京 都 大 学 防 災 研 究 所 年 報 第 50 号 B 平 成 19 年 4 月
Annuals of Disas. Prev. Res. Inst., Kyoto Univ., No. 50 B, 2007

The Spatio-Temporal Predictions of Rainfall-Sediment-Runoff
Based on Lumping of a Physically-based Distributed ModelA P I P*, Takahiro SAYAMA, Yasuto TACHIKAWA* and Kaoru TAKARA*Graduate School of Urban and Environment Engineering, Kyoto University
Synopsis
This paper proposes a lumping method of a distributed rainfall-sediment-runoff
model.

Based on the assumption of steady state conditions, the relationship between

outflow discharges and water storage in a catchment can be derived.
sediment model is developed.

Then a lumped

The maximum sediment storage in a catchment wasmathematically derived as functions of sediment transport capacity and total storage ofwater at each grid-cell. Soil detachment and redeposition represented by the balance
between the actual sediment storage and the maximum sediment storage in a catchment
scale. The performance of lumped model is examined in the Lesti River, Indonesia.
Keywords: lumping, distributed rainfall-sediment-runoff model, erosion, Lesti River basinand sedimentation rates, responding to changes both

1. Introduction

in rainfall volume and intensity, with consequences
Estimations of the changes in total runoff andfortopsoildegradation,

loss

of

agriculture

sediment yield with time in the catchment scale areproductivity and increased export of sediment andrequired for solution of a number of problems.

contaminants especially from agriculture fields.

Design of dams and reservoirs, design of soil

Problems requiring estimate of total runoff,conservation, land-use planning, and water quality

erosion and sediment yield or statistical features ofmanagement are some of the examples.

those variables, can be addressed by model forTheannual sediment loads was correlated highly with

examplessediment transported during the highest floods.

Recent physically event-based rainfall-sediment-

Most of the material sediment is transported during arunoff models include KINEROS (Smith et al., 1995),

few floods in a year (Markus and Demissie, 2006).

WEPP (Nearing, 1999), LISEM (De Roo et. al.,complexphysically-basedmodels.

Global climate change issue is one of the

1996a,b), and EUROSEM (Morgan et al., 1998) has

important topic for recent research, the results suggest

improved steadily of the prediction potential of soilthat climatic instability will increase and resulting inloss rates from the average annual estimations using

greater frequency and intensity of extreme weather

theevents (Senior et al., 2002).

Beside floods, other(Wischmeier and Smith, 1978) to increasingly

potential consequence will be the acceleration erosion

complex rainfall-sediment-runoff models capable of－ 79 －

Universal

Soil

Loss

Equation

approachestimating the consequences of single rainfall events.

can be derivated from the distributed results.

Overall, and despite improved spatial and temporalparameter values of lumped TOPKAPI Model are

discretization

description,

shown to be scale independent and obtainable fromsimulation results are highly variable and very

distributed information of topographic, land-use, andsensitive to input parameters (Jetten et al., 2003).

soil type.

This uncertainty can be related with variability in

be applied at increasing spatial scales without losing

model input data, model structure, and with themodel and parameter physical interpretation.

non-linear nature of the rainfall-sediment- runoff

model is allowed to be suitable for land-use andprocesses.

climate change impact assessment, its extension toandbetterprocessThe

The main advantage of this approach can
The

Although model results can generally be

ungauged catchments, for the extensive simulations,improved by calibration, calibrating for future

and coupling with General Circulation Models

condition is not possible.

(GCMs) (Liu and Todini, 2002).

Therefore, rainfall-

sediment-runoff models for climate change analysis

Another concept proposed lumping a distributed

should be a robust model capable to reasonably

rainfall-runoff model using digital topographic

perform with similar parameters values, including

information to reduce computational burden required

highly dynamic for the widest possible range ofin a long-term runoff simulation. The lumped model

space-time conditions.

Physically based distribu-

version was derived from relationship between

ted model-MEFIDIS was developed to assess the riskstorage volume and discharge over the entire

posed by unusually intense storm events for flooding

catchment by considering spatial distribution ofand land degradation in medium-sized watersheds, in

topographic variables and water content of watershedparticular due to climate and land-use changes.

by assuming that a rainfall-runoff process of the

The robustness of MEFIDIS for climate and land-use

system reaches a steady state by spatially uniformchanges assessment was evaluated using a single

rainfall input.

parameter set, although this still increases thegood simulation results and required less computation

computational power needed to run the model (Nunos

time than the distributed model (Ichikawa and Shiiba,

et al., 2005). It related to the problems that the fully2002).

The lumped model basically gave

distributed models still have limitation in application

This research article presents a method to lump abecause these models require much computation timephysically based distributed rainfall-sediment-runoff

to conduct a simulation especially for large catchment

model as the tool for estimates spatio-temporal

size and long-term simulation.

erosion, runoff, and sediment transport processes in

One of alternative way to propose a robust

order to reduce computation time and computermodel is by reducing the number of non-dimensional

resources to conduct a long-term rainfall-sediment-

parameters to the dimensional parameters which

runoffhavingsub-catchment size.

physicalmeaning.

For

example,simulation

in

a

large

catchmentorThe equations and physicalphysically-based rainfall-runoff-TOPKAPI approach

meaning of the lumped model parameters derived

is a comprehensive distributed-lumped approach.

from distributed approach.

Theoretically it proves that the lumped version of thedivided in two parts. First, lumping of physically

TOPKAPI model can be derived directly from thebased distributed rainfall- runoff model and second,

results of the distributed version and does not require

lumping

additional calibration. At each time step in time therainfall-sediment-runoff model.

number of saturated cells is compared with the total

from the lumping method of physically-based

volume of water stored in the catchment can be

distributed

approximated by a beta distribution function curve,

Ichikawa and Shiiba (2002), the authors have been

and finally the lumped TOPKAPI model parameters

adopted and extended that method with adding new－ 80 －

ofThe lumping method

physicallyrainfall-runoff

based

distributed

As the extension

model

proposedbylumping method for sediment transport processes.

effects. The input rainfall data r(t) is directly added

Because the dominant processes of erosion andto subsurface flow or surface flow according to thesediment transport is effected only by surface flow,

water depth on the area where the rainfall dropped.

lumping of physically-based distributed rainfall-runoff model in initial stage of model development
does not consider subsurface flow processes. To
demonstratethequalityofthelumping

physically-based distributed rainfall-sediment-runoff
approach, the authors did numerical experiments forwater and sediment discharges and compared theresults with water and sediment discharges simulatedby distributed model version.

To investigate theperformance of the distributed model and lumped
model, these models applied in the Lesti River,Indonesia.
2.

Fig. 1 (a) Concept of hillslope soil layer structure

Physically-Based Distributed Rainfall-Sedi-and (b) stage-discharge relationship of CDRMV3.

ment-Runoff Model

The model incorporating soil layer model
structure which consists of a capillary pore which

2.1 Cell Distributed Rainfall-Runoff Model
Since soil erosion and sediment transport bywater is closely related to rainfall and runoff
processes, erosion and sediment transport modellingcannot be separated from the procedures to use runoff
generation

modelhydrological model.

by

using

physically-based

Cell Distributed Rainfall-

Runoff Model Version 3 (CDRMV3) developed at
Innovative Disaster Prevention Technology and
Policy Research Laboratory DPRI-Kyoto University(Kojima et al., 2003) including stage-discharge
relationshipwithsaturated-unsaturated

flow(Tachikawa et al., 2004) already applied for our study
site. The model uses the Lax-Wendroff finite
difference scheme to solves the one-dimensional
kinematic wave equation with the stage-discharge
equation to simulate runoff generation and routing.

unsaturated flow occurs inside and a non-capillarypore in which saturated flow occurs.

In the soil

layer, slow flow and quick flow are modeled as
unsaturated Darcy flow with variable hydraulicconductivity and saturated Darcy flow. According
to this mechanism, surface flow will occurs if thewater levels are higher than the total soil depth. In
the CDRMV3, the horizontal sub-surface and surface
flows, q (discharge with unit width), are calculated by
the approximation equation (1) corresponding tostage-discharge relationship (Fig. 1) defined as:⎧v m d m (h / d m ) β ,0 ≤ h ≤ dm⎪
q = ⎨v m d m + v a (h − d m ),dm ≤ h ≤ da⎪m
⎩v m d m + v a (h − d m ) + α (h − d a ) , d a ≤ h…….. (1)The simulation area is divided into an orthogonal∂h ∂q+= r (t )∂t ∂xmatrix of square cells (250 m x 250 m), assumed torepresent homogenous conditions based on DEM…….. (2)The continuity equation (2) takes into account flowdata.
The model assumes that the flow lines areparallel to the slope, the hydraulic gradient is equal torate at each grid-cell or slope segment according
governing equations at the above.

the slope. The kinematic wave of model does notconsider the vertical water flow like infiltration－ 81 －

⎧vm = k m i⎪v = k ia⎪ a⎨k = k / βa⎪ m⎪α = i / n⎩

WithThe basic assumption of this model is that the
sediment is yielded when surface flow occurs whichcalculated based Kinematic Wave Runoff (KWR)
model.

The concept of spatially distributedmodeling is shown in Fig. 2.

The simulation area

Where i is the slope gradient, km is the saturated

is divided into an orthogonal matrix of square cells,

hydraulic conductivity of the capillary soil layer, ka isassumed to represent homogenous conditions which

the hydraulic conductivity of the non-capillary soil

has the same resolution as DEM (250 m). Runoff

layer, dm is the depth of the capillary soil layer, da isgeneration, soil erosion and deposition are computed

the soil depth, and n is the roughness coefficientfor each grid-cell.

based on the land cover classes.

Soil detachment and transport is handled with

The model parameters to be determined are n

the continuity equation representing soil detachments), ka (m/s), da (m), dm (m), and β. For riverby raindrop (DR) and surface flow (DF) (kg/m2/hr) as

-1/3

(m

flow routing, surface flow with rectangular crosssectionisassumedforkinematicfollows :∂ (hs c ) ∂ (qs c )= e (x , t )
+
∂x
∂t

wave

approximation.

…….. (3)With e(x, t ) = DR + DF
2.2 Coupling of Cell Distributed Rainfall-Runoff

Where C is the sediment concentration in the

Model and Sediment Transport Model

flow (kg/m3), hs is the water level of surface flow (m),qs is the discharge of surface flow (m3/s), e is the net
erosion (kg/m2/hr).
The net erosion is calculated by adding DR+DF.
From observation of rainfall characteristic in ourstudy area, the empirical equation for DR for eachgrid-cell expressed as a linear function (Oishi et al.,2005) as follows :DRi = k KE = k 56.48 ri…….. (4)

Schematic diagram of the physicallyWhere k is the soil detachability (kg/J) (= 0.002), KE

based rainfall-sediment-runoff model at a grid-cellis the total kinetic energy of the net rainfall (J/m2),

scale.

and r is the depth of rainfall intensity (mm/h).

Fig. 2Many distributed hydrological models have beendeveloped and utilized, but there has been a few tries
to couple these with sediment transport and sedimentyield models. A physically-based distributed rainfallsediment-runoff

model

deriving

from

the

integration in space of the kinematic wave model wasdeveloped by authors,

sediment transport algorithm

added to the distributed rainfall-runoff model(CDRMV3)

based on physical processes andinvestigated the characteristics of soil erosion basedForgrid-cell,soil

detachment

anddeposition by flow DF is simulated as a result of
surface flow, following the sediment Transport
Capacity of water (TC).

Surface flow has limitation

for transporting sediment at each grid-cell, therefore
this study considers TC of surface flow at eachgrid-cell in each time step.

If actual suspended

sediment from upper grid-cells is lower than thiscapacity detachment occurs, otherwise excess soil
deposition (Govers, 1990). The equation is :DFi = α (TC i − C i ) hs i

on field observation in our study area (Nakagawa etal., 2005).

each…….. (5)

Where α is the detachment/deposition efficiency－ 82 －

factor, TC is the sediment transport capacity ofrunoff model which has been developed by authors

overland flow (kg/m3/h).

consists of one non-linear model for hydrological

DR+DF are as definedfor equation (3).

response and one linear model for sediment transport

The transportation capacity is calculated byprocesses.

We used the same concept withequation (7), which is based on the Unit Stream

distributed model version to explore how sediment

Power (USP) theory.

load is related to catchment hydrological response,

The grid-cell based KWR

model simulates the mean velocity v and the water

detachment

depth h at each grid-cell. Sediment transport capacity

depositional processes.

is the product of the Ct and h.

hydrologically-driven

Yang (1972) derived

sourcetype

andintensity,

and

It is well known that
detachment

model

must

a relationship between unit stream power and total

faithfully simulate surface responses.

sediment concentration as :sediment-runoff processes in this study are affected

log Ct = I + J log((vi − vcritical i) / ω )….. (6)

TC = Ct h.….. (7)In whichω=by surface water, we have been developed astorage-type sediment-runoff.
For more detail, the lumping of distributed
approach can be solved analytically with thefollowing results:

2
36
36
+
−
3 DSTAR
DSTAR

3.1 Lumping of Physically-Based Distributed
rainfall-Runoff Model2⎛ρ⎞ ⎛ D50 ⎞ NU 2DSTAR = ⎜⎜ s − 1⎟⎟ g ⎜⎟⎝ ρw⎠ ⎝ 1000 ⎠For

distributed
The most

importance feature of this method is that the method

is the unit stream power (m/s), i is the slope gradient(m/m), vcriticali is the critical unit stream power, vcriticalis the critical flow velocity, and ω is the sedimentterminal fall velocity in water (m/s), ρs is thesediment particle density (kg/m3), ρw is the water
density (kg/m3), g is the specific gravity (m/s2), andD50 is the median of grain size (mm).
Physically-Based Lumped Rainfall-SedimentRunoff Model

does not presumptively give a bulk relationshipbetween outflow discharges and storage volume at thecatchment scale but it deductively derives a
relationship from a physically-based approach byconsidering

physically-based distributed rainfall-sediment-runoff
model is to derives equations and physical meaningof the physically-based lumped rainfall-sedimentThe lumped modelare derived by analytically the integra-

spatialdistributionof

topographic

variables.
Based on the assumption of steady state
conditions of rainfall-runoff by spatially uniform
rainfall input, the relationship between total storageof water at the ith grid-cell ( s wi ) and the discharge
flow at the ith grid-cell (Qwi) can be theoreticallyderived.

The main objective of the method to lump aparameters

physically-basedby Ichikawa and Shiiba (2002).

Where Ct is the total sediment concentration (ppm), vi

runoff model parameters.

lumping

rainfall-runoff model we used the method proposed

I = 5.0105
J = 1.363

3.

Therefore, theFlux of Qwi is expressed as the product ofrainfall intensity ( r ) and Upslope contributing areas(U).

Upslope contributing area can be calculated

from a DEM from each grid-cell. The storage ofwater at the catchment scale ( S w ) can be calculated
by adding up the s wi from each grid-cell as a functionof the topographic variables.

tion of the distributed model.
The physically-based lumped rainfall-sediment-－ 83 －

follows :q ( x ) = α h( x ) m…….. (10)In which

α=

sin θ
i
=
n
n

Where i is the slope gradient (m/m), n is theroughness coefficient, m is the exponent constant,which can be shown to be 5/3 from manning’s
equation.
Substituting equation (10) into (9), we obtain

Fig. 3 Schematic diagram of the physically-based

expression for surface water depth with the following

rainfall-runoff model at a grid-cell scale.

result :Flux of water discharge at a grid-cell Q(x) (seeα i hi ( x) m = r (Fig. 3) is expressed as the product of the rainfall
intensity(spatially

uniform)

andtheupslope

1

⎛ ⎛Ui
⎞⎞ m
⎜r ⎜
+ xi ⎟ ⎟w⎠⎟
hi ( x) = ⎜⎜ ⎝
⎟
αi
⎜⎜
⎟⎟
⎝
⎠

contributing area as follows :x

Qi ( x) = Qi (0) + r ∫ wi ( x) dx…….. (8)0Where x is the horizontal distance from the upstream

Ui+ xi )w…….. (11)end of a grid-cell, L is total length of the horizontal

The storage volume of surface water for a grid-cell( s wi ) as the differential functions of water depth, thedistance of the grid-cell.

width of grid-cell and horizontal distance fromThe upslope contributing

area can be generated from water flow accumulation

upstream to downstream end is given as :data at each grid-cell based on DEM data. Becausea grid-cell is an orthogonal matrix of square cell, w(x)is constant in the unit. Then the equation (8) can be
rewritten as :xs wi = ∫ hi ( x) wi ( x) dx
0

1

⎛ ⎛Ui
⎞⎞ m
⎜
r
x
+
⎜
L
i⎟⎟
w
⎝
⎠ ⎟ dx
⎜
= w∫
⎟
⎜
αi
0
⎟
⎜
⎠
⎝

Qi ( x) = Qi (0) + r xi w( x)= r U i + r xi w( x)Flux of water discharge can be rewritten with unit
width becomes :⎛ r ⎞ m 1 ⎛U ⎞= w ⎜⎜ ⎟⎟⎜ xi + i ⎟w⎠⎝ αi ⎠ 1 + 1 ⎝m

Q( x) / w = r U / w + r x w( x) / w
q i ( x ) = r U i / w + r xi

=r(
From

L

1

Ui
+ xi )
w

stage-discharge

relationship….….. (9)thegeneralkinematic wave equation (q∼h) for surface runoff as－ 84 －

1
+1
m

⎤
⎥
⎥
⎥
⎦0

1
1
1
+1
+1 ⎞
⎛
⎛ r ⎞ m 1 ⎜⎛Ui ⎞m
Ui ⎞m ⎟⎛
= w ⎜⎜ ⎟⎟⎜ ⎜ Li + w ⎟ − ⎜ w ⎟ ⎟⎠⎝ ⎠ ⎟⎝ αi ⎠ 1 +1⎜⎝⎠m ⎝

s wi = w r p ki

p +1
p +1
Ui ⎞
1 ⎛⎜ ⎛⎛ U i ⎞ ⎞⎟
+
−
L
⎜ i
⎟
⎜ ⎟ ⎟
p + 1 ⎜⎝ ⎝
w⎠
⎝w⎠ ⎠

……..

N

K =∑
i =1

p +1
p +1
Ui ⎞
w k i ⎛⎜ ⎛⎛ U i ⎞ ⎞⎟
+
−
L
⎜
⎟
⎜
⎟
( A) p p + 1 ⎜⎝ ⎝ i w ⎠⎝ w ⎠ ⎟⎠…….. (15)(12)

3.2 Lumping of Physically-Based Distributed

1mWhere p = 1 , and ⎛⎜ 1 ⎞⎟ = km⎝α ⎠rainfall-Sediment-Runoff Model

Sediment transport capacity at each grid-cell( TC i ) is function of topographic variables and

The total storage volume of surface water in thecatchment system ( S w ) is calculated by adding up thehydrological responses. For each time step sediment

total storage volume from each grid-cell insideconcentration (C) is assumed to be uniform over thecatchment as a function of the topographic variables

catchment and this is the variable of sedimentand rainfall intensity :continuity (see Equation 21). The maximum sedimentstorage of the catchment ( S smax ) can be calculatedN

S w = ∑ s wi

from each grid-cell based on s wi and TCi which are

i =1mathematically derived from Sw.

U ⎞1 ⎛⎜ ⎛= ∑ w r ki
⎜ Li + i ⎟⎜p +1⎝⎝
w⎠
i =1
N

p +1

p

⎛U ⎞
−⎜ i ⎟
⎝w⎠

p +1

⎞
⎟
⎟
⎠

The Maximum Sediment
Catchment ( S smax )StorageoftheThe maximum sediment storage is defined as the

Where N is the total of the grid-cells.
Because we assume that the rainfall-runoff
process reaches steady state with spatially uniform
rainfall input, r can be expressed as functions ofoutlet discharge and total area of catchment system,

total sediment transport capacity in a whole thecatchment for each time step.
Therefore, we
expressed S smax as the function of sediment transportcapacity and total storage of water.

then the above equation becomes :
p +1
p +1
p
Ui ⎞
1 ⎛⎜ ⎛⎛ U i ⎞ ⎞⎟⎛Q⎞
S w = ∑ w ⎜ ⎟ ki
⎜ Li +⎟ −⎜ ⎟w⎠
p + 1 ⎜⎝ ⎝⎝ A⎠
⎝ w ⎠ ⎟⎠
i =1
N…….. (13)Finally, equation (13) describes a relationshipbetween storage volume and outlet discharge of thecatchment
procedures.

systemobtained

from

the

above

Equation (13) can be simplified toexpressing that relationship by using a parameter
which is having physical meaning as follows :Sw = K Q p

Fig. 4 Schematic diagram of the physically-based…….. (14)rainfall-sediment-runoff model at a grid-cell scale.

K as the model parameter having a physical meaning,can be interpreted as the time of concentration for akinematic wave to travel across the system.

Fromequation (13), K is influenced by the length of path,
its slope, roughness coefficient, upper contributing
area, and total area. Its prove that K can be derived
from the integration of distributed equation.

Now

K is dimensional parameter (m6/5s3/5) is defined as :Physically, the maximum sediment storage can
not be calculated by adding up the sediment transportcapacity from each grid-cell ( TCi ) directly, but needto multiply it with total storage water at each grid-cell( s wi ). The physically-based distributed rainfallsediment-runoff approach can be simplified to derive
physical meaning of the model parameter S smax .

－ 85 －

Based on the same assumption with the lumping

for all grid-cells as follows :method of physically-based distributed rainfall-runoff
model, several variables of TCi and s wi can be

S smax = ∑ TCi swi

calculated at each grid-cell (see Fig. 4).
N= ∑ w r p ki

From equation (9) and equation (11), vi has been

i =1

p +1
p +1
U ⎞
1 ⎛⎜ ⎛⎛U ⎞ ⎞⎜ Li + i ⎟ − ⎜ i ⎟ ⎟ TC i
p + 1 ⎜⎝ ⎝w⎠
⎝ w ⎠ ⎟⎠

estimated by :NU ⎞1 ⎛⎜ ⎛⎛Q⎞= ∑ w ⎜ ⎟ ki
⎜ Li + i ⎟⎜p + 1⎝⎝
w⎠
⎝ A⎠
i =1
p

q ( x)vi ( x ) = ihi ( x)⎛U ⎞−⎜ i ⎟
⎝ w⎠

p +1

⎞
⎟ TC i⎟⎠…….. (18)⎛U⎞r ⎜ i + xi ⎟
⎝ w
⎠
=
p
⎛ ⎛Ui
⎞⎞
⎜ r⎜
+ xi ⎟ ⎟
⎠⎟
⎜ ⎝ w
⎜
⎟
αi
⎜⎜
⎟⎟
⎝
⎠

Sediment Concentration (C) :The relationship between detachment andredeposition

representedbyequation(5)

depends to the balance between the actual
sediment storage per unit area ( S s ) (kg/m2/h)

U ⎞⎛r ⎜ xi + i ⎟
w⎠
⎝
=
p
⎛ ⎛
Ui ⎞⎞
⎜⎜ r ⎜ xi + ⎟ ⎟⎟ k i
w ⎠⎠
⎝ ⎝

and the maximum sediment storage per unit
area ( S smax ) (kg/m2/h).

⎛ ⎛
U ⎞⎞
= ⎜⎜ r ⎜ xi + i ⎟ ⎟⎟
w ⎠⎠
⎝ ⎝

Based on the relation

between S s and S w , the value of C from watershed

1− p

ki

outlet for each time can be solved as follows :−1C (t ) =For a grid-cell x = L , then the above equation
rewritten becomes :
1− p

⎛Q ⎛
U ⎞⎞
vi = ⎜⎜ ⎜ Li + i ⎟ ⎟⎟
w ⎠⎠
⎝A⎝

p +1

ki−1

…….. (16)S s (t )S w (t )……. (19)

As a first stage of the development, neglectsubsurface contributions, and assume that all thewater flows as surface flow.

Where vi is independent variable of Unit Stream
Power at each grid-cell ( USPi ) as follows :Finally, the continuity equations of Sw and Ss on
a catchment scale are presented as follows :USPi = vi sin θ i = vi ii

⎛Q ⎛
U ⎞⎞
USPi = ⎜⎜ w ⎜ Li + i ⎟ ⎟⎟
w ⎠⎠
⎝ A ⎝

1

dS w
⎛S ⎞ p= r A−⎜ w ⎟
dt
⎝ K ⎠

1− p

k i−1 ii….. (20)

dS s= DR + DF − QwC
dt

( (

)

)

= k 56.48 r + α S smax − S s − QwC / A

TC for a grid-cell (TCi) has been estimated by :TC i = 10 5.0105 +1.363 log(( USPi −USPcritical ) / ω )….. (21)…….. (17)The maximum sediment storage at the catchment
scale is calculated by adding up TCi multiply to swi

Where A is the total area of the catchment (m2), r is
the actual rainfall intensity (m/hr), K is the catchment
response as the model parameter, α is thedetachment/deposition efficiency factor, and k is the－ 86 －

Table 1. Physically-based distributed rainfall-runoff-soil detachability (kg/J).
Runoff and sediment load calculations are
conducted by using those parameters K and S smax

sediment model parameters used in the Lesti River (3to 6 October, 2003)

obtained from the above procedures and continuity
equations.

The continuity equations of water andsediment

weresolvednumerically

SymbolDescriptiondm

The depth of the capillary soil layer (m)dm = 0.8

usingdaRungge-Kutta method order 4th .

non-capillary soil layer (m)The most important feature that the model hasda = 1.2

two kind model parameters which having physicalmeaning, they are K and S smax . The physicalkahydraulicconductivity

of

the

ka = 0.008

βphysically-based distributed rainfall-sediment-runoff

The non-linear exponent constant for thecapillary soil layer

Model parameter K derived from theβ = 30.0

lumping method of physically-based distributednrainfall-runoff model (CDRMV3), and model
parameter S smax can be derived from coupled of

The roughness coefficient based on theland cover classes (m-1/3s)
The range of n values 0.03-3

CDRMV3 and sediment transport algorithm.
D50

4.

The

non-capillary soil layer (m/s)

meaning of the lumped model parameters K andS smax are resulted from the lumping method ofmodel.

The depth of capillary soil layer +The median grain size (mm)D50 = 0.062

Application of the Physically-based Distribuk

ted Rainfall-Sediment-Runoff Model

The soil detachability (kg/J)
K = 0.001

α

The hydrological dataset in the year 2003 is

α = 0.98

selected as an example for the verification of theρscoupled CDRMV3 and sediment transport algorithm
in the Lesti River catchment.

ρw

from

three

g

automatic

The specific gravity (m/s2)
g = 9.8

raingauges and one turbidity-discharge at the
NUwatershed outlet were available. The areal rainfall

The kinetic viscosity of water (m2/s)
NU = 10-6

distribution was estimated by the Thiessen Polygon
method.

The water density (kg/m3)

ρw = 1000

event with a peak discharge of 54 m3/s occurred.
measurements

The sediment particle density (kg/m3)

ρs = 2650

Since in that year

from 3 to 6 October, 2003, a relatively large flood
HourlyThe detachment or deposition efficiency

Sub-surface and surface flow by using
4.1 The Study Area

equation (1) was incorporated.
The model calibration was performed at 1-hourtime step for the hydrological dataset from 3 to 6

The Lesti River catchment is a small catchment(351.3 km2) inside the Brantas River basin (11,800

October, 2003 with the model parameter set is givenkm2), east Java in Indonesia. The Lesti Riverin Table 1.

catchment is located at the upstream of the Brantas
River basin.

The Lesti River catchment is also

instrumented for rainfall, runoff, erosion, andsediment yield measurement. This area represents a
tropic-volcanic area covered with forest, agriculture,urban, and shrub vegetation.

The Lesti River

catchment is mostly covered with volcanic soil

－ 87 －

originated from Mt. Semeru, an active volcano

shows that the model was capable of locating thelocated at the upstream of the Lesti River. At themain sediment sources and sink within the Lesti

merging point of the Lesti River and the Brantas mainRiver during this flood event. The model was alsoreach, Sengguruh dam was constructed in 1998 forable to assign major erosion features.

water

resources

and

power

generation.

Unexpectedly, most of the gross storage (21.5 millionm3) has been already filled with the large amount of
sedimentation from the Lesti River and the main
reach of the Brantas River.

0
5
10
15
20
56

6000

Obs. Discharge

5000

Comp. Discharge

48

Comp. Sediment

40

4000

3

Discharge (m /s)

Obs. Sediment

32

3000

24
2000
16

Sediment concentration (mg/l)

Rainfall (mm)4.2 Simulation Result

Fig. 6 Erosion and deposition sources distribution inwatershed for 3 to 6 October 2003.
5

Evaluation of The Physically-based Lumped
Rainfall-Sediment-Runoff Model

1000

8
0

0
1

8

15

22

29

36

43

50

57

64

71

Time

Fig. 5Comparison of predicted runoff event andsediment discharge with observed data for 3 to 6
October 2003.

20

20

30

30

40

40

50

50

60

60

70

70

80

80

90

90

100

100

110

One of the simulated flood events is shown in
Fig. 5. The predicted runoff and sediment concen-

110

A

120

130
30

40

50

60

70

80

90

100

110

120

20

20

20

40

40

model efficiencies are : 0.854 (R2 : the coefficient of60

60

determination), 0.251 (RRMSE : Relative Root Mean

80

80

100

100

tration were compared with the observed one.

B

120

130

40

60

80

100

120

140

A

summary of this flood runoff event is given by the

Square Error), and 5.831 (absolute error) (seeAppendix-1). We may be able to recognize that themodel prediction from an example is closer to theobserved value.

120

C

D

120

140

140

30

40

50

60

70

80

90

100

110

120

20

40

60

80

100

120

140

Fig. 7 Lesti River catchment as partioned into four
catchment levels based on the upscalling area where

4.3 Spatial Patterns of Erosion and Deposition

Fig. 6 shows simulated erosion and depositionareas inside watershed for flood runoff event 3 to 6

red lines are the catchment area and blue lines arechannels river.

October 2003. A direct comparison with observed
data is difficult since the threshold erosion rate above
which these areas are mapped is not known. Fig. 6

As

explained

previously,

thelumpedrainfall-sediment-runoff model parameters can bederived directly from the distributed approach of－ 88 －

Fig. 8 Comparison of calculated water discharges using the distributed and lumped models fortwo periods at different catchment sizes (A,B,C and D) inside the Lesti River catchment.

－ 89 －

Fig. 9 Comparisons of calculated sediment concentration using the distributed and lumped
models for two periods at different catchment sizes (A,B,C and D) inside the Lesti River
catchment.

－ 90 －

physically-based distributed rainfall-sediment-runoff

the simulated discharges for the whole catchment sizemodel version.

For simplification, the distributed

criteria and two kinds rainfall data input, generated

neglects the sub-surface flow processes forby the physically-based distributed rainfall-runoff

model

this lumping experiment.

The model which wasmodel and the lumped model.
that

the

These cases

lumped was applied to Lesti River catchment and we

demonstrateevaluate the performance of the lumped model. The

rainfall-runoff

numerical experiments was run for two cases ofreproducing the overall water discharge that reaches

hourly rainfall data input (case 1 : short duration, andthe river network generated by the distributed model.

case 2 : long duration) and four cases catchment size.

These results enhance the lumping method proposed

Four scenarios of the catchment size were delineatedby Ichikawa and Shiiba (2002), the lumped model(see Fig. 7) and its characteristics are given inwhich derived from distributed model basically gave

Table 2.

good simulation results and required less computation

model

lumpedisversionalreadyofthesuitablefortime than the distributed model.
Fig. 9 plots sediment concentration by the

Table 2. The characteristics of four cases catchment

physically-based distributed rainfall-sediment-runoff

size in the Lesti River catchment (Fig. 7)
Catchment

Total grid-cell

Area (km2)

Case A

404

25.3

Case B

1743

108.9

Case C

2114

132.1

Case D

6101

351.3

model and the lumped model.

model show overestimated sediment concentration
for all simulation results to the distributed model
results especially for peak times and low flows, any
way the sediment concentration predictions of time
when a sedimentgraph rises or recesses are good.

A digital topographic model for the Lesti River
catchment was first developed and the area divided
into four catchment areas defined based on the four
scenarios of catchment size and topographic model.
Then the lumped model developed in the previous
section was applied to a catchment area of eachscenario.

The parameter values of the lumped

grid-cell model were as follows : n = 0.2 m-1/3s except
for river channel n = 0.001 m-1/3s, m = 1.667, dm = 0
m, da = 0 m, D50 = 0.062 mm, k = 0.004 kg/J, α =0.98, ρs = 2650 kg/m3, ρw = 1000 kg/m3, g = 9.8 m/s2,
and NU = 10-6 m2/s.
Fig. 8 and Fig. 9 show comparisons of simulated
water and sediment discharges (hydrograph andsedimentgraph) for two periods rainfall data input andfour cases catchment size which calculated by the
distributed and the lumped model version.

The

black lines are simulated water and sedimentconcentration by

The outputs lumped

the distributed model and bluelines are simulated water and sediment concentrationby the lumped model.
Fig. 8 shows only marginal differences between

The lumped model performance qualitative andstatisticforoutflowdischarges

andsediment

discharges based on two periods rainfall data input
called case 1 and case 2 combined with all scenariosof catchment size given in Fig. 10.

That figure

shows that for both cases, the simulation results foroutflow discharge and sediment discharge computedby lumped model basically approximated thesimulation results computed by distributed model inthe spatial and temporal changes. All the values ofcorrelation

coefficient

more

than

0.90

(see

Appendix-2), its mean that the lumped model is in an
acceptable way to reproduce the discharge andsediment discharge which calculated by distributedmodel.

Just as with the case of sediment

discharges the correlation coefficients lower thanoutflow discharges, the most of sediment discharges
are overestimated for the smaller and larger simulated
values calculated by distributed model.

On theother hand, in the case of rainfall data input where
total rainfall accumulation in case 2 is higher thancase 1, the error of the case 2 is larger than case 1.

－ 91 －

(a)(b)

Fig. 10 Comparison of simulated outflow discharges and sediment discharges by lumped model and distributedmodel for two periods rainfall data input : case 1 (a) and case 2 (b). The solid line is 1 to 1.
6

distributed model.

Summary

The model runs efficiently interms of calibrating and running time.
The lumped rainfall-sediment-runoff model is

The main advantage of lumping distributed

developed by lumping physically-based distributed

approach lies in it capability of being applied at large

rainfall-sediment-runoff model without introducing

scales without losing model and parameter physicalany tuning parameter. The distributed model is usedinterpretation. By incorporating hydrological andto identify the mechanism governing the dynamics of

sediment transport processes derived from thethe surface runoff as a function of the total water

distributed model it is possible to reproduce thestorage, and sediment concentration as functions ofrainfall-sediment-runoff mechanism at catchment

the sediment transport capacity of surface flow.

scale preserving physical values of the parameters.

distributed

The lumping physically-based distributed model

rainfall-sediment-runoff model which based on steadyis foreseen to be suitable for large catchments,

state condition and spatially uniform rainfall dataland-use and climate change impact assessment, forbasically gave good simulation results.

extreme flood analysis, for use with General

Lumpingofphysically-based

Discharges

simulated by the lumped model agree well with

Circulation Models (GCMs).

The

Sub-surface processes as path of hydrologicalsediment concentration predictions of time when apathway has still to be introduced ; this was ignored

sedimentgraph rises or recesses are as well fitted toin the initial stage since it was not dominant source indischarges simulated by distributed model.

－ 92 －

the sediment transport processes.

354-361

This objective

may be pursued through the introduction the lumping

Morgan, R.P.C., Quinton, J.N., Smith, R.E., Govers,

method of physically-based distributed rainfall-G., Poesen, J.W.A., Chisci, G., and Torri, D. (1998):

sediment-runoff model by incorporating sub-surface

Theflow and river channel erosion and deposition. In

Favis-Mortlock,

addition, an approach for incorporating reservoirs and

Erosionlakes should be included in further study.

Springer, Berlin, pp. 389-398.

EUROSEMbymodel.
D.T.

Water,

In:

Boardman,J.,(Eds.),

Modelling

Soil

NATO-ASI

SeriesI-55.

Nakagawa, H., Satofuku, Y., Muto, Y., Oishi, S.,
Sayama, T., and Takara, K. : On sediment yield

References

and transport in the Lesti River basin, a tributary of
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LISEM: A single event physicallyfield observations and remotely sensed data -, Proc.

hydrological and soil erosion model for drainage

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Disasters, -Coping with Extreme Events and(1996a):Hydrol.

Regional Diversity-, Kyoto, Japan, 1 - 2 December

Process. 10(8), 1107-1117.

2005, CD-ROM, 2005.

De Roo, A.P.J., Offermans, R.J.E., and Cremers,
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physically hydrological and soil erosion model for

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andapplication.

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transporting capacity of overland flow. Int. Assoc.

Senior, C.A., Jones, R.G., Lowe, J.A., Durman, C.F.,

Hydrol. Sci. Publ. 189, 45-63.
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kinematic wave equation considering field capacity.

precipitation and sea-level rise under climate

Third international conference on water resourceschange. Philos. Trans. R. Soc. Lond. Ser. A: Math.

ndthPhys. Sci. 360 (1796), 1301-1311.

and environmental research. 22 -25 of July 2002

Smith, R.E., Goodrich, D.C., Woolhiser, D.A., andat Dresden University of Technology.

Unkrich, C.L. (1995):

Jetten, V., Govers, G., and Hessel, R. (2003):

KINEROS-a kinematic

Erosion models: quality of spatial predictions.

runoff and erosion model. In: Singh, V.J. (Ed.),

Hydrol. Process. 17, 887-900.

Computer Models of Watershed Hydrology, Water

Kojima, T., and Takara, K.

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weather

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(2003): A grid-cellradar

information

Tachikawa, Y., Nagatani, G., and Takara, K. (2004):
Development

and

of

stage-discharge

relationshipdistributed hydrological modelling, IAHS Publ. No.

equation incorporating saturated – unsaturated flow

282, pp 234-240.

mechanism,

Liu, Z., and Todini, E.

Journalof

HydraulicEngineering, JSCE, 48, 7-12.

(2002):

Towards a

physically-based

rainfall-runoff

Oishi, S., Sayama, T., Nakagawa, H. Satofuka, Y.,

Hydrology and Earth System Sciences,Muto, Y., Sisinggih, D. and Sunada, K. (2005):

comprehensivemodel.

AnnualDevelopment of estimation method for impact

6(5), 859-881.
Markus, M., and Demissie, M. (2006): Predictibilityenergy of raindrop considering raindrop sizeof annual sediment loads based on flood events.

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1084-0699,－ 93 －

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Absolute Error (ABSERR) :n

pp. 1087 - 1092.
Wischmeier,

W.H.,

and

Smith,

D.D.

(1978):∑ Oi − P i

ABSERR = i =1

conservation planning.

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Oi : observed discharge

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n : Number of P or O values

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Appendix-2

Appendix-1

Correlation coefficient :Statistical indexes :Model Efficiency (EF) :
n

(

) (
2

∑ Oi − O − Pi − P

EF = i=1n(∑ Oi − O
i =1

)

R=

2

→ −∞ < EF ≤ 1

)

2

n

∑ ( P i − Oi )

i =1

n

∑ (X − X )(Y − Y )∑ (X − X ) ∑ (Y − Y )
i

i

2

2

i

Where,

Xi : predicted value by lumped model
Yi : predicted value by distributed model

Relative Root Mean Square Error (RRMSE) :RRMSE =

→ 0 ≤ ABSERRn

Predicting rainfall erosion losses, a guide to

X and Y : the mean of the predicted values by

2⋅

lumped model and distributed model1→ 0 ≤ RRMSE
O

物理型分布モデルの集中化による降雨土砂流出現象の時空間予測

ＡＰＩＰ*・佐山敬洋・立川康人*・寶馨* 京都大学工学研究科都市環境工学専攻

要旨

分布型降雨土砂流出モデルの集中化手法を提案する。この方法は，キネマティックウェーブ式にもとづく分布型降雨流出モデルの集中化と，土砂生産・堆積過程を再現する分布型土砂流出モデルの集中化の二つのステップからなる。まず，流出現象の定常性を仮定することにより，流域全体の貯留量と流域下端からの流量との関係を導出する。次に，流域全体の土砂輸送可能量と水の貯留量との関係を，分布型土砂流出モデルの式から解析的に導出する。各時間ステップで状態量として計算する土砂濃度・貯留量と，土砂輸送可能量とのバランスにもとづき，土砂生産・堆積量を時々刻々推定する。提案する集中型モデルを，インドネシアのレスティ川流域に適用し，分布型降雨土砂流出モデルと比較することによって，その性能を検証する。

キーワード：集中化，分布型降雨土砂流出モデル，土壌侵食，レスティ川流域－ 94 －