Statistical Methods in Hydrology and Hydroclimatology

It is oblivious to state the need of statistical methods in any field of engineering and science. In the area of hydrology and hydroclimatology, use of different statistical methods is inevitable due to inherent uncertainty. This chapter starts with some basic definitions and scope in hydrology, climatology, and hydroclimatology. Role of statistical methods in the context of inherent variability and uncertainty is discussed afterward. Organization of the book is also presented at the end of this chapter.

Probability is the measure of chance of occurrence of a particular event. The basic concept of probability is widely used in the field of hydrology and hydroclimatology due to its stochastic nature. The inferences like the expected frequency of events, prediction of hydrologic phenomena based on the dependent variables, risk assessment and modeling require in-depth knowledge of probability theory. This chapter starts with the basic concepts of probability that is required for a clear understanding of random experiment, random variables, events, and assignment of probability to events. The axioms of probability and the fundamental rules are explained with the help of Venn diagrams. Later, the concepts of univariate and bivariate random variables along with their respective forms of probability distribution function, cumulative distribution function, and joint probability distribution are discussed. Application of the probability theories in the field of hydrology and hydroclimatology is illustrated with different examples.

This chapter starts with some basic exploratory statistical properties from sample data. Concept of moment and expectation, and moment-generating and characteristic functions are considered afterwards. Different methods for parameter estimation build the foundation for many statistical inferences in the field of hydrology and hydroclimatology.

Use of probability distributions in hydrology and hydroclimatology is inevitable. This is mostly due to the presence of uncertainty and lack of complete knowledge from the data. In this chapter, two types of probability distributions, namely discrete probability distribution and continuous probability distribution are discussed elaborately. Commonly used distributions with their parameters, properties of the distribution supported by graphical representation, and their plausible applications in hydrology and hydroclimatology are explained. Each distribution is explained in the following order—basics, interpretation of the random variable, parameters, probability mass/density function, description, potential applications, and illustrative examples. This order is expected to help the readers to understand the distribution and to develop the knowledge base for its further applications.

Analysis of extreme events like severe storms, floods, droughts is an essential component of hydrology and hydroclimatology. The extreme events have catastrophic impact on the entire agro-socioeconomic sector of a country as well as of the whole world. It has aggravated in a changing climate. Thus, it has become really important to predict their occurrences or their frequency of occurrences. This chapter focuses on different methods to analyze these extreme events and forecast their possible future occurrences. At the very beginning of the chapter, the concept of return period has been discussed elaborately which is the building block of any frequency analysis. However, identification of the best-fit probability distribution for a sample data is essential for any frequency analysis. Concept of probability paper is important in this regard, and its construction is discussed along with graphical concept of frequency factor. Next, the concept of frequency analysis is discussed using different parametric probability distributions, such as normal distribution, lognormal distribution, log-Pearson type III distribution, Gumbel’s distribution. Basic concepts of risk, reliability, vulnerability, resiliency, and uncertainty are also explained which are inevitable in any kind of hydrologic design based on frequency analysis of extreme events.

It is often required to make some inferences about some parameter of the population on the basis of available data. Such inferences are very important in hydrology and hydroclimatology where the available data is generally limited. This is done through hypothesis testing. However, hypothesis testing requires the knowledge of sampling distribution of different statistics and parameter estimation. Sampling distribution of mean and variance and two types of parameter estimation – point estimation and interval estimation – are discussed at the starting of this chapter. Next, the hypothesis testing is taken up. Different cases are discussed elaborately with illustrative examples. Later, a few statistical tests are discussed that deal with the goodness-of-fit of a probability distribution to the data using the knowledge of hypothesis testing. Some of the commonly used nonparametric tests are also explained along with appropriate examples in the field of hydrology and hydroclimatology.

Many applications in hydrology and hydroclimatology deal with studying the relationship between the associated variables. The target variable is known as dependent variable, whereas other variables are known as independent variables. In statistics, the procedure of developing such relationship between dependent and independent variables is called regression analysis. The fitted statistical model is termed as regression model. Such models can be used for assessment of the dependent variable, knowing the independent variables. There are different types of regression models, and every regression model consists of some mathematical formulation with parameters to relate independent variables to dependent variable. All these types of regression models are discussed in this chapter.

Often many hydroclimatic variables are associated with each other and such associations are complex. Many a times several hydroclimatic variables are required to be analyzed simultaneously. Several techniques related to multiple hydroclimatic variables are discussed in this chapter. Different techniques include principal component analysis, supervised principal component analysis, canonical correlation analysis, empirical orthogonal function, one-way and two-way analysis of variance. All these techniques are explained in this chapter with illustrative examples.

Hydroclimatic variables such as rainfall intensity, streamflow, air temperature vary with space and time, due to different hydrological/climatic phenomena/processes. As these processes are continuously evolving over time, studying the interdependence in hydroclimatic data with proper consideration of temporal information may lead to better insight into the governing processes. Observations of any variable, recorded in chronological order, represent a time series. A time series is generally assumed to consist of deterministic components (results can be predicted with certainty) and stochastic components (results cannot be predicted with certainty as the outcome depends on chance). Analysis of time series helps to get an insight of the time series that in turn may enhance the prediction of the hydroclimatic processes/variables. The objective of this chapter is to introduce different types of time series analysis techniques. This requires an understanding of time series analysis techniques and time series properties like stationarity, homogeneity, periodicity, which is the subject matter of this chapter.

This chapter starts with an introduction to copulas. The copula theory is relatively new to hydrology and hydroclimatology but has already established itself to be highly potential in frequency analysis, multivariate modeling, simulation and prediction. Development of joint distribution between multiple variables is the key to analyze utilizing the potential of copulas. The chapter starts with the mathematical theory of copulas and gradually move on to the application. If the readers are already aware of the background theory and look for application of copula theory, they can directly proceed to Sect. 10.8. Basic mathematical formulations for most commonly used copulas are discussed, and illustrative examples are provided. It will enable the readers to carry out applications to other problems. All the illustrative examples are designed with very few data points. This helps to show the calculation steps explicitly. Please note that any statistical analysis should be done with sufficiently long data. Once the readers understand the steps, computer codes can be written easily for large data sets. Example of MATLAB codes is also provided at the end.