Groundwater is a vital natural resource. Water from this source is used practically worldwide because of its economical price, easy access, reliability, and excellent quality. However, groundwater contamination might make it unsafe to use for agriculture, household, and drinking needs. To solve groundwater management issues, groundwater system modelling is a fundamental requirement. Adopting any cleanup technique requires the identification of undiscovered sources of pollution. The data required for the processes involved in developing Artificial Neural Networks (ANN) models is produced by a groundwater flow and transport simulation model. The concentration breakthrough curves are described by the suggested approach using statistical parameters such as skewness, kurtosis, maximum value, average value, and standard deviation. To identify the sources in terms of their location, magnitudes, and duration of activity, a feed-forward multilayer artificial neural network (ANN) is employed with the defined parameters. Test and training patterns are used in varying numbers during experiments. To determine the source fluxes of groundwater pollution, a developed methodology is illustrated using example problems. To solve the groundwater flow and transport simulation model, specific boundary conditions and input parameters may be needed. Uncertainty in one is understanding of the given data, or epistemic uncertainty, can result in imprecise input parameters via indirect measurements and subjective interpretation. Using the Fuzzy Vertex Method, the epistemic uncertainty in the flow parameters is characterized as a result of imperfect measurement of source flux and one of the flow parameters.
Research Article
Uncertainty Characterization in Contaminated Groundwater system Using Fuzzy Vertex Alpha-cut Technique
https://doi.org/10.21203/rs.3.rs-4376018/v1
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Breakthrough curve (BTC)
Artificial Neural Network (ANN)
Groundwater pollution
Pollution source identification
Groundwater flow and transport
Simulation model
Groundwater contamination is a major environmental issue. Contaminated aquifers provide a significant challenge to scientists working on environmental issues. In addition to having a negative impact on the environment and human health, groundwater pollution entails significant financial expenditures for remediation, legal fees, and penalties. Cleaning up polluted groundwater is a challenging task and expensive to perform. Years after the cause of pollution has been active in the aquifer, pollution of groundwater can frequently be found in a water supply well. Determining contaminated source location in space and time utilizing a collection of previous data from observations from monitoring sites is an issue that frequently arises in health risk assessment investigations and the cleanup of polluted sites. Groundwater pollution treatment and management is one of the toughest environmental management problems. To address groundwater management issues, modelling subsurface flow and pollutant transport mechanisms and their dynamics is a fundamental requirement. A thorough understanding of aquifer systems' hydrogeologic parameters, including hydraulic conductivity, permeability, storativity, porosity, and aquifer recharge, is essential for groundwater modelling. Parameter estimation was carried out by Keidser and Rosbjerg (1991) using a simple trial-and-error approach. Various collections of parameter values were chosen for the model, and the model output was then calculated. Subsequent research on the issue of estimation of parameters for groundwater flow models was carried out by McLaughlin and Townley (1996), who developed several techniques that use a search approach to find the parameter values that reduce the variations between the simulated and true hydraulic head values. Chadalavada et al. (2012), Srivastava and Singh (2013), Jha and Datta (2014), Srivastava and Singh (2014), and Srivastava and Singh (2015) are some of the relevant studies that are attainable in the literature. The aquifer parameters that were considered might not be precise and probably contain estimation errors. The inaccurate input parameters might originate from expert appraisal of the available data, subjective interpretation, and indirect measurements (informal uncertainty). Srivastava and Singh (2015) took into consideration constraints on boundaries, transport parameters, and flow parameter uncertainty. They used fuzzy numbers (also known as the cut approach) to predict potential source flow errors at various levels of parameter estimation uncertainty. Stepping forward, the uncertainty in source flux measurement is taken into account for potential prediction of one of the imperative groundwater flow parameters. Fuzzy numbers (α-cut approach) are used to evaluate the probable inaccuracy in parameter prediction at different levels of unpredictability in source flow estimates. This technique might be useful in identifying the true source of pollution in an aquifer.
According to Rumelhart et al. (1986), the word "ANN" is broad and embraces a wide range of network technologies, the most popular of those being a multilayered perceptron feed-forward network system using back propagation algorithm. The total number of not visible layers and nodes inside those hidden layers are often selected by trial-and-error testing because there doesn't seem an accurate equation that can be applied before training starts. Gradient descent is employed in the backpropagation technique to train a feed-forward multiple-layer perceptron (Bishop, 1995; Hagan, 1994). Comparing groundwater difficulties to other hydrological problems reveals a limited use of artificial neural networks (ANNs). Here are a few examples of these applications: Ranjithan et al. (1993) effectively employed artificial neural networks (ANN) to model pumping indexes for hydraulic conductivity of water prediction in treating groundwater under unpredictability. Rogers et al. (1994) performed similar work to generate a regulatory index for several different types of pumping realization at a polluted site that has several plumes. By employing ANN for groundwater remediation, Rogers et al. (1995) advanced the simulation of regulatory index, remedial index, and cost index. Several artificial neural network (ANN) techniques have been developed in the last ten years to forecast hydrogeologic characteristics (Aziz and Wong, 1992; Balkhair, 2002). Extremely complicated issues with strong non-linear connections can be solved with ANN. Almost every input-output connection may be approximated thanks to the adaptable nature of ANNs. An increasing number of studies in hydrology have used artificial neural networks (ANN) with success since the early 2000s. These studies include modeling rainfall-runoff, stream flow forecasting, precipitation forecasting, groundwater modeling, and water quality and management modeling (ASCE 2000; Morshed and Kaluarachchi, 1998; Maier and Dandy, 2000). ANNs have also been used for estimating aquifer parameters (Zio 1997, Lingireddy 1998, Balkhair, 2002, Garcia and Shigidi 2003, Garcia and Shigidi 2006, Karahan and Ayvaz 2006). Ajmera et al. (2007) used Feed Forward Back Propagation (FFBP) to estimate aquifer parameters without the need for source/sink terms. They illustrated the significance of this technique and examined many training methods. In an attempt to demonstrate the use of various neural network types (FFBP, RBF, and RANN) for aquifer parameter estimation, Ajmera and Rastogi (2008) compared the outcomes using various criteria for performance assessment (expand these R, E, and RMSE) for two distinct scenarios: one with source/sink terms and the other without. On the other hand, comparatively little research has been done on parameter estimates and uncertainty characterization. Aly and Peralta (1999) introduced a combined GA-ANN technique to handle hydraulic conductivity uncertainty in a field-scale situation. The development of the ANN models is predicated on the knowledge of the aquifer's pumping and recharge histories, as well as its parameter values. Nevertheless, in a lot of field scenarios, either none of the parameter values are known, or part of the values are unknown. In these situations, estimating aquifer characteristics is a crucial component of any proposed process for identifying contaminated sources. Srivastava and Singh (2015) provide a methodology for simultaneously estimating flow and transport characteristics and identifying unidentified sources of pollution.
When it comes to solving large-scale complicated issues like recognizing patterns, nonlinear modelling, classification, association, and control, artificial neural networks (ANNs) become effective substitutes for conventional modelling approaches (Tayfur and Singh, 2006). The word "ANN" encompasses a wide range of network architectures, the most often used of which is the multilayer perceptron. Applications demonstrate the universal approximation capabilities and very easy training of just a single not visible layer-feed forward kind of ANN. Nonetheless, it could be preferable to employ an additional hidden layer when a situation calls for a greater number of concealed nodes. The number of hidden layers and nodes in the hidden layers before training begins cannot be established using a precise formula; instead, it is often determined by a trial-and-error process.
The multi-layer perceptron is trained using gradient descent and the back propagation technique, which is then applied to the sum-of-squares error function. The technique employs an iterative process to minimize the error function, whereby weights are adjusted over several phases. Every step has two separate stages. In the first step, mistakes are propagated backward to calculate the error function's derivatives concerning weights. The derivatives are utilized in the second step to calculate the weight modifications that need to be made (Bishop, 1995; Singh et al., 2004). According to ASCE (2000) and Maier and Dandy (2000), artificial neural networks (ANNs) are adaptable mathematical structures that can recognize intricate nonlinear correlations between data inputs and outcome sets. To address the issue of identifying the unknown source of pollution in groundwater, these characteristics of ANN are utilized in this study. In their ANN-based technique, Singh et al. (2004) employed 160 inputs, or 40 concentration measurement data at each of the four observation wells. When concentration measurement data during an initial period of time was unavailable, Singh and Datta (2007) used an (ANN-based technique to discover unknown sources of pollution for missing data scenarios. Instead of utilizing the entire breakthrough curve to decrease the inputs to an ANN, the current study used description of the breakthrough curve. The analysis significantly decreased the total amount of inputs and complexity of the system.
The mathematical framework for the aquifer system consists of equations governing the system with defined beginning and boundary conditions. The aquifer system's physical processes are simulated by solving a mathematical model. Using an Eulerian technique and the principle of conservation of mass, the governing equations reflecting the physics of groundwater flow and pollutant transport are obtained down to an infinitesimal fixed control volume. Comprehensive calculations and conversations may be found in Bear (1972, 1979), as well as Freeze and Cherry (1979).
3.1. Groundwater flow equation
This study applies the suggested technique to groundwater systems with steady-state flow conditions. In Cartesian tensor notation (Pinder and Bredehoeft, 1968), the governing equation for the stable 2-D areal flow of groundwater in a non-homogenous anisotropic and saturated aquifer is as follows:
It is necessary to simulate the groundwater system's flow and transport processes to develop ANN detection models. An essential component of groundwater modelling is the description of the equation that reflects the flow and transport procedures the aquifer's boundary conditions, and the aquifer's beginning conditions. The well-known MOC (Method of Characteristics) groundwater flow and transport simulator, created by U.S.G.S. (Konikow and Bredehoeft, 1978), is used to simulate aquifer transport processes.
This study treats the product of the source of solute and the volume of the liquid disposal rate as a single variable known as source flux or disposal flux. At every possible source site, a groundwater contamination source is considered to stay constant for the duration of a single disposal period. If the location, amount, and duration of a groundwater contamination source activity are stated, the source is fully recognized or described. The variability of the aquifer, the number of source sites and disposal time periods, the sparsity of the observation data available, the position of observation wells in relation to the actual source locations, the errors in parameter estimate within the designated boundary conditions, etc. add to the identification problem's ill-posedness (Datta, 2002) and consequent complexity. Since contaminant fate and transport pathways are irreversible, identifying unknown pollution sources is an ill-posed inverse issue (Aral et al., 2001).
In the topic under discussion, all geology and geochemical characteristics of the aquifer and the pollutant are taken to be known at the outset, and contaminants are regarded as conservative. We study the transport of contaminants in a 2D confined aquifer under constant circumstances. Each unidentified source in this study is regarded as point source.
Here, the capacity of the artificial neural network (ANN) to learn the input-output connection from training and testing patterns is used to simultaneously estimate the flow and transport parameters and identify unidentified sources of groundwater pollution. To figure out the parameter values and identify the unknown source fluxes producing the contamination, measured concentration data are utilized in tandem. Hydraulic conductivity, both transverse (kxx) and transverse (kyy), is taken into consideration for flow characteristics (kxx=kyy). Statistical characterization of the predicted concentration in the aquifer's Breakthrough Curve (BTC) provides the inputs for the ANN models during the training phase. The target or output values for the ANN are the sets of parameters to be evaluated and the designated pollutant injection rates. Figure 1 depicts a graphical depiction of the established approach for various aquifer parameter estimations.
Principal component analysis, hydrograph characterization, and statistical characteristics might all be used by a BTC (PCA). Srivastava and Singh (2014) go into detail on these characterizations. When compared to other characterizations, it was shown that BTC characterization using statistical variables performed better in the source identification challenge. Here, source flow and hydraulic conductivity are simultaneously identified, uncertainty is characterized, and BTC characterisation using four statistical factors is used. The simulated temporal concentration data yield four statistical parameters: mass (the total of concentration per BTC), the mean value, the standard deviation, and skewness values.
An artificial neural network (ANN) model is created for recognizing sources of pollution in a challenging situation where the aquifer traits are unknown. Figure 2 depicts the notional investigation region that includes a section of an aquifer. Groundwater parameter values and percentage measurement data for a large number of specified pollution sources that were selected at random are generated as training data utilizing a groundwater flow and transport simulation model (Konikow and Bredehoeft, 1978). In this case, time-varying pollution sources are taken into account. O1, O2, O3, and O4 are the concentration in order observation locations, while S1 and S2 are additionally thought to be possible source locations (Figure 2). It is believed that there was no pollution in the aquifer at first. Tables 1 and 2 list the true values of source fluxes, the aquifer parameter values, and the sizes of the finite difference grid. Aquifer (b) thickness is estimated to be 30.5 m (Table 1). The source flow is measured in grams per second (g/s).
Table 1. The values of the flow and transport parameters used to simulate the observed data
|
Parameter |
Value |
|
Kxx (m/s) |
0.0001 |
|
Kyy (m/s) |
0.0001 |
|
|
0.20 |
|
aL(m) |
30.5 |
|
aT(m) |
12.2 |
|
b (m) |
30.5 |
|
Δx(m) |
91.5 |
|
Δy(m) |
91.5 |
|
Δt(month) |
3 |
Table 2. Values of the source fluxes used for simulating data that was observed
|
Year |
Source fluxes at S1 |
Source fluxes at S2 |
|
Year 1 Year 2 Year 3 Year 4 Year 5 |
48.8 00.0 10.0 42.0 36.0 |
00.0 00.0 00.0 00.0 00.0 |
To simulate the concentration measurements, a 10-year time domain dissect into 40 equal time steps of three months each is taken into consideration. For the first five years of the ten-year time span, it is assumed that the pollutant source is operational. The study's unit of activity, which is one year, is considered to have constant sources. These models are developed with randomly generated source fluxes for each potential source. The values of the parameters that need to be approximated are likewise produced at random within predetermined limits. In accordance with our previous research, Srivastava and Singh, 2013, 2014 and 2015, 260 patterns were developed in this work. More or less than 260 patterns could exist. It was discovered that the breakthrough characterisation with statistical approach performed better. the whole breakthrough curve was characterized at the observation site in terms of statistical metrics. The inputs for the ANN model are considered as these statistical parameters at each observation site. The outputs used to train the ANN are the source fluxes causing these concentration levels combined with unknown parameters to be predicted. An Artificial Neural Network (ANN) model has been created to simultaneously identify the source of pollution and estimate the flow parameter, or hydraulic conductivity. For performance assessments, previous data in the form of source flux fluctuation within a certain range is integrated. By factoring in 50% uncertainty in the parameter estimating process, the upper and lower limits of the source flux values are determined. For example, after a year of solute transport, the source flux values are given as 24.4 and 73.2, respectively. It is believed that these upper and lower boundaries may be established based on some past data from nearby sites as well as some information on the qualities of the soil, etc. (Mahar, 1996).
These higher and lower boundaries are used to generate randomly dispersed values for these parameters that are evenly distributed. The computerized simulation model then uses these values to simulate the concentration measurement that finally results.
5.1 Performance Evaluation Criteria
The normalized Error (NE) value (Mahar 1996; Singh and Srivastava 2014) is used to quantify the mistakes to assess the prediction accuracy. As a gauge of the methodology's effectiveness, the NE is described as follows:
5.2 Training and Testing Results
By applying the suggested approach, observation from several observation wells is used to determine the possible source site. 30% of the total data sets are utilized for testing, while 70% are used for training. It should be noted that entire data was standardized into the range of 0:1 before the model was trained, and these normalized data were used in the computations.
Four statistical characteristics are used to characterize each breakthrough curve that was collected at four observation wells, for a total of 16 inputs. The outcomes of the ANN Model include source fluxes at two possible sites, which total have 10 source fluxes, and one flow parameter i.e. hydraulic conductivity, in total 11 source fluxes.
The network that included four neurons in each of the first and second hidden layers, sixteen input values, eleven target values per pattern, and 16-4-4-11 represented as 16-4-4-11 performed better for the ANN Model (table 3). This decision is supported by the assessment outcomes for various ANN model designs. Table 3 displays a selection of the 1000 iterations' outcomes. This design performs better in both the training and testing modes. Figure 3 displays the error vs number of iterations plot for the ANN design 16-4-4-11 that was produced using MATLAB (version 7.0). The efficiency of the ANN architecture or the Mean Square Error (MSE) is used to represent the error.
Table 3. Evaluations of performance for the generated ANN Model during training and testing
|
ANN Model Architecture |
Normalized Error (%) | |
|
Training |
Testing | |
|
16-9-11 |
28 |
33 |
|
16-3-3-11 |
35 |
40 |
|
16-4-4-11 |
28 |
30 |
|
16-7-7-11 |
28 |
34 |
5.3 Identification Results
The trained ANN models are then assessed further for the example issue of observed concentration, where the given parameter and source flux magnitudes are taken to be unknown. The normalized error numbers in Table 4 demonstrate that the model's performance is satisfactory (NE=18.57%). As a result, complexity rises with an increase in unknown parameters, which has an immediate impact on model performance. Figure 4 illustrates how the anticipated values of parameters and source fluxes closely match the actual values.
Table 4. Comparison of actual and predicted source fluxes for developed ANN Model
|
Parameter or source duration and location |
Actual value |
Estimated value |
NE(%) |
|
Year 1, S1 Year 1, S2 Year 2, S1 Year 2, S2 Year 3, S1 Year 3, S2 Year 4, S1 Year 4, S2 Year 5, S1 Year 5, S2 Kxx(=Kyy)(m/sec) |
48.8 0.0 0.0 0.0 10.0 0.0 42.0 0.0 36.0 0.0 0.00010 |
49.05 4.22 1.5 2.12 8.3 0.09 34.71 3.96 34.13 2.41 0.0000841
|
18.57 |
Compared to training and testing outcomes, the NE values discovered for prediction results are superior. It may be because there are fewer patterns in the identifying process. The comparison of the outcomes indicates that the quality of the results decreases with increasing problem complexity. When just the source identity problem was resolved, the NE value was reported to be 10.35 (Srivastava and Singh, 2014). Thus, both causes and parameters are discovered at the same time, the results get worse.
The input parameters needed to solve the flow of groundwater and transport simulation model may be not fully understood or are not exact. These inaccurate input parameters may result from subjective interpretation of the given data and indirect measurements, a phenomenon referred to as epistemic uncertainty. Here, the fuzzy vertex method with fuzzy alpha (α) cut methodology is used to characterize the epistemic uncertainty in the source flow.
6.1. Fuzzy Vertex Method in Assessing Uncertainty in Model Parameters
The fuzzy numbers A1, A2, A3………AN is specified on the real line L in the Vertex Method, and the elements of Ai are represented by xi, where i=1, 2,…..,N. The following process may be used to find the answer to the fuzzy number B in y that would match the fuzzy numbers A1 in x1, A2 in x2,…..,AN in xN if x1, x2,…..,xn are associated to a real number y by mappings y = ƒ(xi, x2,……xN):
- The membership range [0, 1] is divided into a limited set of values known as α -cuts, which are represented as 1, 2,... M. The desired level of precision determines how finely the discretization is refined.
- The matching intervals for Ai in xi, i=1, 2, ..., N, are found for each membership value j. These are the j-cut supports for A1, A2,... AN. These intervals' end points are denoted by the notations [a1, b1], [a2, b2] …..., [aN, bN]. In the event where ai and bi are equal, the interval would shrink to a single point.
- The vector (x1, x2,... xN) can have 2N combinations when one end point from each interval is merged to create 2N unique permutations.
- iv. To determine values for y, which are represented as y1, y2……yN, the function ƒ (x1, x2……xN) is calculated for each of the 2N permutations. The interval that is sought for y may be found in [], which also defines the base of the j-cut of B.
- In order to acquire more -cuts of B and the answer to the fuzzy number B, the procedure is repeated for additional -cuts.
- The measure of uncertainty used for the Fuzzy Alpha Cut (FAC) technique is the ratio 0.1-level support to the value for which the membership function is equal to 1 (Abebe et al., 2000.
6.2 Assessing Uncertainty in Source Identification using Fuzzy Vertex Method
By describing source flux uncertainty, the vertex approach is used to determine source identification uncertainty. Like in earlier instances, the procedure begins with the random creation (LHS) of source fluxes at various time intervals at possible source locations. Subsequently, the pollution scenario is simulated using the source magnitudes and parameter values provided. The pollutant concentration at the observation point is simulated using the water flow and transport simulation model MOC (Konikow and Bredehoeft, 1978). Next, for each one of the source magnitudes, the complete breakthrough curve at the observation site is computed using a set of statistical parameters. The ANN model was formed using these patterns in the data. The steps depicted in Figure 7 must be followed to implement the suggested technique.
In the current study, ANN models created for source and parameter detection simultaneously are also used to characterize uncertainty.
6.3 Uncertainty in Source Flux
For the purposes of uncertainty analysis, the Source Flux is regarded as an uncertain parameter. Fuzzy triangular numbers are used to represent the unknown parameter, Source Flux. The deterministic value of Source Flux used in ANN models is 48.8 g/sec, which is also the center value of the triangle fuzzy representation. The upper and lower values (as the remaining two vertices) in the triangle representation correspond to the given ranges (24.4-73.2). As seen in Figure 7, there are two values—low and high—at each -level. Hydraulic conductivity predictions are produced for various Source flux values. The vertex approach is then used to characterize the uncertainty in source identification.
Lastly, for uncertainty characterisation, the ratio of 0.1-level support to the value where the membership function equals 1 is anticipated (Figure 5).
6.4 Discussion of Results for Uncertainty Characterization
For the particular case of observed concentration with undetermined source fluxes, the effectiveness of ANN models designed for simultaneous recognizing source fluxes and flow variables is also assessed. An ANN model for simultaneous detection of sources of pollution and parameter estimation that was previously developed is used to estimate hydraulic conductivity uncertainty assuming a 50% margin of error in source flow estimation. A collection of hydraulic conductivity values is estimated for each α-level. The findings of the uncertainty characterization indicate that the average uncertainty in the prediction of hydraulic conductivity is 11.5% when the source flow is 50% unknown.
The approach offered in this paper can be employed for input reduction and uncertainty characterization in groundwater flow and transport problems. In addition to unidentified source fluxes, instances with uncertain or imprecise source fluxes are additionally taken into account. It is found that with an increasing number of unknown parameters complexities increases and model performance deteriorates.
As the uncertainty description outcomes show with 50% uncertainty in source flux, the average uncertainty in hydraulic conductivity prediction is 11.5%, this reveals that the prediction of hydraulic conductivity, which is one of the most important aspects of groundwater flow and transport problems in groundwater system is less sensitive to source flux estimation.
Author Contribution
"Dr. Divya," wrote the main manuscript text. " Dr. Raj Mohan" contributed to the methodology, and supervision and reviewed the manuscript. Editing ,writing and revieweing contributed by "Pushpanjali".
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No competing interests reported.