Modeling of Fine Resolution Precipitation Data Using GAME Data Product andValidating Against the Spatial Patterns of HUBEX-IOP-EEWB Data
* R. K. Shrestha 1, Y. Tachikawa 2, K. Takara 2(1: Graduate School of Urban and Environmental Engineering, Kyoto University, Japan,
2: Disaster Prevention Research Institute, Kyoto University, Japan)* DPRI, Kyoto University, Japan, 611-0011, Gokasho, Uji, Kyoto, Japane-mail: roshan@rdp.dpri.kyoto-u.ac.jp
Abstract
Multiplicative Random Cascade (RC) method is a stochastic tool to model a fine resolution precipitation data, which isbased on assumption that the precipitation structure is Multifractal. The random process involved within it, which oftencauses the results either to differ largely in between repeated trials or to differ from a measured one, dominates existing
RC method. This weakness is necessary to remove before adopting the method as a reliable tool of modeling the fineresolution precipitation field. Noticing that the spatial rainfall field contains spatial correlation, we attempted to include
this information in a RC method. First, we evaluate a reference matrix, which accounts for the spatial correlation effectof a coarse precipitation field and the distance from the nearest point of coarse grid to the center of a grid at the fineresolution. Then, this reference matrix assists to find the location of a cascade generator by comparing the hierarchicalorder. It also assists to re-allocate the statistically filtered peak values to a proper location. This method, named as
Multiplicative Random Cascade with Hierarchical and Statistical Adjustment (RCHSA) method, was used to model thefine resolution precipitation data (10-arc minute resolution) using GAME Reanalysis 1.25 degree data as the input.
Comparing the spatial patterns of over 40,000 test outputs from the RCHSA method against the spatial patterns of the
HUBEX-IOP-EEWB data, it revealed that the RCHSA method was largely successful to remove the weakness of the
RC method. The overall performance was improved to 0.6 from 0.34 after including the HSA method. This has helped
us to use the GAME 1.25 degree data product in hydrological simulations of catchments as small as of 2000 sq. km.
scale successfully.
Keyword: rainfall downscaling, random cascade method, HSA method, spatial correlation.

1. Introduction
Precipitation data are the important binding forcing tounderstand space time variability of meteorological andhydrological conditions, and to drive small to large-scaleand short to long-term simulations in both sectors.
GAME Re-analysis project, started in 1999 in Japan witha joint effort of Meteorological Research InstituteNumerical Prediction Division / Japan Meteorological
Agency and Earth Observation Research Center, had
aimed to obtain higher quality data using the most
updated assimilation system, the 4DDA, and the off-line
data collected during GAME-IOP period. This project
released the GAME re-analysis data in September, 2000.
Before the release, it conducted the case study of heavy
precipitation in the Yangtze River in addition tocomparison of the re-analyzed data with ECMWF and
NCEP products (Yamazaki et al., 2000). This is an
example of use of the precipitation data `to evaluate theperformance of a numerical model in meteorological dataacquisition system.
Three different sets of the GAME re-analysis productwere released in its version 1.1 package having different
horizontal resolutions, 2.5-degree, 1.25-degree and
0.5-degree including 3-dimensional analyzed fields andtwo dimensional forecasted fields. The data have covered
the summer of 1998. The effort of releasing
multi-resolution data product was appreciable realizing
the fact that what we observe at a scale does notnecessarily be true at another scale, though there
remained an unanswered question “how many different
resolutions are to release?” Provided a finer resolution
data, there is comparatively little trouble in getting acoarser resolution data which have averaged intensity ina wider region and smoother spatial pattern. But toobtain a finer resolution data, in terms of the
precipitation data, from a coarser resolution data is achallenging task as it needs to describe the sub-grid scalevariability, particularly in the subtropical and temperatemonsoon regions of East Asia, where the energy andwater cycle is characterized largely by the Baiu front insummer. Various scales of cloud/precipitation systems
associated with the complex processes between air andland-surface are formed in this frontal zone and they playa major role in this region making it hard to describe thesub-grid scale variability.
Attempts to obtain a sub-grid scale spatial variabilityof the precipitation are headed toward two major
directions. The first direction is mainly developed inmeteorology literature describing the atmospheric
dynamics and thermodynamics. There are some practicaldifficulties to obtain finer resolution outputs over longtimescale due to complexities of nesting boundary
conditions, unknown parameterization for convective
precipitation, and multiscale variability of deep
convection through this direction (Giorgi and Mearns,
1991; Houze, 1997; Chen et al., 1996). Also, there aremany unknowns in describing the microphysics of theintermittent precipitation mechanism in finer resolution.
The second direction is mainly developed in hydrology
literature that leads to the description of spatialorganization of the precipitation in terms of intensity at
multiple scales. In this discipline, the recognition of thehidden sub-grid scale features from the coarse fuzzy
information has become a major key to disaggregate theprecipitation data into finer resolution using probabilistic
models of space-time rainfall.
The second approach is based on the scaling
invariance features of observed spatial rainfall fields

(Schertzer and Lovejoy, 1987; Lovejoy and Schertzer,
1990; Gupta and Waymire, 1990) with extreme
variability and strong intermittence (Georgakakos andKrajewski, 1996), which has yielded a multiplicative
random cascade theory (Lovejoy and Schertzer, 1990;
Gupta and Waymire, 1993). Due to the scaling invarianceor self-similarity concept in this approach of space-time
rainfall modeling, the parameterization is parsimonious
and valid over a wide range of scales (Lovejoy andSchertzer, 1990; Gupta and Waymire, 1993; Over andGupta, 1994; Foufoula-Georgiou and Krajewski, 1995;
Olsson, 1996).
The objective to obtain further finer resolution
precipitation data than that of the currently available one
has multi-purpose attraction either in research or inapplication studies. In this study, we present the resultsof out attempt to model the fine resolution precipitation
data using the GAME 1.25-degree two-dimensional
forecast field. We compare the outcome of the modelwith the spatial patterns of HUBEX IOP EEWB data
(Kozan et al. 2001

2. Description of the model
A multiplicative cascade treatment based on thestatistical theory of turbulence (Mandelbrot, 1974) offersa concrete way of modeling these fields (Schertzer andLovejoy, 1987) as the kinetic energy transfer is seen in
the cascade of turbulent eddies from a large energy scale
to smaller dissipation scales. Similarly, in the cascades ofprecipitation modeling, an area of higher intensity
precipitation is embedded in larger areas of lower
intensity precipitation, which are again a part of evenlarger areas but of even lower intensity. This can be
described either in continuous or in discrete form ofmultiplicative cascades. Debates are ongoing on
suitability of approaches to form the multiplicative
random cascade whether the continuous or discrete. The
continuous form of multiplicative random cascades has
the major advantage of developing cascades over acontinuous interval of scales instead of only a discrete set(Marsan et al., 1996); however, the discrete form ofmultiplicative random cascade has ability to separate
rainy and non-rainy area (Gupta and Waymire, 1993;
Over and Gupta, 1994) and can be adopted to respect thediscrete sub catchment partitioning of the landscape by
the drainage network of a catchment (Gupta et al., 1996;
Over and Gupta, 1996). We choose here a discrete
multiplicative random cascade method, so called the betalognormal model developed by Over and Gupta (1994,
1996) and apply a method to improve the shortcoming of
the model. This is called as the multiplicative randomcascade HSA method (Shrestha et al., 2004)

2.1.

The beta lognormal model

In the beta lognormal model, the cascade construction
process successively divides a two-dimensional region
into equal parts at each step, and during each subdivision(say n = 2) the mass (or volume) of rainfall over theregion obtained at the previous step (n = 1) is distributed
into the b subdivisions (for the case of d = 2; b = 4) bymultiplying by a set of “cascade generators” W, as shown
schematically in Figure 1.

0

For an area at level 0, denoted by ∆ 0 , has the outer

L0 and average precipitation intensity

length scale ofof

R0 . The initial volume µ0 (∆00 ) becomes R0 Ld0 .

µ0 (∆00 ) divides into b = 4

At level 1, the volumei

sub-areas denoted as ∆1 , (i = 1, 2, 3, 4) and the sub-areavolumeµ1 (∆i1 ) is R0 Ld0 b −1W1i , (i = 1, 2, 3, 4). At

level 2, each of the sub-area volume is further subdivided
into b = 4, all together b2= 16 sub sub-area, denotingi
2 , (i = 1, 2, ..., 16) and the corresponding
d −1
i
i
i
is 0 022
12 , (i = 1, 2, ..., 16).

them as ∆volumeµ (∆ )RLb WW

The process of sub-division is continued further until thenth level up to bni

sub-areas, which are denoted as ∆ n ,n(i = 1, 2, ..., b ). At the nth level, the volume in the
sub-areas can be expressed asnµ n (∆in ) = R0 Ld0 b −n ∏W ji ; (i = 1,2, ..., b )
n

(1)

j =1

Fig. 1: Schematic of cascade branching

The cascade generators W are non-negative random
values with E[W] = 1, which is imposed to ensure themass conservation from one discretization level to thenext (see Over and Gupta, 1996). To get the cascade
generator W values, Over and Gupta (1994, 1996) has
proposed a model called beta-lognormal model such that
W = BY(2)
Here, B is a generator from the “beta model” that
separates the rainy and non-rainy zone on the basis ofdiscrete probability mass function and Y is obtained from
lognormal distribution (Gupta and Waymire, 1993) in theform of Y = bσ 2 ln b  σX − 2 , where X is standard normal

random variate and σ

2is a parameter equal to thevariance of log b Y . The value of W is evaluated as0
W =  β +σX −σ 2 ln b
2
b

when
when

P (B = 0 ) = 1 − b − β
P B = b β = b −β

(

)

(3)
This model consists of only two parameters,

β and σ 2 . The parameter estimation method is proposedby Over and Gupta (1994, 1996) by using theMandelbrot-Kahane-Peyriere (MKP) function, named

after Madelbrot (1874) and Kahane and Pyeriere (1976),
which characterizes the fractal or scale-invariant
behavior of the multiplicative cascade process.
This method is largely dependent on the spatial
structure of generators, which is a random process within
this method. The random nature of the generators oftencauses the results either to differ largely in between
repeated trials or to differ from a real one. To remove theweakness, the following method has been employed inthis study.

2.2.

The HSA method

In smaller spatial scale, the precipitation fields havestrong spatial correlation. This phenomenon wasappeared in the GAME 1.25 degree precipitation data too,
although the data is of coarse resolution. Figure 2 shows
the spatial correlation of the precipitation data.

(x − x ) + ( y − y )2

Z nm =
where,

H

m

2

j

m

k

(11)

jk
th
n is the non-negative reference index at n

level for jkth sub-area; Rm is the rainfall of mth neighbor
cell; ρ Z m is the spatial correlation with the mth neighbornviewed from jkth location at nth level, from where thedistance up to the mth neighbor becomes
sub-area, theZ nm . For thex j and y j represents the central pointof the sub-area.
The multiplicative random cascade HSA method

Wni values by preserving the spatial

assigns thecorrelation structure of the precipitation field such that,
where,[]

W [•] = BY

(12)
represents the W with its spatial

W [•]address • . The reference index[]

H njk is used to obtainthe spatial address • using its two dimensional spatialreference j and k inside the sub-dividing region.
At every additional level (n+1), four more randomi'cascade generators, Wn+1 (i' = bn+1-3, bn+1-2, bn+1-1,

bn+1)

∆

Fig. 2: Spatial Correlation of the precipitation dataρZ − αρ Z= 0, for(9)κwhere, Z is distance in kilometers; ρ Z is the spatial
correlation value at Z; Z 0 is the threshold beyond which
Z 0 = anti log λthe spatial correlation remains zero by the use of thelogarithmic spatial correlation function with α , κ andλ parameters.
In the process of cascading down of a two dimensionalb n numbers of sub-areas,n

named as, (i = 1, 2, ..., b ) are obtained at the nth levelLnwith the grid dimension 0 / d . For each of these
sub-areas, a spatial correlation reference index H isevaluated, which works as a spatial guide matrix. The
reference index H is influenced by the average rain
intensities Rm of the surrounding eight coarse scale grids(m = 1, 2, …, 8) and corresponding distances of themsub-area from the referred neighbor grid Z n . For nth
level, the reference index H can be represented as,
8

H njk = ∑ Rm ρ Z m
m =1

n

; for

j = 1,2,..., d n
k = 1,2,..., d n

for

i
n ,

∆

newly

disaggregatednsub-area(i = b ). Their spatial address[•] isdetermined on the basis of comparison between the

The spatial correlation gradually decreases uponincrease of distance. The decaying shape of the spatial
correlation function may be represented by a logarithmic
function (equation 8). This is given byα + κ logλ Z if (ρZ > 0 and Z < Z0 )
ρZ = 
0if (ρZ < 0 or Z > Z0 )
(8)(d = 2) spatial field, theappeari'n +1 from(10)reference indexesi'H nj+'k1' and the random cascadei'generators Wn+1 . The Wn+1 may need to reshuffle its
i'location within ∆ n +1ofH

j 'k '
n +1

in order to attain same hierarchy

locations. This process includes a spatial

correlation structure into the cascade generators insuccessive progress of disaggregation.
The spatial H n field is a smooth gradient surface
and its shape is based on the surrounding coarse grid
average rainfall. The lowest and highest zones of the H nfield are most possibly the non-rainy and rainy zones
respectively. If the peak rainy cells of the µ n (∆ n ) field
are not in accordance with the rainy zones of

H n field

and vice versa, a statistical filter is applied to improve it.
Though these extreme high and low value cell numbers
may not be significant to influence spatial statistics, theymight have practical significance. Therefore at the nthlevel, spatial locations of the extreme µ n (∆ n ) values
are re-adjusted following hierarchical order ofHnfield after statistical separation of extreme high and low
values from both fields. The statistical adjustment isomitted if the correlation of H n field and µ n (∆ n )field is found higher than a target correlation, which isadapted 80% arbitrarily in this case.

(a) GAME 1.25-degree data(b) Fine resolution data obtained by the model

The modeled fine resolution precipitation data may notnecessarily represent a true precipitation pattern eventhough it perfectly match with the HUBEX-IOP EEWB
data because there could be uncertainty within the
HUBEX-IOP EEWB data itself. Their ground based
observation data, however, provides the most detailed
spatial structure of the precipitation data in the study
region. We foresee the possibility of using the modeled
data in hydrological simulation of smaller catchments,
much smaller than the size of a grid-cell that the GAME
1.25-degree data have, if the modeled fine resolution
data can generate similar data as that of the observed one.
The HUBEX-IOP EEWB precipitation data provides asmooth spatial pattern, which is the obvious result ofinterpolation that was employed to make the spatiallydistributed data from a network of point observed
precipitation data (Kozan et al., 2001). The modeled
results are not as smooth as the interpolated field. Even
being the modeled outputs able to preserve the spatial
pattern in terms of separation of rainy zones andnon-rainy zones, they include a higher degree of spatial
heterogeneity. Definitely, this is the effect of the random
generators. The rainfall magnitudes at a particular cell,
however, do not remain exactly same in the repeatedtrials, as they depend upon the generators based on the
cascade of multiplication of the random numbers inside
the model despite their superb capability of preserving
the same spatial patterns. We argue this as a beneficial
point of the suggested method to infer the probability ofuncertainty in rainfall phenomenon, which might help tounderstand the consequences of rainfall uncertainty inrainfall-runoff modeling.

References(c) HUBEX-IOP EEWB data
Fig. 3: Spatial patterns of the precipitation data obtained
from a) GAME 1.25-degree precipitation data; b) output of
the modeling of fine resolution precipitation data using the
GAME data; and c) HUBEX-IOP EEWB precipitation
data used to compare the result of the model

3. Results and Discussion
The experiment was successful to produce a fineresolution precipitation data. Figure 3 shows a typicalresult of this kind. Figure 3a is the GAME 1.25-degree
data and Figure 3b is the 10-minute resolution
precipitation data. The results were compared with thespatial patterns of the precipitation data collected by the
HUBEX-IOP during the same period (Figure 3c).
Comparison of the spatial patterns shows good match in
between the two precipitation data sets. In another test
without the HSA method, so-called as the RC method,
the fine resolution precipitation data were found havingvery different spatial pattern from that of the
HUBEX-IOP EEWB data. While the performance of the
RC method yielded 0.34 in the test of spatial covariance
between the HUBEX-IOP EEWB data and the modeled
fine resolution data using the RC method, the RCHSA
method, as described in this study, yielded 0.6. Thisconfirms the improved ability of modeling fineresolution data using the GAME 1.25-degree product.

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