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Open AccessEditorial

Entropy Applications in Environmental and Water Engineering

by

Huijuan Cui

,

Bellie Sivakumar

and

Vijay P. Singh

Entropy 2018, 20(8), 598; https://doi.org/10.3390/e20080598 - 10 Aug 2018

Cited by 5 | Viewed by 2922

Abstract

n/a Full article

(This article belongs to the Special Issue Entropy Applications in Environmental and Water Engineering)

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Open AccessArticle

Cross Mean Annual Runoff Pseudo-Elasticity of Entropy for Quaternary Catchments of the Upper Vaal Catchment in South Africa

by

Masengo Ilunga

Entropy 2018, 20(4), 281; https://doi.org/10.3390/e20040281 - 13 Apr 2018

Cited by 3 | Viewed by 4388

Abstract

This study focuses preliminarily on the intra-tertiary catchment (TC) assessment of cross MAR pseudo-elasticity of entropy, which determines the impact of changes in MAR for a quaternary catchment (QC) on the entropy of another (other) QC(s). The TCs of the Upper Vaal catchment [...] Read more.

This study focuses preliminarily on the intra-tertiary catchment (TC) assessment of cross MAR pseudo-elasticity of entropy, which determines the impact of changes in MAR for a quaternary catchment (QC) on the entropy of another (other) QC(s). The TCs of the Upper Vaal catchment were used preliminarily for this assessment and surface water resources (WR) of South Africa of 1990 (WR90), of 2005 (WR2005) and of 2012 (WR2012) data sets were used. The TCs are grouped into three secondary catchments, i.e., downstream of Vaal Dam, upstrream of Vaal dam and Wilge. It is revealed that, there are linkages in terms of mean annual runoff (MAR) between QCs; which could be complements (negative cross elasticity) or substitutes (positive cross elasticity). It is shown that cross MAR pseudo-elasticity can be translated into correlation strength between QC pairs; i.e., high cross elasticity (low catchment resilience) and low cross elasticity (high catchment resilience). Implicitly, catchment resilience is shown to be associated with the risk of vulnerability (or sustainability level) of water resources, in terms of MAR, which is generally low (or high). Besides, for each TC, the dominance (of complements or substitutes) and the global highest cross MAR elasticity are determined. The overall average cross MAR elasticity of QCs for each TC was shown to be in the zone of tolerable entropy, hence the zone of functioning resilience. This could assure that water resources remained fairly sustainable in TCs that form the secondary catchments of the Upper Vaal. Cross MAR pseudo-elasticity concept could be further extended to an intra-secondary catchment assessment. Full article

(This article belongs to the Special Issue Entropy Applications in Environmental and Water Engineering)

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Open AccessArticle

Application of Entropy Ensemble Filter in Neural Network Forecasts of Tropical Pacific Sea Surface Temperatures

by

Hossein Foroozand

,

Valentina Radić

and

Steven V. Weijs

Entropy 2018, 20(3), 207; https://doi.org/10.3390/e20030207 - 20 Mar 2018

Cited by 8 | Viewed by 4026

Abstract

Recently, the Entropy Ensemble Filter (EEF) method was proposed to mitigate the computational cost of the Bootstrap AGGregatING (bagging) method. This method uses the most informative training data sets in the model ensemble rather than all ensemble members created by the conventional bagging. [...] Read more.

Recently, the Entropy Ensemble Filter (EEF) method was proposed to mitigate the computational cost of the Bootstrap AGGregatING (bagging) method. This method uses the most informative training data sets in the model ensemble rather than all ensemble members created by the conventional bagging. In this study, we evaluate, for the first time, the application of the EEF method in Neural Network (NN) modeling of El Nino-southern oscillation. Specifically, we forecast the first five principal components (PCs) of sea surface temperature monthly anomaly fields over tropical Pacific, at different lead times (from 3 to 15 months, with a three-month increment) for the period 1979–2017. We apply the EEF method in a multiple-linear regression (MLR) model and two NN models, one using Bayesian regularization and one Levenberg-Marquardt algorithm for training, and evaluate their performance and computational efficiency relative to the same models with conventional bagging. All models perform equally well at the lead time of 3 and 6 months, while at higher lead times, the MLR model’s skill deteriorates faster than the nonlinear models. The neural network models with both bagging methods produce equally successful forecasts with the same computational efficiency. It remains to be shown whether this finding is sensitive to the dataset size. Full article

(This article belongs to the Special Issue Entropy Applications in Environmental and Water Engineering)

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Open AccessArticle

Bayesian Technique for the Selection of Probability Distributions for Frequency Analyses of Hydrometeorological Extremes

by

Lu Chen

,

Vijay P. Singh

and

Kangdi Huang

Entropy 2018, 20(2), 117; https://doi.org/10.3390/e20020117 - 11 Feb 2018

Cited by 24 | Viewed by 4064

Abstract

Frequency analysis of hydrometeorological extremes plays an important role in the design of hydraulic structures. A multitude of distributions have been employed for hydrological frequency analysis, and more than one distribution is often found to be adequate for frequency analysis. The current method [...] Read more.

Frequency analysis of hydrometeorological extremes plays an important role in the design of hydraulic structures. A multitude of distributions have been employed for hydrological frequency analysis, and more than one distribution is often found to be adequate for frequency analysis. The current method for selecting the best fitted distributions are not so objective. Using different kinds of constraints, entropy theory was employed in this study to derive five generalized distributions for frequency analysis. These distributions are the generalized gamma (GG) distribution, generalized beta distribution of the second kind (GB2), Halphen type A distribution (Hal-A), Halphen type B distribution (Hal-B), and Halphen type inverse B (Hal-IB) distribution. The Bayesian technique was employed to objectively select the optimal distribution. The method of selection was tested using simulation as well as using extreme daily and hourly rainfall data from the Mississippi. The results showed that the Bayesian technique was able to select the best fitted distribution, thus providing a new way for model selection for frequency analysis of hydrometeorological extremes. Full article

(This article belongs to the Special Issue Entropy Applications in Environmental and Water Engineering)

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Open AccessFeature PaperArticle

Scaling-Laws of Flow Entropy with Topological Metrics of Water Distribution Networks

by

Giovanni Francesco Santonastaso

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Entropy 2018, 20(2), 95; https://doi.org/10.3390/e20020095 - 30 Jan 2018

Cited by 16 | Viewed by 4190

Abstract

Robustness of water distribution networks is related to their connectivity and topological structure, which also affect their reliability. Flow entropy, based on Shannon’s informational entropy, has been proposed as a measure of network redundancy and adopted as a proxy of reliability in optimal [...] Read more.

Robustness of water distribution networks is related to their connectivity and topological structure, which also affect their reliability. Flow entropy, based on Shannon’s informational entropy, has been proposed as a measure of network redundancy and adopted as a proxy of reliability in optimal network design procedures. In this paper, the scaling properties of flow entropy of water distribution networks with their size and other topological metrics are studied. To such aim, flow entropy, maximum flow entropy, link density and average path length have been evaluated for a set of 22 networks, both real and synthetic, with different size and topology. The obtained results led to identify suitable scaling laws of flow entropy and maximum flow entropy with water distribution network size, in the form of power–laws. The obtained relationships allow comparing the flow entropy of water distribution networks with different size, and provide an easy tool to define the maximum achievable entropy of a specific water distribution network. An example of application of the obtained relationships to the design of a water distribution network is provided, showing how, with a constrained multi-objective optimization procedure, a tradeoff between network cost and robustness is easily identified. Full article

(This article belongs to the Special Issue Entropy Applications in Environmental and Water Engineering)

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Open AccessEditor’s ChoiceArticle

Information Entropy Suggests Stronger Nonlinear Associations between Hydro-Meteorological Variables and ENSO

by

Tue M. Vu

,

Ashok K. Mishra

and

Goutam Konapala

Entropy 2018, 20(1), 38; https://doi.org/10.3390/e20010038 - 09 Jan 2018

Cited by 14 | Viewed by 5443

Abstract

Understanding the teleconnections between hydro-meteorological data and the El Niño–Southern Oscillation cycle (ENSO) is an important step towards developing flood early warning systems. In this study, the concept of mutual information (MI) was applied using marginal and joint information entropy to [...] Read more.

Understanding the teleconnections between hydro-meteorological data and the El Niño–Southern Oscillation cycle (ENSO) is an important step towards developing flood early warning systems. In this study, the concept of mutual information (MI) was applied using marginal and joint information entropy to quantify the linear and non-linear relationship between annual streamflow, extreme precipitation indices over Mekong river basin, and ENSO. We primarily used Pearson correlation as a linear association metric for comparison with mutual information. The analysis was performed at four hydro-meteorological stations located on the mainstream Mekong river basin. It was observed that the nonlinear correlation information is comparatively higher between the large-scale climate index and local hydro-meteorology data in comparison to the traditional linear correlation information. The spatial analysis was carried out using all the grid points in the river basin, which suggests a spatial dependence structure between precipitation extremes and ENSO. Overall, this study suggests that mutual information approach can further detect more meaningful connections between large-scale climate indices and hydro-meteorological variables at different spatio-temporal scales. Application of nonlinear mutual information metric can be an efficient tool to better understand hydro-climatic variables dynamics resulting in improved climate-informed adaptation strategies. Full article

(This article belongs to the Special Issue Entropy Applications in Environmental and Water Engineering)

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Open AccessArticle

Combined Forecasting of Rainfall Based on Fuzzy Clustering and Cross Entropy

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Entropy 2017, 19(12), 694; https://doi.org/10.3390/e19120694 - 19 Dec 2017

Cited by 5 | Viewed by 4055

Abstract

Rainfall is an essential index to measure drought, and it is dependent upon various parameters including geographical environment, air temperature and pressure. The nonlinear nature of climatic variables leads to problems such as poor accuracy and instability in traditional forecasting methods. In this [...] Read more.

Rainfall is an essential index to measure drought, and it is dependent upon various parameters including geographical environment, air temperature and pressure. The nonlinear nature of climatic variables leads to problems such as poor accuracy and instability in traditional forecasting methods. In this paper, the combined forecasting method based on data mining technology and cross entropy is proposed to forecast the rainfall with full consideration of the time-effectiveness of historical data. In view of the flaws of the fuzzy clustering method which is easy to fall into local optimal solution and low speed of operation, the ant colony algorithm is adopted to overcome these shortcomings and, as a result, refine the model. The method for determining weights is also improved by using the cross entropy. Besides, the forecast is conducted by analyzing the weighted average rainfall based on Thiessen polygon in the Beijing–Tianjin–Hebei region. Since the predictive errors are calculated, the results show that improved ant colony fuzzy clustering can effectively select historical data and enhance the accuracy of prediction so that the damage caused by extreme weather events like droughts and floods can be greatly lessened and even kept at bay. Full article

(This article belongs to the Special Issue Entropy Applications in Environmental and Water Engineering)

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Open AccessArticle

Testing the Beta-Lognormal Model in Amazonian Rainfall Fields Using the Generalized Space q-Entropy

by

Hernán D. Salas

,

Germán Poveda

and

Oscar J. Mesa

Entropy 2017, 19(12), 685; https://doi.org/10.3390/e19120685 - 13 Dec 2017

Cited by 4 | Viewed by 3807

Abstract

We study spatial scaling and complexity properties of Amazonian radar rainfall fields using the Beta-Lognormal Model (BL-Model) with the aim to characterize and model the process at a broad range of spatial scales. The Generalized Space q-Entropy Function (GSEF), an entropic measure [...] Read more.

We study spatial scaling and complexity properties of Amazonian radar rainfall fields using the Beta-Lognormal Model (BL-Model) with the aim to characterize and model the process at a broad range of spatial scales. The Generalized Space q-Entropy Function (GSEF), an entropic measure defined as a continuous set of power laws covering a broad range of spatial scales,

S_{q} (λ) \sim λ^{Ω (q)}

, is used as a tool to check the ability of the BL-Model to represent observed 2-D radar rainfall fields. In addition, we evaluate the effect of the amount of zeros, the variability of rainfall intensity, the number of bins used to estimate the probability mass function, and the record length on the GSFE estimation. Our results show that: (i) the BL-Model adequately represents the scaling properties of the q-entropy,

S_{q}

, for Amazonian rainfall fields across a range of spatial scales

λ

from 2 km to 64 km; (ii) the q-entropy in rainfall fields can be characterized by a non-additivity value,

q_{s a t}

, at which rainfall reaches a maximum scaling exponent,

Ω_{s a t}

; (iii) the maximum scaling exponent

Ω_{s a t}

is directly related to the amount of zeros in rainfall fields and is not sensitive to either the number of bins to estimate the probability mass function or the variability of rainfall intensity; and (iv) for small-samples, the GSEF of rainfall fields may incur in considerable bias. Finally, for synthetic 2-D rainfall fields from the BL-Model, we look for a connection between intermittency using a metric based on generalized Hurst exponents,

M (q_{1}, q_{2})

, and the non-extensive order (q-order) of a system,

Θ_{q}

, which relates to the GSEF. Our results do not exhibit evidence of such relationship. Full article

(This article belongs to the Special Issue Entropy Applications in Environmental and Water Engineering)

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Open AccessArticle

Entropy-Based Investigation on the Precipitation Variability over the Hexi Corridor in China

by

Liang Cheng

,

Jun Niu

and

Dehai Liao

Entropy 2017, 19(12), 660; https://doi.org/10.3390/e19120660 - 01 Dec 2017

Cited by 13 | Viewed by 4051

Abstract

The spatial and temporal variability of precipitation time series were investigated for the Hexi Corridor, in Northwest China, by analyzing the entropy information. The examinations were performed on monthly, seasonal, and annual timescales based on 29 meteorological stations for the period of 1961–2015. [...] Read more.

The spatial and temporal variability of precipitation time series were investigated for the Hexi Corridor, in Northwest China, by analyzing the entropy information. The examinations were performed on monthly, seasonal, and annual timescales based on 29 meteorological stations for the period of 1961–2015. The apportionment entropy and intensity entropy were used to analyze the regional precipitation characteristics, including the intra-annual and decadal distribution of monthly and annual precipitation amounts, as well as the number of precipitation days within a year and a decade. The regions with high precipitation variability are found in the western part of the Hexi corridor and with less precipitation, and may have a high possibility of drought occurrence. The variability of the number of precipitation days decreased from the west to the east of the corridor. Higher variability, in terms of both of precipitation amount and intensity during crop-growing season, has been found in the recent decade. In addition, the correlation between entropy-based precipitation variability and the crop yield is also compared, and the crop yield in historical periods is found to be correlated with the precipitation intensity disorder index in the middle reaches of the Hexi corridor. Full article

(This article belongs to the Special Issue Entropy Applications in Environmental and Water Engineering)

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Open AccessArticle

Entropy Parameter M in Modeling a Flow Duration Curve

by

Yu Zhang

,

Vijay P. Singh

and

Aaron R. Byrd

Entropy 2017, 19(12), 654; https://doi.org/10.3390/e19120654 - 01 Dec 2017

Cited by 5 | Viewed by 4765

Abstract

A flow duration curve (FDC) is widely used for predicting water supply, hydropower, environmental flow, sediment load, and pollutant load. Among different methods of constructing an FDC, the entropy-based method, developed recently, is appealing because of its several desirable characteristics, such as simplicity, [...] Read more.

A flow duration curve (FDC) is widely used for predicting water supply, hydropower, environmental flow, sediment load, and pollutant load. Among different methods of constructing an FDC, the entropy-based method, developed recently, is appealing because of its several desirable characteristics, such as simplicity, flexibility, and statistical basis. This method contains a parameter, called entropy parameter M, which constitutes the basis for constructing the FDC. Since M is related to the ratio of the average streamflow to the maximum streamflow which, in turn, is related to the drainage area, it may be possible to determine M a priori and construct an FDC for ungauged basins. This paper, therefore, analyzed the characteristics of M in both space and time using streamflow data from 73 gauging stations in the Brazos River basin, Texas, USA. Results showed that the M values were impacted by reservoir operation and possibly climate change. The values were fluctuating, but relatively stable, after the operation of the reservoirs. Parameter M was found to change inversely with the ratio of average streamflow to the maximum streamflow. When there was an extreme event, there occurred a jump in the M value. Further, spatially, M had a larger value if the drainage area was small. Full article

(This article belongs to the Special Issue Entropy Applications in Environmental and Water Engineering)

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Open AccessArticle

An Extension to the Revised Approach in the Assessment of Informational Entropy

by

Turkay Baran

,

Nilgun B. Harmancioglu

,

Cem Polat Cetinkaya

and

Filiz Barbaros

Entropy 2017, 19(12), 634; https://doi.org/10.3390/e19120634 - 29 Nov 2017

Cited by 8 | Viewed by 3264

Abstract

This study attempts to extend the prevailing definition of informational entropy, where entropy relates to the amount of reduction of uncertainty or, indirectly, to the amount of information gained through measurements of a random variable. The approach adopted herein describes informational entropy not [...] Read more.

This study attempts to extend the prevailing definition of informational entropy, where entropy relates to the amount of reduction of uncertainty or, indirectly, to the amount of information gained through measurements of a random variable. The approach adopted herein describes informational entropy not as an absolute measure of information, but as a measure of the variation of information. This makes it possible to obtain a single value for informational entropy, instead of several values that vary with the selection of the discretizing interval, when discrete probabilities of hydrological events are estimated through relative class frequencies and discretizing intervals. Furthermore, the present work introduces confidence limits for the informational entropy function, which facilitates a comparison between the uncertainties of various hydrological processes with different scales of magnitude and different probability structures. The work addresses hydrologists and environmental engineers more than it does mathematicians and statisticians. In particular, it is intended to help solve information-related problems in hydrological monitoring design and assessment. This paper first considers the selection of probability distributions of best fit to hydrological data, using generated synthetic time series. Next, it attempts to assess hydrometric monitoring duration in a netwrok, this time using observed runoff data series. In both applications, it focuses, basically, on the theoretical background for the extended definition of informational entropy. The methodology is shown to give valid results in each case. Full article

(This article belongs to the Special Issue Entropy Applications in Environmental and Water Engineering)

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Open AccessArticle

Maximum Entropy-Copula Method for Hydrological Risk Analysis under Uncertainty: A Case Study on the Loess Plateau, China

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Entropy 2017, 19(11), 609; https://doi.org/10.3390/e19110609 - 15 Nov 2017

Cited by 15 | Viewed by 5332

Abstract

Copula functions have been extensively used to describe the joint behaviors of extreme hydrological events and to analyze hydrological risk. Advanced marginal distribution inference, for example, the maximum entropy theory, is particularly beneficial for improving the performance of the copulas. The goal of [...] Read more.

Copula functions have been extensively used to describe the joint behaviors of extreme hydrological events and to analyze hydrological risk. Advanced marginal distribution inference, for example, the maximum entropy theory, is particularly beneficial for improving the performance of the copulas. The goal of this paper, therefore, is twofold; first, to develop a coupled maximum entropy-copula method for hydrological risk analysis through deriving the bivariate return periods, risk, reliability and bivariate design events; and second, to reveal the impact of marginal distribution selection uncertainty and sampling uncertainty on bivariate design event identification. Particularly, the uncertainties involved in the second goal have not yet received significant consideration. The designed framework for hydrological risk analysis related to flood and extreme precipitation events is exemplarily applied in two catchments of the Loess plateau, China. Results show that (1) distribution derived by the maximum entropy principle outperforms the conventional distributions for the probabilistic modeling of flood and extreme precipitation events; (2) the bivariate return periods, risk, reliability and bivariate design events are able to be derived using the coupled entropy-copula method; (3) uncertainty analysis highlights the fact that appropriate performance of marginal distribution is closely related to bivariate design event identification. Most importantly, sampling uncertainty causes the confidence regions of bivariate design events with return periods of 30 years to be very large, overlapping with the values of flood and extreme precipitation, which have return periods of 10 and 50 years, respectively. The large confidence regions of bivariate design events greatly challenge its application in practical engineering design. Full article

(This article belongs to the Special Issue Entropy Applications in Environmental and Water Engineering)

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Open AccessArticle

Comparison of Two Entropy Spectral Analysis Methods for Streamflow Forecasting in Northwest China

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Entropy 2017, 19(11), 597; https://doi.org/10.3390/e19110597 - 07 Nov 2017

Cited by 5 | Viewed by 3686

Abstract

Monthly streamflow has elements of stochasticity, seasonality, and periodicity. Spectral analysis and time series analysis can, respectively, be employed to characterize the periodical pattern and the stochastic pattern. Both Burg entropy spectral analysis (BESA) and configurational entropy spectral analysis (CESA) combine spectral analysis [...] Read more.

Monthly streamflow has elements of stochasticity, seasonality, and periodicity. Spectral analysis and time series analysis can, respectively, be employed to characterize the periodical pattern and the stochastic pattern. Both Burg entropy spectral analysis (BESA) and configurational entropy spectral analysis (CESA) combine spectral analysis and time series analysis. This study compared the predictive performances of BESA and CESA for monthly streamflow forecasting in six basins in Northwest China. Four criteria were selected to evaluate the performances of these two entropy spectral analyses: relative error (RE), root mean square error (RMSE), coefficient of determination (R2), and Nash–Sutcliffe efficiency coefficient (NSE). It was found that in Northwest China, both BESA and CESA forecasted monthly streamflow well with strong correlation. The forecast accuracy of BESA is higher than CESA. For the streamflow with weak correlation, the conclusion is the opposite. Full article

(This article belongs to the Special Issue Entropy Applications in Environmental and Water Engineering)

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Open AccessArticle

Spatial Optimization of Agricultural Land Use Based on Cross-Entropy Method

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Entropy 2017, 19(11), 592; https://doi.org/10.3390/e19110592 - 07 Nov 2017

Cited by 17 | Viewed by 4720

Abstract

An integrated optimization model was developed for the spatial distribution of agricultural crops in order to efficiently utilize agricultural water and land resources simultaneously. The model is based on the spatial distribution of crop suitability, spatial distribution of population density, and agricultural land [...] Read more.

An integrated optimization model was developed for the spatial distribution of agricultural crops in order to efficiently utilize agricultural water and land resources simultaneously. The model is based on the spatial distribution of crop suitability, spatial distribution of population density, and agricultural land use data. Multi-source remote sensing data are combined with constraints of optimal crop area, which are obtained from agricultural cropping pattern optimization model. Using the middle reaches of the Heihe River basin as an example, the spatial distribution of maize and wheat were optimized by minimizing cross-entropy between crop distribution probabilities and desired but unknown distribution probabilities. Results showed that the area of maize should increase and the area of wheat should decrease in the study area compared with the situation in 2013. The comprehensive suitable area distribution of maize is approximately in accordance with the distribution in the present situation; however, the comprehensive suitable area distribution of wheat is not consistent with the distribution in the present situation. Through optimization, the high proportion of maize and wheat area was more concentrated than before. The maize area with more than 80% allocation concentrates on the south of the study area, and the wheat area with more than 30% allocation concentrates on the central part of the study area. The outcome of this study provides a scientific basis for farmers to select crops that are suitable in a particular area. Full article

(This article belongs to the Special Issue Entropy Applications in Environmental and Water Engineering)

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Open AccessArticle

A Connection Entropy Approach to Water Resources Vulnerability Analysis in a Changing Environment

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Entropy 2017, 19(11), 591; https://doi.org/10.3390/e19110591 - 06 Nov 2017

Cited by 16 | Viewed by 3900

Abstract

This paper establishes a water resources vulnerability framework based on sensitivity, natural resilience and artificial adaptation, through the analyses of the four states of the water system and its accompanying transformation processes. Furthermore, it proposes an analysis method for water resources vulnerability based [...] Read more.

This paper establishes a water resources vulnerability framework based on sensitivity, natural resilience and artificial adaptation, through the analyses of the four states of the water system and its accompanying transformation processes. Furthermore, it proposes an analysis method for water resources vulnerability based on connection entropy, which extends the concept of contact entropy. An example is given of the water resources vulnerability in Anhui Province of China, which analysis illustrates that, overall, vulnerability levels fluctuated and showed apparent improvement trends from 2001 to 2015. Some suggestions are also provided for the improvement of the level of water resources vulnerability in Anhui Province, considering the viewpoint of the vulnerability index. Full article

(This article belongs to the Special Issue Entropy Applications in Environmental and Water Engineering)

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Open AccessFeature PaperArticle

Entropy Production in Stochastics

by

Demetris Koutsoyiannis

Entropy 2017, 19(11), 581; https://doi.org/10.3390/e19110581 - 30 Oct 2017

Cited by 18 | Viewed by 3931

Abstract

While the modern definition of entropy is genuinely probabilistic, in entropy production the classical thermodynamic definition, as in heat transfer, is typically used. Here we explore the concept of entropy production within stochastics and, particularly, two forms of entropy production in logarithmic time, [...] Read more.

While the modern definition of entropy is genuinely probabilistic, in entropy production the classical thermodynamic definition, as in heat transfer, is typically used. Here we explore the concept of entropy production within stochastics and, particularly, two forms of entropy production in logarithmic time, unconditionally (EPLT) or conditionally on the past and present having been observed (CEPLT). We study the theoretical properties of both forms, in general and in application to a broad set of stochastic processes. A main question investigated, related to model identification and fitting from data, is how to estimate the entropy production from a time series. It turns out that there is a link of the EPLT with the climacogram, and of the CEPLT with two additional tools introduced here, namely the differenced climacogram and the climacospectrum. In particular, EPLT and CEPLT are related to slopes of log-log plots of these tools, with the asymptotic slopes at the tails being most important as they justify the emergence of scaling laws of second-order characteristics of stochastic processes. As a real-world application, we use an extraordinary long time series of turbulent velocity and show how a parsimonious stochastic model can be identified and fitted using the tools developed. Full article

(This article belongs to the Special Issue Entropy Applications in Environmental and Water Engineering)

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Open AccessArticle

Forewarning Model of Regional Water Resources Carrying Capacity Based on Combination Weights and Entropy Principles

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Entropy 2017, 19(11), 574; https://doi.org/10.3390/e19110574 - 25 Oct 2017

Cited by 32 | Viewed by 3940

Abstract

As a new development form for evaluating the regional water resources carrying capacity, forewarning regional water resources of their carrying capacities is an important adjustment and control measure for regional water security management. Up to now, most research on this issue have been [...] Read more.

As a new development form for evaluating the regional water resources carrying capacity, forewarning regional water resources of their carrying capacities is an important adjustment and control measure for regional water security management. Up to now, most research on this issue have been qualitative analyses, with a lack of quantitative research. For this reason, an index system for forewarning regional water resources of their carrying capacities and grade standards, has been established in Anhui Province, China, in this paper. Subjective weights of forewarning indices can be calculated using a fuzzy analytic hierarchy process, based on an accelerating genetic algorithm, while objective weights of forewarning indices can be calculated by using a projection pursuit method, based on an accelerating genetic algorithm. These two kinds of weights can be combined into combination weights of forewarning indices, by using the minimum relative information entropy principle. Furthermore, a forewarning model of regional water resources carrying capacity, based on entropy combination weight, is put forward. The model can fully integrate subjective and objective information in the process of forewarning. The results show that the calculation results of the model are reasonable and the method has high adaptability. Therefore, this model is worth studying and popularizing. Full article

(This article belongs to the Special Issue Entropy Applications in Environmental and Water Engineering)

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Open AccessArticle

Rainfall Network Optimization Using Radar and Entropy

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Entropy 2017, 19(10), 553; https://doi.org/10.3390/e19100553 - 19 Oct 2017

Cited by 20 | Viewed by 4788

Abstract

In this study, a method combining radar and entropy was proposed to design a rainfall network. Owing to the shortage of rain gauges in mountain areas, weather radars are used to measure rainfall over catchments. The major advantage of radar is that it [...] Read more.

In this study, a method combining radar and entropy was proposed to design a rainfall network. Owing to the shortage of rain gauges in mountain areas, weather radars are used to measure rainfall over catchments. The major advantage of radar is that it is possible to observe rainfall widely in a short time. However, the rainfall data obtained by radar do not necessarily correspond to that observed by ground-based rain gauges. The in-situ rainfall data from telemetering rain gauges were used to calibrate a radar system. Therefore, the rainfall intensity; as well as its distribution over the catchment can be obtained using radar. Once the rainfall data of past years at the desired locations over the catchment were generated, the entropy based on probability was applied to optimize the rainfall network. This method is applicable in remote and mountain areas. Its most important utility is to construct an optimal rainfall network in an ungauged catchment. The design of a rainfall network in the catchment of the Feitsui Reservoir was used to illustrate the various steps as well as the reliability of the method. Full article

(This article belongs to the Special Issue Entropy Applications in Environmental and Water Engineering)

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Open AccessArticle

Randomness Representation of Turbulence in Canopy Flows Using Kolmogorov Complexity Measures

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Entropy 2017, 19(10), 519; https://doi.org/10.3390/e19100519 - 27 Sep 2017

Cited by 20 | Viewed by 4573

Abstract

Turbulence is often expressed in terms of either irregular or random fluid flows, without quantification. In this paper, a methodology to evaluate the randomness of the turbulence using measures based on the Kolmogorov complexity (KC) is proposed. This methodology is applied to experimental [...] Read more.

Turbulence is often expressed in terms of either irregular or random fluid flows, without quantification. In this paper, a methodology to evaluate the randomness of the turbulence using measures based on the Kolmogorov complexity (KC) is proposed. This methodology is applied to experimental data from a turbulent flow developing in a laboratory channel with canopy of three different densities. The methodology is even compared with the traditional approach based on classical turbulence statistics. Full article

(This article belongs to the Special Issue Entropy Applications in Environmental and Water Engineering)

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Open AccessArticle

Modeling NDVI Using Joint Entropy Method Considering Hydro-Meteorological Driving Factors in the Middle Reaches of Hei River Basin

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Entropy 2017, 19(9), 502; https://doi.org/10.3390/e19090502 - 15 Sep 2017

Cited by 11 | Viewed by 4340

Abstract

Terrestrial vegetation dynamics are closely influenced by both hydrological process and climate change. This study investigated the relationships between vegetation pattern and hydro-meteorological elements. The joint entropy method was employed to evaluate the dependence between the normalized difference vegetation index (NDVI) and coupled [...] Read more.

Terrestrial vegetation dynamics are closely influenced by both hydrological process and climate change. This study investigated the relationships between vegetation pattern and hydro-meteorological elements. The joint entropy method was employed to evaluate the dependence between the normalized difference vegetation index (NDVI) and coupled variables in the middle reaches of the Hei River basin. Based on the spatial distribution of mutual information, the whole study area was divided into five sub-regions. In each sub-region, nested statistical models were applied to model the NDVI on the grid and regional scales, respectively. Results showed that the annual average NDVI increased at a rate of 0.005/a over the past 11 years. In the desert regions, the NDVI increased significantly with an increase in precipitation and temperature, and a high accuracy of retrieving NDVI model was obtained by coupling precipitation and temperature, especially in sub-region I. In the oasis regions, groundwater was also an important factor driving vegetation growth, and the rise of the groundwater level contributed to the growth of vegetation. However, the relationship was weaker in artificial oasis regions (sub-region III and sub-region V) due to the influence of human activities such as irrigation. The overall correlation coefficient between the observed NDVI and modeled NDVI was observed to be 0.97. The outcomes of this study are suitable for ecosystem monitoring, especially in the realm of climate change. Further studies are necessary and should consider more factors, such as runoff and irrigation. Full article

(This article belongs to the Special Issue Entropy Applications in Environmental and Water Engineering)

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Open AccessArticle

Spatio-Temporal Variability of Soil Water Content under Different Crop Covers in Irrigation Districts of Northwest China

by

Sufen Wang

and

Vijay P. Singh

Entropy 2017, 19(8), 410; https://doi.org/10.3390/e19080410 - 18 Aug 2017

Cited by 7 | Viewed by 4220

Abstract

The relationship between soil water content (SWC) and vegetation, topography, and climatic conditions is critical for developing effective agricultural water management practices and improving agricultural water use efficiency in arid areas. The purpose of this study was to determine how crop cover influenced [...] Read more.

The relationship between soil water content (SWC) and vegetation, topography, and climatic conditions is critical for developing effective agricultural water management practices and improving agricultural water use efficiency in arid areas. The purpose of this study was to determine how crop cover influenced spatial and temporal variation of soil water. During a study, SWC was measured under maize and wheat for two years in northwest China. Statistical methods and entropy analysis were applied to investigate the spatio-temporal variability of SWC and the interaction between SWC and its influencing factors. The SWC variability changed within the field plot, with the standard deviation reaching a maximum value under intermediate mean SWC in different layers under various conditions (climatic conditions, soil conditions, crop type conditions). The spatial-temporal-distribution of the SWC reflects the variability of precipitation and potential evapotranspiration (ET₀) under different crop covers. The mutual entropy values between SWC and precipitation were similar in two years under wheat cover but were different under maize cover. However, the mutual entropy values at different depths were different under different crop covers. The entropy values changed with SWC following an exponential trend. The informational correlation coefficient (R₀) between the SWC and the precipitation was higher than that between SWC and other factors at different soil depths. Precipitation was the dominant factor controlling the SWC variability, and the crop efficient was the second dominant factor. This study highlights the precipitation is a paramount factor for investigating the spatio-temporal variability of soil water content in Northwest China. Full article

(This article belongs to the Special Issue Entropy Applications in Environmental and Water Engineering)

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Open AccessArticle

Modeling Multi-Event Non-Point Source Pollution in a Data-Scarce Catchment Using ANN and Entropy Analysis

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Entropy 2017, 19(6), 265; https://doi.org/10.3390/e19060265 - 19 Jun 2017

Cited by 12 | Viewed by 4640

Abstract

Event-based runoff–pollutant relationships have been the key for water quality management, but the scarcity of measured data results in poor model performance, especially for multiple rainfall events. In this study, a new framework was proposed for event-based non-point source (NPS) prediction and evaluation. [...] Read more.

Event-based runoff–pollutant relationships have been the key for water quality management, but the scarcity of measured data results in poor model performance, especially for multiple rainfall events. In this study, a new framework was proposed for event-based non-point source (NPS) prediction and evaluation. The artificial neural network (ANN) was used to extend the runoff–pollutant relationship from complete data events to other data-scarce events. The interpolation method was then used to solve the problem of tail deviation in the simulated pollutographs. In addition, the entropy method was utilized to train the ANN for comprehensive evaluations. A case study was performed in the Three Gorges Reservoir Region, China. Results showed that the ANN performed well in the NPS simulation, especially for light rainfall events, and the phosphorus predictions were always more accurate than the nitrogen predictions under scarce data conditions. In addition, peak pollutant data scarcity had a significant impact on the model performance. Furthermore, these traditional indicators would lead to certain information loss during the model evaluation, but the entropy weighting method could provide a more accurate model evaluation. These results would be valuable for monitoring schemes and the quantitation of event-based NPS pollution, especially in data-poor catchments. Full article

(This article belongs to the Special Issue Entropy Applications in Environmental and Water Engineering)

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Open AccessArticle

Generalized Beta Distribution of the Second Kind for Flood Frequency Analysis

by

Lu Chen

and

Vijay P. Singh

Entropy 2017, 19(6), 254; https://doi.org/10.3390/e19060254 - 12 Jun 2017

Cited by 17 | Viewed by 7444

Abstract

Estimation of flood magnitude for a given recurrence interval T (T-year flood) at a specific location is needed for design of hydraulic and civil infrastructure facilities. A key step in the estimation or flood frequency analysis (FFA) is the selection of [...] Read more.

Estimation of flood magnitude for a given recurrence interval T (T-year flood) at a specific location is needed for design of hydraulic and civil infrastructure facilities. A key step in the estimation or flood frequency analysis (FFA) is the selection of a suitable distribution. More than one distribution is often found to be adequate for FFA on a given watershed and choosing the best one is often less than objective. In this study, the generalized beta distribution of the second kind (GB2) was introduced for FFA. The principle of maximum entropy (POME) method was proposed to estimate the GB2 parameters. The performance of GB2 distribution was evaluated using flood data from gauging stations on the Colorado River, USA. Frequency estimates from the GB2 distribution were also compared with those of commonly used distributions. Also, the evolution of frequency distribution along the stream from upstream to downstream was investigated. It concludes that the GB2 is appealing for FFA, since it has four parameters and includes some well-known distributions. Results of case study demonstrate that the parameters estimated by POME method are found reasonable. According to the RMSD and AIC values, the performance of the GB2 distribution is better than that of the widely used distributions in hydrology. When using different distributions for FFA, significant different design flood values are obtained. For a given return period, the design flood value of the downstream gauging stations is larger than that of the upstream gauging station. In addition, there is an evolution of distribution. Along the Yampa River, the distribution for FFA changes from the four-parameter GB2 distribution to the three-parameter Burr XII distribution. Full article

(This article belongs to the Special Issue Entropy Applications in Environmental and Water Engineering)

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Open AccessArticle

An Entropy-Based Generalized Gamma Distribution for Flood Frequency Analysis

by

Lu Chen

,

Vijay P. Singh

and

Feng Xiong

Entropy 2017, 19(6), 239; https://doi.org/10.3390/e19060239 - 02 Jun 2017

Cited by 27 | Viewed by 6336

Abstract

Flood frequency analysis (FFA) is needed for the design of water engineering and hydraulic structures. The choice of an appropriate frequency distribution is one of the most important issues in FFA. A key problem in FFA is that no single distribution has been [...] Read more.

Flood frequency analysis (FFA) is needed for the design of water engineering and hydraulic structures. The choice of an appropriate frequency distribution is one of the most important issues in FFA. A key problem in FFA is that no single distribution has been accepted as a global standard. The common practice is to try some candidate distributions and select the one best fitting the data, based on a goodness of fit criterion. However, this practice entails much calculation. Sometimes generalized distributions, which can specialize into several simpler distributions, are fitted, for they may provide a better fit to data. Therefore, the generalized gamma (GG) distribution was employed for FFA in this study. The principle of maximum entropy (POME) was used to estimate GG parameters. Monte Carlo simulation was carried out to evaluate the performance of the GG distribution and to compare with widely used distributions. Finally, the T-year design flood was calculated using the GG and compared with that with other distributions. Results show that the GG distribution is either superior or comparable to other distributions. Full article

(This article belongs to the Special Issue Entropy Applications in Environmental and Water Engineering)

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Open AccessArticle

Entropy-Based Parameter Estimation for the Four-Parameter Exponential Gamma Distribution

by

Songbai Song

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Xiaoyan Song

and

Yan Kang

Entropy 2017, 19(5), 189; https://doi.org/10.3390/e19050189 - 26 Apr 2017

Cited by 9 | Viewed by 4567

Abstract

Two methods based on the principle of maximum entropy (POME), the ordinary entropy method (ENT) and the parameter space expansion method (PSEM), are developed for estimating the parameters of a four-parameter exponential gamma distribution. Using six data sets for annual precipitation at the [...] Read more.

Two methods based on the principle of maximum entropy (POME), the ordinary entropy method (ENT) and the parameter space expansion method (PSEM), are developed for estimating the parameters of a four-parameter exponential gamma distribution. Using six data sets for annual precipitation at the Weihe River basin in China, the PSEM was applied for estimating parameters for the four-parameter exponential gamma distribution and was compared to the methods of moments (MOM) and of maximum likelihood estimation (MLE). It is shown that PSEM enables the four-parameter exponential distribution to fit the data well, and can further improve the estimation. Full article

(This article belongs to the Special Issue Entropy Applications in Environmental and Water Engineering)

Open AccessArticle

Investigation into Multi-Temporal Scale Complexity of Streamflows and Water Levels in the Poyang Lake Basin, China

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Entropy 2017, 19(2), 67; https://doi.org/10.3390/e19020067 - 10 Feb 2017

Cited by 10 | Viewed by 4407

Abstract

The streamflow and water level complexity of the Poyang Lake basin has been investigated over multiple time-scales using daily observations of the water level and streamflow spanning from 1954 through 2013. The composite multiscale sample entropy was applied to measure the complexity and [...] Read more.

The streamflow and water level complexity of the Poyang Lake basin has been investigated over multiple time-scales using daily observations of the water level and streamflow spanning from 1954 through 2013. The composite multiscale sample entropy was applied to measure the complexity and the Mann-Kendall algorithm was applied to detect the temporal changes in the complexity. The results show that the streamflow and water level complexity increases as the time-scale increases. The sample entropy of the streamflow increases when the timescale increases from a daily to a seasonal scale, also the sample entropy of the water level increases when the time-scale increases from a daily to a monthly scale. The water outflows of Poyang Lake, which is impacted mainly by the inflow processes, lake regulation, and the streamflow processes of the Yangtze River, is more complex than the water inflows. The streamflow and water level complexity over most of the time-scales, between the daily and monthly scales, is dominated by the increasing trend. This indicates the enhanced randomness, disorderliness, and irregularity of the streamflows and water levels. This investigation can help provide a better understanding to the hydrological features of large freshwater lakes. Ongoing research will be made to analyze and understand the mechanisms of the streamflow and water level complexity changes within the context of climate change and anthropogenic activities. Full article

(This article belongs to the Special Issue Entropy Applications in Environmental and Water Engineering)

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Review

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Open AccessReview

Tsallis Entropy Theory for Modeling in Water Engineering: A Review

by

Vijay P. Singh

,

Bellie Sivakumar

and

Huijuan Cui

Entropy 2017, 19(12), 641; https://doi.org/10.3390/e19120641 - 27 Nov 2017

Cited by 33 | Viewed by 5168

Abstract

Water engineering is an amalgam of engineering (e.g., hydraulics, hydrology, irrigation, ecosystems, environment, water resources) and non-engineering (e.g., social, economic, political) aspects that are needed for planning, designing and managing water systems. These aspects and the associated issues have been dealt with in [...] Read more.

Water engineering is an amalgam of engineering (e.g., hydraulics, hydrology, irrigation, ecosystems, environment, water resources) and non-engineering (e.g., social, economic, political) aspects that are needed for planning, designing and managing water systems. These aspects and the associated issues have been dealt with in the literature using different techniques that are based on different concepts and assumptions. A fundamental question that still remains is: Can we develop a unifying theory for addressing these? The second law of thermodynamics permits us to develop a theory that helps address these in a unified manner. This theory can be referred to as the entropy theory. The thermodynamic entropy theory is analogous to the Shannon entropy or the information theory. Perhaps, the most popular generalization of the Shannon entropy is the Tsallis entropy. The Tsallis entropy has been applied to a wide spectrum of problems in water engineering. This paper provides an overview of Tsallis entropy theory in water engineering. After some basic description of entropy and Tsallis entropy, a review of its applications in water engineering is presented, based on three types of problems: (1) problems requiring entropy maximization; (2) problems requiring coupling Tsallis entropy theory with another theory; and (3) problems involving physical relations. Full article

(This article belongs to the Special Issue Entropy Applications in Environmental and Water Engineering)

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Open AccessFeature PaperReview

Entropy Applications to Water Monitoring Network Design: A Review

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Entropy 2017, 19(11), 613; https://doi.org/10.3390/e19110613 - 15 Nov 2017

Cited by 43 | Viewed by 5449

Abstract

Having reliable water monitoring networks is an essential component of water resources and environmental management. A standardized process for the design of water monitoring networks does not exist with the exception of the World Meteorological Organization (WMO) general guidelines about the minimum network [...] Read more.

Having reliable water monitoring networks is an essential component of water resources and environmental management. A standardized process for the design of water monitoring networks does not exist with the exception of the World Meteorological Organization (WMO) general guidelines about the minimum network density. While one of the major challenges in the design of optimal hydrometric networks has been establishing design objectives, information theory has been successfully adopted to network design problems by providing measures of the information content that can be deliverable from a station or a network. This review firstly summarizes the common entropy terms that have been used in water monitoring network designs. Then, this paper deals with the recent applications of the entropy concept for water monitoring network designs, which are categorized into (1) precipitation; (2) streamflow and water level; (3) water quality; and (4) soil moisture and groundwater networks. The integrated design method for multivariate monitoring networks is also covered. Despite several issues, entropy theory has been well suited to water monitoring network design. However, further work is still required to provide design standards and guidelines for operational use. Full article

(This article belongs to the Special Issue Entropy Applications in Environmental and Water Engineering)

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