Annual Journal of Hydraulic Engineering, JSCE, Vol.52, 2008, February
Annual Journal of Hydraulic Engineering, JSCE, Vol.52, 2008, February

LUMPING OF A PHYSICALLY-BASED DISTRIBUTEDMODEL FOR SEDIMENT RUNOFF PREDICTION
IN A CATCHMENT SCALE
A P I P 1, Takahiro SAYAMA2, Yasuto TACHIKAWA3 and Kaoru TAKARA4
1Student Member, Graduate student, Dept. of Urban and Environmental Eng., Kyoto University(Kyoto 615-8540, Japan) E-mail: apip@flood.dpri.kyoto-u.ac.jp
2Member of JSCE, Dr. Eng., Assistant Professor, DPRI, Kyoto University (Uji 611-0011, Japan)
3Member of JSCE, Dr. Eng., Associate Professor, Dept. of Urban and Environmental Eng., Kyoto University(Kyoto, 615-8540, Japan)
4Fellow of JSCE, Dr. Eng., Professor, DPRI, Kyoto University (Uji 611-0011, Japan)

This paper proposes a lumping method of a distributed rainfall-sediment-runoff model. The lumping
method is divided into two parts: lumping of the kinematic wave distributed rainfall-runoff model, andlumping of the distributed sediment transportation model that takes into account for erosion anddeposition by raindrop and overland flow. A kinematic wave distributed rainfall-runoff model is lumped
based on the assumption of steady state conditions. Soil detachment and deposition from an entire
catchment are simulated by the balance between the current sediment storage and the maximum sedimentstorage, which is estimated from a distributed rainfall-sediment-runoff model as a function of total surface
water storage of the lumped rainfall-runoff model. The proposed lumping method is examined bycomparing water and sediment discharges simulated by the distributed and the lumped rainfallsediment-runoff models in the Lesti River Basin, Indonesia.
Key Words: Lumping, distributed rainfall-sediment-runoff model, erosion, Lesti River basin.

1. INTRODUCTION
To predict sediment transport and runoff arevery important to solve the problem of water
resources. Design of reservoirs and dams, design ofsoil conservation, land-use planning, water quality
and aquatic habitat management are some of theexamples. Sediment transport is highly dependent
on topography, land use, and soil type. Therefore, itis essential to incorporate physically reasonable
functions in a rainfall-sediment-runoff model.
Complex distributed rainfall-sediment-runoff
models have been developed. KINEROS, LISEM,WEPP, and EUROSEM1) are some of the examples.
These models are normally designed to estimate
sediment runoff during a single rainfall event.
The problem of those fully distributed models
have limitation in application because those models
generally require much computation time to conduct
a simulation especially for large catchments andlong-term simulations. In the past, lumped andsemi-distributed models have been widely preferred
for representing the rainfall-sediment-runoff process
at scales ranging from few to several hundreds
square kilometers. Unfortunately, given lack of

direct physical interpretation of their model
parameters, long term observed data records areneeded for their calibration, which may not always
be available.
From the above considerations, the need is
obvious for creating a new generation of a lumped
model with parameters directly related to physicallymeaningful quantities derived from appropriate
distributed scales. A concept for lumping a
distributed rainfall-runoff model using digital
topographic information to reduce computationalburden required in runoff simulation was proposed2).
The governing equations and the lumped model
parameters are derived from a distributed rainfallrunoff model by assuming that a rainfall-runoff
process of the system reaches a steady state
condition. The advantage of the lumping is that it
can derive the lumped parameters keeping thephysical meanings of the original distributed model
without any additional calibration.
As an extension of the lumping method for aphysically-based distributed rainfall-runoff model2),
the authors extend the method with adding a new
lumping method for sediment transport processes
incorporating surface and subsurface flow in- 43 -

unsaturated and saturated zones. This researchpresents a method to lump a physically-based
distributed rainfall-sediment-runoff model as a toolfor estimating erosion, runoff, and sedimenttransport processes in a catchment scale. The
lumped model is applied in the Lesti River locatedin the upper Brantas River basin, Indonesia.

(DF). The basic assumption of this model is thatthe sediment is yielded when overland flow occurs.
The eroded sediment is transported by overland flow
to river channels.

2. TH E DI STRI BUTED RAI NFALLSEDIMENT-RUNOFF MODEL(1) Physically-based Distributed Rainfall-Runoff
Model, CDRMV3Cell Distributed Rainfall-Runoff Model Version
3 (CDRMV3) as a distributed hydrological model3)
is used as a base of a distributed sediment transportmodel. The model includes a stage-discharge, q-h,
relationship for both surface and subsurface flows4):⎧v m d m (h / d m ) β ,0 ≤ h ≤ dm⎪
q = ⎨v m d m + v a (h − d m ),d m ≤ h ≤ d a (1)⎪m⎩v m d m + v a (h − d m ) + α (h − d a ) , d a ≤ hν m = kmi , ν a = k a i , km = ka / β , α = i / nwhere q is discharge per unit width; h is water
depth; i is the slope gradient; km is the saturated
hydraulic conductivity of the capillary soil layer; ka
is the hydraulic conductivity of the non-capillarysoil layer; dm is the depth of the capillary soil layer;
da is the depths of capillary and non-capillary soil
layer; and n is the roughness coefficient based on
the land cover classes. The continuity equation takes
into account flow rate of each grid-cell or slope as:∂h ∂q+= r (t )
∂t ∂x

(2)

where t and x are time and distance along water flow,
respectively; and r is the rainfall intensity.
The Lax-Wendroff finite difference scheme isused to solves the one-dimensional kinematic wave
equation with the stage-discharge equation tosimulate runoff generation and routing. The
simulation area is divided into an orthogonal matrix
of square grid-cells (250 m x 250 m).
(2) Coupling of CDRMV3 and SedimentTransport Model
The concept of spatially distributed sediment
runoff modeling is shown in Fig. 1. A sedimenttransport algorithm is newly added to the CDRMV3.
Runoff generation, soil erosion and deposition arecomputed for each grid-cell and are routed between
grid-cells following water flow direction. The
sediment transport algorithm includes multiple
sources of sediment transport, which are soil
detachment by raindrop (DR) and hydraulicdetachment or deposition driven by overland flow

Fig.1

Schematic diagram of the physically based
rainfall-sediment-runoff model in the ith grid-cell scale.

Soil detachment and transport is handled withthe continuity equation representing DR and DF as:∂ (hs c ) ∂ (qs c )+= e (x, t )
(3)
∂t
∂x
e( x , t ) = DR + DF
where C is the sediment concentration in theoverland flow (kg/m3); hs is the water depth ofoverland flow (m); qs is the discharge of overland
flow (m3/s); and e is the net erosion (kg/m2/hr).
Soil detachment by raindrop is given by anempirical equation in which the rate is proportional
to the kinetic energy of rainfall and decreases withincreasing hs. From the observation of rainfall
characteristic in the study area5) and dampening soil
detachment rate by hs6), the empirical equation for
DR for the ith grid-cell is expressed as:(4)
DRi = k KE e − b*hsi = k 56.48 ri e − b*hsi
where k is the soil detachability (kg/J); KE is thetotal kinetic energy of the net rainfall (J/m2); and b
is an exponent to be tuned.
Following the theoretical work of EUROSEM6),
the concept of transport capacity is used todetermine sediment transport rates in overland flow.
Sediment transport capacity of overland flow (TC) isdefined as the maximum value of sedimentconcentration to transport, which is estimated foreach grid-cell. Then for the ith grid-cell, DF issimulated as a result of overland flow and functionof TC as follows:(5)
DFi = α (TCi / 1000 − Ci )hs
iwhere α is the detachment/deposition efficiency
factor. Detachment or deposition by flow isassumed to be proportional to the TC deficit.
Following the TC approach; if actual suspended
sediment from upper grid-cells is lower than thiscapacity, detachment or erosion occurs, otherwise
soil deposition excess.
The transportation capacity is calculated based- 44 -

on the Unit Stream Power (USP) theory. The USP
theory contributing to TC is defined as a product of
the overland flow velocity, v, and slope, i, in the ith
grid-cell. A relationship between USP and theupper limit to the sediment concentration in theoverland flow, Ct (ppm), can be derived7) (see Eq.
(6)). TC is the product of Ct as follows:(6)
TC = log C t = I + J log((vi − v critical i ) / ω )in which:
I = 5.435 − 0.386 log(ω D50 / NU ) − 0.457 log(U ∗ / ω )the catchment under the assumption of steady state
conditions of rainfall-runoff by spatially uniform
rainfall input.
The relationship between total storage of waterin the ith grid-cell (swi) and the outflow discharge in
the ith grid-cell (Qi) is theoretically derived. Flux
Qi is expressed as the product of rainfall intensity( r ) and the upslope contributing areas (Ui):J = 1.799 − 0.409 log(ω D50 / NU ) − 0.314 log(U / ω)(7)
where x is the horizontal distance from the upstream end of a grid-cell and w is the width of gridcell. The upslope contributing area can be generated
from water flow accumulation data in each grid-cell.
The storage volume of overland flow for the ith
grid-cell (swi) as a function of h, w, and x is given as:

∗

ω=

2
36
+
−
2
3 ⎛ρ
⎞
D50 ⎞ NU⎛s⎜⎜
− 1⎟⎟ g ⎜⎟⎝ ρ w ⎠ ⎝ 1000 ⎠

36
2

⎛ ρs
⎞ ⎛ D ⎞ NU⎜⎜
− 1⎟⎟ g ⎜ 50 ⎟⎝ ρ w ⎠ ⎝ 1000 ⎠where vi is the unit stream power, m/s (v is flow
velocity in m/s and i is the slope gradient m/m);
vcriticali is the critical unit stream power (vcritical is thecritical flow velocity); ω is the sediment fallvelocity (m/s) calculated by Rubey’s equation; ρs isthe sediment particle density (kg/m3); ρw is the water
density (kg/m3); g is the specific gravity (m/s2); D50
is the median of grain size (mm); and NU is the
kinematic viscosity of the water (m/s2).
U*(= g i hs ) is the average shear velocity (m/s).
This model does not explicitly separate rill andinterrill erosion. Gullying, river bed erosion, riverbank erosion and lateral inflow of sediment to river
channel are not considered.

3. THE LUMPED RAINFALL-SEDIMENTRUNOFF MODEL
The basic motivation in lumping lies in thedevelopment of a sediment runoff model applicablefor a larger catchment scale by considering all thesmaller scale processes and information, in which
the derivation of lumped governing equations areassociated to the physical meaning of model
parameters, and it realizes to reduce computationaltime required in a simulation.
(1) Lumping of Physically-Based Distributed
Rainfall-Runoff Model
The lumping method of physically-based
distributed rainfall-runoff model2) was used andextended. To obtain the lumped sediment runoff
model, the process equation, namely the kinematicwave equation, is integrated over the entire systemof grid-cells describing the catchment. This isdone first by computing the total water volume
stored in the soil, on the surface or in the channel
network by adding up the single grid-cell volumes
as a function of the geomorphology and topology ofx

Qi ( x) = Qi (0) + r ∫ wi ( x) dx = r U i + r xi w( x )0q i ( x) = Qi ( x) / wi ( x) = r U i / wi ( x) + r xi

xs wi = ∫ hi ( x) wi ( x) dx(8)0

By substituting the variable of integration from x toq using the relationship given by (8):q( X )ww
s wi =f ( q ) dq = [F ( q ( x )) − F ( q (0)) ] (9)∫r q (0)r
Figure 2 schematically shows a relationshipbetween q, h, and F. F(q(x)) can be further
estimated depending on the surface soil conditionand q-h relationship for three layers. It is assumed
that q is a function of h and can be analyticallyintegrated with h. If the values of qi(x) using Eq.(7)and hi(x) numerically is obtained using Eq.(1), thenF(q(x)) can be calculated. The storage of water at
the catchment scale, Sw, can be calculated by adding
up swi from each grid-cell. Finally, Q of the outlet islinked to Sw as a function of the topographic andphysical characteristics of each grid-cell, as well as
rainfall intensity. To relate Q of the catchment
outlet and Sw, the Q-Sw relationship was established.
Through variable rainfall intensities, the nonlinear
Q-Sw discrete relationship at the study catchment
was obtained (Fig. 3a). In order to obtain the
dynamic distribution of overland flow in thecatchment, a relationship between the surface water
storage amount, Sws, and Sw was developed (Fig. 3a).
(2) Lumping of Physically-Based Distributed
Rainfall-Sediment-Runoff Model
The same concept is explored to the distributed
rainfall-sediment-runoff model to derive how
sediment load is related to catchment hydrologicalresponse, detachment source type and intensity, anddepositional processes. The sediment-runoff
processes in this study are affected by dynamic
spatial distribution of overland flow.

- 45 -

In the case unsaturated flow (q(x) less than or equal qm) :h(x )F (q ( x )) = q ( x )h( x ) − ∫ g (h ) dh = q (x )h(x ) −0

V m h β +1

β + 1 d m β −1

In the case saturated flow (q(x) less than or equal qa) :
β

h(x )
m
⎛ h ⎞
⎟⎟ dh − ∫ (Vm d m + Va (h − d m )) dh
F (q (x )) = q ( x )h( x ) − ∫ Vm d m ⎜⎜⎝ dm ⎠dm0d= q (x )h( x ) −

2

(

)

Vm d m
− 0.5V a h 2 −d m 2 − d m (h − d m )(V m − V a )β +1

In the case surface flow (q(x) great than qa) :
β

dm

(

d

)

)

h( x)
a
⎛ h ⎞
⎟⎟ dh − ∫ (V m d m + V a (h − d m )) dh − ∫ V m d m + V a (h − d m ) + α (h − d m )m dh
F (q (x )) = q ( x )h( x ) − ∫ V m d m ⎜⎜⎝ dm ⎠0dmda= q (x )h(x ) −()

Vm d mα2(h − d a )m+1 − d m (h − d m )(Vm − Va )− 0.5Va h 2 − d m −
β +1
m +1
2

10000

6.E+07

Q-Sw relationship
Sws-Sw relationship

5.E+078000

4.E+076000

3.E+074000

2.E+072000

1.E+070
0.E+002.E+07

4.E+07

6.E+078.E+07

1.E+080.E+00
1.E+08

Total water storage vol., Sw (m3)

5.3
4.5
3.8

0.9

(b)

0.8

Ssmax.-Sws relationshiphs-avr-Sws relationship

0.7
0.6

3.0

0.5

2.3

0.4
0.3

1.5

0.2
0.8
0.0
0.E+000.1
1.E+072.E+07

3.E+07

4.E+07

5.E+07

6.E+070
7.E+07

Average of surface water depth,
hs-avr (m)7.E+07(a)

12000

Max. sediment storage., Ssmax(kg/m2/hr)

14000

Surface water storage vol., Sws (m3)Outflow discharge, Q (m3/sec)

Fig.2 Schematic drawing of q, h, and F relationship.

Surface water storage vol., Sws (m3)

Fig.3 (a) Plots of discrete relationships between Q-Sw and Sws-Sw; and (b) discrete relationships between S smax -Sws and hs-avr-Sws.

a) The Maximum Sediment Storage ( S smax )The maximum sediment storage is defined as
the total sediment transport capacity of overland
flow in a whole of the catchment for each time step
calculation. Therefore, we expressed the maximum
sediment storage as the function of TC from the ith
grid-cell, surface water storage amount in the ith
grid-cell (swsi), and Sws. The maximum sedimentstorage at catchment scale is calculated by adding
up TCi multiplied to swsi for all grid-cells as:(10)
S smax = ∑ TC i s wsi /( S ws 1000)

TC (ppm) for the ith grid-cell has been estimated by:
TC i = C t i = 10 5.0105 +1.363 log(( USPi −USPcritical ) / ω ) (11)where v in the ith grid-cell is calculated if h > dabased on the q-h relationship as:i(12)
ν i = k a ii + i (hi − d a )m −1nib) Sediment Concentration (C)Based on the relationship between the current
sediment storage (Ss) (kg/m3/hr) and Sw for eachtime step calculation, the value of C from watershed
outlet can be solved as:
C=

Ss
S ws

(13)

For each time-step calculation C is assumed to beuniform over the catchment and this is the variableof sediment continuity (see Eq.(15)).
To relate the sediment transport variables andthe rainfall-runoff variables at the study catchment
scale, a discrete relationship between Sws , S smax , andthe average of overland flow water depth at thecatchment, hs-avr, was resulted as shown in Fig. 3busing the above procedures.
Finally, the continuity equations of runoff (Eq.
(14)) and sediment (Eq.(15)) on a catchment scale:dS w (t )(14)= r A−Qdt
dS s (t )= DR + DF − Q C
dt= k KE e −b*hs −avr + α (S smax − S s )hs−avr − Q C 3600/ A (15)where A is the total area of the catchment (m2).
The relationship between detachment andredeposition represented by Eq.(15) depend on thebalance between Ss and S smax .
The most important feature of this method isthat the relationship between Q, Sw, Sws, hs-avr, andS smax at the catchment scale are deductively derived
as a discrete relationship, which is estimated on the- 46 -

basis of the grid-cell from the distributed model andserves basis to solve the continuity equations of the
lumped model. Thus transforming the distributedmodel into a lumped model. The lumped parameters
maintain the physical meaning of the original
distributed model without any additional calibration.
The continuity equations were solved numericallyusing a fourth-order Runge-Kutta algorithm.

4. EVALUATION OF THE LUMPEDSEDIMENT RUNOFF MODEL
The lumped model was applied to the Lesti
River catchment (351.3 km2), a tributary catchment
in the Upper Brantas River basin (11,800 km2), in
East Java, Indonesia. This area represents a tropicalvolcanic area, where land use types are largely
dominated by agriculture lands. Most of urban
lands, dense forests, and paddy fields are relatively
small and concentrated in several spot areas (see Fig.
4). At the confluence point of the Lesti River andthe Brantas main reach, the Sengguruh dam wasconstructed in 1998. Unexpectedly, most of thegross storage (21.5 million m3) has been alreadyfilled with the large amount of sedimentation fromthe Lesti River.
The parameter values of the lumped sediment
runoff model were: n=0.001-0.3 m-1/3s, da=0.08-1.2
m, dm=0.04-0.8 m, Ka=0.015 m/sec, m=1.667, β =8,D50 =0.062 mm, k=0.004 kg/J, α=0.98, ρs=2650
kg/m3, ρw=103 kg/m3, g=9.8 m/s2, and NU=10-6 m2/s.

Figures 5 and 6 show comparisons of simulated water and sediment concentration at the outlet
for two cases of soil thickness which calculated by
the distributed and the lumped models. The blue andblack lines are simulated water and sedimentconcentration by the distributed model and red lines
are simulated water and sediment concentration by
the lumped model. In the case when soil thickness
is shallow (Case 1), discrepancy between the results
for both the water discharge and sedimentconcentration by the lumped and distributed isgenerally less than in the case surface soil thickness
is more thick (Case 2). The difference increases in
the case when soil thickness is very thick, where thelumped model generally tends to under estimate
water discharge and sediment concentration at therising limb, and over estimate at the falling limb ofhydrograph and sedimentgraph. This difference isdue to the assumed steady state condition inderiving the lumped model. In Case 2, the catchment
system did not reach a steady state due to slow
movement of water for deep soil layer. Thus the
Q-Sw relationship for the lumped model shows somedifference from distributed one. In the distributedmodel, some parts of the catchment are saturated andoverland flow happens thus the rising limb of thehydrograph is faster than the lumped one.
Within the range of possible soil depths, thedifference is small and it does not cause severe
problems for applications. Table 1 shows thedifferences value in cumulative total runoff and

50

100

Rainfallby lumped modelby distributed model

40

75

30

50

20

25

10

0

0
0

10

20

30

40

50

60

70

60

125

Case 2: D=2.0 m, da=1.0 m, dm=0.50 m

50

100

Rainfallby lumped modelby distributed model

40

75
50

20

25

10

0

0
1

80

30

10

19

28

37

46

55

64

Rainfall intensity (mm/hr)

125

150

Water discharge (m3/sec)

Water discharge (m3/sec)

60

Case 1: D=0.5 m, da=0.30 m, dm=0.20 m

Rainfall intensity (mm/hr)

Fig.4 Lesti river catchment and land use in 2003.
150

73

Time from beginning of rain (hr)Time from beginning of rain (hr)Fig.5 Computed water discharges with different values of soil thickness D=0.5 m (Case 1) and D=2.0 m (Case 2).

- 47 -

Case 1: D=0.5 m, da=0.30 m, dm=0.20 m

40000

Rainfallby lumped modelby distributed model

30000

50
40
30

20000

20

10000

10

0

0
1

10

19

28

37

46

55

64

60

50000

Case 2: D=2.0 m, da=1.0 m, dm=0.50 m

50

40000

Rainfallby lumped modelby distributed model

40

30000

30

20000

20

10000

10

0

73

0
1

Time from beginning of rain (hr)

10

19

28

37

46

55

64

Time from beginning of rain (hr)Rainfall intensity (mm/hr)

50000

60000

Sediment concentration
(ppm)

60

Rainfall intensity (mm/hr)

Sediment concentration
(ppm)

60000

73

0
3
6
9
12

1

20

39

58

Computed discharge by distributed
Observed discharge
Computed discharge by lumped
Computed sediment concentration by distributed
Observed sediment concentration

60

12000

Computed sediment concentration by lumped

50

10000

D=0.2 m, da=0.08 m, dm=0.04 m

40

8000

30

6000

20

4000

10

2000

0

0
0

10

20

30

40

50

60

70

80

Sediment concentration (ppm)

Rainfall(mm/hr)(mm/hr)
Outflow discharge (m3/sec) Rainfall

Fig.6 Computed sediment discharges with different combinations soil thickness D=0.5 m (Case 1) and D=2.0 m (Case 2).

Time from beginning of rain (hr)Fig.7 Event sediment-runoff observation and simulation at
Tawangrejeni station in 2003.

sediment yield calculated by lumped and distributedmodels, which are less than 7%.
Table 1 Simulated cumulative runoff and sediment yield for thetwo simulation cases
Distribu- Lumped Difference
Variables
Cases
ted Mod.
Mod.
(%)
Runoff
Case 12724.202776.63
1.56(m3)
Case 22192.482293.90
4.63
Sediment
Case 11517.05
1548.86
2.10
yield (kg)Case 2856.45909.13
6.15

Figure 7 plots computed discharges andsediment concentrations by the distributed andlumped models compared to the observed data.
The simulation results of the lumped model shows
good agreement with the observed one. The
lumpedmodel
successfully
reducedthecomputational burden. Simulation time of the
lumped model was about 1/37 of that of the original
distributed model.

5. CONCLUSIONS
Simulation results computed by the lumped
model agree well with the simulation results
computed by the original distributed model. The
lumped model runs efficiently in terms ofcalibrating and running time. The discrepancy ofthe lumped model and distributed model increases
when the soil thicknesses increase, low
accumulative rainfall amount, and/or spatial

temporal variation of rainfall is large.
The main advantage of lumping distributedmodel lies in the capability of being applied at large
scale catchments without losing model andparameter physical interpretation. By incorporatinghydrological and sediment transport processes
derived from the distributed model, it is possible toreproduce the sediment runoff mechanism atcatchment scale especially for large catchments.
River bed erosion or deposition, river bankerosion and lateral inflow of sediment to river
channel were ignored. Its process will be considered in further developments of the lumped model.
ACKNOWLEDGMENT:This study was supported by the MEXT
Coordination Fund for Promotion of Science andTechnology, Japan Science and Technology Agency(PI: Prof. Kaoru Takara, DPRI, Kyoto Univ.).
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- 48 -(Received September 30, 2007)