Nash Model

71

CHAPTER 4

Nash Model

4.1 Introduction

Water resources development and planning require Hydrologic Transform

Models to assess runoff from a catchment. For catchments having scanty

data the unit hydrograph technique is very useful for this purpose. This part of

research determines a unique pair of hydrologic parameters of Nash Model

using optimization. The model is applied to a catchment with hill torrent flows

in semi-arid region of Pakistan. Computer program is developed to

systematically estimate the related parameters of model. Analysis Group

(NAG) subroutine is used for optimization. Finally the direct surface runoff

(DSRH) is simulated using the model. The data regarding rainfall- runoff was

collected from Punjab Irrigation and Power Department, Pakistan. Model

calibration and validation was made for 15 rainfall runoff events. Ten events

were used for calibration and five for validation. Various objective functions

are tested to find the best solution. The suitability of 4 objective functions is

investigated for developing direct runoff hydrograph (DRH).

It is found that Nash-Sutcliffe coefficient (Nash and Sutcliffe 1970) and

weighted root mean square error (RMSE) are most suitable for determination

of Nash model parameters when full shape of the direct surface runoff

hydrograph is known.

The sensitivity of the Nash model output against variation in hydrologic

parameters (number of linear cascade (n) and storage coefficient (k) is also

72

investigated. It is found that the model output is more sensitive to the

parameter of number of linear cascade (n) as compared to the parameter

of storage coefficient (k). Also uniqueness of Nash Model parameters is

established. The suitability of the Nash model application to un-gauged

catchments is also described.

Hydrologic modeling plays key role in water resources planning and

management. Limited availability of hydrologic data is major hurdle towards

implementation of detailed hydrologic models. In cases of limited data the

simple hydrologic models consisting of minimum number (one or two) of

model parameters are suitable for planning water resources. The simple

conceptual model like the Nash’s Model of linear cascades is very effective

in simulating Hydrologic Transform Model process as its parameters can be

determined indirectly by computations. It is fact that these cannot be

measured physically (Patil 2006) although these represent some physical

phenomenon indirectly. This type of model is useful for flood forecasting and

design purposes (Bardossy 2007). The estimation of hydrologic parameters is

complex and various efforts have been made to simplify it.

The most commonly used method for estimation of hydrologic parameters is

based on calibration of the model, implementing boundary conditions and

simulating the Hydrologic Transform Model process such that the set of

parameter values obeys the imposed constraints.

Nash (1958) proposed estimation of hydrologic parameters (n, k) through

method of moments. This estimation technique depends on temporal

distribution of rainfall and runoff and hence the errors in observed and

computed hydrologic output are high (Dong 2007). Further a different pair of

hydrologic model parameters is obtained for every event. Then a single pair

of parameters representing Hydrologic Transform Model process for all events

of catchment is determined by their average. Hydrologic modeling package

HMS-HEC (2000) contains built in non linear constrained optimization

73

technique with a normalized objective function that was developed in 1998

(Michel 1998).

This technique requires averaging of computed hydrologic parameters.

Rosso (1984) developed equations for Nash Model parameters by relating

them to the geomorphologic descriptors of the catchment. These empirical

equations require estimation of velocity which in turn requires a relationship

for it. Various attempts have been made to estimate this velocity by relating it

to physical characteristics of the catchment (Zalazainski 1986, Al-Wagdany

1997, Sahoo etal 2006). However, this technique requires that the velocity

itself be taken as a calibrating parameter which ultimately increases the

number of hydrologic parameters to be estimated.

Absence of channel translation in Nash Model is one of its drawbacks. Singh

et al (2007) proposed an extended hybrid model for simulation of Hydrologic

Transform Model process based on Nash Model. In this approach, the number

of hybrid units and pair of storage coefficients were prefixed using empirical

equations.

The associated translation is determined through model calibration. Their

technique also produces different set of parameters for various events and

hence the determination of a unique set of parameters still needs to be

further investigated.

Bardossy (2007) also determined more than one set of Nash Model

parameters for estimation of runoff hydrograph at a particular gauge site. In

this study, a methodology is developed by using comparative study of 4

objective functions to arrive at a unique pair of Nash Model parameters using

average value of objective function for calibration events. The pair of

hydrologic parameters represents lumped runoff response of a particular

catchment.

74

4.2 THEORETICAL BACKGROUND

4.2.1 Nash Model of Linear Cascades

There are two analytical versions of Nash model which are used in catchment

routing. The first one uses instantaneous unit hydrograph obtained by

applying continuity equation. In this concept, the IUH flow rate is given by

(Serrano 1997):

n

ktn

IUH kn

eAttQ

)(

2778.0)(

/1

(4.1)

Where )(tQIUH is IUH flow rate at time t in m3/s, t is time from start of excess

precipitation in hours, n & k are hydrologic model parameters, n being unit

less, k being in hours and A is catchment area in km2. )(n is a two parameter

gamma function having no units.

The second analytical form is used by SSARR (Stream flow Synthesis and

Reservoir Regulation) model (USACE, 1986). It is expressed as:

n

t

n

tt

n

t QQQkt

ktAQ 11

1

1 ))/(2(

)/()9/25(

(4.2)

Where tQ represents discharge in m³/s at time t in hour, t is computational

time interval in hour , n & k are Nash Model hydrologic parameters, n having

no units and k is in hour, tQ is mean discharge in m3/s, 25/9 is units conversion

factor and A is catchment area in square kilometer. Equation 4.2 defines

Comment [m9]: Add units of parameters for the above equation here

Comment [m10]: Add units

75

complete shape of the surface runoff hydrograph with excess rain

hyetograph being the input to the model.

In this study, use of equation 4.2 is preferred over equation 4.1 due to the fact

that gamma function for real numbers can be approximated only. Also

equation 4.2 is suitable for simulation relatively easily on a computer.

4.3 Determining the Hydrologic Model Parameters

In equation (4.2), n and k are hydrologic parameters of model where n is

number of linear cascades/reservoirs and k is storage coefficient. It defines

the catchment response by routing excess rain hyetograph through the

hypothetical linear reservoirs at the catchment outlet. Nash model has two

hydrologic parameters and equation (4.2) is redundant by one degree. The

main problem is to find unique pair of parameters that gives representative

catchment response. Determination of parameters of model is an inverse

problem.

These are determined by obtaining a good match between the observed and

simulated results by using optimization techniques. All optimization techniques

are based upon minimizing/maximizing a function called objective function.

Choice of objective function plays an important role in optimization. The

objective functions that were used for Clark’s model (chapter 2, Section 2.2)

have also been employed for Nash model.

4.4 Model Performance

The model efficiency and peak weighted root mean square error were

selected to test the performance of the model as proposed by Nash and

Sutcliffe (1970) and USACE (1998). These have already been explained in

chapter 2, section 2.3.

4.5 SENSITIVITY OF THE MODEL PARAMETERS

76

Sensitivity analyses were conducted to determine the relative importance of

parameter n and k of the Nash model. This was done by using formula for

calculating relative sensitivity given below (James and Burges, 1982 and

Katli et. al, 2005)

12

12

xx

yy

y

xSr (4.3)

Where Sr is relative sensitivity with units of objective function units divided by

units of hydrologic parameter whose sensitivity is being measured, x is

hydrologic parameter and y is the predicted output. x1=x+∆x and x2=x-∆x are

parameter values that result in output of y1 and y2 respectively. In Nash

model parameters are n and k whereas the predicted output is adopted as

certain objective function.

4.6 STUDY AREA

The characteristics of study area were explained in chapter 2, section 2.1.

Nash’s model was applied to Kaha catchment.

4.7 Optimization to identify Nash Model parameters

The optimization routine explained in chapter 2, section 2.2.2 has been

applied for Nash model and different steps involved in estimating n and k are

shown in a flow chart in Figure 4.1.

Comment [m11]: What is Sr and its unit?

77

Fig. 4.1 Schematic of Proposed Method for Nash’s IUH model simulation

4.8 RESULTS AND DISCUSSIONS

4.8.1 Calibration of Nash Model

The results based on the objective function F1 are given in Figure 4.2. In order

to find unique n and k, sum of square errors between observed and

computed runoff hydrographs for all the ten calibration events corresponding

to each calculation of F1 are plotted in Figure 4.3 and numerical values are

given in Table 4.1 and 4.2.

Excess Rain Hyetograph (Calibration Events)

No. of Linear Cascades (n) Storage Coefficient (k) (Trial Value)

Nash Hydrologic Transform Model Transform Model

n

t

n

tt

n

t QQQkt

ktAQ 11

1

1))/(2(

)/()9/25(

Compute Objective Function (Average)

Is Objective Function Minimized

Compute Model Performance

No

Yes

78

-

0.5

1.0

1.5

2.0

2.5

3.0

5 10 15 20 25

Storage Coefficient, k (hours)

Ob

ject

ive

Fu

nct

ion

, F1

n=1 n=2 n=3 n=4 n=5

Fig. 4.2 Storage Coefficient vs Objective Function F1

Table 4.1 gives the optimum k value corresponding to minimum objective

function for different values of n. As observed from this table, the value of n of

3 yields minimum sum of square error. However this single constraint of

minimum sum of square error does not define best n & k pair. It was necessary

to evaluate model performance prior to deciding which pair is the best. For

the purpose NS coefficient and weighted RMSE were calculated and shown

in Table 4.3 & 4.4. The model performance was best at n value of 4, therefore

the best pair of Nash model parameters based on objective function F1 is

found as (n, k)=(4, 9.04).

79

-

10,000

20,000

30,000

40,000

50,000

60,000

70,000

80,000

90,000

- 5 10 15 20 25 30


Su

m o

f S

qu

ares

(fo

r F

1)

n=1 n=2 n=3 n=4 n=5

Fig. 4.3 Sum of Square Error for Objective Function F1

The results based on objective function F2 consisting of absolute sum of errors

are shown in Figure 4.4. The minimum value of the objective function F2 is

observed at n=5. However, the Nash model performance indicators

advocate n=4 (Table 4.3 & 4.4) yielding maximum efficiency and weighted

root mean square error. The best pair therefore is selected as (n, k)=(4, 9.21).

This justifies the importance of present research. It is obvious from this

exercise that only the minimum value of objective function cannot define

best pair of hydrologic model parameters.

The objective function F3 consisting of sum of square errors gives k value of

21.41 hours at n=2 (see Figure 4.5). However at this level, the model efficiency

is as lower as 0.61 and RMSE on higher side having value of 425 (m3/s). As

shown in Table 4.3 and 4.4, the pair (4, 9.11) gives best model performance.

80

Finally, the objective function F4 was employed for seek of unique (n, k) pair

which incorporates in itself another important characteristic of direct runoff

hydrograph (DRH), the time to peak along with peak discharge. The

computed value of F4 corresponding to k is plotted in Figure 4.6. The shape of

the curves indicate slight discontinuities due to selected computational time

interval of one hour as was used in Clark’s model, however minimum value of

F4 is observed at (5,7.09). Despite the fact that minimum weighted root mean

square error is observed at this point, the efficiency of the model is slightly

reduced. The efficiency is maximum at (n=4, k=8.46). Therefore this pair was

selected as unique pair of Nash model parameters.

Table 4.1 Optimum value of Storage Coefficient, k (hours)

For Calibration Events

F1 F2 F3 F4

n Storage Coefficient, k (hours)

1 25 25 25 25

2 14.82 21.84 21.41 20.29

3 10.92 12.75 11.11 10.29

4 9.04 9.21 9.11 8.46

5 7.89 7.35 5.12 7.09

6 7.08 6.15 4.22 6.06

Once it was concluded that the model efficiency defined here by Nash-

Sutcliffe coefficient as the deciding parameter, it was tried to use NS

Coefficient and RMSE as objective functions. These are plotted in Figures 4.7

and 4.8. Both converge at point (4, 9.0). This is the unique pair of Nash model

parameters that represents lumped Hydrologic Transform Model process of

the catchment under study which results in unique DRH at the catchment

outlet with only single input of temporally distributed excess rain hyetograph.

The advantage of Nash model is demonstrated clearly that one can simulate

81

DRH for any design storm hyetograph at the catchment outlet. This is

particularly valuable for un-gauged catchments where limited or scarce

hydrologic data is available.

The various error measures of the Nash Model are given in Table 4.5. The error

in peak discharge varies from -14% to 2% whereas time to peak is under

estimated by -8% to 3%.

Table 4.2 Optimum value of Dimensionless Objective Function

for Calibration Events

F1 (SOSE) F2 F3 F4

n Values of Objective Function

1 708 9962 707 30.26

2 402 3086 299 5.81

3 313 1895 312 2.94

4 1505 1044 864 0.99

5 5283 1012 2492 0.51

6 15735 1756 5014 0.92

82

-

5,000

10,000

15,000

20,000

25,000

- 5 10 15 20 25

Storage Coefficient, k (hour)

ASO

E

n=1 n=2 n=3 n=4 n=5 n=6

Fig. 4.4 Variation of Objective Function F2 with k, the storage coefficient

Table 4.3 NS Coefficient corresponding to optimum, k

n F1 F2 F3 F4

(Nash-Sutcliffe coefficient)

1 -0.90 -0.90 -0.90 -0.90

2 0.42 0.59 0.61 0.64

3 0.83 0.87 0.85 0.76

4 0.94 0.95 0.94 0.93

5 0.85 0.92 0.13 0.93

6 0.61 0.86 -0.06 0.87

83

-

500

1,000

1,500

2,000

2,500

3,000

- 5 10 15 20 25 30

Storage coefficient, k (hours)

Ob

ject

ive

Fu

nct

ion

F2

n=1 n=2 n=3 n=4 n=5

Fig. 4.5 Sum of Square Error for Objective Function F3

The runoff volume is under estimated by 11%. The Nash-Sutcliffe coefficient

varies from 0.93 to 0.96 for ten calibration events showing model efficiency

more than satisfactory. The weighted root mean square error ranges from 56

to 250 whereas the latter being maximum is for extreme event indicating non-

homogeneity in Hydrologic Transform Model process. These results provoke

effectiveness of Nash model usage for simulation of Hydrologic Transform

Model process for catchment under study. The calibration events are plotted

in Figure 4.9 showing a coefficient of determination of 0.98 for observed and

computed flows showing close relation between observed and computed

discharges.

84

Table 4.4 RMSE corresponding to optimum k

n F1 F2 F3 F4

(Weighted Root Mean Square Error [RMSE])

1 887 887 887 887

2 526 432 425 413

3 291 232 276 357

4 135 136 134 181

5 220 157 683 157

6 388 233 763 229

-

0.5

1.0

1.5

2.0

2.5

3.0

3.5

5 10 15 20


Obje

ctiv

e Fun

ctio

n

n=2 n=3 n=4 n=5 n=6

Fig. 4.6 Variation of Objective Function F4 with k, the storage coefficient,

85

-

500

1,000

1,500

2,000

2,500

3,000

- 5 10 15 20 25 30


Wei

ghte

d R

SM

E

n=1 n=2 n=3 n=4 n=5 n=6

Fig. 4.7 Variation of RSME with k, the storage coefficient

-0.10.20.30.40.50.60.70.80.91.0

5 10 15 20 25


Nas

h-S

utc

liff

e C

oeff

icie

nt

n=2 n=3 n=4 n=5 n=6

Fig. 4.8 Variation of NS-Coefficient with k, the storage coefficient

86

R2 = 0.9778

-

500

1,000

1,500

2,000

2,500

3,000

3,500

- 500 1,000 1,500 2,000 2,500 3,000 3,500

Ordinates of Observed Runoff Hydrographs (m3/s)

Ord

inat

es o

f C

omp

ute

d R

un

off

Hyd

rogr

aph

s (m

3 /s)

Fig. 4.9 Observed and Computed Runoff for 10 Calibration Events

4.8.2 Validation of Model

Once the unique pair (n, k)=(4,9) is known, the five validation events are

synthesized and various catchment characteristics are computed as shown

in Table 4.5. The error in peak discharge is maximum by -8%, the peak time is

under estimated by 3%. The error in runoff volume is around 6% except the

event number 14 where it is 11%. The Nash-Sutcliffe coefficient is around 0.94

and RMSE value is 96 maximum. These indicators validate the model for future

use. The observed and computed discharge for the validation events is

plotted in Figure 4.10 yielding satisfactory coefficient of determination. When

all the calibration and validation events are plotted (Figure 4.11), a high

value of coefficient of determination is observed at low and medium

discharge whereas slight deviation is observed at highest discharge values.

87

Table 4.5 Model Errors and Performance

Ca

libra

tion

1 -13 0 -4 0.94 64

2 -6 -8 1 0.95 250

3 -12 -3 2 0.94 113

4 -5 0 9 0.93 56

5 -6 -6 0 0.96 141

6 +2 -6 9 0.95 217

7 -5 -6 2 0.96 143

8 -14 -3 -4 0.94 120

9 -5 -3 5 0.95 82

10 -11 -6 -5 0.95 169

Val

ida

tion

11 -8 -3 3 0.95 84

12 -7 -3 3 0.95 91

13 -4 -3 6 0.95 80

14 0 -3 11 0.94 96

15 -8 0 6 0.93 77

Event No. Error (%) Performance

Peak

Discharge

Peak

Time

Runoff

Volume

NS-

Coefficie

nt

RMSE

88

R2 = 0.9791

-

600

1,200

1,800

- 600 1,200 1,800


Ord

inat

es o

f C

omp

ute

d R

un

off

Hyd

rogr

aph

s (m

3 /s)

Fig. 4.10 Observed and Computed Runoff for 5 Validation Events

R2 = 0.9781

-

500

1,000

1,500

2,000

2,500

3,000

3,500

- 500 1,000 1,500 2,000 2,500 3,000 3,500


Ord

inat

es o

f C

omp

ute

d R

un

off

Hyd

rogr

aph

s (m

3 /s)

Fig. 4.11 Observed and Computed Runoff for all Events

89

4.8.3 Sensitivity of Hydrologic Parameters n & k

The values of n and k are varied and relative sensitivity of the objective

functions is observed. As shown in Figure 4.12, for the equal variation of n and

k, higher value of relative sensitivity is noted when n is varied as compared to

that when k is varied. This shows that Nash Model output is more sensitive to n

value as compared to k value.

-

1

2

3

4

5

6

- 2 4 6 8 10 12

Nash model parameters, k, n

Rel

ativ

e se

nsi

tivi

ty

storsge coefficient, k no. of linear cascades

Fig. 4.12 Sensitivity of Nash Model Parameters

4.9 SUMMARY AND CONCLUSIONS

A method to estimate Number of Linear Cascades (n) and Storage

Coefficient (k) to develop Direct Runoff Hydrograph based on Nash Model is

presented. The value of n is estimated from family of curves drawn between

storage coefficient and certain objective functions. The analytical form of

90

Nash model is adopted from SAAR model. The optimum and unique pair of

(n, k) is determined using iterative method. Direct surface runoff hydrograph

(DSRH) is developed for a large catchment with area of 5598 square

kilometers in a semi-arid region of Pakistan that experiences hill torrent flows.

Four different objective functions are used in optimization to determine the

sensitivity of number of linear cascades and storage coefficient k. The results

during validation are very good with model efficiency of more than 95% and

root mean square error of less than 8%.

The following main conclusions are derived from this study:

It is not always necessary to develop regional equations for estimating

Nash Model parameters. These can be determined from Hydrologic

Transform Model observations for a particular catchment and depend on

the length of data used for determination of these parameters. To get

stable values of n and k, longer Hydrologic Transform Model records are

preferable which is the most crucial for catchments having scanty data.

Value of n and k depend on the size of the catchment, hydrologic

abstractions and temporal distribution of excess rain.

For limited data the objective function based on Nash-Sutcliffe coefficient

and Weighted Root Mean Square Error gives better results as compared to

the objective function based on the Sum of Square Errors.

DSRH shape is more sensitive to n value than that of k showing that runoff

diffusion phenomenon is dominant as compared to translation flow effects

when evaluating hydrologic response of catchments of large size.

Nash’s unit hydrograph generally under estimates runoff volume.

DSRH derived from Nash model gives acceptable accuracy and model

parameters can be easily updated as additional Hydrologic Transform

Model data becomes available. However, updating of the parameters is

possible only for gauged catchments.

Although the proposed method is applied to a catchment having observed

rainfall and peak flow data, it can be applied to un-gauged catchments by

91

simulating hypothetical storms and survey of highest flood marks at the outlet.

Value of n & k determined for single flood event (corresponding to highest

flood marks) can be used to compute different runoff hydrographs for

different design storms.

4.10 References

Al-Wagdany A.S., Rao A.R., 1997, Estimation of the velocity parameter of the

geomorphologic instantaneous unit hydrograph, Water Res. Manag., 11, pp.

1-16

Bardossy, A., 2007, Calibration of hydrologic model parameters for ungauged

catchments, Hydrol. and Earth Sys. Sci., 11, pp. 703-710.

Dong., S.H., 2007, Genetic algorithm based parameter estimation of Nash

Model, J. of Water Resour. Manag., DOI 10.1007/s11269-007-9208-6.

Hydrologic Engineering Centre (HEC), 2000, HEC-HMS user’s manual, Davis,

California.

James, L.D. and S.J. Burges, 1982. Selection, calibration and testing of

hydrologic models. Hydrologic modeling of small watersheds, C.T. Hann, H.P.

Johnson, and D.L. Brakensiek (Editors). ASAE Monograph, St. Joseph,

Michigan, pp. 437-472.

Kati L, Indrajeet C., 2005, Sensitivity analysis, calibration and validation for a

multisite and multivariable SWAT model, J. of the Am. Water Res. Ass., 41(5),

pp. 1077-1089.

92

Michel C. B., 1998, Unit hydrographs derived from the Nash model, J. Am.

Wat. Res. Assoc., Vol. 34, No. 1, pp 167-177.

Nash, J. E. (1957). “The form of instantaneous unit hydrograph.” Int. Assn. Sci.

Hydro. Publ. No. 51, 546-557, IAHS, Gentbrugge, Belgium.

Nash, J. E. and Sutcliffe, J. V. 1970. River flow forecasting through conceptual

models, Part-I: A discussion of principles. J. Hydrol., 10(3), 282-290.

Patil S., Bardossy A., 2006, Regionalization of runoff coefficient and

parameters of an event based Nash-cascade model for predictions in

ungauged basins, Geophy. Resea. Abst., 8(74).

Rosso, R (1984). “Nash model relation to Horton order ratios.” Water Resour.

Res., 20(7), 914-920

Sahoo B., Chandarnath C., Narendra S. R., Rajendra S., Rakesh K., 2006, Flood

Estimation by GIUH based Clark and Nash models., J. of Hydrol. Eng., 11(6),

pp. 515-525.

Serrano, S. E., 1997, Hydrology for Engineers, Geologists and Envirornmental

Professionals, HydroScience Inc., Lexington, Kentucky, pp. 263-268.

Singh P. K., Bhunya P. K., Mishra S. K., Chaube, U. C., 2007 An extended hybrid

model for synthetic unit hydrograph derivation, J. of Hydrol., 336, pp. 347-360.

USACE, 1986, Program Description and User Mannual for SSARR Model,

Portland, Oregon.

93

Zelazinski J., 1986, Application of the geomorphological instantaneous unit

hydrograph theory to development of forecasting models in Poland, Hydro.

Scien., 31(2), pp. 263-270.

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