1. Introduction:
River-aquifer (R-A) exchange influences both quantity and quality aspects of groundwater and river water systems significantly. Consequently, proper and effective quantification and representation of river-aquifer exchange are very important for the management of water resources and aquatic ecosystems. Climate and groundwater pumping for irrigation has caused rapid groundwater depletion in India (Dangar et al.,2021). Several works have been carried out about the identification and the quantification of groundwater and surface water interactions i.e. R-A exchanges (Constanzt, 1998; Sophocleous, 2002; Becker et al., 2004; Anderson, 2005; Kalbus et al., 2006; Keery et al., 2007; Lowry et al., 2007; Yang et al., 2017). Some studies on R-A exchanges also demonstrate the effect of it on river temperature (Westhoff et al., 2007; Burkholder et al., 2008; Hebert et al., 2011) along with the emphasis on exchanges in hyporheic zone (Burkholder et al., 2008).
Simulation-optimization (S-O) techniques are often used in identifying the optimal management practices of groundwater for the selected region. Many challenges are associated with this S-O technique like the selection for proper simulation or optimization algorithm and comparison of the results of various techniques used. In the multi-objective optimization problems, comparison of results becomes more significant as Pareto fronts generated in this approach can have some similarities with some diversity. Jha et al. (2020) statistically evaluated the relationship between groundwater pumping rates and groundwater levels during pre-monsoon and post-monsoon seasons using a numerical groundwater-flow simulation model. The simulation-optimization (S-O) model used linear programming (LP) optimization to handle groundwater hydraulic optimization management problems after convolution techniques were employed to integrate the simulation model with an optimization method. Kamali Asghar (2017) used a mathematical simulation optimization programming model to handle the issue of aquifer management, relying on the stability of water quality and quantity. The simulation-optimization models and the construction of the sustainability index for determining the best point of the Pareto front were used in the study’s modeling. Abd-Elmaboud et al. (2021) introduced a new model that connects geomorphological and hydrogeological data with recharge rates. The MFUSG-PSO model with an indirect simulation-optimization technique for the inversion of the groundwater flow problem was used to calibrate the recharge rates. The second phase involved training a CFNN model to connect the calibrated recharge rates with freely available geomorphological and hydrogeological data to construct interrelationships. For the best possible conservation of water resources, Conant et al. (2019) suggested that it is critical to understand and quantify exchange activities between rivers and groundwater. The exchange between rivers and groundwater is significant in a variety of current concerns, including providing drinking water, characterizing and managing environmental flow regimes, preserving or restoring riverine ecosystem health and functioning, and alleviating toxins. For reaches in plains with flow monitoring data. Li et al. (2020) introduced a cumulative exchange fluxes method based on surface water balance to study GW-SW interactions. The dynamic change processes of GW-SW interactions can be qualitatively and quantitatively judged through a curve of cumulative exchange fluxes by this method. The hyporheic zone, which is usually regarded as a biogeochemical hotspot, is where surface-groundwater interactions are most frequent and according to McClain et al. (2003), these hotspots are the key to manage water resources effectively.
Calandra et al. (2014) proposed an approach to make full use of the simulation algorithms developed by Bayesian Optimization to analyze the quality of Pareto fronts, unlike the existing algorithms which only returned the Pareto solution sets without considering the qualitative assessment. The approach was able to delineate the actual Pareto fronts better in presence of noise and at the same time can also perform sensitivity analysis of the parameters concerning the quality of model output. Lobato et al. (2016) have presented a Bayesian method called PESMO to identify Pareto solution set to Multi-Objective optimization problems. The evaluation points were chosen as such to minimize the entropy of the Pareto set. When compared with existing techniques, PESMO was found to produce better results with a smaller count of evaluations, while the decoupled estimation led to an improved performance particularly when the existing techniques lose efficiency with an increase in objectives. Emmerich et al. (2018) have also discussed the fundamentals and evolutions in the field of Multi-objective optimization. The topics covered in their article include- order-theoretical foundations, scalarization approaches, and optimality conditions. In context to the evolutionary methods, three state-of-the-art techniques were discussed namely NSGA-II, SMS-EMOA, and MOEA/D. NSGA-II representing the Pareto-based approach, SMS-EMOA exemplifies the Indicator-based approach while MOEA/D is an example of a decomposition-based approach. The choice of the correct technique depends on the number of objectives, count of solution sets, desired distribution of the solutions along with the location and shape of the Pareto front. Belakaria et al. (2019) have addressed the optimization of the Multi-Object (MO) black box to determine true solutions of Pareto-set by reducing the count of function evaluations. The study proposed a new algorithm called Max-value Entropy Search for Multi-objective Optimization (MESMO) to ascertain an optimized design that efficiently trades off among performance, power requirement, and area overhead. Their approach used the output matrix entropy function to efficiently select the input parameters, to obtain highly accurate Pareto-set solutions. The algorithm was found to be constantly outperforming the existing state-of-the-art algorithms in computing Pareto set solutions.
Binois et al. (2014) have taken into account Kriging metamodel for estimating the Pareto front and also to quantify bias associated with the solution set at any phase of the multi-objective optimization. The approach taken by them assumed the original dataset to be having Gaussian distribution. The concept of random set theory has been used to compute Vorob’ev deviation to capture the variability of the dataset with non-dominated points. This method applied on several numerical problems yielded satisfactory output in accurately determining of Pareto front. Horn et al. (2016) proposed a model based on Random Forests for model-based multi-objective optimization to determine the Pareto front for mixed hierarchical configuration problems. A two-phase parameter experiment was carried out. The results of the single model experiment were quite promising, however, the solution derived through a multi-model approach could be improved further. Especially, the bias estimation associated with the Random Forest model needed to be resolved.
Cao et al. (2017) have tried to quantify the bias of the estimated solution of multi-objective problems using two versions of the normalized hypervolume. A case study was carried out in a chemical process with three sets of solutions, each having a different optimization goal. Results showed that the normalized hypervolume showed great accordance to the ideal situations while taking into account the degree of the trade-offs between the possibilities considered to compute the front. Bassi et al. (2018) have considered a new approach to evaluate statistical parameters for a set of objects equivalent to surfaces and arcs. Their approach is based on the identification of the most representative member of a family tree and was found useful to address the uncertainties associated with Pareto fronts. The Pareto front was ascertained by minimizing the hypervolume between the front and the reference points. The algorithm was tested on a complex 5 bar truss structure with satisfactory outputs being obtained. Avent et al. (2019) proposed a new approach called DPARETO, to determine the trade-off among various differentially cloistered algorithms. Bayesian Optimization (BO) was used to concurrently optimize both privacy and utility parameters of a Pareto front. Moreover, they also established the effectiveness of BO in creating visualization interfaces helpful in decision making. Bionis et al. (2019) have designed a GPareto library for R which enables the optimization of multi-objective algorithms for functions associated with a black box. Moreover, the package also contained several algorithms for the accurate assessment of the associated bias. The study also proposed several infill criteria in minimizing the bias associated with several optimization models such as efficient global optimization technique.
Marjit R. (2009) has made a noble attempt to increase the efficiency of the Building performance simulation (BPS) tool by combining it with a robust multi-objective optimization algorithm. The thesis aimed at developing a multi-objective algorithm with a meta-model to optimize the simulation of a Pareto front. Asadzadeh et al. (2014) on the other hand, have developed a new technique called Convex Hull Contribution (CHC) to address multi-objective (MO) optimization functions associated with Pareto fronts. They demonstrated the effectiveness of CHC in enhancing the accuracy of Pareto archived multi dimensioned search while addressing the multi-objective problems. Audet et al. (2018) have presented a review of the algorithms devised so far to address the multi-objective-based optimization in the determination of Pareto fronts. An assessment of a large number of performance indicators was carried through various algorithms. Total 57 performance indicators were grouped into four categories by their properties, viz. cardinality, convergence, distribution, and spread. Hollermann et al. (2019) have proposed a new flexible method to automatically identify one optimum design for multi-objective decision-making. Their approach addresses both economic and environmental aspects to identify a viable design. The bias associated with the parameters while predicting a future event could be handled easily by developing an extension of this algorithm.
Present work was carried out to address the challenging issue of R-A exchanges which can be analyzed through numerical models. The output of the numerical models depends on the model domain considered, different hydrogeological parameters used and boundary conditions applied. Considering the R-A exchanges in groundwater resource management problems can help in finding out more efficient and river health inclusive management practices through simulation-optimization models. Meanwhile, selection of the optimization algorithm, model domain demarcation and the comparison of optimum results to provide a physically meaningful management strategy is still a bottleneck in the groundwater management problems. Therefore, different model domain demarcations for the same river system were considered for the development of the groundwater model, and their impact on R-A exchanges was analyzed. In addition, different optimization techniques were also used and their outputs i.e. Pareto fronts were compared. Finally, this set of optimal results were interpreted in terms of groundwater management.