Quantitative approach to floods and flooding

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Quantitative approach to floods and flooding

by Dr. Eric M. Baer, Geology Program, Highline Community College

Floods and flooding quantitatively
Discharge, stage, flood stage, and crest: What's the difference?
Rating a river: the relationship between discharge and stage
Flood stage and inundation area: it's not just how high the river is
What is a one hundred year flood?
Flood forecasting: forecasts based on historic data and the "100-year flood"
An example of taking historical discharges to forecasted inundations
Resources and activities

Quantitiative approach to floods and flooding

Essential Concepts

Generally there are five main concepts that our students struggle with when thinking about floods:
  1. the various ways that we measure the size or flow of a river,
  2. the relationship between discharge and stage,
  3. how a rise in flood stage leads to an area of inundation,
  4. flood forecasts, probabilities, recurrence intervals and how to calculate them,
  5. the meaning of a "100-year flood".


Discharge, stage, flood stage, and crest: What's the difference?
Students often get confused by the variety of ways the size or magnitude of flows in rivers are measured and communicated. It is not that these concepts are difficult, however if these terms are not explicitly defined, students often end up perplexed.

In class, and for most hydrologic needs, the discharge (also called flow) is the critical measure of the river. However, it is difficult to measure discharge, so the most common way flows are reported in the media is by stage. In the case of a flood, news reports will often say that a river is "5 feet above flood stage". Floods and rivers also have a crest which is often reported in relationship to flood stage as in "the river will crest at 7 feet above flood stage tomorrow."

Rating a River
Rating a river: the relationship between discharge and stage

The relationship between discharge and stage is empirical and typically is represented graphically as a rating curve.

rating curve for the Connecticut River

Rating curve for the Connecticut River at Montague City station
The making of a rating curve is difficult, but is crucial to taking measurements (like stage) and turning them into the much more useful discharge information. Ideally, I would take students out to a stream a couple of times and have them measure discharge and stage so they get an idea of these measurements (see Discharge and Sediment Transport in the Field for an example). That is usually not possible, so I lead an open ended discussion of how we can measure the discharge. I start out with the definition of discharge and then have them come up with the methods they would use. I then have them think about how, if they made that measurement multiple times over many years they could develop a rating curve. Rating curves are an excellent topic for class discussion because they often seem simple to students at first glance, but are much more complicated. I often ask things like "what happens to a river when the discharge increases?" and then have students explore the rather complicated answers. For example, rating curves may change with time and there is often significant scatter in the data, which can be used as a launching point for discussing regressions and error.

Flood stage and inundation area: it's not just how high the river is

 Wildhorse Canyon 1988

Often students have a difficult time realizing that the area of inundation for a flood depends not just on the stage, but also on the slope of the ground around the water. This is often expressed as students: 

These same misconceptions are also true for non-stream flooding events such as tsunami and storm surges where they expect coastal damage to be a fixed distance inland. To help with these misunderstandings, instead of going straight to a mapping of flood inundation levels on a map,

 Wildhorse Canyon 2001

I often have students do a couple of cross-sections along differing parts of the stream/river so as to illustrate that the aerial extent of flood inundation is dependent on the topography of the stream channel.

For some students that really have a hard time visualizing inundation area as a function of topography, I have an inexpensive model of a stream made from a plastic food storage container and modeling clay. I have one stream channel narrow and steep sided and the other broad and relatively flat. I then pour water into the model to simulate the rise of water from a flood and see the difference in inundation area. In the steep valley they can see that the flood does not spread far away from its original channel whereas in the broader valley the area under water is much greater.

The impact of a flood that raises the water level of a stream by 10 feet is very dependent on the slopes of the valley. In a steep canyon, there might be no significant impact, while on a broad flood plain the water could cover a great area and do tremendous damage. So the damage in a flood is not only dependent on the discharge, but also the topography around a river. In fact, engineers often build levees to artificially steepen the sides of a river in order to reduce flood damage.

Flood forecasting: forecasts based on historic data and the "100-year flood"

Flood forecasts are determined by examining past occurrences of flooding events, determining recurrence intervals of historical events, and then extrapolating to future probabilities. These calculations are often very difficult for students to understand because of the use difficult graphs to extrapolate data. Typically, the maximum annual discharge is examined and ranked to generate a recurrence interval for historical discharges. The calculations needed to make these forecasts are discussed on the recurrence interval page and the probability pages.

The determination of recurrence intervals has many inherent assumptions that are often false. One assumption is that each flood event is independent of previous flooding events, which is often false. Another assumption is that the occurrences and recurrence intervals of floods in the past is the same as the occurrences in the future. Because drainage basins are changed by human activities and other events, and rainfall may be changing due to local or global climate variations, the extrapolation of past events to the future may be invalid.

Red River Recurrence interval

Once the recurrence intervals of historic data are determined, these are plotted on graph paper in order to extrapolate to events beyond the historical record. Many exercises use log-normal graph paper to make this graph. However, this assumes a log-normal distribution of flood data when there are many possible distributions including log-normal, Gumbel, and Pearson distributions. A correct (at least according to the U.S. government) manner of determining flood frequency is to do a Log-Pearson Type III analysis.
> To learn more about flood frequency distributions, you may want to look at Flood frequency distributions, a .pdf file of a U.S.G.S. document describing various distributions and how to fit them to data or at Log-Pearson Type III analysis, a web page that has step-by step instructions on how to do this analysis for stream data using excel. The extrapolations of recurrence intervals are then to forecast the future probability of a flood of a given discharge.

The probability (P) of an flood with recurrence interval T is

So a flood discharge that has a 100-year recurrence interval has a 1% chance of occurring or being exceeded in a given year. The stage of such a flood can be back-calculated using the rating curve for the river. Once the stage is known, a topographic map can be consulted to examine inundation.

An example of taking historical discharges to forecasted inundations

Rating curve for Wetsfield

Rating curve for the Raging River The town of Wetsfield sits on the Raging River. They are planning to install a new sewage treatment plant and want to know where to put it. Concerned by potential damage from floods, they decide that they need to examine several proposed sites for flood risk. They first gather data from a local stream gauge which shows the maximum annual stage each year. Unfortunately, the gauge has only been operating for 16 years. They then assign a recurrence interval to each of the year’s peak floods.

Year discharge ranking RI
2004 134 3 5.666667
2003 119 8 2.125
2002 118 9 1.888889
2001 137 2 8.5
2000 111 14 1.214286
1999 125 5 3.4
1998 114 12 1.416667
1997 111 15 1.133333
1996 171 1 17
1995 117 10 1.7
1994 130 4 4.25
1993 121 7 2.428571
1992 123 6 2.833333
1991 115 11 1.545455
1990 112 13 1.307692
1989 110 16 1.0625

Graph for projecting recurrence intervals on the Raging river

To find out the stage of such floods, the town consults the Raging river's rating curve. They find that the 25 year flood discharge would be at about 700 feet elevation, while the 50 year flood would be at 710 feet elevation and the 100 year flood would be at 730 feet elevation. So if they site the treatment plant at 730 feet, they would have a 1 percent chance any given year of the plant being flooded.

What is a one hundred year flood?

The term "a one-hundred year flood" is actually a misnomer, and as a result causes a great deal of confusion with students and professionals alike. What is really meant by this term is a flood with recurrence interval of 100 years - one that has a 1% chance of occurring in any given year. There are several ideas and resources for teaching recurrence intervals for floods and other events on the recurrence interval page.

Understanding odds and probability in the geosciences

Jump down to:

Strategies | Materials & Exercises | Resources

Probability is the study of the chance that a particular event or series of events will occur. Typically, the chance of an event or series of events will occur is expressed on a scale from 0 (impossible) to 1 (certainty) or as an equivalent percentage from 0 to 100%.
The probability (Pf) of a favorable outcome is

rolling dice The analysis of many events governed by probability is statistics, covered elsewhere.

Strategies: Ideas from Mathematics
Put quantitative concepts in context

There are a number of geologic contexts in which to introduce and explore probability

lava flow into the ocean

Use multiple representations

Because everyone has different ways of learning, mathematicians have defined a number of ways that quantitative concepts can be represented to individuals. Here are some ways that probabilities can be represented.
Verbal representation Probabilities are often given verbally as a percentage, such as a "50% chance." Other times probabilities are even represented by qualitative terms such as "sure," "unlikely," or "almost certain." For example,
The U.S weather service defines several verbal probabilities:
A anvil cloud of a thunderstorm

Numerical representation One common source of confusion is that probabilities are represented in many differing ways all of which mean the same thing. Probabilities can be represented as a ratio, percentage, fraction or as a decimal; I often point this out to students, so they are alert to the multiple ways we represent odds. This often brings up difficulties and fears with basic mathematics including fractions, percentages and ratios.

Some ways of representing the same probability

Tabular representation Tables are often used to show probabilities across a sample space, called a probability distribution.

Sum probability (fractional) probability (percentage)
2 1/36 2.78%
3 2/36 5.56%
4 3/36 8.33%
5 4/36 11.11%
6 5/36 13.89%
7 6/36 16.67%
8 5/36 13.89%
9 4/36 11.11%
10 3/36 8.33%
11 2/36 5.56%
12 1/36 2.78%
TOTAL 36/36 100%

Graphical representation There are several graphical ways to portray both probabilities and probability distributions.

Graphical ways of showing probabilities

bargraph Bar     piechart Pie chart      NUMBERLINE Number line

Graphical ways of showing probability distributions
 HISTOGRAPH Bar chart     Normal distribution probability graph Distribution probability     cumulative percent probability Cumulative percent   

Model representation There are many models one can use as an introduction to probability, including dice rolls, coin tosses, causes of death, and chances of floods.
One example is the probability of death due to individual causes. For instance, a typical American has a one in three chance of dying in a heart attack, a one in 1/100 chance of being killed in auto accident and an estimated 1 in 20,000 chance of being killed by a meteor strike. These probabilities can then be used to stimulate a discussion if geologic hazards, impact of risk perceptions, and mathematical calculations of probability.

Chance/probability of death from various causes for an average American in a given year.

Cause of death Probability
Aids/HIV infection 1 in 19,000
Airplane Crash 1 in 2,736,000
Alcoholic Liver Disease 1 in 22,000
Car Crash 1 in 6357
Earthquake 0 to 1 in 40,000,000
Falls 1 in 20,688
Flood 1 in 6,700,000
Food Poisoning 1 in 53,000
Heart disease 1 in 387
Homicide 1 in 16439
Suicide 1 in 9350

Algebraic representation The probability (Pf) of a favorable outcome is
where n is the total number of possible outcomes and f is the number of possible outcomes that match the favorable outcome criteria.

Use technology appropriately

Students have any number of technological tools that they can use to better understand quantitative concepts -- from the calculators in their backpacks to the computers in their dorm rooms. Probability and recurrence intervals can make use of these tools to help the students understand this often difficult concept.

Graphing calculators

 graphing calculator

Graphing calculators are an easy way for all students to enter data and to see what a curve of that data looks like. All graphing calculators are slightly different and students may need help with their particular model.


black line drawing of coputer

Probability provides an excellent opening for an introduction to the use of spreadsheet programs. Spreadsheet programs can be used to develop probability distributions and to generate graphs of these distributions. An example to start with might be the probability of rolling a particular sum for 1, 2, 3 and more sets of dice. Students are likely to encounter spreadsheet programs in many of their classes and they are excellent tools for visualizing the shape of an equation.

Work in groups to do multiple day, in-depth problems

Mathematicians also indicate that students learn quantitative concepts better when they work in groups and revisit a concept on more than one day. Therefore, when discussing quantitative concepts in entry-level geoscience courses, have students discuss or practice the concepts together. Also, make sure that you either include problems that may be extended over more than one class period or revisit the concept on numerous occasions.

Probability is a concept that comes up over and over in introductory geoscience: Volcanic eruptions, mass extinctions, earthquakes, floods, confidence intervals, etc. When each new topic is introduced, make sure to point out that they have seen this type of mathematics before and should recognize it.

Recurrence intervals

by Dr. Eric M. Baer, Geology Program, Highline Community College

Jump down to:

randomness | forecasting vs. recurrence | how to calculate | materials exercises

Essential Concepts

Generally, there are four concepts that our students struggle with when studying recurrence intervals:

hydrologic technitian from USGS
  1. the concept of a recurrence interval
  2. the idea that random events that are independent of previous events
  3. the difference between recurrence intervals and forecasting
  4. how to calculate recurrence intervals when data have variable magnitudes and when they don't.

Recurrence interval

When geologic events are random or quasi-random, it is helpful to represent their frequency as an average time between past events, a "recurrence interval" also known as a "return time." For instance, there have been 7 subduction zone earthquakes in the Pacific Northwest in the past 3500 years, giving a recurrence interval of about 500 years.

How do geologists use this concept?

Recurrence intervals occur in a variety of geologic contexts including:

 sign of flood

These many contexts occur throughout an introductory geoscience course and give opportunities to revisit and reinforce this concept.

What is meant by "random"?

Random events have a probability of occurring that do not depend on the past. While this may not always be true, for many geologic events this is a robust model. For more information on teaching and using probability, please look at our probability page.

Students express their misunderstanding of random events in a variety of ways:

rolling dice

To reinforce the idea that geologic events are random, I often relate them to events that they recognize as random such as flipping a coin or rolling a 6 on a die. When we talk about streaks, I use the question "how does the die/coin know what it rolled the last time?" as a way to dispel misperceptions arising out of superstitions. Students can also see this by conducting an experiment flipping coins. I have each student flip a coin until they get three heads or tails in a row. Then I have the entire class flip one more time. They see that half of the class that continues their "streak" and half breaks their "streak."

Recurrence intervals vs. forecasting

Recurrence intervals refer to the past occurrence of random events. Forecasting refers to the future likelihood of random events. These are often confused because the recurrence interval (calculated from past events) is used to gauge the future probability of an event. However, the mathematics used with these two concepts are very different. The confusion between the past-determined recurrence interval and the forecasted probability is reinforced by the widespread use of "a 100-year flood" to mean a "flood with a 1% probability of occurring in any given year."

Recurrence intervals

Determination of the recurrence interval is straight-forward when looking at past events. Where there is no associated magnitude or a limited magnitude (such as pumice-producing volcanic eruptions) the recurrence interval (T) is the number of events (N) divided by the number of years in the record (n). An example of an activity using these calculations is Determining Earthquake Probability and Recurrence.
When there is a magnitude associated with the data (such as discharge with a flood or seismic moment with an earthquake) the recurrence interval (T) is

where n is the number of years of the record and m is the magnitude ranking.
Student activities using these calculations are
Two streams, two stories... How Humans Alter Floods and Streams
 Flood Frequency and Risk Assessment.

 The Los Angeles River at Sepulveda Blvd had the following peak discharges between 1970 and 1979 (data from the U.S.G.S.):

Los angeles river
The L.A. River at Sepulveda Blvd

Year Discharge (cfs)
1970 81,806
1971 123,006
1972 75,806
1973 112,006
1974 99,706
1975 114,006
1976 57,406
1977 95,106
1978 147,006
1979 112,006

The data can be reorganized, and a recurrence interval computed for each discharge. In this case, n is 10, because we are using 10 years of data.

Year Discharge rank (m) recurrence interval (n+1)/m
1976 57,406 10 1.1
1972 75,806 9 1.2
1970 81,806 8 1.4
1977 95,106 7 1.6
1974 99,706 6 1.83
1973 112,006 5 2.2
1979 112,006 4 2.8
1975 114,006 3 3.7
1971 123,006 2 5.5
1978 147,006 1 11

Note that these are not the true recurrence values for the L.A. River since only a selection of available data have been used.


Once we start looking to the future, we are looking at forecasting which is governed by the mathematics of probability. The probability (P) of an event with recurrence interval
The probability that a given event will be equaled or exceeded at least once in the next r-years is:

Illustrating the difference between forecasts and intervals

A flood is a 100-year flood if the discharge has exceeded that value on average once every 100 years in the past. In this case the probability of such a flood occurring in the next year is 1/100 or 1%. Many students would assume that the chance of a 100-year flood occurring in the next 100 years is 100%, but that is not true. It may be counter-intuitive to students that that a 100-year flood has a less than two thirds chance of occurring in 100 years. I explain that the chance of a 100-year flood not happening in the next 100 years is 99%100, or 36.6%. If the probability of an event NOT happening is 36.6%, the probability of it happening is 63.4%. I also explain that that there is a chance that 2 or even 3 100-year events will occur within a given 100 year period. As a result, the average recurrence will drop from these more closely spaced events. Indeed, there is only about a 35% chance that a single 100-year flood will occur in 100 years, an 18% chance that 2 100-year floods will occur, and even a nearly 2% chance that we will get four 100-year events within a given 100 year time period.

Resources and activities