京 都 大 学 防 災 研 究 所 年 報 第 48 号 B 平成 17 年 5 月
Annuals of Disas. Prev. Res. Inst., Kyoto Univ., No. 48 B, 2005

Rainfall Prediction Considering Error Structure of Predicted Rainfall

Sunmin KIM*, Yasuto TACHIKAWA, and Kaoru TAKARA*Graduate school of Urban and Environment Engineering, Kyoto University

Synopsis
Rainfall field is predicted using radar observation translation vector model, andprediction error structure is analyzed with the predicted and observed rainfall fields. With
the analyzed error characteristics, random error fields are simulated by covariance matrix
decomposition method. The simulated random error fields successfully contain the errorstructure. The simulated error fields offer prediction reliability in the form of error range
which is formulated by standard deviation of the generated error values on each grid.
Keywords: Radar rainfall prediction, Error structure, Random error field

1. Introduction

analyzed with past prediction results and effective
error factors are found, the error structure would beCurrently, there are many hydrological models

very useful to the present rainfall prediction. The

which can properly simulate basin characteristic if

analyzed error structure can be used for improving

parameters of the model are optimized to a subject

prediction methods or for figuring out reliability ofbasin. In the case, accuracy of the simulation results

the predictions. Whatever the error structure is for,which usually represent as a discharge hydrograph isimportant point to consider is that the prediction

mostly controlled by accuracy of rainfall data. This

rainfall error has temporal variance and also spatial

important roll of input rainfall data becomes morevariance.

severe when the simulation is on real-time basis andIn this research, radar rainfall is predicted using

so the input data is predicted rainfall data. But, it istranslation vector model, and error structure of thestill hard to say that the accuracy of the predictedpredicted rainfall field is analyzed to get its statistics

rainfall is high enough to give confidence to theandsimulation results.

prediction error fields are simulated as a spatially

spatialcorrelation

coefficient.

AndthenWhen we take into consideration of uncertainty incorrelated random error field according to thepredicted rainfall data, it would be useful to provide

analyzed error structure. The final purpose of thisaccuracy of the predicted rainfall. During short-term

research is to produce extended prediction rainfall

rainfall prediction with any prediction model, if anfield which offer prediction with its reliability in theerror structure of predicted rainfall is properly

form of probable error range.

2. Prediction and Prediction Error
2.1 Translation Model
Translation model (Shiiba et al., 1984) is used forshort-term radar rainfall prediction. In the translation
model, the horizontal rainfall distribution, z(x,y,t)with the spatial coordinate (x,y) at time t is modeled
as:

∂z
∂z
∂z+u +v = w
∂t
∂x
∂ydx
dy
here, u =, v=,dt
dt

(1)

w=dz
dt

Figure 2 shows statistics of the prediction error.
Average of the error is distributed around zero while
standard deviation varies as time pass. The variationof standard deviation has a relation with rainfall
intensity when it is compared with Figure 1. If there
is a correlation between the standard deviation of theerror and the prediction rainfall intensity, theprediction reliability can be estimated in the form ofstandard deviation of the prediction error. Frequency
distributions of Ea show a normal distribution pattern(Figure 3). Here, the statistics and the distributions
are calculated only with non-zero prediction error.

where, u and v are advection velocity along x and y,
and w is rainfall growth-decay rate. The translation
model assumes u, v, and w are having the forms:

u = c1 x + c 2 y + c 3v = c 4 x + c5 y + c 6
w = c 7 x + c8 y + c 9(2)The values of parameters c1, c2, …, c9 are
identified by the least square method using observed

Fig. 1 Observed and predicted rainfall intensities

radar rainfall data and they are updated on real-timebasis. In this research, three observed rainfall fields,
which have 3 km and 5 minutes of observe resolution,
are used to determine u and v. When forecasting
rainfall fields, the u and v are assumed to bespatially constant, but vary temporally. The growthdecay rate w is always assumed to be zero.
Radar rainfall event used here is observed at
Miyama radar station of Kinki area on 1990/9/12 and9/13. Observation is available with 5 minutes interval,
and prediction is done at every 5 minutes for one hourFig. 2 Statistics of prediction error (1hr prediction)ahead. Average rainfall intensity of observed one andprediction one are shown in Figure 1.
2.2 Error Structure Analysis
After 1hr prediction is done by the translation
model, absolute prediction error Eai on grid i is
calculated from difference between predicted rainfalls
Rpi and observed rainfalls Roi on the grid i.
Eai = Roi-Rpi(3)
Fig. 3 Frequency distribution of the Ea

Then a covariance matrix of the vector Y would besame as a square of the matrix B. (Tachikawa et al.,
2003)
E[YYT] = E[BxxTBT] = BE[xxT]BT = BBT(4)

After getting the matrix B from the BBT using
matrix decomposition method, the error field vector Y
can be simulated with the matrix B and the random
value vector x.
Fig. 4 Spatial correlation coeff variance along distance

Figure 6 shows example of simulated random error
field simulated by the covariance matrix
decomposition method. By simulating many random
error fields, probable error range of the prediction oneach grid can be generated as shown in Figure 7.
Statistics and spatial correlation coefficients for thesimulated random error field is using the values of
14:00 on 13th Sep. Averages and standard deviationsof the Figure 7 come from one hundred simulated
random error fields when “-0.5993” for average and“7.7240” for standard deviation are given to therandom error field simulation.

Fig. 5 Spatial correlation coefficient variance along time

Figure 4 and Figure 5 show spatial correlation
coefficients from the prediction error fields. It shows
how much the prediction error is spatially correlated
to each other. Each coefficient is calculated bygrouping each pairs of error which apart one grid for
3km, two grids for 6km, etc. The coefficient variancealong distance (Figure 4) shows that effective rangeof the correlation is within 10 km. The coefficientvariance along time (Figure 5) shows it also has arelation with rainfall intensity. The spatial correlation
coefficients also can be estimated by using predicted
rainfall intensity if we need to consider its timevariance.

Fig. 6 Example of simulated random error field

3. Producing Prediction Reliability Range
3.1 Random Error Fields Simulation
Matrix decomposition method is used to simulaterandom error fields which contails the analyzed error
structure. Let’s assume predicted error field vector Y
is composed of a matrix B, which has a spatial
correlation characteristic, and a random value vector
x; Y=Bx. Here, the random values x are uncorrelated
each other but follows a given probability distribution.

Fig. 7 Average and standard deviation on each gridcalculated from 100 simulated random error fieldObserved Rainfall
Predicted Rainfall
Prediction error range

Fig. 8 Input and simulated values of Average and StDev

Fig. 10 Observed and prediction rainfall with its error range

Main problem is variant amount of prediction error
according to different rainfall intensity. While thesimulated random error fields make uniform error
range over the prediction rainfall field, high amountof prediction error is found at high rainfall intensity
area and low amount or no prediction error is found at
low rainfall intensity area. Furthermore, the uniform
prediction error range would be applied to no rainfall
area and it makes unnecessary error range. AnotherFig. 9 Input and simulated values of spatial corr. coeff.

problem which is also related to uniform error range
is minus rainfall values of extended prediction
rainfall fields. Even though minus values are

As shown in the Figure 8, each random error field
is successfully reflecting input statistic values. Also
Figure 9 shows the generated one hundred random
error fields successfully reflecting input spatial
correlation coefficient data.
3.2 Prediction Reliability Range

unavoidable when we consider the possible error
range, minus rainfall is not acceptable physically andit may cause malfunction if this extended prediction
rainfall fields are applied to hydrologic model for
runoff simulation.
4. Conclusion and Further Research

Deterministic prediction rainfall field from thetranslation model would be extended to probableprediction rainfall fields by joining with the simulated
random error fields. The error range generated bymany error fields would offer prediction reliability
range in the form of standard deviation of theprediction errors. Successfully generated prediction
error range is supposed to include observed rainfall
field in the range.
Several problems occur during the extended
prediction field generation procedure. Figure 10
shows schematic drawing of one rainfall field sectionof prediction with its error range and observed
rainfall intensity.

In this research, radar rainfall field is predictedusing the translation vector model, and prediction
error structure is analyzed spatially and temporallywith the predicted and observed rainfall fields. The
analyzed error structure includes average andstandard deviation of the prediction error, probability
distribution pattern, and spatial correlation of theerror. Then random error fields are simulated by acovariancematrixdecomposition

method.

The

simulated random error fields successfully reflect theanalyzed error structure.
The simulated random error fields offer prediction

reliability in the form of error range which isformulated by standard deviation of the error values
on each grid. But, the error range producing method
introduces several problems to be considered. These
problems are;a. Different error range according to rainfall intensityb. Uniform error range applied to no rainfall areac. Minus rainfall values caused by error range
There are possible solutions to the uniform error
range problem. First one is using accumulated rainfall
field information. As time pass and as prediction error
is accumulated, the accumulated error field shows apattern rather than randomly distributed. There iscertain area which has plus or minus value by theaccumulated prediction error as shown in Figure 13Fig. 12 Accumulated observed rainfall at 24:00 on 12th Sep.

which means those area has been plus or minus
prediction error continuously. Correlation coefficientof accumulated error field (Figure 13) and each error
field (Figure 14) has rather low value as shown in
Figure 11, the accumulated error pattern would be
useful to give flexibility to the prediction error range.
Another possible solution to the uniform error
range problem is to calculate each statistics on eachgrid using prediction error time series of its own grid.
In that case the variant error range according torainfall intensity would be automatically simulated.
Also unnecessary error range on no-rainfall area
would not happen if that area does not have predicted
rainfall and prediction error.

Fig. 13 Accumulated prediction error at 24:00 on 12th Sep.

The extended prediction by random error field
introduced here can offer prediction reliability in theform of probable error range.

Fig. 11 Correlation coefficient of prediction error andaccumulated prediction error on each time step

Fig. 14 Prediction error at 24:00 on 12th Sep.

M. (2003) Stochastic modeling of the error
structure of real-time predicted rainfall andrainfall field generation, in Weather Radar
Information and Distributed HydrologicalModelling (ed. by Y. Tachikawa, B. E. Vieux, K.
P. Georgakakos & E. Nakakita), IAHS Publ., no.
282, pp. 66-73.

References
Shiiba, M., Takasao, T. & Nakakita, E. (1984)
Investigation of short-term rainfall prediction
method by a translation model, Proc. 28th Annual
Conf. on Hydraulics, JSCE, 423-428.
Tachikawa, Y., Komatsu, Y., Takara, K. & Shiiba,

予測の誤差構造を考慮した降雨予測
キム，スンミン*・立川康人・寶馨*

京都大学大学院工学研究科都市環境工学専攻
要旨

移流モデルとレーダー観測値を用いて降雨を予測し，予測した降雨場の予測誤差構造を分析した。次に，予測誤差の空間場を確率場として捉え，分析した誤差特性に従って，共分散行列を分解する手法により誤差の空間場を生成した。模擬発生した誤差の空間場は，実際の予測誤差構造を適切に反映している。この手法は，グリッドごとに誤差分散として予測の信頼度を提供している。

キーワード: 実時間流出予測, 状態更新, 分布型流出モデル