# Recurrence Interval

**Quantitative concepts:**probability# Teaching recurrence intervals

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

*Jump down to:*randomness | forecasting vs. recurrence | How to calculate | Teaching Examples and Resources

## Essential Concepts

Generally, there are four concepts that our students struggle with when studying recurrence intervals:Hydrologic technician for USGS. Details

- 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:High flood level reaches street sign. Details

- flood frequency
- earthquake hazards
- volcanic eruption frequency
- severe weather and storms
- meteor impacts

## 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."

### Teaching 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 number of years in the record (**N**) divided by the the number of events (**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

whereT = (n+1)/m

**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 and 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.):

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 (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.*### Teaching forecasting

Once we start looking to the future, we are looking at forecasting which is governed by the mathematics of probability.
The probability (

The probability

**P**) of an event with recurrence interval**T**isP = 1/T

The probability

**P**that a given event will be equaled or exceeded at least once in the next_{T}**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.## Teaching Examples

**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.**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.**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.

## Resources

**Mount Rainier eruptions (more info)**Data from the U.S.G.S. on Mount Rainier's eruptions in the past 10,000 years could be used to do a simple recurrence interval calculation.**The Earth Impact Database**This database has an updated list of known impact structures, ages and sizes that could be used to examine recurrence intervals of meteor impacts on Earth. The individual data have errors indicated which could stimulate good discussion of errors associated with recurrence interval calculations.