Impulse responses of digital filters for use in separating incident and reflected water waves in a time domain are improved by using a non- linear least square ...
Proceedings of The Twelfth (2002) International Offshore and Polar Engineering Conference Kitakyushu, Japan, May 26 –31, 2002 Copyright © 2002 by The International Society of Offshore and Polar Engineers ISBN 1-880653-58-3 (Set); ISSN 1098-6189 (Set)
An Improved Time Domain Procedure for Separating Incident and Reflected Water Waves Akira Matsumoto1, Masashige Tayasu2, Setsuo Matsuda1, Tue Hald3 and Hans F. Burcharth3 1 2
Environmental Engineering Research Center, Tetra Co., Ltd., Tsuchiura, Ibaraki, Japan
Department of Civil Engineering, Fukui National College of Technology, Sabae, Fukui, Japan, (Ex-Research engineer of Tetra Co., Ltd.) 3
Department of Civil Engineering, Aalborg University, Aalborg, Denmark
the filter responses. However, this type of approach increases the computation time.
ABSTRACT Impulse responses of digital filters for use in separating incident and reflected water waves in a time domain are improved by using a nonlinear least square formulation. The applicability and limitations of the method are discussed. Trial computations using a set of analytical examples with known incident and reflected waves demonstrate the validity and usefulness of the proposed method.
In this study, a new procedure is proposed to determine the optimum impulse responses of digital filters to improve the performance of separation. First, a non-linear least square formulation is derived to acquire the optimum filter responses. Second, the performance of separation is investigated using a set of analytical examples with known incident and reflected waves.
KEY WORDS: reflected wave, time domain, frequency response, digital filter, non-linear least square method.
PRINCIPLE OF ACTIVE WAVE ABSORPTION The active wave absorption is based on the AWASYS system (Frigaard and Christensen, 1994), which was originally based on the simultaneous on-line measurement of the surface elevation at two different points. In this study, a velocity based system is used. The following formulation is quoted from Troch and De Rouck (1999). See their paper for the details.
INTRODUCTION To separate incident and reflected water waves from a composite wave field is of prime importance in the field of coastal engineering. If an accurate and simple method is provided for this purpose, both physical and numerical experiments can be conducted without re-reflected waves from the offshore boundary.
Fig.1 shows the set-up of the wave flume equipped with twocomponent velocity meters (u, w) at (x1, z1) and a wave generatingabsorbing boundary at x0=0. It schematically demonstrates the principle of the system. The water depth in the wave flume is kept d. The correction signal η * that cancels out the reflected wave is determined from superposition of the two filtered velocity signals u*+w*. The digital Finite Impulse Response (FIR) filters are operated using a time domain discrete convolution of the velocities (u, w) and the impulse response hi, where, i=u or w. For the horizontal velocity, the input/output relationship of the FIR filter is expressed as follows:
Goda and Suzuki (1976) presented a frequency domain method for the estimation of irregular incident and reflected waves in random seas. Mansard and Funke (1980) improved the method by Goda and Suzuki using a least square technique. Zelt and Skjelbreia (1992) extended the method to accommodate an arbitrary number of wave gauges. These methods are all based on the frequency domain representation. Recently, Frigaard and Brorsen (1995) proposed a method for separating incident and reflected waves based on digital filters. This method was entirely different from the ones described above because it works in a time domain. The method was applied to both a physical (Frigaard and Christensen, 1994) and numerical (Troch and De Rouck, 1999) wave flume. The performances of active wave absorption characteristics were investigated in detail by Hald and Frigaard (1997). Although the method appeared quite promising, performance of the separation is reduced if the frequencies of the componential wave do not coincide with the discrete frequencies of the digital filter. One way to improve the performance is to increase the number of data points of
J −1
u * [n] = ∑ hu [ j ]u[n − j ]
(1)
j =0
where, J is the number of filter coefficients, u*[n]=u(n ∆ tf) and w*[n]=w(n ∆ tf) is the filter outputs at time t=n ∆ tf, ∆ tf is the filter time interval.
609
cosh k (d + z1 ) cos(ωt + kx1 + ϕ R ) sinh kd sinh k (d + z1 ) wI (x1 , z1 , t ) = −a I ω sin (ωt − kx1 + ϕ I ) sinh kd sinh k (d + z1 ) wR (x1 , z1 , t ) = −a Rω sin(ωt + kx1 + ϕ R ) sinh kd
Having outlined the principle of the active wave absorption, the only remaining problem is to design the corresponding frequency response function H(f) of the filters from which the impulse response function h(t) is derived. Given a desired frequency response function, the corresponding impulse response function is obtained by computing the inverse discrete Fourier transform of the complex frequency response function (Karl, 1989).
u R (x1 , z1 , t ) = −a Rω
Calculation of the paddle displacement correction signal needed for absorption for the reflected waves is done by digital filtering and subsequent superposition of two velocity signals measured in front of the wave paddle.
(7) (8) (9)
and f is frequency, a = a( f ) is wave amplitude, k = k ( f ) is wave number, ω = 2πf is angular frequency, ϕ = ϕ ( f ) is phase angle and indices I and R denote incident and reflected, respectively.
When active absorption is applied, the paddle displacement signal is added to the input paddle displacement signal read from the signal generator causing the wave generator to operate in a combined generation/absorption mode.
At the wave paddle x0=0, the water surface elevation η (x0,t) of the wave component is composed of the incident and reflected surface elevations:
η (x0 , t ) = η I (x0 , t ) + η R (x0 , t ) = a I cos(ωt − kx0 + ϕ I ) + a R cos(ωt + kx0 + ϕ R )
(10)
The wave component η -R(x0,t) that has to be generated at the wave paddle in order to cancel out the reflected component η R(x0,t), has the same amplitude as, and is in opposite phase with η R(x0,t):
η − R (x0 , t ) = aR cos(ωt − kx0 + ϕ R + π )
The amplification factors Cu, Cw and phase shifts ϕ u, ϕ w are introduced into the expressions for u(x1,z1,t) and w(x1,z1,t). The modified signals are denoted by u*(x1,z1,t) and w*(x1,z1,t).
Fig. 1: Principle of wave absorption.
u * (x1 , z1 , t ) = u I (x1 , z1 , t ) + u R (x1 , z1 , t ) *
= Cu a I ω
The complex frequency response Hi(f) for each filter is composed of an amplification factor Ci(f) and a phase ϕ i(f), where i=u or w. (2)
Assuming
ϕu = ϕw − π / 2 the sum of u*(x1,z1,t) and w*(x1,z1,t), which is denoted by
(14)
η *(x0,t), is:
η * (x0 , t ) = u * ( x1 , z1 , t ) + w* ( x1 , z1 , t )
sinh k (d + z1 ) cosh k (d + z1 ) = a I ω Cu − Cw kd sinh sinh kd × cos(ωt − kx1 + ϕ I + ϕ u )
(4) (5)
sinh k (d + z1 ) cosh k (d + z1 ) − aRω Cu + Cw kd sinh sinh kd × cos(ωt + kx1 + ϕ R + ϕ w )
where,
cosh k (d + z1 ) cos(ωt − kx1 + ϕ I ) sinh kd
)
= −C w a I ω
The horizontal and vertical water particle velocity components u(x1,z1,t) and w(x1,z1,t) at location (x1,z1) can be written as the sum of the incident and reflected part, the incident wave propagating away from the wave paddle, and the reflected wave propagating toward the wave paddle:
u I (x1 , z1 , t ) = a I ω
(
(13) sinh (d + z1 ) sin (ωt − kx1 + ϕ I + ϕ w ) sinh kd sinh (d + z1 ) sin (ωt + kx1 + ϕ R + ϕ w ) − C w a Rω sinh kd
(3)
A two-component velocity meter is used to separate the incident and reflected waves. Even though the method works for irregular waves, it will be demonstrated for the case of monochromatic waves in the following.
u (x1 , z1 , t ) = u I (x1 , z1 , t ) + u R (x1 , z1 , t ) w(x1 , z1 , t ) = wI (x1 , z1 , t ) + wR (x1 , z1 , t )
*
(12) cosh(d + z1 ) cos ωt − kx1 + ϕ I + ϕ u sinh kd cosh(d + z1 ) cos(ωt + kx1 + ϕ R + ϕ u ) − C u a Rω sinh kd * * w * (x1 , z1 , t ) = wI (x1 , z1 , t ) + wR (x1 , z1 , t )
FREQUENCY RESPONSE OF DIGITAL FILTERS
Re{H i ( f )} = C i ( f )cos(ϕ i ( f )) Im{H i ( f )} = C i ( f )sin (ϕ i ( f ))
(11)
(6)
It is seen that
610
η *(x0,t) and η -R(x0,t) are identical signal in case:
(15)
1 sinh kd 2ω cosh k (d + z1 ) ϕ u ( f ) = −kx1 + π 1 sinh kd Cw ( f ) = − 2ω sinh k (d + z1 ) ϕ w ( f ) = −kx1 + π + π / 2 Cu ( f ) = −
which coincides with the time length of impulse response (Fig.2).
(16)
Fig.3(b) shows an example which is identical to the example shown in Fig.3(a), except that the wave frequencies fi do not coincide with the discrete filter frequencies, but are placed midway between the frequencies. This example shows significant error after warm up. One way to improve the performance is to increase the number of data points of the filter responses to get a higher resolution. However, this type of approach increases the computation time. To overcome this problem, an optimum filter design method will be discussed in the next chapter.
(17) (18) (19)
DESIGN OF DIGITAL FILTERS The practical design of digital filters in the specified frequency range is conducted according to the criteria given by Troch and De Rouck (1999). Fig. 2 gives an example of the theoretical and realized complex frequency response of horizontal and vertical water particle velocities and corresponding impulse responses. In this example, the following conditions are set: water depth d=0.2m, location of velocity meter (x1, z1)=(2.99m, -0.05m), number of filter coefficient J=100, filter duration T0=10.0s, filter time interval ∆ tf=0.1s, frequency resolution ∆ ff=0.1Hz and the filter sample frequency fsf=10.0Hz.
OPTIMUM DESIGN OF DIGITAL FILTERS BY USING LEAST SQUARE FORMULATION AND THE PERFORMANCE OF WAVE ABSORPTION The optimum filter response was designed in advance for irregular waves composed of 12 componential waves with their frequencies being 4.5 ∆ f, 5.5 ∆ f, 6.5 ∆ f, 7.5 ∆ f, 8.5 ∆ f, 9.5 ∆ f, 10.5 ∆ f, 11.5 ∆ f, 12.5 ∆ f, 13.5 ∆ f, 14.5 ∆ f and 15.5 ∆ f. Denoting the target time series of water surface elevation to absorb the reflected wave by η -R, which is expressed by Eq.(23), the total residual squared is given by Eq.(24):
PERFORMANCE OF WAVE ABSORPTION USING THE CONVENTIONAL DIGITAL FILTERS The performance of absorption was investigated using a set of an analytical example with known incident and reflected waves. In the example, the number of componential waves is set to four. Individual frequencies of componential waves are set to 5 ∆ f, 8 ∆ f, 10 ∆ f and 12 ∆ f (example(a)), and 5.5 ∆ f, 8.5 ∆ f, 10.5 ∆ f and 12.5 ∆ f (example(b)), where ∆ f denotes the frequency interval of the digital filters. The water particle velocities at location (x1,z1) are thus expressed by:
12
η − R (x 0 , t ) = ∑ [Rc × Amp × cos(2πf i t − k i x 0 + π )] M
j =1
2
]
(24)
where, M is the number of data points of the time series. The problem can then be regarded as a non-linear least square problem. The optimum impulse responses of the digital filters are obtained by finding the value of hu(i) and hw(i) which minimize r2.
cosh k i (d + z1 ) u(x1 , z1 , t ) = Amp × ∑ 2πf i cos(2πf i t − k i xi ) sinh k i d i =1 cosh k i (d + z1 ) cos(2πf i t + k i x1 ) ] (20) − Rc × 2πf i sinh k i d
The general description of the least square problem is to find a set of N unknown parameters
4 sinh k i (d + z1 ) w(x1 , z1 , t ) = Amp × ∑ 2πf i cos(2πf i t − k i xi ) sinh k i d i =1 sinh k i (d + z1 ) cos(2πf i t + k i x1 ) ] (21) − Rc × 2πf i sinh k i d
(x1 , x2 ,..., x N )
(25)
which minimizes the value of M
r 2 (x ) = ∑ f j2 (x )
(26)
j =1
in which, Amp is 1.0cm, Rc is 0.5 (50% Reflection). Consequently, wave component η -R is defined as: 4
[
r 2 = ∑ η * ( j ) − η− R ( j )
4
η − R (x 0 , t ) = ∑ [Rc × Amp × cos(2πf i t − k i x 0 + π )]
(23)
i =1
where f j (x1 , x 2 ,..., x N )
(22)
i =1
( j = 1,..., M ) are arbitrary functions of x.
The error is described by the difference between the calculated wave signal η * and the theoretical wave signal η -R.
When r2(x) depends nonlinearly on a set of N unknown parameters xi, the minimization must proceed iteratively. From the given initial trial values for the parameters, the calculation proceeds in a way that improves the trial solution. Such a procedure is then repeated until r2(x) reaches an equilibrium (Press et al., 1992).
Fig.3(a) shows the functionality of the method when the wave frequencies fi of the componential wave all coincide with some frequencies of the discrete filter, i.e., fi=n ∆ f. As expected, the method is exact for signals consisting only of energy placed at the discrete frequencies. It is also seen that during the warm up time T0, errors are present. This warm up time is inevitable. The duration of it is 10 sec,
In this study, Marquardt’s method is suitably employed to solve Eq.(24) because of its stability in computation. The usefulness of such formulation was correctly suggested by Matsumoto and Hanzawa (1996) for the optimum rod amplitude of the segmented wavemaker to acquire a uniform wave field in an experimental wave basin.
611
used for the separation of a wave field. Fig.4 gives the original and modified filter responses hu and hw. Deviations between them are also indicated in the figure. Deformations of filter responses are not so large. It should be noted here, by this formulation, that the exact impulse responses at discrete frequencies are not needed any more because the whole shape of the impulse responses are optimized.
As this investigation is a very limited one, i.e. all the componential waves have the same amplitude which is equal to 1.0cm, and phase shift between the componential wave are not taken into account, a detailed investigation on the applicability of this method for practical use remains for further work. The reliability of the method against unexpected errors in the measuring system has to be also examined.
Fig.5 shows the calculated ( η *) and theoretical ( η -R) time series needed to absorb the reflected waves with the associated errors ( η *η -R) by conventional and the proposed method. The proposed method adopts the modified impulse responses indicated in Fig.4, while the conventional method uses the original ones. In this example, the wave train to be analyzed is composed of 12 componential waves having frequencies 4.5 ∆ f to 12.5 ∆ f. Although it is seen that significant errors are present during warm up of the filter in both methods, the new method clearly improves the accuracy of separation. In this example, the proposed method reduces the value of the variance of error after warm up of the filter up to approximately 1/1000 of the original value. This result is easily expected because the impulse responses hu and hw are designed for this particular wave train.
ACKNOWLEDGEMENT The authors wish to express their gratitude to Dr. Troch from the University of Ghent, Belgium, for his valuable advice at the beginning stage of this study.
REFERENCES Frigaard, P. and Brorsen, M. (1995) A time-domain method for separating incident and reflected irregular waves. Coastal Engineering, Vol.24, Nos.3 and 4, pp.205-215. Frigaard, P. and Christensen, M. (1994) An absorbing wave-maker based on digital filters. Proc. 24th International Conference on Coastal Engineering, Kobe, Vol.1, pp.168-180.
In order to evaluate the applicability of the proposed method, two other wave trains which are composed of four componential wave are examined. The individual wave frequencies for one example are set to 4.5 ∆ f, 7.5 ∆ f, 9.5 ∆ f and 11.5 ∆ f, while the other example uses 5.5 ∆ f, 8.5 ∆ f, 10.5 ∆ f and 12.5 ∆ f. The modified impulse responses indicated in Fig.4 are used again. The results are shown in Figs.6 and 7. These figures show that the proposed method gives favorable results when the wave frequencies are chosen from the frequencies from which the optimized impulse response is designed. In these examples, the proposed method reduces the value of the variance of error up to approximately 1/30 to 1/70 of the original value.
Goda, Y. and Suzuki, Y. (1976) Estimation of incident and reflected waves in random wave experiments. Proc. 15th International Conference on Coastal Engineering, Honolulu, Vol.1, pp.828-845. Hald, T. and Frigaard, P. (1997) Performance of active wave absorption system, Comparison of wave gauge and velocity meter based systems. Proc. 7th International Offshore and Polar Engineering Conference, Vol.3, pp 221-227. Karl, J.H. (1989) An Introduction to Digital Signal Processing, Academic Press., 341p.
These results mean that, once the optimum filter responses are obtained for a sufficient number of fi taking into account of phase shift between componential waves, the proposed method has the possibility to be applied to any wave train.
Mansard, E. and Funke, E. (1980) The measurement of incident and reflected spectra using a least square method. Proc. 17th International Conference on Coastal Engineering, Sydney, Vol.1, pp.154-172.
CONCLUDING REMARKS
Matsumoto, A. and Hanzawa, M. (1996) New optimization method for paddle motion of multi-directional wavemaker. Proc. 25th International Conference on Coastal Engineering, Orlando, Vol.1, pp.479-492.
Impulse responses of digital filters for use in separating incident and reflected water waves in a time domain are improved by using a nonlinear least square formulation. Deviation between the improved impulse response and the original is not so large.
Press, W., Teukolsky, S., Vetterling, W. and Flannery, B. (1992) Numerical recipes in FORTRAN, 2nd Ed., Cambridge University Press, 963p.
Trial computation demonstrates that the improved impulse responses reduces the value of the variance of error after warm up of the filter up to approximately 1/1000 of the original value for the case in which the individual wave frequencies of componential waves of analyzed random waves are identical to the frequencies used to design the improved impulse response.
Troch, P. and De Rouck, J. (1999) An active generating-absorbing boundary condition for VOF type numerical model. Coastal Engineering, Vol.38, No.4, pp.223-247.
The proposed method gives the favorable results when the wave frequencies of the analyzed wave train are chosen from the frequencies from which the optimized impulse response is designed.
Zelt, J. and Skjelbreia, J. (1992) Estimating incident and reflected wave fields using an arbitrary number of wave gauges. Proc. 23th International Conference on Coastal Engineering, Venice, Vol.1, pp.777-789.
Throughout this study, it was clarified that the proposed method will provide a useful tool for designing an optimum impulse response to be
612
0.5
0.5 Theoretical Realized
0.3 0.2
0.4
Gain Cw
Gain Cu
0.4
0.1
Theoretical Realized
0.3 0.2 0.1
0.0 0.0
0.5
1.0
1.5
0.0 0.0
2.0
2
2
1
1
0 -1 -2 0.0
0.5
1.0
1.5
1.5
2.0
-1 -2 0.0
2.0
0.02 0.01 0.00 -0.01 -0.02 -0.03 0 1 2 3 4 5 6 7 8 9 10
0.5
1.0
1.5
2.0
Frequency (Hz) Impulse Response hw
Impulse Response hu
1.0
0
Frequency (Hz)
-0.04
0.5
Frequency (Hz)
Phase ϕ w/π
Phase ϕ u/π
Frequency (Hz)
Time (s)
0.02 0.01 0.00 -0.01 -0.02 -0.03 -0.04
0 1 2 3 4 5 6 7 8 9 10
Time (s)
Fig. 2: Magnitude and phase of the frequency response of a discrete filter associated with corresponding impulse response.
613
η -R
η
*
1 0
*
-1 -2
0
5
10
15
20
25
η -R
2
η , η -R (cm)
*
η , η -R (cm)
2
0 -1
0
5
Time (s)
15
20
25
30
25
30
1.0
η -η -R (cm)
0.5 0.0 -0.5
*
*
η -η -R (cm)
10
Time (s)
1.0
-1.0
*
1
-2
30
η
0
5
10
15
20
25
0.5 0.0 -0.5 -1.0
30
0
5
Time (s)
10
15
20
Time (s)
(a)
(b)
Fig. 3: Comparison between the surface elevations η * and
η -R with associated error using conventional impulse response. (a) f1=5 ∆ f, f2=8 ∆ f, f3=10 ∆ f, f4=12 ∆ f and (b) f1=5.5 ∆ f, f2=8.5 ∆ f, f3=10.5 ∆ f, f4=12.5 ∆ f (a) Variance = 5.63 × 10 −13 cm 2
(b) Variance = 8.26 × 10 −12 cm 2
0.02
0.02 0.01 0.00
hw
hu
0.01 0.00
-0.02
-0.01 -0.02
-0.01
Modified Original 0
2
4
6
8
Modified Original
-0.03 -0.04
10
0
2
0.02 0.01 0.00 -0.01 -0.02
0
2
4
6
4
6
8
10
8
10
Time (s)
hw(modified)-hw(original)
hu(modified)-hu(original)
Time (s)
8
10
Time (s)
0.02 0.01 0.00 -0.01 -0.02
0
2
4
6
Time (s)
Fig. 4: Original and improved impulse response of the digital filters.
614
η -R
η
*
*
η , η -R (cm)
*
η , η -R (cm)
8 6 4 2 0 -2 -4 -6 -8
0
5
10
15
20
25
30
8 6 4 2 0 -2 -4 -6 -8
η -R
0
5
Time (s)
20
25
30
25
30
1.0
η -η -R (cm)
0.5 0.0 -0.5
*
*
η -η -R (cm)
15
*
Time (s)
1.0
-1.0
10
η
0
5
10
15
20
25
0.5 0.0 -0.5 -1.0
30
Time (s)
0
5
10
15
20
Time (s)
(a)
(b)
Fig. 5: Comparison between the surface elevations η * and η -R with associated error using conventional and improved impulse response for the 12 component irregular wave. (a) conventional impulse response was used, (b) improved impulse response was used. (b) Variance = 7.04 × 10 −13 cm 2 (a) Variance = 7.80 × 10 −10 cm 2
615
1.0
0.5
0.5
η -η -R (cm)
0.0
*
*
η -η -R (cm)
1.0
-0.5 -1.0
0
5
10
15
20
25
0.0 -0.5 -1.0
30
0
5
Time (s)
10
15
20
25
30
Time (s)
(a)
(b)
Fig. 6: Comparison between the surface elevations η * and
η -R with associated error using conventional and improved impulse response for the four component irregular wave.( f1=4.5 ∆ f, f2=7.5 ∆ f, f3=9.5 ∆ f, f4=11.5 ∆ f)
1.0
1.0
0.5
0.5
η -η -R (cm)
0.0
*
*
η -η -R (cm)
(a) conventional impulse response was used, (b) improved impulse response was used. (b) Variance = 3.33 × 10 −13 cm 2 (a) Variance = 2.33 × 10 −11 cm 2
-0.5 -1.0
0
5
10
15
20
25
0.0 -0.5 -1.0
30
Time (s)
0
5
10
15
20
25
30
Time (s)
(a)
(b)
η * and η -R with associated error using conventional and improved impulse response for the four component irregular wave.( f1=5.5 ∆ f, f2=8.5 ∆ f, f3=10.5 ∆ f, f4=12.5 ∆ f)
Fig. 7: Comparison between the surface elevations
(a) conventional impulse response was used, (b) improved impulse response was used. (b) Variance = 2.74 × 10 −13 cm 2 (a) Variance = 8.26 × 10 −12 cm 2
616
