Entropy Applications to Water Monitoring Network Design: A Review <s> Soil Moisture and Groundwater Networks <s> Assessment and redesign of water quality monitoring networks is an important task in water quality management. This paper presents a new methodology for optimal redesign of groundwater quality monitoring networks. The measure of transinformation in discrete entropy theory and the transinformation-distance (T-D) curves are used to quantify the efficiency of sampling locations and sampling frequencies in a monitoring network. The existing uncertainties in the T-D curves are taken in to account using the fuzzy set theory. The C-means clustering method is also used to classify the study area to some homogenous zones. The fuzzy T-D curve of the zones is then used in a multi-objective hybrid genetic algorithm-based optimization model. The proposed methodology is utilized for optimal redesign of monitoring network of the Tehran aquifer in the Tehran metropolitan area, Iran. <s> BIB001 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Soil Moisture and Groundwater Networks <s> A discrete entropy-based approach is used to assess the groundwater monitoring network that exists in Kodaganar River basin of Southern India. Since any monitoring system is essentially an information collection system, its technical design and evaluation require a quantifiable measure of information and this measure can be derived using entropy. The use of information-based measures of groundwater table shows that the existing monitoring network contains a sufficient number of wells but is not well designed for the measurement of groundwater level. Entropy-based results show that 15 wells are vital to measure regional groundwater level, not 28 wells which are being monitored effectively in this basin. <s> BIB002 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Soil Moisture and Groundwater Networks <s> Assessment of groundwater quality monitoring networks requires methods to determine the potential efficiency and cost-effectiveness of the current monitoring programs. To this end, the concept of entropy has been considered as a promising method in previous studies since it quantitatively measures the information produced by a network. In this study, the measure of transinformation in the discrete entropy theory and the transinformation-distance (T-D) curves, which are used frequently by other researchers, are used to quantify the efficiency of a monitoring network. This paper introduces a new approach to decrease dispersion in results by performing cluster analysis that uses fuzzy equivalence relations. As a result, the sampling (temporal) frequency determination method also recommends the future sampling frequencies for each location based on certain criteria such as direction, magnitude, correlation with neighboring stations, and uncertainty of the concentration trend derived from representative historical concentration data. The proposed methodology is applied to groundwater resources in the Tehran-Karadj aquifer, Tehran, Iran. <s> BIB003 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Soil Moisture and Groundwater Networks <s> [1] Establishing an adequate hydrometric network to provide accurate and reliable continuous flow information for various users remains a major challenge. This includes the design of new networks or the evaluation of existing networks. This study proposes a combined regionalization and dual entropy-multiobjective optimization (CRDEMO) method for determining minimum network that meets the World Meteorology Organization (WMO) standards, which is considered herein an optimal minimum network. A regionalization approach is used to estimate flows in potential locations for new additional stations, and a dual entropy-multiobjective optimization approach is used to identify optimal trade-offs between the maximum possible information content and the minimum shared information among the stations. The method was examined in two Canadian River basins, namely, the St. John River and St. Lawrence River basins. The St. John River basin has a higher network density compared to the St. Lawrence River basin. Results of the analysis indicate that the St. Lawrence River basin requires a high number of new additional stations to meet the WMO minimum network density. It was also determined that existing stations do not have a significant influence in determining the locations of new stations in the St. Lawrence River basin. Conversely, however, in the St. John River basin, existing stations have significant influence on the determination of locations of new stations due to the higher number of existing stations. Overall, the CRDEMO technique is shown to be robust for designing optimal minimum hydrometric networks and can be a useful tool for evaluating current and proposed networks. <s> BIB004 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Soil Moisture and Groundwater Networks <s> A weighted multi-criteria decision-making framework was designed to identify and prioritize areas for monitoring aquifer hydraulic heads within the Victoria County Groundwater Conservation District. The criteria address salient hydrogeological aspects including groundwater variability, recharge, surface water–groundwater interactions, and groundwater fluxes across district boundaries. Texas water statutes require groundwater conservation districts to quantify these factors and utilize them to manage aquifers. A stakeholder survey instrument was used to prioritize these criteria. A monitoring priority index (MPI), whose value theoretically ranges between 0 and 1 was computed as a weighted average of six mutually exclusive criteria. The MPI value was seen to range between 0.2 and 0.6 within the study area. Lower values of MPI indicate areas where the current level of monitoring is adequate and areas with a higher MPI point to locations where additional monitoring is necessary. In particular, monitoring along district boundaries, particularly along the western and southern sections, is recommended. The developed methodology was seen to provide a transparent and simple-to-use approach to identify and prioritize areas within the district for the purposes of groundwater monitoring. The proposed framework demonstrates the utility of GIS in facilitating a scientifically credible and stakeholder-driven approach for establishing groundwater monitoring networks and can be adapted in other similar applications. <s> BIB005 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Soil Moisture and Groundwater Networks <s> Summary Improving understanding of chemical transport in the subsurface commonly employs evolving groundwater monitoring networks. The objective of this work was to apply the information theory to propose an objective algorithm for augmenting a subsurface monitoring network (SMN) with the purpose of discrimination of conceptually different subsurface flow and transport models. This method determines new monitoring locations where the Kullback–Leibler total information gain is maximized. The latter is computed based on estimates of the uncertainty in modeling results and uncertainty in observations. The method was applied to discriminate models in (1) a synthetic case of groundwater contamination from a point source; (2) the tracer experiment conducted at the USDA-ARS OPE3 research site where a pulse of KCL solution was applied with irrigation water and CL − concentrations were subsequently monitored. Models were compared that included or ignored the effect of subsurface soil lenses on chemical transport. Pedotransfer functions were used to develop the ensemble of models for estimating the uncertainty in modeling results obtained with the numerical 3D flow and transport model. Peak tracer breakthrough concentrations were used to define the information gains. The determination of the new locations to augment existing ones was conducted on a 2-D grid. The information gain peaked in small area, and additional observation locations were very well spatially defined. Well-calibrated models provided a single optimal location, whereas, if models were not calibrated well, the Bayesian estimates of the new observation location depended on the activation sequence assumed for existing locations. The information gain maximization can suggest data collection locations to reduce uncertainties in the conceptual models of subsurface flow and transport. <s> BIB006 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Soil Moisture and Groundwater Networks <s> Many methods have been proposed to select sites for grid-scale soil moisture monitoring networks; however, calibration/validation activities also require information about where to place grid representative monitoring sites. In order to design a soil moisture network for this task in the Great Lakes Basin (522 000 km2), the dual-entropy multiobjective optimization algorithm was used to maximize the information content and minimize the redundancy of information in a potential soil moisture monitoring network. Soil moisture retrieved from the Soil Moisture and Ocean Salinity (SMOS) mission during the frost-free periods of 2010–2013 were filtered for data quality and then used in a multiobjective search to find Pareto optimum network designs based on the joint entropy and total correlation measures of information content and information redundancy, respectively. Differences in the information content of SMOS ascending and descending overpasses resulted in distinctly different network designs. Entropy from the SMOS ascending overpass was found to be spatially consistent, whereas descending overpass entropy had many peaks that coincided with areas of high subgrid heterogeneity. A combination of both ascending and descending overpasses produced network designs that incorporated aspects of information from each overpass. Initial networks were designed to include 15 monitoring sites, but the addition of network cost as an objective demonstrated that a network with similar information content could be achieved with fewer monitoring stations. <s> BIB007 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Soil Moisture and Groundwater Networks <s> Hydrology is rich in methods that use information theory to evaluate monitoring networks. Yet in most existing studies, only the available data set as a whole is used, which neglects the intraannual variability of the hydrological system. In this paper, we demonstrate how this variability can be considered by extending monitoring evaluation to subsets of the available data. Therefore, we separately evaluated time windows of fixed length, which were shifted through the data set, and successively extended time windows. We used basic information theory measures and a greedy ranking algorithm based on the criterion of maximum information/minimum redundancy. The network investigated monitored surface and groundwater levels at quarter-hourly intervals and was located at an artificially drained lowland site in the Spreewald region in north-east Germany. The results revealed that some of the monitoring stations were of value permanently while others were needed only temporally. The prevailing meteorological conditions, particularly the amount of precipitation, affected the degree of similarity between the water levels measured. The hydrological system tended to act more individually during periods of no or little rainfall. The optimal monitoring setup, its stability, and the monitoring effort necessary were influenced by the meteorological forcing. Altogether, the methodology presented can help achieve a monitoring network design that has a more even performance or covers the conditions of interest (e.g., floods or droughts) best. <s> BIB008 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Soil Moisture and Groundwater Networks <s> Abstract This study explores the inclusion of a groundwater recharge based design objective and the impact it has on the design of optimum groundwater monitoring networks. The study was conducted in the Hamilton, Halton, and Credit Valley regions of Ontario, Canada, in which the existing Ontario Provincial Groundwater Monitoring Network was augmented with additional monitoring wells. The Dual Entropy-Multiobjective Optimization (DEMO) model was used in these analyses. The value of using this design objective is rooted in the information contained within the estimated recharge. Recharge requires knowledge of climate, geomorphology, and geology of the area, thus using this objective function can help account for these physical characteristics. Two sources of groundwater recharge data were examined and compared, the first was calculated using the Precipitation-Runoff Modeling System (PRMS), and the second was an aggregation of recharge found using both the PRMS and Hydrological Simulation Program-Fortran (HSP-F). The entropy functions are used to identify optimal trade-offs between the maximum information content and the minimum shared information between the monitoring wells. The recharge objective will help to quantify hydrological characteristics of the vadose zone, and thus provide more information to the optimization algorithm. Results show that by including recharge as a design objective, the spatial coverage of the monitoring network can be improved. The study also highlights the flexibility of DEMO and its ability to incorporate additional design objectives such as the groundwater recharge. <s> BIB009 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Soil Moisture and Groundwater Networks <s> Abstract In this paper, a new data fusion-based methodology is presented for spatio-temporal (S-T) redesigning of Groundwater Level Monitoring Networks (GLMNs). The kriged maps of three different criteria (i.e. marginal entropy of water table levels, estimation error variances of mean values of water table levels, and estimation values of long-term changes in water level) are combined for determining monitoring sub-areas of high and low priorities in order to consider different spatial patterns for each sub-area. The best spatial sampling scheme is selected by applying a new method, in which a regular hexagonal gridding pattern and the Thiessen polygon approach are respectively utilized in sub-areas of high and low monitoring priorities. An Artificial Neural Network (ANN) and a S-T kriging models are used to simulate water level fluctuations. To improve the accuracy of the predictions, results of the ANN and S-T kriging models are combined using a data fusion technique. The concept of Value of Information (VOI) is utilized to determine two stations with maximum information values in both sub-areas with high and low monitoring priorities. The observed groundwater level data of these two stations are considered for the power of trend detection, estimating periodic fluctuations and mean values of the stationary components, which are used for determining non-uniform sampling frequencies for sub-areas. The proposed methodology is applied to the Dehgolan plain in northwestern Iran. The results show that a new sampling configuration with 35 and 7 monitoring stations and sampling intervals of 20 and 32 days, respectively in sub-areas with high and low monitoring priorities, leads to a more efficient monitoring network than the existing one containing 52 monitoring stations and monthly temporal sampling. <s> BIB010 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Soil Moisture and Groundwater Networks <s> This paper presents a new methodology for analyzing the spatiotemporal variability of water table levels and redesigning a groundwater level monitoring network (GLMN) using the Bayesian Maximum Entropy (BME) technique and a multi-criteria decision-making approach based on ordered weighted averaging (OWA). The spatial sampling is determined using a hexagonal gridding pattern and a new method, which is proposed to assign a removal priority number to each pre-existing station. To design temporal sampling, a new approach is also applied to consider uncertainty caused by lack of information. In this approach, different time lag values are tested by regarding another source of information, which is simulation result of a numerical groundwater flow model. Furthermore, to incorporate the existing uncertainties in available monitoring data, the flexibility of the BME interpolation technique is taken into account in applying soft data and improving the accuracy of the calculations. To examine the methodology, it is applied to the Dehgolan plain in northwestern Iran. Based on the results, a configuration of 33 monitoring stations for a regular hexagonal grid of side length 3600 m is proposed, in which the time lag between samples is equal to 5 weeks. Since the variance estimation errors of the BME method are almost identical for redesigned and existing networks, the redesigned monitoring network is more cost-effective and efficient than the existing monitoring network with 52 stations and monthly sampling frequency. <s> BIB011
Soil moisture is a critical water variable as the interface between the atmosphere and subsurface. Unfortunately, the monitoring of soil moisture is very sparse compared to its spatial variability. To design an optimum network for monitoring soil moisture in the Great Lakes Basin, Kornelsen and Coulibaly BIB007 proposed using data from the Soil Moisture and Ocean Salinity (SMOS) satellite  to design a soil moisture monitoring network using the DEMO algorithm of Samuel et al. BIB004 . Grid cells were selected to add monitoring stations that optimally maximized joint entropy while minimizing total correlation using only the satellite data. The ascending and descending overpasses were found to contain different information, and the spatial distribution of a network designed with both overpasses was found to contain complimentary features from both datasets BIB007 . Groundwater monitoring allows for a better understanding of the hydrogeology in an area. This is achieved through groundwater quality and quantity monitoring. Groundwater quality monitoring is used to detect contaminant plumes or for long-term monitoring (LTM) of post remediation effects, and groundwater quantity monitoring is used to determine available water for drinking, irrigation and industry. However, monitoring groundwater is inherently difficult due to physical barriers between observers and the water. Through the understanding of subsurface flow physics and with flow and contaminant transport models such as MODFLOW, MODPATH and MT3D [77]   , we can simulate the behaviour of groundwater. Unfortunately, our simulations are not always accurate, and the models require real-world observations to be calibrated and validated. Due to constraints such as accessibility and financial cost, it is not feasible to monitor at every possible location in an area of interest. It is instead ideal for an optimal monitoring network to be designed to allow for the best placement of monitoring stations and to determine the ideal measurement frequencies. The merit of using information theory entropy has been shown in several cases of groundwater network design BIB008 BIB010 BIB011 BIB009 BIB001 BIB002 BIB005 BIB006 . Various methods that utilize information theory entropy have been developed for use in designing optimal groundwater monitoring networks. These include the use of entropy measures in both single and multi-objective optimization problems and are used in network reduction BIB010 BIB011 , expansion BIB009 BIB006 and redesign BIB001 , as well as have been used to highlight vulnerable areas in an area that should be monitored BIB005 . In identifying vulnerable areas in the Victoria County Groundwater Conservation District (VCGCD) in Texas, USA, Uddameri and Andruss BIB005 developed a monitoring priority index (MPI) based on a weighted stakeholder preference to highlight the areas of interest. They compared kriging standard deviation and marginal entropy as metrics to characterize groundwater variability and found entropy to be the more conservative metric. In areas where there is excessive monitoring, Mondal and Singh BIB002 showed the information transfer index (ITI), the quotient of joint entropy and transinformation, could be used to evaluate the existing monitoring network. Through this evaluation, redundant monitoring stations (wells) could be identified and removed from the groundwater monitoring network. It may also be the case that the existing groundwater monitoring network is not adequate and additional monitoring stations are needed. Yakirevich et al. BIB006 developed a method that utilizes minimum cross entropy (MCE) to sequentially add monitoring stations to a network. MCE was used as a metric to quantify the difference between two variants of a Hydrus-3D model  , and the monitoring stations were added to the network where the difference between models was largest. A multi-objective approach for adding monitoring stations to a groundwater monitoring network was applied by Leach et al. BIB009 , which utilized two entropy measures, total correlation and joint entropy and a metric used to quantify the spatial distribution of annual recharge; the results of which were used to develop maps that highlight areas in which additional monitoring stations should be added. The majority of network design experiments look at the entire available time series when calculating entropy measures; however, Fahle et al. BIB008 showed that using a combination of MIMR and subsets of the data series could be more ideal. The subsets were used to represent the intra-annual variability of groundwater levels. This method identified locations which were consistently important through each subset and found that monitoring stations showed similarities during wet periods and uniqueness during dry periods. Fahle et al. BIB008 also suggest that a consequence of using subsets of data allows for the design of a network, which can be focused on floods or droughts. One issue that can arise with entropy-based methods is the need for lengthy data series to produce accurate measures of entropy. Unfortunately, the area of interest for new monitoring stations will not have available data for all possible locations. To work around this limitation transinformation-distance (T-D) curves have been applied in the design of optimal groundwater monitoring networks BIB001 BIB003 . In these studies, T-D curves were developed for sub areas within the desired study area based in different clustering methods. Additionally, Masoumi and Kerachian BIB001 showed that this method could be applied temporally as transinformation-time curves which could then be used to optimize the temporal sampling frequency of the stations. It should be noted that both previously mentioned studies were applied in the same study area using slightly different methods for clustering monitoring stations, and both produced different groundwater monitoring networks that could be considered optimal. This highlights an issue with optimal monitoring network design in that it can be subjective and does not have a singular solution. A comparison of Hosseini and Kerachian BIB010 BIB011 also illustrates this issue, where through the use of different entropy measures, marginal entropy and Bayesian maximum entropy and optimization techniques, one experiment found the optimal monitoring network included 42 monitoring stations while the other only included 33 stations.