Discharge Magnitude-Frequency Lab: CSULB

Geography 140

Introduction to Physical Geography

Lab 8: Flood Discharge Magnitude-Frequency Relationships

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This lab will familiarize you with magnitude-frequency relationships in risk analysis. Many natural hazards show a pattern where low magnitude events happen pretty frequently, while high magnitude events are rare. Earthquakes, taken as a whole from around the world, fit this pattern. Asteroid and comet impacts with Earth or its atmosphere are another hazard that behaves this way. So do floods along a particular stretch of river.

Floods (and other hazards) can be characterized by magnitude, how large they are. With floods, a common measure of magnitude is discharge, or how much water in the stream bed is flowing past a certain point in a given period of time. For example, discharge could be given in cumecs or cubic meters per second or, as in this lab, as cfs or cubic feet per second.

You can rank each year or event by its discharge. So, the biggest flood would be ranked #1, the second biggest as #2, the third biggest as #3, and so on until you have ranked even the tiniest event.

Once you've ranked your data set, you are in a position to calculate recurrence intervals or frequencies. Recurrence interval means the average number of years between one flood of a given magnitude and the next flood that is as big or bigger. If you were interested in a flood of, say, magnitude 5,000 cumecs, you would figure out the average time it would take to get another flood of at least 5,000 cumecs.

Once you have recurrence interval, you can figure out the probability that a flood of a given magnitude will hit during any particular year. Probability is merely the reciprocal of recurrence interval, that is, 1 divided by the recurrence Interval ( 1 / I ).

Something else you can figure out from recurrence interval is the magnitude of a specified level of flood. In other words, you can estimate the size of the 10 year flood (the flood level that has a 0.10 or 10% chance of happening in any given year: 0.1 is the reciprocal of 10). Planners and insurance companies like to think in nice, round recurrence intervals (e.g., the "100 year flood" or the "500 year flood"), but we're not going to be that ambitious in this lab.

The data used for this lab come from the USGS WATSTORE water data warehouse. They represent a 20 year run of peak discharge data taken from station 11098000 in Arroyo Seco near Pasadena for every flood year from 1983 (2 March 1983 through 28 January 2002). The peak discharge event for each water year is listed by the date it was recorded. Peak discharge is given in cubic feet per second (cfs).

What's a "water year"? It is conventionally defined as the twelve month period from October 1st through September 30th. The water year is named for the calendar year in which it ends (which includes nine of the twelve months). For example, the year ending on September 30th of 2005 is called the "2005 water year." Because of this water year business, you will notice a couple of peak discharge events may have occurred during the same calendar year (e.g., 1996) and some calendar years are missing an event (e.g., 1997)

Lab A: Your Data

It's about time you see your data. You can download them and a handy answer sheet at https://home.csulb.edu/~rodrigue/geog140/labs/floodLAbasic.pdf. When you do, you'll find two columns of data: Date of the biggest flood in each water year is in Column A and Discharge is in Column B. There are three empty columns, headed by Rank, Recurrence Interval, and Probability. These are the columns that you'll be filling in during this lab.
Show Rank as an integer running from 1 for the biggest flood to 20 for the smallest flood. Show Recurrence Interval and Probability to two decimals places of accuracy (and don't forget the proper rules for rounding!).

Lab B: Calculate Ranks, Estimated Recurrence Intervals, and Flood Probabilities

First, you need to rank each flood from highest (1) to lowest (20). Be careful here! One thing to watch out for is a tie. If two water years have the same peak discharge, give both of them the average rank of the two of them. So, if they are tied for 8th, say, you'd give them both a rank of 8.5.
Then, the formula for recurrence interval for each rank is:
I = (n + 1) / r
Where:
I = Interval of recurrence
n = number of years for which you have data
r = rank of a particular magnitude of event, e.g., 509 cfs
To estimate probabilities of recurrence in any one year, the formula is:
P = 1/ I
Where:
P = Probability of recurrence
I = Interval of recurrence

Lab C: Analysis

Calculate the average peak discharge for this stream and put on the answer sheet
Calculate the middle score or median (which will be the average of the 10th ranked flood and the 11th ranked flood)
Averages and medians are statistical measures of central tendency. In normal distributions (like your class grades, but I digress...), the average and the median are very close. In skewed distributions, the average is pulled away from the median by extreme and unbalanced scores. Are the average and the median close in this set of records?
Why might that be? Put your answer on the back of the answer sheet.
On the back of your answer sheet, create a chart with Rank along the X axis and Discharge along the Y axis. Create a line chart (or a bar chart, if you prefer) showing the discharge for each rank. This graph illustrates the magnitude-frequency relationship: The biggest flood has the highest rank (lowest number), which is associated with the least frequent and lowest probability of recurrence.
So, what would be the discharge associated with the first ranked flood?
_______________ cfs

What is the recurrence interval for a flood of 217 cfs?
_______________ years

What is the probability that next year will equal or exceed 1,710 cfs?
_______________
What is the rank of the 2,640 cfs flood?
_______________
What is the discharge associated with the 14th ranked flood?
_______________ cfs
What is the recurrence interval of the flood that has a 0.10 probability of occurring next year?
_______________ years
What is the probability of getting a flood of at least 4,380 cfs next year?
_______________