Advances in Hydro-Science and Engineering, vol. VIII
Proc. of the 8th International Conference on Hydro-Science and Engineering, Nagoya, Japan, Sept. 9-12, 2008

CATCHMENT SCALE COMPARISON ON LUMPED
REPRESENTATION FOR A DISTRIBUTEDSEDIMENT RUNOFF MODELAPIP 1, Yasuto TACHIKAWA2, Takahiro SAYAMA3, 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
2Dr. Eng., Associate Professor, Dept. of Urban and Environmental Eng., Kyoto University
Kyoto, 615-8540, Japan, e-mail: tachikawa@mbox.kudpc.kyoto-u.ac.jp
3Dr. Eng., Assistant Professor, DPRI, Kyoto University
Uji 611-0011, Japan, e-mail: sayama@flood.dpri.kyoto-u.ac.jp
4Dr. Eng., Professor, DPRI, Kyoto University
Uji 611-0011, Japan, e-mail: takara@mbox.kudpc.kyoto-u.ac.jp

ABSTRACTThis study analyses catchment scale effect to a lumped sediment runoff model
performance derived from lumping a physically-based distributed sediment runoff model.
The proposed lumped model takes into account the different sources of erosion and deposition
processes on hillslopes. The eroded soils provided to river channel flow with sedimenttransport mechanism. To investigate catchment scale dependency of the new lumped sediment
runoff model, sediment runoff simulations are conducted using different sizes of thecatchment area and spatially averaged hourly rainfall. Four different sizes of catchment
area are used, representing low-, medium-, and high- resolution. Then, the lumped model isinstalled for each catchment size scenario. The proposed method is examined by comparing
water and sediment discharges simulated by the lumped and the original distributed sediment
runoff models in the Lesti River basin, East Java, Indonesia.
Keywords: Lumping, catchment scale, sediment runoff model, Lesti River basin.
1.

INTRODUCTION

The Lesti River catchment (381.2 km2), a tributary catchment in the Upper Brantas
River basin is selected as study area. The Brantas River, 320 km length with 39 tributaries
and a catchment area about 11,800 km2, is the second largest river located in East Java, has anaverage annual rainfall 2000 mm (80% fall in rainy season). At the confluence point of the
Lesti River and the Brantas main reach, the Sengguruh dam was constructed in 1988 for water
resources and power generation, which has the effect of trapping sediment into the SutamiDam (formerly Karangkates) from the Brantas River and the Lesti River. The Sengguruh
Dam has a catchment area of 1,659 km2. Its original gross storage in 1989 was 22.4 million
m3 and reduced to 5.5 million m3 in 1993. Unexpectedly, most of the gross storage has been
already filled with the large amount of sedimentation from the Lesti River (Takara et al.,1996). The Lesti River transports sediment derived from the lahar (volcanic ashes andsands) of Mountain Semeru.
In order to effectively protect and manage water resources there is need to develop thescience and to assemble the necessary information on which to base decision making.
Herein, estimations of the changes in total runoff and sediment yield as well as anunderstanding of those processes mechanism with time and space in the catchment scale arequite important for solution of a number of problems. Design and operational of dams andreservoirs, design of soil conservation, land-use planning, water quality and aquatic habitat
management are some of the examples.
Mathematical models, including physically-based, can help to make quantified
evaluation, prediction, or to understand the important processes and interactions in sediment
runoff phenomenon. Physically-based distributed sediment runoff model and it lumping for
Lesti River catchment were newly developed by authors (Apip et al., 2008 and Takara et al.,2001). The main propose to lump a distributed model is to produce a new lumped sediment
runoff model version as interest in sediment runoff modelling extends to large catchments
scale and to reduce computational time. Governing lumped model parameters derived
physically by keeping the physical meanings of an original distributed model, which are
obtained from integration of distributed equation according to lumping distributed approachand catchment characteristic from grid-cell based scale to the catchment scale, then new typeof lumped model version is run without any additional calibration.
Earlier research results have shown that the rainfall runoff simulations from a wellvalidated hydrological model scale dependent to catchment area and the complex patterns.
The lumped scale for rainfall runoff model is about 200 km2, however the lumped scale ofsediment runoff model is not clear. This paper illustrates the relation between catchment
area, water discharge (total runoff), and sediment concentration by a combination ofsimulated of those values by an original distributed sediment runoff model and it lumped at
different spatial scales inside study area.
2. DISTRIBUTED SEDIMENT RUNOFF MODEL
The physically-based distributed sediment runoff model has been developed todetermine the runoff hydrograph, sedimentgraph, and total sediment runoff generated fromany temporally-spatially varied rainfall event and continuous rainfall data input.
The
modeling approach is deterministic, physically-based, empirical, spatially distributed anddynamical in time. Dynamic spatial of water movements, erosion patterns and sediment
rates can be predicted at any location inside the catchment as well. The concept ofphysically-based distributed sediment runoff modeling is shown in Figure 1. A sedimenttransport algorithm is newly added to the rainfall runoff model. Sediment runoff simulation
can be divided in two parallel phases: runoff generation and soil detachment.

xyFigure 1 Schematic diagram of the physically-based distributed sediment runoff model within
grid-cell scale.

For a given rainfall event, once the rainfall is directly added to subsurface or surface
flow according to the water depth on the rainfall dropping grid-cell. The model does notconsider the initial rainfall losses due to vertical water flow directly, such as infiltration
effects. Rainfall runoff model effectively simulates lagged subsurface flow with calibrated
hydraulic conductivities and soil layers depth.
The hydrological model considers three principal water flux pathways within acatchment: subsurface flow through unsaturated flow (capillary pore), subsurface flow
through saturated flow (non-capillary pore), and surface overland flow. Using a stagedischarge relationship, after the water depth is greater than the surface soil layer, the net
rainfall will accumulate as surface water and begin to flow as overland flow. Subsurface andsurface flows in both land surface and river channel networks are computed as kinematicwave. The eroded sediment is transported by overland flow to river channels.
The sediment transport algorithm includes multiple sources of sediment transport,
which are soil detachment by raindrop (DR) and hydraulic detachment or deposition driven byoverland flow (DF). Soil detachment for interrill and rill implicitly are simulated
respectively, rain splash and flow detachment. The erosion or deposition rates are calculated
as a function of the hydraulic properties of the flow, the physical properties of the soil and thesurface characteristic. The detachment of soil particles by raindrop impact is function of theenergy imparted to the soil surface by the individual drops. The basic assumption of thismodel is that the sediment is transported and yielded when overland flow occurs. The
transport capacity of the overland flow also needs to be specified, in which suspended
sediment flow is calculated using the transport capacity approach, as it acts as an upper limit
to the potential contribution of each grid-cell to sediment concentrations in saturated areas.
The simulation area is divided into an orthogonal matrix of square cells (250mx250m),
assumed to represent homogenous conditions according to the digital elevation model (DEM).
This allows the use of DEM to derive flow direction map to define the interaction between theobjects which simulate sediment runoff at each grid-cell. Runoff generation, soil erosion or
deposition are computed for each grid-cell and are routed between grid-cells using thekinematic wave model following water flow direction, which defines the routine order for thewater flow and sediment transport propagation. The model uses the one-dimensional
kinematic wave equation for both subsurface and surface flow.
3. LUMPING OF DISTRIBUTED SEDIMENT RUNOFF MODEL
Lumped Sediment Runoff Model Based on Traditional Method
A simple model of catchment response by separating hillslope process and river
channel process (Sivapalan et al., 2002) is adopted and extended to incorporate sedimenttransport processes. This study uses the same principle to explore how sediment yield isrelated to hydrological response, erosion source, transport mechanism and depositional
processes.
On a rainfall event basis, the sediment runoff processes are assumed only affected bysurface runoff without consider the effect of sediment load from subsurface layer.
According with storage-type concept, the model consists of three water stores, it called
rainfall runoff model, and two sediment stores, it called sediment runoff model (Figure 2).
When the water depth larger than the maximum subsurface flow depth of Tank 1,
surface runoff occurs and is added to Tank 2, outflow discharge (Qw) from Tank 3 as afunction of water storage amount (Sw) from each Tanks. After overland flow occurs,
sediment transport mechanism on hillslope is computed (Tank 2). The sediment storage (Ss)in (Tank 2) is supplied by the balance between hillslope soil erosion rate, redeposition rate,and sediment discharge released to river channel. Herein, soil erosion by effective rainfall(DR), soil erosion or redeposition by overland flow (DF) are calculated.
Similarly, at a given time t, the river channel store in Tank 3 is supplied with sedimentmaterial from the hillslope plus the river bed erosion, and only suspended bed material load isconsidered and soil detachment due to rain drop energy is negleted. Some of total sedimentload, as amount of wash load plus suspended material load, in river channel store is
redeposited back into river bed, while another fraction is transported to catchment outlet.
Similarly with the hillslope process, the mass of sediment stored in the river channel isdetermined by the balance between hydraulic erosion rate, redeposition rate, and the releaseof sediment discharge to the catchment outlet. The rate of erosion or redeposition isdepending on the transport capacity of flow and current sediment concentration carried byflow.
rSurface water
storage (SwH)

Tank 1Sub-surface water
storage (SwbH)

Wash Load(DR and DF)
Sediment
storage (SsH)

Hillslope Storage(Wash Load)

QwH
QwHC

Tank 3

Surface water
storage (SwN)River Channel Storage(Bed Material Storage)

Fig. 2

Tank 2

Suspended
Material Load
Sediment
storage (SsN)

QwN
QwNC

Schematic diagram of the lumped sediment runoff model at catchment scale

The continuity equation of runoff and sediment models is represented as follows:
Hillslope process:dS wH= rH AH − QH
dt

(1)

dS sH
= DR + DF H −QH C H
dt

(2)

( (

)

)

max
= k 56.48 rH e −b*hs −avr + α S sH
− S sH hs−avr − QH C H 3600 / AH

River channel process:dS wN= rN AN + QH − QN
dt
dS sN
= YH + DF N −Q N C N
dtmax= QH C H + (α (S sN− S sN )hs −avr − QN C N 3600 )/ AN

(3)
(4)

where k is the soil detachability (kg/J); KE is the total kinetic energy of the net rainfall (J/m2);
and b is an exponent to be tuned; hs-avr is the average of overland flow water depth at thehillslope/river channel, YH is the hillslope sediment yield; α is the erosion/ deposition
efficiency factor; r is the effective rainfall intensity; Ssmax is the maximum storage amount of
sediment concentration; A is the total area; and the subscript of H and N show the hillslope
and river channel section, respectively.
If the effective rainfall intensity is known, Eqs. (1,2,3,4) cannot be solved directly toobtain the outflow hydrograph and sedimentgraph from hillslopes or river channels, because
other variables are unknown. A relationship is needed to relate Q, Sw, Sws, hs-avr, and Ssmax
which are estimated from lumping distributed approach.
Lumping Distributed Rainfall Runoff ModelAs first stage of lumped model development, a method to lump a distributed sediment
runoff model for one layer, in case the sub-surface layer was assumed reached to saturated
condition, was derived (Apip et al., 2007) as an extension from the lumping method proposedby Ichikawa et al., (2000). Herein, lumped rainfall runoff model derived from lumping
distributed rainfall runoff model under steady-state condition is expressed by a non-linear
reservoir, the storage is non-linearly related to outlet water discharge by storage constants K
and p as follows:

Sw = K Q p(5)

by substituting Eq. 5 into Eq. 1 or Eq. 3 becomes:dS wN
S= rN AN + QH − ( w )1 / p
dt
K

(6)

K is the model parameter having a physical meaning, can be interpreted as the time ofconcentration for a kinematic wave to travel across the system.
The value of K is derived from the lumping distributed rainfall runoff approach (Apipet al., 2007). K is influenced by spatially distributed of slope length (L), slope gradient (i),roughness coefficient (n), upper contributing area (U), and total area (A). It proves that K
can be derived from the integration of distributed equation.
In new lumped model, K is
dimensional parameter (m6/5s3/5) is defined as:
N

K =∑
j =1

p +1
p +1
k j ⎛⎜ ⎛Uj ⎞⎛ U j ⎞ ⎞⎟⎜L +⎟ − ⎜⎜⎟⎟w( A) p p + 1 ⎜⎝ ⎜⎝ j w ⎟⎠
⎝
⎠ ⎟⎠

w

1

(7)

sin θ
i
1
⎛ 1 ⎞m=in which p = , k = ⎜ ⎟ , and α =
n
n
m
⎝α ⎠
where i is the slope gradient (m/m), n is the roughness coefficient, m is the exponent constant,which can be shown to be 5/3 from manning’s equation, and j is the number of grid-cell.
Discharge per unit width (q), flow velocity (v), and water storage (sw) for each grid-cell defined as:1⎛ ⎛ Ui⎞⎞ m
1− p⎜ r ⎜ + xi ⎟ ⎟
⎞
⎛
U
Q
⎛
⎞
w
−1
⎠⎟ v = ⎜U⎜ ⎝⎜ Li + i ⎟ ⎟⎟ ki
qi = r ( i + xi ) , hi = ⎜
, i
⎜
⎟
αi
w ⎠⎠
w
⎝A⎝
⎜
⎟
⎝
⎠
1
1
1
+1
+1 ⎞
m
⎛ r ⎞ m 1 ⎛⎜ ⎛Ui ⎞ mU⎟⎛i ⎞swi = w ⎜⎜ ⎟⎟⎜ Li + ⎟ − ⎜ ⎟ ⎟
⎜
w⎠
⎝w⎠ ⎟
⎝ α i ⎠ 1 + 1 ⎜⎝ ⎝⎠m

Total water storage at the hillslope area or river channel area are expressed as:N

S w = ∑ s wi
i =1

Lumping Distributed Sediment Runoff Model
The sediment runoff processes in this study are affected by dynamic spatial
distribution of overland flow. The relationship between detachment and redeposition
represented by Eq. 2 and Eq. 4 depends on the balance between Ss and Ssmax, the depth ofoverland flow or total runoff as well as. Those variables are produced from lumping
distributed approach as:
The Maximum Sediment Storage (Ssmax)
Many, mostly empirical, equations have been developed to predict sediment transportcapacity of flow (TC) as function flow characteristics, slope, and material characteristics.
These equations often use a threshold value of stream power, shear stress, or discharge. In
this study, the transportation capacity is calculated based on the Unit Stream Power (USP)
theory that can be applied for sediment transport in open channels and surface land erosion(Yang, 1973). The USP theory stems from a general concept in physics that the rate ofenergy dissipation used in transporting sediment materials should be related to the rate ofmaterial being transported. Sediment concentration in the water flow must be directly
related to USP. 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. Small particles such as clay and silt move
mostly in suspension and easily carried by the flow while the sand fraction moves as bedmaterial and more difficult to move by flow. This is accomplished that TC depends on theparticle settling velocity, shear velocity, grain size, kinematic viscosity of the water, andwater density. A relationship between USP and the upper limit to the sedimentconcentration in the overland flow, Ct (ppm), can be derived (Yang, 1973 and Yang, 1979).
Hence TC is the product of Ct as:a. Surface Land and Erosion:

TC = log C t = I + J log((vi − v critical i ) / ω )(8)in which:

I = 5.435 − 0.386 log(ω D50 / NU ) − 0.457 log(U ∗ / ω )J = 1.799 − 0.409 log(ω D50 / NU ) − 0.314 log(U ∗ / ω )

ω=

2
36
−
+
2
NU
3 ⎛ρ
⎞
D
⎛
⎞
⎜⎜ s − 1⎟⎟ g ⎜ 50 ⎟⎝ ρ w ⎠ ⎝ 1000 ⎠362⎞ ⎛ D ⎞ NU
⎛ ρs⎜⎜
− 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 gradientm/m); vcriticali is the critical unit stream power (vcritical is the critical flow velocity); ω is thesediment fall velocity (m/s) calculated by Rubey’s equation; ρs is the sediment particle
density (kg/m3); ρw is the water density (kg/m3); g is the specific gravity (m/s2); D50 is themedian 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).

b.

River Channel Erosion
The sediment transport function within river channel has been intended for theestimation of sediment transport rate or concentration at a nonequilibrium condition withdeposition process. When the wash load and concentration of fine material is high,nonequilibrium bed-material sediment transport may occur, and its amount is a function ofwash load. Wash load which depends on the supply from hillslopes has been assumed is highenough to significantly affect the fall velocity of sediment particles, flow viscosity, relative
density of sediment and water.
For flow in river channel and at a nonequilibrium, transport capacity concentration offlow is modeled as a function of modified Yang’s unit stream power, which is an expression
for the total load with high concentration of fine sediment particle. Regarding Eq. 8 when
sediment concentration is not too low, the incipient motion criteria, called critical stream
power, can be neglected. To apply Eq. 8 to a river with a high concentration of finematerials and wash load, the values of viscosity, fall velocity, and relative density have tomodified to consider the influence of high concentration of fine material on those values.
Herein, the modified unit stream power formula proposed by Yang et al. (1979) is expressed:

TC = LogC t = 5.165 − 0.153+ (1.780 − 0.360 log

ωm D50NU m

ω m D50
NU m− 0.297 log− 0.480 log

the coefficients in Eq. 9 are identical to those in Eq. 8.
are modified for sediment transport in sediment-ladenU*ωm) log(U*ωm

γ m vi)γ s − γ m ωm(9)However, the values of ω, γs, and NU

The maximum sediment storage on hillslopes or river channels scale is defined as thetotal sediment transport capacity of overland flow in a whole of the hillslopes 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 sediment storage at hillslope area is calculated by adding up TCi multiplied to swsi
for all grid-cells as:

S smax = ∑ TCi swi /( S w1000)

(10)

th

TC (ppm) for the i grid-cell has been estimated by:
Sediment Concentration (C)Based on the relationship between the current sediment storage (Ss) (kg/m3/hr) and Sw
for each time step calculation, the value of C from hillslope/river channel area can be solved:

C=

Ss
S ws

(11)

for each time-step calculation C is assumed to be uniform over the hillslope/river channel area
and this is the variable of sediment continuity (see Eqs. 2 & 4).
4. COMPARISON OF LUMPED AND DISTRIBUTEDDISCRETIZATION OF CATCHMENT AREA

MODEL

UNDER

The numerical experiments were run in two cases of rainfall event mode with hourly
rainfall data input (Case 1: short duration, and Case 2: long duration) at four cases catchment

size. A digital topographic model for the Lesti River catchment was first developed and thenfour scenarios of the catchment size (Case A, Case B, Case C, and Case D), the high-,
medium-, and low resolution of catchment size in term of lumped mechanism were delineated(see Figure 3) and it characteristics are given in Table 1. For each catchment size, average
rainfall over whole area was used.
The original distributed sediment runoff model parameters were calibrated andvalidated using historical data for the 351.3 km2 of the Lesti River catchment. These
calibration and validation runs suggest that observed data of hydrological condition andsediment transport mechanism is generally amenable to the hydrological and sedimenttransport mechanism based on the original distributed sediment runoff model. The lumped
model algorithm was applied to each synthetic of catchment area and compared withsimulation results using distributed model.
Table 1 The characteristic of four cases
catchment size in the Lesti River catchment
Catchment
Case A
Case B
Case C
Case D

Total grid-cell
404
1743
2114
6101

2

Area (km )
25.3
108.9
132.1
351.3

20

20

30

30

40

40

50

50

60

60

70

70

80

80

90

90

100

100

110

110

A

120
130

130
30

Figure 3
Lesti River catchment waspartioned into four synthetic catchment area,
in which red lines are the catchment areaand blue lines are river channels.

B

120

40

50

60

70

80

90

100

110

120

20

20

20

40

40

60

60

80

80

100

100

120

C

40

60

80

100

140

D

120

140

120

140

30

40

50

60

70

80

90

100

110

120

20

40

60

80

100

120

Figure 4 is plots of the water discharge and sediment concentration by the physicallybased distributed sediment runoff model and the lumped model for the entire period of
rainfall scenario (Case 1 and Case 2). Figure 4 shows the discrepancy of simulation results
between the original distributed model and it lumped model version for both water discharge
and sediment concentration. The lumped model is in an acceptable way to reproduce thedischarge and sediment concentration simulated by distributed model. All the values ofcorrelation coefficient more than 0.90, its mean that the lumped model is in an acceptable way
to reproducing the water discharge and sediment concentration calculated by distributedmodel. Just as with the case of sediment concentrations the correlation coefficients lower
than outflow discharges.
However, in the case water discharge when the catchment size is increase, discrepancy
between simulated values by the lumped and distributed models is generally higher. In thecase of simulated sediment concentrations, the differences are higher than simulated water
discharges, variation depending on the catchment size and characteristics as well as.
The discrepancy between lumped and distributed models for water discharges
simulated in Case D under the rainfall condition Case 1 is higher than Case 2, in which the
lumped model tends to underestimate, is due to the assumed steady state condition in deriving

140

lumped model, total rainfall in Case 1 is less than total rainfall of Case 2.
Conventionally sediment yield is held to decrease as basin area increases. For
drainage areas in the range between small catchments and large basin (>10km2), the effect of
sediment sinks often becomes dominant over sediment sources, resulting in a gradual decline
in sediment yield.
The explanations for this is that large catchments often have more
extensive floodplain development as well as footslopes where sediment are stored, the travel
distance for sediment through catchment is longer and small catchments are more likely torespond directly to event driven flood than large catchments. The characteristics of sedimentyield simulated by lumped model are consistent with these studies (see Case A).

9000
Case A
Case B
Case C
Case D
1 t o 1 line
plus/minus 20% of t he 1 t o 1 line

7500

3

7500

/sec)

Case A
Case B
Case C
Case D
1 t o 1 line
plus/minus 20% of t he 1 t o 1 line

Discharges by distributed model (mDischarges by distributed model (m 3 /sec)

9000

6000

(a) Case 1
4500

3000

1500

Correlation coeficient=0.994
06000(b) Case 2
4500

3000

1500

Correlation coefficient=0.994
0

0

1500

3000

4500

6000

7500

9000

0

1500

Discharges by lumped model (m3 /sec)
140000

4500

6000

7500

9000

140000Case A
Case B

Case A
Case B

Case C
Case D

1200001 t o 1 line
plus/minus 20% of t he 1 t o 1 line
100000(a) Case 1
80000

60000

40000

20000

Correlation coefficient=0.952
0
0

20000

Case C

120000

Sediment concentrations by distributed model (mg/l)

Sediment concentrations by distributed model (mg/l)

3000

Discharges by lumped model (m3 /sec)

40000

60000

80000

100000120000

Sediment concentrations by lumped model (mg/l)

140000Case D
1 t o 1 line
plus/minus of t he 1 t o 1 line

100000(b) Case 2
80000

60000

40000

20000

Correlation coefficient=0.909
0
0

20000

40000

60000

80000

100000120000

Sediment concentrations by lumped model (mg/l)

Figure 4 Comparison of simulated outflow discharges and sediment discharges by lumped
and distributed models for two scenario rainfall data input: (a) Case 1; and (b) Case 2 andcombined with four scenarios of synthetic catchment area.
CONCLUSIONS
Within the range of possible catchment area and soil depth, simulation results
computed by the lumped model for hillslope area and river channel agree well with thesimulation results computed by the original distributed model. For lumped rainfall runoff
model, the discrepancy of the lumped model and distributed model increases when thecatchment area and soil thicknesses increase, low accumulative rainfall amount, and/or spatial
temporal variation of rainfall are large. In the case of simulated sediment concentrations, the140000differences are higher than simulated water discharges, variation depending on the catchment
size and characteristics as well as.
The analyses spatial scale dependency of a lumped sediment runoff model derived
from a physically-based distributed sediment runoff model under land use and rainfall
scenario and it application for large catchments are important areas of further research.
ACKNOWLEDGMENTS
This study was supported by the MEXT Coordination Fund for Promotion of Scienceand Technology, Japan Science and Technology Agency (PI: Prof. Kaoru Takara, DPRI,
Kyoto University).
REFERENCES
Apip., Sayama, T., Tachikawa., Y. and Takara, K. (2008), Lumping of a physically-based
distributed model for sediment runoff prediction in a catchment scale, Annual Journal of
Hydraulic Engineering, JSCE, Vol. 52, pp 43-48.
Apip., Sayama, T., Tachikawa., Y. and Takara, K. (2007), The spatio-temporal predictions ofrainfall-sediment-runoff based on lumping of a physically-based distributed model,
Annuals of Disas. Prev. Res. Inst., Kyoto Univ., No. 50B, pp. 79-94.
Ichikawa, Y., Oguro, T., Tachikawa, Y., Shiiba, M., and Takara, K. (2000), Lumping
general kinematic wave equation of slope runoff system (in Japanese), Annual Journal of
Hydraulic Engineering, JSCE, 44, pp. 145-150, 2000.
Sivapalan, M., C. (2002), Linearity and non-linearity of basin response as a function of scale:Discussion of alternative definitions, Water Reour. Res., Vol. 38(2), 1012,
doi:10.1029/2001WR000482.
Takara, K., Nakayama, D., Tachikawa, Y., Sayama, T., Nakagawa, H., Satofuka, Y., Egashira,S., and Fujita, M. (2007), A Rainfall-Sediment-Runoff model in the Upper Brantas
River, East Java, Indonesia. Annuals of Disas. Prev. Res. Inst., Kyoto Univ., No. 44 B-2,
2007.
Takara, K., Yamashita, T., Egashira, S., Dyah, P.R., Irwan, S., Syamsudin, A.R., and Anton.
(1996), Application of Remote Sensing and GIS to Research on Disasters Caused by
Floods and Sedimentation. In Proceedings of Workshop on Disasters Caused by Floods
and Geomorphological Changes and Their Mitigations (WDFGM-1996); Dyah, P.R.,
Raharjanto, Sudarminto, Agus, S., Isnugroho, and Djamal, H. (Ed.), pp. 62-77.
Yang, C. T. (1973), Incipent motion and sediment transport. J. Hydraul. Div. Am. Soc. Civ.
Eng., Vol. 99, No. HY10, pp. 1679-1704.
Yang, C. T. (1979), Theory the minimum rate of energy dissipation. J. Hydraul. Div. Am. Soc.
Civ. Eng., Vol. 105, No. HY7, pp. 1679-1704.