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Article

Groundwater Contamination Source Recognition Based on a Two-Stage Inversion Framework with a Deep Learning Surrogate

by
Zibo Wang
1,2,3 and
Wenxi Lu
1,2,3,*
1
Key Laboratory of Groundwater Resources and Environment, Ministry of Education, Jilin University, Changchun 130021, China
2
Jilin Provincial Key Laboratory of Water Resources and Water Environment, Jilin University, Changchun 130021, China
3
College of New Energy and Environment, Jilin University, Changchun 130021, China
*
Author to whom correspondence should be addressed.
Water 2024, 16(13), 1907; https://doi.org/10.3390/w16131907
Submission received: 11 June 2024 / Revised: 28 June 2024 / Accepted: 2 July 2024 / Published: 3 July 2024
(This article belongs to the Special Issue Deep Learning-Based Methods for Groundwater Contamination Identification)
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Abstract

:
Groundwater contamination source recognition is an important prerequisite for subsequent remediation efforts. To overcome the limitations of single inversion methods, this study proposed a two-stage inversion framework by integrating two primary inversion approaches—simulation-optimization and simulation-data assimilation—thereby enhancing inversion accuracy. In the first stage, the ensemble smoother with multiple data assimilation method (a type of simulation-data assimilation) conducted a global broad search to provide better initial values and ranges for the second stage. In the subsequent stage, a collective decision optimization algorithm (a type of simulation-optimization) was used for a refined deep search, further enhancing the final inversion accuracy. Additionally, a deep learning method, the multilayer perceptron, was utilized to establish a surrogate of the simulation model, reducing computational costs. These theories and methods were applied and validated in a hypothetical scenario for the synchronous identification of the contamination source and boundary conditions. The results demonstrated that the proposed two-stage inversion framework significantly improved search accuracy compared to single inversion methods, with a mean relative error and mean absolute error of just 4.95% and 0.1756, respectively. Moreover, the multilayer perceptron surrogate model offered greater approximation accuracy to the simulation model than the traditional shallow learning surrogate model. Specifically, the coefficient of determination, mean relative error, mean absolute error, and root mean square error were 0.9860, 9.72%, 0.1727, and 0.47, respectively, highlighting its significant advantages. The findings of this study can provide more reliable technical support for practical case applications and improve subsequent remediation efficiency.

1. Introduction

Groundwater contamination, unlike surface water contamination, is often difficult to detect and, if untreated, poses significant risks to the environment and human health [1,2,3]. Therefore, designing an effective and efficient remediation strategy is critical. Groundwater contamination source recognition (GCSR) is essential for obtaining reliable contamination source information, which is a prerequisite for remediation efforts.
GCSR involves the inversion and solution of the groundwater contamination numerical simulation model based on actual water level and contaminant concentration monitoring data, along with auxiliary information such as site investigations and professional expertise. This process makes groundwater contamination visible by determining relevant contamination source information. Currently, the inversion approaches for GCSR are categorized into three types [4]: simulation-Bayesian inference (S-BI), simulation-data assimilation (S-DA), and simulation-optimization (S-O). The S-BI is a probabilistic method known for its ability to consider the uncertainty of unknown variables. It not only provides point estimates but also offers comprehensive interval estimates. However, the search efficiency of S-BI methods may decrease when dealing with high-dimensional complex problems. S-DA methods, such as the ensemble Kalman filter, ensemble smoother (ES), and their variants, have recently gained popularity in GCSR due to their fast calculation speed and efficiency [5]. Among them, the ensemble smoother with multiple data assimilation (ES-MDA) is an iterative variant of the ES [6,7]. Its core idea is to use the covariance matrix representing unknown variables and system outputs to perform multiple data assimilation with monitoring data, thereby realizing iterative updates of unknown variables. Compared to the ES, the ES-MDA improves the approximation accuracy of nonlinear systems by performing multiple iterations. Studies have demonstrated the advantages of ES-MDA in parameter estimation [8,9,10,11]. However, this method has certain limitations, as it is based on linear estimation theory and the Gaussian assumption. Consequently, as system nonlinearity increases, the refined deep search capability of S-DA may diminish, and the inversion accuracy still needs to be improved. In contrast, the S-O method has strong search capabilities and is the most widely used in GCSR [4,12]. This method minimizes the objective function value by establishing and solving an optimization model until a set of unknown variable values is found [5]. The core of solving the optimization model lies in the optimization algorithms, which are divided into heuristic and non-heuristic types. Heuristic optimization algorithms, in particular, can avoid local optima and improve search efficiency to some extent, making them widely applied in GCSR [13,14,15,16,17,18]. These research results sufficiently demonstrated the potential of heuristic optimization algorithms. The collective decision optimization algorithm (CDOA) [19] is a novel heuristic optimization algorithm that simulates human collective decision-making behavior. The global optimal solution is achieved through information transmission and cooperation among individuals. The advantages of CDOA have been demonstrated in our previous study. Therefore, this work adopted the CDOA to solve the optimization model of GCSR. However, the S-O method has limitations, such as reliance on initial value selection. If the initial value is far from the reference value, it may decrease search efficiency and fall into the local optima, thereby affecting final accuracy [12,20].
Each of the aforementioned methods has its own advantages and disadvantages. Previous studies [21,22,23,24] have usually only used one method separately, making it easy to fall into the limitations of the methods themselves. To better capitalize on the strengths of each method, this work proposed a two-stage inversion framework that combined the two primary inversion techniques (S-O and S-DA), thereby enhancing recognition accuracy. In the first stage, the ES-MDA method took advantage of its fast speed and was employed for a global broad search to quickly obtain point and ensemble estimation results, thus providing better initial values and intervals of each unknown variable for the subsequent stage, effectively avoiding the limitation of the CDOA relying on initial value selection. Based on these results, in the second stage, the CDOA was applied for GCSR based on the optimization model, performing a refined deep search to achieve the final inversion results, to effectively compensate for the shortcomings of the weak refined search ability of the ES-MDA in dealing with complex nonlinear problems. To the best of our knowledge, this research is the first to combine ES-MDA and CDOA for GCSR. This approach breaks the constraints of previous research that relied on a single inversion method, offering a new perspective on inversion techniques, enriching the theoretical foundation of GCSR, and providing more reliable technical support for practical applications, thereby significantly improving subsequent remediation efficiency.
Additionally, whether employing the S-DA or S-O method for GCSR, a substantial number of iterations of the numerical simulation model are required, incurring high costs. To mitigate this, many scholars favor using surrogate models with lower costs to replace numerical simulation models for calculations [25,26,27,28]. Surrogate models are black box models that learn input–output mapping through large datasets. With the rise of artificial intelligence, the modeling methods of surrogate models in GCSR have shifted from traditional shallow learning (SL) methods [18,29,30,31] to deep learning (DL) methods [32,33,34,35]. DL has demonstrated significant advantages in fitting complex nonlinear relationships. Among them, the multilayer perceptron (MLP) has gained a strong reputation in groundwater studies [36,37]. Therefore, this work utilized MLP to establish a surrogate of the simulation model, and the results were compared with those of the traditional SL surrogate model.
In summary, the contamination source information and boundary conditions were treated as unknown variables and were identified using the theories and methods mentioned above in this work. Their effectiveness was tested through a hypothetical case. The main contributions of this work can be summarized as follows: (1) For the first time, combining the ES-MDA and CDOA for GCSR, overcoming the limitations of using a single method and further improving the inversion accuracy. (2) Use the deep learning model in the two-stage inversion framework, effectively improving the approximation accuracy to the simulation model.
The following sections are structured as follows: Section 2 describes the principles of the methods used in this work. Section 3 presents the application and validation of these methods in a hypothetical scenario. Section 4 provides an in-depth discussion on this work. Finally, Section 5 summarizes the related conclusions.

2. Methodology

2.1. Simulation Model

The establishment of a reliable simulation model is fundamental for GCSR, encompassing both groundwater flow and solute transport simulation models. In a 2D steady-state-flow confined aquifer, the partial differential equations governing these models can be expressed as follows:
∂ ∂ x i T i j ∂ h ∂ x j = W   i , j = 1 , 2 ;
∂ θ C ∂ t = ∂ ∂ x i θ D i j ∂ C ∂ x j − ∂ ∂ x j θ C u i + C s W e   i , j = 1 , 2 .
Equations (1) and (2) are linked by Equation (3):
u i = − K i j θ ∂ h ∂ x j   i , j = 1 , 2
where xi and xj are the coordinates; ui is the actual average flow velocity; t is the time; h is the hydraulic head; e is the aquifer thickness; θ is the porosity; W is the volumetric flux per unit area (positive sign for inflow and negative sign for outflow); C is the contaminant concentration; Cs is the contaminant concentration in the sources or sinks; and Kij, Tij, and Dij are the hydraulic conductivity tensor, transmissivity tensor, and hydrodynamic dispersion tensor, respectively.
The above partial differential equations, along with specific initial and boundary conditions (Equation (22)), constituted the groundwater flow and solute transport simulation models.

2.2. Low-Cost Surrogate Models

2.2.1. The KELM

The extreme learning machine (ELM), proposed by Huang et al. [38], is a single-hidden-layer feedforward neural network. When the input sample x j is given, the ELM output can be calculated as follows:
∑ i = 1 L β i · p ( ω i x j + b i ) = f ( x j ) = y j   j = 1 , 2 , … , m
where L represents the number of hidden layer neurons, β i represents the connection weight between the i-th hidden layer neuron and the output layer neuron, p ( · ) is the activation function, p ( ω i x j + b i ) represents the i-th hidden layer neuron output, ω i represents the connection weight between the input layer neuron and the i-th hidden layer neuron, xj represents the j-th input sample, and bi represents the bias of the i-th hidden layer neuron. Equation (4) can also be expressed as follows:
Y = Q β ;
β = Q + T
where β = [ β 1 , β 2 , … , β L ] T , Y = [ y 1 , y 2 , … , y m ] T , Q is the output matrix of the hidden layer, Q + is the Moore–Penrose generalized inverse of Q, and T = [ t 1 , t 2 , … , t m ] T is the target data when the ELM can perform unbiased learning of training samples.
However, the ELM suffers from stability issues. Consequently, Huang [39] introduced a kernel function into the training process, replacing traditional random maps with kernel maps, and, thus, proposed the KELM. Compared to the ELM, the KELM produces more stable output results, as expressed below:
f ( x ) = h ( x ) Q T ( Q Q T + I R e ) − 1 T = K e ( x , x 1 ) K e ( x , x 2 ) ⋮ K e ( x , x m ) T ( Ω E L M + I R e ) − 1 T ,
Ω E L M = Q Q T ,
Ω E L M ( i , j ) = h ( x i ) · h ( x j ) = K e ( x i , x j )
where Re is the regularization coefficient, I is the identity matrix, h(x) is the feature mapping function, Ω E L M is the kernel matrix, and Ke(u,v) is the kernel function.
In this study, the Gaussian RBF kernel function was selected, and the number of hidden layer neurons was set to 200.

2.2.2. The MLP

The MLP is a type of feedforward artificial neural network. Its structure (Figure 1) consists of an input layer, multiple hidden layers, and an output layer, providing strong nonlinear fitting capabilities [37]. The training process of the MLP involves two main stages: forward propagation and backward propagation. During forward propagation, data are inputted into the input layer and processed sequentially through each hidden layer. The prediction result is finally obtained at the output layer. The output of the m-th neuron in the l-th layer y m l can be calculated as follows:
x m l = b m l + ∑ i = 1 k ω i m l − 1 y i l − 1 y m l = f ( x m l )
where x m l is the corresponding input, b m l is the bias, ω i m l − 1 is the connection weight between the m-th neuron in l-th layer and the i-th neuron in (l − 1)-th layer, and f ( · ) is the activation function.
Then, in the backward propagation stage, the error between the network’s prediction result and the target result is calculated. The network parameters ( ω   and   b ) are continuously adjusted until the error is minimized. The update equations for the network parameters are shown as follows:
ω ˜ i m l − 1 = ω i m l − 1 − η ∂ E ∂ ω i m l − 1 b ˜ m l = b m l − η ∂ E ∂ b m l
where η is the learning rate and E is the loss function.
In this study, the MLP model was structured with one input layer, three hidden layers, and one output layer. The hidden layers contained 16, 64, and 128 neurons, respectively, and all used the Logistic Sigmoid activation function. The output layer employed the Purelin activation function. During the training process, the learning rate was set to 0.0025.

2.3. Inversion Framework

2.3.1. The ES-MDA

To enhance robustness in addressing nonlinear problems, the ES-MDA method was proposed. Unlike the ES method, the ES-MDA method performs multiple iterations. The core update equations are as follows:
X ˜ i j = X i j + C i X Y ( C i Y Y + α i M ) − 1 ( Z o b s + α i ε j − Y i j ) ,
C i X Y = 1 N e − 1 ∑ j = 1 N e ( X i j − X ¯ i ) ( Y i j − Y ¯ i ) T ,
C i Y Y = 1 N e − 1 ∑ j = 1 N e ( Y i j − Y ¯ i ) ( Y i j − Y ¯ i ) T
where X i j , X ˜ i j , Y i j are the j-th realization at the i-th iteration in the prior ensemble, posterior ensemble, and predicted ensemble, respectively; ε j is the monitoring error, following the Gaussian distribution (the mean value is 0 and covariance matrix is M); Zobs is the actual monitoring data; C i X Y and C i Y Y are the cross-covariance matrix and auto-covariance matrix, respectively; X ¯ i , Y ¯ i are the mean values of X i , Y i , respectively; Ne is the ensemble capacity; and α i is the inflation factor. More extensive descriptions are provided by Emerick and Reynolds [40], Evensen [41], and Wang et al. [42].
In this work, when we used the ES-MDA for GCSR, the Ne and the number of iterations were set to 800 and 8, respectively.

2.3.2. The CDOA

The CDOA [19] is a population-based intelligent heuristic optimization algorithm, with its specific principles detailed in our previous work. In CDOA, for each individual, multiple candidate positions are generated through information transmission and cooperation within the population (Equations (15)–(21)). These candidate positions are then compared with the corresponding initial positions to select the best positions in the current iteration and update the corresponding population. This process is repeated until the iteration termination condition is satisfied, i.e., the maximum number of iterations T is reached. Finally, the iteration process is terminated, and the optimal solution is output.
1. Experience-based phase
n e w X j 0 = X j ( i ) + a 0 · s t e p ( i ) · d 0 d 0 = φ p − X j ( i )
2. Others-based phase
n e w X j 1 = n e w X j 0 + a 1 · s t e p ( i ) · d 1 d 1 = r 1 · d 0 + r 2 · ( X o ( i ) − X j ( i ) )
3. Group-thinking-based phase
n e w X j 2 = n e w X j 1 + a 2 · s t e p ( i ) · d 2 d 2 = r 3 · d 1 + r 4 · ( X G − X j ( i ) ) X G = 1 N ∑ k = 1 N X k ( i )
4. Leader-based phase
n e w X j 3 = n e w X j 2 + a 3 · s t e p ( i ) · d 3 d 3 = r 5 · d 2 + r 6 · ( X L − X j ( i ) )
Additionally, a random walk strategy was used to change the position of XL (Equation (19)), and the new XL can be obtained by Equation (20).
n e w X q = X L + W q   q = 1 , 2 , 3 , 4 , 5
X L = n e w X k   k = min _ ObjFun ( n e w X q )
5. Innovation-based phase
n e w X j 4 = n e w X j 3 r 7 ≤ M F   x j 4 p = L B p + r 8 · ( U B p − L B p )
where X j ( i ) = [ x j 1 ( i ) , x j 2 ( i ) , … , x j d ( i ) ]   ( j = 1 , … , N ) is the position of the j-th individual in the population N at the i-th iteration; d is the dimension; a0, a1, a2, a3, and Wq are the random vectors in (0, 1), respectively; s t e p ( i ) = 2 − 1.7 · ( i − 1 T − 1 ) is the step size at the i-th iteration; d0, d1, d2, and d3 are the directions of movement, respectively; φ p , Xo(i), XG, and XL are the best position of the individual, the position of a randomly selected superior individual, geometric center position of all individuals, and the position of the best individual in the population, respectively; r1, r3, and r5 are the random numbers in (−1, 1), respectively; r2, r4, and r6 are the random numbers in (0, 2), respectively; r7 and r8 are the random numbers in (0, 1), respectively; MF is the innovation factor, p is randomly generated in the range [1, d]; and UB and LB are the upper and lower bounds, respectively. In CDOA, the objective function was used to evaluate the quality of individuals, and min_ObjFun is an indicator of the minimum objective function value.
When the CDOA was used for GCSR, an individual represented a set of the unknown variables in the inverse problem, and the position of the individual can be seen as the solution of the unknown variable. The process of finding the optimal position was the process of obtaining the optimal solution, i.e., the process of solving the inverse problem. In GCSR, the sum of squared errors between the surrogate model output data and the actual monitoring data was adopted as the objective function (Equation (29)). The smaller the objective function value, the closer the solution is to the reference value. In this work, the key parameters N and T were set to 75 and 900, respectively.

2.3.3. Two-Stage Inversion Framework

In this study, the ES-MDA and CDOA were combined to construct a two-stage inversion framework, improving inversion accuracy. The main steps are as follows:
Step 1. The first-stage inversion process: based on the specific principles provided in Section 2.3.1, the ES-MDA algorithm was employed for a global broad search to quickly obtain point estimation and ensemble estimation results, which served as the initial values and intervals for the next stage.
Step 2. The second-stage inversion process: based on the results of the first-stage inversion process, the S-O method was then applied for GCSR using the optimization model and CDOA, conducting a refined deep search to obtain the final results.
The main parameter settings of the two-stage inversion framework proposed in this study are shown in Section 2.3.1 and Section 2.3.2, and the flowchart is depicted in Figure 2.

3. Site Overview

3.1. Case Description

In this study, a hypothetical numerical case was utilized to verify the effectiveness of the proposed methods and theories. This case was modified from Zhang et al. [43]. As depicted in Figure 3, the simulation area was a 2D confined aquifer with a size of 20 × 10 [L], featuring stable water flow from left to right. In the solute transport model, the left and right boundaries were seen as specified concentration boundaries, with the concentration at the left boundary being unknown. The expressions of initial and boundary conditions are shown in Equation (22). Additionally, to better characterize the uncertainty of hydrogeological parameters, the hydraulic conductivity field K was considered heterogeneous. The logarithmic form Y = ln K was used to represent the field K. The covariance function for any two points (xa, ya) and (xb, yb) is shown in Equation (23). In this work, we adopted the Karhunen–Loève (KL) expansion [44] to describe the reference Y field (Figure 4), as shown in Equation (24).
C x , y t = 0 = C 0 x , y x , y ∈ Ω h x , y Γ 2 , Γ 4 = ϕ x , y , t x , y ∈ Γ 2 , Γ 4 ,   t ≥ 0 ∂ h ∂ n → Γ 1 , Γ 3 = 0 x , y ∈ Γ 1 , Γ 3 ,   t ≥ 0 C x , y Γ 2 , Γ 4 = φ x , y , t x , y ∈ Γ 2 , Γ 4 ,   t ≥ 0 C u → − D g r a d C · n → Γ 1 , Γ 3 = 0 x , y ∈ Γ 1 , Γ 3 ,   t ≥ 0
where Ω is the simulation area, ϕ x , y , t , φ x , y , t are the known functions, n → is the normal of the outer boundary, and C 0 x , y is the known function of the initial concentration.
C Y ( x a , y a ; x b , y b ) = σ Y 2 exp − | x a − x b | l x − | y a − y b | l y ,
Y ( x , y ) ≈ Y ¯ ( x , y ) + ∑ i = 1 N KL τ i s i ( x , y ) ζ i
where NKL is the number of KL items, σ Y 2 , Y ¯ ( x , y ) are the variance and mean function, respectively, lx, ly are the correlation lengths in different directions, τ i , s i ( x , y ) are the eigenvalues and eigenfunctions, respectively, and ζ i is a random variable that follows a Gaussian distribution with a mean of 0 and a standard deviation of 1. In this work, NKL, σ Y 2 , Y ¯ ( x , y ) were set to 20, 0.4, and 2, respectively, and lx and ly were set to 10 and 5, respectively.
In the simulation area, one unknown contamination source and 15 known monitoring wells were present, with the total simulation duration set to 16 [T]. The contamination source continuously released a conservative contaminant throughout the simulation period. The release intensity, which varied over time (t = i: i + 1 [T], for i = 1,…,6), was unknown. There were nine unknown variables in total, including the horizontal coordinate Sx and longitudinal coordinate Sy of the contamination source location, the release intensity RI1~RI6 for six time periods, and the left boundary concentration (BC). Their reference values are presented in Table 1. These reference values were input into the simulation model for forward calculation, yielding monitoring data for water levels and contaminant concentrations at the 15 monitoring wells. These data were considered the actual monitoring data in the hypothetical case, comprising a total of 90 dimensions: 15 dimensions for water level monitoring and 75 dimensions for concentration monitoring at five different times (t = 6, 8, 10, 12, and 14 [T]).

3.2. Application of the Low-Cost Surrogate Model

In this section, we adopted the DL and SL methods, respectively, to establish the low-cost surrogate models of the simulation model. The accuracy of these two surrogate models was then compared to evaluate their respective advantages and applicability.
Step 1. Based on the initial distribution of each unknown variable (Table 1), sampling was conducted within the feasible domain of each variable, resulting in 550 sets of input samples. These samples were inputted into the simulation model for forward calculations, yielding 550 sets of output samples.
Step 2. The first 500 sets of input–output samples were designated as training samples, used to train the KELM and MLP surrogate models, respectively. The parameter settings are shown in Section 2.2.1 and Section 2.2.2.
Step 3. The remaining 50 sets of input–output samples were used as testing samples to evaluate the approximation accuracy of the two surrogate models. Four accuracy indicators were selected for verification in this work: the coefficient of determination (R2), mean relative error (MRE), mean absolute error (MAE), and root mean square error (RMSE):
R 2 = 1 − ∑ k = 1 l ∑ i = 1 s ( y k , i − y ^ k , i ) 2 ∑ k = 1 l ∑ i = 1 s ( y k , i − y ¯ ) 2 ,
M R E = ∑ k = 1 l ∑ i = 1 s y k , i − y ^ k , i / y k , i l · s ,
M A E = ∑ k = 1 l ∑ i = 1 s y k , i − y ^ k , i l · s ,
R M S E = ∑ k = 1 l ∑ i = 1 s ( y k , i − y ^ k , i ) 2 l · s
where l is the output dimension, s is the number of samples, y k , i , y ^ k , i are the k-th dimension in the i-th sample of the simulation model and the low-cost surrogate model, respectively, and y ¯ is the average value of the simulation model output.
The final results of the two surrogate models are presented in Table 2. To illustrate the effectiveness more intuitively, fitting diagrams of the different surrogate models to the simulation model and relative error box plots were drawn, as shown in Figure 5 and Figure 6. In Figure 5, the data points obtained by the MLP surrogate model were closer to the y = x line, indicating a more concentrated and better fit to the simulation model. In Figure 6, data points within the 1.5 IQR range were considered normal, with 25% and 75% representing the lower and upper quartiles, respectively. Compared to the relative error results of the KELM surrogate model, the upper whisker, upper quartile, median line, and mean line of the MLP surrogate model were lower, demonstrating that the MLP surrogate model had higher prediction accuracy. Additionally, as shown in Table 2, each indicator result of the MLP surrogate model was superior to that of the KELM surrogate model. Thus, in this case, the DL surrogate model established by the MLP method exhibited advantages and overall better prediction performance. Therefore, the MLP surrogate model was used for GCSR.

3.3. Application of the Two-Stage Inversion Framework

Based on the MLP surrogate model, a two-stage inversion framework was employed. In the first stage, the fast ES-MDA method was utilized for a global broad search. The point estimation results from the first-stage inversion process are presented in Table 1, while the ensemble estimation results are depicted in Figure 7. It is evident that the variable distribution obtained by the ES-MDA method was generally concentrated and nearly encompassed the reference values. However, a few variables (e.g., RI4~RI6) exhibited a maximum posterior probability distribution that significantly deviated from the corresponding reference values, indicating a need for improved inversion accuracy. Furthermore, based on the obtained ensemble estimation results, the upper and lower limits of each unknown variable were redefined to form a new search interval (i.e., optimized interval) for the second-stage inversion process. As shown in Figure 7, compared to the initial interval in Table 1, the optimized interval obtained from the first-stage inversion process was narrower, potentially providing strong support for the subsequent inversion stage.
In the second-stage inversion process, the S-O method based on the CDOA was employed for a refined deep search. During this process, the inversion information from the first stage was fully utilized. Specifically, the values of each unknown variable were re-constrained within the optimized interval as inequality constraints, while the MLP surrogate model served as an equality constraint. The sum of squared errors between the surrogate model output data and the actual monitoring data was used as the objective function, establishing the optimization model (Equation (29)). Subsequently, we adopted the CDOA to solve the optimization model. Notably, the initial population was generated based on the optimized interval obtained from the first-stage inversion process, and the worst individual in the initial population was replaced by the point estimation results from the first stage. This provided better initial values and intervals, thereby accelerating the search efficiency and improving the inversion accuracy.
min ∑ t = 1 5 ∑ k = 1 15 C k , t − C ˜ k , t 2 + ∑ k = 1 15 h k − h ˜ k 2 s . t . ( h k , C k , t ) = g S x , S y , R I 1 , R I 2 , R I 3 , R I 4 , R I 5 , R I 6 , B C S x L B ≤ S x ≤ S x U B S y L B ≤ S y ≤ S y U B R I 1 L B ≤ R I 1 ≤ R I 1 U B R I 2 L B ≤ R I 2 ≤ R I 2 U B R I 3 L B ≤ R I 3 ≤ R I 3 U B R I 4 L B ≤ R I 4 ≤ R I 4 U B R I 5 L B ≤ R I 5 ≤ R I 5 U B R I 6 L B ≤ R I 6 ≤ R I 6 U B B C L B ≤ B C ≤ B C U B
where h k represents the water level of the k-th monitoring well obtained by the MLP surrogate model, C k , t represents the contaminant concentration of the k-th monitoring well at the t-th time obtained by the MLP surrogate model, and h ˜ k , C ˜ k , t represent the actual water level and contaminant concentration, respectively.
Finally, the identification results obtained through the two-stage inversion framework are presented in Table 1. We employed the MRE and MAE to evaluate the performance. Compared to the first-stage inversion results, the identification results from the two-stage inversion framework were generally closer to the reference values. Further calculations indicated that the MRE and MAE of the inversion results from the first stage were 17.53% and 0.6789, respectively. However, upon completion of the two-stage inversion framework, the MRE and MAE decreased to 4.95% and 0.1756, respectively. This demonstrated that the introduction of the second stage for a refined deep search significantly enhanced the inversion accuracy. Additionally, a comparison was made between the final estimated results and those obtained using the second-stage inversion process alone. The convergence curves of the CDOA search process under these two scenarios are plotted in Figure 8, where a logarithmic scale was used for the vertical axis for better visualization. It is evident that the initial value of the CDOA based on the two-stage inversion framework was significantly better than that using CDOA alone, attributable to the first-stage inversion process. Moreover, the CDOA based on the two-stage inversion framework had a significantly smaller final objective function value compared to CDOA alone. This indicated that the two-stage inversion framework improved the search efficiency of the optimization algorithm and enhanced its ability to avoid local optima to some extent. For inversion accuracy, the degree of approximation of the identification results to the reference values under different inversion scenarios is plotted in Figure 9. From Table 1 and Figure 9, it can be seen that the introduction of the first-stage inversion process reduced the MRE from 13.78% to 4.95%, and the MAE was also reduced from 0.5389 to 0.1756. Compared to using the second-stage inversion process alone, the preliminary global broad search of the first stage proved to be beneficial and effective. The first-stage inversion process provided better initial values and optimized intervals for each unknown variable, thereby enhancing the accuracy of the subsequent inversion stage.
In summary, the two-stage inversion framework proposed in this study significantly improved inversion accuracy compared to the single-stage inversion process, proving to be both effective and feasible.

4. Discussion

Several new insights were gained during the research process. First, it was demonstrated that, compared to the traditional SL surrogate model, the DL surrogate model significantly enhances the approximation accuracy of the simulation model. The result was consistent with that of previous research [45,46]. However, deep neural networks often require the setting of more hyperparameters (such as neurons and network layers), which greatly influence prediction accuracy. In this study, these hyperparameters were manually adjusted, a process that is both time-consuming and labor-intensive. Some existing studies [4,32] have shown that using the hyperparameter optimization strategy can effectively replace previous manual tuning and improve efficiency. While this was not the focus of the current study, it represents a valuable research avenue for future exploration. Furthermore, although the DL method has advantages over the SL method, it entails longer and more complex training processes. If the model input dimensions are not excessively high and the mapping relationship between input and output is not overly complex, the SL surrogate model may suffice. Therefore, an optimal choice between DL and SL surrogate models should be made based on the specific research needs and actual conditions, to better balance training time and approximation accuracy.
For the inversion method, we adopted a two-stage inversion framework for GCSR. The research results indicated that this combination strategy achieved higher inversion accuracy compared to a single method. Therefore, the combination inversion strategy has certain development potential. However, it is important to note that combining different methods may require more time than using a single method. In this study, for example, under equivalent parameter settings on a 2.60 GHz dual-core E5 CPU and 96 GB RAM PC platform, the two-stage inversion framework required approximately 8740s, whereas the CDOA alone required about 8681 s. Although the former takes slightly more time, the difference is not significant. We believe that sacrificing a small amount of time for improved inversion accuracy is acceptable. In practical groundwater contamination scenarios, if the two-stage inversion framework proposed in this work can enhance accuracy, more precise contamination source information can be obtained, allowing for swift remediation and treatment measures. Consequently, the improvement in inversion accuracy plays a crucial role in subsequent groundwater contamination remediation plans, risk assessment, and the accurate determination of contamination responsibilities, thereby significantly enhancing the efficiency of subsequent efforts. This is of great significance for real-world environmental management and remediation efforts.
Additionally, several limitations and uncertainties are present in this study. First, a hypothetical case modified from previous research [43] was adopted. The model assumed the actual monitoring data quality was perfect. However, in real-world scenarios, data quality may be imperfect and contain some errors. Although our previous study [4] demonstrated that the ES-MDA in the two-stage inversion framework has strong noise resistance, it is hypothesized that the two-stage inversion framework may inherit this advantage and have a certain degree of noise resistance. However, this hypothesis was not tested in this study, as robustness testing with monitoring data of varying noise levels was not conducted. The impact of noise on the accuracy of the proposed inversion framework remains uncertain and should be addressed in future research. Secondly, in our case, the K field was known, while the contamination source information and boundary conditions were treated as unknown variables. In practical applications, many variables (such as the K field) cannot be reliably estimated in advance, potentially requiring more variables to be considered unknown. Moreover, actual case simulation models are more complex, posing greater challenges for inversion techniques. Therefore, future work will focus on applying this framework to high-dimensional and complex practical cases to further test its efficacy in addressing real-world problems.

5. Conclusions

In this study, a two-stage inversion framework was proposed to synchronously identify groundwater contamination source information and boundary conditions. The results were compared with those obtained from single methods. To reduce computational costs, an MLP surrogate model was constructed to replace the simulation model in completing the above inversion task. The effectiveness of the MLP surrogate model was further compared and analyzed against that of the KELM surrogate model. These theories and methods were tested using a hypothetical site, leading to the following conclusions:
(1) In the two-stage inversion framework, the global broad search in the first-stage inversion process effectively provided better initial values and optimized intervals for the subsequent stage, accelerating search efficiency and improving identification accuracy. The introduction of the second-stage inversion process for a refined deep search further enhanced the inversion effect. Therefore, compared to using the first and second stages separately, the proposed two-stage inversion framework significantly improved inversion accuracy and effectiveness.
(2) Compared to the KELM surrogate model, the R2, MRE, MAE, and RMSE metrics of the MLP surrogate model were significantly better, demonstrating the improved approximation accuracy of the simulation model and proving its advantages.

Author Contributions

Conceptualization, Z.W. and W.L.; Formal analysis, Z.W.; Methodology, Z.W.; Project administration, W.L.; Resources, W.L.; Software, Z.W.; Supervision, W.L.; Validation, Z.W.; Writing—original draft, Z.W.; Writing—review and editing, W.L. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the National Natural Science Foundation of China (No. 42272283).

Data Availability Statement

Data are available on request from the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. The general structure of the MLP.
Figure 1. The general structure of the MLP.
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Figure 2. The two-stage inversion framework.
Figure 2. The two-stage inversion framework.
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Figure 3. The schematic diagram of the simulation area.
Figure 3. The schematic diagram of the simulation area.
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Figure 4. The reference Y field.
Figure 4. The reference Y field.
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Figure 5. The fitting diagrams of the different surrogate models. (A) The KELM surrogate model and (B) The MLP surrogate model.
Figure 5. The fitting diagrams of the different surrogate models. (A) The KELM surrogate model and (B) The MLP surrogate model.
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Figure 6. The relative error box plots of the different surrogate models.
Figure 6. The relative error box plots of the different surrogate models.
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Figure 7. The ensemble distribution obtained in the first-stage inversion process. (A) Sx, (B) Sy, (C) RI1, (D) RI2, (E) RI3, (F) RI4, (G) RI5, (H) RI6, and (I) BC. The red line represents the corresponding reference value of each variable.
Figure 7. The ensemble distribution obtained in the first-stage inversion process. (A) Sx, (B) Sy, (C) RI1, (D) RI2, (E) RI3, (F) RI4, (G) RI5, (H) RI6, and (I) BC. The red line represents the corresponding reference value of each variable.
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Figure 8. The convergence curves of the different inversion situations.
Figure 8. The convergence curves of the different inversion situations.
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Figure 9. The performance of the different inversion situations. The red dotted line indicates that the identification value is 100% close to the reference value.
Figure 9. The performance of the different inversion situations. The red dotted line indicates that the identification value is 100% close to the reference value.
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Table 1. Reference details of unknown variables.
Table 1. Reference details of unknown variables.
Unknown VariableReference ValueInitial RangeOnly the First-Stage Inversion Process Result Only the Second-Stage Inversion Process ResultThe Two-Stage Inversion Framework Result
Sx [L]3.99U [3, 5]3.953.953.96
Sy [L]4.71U [4, 6]4.724.724.71
RI1 [MT−1]4.90U [0, 8]4.754.724.98
RI2 [MT−1]3.65U [0, 8]3.723.993.46
RI3 [MT−1]2.36U [0, 8]3.162.662.67
RI4 [MT−1]6.70U [0, 8]5.005.466.38
RI5 [MT−1]2.78U [0, 8]4.734.383.15
RI6 [MT−1]7.76U [0, 8]6.416.687.50
BC [ML−3]0.84U [0.6, 1.2]0.880.900.86
Table 2. The performance of the different surrogate models.
Table 2. The performance of the different surrogate models.
IndicatorKELM Surrogate ModelMLP Surrogate Model
R20.95770.9860
MRE (%)13.079.72
MAE [ML−3]0.23790.1727
RMSE [ML−3]0.820.47
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Wang, Z.; Lu, W. Groundwater Contamination Source Recognition Based on a Two-Stage Inversion Framework with a Deep Learning Surrogate. Water 2024, 16, 1907. https://doi.org/10.3390/w16131907

AMA Style

Wang Z, Lu W. Groundwater Contamination Source Recognition Based on a Two-Stage Inversion Framework with a Deep Learning Surrogate. Water. 2024; 16(13):1907. https://doi.org/10.3390/w16131907

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Wang, Zibo, and Wenxi Lu. 2024. "Groundwater Contamination Source Recognition Based on a Two-Stage Inversion Framework with a Deep Learning Surrogate" Water 16, no. 13: 1907. https://doi.org/10.3390/w16131907

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