Chapter 17, Problem 26OFT

a)

To determine

With reference to the graph in Box $17.2$, the recurrence interval of a flood with a discharge of $650\text{cubic feet per second}$.

Answer to Problem 26OFT

From the graph, a flood with a discharge of $650\text{cubic feet per second}$ will have $100\text{years}$ of recurrence interval.

Explanation of Solution

The graph given in Box $17.2$ shows the flood frequency and peak discharge. It relates the discharge in a flood to the recurrence interval. Discharge in cubic feet per second is plotted in the ordinate and recurrence interval in years is plotted on the abscissa.

The graph gives a recurrence interval of $100\text{years}$ for a discharge of $650\text{cubic feet per second}$. This is called a $100-\text{year}$ flood. From the recurrence interval, the annual probability can be found using the following formula:

$\text{Annual probability}=\frac{1}{\text{Recurrence interval}}$

Then, the annual probability of a $100-\text{year}$ flood is $1\raisebox{1ex}{$0$}\!\left/ \!\raisebox{-1ex}{$0$}\right.$.

b)

To determine

With reference to the graph in Box $17.2$, the amount of more likeliness of a flood with a discharge of 200 cubic feet per second in comparision with a flood of 400 cubic feet per second of discharge in a given year.

Answer to Problem 26OFT

A flood with a discharge of 200 cubic feet per second would be about $4$ or $5$ times more probable than a flood of 400 cubic feet per second in a given year, based on the graph.

Explanation of Solution

The graph given in Box $17.2$ shows the flood frequency and peak discharge. It relates the discharge in a flood to the recurrence interval. Discharge in cubic feet per second is plotted in the ordinate and recurrence interval in years is plotted on the abscissa.

The recurrence interval of a flood with a discharge of 200 cubic feet per second would be nearly one year, while that of a flood with a discharge of 400 cubic feet per second is around five. Thus, the likeliness of a flood with a discharge of 200 cubic feet per second may be $4$ or $5$ times more than that of a flood with 400 cubic feet per second of discharge in the given year.

c)

To determine

For the purpose of discussion, imagine that the floodplain of the river that gets totally inundated when a flood with an annual probability of 1/300 befalls. Find whether people will build a new home in the floodplain, whether it makes a difference if the last flood with this probability occurred one year ago, and whether it makes a difference if the last flood with this probability occurred one hundred years ago.

Answer to Problem 26OFT

It is certain that the floodplains are not safer sites for new constructions. Suppose for a $50\text{years}$ of ownership of a home, the annual probability for obtaining a $300-\text{year}$ flood in the interval would be nearly $15\raisebox{1ex}{$0$}\!\left/ \!\raisebox{-1ex}{$0$}\right.$, presuming that the topography does not change further, giving rise to a submerged floodplain under less intense situations. Since floods of a given severity are not uniformly spaced over time, it is not significant whether a $300-\text{year}$ flood took place a year ago or before hundred years.

Explanation of Solution

Given info:

The annual probability of the flood is given as $\frac{1}{300}$.

The annual probability or annual exceedance probability represents the likelihood of occurrence of a flood of a given size or more at a definite area in any given year. There is another term related to the occurrence of a flood, which is called the “recurrence interval of a flood”. The recurrence interval of a flood of a given size refers to the average number of years amid the occurrence of successive floods of at least the specified size.

The annual probability and recurrence interval can be related by the following equation:

$\text{Annual probability}=\frac{1}{\text{Recurrence interval}}$

From the above equation, the recurrence interval can be expressed as follows:

$\text{Recurrence interval}=\frac{1}{\text{Annual probability}}$

Substitute the value of annual probability as $\frac{1}{300}$.

$\begin{array}{c}\text{Recurrence interval}=\frac{1}{\frac{1}{300}}\\ =300\text{years}\end{array}$

Thus, it is a $300-\text{year}$ flood.

The annual probability of the $300-\text{year}$ flood can be written in percentage as $0.3\raisebox{1ex}{$0$}\!\left/ \!\raisebox{-1ex}{$0$}\right.$.

Assume that the home ownership on the floodplain is for $50\text{years}$. Then, the probability of its occurrence can be expressed as follows:

$\text{Probability of occurrence}\text{=annual probability}\times \text{number of years}$

Substitute the value of annual probability as $0.3\raisebox{1ex}{$0$}\!\left/ \!\raisebox{-1ex}{$0$}\right.$ and number of years for which the probability is to be calculated as $50\text{years}$.

$\begin{array}{c}\text{Probability of occurrence}\text{=0}\text{.3}\raisebox{1ex}{$0$}\!\left/ \!\raisebox{-1ex}{$0$}\right.\times 50\text{years}\\ =15\raisebox{1ex}{$0$}\!\left/ \!\raisebox{-1ex}{$0$}\right.\end{array}$

Hence, during the $50\text{years}$ interval, the probability of the occurrence of a $300-\text{year}$ flood is $15\raisebox{1ex}{$0$}\!\left/ \!\raisebox{-1ex}{$0$}\right.$, provided that no changes have affected the land, leading to a submerged floodplain under less dangerous situations. It is to be noted that the floods of the given severity do not possess regular intervals through time.

Due to this, it is not important whether a $300-\text{year}$ flood befell a year ago or prior to hundred years. Floodplains are always not the safer places for constructions, as the predictions about the likelihood of flood occurrence are just a possibility (it may or may not occur). Hence, it would be better not to build homes on these plains.