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Design Flood Estimation

1. Estimation of Peak Discharge

The peak discharge is estimated using the GIS based Clarks Unit Hydrograph Model. Clarks synthetic unit hydrograph methodology involves the application of a unit excess rainfall (1 mm) over the watershed. As mentioned previously, the precipitation is conveyed to the catchment outlet by a translation hydrograph and linear reservoir routing. For this purpose, the time of concentration value (Tc), the storage attenuation coefficient (R) and the time area histogram of the catchment are necessary. The methodology used is described through the following Flowchart (Figure 1).

1.1 Time of concentration

Time of concentration is the time required for excess rainfall to travel from the most remote point of the catchment to the outlet. At the end of this time, entire catchment will be contributing to the flow at the outlet. In literature, several equations are available for calculation of time of concentration (ASCE, 1996). After considering the availability of parameters the following SCS equation is selected for this study.

0.70.8

0.5

1000 9

190cL

CNTS

= (1)

where, Tc is the time of concentration in minutes, L is the longest flow path of the catchment in feet, CN is the average Curve Number value of the watershed and S is the average watershed slope. Longest flow path is the maximum value of Flowlength Grid (FlGrid). As described in the next section, the FlGrid represents travel distance of each cell in the catchment to the outlet. In this study two FlGrids are calculated: for weight condition and no weight condition. Average CN value of the catchment was determined as 78. Average watershed slope is derived from slope grid (SGrid) of the catchment, which is obtained from DEM. Using Eq. (1), the value of Tc is estimated as 7.5 hours.

1.2 Storage attenuation coefficient The storage attenuation coefficient, which represents the storage effect of stream channel, is calculated from an observed flood hydrograph of the catchment. However, when observed discharge data is not available, it can be considered between 2 to 4 hours. Sincce it is small catchment therefore, the value of storage coefficient is cansidered 2 hours.

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Figure 1. Flowchart of the methodology

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1.3 Time area histogram

The area of the catchment is divided into travel time zones. Each zone represents the part of catchment, which drains the unit excess rainfall to the outlet at a certain time interval. The plot of these areas with respect to corresponding time intervals gives the time-area histogram of the catchment. It is the most important parameter of the methodology, since it reflects the runoff response of the catchment to the rainfall at the outlet. The digital elevation model of the catchment is used for determination of time-area histogram. For this purpose, the direction of flow from each cell is found first. Than by tracing the flowdirection, travel distance of flow from each cell to the catchment outlet is calculated. These distances are turned to travel time values. Finally by converting the number of cells to area, the time-area histogram is derived. These steps are explained in the following sections.

1.3.1 Flowlength grid

To obtain the time-area graph of catchment, first the flowlength grid (FlGrid) of the watershed is developed. The FlGrid represents for each cell, the total travel length of the water droplet from that cell to the catchment outlet along the direction of flow. Required inputs for calculation is flowdirection grid (FdGrid). In a square grid environment, each grid cell is surrounded by eight cells. Using GIS tools flowdirection value of each cell in the catchment is calculated from the DEM of the catchment then a grid containing these values is obtained and named as flowdirection grid, FdGrid.

1.3.2 Travel time grid

After computing the travel distance of each cell (FlGrid), the next step is calculating the travel time values. The maximum value of the FlGrid belongs to the remotest cell of the catchment to the outlet. Travel time of flow from that cell to outlet gives the time of concentration value of the catchment. Eq. (2) is used to prorate the values of FlGrid and to convert it to time values (Kull and Feldman, 1998). The travel time grid of the catchment is then determined from Eq. (2) and named as TtGrid.

cTTt Grid = FlGridMax.of cell travel lengths

(2)

In Eq. (2) maximum of the cell travel lengths is the maximum value of FlGrid and Tc is the time of concentration value of the catchment.

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The third parameter of the Clarks methodology, the time-area histogram of the catchment is determined from TtGrid of the catchment. First, histograms of the TtGrid are derived for different time intervals. It is observed that as the interval gets smaller shape of the histogram resembles complex hydrographs, and as it gets bigger the shape roughly looks like a single peaked hydrograph. Aim of trying several interval values is to determine a histogram shape close to a single peaked hydrograph shape with the smallest possible interval.

IUH of the catchment will be derived if the selected time interval is infinitely small. Practically it is impossible to obtain the histogram of TtGrid with infinitely small time interval. So the smallest possible time interval is selected to apply the Clarks technique.

Histogram of TtGrid has time values on the abscissa and number of cells on the ordinate. The time-area histogram of the catchment is calculated from histogram of TtGrid by converting number of cells to area. The time area graph of catchment is shown in Figure 2.

0

5

10

15

20

25

0 60 120 180 240 300 360 420 480 540

Area

(%

)

Time (min)

Figure 2 Time-area histogram of catchment

1.3.4 Translation hydrograph

After determining the three parameters of Clarks methodology, the unit excess rainfall is uniformly distributed over the catchment. Then this precipitation is conveyed to the catchment outlet by a translation hydrograph. For this purpose the time-area histogram of the catchment, which is obtained in the previous step, is used. As presented in Figure 2, the time-area histogram represents the percent of the catchment area contributing to the flow at the outlet in each time interval.

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After instantaneous application of unit rainfall, total volume of water that will be observed at the catchment outlet is determined by multiplying catchment area (57 sq km) with the depth of rainfall (1 mm). Then from the time-area histogram of the catchment, percentage of total volume contributing to the flow at the outlet in each time interval is calculated. Volumes are then converted to discharges for corresponding time intervals. Finally by plotting these values at the mid values of time intervals the translation hydrograph is determined (Figure 3).

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

0 100 200 300 400 500 600

Tra

nsl

atio

n hy

dro

gra

ph (m

3 /s)

Time (min)

Figure 3 The translation hydrograph of catchment

1.4 Linear reservoir routing

As previously mentioned the instantaneous unit excess rainfall is conveyed to the catchment outlet by two components: a translation hydrograph and linear reservoir routing. The translation hydrograph represents the rainfall-runoff relationship of the catchment by means of surface flow only. The effect of stream channel storage on the hydrograph is reflected by linear reservoir routing. The translation hydrograph obtained in the previous section is routed by Eq. (3).

110.5 0.5t t tt tQ I Q

R t R t

= + + + (3)

In Eq. (3), It is the calculated translation hydrograph, R is the storage attenuation coefficient and t is the selected time interval for routing. Qt which is obtained after routing is the

instantaneous unit hydrograph of the catchment. The routing process is continued till an excess flow depth of 1 mm is obtained under the hydrograph. Eq. (3) results the instantaneous unit

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hydrograph for the catchment. Finally the ordinate of the unit hydrograph is calculated from Eq. (4). The derived unit hydrograph for the catchment is shown in Figure 4.

0.5 ( )t t t tU Q Q += + (4)

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0 120 240 360 480 600 720 840 960 1080 1200

UH O

rdin

ate

s (m

3 /s/m

m)

Time (min)

Figure 4 The 1-hour unit hydrograph (1-h UH) of catchment

1.5 Estimation of discharge hydrographs

The discharge is estimated using the convolution technique. The derived UH is convoluted with excess rainfall hyetograph. The rainfall excess is calculated based on the SCS-CN-runoff method (Soil Conservation Services, 1972).

1.5.1 Excess rainfall estimation

When rainfall, P occurs, a small fraction, Ia of rainfall is subjected to abstraction at the surface (i.e. initial abstraction). The runoff does not start until Ia is satisfied. Therefore, the potential runoff will be aP I . During the storm rainfall, the depth of excess rainfall eP (i.e. direct runoff depth) is therefore less or equal to the rainfall P; and the depth of infiltrated water aF is less or equal to the potential maximum retention S. The SCS-CN-runoff relationships are stated as follows.

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( / ) /( )a e aF P S P I= (5)

( / ) /( )a e aF S P P I= (6)

The continuity principle states the followings.

e a aP P I F= + + (7)

From eq. (7), substituting eP into eq. (6) the following results,

( / ) ( ) /( )a a a aF S P I F P I= (8)

Solving eq. (8) for aF the following relationship is obtained,

( ) /( )a a a aF S P I P I S P I= + (9)

Equation (9) gives the time distribution of abstraction. Differentiating and noting that aI and S are constants, the relationship are stated as follows.

2 2/ ( / ) /( )a adF dt S dP dt P I S= + (10)

As P , / 0adF dt , the presence of /dP dt (rainfall intensity) in the numerator suggests that as the rainfall intensity increases, the rate of retention of water within the catchment also tends to increase. This simplified assumption (Ponce and Hawkins, 1996) results in the following runoff equation where the curve number ( 1000 CN ) represents a convenient representation of the potential maximum soil retention (S). For a known CN, potential maximum abstraction S can be computed as follows.

(25400 / ) 254S CN= (11)

Where S is in mm and aI is computed using the following relationships.

0.2 , 0.2, 0.2a

S P SI

P P S

=

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( ),

0 ,

aa

aa

a

S P I P IP I SF

P I

+=

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0

0.5

1

1.5

2

2.5

3

3.5

4

0 5 10 15 20 25 30 35 40

Disc

harg

e (m

3 /s)

Time (hour)

(b) Event Date: 14/08/2010

0

0.5

1

1.5

2

2.5

3

3.5

4

0 5 10 15 20 25 30 35 40

Disc

harg

e (m

3 /s)

Time (hour)

(c) Event Date: 19-20/08/2010

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0

2

4

6

8

10

12

14

16

0 5 10 15 20 25 30 35 40

Disc

harg

e (m

3 /s)

Time (hour)

(d) Event Date: 2/07/2010

0

5

10

15

20

25

30

0 5 10 15 20 25 30 35 40

Disc

harg

e (m

3 /s)

Time (hour)

(e) Event Date: 21/07/2010

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0

10

20

30

40

50

60

70

80

90

0 5 10 15 20 25 30 35 40

Disc

harg

e (m

3 /s)

Time (hour)

(f) Event Date: 25/07/2010

0

5

10

15

20

25

0 5 10 15 20 25 30 35 40

Disc

harg

e (m

3 /s)

Time (hour)

(g) Event Date: 01/09/2010

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0

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20 25 30 35 40

Disc

harg

e (m

3 /s)

Time (hour)

(h) Event Date: 16/06/2011

0

20

40

60

80

100

120

140

160

180

200

0 10 20 30 40 50 60 70 80 90 100

Dis

char

ge (m

3 /s)

Time (hour)

(i) Event Date: 21-23/06/2011

Figure 5 Estimated hydrographs for the catchment

2. Flood Frequency Analysis

Flood frequency analysis is carried out for the annual maximum discharges. The annual maximum discharge is estimated using the Clarks model with daily annual maximum rainfall followed by its hourly distribution. The hourly distribution of daily rainfall is estimated from the observed rainfall hyetographs recorded by RTEI.

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For flood frequency analysis, two types of distribution functions viz. Pearson type 3, EV1 has been used. The parameters of the distribution function were fitted with the Method of Moments and L-Moments techniques. The description of the two distribution function is as follows:

Pearsons distribution

The probability density function (pdf) of PT3 is: 11( ) exp( )

x xf x

=

(15)

The flood quantile can be estimated using the following set of equations (eqs 16 to 21):

2

T Tx K = + + (16) where KT is the frequency factor corresponding to a return period of T years. The Wilson-Hilferty approximation for KT is:

32 1 1 , 0

6 6s s

T ss

C CK u CC

= + >

(17)

where u depends on the return period T. The Wilson-Hilferty approximation is quite accurate for and may be sufficiently accurate for Cs as high as 2.0. The parameter u is represented as follows.

20 1 2

2 31 2 31

c c w c wu w

d w d w d w+ +

=

+ + + (18)

where: c0 = 2.515517, c1 = 0.802853, c2 = 0.010328, d1 = 1.432788, d2 = 0.189269 and d3 = 0.001308.

2ln( )w P= (19)

1P F= (20)

1 1/F T= (21)

The parameters of the distribution using MOM and PWM are given as follows:

MOM: The following set of equation will be used to estimate the parameters of the PT3 using MOM.

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2 (2 / )sC = (22) 0.5

2

( / )m = (23) ' 0.5

1 2

( )m m = (24)

33/ 2

2s

mCm

= (25)

where: Cs is the coefficient of skewness, m1 is the first sample moment with zero mean, m2 and m3 are the second and third moment with respect to the mean.

PWM:

The plotting position estimates for PWMs are given as follows:

1,0,1

1 (1 )

Ns

s s i ii

a M F xN

=

= = (26)

where F is the probability of non-exceedance ( ). The L-moments can be defined in terms of as follows:

1 0

2 0 1

3 0 1 2

4 0 1 2 3

26 612 30 20

l al a al a a al a a a a

=

=

= +

= +

(27)

The L-moment ratios, which are analogous to conventional moment ratios, are defined by Hosking (1990) as:

2 1

2

// , 3r r

t l lt l l r

=

= (28)

Hosking (1986) has given the parameters for Pearson 3 distribution, and are:

For 3 1/ 3t , let 31mt t= and

2 3

2 3(0.36067 0.5967 0.25361 )

(1 2.78861 2.56096 0.77045 )m m m

m m m

t t t

t t t +=

+ (29)

For 3 1/ 3t < , let 2

33mt tpi= and

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2 31 0.2906

( 0.1882 0.0442 )m

m m m

t

t t t +=

+ + (30)

2

( )

1( )

2

l pi

=

+ (31)

1

l = (32)

The extreme value type I (EV1(2)) distribution The pdf of EV1(2) is given as:

1( ) exp expx xf x

=

(33)

The distribution of x can be given as follows:

( ) exp exp xF x

=

(34)

The flood quantile will be estimated using the following relationship:

1

ln ln 1Tx T =

(35)

The parameters of the EV1(2) can be estimated using either MOM or PWM. The required sets of expressions are given as follows:

MOM:

Using the MOM, the parameter can be estimated as follows

2 0.7797 m = (36)

1 2

' 0.45005m m = (37) The associated standard error in estimation of flood quantile is calculated as:

( )22 2 1.15894 0.19187 1.1TS Y YN

= + + (38)

where: ln[ ln( )]Y F=

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PWM:

The PWM of the EV1(2) distribution are the form (Greenwood et al., 1979; Hosking, 1986):

2 / ln(2)l = (39)

1

0.5772157l = (40)

The standard error, of quantile estimate using PWM can be calculated as follows:

( )22 2 1.1128 0.4574 0.8046TS Y YN

= + + (41)

2.1 Results The time series of the estimated peak discharge for the years from 1974 to 2011 is given in Figure 6. The probability plot is shown in Figure 7. The estimated parameters are given in Table 1.

0

100

200

300

400

500

600

1970 1975 1980 1985 1990 1995 2000 2005 2010 2015

Peak

Di

scha

rge

(m

3 /s)

Year

Figure 6 Estimated annual peak discharges for the period from 1974 to 2011

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0

100

200

300

400

500

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Annu

al P

eak

Disc

harg

e (m

3 /s)

Probability of Non-exceedance

Figure 7 Probability plot of annual peak discharges

Table 1 Statistical characteristics of the annual peak discharge time series

MOM PWM and L-moments m1 = 83.8 a0 = 83.788 m2

1/2 = 105.23 a1 = 17.993

m2 = 11071.2 a2 = 7.133 Cs = 2.21 a3 = 3.768 Ck = 5.15 l1 = 83.788

l2 = 47.802 l3 = 18.628 l4 = 6.491 t = 0.571 t3 = 0.390 t4 = 0.136

The estimated values of the parameters for the PT3 and EV1 using the MOM and PWM are given in Table 2. The table also includes the value of PPCC. The graphical comparison of the observed and estimated vales of the maximum flow is shown in Figure 7.

Table 2 Parameters of the PT3 and EV1 using MoM and L-Moment PT3 EV1

PWM PWM MOM 116.796 68.9638 82.0398

0.7276 43.9806 36.4335

-1.1977

PPCC 0.97408 PPCC 0.92766 0.92766

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1

10

100

1000

0 0.2 0.4 0.6 0.8 1

Flo

od

(m3 /s

)

Probability of Non-exceedance (F)

PT3 (PWM)EV1 (PWM)EV1 (MOM)Observed

Figure 7 Comparison of observed and fitted probability plots

Based on the visual comparison and statistical test (i.e. PPCC), it is observed that the PT3 distribution is giving the best results. The quantile estimates using the PT3 and EV1 distribution along with the associated standard error values is given in Table 3.

Table 3 Quantile estimates of the annual maximum flood

Return Period, T (years)

F=1-1/T P=1-F

EV1: MOM EV1: PWM PT3: PWM

XT ST

(Standard error)

XT ST

(Standard error)

XT

25 0.96 0.04 298.84 50.78 264.56 38.88 302.38 50 0.98 0.02 356.55 60.77 313.07 46.03 372.65 100 0.99 0.01 413.83 70.80 361.22 53.21 444.08 250 0.996 0.004 489.25 84.10 424.62 62.75 540.30 500 0.998 0.002 546.20 94.19 472.49 69.99 614.43

1000 0.999 0.001 603.10 104.31 520.33 77.25 689.70 2000 0.9995 0.0005 659.99 114.45 568.15 84.52 766.08