HYDROLOGICAL PROCESSES
Hydrol. Process. 23, 597– 611 (2009)
Published online 15 December 2008 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/hyp.7188

Ensemble flood forecasting with stochastic radar image
extrapolation and a distributed hydrologic model
Sunmin Kim,1 * Yasuto Tachikawa,2 Takahiro Sayama1 and Kaoru Takara1
1 Disaster Prevention Research Institute, Kyoto University, Gokasho, Uji, Kyoto 611-0011, Japan
2 Graduate School of Urban and Environmental Engineering, Kyoto University, C1, Nishikyo-ku, Kyoto 615-8540, Japan

Abstract:
A new attempt on ensemble flood forecasting is introduced by use of a radar image extrapolation along with a stochastic
error field simulation, and a distributed hydrologic model. In contrast to the conventional ensemble simulation that uses initialcondition control to obtain a statistical outcome, stochastic prediction error fields were externally generated to offer probablevariations of deterministic predictions. Firstly, a radar extrapolation model provided deterministic rainfall prediction, and its
prediction error structure was analysed by comparing the observed rainfall fields. Secondly, on the basis of the analysed error
structure, spatially correlated error fields were simulated using a covariance matrix decomposition method on a real-time basis.
The simulated error fields enabled not only the production of probable rainfall field variations for ensemble simulation but
also an improvement in prediction accuracy by offsetting the deterministic prediction error. Finally, the simulated error fields
along with the deterministic fields were tested with a distributed hydrologic model to measure the validity of the ensemble
runoff prediction. Copyright  2008 John Wiley & Sons, Ltd.
KEY WORDS

stochastic flood forecasting; ensemble; external error simulation; radar image extrapolation; distributed hydrologic
model

Received 8 August 2006; Accepted 17 August 2007

INTRODUCTION
Nowcasting of precipitation is an essential prerequisite
for real-time flood forecasting in operational hydrology.
The term ‘nowcasting’ is used to emphasize the specificity and shortness (0 ¾ 3 h) of rainfall forecast largely
by radar image extrapolation, a method going back nearly
50 years (Smith and Austin, 2000; Fox and Wilson,2005). Even though the benefits of short-term precipitation forecasts are well known, it is acknowledged as
being among the most challenging areas in hydrology
and meteorology (Collier and Krzysztofowicz, 2000).
Forecast techniques using radar observations are on thebasis of tracking past movements of rain cells and extrapolating those movements. It does not usually allow for thegrowth and decay of the rainfall intensities or nonlinear
motion of the rainfall band. The main disadvantage ofthis technique is that because of its simplicity, the forecast accuracy decreases rapidly within an hour (Bellon
and Austin, 1984; Wilson et al., 1998). In a study of the
improvement of forecast accuracy, elaborate nonlinear
extrapolation schemes only give negligible improvement
or even worse results than linear extrapolation (Smith andAustin, 2000).
Many hydrologists and meteorologists have conductedvast research efforts over several decades allowing for the
introduction of many new schemes. These new schemes* Correspondence to: Sunmin Kim, Disaster Prevention Research Institute, Kyoto University, Gokasho, Uji, Kyoto 611-0011, Japan.
E-mail: sunmin@flood.dpri.kyoto-u.ac.jp
Copyright  2008 John Wiley & Sons, Ltd.

include mathematical and stochastic models integrated
with a meteorological component (e.g. Georgakakos andBras, 1984; Nakakita et al., 1996) and hybrid models,
which are a combination of numerical weather prediction(NWP) and image extrapolation models (e.g. Golding,
2000; Ganguly and Bras, 2003). In addition, there havebeen complex statistical approaches such as using fractal
generation algorithms (Lovejoy and Schertzer, 1986)
and artificial neural networks (Grecu and Krajewski,2000). However, even though there have been vast
research efforts, it has still proven hard to find a notable
improvement of forecast accuracy.
Extrapolation techniques are still in the mainstream for
nowcasting, as many studies are taking place in order todevelop more accurate extrapolation models (e.g. Kawamura et al., 1997; Georgakakos, 2000; Grecu and Krajewski, 2000). Even though there are many NWP models,these models still have insufficient spatial and temporal
resolution to represent the detailed distribution of precipitation, and furthermore they require sophisticated data,
which in many cases are unavailable (Golding, 2000). For
this reason, radar image extrapolation is still a very powerful nowcasting tool in many practical flood forecasting
situations.
For any natural phenomenon scientists try to forecast,
one should bear in mind that there will always be an initial error in the model at the beginning of simulation andthere will always be additional errors during a simulation because of the imperfection of the model’s structure.
To estimate the effect of the errors on the forecasting,it is necessary to supplement the deterministic forecasts

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S. KIM ET AL.

with detailed information regarding forecast reliability.
For this reason, the stochastic concept has been included
in forecasting, and ensemble simulation has been used
as an effective tool for incorporating stochastic concepts into computer simulation. The ensemble forecast ofhydrographs is also a recent trend away from the conventional simple deterministic forecasts of hydrographs andtowards probabilistic forecasts, which include prediction
uncertainty.
In atmospheric modelling, small perturbations of theinitial condition and/or boundary condition in the beginning of a model simulation have been used as a trigger
for ensemble forecasting (e.g. Du and Mullen, 1997).
Most of the ensemble simulations in the early stages
have been concerned only with the internal growth of
initial error and therefore, have been criticized for underestimating the total uncertainty as not all sources of
uncertainty are accounted for in the ensemble generator (Leith, 1974; Krzysztofowicz, 2001). In considering
the additional growth of prediction error rising from animperfect model structure, external error consideration
should be contemplated.
As a step towards addressing the improvement of forecast accuracy and stochastic forecasting with consideration of external error, this study introduces a new attempt
at ensemble rainfall forecasting using a radar image
extrapolation and a stochastic error field simulation. The
simulated prediction error not only gives probable rainfall field variations for the ensemble simulation but also
improves the accuracy of the deterministic prediction bycorrecting the possible prediction error. Then, stochastic prediction fields are given to a distributed hydrologic
model to achieve ensemble runoff predictions.
The organization of this paper is as follows. First, anillustration is given of a method of deterministic prediction of rainfall using a radar image extrapolation as wellas the analysis of its prediction error pattern. Second,
the main algorithm for the generation of stochastic error
field is given, and ensemble forecasting of rainfall is performed on the basis of the proposed algorithm. Then, therest of this paper evaluates the ensemble forecast results
for both precipitation and runoff discharge. Finally, themain conclusions are outlined.

where, u and v are advection velocity along x and y,
respectively, and w is rainfall growth-decay rate withtime. As with other similar discrete equations for the rainfall intensity distribution, characteristics of the translation
model are defined by the vectors u, v and w, which are
specified on each grid as follows:ux, y D c1 x C c2 y C c3vx, y D c4 x C c5 y C c6wx, y D c7 x C c8 y C c92

As such, the advection velocities can express thepatterns of the non-uniform movement of rainfall, such as
rotation and sheer strain (Takasao et al., 1994). In orderto optimize the parameters c1 ¾ c9 using observed radar
rainfall data, Equation (2) is approximated by the central
difference scheme on the rectangular horizontal area withx ð y grid size and t time resolution.
1
i D 1, Ð Ð Ð , M
xi D i  x,2
1
yi D j  y,j D 1, Ð Ð Ð , N
2
k D K  1, Ð Ð Ð , 0
tk D kt,

3Here, M and N are the number of grids along thex and y-axis, respectively, and K is the number ofrainfall patterns used for the optimization. The parameters
c1 ¾ c9 are sequentially optimized using the square root
information filter in a manner of minimizing
Jc D

1 M1

 N1


v2ijk

4

kDK iD2 jD2

where,






z
z
C c1 xi C c2 yj C c3 C c4 xi
t ijkx ijkzC c5 yj C c6  c7 xi C c8 yj C c9 y ijkvijk D

The translation model by Shiiba et al. (1984) is used inthis study for deterministic predictions of short-term radar
rainfall. In this model, the horizontal rainfall intensity
distribution, zx, y, t with the spatial coordinate (x, y) attime t is defined as

The translation model provides expected rainfall movements under an assumption that the vectors u and v aretime invariant for the next several hours and that there isno growth-decay of rainfall intensities during that time.
In this study, three consecutive observed rainfall fields,
which have a resolution of 3 km and 5 min, are used todetermine u and v. When forecasting rainfall fields, the uand v are assumed spatially uniform and updated every
5 min. Most heavy rainfall events in Japan, which occur
during the rainy and typhoon season, have a frontal rainband over several hundred kilometers, and therefore themovement of the rainfall band can be considered to bespatially uniform within a single radar range that has a
120-km radius.

∂z
∂z
∂z
Cu Cv
Dw
∂t
∂x
∂y
dx
dz
dy
uD
, vD
, wD
dt
dt
dt

Deterministic prediction of precipitation and its error
structure
Radar data used for testing the translation modelare observed at the Miyama radar station located in

NOWCASTING WITH RADAR IMAGE
EXTRAPOLATION AND ITS ERROR STRUCTURE
Introduction of the translation model

Copyright  2008 John Wiley & Sons, Ltd.

1Hydrol. Process. 23, 597–611 (2009)
DOI: 10.1002/hyp

FLOOD FORECASTING WITH STOCHASTIC RADAR IMAGE EXTRAPOLATION

599

Figure 1. Miyama radar station (120-km radius circle) and Yodo-River basin, which includes Ootori, Ieno, and Kamo stations for ensemble discharge
prediction analysis

the central part of Kinki district, Japan (Figure 1). The
observation field of the radar includes all spaces within aradius of 120 km and a height of 15 km (for details on theradar station refer to Nakakita et al., 1990). It takes 5 min
to scan over the entire observation field and therefore
rainfall reflectivity data are produced every 5 min. The
reflectivity data are converted to rainfall intensity of a
3-km spatial resolution by the Z-R relationship, Z D
200R1Ð6 according to Marshall and Palmer (1948). During
the conversion, basic correction for ground clutter andshadow effects is carried out. The converted rainfall
intensity, which is called observed rainfall field in thisstudy, was used as a reference value for the accuracy
improvements of prediction fields.
Two representative rainfall events have been selected
for this study. Those events are of a typical frontal rainband type, which occurred during the rainy season (Juneand July) and the typhoon season (August and September)in Japan. Details of the selected events are illustrated inTable I. One event that took place in August of 1992 hadsevere rainfall intensity with rapid changes in velocity,while the event of June 1993 had a rather steady andslowly moving rainfall band.
Figure 2 shows the characteristics of forecasted rainfall
by the translation model for 60, 120, and 180-min leadtimes. First of all, Figure 2a shows the spatially averaged
rainfall intensities of observation and each prediction
for the August 1992 event. In order to prevent outside
influence from the size and shape of rainfall bands,
the intensity calculation includes every grid within theradar observation domain. Note that there are overall
delays of rainfall intensities, as prediction time elongates.
Table I. Radar data and its characteristics
Title
August 1992 event
June 1993 event

DurationType

92/8/18 ¾ 19
93/6/30 ¾ 31

Typhoon season (frontal)
Rainy season (frontal)

Copyright  2008 John Wiley & Sons, Ltd.

Because the translation model used here only represents
the movement of the rainfall bands without their growthor decay, the model assumes the same amount of current
rainfall intensities lasts until the prediction target time.
In Figure 2b, correlation coefficients (CCs) of theobservation and predictions are shown. The majority ofthe time, CC values are under 0Ð5, which is a rather low
value. As it can be expected, the coefficients with shortlead-time have higher values compared to longer leadtime predictions.
Another method used to measure prediction accuracy isthe critical success index (CSI), which is widely adopted
in forecast verification in the form of
CSI% D

X
ð 100
XCYCZ

5

where, X is the number of correctly forecasted rainfall
cells (i.e. rainfall is observed and also predicted in thegrid), Y is the number of misses (i.e. rainfall is observed,
but not predicted), and Z is the number of false alarms(i.e. rainfall is predicted, but not observed). A thresholdrain-rate for the CC and CSI is over 0Ð0 mm/h in thisstudy.
The CSI show rather high values in most prediction
times even for the 180-min prediction (Figure 2c). In
the figure, the index clearly shows that shorter lead-time
predictions have higher prediction accuracy compared tolonger lead-time. The index appeared to have a positive
relationship to the covering area of rainfall bands. Thisis rather a reasonable phenomenon since the index canbe simply regarded as an overlap ratio of the prediction
rainfall band to the observed band.
Tachikawa et al. (2003) statistically analysed the characteristics of prediction error and relative prediction error
and defined them as shown in Equations (6) and (7).
Ea,i D Ro,i  Rp,i

6Er,i D Ro,i  Rp,i /Rp,i

7

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S. KIM ET AL.

Figure 2. Deterministic prediction results from the translation model. Spatially averaged rainfall intensities (a), performance index [correlation
coefficient (b) and critical success index (c)], and the error characteristics [mean (d) and standard deviation (e) of the error and spatial correlation
coefficient (f)] in the case of August 1992 event

The prediction error Ea,i on a certain grid i is calculated from the difference between predicted rainfall Rp,iand observed rainfall Ro,i on the grid, while the relative prediction error Er,i is the ratio of the prediction
error to its predicted rainfall. Tachikawa et al. (2003)
examined the temporally accumulated error values withvariant spatial resolutions and found that the distributionsof the prediction error and relative prediction error are
respectively close to normal distribution and lognormal
distribution.
This study concentrates on the prediction error Ea,iand simulates the spatially correlated possible error for
future prediction target times on a real-time basis. Basicstatistics of the prediction error examined in this study
include the mean and standard deviations as well as
the probability distribution of the error, which indicatesnormal distribution allowing a slight variation in each
event and prediction case. Figure 2d and e show basic
Copyright  2008 John Wiley & Sons, Ltd.

statistics of 60, 120, and 180-min prediction error for theAugust 1993 event. The spatial correlation coefficients(SCCs)of Ea,i , which shows how much the errors are
spatially correlated to each other, is calculated for every
time step by grouping every pair of errors having thesame distance on each error field. The SCC in Figure 2f
shows high values for close distances and decreases
as the distance increases, and the error from longer
prediction times has higher SCC values than the error
from shorter prediction times. In most prediction cases,
the sccs decrease drastically at a distance of 10 km, as
shown in Figure 2f with the 60-min prediction case.
For reviewing the spatial pattern of the prediction
error, the errors on each grid are accumulated eventby event. For example, if there is a certain spatial
and/or temporal pattern in the prediction error, becauseof perpetual overestimation or underestimation on acertain area during a certain event, the accumulated error
Hydrol. Process. 23, 597–611 (2009)
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FLOOD FORECASTING WITH STOCHASTIC RADAR IMAGE EXTRAPOLATION

601

(a)

(b)

Figure 3. Accumulation of the prediction error during each event (unit: mm/h). There is a specific spatial pattern on each accumulation, and it presents
localized overestimation or underestimation on a certain area

will present those patterns. Otherwise, if the error does
not have any spatiotemporal pattern namely arbitrarily
random errors, the accumulated prediction error on every
grid would compensate each other.
As shown in Figure 3, there is a specific spatial pattern
on each accumulation of prediction error. Note that
although each individual event depicts similar spatial
error patterns within itself, prediction error patterns aredifferent from event to event. The error would havea complex relationship with topography as well as
meteorological conditions of each event, and it is difficult
to define the error beforehand. Thus, in the next section,a method to obtain a spatially variant error pattern on areal-time basis is proposed and it is used for stochastic
prediction in this study.
STOCHASTIC FORECASTING OF PRECIPITATIONWITH PREDICTION ERROR FIELD
Prediction error field simulation algorithm
The main purpose of the algorithm is to simulate possible error fields using the current prediction error structure
including its spatial correlation. This is conducted under
the assumption that a temporal persistence of the error
characteristics continues from the current time to the prediction target time. The proposed scheme uses a certain
time length of previous prediction error data to simulate
future prediction errors as shown in Figure 4.
In Figure 4, the observed rainfall fields, the deterministic prediction fields, and the prediction error fields aresequentially illustrated until the current time t. Although
Copyright  2008 John Wiley & Sons, Ltd.

distinct prediction fields having different lead-time canexist on each time step, for the sake of simplicity only
one prediction lead-time T is considered in the figure.
Again, every prediction field at each time step is the prediction result that is obtained at T before that time. At
the current time t, the translation model carries out prediction for the time t C T; then the probable prediction
error of the prediction is simulated in accordance withthe current error characteristics.
The current characteristics of the prediction error canbe presented by basic statistics under an assumption thatthe time series of the error on each grid follows normal
probability distribution. Here, the basic statistics stand forthe mean and standard deviation values of the most recent
errors in certain time length on each grid. On the basis of
this procedure, the ‘error statistic field’ can incorporate
spatial and temporal characteristics of the current errors
and is updated on a real-time basis.
If the spatiotemporal characteristics of the prediction
error lasts for several hours, and the statistical characteristics of the error on the prediction target time t C T are
similar to the characteristics of the current error, the possible error fields at t C T can be simulated by using theerror statistic field. This assumption regarding the temporal persistency of the error characteristics is evaluatedin the next section.
Figure 5 explains the procedure for the simulation of
the possible prediction error. The ‘error statistic field’
and the mean and standard deviation fields of the current
prediction error (Figure 5b and d), convert the ‘unit
random error field’ (Figure 5c) to the target error field
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S. KIM ET AL.

Figure 4. Schematic drawing of the algorithm for a stochastic error field simulation. At the current time t, the translation model carries out prediction
for the time t C T; then the probable prediction error of the prediction is simulated in accordance with the current error characteristics

Figure 5. Simulation procedure of prediction error field using error statistic field and unit random error field. The ‘error statistic field’, the mean,and standard deviation field of the current prediction error converts the ‘unit random error field’ comprised of spatially correlated random numbersof N(0,1) to the prediction error field(Figure 5a). The unit random error field is a set of random
values, which are spatially correlated to each other andare assumed to follow the normal distribution of N(0,1).
Through numerous generations of the unit random error
field, it becomes possible to acquire various error fields
for the prediction target time. Finally, the deterministic
prediction field by the translation model is extended tomany prospective prediction fields by combining themwith the simulated error fields. Details of this procedure
are discussed in the following section.
Time persistency of the error characteristics
To verify the temporal persistence of the characteristics of prediction error, the modified critical success index(MCSI) in Equation (8) is proposed. The MCSI uses thesame form of CSI, shown in Equation (5), with the exception of range concept. As noted in the previous section,
the error statistic field gives a specific probabilistic range
on each grid by the mean  and the standard deviation
 of the error on its own grid. If the analysed prediction
error of the target time on a certain grid is within therange    and  C  on the grid, it is counted as acorrect value, X, and if the error is out of the range, it iscounted as an incorrect value, Y.
MCSI% D

X
ð 100
XCY

Copyright  2008 John Wiley & Sons, Ltd.

8

Since the probability within the unit variance in astandard normal distribution is 0Ð6826, the expected
MCSI value in this analysis is around 68%. Evaluations
were conducted with three different sets of error statistic
fields generated from different time lengths of error data:
10, 30, and 60 min. Figure 6 shows the MCSI values forthe 60-min lead-time predictions with three different error
statistic fields. In the results from the August 1992 event,the MCSI using the one of 30-min time length starts
with high values around 80%, and lowers as the rainfall
intensity becomes stronger. When it is considered thatmost MCSI values are over 60%, this result is highly
encouraging for adopting the time persistence of theprediction error.
Compared to the MCSI using the 30-min time length oferror statistic field, the 60-min time length one has larger
values, whereas the 10-min time length one indicateslower values. When the error statistic fields of different
time lengths were compared to each other, the mean fieldof error did not show significant differences. However,the longer the time length of an error statistic field is, thelarger the produced standard deviation value is; therefore
the value X can have bigger values in MCSI.
In this study, the 30-min time length of the error
statistic field is used for the error field simulation in thenext section. Tables II and III show the time averaged
MCSI values for different lead-times (60, 120, and 180min prediction) from both events.
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603

FLOOD FORECASTING WITH STOCHASTIC RADAR IMAGE EXTRAPOLATION

Figure 6. MCSI determined by different durations of error statistic field (ESF), which shows highly encouraging results (over 68%) for adopting thetime persistence of the prediction error [from the 60-min prediction of the August 1992 event (left) and the June 1993 event (right)]

Table II. Averaged MCSI values from variant prediction leadtimes (August 1992 event, Unit: % )Lead-time

60-min prediction
120-min prediction
180-min prediction

Time lengths to determine error statistics
10 min30 min60 min

37Ð88
35Ð43
35Ð83

53Ð92
50Ð52
50Ð59

64Ð13
60Ð48
60Ð09

Table III. Averaged MCSI values from variant prediction leadtimes (June 1993 event, Unit: % )Lead-time

60-min prediction
120-min prediction
180-min prediction

Time lengths to determine error statistics
10 min30 min60 min

25Ð69
21Ð75
23Ð30

39Ð46
32Ð90
34Ð12

49Ð95
42Ð66
42Ð48

Spatially correlated random error field
The simulation of the unit random error field is on
the basis of the decomposition of a matrix that includes
spatial correlation characteristics of the prediction error
in a covariance matrix form. The matrix is decomposed
approximately into its square root matrix with the matrix
factorization technique and the Chebyshev polynomials(Davis, 1987; Tachikawa and Shiiba, 2000). Multiplying
the square root matrix by a random vector N(0,1) gives a
non-conditional simulation of the unit random error field.
Davis (1987) proved a symmetric matrix B that satisfies K D BB could be found when K is symmetric andpositive-definite. The random vector Y (in this study, thespatially correlated unit random error) can be written as
Y D Bw
Where, w is the uncorrelated random vector N(0,1).
The expected value of the matrix YYT n ð n is givenby
E[YYT ] D E[BwwT BT ] D BE[wwT ]BT
Because w is a vector of independent random numbers,
E[wwT ] D I, and thus,
E[YYT ] D BIBT D K
Copyright  2008 John Wiley & Sons, Ltd.

The SCCs, which are obtained from the prediction
error Ea , make up the covariance matrix K.
scc0
 scc1

scc
KD
 .2
 .
.

scc1
scc0
scc1
..
.

scc2
scc1
scc0
..
.

sccn

sccn1

sccn2Ð Ð Ð sccn Ð Ð Ð sccn1 Ð Ð Ð sccn2 
.. 
..

.
.
Ð Ð Ð scc0

9

Equation (9) shows the basic form of the matrix K.
Under an assumption of ergodicity on scc, the sccj
is prepared using two error groups, which are j grids
separate from each other within one error field. Since thevariation of each scci does not vary with time (Figure 2f),time averaged scci from the 30-min time length of error
statistic fields is used for the matrix K in this study.
The matrix K is decomposed into a symmetricmatrix B approximately by the Chebyshev polynomials.
Vector Y (or unit random error field), which is a nonconditional simulation of spatially correlated random vectors, can be generated continuously by multiplying thematrix B by an uncorrelated random vector w.
The error statistic field converts the unit random error
field to the error fields as shown in Equation (10):
E





Es,n
 
y



sd0
 Es,2   0  Es,3  D  0
 .   .
 .   ..
.
s,1

0



m1
1
 y 2   m2    y 3  C  m3  .   . 
 .   . 
.
.
mn
yn


0
0 

0 
.. 
. 

0
sd1
0
..
.

0
0
sd2
..
.

ÐÐÐ
ÐÐÐ
ÐÐÐ
..
.

0

0

Ð Ð Ð sdn

10Here, the mi and sdi are the mean and standard
deviations of the current prediction error on grid i.
The yi is the unit random error of the vector Y, andthe Es,i is the simulated error for the prediction target
time. Equation (10) is a linear equation, thus the spatial
correlation structure of Y, which is obtained from Ea , ismaintained in Es . The form of Equation (10) is identicalto Figure 5, and the total grid number of the Miyama
radar image is 80 ð 80, thus the n in Equation (10) is
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S. KIM ET AL.

6400. Fifty sets of error fields at each time increment
were generated for an ensemble simulation.
Generation of stochastic prediction fields
Deterministic predictions from the translation modelare extended to many stochastic prediction fields by combining them with the simulated error fields as follows:
Rs,i D Rp,i C Es,i

³Ro,i D Rp,i C Ea,i 

11

where, Es,i is the simulated prediction error value on gridi, Rp,i is the prediction from the translation model, andRs,i is the stochastic prediction. Because the simulated
prediction error contains the error statistics of the prediction error (Es,i ³ Ea,i ), the stochastic prediction can
be close to the observed rainfall on the prediction target
time (Rs,i ³ Ro,i ).
Because some values on the simulated error field yield
negative values that can be larger than the predicted
rainfall values at that point, negative values could occur
on the stochastic prediction fields. These negative rainfall
values are set to zero, and the same amount of negativevalues is subtracted from the positive rainfall values so
as to keep the total rainfall amount as
R0 s,i D Rs,i 1 C r
D 0Ð0



if Rs,i ½ 0Ð0

if Rs,i < 0Ð0

field to the total positive rainfall amount. The total
amount of negative rainfall generally is 10–20% of thetotal positive rainfall amount; therefore, r varies from0Ð1 to 0Ð2. In addition to tallying total rainfall amount,
this procedure gives a smoothing effect on the stochastic
prediction fields, so that it decreases abnormally highprediction values.

12

NegativeR
here, r D 
PositiveR
Here, the value r stands for the ratio of the total
negative rainfall amount on each stochastic prediction

RESULTS AND EVALUATION OF THE
STOCHASTIC PREDICTION FIELDS
Spatially averaged rainfall intensities are checked asshown in Figure 7a with the 60-min prediction of theAugust 1992 event. The intensities from the stochastic
prediction fields make a certain range and show similar patterns of intensities in the deterministic prediction. If the bias correction mentioned in the previous
section works properly, the stochastic prediction fields
should have values that are more accurate. As shown in
Figure 7b, the spatial mean of absolute error at each time
step from the stochastic prediction have smaller values
than the error from the deterministic prediction. The CCsfrom the stochastic prediction also show improved resultsin most prediction times (Figure 7c). In most prediction times, the coefficients from the stochastic prediction
fields have higher values compared to the values from thedeterministic prediction. Yet, the CSI values in Figure 7d
do not show a vast difference between the stochastic
prediction and the deterministic prediction. Because theerror field simulation is on the basis of the most recent
prediction results, theoretically, the simulated error field

Figure 7. Verification of stochastic prediction of precipitation (August 1992 event). The stochastic prediction fields show improved accuracy [smaller
spatial mean of absolute prediction error (b) and higher correlation coefficient (c)] with its reliability range (d) critical success index
Copyright  2008 John Wiley & Sons, Ltd.

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FLOOD FORECASTING WITH STOCHASTIC RADAR IMAGE EXTRAPOLATION

contains the same area as the prediction fields, and therefore it does not significantly alter the CSI values. Moredetailed verification will be discussed in the following
sections.
Forecast verification
For an overall forecast verification, several performance indices were adopted, such as accumulated rainfall
amounts (ACRAs), root mean square error (RMSE) andmean of absolute error (MAE). Firstly, ACRA is total
amount of the observation and prediction (whether it isdeterministic or stochastic) given as
ACRA D

nt


s
1 
Ot,s
ns sD1

n

tD1

D

nt


or

s
1 
Ft,s
ns sD1

n

tD1

13

where, ns and nt are the numbers of rainfall grids andtime steps. Ot,s and Ft,s are observed and predicted values
at time t on point s. Note that the observed values Ot,smay be different from the ground gauge observation,
since the radar observation used in this study is not
adjusted to the ground gauge data. However, this radarobservation Ot,s is regarded as the reference rainfall valueof the prediction. For the ensemble forecasting results,
which has 50 sets of the stochastic prediction field,
the mean and standard deviations of the ACRA were
calculated, and the minimum as well as maximum of the
ACRA values was also examined.
Secondly, the performance indices adopted to obtain asingle overall score are RMSE and MAE as expressed in
Equations (14) and (15).



nt
ns
1 
1 

RMSE D
Ot,s  Ft,s 2  14
nt tD1
ns sD1t
1 
nt tD1

n

MAE D

s
1 
jOt,s  Ft,s j
ns sD1

n

15

605

Verification results and discussion
Figure 8 and Table IV show the comparison of ACRA
values of observation, deterministic and stochastic prediction from the variant prediction lead-time of the August
1992 event. Better results having closer ACRA value
to the observed one are marked with bold character inTable IV. In the figure, ACRA value of the observation
is presented with a solid line parallel with the x-axis,
which represents the prediction lead-time. The ACRA
values from the deterministic and stochastic prediction
are expressed with points and error-bars.
The ACRA values from the deterministic prediction
decrease as prediction lead-time get longer, showing
some differences to the observed one. The reason of
the decrease can be found in the simulation behaviourof the translation model. When the model performs asimulation, the optimized u and v vectors transfer thecurrent rain bands, and therefore some rainfall area islocated outside of the radar range at the prediction target
time. In addition, because new rain band that comes in
the radar range during the simulation is not counted
in the prediction results, the deterministic prediction
shows a trend of decreasing rainfall area as well as
the ACRA values. However, the error field simulationof this study incorporates the information of the newly
added rain bands, and as a result, the stochastic prediction
shows relatively steady ACRA values regardless of theprediction lead-time.
The reliability range of the stochastic forecasting(whether the standard deviation, or the maximum andminimum values) clearly expresses the uncertainty of theforecasted values; the larger the prediction lead-time isextended, the wider the range becomes.
Figure 9 and Table V show the comparison of ACRA
values from the June 1993 event. In the figure, ACRA
values of the deterministic prediction show a larger discrepancy to the observation as the lead-time is longer.
While the mean values of the stochastic prediction also
show some discrepancy to the one from the radar observation, the range of the stochastic prediction successfully
encompasses the observation values.

Lastly, time averaged CC and CSI were examined.
These are simply the mean of the CC and CSI values’
time series for a perspective comparison of variant
prediction lead-time.
MCC D

t
1 
CCt
nt tD1

16

MCSI Dt
1 CSIt
nt tD1

17

nn

For an ensemble simulation result, which has fifty
stochastic predictions, mean of RMSE, MAE, MCC andMCSI values are calculated and compared with the values
from the deterministic prediction.
Copyright  2008 John Wiley & Sons, Ltd.

Figure 8. Accumulated rainfall amounts (ACRA) for August 1992 event(high rainfall intensity). The stochastic prediction shows relatively steady
ACRA values regardless of the prediction lead-time, and the reliability
range (the standard deviation, or the maximum and minimum values)
clearly expresses the uncertainty of the forecasting
Hydrol. Process. 23, 597– 611 (2009)
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606

S. KIM ET AL.

Table IV. Accumulated rainfall amount (ACRA) values (August 1992 event, Unit: mm/h)
Lead-time

Observation

60-min prediction
120-min prediction
180-min prediction

Deterministic prediction

217Ð38217Ð38217Ð38226Ð64215Ð94
192Ð17

Figure 9. Accumulated rainfall amounts comparison for June 1993 event(low rainfall intensity). While the mean values of the stochastic prediction
show some discrepancy to the radar observation, the ranges of thestochastic prediction encompass the observation values

Tables VI and VII show the RMSE and MAE from thevariant forecast lead-times of the August 1992 event andJune 1993 event. The values for the ‘stochastic’ in thetable are averaged values from the 50 sets of stochastic

Stochastic prediction
MeanšStDev

Min¾Max

224·79 š 28Ð51
226Ð29 š 38Ð29
228·97 š 39Ð74

165Ð09 ¾ 290Ð06
147Ð15 ¾ 314Ð78
149Ð04 ¾ 321Ð40

prediction results. The August 1992 event clearly illustrates that the stochastic prediction gives much smaller
error values than the deterministic prediction in all leadtimes. Furthermore, both scoring values become larger
for longer lead-times, which show higher uncertainty for
longer lead-time predictions. On the other hand, the June
1993 event resulted in better values for the deterministic
prediction than for the stochastic prediction. This may
be because the June 1993 event has low rainfall intensities and therefore, the stochastic predictions are much
more influenced by the negative rainfall handling during the stochastic prediction field simulation. However,undefined spatial and/or temporal characteristics in a certain rainfall pattern can be there during the June 1993
event and therefore, more detailed study with various
event types is required in order to generalize the proposed
stochastic prediction method.
The mean of correlation coefficient (MCC) and MCSI
for the August 1992 event and June 1993 event are
respectively presented in Tables VIII and IX. The values
for the ‘stochastic’ in the table are averaged values from

Table V. Accumulated rainfall amount (ACRA) values (June 1993 event, Unit: mm/h)
Observation

60-min prediction
120-min prediction
180-min prediction

Deterministic prediction

35Ð14
35Ð14
35Ð14

34Ð81
32Ð01
26Ð42

Stochastic prediction
MeanšStDev

Min¾Max

36Ð88 š 2Ð30
40Ð98 š 3Ð85
39·06 š 4Ð67

31Ð97 ¾ 42Ð28
32Ð85 ¾ 49Ð97
29Ð39 ¾ 50Ð06

Table VI. Root mean square error (RMSE) and mean of absolute error (MAE) (August 1992 event, Unit: mm/h)
Lead-time

RMSE

MAE

Deterministic

Stochastic

Deterministic

Stochastic

12Ð97
14Ð01
14Ð06

11Ð50
12Ð17
12Ð59

5Ð96
6Ð60
6Ð58

5Ð36
5Ð86
6Ð19

60-min prediction
120-min prediction
180-min prediction

Table VII. Root mean square error (RMSE) and mean absolute error (MAE) (June 1993 event, Unit: mm/h)
Lead-time

RMSE

MAE

Deterministic

Stochastic

Deterministic

Stochastic

5Ð22
5Ð33
5Ð41

5Ð25
5Ð38
5Ð42

3Ð26
3Ð36
3Ð41

3Ð35
3Ð48
3Ð47

60-min prediction
120-min prediction
180min prediction

Copyright  2008 John Wiley & Sons, Ltd.

Hydrol. Process. 23, 597–611 (2009)
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607

FLOOD FORECASTING WITH STOCHASTIC RADAR IMAGE EXTRAPOLATION

Table VIII. Mean correlation coefficient (MCC) and mean critical success index (MCSI) values comparison (August 1992 event)Lead-timeMCC

MCSI (%)

Deterministic

Stochastic

Deterministic

Stochastic

0Ð164
0Ð063
0Ð052

0Ð252
0Ð173
0Ð143

50Ð5
42Ð1
36Ð6

49Ð1
46Ð6
45Ð7

60-min prediction
120-min prediction
180-min prediction.

Table IX. Mean correlation coefficient (MCC) and mean critical success index (MCSI) values comparison (June 1993 event)Lead-timeMCC

MCSI (%)

Deterministic

Stochastic

Deterministic

Stochastic

0Ð216
0Ð090
0Ð070

0Ð229
0Ð123
0Ð072

74Ð03
65Ð15
56Ð51

69Ð45
66Ð73
64Ð86

60-min prediction
120-min prediction
180-min prediction

the 50 sets of stochastic prediction results. From boththe tables, the values represent that stochastic prediction
gives improved accuracy compared to the deterministic
predictions while the accuracy decreases as prediction
lead-time gets longer.
ENSEMBLE RUNOFF SIMULATION WITH A
DISTRIBUTED HYDROLOGIC MODEL
Model introduction
For real-time flood forecasting, there has been considerable interest in utilizing weather radar and distributedhydrologic models, as it can provide continuous spatiotemporal measurements and outputs that are immediately available at any location in a catchment. Froma hydrological point of view, runoff responses of forecasted rainfall throughout a hydrologic system are valuable information for checking the validity of the input
data during operational usage. This study assesses thevalidity of stochastic prediction fields using a distributedhydrologic model, which is developed for the Yodo-Riverbasin located in the Miyama radar observation range.
The Yodo-River model (Sayama et al., 2006) used here
solves kinematic wave equations for both subsurface flow
and surface flow using the Lax-Wendroff scheme. Discharge and water depth diffuse node to node according toa predefined routine order, which is determined in accordance with Digital Elevation Model (DEM) and river
channel network data. The eight-direction flow map isprepared for defining the routing order for water flow diffusion as shown in Figure 10. One characteristic of the
Yodo-River model is a specific stage-discharge relationship, which incorporates saturated and unsaturated flowmechanisms as Equation (18):vc dc h/dc ˇ , 0  h < dc qh D vc dc C va h  dc , dc  h < ds vc dc C va h  dc  C ˛h  ds m , ds  h
18In the equation, when water depth h is less than thedepth of the capillary pore layer dc 0  h < dc , flow
Copyright  2008 John Wiley & Sons, Ltd.

Figure 10. Flow direction map included in the Yodo-River model. Using
the Kinematic wave equation, discharges and water depths diffuse node
to node according to a predefined routing order, which is determined inaccordance with DEM and river channel network data

is described by Darcy’s law with a degree of saturation
h/dc ˇ and a saturated velocity vc D kc ð i. Here, ˇ isthe degree of saturation ratio, kc is saturated hydraulicconductivity in the capillary layer, and i is the slope
gradient. If h increases (dc  h < ds ), the velocity of flowfrom the non-capillary pore layer is expressed as va D
ka ð i, where ka is the saturated hydraulic conductivity inthe non-capillary layer. When the water depth is greaterthan that of the soil layer (ds  h), overland flow isadded using Manning’s resistance law using ˛ D i1/2 /n,m D 5/3 and the roughness coefficient n.
Model parameters in the stage–discharge relationshipare dc , ds , kc , ks and n, and ˇ D va /vc  is determinedby the continuity condition on the wave celerity. This
specific stage-discharge relationship characterizes themodel and as a result, each cell has its own stagedischarge relationship determined by topography, landuse, and soil type (Tachikawa et al., 2004). Details on the
Yodo-River model are given by Sayama et al. (2006).
Simulation methodology
The runoff simulation is carried out in three different catchments located within the observation range ofthe Miyama radar: Ootori (156 km2 ), Ieno (476 km2 )Hydrol. Process. 23, 597– 611 (2009)
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608

S. KIM ET AL.

and Kamo (1469 km2 ) (Figure 1). The parameters were
calibrated prior to performing the ensemble runoff simulation, using the Miyama radar observation and observed
discharge data for each outlet. The purpose of this runoff
simulation is to examine the accuracy and reliability
range of the stochastic rainfall prediction from a hydrological point of view. Thus, the discharge simulated using
the deterministic prediction and the ensemble dischargeusing the stochastic prediction is compared.
For each lead-time prediction from both events, the
50 sets of stochastic prediction fields generate an ensemble runoff simulation through the distributed hydrologic
model. Each stochastic prediction field among the 50 setsof input data was assigned to each runoff simulation
independently. Firstly, preliminary simulation was conducted with the observed rainfall data until 00 : 00 h on
August 18 for the August 1992 event and until 06 : 00 h
on June 30 for the June 1993 event. The deterministic andstochastic prediction data were inputted after the abovementioned times. The ensemble simulation was carriedout until 00 : 00 h on August 20 (August 1992 event)and until 18 : 00 h on June 30 (June 1993 event). After
those periods, the observed rainfall data were equally
set to every 50-ensemble simulations, and the simulation
was shortly continued in order to observe the remaining
effect of the input rainfall on the runoff. The rainfall data,
whether observed or predicted, were given every 5 minfor the distributed hydrologic model, which produced ahydrograph every 10 min.
Runoff simulation results and discussion
Figure 11 shows the ensemble runoff simulation results(60-min prediction) in the August 1992 event and
the June 1993 event respectively, for all three subject
basins: Ootori (156 km2 ), Ieno (476 km2 ) and Kamo
(1469 km2 ). The black thick line stands for the discharge
from the observed radar rainfall, which is the reference
discharge regarded as the actual value. The thick greyline represents the discharge from the original deterministically predicted rainfall, and the 50 thin grey lines showeach discharge from the stochastic prediction fields.

Figure 11. Discharge hydrograph from the radar observation, deterministic, and stochastic prediction [using 60-min prediction rainfall data of August
1992 event (left) and June 1993 event (right)]. Fifty hydrographs from the stochastic rainfall data produce an improved ensemble simulation ofdischarges
Copyright  2008 John Wiley & Sons, Ltd.

Hydrol. Process. 23, 597–611 (2009)
DOI: 10.1002/hyp

609

FLOOD FORECASTING WITH STOCHASTIC RADAR IMAGE EXTRAPOLATIONIn the results of the August 1992 event (Figure 11a,b and c), the discharges from the stochastic rainfall data
show closer values to the reference discharges, suggesting that the prediction accuracy of the stochastic prediction has improved. The ensemble simulation using thestochastic rainfall prediction clearly shows the improved
prediction accuracy with the reduced discharge comparedto the output from the deterministic rainfall data. While
the forecast accuracy improvement is encouraging, it ishard to determine whether the band, which stands for the
reliability range of the stochastic prediction, is acceptable. In the case of the June 1993 event (Figure 11d,e and f), the ensemble runoff simulation results do notshow clear improvement compared to the discharges fromthe deterministic prediction. As discussed in the forecast
verification section, the June 1993 event has low rainfall intensities and the accuracy improvement through theerror simulation may not work properly for this event.
Note that the runoff simulation results were given by
the continuous application of the stochastic prediction
data; therefore, the discharge output accumulated theprediction error. In practical application of real-time
rainfall forecasting, the rainfall input data will be given inaccordance with the most updated forecast for each leadtime; therefore the accumulated prediction error will not
be included in the discharge output.
For an overall comparison of the runoff simulation,
peak discharges of each case from both events areextensively examined, and those values are presented inTables X and XI. In the case of the August 1992 event,the peak discharges form the deterministic prediction
give variant values showing large overestimation inmost cases. However, the stochastic prediction using theerror simulation model decreases the overestimation ofdischarge in most cases. The closer values that standfor accuracy improved results are marked using boldcharacters in those tables.
In the results from the June 1993 event (Table XI),
about half of the runoff simulation results using thestochastic rainfall data do not produce positive resultswith a reasonable reliability range. For Ieno and Kamobasins, the deterministic prediction already provides
highly accurate peak discharge values, and the ensemble
forecasting using the stochastic rainfall data fails to offer
much more accurate and reasonable reliability. This result
may be because of a certain characteristic in the June
1993 event as mentioned in the earlier section including
low rainfall intensities.
The ACRA values are calculated using the rainfall
data on each testing basin, and the values are presented
in Tables XII and XIII. The ACRA values of eachbasin have similar patterns to the earlier analysis using
the overall rainfall amount within the radar range; thevalues from the stochastic rainfall data provide not only
improved results but also an improved reliability range.
The ACRA in the August 1992 event is rather large when

Table X. Peak discharge comparison of August 1992 event (Unit: m3 /s)
Basin

Prediction lead-timeObservation

Deterministic prediction

Stochastic prediction
Mean

Min¾Max

Ootori (156 km2 )60 min
120 min180 min

610Ð3
610Ð3
610Ð3

358Ð9
913Ð4
1096Ð2

432Ð1
622Ð5
866Ð9

378Ð0 ¾ 520Ð3
542Ð7 ¾ 694Ð3
664Ð1 ¾ 1040Ð3

Ieno (476 km2 )60 min
120 min180 min1426Ð71426Ð71426Ð73838Ð62534Ð7454Ð32108Ð41947Ð41259Ð91882Ð5 ¾ 2348Ð5
1728Ð6 ¾ 2175Ð21095Ð5 ¾ 1463Ð2

Kamo (1469 km2 )60 min
120 min180 min3836Ð13836Ð13836Ð17362Ð87271Ð04470Ð34474Ð85385Ð64718Ð84063Ð2 ¾ 4743Ð44939Ð0 ¾ 6034Ð54343Ð5 ¾ 5271Ð2

Table XI. Peak discharge comparison of June 1993 event (Unit: m3 /s)
Basin

Prediction lead-timeObservation

Deterministic prediction

Stochastic prediction
Mean

Min¾Max

Ootori (156 km2 )60 min
120 min180 min

181Ð8
181Ð8
181Ð8

190Ð2
120Ð7
111Ð3

218Ð6
201Ð2
188Ð9

191Ð5 ¾ 265Ð3
179Ð0 ¾ 228Ð1
161Ð6 ¾ 213Ð1

Ieno (476 km2 )60 min
120 min180 min

496Ð2
496Ð2
496Ð2

484Ð3
479Ð2
430Ð3

504Ð6
552Ð2
501Ð5

487Ð4 ¾ 527Ð3
538Ð5 ¾ 569Ð4
481Ð5 ¾ 522Ð9

Kamo (1469 km2 )60 min
120 min180 min1073Ð61073Ð61073Ð61067Ð21043Ð8918Ð01104Ð51190Ð51083Ð21087Ð2 ¾ 1131Ð21162Ð9 ¾ 1218Ð61052Ð9 ¾ 1112Ð5

Copyright  2008 John Wiley & Sons, Ltd.

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610

S. KIM ET AL.

Table XII. Accumulated rainfall amount (ACRA) values on each basin (August 1992 event, Unit: mm/h)
Basin

Prediction lead-timeObservation

Deterministic prediction

Stochastic prediction
MeanšStDev

Min¾Max

Ootori (156 km2 )60 min
120 min180 min

273Ð2
273Ð2
273Ð2

341Ð6
442Ð8
593Ð8

331·3 š 24Ð2
357·9 š 27Ð0459·3 š 30Ð9284Ð7 ¾ 376Ð8
281Ð0 ¾ 416Ð5
390Ð7 ¾ 507Ð0Ieno (476 km2 )60 min
120 min180 min

362Ð7
362Ð7
362Ð7

818Ð9
437Ð3
80Ð0

487·5 š 27Ð2413·4 š 26Ð4
300·3 š 18Ð9424Ð4 ¾ 537Ð7
350Ð4 ¾ 477Ð4
258Ð4 ¾ 352Ð1

Kamo (1469 km2 )60 min
120 min180 min

319Ð6
319Ð6
319Ð6

574Ð4
533Ð5
238Ð9

389·2 š 14Ð7387·4 š 18Ð9
312·4 š 13Ð3

357Ð8 ¾ 418Ð4
340Ð5 ¾ 421Ð1
258Ð9 ¾ 336Ð4

Table XIII. Accumulated rainfall amount (ACRA) values on each basin (June 1993 event, Unit: mm/h)
Basin

Prediction lead-timeObservation

Deterministic prediction

Stochastic prediction
MeanšStDev

Min¾Max

Ootori (156 km2 )60 min
120 min180 min

58Ð2
58Ð2
58Ð2

55Ð2
27Ð6
39Ð7

60·5 š 7Ð1
51·1 š 6Ð0
63·2 š 7Ð9

46Ð1 ¾ 80Ð2
39Ð9 ¾ 66Ð0
48Ð5 ¾ 88Ð1

Ieno (476 km2 )60 min
120 min180 min

41Ð5
41Ð5
41Ð5

50Ð0
53Ð0
44Ð4

53Ð8 š 3Ð1
62Ð1 š 3Ð5
59Ð0 š 4Ð5

47Ð5 ¾ 60Ð2
54Ð0 ¾ 71Ð6
45Ð4 ¾ 68Ð0

Kamo (1469 km2 )60 min
120 min180 min

41Ð7
41Ð7
41Ð7

51Ð4
47Ð1
36Ð8

54Ð2 š 2Ð3
58Ð7 š 2Ð9
54Ð7 š 3Ð5

50Ð3 ¾ 60Ð6
49Ð9 ¾ 63Ð5
47Ð8 ¾ 62Ð1considering that it is within two days’ accumulation,
while the amount in the June 1993 event is small.
The proposed algorithm for the stochastic forecasting of
precipitation has some possibility to work more properly
when there is a severe rainfall event, which is morepractical for the purpose of flood forecasting.

CONCLUSIONSFor forecast accuracy improvement and ensemble flood
forecasting with an external error consideration, thisstudy introduced ensemble rainfall forecasting using astochastic error field simulation along with a runoff simulation using a distributed hydrologic model. The proposed algorithm is for offering probable variation of
the deterministic prediction results from the extrapolation model, as well as for improving its forecast accuracy.
The translation model predicted the radar rainfall field,
and the prediction error structure was thus spatially andtemporally analysed. The random error fields were simulated using the error structure, the stochastic prediction field, which is the combination of the deterministic
rainfall and the simulated error, was generated, and its
stochastic validity was examined. The stochastic prediction fields not only gave probable reliability with variant
form of rainfall fields but also improved the accuracy ofthe deterministic prediction.
Copyright  2008 John Wiley & Sons, Ltd.

The validation of the stochastic prediction fields wascompleted from two different perspectives: checking
the performance of the rainfall prediction using various
indexes, and comparing hydrographs simulated througha distributed hydrologic model. Firstly, for an overall
forecast verification, several performance indices, suchas ACRA, RMSE and MAE were adopted. From the
ACRA testing, the reliability ranges of the ensemble
forecasting using stochastic rainfall data clearly expressed
the uncertainty of the forecasted values; the greater theprediction lead-time is extended, the wider the range
becomes. RMSE and MAE indices also demonstrateaccuracy improvement from the stochastic prediction.
Secondly, the ensemble runoff simulation with the
Yodo-River model verified the hydrologic effectivenessof the stochastic prediction fields. While the ensemble runoff simulation using the stochastic rainfall data
showed highly encouraging results, the reliability rangeof the stochastic prediction needs to be more carefully
verified on the basis of various event types in order toconfirm the proposed ensemble forecasting method.

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