th

7 International Conference on Hydroinformatics
HIC 2006, Nice, FRANCESTOCHASTIC REAL-TIME FORECASTING WITH
RADAR IMAGE EXTRAPOLATION AND
A DISTRIBUTED HYDROLOGIC MODELSUNMIN KIM
Dept. of Urban and Environmental Engineering, Kyoto University
Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501, Japan
YASUTO TACHIKAWA
Disaster Prevention Research Institute, Kyoto University
Gokasho, Uji Kyoto611-0011, JapanKAORU TAKARA
Disaster Prevention Research Institute, Kyoto University
Gokasho, Uji Kyoto 611-0011, JapanA new attempt of ensemble rainfall-runoff prediction is presented with probabilistic radar
rainfall forecasts and recursive measurement update on a distributed hydrologic model.
Different to a conventional ensemble simulation that uses initial condition control to
obtain statistical outcome, error fields of spatiotemporal rainfall predictions are generated
to offer probable variation of deterministic predictions. The error field simulation uses a
random field generation method based on an analyzed error structure of the current time
rainfall prediction. Then, the probable rainfall fields with generated error fields are given
to a distributed hydrologic model to achieve an ensemble runoff prediction. The
distributed model is coupled with the Kalman filter to utilize online hydrologic
information by several techniques including Monte Carlo simulation scheme.
INTRODUCTION
Recent trends of flood forecast are away from the conventional simple deterministic
forecasts of hydrographs toward offering probabilistic forecasts, which include its
prediction uncertainty. Deterministic flood forecast specifies a point estimate of thepredicted values, such as precipitation and river stages/discharges. On the other hand, aprobabilistic forecast specifies a certain probability distribution function of the predictedvalues. The predictive probability in a probabilistic forecast is a numerical measure of thecertitude degree about the intensity of a flood event, based on all meteorological orhydrological information utilized in the forecasting process (R. Krzysztofowicz, 2001).
This study presents a real-time flood forecast algorithm, which is built in aprobabilistic way, with a radar observation and a distributed hydrologic model. The
algorithm mainly consists of two parts; probable rainfall forecast with a radar image
extrapolation model and state variables update in a distributed hydrologic model.
1First, a new attempt of ensemble rainfall forecast is carried out with radar rainfall
prediction and spatial random error field simulation (Kim et al., 2006). The radar
extrapolation model gives a deterministic rainfall prediction,; then 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, which successfully keep theanalyzed error structure, improve the accuracy of the deterministic rainfall prediction.
Then, the random error fields with the deterministic fields are given to a distributedhydrologic model to achieve an ensemble runoff prediction.
Second, a Kalman filter is coupled with a distributed hydrologic model to update
spatially distributed state variables and to incorporate the uncertainty of rainfall forecast
data (Kim et al., 2005). Here, rather than attempting an impractical algorithm formulation,
several techniques are newly adopted. In the measurement update algorithm, thedischarge and storage amount relationship (Q–S relationship) is used as the observation
equation, and the ratio of total storage amount was applied for setting the water stage foreach cell in the distributed hydrologic model. For the prediction algorithm, a Monte Carlosimulation is adopted to propagate the state variable and its error covariance.
Succeeding this study, the developed algorithm is to be applied on the Gamcheon
Basin, South Korea with Typhoon Rusa flood event in 2002, which was one of the mostdisastrous flood disasters in South Korea.
SHORT-TERM RAINFALL FORECAST USING WEATHER RADAR
Translation model (Shiiba et al., 1984), which is one of radar image extrapolation models,simulates short-term rainfall forecasts in this study. In this model, the horizontal rainfall
intensity distribution, z(x,y,t) with spatial coordinate (x,y) at time t is represented as:

wz
wz
wz
+u
+ v = w , here u
wt
wx
wy

dx
, v
dt

dy
, w
dt

dz
dt

(1)

where, u and v are advection velocity along x and y, respectively, and w is rainfall
growth-decay rate along time. Characteristic of the translation model is that the vector u,v, and w are specified on each grid in a manner of:u ( x, y )c1 x  c2 y  c3

v ( x, y ) c 4 x  c 5 y  c 6
w( x, y ) c7 x  c8 y  c9(2)

so that the advection velocities can express the patterns of non-uniform movement of
rainfall, such as rotation and sheer strain (Takasao et al., 1994). The parameters c1~c9 aresequentially optimized using observed rainfall data by the square root information filter.

2

7th International Conference on Hydroinformatics – HIC 2006

Prediction error structure analysis and simulation of the error
Similar to the other extrapolation models, the translation model usually ignores thegrowth and decay of the rainfall intensities or nonlinear motion of rainfall bands. To
forecast precipitation accurately, however, it needs to understand not only the exact rainband movement but also the generation, growth, and decay of rain cell, particularly inmountainous regions such as in Japan.
For checking of the relationship between topographic pattern and rainfall prediction
error, which is mainly caused by the ignorance of the growth and decay of rain cell,absolute prediction error was calculated and accumulated on every grid. The absolute
prediction error Ea,i on grid i was calculated from the difference between predicted
rainfalls Rp,i and observed rainfalls Ro,i on the grid (Ea,i = Ro,i - Rp,i). As shown in Figure 1,
there were certain spatial patterns of prediction error on each accumulation. It was also
found that different wind direction gives different spatial pattern of the prediction error
through the same test with other rainfall events. The frequency distribution of theabsolute error follows a normal distribution.
Even though the accumulated prediction error shows a specific spatial pattern of theerror, it is not appropriate for real-time simulations, since the accumulated error
information is available only when enough error is accumulated to be analyzed. One
solution of this problem is using certain duration of the error data as shown in Figure 2.
In this figure, sequential observed fields until time t (Let us assume the t is current time
when new forecast is carried.), previous prediction fields for lead-time 't, and prediction
error fields on each time are illustrated. At time t, the translation model carries anotherprediction for t+'t, and probable prediction errors for t+'t are to be simulated with thecurrent error characteristics. The current characteristics of the prediction error can bepresented by basic statistics of the error, referred to here as the statistic field. The statistic
field, the mean and standard deviation field of error, make it possible to compromise thespatial and temporal pattern of the most recent prediction errors, and it can be updated on

Figure 1. Accumulated 60min (left) and 120min (right) prediction error (unit: mm/hr,
Observed at Miyama radar station, Japan on June 1993).

7th International Conference on Hydroinformatics – HIC 2006

3

Figure 2. Schematic drawing of the probabilistic error field simulation using statistic
error field and error persistency assumption.
a real-time basis. The statistic field was then converted to spatially correlated random
values to the probable error field by Equation 3, which is the goal of the error filed
simulation,.

ª E s ,1 º«E »« s,2 »
« E s ,3 »«»« 0»
« E s ,n »¬¼

0
ª sd 1 0« 0 sd02«« 00 sd 3«« 0 0 0
«¬ 0
0
0

/
/
/
2
/

0 º ª y1 º ª m1 º
0 »» «« y 2 »» «« m2 »»0 » « y 3 »  « m3 »»« » « »0 » « 0» « 0»
sd n »¼ «¬ y n »¼ «¬mn »¼(3)Here, the mi and sdi are the mean and standard deviation of the current prediction error on
the grid i, respectively. The yi is the unit random error of the vector Y, which is a set of
spatially correlated random values with zero mean and unit standard deviation; N(0,1).
The vector Y was generated using spatial correlation of the current error by thecovariance matrix decomposition method of Davis (1980). The Es,i is the simulated error
for the prediction target time. Equation 3 is a linear equation, thus the spatial correlation
structure of Y, which was obtained from the Ea, was maintained in the Es. Generation ofmany sets of Y made it possible to get many target error fields.
Generation of extended prediction fields
Deterministic prediction from the translation model is extended to many possible
prediction rainfall fields by combination with the simulated error fields in a form of:

Re ,i

4

R p ,i  E s , i

7th International Conference on Hydroinformatics – HIC 2006

(4)

Figure 3. Correlation coefficient to observation of deterministic and extended prediction(Results are from the testing with the Miyama radar station data).
where, Es,i is the simulated prediction error value on grid i, Rp,i is the deterministic
prediction from the translation model, and Re,i is the extended prediction. Because thesimulated error keeps the error statistics of the absolute prediction error (Es,i § Ea,i), theextended prediction can be close to the observed rainfall on the prediction target time. In
other words, the properly simulated prediction error can improve the accuracy of thedeterministic prediction. Negative values could occur on the extended prediction field,
since some values on the simulated error field could have a negative value, which can belarger than the deterministic prediction rainfall value on that point. This negative rainfall
set to zero, and the same amount of negative values compensated the positive rainfall
values for keeping the total rainfall amount.
Evaluation with correlation coefficient of observation and extended prediction using
the extended 60min prediction fields are presented in Figure 3. The coefficient values
from the extended prediction show more improved results than from the deterministic
prediction.
DISTRIBUTED HYDROLOGIC MODEL WITH KALMAN FILTER
The objective of the real-time update algorithm was to couple the Kalman filter to aphysically based distributed model for recursive state variables updating and forincorporating of rainfall input data uncertainty into the simulated discharge output data.
The model used here is the Cell-based Distributed Runoff Model Version 3 (CDRMV3,
Kojima et al., 2003). The model solves the one-dimensional kinematic wave equations
for both subsurface flow and surface flow using the Lax–Wendroff scheme on every

7th International Conference on Hydroinformatics – HIC 2006

5

computational node in a cell. Discharge and water depth propagate to the steepest
downward adjacent cell according to a flow direction map generated from DEM data.
The flow direction map that defines the routine order for water flow propagation in
CDRMV3 is prepared by the conventional eight-direction method. A specified stagedischarge relationship, which incorporates saturated and unsaturated flow mechanism,
was included in each cell (Tachikawa et al., 2004). The stage-discharge relationship isexpressed by three equations corresponding to the water levels divided into three layers.
For the incorporation of the filtering concept into the distributed model, there wereseveral barriers to overcome. In the measurement update algorithm, as an alternative tothe linear observation function with the system states, this study introduced an external
relationship of observed data and the internal state variables of the hydrologic model.
Here, the observed data was outlet discharge and the state variable in the Kalman filter
algorithm was the total amount in storage in the basin. Rather than inputting a linear
function of the observation and the system states into the filter, a table of those two sets
of values successfully defined the nonlinear interaction in the updating algorithm. The
other problem in the measurement update algorithm was how a very large number of statevariables, which are usually based on the fine grid cells of a distributed hydrologic
model, can be updated at the same time without excessive computational burden. A
simple but very efficient method using a ratio of the state variables made it possible tosolve this restriction on the application of the filter with a distributed hydrologic model.
The Kalman filter algorithm updates the total amount in storage in the basin, and a ratioof the updated and simulated storage amount is calculated and applied to each of theinternal state variables on a fine grid cell.
In the time update algorithm of the Kalman filter, linearized equations for the system
dynamics were necessary for projecting the state variables and their error covariance inthe algorithm. However, use of the Monte Carlo simulation method (see Figure 4) made it
possible to project the nonlinear variation of system states and their error covariancewithout the need for linearized system equations. Evensen (1994) showed that Monte
Carlo methods permit the derivation of forecast error statistics in the Kalman filter
algorithm, and thus, the inefficiency involved in the linearization of system states can beeliminated.

Figure 4. Schematic drawing of time update algorithm (a) of the conventional Kalmanfilter concept and (b) using Monte Carlo simulation methods.

6

7th International Conference on Hydroinformatics – HIC 2006

Figure 5. Observed and feedback through the Kalman filter under different system andobservation noise conditions (The results are from testing atVJGKamishiiba Basin,
Japan). SN 30 and ON 30 denote the given error covariance in the form of standard
deviation of noise in discharge, ±30 m3/s, and SN 00 and ON 00 mean that no error
covariance is given for system and observation, respectively.

Figure 6. Location map (left) of Kunsan radar station and Gam-cheon Basin, Korea, andthe produced flow direction map (right) for CDRMV3 simulation.
The Kalman filter-coupled with CDRMV3 was tested on the Kamishiiba basin undervarious error covariance conditions. Figure 5 shows the feedback through the algorithm
under the three different error conditions. The filter-coupled CDRMV3 yielded better
results than off-line simulations and can thus be used as a probabilistic forecast algorithm.
Furthermore, the developed algorithm can incorporate the uncertainty of input and output
measurement data as well as the uncertainty in the model itself.

7th International Conference on Hydroinformatics – HIC 2006

7

FURTHER RESEARCH
The past section detailed the real-time flood forecast algorithm with radar observationand distributed hydrologic model. Newly suggested ensemble simulation shows
encouraging results on accuracy improvement, however it needs more consideration on
the reliability range of the simulation results.
The developed real-time forecasting algorithm is to be applied on the Gam-cheon
Basin, South Korea with Typhoon Rusa flood events in 2002, which was one of the mostdisastrous flood disasters in South Korea. Radar data of the Kunsan radar station andgeographic data of the Gam-cheon Basin is under processing for applying the developed
algorithm (see Figure 6).
REFERENCES[1] Davis, M. W.: Generating large stochastic simulation-The matrix polynomial
approximation method, Mathematical Geology, Vol. 19, No. 2, 99-107, 1987
[2] Evensen, G.: Sequential data assimilation with a nonlinear quasi-geostrophic modelusing Monte Carlo methods to forecast error statistics, Journal of GeophysicalResearch, 99, 10,143-10,162, 1994
[3] Kim, S., Tachikawa, Y., and Takara,K.: Real-time prediction algorithm with a
distributed hydrological model using Kalman filter, Annual journal of hydraulicengineering, JSCE, No. 49, 163-168, 2005
[4] Kim, S., Tachikawa, Y., and Takara,K.: Ensemble rainfall-runoff prediction withradar image extrapolation and its error structure, Annual journal of hydraulicengineering, JSCE, No. 50, 2006 (in printing)[5] Kojima, T. and Takara, K.: A grid-cell based distributed flood runoff model and its
performance, Weather radar information and distributed hydrological modeling(Proceedings of HS03 held during IUG2003 at Sapporo, July 2003), IAHS Publ. No.
282, 234-240, 2003 (CDRMV3 can be downloaded from http://flood.dpri.kyotou.ac.jp/product/cellModel/cellModel.html)[6] Krzysztofowicz, R.: The case for probabilistic forecasting in hydrology, Journal of
Hydrology, 249, 2-9, 2001.
[7] Shiiba, M., Takasao, T. and Nakakita, E.: Investigation of short-term rainfall
prediction method by a translation model, Jpn. Conf. on Hydraul., 28th, 423-428.,
1984[8] Tachikawa, Y., Nagatani, G., and Takara, K.: Development of stage-discharge
relationship equation incorporating saturated -unsaturated flow mechanism, Annual
Journal of Hydraulic Engineering, JSCE, 48, 7-12, 2004 (Japanese with English
abstract)[9] Takasao, T., Shiiba, M. and Nakakita, H.: A real-time estimation of the accuracy ofshort-term rainfall prediction using radar, Stochastic and Statistical Methods inHydrol. and Environm. Eng., Vol.2, 339-351, 1994

8

7th International Conference on Hydroinformatics – HIC 2006