Nourani-GeomorphologicalRainfallRunoff-Catena2009

Catena 76 (2009) 206–214

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Three geomorphological rainfall–runoff models based on the linear reservoir concept

Vahid Nourani a,⁎, Vijay P. Singh b, Hadi Delafrouz a

a Faculty of Civil Eng., Univ. of Tabriz, Tabriz, Iranb Faculty of Biological and Agricultural Eng., Texas A & M Univ., Texas, USA

⁎ Corresponding author. Tel.: +98 411 339 2409; fax:E-mail addresses: [email protected], nourani@ta

0341-8162/$ – see front matter © 2008 Elsevier B.V. Aldoi:10.1016/j.catena.2008.11.008

a b s t r a c t

Article history:

Keywords:Rainfall–runoff modelingGeomorphological modelLinear reservoirGISAmameh watershed

dels, based on linear reservoirs cascade, are developed; two of the models areunit hydrograph (UH) models and one contains a non-linear routing approach. In the first UH model, eachsub-basin is represented as a cascade of equal linear reservoirs (Nash's model) which directly discharges tothe watershed outlet through a linear channel. In the second UH model each sub-basin is represented as alinear reservoir arranged in a sequence based on the drainage network. For both unit hydrograph models,parameters are explicitly derived, which make the models simple and applicable for real world problems. Inthe third model, output hydrographs of the sub-basins are calculated by the Nash model but with considera-tion of geomorphological properties of the sub-basins. Then, the obtained hydrographs are routed throughthe main channel using a non-linear kinematic wave model. Combination of the kinematic wave routingmodel (a non-linear routing model) and the Nash model (a linear lumped model) makes this third modelmore appropriate for runoff modeling.Two important properties of these three models are that they consider the effect of watershed geomorphol-ogy and include only one parameter which can be estimated using observed rainfall–runoff data. GIS tools areused for determining the watershed geomorphological parameters. The results of these models are comparedwith the Nash's black box and the geomorphological SCS models for the Amameh watershed, Iran. Althoughthe results show that the proposed models yield good efficiency in rainfall–runoff modeling, the semi-distributed routing ability in the second and third models produces better results.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

A multitude of linear and nonlinear rainfall–runoff models havebeen proposed since 1930 and a comprehensive classification of thesemodels has been presented by Nourani et al. (2007). According to thecomplexity of the rainfall–runoff relationship, conceptual models areemployed for analysis and simulation of this relationship. The linearreservoir model presented by Zoch in 1934 is one of the oldest andsimplest models for simulation of the rainfall–runoff relationship andit constitutes the basis of many conceptual models (Chow et al., 1988).

Nash (1957) proposed a conceptual model composed of a cascadeof linear reservoirs with equal storage coefficients. This has been oneof the most popular models, because it provides an explicit equationfor the Instantaneous Unit Hydrograph (IUH) of a watershed whereinreservoirs have a quasi-physical meaning. Dooge (1959) extended theNash model by considering the effect of flow transition and includingthe concept of a linear channel. For complex rainfall distributions andwatershed geomorphology, it is not easy to derive an explicit expres-sion for the IUH. Therefore, simplifiedmodels have been proposed. For

+98 411 334 4287.brizu.ac.ir (V. Nourani).

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instance, Wang and Chen (1996) and Jeng and Coon (2003) presentedmodels using linear reservoir cascades. A number of computer water-shedmodels, introduced in recent years, employ linear and non-linearreservoirs (Singh andWoolhiser, 2002). Most fully lumped conceptualmodels contain large numbers of parameters; however usually theseparameters cannot be related to physical watershed characteristicsand will not have a clear physical meaning, so they must be estimatedby calibration using observed data and usually these models calculatethe runoff hydrograph only at the watershed outlet.

Since the introduction of the first blueprint of a distributed model(Freez and Harlan,1969) the desire to developmore physically realisticdistributed models has been motivated by the need to make hydrol-ogic predictions in ungaged basins. However amodelmay be physicallybased in theory but not consistent with observations because of themismatch in scales between the scale of observable state variable andthe scale of applications (Pradhan et al., 2004). Furthermore, these fullydistributed models have usually many physically based parameterswhich must be defined using huge amount of the field data. For in-stance, TOPMODEL (Beven and Kirkby, 1979), which is a populardistributed and topographic based model, has seven physically basedparameters (such as soil transmissivity) and the estimation of theseparameters is not easy and requires a large amount of filed data.SHE (Abbott et al., 1986) is another distributed model that solves

207V. Nourani et al. / Catena 76 (2009) 206–214

simplifications of the St Venant equations for runoff routing and has acomplex framework with an enormous data requirement.

In order to overcome the problems mentioned about fully distrib-uted models, the next generation of models, semi-distributed models,has been developed and is still being developed (Nourani and Mano,2007).

Many of these models introduce routing on the basis of thebasin geomorphology and lead to the Geomorphological Unit Hydro-graph (GUH). Boyd (1978) developed a conceptual model using water-shed geomorphological properties. Using a probabilistic framework,Rodriguez-Iturbe andValdés (1979) andGupta et al. (1980) presented ageomorphological instantaneous unit hydrograph with the exponen-tial probability distribution for the time of travel of water drops whichis essentially equivalent to using a linear reservoir. Rosso (1984) de-rived the Nash IUH parameters as functions of Horton's ratios. Karnieliet al. (1994), Hsieh and Wang (1999), Nourani and Monadjemi (2006)developed geomorphological runoff models using a method similar toBoyd's. Yen and Lee (1997) presented a geomorphologicalmodelwhichwas applicable not only to those watersheds lacking enough statisticaldata but also lacking geomorphological data. The advantage of incor-porating geomorphology in rainfall–runoff models is that the modelscanbeapplied towatershedswith lackof observations andmodel param-eters can be determined fromwatershed geomorphology.

Recently, Agirre et al. (2005) and López et al. (2005) presented unithydrograph models using watershed geomorphology and linear re-servoirs, with a constant calibrated lag time for all reservoirs. Thewatershed morphology was employed just for the determination ofsub-basins (linear reservoirs) without any consideration of the sub-basin physiographical properties. Nourani and Mano (2007) usedTOPMODEL and the kinematic wave approach, wherein all modelparameters, except one, were linked to the geomorphologic proper-ties. With the development of Geographical Information System (GIS)tools, it is now possible to determine hydrological and watershed geo-morphological parameters using Digital Elevation Models (DEMs)(Jenson and Domingue, 1988; Maidment et al., 1996; Olivera andMaidment, 1999; Maidment, 2002).

Providing a brief review of the Nash and Soil Conservation Service(SCS)models, the objective of this paper is to develop three UHmodels,test them on a small watershed, called Amameh, in central Iran, andcompare their results with the results of the Nash and SCSmodels. Theparameters of the UH models are determined using explicitly geo-morphological based equations, but the direct search method is usedfor the non-linear routing model parameter. All of the geomorpholo-gical parameters are determinedbyGIS tools and only one hydrologicalparameter of each model is determined and calibrated using observedrainfall–runoff data sets. The developed and evaluated models in thispaper use relatively simple equations, require a limited number ofinput variables and parameters, and are semi-distributed. As thesemodels require limited data, they are potentially useful for applicationin data sparse areas and ungaged basins. Also, theywill have very shortrun times, allowing application at the global scale.

Fig. 1. Geomprphological Unit Hydrograph

2. Nash model

The Nash IUH (Nash, 1957) was obtained under the assumptionthat the response of a watershed to rainfall can be represented by acascade of n equal linear reservoirs, wherein rainfall is input to theuppermost reservoir. Denoting a linear reservoir as:

S tð Þ = kQ tð Þ ð1Þ

where k is the storage constant or lag time which has the dimensionof time and Q (with dimension [L3 /T]), S (with dimension [L3]) areoutflow rate and reservoir storage at time t. Combining Eq. (1) withthe continuityequation for a reservoir, representing the inflowrainfall-excess (I(t) with dimension [L /T]) as a Dirac Delta function (δ(t)), com-puting the reservoir outflowand then routing this outflow through theremainder of the cascade (Singh, 1988; Chow et al., 1988), one obtainsthe Nash model:

h tð Þ = δ tð ÞkD + 1ð Þn =

e−tk

kC nð Þtk

� �n−1ð2Þ

where h(t) is the IUH of the Nash model, Γ(n) is the Gamma functionand D is differential operation D = d

dt

� �. Using the method of moments,

the model parameters n and k can be determined as (Singh, 1988;Chow et al., 1988):

M1 Qð Þ−M1 Ið Þ = nkM2 Qð Þ−2M1 Ið ÞM1 Qð Þ +M2 Ið Þ = nk2 n−1ð Þ ð3Þ

where M1 and M2 are the first and the second moments of the quan-tities within parentheses. For British catchments, Nash establishedexperimental relations betweenwatershed physical properties and theIUH parameters and presented a synthetic IUH (Singh, 1988).

3. SCS model

The method of hydrograph synthesis employed by the Soil Conver-sion Service (SCS), U. S. Department of Agriculture, uses an averagenumber of natural UHs for watersheds varying widely in size andgeographical locations (Chow et al., 1988). In the SCS model, the lagtime, tl, can be determined from watershed physical properties, suchas area, main river length, average slope, CN (Curve Number), after-wards the synthetic unit hydrograph can be computed. The SCS modelpermits computing the UH for a watershed that has insufficient ob-served rainfall–runoff data.

4. Proposed models

4.1. Geomorphological Unit Hydrograph based on Nash's Model (GUHN)

In the geomorphological unit hydrograph using the Nash model(GUHN), a watershed is divided into different sub-basins, on the basis

model based on Nash's model (GUHN).

Fig. 2. A schematic of Geomorphological Unit Hydrograph model based on Cascade of linear Reservoirs (GUHCR).

208 V. Nourani et al. / Catena 76 (2009) 206–214

of approximately the same interval lines passing through the mainchannel reach joints, as shown in Fig. 1. Then each sub-basin, accom-panied by its main channel, is represented by a cascade of linear re-servoirs and a linear channel (of course the most downstream sub-basin does not need this channel because its outlet is exactly thewatershed outlet). The total excess rainfall of the watershed is dividedin proportion to the sub-basin area. In this manner, wemay have Nashmodel with distributed rainfall excess. Fig. 1 shows the schematic ofthe watershed (N=5).

Assuming the rainfall excess distribution with the watershedarea, entering linear channel concept into Eq. (2) and considering theprinciple of superposition, the IUH of the watershed can be expressedas:

h tð Þ =c1A δ t−T1ð Þ1 + k1Dð Þn1

+c2A δ t−T2ð Þ1 + k2Dð Þn2

+: : : +cNA δ t−0ð Þ1 + kNDð ÞnN

Zh tð Þ = ∑i = N

i = 1

ciA δ t−Tið Þ1 + kiDð Þni

� �ð4Þ

where subscript i=1,2,…,N (N is the number of sub-basins) denotesthe sub-basin number, (ni, ki) are parameters of Nash's model for theith sub-basin, Ti is the linear channel lag time, h(t) is the IUH of thewatershed, ci is ith sub-basin area, and A is the watershed area.

Parameters (ni, ki, Ti) of Eq. (4) must be computed for everysub-basin and can be defined using Nash's synthetic model as (Singh,1988):

ki = 0:79L−0:1i c0:3i S−0:3

˚ið5Þ

ni =kNi inwhich

kNi = L0:1i ð6Þ

Where S°is the average overland slope and L, with the dimension

[L], is the longest flow path in the drainage network. However, in theGUHN, Eq. (6) was modified as:

ni = nkNi ð7Þ

And Ti (hour) was computed using the Manning equation for afictitious linear channel with unit hydraulic radius as:

Ti = ∑j = N

j = i + 1

LRj×nmj

3600ffiffiffiffiffiffiSRj

q for i = 1 to N−1 and TN = 0: ð8Þ

SR is the channel slope; nm is the Manning coefficient and LR, withdimension [L], is the length of the channel. The values of c, S

°, L, SR and

LR can be determined using GIS tools. Now the GUHN model has onlyone unknown parameter, n–, with the dimension [L−0.1], that must becalibrated using observed rainfall–runoff data.

Parameter, n–, can be calculated using the method of moments(Singh, 1988). By calculating the first moment of the IUH (Eq. (4)) andthen applying Eq. (7), the following equation can be obtained:

M1 hð Þ = −1ð ÞdH1 sð Þds j

s = 0= ∑

i = N

i = 1

ciA

niki + Tið Þ

= n ∑i = N

i = 1

ciA

kNiki !

+ ∑i = N

i = 1

ciATi

ð9Þ

where H1(s) is the Laplace transform of IUH (Eq. (4)). Considering thetheorem of moments (Singh, 1988):

M1 hð Þ =M1 Qð Þ−M1 Ið Þ ð10Þ

and combining Eqs. (9) and (10):

n =M1 Qð Þ−M1 Ið Þ− ∑

i = N

i = 1

ciA Ti� �

∑i = N

i = 1

ciA

kNiki ! : ð11Þ

Therefore, the GUHN model parameter can be explicitly calculatedusing watershed geomorphological factors. Eq. (11) shows that themodel parameter (i.e., n–) acts as a modifier of the estimated values ofni,ki,Tiwhichareextracted fromsub-basingeomorphologyusingexper-imental relations (Eqs. (5), (6) and (8)) andmay contain some degree ofuncertainty.

4.2. Geomorphological Unit Hydrograph based on Cascade of linearReservoirs (GUHCR)

The GUHN model calculates the hydrograph just at the watershedoutlet without any information about hydrological behavior of thewatershed interior points. In order to overcome this shortage, GUHCRmodel is presented. The GUHCR model also employs a linear reservoircascade. In this model, awatershed is divided into sub-basins based onthe basin topography, similar to the GUHN model, and then each sub-basin is represented by a linear reservoir. In this way, a watershed isrepresented by a sequence of reservoirs distributed according to thewatershed geomorphology. Themain excess rainfall is divided propor-tionally between sub-basins on the basis of their areas, as illustrated inFig. 2. In this way, an unequal linear reservoir cascade model withdistributed excess rainfall results (Singh, 1988).

Distributing excess rainfall in proportion to sub-basin areas, onecan calculate the IUH vector as (Singh, 1988):

h tð Þ½ � = 1k½ � e

t a½ � C½ � where ð12Þ

Fig. 3. A schematic of Semi-distributed Linear Reservoir Cascade (SLRC) model.

209V. Nourani et al. / Catena 76 (2009) 206–214

c1c2

26

37 k1

k2

26

37

−a1 ˚ ˚ ˚a1 −a2 ˚ ˚

a2 −a3

2666

3777

1ki; C½ � = �

��cN

666664777775; k½ � = :

::kN

666664777775

and a½ � =˚ ˚˚ ˚ ˚ ˚: : : : :: : : : :: : : : :

˚ ˚ ˚ aN−1 −aN

66666664

77777775:

Eq. (12) gives the IUH of each sub-basin (hi(t)) and consequentlythe related outflow hydrograph can be computed. Using this semi-distributed GUHCR model, it is possible to incorporate hydrologicalconditions of the interiorwatershed parts.Moreover in GUHNeach sub-basin accompanied by its main channel was represented by a cascadeand a linear channel which directly discharges to the outlet and onlyrainfall was imposed on each sub-basin as input (without consideringprevious reservoir output as input). Therefore the GUHNmodel may beclassified as a lumped model. Here, when a small watershed is dividedinto much smaller sub-watersheds and then each sub-watershed isrepresented by an independent cascade without considering theassumption of direct discharge to the outlet, the smallness of the sub-watershed leads to a very low value for ni (less than one). Hence inGUHN, it is compulsorily assumed that each sub-basin dischargesdirectly to the outlet without having any semi-distributed character.

One can obtain the IUH of GUHCR in another way, as shown inAppendix A in the form of:

hi tð Þ = ∑i1 = i

i1 = 1

ci1A δ tð Þ

∏j = i1

j = 1kjD + 1� � i = 1;2;3; N ;N: ð13Þ

In the classic model of cascade of unequal linear reservoirs (Singh,1988), the storage parameter (lag time) of any sub-basin (ki), re-presented by a reservoir, should be determined by calibration. How-ever in GUHCR, ki can be related to the geomorphological properties ofthe sub-basin and just one unknown parameter (k

–) by using Eq. (5) as:

ki = k Kið Þinwhich

ð14Þ

Ki = L−0:1i c0:3i S−0:3˚i

ð15Þ

where k–is the model parameter with dimension [TL−1/2].

The method of moments can be used for determining parameter k–.

The first moment of watershed IUH using Eqs. (13) and (14) can bewritten as:

M1 hð Þ = −1ð Þ dH2 sð Þds js = 0

= ∑i = N

i = 1

ciA

∑j = i

j = 1ki

!= k ∑

i = N

i = 1

ciA

∑j = i

j = 1Ki

! !ð16Þ

where H2(s) is the Laplace transform of Eq. (13). Therefore, the water-shed parameter in GUHCR can be determined using the theorem ofmoments as:

M1 hð Þ =M1 Qð Þ−M1 Ið ÞZk =M1 Qð Þ−M1 Ið Þ

∑i = N

i = 1

ciA ∑

j = i

j = 1Ki

! ! : ð17Þ

Eq. (17) shows that the model parameter, k–, is explicitly linked

to geomorphological properties of the sub-basins. In modeling byGUHCR, just one parameter (i.e., ki) is computed by the experimentalequation (Eq. 14). This is in contrary to GUHNwhere three parameters(i.e., ni, ki, Ti) must be determined using Eqs. (5), (6) and (8).

4.3. SLRC model

Both GUHN and GUHCR are inherently linear models. Howevera watershed sometimes may show non-linear behavior, especially athigh discharge (Pilgrim, 1976).

A Semi-distributed Linear Reservoir Cascade (SLRC) model is asemi-distributed version of the linear reservoir cascade. In this model,a watershed is divided into subdivisions based on topography, andthen each sub-basin is represented by one linear reservoirs cascade.Thus, the watershed is represented by reservoirs that are distributedaccording to the watershed morphology. Finally a nonlinear equa-tion (i.e., kinematic wave model) is used for flow routing through thewatershed main channel. The precipitation is divided proportionallybetween sub-basins on the basis of areas.

In this model, the outflow of each sub-watershed is determinedusing geomorphologic properties and Nash's synthetic UH (Eqs. (5)and (6)). Then, the determined runoff applies to the mainriver momentarily as upstream flow. Thereafter flow is routed in themain channel using the kinematic wave equation; because thekinematic wave equation can be considered as a reliable routingmodel for the watersheds which have high land slopes, such asAmameh watershed. The circumstance of the model operation isillustrated by Fig. 3.

Using the Manning formula, the kinematic wave equation can beobtained as (Chow et al., 1988):

AQAx

+ αβQβ−1 AQAt

� �= q ð18Þ

with coefficients:

β = 0:6 ; α = nmB2=3=ffiffiffiffiffiSR

p 0:6ð19Þ

210 V. Nourani et al. / Catena 76 (2009) 206–214

Where q is the lateral flow, SR is the channel slope, nm is theManning coefficient, and B, with dimension [L], is the channel width.

Fig.

4.a:

Situationmap

,b:DEM

,c:Veg

etationco

verage

,d:Watersh

eddivision

.

Solving Eq. (18), the flow discharge can be obtained through thewatershed main channel at any time (t) and distance (x).

However, in the SLRC model, Eq. (18) is used as the adjusted formof:

AQAx

+ ααβQβ−1 AQAt

� �= q ð20Þ

Where α–is a correction coefficient to account for the assumptions

applied to the model.In SLRC, the Digital Elevation Model (DEM) is used for determining

the channel slope. The Manning coefficient can be obtained from theland use and plant coverage of the watershed. The use of large scalemaps for extracting hydrological properties, such as channel width, isnot suitable and the use of small scale maps for hydrological modelingis not economic. An equation, proposed by Bandaragoda et al. (2004),was used to estimate the channel cross section width as:

B = aAbupstream ð21Þ

where Aupstream is the upstream drainage area (km2), a=0.0011,b=0.518, and B is the channel width (m). Eq. (20) has been found tobe adequate (Nourani and Mano, 2007) but unlike b, a may signif-icantly change from one watershed to the other. Aupstream is calculatedat each point along themain river according to the accumulated drain-age area upstream of the point. α

–, a watershed parameter with no di-

mension, is the only parameter of the SLRC model which should becalibrated using rainfall–runoff data. As a matter of fact, α

–is used as a

correction for assumptions made and the uncertainty in the estimatedparameters, such as nm and a.

An implicit finite difference method (Chow et al., 1988) is usedin the SLRC model for solving Eq. (20). All geomorphological param-eters are extracted using GIS tools. In order to determine the modelparameter (α

–), the direct search method (Yue and Hashino, 2000),

among the other optimization schemes, is used.

5. Model performance measures

For assessing the performance of the proposed models, the Nashand Sutcliffe index (E) (1970), correlation coefficient between ob-served data and calculated data (R), and the ratio of absolute error ofpeak flow (RAEP (%)) were used. These measures are defined as:

E = 1−∑No

i = 1Qi;obs−Qi;sim� �2

∑No

i = 1Qi;obs−Q obs

2 ð22Þ

R =∑No

i = 1Qi;obs−Oobs

Qi;sim−Q sim

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi∑No

i = 1Qi;obs−Oobs

2∑No

i = 1Qi;sim−Q sim

2s ð23Þ

RAEP kð Þ = jQP obs−QP simjQP obs

×100 ð24Þ

where Qi,obs is the observed discharge at t= i; Qi,sim is the simulateddischarge at t= i; No. is the number of observed data; and QP.obs, QPsim

are observed and simulated peak discharges, respectively.

6. Study area

The Amameh watershed, one of the sub-basins of the Jajrood riverbasin, upstream of Latian dam, is located in the southern area of

Table 1Data extracted from rainfall–runoff events

No.Event Date hrainfall hrunoff ϕ hrunoff/hrainfallmm mm mm/h

1 8/10/1993 8.05 0.20 6.40 0.0252 3/27/1992 5.45 2.78 1.57 0.5113 7/04/1992 6.35 2.63 1.51 0.4154 9/05/1992 11.5 0.20 0.83 0.0175 7/04/1997 7.65 0.28 0.07 0.0366 4/17/1997 6.25 2.82 5.17 0.4517 7/13/1996 10.50 1.85 1.20 0.1768 7/18/2004 4.95 2.74 1.11 0.553

211V. Nourani et al. / Catena 76 (2009) 206–214

central Alborz, near Tehran (capital of Iran), with an area of 37.2 km2

and the watershed elevation ranges from 1900 m to 3868 m above sealevel. Fig. 4(a) shows an aerial photograph of the Amameh watershed.Fig. 4(b) shows a raster Digital Elevation Model (DEM) of the Amamehwatershed with 50 m resolution which was used for extracting geo-morphological properties of the watershed. This map was obtainedfrom the topographical map of the watershed (scale: 1/25,000). Thewatershed was discretized in a raster-based GIS environment into anarray of grid cells with 50 m resolution. Each cell represents an areawith average properties. The direction of flow from one cell to otherneighboring cells was determined by the using eight-direction pointpour algorithm in which the algorithm chooses the direction of thesteepest descent among the eight permitted choices (Maidment, 2002).Once the algorithm identified the flow direction in each cell, a cell-to-

Table 2Geomorphological factors of GUHN, GUHCR and SLRC models

Sub-basin

ci Li S0iLRi

SRiki N

ki Ti Ki Bi

(m2) (m) % (m) % (h) (h) (m)

1 6,349,057 3047.08 49.90 2141 15.71 0.76 2.23 0.273 60.67 3.342 9,259,785 4325.05 66.61 3329 25.33 0.75 2.3 0.23 60.16 5.323 7,378,772 4119.52 37.97 2559 6.56 0.84 2.29 0.162 68.85 6.514 7,106,051 5460.24 53.52 2140 4.30 0.73 2.36 0.093 57.97 7.485 7,294,662 4664.72 60.12 3074 4.84 0.72 2.32 0 57.32 8.37

cell flow path was determined to the watershed outlet (Maidment,2002). Next, by studying the flow accumulation pattern among thecells, the computational drainage path and network were generated.The calculation was done starting at the most upstream cells deter-mined through DEM analysis. The two-dimensional array of DEM cellswas re-arranged into a one-dimensional array as a cascading system(Maidment, 2002). Also, the land slope of aggregated DEM grids wascalculated by averaging the slope values of the original cells formingthe aggregated grid.

About 200 ha of the vegetation coverage includes gardens and grass,and the remainder has vegetation coverage of bushes. The vegetationcoverage of thewatershed extracted from aerial photographs is shownin Fig. 4(c). For rainfall–runoff simulation, the watershed was dividedinto 5 sub-basins using GIS tools as shown in Fig. 4(d). Some of the

Table 3Calibration results of Nash, SCS, GUHN, GUHCR and SLRC models

Event Nash model SCS model GUHN m

k n E R RAEP tL(min) E R RAEp n–

1 0.829 1.78 0.78 0.93 19.66 45 0.53 0.95 22.10 0.742 1.001 4.79 0.87 0.93 26.57 195 0.32 0.90 61.74 2.633 1.421 2.49 0.89 0.95 18.08 120 0.53 0.97 38.49 1.924 1.421 2.52 0.89 0.95 18.21 125 0.55 0.98 40.81 1.935 1.027 4.50 0.87 0.93 26.20 185 0.34 0.91 59.95 2.5336 1.315 2.38 0.91 0.95 13.84 110 0.54 0.98 38.71 1.69Ave. 1.169 3.08 0.87 0.94 20.43 130 0.47 0.95 43.63 1.907

stormevents for thiswatershed have been registered since 1990with atime interval of 30 min and were used to verify the models.

7. Results and discussion

Due to the lack of registered data, 8 storm events were used forcomparing simulated hydrographs; 6 events for model calibration and2 events for verification. In this way, the direct hydrograph of anyevent was specified by the fixed gradient method first and then infi-ltration of each event was calculated. Thereafter, using the ϕ indexmethod the excess hyetograph of the event was obtained from theobserved hyetograph (Chow et al., 1988). In this way any rainfall priorto the beginning of direct runoff was taken as initial abstraction.Specifications of storm events are provided in Table 1.

Geomorphological factors extracted by GIS and necessary para-meters for calculating parameters of the GUHN, GUHCR and SLRCmodels are shown in Table 2.

The average Manning coefficient values were nm=0.024 for sub-watersheds 1, 2, 4, and 5 and nm=0.03 for sub-watershed 3, whichwere obtained considering vegetation coverage extracted from aerialphotography using ERDAS IMAGINE software (ERDAS, 2000). Modelparameters were determined using these quantities and 6 stormevents data. The calculated parameters of the Nash model (n, k for thewhole watershed), GUHN model (n

–), GUHCR model (k

–) and also SLRC

model (α–) are given in Table 3.

A watershed is considered as a single piece with no sub-basinfor modeling by the SCS and Nash methods. The Curve Number (CN)was chosen as 79 for the watershed vegetation coverage. The lag timeparameter was chosen as the SCS variable parameter and an optimumvalue of parameter for each stormwas calculatedusing the direct searchmethod (Yue and Hashino, 2000) and the results are given in Table 3.

The results of calibration of the Nash, GUHN, GUHCR models withparameters calculated using the method of moments and SCS andSLRC models with parameters calculated by the direct search methodare shown in Table 3 and are graphed in Fig. 5. Thereafter, the modelswere verified for two other event data sets and the results are shownin Table 4. The average values of parameters obtained from calibrationwere used for verification and the results are shown in Fig. 6. The re-sults of the SCS model show that the use of inadequate model param-eters leads to unreliable results.

The Nash model has two degrees of freedom, so the calibrationresults might be better because a two-parameter model may better fitthe observed data. However, two parameters increase more depen-dency on the accuracy of the determined values of the parameters andprobable error existing in the determined parameters sometimes causesa decrease in the accuracy of the model in the verification process.Considering Tables 3 and 4, the average values of the efficiency criteriafor one-parameter GUHCR, GUHN and SLRC models show their capa-bilities so that there is not a significant difference between the calibra-tion and verification results.

From the results of both calibration and verification (Figs. 5 and 6),it can be seen that the rising limbs of hydrographs which are mostlyrelated to stormproperties aswell as recession limbswhich are usuallyrelated to the watershed morphology can be properly simulated by

odel GUHCR model SLRC model

E R RAEp k–

E R RAEp α– E R RAEp

0.85 0.95 21.91 0.008 0.74 0.95 0.34 0.43 0.70 0.81 8.810.90 0.97 18.67 0.026 0.91 0.87 23.20 1.5 0.85 0.95 34.090.87 0.96 0.98 0.019 0.77 0.88 17.56 0.74 0.94 0.99 1.560.87 0.96 0.88 0.02 0.78 0.89 21.7 0.81 0.94 0.98 1.040.90 0.95 17.63 0.025 0.83 0.92 24.59 1.38 0.87 0.96 31.470.91 0.98 2.17 0.017 0.83 0.92 9.81 0.52 0.93 0.99 5.870.88 0.96 10.37 0.019 0.81 0.91 16.2 0.9 0.87 0.95 13.81

Fig. 5. Calibration graphs of Nash, SCS, GUHN, GUHCR and SLRC models.

212 V. Nourani et al. / Catena 76 (2009) 206–214

GUHN, GUHCR and SLRC models. Simultaneous consideration of geo-morphological and calibrated parameters in the formulation of theproposed models may give this capability to the models.

For high discharges, a watershed usually shows a linear behavior(Pilgrim, 1976) and linear models, such as GUHN and GUHCR, canperformwell in such situations, as indicated by RAEp in Tables 3 and 4.Also, taking advantage of the non-linear regression betweenmodel pa-rameters and watershed physical features (Eqs. (5), (6), (8), and (14))

may help GUHN and GUHCR models to simulate low dischargesappropriately as well. Similarly, in the SLRC model the linear Nashmodel leads to a suitable estimation of peaks as well as low discharges.

In comparison of two linear UH models (i.e., GUHN and GUHCR),the efficiency of the fully lumped GUHN model which uses threegeomorphological relations (i.e., Eqs. (5), (6), (8)) is satisfactory,especially in the verification phase. However, the semi-distributedbackground of the GUHCR model leads to the output hydrographs of

Table 4Verification results of Nash, SCS, GUHN, GUHCR and SLRC models

Event Nash model SCS model GUHN model GUHCR model SLRC model

k n E R RAEp tL(min) E R RAEp n– E R RAEp k–

E R RAEp α– E R RAEp

7 1.169 3.08 0.81 0.93 29.64 130 0.41 0.97 53.98 1.91 0.80 0.89 14.75 0.019 0.79 0.88 17.98 0.90 0.82 0.93 21.148 1.169 3.08 0.78 0.91 33.54 130 0.47 0.96 55.68 1.91 0.93 0.98 17.93 0.019 0.74 0.85 1.74 0.90 0.90 0.95 23.22Ave. 1.169 3.08 0.8 0.92 31.59 130 0.44 0.97 54.83 1.91 0.87 0.94 16.34 0.019 0.77 0.87 9.86 0.9 0.86 0.94 22.18

213V. Nourani et al. / Catena 76 (2009) 206–214

sub-basins using sub-basins IUHs (Eq. (13)). Furthermore, inclusion ofa non-linear distributed routing model (i.e., kinematic wave) with alinear rainfall–runoff model (i.e., Nash's model) makes the SLRCmodelsuitable as a semi-distributedmodel with relatively high performanceefficiency.

The lag time of each sub-basin represented by a linear reservoir, ki,in the unequal cascade of linear reservoirs with distributed excessrainfall may be different from one sub-basin to the other (Singh,1988).Therefore, calculation of parameters is hard and almost undetermin-able due to the great number of parameters. However, there is only oneparameter, k

–, in the GUHCR model. Similarly, the GUHR (Geomorpho-

logical Unit Hydrograph of Reservoirs)model (Agirre et al., 2005; Lópezet al., 2005), which uses cascaded linear reservoirs, makes use of onlyone parameter but it cannot consider changes of storage parameters(ki) efficiently, because it does not use physical properties of sub-basinsin the parameter computation. However, the lag timeof each sub-basinin the GUHCR model is related to geomorphological properties and isdifferent from one sub-basin to the other and also the number ofparameters reduces to one, while considering the changes of lag timeparameters. Therefore, the GUHCR model is similar to GUHR andunequal linear reservoir cascade models but considers geomor-phological properties of the sub-basin in parameter formulation.Unlike the fully lumped models such as Nash and GUHCR, the semi-distributed GUHCR model can also reflect the hydrological conditionsof the interior parts of the watershed.

8. Concluding remarks

From this study, the following conclusions are drawn about theproposed GUHN, GUHCR and SLRC models:

a) Themodels are geomorphological and are constructed on the basisof watershed physical properties.

b) The models have only one parameter. Although the proposedmodels have just one parameter which must be calibrated, incor-poration of geomorphological characteristics in the parameter esti-mation leads us to consider them as promisingmodels, particularly

Fig. 6. Verification graphs of Nash, SCS

for large watersheds which have different sub-basins with variantgeomorphologic properties or for such watersheds having largeheterogeneous physical factors, like Amameh watershed whereland elevation variation is significant (from 1900 m to 3868 m).

The GUHN model is a lumped model like Nash's; however thewatershed division into some small fragments with own physical andgeomorphological properties converts this black box model to a rea-sonable model for conversion of rainfall into runoff.

In addition to the spatial dynamic character of model parameters,derived explicit equations for parameters in the presented UHmodels,make the application of the models more straightforward, especiallyfor calibration.

In spite of adjusting Nash's synthetic model for Amameh water-shed by applying model parameters, some errors may be exist in themodel because the experimental equations were established on thebasis of British catchment data sets. Therefore, it is recommended toderive similar formulas for related regions in Iran and then apply themodels for more watersheds with similar morphological conditions.Of course, implementation of this recommendation is difficult in devel-oping countries such as Iranwhich generally suffer from data shortage.

In the face of the good efficiency of the presented models, they arevery sensitive to the used topography data and DEM resolution and themodels efficienciesmay decay due toman-made (e.g., urbanization) ornatural (e. g., erosion) changes of these data. Furthermore, conceptua-lizations of the homogeneity of the hydrological quantities inside thegrid of a DEM result in different performances in the models them-selves with variations in the assumed scale.

The proposed models were applied to a watershed which wasdivided on the basis of the same interval lines, but thesemodels can beused for watersheds which are divided into sub-watersheds on thebasis of stream junctions in the drainage network. It is also suggestedto contemplate seasonality and temporal effects accompanied byphys-ical and geomorphological factors in the formulation of the proposedmodels. This consideration and using more storm events data mayextend the model capability and efficiency; this proposal can be con-sidered as a new research plan for the future work.

, GUHN, GUHCR and SLRC models.

214 V. Nourani et al. / Catena 76 (2009) 206–214

Acknowledgments

This paper is supported by a research grant of the University ofTabriz. Also, two anonymous reviewers are gratefully acknowledgedfor providing useful comments on an earlier version of this paper.

Appendix A

For GUHCR model with the assumption of distributing excessrainfall in proportion to areas and applying Eq. (1) and continuityequation (Chow et al., 1988) (I tð Þ−Q tð Þ = ds

dt) to all the reservoirs withrainfall and outflow of prior reservoir as inflow, the system equationsobtained are (Singh, 1988):

1 + k1Dð ÞQ 1 tð Þ = c1AI ðA� 1Þ

1 + kiDð ÞQ i tð Þ = ciAI +Qi−1 i = 2;3; :::;N:

Hence, the outflow of the first reservoir will be:

Q1 tð Þ =c1A I

1 + k1Dð Þ : ðA� 2Þ

For the second reservoir, when the inflow is constituted by theexcess rainfall plus discharge flow from the first reservoir, the outflowwill be:

Q2 tð Þ =c1A I

1 + k1Dð Þ +c2A I

1 + k1Dð Þ 1 + k2Dð Þ : ðA� 3Þ

Likewise, system equations for the instantaneous unit precipita-tion are:

hi tð Þ = ∑i1 = i

i1 = 1

ci1A δ tð Þ

∏j = i1

j = 1kjD + 1� � i = 1;2;3; N ;N: ðA� 4Þ

The hi(t) vector in Eq. (A-4) is a general solution for IUH and fori=N, hN(t)=h(t) will be the watershed IUH.

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