Urban drainage model uncertainty control method and device based on heterovariance error model

Abstract

The invention discloses an urban drainage model uncertainty control method based on a heterovariance error model, and the method comprises the following steps: obtaining historical data of an area where an urban drainage model is located, the historical data comprising rainfall monitoring data and water level monitoring data; performing simplified expression processing on the urban drainage model to obtain a simplified function composed of simulation values and error terms, and setting that the error terms obey normal distribution and are mutually independent; and constructing a verification target function based on the simplified function, and inputting the historical data as an analog value to perform iteration so as to update model parameters of the urban drainage model and obtain an optimal urban drainage model. The invention further provides an urban drainage model uncertainty control device. According to the method provided by the invention, the problem of unreasonable prediction distribution caused by unreasonable model error hypothesis of a traditional least square method can be solved, so that the accuracy of probability prediction of the urban drainage model is improved.

Description

Technical Field

[0001] The invention belongs to the field of municipal engineering design, and in particular relates to an uncertainty control method and device for an urban drainage model based on a heteroscedastic error model. Background Art

[0002] In the actual application process, urban drainage models are affected by various uncertainties, which will affect the final model prediction. Various uncertainties can be divided into: input uncertainty, structural uncertainty, parameter uncertainty and measurement uncertainty according to their sources. At present, the parameter uncertainty of urban drainage models has been widely recognized by urban drainage model researchers, but the uncertainty from other sources has rarely been taken seriously by researchers. The current method for dealing with the uncertainty of urban drainage models is mainly the generalized likelihood uncertainty estimation method (GLUE). The GLUE method has been criticized by some scholars for being too subjective and not statistically rigorous in practical applications.

[0003] Due to its rigorous statistics, the formal Bayesian method is widely used in the uncertainty quantification of the conceptual rainfall-runoff model in non-urban watersheds. However, the traditional formal Bayesian method usually adopts the least squares method, which assumes that the model error obeys a normal distribution with a mean of 0 and a constant variance. However, this usually results in the model prediction uncertainty being too small at high flow (water level) and too large at low flow (water level), which does not conform to the actual application of urban drainage models.

[0004] Patent document CN117649058A discloses a water conservancy guidance optimization method, system and storage medium for urban drainage, including: obtaining terrain data and drainage distribution data of the urban area from the urban terrain database; establishing an urban drainage model based on the terrain data and drainage distribution data of the urban area; analyzing the urban drainage model, obtaining the drainage distribution in the urban drainage model, and delineating the optimization area; guiding and optimizing urban drainage in the optimization area, and delineating the drainage area to be optimized and the drainage route to be optimized.

[0005] Patent document CN109657841A discloses a method for extracting the depth of urban waterlogging caused by heavy rain, including establishing an urban terrain model, obtaining the waterlogging contribution area of ​​urban depressions, establishing an urban rainfall model and an urban drainage model, obtaining the total surface runoff, establishing an urban surface waterlogging model, and obtaining the waterlogging depth. Summary of the invention

[0006] The purpose of the present invention is to provide a method and device for controlling uncertainty of an urban drainage model based on a heteroscedastic error model. The method can solve the problem that the traditional least squares method has unreasonable assumptions about model errors, which leads to unreasonable prediction distribution, so as to improve the accuracy of probability prediction of the urban drainage model.

[0007] In order to achieve the first object of the present invention, the following technical solution is provided: an uncertainty control method of an urban drainage model based on a heteroscedastic error model, comprising the following steps:

[0008] Acquire historical data in the area where the urban drainage model is located, wherein the historical data includes rainfall monitoring data and water level monitoring data;

[0009] Simplifying the urban drainage model to obtain a simplified function consisting of a simulation value and an error term, and setting the error term to obey a normal distribution and be independent of each other;

[0010] A verification objective function is constructed based on the simplified function, and the historical data is used as a simulation value input for iteration to update the model parameters of the urban drainage model, thereby obtaining an optimal urban drainage model.

[0011] The present invention adds multiplicative Gaussian noise to the error assumption of the urban drainage model to provide model parameters that are more in line with reality, thereby improving the accuracy of the probability prediction of the urban drainage model.

[0012] Specifically, the expression of the simplified function is as follows:

[0013] Y t =f(X t ,β f )+e t

[0014] Among them, Y t represents the observed value, f(·) represents the urban drainage model, X t represents the simulated value of the model input, β f represents the model parameters of the urban drainage model, e t represents the error term.

[0015] Specifically, the error term includes multiplicative Gaussian noise and additive Gaussian noise.

[0016] Specifically, the expression of the error term is as follows:

[0017] e t =u t f(X t ,β f )+v t

[0018]

[0019] Among them, u t represents multiplicative Gaussian noise, represents the maximum value of multiplicative Gaussian noise, v t represents additive Gaussian noise, Represents the maximum value of additive Gaussian noise.

[0020] Specifically, the urban drainage model is constructed using the SWMM model framework.

[0021] Specifically, the verification objective function is constructed using the formal Bayesian method and solved by Markov Monte Carlo. Its specific expression is as follows:

[0022] Among them, p(β f ,β e ) represents the prior distribution of the model and error term, n represents the total number of observations, f(·) represents the urban drainage model, X t represents the simulated value of the model input, β f represents the model parameters of the urban drainage model, represents the maximum value of multiplicative Gaussian noise, Represents the maximum value of additive Gaussian noise.

[0023] Specifically, the Markov chain reaching a stable distribution is used as the termination condition of the iteration.

[0024] In order to achieve the second purpose of the present invention, the following technical solution is provided: an urban drainage model uncertainty control device, which is implemented by the above-mentioned urban drainage model uncertainty control method based on heteroscedastic error model to obtain the optimal urban drainage model.

[0025] Compared with the prior art, the present invention has the following beneficial effects:

[0026] An optimization method for the urban drainage model is provided. Its multiplicative Gaussian noise is added to the error term of the urban drainage model, and a more reasonable assumption is made on the error term structure. In combination with the formal Bayesian method, the problem of the traditional least squares method having too small uncertainty when the model response is high and too large uncertainty when the model response is low is solved, thereby improving the accuracy of the probabilistic prediction of the urban drainage model. BRIEF DESCRIPTION OF THE DRAWINGS

[0027] Figure 1 A flow chart of the uncertainty control method of the urban drainage model based on the heteroscedastic error model provided in this embodiment;

[0028] Figure 2The layout diagram corresponding to the urban drainage model provided in this embodiment;

[0029] Figure 3 A comparison chart of the model optimized by the method provided in this embodiment and the model optimized by the traditional method. DETAILED DESCRIPTION

[0030] In order to make the purpose, technical scheme and advantages of the embodiments of the present invention clearer, the technical scheme in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. The components of the embodiments of the present invention generally described and shown in the drawings here can be arranged and designed in various different configurations. Therefore, the following detailed description of the embodiments of the present invention provided in the drawings is not intended to limit the scope of the claimed invention, but merely represents selected embodiments of the present invention. Based on the embodiments in the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention.

[0031] like Figure 1 As shown, the uncertainty control method of the urban drainage model based on the heteroscedastic error model provided in this embodiment includes the following steps:

[0032] The historical data in the area where the urban drainage model is located is obtained, and the historical data includes rainfall monitoring data and water level monitoring data. In this embodiment, a rain gauge and a pipe network water level sensor are used to obtain the corresponding historical data.

[0033] The urban drainage model is simplified to construct a simplified function consisting of the simulated value plus the additional error term to express the monitoring value. The expression is as follows:

[0034] Y t =f(X t ,β f )+e t ;

[0035] where Y t represents the observed value, f() represents the urban drainage model, X t represents the model input (e.g. rainfall), β f represents the model parameters, e t Represents the error term caused by various sources of uncertainty.

[0036] Considering that the heteroscedasticity of errors can affect the parameters and prediction uncertainty of the model, it is assumed that the error terms include:

[0037] e t =u t f(Xt ,β f )+v t ;

[0038]

[0039] Among them, u t represents multiplicative Gaussian noise, represents the maximum value of multiplicative Gaussian noise, v t represents additive Gaussian noise, Represents the maximum value of additive Gaussian noise.

[0040] The simplified function is transformed by using the formal Bayesian method, and the Markov Monte Carlo method is used to solve the verification objective function. The specific process is as follows:

[0041] The likelihood function formula for a given step length is derived based on the simplified function, which is expressed as follows:

[0042] p(Y t |X t ,β f ,β e )=N(f(X t ,β f ),R t )

[0043]

[0044] Among them, the likelihood function value reflects the similarity level between the simulated value and the observed value. The higher the likelihood function value, the more similar the simulated value and the observed value.

[0045] Assuming that the error terms are independent of each other, the likelihood function of multiple step sizes is the product of the likelihood functions of the individual step sizes, which can be rewritten as Where n is the total number of observations.

[0046] Based on the Bayesian formula, the posterior probability function of the urban drainage model is rewritten as:

[0047] p(β f ,β e |X t＝1:n ,Y t＝1:n )∝p(Y t＝1:n |X t＝1:n ,β f ,β e )p(β f ,β e )

[0048] Among them, p(β f ,β e) is the prior distribution of the model and error parameters, usually assumed to be uniform, reflecting the possible range of parameter values.

[0049] By maximizing the logarithmic form of the posterior probability function of the equation, the urban drainage model and the error term parameters are estimated, and the following equation can be obtained:

[0050]

[0051] Then reverse the above formula to obtain the final verification objective function:

[0052] Among them, p(β f ,β e ) represents the prior distribution of the model and error term, n represents the total number of observations, f(·) represents the urban drainage model, X t represents the simulated value of the model input, β f represents the model parameters of the urban drainage model, represents the maximum value of multiplicative Gaussian noise, Represents the maximum value of additive Gaussian noise.

[0053] The relative least squares algorithm ignores the multiplicative part, which makes the final model less responsive. The final function expression of the least squares form is min:

[0054] This embodiment also provides an urban drainage model uncertainty control device, which is implemented by the urban drainage model uncertainty control method based on the heteroscedastic error model provided in the above embodiment to obtain the optimal urban drainage model.

[0055] In order to better illustrate the technical effect of the present invention, based on Figure 2 The urban drainage model layout diagram shown in the figure compares the heteroscedastic error model method and the traditional least squares method to analyze the probability prediction distribution of the complex urban drainage model.

[0056] SWMM was used to establish the drainage model in the city, with a total of 538 subcatchments, 684 nodes, and 696 pipe segments. In this example, 11 hydrological parameters and 2 error model parameters were calibrated. In order to balance the heterogeneity and computational burden between subcatchments, all hydrological parameters were assumed to change synchronously. The uncertainty ranges of the hydrological parameters and error term parameters are shown in Table 1.

[0057] In this study of urban drainage model, rain gauges and water level sensors provide the required input and calibration data. Through the formal Bayesian method, the parameters of urban drainage model and error model are calibrated simultaneously to obtain the posterior distribution of model and error parameters. After obtaining the posterior distribution of model and error parameters, the expression is: (Y≤Y α |X)=p([f(X,β f )+e(β e )] 1…M ≤Y α |X)=α.

[0058] The 95% confidence interval of the urban drainage model prediction can be obtained by the formula. Figure 3 It can be seen from the figure that the uncertainty range changes after the formula is modified (light gray part), such as Figure 3 a and Figure 3 Compared with b in Fig. 2, at high water levels (50-100th step length), the light gray area is wider, while at low water levels (0-50, 150 onwards), the light gray area is narrower, which shows that a larger uncertainty range is given to large water levels, that is, when predicting flooding, the range changes from, for example, 1-2m to 0.5-3m. At low water levels, this range is narrower.

[0059] and Figure 3 c and Figure 3 Similarly, when the water level is high, the light gray area contains more observations, and the water level range of the waterlogging forecast is more reasonable. When the water level is low, the light gray area is more compact and reasonable.

[0060] That is, the model prediction interval obtained by the heteroscedastic model is more reasonable than the prediction interval obtained by the homoscedastic model. The uncertainty of the traditional least squares method (homoscedastic model) is too small at high flow (water level), and some observations fall out of the confidence interval. At low flow (water level), the uncertainty is too large and far from the observations. In contrast to the traditional least squares method (homoscedasticity), the heteroscedastic model includes more observations at high flow (water level), and becomes more compact and closer to the observations at low flow (water level).

[0061] Table 1

[0062] parameter Minimum Maximum Urban drainage model parameters Impermeability 50％ 150％ Feature Width 25％ 175％ Subcatchment slope 25％ 175％ Manning's coefficient of impervious area 50％ 150％ Manning coefficient of permeable area 25％ 175％ Depth of depression in impervious area 50％ 150％ Depth of depression in permeable area 25％ 175％ Percentage of impervious area without depressions 50％ 150％ Maximum infiltration rate in Horton infiltration 50％ 150％ Minimum infiltration rate in Horton infiltration 50％ 150％ Decay constant in Horton's infiltration 50％ 150％ Error model parameters Multiplicative noise standard deviation 0 1 Additive noise standard deviation (heteroskedastic model) 0 1 Additive noise standard deviation (homoskedastic model) 0 1

[0063] In summary, the present invention overcomes the defect of the traditional least squares-based model uncertainty analysis method that assumes that the variance of the error is constant, and makes a more reasonable assumption on the error model structure; through the formal Bayesian method, a more reasonable model prediction distribution can be obtained, which solves the problem that the uncertainty of the traditional least squares method is too small when the model response is high and the uncertainty is too large when the model response is low, thereby improving the accuracy of the probability prediction of the urban drainage model.

[0064] In addition, the terms "upper", "lower", "inner", "outer", "front", "rear" are used for descriptive purposes only and should not be understood as indicating or implying relative importance. Unless otherwise specifically stated, the relative steps, numerical expressions and values ​​of the components and steps described in these embodiments do not limit the scope of the present invention.

[0065] Of course, the above description is only a specific embodiment of the present invention and is not intended to limit the scope of implementation of the present invention. All equivalent changes or modifications made according to the structure, characteristics and principles described in the patent application scope of the present invention should be included in the patent application scope of the present invention.

[0066] Finally, it should be noted that the above-described embodiments are only specific implementations of the present invention, which are used to illustrate the technical solutions of the present invention, rather than to limit them. The protection scope of the present invention is not limited thereto. Although the present invention is described in detail with reference to the above-described embodiments, ordinary technicians in the field should understand that any technician familiar with the technical field can still modify the technical solutions recorded in the above-described embodiments within the technical scope disclosed by the present invention, or can easily think of changes, or make equivalent replacements for some of the technical features therein; and these modifications, changes or replacements do not make the essence of the corresponding technical solutions deviate from the spirit and scope of the technical solutions of the embodiments of the present invention, and should be included in the protection scope of the present invention. Therefore, the protection scope of the present invention shall be subject to the protection scope of the claims.

Claims

1. A method for controlling uncertainty of an urban drainage model based on a heteroscedastic error model, characterized in that: The following steps are involved: Acquire historical data in the area where the urban drainage model is located, wherein the historical data includes rainfall monitoring data and water level monitoring data; Simplifying the urban drainage model to obtain a simplified function consisting of a simulation value and an error term, and setting the error term to obey a normal distribution and be independent of each other; A verification objective function is constructed based on the simplified function, and the historical data is used as a simulation value input for iteration to update the model parameters of the urban drainage model, thereby obtaining an optimal urban drainage model.

2. The uncertainty control method of urban drainage model based on heteroscedastic error model according to claim 1 is characterized in that: The expression of the simplified function is as follows: Y t =f(X t ,b f )+e t Among them, Y t represents the observed value, f(·) represents the urban drainage model, X t represents the simulated value of the model input, β f represents the model parameters of the urban drainage model, e t represents the error term.

3. The uncertainty control method of urban drainage model based on heteroscedastic error model according to claim 1 or 2 is characterized in that: The error term includes multiplicative Gaussian noise and additive Gaussian noise.

4. The uncertainty control method of urban drainage model based on heteroscedastic error model according to claim 3 is characterized in that: The expression of the error term is as follows: e t =u t f(X t ,β f )+v t Among them, u t represents multiplicative Gaussian noise, represents the maximum value of multiplicative Gaussian noise, v t represents additive Gaussian noise, Represents the maximum value of additive Gaussian noise.

5. The uncertainty control method of urban drainage model based on heteroscedastic error model according to claim 1 is characterized in that: The urban drainage model is constructed using the SWMM model framework.

6. The uncertainty control method of urban drainage model based on heteroscedastic error model according to claim 1 is characterized in that: The verification objective function is constructed using the formal Bayesian method and solved by Markov Monte Carlo. Its specific expression is as follows: Among them, p(β f ,β e ) represents the prior distribution of the model and error term, n represents the total number of observations, f(·) represents the urban drainage model, X t represents the simulated value of the model input, β f represents the model parameters of the urban drainage model, represents the maximum value of multiplicative Gaussian noise, Represents the maximum value of additive Gaussian noise.

7. The uncertainty control method of urban drainage model based on heteroscedastic error model according to claim 1 is characterized in that: The termination condition of the iteration is when the Markov chain reaches a stationary distribution.

8. An urban drainage model uncertainty control device, characterized in that: This is achieved by the uncertainty control method of the urban drainage model based on the heteroscedastic error model as described in any one of claims 1 to 7.

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