Annual Journal of Hydraulic Engineering, JSCE, Vol.51, 2007, February
Annual Journal of Hydraulic Engineering, JSCE, Vol.51, 2007, February

RECURSIVE UPDATING OF DISTRIBUTED STATE
VARIABLES USING OBSERVED DISCHARGES
THROUGH KALMAN FILTER
Sunmin KIM 1, Yasuto TACHIKAWA2 and Kaoru TAKARA3
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)
3Fellow of JSCE, Dr. Eng., Professor, DPRI, Kyoto University (Gokasho, Uji 611-0011, Japan)

By means of Kalman filter, this study examines recursive updating of a distributed model’s statevariables, which were the water depths on a fine grid cell, using observed discharges at the basin outlet.
The measurement update algorithm of the filter decides the optimal discharge, and the ratio of the optimal
discharge to the simulated discharges was applied to every discharge from each cell for a reformation of
the distributed state variables with a specific stage-discharge relationship. For the time update algorithmof the filter, Monte Carlo simulation was adopted for the prediction algorithm of the filter. The proposed
algorithm offers efficient state variables updating in a distributed hydrologic model.
Key Words: Kalman Filter, Distributed Hydrologic Model, Updating of Distributed State Variables

1. INTRODUCTION
The advances in knowledge and technology up to
now have increased the performance of flood
forecasting. Hydrologists have understood thedetailed physics of water movement with various
analytical and empirical equations. Data collecting
as well as increased computational technology haveharnessed more accurate flood forecasting on areal-time basis.
However, over the last several decades, there has
been more progress in computing and developmentsof distributed models, than in field measurement
techniques1). Current hydrology suffers from ameasurement problem especially with the
distributed properties of catchment: such as waterlevels and soil moisture at every point in a basin. It
is unrealistic to define the detailed spatial
distribution of catchment properties throughextensive field measurement, as such a process is
too costly and time-consuming. Except for few
specially designed catchments studied for a specificpurpose, outlet discharge is the only commonly
available observation in a given catchment. To
reflect a low-resolution observation into a detailed
spatial resolution for a distributed hydrologic model,
an effective method is needed to estimate andupdate the distributed catchment properties.

To minimize the discrepancy between simulationand observation, correcting the model internal statevariables is the most commonly used updating
scheme in a real-time simulation. However, in
updating the measurement for a distributed model,
not only the magnitude of the state variable but also
its spatial distribution pattern should be considered.
During a rainfall-runoff simulation, inappropriate
rearrangement of the spatial distribution of statevariables produces obvious effects on the simulation
results2),3).
Kim et al.3) showed an efficient way of updating
distributed state variables and successfully applied
the Kalman filter into a distributed model. The study
introduced an external relationship of observed data
and the internal state variables of the hydrologic
model, as an alternative to the linear observation
function of the Kalman filter4). In the relationship,
which is named Q-S relatiohship, the observed data
is defined as outlet discharge and the state value inthe Kalman filter is the total amount in storage inthe basin. In the time update algorithm, Monte Carlosimulation method made it possible to project anonlinear variation of system states and their error
covariance without the need for linearized systemequations. The developed Kalman filter coupledwith the physically based distributed hydrologic
model can incorporate the uncertainty of input and- 67 -

output measurement data as well as the uncertainty
in the model’s structure.
However, one considering point still remains on
the use of the Q-S relationship. The relationshiprepresents the overall link of the storage amount in acatchment to the outlet discharge under a steadystate assumption, which would hardly happen in thenatural phenomena. By this reason, using the Q-S
relationship to calculate total storage amount causes
additional uncertainty, which is not simple toestimate, in the measurement update algorithm.
This study introduces another efficient way of
Kalman filter application into a distributed model,
and its adaptability is examined. Newly introduced
method uses a ratio of observed discharge (orupdated discharge) to simulated discharge for themeasurement update algorithm and updates the
distributed state variables, water depths, efficiently.
For the time update algorithm, the same as theprevious study, Monte Carlo simulation offers theevolution of the state variables and its error variance
until the next update time.
The organization of this paper is as follow:
Section 2 briefly illustrates the previous update
method (S-ratio method) of Kim et al.3) andintroduced another ratio method using an outlet
discharge (Q-ratio method). In Section 3, Kalmanfilter application using the Q-ratio method is given
with a reviewing on the previous filter application.
Discussions are on the efficiency of the new method
through the feedback results comparison. Finally,
Section 4 concludes the paper by summarizing thisstudy.

2. UPDATING OF DISTRIBUTED STATE
VARIABLES USING RATIO METHODS(1) Reviewing the Distributed Model, CDRMV3
The distributed hydrologic model used here is
known as the Cell-based Distributed Runoff Model
Version 3 (CDRMV3). CDRMV3 is a physically
based model developed at the Disaster Prevention
Research Institute of Kyoto University, Japan5). The
model solves the one-dimensional kinematic wave
equations for both subsurface flow and surface flowusing the Lax–Wendroff scheme on every
computational node in a cell. Discharges and water
depthes diffuse to the steepest downward adjacentcell according to a flow direction map generated
from DEM data. The CDRMV3 adopts specificstage–discharge relationship incorporating thesaturated and unsaturated subsurface flows as wellas the surface flow6). This relationship is expressedby three equations as shown in Figure 1, one
applying to each water level.
The Kalman filter coupled CDRMV3 is applied
to Kamishiiba catchment in Kyushu, Japan, thesame as the previous study3). The identical flood

αβFig. 1 Relationship between unit width discharge andwater depth in the CDRMV3.

Fig. 2 Resetting of state variables using the ratio of
total storage amounts.

events and the parameter set are selected to compare
the feedback results and efficiency with the previous
study. More details for the CDRMV3, the subject
basin, selected flood events and parameter set,please refer to Kim et al.3).
(2) S-ratio Method
In the study of Kim et al.3), to avoid anunpredictable collapse of the internal model state
during a simulation, the update method maintains
the spatial distribution pattern of the state variables
before and after updating. As shown in Figure 2, by
multiplying the variables by a specific factor, it wasable to update the total storage amount while
retaining the spatial distribution pattern of theinternal state variables. This factor was calculated
from the ratio of the total storage amount, estimatedfrom observed discharge, and the simulated total
storage amount.
Even though the S-ratio method offers effective
updating results, it is not easy to estimate theobserved storage amount. Since the total storage
amount in any given basin cannot be measured
directly, the corresponding total storage amountmust be estimated from the observed discharge,
assuming there is a specific discharge–storage
relationship. To relate discharge at the basin outlet
and the total storage amount, the Q–S relationshipunder a steady-state assumption was established byapplying constant rainfall intensities over the study
basin, and variant pairs of total storage amount anddischarge was acquired from the CDRMV3.
The critical point on the S-ratio method is that 1)
the Q-S relationship is produced under steady state
assumption and 2) it should be replaced for every
different catchment and even for every different
parameter set3).

- 68 -

Q

Discharge profile of
one dimensional basin
Qa

H

Qb

R×Qa

R×Qb

A

B

Stage-Discharge Relationshipof point A
of point B
Q

Qs
Qo

Qb
R×Qb

Distance

Water Depth profile of
one dimensional basin

Qa
R×Qa

Ha
H’a

Hb
H’b

Ha Hb
H’ a H’b

HA
B
Distance
Fig. 3 Schematic illustration of the Q-ratio application into the ideal one-dimensional basin.

Fig. 4 Updated results from the ratio-methods for Event 979 (left) and Event 996 (right),
The vertical line stands for the updating time.

(3) Q-ratio Method
Newly adopted method in this study to reset the
distributed storage amount is by applying a ratio of
discharge difference at the outlet to the discharge ofevery point in a basin. This concept is under asteady state assumption such that every discharge ina basin would increase or decrease with the sameratio according to the discharge change in the outlet.
In the CDRMV3, water depth and discharge on
every cell are simulated based on the kinematicwave equation, and those two values are convertibleat each calculation time step by the specified
stage-discharge relationship on each grid cell. If
every discharge is reset by the ratio of simulated andobserved discharge at the basin outlet (R=Qs/Qo),each water depth would also be reset by its own
stage discharge relationship. The updating of thedischarge means the water depth is also updated,which finally results in the storage amountreformation.
Figure 3 shows the schematic drawing ofdischarge and water depth profile at the ideal one
dimensional basin, and an updating procedure with
the Q-ratio and the stage discharge relationship.
While every updated discharge has the same ratio of
discharge before and after the updating, the ratio ofwater depth will be different by each point based on

the variant topographic and physical characteristics
on that cell, such as slope and roughness coefficient.
The Q-ratio method has several merits comparedto S-ratio method: 1) a characteristic stage discharge
relationship of each cell can reflect topographic andphysical characteristics of each individual cell, 2)
steady state assumptions of Q-S relationship are notneeded to acquire an observed storage amount and3) can efficiently get the ratio and update.
Figure 4 shows the updated results from the two
ratio methods and the steady-state method. The
steady-state method is resetting the state variables
uniformly as the initial condition based on the outlet
discharge at the updating time. The reset of thespatial variables causes the poorest simulation result
as discussed in the previous study2),3). The Q-ratio
method gives positive updating results as good as
the results from the S-ratio method. To check thequantitative updating efficiency, root mean squareerror (RMSE) is calculated and presented in Table 1.

- 69 -

Table 1. RMSE after the updating point (unit: m3/sec).
Steady-state

S-Ratio

Q-Ratio

Event 979

63.04641.74039.036

Event 996

36.14229.15228.677

3. KALMAN FILTER APPLICATION
INTO THE DISTRIBUTED MODEL(1) Measurement Update Algorithma) Using the S-ratio Method
In the measurement update algorithm of the
Kalman filter, an observation vector yk at time step k
is described as a linear vector function of a statevector xk, and observation noise vector wk assuming
white noise is included. In the previous study of
Kim et al.3), the observation equation was the Q–Srelationship; thus, the gradient of the relationshiprepresents the conversion matrix Hk at the updatingtime step. In the Kalman filter using S-ratio method(hereafter KFSM), the observation is the basin outlet
discharge and the state value is total storage amount,and those two values are related by the Q-S
relationship.
As illustrated in Figure 5 (a), the measurement
update algorithm was used to update the total
storage amount and its error variance of the subject
basin. With the updated storage amount, the S-ratio
method is applied to update the spatially distributed
water depth in the distributed hydrologic model.
Then the updated state variables are given to themodel simulation for the time update algorithm. The
procedure of the measurement update algorithm
using the Q-S curve and S-ratio method is presented
in Figure 5 (a).
b) Using the Q-ratio Method
In the Kalman filter using Q-ratio method(hereafter KFQM), the observation and the statevalue of the filter have the same form as the outlet
discharge (or discharges at several points in a basin
if the basin has multiple gauge station). Both KFSM
and KFQM are basically for updating the distributed
storage amount in the basin. However, each method
has a different state value in the Kalman filter
algorithm: total storage amount for the KFSM, anddischarge at the outlet for the KSQM.
Since the state value in the KFQM is the same as
the observation (discharges at the basin outlet), themeasurement update algorithm directly estimates
the best discharges minimizing its error statistically.
The results from the update algorithm are theupdated discharge with its error variance as shownin Figure 5 (b).
The updating of spatially distributed statevariables is carried by Q-ratio method; the ratio of
updated discharge to the simulated discharge isapplied to every discharge on computation node,
then the discharges converted to the water depth
through the specific stage-discharge relationship oneach grid cell. The procedure of the measurement
update algorithm using the Q-ratio method ispresented in Figure 5 (b).

(a) using the S-ratio method(b) using the Q-ratio method
Fig. 5 Illustration of the measurement update algorithm.

xˆ ( k k ), σ k(σ k ) 2 = P(k k )Hydrologic
SYSTEMxˆ (k + 1 k ), σ k +1
(σ k +1 ) 2 ≈ Fk P(k k ) FkT

P(k + 1 k ) = (σ k +1 ) 2 + Vk

Fig. 6 Schematic drawing of the time update algorithm
using Monte Carlo simulation technique.

(2) Time Update Algorithm
In the CDRMV3, a complicated relationshipexists between the present and the next time-stepstate variable. The current water depth at each cell
responds interdependently to the next step’s water
depth according to the current spatial distribution ofwater depth and rainfall input. It is impractical todefine the system matrix Fk to express the process
formally from the hydrologic system equations;instead, a Monte Carlo simulation (Figure 6) wasapplied to solve this problem in the previous study3).
Many random variables (one hundred total storage
amounts in the previous study, KFSM) values weregenerated within a range of the normal distribution,N( xˆ (k k ) , σk), where xˆ (k k ) is the updated statevariables and σk is the square root of the error
variance P(k│k) at time step k.
For KFQM in this study, one hundred discharge
values are generated for the Monte Carlo simulation.
Then the Q-ratio method is applied to reset the
distributed state variables using the ratio of eachgenerated value to the updated value xˆ (k k ) . Thenthe multiple input sets are repeatedly given to therunoff simulation to calculate its evolution throughthe system until the next update time step k+1.
Finally, the propagated probability distribution,N( xˆ ( k + 1 k ) ,σk+1), is acquired from the one hundred
simulation results assuming that the results follow
normal distribution.

- 70 -

Table 2. Comparison of the Monte Carlo Simulation (M.C.S.) Procedure for KFSM and KFQM
Procedures

KFSM

KFQM

1. Measurement Update Results

Sk, σs,kQk, σq,k

2. Generated Value for M.C.S.

{Sk,1,…Sk,i, …Sk,100} ~ N(Sk, σs,k){Qk,1,…Qk,i, …Qk,100} ~ N(Qk, σq,k)3. Ratio Calculation

Rs,i = Sk,i / Sk,oRq,i = Qk,i / Qk,o

4. Ratio Multiplies by

Updated Water Depths of each cell

Updated Discharges of each cell

N(Sk+1, σs,k+1)

N(Qk+1, σq,k+1)

5. M.C.S.(Time Update) Results

Fig. 7 Observed and feedback values through the Kalman filter using the S-ratio and Q-ratio method (Event 998).

Now, the estimated state xˆ ( k + 1 k ) is mean valueof the one hundred state variables evolved throughthe model, and the variance σk+12 is regarded as
FkP(k│k)FkT. Adding the system error variance Vk
completes the error variance P(k+1│k) at time step
k+1 (see Figure 6). The estimated error variance
FkP(k│k)FkT stands for a evolution of the error
variance P(k│k) until time step k+1 through thesimulation, and Vk stands for new generated or
added system variance during that simulation. The
newly added system variance is caused by system
structure or input data such as rainfall.
Table 2 explains the input, output data and its
procedure for the Monte Carlo simulation for both
KFSM and KFQM. In the procedure of the KFQM,
firstly, Qk and σq,k represent the updated discharge
and its standard deviation at time step k, which are
the output from the measurement update algorithmof the filter. Following the probability distribution,
N(Qk, σq,k), one hundred possible discharges(Qk,1,…Qk,i, …Qk,100) are generated and each Q-ratio
is calculated (Rq,1,…Rq,i, …Rq,100) with an updated
discharge Qk,o. After one hundred ratio values areapplied to the updated discharges on each cell, one
hundred input sets for the Monte Carlo simulation
are ready. From the Monte Carlo simulation results,
which are assumed following a normal distribution,
the mean value of discharge, Qk+1 and its standard
deviation, σq,k+1 are acquired for the time step k+1.
Adding the system error variance Vk completes thetime update algorithm.
In the case of the KFSM, the difference is in thegeneration of storage amount values and applying
the storage amount ratio to the updated water depths
on each cell. For the details of KFSM, please refer
to Kim et al.3).
(3) Application Results and Discussions
The Kalman filter coupled distributed hydrologic
model CDRMV3 was tested on the Kamishiiba
basin and the feedback results from the both
application are compared. Both system noise andobservation noise in the filter are assumed timeinvariant and having 30m3/sec amount of standard
deviation following the previous study3). For the
KFSM, the system noise in discharge form is- 71 -

converted to the form of storage amount using the
Q-S relationship before it is applied in the filter(refer to Kim et al.3) for more details). Utilizing the
Q-S relationship to convert the noise term, eventhough it is not significant, offers another additional
error source. However, the system noise ofdischarge form is directly utilized in the KFQM,
since the state value in the filter is the outlet
discharge itself, which is another benefit of this
newly introduced method.
Figure 7 shows the comparison of the feedback
results from the both Kalman filter applications in
the form of discharge hydrograph. The feedback
from the KFSM and KFQM both show the
improved simulation results compared to the offline
simulation result. When RMSE to the observation
was examined, it was found that the Kalman filter
using Q-ratio method gives more improved results
as shown in Table 3.
Table 3. RMSE of each simulation (unit: m3/sec).
EVENT

Offline

KFSM

KFQM

Event 979

36.59

31.04

30.32

Event 996

23.84

8.43

6.55

Event 998

21.73

12.97

11.55

Event 999

25.21

22.11

16.52

The computation time for KFQM requires about
the same for the KFSM calculation. The most time
consumption of the both method is on the Monte
Carlo simulation. In this study, one hundred
simulations were carried out to assess the evolutionof the state variable and its error variance at every
time increment: every one hour. Time elapsing will
be variable under different model running
conditions, such as catchment size, updating
frequency and simulation number for the Monte
Carlo simulation. Reasonable model condition
should be carefully designed based on operating
environments.

4. CONCLUSIONAs shown already in the previous research, theS-Ratio method showed improved simulation results
upon updating. However, the Q-S relationship undera steady state assumption was found necessary inorder to produce a storage amount corresponding to
an observed discharge, which may also lead to
unexpected error in the S-Ratio method. On theother hand, with the Q-ratio method, a successful
updating of the state variables becomes simple andeffective, as well as the stage-discharge relationshipof each cell reflects topographic and physicalcharacteristics of each cell during the update.
The Kalman filter was successfully coupled tothe distributed hydrological model, CDRMV3, to
update internal distributed state variables using the
Q-ratio method. In the Kalman filter application,
both the observation and state variable to be updated
are outlet discharge, therefore there is no necessary
to formulate the observation equation. The
measurement update algorithm decide the optimal
outlet discharge based on the given noise terms, andthe ratio of updated to simulated discharge isapplied for setting the discharges for each cell in the
distributed hydrologic model. In the CDRMV3, thereset of discharges is directly connected to the resetof water depths on each grid cell by the stagedischarge relationship.
For the prediction algorithm, a Monte Carlosimulation was adopted to diffuse the state variable
and its error covariance. The CDRMV3 using the
Kalman filter yielded more effective results thanoff-line simulations and can thus be used as aprobabilistic forecast algorithm. The developed
algorithm can incorporate the uncertainty of input
and output measurement data as well as theuncertainty of the model itself.
Further work will include developing a method
to apply several observations for the updating
method to fully utilize the properties of a distributedhydrologic model. One way would be to define subbasins according to the location of observation
stations, and to apply different factors to each subbasin for updating the spatial pattern of water depth.
For nested river basins with different observation
stations, sophisticated methods to define the ratio
would be required.
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- 72 -(Received September 30, 2006)