
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Application publication date: 20181123 |