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7 Methods of Analyzing Variability

Part 1. Theory, Organization and Scale

  1. Lars Gottschalk

Published Online: 15 APR 2006

DOI: 10.1002/0470848944.hsa006

Encyclopedia of Hydrological Sciences

Encyclopedia of Hydrological Sciences

How to Cite

Gottschalk, L. 2006. Methods of Analyzing Variability. Encyclopedia of Hydrological Sciences. 1:7.

Author Information

  1. University of Oslo, Department of Geosciences, Oslo, Norway

Publication History

  1. Published Online: 15 APR 2006


In an introductory part basic concepts from probability theory, and specifically from the theory of random processes, are introduced as a basis for the characterization of variability of hydrological time series, space processes and time-space processes. A partial characterization of the random process under study is adopted in accordance with three different schemes:

  1. Characterization by distribution function (one dimensional),

  2. Second moment characterization, and

  3. Karhunen–Loève expansion, that is, a series representation in terms of random variables and deterministic functions of a random process.

The article follows the same division into three major sections. In the first one, distribution functions of frequent use in hydrology are shortly described as well as the flow duration curve. The treatment of second-order moments includes covariance/correlation functions, spectral functions and semivariograms. They allow establishing the structure of the data in space and time and its scale of variability. They also give the possibility of testing basic hypothesis of homogeneity and stationarity. By means of normalization and standardization, data can be transformed into new data sets owing these properties.

The section on Karhunen–Loève expansion includes harmonic analysis, analysis by wavelets, principal component analysis, and empirical orthogonal functions. The characterization by series representation in its turn assumes homogeneity with respect to the variance–covariance function. It is as such a tool for analyzing spatial-temporal variability relative to the first- and second-order moments in terms of new sets of common orthogonal random functions.


  • random variable;
  • process, vector;
  • persistence;
  • time series;
  • integral scale;
  • distribution function;
  • correlation function;
  • semivariogram;
  • Karhunen–Loève expansion