Coefficient Of Variation For The Annual Stream Flow

Published Date: 02 Nov 2017

INTRODUCTION

The whole purpose of this chapter is to analyse extensively the historical data obtained from the Department of Irrigation and Drainage.

The Sungai Langat River basin is an important agro-hydrological watershed in the state of Selangor, it is considered as a major source of raw water supply which meets the demand of large scale irrigation scheme, water treatment plant and other useful purposes. Appropriate distribution and disposal of water and also conservation of water and the environment is very important in this hydrological basin. It’s is based on the mentioned fact above that many hydraulic structures have been constructed and implemented rightly in order to achieve the mentioned goal. Some commonly used low flow frequency distributions were used in the analysis and evaluation of the basin in order to achieve the most appropriate low flow frequency technique which is done by using the 50 years of historical streamflow data of the basin, which started from 1962 to 2012. The recording of this data was done by the Department of Irrigation and Drainage (DID).

This chapter illustrates detailed and explanatory analysis with the deliberated results. The data obtained will be expressed in the form of tables, graphs, charts and many more. It is from these easily expressed forms of presentation that the results will be analysed and discussed.

4.1 CHARACTERISTICS OF THE HISTORICAL DATA

The objective of this study is to analyse the minimum flow (low flow) at the Sungai Langat river basin. For more efficient low flow frequency analysis engineers and hydrologist began to use historical data because it aided in the creation of much more accurate analysis. The data which has been collected from the Department of Irrigation and Drainage will be used to produce the historical data.

For each year of historical data the lowest flow, mean, standard deviation and coefficient of variation was determined, this can be visibly seen in the Table 4.1 which is the descriptive statistics of the annual streamflow data. In the Table 4.2 the mean, median, variance, minimum flow, range and others more where calculated with respect to the lowest flow obtained from the year 1962-2012, this shows a descriptive statistics for the annual minimum flow data for the 50 years duration.

Table 4.1: Descriptive Statistics of the Annual Streamflow Data at the Sg. Langat River Basin (Station No. 2816441)

Year

Lowest flow (m3/s)

Mean (m3/s)

Standard Deviation (m3/s)

Coefficient of Variation

1962

5.15

6.49

0.69

0.11

1963

4.77

5.79

0.83

0.14

1964

5.24

6.11

0.5

0.08

1965

5.06

6.23

0.75

0.12

1966

5.21

6.3

0.57

0.09

1967

5.2

6.42

0.59

0.09

1968

5.05

6.05

0.53

0.09

1969

4.8

5.57

0.48

0.09

1970

4.12

4.71

0.42

0.09

1971

4.1

4.68

0.45

0.10

1972

4

4.48

0.37

0.08

1973

3.85

4.68

0.49

0.10

1974

3.72

4.27

0.34

0.08

1975

3.79

4.38

0.33

0.08

1976

3.42

3.99

0.38

0.10

1977

3.14

3.56

0.32

0.09

1978

2.9

3.53

0.3

0.08

1979

3.06

3.65

0.36

0.10

1980

2.97

3.52

0.36

0.10

1981

3.12

3.64

0.38

0.10

1982

3.08

3.8

0.57

0.15

1983

3.27

3.7

0.3

0.08

1984

3.14

3.98

0.54

0.14

1985

3.15

3.84

0.53

0.14

1986

3.04

3.73

0.51

0.14

1987

2.96

3.58

0.57

0.16

1988

3.22

3.85

0.45

0.12

1989

3.33

4.12

0.52

0.13

1990

3.11

3.72

0.42

0.11

1991

2.98

4.25

0.95

0.22

1992

3.16

4.08

0.88

0.22

1993

2.95

3.93

0.69

0.18

1994

2.55

3.86

0.66

0.17

1995

2.83

3.94

0.86

0.22

1996

3.02

3.71

0.52

0.14

1997

3.02

4.08

0.46

0.11

1998

2.75

3.81

0.45

0.12

1999

2.47

3.69

0.65

0.18

2000

2.47

3.54

0.58

0.16

2001

2.53

3.36

0.49

0.15

2002

2.66

3.16

0.51

0.16

2003

2.85

3.5

0.55

0.16

2004

2.83

3.58

0.73

0.20

2005

2.9

3.31

0.4

0.12

2006

3.03

3.8

0.55

0.14

2007

3.06

3.76

0.46

0.12

2008

2.2

3.68

0.56

0.15

2009

3.01

3.66

0.55

0.15

2010

3.19

3.67

0.41

0.11

2011

3.2

3.75

0.43

0.11

2012

3.15

3.78

0.66

0.17

From the Table 4.1 above shows the descriptive statistics of the annual streamflow data at the Sg. Langat river basin. The lowest flow, mean, standard deviation and coefficient of variation for each year were obtained or calculated from the historical data. From the data ranging from 1962 – 2012. It is seen that the lowest flow of the river basin occurs in 2008 where by the average daily low flow of 2.2 m3/s was recorded on a particular day of that year.

Table 4.2: Statistics for The Annual Minimum flow (1962-2012)

Variables

Values

Mean

3.41

Median

3.12

Mode

3.14

Variance

0.67

Standard deviation

0.82

Skewness coefficient

1.18

Coefficient of variation

0.24

Min flow

2.20

Max flow

5.24

Range

3.04

From the Table 4.2 the descriptive statistics have been calculated with relations to the lowest flow (minimum flow) of each year. The skewness coefficient was calculated to be 1.18. Since 1.18 is a positive value the theory that, if the mean is greater than the mode, the skewness coefficient is positive was fully confirmed.

4.2 GRAPHICAL ANALYSIS OF THE DESCRIPTIVE STATISTICS OF THE ANNUAL STREAMFLOW DATA AT THE SG. LANGAT RIVER BASIN

Using the Table 4.1 graphs were plotted to give a descriptive analysis of the annual stream flow. The annual minimum flow (lowest flow), average flow (mean), standard deviation and coefficient of variation were plotted against the year duration. This is represented in the figures 4.1 to 4.4.

4.2.1 Annual Minimum flow

Figure 4.1: Annual minimum flows at the Dengkil gauging station in the Sungai Langat river basin (1962-2012)

The figure 4.1 shows the annual minimum flow which has been plotted graphically. It can be visibly seen that from the period of 1962-1969 a low flow of about 5.24 – 4.8 m3/s was maintained. It was in 1970 that a low flow of 4.12 m3/s was recorded which showed a visible drop compared to the past years.

In 1978 it dropped to another minimum of 2.9 m3/s. A steady low flow was maintained between 2.9 m3/s and 3.27 m3/s from 1978 to 1988, after which a high low flow was recorded at 3.33 m3/s in 1989. From 1989 to 2012 low flow has been in the range of between 3.33 m3/s and 2.47 m3/s, with the exception of 2008 which had a record low flow of 2.2 m3/s.

It is worth noting that the highest and lowest minimum flow recorded in the river basin is 5.24 m3/s recorded in 1964 and 2.2 m3/s recorded in 2008 respectively, also the average minimum flow value is 3.41 m3/s.

4.2.2 Annual Average flow (mean flow)

Figure 4.2: Mean for the Annual Daily stream flows at the Dengkil gauging station in the Sungai Langat river basin (1962-2012)

From the graphical figure 4.2 it is visible that the highest mean flow of 6.49 m3/s happens to be recorded in 1962. From 1963 to 1967 the average flow was between 6.42 m3/s and 5.79 m3/s. A slump downwards from 1968 is visible until a steady mean flowing between 3.16 m3/s and 4.71 m3/s was recorded from 1970 – 2012.

It is worth noting that the highest and lowest average flow recorded in the river basin is 6.49 m3/s recorded in 1962 and 3.16 m3/s recorded in 2002 respectively, also the mean value of the average flow is 4.20 m3/s.

4.2.3 Annual Standard Deviation Flow

Figure 4.3: Standard Deviation for the Annual Daily stream flows at the Dengkil gauging station in the Sungai Langat river basin (1962-2012)

The figure 4.3 shows the standard deviation for 51 years of stream flow data which has been graphically plotted. The plotted values mostly go in a zigzag manner (therefore there is much high to low transition) with the exception of, from 1967 to 1970, where a visible downward slump is seen between the range of 0.59 – 0.42 m3/s.

It is worth noting that the highest and lowest flow of standard deviation recorded in the river basin is 0.95 m3/s recorded in 1991 and 0.3 m3/s recorded in 1978 and 1983 respectively, also the average value of the flow is 0.53 m3/s.

4.2.4 Coefficient of Variation For The Annual Stream flow

Figure 4.4: Coefficient of Variation for the Annual Daily stream flows at the Dengkil gauging station in the Sg. Langat river basin (1962-2012)

The coefficient of variation can be simply described as the ratio of standard deviation to the mean. In the figure 4.4 the coefficient of variation has been calculated for each year from 1962 to 2012 and the graphical illustration has been shown.

It is worth noting that the highest and lowest coefficient of variation recorded in the river basin is 0.224 m3/s recorded in 1991 and 0.075 m3/s recorded in 1975 respectively, also the average value of coefficient of variation is 0.128 m3/s.

4.3 ESTIMATION OF LOW FLOW FREQUENCY ANALYSIS

For the estimation of low flow frequency analysis, two methods will be utilized, which are namely ; Gumbel distribution and the flow duration method.

For the Gumbel method the long period of recorded data obtained from the stream flowing gaging station, will be used to develop a regional regression equation. This regression equation aids in the development of an accurate stream flowing statistics of natural flow conditions. The regression equations developed will belong to the 1, 2, 3, 5, 7, 10, 15 and 30 consecutive day’s low flow with recurrence interval of 51 years.

A flow duration curve is a tool which describes the relationship between the magnitude and frequency of daily stream flow of the river basin. It will provide an estimation representing the percentage of time a given stream flow which was equalled or exceeded over the 51 years period.

4.3.1 Gumbel Distribution

4.3.1.1 Low flow estimation for 1LQ (1 DAY LOW FLOW)

Figure 4.5: Exceedance probability for 1LQ (1 day low flow) Analysis for the Sg. Langat river basin (1962-2012)

The 1 day low flow signifies the average 1 day minimum flow that can be obtained in the river basin annually

From the figure 4.5 the non-exceedance is plotted versus the discharge. It is seen from the linear trend-line, that as the non-exceedance probability increases the discharge also increases, therefore it can be said that the graph is a directly proportional graph. From the graph we can also see that a linear equation has been derived which is

Y= 0.0256X + 2.1267

Where by X = Non-exceedance probability (%)

Y = Discharge (m3/s)

The equation formed highlights the main bases of the Gumbel distribution. This equation will be used to estimate the magnitudes of low flows with different reliabilities, simply meaning that, the probability that a given discharge will not be equalled or exceeded.

An example is, if the non exceedance probability is 60%, it means that 60% is the probability that a discharge of 3.66m3/s will not be equalled or exceeded in the river basin. The application of low flow is very essential in water supply ad management.

4.3.1.2 Low flow estimation for 2LQ (2 DAYS LOW FLOW)

Figure 4.6: Exeedence probability for 2LQ (2 day low flow) Analysis for the Sg. Langat river basin (1962-2012)

The 2 day low flow signifies the average 2 consecutive days minimum flow that can be obtained in the river basin annually

From the figure 4.6 the non-exceedence is plotted versus the discharge. It is seen from the linear trend-line, that as the non-exceedence probability increases the discharge also increases, therefore it can be said that the graph is a directly proportional graph. From the graph we can also see that a linear equation has been derived which is

Y= 0.0248X + 2.2025

Where by X = Non-exceedance probability (%)

Y = Discharge (m3/s)

The equation formed highlights the main bases of the Gumbel distribution. This equation will be used to estimate the magnitudes of low flows with different reliabilities, meaning that, the probability that a given discharge will not be equalled or exceeded.

An example is, if the non exceedance probability is 70%, it means that 70% is the probability that a discharge of 3.94m3/s will not be equalled or exceeded in the river basin.

4.3.1.3 Low flow estimation for 3LQ (3 DAYS LOW FLOW)

Figure 4.7: Exceedance probability for 3LQ (3 day low flow) Analysis for the Sg. Langat river basin (1962-2012)

The 3 day low flow signifies the average 3 days consecutive minimum flow that can be obtained in the river basin annually

From the figure 4.7 the non-exceedence is plotted versus the discharge. It is seen from the linear trend-line, that as the non-exceedence probability increases the discharge also increases, therefore it can be said that the graph is a directly proportional graph. From the graph we can also see that a linear equation has been derived which is

Y= 0.0247X + 2.2278

Where by X = Non-exceedance probability (%)

Y = Discharge (m3/s)

The equation formed highlights the main bases of the Gumbel distribution. This equation will be used to estimate the magnitudes of low flows with different reliabilities, meaning that, the probability that a given discharge will not be equalled or exceeded.

An example is, if the non exceedance probability is 80%, it means that 80% is the probability that a discharge of 4.20m3/s will not be equalled or exceeded in the river basin.

4.3.1.4 Low flow estimation for 5LQ (5 DAYS LOW FLOW)

Figure 4.8: Exeedence probability for 5LQ (5 day low flow) Analysis for the Sg. Langat river basin (1962-2012)

The 5 day low flow signifies the average 5 consecutive days minimum flow that can be obtained in the river basin annually

From the figure 4.8 the non-exceedence is plotted versus the discharge. It is seen from the linear trend-line, that as the non-exceedence probability increases the discharge also increases, therefore it can be said that the graph is a directly proportional graph. From the graph we can also see that a linear equation has been derived which is

Y= 0.024X + 2.3018

Where by X = Non-exceedance probability (%)

Y = Discharge (m3/s)

The equation formed highlights the main bases of the Gumbel distribution. This equation will be used to estimate the magnitudes of low flows with different reliabilities, meaning that, the probability that a given discharge will not be equalled or exceeded.

An example is, if the non exceedance probability is 90%, it means that 90% is the probability that a discharge of 4.46m3/s will not be equalled or exceeded in the river basin.

4.3.1.4 Low flow estimation for 7LQ (7 DAYS LOW FLOW)

Figure 4.9: Exeedence probability for 7LQ (7 day low flow) Analysis for the Sg. Langat river basin (1962-2012)

The 7 day low flow signifies the average 7 consecutive days minimum flow that can be obtained in the river basin annually. It can also be described as the average weekly flow expected at the river basin annually.

From the figure 4.9 the non-exceedence is plotted versus the discharge. It is seen from the linear trend-line, that as the non-exceedence probability increases the discharge also increases, therefore it can be said that the graph is a directly proportional graph. From the graph we can also see that a linear equation has been derived which is

Y= 0.0243X + 2.3057

Where by X = Non-exceedance probability (%)

Y = Discharge (m3/s)

The equation formed highlights the main bases of the Gumbel distribution. This equation will be used to estimate the magnitudes of low flows with different reliabilities, meaning that, the probability that a given discharge will not be equalled or exceeded.

An example is, if the non exceedance probability is 50%, it means that 50% is the probability that a discharge of 3.52m3/s will not be equalled or exceeded in the river basin.

4.3.1.5 Low flow estimation for 10LQ (10 DAYS LOW FLOW)

Figure 4.10: Exeedence probability for 10LQ (10 day low flow) Analysis for the Sg. Langat river basin (1962-2012)

The 10 day low flow signifies the average 10 consecutive days minimum flow that can be obtained in the river basin annually

From the figure 4.10 the non-exceedence is plotted versus the discharge. It is seen from the linear trend-line, that as the non-exceedence probability increases the discharge also increases, therefore it can be said that the graph is a directly proportional graph. From the graph we can also see that a linear equation has been derived which is

Y= 0.0243X + 2.3327

Where by X = Non-exceedance probability (%)

Y = Discharge (m3/s)

The equation formed highlights the main bases of the Gumbel distribution. This equation will be used to estimate the magnitudes of low flows with different reliabilities, meaning that, the probability that a given discharge will not be equalled or exceeded.

An example is, if the non exceedance probability is 60%, it means that 60% is the probability that a discharge of 3.79m3/s will not be equalled or exceeded in the river basin.

4.3.1.6 Low flow estimation for 15LQ (15 DAYS LOW FLOW)

Figure 4.11: Exeedence probability for 15LQ (15 day low flow) Analysis for the Sg. Langat river basin (1962-2012)

The 15 day low flow signifies the average 15 consecutive days minimum flow that can be obtained in the river basin annually

From the figure 4.11 the non-exceedence is plotted versus the discharge. It is seen from the linear trend-line, that as the non-exceedence probability increases the discharge also increases, therefore it can be said that the graph is a directly proportional graph. From the graph we can also see that a linear equation has been derived which is

Y= 0.0244X + 2.3787

Where by X = Non-exceedance probability (%)

Y = Discharge (m3/s)

The equation formed highlights the main bases of the Gumbel distribution. This equation will be used to estimate the magnitudes of low flows with different reliabilities, meaning that, the probability that a given discharge will not be equalled or exceeded.

An example is, if the non exceedance probability is 40%, it means that 40% is the probability that a discharge of 3.35m3/s will not be equalled or exceeded in the river basin.

4.3.1.7 Low flow estimation for 30LQ (30 DAYS LOW FLOW)

Figure 4.12: Exeedence probability for 30LQ (30 day low flow) Analysis for the Sg. Langat river basin (1962-2012)

The 30 day low flow signifies the average 30 consecutive days minimum flow that can be obtained in the river basin annually. It can also be described as the average monthly flow expected at the basin annually

From the figure 4.12 the non-exceedence is plotted versus the discharge. It is seen from the linear trend-line, that as the non-exceedence probability increases the discharge also increases, therefore it can be said that the graph is a directly proportional graph. From the graph we can also see that a linear equation has been derived which is

Y= 0.0247X + 2.4527

Where by X = Non-exceedance probability (%)

Y = Discharge (m3/s)

The equation formed highlights the main bases of the Gumbel distribution. This equation will be used to estimate the magnitudes of low flows with different reliabilities, meaning that, the probability that a given discharge will not be equalled or exceeded.

An example is, if the non exceedance probability is 70%, it means that 70% is the probability that a discharge of 4.18m3/s will not be equalled or exceeded in the river basin.

Table 4.3: Linear Equations For The Different Low Flow Estimations

Low flow estimation

Linear equation

1LQ

Y= 0.0256X + 2.1267

2LQ

Y= 0.0248X + 2.2025

3LQ

Y= 0.0247X + 2.2278

5LQ

Y= 0.024X + 2.3018

7LQ

Y= 0.0243X + 2.3057

10LQ

Y= 0.0243X + 2.3327

15LQ

Y= 0.0244X + 2.3787

30LQ

Y= 0.0247X + 2.4527

Where ‘Y’ is the discharge value and ‘X’ is the non-exceedance probability in percentage, A linear equation was formed for each low flow estimation

Figures 4.3.1.1 - 4.3.1.7 shows the relationship between the discharge and the non-exceedance probability for each low flow estimation, it can be noticed that the linear trend lines are almost the same and this can be seen in the similarity of the equations for estimation. This linear equations are essential in aiding to find different discharge availability in the river basin.

4.3.2 Estimation Of River Discharge Availability For Different Low Flow Patterns

From the different low flow estimations, (i.e. the 1, 2, 3, 5, 7, 10, 15, 30 days low flow) valuable calculations for various discharge availability were made from the linear equations obtained in the table 4.3. the table 4.4 shows the result of this calculations

Table 4.4: Discharge Availability For Different Low Flow Patterns

Reliability(%)

1LQ (m3/s)

2LQ (m3/s)

3LQ (m3/s)

5LQ (m3/s)

7LQ (m3/s)

10LQ (m3/s)

15LQ (m3/s)

30LQ (m3/s)

50

3.41

3.44

3.46

3.50

3.52

3.55

3.60

3.69

60

3.66

3.69

3.71

3.74

3.76

3.79

3.84

3.93

70

3.92

3.94

3.96

3.98

4.01

4.03

4.09

4.18

80

4.17

4.19

4.20

4.22

4.25

4.28

4.33

4.43

90

4.43

4.43

4.45

4.46

4.49

4.52

4.57

4.68

Figure 4.13: Discharge availability for different low flow conditions (estimations)

From the figure 4.13 the different low flow estimations are shown in the graph. For each estimation, the probability of discharge can be obtained from the graph. Basically it can be seen that for a certain probability e.g. 60% non exceedance probability, the discharge in the river increases as the low flow estimation increases (i.e. the increase from average 3 days consecutive low flow to average 5 days consecutive low flow). from the figure the discharge availabilities can also be predicted. For example, for the average 5 consecutive days low flow over a period of 51years, there is 70% probability that the flow in the river will not exceed 3.98m3/s, and for the average weekly consecutive days low flow (7 days) there is 70% probability that the flow in the river will not exceed 4.01m3/s. This calculated prediction helps water supply and management immensely and effectively.

4.4 FLOW DURATION CURVE METHOD

Flow duration curve gives the percentage of time that a specified stream flow discharge is exceeded or equalled during a given period of time. The duration curves are a ranked representation of frequency curves. Flow duration curves are designed by sorting out the daily mean flows for the period recorded from largest to smallest (ranking), then class boundaries are assigned in the same descending order, thereby producing a histogram from this histogram, the cumulative frequency curve (flow duration curve or frequency of non exceedance curve) can be produced. The figures and tables below will give more detailed explanation on how this is carried out.

Table 4.5: Frequency of values obtained for each class interval

class interval (m3/s)

Frequency

0.0 – 3.0

500

3.0 - 3.5

4274

3.5 – 4.0

4073

4.0 - 4.5

2550

4.5 – 5.0

1212

5.0 - 5.5

959

5.5 – 6.0

840

6.0 - 6.5

650

6.5 – 7.0

542

7.0 - 7.5

351

7.5 – 8.0

45

8.0 - 8.5

1

8.5 – 9.0

1

9.0 - 9.5

1

It is known that the lowest and highest low flow discharge in the Sungai Langat river basin over a period of 51 years is 2.2m3/s and 9.17m3/s respectively. From this knowledge 14 class intervals were made. For each class interval discharge the frequency of occurrence was noted, for example, a river discharge of equal or greater than 5.0 m3/s but less than 5.5 m3/s (5.0 – 5.5) occurred 959 times in the river basin. Based on the table 4.5 a histogram is constructed.

Figure 4.14: A histogram showing the magnitude of flow and its frequency of occurrence over a period of 51 years in the Sg. Langat river basin.

The histogram shows a proper representation of the distribution of data which has to do with the flow in the river and its occurrences. A flow duration curve can be obtained after constructing the histogram .

Table 4.6: Flow duration table

class interval (m3/s)

Midpoint (m3/s)

frequency

P (%)

Cumulative P (%)

9.00 - 9.50

9.25

1

0.01

0.01

8.50 - 9.00

8.75

1

0.01

0.01

8.00 - 8.50

8.25

1

0.01

0.02

7.50 - 8.00

7.75

45

0.28

0.30

7.00 - 7.50

7.25

351

2.19

2.49

6.50 - 7.00

6.75

542

3.39

5.88

6.00 - 6.50

6.25

650

4.06

9.94

5.50 - 6.00

5.75

840

5.25

15.19

5.00 - 5.50

5.25

959

5.99

21.19

4.50 - 5.00

4.75

1212

7.58

28.76

4.00 - 4.50

4.25

2550

15.94

44.70

3.50 - 4.00

3.75

4073

25.46

70.16

3.00 - 3.50

3.25

4274

26.71

96.87

0.00 - 3.00

1.50

500

3.13

100.00

TOTAL

15999

100.00

Figure 4.15: Flow duration curve for the year 1962-2012

From the flow duration curve the time of exceedance for each discharge can be determined, that is, the flow discharge which can be expected to be exceeded in a percentage of time. The table 4.7 will be showing this percentiles and the various discharges, which are expected to be exceeded

Table 4.7: Exceedance percentiles and its discharge

PERCENTILES (%)

DISCHARGE (m3/s)

1

7.27

2

7.1

5

6.64

10

6

20

5.11

30

4.45

50

3.88

80

3.36

90

3.19

95

3.07

98

2.95

99

2.84

The table 4.7 illustrate the exceedance percentile and its discharge. This table is used to predict the percentage time that a flow will be equalled or exceeded.

For example, The exceedance percentile 80% can be interpreted as, 3.36m3/s river discharge which is expected to be exceeded 80% of the time in the Sungai Langat river basin, Therefore the flow in the river basin will exceeded 3.36m3/s, 80% of the time over a period of 50 years. it is best to work with the 95th percentile, so from the graph the flow in the river basin will exceed 3.07m3/s , 95% of the time.