Question

please show all work.

The maximum annual peak discharges of the Wonder River for eleven years are given in the following table. a) Prepare a flood-frequency curve using plotting position empirical method. Use the graph provided below to plot the flood-frequency curve. b) According to the flood-frequency curve you constructed: - What would be the discharge during a 20-year flood? - What is the recurrence interval for a flood with a discharge of 1800 cfs? c) A small dam on this river is designed for 20-year exceedance flood. What is the probability of the design flood occurring exactly twice in the next 50 years? Year 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 Discharge (eſs) 800 1080 692 1380 1280 889 2030 890 1480 1600 1330 3000 2500 2000 Discharge (cis) 1500 1000 500 1.0 1.1 1.3 50 100 2 3 4 5 10 15 20 30 Recurrence Interval (years) 2

Answer 1

The exceedance probability of the event is obtained by the use of empirical formula known as plotting position. Several plotting formulas were developed and some of them are given below:

Method	P (probability)
California	$\frac{m}{N}$
Hazen	$\frac{m-0.5}{N}$
Weibull	$\frac{m}{N+1}$
Chegodayev	$\frac{m-0.3}{N+0.4}$
Blom	$\frac{m-0.44}{N+0.12}$

m = rank assigned to the data after arranging them in the descending order of magnitude

N = number of records

Weibull formula is the most widely used empirical plotting position formula

(a) Discharge is arranged in descending order and ranks (m) has been assigned. Probability is calculated using Weibull's formula, P= m/N+1, where N= 11 in this case. Reccurence Interval T = 1/P.

m	Discharge (cfs)	P=m/N+1	T=1/P
1	2030	0.083333333	12
2	1600	0.166666667	6
3	1480	0.25	4
4	1380	0.333333333	3
5	1330	0.416666667	2.4
6	1280	0.5	2
7	1080	0.583333333	1.714285714
8	890	0.666666667	1.5
9	889	0.75	1.333333333
10	800	0.833333333	1.2
11	692	0.916666667	1.090909091

The flood frequency curve has been plotted as shown in figure:

3000 2500 2000 Discharge (cfs) 1500 1000 500! 1.0 100 1.1 1.3 2 3 4 5 10 15 20 30 50 Recurrence Interval (years)

(b) For a flood having return period T = 20 years, From the figure, the corresponding discharge value is obtained as Q = 2430 cfs (The graph has been extrapolated)

3000 2500 24301 2000 Discharge (cfs) 1500 1000 500! 1.0 1.1 1.3 100 2 3 4 5 10 15 20 30 50 Recurrence Interval (years)

For a flood having discharge Q = 1800cfs, From the figure, the corresponding reccurence interval = 8 years

3000 2500 Discharge (cfs) 2000 1800 1500 1000 500! 1.0 1.1 1.3 100 2 3 4 5 8 10 15 20 30 50 Recurrence Interval (years)

(c) Reccurence Interval T = 20 years

Probability of occurence of event p = 1 / Reccurence Interval = 1/T = 1/20

q = 1- p = 1- 1/20 = 19/20

The probability(P) of occurence of the event r times in n successive years is given by:

$P_{r,n}=^{n}C_{r}p^{r}q^{n-r}$

The probability of occurence of the flood exactly twice in next 50 years :

$P_{2,50}=^{50}C_{2}\left ( \frac{1}{20} \right )^{2}\left ( \frac{19}{20} \right )^{50-2}= \mathbf{0.2611}$