Entropy Applications to Water Monitoring Network Design: A Review <s> Streamflow and Water Level Networks <s> Abstract This paper extends the original use of entropy by Amorocho and Espildora as a measure of uncertainty in hydrologic data and the reduction in that uncertainty due to application of a model. In many situations calculations of entropy are better based on a proportional rather than a fixed class interval. General equations are given for the proportional class interval, are solved for assumed log-normal and gamma distributions, and extended to data series with zero values. The solutions are shown to be independent of the units of measurement of the original data. The results are applied to comparative evaluation of hydrologic models using the same and different data sets, and a new criterion of model performance is suggested for the latter case. <s> BIB001 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Streamflow and Water Level Networks <s> Approaches to modeling the continuous hydrologic response of ungauged basins use observable physical characteristics of watersheds to either directly infer values for the parameters of hydrologic models, or to establish regression relationships between watershed structure and model parameters. Both these approaches still have widely discussed limitations, including impacts of model structural uncertainty. In this paper we introduce an alternative, model independent, approach to streamflow prediction in ungauged basins based on empirical evidence of relationships between watershed structure, climate and watershed response behavior. Instead of directly estimating values for model parameters, different hydrologic response behaviors of the watershed, quantified through model independent streamflow indices, are estimated and subsequently regionalized in an uncertainty framework. This results in expected ranges of streamflow indices in ungauged watersheds. A pilot study using 30 UK watersheds shows how this regionalized information can be used to constrain ensemble predictions of any model at ungauged sites. Dominant controlling characteristics were found to be climate (wetness index), watershed topography (slope), and hydrogeology. Main streamflow indices were high pulse count, runoff ratio, and the slope of the flow duration curve. This new approach provided sharp and reliable predictions of continuous streamflow at the ungauged sites tested. <s> BIB002 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Streamflow and Water Level Networks <s> In recent years, climate change impacts on water resources sectors have been extensively documented. Anticipated changes range from more severe storms to more frequent floods and drought at regional scale. Pressure on water resources and hence on the environment will probably increase with the need of quality data for sustainable design of water resources projects. Therefore, for an optimal network design, hydrometric networks are to be reviewed periodically based on the information needs of the various end users and the perspective for future water resources development. This paper evaluates hydrometric networks to identify essential streamflow stations and critical areas (highly poor network density) within major watersheds across the different provinces of Canada using entropy theory. The method is applied to daily streamflow data and information coefficients such as marginal entropy, joint entropy and transinformation index are used for the identification of important stations as well as critical areas in the basin. The analysis results show that almost all Canadian main watersheds contain deficient hydrometric networks. The most deficient streamflow networks are identified in Alberta (North Saskatchewan, Oldman, and Red Deer basins), Northern Ontario (Hudson Bay basin), and the Northwest Territories. The information might prove to be helpful for decision makers to undertake cost-benefit analyses for hydrometric network updating in each region. <s> BIB003 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Streamflow and Water Level Networks <s> [1] Data collection is a critical activity in the management of water systems because it supports informed decision making. Data are collected by means of monitoring networks in which water level gauges are of particular interest because of their implications for flood management. This paper introduces a number of modifications to previously published methods that use information theory to design hydrological monitoring networks in order to make the methods applicable to the design of water level monitors for highly controlled polder systems. The new contributions include the use of a hydrodynamic model for entropy analysis, the introduction of the quantization concept to filter out noisy time series, and the use of total correlation to evaluate the performance of three different pairwise dependence criteria. The resulting approach, water level monitoring design in polders (WMP), is applied to a polder in the Pijnacker region, Netherlands. Results show that relatively few monitors are adequate to collect the information of a polder area in spite of its large number of target water levels. It is found, in addition, that the directional information transfer DITYX is more effective in finding independent monitors, whereas DITXY is better for locating sets of monitors with high joint information content. WMP proves to be a suitable and simple method as part of the design of monitoring networks for polder systems. <s> BIB004 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Streamflow and Water Level Networks <s> Hydrologic similarity between catchments, derived from similarity in how catchments respond to precipitation input, is the basis for catchment classification, for transferability of information, for generalization of our hydrologic understanding and also for understanding the potential impacts of environmental change. An important question in this context is, how far can widely available hydrologic information (precipitation-temperature-streamflow data and generally available physical descriptors) be used to create a first order grouping of hydrologically similar catchments? We utilize a heterogeneous dataset of 280 catchments located in the Eastern US to understand hydrologic similarity in a 6-dimensional signature space across a region with strong environmental gradients. Signatures are defined as hydrologic response characteristics that provide insight into the hydrologic function of catchments. A Bayesian clustering scheme is used to separate the catchments into 9 homogeneous classes, which enable us to interpret hydrologic similarity with respect to similarity in climatic and landscape attributes across this region. We finally derive several hypotheses regarding controls on individual signatures from the analysis performed here. <s> BIB005 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Streamflow and Water Level Networks <s> As a key contribution to Canada's Ecosystem Status and Trends (ESTR) national assessment, the goal of our study was to utilize available flow data as a surrogate of habitat suitability for aquatic ecological communities, and examine temporal trends in hydroecological variables over the 1970–2005 period. Daily flow data were extracted from the Reference Hydrological Basin Network, and an agglomerative hierarchical classification method was used to identify homogenous regions with similar seasonality of the flow regime. Six regime regions were identified reflecting the timing of the annual peaks and low flows in addition to the patterns in the rising and falling limbs. For each of the gauging station sites, the magnitude, duration, timing, frequency, and rate of change of annual hydrological events were quantified through 32 ecologically important hydrological variables. Long-term patterns in the hydroecological variables were quantified using the nonparametric Mann-Kendall trend statistic. The results revealed more trends than would be expected to occur by chance for most variables. Clear regional trend patterns were observed within individual regime groups demonstrating the often differing response to environmental variability within the different regions. Results at the national scale were highly variable, but trends towards increased variability in river flows were observed with a predominant increasing trend in the number of flow reversals over the water year and decreasing trend in the annual low-flow indices. The identified river regime regions offer an initial framework for scientific investigation of hydroecological patterns and an opportunity to move towards a more predictive approach to environmental flows assessment in sustainable resource management and planning. Copyright © 2011 John Wiley & Sons, Ltd. <s> BIB006 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Streamflow and Water Level Networks <s> [1] Hydrometric information constitutes the fundamental input for planning, design, operation, and management of water resources systems. How to optimally site monitoring gauges such that they are effective and efficient in gathering the hydrometric information or data has received considerable attention. This paper presents a generic approach for the design (or evaluation) of hydrometric networks. First, an entropy theory-based criterion, named as maximum information minimum redundancy (MIMR), is proposed. The MIMR criterion maximizes the joint entropy of stations within the optimal set, and the transinformation between stations within and outside of the optimal set. Meanwhile, it insures that the optimal set contains minimum duplicated information. An easy-to-implement greedy ranking algorithm is developed to accomplish the MIMR selection. Two case studies are presented to illustrate the applicability of MIMR in hydrometric network evaluation and design. We also compare the MIMR selection with another entropy-based approach. Results illustrate that MIMR is apt at finding stations with high information content, and locating independent stations. The proposed approach is suitable for design (or evaluation) of any type of hydrometric network. <s> BIB007 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Streamflow and Water Level Networks <s> The acquisition of good hydrologic information is an important issue in water management since it is the basis of decisions concerning the allocation of water resources to different users. However, sufficient data are often not available to describe the behaviour of such systems, especially in developing countries, where monitoring networks are inappropriately designed, poorly operated or are inadequate. Therefore, it is of interest to design and evaluate efficient monitoring networks. This paper presents two methodologies to design discharge monitoring networks in rivers using the concepts of Information Theory. The first methodology considers the optimization of Information Theory quantities and the second considers a new method that is based on ranking Information Theory quantities with different possible monitor combinations. The methodologies are tested for the Magdalena River in Colombia, in which the existing monitoring network is also assessed. In addition, the use of monitors at tributaries is explored. It is demonstrated that the ranking method is a promising way of finding the extremes of Pareto fronts generated during multi-objective optimization processes and that better (more informative and less redundant) monitoring network configurations can be found for the Magdalena River. <s> BIB008 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Streamflow and Water Level 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> BIB009 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Streamflow and Water Level Networks <s> There has been a recent resurgence of interest in the application of Information Theory to problems of system identification in the Earth and Environmental Sciences. While the concept of entropy has found increased application, little attention has yet been given to the practical problems of estimating entropy when dealing with the unique characteristics of two commonly used kinds of hydrologic data: rainfall and runoff. In this paper, we discuss four important issues of practical relevance that can bias the computation of entropy if not properly handled. The first (zero effect) arises when precipitation and ephemeral streamflow data must be viewed as arising from a discrete-continuous hybrid distribution due to the occurrence of many zero values (e.g., days with no rain/no runoff). Second, in the widely used bin-counting method for estimation of PDF's, significant error can be introduced if the bin width is not carefully selected. The third (measurement effect) arises due to the fact that continuously varying hydrologic variables can typically only be observed discretely to some degree of precision. The Fourth (skewness effect) arises when the distribution of a variable is significantly skewed. Here we present an approach that can deal with all four of these issues, and test them with artificially generated and real hydrological data. The results indicate that the method is accurate and robust. <s> BIB010 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Streamflow and Water Level Networks <s> AbstractHydrologic extremes such as severe storms, floods, and droughts are inherently seasonal in nature and remain the main concern in designing hydrometric networks. In general, hydrometric networks have been designed without paying particular attention to the effect of seasonal streamflow information (SSI) at gauging stations on the efficiency of the hydrometric networks. This paper evaluates the effect of SSI on streamflow networks based on nonparametric implementation of entropy theory using the kernel density approach for estimating the mutual information between gauging stations on a seasonal basis. Overall, it is shown that the SSI of individual stations is season dependent and the efficiency of the streamflow network is also season dependent, therefore the effect of seasonality should be incorporated in future hydrometric network design. This methodology was applied at five river basins in Canada and its role for network design is discussed. <s> BIB011 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Streamflow and Water Level 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> BIB012 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Streamflow and Water Level Networks <s> Summary This study highlights the impacts that the selected hydrologic characteristics of a basin have on the spatial variability of hydrometric networks. The study was conducted using streamflow monitoring networks in two Canadian basins, specifically in the Hamilton, Halton, Credit Valley basins of Ontario and the Columbia River basin of British Columbia. The Dual Entropy-Multiobjective Optimization (DEMO) model which has been demonstrated to be sufficiently robust for designing optimum networks in a large basin was used in these analyses. In addition to the entropy functions, the spatial variability of flow networks was examined by either excluding or including hydrologic signatures and indicators of hydrologic alteration (IHA) of a basin. The entropy functions are used to identify optimal trade-offs between the maximum possible information content and the minimum shared information among stations. The hydrologic signatures are used to quantify hydrological response characteristics among sub-basins, and the IHAs, which are normally used to determine how a hydrologic regime has been altered by an impact, are instead used to quantify differences between sub-basins. It was found that the optimal locations for the new stations were well captured by the objective functions. Also, new stations covered a wider area when hydrological signatures and IHAs were considered, enhancing the objective functions. The inclusion of streamflow signatures increased emphasis on the headwaters whereas the inclusion of IHAs increased emphasis on the downstream and disturbed regions. Accounting for such spatial variability in designing hydrometric networks is crucial in obtaining an optimal network. <s> BIB013 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Streamflow and Water Level Networks <s> Abstract Using joint permutation entropy we address the issue of minimizing the cost of monitoring, while minimizing redundancy of the information content, of daily streamflow data recorded during the period 1989–2016 at twelve gauging stations on Brazos River, Texas, USA. While the conventional entropy measures take into account only the probability of occurrence of a given set of events, permutation entropy also takes into account local ordering of the sequential values, thus enriching the analysis. We find that the best cost efficiency is achieved by performing weekly measurements, in comparison with which daily measurements exhibit information redundancy, and monthly measurements imply information loss. We also find that the cumulative information redundancy of the twelve considered stations is over 10% for the observed period, and that the number of monitoring stations can be reduced by half bringing the cumulative redundancy level to less than 1%. <s> BIB014 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Streamflow and Water Level Networks <s> Adequate and accurate hydrologic information from optimal hydrometric networks is an essential part of effective water resources management. Although the key hydrologic processes in the water cycle are interconnected, hydrometric networks (e.g., streamflow, precipitation, groundwater level) have been routinely designed individually. A decision support framework is proposed for integrated design of multivariable hydrometric networks. The proposed method is applied to design optimal precipitation and streamflow networks simultaneously. The epsilon-dominance hierarchical Bayesian optimization algorithm was combined with Shannon entropy of information theory to design and evaluate hydrometric networks. Specifically, the joint entropy from the combined networks was maximized to provide the most information, and the total correlation was minimized to reduce redundant information. To further optimize the efficiency between the networks, they were designed by maximizing the conditional entropy of the streamflow network given the information of the precipitation network. Compared to the traditional individual variable design approach, the integrated multivariable design method was able to determine more efficient optimal networks by avoiding the redundant stations. Additionally, four quantization cases were compared to evaluate their effects on the entropy calculations and the determination of the optimal networks. The evaluation results indicate that the quantization methods should be selected after careful consideration for each design problem since the station rankings and the optimal networks can change accordingly. <s> BIB015 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Streamflow and Water Level Networks <s> Water resources managers commonly rely on information collected by hydrometric networks without clear knowledge of their efficiency. Optimal water monitoring networks are still scarce especially in the Canadian context. Herein, a dual entropy multi-objective optimization (DEMO) method uses information theory to identify locations where the addition of a hydrometric station would optimally complement the information content of an existing network. This research explores the utility of transinformation (TI) analysis, which can quantitatively measure the contribution of unique information from a hydrometric station. When used in conjunction, these methods provide an objective measure of network efficiency, and allow the user to make recommendations to improve existing hydrometric networks. A technique for identifying and dealing with regulated basins and their related bias on streamflow regionalization is also examined. The Ottawa River Basin, a large Canadian watershed with a number of regulated hydroelectric dams, was selected for the experiment. The TI analysis approach provides preliminary information which is supported by DEMO results. Regionalization was shown to be more accurate when the regulated basin stations were omitted using leave one out cross validation. DEMO analysis was performed with these improvements and successfully identified optimal locations for new hydrometric stations in the Ottawa River Basin. <s> BIB016
Water quantity monitoring, such as streamflow rates and water level, is one of the essential tasks for water management to prevent damage to nature and human beings from flooding. A successful floodplain management or flood forecasting and warning system can be feasible through expert forecasters who implement well-calibrated models and reliable tools using quality data  . The design of water quantity monitoring network has been well implemented because of not only the good performance of entropy-based methods, but also the unaffectedness by the zero effect, which is caused by discontinuity of probability density function due to zero values in data, except for the ephemeral or intermittent streams. To deal with the zero effect in entropy calculations, Chapman BIB001 and Gong et al. BIB010 separated the marginal entropy Equation (3) to nonzero terms and zero values, which are certain. While Gong et al. BIB010 summarized the possible issues in entropy calculations from hydrologic data as effects due to zero values, histogram binning including skewness consideration and measurement errors, some studies noticed that the length and the location of time window also affect entropy calculations and the corresponding network design. Fahle et al. BIB012 observed the temporal variability of station rankings by shifting the time window for the design of water level network of a ditch system in Germany. Mishra and Coulibaly BIB011 also found the dependency of the seasonality on the efficiency of hydrometric networks. Stosic et al. BIB014 found an inverse relationship between the network density and sampling time interval as the larger number of monitoring stations is required if the time interval is shorter and vice versa. Keum and Coulibaly  analyzed the temporal changes of entropy measures and optimal networks by applying daily time series for streamflow network design. They found that the information gain of a monitoring network is not significant when the length of time series is longer than 10 years, and the total correlation tends to stabilize within five years of data. The optimal networks using the data lengths of 5, 10, 15 and 20 years also show that there are no significant differences in the results from 10 years or longer while the optimal network using five years of data was evidently different from others. Werstuck and Coulibaly  analyzed scaling effects by considering two study areas. Specifically, one study area is a small watershed, which is a part of another study area. After applying the transinformation analysis and the multi-objective optimization, they concluded that the optimal networks tend to be affected by scaling while transinformation index does not. Mishra and Coulibaly BIB003 evaluated the effects of the class intervals and the infilling missing data by applying the linear regression method to daily time series and concluded that the station rankings based on the transinformation values were not significantly changed. Li et al. BIB007 also investigated the changes of station rankings based on the maximum information minimum redundancy (MIMR) approach and obtained the similar conclusion. However, Fahle et al. BIB012 and Keum and Coulibaly BIB015 drew the opposite opinion that station rankings can be affected by the binning method that defines the class intervals. The conflict comes from the selection of the binning methods compared. The former group applied different parameters to a single binning method, the mathematical floor function. However, the latter group compared other binning methods with the floor function. Considering that Alfonso et al. BIB004 found that the design solutions were not common in some cases from the sensitivity analysis of the parameter of the mathematical floor function, it is not recommended to use a specific binning method without any consideration. As discussed in the review of precipitation networks in Section 3.1, network redesign and network expansion require data at candidate locations, which are ungauged. Alfonso et al. BIB008 applied a one-dimensional hydrodynamic model to generate the discharge time series. The model estimated discharge at each segment, which divides rivers with approximately 200 m increments longitudinally. The use of hydrodynamic model enabled to determine the critical monitoring locations in the main stream and its tributaries. On the other hand, Samuel et al. BIB009 combined regionalization techniques with entropy calculation in order to estimate the discharge at candidate locations. They compared the performance of various regionalization methods including not only a conceptual hydrologic model, but also spatial proximity, physical similarity and their combinations with drainage area ratio. Based on the performance statistics by applying multiple basins, inverse distance weighting coupled with drainage area ratio performed the best, and this conclusion has been adopted in several studies  BIB015 BIB013 BIB016 . Some studies have extended the entropy applications for the streamflow monitoring network design. Stosic et al. BIB014 proposed the concept of permutation entropy, which is able to differentiate based on the order of sequential observations, as well as the histogram frequency in basic Shannon entropy measures. Even though histograms from two different observations are the same, the permutation entropy value tends to be higher if there are more variations between time steps. However, the network design studies using the permutation entropy are still limited. On the other hand, Leach et al. BIB013 applied additional features to the network design. While the common objectives in water monitoring network design using an optimization technique are to maximize the information and to minimize the redundancy in the network, they additionally considered the physical properties of watersheds, such as the streamflow signatures BIB002 BIB005 and the indicators of hydrologic alterations (IHAs)  BIB006 . After the comparison of the optimal streamflow monitoring networks with and without considerations of the streamflow signatures and IHAs, it was concluded that inclusion of basin physical characteristics yielded a better coverage of the selected locations of the optimal networks.