Annual Journal of Hydraulic Engineering, JSCE, Vol.50, 2006, February

ENSEMBLE RAINFALL-RUNOFF PREDICTION WITH
RADAR IMAGE EXTRAPOLATION AND
ITS ERROR STRUCTURE
Sunmin KIM 1, Yasuto TACHIKAWA2, Takahiro SAYAMA3 and Kaoru TAKARA4
1Student Member, Graduate student, Dept. of Urban and Environmental Eng., Kyoto University(Kyoto 606-8501, Japan) E-mail:sunmin@rdp.dpri.kyoto-u.ac.jp
2Member of JSCE, Dr. Eng., Associate Professor, DPRI, Kyoto University (Gokasho, Uji 611-0011, Japan)
3Member of JSCE, M. Eng., Assistant Professor, DPRI, Kyoto University (Gokasho, Uji 611-0011, Japan)
4Fellow of JSCE, Dr. Eng., Professor, DPRI, Kyoto University (Gokasho, Uji 611-0011, Japan)

New attempt of ensemble rainfall-runoff prediction is presented with radar rainfall prediction andspatial random error field simulation. A radar image extrapolation model gives deterministic rainfall
predictions, and its prediction error structure is analyzed by comparing with the observed rainfall fields.
With the analyzed error characteristics, spatial random error fields are simulated using covariance matrix
decomposition method. The simulated random error fields successfully keep the analyzed error structure
and improve the accuracy of the deterministic rainfall predictions; then the random error fields with thedeterministic fields are given to a distributed hydrologic model to achieve an ensemble runoff prediction.
Key Words: radar rainfall prediction, error structure, random error field, ensemble simulation

1. INTRODUCTION
Nowcasting of precipitation is an essential
prerequisite for real-time flood forecasting inoperational hydrology. The term “nowcasting” isused to emphasize the specificity and shortness(0~3hrs) of rainfall forecast largely by radar image
extrapolation, which has a history of nearly 50
years1). Even though the benefits of short-term
precipitation forecasts are well known, this is
acknowledged to be among the most challenging
areas in hydrology and meteorology2).
The early stage of the extrapolation methods
takes the simple techniques in the extremes. It does
not usually allow for the growth and decay of therainfall intensities or nonlinear motion of the rainfall
band. Main shortage of the techniques is that mainly
because of its simplicity, the forecast accuracy
decreases very rapidly within an hour3). In a study ofthe forecasts accuracy improvement, elaborate
nonlinear extrapolation schemes only give
negligible improvement or even worse results thanlinear position extrapolation4).
While the simple extrapolation techniques havebeen the mainstream of nowcasting until recently,
vast research efforts by many hydrologists andmeteorologists have been done over several decadeswith introductions of many new schemes. These
new schemes include mathematical and stochastic
models integrated with a meteorological component
5),6)
and hybrid models, which are combination of
Numerical Weather Prediction (NWP) models andimage extrapolation models7),8). In addition, there
have been complex statistical approaches like withfractal generation algorithms 9) or artificial neuralnetworks10).
Even with the vast research efforts, it has been
hard to find a notable improvement of the forecast
accuracy from the chaotic nature. As an alternative
to the accuracy improvement, numerical models
have been run in an ‘ensemble mode’ for the last
decade or so. Statistical outcome is obtained fromthe ensemble simulations mostly by adopting small
perturbation in the initial values and initial condition
11). The ensemble forecast of hydrographs is also a
recent trend away from the conventional simple
deterministic forecasts of hydrographs toward
offering probabilistic forecasts, which include its
prediction uncertainty.
As a step towards addressing the forecast
accuracy improvement and ensemble hydrologic
prediction, this study introduces a new attempt ofensemble rainfall-runoff simulations with a spatial
random error field (SREF). Different to a

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conventional ensemble simulation, which uses
initial condition control to obtain statisticaloutcome, the SREF offers probable prediction error
fields for the deterministic prediction results of arainfall-forecasting model. The characteristics of
SREF is based on an analyzed error structure of thespatial rainfall prediction, and it is simulated using
the random field generation method12).
The organization of this paper is as follow:
section 2 illustrates the deterministic prediction of
rainfall field using a radar extrapolation model. The
prediction error fields are examined to extract thecharacteristics of prediction error structure, and the
SREF is simulated based on the analyzed error
structure. In section 3, probable extended prediction
fields are formulated using the deterministic rainfall
filed and the SREF. The extended prediction fields
offer ensemble rainfall-runoff simulations with a
distributed hydrological model. Discussions are on
the accuracy of the rainfall prediction and its
propagation to the discharge hydrograph through the
distributed hydrological model in section 4.

2. SHORT-TERM RAINFALL FORECAST
AND RANDOM ERROR FIELD(1) Radar Image Extrapolation Model
The translation model by Shiiba et al.14)
simulates short-term radar rainfall predictions in thisstudy. In the model, the horizontal rainfall intensity
distribution, z(x,y,t) with the spatial coordinate (x,y)
at time t is modeled as:
∂z
∂z
∂z
(1)
+u +v = w
∂t
∂x
∂y
dx
dy
dz
here, u =, v= , w=
dt
dt
dt
where, u and v are advection velocity along x and y,
and w is rainfall growth-decay rate. The vector u, v,
and w are identified by the least square method
using observed radar rainfall data and are updated
on a real-time basis. Shiiba et al.14) and Takasao etal.15) give more details of the translation model andthe function of the vector u, v, and w.
In this research, three consecutive observed
rainfall fields, which have 3 km and 5 minutes ofresolution, are used to determine u and v. Whenforecasting rainfall fields, the u and v are assumed
spatially constant in each rainfall fields, but updated
on a real-time basis. The growth decay rate w isalways assumed to be zero.
Radar rainfall events used here are observed at
the Miyama radar station located in the central partof Kinki District, Japan on 30th and 31st June 1993.
Operated by the Japan Ministry of Land,
Infrastructure, and Transportation, the radar data isconverted to rainfall intensity covering a 240km×240km area with 3 km spatial resolution. A moredetailed description of the Miyama radar is given by
Nakakita et al.6).
(2) Prediction Error Structure Analysis
Tachikawa et al.16) statistically analyzed thecharacteristics of absolute error and relative error
defined as Equations 2 and 3.
(2)
E a ,i = Ro ,i − R p ,i
(3)
Er ,i = ( Ro ,i − R p ,i ) / R p ,iThe absolute prediction error Ea,i on the grid i is
calculated from the difference between predicted
rainfalls Rp,i and observed rainfalls Ro,i on the grid,
while the relative prediction error Er,i is the ratio of
the absolute prediction error to its predicted rainfall.
Tachikawa et al.16) examined the time accumulated
error values with variant spatial resolutions andfound that the distribution of absolute and relative
error are close to normal distribution and lognormal
distribution respectively. Here we concentrate on
the absolute prediction error Ea,i and simulate thespatial random error fields in real-time basis.
Figure 1 shows spatially averaged rainfall
intensities, on each time step, of the observed andpredicted for 30, 60, and 120 min. In order to avoid
an influence of non-rainfall area to the intensity
calculation, the calculation only includes those grids
in which rainfall was either predicted or observed
within the study area. Note that there are overall
delays of rainfall intensities as prediction time getslonger. Because the translation model used here
only represents the movement of the rainfall bands
without their growth or decay, the model assumes
the same amount of current rainfall intensities lasts
to the prediction target time. The slight differencesof the intensities are only because of new included
or excluded rainfall cells in the study area.
Two kinds of prediction performance measuring
values are adopted: Correlation Coefficient (CC) of
the observed and the predicted rainfall, and Critical
Success Index (CSI). The CSI is given as:
X(4)
CSI (%) =× 100
X +Y + Z
where, X is the number of correct forecasts rainfall
cells (i.e. rainfall is observed and also predicted in
the grid), Y is the number of misses (i.e. rainfall isobserved, but not predicted), and Z is the number offalse alarms (i.e. rainfall is predicted, but notobserved). The CSI is especially appropriate as asummary measure of forecast skill for the case of
extreme events, since the index value decreases withincreasing both the number of misses and false
alarms4). A threshold rain-rate for the CSI in thisresearch is 0.0 mm/hr within the radar range.
The CC values of the observation and prediction
are under 0.5 even for the 30 min prediction andworse for the 60 min and 120 min predictions (seeFig. 2). On the other hand, the CSI values are- 44 -

mostly over 50% and almost 100% for some times(see Fig. 3). The high CSI values are given by wide
spread and slowly moving rainfall bands of thesubject event. When a check of other events having
variant rainfall area size is conducted, the CSI
values are somewhat related to the size of rainfall
area.
Spatial Correlation Coefficients (SCC) of theprediction error, which shows how much the error is
spatially correlated to each other, for 60 min
prediction is presented in Figure 4. The SCC iscalculated by grouping each pairs of error values
which apart one grid for 3 km, two grids for 6 km,
etc. The SCC shows high values for short distanceand decrease with long distance. The frequency
distribution of the prediction error follows a normal
distribution as the same pattern of Tachikawa etal.16).

Fig.1 Average rainfall intensity of observation and prediction(3) Prediction Error Field Simulationa) Outline of the Error Field Simulation
The main part of this study is to simulatepossible error fields of the future prediction using
the current prediction error structure, assuming
temporal persistency of the error characteristics
from the current time to the prediction target time.
Figure 5 explains the procedure for possible error
field simulation. The current characteristics of theprediction error are represented in the mean andstandard deviation field of error (see Fig. 5(b) and(d)). The mean and standard deviation of the error
on each grid formulate the mean and SD field,
hereafter statistic fields, using certain duration ofrecent prediction error. The statistic fields convert
Unit Random Error Field (UREF; Fig. 5(c)) to thetarget error fields (Fig. 5(a)), which are the aim of
the error filed simulations. The UREF is a spatiallycorrelated random field with the N(0,1), and thespatial correlation for the UREF is determined by
the current error fields. Generation of many UREF
make it possible to get many target error fields.
Finally, the deterministic prediction field obtainedby the translation model is extended to many
possible prediction fields by combining with thesimulated prediction error fields.
b) Persistency of the Error Characteristics
To confirm the validity of the method, it wasnecessary to check how much the current statistic
fields are close to the prediction error fields on thetarget time. Tests were conducted with three sets ofstatistic fields calculated using 10 min, 30 min, and60 min of duration. After each statistic field was
prepared from the current time and prediction target
time, modified CSI values were calculated with eachpair of the statistic field. The modified CSI wasusing the same form of Equation 4 only except arange concept. If the mean (or standard deviation)

Fig.2 Correlation coefficients of observation and prediction

Fig.3 Critical success index (%) for observation and prediction

Fig.4 Spatial correlation coefficients of 60min prediction

value of future target time on a certain grid waswithin a certain range of the current mean (or SD)
value on that grid, it was counted as correct forecast.
If the values are out of range, it was counted as
wrong forecast.
Figure 6 shows the modified CSI values witheach pare of 10 min, 30 min, and 60 min statistic
fields with the range of ±3mm. The CSI was over
60% for the mean field and over 80% for standard
deviation field. With variant range, it was possible
to check that the current statistic fields were goodfor simulating the prediction error on the target time

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=×+(a) Prediction error field(b) StDev field of error(c) Unit random error field(d) Mean field of error
Fig.5 Simulation procedure of prediction error field using the statistic fields and UREF

Fig.6 Modified CSI (%) for statistic fields of prediction error

The duration for calculating statistic fields does notgive much effect to the CSI, thus the 30 min statistic
fields were used.
The high CSI values may come from thecharacteristics of rainfall distribution in highly
mountainous areas. Dynamics and microphysics ofatmosphere are highly related to the topography for
generation, growth, and decay of rainfall6). Because
the translation model does not account this specifictopography information, the prediction from themodel contains inevitable errors, which relate to
topographic information. The statistic fields made it
possible to compromise the spatial and temporal
pattern of these errors. Moreover, the information of
the current prediction error can be updated inreal-time basis with the statistic fields.
c) Prediction Error Field Simulation Algorithm
The UREF simulation uses the decomposition
method of a matrix that includes spatial correlation
characteristics of the prediction error. The matrix is
decomposed approximately into its square root
matrix with the matrix factorization technique12),13).
Multiplying the square root matrix by a random
vector N(0,1) gives a non-conditional simulation of
the UREF.
Davis13) proved a symmetric matrix B thatsatisfies K=BB could be found when K issymmetric and positive-definite. Considering therandom vector Y; in this study, spatially correlated
unit random error vector:
Y=Bw
where w is uncorrelated random vector N(0,1). The
expected value of the matrix YYT (n×n) is given by
E[YYT] = E[BwwTBT] = BE[wwT] BT
Because w is a vector of independent random
numbers, E[wwT] = I, thus

E[YYT] = B I BT= K
A covariance matrix K is made out by using the
spatial correlation coefficient, which is obtained
from the absolute error Ea,i. The matrix K is
decomposed into the symmetric positive-definite
matrix B approximately by the Chebyshev
polynomials12),16). A vector Y (or UREF), which is a
non-conditional simulation of spatially correlated
random vector, can be generated by multiplying B
by an uncorrelated random vector w. Figure 5(c)shows one example of UREF.
The statistic fields convert the UREF to theprediction error fields as Equation 5:⎡ E S ,1 ⎤ ⎡ sd1 0 L 0 ⎤ ⎡ y1 ⎤ ⎡ m1 ⎤⎢E ⎥ ⎢⎥⎢ ⎥ ⎢ ⎥⎢ S , 2 ⎥ = ⎢ 0 sd 2 L 0 ⎥ ⎢ y 2 ⎥ + ⎢m2 ⎥ (5)⎢ M ⎥ ⎢ M
M O M ⎥⎢ M ⎥ ⎢ M ⎥⎢⎥ ⎢⎥⎢ ⎥ ⎢ ⎥0 L sd n ⎦ ⎣ y n ⎦ ⎣mn ⎦⎣ E S ,n ⎦ ⎣ 0Here, the mi and sdi are the mean and standard
deviation of the current prediction error on the gridi. The yi is the unit random error of the vector Y, andthe Es,i is the simulated error for the prediction target
time. Equation 5 is linear equation, thus thecorrelation structure of Y, which is obtained fromthe Ea,i, is maintained in the Es,i. The form ofEquation 5 is identical with Figure 5. The total grid
number of the Miyama radar image is 80×80, thus
the n of Equation 5 is 6400. This simple procedureallows each random error value yi to have statisticalcharacteristics of its grid. Fifty sets of prediction
error fields for each time step were generated.

3. ENSEMBLE RAINFALL-RUNOFFSIMULATION(1) Generation of Extended Prediction Fields
Deterministic prediction rainfall field from thetranslation model is extended to many possible
prediction rainfall fields by combining with thesimulated prediction error fields as:(6)
Re ,i = R p ,i + E s ,iwhere, Es,i is the simulated prediction error value on
grid i, Rp,i is the prediction from the translation
model, and Re,i is the extended prediction. Because
the simulated prediction error keeps the error
statistics of the absolute prediction error (Es,i≈Ea,i),- 46 -

the extended prediction can be close to the observed
rainfall on the prediction target time as:(7)
Re ,i ≈ Ro ,i = R p ,i + E a ,iIn other words, the properly simulated prediction
error improves the forecast accuracy of thedeterministic prediction.
Negative values could occur on the extended
prediction fields, because the values on some pointsof the prediction error field could have larger minus
value than the predicted rainfall value on that point.
These minus rainfall set to zero, and the sameamount of minus values compensate from the plus
rainfall values for keeping the total rainfall amountas:
R’e,i= Re,i(1+r)(if R e,i≥0.0)
(8)
= 0.0
(if R e,i<0.0)
MinusRe ,ihere, r = ∑∑ PlusRe,iThe value r in Equation 8 stands for the ratio of total
minus rainfall amount on each extended prediction
field to the total plus rainfall amount on it. The total
amount of minus rainfall generally has 10% to 20%of the total plus rainfall amount, and the value r
varies from -0.1 to -0.2.
The same error structure analysis described at
2.(2) is carried out for the extended prediction
fields, and the result is compared with the originalerror structure of its corresponding time. Figure 7
shows the rainfall intensities of observation,
deterministic prediction, and fifty sets of theextended predictions. The CC values in Figure 8 andCSI in Figure 9 show each value from the extended
prediction fields. Average values (Fig. 10) and thestandard deviation (Fig. 11) of prediction error, andSCC (Fig. 12) of the simulated prediction error
show that the error characteristics were successfully
kept through the whole procedure.

Fig.7 Average rainfall intensities

Fig.8 Correlation coefficients

Fig.9 Critical success index (%)

Fig.10 Average of prediction error(2) Runoff Simulation Results and Discussion
The fifty sets of simulated extended prediction
fields offer ensemble rainfall-runoff simulation witha distributed hydrological model. The model thatwas used here is the Yodo-river model using
Ohymos system17). The simulation is carried out for
two different catchments located in the range of the
Miyama radar; Ootori (156 km2) and Ieno (476km2).
The fifty extended prediction fields for fiftyensemble simulations were given at every 5 minduring 6:00 and 18:00 on 30th Jun, 1993. Eachextended prediction field was assigned to eachrunoff simulation independently. After 18:00 on
30th, the deterministic prediction was set to every
fifty-ensemble simulations and continues thesimulation for a while to observe the remaining
effect of the extended prediction fields on therunoff.

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Fig.11 Standard deviation of prediction error

Fig.12 Spatial correlation coefficientFig.13 Rainfall-runoff Simulation Results for Ootori (156km2)

Fig.14 Rainfall-runoff Simulation Results for Ieno (476km2)

The discharges from the extended rainfall
prediction were compared with the discharges fromthe original deterministic rainfall prediction torealize stochastic ensemble discharge simulations.
In addition, we regarded the simulated dischargeusing the observed radar rainfall as the reference
discharge to validate the improvement of theprediction accuracy by the extended prediction
fields. Figures 13 and 14 show the ensemble runoff
simulation results for two subject basins. The thicksolid line stands for the discharge from the observed
radar rainfall, and the other solid line is thedischarge results from the original deterministic
predicted rainfall for 60 min ahead at every time
step. In the both simulation results, the discharges
from the ensemble simulations (fifty dot lines) were
not showing noticeable improvements. However,this results shows reliability of the prediction resultswith possible discharge bands.
It needs more study to validate carefully thereliability band of the ensemble simulation results
using variant prediction lead-time. The event used inthis research has rather low rainfall intensities
compare to other events. Other rainfall event, which
has higher rainfall intensities, for example typhoon
event, need to be examined for checking the validityof the extended prediction field.

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4. CONCLUSION
In this study, the translation model predictedradar rainfall field, and the prediction error structure
was analyzed spatially and temporally. The random
error fields were simulated using the error structure,and the extended rainfall prediction fields weregenerated. The extended prediction fields not only
improved the accuracy of the original prediction by
the translation model but also gave reliability withvariant form of rainfall fields. The results ofrainfall-runoff simulation with the distributedhydrologic model verified the hydrologic usefulnessof the extended prediction.
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