Proc. of the 2nd Asia Pacific Association of Hydrology and Water Resources (APHW) Conference,
July 5-8, 2004, Suntec Singapore International Convention Exhibition Centre, Singapore,vol. 2, pp. 617-624, 2004SPATIAL RAINFALL FIELD SIMULATION WITH RANDOM
CASCADE INTRODUCING OROGRAPHIC EFFECTS
ON RAINFAL
Yasuto TACHIKAWA
Disaster Prevention Research Institute, Kyoto University
Gokasho, Uji, 611-0011, Japan
Masahiro HIWASA
Graduate School of Civil Engineering, Kyoto University
Yoshida Honmachi, Sakyo-ku, Kyoto, 606-8501, Japan
Kaoru TAKARA
Disaster Prevention Research Institute, Kyoto University
Gokasho, Uji, 611-0011, JapanA space rainfall model is constructed to generate spatial rainfall fields. The model isbased on a random cascade process which disaggregates a regionally averaged
rainfall amount to generate spatial distributions of rainfall. To include a structure of
rainfall field heterogeneity in rainfall simulations, a deterministic factor isintroduced in a random cascade generator. The factor represents an orographic
effect on spatial distributions of rainfall, which is determined by using thedependence line on topographic elevations (DLTE) developed by Nakakita et al. [1]and Suzuki et al. [2]. The model is applied to a 192 km × 192 km area covered by
the Miyama radar rain gauge in the Kinki region of Japan. It is found that simulation
results well preserve the characteristics of orographic effects represented by the
DLTE. This suggests that the proposed method has the ability to capture statisticalcharacteristics of topography on rainfall fields found in radar rainfall data.
INTRODUCTION
A downscaling technique to generate local scale rainfall distributions from global scale
information is newly devised for local scale watershed managements in ungauged basins.
In the method, a statistical downscale technique using a random cascade theory
considering orographic effects on rainfall distributions is proposed.
The WGR model based on the point process theory developed by Waymire et al. [3]is a well-known stochastic model to generate a space-time rainfall distribution; however
it is quite difficult to identify the model parameters. On the other hand, random cascade
models, which assume self-similarity in spatial rainfall distributions, have recently used
as a rainfall field generator, for example, see Over and Gupta [4]. These models have few
model parameters which are easily obtained using observed spatially distributed rainfall
information. To represent non-homogeneous spatial features of rainfall patterns,

1Proc. of the 2nd Asia Pacific Association of Hydrology and Water Resources (APHW) Conference,
July 5-8, 2004, Suntec Singapore International Convention Exhibition Centre, Singapore,vol. 2, pp. 617-624, 2004

2
Jothityangkoon et al. [5] introduced the regional characteristics of rainfall into thecascade generator with the beta-lognormal model developed by Over and Gupta [4].
In this study, a topographic effect on rainfall spatial distributions is incorporated intoa random cascade generator to realize more realistic rainfall field simulations. It is known
that a spatial distribution of accumulated rainfall is well related to a topographic elevation
pattern. To include the topographic effects into simulated rainfall distributions, thedependence line on topographic elevations (DLTE), which represents the dependence
properties of spatial rainfall distributions on topographic elevations developed by
Nakakita et al. [1] and Suzuki et al. [2], is incorporated into a random cascade generator.
DLTE is a regression line between the spatial averages of accumulated rainfall andtopographic elevations. Nakakita et al. [1] and Suzuki et al. [2] found that the correlation
coefficient of the relationship obtained using radar rainfall data was more than 0.9. Using
the DLTE, deterministic topographic multiplier matrices are introduced into a randomcascade generator using the beta-lognormal model.
The method is applied to a 192 km × 192 km region in the Kinki area of Japan. The
parameters of the random cascade generator are obtained using radar rainfall data. The
values of the deterministic topographic multiplier matrices for each downscale level are
determined with the DLTE and average elevation of each sub-area. By using the cascade
generator, spatial rainfall distributions with 3km resolution are simulated over the region.
Then statistical characteristics of the generated data are examined. It is found that theaccumulated values of generated rainfall fields with 3 km grid resolution shows a linear
relationship with the elevations, which is in good agreement with the relationshipintroduced by the DLTE. This suggests that the proposed method has the ability tocapture statistical characteristics of topography on rainfall fields found in radar rainfall
data.
RANDOM CASCADE MODEL
A random cascade process starts spatial disaggregation of rainfall with a given volume ofrainfall in an initial area (level 0). The area is subsequently subdivided into sub-areas(four areas in this case) at each disaggregation step and the volume given at the previous
step is assigned into the four sub-areas. The rainfall volume in the sub-area i at the n thlevel of disaggregation step is given bynµ ni = R 0 L 0 b − n ∏ W ji

(1)

j =1

where R0 is an average rainfall intensity (mm/h) of the initial area, L0 is the length of thestudy area (in this case 192 km), b is a branching number (b = 2d), and d is a dimension
constant (d = 2). The multipliers Wji are random cascade generators which are weights toassign average rainfall intensity to four sub-areas. As the cascade generator Wji, we adopt
the beta-lognormal model proposed by Over and Gupta [4]. In the model, W is a productof independent random variables B and Y, that is W=BY. B and Y are represented asProc. of the 2nd Asia Pacific Association of Hydrology and Water Resources (APHW) Conference,
July 5-8, 2004, Suntec Singapore International Convention Exhibition Centre, Singapore,vol. 2, pp. 617-624, 20043
B = 0 with probability P ( B = 0 ) = 1 − b − β , B = b − β with probability P ( B = b − β ) = b − β (2)Y = b − ( σ ln b ) / 2 + σ X
2(3)

where X is a standard normal random variate and β and σ 2 are parameters of the betalognormal model. The expectation and variance of B and Y are probed to be
E [ B ] = 1,

Var [ B ] = b β − 1

E [Y ] = 1,

( σ ln b ) 2

Var [Y ] = e

(4)
−1

(5)

From the Eqs. (4) and (5), E[W] = 1 is satisfied. The random variate B expresses thecondition whether it rains in the sub-area or not, and the random variate Y represents theweight for assigning a rainfall volume into the four sub-areas. Figure 1 shows an example
of generated rainfall fields at each disaggregation step (from step1 to step 6) by using thebeta-lognormal model. The study area is a 192 km × 192 km area covered by the
Miyama radar rain gauge in the Kinki region of Japan.

rainfall intensity(mm/h)

10
9
8
7
6
5
4
3
2
1
0

3
2.5
2
1.5
1
0.5
0

2
1.5
1
0.5
0

0

30

60

90
90

120

x (km)

60
150

30
180

180
150
120

y (km)

0

30

0

60

90
90

120

x (km)

60
150

30
180

180
150
120

rainfall intensity(mm/h)

90
90x (km)

120

60
150

30
180

0

.GXGN MOTGUQNWVKQP

180
150
120

y (km)

0

60

90
90

120

x (km)

60
150

30
180

180
150
120

y (km)0.GXGN MOTGUQNWVKQP
rainfall intensity(mm/h)

90
80
70
60
50
40
30
20
10
0

60

30

0

rainfall intensity(mm/h)

18
16
14
12
10
8
6
4
2
0

30

0

y (km).GXGN MOTGUQNWVKQP.GXGN MOTGUQNWVKQP

0

rainfall intensity(mm/h)

rainfall intensity(mm/h)

2.5

80
70
60
50
40
30
20
10
0

30

60

90
90

x (km)

120

60
150

30
180

0

.GXGN MOTGUQNWVKQP

180
150
120

y (km)

0

30

60

90
90

x (km)

120

60
150

30
180

180
150
120

y (km)0.GXGN MOTGUQNWVKQP

Figure 1. An example of disaggregated rainfall fields from level 1 to level 6 using thebeta-lognormal model applied at a 192 km × 192 km area covered by the Miyama radarrain gauge in the Kinki region of Japan.

PARAMETER ESTIMATION OF BETA-LOGNORMAL MODEL
The values of model parameters β and σ 2 are identified using the data observed at the
Miyama radar rain gauge. The observation range is the inside of the radar radius 120 km.

Proc. of the 2nd Asia Pacific Association of Hydrology and Water Resources (APHW) Conference,
July 5-8, 2004, Suntec Singapore International Convention Exhibition Centre, Singapore,vol. 2, pp. 617-624, 2004

4
The study area is a 192 km × 192 km square area within the radar observation range. The
space resolution of the radar rainfall data is 3 km. Thus, the study area is composed of a
64 × 64 grid system. The time resolution is 5 minute. Radar data of 1000 observations
during August 1989 to September 1989 were used to identify the values of model
parameters.

0.9

0.18

β = 0.3089 R

0.8

0.1691

0.7

0.14

0.6

0.12

0.5

σ2

β

0.1

0.4

0.08

0.3

0.06

0.2

0.04

0.1

0.02

0
0.01

2

σ 2 = exp [-0.0875( ln R ) -0.0963( ln R)-2.9380]

0.16

0.1

1

Average rainfall intensity, R (mm/h)

10

0
0.01

0.1

1

10

Average rainfall intensity, R (mm/h)(a) β(b) σ 22
Figure 2. Estimated model parameters of β (a) and σ (b) as functions of average
rainfall intensity R.

The method to identify the parameters is the same as shown in Jothityangkoon et al.
[5]. In Figure 2, 1000 values of obtained parameters are plotted as a function of spatial
average rainfall intensity R (mm/h). The solid lines in the figures are regression curves
derived by a least square method. The empirical functions of the regression curves
adopted here are the same function type used by Jothityangkoon et al. [5].
It is found that when the value of average rainfall intensity R increasing, the value ofβ decreases. This implies when the regional average of rainfall intensity is higher, thepossibility that rainfall occurs at somewhere in the area becomes large. For the
relationship between σ 2 and R, it shows positive correlation when the value of R is lessthan several mm/hr. This shows when the regional average of rainfall intensity is higher
in the extent, the variance of Y becomes large, which leads to large variations of spatial
rainfall intensity. These relationships are similar to the results obtained by Over andGupta [4] and Jothityangkoon et al. [5].
INTRODUCING DEPENDENCE PROPERTIES OF SPATIAL RAINFALL
DISTRIBUTION ON TOPOGRAPHY INTO CASCADE GENERATOR
To incorporate a structure of rainfall field heterogeneity in rainfall simulations, adeterministic factor is introduced in a random cascade generator. The factor represents
an orographic effect on spatial distributions of rainfall, which is determined by using theProc. of the 2nd Asia Pacific Association of Hydrology and Water Resources (APHW) Conference,
July 5-8, 2004, Suntec Singapore International Convention Exhibition Centre, Singapore,vol. 2, pp. 617-624, 2004

5
dependence line on topographic elevations (DLTE) developed by Nakakita et al. [1] andSuzuki et al. [2]. Nakakita et al. [1] and Suzuki et al. [2] presented an obvious
dependence relation between accumulated rainfall distributions and topographic
elevations. They examined the relationship between the spatial mean values ofaccumulated rainfall obtained in the classes divided by 100 m interval of topographic
elevations and their mean topographic elevations. They found a high value of thecorrelation coefficient over 0.9 between the logarithmic values of average accumulated
rainfall amount in each class and their mean topographic elevations. The relation (theDependence Line on Topographic Elevation, DLTE) is represented aslog (µ k (T ) / µ (T ) ) = a × z k + b(6)where µ (T ) (mm) is an accumulated value of spatially averaged rainfall over the entire
study region during the accumulation time scale T; µ k (T ) (mm) is the spatial averagevalue of accumulated rainfall amount in the elevation class k; zk (m) is the spatial averageof topographic elevation in the elevation class k; and a, b are parameters. Suzuki et al. [5]obtained the DLTEs for various µ (T ) in the study area by using the radar rainfall data.
When µ (T ) = 300 (mm) and setting y = µ k (T ) / µ (T ) , the DLTE is obtained aslog y = 0 . 00027175 z + log 0 . 83099(7)

By using this relationship, deterministic topographic multiplier matrices G are
introduced into a random cascade generator. For the subdivision area i (i = 1, 2,…, 22n) at
the disaggregation level n (n=1, 2,…, 6), the topographical weight Gni is incorporated
according to Jothityangkoon et al. [5] as
W = BYG(8)

with the condition that the average value of G over the sub-areas is equal to 1 at eachdisaggregation step. Gni is given by
G = µ /µ ,
i
n

i
n

i
n

µ = e µ (T ),
i
n

y ni

e y n µ (T )
µ =∑
2 2n
i =1
22n

i
n

i

(9)

where yni is the value of y calculated using Eq. (7) at the n th level of disaggregation in
the i th sub-area. Thus the values of the topographic multiplier matrices G are determinedby the DLTE and topographic elevation information.
Figure 3 shows simulated rainfall fields for the 192 km × 192 km region withspatial resolution of 3 km at the sixth level of disaggregation using the cascade generator
in Eq. (8) for the rainfall field at 12:40 on September 5 in 1989.

Proc. of the 2nd Asia Pacific Association of Hydrology and Water Resources (APHW) Conference,
July 5-8, 2004, Suntec Singapore International Convention Exhibition Centre, Singapore,vol. 2, pp. 617-624, 2004

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Figure 3. Simulated rainfall fields for the 192 km × 192 km region with spatial
resolution of 3 km at the sixth level of disaggregation using the cascade generator
introducing topographic effects for the rainfall field at 12:40 on September 5 in 1989.

To examine the effect of the topographic multiplier matrices G, 100 realizations ofrainfall distributions were simulated. In the simulations, the initial rainfall intensity forthe 192 km × 192 km area at the level 0 was set to 1 (mm/h). Figure 4 shows the relation
between the accumulated rainfall volume and the elevation using the simulated rainfall at
the sixth level (spatial resolution 3km). The figures (a) and (b) are the cases of 50 and
100 times accumulations respectively. The solid line represents the DLTE set in thesimulations.
The number of times of accumulation increases, the more clear correlation
relationship between topographic elevations and the amounts of accumulated rainfall
appears. This shows the topographic effects on spatial rainfall distributions described byusing the DLTE are well incorporated into the random cascade rainfall simulations
through the deterministic topographic multiplier matrices G.
However, most of results in Figure 4 are plotted over the DLTE and the gradientcalculated using the plotted points is steeper than the one of the DLTE. In the simulations
presented in Figure 4, G was incorporated at all disaggregation levels. For the reason, thetopographic dependence on rainfall distributions appears strongly.

Proc. of the 2nd Asia Pacific Association of Hydrology and Water Resources (APHW) Conference,
July 5-8, 2004, Suntec Singapore International Convention Exhibition Centre, Singapore,vol. 2, pp. 617-624, 2004

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To avoid the overdependence, the simulations that G was incorporated only at the 6th
disaggregation level were carried out. Figure 5 shows the results of the simulations. In
both cases of 50 times and 100 times accumulations, the results are well plotted aroundthe DLTE, which are better results than shown in Figure 4.

10Σ R / 100 (mm/hr)

Σ R / 50 (mm/hr)

101

y = 101

0.00027175x + loglog(0.83099)(0.83099)

y = 100.1

0.00027175x + loglog(0.83099)(0.83099)

0.1
0

200

400

600

800

1000

1200

1400

1600

0

200

400

600

altitude(m)

800

1000

1200

1400

1600

altitude(m)(a) 50 times accumulations(b) 100 times accumulations
Figure 4. The relation between the accumulated rainfall volume and the elevation using
the simulated rainfall at the sixth level (spatial resolution 3km). In the cases, G isincorporated at all disaggregation levels. The figures (a) and (b) are the cases of 50 and
100 times accumulations respectively. The solid line represents the DLTE set in thesimulations.

10Σ R / 100 (mm/hr)

Σ R / 50 (mm/hr)

101

y = 101

0.00027175x + loglog(0.83099)(0.83099)

y = 100.00027175x + loglog(0.83099)(0.83099)

0.1
0.

0.1
0

200

400

600

800
altitude(m)

1000

1200

1400

1600

0

200

400

600

800

1000

1200

1400

1600

altitude(m)(a) 50 times accumulations(b) 100 times accumulations
Figure 5. The relation between the accumulated rainfall volume and the elevation using
the simulated rainfall at the sixth level (spatial resolution 3km). In the cases, G isincorporated only at the 6th disaggregation level. The figures (a) and (b) are the cases of
50 and 100 times accumulations respectively. The solid line represents the DLTE set in
the simulations.

Proc. of the 2nd Asia Pacific Association of Hydrology and Water Resources (APHW) Conference,
July 5-8, 2004, Suntec Singapore International Convention Exhibition Centre, Singapore,vol. 2, pp. 617-624, 2004

8

CONCLUSION
In this study, a topographic effect on rainfall spatial distributions was incorporated into arandom cascade generator to realize more realistic rainfall field simulations. To include
the topographic effects into generated rainfall distributions, the dependence line on
topographic elevations (DLTE) of rainfall distributions developed by Nakakita et al. [1]and Suzuki et al. [2] was incorporated into the random cascade generator. Using theDLTE, deterministic topographic multiplier matrices were introduced into a randomcascade generator using the beta-lognormal model.
The method was applied to a 192 km × 192 km region in the Kinki area of Japan.
The parameters of the random cascade generator were obtained using radar rainfall data.
By using the cascade generator, spatial rainfall distributions with 3 km resolution were
simulated over the region. Two kinds of random cascade generators were tested; one is a
generator incorporating the deterministic topographic multiplier matrices G at every
disaggregation step, and another is the one with introducing G only at the 6th level ofdisaggregation. Then statistical characteristics of the simulated rainfall data were
examined. It was found that the accumulated values of generated rainfall fields with 3 km
grid resolution showed a linear relationship with the elevations, which was in good
agreement with the relationship introduced by the DLTE, especially when using the latterrandom cascade generator. The results suggest that the proposed method has the ability tocapture statistical characteristics of topography on rainfall fields found in rainfall data.
Acknowledgments: The authors thank the Yodo River Dams Control Office, Ministry ofLand, Infrastructure and Transport, Japan for providing radar rainfall data, and Dr.
Suzuki at Department of Civil Engineering, Utsunomiya University for making
comments about DLTE.
REFERENCES[1] Nakakita, E., Y. Suzuki, and S. Ikebuchi (2001) Hierarchical time-scale structure in

[2]

[3]
[4]
[5]

the dependence of rainfall distribution on topography, Journal of Hydroscience andHydraulic Engineering, Japan Society Civil Engineers, 19 (2), pp. 1-10.
Suzuki, Y., E. Nakakita, and S. Ikebuchi (2002) A study of dependence properties ofrainfall distribution on topographic elevation, Journal of Hydroscience andHydraulic Engineering, Japan Society Civil Engineers, 20 (1), pp. 1-11.
Waymire, E., V. K. Gupta and I. Rodriguez-Iturbe (1984) A spectral theory of
rainfall Intensity at the meso-beta Scale, Water Resour. Res., 20(10), pp. 1453-1465.
Over, T. M., and V. K. Gupta (1996) A space-time theory of mesoscale rainfall using
random cascades, Jour. Geophys. Res., 101(D21), pp. 26,319-26,331.
Jothityangkoon, C., M. Sivapalan, and N. R. Viney (2000) Tests of a space-timemodel of daily rainfall in Southwestern Australia based on nonhomogeneous randomcascades, Water Resour. Res., 36(1), pp. 267-284.