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Simple linear regression curve for maximum non-redundant information curve for summer and winter seasons (above) and annual data (below) on a semi-log scale. The equation and the coefficient of determination (R 2 ) of each regression curve are given in the box below each plot.
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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 essen...
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... both seasons and for the annual aggregate data, the latter value of the nontransferred information index is plot- ted against the corresponding sampling time interval on a semi-log scale (Fig. 5). The plot shows scale-invariance. The relation between the two variables is linear on the semi-log scale. It can, thus, be inferred that once this curve is known for a given network, the maximum value of nontransferred information can be obtained for the other sampling times. ...
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The aim of this study is to assess the influence of sensor locations and varying observation accuracy on the assimilation of distributed streamflow observations, also taking into account different structures of semi-distributed hydrological models. An ensemble Kalman filter is used to update a semi-distributed hydrological model as a response to me...
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... In many studies, the rain gauge station with the maximum information entropy is used as the first station. Among the remaining stations, the most important station is then determined based on a certain objective optimization rule (e.g., maximum information entropy, minimum transfer information, or minimum total correlation) [9,[41][42][43]. ...
Optimized rain gauge networks minimize their input and maintenance costs. Satellite precipitation observations are particularly susceptible to the effects of terrain elevation, vegetation, and other topographical factors, resulting in large deviations between satellite and ground-based precipitation data. Satellite precipitation observations are more inaccurate where the deviations change more drastically, indicating that rain gauge stations should be utilized at these locations. This study utilized satellite precipitation observation data to facilitate rain gauge network optimization. The deviations between ground-based precipitation data and three types of satellite precipitation observation data were used for entropy estimation. The rain gauge network in the Oujiang River Basin of China was optimally designed according to the principle of maximum joint entropy. Two optimization schemes of culling and supplementing 40 existing sites and 35 virtual sites were explored. First, the optimization and ranking of the rain gauge station network showed good stability and consistency. In addition, the joint entropy of deviation was larger than that of ground-based precipitation data alone, leading to a higher degree of discrimination between rain gauge stations and enabling the use of deviation data instead of ground-based precipitation data to assist network optimization, with more reasonable and interpretable results.
... The schematic of the procedure to calculate the two proposed scores.Masoumi and Kerachian (2010),Mogheir et al. (2003), andRidolfi et al. (2011Ridolfi et al. ( , 2016 M4 NC = Int( √ ) Hao and Singh (2013) and Montgomery and Runger (2014, p. 135) M5 The method based on maximum a posteriori estimation and Bayes' theorem (also called the Generalized Knuth method) Gencaga et al. (2015), Knuth (2013), and Tovo et al. (2016) Note. : the bin width; NC : the number of classes (bins); : the standard deviation of the distribution; : the number of available samples belonging to the distribution; Int(⋅) : the decimal integer function; The MATLAB code of M5 can be found in Knuth (2013). ...
The very nature of weather forecasts and verifications and the way they are used make it impossible for one single or absolute standard of evaluation. However, little research has been conducted on verifying deterministic multi‐category forecasts, which is based on the attribute of uncertainty. The authors propose a new approach using two mutual information theory‐based scores for assessing the comprehensive uncertainty of all categories and the uncertainty for a certain category in deterministic multi‐category precipitation forecasts, respectively. Specifically, the comprehensive uncertainty is defined as the average reduction in uncertainty about the observations resulting from the use of a predictive model to provide all categories forecasts; the uncertainty of a certain category is defined as the reduction in uncertainty about the observations resulting from the use of a predictive model to provide a certain category forecast. By applying the proposed approach and traditional verification methods, the four precipitation forecasting products from the China Meteorological Administration, European Centre for Medium‐Range Weather Forecasts, National Centers for Environmental Prediction, and United Kingdom Meteorological Office were verified in the Dahuofang Reservoir Drainage Basin, China. The results indicate that: (a) the proposed approach can better capture the changing patterns of uncertainties with lead times and distinguish the forecasting performance among different forecast products; (b) the proposed approach is resistant to the extreme bias; (c) the proposed approach needs a careful choice of bin width; and (d) the bias analysis is necessary before verifying the uncertainties in precipitation forecasts.
... The concept of Shannon's information entropy (1948) [16], which is very widely used in hydrology [2,9,[54][55][56][57][58][59][60][61][62][63], was also applied in the present work. ...
The purpose of this study was to obtain synergistic information and details in the time–frequency domain of the relationships between the Palmer drought indices in the upper and middle Danube River basin and the discharge (Q) in the lower basin. Four indices were considered: the Palmer drought severity index (PDSI), Palmer hydrological drought index (PHDI), weighted PDSI (WPLM) and Palmer Z-index (ZIND). These indices were quantified through the first principal component (PC1) analysis of empirical orthogonal function (EOF) decomposition, which was obtained from hydro-meteorological parameters at 15 stations located along the Danube River basin. The influences of these indices on the Danube discharge were tested, both simultaneously and with certain lags, via linear and nonlinear methods applying the elements of information theory. Linear connections were generally obtained for synchronous links in the same season, and nonlinear ones for the predictors considered with certain lags (in advance) compared to the discharge predictand. The redundancy–synergy index was also considered to eliminate redundant predictors. Few cases were obtained in which all four predictors could be considered together to establish a significant information base for the discharge evolution. In the fall season, nonstationarity was tested through wavelet analysis applied for the multivariate case, using partial wavelet coherence (pwc). The results differed, depending on the predictor kept in pwc, and on those excluded.
... Recently, information theories have been widely applied in hydrometeorology research to quantify the precipitation variability and complexity observed in recent decades (Ridolfi et al., 2011;Sang et al., 2016;Wang et al., 2018). Among these approaches, entropy theory can provide a primary measurement of the degree of variability, such as disorderliness, randomness, and irregularity (Mishra et al., 2009;Papalexiou and Koutsoyiannis, 2012;Hao and Singh, 2013). ...
Precipitation plays a fundamental role in terrestrial modelling and is extremely vulnerable to climate change. It is particularly challenging to describe precipitation due to its intermittency and fluctuation at nearly all temporal and spatial scales. Precipitation variability represents the degree of unevenness in its distribution or its lack of uniformity over different scales. This study examined the seasonal variability of meteorological station precipitation observations during 1961–2018 over mainland China using entropy theory. The intensity disorder index (IDI) and apportionment disorder index (ADI) were used to characterize the distribution unevenness, including the number of precipitation days, and precipitation amounts. Six special regions were selected according to their geographical location and climate conditions to show the regional differences of the disorder indices. At the daily scale, the regional mean IDI was 0.4081 for the eastern Tibetan Plateau and the regional mean ADI was 0.9984 for north China, which were the largest value among the six regions investigated. The regional mean IDI was 0.1128 and the regional mean ADI was 0.4071 both for the Yangtze River Basin, which were the smallest value among the six regions investigated. Temporal variations of the DIs were also detected. The IDI have decreased not significantly in most regions, except increasing by 0.0223/decade and 0.0129/decade in southwest China during nighttime and daytime, respectively. The ADI decreased by −0.0191, −0.0275, −0.0229, and −0.0203/decade and passed the significant test at .05 level for northwest, northeast, north China, and eastern Tibetan Plateau at the daily scale, respectively. The results quantify the precipitation unevenness over mainland China, which can be treated as a reference for water resources management and land surface model evaluation.
... Indeed, entropy in random processes is an uncertainty measure which indirectly reflects the temporal and spatial features of the process (Montesarchio and Napolitano 2010). This model is suitable for usage in the ecological contexts (Wei et al. 2018;Kumar and Stohlgren 2009;Phillips et al. 2006;Harte, 2011), urban geography and archaeological research (Bevan and Wilson 2013;Davies et al. 2014;Howey et al. 2016), rainfall-runoff modeling (Agrawal et al. 2005), design and evaluation of water quality monitoring networks (Jha and Singh, 2008), design of rain-gauge and hydrometric stations network (Ridolfi et al. 2011;Mishra and Coulibaly. 2010).This paper recommends applying MaxEnt as a tool for producing an urban flood hazard map (Phillips et al. 2010(Phillips et al. , 2018. ...
Rapid urban development, increasing impermeable surfaces, poor drainage system and changes in extreme precipitations are the most important factors that nowadays lead to increased urban flooding and it has become an urban problem. Urban flood mapping and its use in making an urban development plan can reduce flood damages and losses. Constantly producing urban flood hazard maps using models that rely on the availability of detailed hydraulic-hydrological data is a major challenge especially in developing countries. In this study, urban flood hazard map was produced with limited data using three machine learning models: Genetic Algorithm Rule-Set Production, Maximum Entropy (MaxEnt), Random Forest (RF) and Naïve Bayes for Kermanshah city, Iran. The flood hazard predicting factors used in modeling were: slope, land use, precipitation, distance to river, distance to channel, curve number (CN) and elevation. Flood inventory map was produced based on available reports and field surveys, that 117 flooded points and 163 non-flooded points were identified. Models performance was evaluated based on area under the receiver-operator characteristic curve (AUC-ROC), Kappa statistic and hits and miss analysis. The results show that RF model (AUC-ROC = 99.5%, Kappa = 98%, Accuracy = 90%, Success ratio = 99%, Threat score = 90% and Heidke skill score = 98%) performed better than other models. The results also showed that distance to canal, land use and CN have shown more contribution among others for modeling the flood and precipitation had the least effect among other factors. The findings show that machine learning methods can be a good alternative to distributed models to predict urban flood-prone areas where there are lack of detailed hydraulic and hydrological data.
... These studies apply informationtheoretical measures on multiple time series from a set of sensors, to identify optimal subsets. Jointly, these papers (Alfonso et al., 2010a, b;Li et al., 2012;Ridolfi et al., 2011;Samuel et al., 2013;Stosic et al., 2017;Keum and Coulibaly, 2017;Banik et al., 2017;Wang et al., 2018;Huang et al., 2020;Khorshidi et al., 2020) have proposed a wide variety of different optimization objectives. Some have suggested that either a multi-objective approach or a single objective derived from multiple objectives is necessary to find an optimal monitoring network. ...
... 10). These methods were chosen since they are highly cited methods in this field, and more importantly, recent new approaches in the literature have mostly been built on one of these methods with additional objectives (Alfonso et al., 2010a, b;Ridolfi et al., 2011;Li et al., 2012;Samuel et al., 2013;Stosic et al., 2017;Keum and Coulibaly, 2017;Banik et al., 2017;Wang et al., 2018;Huang et al., 2020). ...
This paper concerns the problem of optimal monitoring network layout using information-theoretical methods. Numerous different objectives
based on information measures have been proposed in recent literature,
often focusing simultaneously on maximum information and minimum
dependence between the chosen locations for data collection stations. We discuss
these objective functions and conclude that a single-objective optimization of joint entropy suffices to maximize the collection of information for a given number of stations. We argue that the widespread notion of minimizing redundancy, or dependence between monitored signals, as a secondary objective is not desirable and has no intrinsic justification. The negative effect of redundancy on total collected information is already accounted for in joint entropy, which measures total information net of any redundancies. In fact, for two networks of equal joint entropy, the one with a higher amount of redundant information should be preferred for reasons of robustness against failure.
In attaining the maximum joint entropy objective, we investigate exhaustive optimization, a more computationally tractable greedy approach that adds one station at a time, and we introduce the “greedy drop” approach, where the full set of stations is reduced one at a time. We show that no greedy approach can exist that is guaranteed to reach the global optimum.
... In fact, entropy in random processes is a measure of uncertainty that indirectly reflects the temporal and spatial characteristics of the process [35]. Maximum Entropy Model (MaxEnt) is often well applied in ecological contexts [36][37][38][39], urban geography and archaeological researches [40][41][42][43] and Runoff-Rainfall Modeling [44], design of water quality monitoring networks [45], locating rain gauges and hydrometric stations networks [46,47]. This paper recommends applying MaxEnt as a tool for producing an urban flood hazard map [22,48]. ...
Urban flood risk mapping plays a decisive role in urban management and planning, especially in reducing flood damages. In this study, a flood risk map was produced for Kermanshah city (Iran) by combining flood hazard and flood vulnerability maps. Based on effective factors of urban flooding, flood hazard maps were generated using two machine learning models: Maximum Entropy (MaxEnt) and Genetic Algorithm Rule-Set Production (GARP). These models were developed on 117 flood sites which were identified on the reports and field surveys for Kermanshah city, and 163 non-flooded points. Economic, social, and infrastructure criteria were considered to analyze flood vulnerability. The sub-criteria were defined based on social and cultural structure in developing countries and urban infrastructure facilities involved in the floods. Fuzzy Analytical Hierarchical Process method (FAHP) was applied to determine the overall weight vector. The results demonstrated that the MaxEnt model had a better performance than the GARP model based on two common indices which are the area under the receiver-operator characteristic curve (AUC-ROC = 96.76–98.32%) and kappa statistic (Kappa = 0.82-0.86). These findings showed that machine learning models provide reliable results for areas where data access is challenging, especially in developing countries. The results also indicated that infrastructure criterion has the highest impact weight on the vulnerability. In general, population, urban texture, and distance to the major drainage channels are the most important factors in increasing flood risk. Also, the vulnerability of an urban neighborhood greatly increases the risk of flooding.
... The concept of the Shannon (1948) information entropy was very widely used in hydrology (e.g. Harmancioglu and Yevjevich 1985, Singh 1997, Krasovskaia and Saelthun 1997, Schreiber 2000, Markus et al. 2003, Khan et al. 2006, Pechlivanidis et al. 2010, Weijs et al. 2010, Ridolfi et al. 2011, Wrzesiński 2013, Gong et al. 2013, Foroozand and Weijs 2017, Paluš 2018, Vu et al. 2018. ...
The aims of this study are to investigate the influence of large-scale atmospheric circulation quantified by indices such as the North Atlantic Oscillation index (NAOI), the Greenland-Balkan Oscillation index (GBOI) and blocking-type indices on the Lower Danube discharge. We separately analysed each season for the 1948–2000 period. In addition to the statistical linear procedure, we applied methods to quantify nonlinear connections between variables, as mutual information between predictors and predictand, using Shannon’s information entropy theory. The nonlinear correlation information between climate indices and discharge is higher than that obtained from the linear measure, providing more insight into real connections. Also, the non-stationarity of the link between variables is highlighted by spectral coherence based on wavelet analysis. For the physical interpretation, we analyse composite maps over the Atlantic-European region. The most significant influence on the discharge of the Lower Danube Basin is given by the GBOI and blocking-type atmospheric circulation over Europe.
... Rainfall networks provide basic data for the study of hydrological systems, and these data are fundamental for water resource planning and management. The question of designing an optimal rainfall network has been discussed since the 1960s ( Chacon-Hurtado et al., 2017 ), and there are different design methods, including geostatistical methods ( Pardo-Igúzquiza, 1998 ;Cheng et al., 2008 ;Shafiei et al., 2013 ), information theory-based methods ( Husain, 1989 ;Krstanovic and Singh, 1992 ;Yang and Burn, 1994 ;Ridolfi et al., 2011 ;Yeh et al., 2017 ), methods based on expert recommendations ( Bleasdale, 1965 ;Laize, 2004 ), and other methods such as value of information and network theory ( Alfonso and Price, 2012 ;Halverson and Fleming, 2015 ). The design of a network is often site-specific or case-specific and may need to consider a multitude of factors to address practical needs. ...
... proposed a multi-objective optimization approach for simultaneously maximizing information, minimizing redundancy of the hydrometric network, and maximizing conditional entropy that indicates information contained in a streamflow network that cannot be obtained from the precipitation network. Previous studies have focused on different types of monitoring networks, such as precipitation networks ( Ridolfi et al., 2011 ;Wei et al., 2014 ), streamflow and waterlevel networks ( Li et al., 2012 ;Fahle et al., 2015 ), soil moisture and groundwater networks ( Kornelsen and Coulibaly, 2015 ;Mondal and Singh, 2011 ), and water quality networks ( Lee, 2013 ). However, the integrated design of networks based on entropy had not received much attention until the work by was conducted. ...
... . In general, the multi-objective optimization problem does not yield a unique solution that simultaneously optimizes each objective ( Pareto, 1964 ). Instead, it is necessary to obtain a Pareto set of solutions that achieve a compromise between different objective functions. ...
Rainfall data are needed for water resource management and decision making and are obtained from rainfall networks. These data are especially important for streamflow simulations and forecasting the occurrence of intense rainfall during the flood season. Therefore, rainfall networks should be carefully designed and evaluated. Several methods are used for rainfall network design, and information theory-based methods have recently received significant attention. This study focuses on the design of a rainfall network, especially for streamflow simulation. A multi-objective design method is proposed and applied to the Wei River basin in China. We use the total correlation as an indicator of information redundancy and multivariate transinformation as an indicator of information transfer. Information redundancy refers to the overlap of information between rainfall stations, and information transfer refers to the rainfall-runoff relationship. The outlet streamflow station (Huaxian station in the Wei River basin) is used as the target station for the streamflow simulation. A non-dominated sorting genetic algorithm (NSGA‐II) was used for the multi-objective optimization of the rainfall network design. We compared the proposed multi-objective design with two other methods using an artificial neural network (ANN) model. The optimized rainfall network from the proposed method led to reasonable outlet streamflow forecasts with a balance between network efficiency and streamflow simulation. Our results indicate that the multi-objective strategy provides an effective design by which the rainfall network can consider the rainfall-runoff process and benefit streamflow prediction on a catchment scale.
... Entropy-based approaches have been widely used in the evaluation and https://doi.org/10.1016/j.envres.2019.108813 Received 12 September 2018; Received in revised form 4 October 2019; Accepted 7 October 2019 design of hydrometric networks, including rainfall gauge networks (Chebbi et al., 2011;Chen et al., 2008;Ridolfi et al., 2011;Su and You, 2014;Volkmann et al., 2010), streamflow gauge networks (Alfonso et al., 2012;Leach et al., 2015;Li et al., 2012;Mishra and Coulibaly, 2010;Mishra and Coulibaly, 2014;Samuel et al., 2013), water level monitoring networks (Alfonso et al., 2010(Alfonso et al., , 2014Leach et al., 2016), and water quality monitoring networks (Mogheir and Singh, 2002;Mogheir et al., 2009;Ozkul et al., 2000). Keum et al. (2017) summarized the common entropy terms and recent entropy applications to water monitoring network design. ...
Hydrometric information collected by monitoring networks is fundamental for effective management of water resources. In recent years, entropy-based multi-objective criterions have been developed for the evaluation and optimization of hydrometric networks, and copula functions have been frequently used in hydrological frequency analysis to model multivariate dependence structures. This study developed a dual entropy-transinformation criterion (DETC) to identify and prioritize significant stations and generate candidate network optimization solutions. The criterion integrated an entropy index computed with mathematical floor function and a transinformation index computed with copula entropy through a tradeoff weight. The best fitted copula models were selected from three Archimedean copula families, i.e., Gumbel, Frank and Clayton. DETC was applied to a streamflow monitoring network in the Fenhe River basin and two rainfall monitoring networks in the Beijing Municipality and the Taihu Lake basin, which covers different network classification, network scale, and climate type. DETC was assessed by the commonly used dual entropy-multiobjective optimization (DEMO) criterion and was compared with a minimum transinformation (MinT) based criterion for network optimization. Results showed that DETC could effectively prioritize stations according to their significance and incorporate decision preference on information content and information redundancy. Comparison of the isohyet maps of two rainstorm events between DETC and MinT showed that DETC had advantage of restoring the spatial distribution of precipitation.
