Tsallis Entropy Theory for Modeling in Water Engineering: A Review
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paper_title: Entropy Theory in Hydraulic Engineering: An Introduction

paper_content:

\IEntropy Theory in Hydraulic Engineering: An Introduction\N is the first book to explain the basic concepts of entropy theory from a hydraulic perspective and demonstrate the theory’s application in solving practical engineering problems. In the hydraulic context, entropy is valuable as a way of measuring uncertainty or surprise —or even disorder or chaos— as a type of information. As hydraulic systems become more complex, entropy theory enables hydraulic engineers to quantify uncertainty, determine risk and reliability, estimate parameters, model processes, and design more robust and dependable water hydraulic systems.
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paper_title: THE USE OF ENTROPY IN HYDROLOGY AND WATER RESOURCES

paper_content:

Since the development of the entropy theory by Shannon in the late 1940s and of the principle of maximum entropy (POME) by Jaynes in the late 1950s there has been a proliferation of applications of entropy in a wide spectrum of areas, including hydrological and environmental sciences. The real impetus to entropy-based hydrological modelling was provided by Amorocho and Espildora in 1972. A great variety of entropy applications in hydrology and water resources have since been reported, and new applications continue to unfold. This paper reviews the recent contributions on entropy applications in hydrology and water resources, discusses the usefulness and versatility of the entropy concept, and reflects on the strengths and limitations of this concept. The paper concludes with comments on its implications in developing countries. #1997 by John Wiley & Sons, Ltd.
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paper_title: Entropy Theory in Hydrologic Science and Engineering

paper_content:

1. Entropy Theory 2. Morphological Analysis 3. Evaluation and Design of Sampling and Measurement Networks 4. Precipitation Variability 5. Rainfall Frequency Distributions 6. Assessment of Potential Water Resources Availability 7. Evaluation of Precipitation Forecasting Schemes 8. Infiltration 9. Soil Moisture 10. Groundwater Flow 11. Rainfall-Runoff Modeling 12. Hydrologic Frequency Analysis 13. Steamflow Simulation 14. Long-Term Streamflow Forecasting 15. Flood Forecasting: Bivariate Analysis 16. River Regime Classification 17. Sediment Yield Eco Index
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paper_title: A toy model of climatic variability with scaling behaviour

paper_content:

Abstract It is demonstrated that a simple deterministic model in discrete time can reproduce the scaling behaviour of hydroclimatic processes at timescales coarser than annual, a behaviour more widely known in hydrology as the Hurst phenomenon. This toy model is based on a generalised ‘chaotic tent map’, which may be considered as the compound result of a positive and a negative feedback mechanism, and involves two degrees of freedom. The model is not a realistic representation of a climatic system, but rather a radical simplification of real climatic dynamics. However, its simplicity helps understand the physical mechanisms that cause the scaling behaviour and simultaneously enables easy implementation and convenient experimentation. Application of the toy model gives traces that can resemble historical time series of hydroclimatic variables, such as temperature and river flow. In particular, such traces exhibit scaling behaviour with a Hurst coefficient greater than 0.5 and their statistical properties are similar to that of observed time series. Moreover, application demonstrates that large-scale synthetic ‘climatic’ fluctuations (like upward or downward trends) can emerge without any specific reason and their evolution is unpredictable, even when they are generated by this simple fully deterministic model with only two degrees of freedom. Thus, the model emphasises the large uncertainty associated with the scaling behaviour, rather than enhances the prediction capability, despite the simple deterministic dynamics it uses, which obviously, are only a caricature of the much more complex dynamics of the real climatic system.
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paper_title: Possible generalization of Boltzmann-Gibbs statistics

paper_content:

With the use of a quantity normally scaled in multifractals, a generalized form is postulated for entropy, namelyS q ≡k [1 – ∑ i=1 W p i q ]/(q-1), whereq∈ℝ characterizes the generalization andpi are the probabilities associated withW (microscopic) configurations (W∈ℕ). The main properties associated with this entropy are established, particularly those corresponding to the microcanonical and canonical ensembles. The Boltzmann-Gibbs statistics is recovered as theq→1 limit.
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paper_title: The constrained entropy and cross-entropy functions

paper_content:

This study introduces the constrained forms of the Shannon information entropy and Kullback–Leibler cross-entropy functions. Applicable to a closed system, these functions incorporate the constraints on the system in a generic fashion, irrespective of the form or even the number of the constraints. Since they are fully constrained, the constrained functions may be “pulled apart” to give their partial or local constrained forms, providing the means to examine the probability of each outcome or state relative to its local maximum entropy (or minimum cross-entropy) position, independently of the other states of the system. The Shannon entropy and Kullback–Leibler cross-entropy functions do not possess this functionality. The features of each function are examined in detail, and the similarities between the constrained and Tsallis entropy functions are examined. The continuous constrained entropy and cross-entropy functions are also introduced, the integrands of which may be used to independently examine each infinitesimal state (physical element) of the system.
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paper_title: On the fractal dimension of orbits compatible with Tsallis statistics

paper_content:

In a previous paper [A. Carati, Physica A 348 (2005) 110–120] it was shown how, for a dynamical system, the probability distribution function of sojourn-times in phase-space, defined in terms of the dynamical orbits (up to a given observation time), induces unambiguously a statistical ensemble in phase-space. In the present paper, the p.d.f. of the sojourn-times corresponding to a Tsallis ensemble is obtained (this, by the way, requires the solution of a problem of a general character, disregarded in paper [A. Carati, Physica A 348 (2005) 110–120]). In particular some qualitative properties, such as the fractal dimension, of the dynamical orbits compatible with the Tsallis ensembles are indicated.
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paper_title: Entropic nonextensivity: a possible measure of complexity

paper_content:

An updated review [1] of nonextensive statistical mechanics and thermodynamics is colloquially presented. Quite naturally the possibility emerges for using the value of q-1 (entropic nonextensivity) as a simple and efficient manner to provide, at least for some classes of systems, some characterization of the degree of what is currently referred to as complexity [2]. A few historical digressions are included as well.
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paper_title: I. Nonextensive Statistical Mechanics and Thermodynamics: Historical Background and Present Status

paper_content:

The domain of validity of standard thermodynamics and Boltzmann-Gibbs statistical mechanics is focused on along a historical perspective. It is then formally enlarged in order to hopefully cover a variety of anomalous systems. The generalization concerns nonextensive systems, where nonextensivity is understood in the thermodynamical sense. This generalization was first proposed in 1988 inspired by the probabilistic description of multifractal geometry, and has been intensively studied during this decade. In the present effort, we describe the formalism, discuss the main ideas, and then exhibit the present status in what concerns theoretical, experimental and computational evidences and connections, as well as some perspectives for the future. The whole review can be considered as an attempt to clarify our current understanding of the foundations of statistical mechanics and its thermodynamical implications.
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paper_title: Two-Dimensional Velocity Distribution in Open Channels Using the Tsallis Entropy

paper_content:

AbstractAssuming time-averaged velocity as a random variable, a two-dimensional (2D) velocity distribution in open channels was derived by maximizing the Tsallis entropy, subject to mass conservation. The derived distribution was tested using field data and was also compared with other two-dimensional velocity distributions. The Tsallis entropy–based velocity distribution was found to predict the velocity near the boundary well.
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paper_title: Possible generalization of Boltzmann-Gibbs statistics

paper_content:

With the use of a quantity normally scaled in multifractals, a generalized form is postulated for entropy, namelyS q ≡k [1 – ∑ i=1 W p i q ]/(q-1), whereq∈ℝ characterizes the generalization andpi are the probabilities associated withW (microscopic) configurations (W∈ℕ). The main properties associated with this entropy are established, particularly those corresponding to the microcanonical and canonical ensembles. The Boltzmann-Gibbs statistics is recovered as theq→1 limit.
---
paper_title: Entropy Theory in Hydrologic Science and Engineering

paper_content:

1. Entropy Theory 2. Morphological Analysis 3. Evaluation and Design of Sampling and Measurement Networks 4. Precipitation Variability 5. Rainfall Frequency Distributions 6. Assessment of Potential Water Resources Availability 7. Evaluation of Precipitation Forecasting Schemes 8. Infiltration 9. Soil Moisture 10. Groundwater Flow 11. Rainfall-Runoff Modeling 12. Hydrologic Frequency Analysis 13. Steamflow Simulation 14. Long-Term Streamflow Forecasting 15. Flood Forecasting: Bivariate Analysis 16. River Regime Classification 17. Sediment Yield Eco Index
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paper_title: Entropy based derivation of probability distributions: A case study to daily rainfall

paper_content:

The principle of maximum entropy, along with empirical considerations, can provide consistent basis for constructing a consistent probability distribution model for highly varying geophysical processes. Here we examine the potential of using this principle with the Boltzmann-Gibbs-Shannon entropy definition in the probabilistic modeling of rainfall in different areas worldwide. We define and theoretically justify specific simple and general entropy maximization constraints which lead to two flexible distributions, i.e., the three-parameter Generalized Gamma (GG) and the four-parameter Generalized Beta of the second kind (GB2), with the former being a particular limiting case of the latter. We test the theoretical results in 11,519 daily rainfall records across the globe. The GB2 distribution seems to be able to describe all empirical records while two of its specific three-parameter cases, the GG and the Burr Type XII distributions perform very well by describing the 97.6% and 87.7% of the empirical records, respectively.
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paper_title: Just two moments! A cautionary note against use of high-order moments in multifractal models in hydrology

paper_content:

The need of understanding and modelling the space-time variability of natural processes in hydrological sciences produced a large body of literature over the last thirty years. In this context, a multifractal framework pro- vides parsimonious models which can be applied to a wide- scale range of hydrological processes, and are based on the empirical detection of some patterns in observational data, i.e. a scale invariant mechanism repeating scale after scale. Hence, multifractal analyses heavily rely on available data series and their statistical processing. In such analyses, high order moments are often estimated and used in model identi- fication and fitting as if they were reliable. This paper warns practitioners against the blind use in geophysical time series analyses of classical statistics, which is based upon indepen- dent samples typically following distributions of exponential type. Indeed, the study of natural processes reveals scaling behaviours in state (departure from exponential distribution tails) and in time (departure from independence), thus im- plying dramatic increase of bias and uncertainty in statisti- cal estimation. Surprisingly, all these differences are com- monly unaccounted for in most multifractal analyses of hy- drological processes, which may result in inappropriate mod- elling, wrong inferences and false claims about the prop- erties of the processes studied. Using theoretical reasoning and Monte Carlo simulations, we find that the reliability of multifractal methods that use high order moments (> 3) is questionable. In particular, we suggest that, because of esti- mation problems, the use of moments of order higher than two should be avoided, either in justifying or fitting models. Nonetheless, in most problems the first two moments provide enough information for the most important characteristics of the distribution.
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paper_title: Downstream hydraulic geometry relations: 1. Theoretical development: DOWNSTREAM HYDRAULIC GEOMETRY RELATIONS, 1

paper_content:

An edited version of this paper was published by AGU. Copyright 2003 American Geophysical Union.
---
paper_title: Entropy Theory in Hydraulic Engineering: An Introduction

paper_content:

\IEntropy Theory in Hydraulic Engineering: An Introduction\N is the first book to explain the basic concepts of entropy theory from a hydraulic perspective and demonstrate the theory’s application in solving practical engineering problems. In the hydraulic context, entropy is valuable as a way of measuring uncertainty or surprise —or even disorder or chaos— as a type of information. As hydraulic systems become more complex, entropy theory enables hydraulic engineers to quantify uncertainty, determine risk and reliability, estimate parameters, model processes, and design more robust and dependable water hydraulic systems.
---
paper_title: Downstream hydraulic geometry relations: 1. Theoretical development: DOWNSTREAM HYDRAULIC GEOMETRY RELATIONS, 1

paper_content:

An edited version of this paper was published by AGU. Copyright 2003 American Geophysical Union.
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paper_title: An entropy-based morphological analysis of river basin networks

paper_content:

Under the assumption that the only information available on a drainage basin is its mean elevation, the connection between entropy and potential energy is explored to analyze drainage basins morphological characteristics. The mean basin elevation is found to be linearly related to the entropy of the drainage basin. This relation leads to a linear relation between the mean elevation of a subnetwork and the logarithm of its topological diameter. Furthermore, the relation between the fall in elevation from the source to the outlet of the main channel and the entropy of its drainage basin is found to be linear and so is also the case between the elevation of a node and the logarithm of its distance from the source. When a drainage basin is ordered according to the Horton-Strahler ordering scheme, a linear relation is found between the drainage basin entropy and the basin order. This relation can be characterized as a measure of the basin network complexity. The basin entropy is found to be linearly related to the logarithm of the magnitude of the basin network. This relation leads to a nonlinear relation between the network diameter and magnitude, where the exponent is found to be related to the fractal dimension of the drainage network. Also, the exponent of the power law relating the channel slope to the network magnitude is found to be related to the fractal dimension of the network. These relationships are verified on three drainage basins in southern Italy, and the results are found to be promising.
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paper_title: Tsallis Entropy Theory for Derivation of Infiltration Equations

paper_content:

An entropy theory is formulated for deriving infiltration equations for the potential rate (or capacity) of infiltration in unsaturated soils. The theory is comprised of five parts: (1) Tsallis entropy, (2) principle of maximum entropy (POME), (3) specification of information on the potential rate of infiltration in terms of constraints, (4) maximization of entropy in accordance with POME, and (5) derivation of the probability distribution of infiltration and its maximum entropy. The theory is illustrated with the derivation of six infiltration equations commonly used in hydrology, watershed management, and agricultural irrigation, including Horton, Kostiakov, Philip two-term, Green-Ampt, Overton, and Holtan, and the determination of the least biased probability distributions underlying these infiltration equations and the entropies thereof. The theory leads to the expression of parameters of the derived infiltration equations in terms of three measurable quantities: initial infiltration capacity (potential rate), steady infiltration rate, and soil moisture retention capacity. In this sense, these derived equations are rendered nonparametric. With parameters thus obtained, infiltration capacity rates are computed using these six infiltration equations and are compared with field experimental observations reported in the hydrologic literature as well as the capacity rates computed using parameters of these equations obtained by calibration. It is found that infiltration capacity rates computed using parameter values yielded by the entropy theory are in reasonable agreement with observed as well as calibrated infiltration capacity rates.
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paper_title: Entropy and 2-D Velocity Distribution in Open Channels

paper_content:

Equations based on the entropy concept have been derived for describing the two‐dimensional velocity distribution in an openchannel cross section. The velocity equation derived is capable of describing the variation of velocity in both the vertical and transverse directions, with the maximum velocity occurring on or below the water surface. Equations for determining the location of mean velocity have also been derived along with those, such as the entropy function, that can be used as measures of the homogeneity of velocity distribution in a channel cross section. A dimensionless parameter of the entropy function named the M number has been found useful as an index for characterizing and comparing various patterns of velocity distribution and states of open‐channel flow systems. The definition and demonstrated usefulness of this parameter indicate the importance and value of the information given by the location and magnitude of maximum velocity in a cross section, and suggest the need for future experime...
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paper_title: Entropy-based parameter estimation in hydrology

paper_content:

preface. 1. Entropy and Principle of Maximum Entropy. 2. Methods of Parameter Estimation. 3. Uniform Distribution. 4. Exponential Distribution. 5. Normal Distribution. 6. Two-Parameter Lognormal Distribution. 7. Three-Parameter Lognormal Distribution. 8. Extreme Value Type I Distribution. 9. Log-Extreme Value Type I Distribution. 10. Extreme Value Type III Distribution. 11. Generalized Extreme Value Distribution. 12. Weibull Distribution. 13. Gamma Distribution. 14. Pearson Type III Distribution. 15. Log-Pearson Type III Distribution. 16. Beta Distribution. 17. Two-Parameter Log-Logistic Distribution. 18. Three-Parameter Log-Logistic Distribution. 19. Two-Parameter Pareto Distribution. 20. Two-Parameter Generalized Pareto Distribution. 21. Three-Parameter Generalized Pareto Distribution. 22. Two-Component Extreme Value Distribution. Subject Index.
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paper_title: A real-time flood forecasting model based on maximum-entropy spectral analysis. II: Application

paper_content:

The MESA-based model, developed in the first paper, for real-time flood forecasting was verified on five watersheds from different regions of the world. The sampling time interval and forecast lead time varied from several minutes to one day. The model was found to be superior to a state-space model for all events where it was difficult to obtain prior information about model parameters. The mathematical form of the model was found to be similar to a bivariate autoregressive (AR) model, and under certain conditions, these two models became equivalent.
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paper_title: Entropy theory for movement of moisture in soils

paper_content:

[1] An entropy theory is formulated for one-dimensional movement of moisture in unsaturated soils in the vertically downward direction. The theory is composed of five parts: (1) Tsallis entropy, (2) principle of maximum entropy, (3) specification of information on soil moisture in terms of constraints, (4) maximization of the Tsallis entropy, and (5) derivation of the probability distributions of soil moisture. The theory is applied to determine the soil moisture profile under three conditions: (1) the moisture is maximum at the soil surface and decreases downward to a minimum value at the bottom of the soil column (it may be near the water table); (2) the moisture is minimum at the soil surface and increases downward to a maximum value at the end of the soil column (this case is the opposite of case 1); and (3) the moisture at the soil surface is low and increases downward up to a distance and then decreases up to the bottom (this case combines case 2 and case 1). The entropy-based soil moisture profiles are tested using experimental observations reported in the literature, and properties of these profiles are enumerated.
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paper_title: Downstream hydraulic geometry relations: 2. Calibration and testing

paper_content:

An edited version of this paper was published by AGU. Copyright 2003 American Geophysical Union.
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paper_title: A model of evapotranspiration based on the theory of maximum entropy production

paper_content:

[1] Building on a proof-of-concept study of energy balance over dry soil, a model of evapotranspiration is proposed based on the theory of maximum entropy production (MEP). The MEP formalism leads to an analytical solution of evaporation rate (latent heat flux), together with sensible and ground heat fluxes, as a function of surface soil temperature, surface humidity, and net radiation. The model covers the entire range of soil wetness from dry to saturation. The MEP model of transpiration is formulated as a special case of bare soil evaporation. Test of the MEP model using field observations indicates that the model performs well over bare soil and canopy.
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paper_title: Downstream hydraulic geometry relations: 1. Theoretical development: DOWNSTREAM HYDRAULIC GEOMETRY RELATIONS, 1

paper_content:

An edited version of this paper was published by AGU. Copyright 2003 American Geophysical Union.
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paper_title: Entropy-based grouping of river flow regimes

paper_content:

Abstract In the context of changing environment, and specifically a possible climate change, a river flow regime type turns from being a merely general, average, descriptive characteristic to a tool for monitoring changes in flow seasonality both in time and space. Utilization of river flow regimes as a diagnostic tool for the output of climate models and in flow sensitivity studies demands an objective grouping of flow series into regime types. A hierarchical aggregation of monthly flow series into flow regime types, satisfying chosen discriminating criteria, is effectively performed by means of minimization of an entropy-based objective function. This function is based on the concept of an `information loss' resulting from such an aggregation and describes the difference between the series aggregated into one group, i.e. inaccuracy of aggregation. The main advantage of the approach, operating on river flow series for individual years (and not only on long-term means), is its ability to consider also the temporal regularity of the seasonal flow patterns, neglected, as a rule, in other approaches. Meanwhile, both flow volumes and seasonal patterns of flow are sensitive to climate fluctuations. A strict formulation of the `stopping rules' in the hierarchical flow regime grouping, directly related to the aggregation principles, is suggested. The approach allows different formulations of criteria for discriminating flow regime types. It is illustrated on a regional river flow sample for Scandinavia for two different formulations of discriminating criteria.
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paper_title: Unit Stream Power and Sediment Transport

paper_content:

A thorough study of the existing applicable data reveals the basic reason that previous equations often provide misleading predictions of the sediment transport rates. The error stems from the unrealistic assumptions made in their derivations. Unit stream power, defined as the time rate of potential energy expenditure per unit weight of water in an alluvial channel, is shown to dominate the total sediment concentration. Statistical analyses of 1,225 sets of laboratory flume data and 50 sets of field data indicate the existence and the generality of the linear relationship between the logarithm of total sediment concentration and the logarithm of the effective unit stream power. The coefficients in the proposed equation are shown to be related to particle size and water depth, or particle size and width-depth ratio. An equation generalized from Gilbert's data can be applied to natural streams for the prediction of total sediment discharge with good accuracy.
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paper_title: Physics of uncertainty, the Gibbs paradox and indistinguishable particles

paper_content:

Abstract The idea that, in the microscopic world, particles are indistinguishable, interchangeable and without identity has been central in quantum physics. The same idea has been enrolled in statistical thermodynamics even in a classical framework of analysis to make theoretical results agree with experience. In thermodynamics of gases, this hypothesis is associated with several problems, logical and technical. For this case, an alternative theoretical framework is provided, replacing the indistinguishability hypothesis with standard probability and statistics. In this framework, entropy is a probabilistic notion applied to thermodynamic systems and is not extensive per se. Rather, the extensive entropy used in thermodynamics is the difference of two probabilistic entropies. According to this simple view, no paradoxical behaviors, such as the Gibbs paradox, appear. Such a simple probabilistic view within a classical physical framework, in which entropy is none other than uncertainty applicable irrespective of particle size, enables generalization of mathematical descriptions of processes across any type and scale of systems ruled by uncertainty.
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paper_title: Entropy based derivation of probability distributions: A case study to daily rainfall

paper_content:

The principle of maximum entropy, along with empirical considerations, can provide consistent basis for constructing a consistent probability distribution model for highly varying geophysical processes. Here we examine the potential of using this principle with the Boltzmann-Gibbs-Shannon entropy definition in the probabilistic modeling of rainfall in different areas worldwide. We define and theoretically justify specific simple and general entropy maximization constraints which lead to two flexible distributions, i.e., the three-parameter Generalized Gamma (GG) and the four-parameter Generalized Beta of the second kind (GB2), with the former being a particular limiting case of the latter. We test the theoretical results in 11,519 daily rainfall records across the globe. The GB2 distribution seems to be able to describe all empirical records while two of its specific three-parameter cases, the GG and the Burr Type XII distributions perform very well by describing the 97.6% and 87.7% of the empirical records, respectively.
---
paper_title: Entropic nonextensivity: a possible measure of complexity

paper_content:

An updated review [1] of nonextensive statistical mechanics and thermodynamics is colloquially presented. Quite naturally the possibility emerges for using the value of q-1 (entropic nonextensivity) as a simple and efficient manner to provide, at least for some classes of systems, some characterization of the degree of what is currently referred to as complexity [2]. A few historical digressions are included as well.
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paper_title: Application of Information Theory to Groundwater Quality Monitoring Networks

paper_content:

Using the criteria of maximizing information and minimizing cost,a methodology is developed for design of an optimal groundwater-monitoring network for water resources management. A monitoring system is essentially an information collection system. Therefore, its technical design requires a quantifiablemeasure of information which can be achieved through applicationof the information (or entropy) theory. The theory also providesinformation-based statistical measures to evaluate the efficiencyof the monitoring network. The methodology is applied to groundwater monitoring wells in a portion of Gaza Strip in Palestine.
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paper_title: An entropy approach to data collection network design

paper_content:

Abstract A new methodology is developed for data collection network design. The approach employs a measure of the information flow between gauging stations in the network which is referred to as the directional information transfer. The information flow measure is based on the entropy of gauging stations and pairs of gauging stations. Non-parametric estimation is used to approximate the multivariate probability density functions required in the entropy calculations. The potential application of the approach is illustrated using extreme flow data from a collection of gauging stations located in southern Manitoba, Canada.
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paper_title: Assessing the Reliability of Water Distribution Networks Using Entropy Based Measures of Network Redundancy

paper_content:

A qualitative approach to assessing redundancy of water distribution networks using entropy theory is proposed. The redundancy measures derived from this approach are able to assess redundancy of supply for individual nodes and for the network as a whole. The measures themselves are based on the fundamental entropy function of Shannon modified to include such network features relevant to redundancy as the number of paths available to supply flow to each demand node, the capacities of these alternate paths, the degree of dependence among the paths, the possibility of flow reversal, and the desirability of having links of equal capacity incident on demand node. The measures are demonstrated by application to eight network layouts subject to the same demand conditions.
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paper_title: Cross Entropy multiobjective optimization for water distribution systems design

paper_content:

[1] A methodology extending the Cross Entropy combinatorial optimization method originating from an adaptive algorithm for rare events simulation estimation, to multiobjective optimization of water distribution systems design is developed and demonstrated. The single objective optimal design problem of a water distribution system is commonly to find the water distribution system component characteristics that minimize the system capital and operational costs such that the system hydraulics is maintained and constraints on quantities and pressures at the consumer nodes are fulfilled. The multiobjective design goals considered herein are the minimization of the network capital and operational costs versus the minimization of the maximum pressure deficit of the network demand nodes. The proposed methodology is demonstrated using two sample applications from the research literature and is compared to the NSGA-II multiobjective scheme. The method was found to be robust in that it produced very similar Pareto fronts in almost all runs. The suggested methodology provided improved results in all trails compared to the NSGA-II algorithm.
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paper_title: Current and future use of systems analysis in water distribution network design

paper_content:

Abstract Computer use in the design of water distribution networks was inititated through the use of network analysis techniques to determine system performance in terms of heads and flows. The last fifteen years, however, have seen the introduction of systems analysis optimization techniques to the range of computer models available for network design purposes. These optimization models differ markedly from the ‘traditional’ network analysis models in that they ‘design’ systems for specified loading conditions rather than just analysing the performance of predetermined systems under given loading conditions. Cost was the primary or only objective in almost all these early optimization models. Water distribution network design has, however, a number of other important objectives, such as maximizing reliability. Issues related to reliability concern include probability of component failure, probability of actual demands being greater than design values, and the system redundancy inherent within the layout ...
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paper_title: The constrained entropy and cross-entropy functions

paper_content:

This study introduces the constrained forms of the Shannon information entropy and Kullback–Leibler cross-entropy functions. Applicable to a closed system, these functions incorporate the constraints on the system in a generic fashion, irrespective of the form or even the number of the constraints. Since they are fully constrained, the constrained functions may be “pulled apart” to give their partial or local constrained forms, providing the means to examine the probability of each outcome or state relative to its local maximum entropy (or minimum cross-entropy) position, independently of the other states of the system. The Shannon entropy and Kullback–Leibler cross-entropy functions do not possess this functionality. The features of each function are examined in detail, and the similarities between the constrained and Tsallis entropy functions are examined. The continuous constrained entropy and cross-entropy functions are also introduced, the integrands of which may be used to independently examine each infinitesimal state (physical element) of the system.
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paper_title: Geometry of river channels

paper_content:

Hydraulic principles, as developed in firm boundary channels, appear insufficient to explain the form and profile of river channels. In nature, stream channels attain a most probable state that must fulfill the necessary hydraulic laws, but in addition, fulfills its degrees of freedom by tendency to equal distribution among velocity, depth, width, and slope. This principle is tested by the use of three examples. Another example explores the accommodation of a river channel to changing discharge. The last example is that of a river free to adjust its profile, velocities, depths, and widths to accommodate the downstream increase in discharge. Each of these examples appears to satisfy the postulate of this paper. The adjustment is toward minimum variance among the components of stream power.
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paper_title: Hydraulic geometry and minimum rate of energy dissipation

paper_content:

The theory of minimum rate of energy dissipation states that a system is in an equilibrium condition when its rate of energy dissipation is at its minimum value. This minimum value depends on the constraints applied to the system. When a system is not at equilibrium, it will adjust in such a manner that the rate of energy dissipation can be reduced until it reaches the minimum and regains equilibrium. A river system constantly adjusts itself in response to varying constraints in such a manner that the rate of energy dissipation approaches a minimum value and thus moves toward an equilibrium. It is shown that the values of the exponents of the hydraulic geometry relationships proposed by Leopold and Maddock for rivers can be obtained from the application of the theory of minimum rate of energy dissipation in conjunction with the Manning-Strickler equation and the dimensionless unit stream power equation for sediment transport proposed by Yang. Theoretical analysis is limited to channels which are approximately rectangular in shape. The width and depth exponents thus derived agree very well with those measured in laboratory experiments by Barr et al. Although the theoretical width and depth exponents are within the range of variations of measured data from river gaging stations, the at-station width adjustment of natural rivers may also depend on constraints other than water discharge and sediment load.
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paper_title: Downstream hydraulic geometry relations: 2. Calibration and testing

paper_content:

An edited version of this paper was published by AGU. Copyright 2003 American Geophysical Union.
---
paper_title: Downstream hydraulic geometry relations: 1. Theoretical development: DOWNSTREAM HYDRAULIC GEOMETRY RELATIONS, 1

paper_content:

An edited version of this paper was published by AGU. Copyright 2003 American Geophysical Union.
---
paper_title: An extremum principle of evaporation

paper_content:

[1] It is proposed, on the basis of an argument of thermodynamic equilibrium, that land-atmosphere interactive processes lead to thermal and hydrologic states of the land surface that maximize evaporation in a given meteorological environment. The extremum principle leads to general equations linking surface energy fluxes to surface temperature and soil moisture. The hypothesis of maximum evaporation has been tested with data from three field experiments. We found strong evidence suggesting that evaporation is maximized and furthermore that it is determined by the state variables (temperature, soil moisture, and sensible heat flux into the atmosphere) and relatively insensitive to water vapor pressure deficit. The theory allows an independent estimate of the coefficient in the Priestley-Taylor formula for potential evaporation, which is consistent with the widely accepted value of 1.26.
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paper_title: On the Cumulative Distribution Function for Entropy-Based Hydrologic Modeling

paper_content:

In spatial or temporal physically based entropy-based modeling in hydrology and water resources, the cumulative distribution function (CDF) of a design variable (e.g., flux, say discharge) is hypothesized in terms of concentration (e.g., stage of flow). Thus far, a linear hypothesis has been employed when applying entropy to derive relationships for design variables, but without empirical evidence or physical justification. Examples of such relationships include velocity distribution as a function of flow depth, wind velocity as a function of height, sediment concentration profile along the flow depth, rating curve, infiltration capacity rate as a function of time, soil moisture profile along the depth below the soil surface, runoff as a function of rainfall amount, unit hydrograph, and groundwater discharge along the horizontal direction of flow. This study proposes a general nonlinear form of the CDF that specializes into commonly used linear forms. The general form is tested using empirical data on velocity, sediment concentration, soil moisture, and stage-discharge and compared with those reported in the literature. It is found that a simpler form of the general nonlinear hypothesis seems satisfactory for the data tested, and it is quite likely that this simple form will suffice for other data as well. The linear hypothesis does not seem to hold for the data employed in the study.
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paper_title: Entropy and 2-D Velocity Distribution in Open Channels

paper_content:

Equations based on the entropy concept have been derived for describing the two‐dimensional velocity distribution in an openchannel cross section. The velocity equation derived is capable of describing the variation of velocity in both the vertical and transverse directions, with the maximum velocity occurring on or below the water surface. Equations for determining the location of mean velocity have also been derived along with those, such as the entropy function, that can be used as measures of the homogeneity of velocity distribution in a channel cross section. A dimensionless parameter of the entropy function named the M number has been found useful as an index for characterizing and comparing various patterns of velocity distribution and states of open‐channel flow systems. The definition and demonstrated usefulness of this parameter indicate the importance and value of the information given by the location and magnitude of maximum velocity in a cross section, and suggest the need for future experime...
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paper_title: Entropy Theory for Distribution of One-Dimensional Velocity in Open Channels

paper_content:

Assuming time-averaged velocity as a random variable, this study develops an entropy theory for deriving the one-dimensional distribution of velocity in open channels. The theory includes five parts: (1) Tsallis entropy; (2) the principle of maximum entropy (POME); (3) the specification of information on velocity for constraints; (4) the maximization of entropy; and (5) the probability distribution of velocity and its entropy. An application of the entropy theory is illustrated by deriving a one-dimensional velocity distribution in open channels in which the dimension is vertical or the flow depth. The derived distribution is tested with field and laboratory observations and is compared to Chiu’s velocity distribution derived from Shannon entropy. The agreement between velocity values are computed with the entropy-based distribution.
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paper_title: On the Cumulative Distribution Function for Entropy-Based Hydrologic Modeling

paper_content:

In spatial or temporal physically based entropy-based modeling in hydrology and water resources, the cumulative distribution function (CDF) of a design variable (e.g., flux, say discharge) is hypothesized in terms of concentration (e.g., stage of flow). Thus far, a linear hypothesis has been employed when applying entropy to derive relationships for design variables, but without empirical evidence or physical justification. Examples of such relationships include velocity distribution as a function of flow depth, wind velocity as a function of height, sediment concentration profile along the flow depth, rating curve, infiltration capacity rate as a function of time, soil moisture profile along the depth below the soil surface, runoff as a function of rainfall amount, unit hydrograph, and groundwater discharge along the horizontal direction of flow. This study proposes a general nonlinear form of the CDF that specializes into commonly used linear forms. The general form is tested using empirical data on velocity, sediment concentration, soil moisture, and stage-discharge and compared with those reported in the literature. It is found that a simpler form of the general nonlinear hypothesis seems satisfactory for the data tested, and it is quite likely that this simple form will suffice for other data as well. The linear hypothesis does not seem to hold for the data employed in the study.
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paper_title: Entropy and 2-D Velocity Distribution in Open Channels

paper_content:

Equations based on the entropy concept have been derived for describing the two‐dimensional velocity distribution in an openchannel cross section. The velocity equation derived is capable of describing the variation of velocity in both the vertical and transverse directions, with the maximum velocity occurring on or below the water surface. Equations for determining the location of mean velocity have also been derived along with those, such as the entropy function, that can be used as measures of the homogeneity of velocity distribution in a channel cross section. A dimensionless parameter of the entropy function named the M number has been found useful as an index for characterizing and comparing various patterns of velocity distribution and states of open‐channel flow systems. The definition and demonstrated usefulness of this parameter indicate the importance and value of the information given by the location and magnitude of maximum velocity in a cross section, and suggest the need for future experime...
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paper_title: Entropy Theory for Two-Dimensional Velocity Distribution

paper_content:

Assuming time-averaged velocity as a random variable, this study develops an entropy theory for deriving two-dimensional (2D) distribution of velocity in open channels. The theory comprises five parts: (1) Tsallis entropy; (2) principle of maximum entropy (POME); (3) specification of information on velocity in terms of constraints; (4) maximization of entropy; and (5) derivation of the probability distribution of velocity. The entropy theory is then combined with a hypothesis on the cumulative distribution function of velocity in terms of flow depth to derive a 2D velocity distribution. The derived distribution is tested using field as well as laboratory observations reported in the literature and is compared with known velocity distributions. Agreement between velocity values computed using the entropy-based distribution and observed values is found satisfactory. Also, the derived distribution compares favorably with known distributions.
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paper_title: Entropy Theory in Hydrologic Science and Engineering

paper_content:

1. Entropy Theory 2. Morphological Analysis 3. Evaluation and Design of Sampling and Measurement Networks 4. Precipitation Variability 5. Rainfall Frequency Distributions 6. Assessment of Potential Water Resources Availability 7. Evaluation of Precipitation Forecasting Schemes 8. Infiltration 9. Soil Moisture 10. Groundwater Flow 11. Rainfall-Runoff Modeling 12. Hydrologic Frequency Analysis 13. Steamflow Simulation 14. Long-Term Streamflow Forecasting 15. Flood Forecasting: Bivariate Analysis 16. River Regime Classification 17. Sediment Yield Eco Index
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paper_title: Entropy and 2-D Velocity Distribution in Open Channels

paper_content:

Equations based on the entropy concept have been derived for describing the two‐dimensional velocity distribution in an openchannel cross section. The velocity equation derived is capable of describing the variation of velocity in both the vertical and transverse directions, with the maximum velocity occurring on or below the water surface. Equations for determining the location of mean velocity have also been derived along with those, such as the entropy function, that can be used as measures of the homogeneity of velocity distribution in a channel cross section. A dimensionless parameter of the entropy function named the M number has been found useful as an index for characterizing and comparing various patterns of velocity distribution and states of open‐channel flow systems. The definition and demonstrated usefulness of this parameter indicate the importance and value of the information given by the location and magnitude of maximum velocity in a cross section, and suggest the need for future experime...
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paper_title: Entropy Theory for Two-Dimensional Velocity Distribution

paper_content:

Assuming time-averaged velocity as a random variable, this study develops an entropy theory for deriving two-dimensional (2D) distribution of velocity in open channels. The theory comprises five parts: (1) Tsallis entropy; (2) principle of maximum entropy (POME); (3) specification of information on velocity in terms of constraints; (4) maximization of entropy; and (5) derivation of the probability distribution of velocity. The entropy theory is then combined with a hypothesis on the cumulative distribution function of velocity in terms of flow depth to derive a 2D velocity distribution. The derived distribution is tested using field as well as laboratory observations reported in the literature and is compared with known velocity distributions. Agreement between velocity values computed using the entropy-based distribution and observed values is found satisfactory. Also, the derived distribution compares favorably with known distributions.
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paper_title: Two-Dimensional Velocity Distribution in Open Channels Using the Tsallis Entropy

paper_content:

AbstractAssuming time-averaged velocity as a random variable, a two-dimensional (2D) velocity distribution in open channels was derived by maximizing the Tsallis entropy, subject to mass conservation. The derived distribution was tested using field data and was also compared with other two-dimensional velocity distributions. The Tsallis entropy–based velocity distribution was found to predict the velocity near the boundary well.
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paper_title: Unit Stream Power and Sediment Transport

paper_content:

A thorough study of the existing applicable data reveals the basic reason that previous equations often provide misleading predictions of the sediment transport rates. The error stems from the unrealistic assumptions made in their derivations. Unit stream power, defined as the time rate of potential energy expenditure per unit weight of water in an alluvial channel, is shown to dominate the total sediment concentration. Statistical analyses of 1,225 sets of laboratory flume data and 50 sets of field data indicate the existence and the generality of the linear relationship between the logarithm of total sediment concentration and the logarithm of the effective unit stream power. The coefficients in the proposed equation are shown to be related to particle size and water depth, or particle size and width-depth ratio. An equation generalized from Gilbert's data can be applied to natural streams for the prediction of total sediment discharge with good accuracy.
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paper_title: Suspended Sediment Concentration in Open Channels Using Tsallis Entropy

paper_content:

AbstractConcentration of suspended sediment is of fundamental importance in environmental management, assessment of best management practices, water quality evaluation, reservoir ecosystem integrity, and fluvial hydraulics. Assuming time-averaged sediment concentration along a vertical as a random variable, a probability distribution of suspended sediment concentration is derived by maximizing the Tsallis entropy subject to the constraint given by the mean concentration and under the assumption that the sediment concentration is zero at the water surface. For deriving the sediment concentration profile along the vertical, a nonlinear cumulative distribution function is hypothesized and verified with observed data. The derived sediment concentration profile is tested using experimental and field data; however, the clear water surface assumption does not seem to be valid for field data. The Tsallis entropy-based concentration profile method is compared with three sediment concentration profile methods. Comp...
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paper_title: Current and future use of systems analysis in water distribution network design

paper_content:

Abstract Computer use in the design of water distribution networks was inititated through the use of network analysis techniques to determine system performance in terms of heads and flows. The last fifteen years, however, have seen the introduction of systems analysis optimization techniques to the range of computer models available for network design purposes. These optimization models differ markedly from the ‘traditional’ network analysis models in that they ‘design’ systems for specified loading conditions rather than just analysing the performance of predetermined systems under given loading conditions. Cost was the primary or only objective in almost all these early optimization models. Water distribution network design has, however, a number of other important objectives, such as maximizing reliability. Issues related to reliability concern include probability of component failure, probability of actual demands being greater than design values, and the system redundancy inherent within the layout ...
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paper_title: Two-Dimensional Velocity Distribution in Open Channels Using the Tsallis Entropy

paper_content:

AbstractAssuming time-averaged velocity as a random variable, a two-dimensional (2D) velocity distribution in open channels was derived by maximizing the Tsallis entropy, subject to mass conservation. The derived distribution was tested using field data and was also compared with other two-dimensional velocity distributions. The Tsallis entropy–based velocity distribution was found to predict the velocity near the boundary well.
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paper_title: Computation of Suspended Sediment Discharge in Open Channels by Combining Tsallis Entropy–Based Methods and Empirical Formulas

paper_content:

AbstractSediment discharge is computed by using different combinations of entropy-based and empirical methods of channel cross-section velocity and suspended sediment concentration distribution, and the results of these different methods are compared. The comparison shows that the entropy-based methods are more accurate than the empirical method and that the Tsallis entropy–based method is more accurate than the one based on Shannon entropy. The accuracy of the computation for all methods can generally be improved by introducing a correction factor; however, the fully entropy-based methods still remain the most accurate.
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paper_title: Suspended Sediment Concentration in Open Channels Using Tsallis Entropy

paper_content:

AbstractConcentration of suspended sediment is of fundamental importance in environmental management, assessment of best management practices, water quality evaluation, reservoir ecosystem integrity, and fluvial hydraulics. Assuming time-averaged sediment concentration along a vertical as a random variable, a probability distribution of suspended sediment concentration is derived by maximizing the Tsallis entropy subject to the constraint given by the mean concentration and under the assumption that the sediment concentration is zero at the water surface. For deriving the sediment concentration profile along the vertical, a nonlinear cumulative distribution function is hypothesized and verified with observed data. The derived sediment concentration profile is tested using experimental and field data; however, the clear water surface assumption does not seem to be valid for field data. The Tsallis entropy-based concentration profile method is compared with three sediment concentration profile methods. Comp...
---
paper_title: On the Cumulative Distribution Function for Entropy-Based Hydrologic Modeling

paper_content:

In spatial or temporal physically based entropy-based modeling in hydrology and water resources, the cumulative distribution function (CDF) of a design variable (e.g., flux, say discharge) is hypothesized in terms of concentration (e.g., stage of flow). Thus far, a linear hypothesis has been employed when applying entropy to derive relationships for design variables, but without empirical evidence or physical justification. Examples of such relationships include velocity distribution as a function of flow depth, wind velocity as a function of height, sediment concentration profile along the flow depth, rating curve, infiltration capacity rate as a function of time, soil moisture profile along the depth below the soil surface, runoff as a function of rainfall amount, unit hydrograph, and groundwater discharge along the horizontal direction of flow. This study proposes a general nonlinear form of the CDF that specializes into commonly used linear forms. The general form is tested using empirical data on velocity, sediment concentration, soil moisture, and stage-discharge and compared with those reported in the literature. It is found that a simpler form of the general nonlinear hypothesis seems satisfactory for the data tested, and it is quite likely that this simple form will suffice for other data as well. The linear hypothesis does not seem to hold for the data employed in the study.
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paper_title: Flow Duration Curve Using Entropy Theory

paper_content:

AbstractUsing the entropy theory, this study derives a function for modeling the flow duration curve (FDC) based on two equations that form simple constraints: (1) the total probability of all flow discharges, and (2) the mean discharge. Parameters of the derived curve are determined from the entropy theory with the use of these two constraints. For deriving the flow duration curve by the entropy theory, a nonlinear cumulative distribution function is assumed, which is tested using observed flow data. The derived flow duration curves are tested using field data and are found to be in agreement with observed curves. With the entropy parameter determined for each station, the flow duration curve can also be forecasted for different recurrence intervals. The main advantage of the use of the entropy theory–based FDCs is that the parameters are based on observations, and hence no fitting is needed. Second, the theory permits a probabilistic characterization of the flow duration curve and hence the probability ...
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paper_title: Intermittent Motion, Nonlinear Diffusion Equation and Tsallis Formalism

paper_content:

We investigate an intermittent process obtained from the combination of a nonlinear diffusion equation and pauses. We consider the porous media equation with reaction terms related to the rate of switching the particles from the diffusive mode to the resting mode or switching them from the resting to the movement. The results show that in the asymptotic limit of small and long times, the spreading of the system is essentially governed by the diffusive term. The behavior exhibited for intermediate times depends on the rates present in the reaction terms. In this scenario, we show that, in the asymptotic limits, the distributions for this process are given by in terms of power laws which may be related to the q-exponential present in the Tsallis statistics. Furthermore, we also analyze a situation characterized by different diffusive regimes, which emerges when the diffusive term is a mixing of linear and nonlinear terms.
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paper_title: On some properties of Tsallis hypoentropies and hypodivergences

paper_content:

Both the Kullback–Leibler and the Tsallis divergence have a strong limitation: if the value zero appears in probability distributions (p1, ··· , pn) and (q1, ··· , qn), it must appear in the same positions for the sake of significance. In order to avoid that limitation in the framework of Shannon statistics, Ferreri introduced in 1980 hypoentropy: “such conditions rarely occur in practice”. The aim of the present paper is to extend Ferreri’s hypoentropy to the Tsallis statistics. We introduce the Tsallis hypoentropy and the Tsallis hypodivergence and describe their mathematical behavior. Fundamental properties, like nonnegativity, monotonicity, the chain rule and subadditivity, are established.
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paper_title: Measures of Qualitative Variation in the Case of Maximum Entropy

paper_content:

Asymptotic behavior of qualitative variation statistics, including entropy measures, can be modeled well by normal distributions. In this study, we test the normality of various qualitative variation measures in general. We find that almost all indices tend to normality as the sample size increases, and they are highly correlated. However, for all of these qualitative variation statistics, maximum uncertainty is a serious factor that prevents normality. Among these, we study the properties of two qualitative variation statistics; VarNC and StDev statistics in the case of maximum uncertainty, since these two statistics show lower sampling variability and utilize all sample information. We derive probability distribution functions of these statistics and prove that they are consistent. We also discuss the relationship between VarNC and the normalized form of Tsallis (α = 2) entropy in the case of maximum uncertainty.
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paper_title: The Legendre Transform in Non-additive Thermodynamics and Complexity

paper_content:

We present an argument which purports to show that the use of the standard Legendre transform in non-additive Statistical Mechanics is not appropriate. For concreteness, we use as paradigm, the case of systems which are conjecturally described by the (non-additive) Tsallis entropy. We point out the form of the modified Legendre transform that should be used, instead, in the non-additive thermodynamics induced by the Tsallis entropy. We comment on more general implications of this proposal for the thermodynamics of “complex systems”.
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paper_title: Tsallis Wavelet Entropy and Its Application in Power Signal Analysis

paper_content:

As a novel data mining approach, a wavelet entropy algorithm is used to perform entropy statistics on wavelet coefficients (or reconstructed signals) at various wavelet scales on the basis of wavelet decomposition and entropy statistic theory. Shannon wavelet energy entropy, one kind of wavelet entropy algorithm, has been taken into consideration and utilized in many areas since it came into being. However, as there is wavelet aliasing after the wavelet decomposition, and the information set of different-scale wavelet decomposition coefficients (or reconstructed signals) is non-additive to a certain extent, Shannon entropy, which is more adaptable to extensive systems, couldn’t do accurate uncertainty statistics on the wavelet decomposition results. Therefore, the transient signal features are extracted incorrectly by using Shannon wavelet energy entropy. From the two aspects, the theoretical limitations and negative effects of wavelet aliasing on extraction accuracy, the problems which exist in the feature extraction process of transient signals by Shannon wavelet energy entropy, are discussed in depth. Considering the defects of Shannon wavelet energy entropy, a novel wavelet entropy named Tsallis wavelet energy entropy is proposed by using Tsallis entropy instead of Shannon entropy, and it is applied to the feature extraction of transient signals in power systems. Theoretical derivation and experimental result prove that compared with Shannon wavelet energy entropy, Tsallis wavelet energy entropy could reduce the negative effects of wavelet aliasing on accuracy of feature extraction and extract transient signal feature of power system accurately.
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Title: Tsallis Entropy Theory for Modeling in Water Engineering: A Review

Section 1: Introduction
Description 1: Introduce the significance of water resources systems, their historical development, and the contemporary integration of engineering and non-engineering aspects. Provide an overview of the paper's objectives and structure.

Section 2: Definition of Entropy
Description 2: Define the concept of entropy and its interpretations in various fields, particularly in water engineering. Introduce Shannon and Tsallis entropy formulations.

Section 3: Properties of Tsallis Entropy
Description 3: Summarize the key properties of Tsallis entropy, including m-entropy, maximum value, concavity, additivity, composability, and interaction between sub-systems.

Section 4: Principle of Maximum Entropy
Description 4: Discuss the principle of maximum entropy (POME) and its application steps in deriving probability distributions for random variables.

Section 5: Specification of Constraints
Description 5: Explain how to define appropriate constraints for deriving probability density functions using POME, focusing on considerations like simplicity, physical meaningfulness, and statistical moments.

Section 6: Entropy Maximizing Using Lagrange Multipliers
Description 6: Outline the process of maximizing Tsallis entropy subject to constraints using the method of Lagrange multipliers, leading to the determination of probability distributions and Lagrange multipliers.

Section 7: Applications in Water Engineering: Overview
Description 7: Provide an overview of the types of problems in water engineering that can be addressed using Tsallis entropy, categorized into three groups: those requiring entropy maximization, coupling with another theory, and deriving physical relations.

Section 8: Problems Requiring Entropy Maximization
Description 8: Detail specific water engineering problems that involve entropy maximization, such as deriving frequency distributions and network evaluation.

Section 9: Problems Requiring Coupling with another Theory
Description 9: Describe problems where Tsallis entropy must be coupled with other theories, such as hydraulic geometry and evaporation, to achieve solutions.

Section 10: Problems Involving Physical Relations
Description 10: Explain how to derive physical relationships in water engineering by concatenating the probability and physical domains, illustrated with examples like velocity distribution and sediment concentration.

Section 11: Hypotheses on Cumulative Probability Distribution Function
Description 11: Discuss the formulation of cumulative probability distribution functions (CDFs) for different types of design variables and their applications in deriving relationships.

Section 12: One-Dimensional Velocity Distribution
Description 12: Explain the derivation of one-dimensional velocity distribution using Tsallis entropy, considering the constraints and assumed CDFs.

Section 13: Two-Dimensional Velocity Distribution
Description 13: Describe the derivation of two-dimensional velocity distribution, transforming coordinates and utilizing the Tsallis entropy for characterizing flow in channel cross-sections.

Section 14: Suspended Sediment Concentration
Description 14: Illustrate the process of deriving suspended sediment concentration distributions using Tsallis entropy, including defining constraints and determining dimensionless parameters.

Section 15: Sediment Discharge
Description 15: Discuss the computation of sediment discharge using combinations of entropy-based and empirical methods for velocity and sediment concentration distributions.

Section 16: Flow-Duration Curve
Description 16: Detail the derivation of flow-duration curves (FDC) using Tsallis entropy, defining discharge as a random variable and determining the PDF and dimensionless parameters.

Section 17: Conclusions
Description 17: Summarize the potential and advantages of Tsallis entropy theory in water engineering, citing its ability to combine statistical information with physical laws and derive probability distributions and physical relations. Discuss the scope for future research.