Entropy Applications to Water Monitoring Network Design: A Review <s> Conclusions and Recommendations <s> [1] Ecohydrological systems may be characterized as nonlinear, complex, open dissipative systems. Such systems consist of many coupled processes, and the couplings change depending on the system state or scale in space and time at which the system is analyzed. The arrangement of couplings in a complex system may be represented as a network of information flow and feedback between variables that measure system processes. The occurrence of feedback on such a network provides sufficient conditions for self-organized and nonlinear behaviors to emerge. We adapt an information-theoretic statistical method called transfer entropy for the purposes of robustly measuring the directionality, relative strength, statistical significance, and time scale of information flow between pairs of ecohydrological variables using time series data. A process network may be delineated where variables are cast as nodes and information flows as weighted directional links between them. The process network captures key couplings and time scales and represents the state of the complex system as a whole, including functional groups of variables (subsystems) and synchronization resulting from feedbacks. It is therefore able to identify interactions which are not detectable using methods which examine the system using one relationship at a time. We assemble an information flow process network using July 2003 FLUXNET data for a Midwestern corn-soybean ecohydrological system in a healthy, peak growing season state and compare the results with those using July 2005 data for the same site during a severe drought. We find that the process network during drought is substantially decoupled, and regional-scale information feedback is reduced during the drought. We conclude that the proposed process network methodology is able to identify the differences between two states of an ecohydrological system on the basis of variations in the pattern of feedback coupling on the network. <s> BIB001 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Conclusions and Recommendations <s> [1] Ecohydrological systems are complex, open dissipative systems characterized by couplings and feedback between subsystems at many scales of space and time. The information flow process network approach is developed to analyze such systems, using time series data to delineate the feedback, time scales, and subsystems that define the complex system's organization. Network statistics are used to measure the statistical feedback, entropy, and net and gross information production of subsystems on the network to study monthly process networks for a Midwestern corn-soybean ecosystem for the years 1998–2006. Several distinct system states are identified and characterized. Particularly interesting is the midsummer state that is dominated by regional-scale information feedback and by information flow originating from the ecosystem's photosynthetic activity. In this state, information flows both “top-down” from synoptic weather systems and “bottom-up” from the plant photosynthetic activity. A threshold in air temperature separates this summer state where increased organization appears from other system states. The relationship between Shannon entropy and information flow is investigated. It is found that information generally flows from high-entropy variables to low-entropy variables, and moderate-entropy variables participate in feedback. <s> BIB002 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Conclusions and Recommendations <s> [1] Hydrometric network design for surface water monitoring is employed to address a wide range of environmental and water resources problems. Historical overview of hydrometric network design is provided along with a discussion on new developments and challenges in the design of optimal hydrometric networks. This review starts with precise examples of decline in hydrometric network density, then highlights the increasing requirement of optimal network design in a context of climate and land use changes. An extensive survey of methodological development in hydrometric network design is provided along with discussion on the issue of uncertainty in hydrometric network design and the evolution in data collection techniques and technology. Finally, some conclusions are drawn on the future of hydrometric network design. <s> BIB003 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Conclusions and Recommendations <s> Summary Information theory metrics are applied to ascertain the spatial and temporal variability of daily precipitation over the continental United States. Entropy is computed of the daily precipitation records and of the precipitation event size distribution. In addition, relative entropy is also computed between the two records and the Hurst exponent is calculated for comparison purposes. These values show little correlation with mean annual precipitation, number of rain days or station elevation. However, there is a clear spatial pattern in the precipitation and relative entropy fields corresponding to a breakpoint at approximately −95° longitude, which the Hurst exponent is not capable of resolving. Using wavelet multiresolution analysis and relative entropy, this breakpoint is shown to correspond to higher temporal variability (sub-monthly) in the western United States. In addition, scalewise mutual information content illustrates the general feature of three distinct scaling regions for precipitation: (1) synoptic (2–10 days), (2) monthly to annual times and (3) interannual variability. This analysis demonstrates the usefulness of an information theory approach to assessing the dynamics of precipitation. <s> BIB004 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Conclusions and Recommendations <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> BIB005 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Conclusions and Recommendations <s> Because of high variation in mountainous areas, rainfall data at different spatiotemporal scales may yield potential uncertainty for network design. However, few studies focus on the scaling effect on both the spatial and the temporal scale. By calculating the maximum joint entropy of hourly typhoon events, monthly, six dry and wet months and annual rainfall between 1992 and 2012 for 1-, 3-, and 5-km grids, the relocated candidate rain gauges in the National Taiwan University Experimental Forest of Central Taiwan are prioritized. The results show: (1) the network exhibits different locations for first prioritized candidate rain gauges for different spatiotemporal scales; (2) the effect of spatial scales is insignificant compared to temporal scales; and (3) a smaller number and a lower percentage of required stations (PRS) reach stable joint entropy for a long duration at finer spatial scale. Prioritized candidate rain gauges provide key reference points for adjusting the network to capture more accurate information and minimize redundancy. <s> BIB006 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Conclusions and Recommendations <s> Summary Rain gauge networks are used to provide estimates of area average, spatial variability and point rainfalls at catchment scale and provide the most important input for hydrological models. Therefore, it is desired to design the optimal rain gauge networks with a minimal number of rain gauges to provide reliable data with both areal mean values and spatial–temporal variability. Based on a dense rain gauge network of 185 rain gauges in Xiangjiang River Basin, southern China, this study used an entropy theory based multi-criteria method which simultaneously considers the information derived from rainfall series, minimize the bias of areal mean rainfall as well as minimize the information overlapped by different gauges to resample the rain gauge networks with different gauge densities. The optimal networks were examined using two hydrological models: The lumped Xinanjiang Model and the distributed SWAT Model. The results indicate that the performances of the lumped model using different optimal networks are stable while the performances of the distributed model keep on improving as the number of rain gauges increases. The results reveal that the entropy theory based multi-criteria strategy provides an optimal design of rain gauge network which is of vital importance in regional hydrological study and water resources management. <s> BIB007 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Conclusions and Recommendations <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> Conclusions and Recommendations <s> Sensors and sensor networks play an important role in decision-making related to water quality, operational streamflow forecasting, flood early warning systems, and other areas. In this paper we review a number of existing applications and analyse a variety of evaluation and design procedures for sensor networks with respect to various criteria. Most of the existing approaches focus on maximising the observability and information content of a variable of interest. From the context of hydrological modelling only a few studies use the performance of the hydrological simulation in terms of output discharge as a design criterion. In addition to the review, we propose a framework for classifying the existing design methods, and a generalised procedure for an optimal network design in the context of rainfall-runoff hydrological modelling. <s> BIB009 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Conclusions and Recommendations <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> BIB010 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Conclusions and Recommendations <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> BIB011
It is evident that successful water management cannot be achieved without proper water monitoring networks. Although there has been much progress in network design methods and applications, a standardized design methodology has not yet emerged. After the pioneering invention of information theory in the 1940s, entropy concepts have been applied in various applications with recent efforts on network design problems. The unique benefit of this approach is that a water monitoring network can be evaluated or designed based on the information the network monitors, which is in contrast to the set station densities proposed by WMO guidelines; the advantage of the former being that a network could be better tailored to specific applications or optimized to provide the most gain at densities lower than those suggested in WMO guidelines. In addition, when combined with multi-objective optimization techniques, users' specific criteria can be included in the optimal network design process. This manuscript provides a comprehensive review of the recent research attainments and their applications in entropy-based water monitoring network design. The literature has demonstrated the use of various information theory measures and adaptations thereof for use in network design with an emerging consensus that the goal of these network design methods is to select the stations that provide the most information to the monitoring network while simultaneously being independent of each other. Through rigorous testing, information theory has proven to be a robust tool to use when evaluating and designing an optimal water monitoring network. However, when it comes to evaluating the optimal design, there are still issues that need to be addressed. The first is that an optimal monitoring network design can be found based on specified design criteria; however, the practical application of the new optimal monitoring network is rarely evaluated in a hydrologic or other model BIB009 BIB007 . This type of numerical experiment is an important requirement to evaluate the utility of a network rather than just identifying its optimality or information content. Further, it is an important exercise to identify the benefits of entropy-based network designs in order to convince decision makers of the importance of adopting entropy approaches. Another issue with the optimal network is that it can be subjective, based on choices made in the calculation of entropy and the design method chosen, especially when additional objective functions are considered in the design. This extends to the method selected for finding the optimal monitoring network, whether it is found using an iterative method where one station is added at a time or a collection of stations is added all at once. Research has also shown that data length, catchment scale and ordering can influence the design of an optimal network BIB008  BIB010 . Finally, when using discrete entropy, the binning method has been shown to influence the final network design BIB011 . The influence of binning on entropy calculation has received greater attention in other geophysical network design applications BIB001 BIB002  , and similar consideration should be given in the field of water resources, particularly owing to the unique and difficult nature of water variables (e.g., streamflow, precipitation) spatial and temporal distribution BIB004 BIB005 . Thus, explicit consideration is needed when choosing the bins based on the intended application of the monitoring network and further research to provide guidance specific to water monitoring networks. Therefore, despite the possibility of finding an optimal network design in a formal sense, the subjectivity induced by the designer's choices, and the lack of standardized design methods, must be recognized. Future research should focus on comparative studies among multiple entropy design methods, discretization approaches and data characteristics. The current literature provides many novel entropy design approaches and the evolution of concepts, but rigorous comparisons are critical to provide generic guidelines for network design. Despite the potential sources of subjectivity identified, entropy methods remain one of the most objective approaches for network design. In particular, more work is needed on spatial and temporal scaling of data for entropy calculation to provide robust guidance to decision makers. Many new methods and optimization techniques have been reviewed herein, but few examples were found in the literature that explored the data characteristics used in those techniques. Further research is required to provide guidance on the proper length of data in water monitoring network design  , the sampling frequency of the data BIB006 and the spatial scale at which information should be measured for various monitoring network applications. The aforementioned issues are considered crucial gaps that need to be filled to enable practical recommendations or guidelines for a widespread adoption of entropy approaches for designing optimal water monitoring networks. In addition, the comparative studies of entropy-based methods reviewed herein should be robustly compared to network design methods from other disciplines, such as geostatistics, to identify areas of equivalence and disparity BIB003 . Considerable advances have occurred over the past decade as reviewed herein, and measures derived from Shannon's base equation  have reached a high level of maturity for the task of network design. We challenge the research community to put a similar creativity into the joint consideration of the nexus of data characteristics, network design and applications, all of which are intricately linked.