Journal of Japan Society of Civil Engineers, Ser.
Annual Journalof HydraulicEngineering,
2014, February
B1 (HydraulicEngineering),
Vol. JSCE,
70, No.Vol.58,
4, I_901-I_906,
2014.

STATISTICAL DOWNSCALING OF PRECIPITATIONWITH A FORMATTED REGRESSION FRAME
Sunmin KIM 1, Eiichi NAKAKITA2, Yasuto TACHIKAWA3,
Michiharu SHIIBA4 and Masataka INOUE5
1Member of JSCE, Dr. Eng., Associate Professor, DPRI, Kyoto University(Gokasho, Uji, Kyoto 611-0011, Japan) E-mail:sunmin@hmd.dpri.kyoto-u.ac.jp
2Member of JSCE, Dr. Eng., Professor, Disaster Prevention Research Institute, Kyoto University
3Member of JSCE, Dr. Eng., Professor, Dept. of Civil and Earth Resources Eng., Kyoto University
4Member of JSCE, Dr. Eng., Emeritus Professor, Dept. of Civil and Earth Resources Eng., Kyoto University
5Member of JSCE, Master of Eng., Staff, Nagoya Station, Central Japan Railway Company

This paper illustrates a statistical downscaling technique considering the spatial correlation structureof precipitation. Downscaling target is 60-km resolution of daily precipitation for 20-km resolution data.
We have considered a window having (3x60-km)x(3x60-km) of area, and the downscaling target is the
3x3 of 20-km resolution grids in the center of the downscaling window. For the evaluation of the
proposed method, we have prepared 15 years (1979-1993) of observation data, and identify theparameters with the square root information filter scheme. We optimize the parameters on a monthly
basis, and apply the regression model to 10 more years of testing period (1994-2004). The proposed
regression model provides very effective and efficient results with a certain level of estimation error.
Key Words: Statistical downscaling, precipitation, spatial correlation, regression model

1. INTRODUCTION
The downscaling issue has been taking animportant role to bridge research in climate change
and impact assessment1),2). Though many generalcirculation models (GCMs) have been developedand improved for the last several decades, their
output is still too coarse for local impact assessment
research. For example, many GCMs in the recent
CMIP53) (Coupled Model Inter-comparison Project,phase 5) provide spatial resolution that is larger than
100-km in longitudinal grid space at 35°N.
In the meantime, Japan Meteorological Agency(JMA) and the Meteorological Research Institute(MRI) of Japan has developed a couple ofAtmospheric General Circulation Models (AGCMs)
with 20-km and 60-km spatial resolutions4), within
two nationwide climate change research projects,
Kakushin (FY2007 - FY2011) and Sousei (FY2012- FY2016 planned) Projects. Though these models
focus on atmospheric simulation only, because of
their ultra-fine resolution in the global sense, the20-km AGCM has provided only two sets of output
so far. (The ongoing Sousei project may produce
more.) On the other hands, the rather coarse
resolution of 60-km AGCM provides 24 sets of

output with variant boundary conditions and model
parameters, which gives us a glimpse of theuncertain future of climate projection5). The 20-km
AGCM output is able to provide reliable hydrologic
impact assessment results in the major river basinsof Japan6),7). The goal of our research is to develop a
simple yet efficient statistical downscaling (SDS)
method to downscale 60-km AGCM output into
20-km resolution for the precipitation data.
The SDS issue has a long history of research anddevelopment in the field of hydrology1) and several
types of SDS methods are already successful inother applications, such as SDSM8). Basic categories
of SDS include regression models (e.g. canonicalcorrelation analysis by Schmidli et. al.9)), weather
generators (e.g. Markov chain model by Wilks10))
and weather typing schemes (e.g. cluster analysis by
Fowler et. al.11)).
The main advantage of SDS compared to DDS(dynamic downscaling) is that it does not take highcomputing resources, and can easily apply to any
place with a minimum of observation data
available12). Even though the DDS method with aregional climate model (RCM) provides stable andreasonable output based on physical backgrounds,
RCMs demand many computing resources,

I_901

additional information, and difficult initial setup13).
However, SDS also has limitations. Somestatistical relationships between model variables are
not strong enough to build a stable SDS model12).
Most critically, we can not sure whether thestatistical relationship developed with the present
climate data can simulate the statistical relationshipof the future climate. We do not have future datawith which to evaluate the present statistical
relationship or to establish the correct future one.
We have been developing an SDS method that
can avoid the critical issue of the conventional SDS
method and take as many advantages of DDS as
possible, based on analyzing two different spatial
resolutions of AGCM outputs, 20-km and 60-km.
By establishing a statistical relationship between the
60-km and 20-km output for both present and future
separately, and by applying the relationship to theensemble output of 60-km AGCM, it can produce
ensemble output at 20-km spatial resolution.
In this paper, we introduce an SDS method withthe basic concept and evaluation results with theobserved precipitation data, before it is applied to
the AGCM output. Section 2 describes the basic
concept of the proposed methodology and the data
used in the experiments. Section 3 introduces thedeveloped SDS method. Section 4 provides theresults and related discussions. The last section
concludes this paper with prospective application
method to the 60-km AGCM output directly.

2. METHODOLOGYThe utilized observation data in this study is thegridded daily precipitation data of Japan, APHR_JP.
The original spatial resolution of the data is 0.05
degrees in both longitude and latitude, and it was
up-scaled into 0.2 degrees (around 20 km) and again
into 0.6 degrees (around 60 km).

Fig. 1 Location map of the sample data. Each region iscomposed of 9x9 grids of 0.2 degrees (around 20-km).

Fig. 2 Schematics of the downscaling target (20-km; r1… r9) using the surrounded grids (60-km; R1 … R9).

Fig. 3 Spatial correlation of the 9 grids (20-km) in thecenter with the surrounded grids (60-km) in the case ofApril and August of the region B in the Fig. 1. The
correlation values are based on 25 years of observation(1979-2003).

We selected three representative regions to test
the proposed method (Fig. 1). Each region iscomposed of 9x9 grids with 20-km resolution foreach grid, or in other words, 3x3 grids with 60-km
resolution (see Fig. 2). The target of downscaling is
the center of the subject regions.
The basic concept of the developed downscaling
method is to consider the spatial correlation patternof precipitation, and to estimate the amount of acertain 20-km gridded precipitation (for example, r1
in Fig. 2) based on its relationship with thesurrounding 60-km gridded precipitation amount(for example, R1, R2 … R9 in Fig. 2). To validate thebasic concept of the method, we checked spatial
correlation of rk (k=1…9) with the surrounding Rk(k=1…9) (result from region B is shown in Fig. 3).
Here, spatial correlation means the correlation ofdaily time series of precipitation.
As shown, rk (k=1…9) correlates most strongly
with R5, which covers its own area, then withadjacent grids with certain tendencies. In region B,
South-East directions show higher correlation thanNorth-West directions, and the April precipitation
pattern shows higher spatial correlation than theAugust one. These tendencies reflect regional,
seasonal, and topographic characteristics, which
would be sophisticated and variant in many cases.

I_902

Rather than analyzing the tendencies directly, we
are focusing on generalizing a relationship thatnaturally includes the variant spatial correlation
pattern and thus can easily apply to different regionswith the same format.
3. FORMATTED REGRESSION FRAME
Considering the correlation of precipitation, we
can estimate daily precipitation amount rk,i on acertain 20-km grid k, on a certain day i, using thesurrounding 60-km gridded precipitation amount
R1,i, R2,i … R9,i and a multi-variable regression
equation like Eq. 1.
rk,i = Ck,1R1,i + Ck,2R2,i + … + Ck,9R9,i + Ɛk,i

results for every month and every region (see thebolded number in the Table 1).
The summer season, especially August, shows
poorer efficiency than the rest of the year. But evenin August, we successfully estimated the parameter,as all nine grids averaged RMSE of 0.26 mm/day
for region A, 0.35 for region B, and 0.17 for region
C. In the case of correlation coefficients (CC), every
region shows very high correlation, even in August,as the 9 grids averaged CC of 0.95 for region A,
0.94 for region B, and 0.98 for the region C.
Table 1. Estimation efficiency of the Region B(1)Here, Ck,1… Ck,9 are the regression coefficients for thetarget grid k, and Ɛk,i is the regression residual. To
estimate the regression coefficients, we can gather ndays of data and reformulate the equations as a matrix,
as shown in Eq. 2 or simple form as Eq. 3.
R2,1  R9,1  k ,1 R2, 2  R9, 2   Ck ,1   k , 2 
    Ck , 2     
R2,i  R9,i       k ,i     Ck ,9     R2,n  R9,n  k ,n  rk ,1   R1,1
r   R
 k , 2   1, 2
     
 rk ,i   R1,i      rk ,n   R1,n

Z  Ax  v(2)

RMSE (unit: mm/day)
June
AugustOctober

Grids

April

r1
r2
r3
r4
r5
r6
r7
r8
r9

0.10
0.19
0.10
0.08
0.04
0.10
0.09
0.13
0.10

0.11
0.21
0.15
0.15
0.08
0.12
0.14
0.17
0.15

0.31
0.55
0.41
0.33
0.23
0.34
0.35
0.31
0.27

0.09
0.19
0.16
0.09
0.05
0.09
0.09
0.13
0.12

MMP*

109

179

265

146

*Monthly Mean Precipitation (MMP; mm) of the R5
grid for the 25 years is shown in the table toconsider the relative magnitude of the RMSE values.
(3)

4. RESULTS AND DISCUSSIONS
Here, Z is the matrix with the n days of rk values, A is
the matrix with the n days of R1…9, x represents theregression coefficient C1…9, and v is the residuals.
Our remaining job is to estimate the x that is
minimizing v, and it can be rewritten as Eq. 4, which isa typical form of the square root information filter(SRIF).
nJ ( x)   vi  Z  Ax  Z  Ax 2

T

We evaluated the regression model on its
applicability with the validation period of
1994-2004 by comparing simulated (downscaled)results with observed (true) values, both directly andby statistical characteristics.
Table 2. Simulation efficiency of the Region B

(4)

i 1

We estimated regression coefficients (orparameters) on a monthly basis using the household
transformation technique on the calculation matrix
for the observation data of 15 years (1979-1993),
and the developed regression model was applied forthe other 10 years of data from 1994 to 2003.
The efficiency of the parameters estimation waschecking with the root mean square error (RMSE),
and the RMSE was calculated including 0 mm/day
value. Table 1 summarizes some of the efficiency
values from region B, and shows successful
estimation results with very small RMSE values.
The most centered grid, r5, generally shows the best

Grids

April

r1
r2
r3
r4
r5
r6
r7
r8
r9

1.81
3.33
1.76
1.50
0.78
1.83
1.62
2.30
1.83
109

MMP*

RMSE (unit: mm/day)
June
AugustOctober
2.01
3.74
2.73
2.73
1.37
2.09
2.45
2.92
2.74
179

5.55
9.63
7.29
5.91
4.11
5.92
6.09
5.49
4.80
265

1.63
3.32
2.83
1.64
0.86
1.62
1.56
2.22
2.04
146

* Monthly Mean Precipitation (MMP; mm) is shownas the Table 1.

I_903

Region A

Region B

Region C

Fig. 5 RMSE of the SDS results for the observed values
for the region A (top), B (middle) and C (bottom).

Fig. 4 Scatter plot of simulated 20-km daily precipitationcompared with true values for region B, r5 (the center
grid) for April (top), which is one of the best results, andAugust (bottom), which is one of the worst results.

Figure 4 shows the representative results forregion B, r5 (the most centered grid) for April andAugust. In the figure, results for April shows quite
successful results with 0.78 mm/day of RMSE, andthe results from the August show rather dispersed
pattern (RMSE: 4.11 mm/day).
Table 2 and Fig. 5 show summarized results forother seasons and regions. Table 2 shows therepresentative RMSE from the region B with thevariation of each grid. In Table 2, the most centered
grid shows the best results for every month, but
region A of Figure 5 trends differently. Efficiencyof the regression model introduced here is largely
affected by correlation with the adjacent area’s
precipitation pattern. Some grids or regions might
have a good pattern relationship with theprecipitation of surrounding areas, making theregression model work better for them than others,
due to particular topographic shape and/or
atmospheric behavior of the region, and vice versa.

August shows the worst results in most cases.
One reason for this is the poor spatial correlation ofthis month as shown in Fig. 3 (right). Another is thelarge amount of precipitation in August, which is in
the middle of summer in Japan. As shown in Table
2, the RMSE magnitude is correlated with themonthly precipitation amount. Lastly, the pattern of
the RMSE from the simulated results (Table 2) isvery similar to the pattern of the parameter
estimation efficiency (Table 1). In other words, thesimulation efficiency in RMSE can be imaginedwith the parameter estimation efficiency.
In this study, we have developed an SDS method
to downscale 60-km precipitation information into
20-km resolution on a daily basis. Simulation results
do not need to match observation values perfectly. Ifthey are good enough to show the statisticalcharacteristics of the observation, we can approve
the proposed regression model. To evaluate thesestatistical characteristics, we have checked thefrequency of rainfall intensity with histograms (Fig.
6), and error percentage of monthly precipitation
amount (Fig. 7). The rainfall intensity interval in
Figure 6 is basically 2 mm/day except for the firstbar of the histogram with the percentage of the 0mm precipitation amount.

I_904

Region A

Region B

Region C
Fig. 6 Histogram of the SDS 20-km daily precipitation
amount comparing with the true values, for region B, r5(the center grid) for April (top) and August (bottom).
Note that the first bar is the frequency of 0 mm/day, andthe others are the frequencies of each 2 mm/day interval.

In Figure 6, the results from August show a very
good match with observed rainfall intensities interms of frequency, and the results from April also
shows good match except for frequencies lower than
2 mm/day. This is an encouraging result, but thediscrepancy of frequency under 2mm/day can makea significant difference in accumulated values over along duration of simulation. Thus, we surveyed theerror ratio of monthly precipitation and summarized
the results (Fig. 7). For a downscaled grid i, for acertain month j, the error ratio Eij is
Eij 

S ij  Oij
Oij

 100(%)

Fig. 7 Error percentage of monthly precipitation amounts
for the region A (up), B (middle) and C (down).

Finally, representation of spatial pattern was
evaluated with correlation of the simulated monthly
mean precipitation with the observed one in thedownscaled 9 grids. As shown in Table 3, the spatial
pattern is successfully well represented in thedownscaled results with high correlation coefficient.
Table 3. Correlation coefficients of simulatedmonthly mean precipi. with the observation(5)

where, Sij is the simulated monthly precipitation
amount and Oij is the observed one, during theexamination periods (1994-2003).
The normalized error ratio with the monthly
precipitation amount does not depend on season ormonth, and most of it is within ±10%. Morespecifically, the error ratio Eij is showing a normal
distribution around zero and the standard deviations
are 4.4%, 5.4% and 5.1% for regions A, B and C,
respectively. Surely there is much room for
improvement, but this is a plausible result when we
consider the simplicity of the proposed method.

I_905

Month

Region A

Region B

Region C

Jan.
Feb.
Mar.
Apr.
May.
Jun.
Jul.
Aug.
Sep.
Oct.
Nov.
Dec.

0.828
0.912
0.992
0.944
0.973
0.963
0.955
0.944
0.964
0.973
0.971
0.932

0.998
0.993
0.995
0.984
0.983
0.964
0.917
0.963
0.989
0.978
0.982
0.985

0.925
0.929
0.959
0.976
0.988
0.941
0.946
0.981
0.970
0.974
0.940
0.963

5. SUMMARY AND
FURTHER RESEARCHESAn SDS technique is proposed with a formalized
regression frame considering spatial correlation
structure of precipitation. Downscaling windows in
a 3x3 60-km grid were considered to downscale thecentered 60-km grid into 9 grids of 20-km spatial
resolution. For the evaluation of the proposed
method, we used 25 years of gridded observation
data to calibrate regression model parameters(1979-1993) and evaluate downscaled results(1994-2004). We optimized the parameters onmonthly basis, and found that the regression model
provides very effective and efficient results except
for the summer season in terms of root mean squareerror. However, statistical characteristics, such as
the frequencies of rainfall intensities, show stable
patterns for every season and region, with a certain
level of mismatch. Overall there was around ±5%
margin of error in the monthly precipitation amount.
The proposed SDS technique can be applied to
the 60-km AGCM of MRI (officially, MRI-AGCM
3.1H & 3.2H) to downscale it into 20-km spatial
resolution based on the analysis of the original
20-km resolution AGCM (MRI-AGCM3.1S &3.2S). Application method can be 1) upscaling theoriginal 20-km AGCM output into 60-km
resolution, 2) estimating the regression parameters
for both the present and future climate using theoriginal 20-km one and the upscaled 60-km one, 3)
applying the estimated parameters to the original
60-km ensemble output from the MRI-AGCM3.1H& 3.2H and downscaling them.
However, statistical characteristics and spatial
pattern can differ between the original 60-km
AGCM output and the upscaled one from the 20-km
AGCM output. One simple solution is to modify theoriginal 60-km data to have the characteristics of the
upscaled one. In this case, we need an additional
consideration while we handle the ensemble outputof the 60-km AGCM: to preserve the original
ensemble characteristics.
Secondly, a certain estimation bias from the
proposed SDS method should be carefully
considered. This may not be avoidable, however,information about the bias range and amount should
be checked before it is applied to the secondary
usage such as for impact assessment research.
For more generalized application of the proposed
method, we are now considering some variation of
the method. For example, the SDS method
illustrated here can be modified to apply to different
spatial and different temporal resolution of data. The
downscaling window (3x3) can be modified forother dimensions, such as (5x5). Finally, theproposed method should be evaluated for its
applicability to different regions and different
climate conditions.

ACKNOWLEDGMENTS
This work was done under the framework of the“Precise impact assessments on climate change”
SOUSEI Program of MEXT, Japan and supported by
the Wakate-B of JSPS (Project No: 23760459).
REFERENCES
1) Fowler, H., Blenkinsop, S. and Tebaldi, C.: Linking climatechange modelling to impacts studies: recent advances indownscaling techniques for hydrological modelling,
International Journal of Climatology, Vol. 27, pp.15471578, 2007.
2) Charles, S., Bari, M., Kitsios, A. and Bates, B.: Effect of
GCM bias on downscaled precipitation and runoff
projections for the Serpentine catchment, Western Australia,
International Journal of Climatology, Vol. 27, pp.16731690, 2007.
3) Taylor, K.E., R.J. Stouffer, G.A. Meehl: An Overview of
CMIP5 and the experiment design, Bulletin of the American
Meteorological Society, Vol. 93, pp. 485-498, doi:10.1175/
BAMS-D-11-00094.1, 2012.
4) Kitoh, A., Ose, T., Kurihara, K., Kusunoki, S., Sugi M. and
KAKUSHIN Team-3 Modeling Group: Projection ofchanges in future weather extremes using super-highresolution global and regional atmospheric models in the
KAKUSHIN Program: Results of preliminary experiments,
Hydrological Research Letters, Vol. 3, pp.49-53, 2009.
5) Kusunoki, S., Mizuta, R. and Matsueda, M.: Future changes
in the East Asian rain band projected by global atmosphericmodels with 20-km and 60-km grid size, Climate Dynamics,
doi:10.1007/s00382-011-1000x, 2011.
6) Kim, S., Tachikawa, Y., Nakakita, E. and Takara, T.:
Hydrologic evaluation on the AGCM20 output using
observed river discharge data, Hydrological Research
Letters, Vol. 4, pp.35-39, 2010.
7) Kim, S., Tachikawa, Y., Nakakita, E., Yorozu, K.. andShiiba, M.: Climate change impact on river flow of the tone
river basin, Japan, Journal of Japan Society of Civil
Engineers, Ser. B1 (Hydraulic Engineering), Vol. 67, No. 4,
pp.I_85-I_90, 2011.
8) Wilby, R.L. and Dawson, C.S.: The statistical DownScaling
Model (SDSM): Insights from one decade of application,
International Journal of Climatology, Vol. 33, pp.17071719, 2013.
9) Schmidli, J., Goodess, C.M., Frei, C., Haylock, M.R.,
Hundecha, Y., Ribalaygua, J. and Schmith, T.: Statisticaland dynamical downscaling of precipitation: An evaluationand comparison of scenarios for the European Alps, Journalof Geophysical Research, Vol. 112, doi: 10.1029/
2005JD007026, 2007.
10) Wilks, D.S. and Wilby, R.L.: The weather generation game:a review of stochastic weather models, Progress in PhysicalGeography, Vol. 23 pp. 329-358, 1999.
11) Fowler, H.J., Kilsby, C.G., C’Connell, P.E. and Burton, A.:A weather-type conditioned multi-site stochastic rainfall
model for the generation of scenarios of climatic variability
and change, Journal of Hydrology, Vol. 308, pp. 50-66,
2005.
12) Wilby, R.L. and Wigley, T.M.L.: Downscaling generalcirculation model output: a review of methods andlimitations, Progress in Physical Geography, Vol. 21 pp.
530-548, 1997.
13) Wilby, R.L., Hay, L.E., Gutowski Jr., W.J., Arritt, R.W.,
Takle, E.S., Pan, Z., Leavesley, G.H. and Clark, M.P.:
Hydrological responses to dynamically and statisticallydownscaled climate model output, Geophysical Research
Letters, Vol. 27, pp. 1199-1202, 2000.

I_906(Received September 30, 2013)