## Abstract

Seawater intrusion monitoring is quite different from the conventional monitoring of groundwater pollution. In this study, a new optimization method for the seawater intrusion monitoring scheme in the transitional zone was proposed. The objective of optimization was to maximize effective information monitored. The structural similarity index method (SSIM) of the image quality assessment was innovatively used to establish a mathematical expression for the effective monitored information, and an optimization model was constructed based on this. Taken the Longkou city of China as the study area, a numerical simulation model of variable density groundwater was constructed. The Monte Carlo method was used to consider the influence of the sensitivity parameters uncertainty on the monitoring scheme design. To avoid repeatedly calling of simulation models in the process of Monte Carlo experiments, a surrogate model was constructed by using the kernel extreme learning machine (KELM). Finally, the optimization model was solved by the genetic algorithm to obtain the optimal monitoring scheme. The results showed that the input-output relationship of the numerical simulation model for variable-density groundwater can be well approximated by the KELM surrogate model. The monitoring scheme optimized by the above method can well reflect the real state of seawater intrusion. This study expands the method on the scheme designs for seawater intrusion monitoring.

## Introduction

Seawater intrusion is a common problem in densely populated coastal areas. Reasonable groundwater monitoring in coastal areas can provide data support for formulating prevention and control strategies for seawater intrusion. The optimal design of a seawater intrusion monitoring scheme aims to obtain the most valuable monitoring information at the lowest cost.

The simulation-optimization method is commonly used in groundwater monitoring network optimization. It can not only follow the transport law of complex groundwater systems but also determine the optimal monitoring under a given objective function (Wagner 1995). Meyer et al. (1994) established a multi-objective optimization for the monitoring wells location. Montas et al. (2000) proposed to minimize the quality assessment error of pollution plumes as the goal of monitoring well location optimization. Zhang et al. (2005) used a stochastic simulation method to consider the uncertainty of the groundwater simulation model and combined genetic algorithm with Kalman filtering method to solve the design of the groundwater quality monitoring network. Datta et al. (2009) proposed an iterative method to optimize the location of monitoring wells and identify the source of pollution. By optimizing the position of the monitoring wells, the inversion accuracy of the pollution source can be gradually improved. After that, Tryby et al. (2010) and Zhang et al. (2020) strengthened the research on the location optimization of monitoring wells for pollution source inversion identification. Masoumi and Kerachian (2010) used discrete entropy theory to redesign a monitoring well network in Iran. Reed et al. (2013) compared several multi-objective evolutionary algorithms that have been applied in the optimization of the groundwater monitoring network. Luo et al. (2016) proposed to minimize the monitoring cost and maximize the spatial representation accuracy of the pollution plumes as the optimization goal and used a new probabilistic Pareto genetic algorithm to solve the optimization model. Bode et al. (2018) discussed practical problems in the design of a pollution monitoring network for drinking water source areas and made a multi-objective optimization for the monitoring wells location. Ohmer et al. (2019) studied the influence of six different density monitoring network designs on the accuracy of groundwater surfaces spatially interpolated. Kumari et al. (2019) constructed an uncertainty-based dynamic sampling framework, which considered fuzzy variance reduction and mass estimation error reduction with maximization of spatial coverage.

However, there have been few reports on the optimization of seawater intrusion monitoring schemes for a long time. Melloul and Goldenberg (1997) summarized the monitoring methods of seawater intrusion and suggested that geomagnetic methods and direct wells construction should be combined to monitor seawater intrusion. Sreekanth and Datta (2013) proposed a method for designing a compliance monitoring network for implementing seawater intrusion management schemes. Roy and Datta (2019) designed a monitoring network for the 3D coastal groundwater system to evaluate the compliance of the implemented strategies with the prescribed management goals. Regarding the optimization of the seawater intrusion monitoring program, there are still many problems to be solved. For example, the transition zone is sensitive to changes in seawater intrusion, so more monitoring wells should be deployed. So far, there has been no report on optimizing the location of monitoring wells for the transition zone of seawater invasion. Furthermore, the monitoring of seawater intrusion is quite different from the monitoring of general groundwater pollution. First, the seawater intrusion simulation needs to consider the influence of density changes caused by concentration changes on the results of the model solution. Secondly, the general groundwater quality pollution is usually a point source or surface source pollution. But the source of seawater invasion is the tortuous coastline, which makes the shape and conditions of the invasion more complicated. Conventional groundwater monitoring network design methods are not completely suitable for seawater intrusion monitoring.

This paper proposed a new optimization method of seawater intrusion monitoring scheme focus on the transition zone, aiming to maximize the effective monitored information. The structural similarity index method (SSIM) of image quality assessment was innovatively using to establish mathematical expressions for the amount of monitored effective information. Taking the Longkou city of China as the study area, a numerical simulation model of variable density groundwater was established. And the sensitivity parameters of the simulation model were selected through local sensitivity analysis. The Monte Carlo method was used to consider the influence of the sensitivity parameters uncertainty on the monitoring wells layout. To avoid the calculation load caused by repeatedly calling the simulation model in the Monte Carlo experiments, the KELM surrogate model of the variable density groundwater simulation model was constructed. Taken the surrogate model as one of the constraints, the simulation-optimization model for seawater intrusion monitoring scheme optimization was established and solved by the genetic algorithm. This study provides a stable and reliable method for the optimal design of seawater intrusion monitoring schemes.

## Methodology

### The KELM Surrogate Model

When using the Monte Carlo method to consider the influence of the simulation model uncertainty on the optimization results, it is inevitable to run the simulation model multiple times in the process of calculating the optimization model. The surrogate model can obtain the input-output relation similar to the simulation model with less calculation. Making the surrogate model becomes one of the constraint condition of the optimization model instead of the simulation model so as to realize the coupling of the simulation model and the optimization model. Therefore, it is necessary to find an effective way to establish a surrogate model of the groundwater simulation model. KELM is an effective method for constructing surrogate models (Jiang et al. 2015; Song et al. 2018).

KELM is an improvement of the Extreme learning machine (ELM), which is a fast learning method based on single-hidden layer feedforward neural networks (SLFNs) (Huang et al. 2006). The configuration of the ELM model was shown in Fig. 1. It is characterized by the random selection of hidden layer nodes and corresponding node parameters of the neural network. During the training process, it is only necessary to adjust the output weights of the network through the regularized least-squares algorithm. Therefore, it can obtain good network generalization performance at an extremely fast learning speed.

For SLFNs with L hidden layer nodes, given N sets of training sample data sets\(\left( {{{\mathbf{x}}_j},{{\mathbf{t}}_j}} \right)\), \({{\mathbf{x}}_j}={\left[ {x{}_{{j1}},{x_{j2}}, \cdots ,{x_{jn}}} \right]^{\text T}}\), \({{\mathbf{t}}_j}={\left[ {{t_{j1}},{t_{j2}}, \cdots ,{t_{jm}}} \right]^{\text T}}\). The output of its network node is

In the formula, \(G\left( \cdot \right)\) is the activation function. The input node is connected to the *i*^{th} hidden neuron through the weight vector\({{\mathbf{\omega }}_i}\). \({b_i}\) is the bias of the *i*^{th} hidden node. \({\beta _i}\) is the weight of the *i*^{th} hidden neuron connecting with the output neuron. If SLFNs can approximate the training samples with zero error, that is, \(\sum\limits_{{j=1}}^{N} {\left\| {{o_j} - {t_j}} \right\|} =0\), then there are\({\beta _i}\),\({{\mathbf{\omega }}_i}\),\({b_i}\), so that

Rewritten to matrix form:

Among them, *B* is the output weight matrix; *H* is the output matrix of the hidden layer node. ELM training is equivalent to solving the least-squares solution *B* of the linear system *HB = T*, that is:

To improve the stability and generalization ability of the network, according to the idea of Tikhonov regularization, given the regularization coefficient \(\eta\), the least-squares solution of the output weight is:

The corresponding output function of ELM is \(y\left( {\mathbf{x}} \right)=h\left( {\mathbf{x}} \right)\hat {B}\).

The hidden layer feature map \(h\left( {\mathbf{x}} \right)\) is unknown in ELM. If it is replaced by a kernel function, the kernel extreme learning machine (KELM) method can be formed (Huang 2014). The advantage is that it is not necessary to determine the number of hidden layer nodes in advance. In the training process of the network, the KELM method only needs to select appropriate kernel parameters and regularization coefficients, the output weights of the network can be obtained through matrix operations.

Define the kernel matrix \({\Omega _{ELM}}=H{H^T}\). Its elements are \({\Omega _{ELM}}\left( {i,j} \right)=h\left( {{{\mathbf{x}}_i}} \right)h\left( {{{\mathbf{x}}_j}} \right)=K\left( {{{\mathbf{x}}_i},{{\mathbf{x}}_j}} \right)\). Then the network output can be expressed as:

In formula 7, the type of kernel function \(K\left( {{{\mathbf{x}}_i},{{\mathbf{x}}_j}} \right)\) is usually chosen as the radial basis function kernel.

Where \(\gamma\) is the nuclear parameter, which represents the kernel width of the radial basis kernel function.

In this paper, the relative error (*RE*) was introduced to evaluate the accuracy of the surrogate model.

Where *y*_{n} represents the output of the simulation model. *y*_{an} denotes the output of the surrogate model. *RE*_{n} is the relative error between the *n*^{th} output of the surrogate model and that of the simulation model.

### Optimization Model of the Monitoring Scheme

How to express the amount of monitored effective information is an important problem in monitoring scheme optimization. In this study, the spatial distribution of seawater intrusion is defined as the seawater intrusion plume, which can be interpolated by the groundwater quality monitoring data. We should explore how to deploy monitoring wells to make seawater intrusion plumes more consistent with real plumes using a given monitoring cost. The more similar the two plumes are, the more effective information the monitoring network obtained. In practice, the state of the real seawater plume is always unknown. Based on actual hydrogeological conditions, a groundwater simulation model can be built, and the state of seawater intrusion plume will be obtained by simulation. The simulated plume can be used as a reference for the real seawater intrusion plume. So how to deploy monitoring wells to make the plume obtained by interpolating monitoring data more alike to the simulated plume?

As a prerequisite, we need to describe the similarity degree of the two intrusion plumes. In this study, the SSIM was introduced to evaluate the approximation between two intrusion plumes. The SSIM is a mainstream full reference image quality assessment method based on structural similarity (Wang et al. 2004). The full reference image quality assessment method can obtain all information from the reference image and assess the image quality by comparing the distorted image with the reference image. In this study, the reference image was the real seawater intrusion plume, while the distorted image was the seawater intrusion plume interpolated by monitoring data. The SSIM measures image similarity from brightness, contrast, and structure. Similarly, in this study, the SSIM measures the approximation degree between two plumes from the chloride concentration, concentration gradient, and the shape and structure of the seawater intrusion plume.

Brightness (chloride ion concentration) comparison:

Contrast (chloride ion concentration gradient) comparison:

The contrast of structure (shape and structure of the plume):

where \({u_M}\) and \({u_J}\) represent the mean value of chloride concentration in the real seawater intrusion plume and the approximate plume interpolated by the *J* monitoring scheme. \({\sigma _M}\) and \({\sigma _J}\) are the standard deviation of the chloride concentration in these two seawater intrusion plumes. \({\sigma _M}^{2}\) and \({\sigma _J}^{2}\) are the variance. \({\sigma _{MJ}}\) represents the covariance of the corresponding concentrations of the two seawater intrusion plumes. *C*_{1}, *C*_{2} and *C*_{3} are constants that stabilize the denominator. Generally,\({C_1}={\left( {{K_1} \times L} \right)^2}\), \({C_2}={\left( {{K_2} \times L} \right)^2}\), \({C_3}={{{C_2}} \mathord{\left/ {\vphantom {{{C_2}} 2}} \right. \kern-0pt} 2}\). *L* is the dynamic range of pixel value, which is generally 255. In this study, we set *K*_{1} was 0.01, *K*_{2} was 0.03.

The formulas for calculating \({u_M}\), \({\sigma _M}\) and \({\sigma _{MJ}}\)are as follows:

where \({M_i}\) is the chloride ion concentration in the *i*^{th} grid. *I* is the total number of grids.

Formula (16) is to calculate the similarity between the two seawater intrusion plumes. The SSIM values can be distributed in the range of [0,1] by reasonable values of *C*_{1}, *C*_{2} and *C*_{3}. When the two plumes are identical, the SSIM value is 1; when the two plumes are completely different, the SSIM value is 0.

The objection of the optimization model is to maximize the effective monitored information, and the decision variable is the location of monitoring wells. The optimization model is as follows:

where *n* represents the *n*^{th} Monte Carlo experiment, and *N* is the total number of Monte Carlo experiments. *t* is the monitoring period. The first constraint condition characterizes the influence of the decision variable on the seawater intrusion plume obtained by interpolation. The second is the groundwater simulation model or the surrogate model. The third term is the total monitoring cost constraint. *E* is the total upper limit of the cost of the construction and sampling of the well. *P* is the number of monitoring wells, and *T* represents the total monitoring time. \(\upsilon\) represents the monitoring frequency. \(\left\lfloor {} \right\rfloor\) is rounding symbol. \(\left\lfloor {\frac{T}{\upsilon }} \right\rfloor\) is the number of sampling times. *e*_{w} represents the cost of building a monitoring well. *e*_{s} is the cost of sampling and testing of a single well. The fourth constraint is the composition of the decision vector. The fifth restriction is the number of monitoring wells. The last constraint is the value constraint of the decision variables, which is a 0–1 integer variable.

## Case Study

### Site Overview

The study area located in Longkou, Shandong Province, China (Fig. 2a), which is approximately 221 km^{2}. The overall flow of groundwater is from southeast to northwest. Figure 2b is a schematic diagram of the hydrogeology conceptual model. There are three layers with a total depth of 90 m (Fig. 2c). The first layer is the phreatic aquifer with fine sand or medium sand with a depth of about 30 m. Based on the permeability, it can be divided into eight zones (Zones 1–8). The second layer (C2) is silt or clay with poor permeability. It has a thickness variation from 0 to 30m. The third layer (C3) is sandstone with very low hydraulic conductivity. The finite difference method was used for the rectangular division of the study area. The X and Y directions were divided into squares, and the side length was 50 m. The Z-direction was divided into three layers according to the condition of the stratum. The total number of cells is 265,200.

Since the first layer is the main aquifer for application, the following seawater intrusion states were all directed at the first aquifer. The precipitation data used in the model was from the Longkou monitoring point of China National Meteorological Data Center. The upstream groundwater lateral runoff recharge was estimated by Darcy’s law, and the total amount was about 14880 m^{3}/d. The rainfall infiltration coefficient, evaporation intensity, and artificial exploitation intensity of each zone were shown in Table 1. The agricultural irrigation period is from April to July, and the groundwater consumption is relatively large in this period. The precipitation infiltration recharge, evaporation and artificial exploitation were all input of the simulation model in the form of source and sink terms, and the lateral runoff recharge was input in the form of boundary conditions.

According to the conceptual model, the variable density groundwater simulation model was constructed. The simulation model was solved by SEAWAT software compiled by the US Geological Survey.

Taking the measured monitoring data in January 2017 as the initial condition, the simulation model was calibrated by measured data of June 2017 and was validated by the data measured in January 2018. The result was shown in Fig. 3c and d. Obviously, the fitting accuracy is high. June 2017 was the wet season, and January 2018 was the dry season. The stages of calibration and validation covered the wet and dry periods of the year. Therefore, we think that the data in these two periods is sufficient to calibrate and validate the model. After calibration and validation, we ascertained the value of hydrogeological parameters in the simulation model (showing in Table 1).

Taking the average precipitation of the past 50 years as the precipitation of the next 20 years. The groundwater pumping amount in 2017 was used as the extraction amount in the future. According to the China Sea Level Bulletin of 2017, the sea level rise in the Longkou area will be 80 mm by 2039, assuming that the annual increase is the same. The remaining boundary conditions and hydrogeological parameters remained unchanged. The state of seawater intrusion in June 2039 was predicted by the simulation model (Fig. 3a). After 20 years, the degree of seawater intrusion in the southwest part of the study area increases, while the degree in the northeastern region decreases. Taking the chloride concentration of 250 mg/L as a sign of seawater intrusion, the whole area of seawater intrusion will increase after 20 years. The 250 mg/L iso-concentration line moved a maximum of 403 m inland, retreated 217 m at the minimum, and invaded approximately 140 m on average.

The most sensitive area of seawater intrusion monitoring is the transitional zone, and the monitoring wells location should be concentrated in this zone. The chloride concentration in the transition zone was set at 200 mg/L-350 mg/L. There were 14 monitoring wells in the study area and 42 potential monitoring wells in and on both sides of the transition zone (showing in Fig. 3b). The total investment for the monitoring scheme was 750,000 yuan, which requires 20 years of monitoring. The cost of building a well was approximately 15,000 yuan, and the cost of sampling and testing for a single well was approximately 450 yuan.

### Sensitivity Analysis of the Simulation Model

To consider the influence of the sensitivity parameters uncertainty on the optimization results of the monitoring schemes, the sensitivity parameters in the simulation model need to be selected first. In this study, seven parameters were compared, including hydraulic conductivity, dispersivity, specific yield, porosity, height of sea level rise, precipitation and groundwater exploitation quantity. Three of these parameters were selected by local sensitivity analysis, and their uncertainties were considered. The location of the three test points for sensitivity analysis was shown in Fig. 3b.

### Construction of the KELM Surrogate Model

We used the Latin hypercube sampling method to obtain 110 samples in the feasible region of the sensitivity parameters. The 110 sets of sensitivity parameters were input into the simulation model, and the corresponding outputs were the chloride ion concentration at all existing monitoring wells and potential monitoring wells. Then 110 groups of input-output data sets were obtained by the simulation model run. Among them, 100 groups of input-output data were used as the training data of the surrogate model. The other 10 groups were used as the test data to test the fitting accuracy of the surrogate model. Same to the simulation model, the input of the surrogate model was the value of the sensitivity parameters, and the output was the chloride ion concentration at the existing monitoring wells and potential monitoring wells.

Based on the theory in Sect. 2.1, the KELM surrogate model was constructed in MATLAB. The surrogate model was trained and tested using the above data. If the accuracy can meet the requirement, the 1000 groups of Monte Carlo experiments were carried out by using the surrogate model to consider the influence of the sensitivity parameters uncertainty on the optimization results.

### Solution of the Optimization Model

The optimization model for monitoring well location selection was constructed based on the theory in Sect. 2.2. The optimization model was solved by the genetic algorithm. First, a certain number of new monitoring wells were randomly selected to form a monitoring network with existing monitoring wells. The seawater intrusion plume can be obtained by interpolating the monitoring data. Compare it with the simulated plume to calculate the SSIM value. Through the calculation of the algorithm, the monitoring scheme was continuously improved to increase the SSIM value. Finally, the SSIM value obtained by the optimal monitoring scheme was maximized, and the most suitable new monitoring wells were selected.

## Results Analysis

### Results of the Sensitivity Analysis

Seven parameters were selected to participate in the sensitivity analysis. The results (Fig. 4a, b and c) showed that the sensitivity of precipitation and the groundwater exploitation quantity was high, followed by hydraulic conductivity, and the rest parameters were relatively low. Therefore, the study took into account the impact of the uncertainties of precipitation, groundwater exploitation and hydraulic conductivity on the monitoring scheme optimization.

The precipitation and groundwater extraction generally follow a normal distribution, and the hydraulic conductivity generally follows a lognormal distribution (Dokou and Pinder 2009). When considering the parameter uncertainty, this study set the variation range of each sensitive parameter to be plus or minus 20%.

### Fitting Accuracy of the Surrogate Model

Figure 4d shows the fitting accuracy of the surrogate model. The maximum relative error of each test group is less than 15%, and the average relative error of each test group is less than 2.5%. The average relative error of the 10 test groups is 0.92%. Therefore, it is feasible to construct a surrogate model for the simulation of variable density groundwater by using the KELM method.

### Solution Results of the Optimization Model

The results of the optimization model were shown in Fig. 5g, and the optimal locations of the monitoring wells for each scheme were shown in Fig. 5a, b and c. Status quo is the scheme that no additional monitoring wells, which means all funds were used for sampling. Due to the limitation of total funds, the monitoring schemes need to tradeoff between the monitoring frequency and the number of monitoring wells to be built. Obviously, the higher the monitoring frequency is, the higher the objective function value is. However, the fitting accuracy between the seawater intrusion plume obtained by interpolating monitoring data and that of the real situation is lower. The more the new added monitoring wells, the more similar of the two seawater intrusion plumes. When 26 wells were added, the SSIM value reached 0.977, indicating that the plumes were almost the same.

Scheme 1 focuses on monitoring the dynamic changes in seawater intrusion. And the small number of new monitoring wells means that the control accuracy of the seawater intrusion throughout the scope of the whole region is relatively low. But compared with the status quo scheme, it can be found that scheme 1 can significantly improve the SSIM value for each monitoring period. Scheme 2 was a balanced monitoring scheme. The SSIM value of each monitoring period reached 0.954. This means that using the monitoring network in Scheme 2, the similarity between the seawater intrusion plume obtained by interpolating monitoring data and the real state plume was greater than 95%. Scheme 3 focused more on the accuracy of a single monitoring period to determine the range of seawater intrusion throughout the whole region. The optimized monitoring well network can achieve a high-precision reflection of seawater intrusion. At the same time, a large number of newly added monitoring wells reduced the monitoring frequency and can only monitor the changes in seawater intrusion during the years. The decision-maker can choose the scheme according to the actual demand.

By comparing Fig. 5a, b and c, it can be seen that the newly added monitoring wells are mostly located in the northeast and southwest regions of the study area, while there are fewer wells in the central area. Combined with the analysis of Fig. 3a, it can be seen that the coastal zone in the northeast and southwest regions of the study area is close to the transition zone, and the concentration of seawater intrusion changes greatly. In the central region, there is a narrow peninsula in the northwest of the study area, which buffers the changes in seawater intrusion. Therefore, the concentration change is minimal, and the number of wells in the optimization results is also small. This shows that the monitoring scheme designed in this study can fully consider the dynamic changes in seawater intrusion and make a reasonable location allocation for monitoring wells.

This paper took the prediction of seawater intrusion in June 2029 (10 years later) as an example to test the performances of monitoring schemes. As shown in Fig. 5d, e and f, the dark blue dotted line is an iso-concentration curve interpolated by monitoring data, and the approximation degree of the two intrusion plumes of seawater can be obtained by comparing the background image with the iso-concentration curve. Scheme 1 achieved a certain degree of approximation when the number of monitoring wells is small. The fitting accuracy of scheme 2 reached a high level. Using the monitoring data of scheme 3 for interpolation, the seawater intrusion plume was very similar to the background image. This indicates that it is feasible to use the SSIM to characterize the approximate degree of similarity between two intrusion plumes. SSIM can establish a mathematical expression for the effective information monitored. At the same time, in the optimization process of the monitoring scheme, this study considered the changes of seawater intrusion in different periods and the impact of the model parameters uncertainty. Therefore, for each period in the next 20 years, the real seawater intrusion situation in the whole area can be accurately obtained by using the optimized monitoring wells network.

## Discussion

This study proposed an optimization method for the seawater intrusion monitoring scheme in the transition zone, which aimed to maximize the amount of effective monitored information and used the SSIM method to measure the amount of effective information for the first time. Compared with the traditional methods, the SSIM is more suitable for the complex situation of seawater intrusion.

The traditional location optimization objectives of groundwater quality monitoring wells can be divided into two categories. One is to minimize monitoring costs, and the other is to maximize the amount of effective monitored information. The expression of effective information can be divided into three categories: (1) maximize the detection probability of pollutants (Meyer et al. 1994; Bierkens 2006; Ayvaz and Elçi 2018; Bode et al. 2018); (2) minimize the variance of pollution concentration prediction (Datta et al. 2009; Sreekanth and Datta 2013); and (3) minimize quality assessment errors of pollution plumes (Montas et al. 2000; Wu et al. 2006; Kollat and Reed 2007; Reed et al. 2013; Narany et al. 2015; Luo et al. 2016). The above optimization objectives have played important roles in the location optimization of conventional groundwater pollution monitoring wells. However, applying these effective information expression methods directly to the field of seawater intrusion will cause some problems.

Maximizing the detection probability of pollutants is suitable for monitoring high-risk pollutants but not for the seawater intrusion problem. Using this optimization target will cause the dense distribution of monitoring wells along the coast. Minimizing the variance of pollution concentration prediction will make the monitoring well focus on the place with a large concentration gradient, but ignore the entire seawater invasion area. To minimize the quality assessment errors of pollution plumes, image moments are often used to describe the spatial state of pollution plumes and quantify the monitored information. However, the image moment method is more suitable for the case where the pollution center is clear, and the shape of the pollution plume is regular. There are still some deficiencies in using this method to describe the intrusion state of seawater with a complex shape. Besides, using the image moment method requires the zero-order moment, first-order moment and second-order moment, a total of three indicators to measure the effective information. Therefore, multi-objective optimization algorithms are often required, and the calculation is relatively complicated.

This study proposed to use SSIM to measure the amount of effective monitored information. The calculation is simple. SSIM can comprehensively consider the concentration of pollutants in the whole area, the concentration gradient and the shape and structure of seawater intrusion plumes. Using SSIM as a measurement index to optimize the monitoring scheme can make the monitoring network cover the entire seawater intrusion transition zone as much as possible, and encrypt in areas with large concentration and large concentration gradient. Therefore, SSIM is a good method to measure the amount of monitored effective information.

In this study, the use of the surrogate model greatly reduced the computational load. It took about 5 min to run the simulation model once. Obtaining the training and testing data required to run the simulation model 110 times, which took 9.2 h. The calculation of the surrogate model only took 3 s and can be ignored. If the surrogate model was not used, the simulation model needed to be called 1000 times, which would take 83.3 h. Therefore, using the KELM surrogate model saved a total of 74.1 h of computing time and greatly reduced the computational load.

## Conclusion

An optimization method for seawater intrusion monitoring schemes in transitional zones was proposed in this paper. To maximize the amount of effective monitored information, the study innovatively used SSIM to establish a mathematical expression for the amount of effective monitored information. The sensitivity analysis was used to select the sensitive parameters of the simulation model. And we considered the influence of the uncertainty of these parameters on the optimization results. To avoid the repeatedly calling of simulation models in the multiple Monte Carlo experiments, the KELM surrogate model was constructed. The results showed that the average relative error of the KELM surrogate model output was less than 2.5% after training. The fitting accuracy was high. It is effective to construct a surrogate model of the numerical simulation model for variable-density groundwater by using the KELM method.

The study took the Longkou city of China as an example to verify the effectiveness of the abovementioned methods. From the final optimization results, the monitoring scheme optimized can fully reflect the real state of seawater intrusion in the study area. The seawater intrusion plume obtained by interpolating monitoring data can reach a high degree of consistency with the plume in the real state. It is feasible to use the SSIM to establish a mathematical expression for the amount of effective monitored information. This study provided a reliable way for the design of monitoring schemes in seawater intrusion areas.

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## Acknowledgements

This study was supported by the National Key Research and Development Program of China (No.2016YFC0402800), the National Nature Science Foundation of China (No.41672232), and the Key Laboratory of Groundwater Resources and Environment, Ministry of Education, Jilin University, Changchun, China.

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Fan, Y., Lu, W., Miao, T. *et al.* Optimum Design of a Seawater Intrusion Monitoring Scheme Based on the Image Quality Assessment Method.
*Water Resour Manage* **34, **2485–2502 (2020). https://doi.org/10.1007/s11269-020-02565-w

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### Keywords

- Seawater intrusion
- Groundwater monitoring
- Simulation-optimization
- Surrogate model
- Image quality assessment