Basic Hydrologic Science Course

Flood Frequency Analysis

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The statistical representation of flood events is affected by both length of the period of record and consistency of the hydrologic conditions within the drainage basin. Unrepresentative data for either of these reasons can lead to inaccurate guidance from flood frequency analysis. This section will introduce basic concepts used in flood frequency analysis and demonstrate calculation of flood statistics. |

In this section you will learn to:

- Explain how the period of record impacts flood frequency guidance.
- Calculate the probability of exceedance or non-exceedance for a given flood magnitude over a specified duration.
- Understand how basin changes may impact the behavior and frequency of floods, thus reducing the length of the period of record.

Topics in this section include:

Period of Record Issues

The Colorado River Compact

Confidence in Return Period Estimates

Exceedance Probability

Example of Exceedance Probability

Independent and Homogeneous Data

Impact of Basin Alterations

Review Questions

Choose a section above by clicking it, or scroll down to continue the module.

Sometimes the record length of river flows may be insufficient for representing the river's entire flow history. Floods or droughts that occur infrequently may be under-represented in a limited streamflow record.

For example, the Mississippi flood of 1993 was a rare event that may only occur once every 100 years. Thus, if the period of record is only 40 years, there is a good chance that a flood like 1993 is not represented in the statistics. On the other hand, if the period of record is 300 years, it is likely that more than one flood like 1993 is in the flood frequency statistics. Thus, a longer period of record results in more representative flood statistics.

For accurate and reliable statistical hydrologic guidance about possible events, it is necessary to use a dataset that includes a representative sample of as many different events as possible.

The longer the period of record, the better the likelihood of capturing the range of possible events.

There are several different return periods traditionally used by hydrologists.

Common return periods include the 2-, 10-, 25-, 50-, 100-, and even 500-year flood. Values for each of these return periods can be calculated based on the statistics of the flow record. However, there is a question of just how representative extreme values, such as the 500-year flood value, might be.

If possible, it is best to avoid estimating return period flood values that are greater than twice the record length. So you might not want to put much faith in your 500-year flood estimate unless you have at least 250 years of data.

One dramatic example on the impact of record length is illustrated by The Colorado River Compact.

In 1922 an agreement was signed between seven states in the western United States governing the use of the water of the Colorado River.

This agreement calls for the river water as measured at Lees Ferry, Arizona to be distributed as follows:

- Over any ten year period, an average annual volume of 7.5 million acre-feet (9,258 cubic hectares) is allocated for the upper basin states (including Colorado, Utah, Wyoming, New Mexico, and parts of Arizona).
- 7.5 to 8.5 million acre-feet is allocated for the lower basin states (including California, Arizona, and Nevada).
- The Republic of Mexico has a separate allocation of 1.5 million acre-feet.
- Together these water allocations total 17.5 million acre-feet.

The problem is that studies have shown that the Colorado River long-term average is much less than the water allocation total. In other words, the river usually does not have enough water for all the allocations.

Original flow estimates were based on a very short period of record, during a relatively high period of flow. Actually the river's long term flow average is about 13 million acre-feet per year. The result is that the river is over-allocated by about 4.5 million acre-feet per year for an average year!

In truth, the annual river flow volumes are highly erratic, ranging from 4.4 million acre-feet to over 22 million acre-feet per year.

Estimates of flood return periods can be made with relatively short periods of record. But the associated confidence level in the flood frequency statistics is much higher with a longer period of data. For example, to estimate a 10-year flood with no more than a ±10 percent error, one would need 90 years of record. If you are willing to accept a ±25 percent error, then only 18 years of record is needed. Here we can see the length of data record needed to be within either ±10 percent or ±25 percent errors for the 10-, 25-, 50-, and 100-year floods.

Sometimes a hydrologist may need to know what the chances are over a given time period that a flood will reach or exceed a specific magnitude. This is called the probability of occurrence or the exceedance probability.

Let's say the value "p" is the exceedance probability, in any given year. The exceedance probability may be formulated simply as the inverse of the return period. For example, for a two-year return period the exceedance probability in any given year is one over two = 0.5, or 50 percent.

**Exceedance probability = 1 - (1 - p) ^{n} **

But we want to know how to calculate the exceedance probability for a period
of years, not just one given year. To do this, we use the formula

1- (1-p)^{n} .** **

In this formula we consider all possible flows over the period of interest "n" and we can represent the whole set of flows with "1." Then (1-p) is the chance of the flow not occurring, or the non-exceedance probability, for any given year.

(1-p)^{n} is all the flows that are less than our flood of interest for the whole time period.

Finally, "1," all possible flows, minus (1-p)^{n}, all flows during
the time period than are lower than our flood of interest, leaves us with
1 - (1-p)^{n}, the probability of those flows of interest occurring
within the stated time period.

This table shows the relationship between the return period, the annual exceedance probability and the annual non-exceedance probability for any single given year.

So, if we want to calculate the chances for a 100-year flood (a table value of p = 0.01) over a 30-year time period (in other words, n = 30), we can then use these values in the formula for the exceedance probability.

We can also use these same values of p and n to calculate the probability of the event *not * occurring in a 30-year period, or the non-exceedance probability.

Let's say you want to know what the probability is for a 50-year flood over a 50-year period. It's not 100 percent!

**Calculation for Probability of 50-Year Flood Over 50-Year Period **

**1 - (1 - p) ^{n}**

**n = 50 **

**p = 0.02 **

We know that n = 50 since we are looking at a 50-year period of time and using the probability of occurrence table we see that p=0.02 for a 50-year return period.

**1 - (1 - 0.02) ^{50} **

**= 1 - (0.98) ^{50}**

So, applying these values in the equation, the (1-p) value is (1-0.02), or 0.98.

**= 1 - 0.36 **

** = 0.64 or 64%**

(1-p) to the n is 0.98 raised to the 50^{th} power. That comes out to 0.36.

Now we have (1-0.36), which is 0.64.

There is a 64 percent chance of a 50-year flood in a 50-year period. That means there is a 36 percent chance we won't see a 50-year flood in the 50-year period.

Now let's determine the probability of a 100-year flood occurring over a 30-year period of a home mortgage where the home is within the 100-year floodplain of a river.

**Calculation for Probability of 100-Year Flood Over 30-Year Period **

**1 - (1 - p) ^{n}**

**n = 30 **

**p = 0.01**

n=30 and we see from the table, p=0.01 .

**1 - (1 - 0.01) ^{30} **

**= 1 - (0.99) ^{30} **

** = 1 - 0.74 **

** (probability of non-occurrence = 0.74) **

** = 0.26 or 26% probability of occurrence **

The 1-p is 0.99, and .99^{30} is 0.74.

There is a 0.74 or 74 percent chance of the 100-year flood not occurring in the next 30 years.

** **But 1-0.74 is 0.26, which shows there is a 26 percent chance
of the 100-year flood in that time.

Flood frequency analysis requires that data be independent and homogeneous. The independence requirement means that floods occur individually and do not influence each other. For example, two peak flows that are above flood stage are independent floods if the flow completely returned to baseflow in between those two events.

On the other hand, if there is not a return to baseflow level between the peaks, the floods are not independent because the first flood has influenced the second flood.

The homogeneous requirement means that each flood needs to occur under the same type of conditions. Two flood events are homogeneous if both are caused by rainfall only.

An example of non-homogeneity is when one flood is caused by rainfall while another flood is caused by a dam failure.

Basin changes such as land development may change the hydrologic properties of a drainage basin over time, thus changing the way a river responds to storms. | |

Such trends may result in future flow behavior that is different from that observed in the past. |

Significant basin alterations violate the requirement of homogeneity for flood frequency analysis.

Basin alterations include urbanization or other land use changes, water diversions, or the construction of reservoirs.

Such changes could affect the behavior and frequency of floods. Consequently, flood frequency statistics generated prior to the basin changes no longer apply.

This can have the effect of dramatically reducing the length of the period of record where homogeneous conditions exist. In other words, if you have a 100-year record, but major urbanization took place 20 years ago, then you really only have a 20-year homogeneous record for your now urbanized basin.

(Choose the best answer.)

a) True

b) False

(Choose the best answer.)

a) 18 years

b) 39 years

c) 50 years

d) 110 years

(Choose the best answer.)

a) True

b) False

(Choose the best answer.)

a) Less than 0.01

b) 0.01

c) 0.04

d) 0.25

(Choose the best answer.)

a) non-exceedance probability

b) exceedance probability

c) null frequency probability

d) exceedance non-probability

(Choose the best answer.)

a) return period

b) homogeneity

c) confidence limits

d) independence

(Choose the best answer.)

a) True

b) False

The correct answer is a).

(Choose the best answer.)

a) 18 years

b) 39 years

c) 50 years

d) 110 years

The correct answer is b).

(Choose the best answer.)

a) True

b) False

The correct answer is a).

(Choose the best answer.)

a) Less than 0.01

b) 0.01

c) 0.04

d) 0.25

The correct answer is c). The 25-year flood has a 1/25 = 0.04, or 4% chance of occurring in any given year.

(Choose the best answer.)

a) non-exceedance probability

b) exceedance probability

c) null frequency probability

d) exceedance non-probability

The correct answer is a).

(Choose the best answer.)

a) return period

b) homogeneity

c) confidence limits

d) independence

The correct answer is d).

**End of Section Two: Statistical Representation of Floods**