Additional exercises

import matplotlib
if not hasattr(matplotlib.RcParams, "_get"):
    matplotlib.RcParams._get = dict.get

7.4. Additional exercises#

For Exercise 7.7, you have two options: work with Python or with Excel. When applicable, python code is provided below each sub-question on this page. If you prefer working with Excel, please use the Python code below to generate an Excel file with the required data and answer the sub-questions in the chronological order. To generate the Excel file, simply copy the Python code and paste it into a notebook on your laptop. Remove the ‘#’ before the last code line to activate the generation of the Excel file. Once this is done, open the generated Excel file (located in the same folder as your notebook) in Excel.

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import numpy as np
import pandas as pd

years = np.array([1982, 1983, 1984, 1985, 1986])
QmaxY = np.array([211, 64, 99, 122, 165])
data = pd.DataFrame(index = years)
data['Qmax [m3/s]'] = QmaxY
nan_array = np.array([np.nan]*len(data['Qmax [m3/s]']))
data['Rank'] = nan_array
data['Probability of Exceedance'] = nan_array
data['Probability of non-Exceedance'] = nan_array
data['Reduced variate'] = nan_array
data['Return period [years]'] = nan_array
display(data)
#data.to_csv('Data.csv', index=False)

	Qmax [m3/s]	Rank	Probability of Exceedance	Probability of non-Exceedance	Reduced variate	Return period [years]
1982	211	NaN	NaN	NaN	NaN	NaN
1983	64	NaN	NaN	NaN	NaN	NaN
1984	99	NaN	NaN	NaN	NaN	NaN
1985	122	NaN	NaN	NaN	NaN	NaN
1986	165	NaN	NaN	NaN	NaN	NaN

Exercise 7.7

Extreme value analysis

You are asked to apply the Gumbel theory for extreme values to determine the design flow of a hypothetical river. No recent data is available, but you have found a very short time-series spanning over a few years in the 1980s, from which you have calculated the yearly maxima.

	Qmax [m3/s]
1982	211
1983	64
1984	99
1985	122
1986	165

a) First of all, fill in the table below.

	Qmax [m3/s]	Rank	Probability of Exceedance	Probability of non-Exceedance	Reduced variate	Return period [years]
1982	211	NaN	NaN	NaN	NaN	NaN
1983	64	NaN	NaN	NaN	NaN	NaN
1984	99	NaN	NaN	NaN	NaN	NaN
1985	122	NaN	NaN	NaN	NaN	NaN
1986	165	NaN	NaN	NaN	NaN	NaN

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years = np.array([1982, 1983, 1984, 1985, 1986])
QmaxY = np.array([211, 64, 99, 122, 165])
data = pd.DataFrame(index = years)
data['Qmax [m3/s]'] = QmaxY
nan_array = np.array([np.nan]*len(data['Qmax [m3/s]']))
data['Rank [-]'] = data['Qmax [m3/s]'].rank(ascending=False)
data = data.sort_values('Rank [-]')
data['Probability of Exceedance [1/year]'] = data['Rank [-]']/(len(data['Rank [-]'])+1)
data['Probability of non-Exceedance [1/year]'] = 1 - data['Probability of Exceedance [1/year]']
data['Reduced variate [-]'] = -np.log(-np.log(data['Probability of non-Exceedance [1/year]']))
data['Return period [years]'] = (len(data.index.values) + 1) / data['Rank [-]']
data = data.round(3)
display(data)

	Qmax [m3/s]	Rank [-]	Probability of Exceedance [1/year]	Probability of non-Exceedance [1/year]	Reduced variate [-]	Return period [years]
1982	211	1.0	0.167	0.833	1.702	6.0
1986	165	2.0	0.333	0.667	0.903	3.0
1985	122	3.0	0.500	0.500	0.367	2.0
1984	99	4.0	0.667	0.333	-0.094	1.5
1983	64	5.0	0.833	0.167	-0.583	1.2

b) Next, make the Gumbel graph used to estimate the river flow for very large return periods. The code cell below provides you with a template for plotting the Gumbel graph.

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import matplotlib.pyplot as plt

years = np.array([1982, 1983, 1984, 1985, 1986])
QmaxY = np.array([211, 64, 99, 122, 165])
data = pd.DataFrame(index = years)
data['Qmax [m3/s]'] = QmaxY
nan_array = np.array([np.nan]*len(data['Qmax [m3/s]']))
data['Rank [-]'] = data['Qmax [m3/s]'].rank(ascending=False)
data = data.sort_values('Rank [-]')
data['Probability of Exceedance [1/year]'] = data['Rank [-]']/(len(data['Rank [-]'])+1)
data = data.round(3)
data['Probability of non-Exceedance [1/year]'] = 1 - data['Probability of Exceedance [1/year]']
data['Reduced variate [-]'] = -np.log(-np.log(data['Probability of non-Exceedance [1/year]']))
data['Return period [years]'] = (len(data.index.values) + 1) / data['Rank [-]']

X = data['Reduced variate [-]']
y = data['Qmax [m3/s]']

# Computing the coefficients of the regression line (y = mx + b)
X_mean = np.mean(X)
y_mean = np.mean(y)

# Calculating the slope (m)
numerator = np.sum((X - X_mean) * (y - y_mean))
denominator = np.sum((X - X_mean) ** 2)
slope = numerator / denominator

# Calculating the intercept (b)
intercept = y_mean - slope * X_mean

# Plot
X_plot = np.linspace(-2, 7, 100)
fig, ax1 = plt.subplots(figsize=(8,6))
ax1.set_title('Gumbel graph', fontsize='16')
ax1.plot(X_plot, slope * X_plot + intercept, color='red', label='Regression Line')
ax1.plot(data['Reduced variate [-]'], data['Qmax [m3/s]'], '*')
ax1.set_xlim(-2, 7)
ax1.set_ylim(0,500)
ax1.set_ylabel('Discharge [m3/s]', fontsize='12')
ax1.set_xlabel('Reduced variate [-]', fontsize='12')
plt.grid()

# Set scond x-axis
ax2 = ax1.twiny()

# Decide the ticklabel position in the new x-axis,
# then convert them to the position in the old x-axis
newlabel = [1.01, 1.2, 1.5, 2., 5., 10., 20., 50., 100., 200., 500., 1000., 2000.] # labels of the xticklabels: the position in the new x-axis
k2_convert = lambda z: -np.log(-np.log(1-1/z))
newpos   = [k2_convert(z) for z in newlabel]   # position of the xticklabels in the old x-axis
ax2.set_xticks(newpos)
ax2.set_xticklabels(newlabel)

ax2.xaxis.set_ticks_position('bottom') # set the position of the second x-axis to bottom
ax2.xaxis.set_label_position('bottom') # set the position of the second x-axis to bottom
ax2.spines['bottom'].set_position(('outward', 36))
ax2.set_xlabel('Return period [years]',fontsize='14')
ax2.set_xlim(ax1.get_xlim());

../_images/9b52033607450f0063d129d3155f6f37241455cca98f6b332122db6a850df3bf.png

c) Consider only one year. According to the table (i.e. without applying a Gumbel fit), what is the probability that the annual maximum river flow exceeds the maximum flow measured in 1985?

d) Consider only one year. According to the table (i.e. without applying a Gumbel fit), what is the probability that the annual maximum river flow is lower than the maximum flow measured in 1984?

e) Based on the Gumbel distribution, estimate the river flow corresponding to a return period of 55.1 years.

Exercise 7.8

True or false?