Proc. Fifth International Symposium on Hydrological Applications of Weather Radar -Radar Hydrology-,
pp. 439-444, Nov. 19-22, 2001, Kyoto Japan

Investigation of Rainfall Spatial Variation Scale to be Considered in
Runoﬀ Simulation Using Distributed Model and Radar Data
Yutaka Ichikawa1 , Yasuto Tachikawa2 and Michiharu Shiiba1
12

Dept. of Civil Engineering Systems, Kyoto Univ., Kyoto, Sakyo-ku, Kyoto 606-8501, Japan
Disaster Prevention Research Institute, Kyoto Univ., Uji, Kyoto 611-0011, Japan

ABSTRACT: A rainfall spatial variation scale to be considered in a runoﬀ simulation will be investigatedusing mesh-type rainfall data and a distributed runoﬀ model. At ﬁrst, a runoﬀ simulation is conductedby using a mesh-type rainfall data as it is, and the result is regarded as a standard result. As thenext step, the rainfall data is averaged in some area and a runoﬀ simulation is conducted by using thisaveraged rainfall data. If we have little diﬀerence between the standard result and the result obtained byusing the averaged data, we do not have to explicitly consider the rainfall spatial variability in the area
where the data is averaged, and we can use the average rainfall data as an input of the runoﬀ simulation.
Conversely, if the diﬀerence is large, we have to consider the rainfall variability in the averaging area when
conducting a runoﬀ simulation. These numerical experiments indicate that it is important to ﬁnd out
a scale of rainfall spatial variability which aﬀects the results of runoﬀ simulations and the scale changes
with the size of drainage area.
1

Introduction

Which extent is the scale of rainfall spatial variation to be considered in a runoﬀ simulation? For example,
which extent is the spatial resolution of radar rainfall data which might be necessary to reproduce thedischarge hydrograph for a basin of 1000 km2 ? How many meters should we choose as the interval of
the rain gauge in order to reproduce the discharge hydrograph in a small experimental basin? We do
not have any clear answers to these questions. Also, we have developed a method to derive a lumped
relationship between storage volume and outﬂow discharge of basin slope systems (Ichikawa et al., 1999;
2000; in submitting). This method is based on the assumption that the rainfall intensity is spatiallyconstant. It then means that this method should be applied to an extent in which the rainfall intensity
can be regarded as constant and that we have to know how the extent is.
In this study, the rainfall spatial variation scale to be considered in a runoﬀ simulation will be investigated using a radar rainfall data and a distributed runoﬀ model.
2

Methodology

The methodology of the investigation is as follows. At ﬁrst, a runoﬀ simulation is conducted by usinga mesh-type rainfall data as it is, and the result is assumed to be a standard result. As the next step,
the rainfall data is averaged in some area and a runoﬀ simulation is conducted by using this averaged
rainfall data. If we have little diﬀerence between the standard result and the result obtained by using the
averaged data, we do not have to explicitly consider the rainfall spatial variation in the area where thedata is averaged, and we can use the average rainfall data as an input of the runoﬀ model. Conversely, if
the diﬀerence is large, we have to consider the rainfall variation in the averaging area when conducting
a runoﬀ simulation.
As a preliminary analysis, numerical experiments will be conducted using mesh-type rainfall data
generated by a random ﬁeld generation computer program. Because this program can control statisticalcharacteristics of generated rainfall data, we can investigate a relationship between the characteristics
and results of runoﬀ simulations. After this analysis, a similar analysis will be conducted using observed
radar rainfall data.
The distributed runoﬀ model used in this study represents a hill slope system as a set of small
rectangular planes (Shiiba et al., 1999), which are called slope units, and routes water ﬂow from upstream
slope units to downstream slope units by applying kinematic wave model considering ﬁeld capacity of soil
layer to each slope unit (Ichikawa et al., in printing). The input data of this runoﬀ model is a mesh-type
rainfall data. When the model calculates the water ﬂow of a slope unit, it uses a rainfall intensity of themesh which overlaps with the slope unit.

(m)
12000

rainfall15
105

rainfall intensity(mm/hour)
35
30
25
20
15
10
5
0

6000

lognormal distribution

0.2
density

9000

0.25

0.15
0.1
0.05

12000
9000
0

3000

3000
x (m)

0
0

3000

6000

9000 (m)Figure 1. Study area

correation coefficient

0

4
8
12
16
rainfall intensity (mm/hour)

20

90000

Figure 3. Histogram of rainfall intensity

Table 1. Parameter values for rainfall
ﬁeld generation

0.8
0.6
0.4
0.2
0
-0.2

0

1000

2000
3000
length (m)

4000

5000

Figure 4. Spatial correlationof generated rainfall ﬁeld

3

6000

0

y (m)

Figure 2. Example of generated rainfall ﬁeld (µ = 5.0, δ = 0.5, α = 1500.0)

simulated correlation coefficientcorrelation function

1

6000
3000

case
1
2
3
4
5
6
7
8

µ (mm/hour)
5.0
5.0
5.0
5.0
10.0
10.0
10.0
10.0

δ
1.0
1.0
0.5
0.5
1.0
1.0
0.5
0.5

α(m)
500.0
1500.0500.0
1500.0500.0
1500.0500.0
1500.0Analysis using generated rainfall data

In this section, numerical experiments will be conducted in order to investigate a relationship among
rainfall averaging scale, characteristics of rainfall ﬁeld (average rainfall intensity, coeﬃcient of variation,correlation length) and runoﬀ simulation error.
3.1 Generating mesh-type rainfall data
Figure 1 shows a study area of this analysis (33 km2 ). The mesh-type rainfall data will be generated over
the rectangular area (9000 m × 12000 m) which covers the study area. The rainfall data are generatedusing a random ﬁeld generation program coded by Tachikawa and Shiiba (2000). The size of each mesh is
250 m × 250 m. It means 36 × 48 mesh data will be generated over the rectangular area. Each generated
data has log normal distribution and also has spatial correlation expressed by Gauss function.
There are three parameters to generate random mesh-type rainfall data, which are spatial average
rainfall intensity, µ [mm/hour], coeﬃcient of variation of rainfall intensity, δ, and correlation length, α[m]. α is the length where the spatial correlation coeﬃcient is 1/e and can be regarded as a representative
scale of spatial variation of rainfall ﬁeld. Figure 2 shows the generated rainfall data using µ = 5.0, δ = 0.5,
and α = 1500.0. Figure 3 and Figure 4 show the histogram of rainfall intensity and the spatial correlationcoeﬃcient of the generated data, respectively.
The rainfall data are generated using the parameter values shown in Table 1. For each case, ﬁve data
are generated in order to eliminate a chance factor. The data generated in this process are referred to as
the data of level 1.
3.2 Averaging rainfall ﬁeld
The rainfall data of level 1 are spatially averaged over: 750m × 750m (referred to as level 2), 1500m ×1500m (level 3), 3000m × 3000m (level 4) and 9000m × 12000m (level 5).
3.3 Runoﬀ simulation
Runoﬀ simulations are conducted using the rainfall ﬁeld data generated in 3.1 and 3.2 . First of all, runoﬀ
simulations are conducted using the level 1 rainfall ﬁeld data for each case shown in Table 1. For eachrainfall
35
30
25
20
15
10
5

rainfall intensity(mm/hour)
35
30
25
20
15
10
5
0

rainfall
20
15
10
5

rainfall intensity(mm/hour)
35
30
25
20
15
10
5
0

12000

12000

9000
0
3000
x (m)

6000
6000

3000

3000
x (m)

case 1

6000
3000

6000

80
60
40

3000
x (m)

90000

80
60
40

12000

6000

3000

3000
x (m)

90000

3000
x (m)

90000

case 5

6000
3000

6000

6000

rainfall
30
25
20
15
10
5

80

60
40

60
40
20
0
12000

12000

9000
0

y (m)

y (m)

90000

rainfall intensity(mm/hour)

80

12000
0

6000
3000

case 4
rainfall
40
30
20
10

9000

y (m)

9000
0

y (m)

20
0

9000
6000

6000

rainfall intensity(mm/hour)

20
0

0

6000
3000

case 3
rainfall
80
70
60
50
40
30
20
10

rainfall intensity(mm/hour)

20
0

12000

9000
0

y (m)

case 2
rainfall
80
70
60
50
40
30
20
10

rainfall15
105

rainfall intensity(mm/hour)
35
30
25
20
15
10
5
0
12000

9000
0

y (m)

90000

rainfall intensity(mm/hour)

3000
x (m)

rainfall15
105

rainfall intensity(mm/hour)
35
30
25
20
15
10
5
0

3000
x (m)

90000

case 6

6000
6000

3000

9000
0

y (m)

3000
x (m)

90000

case 7

6000
6000

3000

y (m)

90000

case 8Figure 5. Generated rainfall ﬁelds

1
0

3
2
1
0

6

12

18
24
time (hour)

30

36

0

6

case 1

6

12

18
24
time (hour)

10level 1level 2level 3level 4level 5

8

4
2
0

30

36

6

12

18
24
time (hour)case 5

1

6

30

36

4
2

6

12

18
24
time (hour)

30

36

6

12

18
24
time (hour)case 6

2
1

0

6

6

30

36

18
24
time (hour)

10level 1level 2level 3level 4level 5

8

12

30

36

30

36

case 4

4
2
0

0

3

0
0

10

level 1level 2level 3level 4level 58level 1level 2level 3level 4level 5

4case 3002case 2discharge (mm/hour)

discharge (mm/hour)

10

3

0

discharge (mm/hour)05level 1level 2level 3level 4level 54discharge (mm/hour)2discharge (mm/hour)35level 1level 2level 3level 4level 54discharge (mm/hour)5level 1level 2level 3level 4level 54discharge (mm/hour)

discharge (mm/hour)5level 1level 2level 3level 4level 5

8
6
4
2
0

0

6

12

18
24
time (hour)case 7

30

36

0

6

12

18
24
time (hour)case 8Figure 6. Runoﬀ simulation resultscase, there are ﬁve rainfall ﬁeld data, then it means that the runoﬀ simulations are conducted ﬁve times.
The results of these simulations are considered as the standard results for evaluating the simulation error
due to rainfall averaging. Next, runoﬀ simulations are conducted using the level 2 rainfall ﬁeld data inthe same way as the simulations using the level 1 data. Runoﬀ simulations using the rest of the rainfall
data (level 3, 4 and 5) are also conducted. The rainfall inputs are given for 80000 seconds at constantintensity.
3.4 Evaluating runoﬀ simulation error
Runoﬀ simulation error r is evaluated by comparing the simulation results of level 2 - 5 with the resultsof level 1, which is considered as the standard result. r is written as:


N i


5
i
i
2


1
j=1 (Q1,j − Qk,j )
i
(1)
r =
/R
m


5
Ni


i=1

i
where i shows a rainfall data and a simulation result for each case, j shows calculation time, Rm
is theaverage rainfall intensity of ith rainfall data, N i is the number of calculation steps of the ith simulation
result, Qik,j is runoﬀ height at time j of the ith simulation result obtained by using level k data.

3.5 Simulation results and discussion
Figure 5 shows the rainfall ﬁeld data of level 1 of each case and Figure 6 shows the runoﬀ simulation
results of each case.

0.02

0.04

0.02

0
2

3

4

0.04

3

4

5

µ = 5.0µ = 10.0

0.04

0.02

0
2

level

δ = 0.5, α = 1500

0.06

µ = 5.0µ = 10.0

0.02

0

5

δ = 0.5, α = 500

0.06µ = 5.0µ = 10.0

error

0.04

error

error

δ = 1.0, α = 1500

0.06

µ = 5.0µ = 10.0

error

δ = 1.0, α = 500
0.06

0
2

3

level

4

5

2

3

level

4

5

level

Figure 7. Comparisons of r between µ = 5.0 and 10.0

0.02

0.04

0.02

0
3

4

5

0.04

3

4

5

δ = 1.0
δ = 0.5

0.04

0.02

0
2

level

δ = 10, α = 1500

0.06

δ = 1.0
δ = 0.5

0.02

02

δ = 10, α = 500

0.06

δ = 1.0
δ = 0.5

error

0.04

error

error

µ = 5, α = 1500

0.06

δ = 1.0
δ = 0.5errorµ = 5, α = 500

0.06

0
2

3

level

4

5

2

3

level

4

5

level

Figure 8. Comparisons of r between δ = 1.0 and 0.5

0.02

0.04

0.02

0
3

4
level

5

0.04

3

4
level

5

α = 500
α = 1500

0.04

0.02

0
2

µ = 10, δ = 0.5

0.06

α = 500
α = 1500

0.02

0
2

µ = 10, δ = 1.00.06

α = 500
α = 1500

error

0.04

error

error

µ = 5, δ = 0.5

0.06

α = 500
α = 1500

errorµ = 5, δ = 1.0

0.06

0
2

3

4

5

level

2

3

4

5

level

Figure 9. Comparisons of r between α = 500 and 1500
Figure 7 shows comparisons of r between the case of µ = 5.0 and 10.0. The vertical axis of the graphs
is the calculation error (r ) and the horizontal axis is the level of rainfall averaging. For all of the cases,
the calculation errors become greater as the rainfall averaging area is extended. Also, there are no large
diﬀerences of r between the cases being compared. Because r is an index standardized by dividing by
the rainfall intensity, the diﬀerences between the calculation results (diﬀerences of runoﬀ height) are inproportion to the rainfall intensity.
Figure 8 shows comparisons of r between the case of δ = 0.5 and 1.0. From this ﬁgure, it can be said
that the calculation errors for the case of δ = 1.0 are greater than the case of δ = 0.5.
Figure 9 shows comparisons of r between the case of α = 500 and 1000. In the case of α = 500,
the calculation errors due to spatial averaging of rainfall appear constantly in the process of extending
the averaging area, because the spatial variation scale of rainfall ﬁeld is relatively small compared tothe size of the study area. In the case of α = 1500, the calculation errors are relatively small when thesize of averaging area is 750 m (level 2). However they increase much when the size of the averaging
area exceeds the spatial correlation length (1500 m). This fact implies that the essential scale of rainfall
spatial variation is 1500 m for this case and that it is important to ﬁnd out a scale of rainfall spatial
variability which aﬀects the results of runoﬀ simulations.
4

Analysis using observed radar rainfall data

In this section, numerical experiments using observed radar rainfall data will be conducted in the same
way as the previous section. A runoﬀ simulation is conducted by using an observed rainfall radar data
as it is, and the result is assumed to be a standard result. Then the observed radar data is spatiallyaveraged and runoﬀ simulations are conducted by using the averaged radar data. By comparing thesesimulation results, a relationship among spatial averaging scale, size of drainage area and calculation
error is discussed.

15

20

2x2
3x3
4x4
standard
average rainfall

discharge [mm/hour]

discharge [mm/hour]

20

10
5
0
09/02 09/02 09/03
00:00 12:00 00:00

2x2
3x3
4x4
standard
average rainfall

15
10
5
0
09/02 09/02 09/03
00:00 12:00 00:00

09/03 09/04 09/04 09/05
12:00 00:00 12:00 00:00
time

time

Figure 10. Drainage area of 74.5 km2

15

20

3x3
4x4
standard
average rainfall

10
5
0
09/02 09/02 09/03
00:00 12:00 00:00

Figure 11. Drainage area of 278.4 km2
discharge [mm/hour]

discharge [mm/hour]

20

09/03 09/04 09/04 09/05
12:00 00:00 12:00 00:00

09/03 09/04 09/04 09/05
12:00 00:00 12:00 00:00

3x3
4x4
standard
average rainfall

15
10
5
0
09/02 09/02 09/03
00:00 12:00 00:00

time

09/03 09/04 09/04 09/05
12:00 00:00 12:00 00:00
time

Figure 12. Drainage area of 582.0 km2

Figure 13. Drainage area of 710.2 km2

4.1 Study basin and radar data
The study basin is Katsura River basin (710.2 km2 ). Runoﬀ simulations for this basin are conducted using
the rainfall data observed at Miyama radar site from September 2 to 5, 1989. The radar reﬂectivity (Z)was converted to rainfall intensity (R) by Z-R relationship (Z = BRβ ; B = 80, β = 1.7). Furthermore,
the rainfall intensity was calibrated with rain gauge. The spatial resolution of the radar data is 3 km.
4.2 Results and discussions
The simulation results are shown in Figure 10 - 13. The lines labeled as “standard” in the ﬁgures show
the standard results, that is, the simulation results obtained by using the observed radar data as it is.
The lines labeled as “2×2”, “3×3” and “4×4” show the simulation results using the radar data averaged
over (A) 2 × 2 radar meshes (6 km × 6 km), (B) 3 × 3 radar meshes (9 km × 9 km) and (C) 4 × 4 radar
meshes (12 km × 12 km), respectively.
From these results, the runoﬀ calculation error due to spatial averaging of the rainfall data, , isevaluated as:
={

te

ts

|Q(t) − Qs (t)|dt}/{tets

Qs (t)dt}(2)

where ts is integration starting time, te is integration ending time, Qs (t) is calculated discharge using theobserved radar data, and Q(t) is calculated discharge using the averaged radar data.
Figure 14 shows the relation between  and the drainage area. The curved lines in this ﬁgure are theenvelope curves of the calculation errors ((A): broken line, (B): dotted line, (C): solid line). The envelope
curves are given by  = exp(axc +b) where x is drainage area (km2 ), a, b and c are constants. In Figure 14
the straight line is also drawn to denote  = 0.05. The drainage areas where this line intersects with theenvelope curves are (A): 52 km2 , (B): 202 km2 and (C): 465 km2 . For case (A), in which the radar data
is averaged over 6 km × 6 km, the minimum size of drainage area at which the error of runoﬀ simulation
can be regarded as relatively small is about 50 km2 . In other words, the radar resolution of 6 km is
required in order to precisely simulate outﬂow discharge from about 50 km2 drainage area. Similarly,
from case (B) and (C), it can be said that the radar resolutions of 9 km and 12 km are required in orderto obtain precise discharge at the outlets of drainage areas of 200 km2 and 450 km2 , respectively. Theseexperiments indicate the importance to ﬁnd out a scale of rainfall spatial variability which aﬀects theusing 2x2 averaged dataenvelope curve of 2x2 result
using 3x3 averaged data
e. c. 3x3 result
using 4x4 averaged data
e. c. 4x4 result
0.05

calculation error

0.5

0.1
0.05

0.01
0.005

0

100

200

300
400
500
drainage area [km]

600

700

800

Figure 14. calculation error and drainage area

accuracy of runoﬀ simulations, which is already pointed out in the previous section. Furthermore, they
imply that the scale changes with the size of drainage area.
These analyses, however, do not have universal applications because they are based on one rainfall
event of one river basin, and implicitly assume that the spatial variability of rainfall ﬁeld is suﬃciently
captured by the radar system used in this study. Further investigations based on various rainfall eventsand river basins will be required.
5

Conclusion

This study investigated a rainfall spatial variation scale to be considered in a runoﬀ simulation by usingmesh-type rainfall data and a distributed runoﬀ model. As a preliminary analysis, numerical experiments
were conducted using generated rainfall data in order to investigate a relationship among rainfall averaging
scale, characteristics of rainfall ﬁeld and runoﬀ simulation error. This analysis showed that (1) the runoﬀ
simulation error is in proportion to the rainfall intensity, (2) the error becomes greater as the coeﬃcient ofvariation of rainfall increases, and (3) it is important to ﬁnd out a scale of rainfall spatial variability whichaﬀects the results of runoﬀ simulations. Next, numerical experiments using observed radar rainfall data
were conducted in the same way as the preliminary analysis. These experiments indicated the importanceto ﬁnd out a scale of rainfall spatial variability which aﬀects the accuracy of runoﬀ simulations, which
was already pointed out in the previous analysis. Furthermore, they implied that the scale changes withthe size of drainage area.
References
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1, pp. 113–118.
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145–150 (in Japanese with English summary).
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system based on a new digital topographic model” Journal of Hydraulic, Coastal and Environmental
Engineering, JSCE, in printing (in Japanese with English summary).
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