Entropy Applications to Water Monitoring Network Design: A Review <s> Precipitation Networks <s> Entropy expressions for several continuous multivariate distributions are derived. Point estimation of entropy for the multinormal distribution and for the distribution of order statistics from D.G. Weinman's (Ph.D dissertation, Ariz. State Univ., Tempe, AZ, 1966) exponential distribution is considered. The asymptotic distribution of the uniformly minimum variance unbiased estimator for multinormal entropy is obtained. Simulation results on convergence of the means and variances of these estimators are provided. > <s> BIB001 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Precipitation Networks <s> Abstract The Xinanjiang model is a rainfall-runoff, distributed, basin model for use in humid and semi-humid regions. The evapotranspiration component is represented by a model of three soil layers. Runoff production occurs on repletion of storage to capacity values which are assumed to be distributed throughout the basin. Prior to 1980, runoff was separated into surface and groundwater components using Horton's concept of infiltration. Subsequently, the concept of hillslope hydrology was introduced with an additional component, interflow, being identified. Runoff concentration to the outflow of each sub-basin is represented by a unit hydrograph or by a lag and route technique. The damping or routing effects of the channel system connecting the sub-basins are represented by Muskingum routing. There are fifteen parameters in all, of which the model is particularly sensitive to six. Optimization of the parameters is achieved with different objective functions according to the nature of each parameter. The model has been widely used in China since 1980, mainly for flood forecasting, though more recently it is also being used for other purposes. <s> BIB002 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Precipitation Networks <s> A conceptual, continuous time model called SWAT (Soil and Water Assessment Tool) was developed to assist water resource managers in assessing the impact of management on water supplies and nonpoint source pollution in watersheds and large river basins. The model is currently being utilized in several large area projects by EPA, NOAA, NRCS and others to estimate the off-site impacts of climate and management on water use, nonpoint source loadings, and pesticide contamination. Model development, operation, limitations, and assumptions are discussed and components of the model are described. In Part II, a GIS input/output interface is presented along with model validation on three basins within the Upper Trinity basin in Texas. <s> BIB003 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Precipitation Networks <s> Summary Employing the entropy concept spatial and temporal variability of precipitation time series were investigated for the State of Texas, USA. Marginal entropy was used to investigate the variability associated with monthly, seasonal and annual time series. Also, apportionment entropy and intensity entropy were used for investigating the intra-annual and decadal distributions of monthly and annual precipitation amounts and numbers of rainy days within a year and decade respectively. Finally, the Hurst exponent and the Mann–Kendall test were used to evaluate the long-term persistence and trend in the variability of precipitation. Distinct spatial patterns in annual series and different seasons were observed. The variability of precipitation amount as well as number of rainy days within a year increased from east to west of Texas. The results also indicated that highly disorderliness in the amount of precipitation and number of rainy days caused severe droughts during the 1950’s in whole of Texas. <s> BIB004 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Precipitation Networks <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> BIB005 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Precipitation Networks <s> Abstract. Hydrological models are the basis of operational flood-forecasting systems. The accuracy of these models is strongly dependent on the quality and quantity of the input information represented by rainfall height. Finer space-time rainfall resolution results in more accurate hazard forecasting. In this framework, an optimum raingauge network is essential in predicting flood events. This paper develops an entropy-based approach to evaluate the maximum information content achievable by a rainfall network for different sampling time intervals. The procedure is based on the determination of the coefficients of transferred and nontransferred information and on the relative isoinformation contours. The nontransferred information value achieved by the whole network is strictly dependent on the sampling time intervals considered. An empirical curve is defined, to assess the objective of the research: the nontransferred information value is plotted versus the associated sampling time on a semi-log scale. The curve has a linear trend. In this paper, the methodology is applied to the high-density raingauge network of the urban area of Rome. <s> BIB006 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Precipitation Networks <s> For monitoring hydrological events characterized by high spatial and temporal variability, the number and location of recording stations must be carefully selected to ensure that the necessary information is collected. Depending on the characteristics of each natural process, certain stations may be spurious or redundant, whereas others may provide most of the relevant data. With the objective of reducing the costs of the monitoring system and, at the same time, improving its operational effectiveness, three procedures were applied to identify the minimum network of rain gauge stations able to capture the characteristics of droughts in mainland Portugal. Drought severity is characterized by the standardized precipitation index applied to the timescales of 1, 3, 6 and 12 consecutive months. The three techniques used to reduce the dimensionality of the network of rain gauges were as follows: (i) artificial neural networks with sensitivity analysis, (ii) application of the mutual information criterion and (iii) K-means cluster analysis using Euclidean distances. The results demonstrated that the best dimensionality reduction method was case dependent in the three regions of Portugal (northern, central and southern) previously identified by cluster analysis. All the reduction techniques lead to the selection of a subset of rain gauges capable of reproducing the original temporal patterns of drought. For specific severe drought events in Portugal in the past, the comparison between drought spatial patterns obtained with the original stations and the selected subset indicated that the subset produced statistically satisfactory results (correlation coefficients higher than 0.6 and efficiency coefficients higher than 0.5). Copyright © 2012 John Wiley & Sons, Ltd. <s> BIB007 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Precipitation 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> BIB008 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Precipitation Networks <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> BIB009 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Precipitation Networks <s> Summary This study proposed a spatial information estimation model for the analysis of precipitation gauge networks, to improve on previous methods based on information theory. The proposed model employs a two-dimensional transinformation–distance ( T – D ) relationship in conjunction with multivariate information approximation to estimate transinformation to ungauged locations from existing stations, while taking into consideration the influence of multiple stations and anisotropy. The proposed model is used to evaluate the spatial distribution of precipitation data and the characteristics of information transfer, which are then applied in a spatial optimization algorithm for the selection of additional station locations. This framework was implemented to investigate temporal and spatial patterns in information content in the Shihmen Reservoir watershed. The results demonstrate obvious anisotropy associated with the delivery of information. By comparing different cases, it was determined that the efficiency of information delivery dominates the spatial distribution of the information content, such that eccentricity is merely supplemental. Efficiency in information delivery is also heavily influenced by temporal scale. For data covering long intervals (monthly and annual), efficiency in the delivery of information is relatively high, while the uncertainty or heterogeneity of hourly or daily time series produces low spatial correlations due to the inefficient delivery of information. The proposed spatial optimization algorithm confirmed that the optimal location for new stations lies close to the center of low information zones. Additional stations could improve information content considerably; however, the margin of improvement decreases with the number of stations. <s> BIB010 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Precipitation Networks <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> BIB011 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Precipitation Networks <s> There is a high level of spatio-temporal variability in rainfall events over a region. Rain gauges provide point measurements that are interpolated to estimate the rainfall in other points of a region or in order to estimate the regional depth of rainfall. Due to limited numbers of rain gauges that can be deployed, it is important to plan for the optimal number and locations of rain gauges in a region. Because of several factors that can affect the optimal number of stations as well as their locations, development of optimisation models for determining the number of rain gauges and their locations within a region is necessary. In this study, an optimisation model for determining the number and location of the rain gauges is proposed. The objectives of the model are maximising the minimum transformation entropy and minimising the rainfall estimation error. To determine the optimal number of new rain gauges, the candidate points, which are those with the maximum variance of Kriging error and the minimum inf... <s> BIB012 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Precipitation Networks <s> ABSTRACT Using the Shannon entropy, the space–time variability of rainfall and streamflow was assessed for daily rainfall and streamflow data for a 10-year period from 189 stations in the northeastern region of Brazil. Mean values of marginal entropy were computed for all observation stations and entropy maps were then constructed for delineating annual and seasonal characteristics of rainfall and streamflow. The Mann-Kendall test was used to evaluate the long-term trend in marginal entropy as well as relative entropy for two sample stations. The marginal entropy values of rainfall and streamflow were higher for locations and periods with the highest amounts of rainfall. The entropy values were higher where rainfall was higher. This was because the probability distributions of rainfall and the resulting streamflow were more uniform and less skewed. The Shannon entropy produced spatial patterns which led to a better understanding of rainfall and streamflow characteristics throughout the northeastern region of Brazil. The total relative entropy indicated that rainfall and streamflow carried the same information content at annual and rainy season time scales. <s> BIB013 </s> Entropy Applications to Water Monitoring Network Design: A Review <s> Precipitation 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> BIB014
The design of a representative precipitation monitoring network is an important and still challenging task for which an entropy approach is well suited. High quality precipitation information is necessary for streamflow and flood forecasting, surface water management, agricultural management, climate process understanding and many other applications. However, precipitation is well known to be highly variable in both space and time BIB004 and often statistically represented by highly skewed distributions BIB008 making the application of parametric analysis methods difficult. These challenges also extend to entropy-based approaches for precipitation monitoring. For example, the marginal entropy has been found to be well correlated with total precipitation in northern Brazil because the probability distribution in regions with higher rainfall tended to be more uniform and less skewed BIB013 . In contrast, Mishra et al. BIB004 found that the marginal disorder index (MDI), which is the ratio of observed entropy to the maximum possible entropy at a given site, was inversely related to mean annual rainfall in the U.S. state of Texas, where MDI was found to vary seasonally. Brunsell BIB005 studied the entropy from monitoring stations across the United States where little correlation was found between precipitation and marginal entropy with the exception of a breakpoint in entropy at −95 • longitude corresponding to high temporal variability precipitation patterns. It has also been noted by several studies that the temporal sampling of precipitation is an important consideration for calculating entropy and for designing precipitation networks BIB009 BIB004 . At finer timescales (hourly to daily), precipitation is highly variable resulting in higher overall entropy, whereas longer time periods (monthly to annual) have less variability resulting in lower marginal entropy BIB005 BIB010 BIB004 . The dependence on spatial and temporal scales has also been identified in a network design application. Wei et al. BIB009 prioritized potential stations in Central Taiwan to maximize the joint entropy of the network at hourly, monthly and annual temporal scales, as well as 1-, 3-and 5-km spatial scales. They found that priority stations changed with both spatial and temporal scales, where changes in temporal scales resulted in more significant changes in station priority than spatial-rescaling. The decrease in entropy at longer timescales also had an impact on station density where fewer stations were required to reach a stable joint entropy value for longer time scales BIB009 . These findings demonstrate the important first consideration of network objectives when determining the spatial and temporal sampling used to calculate entropy. However, the research on this topic is still limited, and more work is needed to provide robust guidance on sampling strategies. Several approaches have been proposed to design or redesign a precipitation monitoring network using one or more entropic measures. Many of these approaches are initialized by building a network around a central station usually selected as the station with the highest marginal entropy BIB012 BIB006   . In urban Rome, Ridolfi et al. BIB006 selected stations for the precipitation network by sequentially finding the next station that minimized the conditional entropy of the network and adding that station to the network. A similar approach was taken by Yeh et al.  to expand a precipitation network in Taiwan. Hourly rainfall data were normalized with a Box-Cox transform and kriging used to interpolate rainfall to candidate grid cells. The joint entropy of the network was calculated using an analytic equation for joint entropy valid for normal data BIB001 , and stations were added sequentially that had the lowest conditional entropy with the rest of the network. The final number of stations needed by the network was accepted when 95% of the network information was captured  . Awadallah  applied multiple entropy measures sequentially to add stations to a precipitation network. The first new stations were selected as those with the highest entropy. The second station was chosen to minimize the mutual information and the third as the station that maximized conditional entropy. The aforementioned approaches all sequentially add single stations to a monitoring network based on a single criterion. Mahmoudi-Meimand et al. BIB012 presented a methodology to add stations to a network based on a multi-variate cost function. Precipitation data were spatially interpolated from existing stations using the kriging approach where the kriging error associated with the rainfall estimation is calculated as the kriging error variance. Their method selected the station that maximized transinformation entropy and minimized error variance using a weighted average of both measures as an objective BIB012 . This approach balanced the information content in the network with the errors in the interpolation method. Xu et al. BIB011 used a multi-objective approach to simultaneously select a subset of stations that minimized the sum of pairwise mutual information, minimized bias and maximized Nash-Sutcliffe efficiency. Solutions were generated via Monte Carlo sampling, and network solutions falling along the Pareto front were found as compromise solutions. Coulibaly and Keum  and Samuel and Coulibaly  also used a multi-objective approach to add stations to snow monitoring networks in Canada. Their approach used a genetic algorithm to find networks that maximized the joint entropy and minimized the total correlation of the network to form a Pareto front of optimal network designs, some of which also included network cost in the optimization BIB014  . A challenge to an entropy-based approach to adding stations to a precipitation network is the requirement to have data available for candidate points. For precipitation, this can be challenging because data at shorter time scales in particular are well known to be non-normal. Most studies use the kriging approach for interpolation BIB012   and address the need for normally distributed data using a Box-Cox transform. Samuel and Coulibaly  addressed the interpolation problem by using the external data from the Snow Data Assimilation System (SNODAS) for candidate stations. Su and You BIB010 presented a unique approach to adding stations that maximized the information content of the network. In most literature cases, entropic measures at ungauged sites are determined by interpolating observations of precipitation across a watershed. Su and You BIB010 calculated the transinformation between neighbouring stations to develop a 2D transinformation-distance relationship. In contrast to transferring data to ungauged stations, this approach transferred transinformation to ungauged stations and selected a site with the maximum transinformation. This approach should be further tested and contrasted with the data transfer approach. As previously stated, precipitation data are of critical importance for a variety of applications. Despite this, few studies have explored the impact of precipitation networks designed with an entropy approach for actual water resource applications. Applications found in the literature have taken the reasonable approach of using entropy to reduce network density for comparison to a network that included all stations. In Portugal, Santos et al. BIB007 compared artificial neural networks, K-means clustering and mutual information (MI) criteria for reducing the density of a precipitation network for drought monitoring at different time scales. They found the best performing reduction method was case dependent depending on the region and time scale applied, but noted that all methods performed well. They also found that all subset networks could reliably reproduce the spatial precipitation pattern. Xu et al. BIB011 used the multi-objective approach previously described to select a subset of precipitation stations from a dense network in the Xiangjiang River Basin in China. Rainfall from the subset networks was used to force the lumped Xinanjiang hydrological model BIB002 and the distributed SWAT hydrological model BIB003 . The author's found that lumped model performance became stable with a subset of 20 to 25 stations, whereas the distributed model's performance continued to increase as more stations were added to the network BIB011 . These analyses are important to demonstrate the utility of precipitation networks and the advantages of entropy-based approaches in designing precipitation networks.