Prediction of propagated wave profiles based on point measurement

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  • ABSTRACT

    This study presents the prediction of propagated wave profiles using the wave information at a fixed point. The fixed points can be fixed in either space or time. Wave information based on the linear wave theory can be expressed by Fredholm integral equation of the first kinds. The discretized matrix equation is usually an ill-conditioned system. Tikhonov regularization was applied to the ill-conditioned system to overcome instability of the system. The regularization parameter is calculated by using the L-curve method. The numerical results are compared with the experimental results. The analysis of the numerical computation shows that the Tikhonov regularization method is useful.


  • KEYWORD

    Wave profile , Singular value decomposition , Ill-conditioned matrix , Tikhonov regularization , Regularization parameter , L-curve , Gaussian wave packet

  • INTRODUCTION

    Sea loads considerably influence on offshore structures’ safety, safety for workers, and efficiency of work in the ocean. Sea loads are the major influencing factors that have to be carefully considered when the structures are in operation. Many studies on sea loads are in progress. Especially, wave induced loads are the dominant factors to be considered. In order to estimate the wave induced loads, the accurate prediction of wave itself is critical.

    In order to predict accurate wave field around offshore structure, the mechanism of wave propagation should be established. Wave propagation prediction has many practical applications. Just name a few of them, prediction of wave near floating structures are necessary for the safe operation of the structures. We can get benefited from estimating the safe time period for landing of helicopters on offshore structures or on warships. The damage caused by sloshing can be avoided if one can estimate the sloshing load from wave prediction by taking countermeasures.

    At present, the usual practice of wave measurement is using wave gauge at a fixed point. Using the linear wave theory and the principle of superposition, the integral equation can be derived from the time series data at the fixed point. The generated integral equation is called Fredholm integral equation of the first kind. With integral equation discretized by wave number and time, the equation which relates wave amplitude and wave spectrum can be formulated. Wave amplitude spectrum can be obtained by solving the constructed matrix and using wave amplitude spectrum. Wave elevation at arbitrary points and time can be estimated. The discretized integral equation is mostly ill-conditioned which leads to unstable solutions. In order to estimate accurate wave elevation at target point, Tikhonov regularization method is introduced (Fridman, 1956; Isakov, 1998; Kammerer and Nashed, 1972; Tikhonov, 1963). When it comes to the Tikhonov regularization method, choosing optimized regularization parameter is essential to ensure the stability and accuracy of matrix. Present study employed the L-curve method.

    In this study, Gaussian wave packet is introduced to realize the incoming waves. Numerical simulation of the Gaussian wave has been conducted. The accuracy was evaluated by performing numerical and analytic works on this. Furthermore, the wave propagation experiment in wave tank is conducted. The wave time series has been measured at a fixed point. By analyzing the data, the wave field at the target point is estimated. Again, the ill-posed system is solved by applying Tikhonnov regularization method. The regularization parameter was estimated by the L-curve criterion which was successfully introduced by Lawson and Hansen (1974). The estimated wave time series was compared with the measured time series. The present scheme can be extended to multidirectional short crested sea if a spreading function is introduced. The present study is limited to linear dispersive waves due to its mathematical formulation. Therefore one should come up with non-dispersive scheme to deal with nonlinear wave field.

    WAVE PROPAGATION MODEL

    Assuming an ideal fluid, with its motion irrotational for a small amplitude wave, the wave elevation can be written as

    image

    where a is the amplitude of the wave, k is the wave number, and ω is the wave frequency. The linear superposition of the elementary solutions gives us an integral form as follows

    image

    where a(k) is wave amplitude spectrum. The wave elevations in different positions and time can be obtained. If the amplitude spectrum, a(k) is given in Eq. (2). However, it is not always straightforward to obtain the analytic closed form solution to the integral defined in Eq. (2) although the function a(k) is specified. When a finite depth of water is considered, it is not possible to obtain a closed form analytic solution to the integral due to the dispersion relation, i.e., the characteristic relation between the wave number and the frequency of the waves. The well-known linear dispersion relation is

    image

    where h is the depth of water. Since the closed form solution is not available for this case, we can seek an approximate solution.

      >  A fredholm integral equation of the first kind; ill-posed problem

    Eq. (2) is a Fredholm integral equation of the first kind. The wave elevation shown in the left hand side of the Eq. (2) will be a given function from measurement. The term a(k) is an unknown function we are after. It is well known that the Fredholm integral equations of the first kind are ill-posed. It is in order to explain the concept of well-posed problem first before we get to the ill-posed problem. The problem given is well-posed if (1) The solution exists (2) The solution is unique (3) The small change in the right hand side of Eq. (2) causes small change in the solution.

    If the given problem is not well-posed, it is said to be ill-posed. Therefore the problem we have formulated is an ill-posed problem. The remedy for the ill-posed problem is regularization which will be covered in a latter chapter. If we discretize the Eq. (2) the matrix we encounter becomes ill-conditioned matrix.

      >  Discretization of the integral equation

    A typical and most often used method of approximating an integral is first to discretize the integrand given in a continuous function into a finite number of segments and then the original integral is reduced to a sum of the integrals of the segments. Then the wave elevation in Eq. (2) can be approximated by

    image

    where xMP is the coordinate at a measuring point and the l and j components are discretized in time and amplitude spectrum, respectively. The measured wave elevation is

    image

    Rewriting the above equation as :

    image

    where X and Y are the wave amplitude spectrum and column vector of wave elevation, respectively. Gaussian wave packet can be written as

    image

    where a0, s and k0 are the maximum amplitude of the amplitude spectrum, the standard deviation of the Gaussian function, and modal wave number of the amplitude spectrum, respectively. The tested simulation parameters are given in Table 1 and the corresponding wave amplitude spectrum is shown in Fig. 1. The time series of the wave is shown in Fig. 2. At the initial stage of the simulation, multiple waves are overlapped. Since the propagation speed on each wave is different, dispersion of the propagating wave can be noticed from the figure.

    REGULARIZATION OF WAVE PROPAGATION MODEL

      >  Ill-conditioned system

    Before solving Eq. (6), the stability of matrix has to be examined. In order to verify the stability, condition number of the matrix should be checked. By singular value decomposition, matrix A can be expressed as follows:

    image

    where U is orthonormal eigenvector of AAT, Σ is square root of the eigenvalues of ATA, V is orthonormal eigenvector of ATA, and VT is transpose matrix of V (Strang, 1980; Groetsch, 1993; Vogel, 2002).

    The condition number of matrix A in Eq. (8) can be written as,

    image

    The matrix which has a small condition number is called a well-conditioned matrix. However, the matrix with a large condition number is called an ill-conditioned matrix which is hard to acquire stable solution. Fig. 3 shows the condition number of the kernel of the Eq. (6) for various number of wave number segments. As shown in Fig. 3, the matrix turned into ill-condition as the number of segments is increased. The small number of segments helps the stability of the matrix. On the other hand, the number of segments has to be increased to describe wave information more accurately. A method has to be introduced to increase the number of segments and stabilize the matrix simultaneously.

      >  Regularization of wave propagation model

    By using least square method, the amplitude spectrum and wave elevation can be expressed as:

    image

    where AT is the transpose of matrix A. The inverse matrix of ATA exists, Eq. (10) can be expressed by its inverse. However, if the matrix is ill-conditioned, Eq. (10) is unstable in most cases. In other words, a small error in Y can cause a large change in X. In order to minimize the unstable responses, Eq. (10) can be written by using regularization parameter α. (Fridman, 1956; Isakov, 1998; Kammerer and Nashed, 1972; Tikhonov, 1963).

    image

    where I denotes the identity matrix. By selecting an appropriate value of α, the matrix can be stabilized. (Kwon et al., 2007; Tikhonov, 1963)

      >  L-curve

    Choosing an appropriate regularization parameter is crucial in obtaining a stable solution when Tikhonov regularization method is used. The success of the regularization depends on a proper choice of the regularization parameter α. In general, with a large value of regularization parameter, the matrix becomes stable but the solution gets inaccurate. In contrast, with a small value of the parameter, the solution becomes accurate but matrix gets unstable. Therefore, selecting optimum regularization parameter is critical to increase stability of the matrix and accuracy of solutions. There are several methods to select regularization parameter. In this study, the L-curve method is used to select optimum regularization parameter α . Since the residual norm and solution norm plotted in logarithmic scale yield the shape of the letter L, the method is called the L-curve. The residual norm and solution norm ξ(α) and ζ(α) are defined in Eqs. (12) and (13). (Calvettia et al., 2000; Hanke, 1996; Johnston and Gulrajani, 2000; Lawson and Hanson, 1995)

    image
    image

    Also, X in Eq. (11) can be written as follows:

    image

    Substituting X in Eqs. (12) and (13), ξ ( α ) and ζ ( α ) can be expressed as:

    image
    image

    where ξ ( α ) represents the residual norm and ζ ( α ) denotes the solution norm. As the value of the regularization parameter increases, the stability of the matrix also increases but the accuracy of solution decreases. With ξ ( α ) and ζ ( α ) , curve regularization parameters, the regularization parameter was chosen at the point of maximum curvature by trading-off between the residual norm and the solution norm. It is a very nice tool in choosing an appropriate the value of regularization parameter. Maximum curvature was calculated using the following Eq. (17). (Johnston and Gulrajani, 2000)

    image

    By using the L-curve method, the desired regularization parameter is obtained from the point of maximum curvature in the L-curve and this point is marked in red circle in Fig. 4.

    EXPERIMENT

      >  Experimental facility

    In order to verify the wave propagation scheme, the experiment is performed in wave tank. The dimension of the wave tank is 85m × 10m × 3.5m (L × B × D). The wave tank is shown in Fig. 5. The wave maker is multi-plunger type and shown in Fig. 6. The specifications of the wave maker are presented in Table 2. Generated Gaussian wave was measured by wave gauges which are shown in Fig. 7 and the specifications of the wave gauge are given in Table 3.

      >  Wave condition and experimental set-up

    The wave spectrum with Gaussian distribution was used to generate the wave time series. As shown in Table 4, the experiment is performed with the four different maximum values of the amplitude spectrum. The water depth, the modal wave number, and the standard deviations are 3.5m, 3.26rad/m, and 1.2rad/m, respectively. The number of wave number segments is 200. The time series of wave maker displacement of the Case 2 is shown in Fig. 8.

    The installed position of the wave gauges is shown in Fig. 9. The measuring point and target point are located at 18.0m and 30.5m away from the wave maker, respectively. Two wave gauges are used to validate wave prediction method. We will compare the wave profile of the predicted from the measured point with measurement of the target point.

    RESULTS AND ANALYSIS

    To verify the numerical method in the study, the various Gaussian wave packets are generated and measured in wave tank. The condition of the Gaussian wave packet is given in Table 4. The experimental results with our prediction model results for various Gaussian wave packets are given in Figs. 10-13. The measured wave time history is given in (a). From the measured wave profile of the front wave gauge, the amplitude spectrum can be generated. The comparison of the target amplitude spectrum with measured spectrum is shown in (b). From the spectrum we can generate wave profile at any point. To validate our model, we compared predicted wave profile with measured wave profile at target point. The comparison is shown in (c). The wave data predicted from the measured wave information agrees well with the measured result at target point. We can notice that the accuracy of the wave prediction was decreased as the wave amplitude was increased. Since the wave propagation model was based on the principle of linear superposition, we deduce that the linear characteristic influences the error of the predicted wave profile with target measurement. However, the phase of the wave profile shows good agreement for all test cases.

    CONCLUSIONS

    The wave prediction scheme based on the measured wave information is proposed. Since obtaining the wave spectrum from the measured wave data is closely related to solving an ill-conditioned matrix. Thus the solution becomes unstable. To remedy this difficulty, the ground-breaking Tikhonov regularization method is used. The Tikhonov regularization method by introducing a regularization parameter is a very efficient and reliable procedure for the stability of the final matrix and the accuracy of the numerical solution. However, there is a still another difficulty on how to choose optimized value of the regularization parameter. To overcome this difficulty, we use the L-curve method where the optimum value of the parameter is taken at the maximum-curvature point in the L-curve. The wave elevation at target point was predicted based on the wave information at measuring point to validate the proposed wave prediction scheme. It was shown that the Tikhonov regulation method using the L-curve parameter estimation give substantially reliable results. Furthermore, the proposed wave prediction scheme can be used as a valuable tool and shed some light on the research field of ocean wave problems formulated in an ill-posed problem.

  • 1. Calvettia D., Morigib S., Reichelc L., Sgallarid F. 2000 Tikhonov regularization and the L-curve for large discrete ill-posed problems [Journal of Computational and Applied Mathematics] Vol.123 P.423-446 google doi
  • 2. Fridman V. 1956 A Method of successive approximations for fredholm integral equations of the first kind [Uspekhi Matematicheskikh Nauk] Vol.11 P.233-234 google
  • 3. Groetsch C.W. 1993 Inverse problems in the mathematical sciences google
  • 4. Hanke M. 1996 Limitations of the L-curve method in ill-posed problems [BIT Numerical Mathematics] Vol.36 P.287-301 google doi
  • 5. Isakov V. 1998 Inverse problems for partial differential equations google
  • 6. Johnston P.R., Gulrajani R.M. 2000 Selecting the corner in the L-Curve approach to tikhonov regularization [IEEE Transactions on Biomedical Engineering] Vol.47 P.1293-1296 google doi
  • 7. Kammerer W.J., Nashed M.Z. 1972 Iterative methods for best approximate solutions of linear integral equations of the first and second kinds [Journal of Mathematical Analysis and Applications] Vol.40 P.547-573 google doi
  • 8. Kwon S.H., Kim C.H., Jang T.S. 2007 An identification of wave propagation based on a single-point measurement [Ocean Engineering] Vol.34 P.1405-1412 google doi
  • 9. Lawson C.L., Hanson L.J. 1995 Solving least squares problems google
  • 10. Strang G. 1980 Linear algebra and its applications google
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  • 12. Vogel C.R. 2002 Computational Methods for Inverse Problems google
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  • [Table 1] Simulation parameters.
    Simulation parameters.
  • [Fig. 1] Amplitude spectrum of Gaussian wave packet.
    Amplitude spectrum of Gaussian wave packet.
  • [Fig. 2] Space-time plot of Gaussian wave packet.
    Space-time plot of Gaussian wave packet.
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  • [Fig. 3] Variation of condition number.
    Variation of condition number.
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  • [Fig. 4] The L-curve.
    The L-curve.
  • [Fig. 5] Wave tank.
    Wave tank.
  • [Fig. 6] Wave maker.
    Wave maker.
  • [Table 2] Specifications of wave maker.
    Specifications of wave maker.
  • [Fig. 7] Wave gauge and amplifier (KENEK, TK6-50).
    Wave gauge and amplifier (KENEK, TK6-50).
  • [Table 3] Specifications of wave gauge.
    Specifications of wave gauge.
  • [Table 4] Experimental condition of the Gaussian wave packet.
    Experimental condition of the Gaussian wave packet.
  • [Fig. 8] Time series of wave maker displacement.
    Time series of wave maker displacement.
  • [Fig. 9] Installation position of the wave gauges.
    Installation position of the wave gauges.
  • [Fig. 10] Experimental and predicted results : Case 1.
    Experimental and predicted results : Case 1.
  • [Fig. 11] Experimental and predicted results : Case 2.
    Experimental and predicted results : Case 2.
  • [Fig. 12] Experimental and predicted results : Case 3.
    Experimental and predicted results : Case 3.
  • [Fig. 13] Experimental and predicted results : Case 4.
    Experimental and predicted results : Case 4.