Review of Systems of Frequency Distributions for Water and Environmental Engineering by Vijay P. Singh and Lan Zhang

Probability distributions are inherent to the analysis of hydrometeorological and environmental data. Most students are introduced to probability distributions through a statistical or stochastic hydrology class. Some, who wish to delve deeper, may take additional classes from the statistics department. While useful, students are not always able to see how what is taught in these classes can be applied to their work, because the textbooks used in these courses are written by statistics and mathematics professors and the examples are seldom relevant to hydrology. The many volumes by Samuel Kotz, N. Balakrishnan, Normal L. Johnson, and Adrienne W. Kemp (Johnson et al. 1994, 1995, 1997, 2008; Kotz et al. 2004) provide a comprehensive treatment of discrete and continuous univariate and multivariate probability distributions, but few are aware of their existence or have access to them. Moreover, they are so extensive that it is easy for a hydrologist or engineer to feel overwhelmed when looking for the right distribution for their data set. In Systems of Frequency Distributions for Water and Environmental Engineering, Singh and Zhang have compiled, in an accessible volume, the most important probability distributions for hydrometeorological and environmental engineering applications.

The book is divided into 12 chapters. Chapter 1, the introductory chapter, provides a brief overview of the main components of the hydrological cycle and processes, supported by illustrations demonstrating how they each follow markedly different distributions. The second part of this chapter lists the main systems of frequency distributions that are presented in the book.

Chapter 2 is an extensive exposé of the Pearson system of frequency distribution. Most hydrologists are familiar with the exponential, gamma, Pearson III, and log-Pearson III distributions because they are commonly employed in the modeling of rainfall, streamflow, and drought. The log-Pearson III, for example, is the recommended distribution for flood frequency analysis in the United States (England et al. 2019). In this chapter, the authors present the 12 distributions in the Pearson family including the underlying hypothesis and derivatives followed by a detailed example of how it is applied.

Chapter 3 focusses on the Burr system of frequency distributions. Just like in the Pearson family, there are 12 frequency distributions in the Burr system, of which Burr XII is the most commonly encountered in hydrologic analysis. Starting with the underlying hypothesis, this chapter presents all 12 distributions and expands on Burr’s theory of cumulative moments for parameter estimation. The chapter concludes with several examples of applications to peak flow, annual rainfall amount, monthly sediment yield, and maximum daily precipitation.

Chapter 4 is devoted to the D’Addario system of frequency distributions. The D’Addario system as such is not very popular in hydrology or environmental engineering, but special cases of distribution springing from this system are commonly employed. This chapter starts with a general description of the D’Addario system and explains how the Pareto Type I and II distributions and the two- and three-parameter lognormal distributions can be derived. It further explains how special cases of the Amoroso distribution lead to 11 well-known distributions including exponential, semi-normal, normal, and Weibull. Examples of application include peak flow, monthly discharge, and maximum daily precipitation.

Chapter 5 presents the Dagum system of distributions. The Dagum distributions are popular in economics and actuarial science because they are often employed to model income distribution and income elasticity. While practitioners may not be aware of this family, which includes 11 distributions, some are commonly employed in modeling hydrology, hydrometeorology, hydraulics, and environmental engineering data sets. In this chapter the authors explain the basis for this family of distributions and present each of the 11 distributions, which include the Pareto I, II, and III, Benini, Weibull, log-Gompertz, Fisk (commonly referred to as the log-logistic distribution), Singh-Maddala, and Dagum I, II, and III distributions.

The Stoppa system of frequency distributions is discussed in Chapter 6. This system leads to a number of generalized distributions including the generalized power distribution, generalized exponential distribution, and generalized Pareto distribution. The authors also demonstrate how the Stoppa system is related to the Burr and Dagum systems. Examples of applications include monthly suspended sediment, annual rainfall amount and daily precipitation maximum, peak flow, and drought.

Chapter 7 discusses the Esteban system of frequency distributions. This system of distribution, like Dagum, originates from work on income elasticity. It leads to a number of distributions that are used in hydrology, hydraulics, environmental, and water resources engineering, such as the generalized gamma distribution, generalized beta distribution of the first kind, and generalized beta distribution of the second kind. Special cases of these generalized distributions along with applications using water quality data, peak flow, drought, and annual rainfall are presented.

In Chapter 8, the authors present the Singh system of frequency distributions. The authors explain how several distributions commonly used in hydrology and hydraulics do not have a closed form and need to be solved numerically. Through empirical data, it is hypothesized that a relation exists between the probability density function and the cumulative density function. Based on this relationship, an exhaustive set of distributions is derived.

The system of frequency distributions using Bessel functions and cumulants is presented in Chapter 9. The first four moments of the Bessel function distribution and the Bessel function line are introduced, followed by the inverse Gaussian and other distributions. The second part of this chapter centers on frequency distribution by series approximation using the Gaussian distribution as the baseline density function. Finally, the chapter concludes with an application of the inverse Gaussian distribution to hydrologic and water quality data.

Chapter 10 is a condensed version of frequency distributions by entropy maximization. Those familiar with the work of Professor Vijay P. Singh would recall his seminal contribution in the application of entropy theory in hydrology and environmental engineering. His numerous books in this field lay the groundwork for a multitude of applications. Readers interested in this topic are encouraged to read this chapter and refer to his more comprehensive book on Entropy-Based Parameter Estimation in Hydrology (Singh 1998). Those who wish to dwell further should consult his recent books, which focus on a range of applications, including environmental and water engineering (Singh 2013), hydrology (Singh 2015), and hydraulic engineering (Singh 2014).

Chapter 11 explains the concept of transformations, where starting with a basic distribution, a distribution with more parameters or a generalized distribution leading to special cases can be derived. The transformation of the normal distribution; transformation based on the first law of Laplace; and transformation of beta, gamma, and Student’s $t$-distributions are presented followed by applications to peak flow, monthly sediment, daily maximum precipitation, and annual rainfall.

Finally, Chapter 12 briefly discusses the genetic theory of frequency, after introducing the concept of elementary errors. The Charlier system is presented followed by extensions proposed by Wicksell.

The material presented in this book is no doubt extremely technical. Readers may need a good background in distribution theory to digest the content. I have learned quite a lot while going through it and I am sure that anyone who will take the time to do so will appreciate the effort that the authors have put into compiling this volume. This book is definitely an important addition to any serious research library.