In the case of Bayesian statistics, all unknown parameters are regarded as random variables, and their distributions are obtained from the known information (Vrugt, 2016). Therefore, Bayesian statistics provides a rigorous method for uncertainty analysis and can provide key information for management decision-making. Bayesian reasoning is based on the following formula (Martino and Elvira, 2018): p X | Y = p X p Y | X p Y = p X p Y | X ∫ X p X p Y | X d X ∝ p X p Y | X , (1) where p X | Y is the a posteriori probability distribution function of X, indicating the obtained observation value, and Y is the distribution law of parameter X; p X refers to the a priori probability distribution function, which represents the distribution law of parameter x obtained from data and experience; p Y | X is the likelihood function, which indicates the fitting degree between the model parameters and the observed data (Zhang et al., 2016).

The optimal design of the drainage network monitoring scheme mainly includes the optimization of the number and location of the monitoring well points (Jiang et al., 2021; Sambito and Freni, 2021). Assuming that the initial monitoring time is t and the monitoring value obtained from the monitoring scheme s is still recorded as y, then the Bayesian formula can be rewritten as the following: p x | y , S = p x | S p y | x , S ∫ p x | S p y | x , S d x , (2)

Since the prior distribution p x | S of the parameters represents a preliminary understanding of the unknown parameters and is not affected by the monitoring design scheme s, that is, p x | S = p x , Equation 2 becomes: p x | y , S = p x p y | x , S ∫ p x p y | x , S d x , (3)

The normalization constant ∫ p x p y | x , S d x represents the probability of the occurrence of monitoring data y obtained from the monitoring design scheme s, which can be abbreviated as p y | S , that is: p y | S = ∫ p x p y | x , S d x , (4)

In the information theory, information entropy is defined as the measure of uncertainty, which has been widely used to estimate the uncertainty of variables (Li et al., 2012). It can be regarded as the negative expected value of the logarithm of the probability density function of a variable (Parvan, 2010). Among them, the uncertainty definition of a random variable is used to describe the amount of information of the variable. The more information required to describe the variable, the greater the uncertainty of the variable and the greater the information entropy (Monache et al., 2008; Retzer et al., 2009; Mooselu et al., 2020). Based on the above research progress, this paper establishes the SWMM pipe network model; uses the SWMM model to simulate the result under the condition of determining the initial monitoring time, monitoring interval, and monitoring times; takes the information entropy as the evaluation index to calculate the optimal monitoring schemes under different monitoring types; screens and selects the monitoring scheme that takes into account the monitoring cost and monitoring accuracy; and finally, operates with the selected monitoring scheme To summarize, the optimal design of a drainage pipe network monitoring network that can provide accurate information is very important to improve our understanding of complex pipe network systems and reduce heavy and unnecessary capital expenditure (Zhang et al., 2015; Yang et al., 2020). The mathematical expression of information entropy is as follows: H x = − ∫ a b f x ln ⁡ f x d x , (5) where x is the random variable; f x is the probability density function of a random variable with a priori range of [a, b].

The specific coupling calculation process of Python programming of a monitoring point optimization model based on information entropy includes the following five steps: generation of unknown parameters,* inp file modification, extraction and storage of the time series, calculation of the conditional probability function, and calculation of information entropy. The automatic modification of the SWMM model and the extraction of the time series are directly realized in Python program without opening SWMM software. The optimization model is also the coupling calculation of the information entropy algorithm and the SWMM model. The time series extracted by the SWMM model is directly used for the calculation diagram of the likelihood functions p y i | x i and p y | x i . The specific calculation principle of information entropy of the monitoring nodes of a pipe network is as follows:

Based on the above coupling principle, Python language programming is used to directly calculate the SWMM model to extract the pollutant concentration monitoring value y, and the inversion model is used to invert the unknown parameter x, where the posterior probability density function of x is p x | y , S , s is the monitoring scheme, and the information entropy of the posterior distribution of parameter X is similarly defined as follows: H S , y = − ∫ p x | y , S ln ⁡ p x | y , S d x , (6)

The left end of Equation 6 contains the monitoring value y, but due to the influence of the actual conditions, the actual monitoring value y cannot be obtained in the optimization node stage of the monitoring points. Therefore, it is assumed that the monitoring value y is a random variable, and the corresponding probability density function is p y | S (Kaynar and Ridder, 2010). Therefore, in order to obtain the calculated information entropy only affected by the monitoring scheme, both sides of Equation 5 are multiplied by p y | S , and then y is integrated to obtain the expectation of information entropy H S , y : E H S , y = − ∫ ∫ p x | y , S ln ⁡ p x | y , S d x p y | S d y = − ∫ ∫ p x | y , S p y | S ln ⁡ p x | y , S d x d y , (7)

According to Equation 7, E H S , y is only affected by the monitoring scheme S, which is recorded as E S . By understanding the relevant principles of information entropy, we can solve E S . According to the concept of information entropy, the unknown parameter x is inversed when using the pollutant concentration monitoring value y* obtained from s*. At this time, the information entropy of x posterior distribution is the smallest, indicating that the uncertainty of X is also the smallest, and the inversion effect is the best (Liu et al., 2013; Wang et al., 2016). In Equation 7, the calculation of the solution algorithm is complex, and it is difficult to calculate the explicit result of this expression. Therefore, this paper uses the Monte Carlo method to approximate the solution. First, Equation 7 is rewritten as follows using Equation 2: E H S , y = − ∫ ∫ p x | y , S ln ⁡ p x | y , S d x p y | S d y = − ∫ ∫ p x | y , S p y | S ln ⁡ p x | y , S d x d y , (8)

The prediction uncertainty of the expected value of information entropy can be divided into two parts as follows: E S = U C + U S = − ∬ p y | x , S p x ln ⁡ p x d x d y + − ∬ p y | x , S p x ln ⁡ p y | x , S − ⁡ ln ⁡ p y | S d x d y , (9) where E S represents the total uncertainty of model prediction, U C is the parameter uncertainty derived from the inside of the model, and U S is the conceptual model uncertainty derived from among the models.

In Equation 9, the prior distribution information entropy of the inversion parameter x is as follows: U C = − ∬ p y | x , S p x ln ⁡ p x d x d y = − ⁡ ln ⁡ p x , (10)

According to Equation 10, the larger the prior range of inversion parameter x, the larger the calculated value of information entropy of parameter x, and the greater the uncertainty of the inversion result of parameter X. When the prior probability distribution p x remains unchanged, the information entropy − ⁡ ln ⁡ p x remains unchanged. Therefore, in order to obtain the minimum value of E S , the second half is extracted. As shown in Equation 1, only the minimum value of U S needs to be calculated to obtain the minimum value of information entropy of the corresponding monitoring scheme (Ainslie et al., 2009; Yan and Zhou, 2019). U S = − ∬ p y | x , S p x [ ⁡ ln ⁡ p y | x , S − ⁡ ln ⁡ p y | S ] d x d y , (11)

According to Keum et al. (2018), the value of U(S) is less than 0 in equation 11. Therefore, according to the concept of information entropy and formula 8, the use of monitoring points in the Bayesian formula reduces the uncertainty of the inversion parameter X.

Equation 12 is solved by the Monte Carlo method as follows: U S ≈ − 1 N ∑ i = 1 N ⁡ ln ⁡ p ( y i x i , S ) − ⁡ ln ⁡ p y i | S , (12)

First, n samples are randomly selected from the unknown parameter x a priori distribution P X , which are recorded as x i (i = 1, 2, … , N). For each i∈N, using Python program to modify the unknown parameters of the SWMM model with samples, the simulation is used to extract the conditional probability density function under the condition of the monitoring scheme s, and then one sample y i is randomly selected from the function p y | x i , S , with a total of n samples. Each group of x i and y i is replaced into Equation 1, and p y i | x i , S is calculated in the conditional probability density function of Equation 12. It can be seen from Equation 5 that in Equation 12, where p y i | S = ∫ p x p y i | x , S d x , the Monte Carlo method is still used to calculate this integral. p y i | S ≈ − 1 N ∑ j = 1 N p y i | x i , S , (13)

Therefore, as long as the monitoring design scheme s is fixed, the approximate information entropy of this monitoring scheme can be obtained through equations 8–11.

The Metropolis–Hastings (MH) algorithm is a typical Markov chain Monte Carlo (MCMC) sampling method. In this study, the Metropolis–Hastings algorithm will be used. Its essence is to construct an appropriate Markov chain and generate the next state under a given state (Abellán et al., 2011; Wang and Chen, 2013). Assuming that the target distribution is f x , a sample with sample size T will be generated from this distribution. The iterative steps of the Metropolis–Hastings algorithm are shown in Figure 1. x t is a vector, which represents the value of the tth iteration in the algorithm (Wang and Jin, 2013; Zeng et al., 2016).

It is assumed that the unknown parameters of the model are days, where C ¯ i , C i θ , and p C ¯ i θ are the measured value, predicted value, and likelihood function of the ith measuring point; ε = C i θ − C ¯ i is the measurement error; and i = 1–6. It is assumed that ε obeys the mean value 0, and the standard deviation is σ. The likelihood function of pollutant traceability, i.e., the conditional probability density function, can be expressed as follows: p C ¯ i | θ = 1 2 π σ 2 6 / 2 exp ∑ i = 1 6 − C i θ − C ¯ i 2 2 σ 2 , (14)

According to the Bayesian theorem, the posterior probability density function of the model parameters is as follows: p θ | C ¯ i = p θ 2 π σ 2 6 / 2 exp ∑ i = 1 6 − C i θ − C ¯ i 2 2 σ 2 , (15)

It is assumed that the value of model parameters α=(Jx, M, T)=(15, 950000, 60) is true, where Jx is the node number, M denotes pollutant loads, and T denotes time. We use the three monitoring schemes selected in the previous section and the MH algorithm to realize the inversion of pollution source parameters, in which the length of each Markov chain is 10,000. According to the calculation, the posterior distribution range of each monitoring scheme is shown in Table 4. The distribution histogram is shown in Figures 4–6.

It can be seen from Tables 4 and 5 and Figures 3 and 5 that the relative error decreases with the increase in the number of monitoring points, and the mean error of the parameters Jx, M, and T at the two monitoring points decreases by 6.774%, 0.177%, and 0.441%, respectively, compared with that of one monitoring point. The mean value errors of parameters JX, m, and t at the three monitoring points are reduced by 0.993%, 0.012%, and 0.224%, respectively, compared with that of the two monitoring points. The median error of the parameter m at two monitoring points is 0.265% lower than that at one monitoring point. The median error of the parameter m at the three monitoring points is 0.056% lower than that at the two monitoring points. It can be seen that the error of the two monitoring points decreases more than that of one monitoring point, and the error of three monitoring points decreases more slowly than that of two monitoring points, which is consistent with the trend of information entropy, indicating that information entropy can accurately describe the uncertainty of the model.

Among them, the prior range of inversion nodes and inversion time from scheme 1 to scheme 3 is also gradually reduced. Due to the high pollution value, the posterior distribution range is not significantly reduced. However, the mean value error is reduced to a certain extent, which is consistent with the change trend of monitoring scheme information. It can be seen from Table 5 that the real value probabilities of the parameters Jx and T in scheme 2 are increased by 7.78% and 10.03%, respectively, compared with the two parameter probabilities in scheme 1, and the real value probabilities of the parameters Jx and T in scheme 3 are increased by 3.79% and 4.24%, respectively, compared with the two parameter probabilities in scheme 2. It is further determined that the smaller the information entropy of the monitoring scheme, the higher the inversion accuracy and the smaller the uncertainty of the posterior distribution. However, the more the number of monitoring nodes, the more the cost and the less the cost of single-well monitoring. The information entropy difference between scheme 2 and scheme 3 is very small, and the average value of the relative error of the posterior sample mean of the three parameters is also approximately equal. Adding a monitoring point for monitoring will significantly increase the cost. For this case, if it is necessary to comprehensively consider the monitoring cost and the posterior distribution range of the parameters, scheme 2 is considered to be the best monitoring scheme.