Frequency Analysis of Point Rainfall

Storms of high intensity and varying durations occur from time to time.  However, the probability of these heavy rainfalls varies with locality. The first step in designing engineering projects dealing with flood control, gully control etc. is to determine the probability of occurrence of a particular extreme rainfall. This information is determined by the frequency analysis of point rainfall data. Frequency analysis deals with the chance of occurrence of an event over a specified period of time. Suppose, P is the probability of occurrence of an event (rainfall) whose magnitude is equal to or in excess of a specified magnitude X.  The recurrence interval (return period) is related to P as follows:

Plotting Position Formulae

There are two methods for performing frequency analysis

·         Empirical method

·         Frequency factor method

The excedence probability of the event is obtained by the use of empirical formula, known as plotting position.

 Several plotting position formulas were developed and some of them are given below:

  

Table 10.1. Plotting position formulae (Source: Subramanya, 2006)

Description: Description: Description: 101.webp

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

Thus the maximum value is assigned m =1, the second largest value (m =2), and the lowest value m =N. 

N = number of records. 

 Weibull formula is the most commonly used plotting position formula.

·         Having calculated P and T for all the events in the series, the variation of rainfall magnitude is plotted against the corresponding T on semi-log or log-log paper.  

·         The rainfall magnitude for any recurrence interval can be determined by extrapolating the plot between magnitude and recurrence interval. 

·         Empirical procedures can give good results for small extrapolations but the errors increased with the amount of extrapolation.  

·         For more accurate results, analytical methods using frequency factor are used.

 Example

For a station A, the recorded annual 24 h maximum rainfall is given below.

Year

Rainfall (cm)

Year

Rainfall (cm)

Year

Rainfall (cm)

1950

13.0

1957

12.5

1964

8.5

1951

12.0

1958

11.2

1965

7.5

1952

7.6

1959

8.9

1966

6.0

1953

14.3

1960

8.9

1967

8.4

1954

16.0

1961

7.8

1968

10.8

1955

9.6

1962

9.0

1969

10.6

1956

8.0

1963

10.2

1970

8.3

 

 

 

 

1971

9.5

 

(a)  Estimate the 24 h maximum rainfall with return periods of 13 and 50 years.

(b) What would be the probability of a rainfall of magnitude equal to or exceeding 10 cm occurring in 24 h at station A.

 Solution

m

Rainfall, cm

P =m/n+1

T=1/P

m

Rainfall, cm

P =m/n+1

T=1/P

1

16

0.043

23.00

12

9

0.522

1.92

2

14.3

0.087

11.50

13

8.9

 

 

3

13

0.130

7.67

14

8.9

0.609

1.64

4

12.5

0.174

5.75

15

8.5

0.652

1.53

5

12

0.217

4.60

16

8.4

0.696

1.44

6

11.2

0.261

3.83

17

8.3

0.739

1.35

7

10.8

0.304

3.29

18

8

0.783

1.28

8

10.6

0.348

2.88

19

7.8

0.826

1.21

9

10.2

0.391

2.56

20

7.6

0.870

1.15

10

9.6

0.435

2.30

21

7.5

0.913

1.10

11

9.5

0.478

2.09

22

6

0.957

1.05

Rainfall Frequency Curve

Description: Description: Description: 1025.webp

(a)  After interpolating and extrapolating the above graph, we can determine rainfall magnitude for 13 and 50 year return period respectively.

 

          13 year RI   = 14.55 cm

          50 year RI   = 18.00 cm

 

(b)  For Rainfall = 10 cm, T =2.4 years and P = 0.417.

 

Intensity-duration-frequency (IDF) Relationship

In many studies related to watershed management, such as runoff disposal and erosion control, it is necessary to know the rainfall intensities of different durations and return periods.

 The relationship between intensity (i, cm/hr), duration (d, hour) and return period (T, years) can be expressed as follows:

 

    Description: Description: Description: 103.webp                           (10.1)                            

Where K, x, a and n are constants for a given catchment.

Description: Description: Description: 104.webp


                             Intensity –Duration-Frequency Curve

 

Typical values of constants K,, x, a, and n for a few selected cities are given below:

Table – Values of constants in equation (29)

(Source: CSWCRTI- Dehradun)

City

K

x

A

n

Bhopal

6.93

0.189

0.50

0.878

Nagpur

11.45

0.156

1.25

1.032

Chandigarh

5.82

0.160

0.40

0.750

Bellary

6.16

0.694

0.50

0.972

Raipur

4.68

0.139

0.15

0.928

 

Example: Compute 10 year, 1 hour design rainfall intensity for Bhopal and Nagpur.

Solution:

For Bhopal

 

Description: Description: Description: 105.webp          cm/hr

 

For Nagpur

 

Description: Description: Description: 106.webp          cm/hr

 

Example

Perform the frequency analysis using different graphical methods for the following annual rainfall data. Estimate annual rainfall amount with 100 and 50 years return periods.

Year

Annual rain (mm)

Year

Annual rain (mm)

Year

Annual rain (mm)

1950

115

1959

71.4

1968

68.6

1951

96.5

1960

83

1969

67

1952

78

1961

96.5

1970

93

1953

89.5

1962

88.3

1971

108.8

1954

94.7

1963

70.6

1972

104.2

1955

73.3

1964

84.5

1973

89

1956

79.3

1965

92.7

1974

86

1957

87.1

1966

101.8

 

 

1958

124.8

1967

76.4

 

 

 

Solution

 

Year

Annual rain (mm)

Data in Descending Order

Rank (m)

PCa= (m/N)

TCa= 1/Pca

PH = (m-0.5)/N

TH= 1/PH

PW= m/ (N+1)

TW= 1/PW

California

Hazen

Weibull

1950

115

124.8

1

0.04

25.00

0.02

50.00

0.038

26.00

1951

96.5

115

2

0.08

12.50

0.06

16.67

0.077

13.00

1952

78

108.8

3

0.12

8.33

0.1

10.00

0.115

8.67

1953

89.5

104.2

4

0.16

6.25

0.14

7.14

0.154

6.50

1954

94.7

101.8

5

0.2

5.00

0.18

5.56

0.192

5.20

1955

73.3

96.5

6

0.24

4.17

0.22

4.55

0.231

4.33

1956

79.3

96.5

7

0.28

3.57

0.26

3.85

0.269

3.71

1957

87.1

94.7

8

0.32

3.13

0.3

3.33

0.308

3.25

1958

124.8

93

9

0.36

2.78

0.34

2.94

0.346

2.89

1959

71.4

92.7

10

0.4

2.50

0.38

2.63

0.385

2.60

1960

83

89.5

11

0.44

2.27

0.42

2.38

0.423

2.36

1961

96.5

89

12

0.48

2.08

0.46

2.17

0.462

2.17

1962

88.3

88.3

13

0.52

1.92

0.5

2.00

0.500

2.00

1963

70.6

87.1

14

0.56

1.79

0.54

1.85

0.538

1.86

1964

84.5

86

15

0.6

1.67

0.58

1.72

0.577

1.73

1965

92.7

84.5

16

0.64

1.56

0.62

1.61

0.615

1.63

1966

101.8

83

17

0.68

1.47

0.66

1.52

0.654

1.53

1967

76.4

79.3

18

0.72

1.39

0.7

1.43

0.692

1.44

1968

68.6

78

19

0.76

1.32

0.74

1.35

0.731

1.37

1969

67

76.4

20

0.8

1.25

0.78

1.28

0.769

1.30

1970

93

73.3

21

0.84

1.19

0.82

1.22

0.808

1.24

1971

108.8

71.4

22

0.88

1.14

0.86

1.16

0.846

1.18

1972

104.2

70.6

23

0.92

1.09

0.9

1.11

0.885

1.13

1973

89

68.6

24

0.96

1.04

0.94

1.06

0.923

1.08

1974

86

67

25

1

1.00

0.98

1.02

0.962

1.04

 

Year

Annual rain (mm)

Data in Descending Order

Rank (m)

PCh=

(m-0.3)/(N+0.4)

TCh= 1/PCh

PB=

(m-0.44)/(N+0.12)

TB= 1/PB

PG=

(m-3/8)/(N+1/4)

TG= 1/PG

Chegodayev

Blom

Gringorten

1950

115

124.8

1

0.03

36.29

0.02

44.86

0.02

40.40

1951

96.5

115

2

0.07

14.94

0.06

16.10

0.06

15.54

1952

78

108.8

3

0.11

9.41

0.10

9.81

0.10

9.62

1953

89.5

104.2

4

0.15

6.86

0.14

7.06

0.14

6.97

1954

94.7

101.8

5

0.19

5.40

0.18

5.51

0.18

5.46

1955

73.3

96.5

6

0.22

4.46

0.22

4.52

0.22

4.49

1956

79.3

96.5

7

0.26

3.79

0.26

3.83

0.26

3.81

1957

87.1

94.7

8

0.30

3.30

0.30

3.32

0.30

3.31

1958

124.8

93

9

0.34

2.92

0.34

2.93

0.34

2.93

1959

71.4

92.7

10

0.38

2.62

0.38

2.63

0.38

2.62

1960

83

89.5

11

0.42

2.37

0.42

2.38

0.42

2.38

1961

96.5

89

12

0.46

2.17

0.46

2.17

0.46

2.17

1962

88.3

88.3

13

0.50

2.00

0.50

2.00

0.50

2.00

1963

70.6

87.1

14

0.54

1.85

0.54

1.85

0.54

1.85

1964

84.5

86

15

0.58

1.73

0.58

1.73

0.58

1.73

1965

92.7

84.5

16

0.62

1.62

0.62

1.61

0.62

1.62

1966

101.8

83

17

0.66

1.52

0.66

1.52

0.66

1.52

1967

76.4

79.3

18

0.70

1.44

0.70

1.43

0.70

1.43

1968

68.6

78

19

0.74

1.36

0.74

1.35

0.74

1.36

1969

67

76.4

20

0.78

1.29

0.78

1.28

0.78

1.29

1970

93

73.3

21

0.81

1.23

0.82

1.22

0.82

1.22

1971

108.8

71.4

22

0.85

1.17

0.86

1.17

0.86

1.17

1972

104.2

70.6

23

0.89

1.12

0.90

1.11

0.90

1.12

1973

89

68.6

24

0.93

1.07

0.94

1.07

0.94

1.07

1974

86

67

25

0.97

1.03

0.98

1.02

0.98

1.03

 

Example

Develop intensity-duration-frequency curve for given data

5 min

15 min

60 min

120 min

Year

Rainfall

Year

Rainfall

Year

Rainfall

Year

Rainfall

 

mm

 

mm

 

mm

 

mm

1908

0.79

1910

1.34

1916

2.09

1917

2.91

1910

0.71

1914

1.13

1914

1.87

1914

2.58

1918

0.67

1916

1.05

1910

1.64

1911

2.28

1914

0.66

1918

0.97

1915

1.39

1908

2.06

1916

0.6

1909

0.91

1908

1.34

1916

1.77

1912

0.56

1913

0.86

1912

1.27

1913

1.58

1917

0.45

1915

0.84

1918

1.19

1910

1.49

1915

0.39

1917

0.76

1917

1.14

1909

1.45

1913

0.3

1911

0.61

1909

1.08

1912

1.4

1911

0.22

1912

0.56

1911

1.05

1915

1.35

1909

0.15

1908

0.46

1913

1.03

1918

1.28

 

Solution

5 min

15 min

60 min

120 min

Year

Rainfall Intensity

Year

Rainfall Intensity

Year

Rainfall Intensity

Year

Rainfall Intensity

cm/h

cm/h

cm/h

cm/h

1908

0.948

1908

0.536

1908

0.209

1915

0.1455

1921

0.852

1915

0.452

1904

0.187

1908

0.129

1915

0.804

1904

0.42

1915

0.164

1904

0.114

1934

0.792

1921

0.388

1926

0.139

1921

0.103

1929

0.72

1926

0.364

1921

0.134

1926

0.0885

1926

0.672

1934

0.344

1914

0.127

1917

0.079

1931

0.54

1929

0.336

1931

0.119

1914

0.0745

1904

0.468

1931

0.304

1934

0.114

1931

0.0725

1917

0.36

1911

0.244

1929

0.108

1934

0.07

1914

0.264

1917

0.224

1911

0.105

1929

0.0675

1911

0.18

1914

0.184

1917

0.103

1911

0.064

 

 

For 5 min duration

Year

Rainfall

Rainfall Intensity

Rank

P= (m/(n+1))

T= (1/P)

 

mm

cm/h

 

 

 

1908

0.79

0.948

1

0.0833

12.00

1921

0.71

0.852

2

0.1667

6.00

1915

0.67

0.804

3

0.2500

4.00

1934

0.66

0.792

4

0.3333

3.00

1929

0.6

0.72

5

0.4167

2.40

1926

0.56

0.672

6

0.5000

2.00

1931

0.45

0.54

7

0.5833

1.71

1904

0.39

0.468

8

0.6667

1.50

1917

0.3

0.36

9

0.7500

1.33

1914

0.22

0.264

10

0.8333

1.20

1911

0.15

0.18

11

0.9167

1.09

 

For 15 min duration

Year

Rainfall

Rainfall Intensity

Rank

P= (m/(n+1))

T= (1/P)

mm

cm/h

1908

1.34

0.536

1

0.0833

12.00

1921

1.13

0.452

2

0.1667

6.00

1915

1.05

0.42

3

0.2500

4.00

1934

0.97

0.388

4

0.3333

3.00

1929

0.91

0.364

5

0.4167

2.40

1926

0.86

0.344

6

0.5000

2.00

1931

0.84

0.336

7

0.5833

1.71

1904

0.76

0.304

8

0.6667

1.50

1917

0.61

0.244

9

0.7500

1.33

1914

0.56

0.224

10

0.8333

1.20

1911

0.46

0.184

11

0.9167

1.09

 

 

 

For 60 min duration

Year

Rainfall

Rainfall Intensity

Rank

P= (m/(n+1))

T= (1/P)

mm

cm/h

1908

2.91

0.209

1

0.0833

12.00

1921

2.58

0.187

2

0.1667

6.00

1915

2.28

0.164

3

0.2500

4.00

1934

2.06

0.139

4

0.3333

3.00

1929

1.77

0.134

5

0.4167

2.40

1926

1.58

0.127

6

0.5000

2.00

1931

1.49

0.119

7

0.5833

1.71

1904

1.45

0.114

8

0.6667

1.50

1917

1.4

0.108

9

0.7500

1.33

1914

1.35

0.105

10

0.8333

1.20

1911

1.28

0.103

11

0.9167

1.09

 

 

For 120 min duration

Year

Rainfall

Rainfall Intensity

Rank

P= (m/(n+1))

T= (1/P)

mm

cm/h

1908

3.06

0.1455

1

0.0833

12.00

1921

2.73

0.129

2

0.1667

6.00

1915

2.43

0.114

3

0.2500

4.00

1934

2.21

0.103

4

0.3333

3.00

1929

1.92

0.0885

5

0.4167

2.40

1926

1.73

0.079

6

0.5000

2.00

1931

1.64

0.0745

7

0.5833

1.71

1904

1.6

0.0725

8

0.6667

1.50

1917

1.55

0.07

9

0.7500

1.33

1914

1.5

0.0675

10

0.8333

1.20

1911

1.43

0.064

11

0.9167

1.09

  

Recurrence Interval

Rainfall Intensity (cm/h)

T= (1/P), Year

5

15

60

120

min

min

min

min

1.090909091

0.18

0.184

0.103

0.064

1.2

0.264

0.224

0.105

0.0675

1.333333333

0.36

0.244

0.108

0.07

1.5

0.468

0.304

0.114

0.0725

1.714285714

0.54

0.336

0.119

0.0745

2

0.672

0.344

0.127

0.079

2.4

0.72

0.364

0.134

0.0885

3

0.792

0.388

0.139

0.103

4

0.804

0.42

0.164

0.114

6

0.852

0.452

0.187

0.129

12

0.948

0.536

0.209

0.1455

 

 Description: Description: Description: 107.webp