Application of wavelet-based multi-model Kalman filters to real-time flood forecasting

HYDROLOGICAL PROCESSESHydrol. Process. 18, 987–1008 (2004)Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/hyp.1451

Application of wavelet-based multi-model Kalman filtersto real-time flood forecasting

Chien-Ming Chou1 and Ru-Yih Wang2*1 Department of Information Management, Ming Dao University, Changhau 523, Taiwan, ROC

2 Department of Bioenvironmental Systems Engineering, National Taiwan University, Taipei 106, Taiwan, ROC

Abstract:

This paper presents the application of a multimodel method using a wavelet-based Kalman filter (WKF) bank tosimultaneously estimate decomposed state variables and unknown parameters for real-time flood forecasting. Applyingthe Haar wavelet transform alters the state vector and input vector of the state space. In this way, an overall detail plusapproximation describes each new state vector and input vector, which allows the WKF to simultaneously estimateand decompose state variables. The wavelet-based multimodel Kalman filter (WMKF) is a multimodel Kalman filter(MKF), in which the Kalman filter has been substituted for a WKF. The WMKF then obtains M estimated statevectors. Next, the M state-estimates, each of which is weighted by its possibility that is also determined on-line, arecombined to form an optimal estimate. Validations conducted for the Wu-Tu watershed, a small watershed in Taiwan,have demonstrated that the method is effective because of the decomposition of wavelet transform, the adaptationof the time-varying Kalman filter and the characteristics of the multimodel method. Validation results also revealthat the resulting method enhances the accuracy of the runoff prediction of the rainfall–runoff process in the Wu-Tuwatershed. Copyright 2004 John Wiley & Sons, Ltd.

KEY WORDS wavelet transform; multimodel Kalman filter; rainfall–runoff; flood forecasting

INTRODUCTION

As Taiwan is located in the major typhoon track in the western Pacific Ocean, typhoons are an influentialweather phenomenon in Taiwan, where short, steep upstream channels characterize all watersheds. Floodingassociated with heavy rainfall is the disaster that causes the greatest loss of property and life in the area. Itis therefore essential to determine the relationship of the rainfall and runoff processes and develop a floodforecasting system to provide protection and warning systems.

Implicit or explicit error updating procedures compensate for the errors between the simulated and theobserved discharge hydrographs in the case of real-time forecasting, where the hydrological simulation modeloperates on-line by applying the most recent data available. These updating procedures vary in detail butgenerally modify one or more of the inputs, outputs, model parameters and state variables of the model(Shamseldin and O’Connor, 1999).

Ahsan and O’Connor (1994) critically reviewed some applications of the Kalman filtering technique in riverflow forecasting. As Ahsan and O’Connor demonstrated algebraically, recognition and acknowledgementof that uncompromising flood forecasting objective (which either implicitly or explicitly assumed themeasurement errors to be zero), in the context of model forecast error updating, causes the special limitingcase of the ‘pure prediction’ scenario, and results in a greatly simplified and degenerate form of the Kalmanfilter. The linear transfer function, or ARX, model is as efficient as the Kalman filter in this form in real-timeriver flow forecasting (Ahsan and O’Connor, 1994).

* Correspondence to: Ru-Yih Wang, Department of Bioenvironmental Systems Engineering, National Taiwan University, Taipei 106, Taiwan,ROC. E-mail: [email protected]

Received 4 June 2001Copyright 2004 John Wiley & Sons, Ltd. Accepted 9 May 2003

988 C.-M. CHOU AND R.-Y. WANG

Rainfall–runoff systems are generally non-linear and time-variant. Their parameters vary in time andspace, and when they are assumed constant they are only so by assumption. The inadequacy of the modelitself, parameter uncertainty, errors in the data used for parameter estimation and inadequate understandingof the rainfall–runoff process, owing in part to randomness, may cause errors in a rainfall–runoff model.Incorporating the Kalman filter in a rainfall–runoff model may reduce the error in the runoff prediction arisingfrom the uncertainty caused by the physical process, the model and the input data (Lee and Singh, 1998). Leeand Singh also reviewed in detail several applications of the Kalman filter to hydrology.

The combination of the Kalman filter with the ARX-type models used in flow forecasting can be applied inmany ways. The Kalman filter provides a method for recursively updating the coefficients of ARX-type modelsby treating them as the state variables. However, the two sources of error (inadequacy of the flow modeland measurement error) are indistinguishable as they are lumped together as vk in the observation equation.Treating the river flow discharge as the state variables, the above two sources of errors can be separated(Ngan and Russell, 1985). The proposed wavelet-based multimodel Kalman filter (WMKF), working on thedecomposition components of the river flow discharge at different levels, can provide the detailed model errorat different levels in order to estimate the state variables more precisely than conventional techniques operatingat only one resolution level. An adaptive Kalman filter approach is used to specify the state covariance matrixand the measurement covariance matrix.

Wavelets are becoming an increasingly important tool for image and signal processing. Wavelets effectivelyextract both time and frequency-like information from a time-varying signal. Wavelets and Kalman filterscan handle the non-stationary signal; many applications combine these two tools. Hong (1994) developed amultiresolutional multiple-model target tracking algorithm, which uses the wavelet to transform data betweendifferent resolution levels. Cheng and Sun (1996) used the theory of wavelet packet analysis to decomposethe primitive measurements into two parts, namely the ‘trend’ and ‘fluctuation’ of the measurement. Honget al. (1998) utilize the discrete wavelet transform to decompose the state variables of the Kalman filter intodifferent components at the desired resolution level, and then process the prediction, correction and updateprocedure using the decomposed state variables. The term ‘resolution level’ refers to the available level ofdecomposition of data and is determined by the length of the data set (e.g. state vectors). The significance toriver flow forecasting of the term resolution level is that it maximizes the advantage of separating noise fromsignals by wavelet transform, to solve the problems produced by noise.

In the field of hydrology, wavelets essentially have been used in order to identify coherent convectivestorm structures and characterize their temporal variability (Szilagyi et al., 1999), and also in order toexplain the non-stationarity of karstic watersheds (Labat et al., 2000). Wavelet analysis of rainfall rates andrunoffs and wavelet rainfall–runoff cross-analysis gives meaningful information on the temporal variabilityof the rainfall–runoff relationship. Chou and Wang (2002) applied the discrete wavelet transform (DWT) todecompose and compress the unit hydrograph. Moreover, a wavelet-based linearly constrained least meansquares (WLCLMS) algorithm is also used to estimate on-line the wavelet coefficients of the unit hydrograph.The updated wavelet coefficients of the unit hydrograph, convoluted with effective rainfall input in the waveletdomain, allow for accurate prediction of one-step-ahead runoff in the time domain. The proposed approachallows the unit hydrographs to vary in time and accurately predicts runoff from a basin, thus making it highlypromising for flood forecasting.

The hydrological parameters or state variables of the state space methods can be assumed to be randomsignals. A general approach in random signal estimation and decomposition is first to estimate the unknownsignal based on its measurement data and then to decompose the estimated signal according to the resolutionrequirement. This two-step approach best suits off-line procedure and is often inappropriate for real-timeapplications. Chui and Chen (1999) have developed an alternative to the two-step approach. This optiondetermines the estimated signal in one step so that the resulting signal naturally possesses the desireddecomposition.

In the context of electric power systems, Zheng et al. (2000) presented a wavelet transform method forload forecasting. The decomposition scheme of multiresolution analysis (MRA) studied the stochastic nature

Copyright 2004 John Wiley & Sons, Ltd. Hydrol. Process. 18, 987–1008 (2004)

REAL-TIME FLOOD FORECASTING 989

of the wavelet coefficients for the daily load variation. Their study indicates that the stochastic process ofthe wavelet coefficients can be modelled as a random walk process. The wavelet coefficients are thereforemodelled as the state variables of Kalman filters. The recursive Kalman filter algorithm best estimates thewavelet coefficients. The predicted daily load is the inverse of the predicted wavelet coefficients.

Ren et al. (2000) offered a multimodel Kalman filter (MKF) method for stochastic systems. The methodutilizes a group of linear models, for instance M models, to represent a non-linearly stochastic system. Itutilizes Kalman filters associated with the group of models to obtain the M estimated states. The M state-estimates, each of which is weighted by its probability that is also calculated on-line, are then combined toform an optimal estimate. The results of their simulation demonstrate that the method is applicable to manyengineering problems for state estimation and parameter identification for non-linear stochastic systems.

The central idea of the present investigation is that the original state vector and input vector, based on thefour previously measured rainfall and runoff values, can be translated into the new state vector and inputvector in the wavelet domain. The resulting multiresolutional state estimation performs more efficiently than aconventional one at only one resolution level (Chui and Chen, 1999). Armed with this simultaneous estimationmethod for state variables and unknown parameters, along with an effective multiresolutional approach, thepresent authors use the combination of both and achieve novel results.

The rest of this investigation is organized as follows. First, the discrete wavelet transform (DWT) isintroduced. Next, the essential elements of the conventional Kalman filter, wavelet-based Kalman filters(WKF) and wavelet-based multimodel Kalman filters (WMKF), respectively, are presented. Then, to illustratethe proposed method, a case study on a small watershed in Taiwan is described. Finally, the results arediscussed and conclusions are presented.

DISCRETE WAVELET TRANSFORM AND FILTER BANKS

The purpose of DWT is to maximize the benefit of separating noise from signals using a wavelet transform,to solve the problems produced by noise. Many new methods and models for analysing and simulatinghydrological time-series have been presented. However, most of the methods and models apply to hydrologicaltime-series. In practice, studying hydrological time-series is difficult because these series are affected bycomplex factors. Each hydrological time-series contains several frequency components, with restricted factorsand development rules. Using only one resolution component to model the hydrological time-series doesnot easily clarify the internal mechanism. Wavelet-based multiresolution analysis could be applied to modelhydrological time-series.

The DWT is similar to a Fourier series but, in many ways, is much more flexible and informative. It canbe made periodic, like a Fourier series, to represent periodic signals efficiently. However, unlike a Fourierseries, it can be applied directly to non-periodic transient signals with excellent results (Burrus et al., 1998).Where the Fourier series maps a one-dimensional function of a continuous variable onto a one-dimensionalsequence of coefficients, the wavelet expansion maps it onto a two-dimensional array of coefficients. Thetwo-dimensional representation allows the signal to be localized in both time and frequency.

Wavelet transform has been applied in two ways to modelling hydrological systems at various resolutionlevels. One is the application of redundant wavelet transforms to the observed effective rainfall and directrunoff time-series, to obtain the wavelet coefficients at each resolution level. Then, these wavelet coefficientsare regarded as the input vectors to hydrological forecasting models. The multiresolution analysis usinga wavelet transform also can decompose the complex hydrological time-series into several simple time-series. Then, the hydrological forecasting models can be used to carry out combination forecasting and thusimprove the accuracy of flow forecasting. The other application of wavelet transform to model hydrologicalsystems involves their application to observed effective rainfall and hydrological systems, to obtain the waveletcoefficients at each resolution level. Ordinary direct runoff time series are regarded as the output of systems.Applying the Haar wavelet transform alters the state vector and input vector of the state space. In this way,



overall detail plus approximation describe each new state vector and input vector, allowing the wavelet-basedmultimodel Kalman filters to simultaneously estimate and decompose state variables and estimate parameters.

For a low-pass filter, the frequency components above the cut-off frequency will be eliminated; onlythe low-frequency component remains. In contrast to a low-pass filter, for a high-pass filter, the frequencycomponents that are lower than the cut-off frequency will be omitted, and only the high-frequency componentsremain. The lower resolution coefficients can be calculated from the higher resolution coefficients using a tree-structured algorithm called a ‘filter bank’ (Burrus et al., 1998). This supports the very efficient calculation ofthe expansion coefficients and is also known as the discrete wavelet transform, and relates wavelet transformsto hydrological forecasting.

An excellent description of the DWT used in this study can be found in Chui and Chen (1999) but a summaryis presented here for convenience. The algorithm can be described as follows. The DWT of a discrete signalxk is the decomposition of a family of functions k (wavelet function) and �k (scaling function). It is revealedthat they have an orthonormal basis (Mallat, 1989). By using a low-pass filter fhkg, which derives from thescaling function �k , and a high-pass filter fgkg, which derives from the corresponding wavelet function k ,the wavelet decomposition is represented by (Chui and Chen, 1999)

xN�1kL

D1∑

jD�1h2k�jxNj �1�

xN�1kH

D1∑

jD�1g2k�jxNj �2�

where N is a fixed resolution level. The wavelet coefficients, as a complement to xN�1kL

, are denoted by fxN�1kH

g.The original signal fxNk g can be recovered from the two filtered and downsampled (lower resolution) signalsfxN�1kL

g and fxN�1kH

g as (Chui and Chen, 1999)

xNk D1∑

jD�1h2j�kxN�1

jLC

1∑jD�1

g2j�kxN�1jH

�3�

The operation defined by Equations (1) and (2) is designated the DWT, and the inverse discrete wavelettransform (IDWT) is defined by Equation (3).

Consider a sequence of signals at resolution level N of length L

XNk D [xNk , xNk�1, . . . , x

Nk�LC1]T �4�

The following matrix operator forms (Chui and Chen, 1999) can express Equations (1) and (2)

XN�1kL D H N�1XNk �5�

andXN�1kH D GN�1XNk �6�

where the operators H N�1 and GN�1 comprise low-pass and high-pass filter responses mapping from levelN to level N� 1. Similarly, when mapping from level N� 1 to level N, Equation (3) can be expressed inoperator form as (Chui and Chen, 1999)

XNk D �H N�1�TXN�1kL C �GN�1�TXN�1

kH �7�

Meanwhile, the orthogonality constraints in operator form are expressed as (Chui and Chen, 1999)

�H N�1�TH N�1 C �GN�1�TGN�1 D I �8�



and [H N�1�H N�1�T H N�1�GN�1�T

GN�1�H N�1�T GN�1�GN�1�T

]D

[I 00 I

]�9�

with I being the unit identity matrix.A filter bank can simultaneously decompose the signal into multiple levels. For instance, the following

composite transform can be applied to decompose XNk into two levels (Chui and Chen, 1999):XN�2kL

XN�2kH

XN�1kH

D TN�2jNXNk �10�

where

TN�2jN D[ H N�2H N�1

GN�2H N�1

GN�1

]�11�

is an orthogonal matrix, simultaneously mapping XNk onto both levels of the filter bank. An example of thetwo-tap Haar wavelet transformation which is applied throughout the investigation is now examined, in whichthe low-pass and high-pass filters are (Hong et al., 1998)

H D[ p

22

p2

2

]and G D

[�

p2

2

p2

2

]�12�

The corresponding two-level transform matrix is given by

TN�2jN D

1/2 1/2 1/2 1/2�1/2 �1/2 1/2 1/2

�p2/2

p2/2 0 0

0 0 �p2/2

p2/2

�13�

which is clearly an orthogonal matrix. The reason for using the Haar transformation method rather thanthe other transformation methods is that the simplest filters correspond to the Haar wavelet, providing theadvantage of orthogonality (self-duality) (Nikolaou and Mantha, 2000).

WAVELET-BASED KALMAN FILTER

Conventional Kalman filter

The Kalman filter, a linear dynamic system, imitates the behaviour of an observed natural process. TheKalman filter accomplishes this by optimally estimating the current state of the observed process at eachinstant. This state estimate is coupled with an internal model of the observed process to predict the nextoutput. When the next output occurs, the filter calculates the difference between its prediction and the actualoutput, and uses this residual to adjust (or update) its estimate of the state. The new corrected estimate canthen predict the next state, thus completing one full iteration of the filter. The standard Kalman filter recursionsare formulated as (Minkler and Minkler, 1993)

xkC1 D AkC1,k xk C Bkuk C kwk �14�

zk D Ckxk C vk �15�

OxkC1/k D AkC1,k Oxk/k C Bkuk �16�

PkC1,k D AkC1,kPk,kAkC1,kT C kQkk

T �17�



OzkC1/k D CkC1 OxkC1/k �18�

KkC1 D PkC1,kCkC1T[CkC1PkC1,kCkC1

T C RkC1]�1 �19�

OxkC1/kC1 D OxkC1/k CKkC1�zkC1 �CkC1 OxkC1/k� �20�

PkC1,kC1 D �I�KkC1CkC1�PkC1,k �21�

where xk , uk and zk are the state variables, deterministic control inputs and measurement vectors, respectively,AkC1,k is the transfer matrix from k to k C 1, Bk , k and Ck are the coefficient matrices, wk and vk are thesystem noise and measurement noise vectors respectively, the corresponding covariance matrices being Qkand Rk , and OzkC1/k , OxkC1/k and OxkC1/kC1 are the a priori estimate of zkC1, the a priori estimate of xkC1 and thea posteriori estimate of xkC1, respectively.

Adaptive Kalman filter

Implementing the Kalman filter requires the first- and second-order moments of the noise processes tobe known a priori. However, the precise a priori statistics for the stochastic disturbances in both the stateand measurement processes are not known, owing to modelling errors and the presence of outliers in themeasurement sequence. Therefore, the unknown a priori noise statistics must be estimated adaptively withthe system states simultaneously.

A situation where AkC1,k , Bk , k and Ck are provided is now discussed. In this situation, an adaptive Kalmanfilter for estimating the state and the noise variance matrix may be termed a noise-adaptive filter. Althoughseveral algorithms exist for achieving this (Chui and Chen, 1999), there is still no algorithm available that isderived from the truly optimal criterion. For simplicity and to account for a complex of correlated noise, suchas the decomposition of system dynamic noise, the situation where AkC1,k , Bk , k , Ck and Qk are providedis discussed herein.

Some studies have determined Qk for hydrological forecasting. Theoretically, Qk is the matrix of covarianceof the model error wk . Model error wk is a non-stationary random process, so Qk is time-variant. However, onlyan average value can be determined in practice in hydrological forecasting. Generally, the 5–10% inaccuracy,ε, in the model, yields Qk D υI, where I is a diagonal matrix and υ D ε2 (Lee et al., 1998). In this study, Qkwas determined from the 5–10% inaccuracy, ε, of the decomposed state vectors, yielding Qk D υI.

Hence, only Rk has to be estimated. The innovation approach (Chui and Chen, 1999) seems to work veryefficiently for this situation. From the incoming data information zk and the optimal prediction Oxk/k�1 obtainedin the previous step, the innovation sequence is defined as

yk D zk �Ck Oxk/k�1 �22�

Obviouslyyk D Ck�xk � Oxk/k�1�C vk �23�

which is a zero-mean Gaussian white noise sequence. By taking variances on both sides yields

Sk D Var�yk� D CkPk,k�1CkT C Rk �24�

This yields an estimate of Rk , i.e.ORk D OSk �CkPk,k�1Ck

T �25�

where OSk is the statistical sample variance estimate of Sk given by

OSk D 1

k � 1

k∑iD1

�yi � yi��yi � yi�T �26�



and yi is the statistical sample mean defined by

yi D 1

i

i∑jD1

yj �27�

Decomposition of the state vector and input vector

The multiresolution decomposition seems to separate components of a signal in a manner that is superiorto most other methods for analysis, processing or compression. Wavelets have been called ‘the MathematicalMicroscope’ because of the ability of the discrete wavelet transform to decompose flexibly a signal intodifferent, independent scales. Given this powerful and flexible decomposition, the linear and non-linearprocessing of signals in the wavelet transform domain offer new methods for signal detection, filtering andcompression.

The DWT of a vector is the outcome of a linear transformation that produces a new vector of equaldimensions to that of the primal vector. Mallat’s MRA algorithm (Mallat, 1989) can efficiently accomplishthis transformation, which is also called the process of decomposition.

Two discrete filters are necessary to compute the DWT: the decomposition low-pass filter H and thedecomposition high-pass filter G , both of which are associated with the primal wavelet. Notably, computingthe IDWT requires the corresponding reconstruction low-pass filter QH and the reconstruction high-pass filterQG associated with the dual wavelet (Nikolaou and Mantha, 2000). The two wavelets are biorthogonal (Strangand Nguyen, 1996).

Figure 1 schematically clarifies the use of the DWT to the decomposition of three levels (Nikolaou andMantha, 2000). Let the vector, h , be a vector in two-dimensional space <2r whose DWT is to be computed.This vector represents the approximation (or signal) at resolution level r, denoted as hr . Filtering this vectorwith the low-pass filter H leads to a vector that is of identical dimension. In addition, downsampling this

h8 h8

h7

h6

h5

h3

h1

h2

h4

h7

h6

h5

h4

h3

h2

h1

∗

∗

∗

∗ ∗

∗

Resolutionlevel 3

Resolutionlevel 2

Resolutionlevel 1

Resolutionlevel 0

lowpassfilter

lowpassfilter

lowpassfilter

highpassfilter

highpassfilter

highpassfilter

detail atlevel 2

detail atlevel 0

signal atlevel 0

detail atlevel 1

Figure 1. Schematic illustration of computing the DWT (Nikolaou and Mantha, 2000)



vector creates the vector hr�1 of half the original length, i.e. of dimension 2r�1 ð 1. This vector is labelled ‘theapproximation at resolution level r � 1’. Likewise, filtering hr with the high-pass filter G , and downsamplingproduces the vector i r�1, which is half the length of hr , and is called the ‘detail at level r � 1’. The detaili r�1 is kept aside, and the approximation hr�1 is again filtered and downsampled to yield two vectors hr�2

and i r�2, which are each half the length of hr�1. These values are the approximation and detail, respectively,at resolution level r � 2. Proceeding in this manner, down to resolution level 0, the DWT vector, h 2 <2r ,with respect to the primal wavelet, is the final aggregate (Nikolaou and Mantha, 2000)

h D [�h0�T �i 0�T �i 1�T Ð Ð Ð �i r�1�T

]T�28�

It preserves the dimension of the original vector h , and consists of the approximation at the lowest resolutionlevel the details at that level and all the details at higher levels.

Estimation and decomposition using the WKF

Consider a deterministic, linear and discrete time system represented by the linear difference equation (Kuo,1999)

yk D a1yk�1 C a2yk�2 C Ð Ð Ð C anyk�n C b1uk�1 C b2uk�2 C Ð Ð Ð C bmuk�m �29�

where yk is the system output at time index k and uk is the corresponding system input. The terms n and mspecify the number of the past output and input values required to model the present output. The coefficients(i.e. the parameters) a1, a2, . . . , an and b1, b2, . . . , bm correspond to output and input variables. The statespace form can represent the above system as follows (Kuo, 1999)

xkC1 D AkC1,kxk C Bkuk �30�

yk D Ckxk �31�

where xk are the state variables at time index k. For instance, consider the case of n D m D 4, at the originalcoarse resolution level N, for which the difference equation has the form below

yNkC1 D a1yNk C a2y

Nk�1 C a3y

Nk�2 C a4y

Nk�3 C b1u

Nk C b2u

Nk�1 C b3u

Nk�2 C b4u

Nk�3 �32�

The state space form representing the above system is as follows:ykC1

N

ykN

yk�1N

yk�2N

D

a1 a2 a3 a4

1 0 0 00 1 0 00 0 1 0

ykN

yk�1N

yk�2N

yk�3N

C

b1 b2 b3 b4

0 0 0 00 0 0 00 0 0 0

ukN

uk�1N

uk�2N

uk�3N

�33�

[zkN] D [ 1 0 0 0 ]

ykN

yk�1N

yk�2N

yk�3N

�34�

These can be expressed simply as

XNkC1 D AXkN C BUk

N �35�

zNk D CXkN �36�

Applying the Haar wavelet transform alters the state vector and input vector of the state space. In this way,each new state vector and input vector is described by an overall ‘detail’ plus ‘approximation’, which allowsthe WKF to estimate and decompose simultaneously.



Two-level decomposition is then performed, by substituting Equation (35) into Equation (10). This results inXN�2kC1L

XN�2kC1H

XN�1kC1H

D A

XN�2kL

XN�2kH

XN�1kH

C B

UN�2kL

UN�2kH

UN�1kH

�37�

where A D TN�2jNA�TN�2jN�T and B D TN�2jNB�TN�2jN�T.The above equation describes a dynamic system for the decomposition quantities. Substituting Equation (36)

into Equation (10) yields its associated measurement equation,

zNk D C

XN�2kL

XN�2kH

XN�1kH

�38�

where C D C�TN�2jN�T.If noise affects the system, the system can be expressed as

XN�2kC1L

XN�2kC1H

XN�1kC1H

D A

XN�2kL

XN�2kH

XN�1kH

C B

UN�2kL

UN�2kH

UN�1kH

C

wN�2kL

wN�2kH

wN�1kH

�39�

zNk D C

XN�2kL

XN�2kH

XN�1kH

C vNk �40�

where wN�2kL

, wN�2kH

, wN�1kH

are the system noise for the ‘approximation’ and ‘detail’ at the different levels,and vNk is the measurement noise. Letting Wk D [wN�2

kLwN�2kH

wN�1kH

]T, then EfWkg D 0, EfWk�Wk�Tg D

Q, EfvNk g D 0, and EfvkN�vkN�Tg D R.The system and measurement equations, Equations (39) and (40), for the decomposed quantities have now

been obtained. These quantities must next be estimated from measurement data. The Kalman filter is readilyapplied to Equations (39) and (40) to provide optimal estimates for these decomposed quantities, resulting inthe WKF. Note that the WKF works on the wavelet decompositions of the time-series at different resolutions,in contrast to the ordinary Kalman filter, which operates directly on the observed time-series. Chui and Chen(1999) demonstrated that the above described simultaneous approach out performs the ordinary direct Kalmanfilter, even for simple two-level decomposition and estimation. The difference is accentuated as the numberof levels is increased.

WAVELET-BASED MULTIMODEL KALMAN FILTERS (WMKF)

Ren et al. (2000) presented a multimodel Kalman filter method for stochastic systems based on a thoroughinvestigation of the Kalman filter. Their method utilizes a group of linear models, for instance M models,to represent a non-linear stochastic system. This system uses Kalman filters associated with the group ofmodels to obtain M estimated states. Each of the M state-estimates is weighted by its probability that is alsocalculated on-line. The estimates are then combined to form an optimal estimate. Their simulation resultsreveal that the method can be applied to a number of engineering problems for state estimate and parameteridentification for non-linear stochastic systems. Thus, in the present study, the concept of the multimodelmethod is adopted, resulting in the WMKF rather than the WKF.



Rewriting Equations (39) and (40) and taking into consideration the deterministic input, the linear stochasticsystems models have the form

XN�2kC1L

XN�2kC1H

XN�1kC1H

D A��

XN�2kL

XN�2kH

XN�1kH

C B��

UN�2kL

UN�2kH

UN�1kH

C

wN�2kL

wN�2kH

wN�1kH

�41�

zNk D C ��

XN�2kL

XN�2kH

XN�1kH

C vNk �42�

These equations can be expressed more concisely as

xkC1 D A��xk C B��uk C wk �43�

zk D C ��xk C vk �44�

where xk D [XN�2kL

XN�2kH

XN�1kH

]T, uk D [UN�2kL

UN�2kH

UN�1kH

]T, wk D [wN�2kL

wN�2kH

wN�1kH

]T,and A��,B�� and C �� are the coefficient matrices that are functions of the given parameters �. Thesecoefficient matrices represent the characteristics of dynamic systems and depend on the parameters � 2 RM.Generally, an approximate value of the parameters, �, can be obtained. Equations (43) and (44) can alsosignify a number of non-linear system models (Ren et al., 2000). For example, the non-linear systems atM operator points can be linearized to allow the M linear equations to approximate the original non-linearone. The choice of parameters � 2 f�1, �2, . . . , �Mg adopted could be discrete values. The combination ofthe M subsystems structured from these M discrete values would then comprise the non-linear system withthe advance towards the solution of the non-linear system by the deep investigation of the linear system.In this study, however, the parameters, �, are regarded as random variables and the probability of �i fromthe observed data and ith subsystem is calculated. Following this procedure, the sum of multiples of the Mstate-estimates with its possibility constitutes the state-estimates of the system.

In general, the M models and the linearization points are determined by the points of operation. However,as Ren et al. (2000) pointed out, the true operation point of a system is unknown, so the parameters � can beconsidered to be random variables. The probability that a system operates in �i can be computed recursivelyfrom the difference between the true system output and the M models’ outputs. Then, an estimate of theproduct of the state of the M models with the above probability yields the state estimate of the system. Thisis the fundamental idea that underlies the multimodel Kalman filter. Therefore, in this study, the M modelswere adopted in accordance with data available.

Let �i be one arbitrary value of the parameters, �. Redefining Equations (43) and (44) as follows (Renet al., 2000)

A��i� D A�i�, B��i� D B�i�, C ��i� D C �i�, i D 1, 2, . . . ,M �45�

the M discrete wavelet transform-based stochastic linear equation can then be described as (Ren et al., 2000)

xkC1 D A�i�xk C B�i�uk C wkC1, i D 1, 2, . . . ,M �46�

zk D C �i�xk C vk, i D 1, 2, . . . ,M �47�

The parameters �i D f�1, �2, . . . , �Mg represent the uncertainty or non-linearity of the system. The proposedmethod utilizes the WMKF to acquire the parallel estimation of the M linear subsystems. The Bayes methodweighs each of the M state-estimates by its probability, and combines them to form an optimal estimate.

Assume the input–output data at time index k, for given �i is defined as

Yk��i� D fZk,Uk, �ig �48�



where Zk D fz1, z2, . . . , zkg,Uk D fu1,u2, . . . ,uk�1g. When the value of Yk��i� is known, the conditionalmean and the covariance matrix are represented as (Ren et al., 2000)

Oxk/k�i� D Efxk/Yk��i�g �49�

k,k�i� D Ef Qxk/k�i� QxTk/k�i�/Yk��i�g �50�

where Qxk/k�i� D xk � Oxk/k�i�. Applying the elementary formula of the WKF algorithm, the following equationsare obtained (Ren et al., 2000)

OxkC1/kC1�i� D A�i� Oxk/k�i�C B�i�uk CKkC1�i�fzkC1 � C �i�[A�i� Oxk/k�i�C B�i�uk]g �51�

kC1,kC1�i� D [I�KkC1�i�C �i�

] [A�i�k,k�i�AT�i�C Q

kC1

]�52�

where A��i�,B��i�,C ��i� are the system matrices, given �i, and KkC1�i� is the Kalman gain, for given �i,which satisfies the equation (Ren et al., 2000)

KkC1�i� D [A�i�k,k�i�AT�i�C QkC1

]C T�i�fC �i�[A�i�k,k�i�AT�i�C QkC1

]C T�i�C RkC1g�1 �53�

Applying the WKF to each subsystem, the M estimation of the state variables can be obtained. Assumingthat P��i/YkC1� represents the conditional probability for the ith subsystem at the time index k, and applyingthe Bayes formula, produces the equation below (Ren et al., 2000)

P��i/YkC1� D L�k/�i�N∑jD1

L�k/�j�P��j/Yk�

P��i/Yk� �54�

where

L�k/�i� D∣∣∣ QkC1,k�i�

∣∣∣�1/2exp

{�1

2QzkC1 QkC1,k�i�QzTkC1

}�55�

QkC1,k�i� D C �i�kC1,k�i�C �i�T C RkC1 �56�

QzkC1 D zkC1 � C �i� OxkC1/k�i� �57�

In the proposed WMKF model, theM state-estimates, each of which is weighted by its probability calculatedon-line, are combined to generate an optimal estimate. Equation (54) was introduced to explain the probabilityweighting of each estimate.

The optimal state estimation of the WMKF is given as (Ren et al., 2000)

Oxk/k DN∑iD1

P��i/Yk� Oxk/k�i� �58�

Similarly, the state prediction of the WMKF is given as

OxkC1/k DN∑iD1

P��i/Yk� OxkC1/k�i� �59�

APPLICATION OF THE WMKF

Rainfall–runoff modelling

The autoregressive model with exogenous input (ARX). The autoregressive model with exogenous input(ARX), also known as the linear transfer function model, is a commonly used dynamic system representation.



This investigation considers black box modelling of the rainfall–runoff relationship using the ARX model tobe the conventional value or base-line method against which the proposed WMKF is compared. The ARXmodel relates the system inputs to the system outputs with a constant-coefficient linear difference equation

qk D a1qk�1 C a2qk�2 C Ð Ð Ð C anqk�n C b1rk�1 C b2rk�2 C Ð Ð Ð C bmrk�m C ek �60�

where qk is the direct runoff at time k, rk is the effective rainfall at time k, ek is the noise term at timek, a1, a2, . . . , an and b1, b2, . . . , bm are the coefficients, respectively. The ARX can be expressed in linearregression form, in which the regressors can have different dimensions (Wing et al., 1990), and identifiedusing the well-known ordinary least squares (OLS) method.

The proposed WMKF method.. Only the four previously measured rainfall and runoff values are used toconstruct the state space model because the watershed area is steep and small, with the upstream flow travellingdownstream in just a few hours. Past experience suggests that these data can satisfactorily represent the waterimpulse and hydrological persistence of the present flow. That is, the order of the ARX Model is determinednot only from past experience but also by the AIC criterion.

Consider the flow discharge sequence at the highest resolution level (level N) XNk D [xkNxk�1Nxk�2

Nxk�3N]T

D [qkNqk�1Nqk�2

Nqk�3N]T as raw state variables, and consider also the added rainfall input term UNk D

[ukNuk�1Nuk�2

Nuk�3N]T D [rkNrk�1

Nrk�2Nrk�3

N]T. Two-level decomposition is then performed, leading tonew state vector and input terms, having the form

TN�2jN

qkN

qk�1N

qk�2N

qk�3N

D TN�2jNXNk D

XN�2kL

XN�2kH

XN�1kH

D

qN�2kL

qN�2kH

qN�1kH1

qN�1kH2

�61�

and

TN�2jN

rkN

rk�1N

rk�2N

rk�3N

D TN�2jNUNk D

UN�2kL

UN�2kH

UN�1kH

D

rN�2kL

rN�2kH

rN�1kH1

rN�1kH2

�62�

where qN�2kL

, qN�2kH

, qN�1kH1

and qN�1kH2

are the different components of the decomposition of the block riverdischarge data at levels N� 2 and N� 1. Similarly, rN�2

kL, rN�2kH

, rN�1kH1

and rN�1kH2

are different components ofthe decomposition of the block rainfall data at levels N� 2 and N� 1.

Regarding the wavelet transform of the original state vector and input vector to the new state vector andinput vector, that is substituting Equations (41) and (42) for Equations (61) and (62) respectively, results inthe WKF. The coefficient matrices A��,B�� and C �� are selected from the results of the calibration of theARX model by the OLS method for 12 typhoon events, i.e. M D 12, i.e. the result of the calibration of eachtyphoon event constitutes the coefficient matrices A�i�, B�i�, C �i�, resulting in the proposed WMKF. Thesatisfactory validating results illustrate the suitability of the proposed model and the justification for usingthe 12 ARX models. The proceeding WMKF is then activated to simultaneously obtain parameter estimation,recursive estimation and decomposition of the river discharge series.

Study basin

Selecting the upstream Wu-Tu watershed, located in the north of Taiwan, as the study area demonstratesthe feasibility of applying the WMKF. Figure 2 presents a map of the whole watershed, the shaded area ofwhich defines the upstream watershed of Wu-Tu, having an area of 198 km2. Eighteen typhoon events over



the Wu-Tu watershed were analysed for the case study. The data interval is 1 h. Twelve of these events wereused for calibration to determine the parameters of the ARX, and the other six were used for verification ofthe performance of both the ARX and the proposed WMKF model. The detailed information of all relevanttyphoon events was provided in Table I.

Figure 2. The map of Wu-Tu watershed showing the study area near Taipei, Taiwan

Table I. The detailed information of all relevant typhoon events

Events Year Duration(hours)

Shape of peak Peak discharge(m3/s)

Note

Abby 1986 84 Multiple 586 ValidationAbe 1990 48 Single 780 CalibrationAmber 1997 48 Single 958 ValidationBess 1985 58 Single 596 CalibrationDoug 1994 47 Multiple 341 CalibrationFread 1994 48 Single 240 CalibrationFred 1994 42 Single 217 CalibrationGladys 1994 38 Single 434 CalibrationHerb 1996 38 Single 1026 CalibrationIrving 1979 24 Single 697 CalibrationLynn 1987 144 Multiple 1947 ValidationMaury 1981 49 Single 1229 ValidationNelson 1985 44 Single 1219 CalibrationRuth 1991 24 Single 450 CalibrationSeth 1994 71 Multiple 454 CalibrationWayne 1981 81 Multiple 727 ValidationWinnie 1997 38 Single 1038 ValidationZane 1996 48 Single 650 Calibration



Comparison of model performances

The results were predicted 1 h ahead and were evaluated based on four different kinds of criteria toquantitatively compare the ARX and WMKF, as illustrated below.

1. Coefficient of efficiency, CE, is defined as

CE D 1 �

T∑iD�nC1�

[q�i�� Oq�i�]2

T∑iD�nC1�

[q�i�� q]2

�63�

where Oq�i� denotes the discharge of the simulated hydrograph for time period i (m3/s), q�i� is the dischargeof the observed hydrograph for time period i (m3/s) and q represents the average discharge of the observedhydrograph for time period i (m3/s). The better the fit, the closer CE is to 1.

2. Error of total volume (EV)

EV D

T∑iD�nC1�

[Oq�i�� q�i�]

T∑iD�nC1�

q�i�

�64�

3. The error of peak discharge, EQp�%�, is defined as

EQP�%� D OqP � qP

qpð 100% �65�

where OqP denotes the peak discharge of the simulated hydrograph (m3/s) and qP is the peak discharge ofthe observed hydrograph (m3/s).

4. The error of the time for the peak to arrive, ETp, is defined as

ETP D OTP � TP �66�

where OTP denotes the time for the simulated hydrograph peak to arrive (hours) and TP represents the timerequired for the observed hydrograph peak to arrive (hours).

RESULTS AND DISCUSSION

The number of levels of the desired decomposition determines the length of the data block. To comparewith the proposed WMKF, the chosen order of the ARX(n,m) model determined not only by past studybut also by AIC criterion, corresponding to the length of the data block, is n D m D 4. Table II lists thecoefficients obtained from the calibration of the ARX model using the OLS method for the 12 typhoon eventsduring 1979 and 1996. The average values of the estimated parameters are � Oa1, Oa2, Oa3, Oa4, Ob1, Ob2, Ob3, Ob4� D�1Ð3920,�0Ð6075, 0Ð1520,�0Ð0257, 1Ð0477, 1Ð4946, 1Ð8144, 0Ð8413�, which were also used for the validation



Table II. The coefficients obtained by calibration of the ARX model

Events Year a1 a2 a3 a4 b1 b2 b3 b4

Abe 1990 1Ð5223 �0Ð8624 0Ð2390 0Ð0105 �1Ð3076 3Ð7968 2Ð2147 0Ð8936Bess 1985 1Ð8043 �0Ð9800 0Ð1467 0Ð0010 1Ð2153 �0Ð9491 0Ð7420 0Ð5439Doug 1994 1Ð5422 �0Ð7566 0Ð0900 0Ð0666 0Ð7183 1Ð0655 0Ð0110 2Ð1162Fread 1994 1Ð1186 �0Ð2503 0Ð2176 �0Ð1693 �0Ð0621 2Ð1561 1Ð5415 0Ð8153Fred 1994 0Ð9621 0Ð3905 �0Ð5352 0Ð0974 1Ð1278 �0Ð4428 1Ð7718 2Ð5455Gladys 1994 1Ð0203 �0Ð5455 0Ð5555 �0Ð1730 1Ð4621 2Ð3576 1Ð6572 2Ð5839Herb 1996 1Ð3395 �0Ð4836 0Ð0331 0Ð0153 4Ð7214 �0Ð7720 5Ð6089 �3Ð6162Irving 1979 1Ð7083 �1Ð0311 0Ð2082 0Ð0043 3Ð1823 2Ð1289 0Ð4328 1Ð2058Nelson 1985 1Ð9933 �1Ð5555 0Ð6290 �0Ð1286 0Ð3649 1Ð2827 2Ð2145 �0Ð0310Ruth 1991 0Ð8062 0Ð0930 �0Ð0973 0Ð0175 0Ð2928 3Ð6432 3Ð8236 1Ð6317Seth 1994 1Ð3331 �0Ð4296 0Ð0824 �0Ð0578 0Ð1548 1Ð5626 1Ð8541 0Ð2725Zane 1996 1Ð5532 �0Ð8788 0Ð2551 0Ð0072 0Ð7023 2Ð1055 �0Ð0989 1Ð1341Average 1Ð3920 �0Ð6075 0Ð1520 �0Ð0257 1Ð0477 1Ð4946 1Ð8144 0Ð8413

Table III. The efficiency results of the calibration for the 12 ARX modelsa

Events Year EV (%) CE EQP (%) ETP (h)

Abe 1990 �0Ð54 0Ð995 0Ð61 0Bess 1985 �1Ð08 0Ð964 4Ð79 1Doug 1994 �0Ð82 0Ð982 5Ð04 0Fread 1994 0Ð37 0Ð998 0Ð03 0Fred 1994 �2Ð76 0Ð939 8Ð87 1Gladys 1994 0Ð56 0Ð994 0Ð84 0Herb 1996 1Ð44 0Ð979 �2Ð71 �1Irving 1979 �1Ð41 0Ð982 0Ð34 1Nelson 1985 �0Ð45 0Ð975 2Ð67 1Ruth 1991 0Ð23 0Ð997 1Ð04 0Seth 1994 1Ð16 0Ð976 �3Ð43 1Zane 1996 �1Ð07 0Ð977 4Ð94 �1Average 1Ð00 0Ð980 2Ð94 0Ð58

a The columns for EV, EQP and ETP all contain negative values, and these columns present the averagesof the absolute values.

of the ARX model. Table III presents the calibration results of the ARX model, based on the four modelefficiency criteria outlined in the previous section from Equation (63) to Equation (66).

The WMKF was applied to the ARX discrete linear transfer function model. Applying the calibrationresults to six verification time typhoon events recorded on that watershed, one step ahead (i.e. 1 h) forecastsare produced in every case. Various validations have been implemented to compare the performance of theARX model (having fixed average coefficients) with that of the proposed WMKF. Table IV displays thequantitative comparisons of these two models based on the four selected kinds of model efficiency criteria.The validation results reveal the significant superiority of the CE for the WMKF over that of the ARXmodel.

The WMKF slightly outperforms the ARX model based on the criterion of the EV. However, most notably,the conventional ARX model ineffectively predicts the peak discharge and the time to peak discharge.Regarding flood forecasting, this phenomenon falls short of the demand for protection and flood warningsystems. The proposed WMKF has improved estimation and adaptation in contiguous environments andconsequently predicts the peak discharge and the time to peak discharge more precisely.



Table IV. The results of the validationa

Events Year EV (%) CE EQP (%) ETP (h)

ARX WMKF ARX WMKF ARX WMKF ARX WMKF

Abby 1986 �1Ð92 0Ð54 0Ð967 0Ð995 10Ð30 5Ð15 0 0Amber 1997 �3Ð60 �1Ð17 0Ð955 0Ð987 16Ð76 3Ð60 0 0Lynn 1987 �0Ð66 0Ð21 0Ð985 0Ð994 2Ð12 �0Ð46 0 0Maury 1981 �1Ð79 �0Ð27 0Ð966 0Ð990 4Ð61 �1Ð82 1 0Wayne 1986 �1Ð74 1Ð27 0Ð969 0Ð995 10Ð15 0Ð16 0 0Winnie 1997 �0Ð81 �0Ð13 0Ð987 0Ð997 7Ð01 1Ð02 0 0Average 1Ð75 0Ð60 0Ð972 0Ð993 8Ð49 2Ð04 0Ð17 0

a The columns for EV, EQP and ETP all contain negative values, and these columns present the averages of the absolute values.

Figures 3a and 4a depict two representative results of validation. The ARX model can match the observeddata accurately according to previously measured values of rainfall and runoff data, but it is insensitive tocases of abrupt change, such as peak discharge. Overall, the WMKF functions more efficiently than the ARXmodel, because of the decomposition ability of wavelet transform, the adaptation of time-varying Kalmanfilters and the characteristics of the multimodel method based on the four kinds of model efficiency criteriaoutlined above.

The performance of the WMKF could be compared with that of other traditional forecast updatingprocedures. This comparison is essential for testing whether the complexity of the WMKF yields significantimprovement in the performance over that of the traditional updating procedures. The Kalman filter algorithmis one of the procedures for updating forecasts, so a comparison between the performances of the WMKFwith that of the Kalman filter based on the criterion of the CE is shown in Table V. Table V demonstratesthat based on the criterion of the CE the WMKF outperforms the direct Kalman filter.

Additionally, a comparison between the performances of the WKF with that of the Kalman filter, to evaluatethe improvement in performance by combining tiles with the WKF is shown as Table V. Table V demonstratesthat based on the criterion of the CE the WKF outperforms the direct Kalman filter. Based on the criterionof the CE, the comparison of the performance of the WMKF with that of the MKF to clarify the role of thewavelet in the overall performance of the WMKF is also shown in Table V. Table V also demonstrates thatbased on the criterion of the CE the WMKF outperforms the MKF.

The four previously measured rainfall and runoff values consist of the original state vector and the inputvector. Wavelet analysis theory can decompose the state vector and input vector into two parts at a fixedresolution level. Two-level decomposition is chosen for the original state variables and input vector with thedimension equal to 4. Applying the DWT to the original state vector and input vector causes the new statevector and input vector in the wavelet domain to have the same dimension of the original vectors. The newstate vector comprises different components of decomposition, such as the approximation at level 0 and thedetail at levels 0 and 1. Figures 3b and 4b display the decomposed components at different levels of blockriver discharge data at different times. These figures also indicate the variation of the new state variables withtime.

The calibrated coefficients of the ARX model, using the OLS method for 12 typhoon events as Table IIdemonstrates, yield the given value of the coefficient matrices in the multimodel method, resulting in 12 WKFat each time index. The proposed WMKF was applied to estimate the state variables and unknown parameterssimultaneously. Applying the concept of probability combines the given parameters � for the group of 12,each of which is weighted by its probability to form the estimation of the coefficients at each time index.Two representative results of the variation of the coefficients are shown as Figures 3c, 3d, 4c and 4d.

When the range of variation of parameters � is small, the linear Kalman filter can solve the filtering problemof the system. When the range of variation of parameters � is classed as medium, the extended Kalman filter



0 10 20 30 40 50 60 70 80 90

Time (hours)

Dis

char

ge (c

ms)

1400

1300

1200

1100

1000

900

800

700

600

500

400

300

200

100

0

−100

−200

(b)

Signal at level 0

Wayne 1986

Detail at level 0

Detail at level 1 (#1)


0 10 20 30 40 50 60 70 80 90

Time (hours)

0

100

200

300

400

500

600

700

800

Dis

char

ge (c

ms)

0 10 20 30 40 50 60 70 80 900

10203040506070

Rai

nfal

l (m

m/h

r)

(a)

Obs. discharge

Wayne 1986

Est. discharge (WMKF)

Est. discharge (ARX)

Figure 3. (a) The forecasts of the discharge hydrograph for Typhoon Wayne, 1986. (b) The decomposed components at different levels ofblock river discharge data at different times for Typhoon Wayne, 1986. (c) The time-varying coefficients (a1, a2, a3, a4) for Typhoon Wayne,

1986. (d) The time-varying coefficients (b1, b2, b3, b4) for Typhoon Wayne, 1986



0 10 20 30 40 50 60 70 80 90

3

2

1

0

−1

−2

−3

Val

ue o

f coe

ffic

ient

s a1

, a2,

a3,

a4

Value of a1

Wayne 1986

Value of a2

Value of a3

Value of a4

Time (hours)

(c)

−6

Val

ue o

f coe

ffic

ient

s b1

, b2,

b3,

b4

Value of b1

Wayne 1986

Value of b2

Value of b3

Value of b4

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

Time (hours)0 10 20 30 40 50 60 70 80 90

(d)

Figure 3. (Continued )



Obs. discharge

Winnie 1997

Est. discharge (WMKF)

Est. discharge (ARX)

0

40

30

20

10

00 10 20 30 40

100

200

300

400

500

600

700

800

900

1000

1100

1200

Dis

char

ge (c

ms)

Rai

nfal

l (m

m/h

r)

0 10 20 30 40

Time (hours)

(a)

Dis

char

ge (c

ms)

−400

−200

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 10 20 30 40

Time (hours)

(b)

Signal at level 0

Winnie 1997

Detail at level 0



Figure 4. (a) The forecasts of the discharge hydrograph for Typhoon Winnie, 1997. (b) The decomposed components at different levelsof block river discharge data at different times for Typhoon Winnie, 1997. (c) The time-varying coefficients (a1, a2, a3, a4) for Typhoon

Winnie, 1997. (d) The time-varying coefficients (b1, b2, b3, b4) for Typhoon Winnie, 1997



−3

−2

−1

0

1

2

3

Val

ue o

f coe

ffic

ient

s a1

, a2,

a3,

a4

0 10 20 30 40

Time (hours)

(c)

Value of a1

Winnie 1997

Value of a2

Value of a3

Value of a4

Value of b1

Winnie 1997

Value of b2

Value of b3

Value of b4

−6

Val

ue o

f coe

ffic

ient

s b1

, b2,

b3,

b4

Time (hours)0 10 20 30 40

(d)

−5

−4

−3

−2

−1

0

1

2

3

4

5

6

Figure 4. (Continued )



Table V. The comparison based on the criterion of the CE between the performances of different models

Events Year CE

ARX Kalman filter MKF WKF WMKF

Abby 1986 0Ð967 0Ð973 0Ð981 0Ð985 0Ð995Amber 1997 0Ð955 0Ð957 0Ð972 0Ð970 0Ð987Lynn 1987 0Ð985 0Ð988 0Ð990 0Ð991 0Ð994Maury 1981 0Ð966 0Ð971 0Ð980 0Ð978 0Ð990Wayne 1986 0Ð969 0Ð974 0Ð983 0Ð985 0Ð995Winnie 1997 0Ð987 0Ð990 0Ð992 0Ð992 0Ð997Average 0Ð972 0Ð976 0Ð983 0Ð984 0Ð993

can generally solve the filtering problem of the system. When the range of variation of parameters � isconsiderable, the multimodel Kalman filter can be suggested to solve the filtering problem of the system(Ren et al., 2000). Table II demonstrates the substantial range of considerable variation of the parameters �obtained from the results of the calibration of the ARX model. Thus the proposed WMKF is appropriate forthe parameter estimations in the case study presented here. Figures 3c, 3d, 4c and 4d display the parametervariations of the validation results.

This investigation grouped the first four sampled data into a processing unit and conducted the process ofsimultaneous estimation and decomposition of random signals within the unit. From the fifth sampled datavalue, the first data value in the unit is released, and the fifth value is accepted into the unit to operate anotherround in the process of simultaneous estimation and decomposition of random signals. This choice of data isappropriate with short fast-moving typhoons.

CONCLUSIONS

In this investigation, the WMKF was applied to simultaneously estimate the decomposed state variables andunknown parameters from noisy measurement. The wavelet transform of hydrological parameters, such as thedecomposition of the discharge of river flow at different resolution levels, are considered the time-varyingstate variables. The proposed method recursively yielded state variables decomposition and estimation andalso parameter estimation that led to a simultaneously decomposed estimation problem which, when solved,provides real-time runoff prediction.

A two-level wavelet decomposition is chosen, which establishes the data length as 4. Applying the wavelettransform to the original state vector, which consists of the four previously measured rainfall and runoffvalues, results in a new state vector having the same dimension. The new state vector consists of differentcomponents of decomposition, such as the ‘approximation’ at level 0, the ‘detail’ at levels 0 and 1, whichcan represent the more detailed characteristics of the flow discharge. The recommended WKF working on thedecomposition components of the river flow discharge at different levels can provide detailed model errors atdifferent levels in order to estimate the state variables more accurately than by using conventional techniquesat only one resolution level. Applying a group of WKFs combines the M state-estimates, each of which isweighted by its probability, to form an improved and optimal estimate.

The ARX and WMKF are also compared, based on the four model efficiency criteria outlined above.This approach enhances the predictive accuracy of the conventional one obtained by the ARX, owingto the decomposition of wavelet transform, the adaptation of time-varying Kalman filters and the specialcharacteristics of the multimodel method. These case study results reveal that the proposed method improvesthe accuracy of one-hour-ahead runoff prediction of the rainfall–runoff process of rivers in Taiwan, suggestingpossible applications for this proposed approach in real-time flood forecasting.



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Application of wavelet-based multi-model Kalman filters to real-time flood forecasting