1

CIVE 646 - Probabilistic Methods in Hydroscience

Fall 2015

Homework 1

Mohammad Ashifur Rahman

CLID: mxr2467

Problem 1.2

Mean annual maximum flow = 5407.9 m3/s

Standard deviation = 1741.9 m3/s

Histogram:

Cumulative relative frequency diagram:

2

1st quartile = 4002.5 m3/s

Median = 5400 m3/s

3rd quartile = 6815 m3/s

Boxplot:

Probability that flow will exceeded 5000 m3/s during a 12-month period =

1 ( 5000

61)2

= 1 0.59022

= 0.6517

3

Problem 1.3

Mean annual maximum flow = 2837.5 m3/s

Standard deviation = 1314.1 m3/s

Histogram:

Cumulative relative frequency diagram:

1000 2000 3000 4000 5000 6000 7000 8000 90000

5

10

15

20

25

bin

frequency

1000 2000 3000 4000 5000 6000 7000 8000 90000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

4

1st quartile = 1980 m3/s

Median = 2410 m3/s

3rd quartile = 3245 m3/s

Boxplot:

The distribution in problem 1.2 is almost symmetric, but distribution in problem 1.3 is skewed to the

left.

1000

2000

3000

4000

5000

6000

7000

8000

1

5

Problem 1.9

Histogram:

The distribution is skewed to the right.

There are on primary peak at around 70 minute and another secondary peak at around 90 mins.

Probably due to the perception of the speed limit or required speed, the two peaks occurred. The

reason of second peak could also be from rough weather (for example, fog), where many cars traveled

in low speed.

The difference between two peaks is about 20 mins. It took 20 minutes more if the drivers had to lower

the speed due to any disturbance like foggy weather.

40 60 80 100 120 140 160 1800

20

40

60

80

100

120

140

mean time

frequency

6

Problem 1.11

Stem and leaf diagram:

Stem Leaf

80 7

81

82

83

84 6

85

86

87 6

88 5 6

89 6

90 9

91 3

92 2

93

94 0 8

95 0 9

96 9

97 8

98 6 7

99 3 5 7 9

100

101 1 5 7

102 6 8 9

103

104 6 6

105 1

106

107

108

109 0 6

110 0 0

111 2

112 8

113 3 3

7

114 2

115 9

116

117 1

118

119 6 7

120

121 5 8

122 8

123

124

125 9

126 4

127

128

129 0

130

131 8

132 3

133

134 5 9

135 6

136 2

137

138

139

140

141

142 2

143

144

145

146

147

148

149 6

150 1

151

8

152 9

153

154

155

156 4

157

158

159

160

161

162

163

164

165 4

Boxplot:

The distribution is skewed to the right.

800

900

1000

1100

1200

1300

1400

1500

1600

1

9

Problem 1.12

Number of class width, nc = 1 + 3.3 log10 n

N = 123, class width (Sturges) nc = 7.9 8

Number of class width, =

13

2

r = 60, n = 123, iqr = 5.76

Class width (Freedman and Diaconis) = 25.9 26

Histograms showed that the distribution is skewed to the right. Second histogram gives better

illustration of the data.

Boxplot:

The mean (12.4) and median (11.3)

are very close. If only 5 outliers are

removed, it becomes a good

distribution. Therefore, the

contractors claim is not justified.

0 10 20 30 40 50 60 700

10

20

30

40

50

60

bins (Sturges)

Fre

qu

en

cy

0 10 20 30 40 50 60 700

5

10

15

20

25

30

bins (Freedman and Diaconis)

Fre

qu

en

cy

0

10

20

30

40

50

60

1

10

Problem 1.15

Chloride Phosphate

Mean, mg/L 65.2 1.823 Standard Deviation, mg/L 3.3363 0.1992 Coefficient of variation

0.05 0.11

From the coefficient of variation value, it can be seen that Phosphate has more variability in

concentration than Chloride.

Scatter plot:

Correlation coefficient, r = 0.0271, indicates very poor correlation. It is very difficult to predict from this

poor correlation but further analysis of variance could be checked whether other relationships between

these two have any influence on prediction.

54 56 58 60 62 64 66 68 70 72 741.3

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3

Chloride

Phosphate

11

Problem 1.18

Line diagram:

The pattern of the two line diagram is not very different. Hurricane occurrences are more than floods.

There was an increase in 6 number of floods.

0 1 2 3 4 5 6 7 8 90

2

4

6

8

10

12

14

16

18

20

Number of hurricanes

freq

uenc

y

12

Problem 1.19

NO2 CO

Mean 122.25 g/m3 4.5125 mg/ m3 Standard Deviation 16.0779 g/ m3 1.4317 mg/ m3 Coefficient of variation

0.1315 0.3173

From the coefficient of variation value, it can be seen that CO has more variability in concentration than

NO2.

Scatter plot:

Correlation coefficient, r = -0.1522, does not indicate a very strong correlation.

90 100 110 120 130 140 1502.5

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5

NO2 concentration (microgram per cubic meter)

CO

concentr

ation (

mill

igra

m p

er

cubic

mete

r)

13

Problem 1.20

Minutes 5 10 20 30 40 50 60 120 180

Mean 13.48 20.44 31.95 38.66 45.51 52.47 57.9 74.81 83.66

Standard Deviation 5.94 9.15 16.11 20.44 26 31.74 36.83 46.28 46.18

Coefficient of skewness, g1 = 3 ( )

1.23 0.93 0.78 1.15 1.38 1.36 1.45 1.19 1.20

Although mean and median depth of rainfall show continuous increase with time, coefficient of

skewness of depth shows initial decrease, increase and then decrease.

0 20 40 60 80 100 120 140 160 18010

20

30

40

50

60

70

80

90

duration (minutes)

mean

14

0 20 40 60 80 100 120 140 160 1805

10

15

20

25

30

35

40

45

50

duration (minutes)

Sta

ndard

Devia

tion

0 20 40 60 80 100 120 140 160 1800.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

duration (minutes)

Coeff

icie

nt

of

Skew

ness

15

Short storms are localized and depths are predictable but long storms are less predictable. Therefore

mean and standard deviations are increasing with durations but it is difficult to analyze the pattern of

coefficient of skewness with duration since it involves the median as well.

Problem 1.23

Coefficient correlation of

Observed values = 0.6962

Calibration 1 = 0.9946

Calibration 2 = 0.9847

Equation for calibration 1: y = 1.1358x + 3.4508

Equation for calibration 2: y = 1.0938x + 3.6048

1.5 2 2.5 3 3.5 4 4.5 5 5.54

4.5

5

5.5

6

6.5

7

7.5

8

8.5

9

Height (m)

Period (

s)

Observed

Calibration 1

Calibration 2

16

The table has been prepared using those equations.

Observed Height

(m)

Observed Period

(s)

Calibration 1 period

(s)

Calibration 1 period

(s)

Deviation 1 (s)

Deviation 2 (s)

2.26 6.1 6.017708 6.076788 0.082292 0.023212

3.1 4.3 6.97178 6.99558 2.67178 2.69558

3.22 5.7 7.108076 7.126836 1.408076 1.426836

3.84 7.7 7.812272 7.804992 0.112272 0.104992

2.56 5.3 6.358448 6.404928 1.058448 1.104928

2.74 5.7 6.562892 6.601812 0.862892 0.901812

2.28 4.9 6.040424 6.098664 1.140424 1.198664

3.88 6.7 7.857704 7.848744 1.157704 1.148744

2.49 5 6.278942 6.328362 1.278942 1.328362

4.22 6.9 8.243876 8.220636 1.343876 1.320636

2.01 5 5.733758 5.803338 0.733758 0.803338

2.77 5.9 6.596966 6.634626 0.696966 0.734626

3.61 6.5 7.551038 7.553418 1.051038 1.053418

3.51 7.4 7.437458 7.444038 0.037458 0.044038

2.52 5 6.313016 6.361176 1.313016 1.361176

2.12 5.1 5.858696 5.923656 0.758696 0.823656

2.73 6.5 6.551534 6.590874 0.051534 0.090874

3.3 5.4 7.19894 7.21434 1.79894 1.81434

Calibration 1 Calibration 2

Mean of deviations 0.9755 0.9988 Standard Deviation of deviations

0.6699 0.6766

Coefficient of variation of deviations

0.6868 0.6774

The calibration 1 is better than calibration two since calibration 1 has lower Mean of deviations and

Standard Deviation of deviations from the observed value than calibration 2, although Coefficient of

variation of deviations is bigger for calibration 1. Correlation coefficient of calibration 1 is closer to 1

than calibration 2. It will probably be wise to go for calibration 1.