Mathematics and flood prediction

Summary:Engineering has always been engaged with flood protection and the containment of floodwaters; mathematics is also used to predict flooding.

Although some floods occur with little to no warning, overall patterns of flooding along rivers or streams can be determined based on measurement and the statistical analysis and extrapolation of gathered data, such as a river’s historical and current discharge, stage, and flood-stage levels. The resulting data can be used to find the probability of future flooding. Mathematicians and engineers are actively engaged in developing systems to model, predict, and control floods, especially for low-lying areas of the world like the Netherlands and the Mississippi Valley in the United States. Flood prediction and flood control are vital because of floods’ potentially devastating impacts—floods are among the leading natural disasters in terms of loss of life and property damage.

Flood Prediction

One of the first steps in flood prediction is the measurement of a river’s discharge, stage, and flood stage. The size and flow of rivers are measured using a variety of different methods. Key determinations include the discharge or flow, which measures the volume of water passing through a section of the river in a particular time frame, such as cubic feet per second; the stage, or water surface level over a set criteria, such as sea level; and the flood stage, when a river’s overflow will result in widespread inundation or heavy impacts on life and property. Determination of the area of inundation during a flood stage must also take into consideration the topography of the nearby area, such as its slope. During a particular flood, analysts also determine the peak or crest, when the river reaches its highest stage.

Scientists then create flood forecasts based on calculations determined from the statistical analysis of the gathered data. The mathematical calculation of the relationship between an area’s precipitation levels and the discharge of nearby rivers and streams relies on a number of complex factors. Geographical factors can include the topography; types of bedrock, soil, and vegetation; and area of the drainage basin. Meteorological factors can include the intensity and duration of precipitation on average, as well as before and during a particular storm. Because of the complexity of the data, forecasters rely on calculating probability based on historical data of peak discharge frequency.

Statistical analysis of the probability of exceeding the average annual peak discharge in a specified time frame can be made for drainage basins for which a series of records of maximum annual discharges (peak flow) are available and ranked from largest to smallest. The calculated probabilities include the probability that a peak flow will be equaled or exceeded within one year, known as the “exceedence probability” and expressed as a decimal fraction; and the recurrence interval, which is the average number of years between past events. The recurrence interval can also be defined as the number of years in which analysts expect a one-time flow that will equal or exceed a peak flow.

The recurrence interval for a particular location can be used to determine the probability of a flood at that location, expressed by the formula

where P is the probability of a flood and T is the recurrence interval. For example, a 100-year recurrence interval would produce a 1% probability of a flood of equal or greater magnitude in a given year. Engineers, scientists, forecasters, and the public must be aware, however, that the resulting probability is an average. For example, a 100-year flood is not statistically expected to occur exactly once every 100 years and two such floods may occur in close proximity.

Graphing and Modeling Floods

Analysts use these statistics in the construction of graphs and tables known as “frequency distributions,” which show the probability of various discharges for particular locations and thus the probability of a flood in a particular area. Analysts can utilize a variety of mathematical equations to carry out the statistical analysis needed to create frequency distributions. The most common equations include Normal Distribution, Log-Normal Distribution, Gumbel Distribution, and Log-Pearson Type III Distribution.

Different mathematical methods are used to determine frequency distributions in those locations where recorded data of discharge is unavailable or incomplete. In some cases, analysts use flood frequency estimates from nearby or similar areas with complete data to create estimates for areas that lack data. One commonly used method is the rational method, which utilizes the relationship between peak discharge and the product of drainage basin area, precipitation intensity level, and a standard coefficient based on the drainage basin’s land use or ground cover. Other methods allow for the incorporation of changes in a river’s discharge over time as well as its peak discharge. The increasing availability of flood-modeling software allows analysts to input data into computers, which then produce flood probabilities and frequency distributions as well as the effects of environmental impacts, such as deforestation and global climate pattern changes, on future flood patterns.

Applications of Flood Models

Meteorologists use flood probabilities and frequency distributions to aid in the issuance of flood watches and warnings. Engineers use flood probability estimates of both magnitude and frequency when constructing and managing flood control structures, such as dams and levees, as well as nearby structures, such as roads and bridges.

The information is also useful when planning to divert or change the course of rivers or streams that frequently flood, increase the slope of the surrounding topography to lessen inundation, create floodway channels, or determine when to lower dam reservoir levels. Governments and other groups use flood probabilities and frequency distributions when planning the location of residences, towns, and industries along rivers and streams.

Bibliography

Baker, Victor R., and R. Craig Kochel. Flood Geomorphology. Hoboken, NJ: Wiley, 1988.

Bedient, Philip B., and Wayne C. Huber. Hydrology and Floodplain Analysis. Upper Saddle River, NJ: Prentice Hall, 2002.

Bhaskar, Nageshwar Rao. Regionalization of Flood Data Using Probability Distributions and Their Parameters. Lexington: University of Kentucky, Water Resources Research Institute, 1989.

Miller, E. Willard, and Ruby M. Miller. Natural Disasters: Floods: Santa Barbara, CA: ABC-CLIO, 2000.

Purseglove, Jeremy. Taming the Flood: A History and Natural History of Rivers and Wetlands. New York: Oxford University Press, 1988.