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2013,25(4):564-571 DOI: 10.1016/S1001-6058(11)60397-2

Dual state-parameter optimal estimation of one-dimensional open channel model using ensemble Kalman filter*

LAI Rui-xun (赖瑞勋), FANG Hong-wei (方红卫), HE Guo-jian (何国建) Department of Hydraulic Engineering and State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084, China Yellow River Institute of Hydraulic Research, Zhengzhou 450003, China, E-mail: lairuixun@163.com YU Xin (余欣), YANG Ming (杨明), WANG Ming (王明) Yellow River Institute of Hydraulic Research, Zhengzhou 450003, China (Received August 25, 2012, Revised November 2, 2012) Abstract: In this paper, both state variables and parameters of one-dimensional open channel model are estimated using a framework of the Ensemble Kalman Filter (EnKF). Compared with observation, the predicted accuracy of water level and discharge are impro- ved while the parameters of the model are identified simultaneously. With the principles of the EnKF, a state-space description of the Saint-Venant equation is constructed by perturbing the measurements with Gaussian error distribution. At the same time, the rough- ness, one of the key parameters in one-dimensional open channel, is also considered as a state variable to identify its value dynamica- lly. The updated state variables and the parameters are then used as the initial values of the next time step to continue the assimilation process. The usefulness and the capability of the dual EnKF are demonstrated in the lower Yellow River during the water-sediment regulation in 2009. In the optimization process, the errors between the prediction and the observation are analyzed, and the rationale of inverse roughness is discussed. It is believed that (1) the flexible approach of the dual EnKF can improve the accuracy of predi- cting water level and discharge, (2) it provides a probabilistic way to identify the model error which is feasible to implement but hard to handle in other filter systems, and (3) it is practicable for river engineering and management. Key words: Ensemble Kalman Filter (EnKF), lower Yellow River, water-sediment regulation, inverse problem

Introduction The state variables and the parameters are two

essential factors to determine the accuracy of a hydro- dynamic model. The state variables which are in a state of flux are propagated over time controlled by the numerical model while the parameters describe the dynamic characteristics of a river, such as bed rough- ness. The numerical models, however, with some li- mitations or hypotheses or both, are imperfect to de-

* Project supported by the National Basic Research and Development Program of China (973 Program, Grant No. 2011CB403306), the Ministry of Water Resources’ Special Funds for Scientific Research on Public Causes (Grant No. 200901023), and the Central Scientific Institutes Foundation for Public Service (Grant No. HKY-JBYW-2012-5). Biography: LAI Rui-xun (1981-), Male, Ph. D. Candidate Senior Engineer

scribe all aspects of a natural river as its complexity can be controlled not only with hydraulics but also with geological theories or sedimentation[1]. In recent decades, the improvements in prediction accuracy of water levels and discharge, for example, can be classi- fied in three ways: one is to estimate the parameters using sets of algorithms[2-6], another is to construct an optimal scheme by coupling the one- and two-dimen- sional models or by coupling the hydrodynamic model with the hydrology model[7,8], and the third is to train the optimal algorithms using historical data sets[9].There are many sources of uncertainties asso- ciated with these applications which are essential to affect the model output. Although parts of these stu- dies contain the effects of parameter uncertainties on generating accurate forecasts, other sources of errors, such as input, output, model structure, and observa- tions, have not yet been considered. In other words,

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Fig.1 The flowchart of dual state-parameter estimation using EnKF the uncertainties of the model structure are added to parameters’ error.

The Kalman filter, which, in a linear system, takes into account all sources of the uncertainties from both the model structure and observations, is one of the widely-used sequential assimilation techniques. Although much research using the Kalman filter has obtained optimal roughness parameters[10], their draw- backs can be listed as follows: (1) some studies con- sidered the parameters to be of static values or the assimilation is applied only to state variables, (2) the Kalman filter is limited to the linear system, and the filter process may result in computational instability as the hydrodynamic equations is a nonlinear system. Furthermore, the accuracy of the predicted value would decrease if either the observations are scarce or the measure error is large, (3) it is difficult to define the hydrodynamic model error. To deal with the draw- backs of the Kalman filter, a Monte Carlo based Ense- mble Kalman Filter (EnKF) was introduced. Compa- red to the traditional Kalman filter, one of the adva- ntages of the EnKF is that it need not define the model error directly.

The series of Kalman filters, including the EnKF, was originally developed for dynamic systems while

in this paper a strategy of dual state-parameter assimilation is introduced for hydrodynamic systems. With observation, the updated states and parameters are considered as the initial values of the next time step to continue the assimilation process. The usefulness and the capability of the dual ensemble Kalman filter are demonstrated in studying the lower Yellow River during the water-sediment regulation in 2009. 1. Data assimilation and equations 1.1 Ensemble Kalman Filter (EnKF)

Apart from the traditional Kalman filter, the EnKF represents the model uncertainty by perturbing the forcing data and parameters:

+1 +1 +1 +1 +1= + , (0, )i i it t t t tU U N R , = 1, ...,i n (1)

+1 +1 +1= +i it t t , = 1, ...,i n (2)

where +1itU is the thi ensemble member of forcing

data at time +1t and +1it is the thi ensemble of para-

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meter at time +1t . The model error is displayed by

+1it and +1(0, )iN R refers to the normal distribution

with a mean of 0 and a covariance of R . Driven by the forcing data, the fluxes of both

state variables and parameters can be described by the theory of nonlinear stochastic-dynamic processes. The model forecast of each ensemble member is as follows

++1 +1 +1= ( , , ) +i i i i

t t t tX f X U (3)

where X represents the state variable, +1

itX is the

thi forecast ensemble at present time +1t , +i

tX is the

thi optimal ensemble at the prior time t and +1it is

the error of thi ensemble at time +1t . Similar to the forcing data, the ensemble of the

observation is generated by perturbing the measure- ments

+1 +1 +1 +1 +1= + , (0, )i i it t t t tY H X N S (4)

where +1

itY is the thi trajectory of the observation by

adding the error with the covariance of S . Genera- lly, the noise of the observation can be considered as a normal distribution.

Then the prior error covariance of the model can be approximately equal to

T+1 +1 +1 +1 +1

=1

1( )( )

1

ni i

t t t t ti

P X X X Xn

(5)

+1 +1=1

1=

ni

t ti

X Xn

(6)

where +1tX

is the mean value of an ensemble member.

Finally, the optimal estimation of the state variable for each ensemble member can be described as

++1 +1 +1 +1 +1= + ( )i i i i

t t t t tX X K Y H X (7)

T T 1

+1 +1 +1 +1= ( + )t t t tK P H HP H S (8) where +1tK is the Kalman gain matrix and H is the

transitional operator weighting the value between ob- servation and model. 1.2 State-space description of equations

The Saint-Venant equation considering the lateral inflow can be written as

+ = l

Z QB q

t x

(9)

2

2+ + + = 0

Q QQ Q ZgA g

t x A x C AR

(10)

where B is the width of the cross section, Q total

volume discharge, Z the water level, lq the side flow

discharge per unit channel length, R the hydraulic radius and C the Chézy coefficient which can be cal- culated as

1/ 61=C R

n (11)

where n is the roughness. Using the Preissmann sche- me, the state-space description of the Saint-Venant equation can be constructed as follows:

+1

+1|= + +

t t

j jt t t tQZ QZ QZ

j j

Q QU

Z Z

,

+1 +1

+1 +1= +

t t

Q j jt tQZ QZ

Z j j

y QH

y Z

(12)

where tQZ is the model error, +1t

QZ the observation

error of water level and discharge, +1tQZH the observa-

tion operator and its value is equal to 1 once there is observational data, t

QZU the control matrix and +1/t t

QZ

the transport matrix. Table 1 Error distributions of observations and initial

values of parameters

State or parameter

Description Error distribution Initialvalue

Z Water level

(m) 2(0, 0.015 )N -

Q Discharge

(m3/s) 2

obs.[0, (0.025 ) ]N Q -

n Roughness - 0.03-0.01

Note: 2(0, 0.015 )N denotes the normal distribution with mean

of 0 and covariance of 0.015.

The purpose of the roughness inverse problem is not only for accurate prediction, but also one of the ways to deepen understanding of the system characte- ristics[11]. For the roughness inverse problem, observa- tion is needed. However, the bed roughness can not be measured directly. This paper constructs the observa- tion of roughness using the difference of the water level between two cross sections[12]. The roughness parameter which is only included in the momentum equation and its finite difference equation can be yie- lded from Eqs.(10) and (11):

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Fig.2 Plot of the lower Yellow River with tributaries and hydrological stations

+1 +12 2

+1 +1 +11 1+1

+1 +11 1

+ + +

t t

t t t tj j j j j jt

jt tj j

Q Q

A AQ Q Z ZgA

t x x

+1

+1 2+1 +1 4 / 3

( )( ) = 0

( )

tj t

jt tj j

Q Qg n

A R (13)

where t is the time step and x is the distance

between cross sections. Then the difference of the water level can be written as

+1 +1 +1 +1 21= = ( ) +t t t t

j j j jZ Z Z n (14)

where

+1

1+1 2 +1 4 / 3

( )=

( ) ( )

tj

jt tj j

Q Qx

A R ,

+1 +12 2

1 1+1+1 +1

= ( )

t t

j j jt tj jt t

j j

Q Q

A AxQ Q

g tA gA

Using the Taylor series expansion and neglecting the second-order term, the observation for roughness in Eq.(14) can be rewritten as

+1 +1 +1= ( ) + ( ) = +tj

t t t t tj j j j jn

fZ f n n n n

n

(15)

where

= 2 tjn , 2 2= ( ) + 2 ( )t t

j jn n

So the state-space form of the roughness can be given as

+1 =t tj jn n , +1 +1= +t t

j jZ n (16)

where refers to the initial difference of roughness

from the previous time to the present time.

Fig.3 Root Mean Square Errors (RMSEs) with and without data

assimilation (DA in the legend refers to data assimilation) 2. Assimilation process of dual state-parameter

Traditionally, in the optimal process of stochastic systems, the parameters are considered to be time in- variant. Within the framework of the dual state-para- meter system, both the state variables and parameters are calculated or predicted simultaneously. One method for such combined state variables and parame- ters is provided by joint estimation in that both state

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Fig.4 Simulated discharge with and without assimilation against observations at the six hydrological stations from Huayuankou to

Luokou and parameter vectors are concatenated to form a single vector. The drawback of this method is that it will result in unstable estimation when the number of the states or parameters increases, especially in the nonlinear system[13]. Another way to cope with these limitations is to estimate the state variable and the pa- rameter separately, in which two interactive filters are designed either to estimate the optimal states or to solve the inverse problem of parameters[14].

Figure 1 illustrates the process of dual state-para- meter estimation using the EnKF with its application in one-dimensional hydrodynamic model. The total number of the ensemble is estimated first at time t , and then the observations such as the water level, dis- charge and roughness parameters are perturbed. The normal distribution of observations and the rational range of parameters are listed in Table 1. According to the regulation of river measurement, the measurement uncertainty of water level is less than three centime- ters. The stochastic error of the discharge is restricted to normal distribution with the confidence level of 95%. So the error of discharge accounts for about 5%

of its total volume. The initial value of roughness is from 0.03 in the input and generally decreases to 0.01 in the output.

Using Eq.(3), the ensemble of the water level or discharge is propagated and each member represents one realization of the model replicates. Then, the prior covariance of the model is calculated using Eq.(5) and, once the observation is available, the optimal states can be yielded with Eq.(7). Next, using Eq.(15), such optimal water level and discharge are adopted to cal- culate observed value of roughness. After that, inverse roughness, which is considered to be the optimal value at this time, is evolved using the Kalman filter algori- thm. Finally, the updated states and roughness values are taken as the initial values in the next time step to continue the process of assimilation. 3. Results and analysis

One of the goals of regulating water and sedi- ment processes is to flush out the sediment and to en- hance flow conveyance capacity of the main channel.

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Figure 2 shows the study area, which is located in the lower Yellow River, from Xiaolangdi to Lijin crossing Henan and Shandong provinces, about 650 km, with eight hydrological stations along the river: Xiaolangdi, Huayuankou, Jiahetan, Gaocun, Sunkou, Aishan, Luokou and Lijin, while Xiaolangdi and Lijin acting as the input and output boundaries respectively.

During the 7th regulation in 2009, the data assi- milation scheme described above was adopted from 8 am of June 15 to 8 am of July 13, a total of 672 h. To make the simulated close to the true process of a natural flood, lateral inflow, irrigation, evaporation and infiltration was calculated in the numerical model. During the regulation, the total inflow of 51 m3/s from Qinhe and Yiluohe rivers, the outflow for irrigation of 388 m3/s, and the lose due to evaporation and infiltra- tion of 120 m3/s were taken into account. 3.1 Error analysis for water level and discharge

Water level and discharge are two most impo- rtant state variables in one-dimensional hydrodynamic model. The RMSE is used to measure the accuracy of predicted state variables. Figure 3(a) plots the RMSEs of discharge with and without data assimilation. With- out data assimilation, the discharge RMSE decreased slightly from 195.6 in the Huayuankou Station to 168.1 in the Jiahetan Station and increased greatly to about 270 in the Lijin Station. However, with data assimilation, the RMSE dropped obviously from 106.5 in the Huayuankou Station to 76.4 in the Lijin Station.

Figure 3(b) shows the result of RMSEs of water levels with and without data assimilation. Without the assimilation method, the RMSE decreased gradually from 0.44 in the Huayuankou Station to 0.24 in the Sunkou station, and it remained almost stable at about 0.4 from the Aishan Station to the Luokou Station. Sources of error were attributed primarily to the un- changed roughness value and, secondly, to the abse- nce of sediment calculation. However, with data assi- milation method, the RMSE dropped dramatically from 0.04 in the Huayuankou station to 0.09 in the Lijin Station. 3.2 Analysis of predicted discharge

During the regulation of flow and sediment pro- cesses in 2009, the simulated peak flow of the Huayuankou station was 4 220 m3/s, almost equal to the bankfull discharge, and maintained for a short time. For this reason, flood was controlled in the main cha- nnel.

Figure 4 shows the measured discharge in com- parison with the predicted result. Without data assimi- lation, the predicted processes of peak flow and its attenuation were correspondent with the observations. As for the simulated value at each hydrological station, the predicted discharge values at Huayuankou and

Jiahetan was in accordance with the observations since the lateral inflow and outfow were taken into consideration. For the stations from Gaocun to Lijin, however, the predicted discharges were much higher than the observations, and those discrepancies can be ascribed to (1) the inconsistence of the irrigation lose calculated in the model with the real process and (2) the errors from numerical model.

It is proved that, with assimilation system accom- panied by observed discharge, a prediction of high accuracy has been gained.

Fig.5 Results of sediment erosion and deposition after the 7th

water-sediment regulation (the negative number means erosion while the positive one means deposition E and D means erosion and deposition, X-H means river from Xiaolangdi to Huayuankou, H-J means Huayuankou to Jiahetan, J-G means Jiahetan to Gaocun, G-S means Gaocun to Sunkou, S-A means Sunkou to Aishan, A-L means Aishan to Luokou and L-L means Luokou to Lijin)

3.3 Analysis of water level and inverse roughness

Since the sediment was not calculated in the cur- rent model, it is necessary to understand the results of erosion and deposition before analyzing the predicted water level. Figure 5 illustrates the results of erosion and deposition after the 7th water-sediment regulation. It shows that, along the lower Yellow River, except for a small amount of sediment deposition from Aisha to Luokou stations, large amount of erosion happened and the number went even up to 9.96 ×106 t from Gaocun to Sunkou stations.

Figure 6 shows the predicted water level compa- red with the inverse roughness. Without data assimila- tion, the simulated water level was a little higher than the observations from Huayuankou to Aishan while it was lower at the Luokou station. One reason for the discrepancies was unreasonable initial value of rough- ness, and another was that there is no sediment calcu- lation adopted to adjust the change of bedform. Com- pared with the results of sediment transport plotted in Fig.5, it is believed that, if the sediment calculation is taken into the model, there will be a better simulated result in which the predicted water level will decrease from the Huayunkou to Aishan stations in the course of eroding while increase at the Luokou Station in the course of deposition.

It is illustrated that, with assimilation system ac- companied by observed water level, a more accurate

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Fig.6 Roughness using inverse method at the six hydrological stations prediction has been gained.

The inverse roughness, which determines the ac- curacy of the calculated water level and depends on a number of factors, was estimated using Kalman filter algorithm. Three conclusions about inverse roughness could be reached. Firstly, as the discharge increases, the inverse roughness drops simultaneously. This phe- nomenon, which is identical to the empirical relations between roughness and discharge, is obviously illu- strated at the four stations including those at Huayuankou, Gaocun, Sunkou and Aishan. Secondly, without the assimilation, the roughness parameter is a unique value which is 0.025 from Xiaolangdi to Jiahetan, 0.015 from Gaocun to Sunkou and 0.01 from Aishan to Lijin. But the inverse roughness values, which varies for each cross section and changes with water level and discharge, are gained using assimila- tion method. Such inverse roughness values for each cross section at each time are beneficial to getting a better predicted result. Thirdly, the inverse roughness values drop gradually from Xiaolandi to Lijin, which is similar to the changes of the roughness values with- out data assimilation.

Observations are essential to the EnKF. Unfortu- nately, the data obtained by current measurements always lags far behind the process of assimilation. Therefore, to achieve an ideal system of real time esti- mation, much research is still needed. 4. Conclusion

In this paper, a data assimilation framework using the dual EnKF for one-dimensional hydrodyna- mic model has been constructed, and its usefulness and applicability have been demonstrated in the predi- ction of water level and discharge in the downstream areas of the Yellow River. During the water-sediment regulation in 2009, the errors both with water level and discharge, and of the difference between observa- tion and prediction, are analyzed. The rational value of the inverse roughness is discussed considering the whole process of flooding. It is believed that (1) the flexible approach of the dual EnKF can improve the accuracy of predicted water level and discharge, (2) it provides a probabilistic way to identify the model error which is feasible to implement but hard to

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handle in other filter systems, (3) the inverse problem of roughness can be used in order to understand its activity in such a dynamic system in greater depth, and (4) the algorithm need not store all the past infor- mation in order to be practicable for river engineering and management. References [1] SCHUMM S. A. River variability and complexity[M].

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