One of the basic questions in many hydrosystems infrastructural designs that an engineer must answer is, “What should be the capacity or size of a system?” The planning goal is not to eliminate all hydro-hazards but to reduce the frequency of their occurrences and thus the resulting damage. If such planning is to be correct, the probabilities of flooding must be evaluated correctly. The problem is made more complex because in many cases the “input” is controlled by nature rather than by humans. For example, variations in the amount, timing, and spatial distribution of precipitation are the underlying reasons for the need for probabilistic approaches for many civil and environmental engineering projects. Our understanding and ability to predict precipitation and its resulting effects such as runoff are far from perfect. How, then, can an engineer approach the problem of design when he or she cannot be certain of the hydrologic load that will be placed on the infrastructure under consideration?
An approach that is used often is a statistical or probabilistic one. Such an approach does not require a complete understanding of the hydrologic phenomenon involved but examines the relationship between magnitude and frequency of occurrence in the hope of finding some statistical regularity between these variables. In effect, the past is extrapolated into the future. This assumes that whatever complex physical interactions control nature, the process does not change with time, and so the historical record can be used as a basis for estimating future events. In other words, the data are assumed to satisfy statistical stationarity by which the underlying distributional properties do not change with time, and the historical data series is representative of the storms and watershed conditions to be experienced in the future. An example that violates this statistical stationarity is the progressive urbanization within a watershed that could result in a tendency of increasing peak flow over time.
The hydrologic data most commonly analyzed in this way are rainfall and stream flow records. Frequency analysis was first used for the study of stream flow records by Herschel and Freeman during the period from 1880 to 1890
(Foster, 1935). The first comprehensive study was performed by Fuller (1914). Gumbel (1941, 1942) first applied a particular extreme-value probability distribution to flood flows, whereas Chow (1954) extended the work using this distribution. A significant contribution to the study of rainfall frequencies was made by Yarnell (1936). The study analyzed rainfall durations lasting from 5 minutes to 24 hours and determined their frequency of occurrence at different locations within the continental United States. A similar study was performed by the Miami Conservancy District of Ohio for durations extending from 1 to 6 days (Engineering Staff of Miami Conservancy District, 1937). An extremal probability distribution was applied to rainfall data at Chicago, Illinois, by Chow (1953), and more recent frequency analysis of rainfall data was performed by the U. S. National Weather Service (Hershfield, 1964; U. S. Weather Bureau, 1964; Miller et al., 1973; Frederick et al., 1977, Huff and Angel, 1989, 1992). Low stream flows and droughts also were studied statistically by Gumbel (1954, 1963), who applied an extremal distribution to model the occurrences of drought frequencies. In the United Kingdom, hydrologic frequency analysis usually follows the procedures described in the Flood Studies Report of 1975 (National Environment Research Council, 1975). In general, frequency analysis is a useful analytical tool for studying randomly occurring events and need not be limited to hydrologic studies. Frequency analysis also has been applied to water quality studies and to ocean wave studies.
Basic probability concepts and theories useful for frequency analysis are described in Chap. 2. In general, there is no physical rule that requires the use of a particular distribution in the frequency analysis of geophysical data. However, since the maximum or minimum values of geophysical events are usually of interest, extreme-value-related distributions have been found to be most useful.
