# Entropy Applications to Water Monitoring Network Design: A Review

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## Abstract

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## 1. Introduction

## 2. Definitions of Entropy Terms as Applied to Water Monitoring Networks

#### 2.1. Entropy Concept

#### 2.2. Marginal Entropy

#### 2.3. Multivariate Joint Entropy

#### 2.4. Conditional Entropy

#### 2.5. Transinformation

#### 2.6. Total Correlation

#### 2.7. Other Entropy Terms

## 3. Applications of Entropy to Water Monitoring Network Design

#### 3.1. Precipitation Networks

#### 3.2. Streamflow and Water Level Networks

#### 3.3. Soil Moisture and Groundwater Networks

#### 3.4. Water Quality Networks

#### 3.5. Integrated Network Design

## 4. Conclusions and Recommendations

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Table 1.**Summary of significant contributions to water monitoring network design using entropy (author alphabetical order).

Authors/Year | Network Types | Study Areas | Methods/Entropy Measures | Key Findings |
---|---|---|---|---|

Alameddine et al., 2013 [24] | Water quality | Neuse River Estuary, NC, USA | -Total system entropy -Standard violation entropy -Multiple attribute decision making process -Analytical hierarchical process | -Networks designed using total system entropy and violation entropy of dissolved oxygen were similar -When measured water quality parameters have a low probability of violating water quality standards, their violation entropy is less informative |

Alfonso et al., 2010 [25] | Water level | Pijnacker Region, The Netherlands | -Directional information transfer (DIT) | -Introduced total correlation for determining multivariate dependence in water monitoring network design -Information content and redundancy is dependent on the DIT between monitoring stations (DIT _{XY} or DIT_{YX}) |

Alfonso et al., 2010 [26] | Water level | Pijnacker Region, The Netherlands | -Max(Joint Entropy) min(Total Correlation) -Non-dominated Sorting Genetic Algorithm II (NSGA-II) | -Total correlation should be combined with joint entropy to get most information out of monitoring network |

Alfonso et al., 2013 [27] | Streamflow | Magdalena River, Colombia | -Max(Joint Entropy) min(Total Correlation) -Rank-based iterative approach | -Rank method is useful in finding extremes on Pareto front -When iteratively selecting stations, the information content of the network is not guaranteed to be maximum if the network contains the station with the most information |

Alfonso et al., 2014 [28] | Water level | North Sea, The Netherlands | -Max(Joint Entropy) min(Total Correlation) -Ensemble entropy -NSGA-II | -By creating an ensemble of solutions through varying the bin size of the initial Pareto optimal solution set, the authors highlight the uncertainty related to choosing bin size |

Boroumand and Rajaee, 2017 [29] | Water quality | San Francisco Bay, CA, USA | -Transinformation-distance (T-D) curve | -Using T-D curve they were able to reduce the network from 37 to 21 monitoring stations. -New network covered entire study area without having redundant data |

Brunsell, 2010 [30] | Precipitation | Continental United States | -Relative entropy -Wavelet multi-resolution analysis | -The temporal scaling regions identified (1) synoptic, (2) monthly to annual, (3) interannual patterns -Little correlation between relative entropy and annual precipitation except for breakpoint at 95° W Lat |

Fahle et al., 2015 [31] | Water level/Groundwater level | Spreewald region, Germany | -MIMR, max(Joint Entropy + Transinformation − Total Correlation) -Subsets of time series data | -Found using subsets of the available time series data could better identify important stations -Showed water levels across network react similarly during high precipitation and are more unique during dry periods -Consequently method can allow for design of network which focuses on floods or droughts |

Hosseini and Kerachian, 2017 [32] | Groundwater level | Dehgolan plain, Iran | -Marginal entropy -Data fusion of spatiotemporal kriging and ANN model -Value of information (VOI) | -Network reduction from 52 to 42 (35 high priority and 7 low priority) stations while standard deviation of average estimation error variance stayed the same -Found sampling frequency of high priority stations should be every 20 days and low priority should be every 32, based on analysis of stations selected using VOI |

Hosseini and Kerachian, 2017 [33] | Groundwater level | Dehgolan plain, Iran | -Bayesian maximum entropy (BME) -Multi-criteria decision making based on ordered weighted averaging | -Network reduction from 52 to 33 stations while standard deviation of average estimation error variance stayed the same -Sampling frequency increased from 4 weeks to 5 weeks |

Keum and Coulibaly, 2017 [34] | Precipitation/Streamflow | Columbia River basin, BC, Canada. Southern Ontario, Canada | -Dual Entropy and Multiobjective Optimization (DEMO) to max(Joint Entropy) and min(Total Correlation) | -Found that networks obtain significant amount of information from 5 to 10 years of data periods, and total correlation tends to be stabilized within 5 years by applying daily time series -Recommended minimum 10 years data periods for designing precipitation or streamflow networks using daily time series |

Keum and Coulibaly, 2017 [35] | Integrated | Southern Ontario, Canada | -DEMO to max(Joint Entropy), min(Total Correlation), and max(Conditional Entropy) -Sturge, Scott and rounding binning methods | -Precipitation and streamflow networks were designed simultaneously. -Binning methods were compared and concluded that the optimal networks can be altered due to the binning methods |

Kornelsen and Coulibaly, 2015 [36] | Soil Moisture | Great Lakes Basin, Canada-USA | -DEMO to Max(Joint Entropy) min(Total Correlation) -SMOS satellite data | -Optimum networks were different for ascending and descending overpasses -Combining overpass data resulted in complimentary spatial distribution of stations |

Leach et al., 2015 [37] | Streamflow | Columbia River basin, BC, Canada. Southern Ontario, Canada | -DEMO to Max(Joint Entropy) min(Total Correlation) -Streamflow signatures -Indicators of hydrologic alteration (IHA) | -Found that including streamflow signatures as design objective increases network coverage in headwater areas. -Found including IHAs increases network coverage in downstream and urban areas. |

Leach et al., 2016 [38] | Groundwater level | Southern Ontario, Canada | -DEMO to Max(Joint Entropy) min(Total Correlation) -Annual recharge | -Found that considering spatial distribution of annual recharge can improve network coverage |

Lee, 2013 [39] | Water quality | Hagye Basin, South Korea | -Marginal entropy analogous cost function -Genetic algorithm | -Developed computationally efficient way to design a monitoring network in an ungauged basin |

Lee et al., 2014 [40] | Water quality | Sanganmi Basin, South Korea | -Multivariate transinformation -Genetic algorithm | -Developed method based on maximizing information content to design a water quality monitoring network in a sewer system |

Li et al., 2012 [41] | Streamflow/Water level | Brazos River basin, USA. Pijnacker, The Netherlands | -MIMR, max(Joint Entropy + Transinformation − Total Correlation) | -Developed maximum information minimum redundancy method (MIMR) -Found it to better at locating high information content stations for a monitoring network |

Mahjouri and Kerachian, 2011 [42] | Water quality | Jajrood River, Iran | -Information transfer index (ITI) distance and time curves -Micro genetic algorithm (MGA) | -The MGA was used to find the optimal combination of monitoring stations which minimize the temporal and spatial ITI -Found that the sampling frequency and number of stations could be increased in the monitoring network |

Mahmoudi-Meimand et al., 2016 [43] | Precipitation | Karkheh, Iran | -Transinformation entropy -Kriging error variance -Weighted cost function to select from Monte Carlo generated networks | -Consideration of spatial analysis error and transinformation entropy improved network design |

Masoumi and Kerachian, 2010 [44] | Groundwater quality | Tehran, Iran | -Transinformation-distance (T-D) curve -Transinformation-time (T-T) curve -C-mean clustering -Hybrid genetic algorithm (HGA) | -Developed different T-D curves based on homogeneous clusters of existing monitoring stations -Used HGA to find optimal network with maximum spatial coverage and minimum transinformation -Showed that sampling frequency could be optimized in the same way |

Memarzadehet al., 2013 [45] | Water quality | Karoon River, Iran | -Information transfer index (ITI) distance curve -Homogenous zone clustering -Dynamic factor analysis (DFA) | -Increased monitoring network without increasing redundant information |

Mishra and Coulibaly, 2010 [46] | Streamflow | Selected basins across Canada | -Transinformation index -Marginal, joint, and transinformation | -Used information theory to highlight critical areas across Canada in need of monitoring -Found that several watersheds are information deficient and would benefit from increased monitoring |

Mishra and Coulibaly, 2014 [47] | Streamflow | Selected basins across Canada | -Transinformation index -Seasonal streamflow information (SSI) | -Evaluated and highlighted the effects of seasonal climate on streamflow network design |

Mondal and Singh, 2012 [48] | Groundwater level | Kodaganar River basin, India | -Marginal entropy, joint entropy, transinformation -Information transfer index (ITI) | -Identified high priority monitoring stations using marginal entropy -ITI was used to evaluate monitoring network, showed that it could be reduced |

Samuel et al., 2013 [49] | Streamflow | St. John and St. Lawrence River basins, Canada | -Combined Regionalization-DEMO -Max(Joint Entropy) min(Total Correlation) | -Proposed combined regionalization dual entropy multi-objective optimization approach to design of minimum optimal network that meets World Meteorological Organization (WMO) guidelines -Found that the location of new monitoring stations added to a network depends on the current network density |

Santos et al., 2013 [50] | Precipitation | Portugal | -ANN sensitivity analysis -Mutual Information criteria -K-means with Euclidean distance | -Compared three clustering methods to reduce station density -Best method was case dependent -All subset networks reproduced spatial precipitation pattern |

Stosic et al., 2017 [51] | Streamflow | Brazos River, TX, USA | -Joint permutation entropy | -Used joint permutation entropy to account for ordering of time series data to better account for station information -Found that the most efficient measurement window was seven days when compared to daily and monthly |

Su and You, 2014 [52] | Precipitation | Shihmen Reservoir Taiwan | -Developed 2D transinformation-distance (T-D) model -T-D model used to interpolate network information | -Network designed by maximizing additional information provided by station given regionalized transinformation -Temporal scale has significant influence on information delivery |

Uddameri and Andruss, 2014 [53] | Groundwater level | Victoria County Groundwater Conservation District, TX, USA | -Marginal entropy -Monitoring priority index (MPI) | -Compared MPI found using kriging to MPI found using marginal entropy -Showed entropy derived MPI to be more conservative measure |

Wei et al., 2014 [54] | Precipitation | Taiwan University Experimental Forest, Taiwan | -Joint Entropy of hourly, monthly, dry/wet months and annual rainfall at 1, 3, 5 km grids | -Station priority changes at different spatiotemporal scales -Temporal scales have more significant changes on joint entropy values than spatial scales -Long time and short spatial scales require fewer stations for stable joint entropy |

Werstuck and Coulibaly, 2016 [55] | Streamflow | Ottawa River Basin, Canada | -Transinformation index -DEMO to Max(Joint Entropy) min(Total Correlation)-Streamflow signatures -Indicators of hydrologic alteration (IHA) | -Compared regionalized data from McMaster University-Hydrologiska Byråns Vattenbalansavdelnin (MAC-HBV) and Inverse Distance Weighting—Drainage Area Ratio (IDW-DAR) and found IDW-DAR to be more adequate for generating synthetic time series for potential monitoring stations -Critical areas highlighted by TI index method were the same areas where additional stations were added using DEMO method |

Werstuck and Coulibaly,2017 [56] | Streamflow | Ottawa River Basin, Canada | -Transinformation index -DEMO to max(Joint Entropy) min(Total Correlation) | -Transinformation index analysis is not significantly affected by scaling -Scaling effects are noticeable when DEMO method was applied |

Xu et al., 2015 [57] | Precipitation | Xiangjiang River Basin, China | -Mutual Information (MI) of rain gauges -Designed network by min(Σ[MI]), min(bias), max(NSE) -Resampled rainfall used in Xinanjiang and SWAT models | -Lumped model performance was stable with different Pareto optimal networks -Distributed model performance improves with number of stations |

Yakirevich et al., 2013 [58] | Groundwater quality | OPE3 research site, Maryland, USA | -Principle of minimum cross entropy (POMCE) -Hydrus-3D | -Using POMCE with two variants of Hydrus-3D, additional monitoring stations were added where the difference between the models was greatest |

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**MDPI and ACS Style**

Keum, J.; Kornelsen, K.C.; Leach, J.M.; Coulibaly, P. Entropy Applications to Water Monitoring Network Design: A Review. *Entropy* **2017**, *19*, 613.
https://doi.org/10.3390/e19110613

**AMA Style**

Keum J, Kornelsen KC, Leach JM, Coulibaly P. Entropy Applications to Water Monitoring Network Design: A Review. *Entropy*. 2017; 19(11):613.
https://doi.org/10.3390/e19110613

**Chicago/Turabian Style**

Keum, Jongho, Kurt C. Kornelsen, James M. Leach, and Paulin Coulibaly. 2017. "Entropy Applications to Water Monitoring Network Design: A Review" *Entropy* 19, no. 11: 613.
https://doi.org/10.3390/e19110613