CN107341318B - Simulation method of full-river-based monthly runoff time displacement two-dimensional matrix - Google Patents

Simulation method of full-river-based monthly runoff time displacement two-dimensional matrix Download PDF

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CN107341318B
CN107341318B CN201710580219.7A CN201710580219A CN107341318B CN 107341318 B CN107341318 B CN 107341318B CN 201710580219 A CN201710580219 A CN 201710580219A CN 107341318 B CN107341318 B CN 107341318B
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CN107341318A (en
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王文卓
董增川
贾一飞
钟敦宇
张永杰
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Hohai University HHU
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Abstract

The invention discloses a full river based simulation method of a monthly runoff time displacement two-dimensional matrix, which is characterized by comprising the following steps: (1) constructing a three-layer simulation model based on Archimedes Copula and an artificial neural network; (2) constructing a single-site monthly runoff time sequence simulation model based on the symmetrical Archimedes Copula and a multi-site monthly runoff joint distribution simulation model based on the asymmetrical Archimedes Copula; (3) calculating parameters by a Kendall coefficient through a conversion method; (4) and constructing a simulation model of the runoff displacement sequence along the river and the moon based on the artificial neural network. The advantages are that: the model base of the medium-and-long-term hydrological forecasting is enriched, the model can simulate the monthly runoff time sequence of a single site and the joint distribution of the monthly rows of multiple sites, and can also simulate a monthly runoff two-dimensional matrix with double variables of time displacement; the temporal and spatial correlation structure is reasonably described.

Description

Simulation method of full-river-based monthly runoff time displacement two-dimensional matrix
Technical Field
The invention relates to a method for simulating monthly runoff, in particular to a method for simulating a monthly runoff time displacement two-dimensional matrix at any point of a whole river.
Background
The united dispatching of the watershed water resources firstly needs to clear the risks and the uncertainty of a complex hydrological system, and the uncertainty of the hydrological system, including randomness, ambiguity, grayness and uncertainty, is always a hotspot of hydrological research. The river water process plays a crucial role in reservoir scheduling, and the randomness is just one of hydrologic uncertainty. The theory of the random process and the time sequence analysis technology can be used for researching the statistical variation characteristics of the runoff process, provide a theoretical basis for planning and scheduling of a hydrological water resource system, and are one of the main modes for recognizing, designing and managing the complex hydrological system.
The runoff simulation sequence is used as an input condition for unified scheduling of water resources, water supply information is provided for a reservoir group, water supply information is also provided for each water intake point along a river, and only a small part of the points have runoff data information. The existing multi-site runoff simulation model is used for carrying out runoff simulation on established hydrological sites by means of existing historical data, the requirement of unified scheduling of water resources cannot be met, and therefore a new runoff simulation model needs to be developed, and the runoff time sequence of any point of the whole river is simulated by analyzing the autocorrelation relation and the cross-correlation relation among the sites and reducing the order by a certain method.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provide a time displacement two-dimensional monthly runoff simulation model which is suitable for nonlinear and skewed distribution under the current climate change condition, the model can simulate the monthly runoff time sequence of a single site and the monthly runoff joint distribution of multiple sites, and can simulate a full river time displacement bivariate monthly runoff two-dimensional matrix, thereby providing a more powerful basis for the uniform dispatching of watershed water resources.
In order to solve the technical problem, the invention provides a full river-based simulation method of a monthly runoff time displacement two-dimensional matrix, which is characterized by comprising the following steps:
(1) constructing a three-layer simulation model based on Archimedes Copula and an artificial neural network;
(2) constructing a single-site monthly runoff time sequence simulation model based on the symmetrical Archimedes Copula and a multi-site monthly runoff joint distribution simulation model based on the asymmetrical Archimedes Copula;
(3) calculating parameters by a Kendall coefficient through a conversion method;
(4) and constructing a simulation model of the runoff displacement sequence along the river and the moon based on the artificial neural network.
Further, the step (1) constructs a three-layer simulation model based on Archimedes Copula and an artificial neural network:
adopting Archimedes Copula to simulate the sample runoff value, adopting an artificial neural network to reduce the order, and solving the relation function of the model parameter and the displacement so as to obtain the two-dimensional runoff simulation function of the whole river:
Figure BDA0001352071660000021
in the formula Fi，jWherein x is runoff marginal distribution, C x is Copula function, x is time and displacement two-dimensional variable,
Figure BDA0001352071660000022
the runoff size of the monthly runoff is shown, i represents the time of the monthly runoff, and j represents the site of the monthly runoff.
Further, a runoff variable matrix of each station in each month is established according to the number of hydrologic stations in the research area and the length of research data
Figure BDA0001352071660000023
Represents a month; j ═ 1,2, …, n, n represents the specific number of sites;
Figure BDA0001352071660000024
represents Yi,jThe runoff value of a variable, k ═ 1,2, …, m, m represents the length of time.
Further, the step (2) constructs a single-site monthly runoff time sequence simulation model based on the symmetrical Archimedes Copula and a multi-site monthly runoff joint distribution simulation model based on the asymmetrical Archimedes Copula:
the month runoff autocorrelation structure has no obvious nesting characteristic, and if the current month runoff and the previous two months runoff have a dependency relationship, a symmetrical Archimedes Copula structure is adopted to calculate the runoff joint distribution of adjacent three months:
Figure BDA0001352071660000031
in the formula ui,j＝Fi,j(. x); p represents a probability.
According to the relevant characteristic that the runoff of the downstream site and the runoff of other sites at the upstream have a dependency relationship, a fully nested asymmetric Archimedes Copula is selected to describe a dependency structure of the joint distribution of the monthly runoff of each site and the monthly runoff of all the sites at the upstream, and the joint distribution is as follows:
Figure BDA0001352071660000032
in the formula
Figure BDA0001352071660000033
Generators of Copula functions for different layers, n*Number indicating station, n*Is less than or equal to n, and is,
Figure BDA0001352071660000034
is composed of
Figure BDA0001352071660000035
Is the inverse function of (c).
Further, the parameters are calculated in the step (3) by adopting a Kendall coefficient conversion method:
by way of variation, the Copula runoff combined distribution for three consecutive months of formula (1.2) can be expressed as:
Figure BDA0001352071660000036
thereby converting the three-dimensional Copula function into a binary Copula function for calculation, and the fully nested asymmetric Archimedes Copula is formed by nesting a string of binary Copula,
therefore, the model parameters can be calculated by adopting a Kendall coefficient conversion method,
copula function parameters are related to Kendall coefficients as follows:
Figure BDA0001352071660000041
the Kendall coefficient calculation equation is as follows:
Figure BDA0001352071660000042
in the formula
Figure BDA0001352071660000043
Figure BDA0001352071660000044
τ1 ijFor time layer, τ2 ijFor displacing the layer, and:
Figure BDA0001352071660000045
further, the step (4) constructs a simulation model of the runoff displacement sequence along the river and the moon based on the artificial neural network:
respectively carrying out error analysis on simulation results of different Archimedes Copula functions including Clayton Copula, Gumbel Copula and Frank Copula in the single-site monthly runoff time sequence simulation model and the multi-site monthly runoff combined distribution simulation model by adopting an AIC (advanced information center) rule, and selecting an optimal Copula function type:
Figure BDA0001352071660000046
wherein V is the number of parameters,
Figure BDA0001352071660000047
is the mean square error, W is the number of samples.
Reducing the selected multi-site simulation sequence of the optimal Copula function, and converting the multi-site monthly runoff joint distribution condition and the relation function of each point along the river into the relation function of the parameter and the displacement of each point from the river source in the model:
ψχ＝gk(f2) (1.9) formula (I) wherein f2Is the output of the artificial neural network, gk(. to a transfer function, ψχX is the number of neurons for the model parameters found based on the corresponding number of neurons,
selecting a BP neural network of a hidden layer to describe the relationship, obtaining the number of the best neurons of the hidden layer by adopting a cross-check method, obtaining parameters of corresponding pairs of any points along a river after determining the structure of the neural network so as to obtain a runoff simulation value,
formula (1.1) can thus be expressed as:
Figure BDA0001352071660000051
the invention achieves the following beneficial effects:
the model base of the medium-and-long-term hydrological forecasting is enriched, the model can simulate the monthly runoff time sequence of a single site and the joint distribution of the monthly rows of multiple sites, and can also simulate a monthly runoff two-dimensional matrix with double variables of time displacement; the symmetric Archimedes Copula function and the non-Archimedes Copula function adopted in the model can reasonably describe time and space related structures respectively; and converting the joint distribution structure of each point along the river into the relation between the calculation distribution parameters and the displacement, and constructing a relation function of the relation through a BP (back propagation) neural network.
Drawings
FIG. 1 is a flow chart of the present invention;
FIG. 2 is a diagram of symmetric and asymmetric Copula function structures;
fig. 3 is a case watershed diagram;
FIG. 4 is a graph of the results of a moonflow simulation at the site of Thangyh;
FIGS. 5-1 to 5-12 are graphs of the α parameter downscaling simulation results for 1 to 12 months;
FIGS. 6-1 to 6-12 are graphs of the β parameter downscaling simulation results for 1 to 12 months;
FIGS. 7-1 to 7-12 are graphs of the simulation results of theta parameter downscaling by 1 to 12 months;
FIG. 8 is a diagram of the space-time two-dimensional simulation results of the yellow river dry flow monthly runoff.
Detailed Description
The invention is further described below with reference to the accompanying drawings. The following examples are only for illustrating the technical solutions of the present invention more clearly, and the protection scope of the present invention is not limited thereby.
The invention discloses a full river based simulation method of a monthly runoff time displacement two-dimensional matrix, which comprises the following specific implementation modes as shown in a figure 1:
(1) and constructing a fitting model of the monthly runoff marginal distribution.
Establishing a runoff variable matrix of each station in each month according to the number of hydrological stations in a research area and the length of research data
Figure BDA0001352071660000061
Represents a month; j ═ 1,2, …, n, n represents the specific number of sites;
Figure BDA0001352071660000062
represents Yi,jThe runoff value of the variable, k is 1,2, …, m, m represents the time length, PIII distribution is selected to fit the monthly runoff sequence of each site, and the maximum likelihood method is adopted to calculate the parameters:
Figure BDA0001352071660000063
in the formula, theta is a parameter vector. And then, the variable values in the matrix are obtained.
(2) And constructing a single-site monthly runoff time series simulation model.
River runoff under the climate change condition has the characteristics of nonlinearity, skewed distribution and the like, a Copula function can be adopted to construct a combined distribution function month by month, the edge distribution and the correlation structure of variables are respectively researched, and information distortion is not generated. According to the Sklar theorem, if H (-) is a joint distribution function with edge distributions F (-) and G (-) which are continuous, there is only one Copula function C (-) satisfying:
Figure BDA0001352071660000064
the month runoff autocorrelation structure has no obvious nesting characteristic, and a symmetrical Archimedes Copula structure shown in figure 2 can be adopted to calculate runoff joint distribution of adjacent three months, such as January and February, December and the following month and February. The high-dimensional Archimedes Copula only contains one parameter, but the calculation process is complex, and the parameter solution can be simplified by converting the high-dimensional Archimedes Copula into binary Copula:
Figure BDA0001352071660000065
the runoff union distribution for three consecutive months can be expressed as:
Figure BDA0001352071660000071
the specific function types of Copula function C (·, ·), the most common of which is the archimedes Copula function because of having a complete generator, are diverse, including Clayton Copula, Gumbel Copula, Frank Copula, and others. And respectively adopting the three binary Archimedes Copula functions to calculate the joint distribution among variables so as to calculate the runoff joint distribution for three continuous months. The joint distribution expression of the binary Clayton Copula is as follows:
Figure BDA0001352071660000072
the joint distribution expression of the binary Gumbel Copula is as follows:
Figure BDA0001352071660000073
the joint distribution expression of binary Frank Copula is:
Figure BDA0001352071660000074
known u1,u2,…,ui-1The conditional distribution of the joint distribution can be obtained:
Figure BDA0001352071660000075
in the runoff matrix, random numbers are randomly generated and uniformly distributed according to (0, 1)tAnd known u1,1U can be estimated from the formula (2.8)2,1,u3,1And repeating the steps to sequentially calculate the rest monthly runoff.
(3) And constructing a multi-site monthly runoff joint distribution simulation model.
Still adopting the fitting result of the marginal distribution of each monthly runoff in each site in the step (1), and according to the relevant characteristic that the runoff of the downstream site has a dependency relationship with the runoff of other sites upstream thereof, selecting a fully nested asymmetric Archimedes Copula layer as shown in FIG. 2 to describe the dependency structure of the joint distribution of each monthly runoff in each site and all the sites upstream thereof, for example, the joint distribution of site 1 and site 2, and site 3 and site 1 and site 2:
Figure BDA0001352071660000081
in the formula
Figure BDA0001352071660000082
A generator that is a function of Copula at different layers.
In the runoff matrix, random numbers are randomly generated and uniformly distributed according to (0, 1)tAnd known u1,1U can be estimated from the formula (2.8)1,2U can be obtained from the following formulae (2.5), (2.6) and (2.7)1,1,u1,2Then to calculate u1,3And sequentially calculating the monthly runoff of other stations.
(4) And calculating parameters of the single-site monthly runoff time sequence simulation model and the multi-site monthly runoff combined distribution simulation model.
There are many methods for calculating Copula function parameters, and the method for solving parameters by using Kendall coefficients is the simplest method, but the method is only suitable for the case of binary Copula. The structure of the model or the structure of the model can meet the use condition of a Kendall conversion method through conversion. The relation function of the parameters and Kendall coefficients in the Clayton Copula function is as follows:
Figure BDA0001352071660000083
the relationship function of the parameters and Kendall coefficients in the Gumbel Copula function is as follows:
Figure BDA0001352071660000084
the function of the relationship between the parameters and Kendall coefficients in the Frank Copula function is:
Figure BDA0001352071660000085
the Kendall coefficient of a data sample can be obtained by equation (2.12):
Figure BDA0001352071660000086
in the formula
Figure BDA0001352071660000087
Figure BDA0001352071660000088
τ1 i,jFor time layer, τ2 i,jFor displacing the layer, and
Figure BDA0001352071660000091
(5) selecting the optimal Copula type simulating the joint distribution of single-site monthly runoff time sequence and multi-site monthly runoff
And respectively carrying out error analysis on simulation results of different Archimedes Copula functions in the single-site monthly runoff time sequence simulation model and the multi-site monthly runoff combined distribution simulation model by adopting an AIC (advanced analytical instrumentation) rule, and selecting an optimal Copula function type:
Figure BDA0001352071660000092
wherein V is the number of parameters,
Figure BDA0001352071660000093
is the mean square error, W is the number of samples.
(6) And constructing a simulation model of runoff displacement sequences along rivers and months.
And reducing the order of the selected multi-site simulation sequence of the optimal Copula function, and converting the multi-site monthly runoff joint distribution condition and the relation function of each point along the river into the relation function of the parameter in the model and the displacement of each point from the river source. Selecting a BP neural network of a hidden layer to describe the relation, wherein the hidden layer adopts a hyperbolic tangent function:
f1 l＝tansig(w1 lx+b1 l) (2.16)
the output layer uses a linear transfer function:
Figure BDA0001352071660000094
RMSE was used as an error function:
Figure BDA0001352071660000095
(7) the number of hidden layer neurons is determined.
The number of the hidden layer optimal neurons is obtained by adopting a cross-checking method, a sample is divided into a plurality of groups, each group is used as a checking group, other groups are used as training groups, results under different neuron numbers are checked, and the number of the neurons under the condition of the minimum error is selected.
And then determining a neural network structure and then obtaining parameters corresponding to any point along the river so as to obtain a runoff simulation value of the runoff.
In the case of the implementation, the method,
the yellow river dry flow monthly runoff is selected for two-dimensional simulation in the embodiment. With the monthly runoff data from 1956 to 2010 of the 11 sites as shown in fig. 3, the basic data of the 11 sites is shown in the following table.
Figure BDA0001352071660000101
(1) And constructing a fitting model of the monthly runoff marginal distribution.
Establishing runoff matrix of each month of each site
Figure BDA0001352071660000102
And selecting P III distribution to fit the monthly runoff sequence of each site, and calculating parameters according to the formula (2.1).
(2) And constructing a single-site monthly runoff time series simulation model.
And calculating the runoff combined distribution of three consecutive months from january and february to march, february and march and april, march and april, april and may, april and may and june, may and june and july, june and july and august, july and august and september, august and september and october, september and october and november, october and november, december and november and decennial and february in turn according to the formulas (2.3) to (2.8).
Random generation of uniformly distributed random numbers obeying (0, 1)tAnd known u11U can be estimated from the formula (2.8)21,u31And repeating the steps to sequentially calculate the rest monthly runoff.
(3) And constructing a multi-site monthly runoff joint distribution simulation model.
And (3) calculating a dependent structure of joint distribution of the monthly runoff of each site and all sites upstream of the site according to the formula (2.9), for example, joint distribution of site 1 and site 2, and joint distribution of site 3 and site 1 and site 2.
Random generation of uniformly distributed random numbers obeying (0, 1)tAnd known ui1U can be estimated from the formula (2.8)i2U can be obtained from the following formulae (2.5), (2.6) and (2.7)i1,ui2Then to calculate ui3And sequentially calculating the monthly runoff of other stations.
(4) And calculating parameters of the single-site monthly runoff time sequence simulation model and the multi-site monthly runoff combined distribution simulation model.
Respectively calculated according to the formula (2.13) and the formula (2.14)
Figure BDA0001352071660000111
Kendall coefficients between the respective data, parameters of the two models were calculated as in equations (2.10) to (2.12). Substituting the parameters into step three to obtain simulated values of month runoff at the site of tanhemia is shown in fig. 4.
(5) Selecting the optimal Copula type simulating the joint distribution of single-site monthly runoff time sequence and multi-site monthly runoff
The AIC method is adopted to carry out error analysis on simulation results of different Archimedes Copula functions in the single-site monthly runoff time series simulation model, and the optimal Copula function type is Clayton Copula as shown in the following table:
Figure BDA0001352071660000112
Figure BDA0001352071660000121
error analysis is carried out on simulation results of different Archimedes Copula functions in the multi-site monthly runoff joint distribution simulation model, and the optimal Copula function type is selected as Gumbel Copula:
Figure BDA0001352071660000122
(6) and constructing a simulation model of runoff displacement sequences along rivers and months.
And reducing the order of the selected multi-site simulation sequence of the optimal Copula function, and converting the multi-site monthly runoff joint distribution condition and the relation function of each point along the river into the relation function of the parameter in the model and the displacement of each point from the river source. Selecting a BP neural network of a hidden layer to describe the relation, wherein the hidden layer adopts a hyperbolic tangent function, an output layer adopts a linear transfer function, the number of the optimal neurons of the hidden layer is 25 by adopting a cross-over test method, and then parameters corresponding to any point along a river are obtained as shown in figures 5, 6 and 7, so that the two-dimensional runoff simulation value of the yellow river dry flow is obtained as shown in figure 8.
The above description is only a preferred embodiment of the present invention, and it should be noted that, for those skilled in the art, several modifications and variations can be made without departing from the technical principle of the present invention, and these modifications and variations should also be regarded as the protection scope of the present invention.

Claims (3)

1. A full river based simulation method of a monthly runoff time displacement two-dimensional matrix is characterized by comprising the following steps:
(1) constructing a three-layer simulation model based on Archimedes Copula and an artificial neural network, simulating a sample runoff value by adopting the Archimedes Copula, reducing the order by adopting the artificial neural network, and obtaining a relation function between model parameters and displacement so as to obtain a two-dimensional runoff simulation function of the whole river:
Figure FDA0002427952890000011
in the formula Fi,jWherein x is runoff marginal distribution, C x is Copula function, x is time and displacement two-dimensional variable,
Figure FDA0002427952890000012
representing the runoff size of the month runoff, i represents the month of the month runoff, and j represents the site of the month runoff;
establishing a runoff variable matrix of each station in each month according to the number of hydrological stations in a research area and the length of research data
Figure FDA0002427952890000013
1,2, …, 12; j ═ 1,2, …, n, n represents the specific number of sites;
Figure FDA0002427952890000014
represents Yi,jThe runoff value of a variable, k ═ 1,2, …, m, m represents the length of time;
(2) constructing a single-site monthly runoff time sequence simulation model based on the symmetrical Archimedes Copula and a multi-site monthly runoff joint distribution simulation model based on the asymmetrical Archimedes Copula,
the month runoff autocorrelation structure has no obvious nesting characteristic, and if the current month runoff and the previous two months runoff have a dependency relationship, a symmetrical Archimedes Copula structure is adopted to calculate the runoff joint distribution of adjacent three months:
Figure FDA0002427952890000021
in the formula ui,j＝Fi,j(. x); p represents a probability;
according to the relevant characteristic that the runoff of the downstream site and the runoff of other sites at the upstream have a dependency relationship, a fully nested asymmetric Archimedes Copula is selected to describe a dependency structure of the joint distribution of the monthly runoff of each site and the monthly runoff of all the sites at the upstream, and the joint distribution is as follows:
Figure FDA0002427952890000022
in the formula
Figure FDA0002427952890000023
Generators of Copula functions for different layers, n*Number indicating station, n*Is less than or equal to n, and is,
Figure FDA0002427952890000024
is composed of
Figure FDA0002427952890000025
The inverse function of (d);
(3) calculating parameters by adopting a Kendall coefficient conversion method;
(4) and constructing a simulation model of the runoff displacement sequence along the river and the moon based on the artificial neural network.
2. The simulation method of the full river-based monthly runoff time displacement two-dimensional matrix as claimed in claim 1, wherein the parameters are calculated in the step (3) by using a Kendall coefficient conversion method:
by way of variation, the Copula runoff combined distribution for three consecutive months of formula (1.2) can be expressed as:
Figure FDA0002427952890000031
thereby converting the three-dimensional Copula function into a binary Copula function for calculation, and the fully nested asymmetric Archimedes Copula is formed by nesting a string of binary Copula,
therefore, the model parameters can be calculated by adopting a Kendall coefficient conversion method,
copula function parameters are related to Kendall coefficients as follows:
Figure FDA0002427952890000032
the Kendall coefficient calculation equation is as follows:
Figure FDA0002427952890000033
in the formula
Figure FDA0002427952890000034
Figure FDA0002427952890000035
For time layer, τ2 i,jFor displacing the layer, and:
Figure FDA0002427952890000036
3. the simulation method of the full river-based monthly runoff time displacement two-dimensional matrix as claimed in claim 1, wherein the step (4) is to construct a simulation model of the river-based monthly runoff displacement sequence based on an artificial neural network, and comprises the following steps:
respectively carrying out error analysis on simulation results of different Archimedes Copula functions including Clayton Copula, Gumbel Copula and Frank Copula in the single-site monthly runoff time sequence simulation model and the multi-site monthly runoff combined distribution simulation model by adopting an AIC (advanced information center) rule, and selecting an optimal Copula function type:
Figure FDA0002427952890000041
wherein V is the number of parameters,
Figure FDA0002427952890000042
is the mean square error, W is the number of samples,
reducing the selected multi-site simulation sequence of the optimal Copula function, and converting the multi-site monthly runoff joint distribution condition and the relation function of each point along the river into the relation function of the parameter and the displacement of each point from the river source in the model:
ψχ＝gk(f2) (1.9)
in the formula f2Is the output of the artificial neural network, gk(. to a transfer function, ψχX is the number of neurons for the model parameters found based on the corresponding number of neurons,
selecting a BP neural network of a hidden layer to describe the relationship, obtaining the number of the best neurons of the hidden layer by adopting a cross-check method, obtaining parameters of corresponding pairs of any points along a river after determining the structure of the neural network so as to obtain a runoff simulation value,
formula (1.1) can thus be expressed as:
Figure FDA0002427952890000043
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