Generally there are five main concepts that our students struggle with
when thinking about floods:
the various ways that we measure the size or flow of a river,
the relationship between discharge and stage,
how a rise in flood stage leads to an area of inundation,
flood forecasts, probabilities, recurrence intervals and how to
calculate them,
the meaning of a "100-year flood".
Discharge
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.
Discharge is the volume of water that passes through a given cross
section per unit time, usually measured in cubic feet per second (cfs)
or cubic meters per second (cms).
Stage is the level of the water surface over a datum (often sea
level). As discharge increases, stage increases, however the
relationship is not linear.
Flood stage is the stage at which overbank flows are of sufficient
magnitude to cause considerable inundation of land and roads and/or
significantly threaten life and property. Students often confuse this
with overbank flow which is when water overtops the channel and has
geomorphic impacts.
Crest (or peak) is the highest stage reached during a hydrologic
event (such as a flood).
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 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
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:
wanting the flood to spread a fixed distance from the stream,
independent of topographic changes
viewing floods as primarily a lateral movement of water rather than
a vertical change in the water level (they sometimes think they will
get pushed perpendicular to the flow instead of downstream)
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,
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.
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
P = 1/T
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 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
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 (P_{f}) of a favorable outcome is
P_{f} = f/n
where:
n is the total number of possible outcomes and
f is the number of possible outcomes that match the favorable
outcome criteria.
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
Hazard assessment including earthquakes, floods
, and volcanic eruptions
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:
slight chance - 20% probability
chance - 30-50 % probability
likely - 60-70 % probability
no qualifier - 80-100% probability
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
ratio - written as 1:3 or 1 in 3.
fraction - 1/3 or sometimes even as an unreduced fraction 3/9
percentage - 33%
decimal - 0.33
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
Bar Pie chart Number line
Graphical ways of showing probability distributions
Bar chart
Distribution 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 (P_{f}) of a favorable
outcome is
P_{f} = f/n
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 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.
Computers
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
Generally, there are four concepts that our students struggle with when
studying recurrence intervals:
the concept of a recurrence interval
the idea that random events that are independent of previous events
the difference between recurrence intervals and forecasting
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:
flood frequency
earthquake hazards
volcanic eruption frequency
severe weather and storms
meteor impacts
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:
People often talk of events being "due," usually when the time
without an event exceeds the recurrence interval.
Students think that because an event has occurred recently it cannot
happen again.
Even clearly random events are often not perceived as truly random;
many people are convinced that streaks exist ("the dice are hot so I
should keep going") or in luck ("I have lost the last 10 times, so my
luck HAS to change.") These superstitions may be transferred to
geologic processes.
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.
T = N/n
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
The Los Angeles River at Sepulveda Blvd had the following peak
discharges between 1970 and 1979 (data from the U.S.G.S.):
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.
Forecasting
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
T
is
P = 1/T
The probability
P_{T}
that a given event will be equaled or exceeded at least once in the next
r-years is:
P_{T} = 1 – P^{r}
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
Two
streams,
two stories... How Humans Alter Floods and Streams In this
exercise, students look at data from two streams in the Seattle metro
area. They calculate the 100-year flood on each stream for two
different time periods and note the changes wrought by urbanization
and damming. This is designed to be worked in groups and can be done
in-class or at home.
Chemung
River
Flood Recurrence Intervals. A WWW and spreadsheet-based exercise
on flood-frequency analysis, from the Journal of Geoscience Education,
v. 46, 1998. In MS Word form.
Flood
Frequency
and Risk Assessment In this lab, students calculate recurrence
intervals for various degrees of flooding on a portion of the Des
Moines River in Iowa, based on historical data. Then they plot these
calculations on a flood frequency curve. Combining flood frequency
data with a topographic map of the region, students do a risk
assessment for the surrounding community.
Rethinking Flood Prediction: Why the Traditional Approach Needs to
Change Gosnold, W. D., et al., 2000. , Geotimes, v. 45, n. 5, p.
20-23. This article, not freely available on-line, discusses many of
the problems with flood forecasting.
Discharge
and
Sediment Transport in the Field In this field activity, students
collect field data on channel geometry, flow velocity, and bed
materials. Using these data, they apply flow resistance equations
(Manning and the depth slope product) and sediment transport relations
(Shields curve) to estimate the bankfull discharge and to determine if
the flow is sufficient to mobilize the bed. This activity requires
students to utilize theoretical and empirical equations derived in
class in the context of a field problem.
Significant
Floods
in the United States During the 20th Century - USGS Measures a
Century of Floods lists thirty-two of the most significant
floods (in terms of number of lives lost and (or) property damage) in
the United States during the 20th century according to the various
types of floods. Internet sites for acquiring near-real-time
streamflow data and other pertinent flood information are provided.
Water resources data from
the U.S.G.S. This page provide access to water-resources data
collected at more than 1 million sites in all 50 States, the District
of Columbia, and Puerto Rico.
Probability
Determining
Earthquake
Probability and Recurrence In this exercise students gather real
web-based data on earthquakes in the Pacific Northwest, calculate
recurrence intervals on these events and make estimations of
recurrence intervals on future events. Includes analysis of error and
societal implication discussions. Designed as a combination in-class
and web-based homework assignment. This could easily be altered to
look at earthquake possibilities for any geographic area.
Chance
news This site contains materials to help teach a Chance course.
Chance is a quantitative literacy course. It includes "The Chance
News" a newsletter that includes lesson plans and articles with
discussion questions from the Chance magazine. A source for
supplementary material for teaching or introducing basic probability.
The
Math Forum has extensive links to classroom resources. Look in
Exploring Data for finding and displaying data sets. Also included are
an Internet Mathematics Library, Course Materials & Lesson Plans,
Collections of Course Materials & Lesson Plans, Problems and
Puzzles, Probability and statistics software, internet projects, and
Reference Materials