Back to the on-screen lesson ·
Recurrence intervals turn a river's record into yearly odds, a 100-year flood has about a one-in-four chance over a mortgage, and hard and soft engineering reduce the damage.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
By the end of this lesson you will be able to find a flood's recurrence interval and yearly chance, work out its chance over many years, and compare ways of managing floods.
You can read a storm hydrograph and explain why some rivers flood faster than others. This lesson asks how often big floods come, what the odds mean for people who live near rivers, and how communities reduce the damage.
| Term | What it means |
|---|---|
| Recurrence interval | The average number of years between floods of a given size; also called the return period. |
| 100-year flood | A flood with a one percent chance of being equaled or exceeded each year. |
| Hard engineering | Structures that change the river, such as levees, dams and channels. |
| Soft engineering | Measures that work with the river, such as zoning, buyouts, wetlands and ponds. |
| Floodplain zoning | Rules that limit building on land that floods often. |
| Detention pond | A basin that stores storm water and releases it slowly. |
A long gauge record lets hydrologists rank floods and estimate how often each size comes:
$$\text{recurrence interval} = \dfrac{\text{years of record} + 1}{\text{rank}}.$$
Another way: picture
Picture rolling a hundred-sided die once every year, and flooding whenever it lands on one. You might roll a one twice in three years, or go two centuries without one. Averaged over a very long time, ones come about once in a hundred rolls: that average is the recurrence interval.
Another way: steps
Risk comparisons need both a chance and a consequence for a stated interval. Suppose a hypothetical annual damaging flood has probability 0.04 and causes 100 loss units if it occurs. Its simplified expected annual loss is 4. A barrier reduces probability to 0.02 but leaves the exposed assets in place: expected loss becomes 2 if failure losses remain 100. Relocation instead reduces exposed loss to 30 while probability stays 0.04: expected loss becomes 1.2. These products compare one hazard model; they do not price all social consequences.
Test uncertainty before declaring a winner. If barrier failure losses are underestimated, or future floods become more likely, its expected loss changes. Relocation may disrupt livelihoods or social networks even when it reduces physical exposure. State which objective is being compared, show a plausible range, and request missing costs and distributional effects. A decision can favor one plan on modeled physical loss without claiming it is best in every respect.
Each recurrence interval is also a chance each year.
| Flood | Chance each year | Expected in 100 years |
|---|---|---|
| 10-year | 10% | about 10 |
| 50-year | 2% | about 2 |
| 100-year | 1% | about 1 |
| 500-year | 0.2% | about 0.2 |
These are averages. A town can see two 100-year floods in a decade, and many American towns have.
Suppose a gauge has $39$ years of record. Hydrologists list the biggest flood of each year and rank them, biggest first. The flood ranked fourth has an interval of $(39 + 1) \div 4 = 10$ years: the 10-year flood.
The biggest flood in a short record can only be given a short interval, because the record is too short to know how rare it is. Estimating the 100-year or 500-year flood takes long records, or statistics that stretch shorter ones.
People often think a 100-year flood gives a century of safety. But over thirty years, the length of a typical mortgage, the chance of at least one 100-year flood is $1 - 0.99^{30}$, about $26$ percent: roughly one in four.
Simply adding thirty years of one percent chances would give thirty percent, which is too high, because it counts the years with two floods twice.
Levees and floodwalls raise the banks so floodwater stays in the channel. Dams hold floodwater in reservoirs and release it slowly. Channels are straightened, deepened or lined with concrete to carry water away faster.
These measures protect large areas and are easy to see. But they are costly, need maintenance, can fail when a flood exceeds their design, and can move the problem downstream by sending water there faster.
Floodplain zoning keeps new homes off land that floods often. Buyouts move people out of the riskiest places, turning their land into parks that can flood safely. Restored wetlands and forests soak up and slow runoff, and detention ponds store storm water and let it out gradually.
These measures give the river room instead of fighting it. They are often cheaper over time and cannot fail all at once, but they may ask people to leave homes or give up land.
The Federal Emergency Management Agency, FEMA, maps the land that the 100-year flood would cover, called the special flood hazard area. Homes there with federally backed mortgages must carry flood insurance, because ordinary homeowners' insurance does not cover floods.
The maps are updated as rivers, basins and climate change. Many damaging floods reach homes outside the mapped area, which is why FEMA encourages flood insurance there too.
A detention pond with a flat floor and steep sides stores its floor area times the water's depth. A pond of $5000$ square meters holds $5000$ cubic meters for every meter of depth.
By storing the peak of a storm's runoff and releasing it slowly through a small outlet, a pond lengthens the lag time and lowers the peak discharge downstream, the opposite of what paving does.
Recurrence intervals come from past records. If a basin is paved over, its floods come faster and higher. If the climate brings heavier downpours, as it is doing in much of the United States, yesterday's 100-year flood may become a 50-year flood.
Engineers increasingly design for the floods expected over a structure's life, not only those recorded in the past.
Checking an answer. A bigger flood has a smaller rank and a longer interval. A chance over many years must be less than one hundred percent, and less than adding the yearly chances.
Estimating an interval from a ranking is allowed because, over a long record, a flood ranked $m$ in $n$ years has been matched or beaten about $m$ times in $n + 1$ years. Multiplying the chances of no flood in each year is allowed because floods in different years are treated as independent.
Subtracting from one is allowed because at least one flood and no flood at all are the only two possibilities.
Most communities combine measures. A river town might keep its levees, buy out the homes flooded most often, restore wetlands upstream, require new buildings to be raised, and build detention ponds in new subdivisions.
Each measure lowers a different part of the risk: the height of the flood, who lives in its way, or how badly their homes are damaged.
The most common slip is thinking a 100-year flood can only come once a century, or that a recent one makes another unlikely. Another is forgetting to add one to the years of record.
A third is adding yearly chances to get the chance over many years. A fourth is trusting a levee completely: a flood bigger than its design will overtop it.
Tulsa, Oklahoma, once had one of the worst flood records in the United States. On Memorial Day 1984, heavy rain sent creeks through neighborhoods, killing fourteen people and damaging thousands of buildings.
Instead of relying only on levees and channels, the city bought out and removed hundreds of homes in the most flood-prone areas, turned the land into parks and greenways, built detention ponds, and required new building to stay out of the floodplain. It also set up a single agency to manage storm water across the city.
Tulsa went on to earn some of the best flood-management ratings in the country, and its residents pay less for flood insurance as a result. The approach shows soft engineering at work: the floods still come, but far fewer people and buildings are in their way.
In April 1997, the Red River of the North rose far above its predicted crest at Grand Forks, North Dakota. The river topped the city's levees and flooded most of the town; a fire downtown burned buildings that firefighters could not reach through the water. Nearly the whole population had to leave.
The river's forecast had given residents a false sense of safety, because the crest came higher than the levees were built to hold. After the flood, Grand Forks and East Grand Forks, Minnesota, built a much larger system of levees and floodwalls, and bought out and removed hundreds of homes along the river, turning the land into a greenway park.
The combination, higher defenses and fewer homes in the way, reflects the lesson's main point: no structure is proof against every flood, so the safest plans also move people out of harm's way.
The name suggests a timetable: one flood, then a hundred quiet years. But the 100-year flood is simply the flood with a one percent chance every year, and the river does not remember last year's flood. Two can come in a row.
Over a thirty-year mortgage the chance of at least one is about one in four. Treat recurrence intervals as odds for each year, not as a schedule.
A gauge has $49$ years of record, and a flood ranks second. Add one to the years.
$49 + 1 = 50$
The ranking formula.
Divide by the rank.
$\dfrac{50}{2} = 25\ \text{years}$
The interval.
Find the annual chance.
$\dfrac{100}{25} = 4\%$
Each year.
Say what it does not mean.
$\text{exactly every 25 years}$
Only on average.
Find the expected number of 25-year floods in $100$ years.
$\dfrac{100}{25} = 4$
Years over interval.
Find it for 10-year floods in $50$ years.
$\dfrac{50}{10} = 5$
Years over interval.
Find it for 100-year floods in $30$ years.
$\dfrac{30}{100} = 0.3$
Less than one.
Say what 0.3 means.
$\text{about one span in three has one}$
Roughly.
Name the exact chance of at least one.
$1 - 0.99^{30} \approx 26\%$
A little less than 30.
A pond has a floor of $8000$ square meters. Write its storage for depth $d$.
$8000d$
Area times depth.
Find the storage at $1.5$ m.
$8000 \times 1.5 = 12000\ \text{m}^3$
Cubic meters.
A storm sends $15000$ cubic meters of runoff. Find the depth needed.
$\dfrac{15000}{8000} \approx 1.9\ \text{m}$
Volume over area.
The pond is $2$ m deep. Judge it.
$\text{it can hold this storm}$
1.9 is less than 2.
Find what a bigger storm of $20000$ would need.
$\dfrac{20000}{8000} = 2.5\ \text{m}$
Too deep for this pond.
Say what happens then.
$\text{it overflows its spillway}$
Every design has a limit.
Add one to the years.
$59 + 1 = 60$
The ranking formula.
Divide by the rank.
$\dfrac{60}{3} = 20\ \text{years}$
The interval.
Find the annual chance.
A river's gauge has $79$ years of record. Ranked from biggest to smallest, one flood comes in at rank $2$. What is its recurrence interval, in years?
Complete the worked solution: a gauge has $29$ years of record, and a flood ranks number $3$ in size. Find its recurrence interval and its chance, as a percent, of being equaled or exceeded in any year.
Find the recurrence interval.
$\dfrac{\text{years} + \text{one}}{\text{rank}} =$ t
Years, on average.
Find the annual chance.
$\dfrac{\text{one hundred}}{\text{interval}} =$ p
Percent each year.
Say why records matter.
$\text{longer records, better estimates}$
Rare floods need long records.
Say what could change the numbers.
$\text{a changing climate or basin}$
The past may not repeat.
Match each flood-management measure to how it reduces flood damage.
| raises the riverbank so floodwater stays in the channel | holds back floodwater upstream and releases it slowly | keeps new homes off land that floods often | soaks up and slows water before it reaches the river | |
|---|---|---|---|---|
| a levee | ||||
| a dam and reservoir | ||||
| floodplain zoning | ||||
| a restored wetland |
Fill in the chance, as a percent, that each flood is equaled or exceeded in any one year.
| chance each year (%) | |
|---|---|
| the 10-year flood | |
| the 50-year flood | |
| the 100-year flood | |
| the 500-year flood |
A detention pond beside a new subdivision has a flat floor of $8000$ square meters and steep sides. Write the volume of storm water it holds, in cubic meters, as a function of the water's depth $d$ in meters.
Answer:
On average, how many $25$-year floods would you expect in $100$ years?
Answer: floods
Suppose a family in North Carolina buys a house on land that floods in the $100$-year flood, with a $15$-year mortgage. What is the chance, as a percent, that at least one such flood comes before the mortgage is paid off?
Answer: %
Two plans cost the same. A leaves 100 homes exposed with annual damaging-flood probability between 1 and 3 percent; B leaves 20 homes with probability between 2 and 4 percent. Each exposed home suffers one loss unit per damaging event. 9 households request relocation help. Mark every supported sentence for a comparison, including the limit of the modeled recommendation.
This task has no paper form; do it on a device.
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A detention pond beside a new subdivision has a flat floor of $3500$ square meters and steep sides. Write the volume of storm water it holds, in cubic meters, as a function of the water's depth $d$ in meters.
Answer:
You can reason about flood odds. Explain why a town that just had a 100-year flood could have another next year.
24. Your turn: a gauge has $59$ years of record and a flood ranks third. What is its recurrence interval?, step 3
$\dfrac{100}{20} = 5\%$
Each year.