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 illconditioned system. Tikhonov regularization was applied to the illconditioned system to overcome instability of the system. The regularization parameter is calculated by using the Lcurve method. The numerical results are compared with the experimental results. The analysis of the numerical computation shows that the Tikhonov regularization method is useful.
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 illconditioned 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 Lcurve 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 illposed system is solved by applying Tikhonnov regularization method. The regularization parameter was estimated by the Lcurve 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 nondispersive 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
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
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 wellknown linear dispersion relation is
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; illposed 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 illposed. It is in order to explain the concept of wellposed problem first before we get to the illposed problem. The problem given is wellposed 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 wellposed, it is said to be illposed. Therefore the problem we have formulated is an illposed problem. The remedy for the illposed problem is regularization which will be covered in a latter chapter. If we discretize the Eq. (2) the matrix we encounter becomes illconditioned 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
where
x_{MP}
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
Rewriting the above equation as :
where
X
and
Y
are the wave amplitude spectrum and column vector of wave elevation, respectively. Gaussian wave packet can be written as
where
a
_{0}
,
s
and
k
_{0}
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.
Simulation parameters.
Amplitude spectrum of Gaussian wave packet.
Spacetime plot of Gaussian wave packet.
REGULARIZATION OF WAVE PROPAGATION MODEL
 Illconditioned 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:
where
U
is orthonormal eigenvector of
AA^{T}
, Σ is square root of the eigenvalues of
A^{T}A
,
V
is orthonormal eigenvector of
A^{T}A
, and
V^{T}
is transpose matrix of
V
(
Strang, 1980
;
Groetsch, 1993
;
Vogel, 2002
).
The condition number of matrix
A
in Eq. (8) can be written as,
The matrix which has a small condition number is called a wellconditioned matrix. However, the matrix with a large condition number is called an illconditioned 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 illcondition 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.
Variation of condition number.
 Regularization of wave propagation model
By using least square method, the amplitude spectrum and wave elevation can be expressed as:
where
A^{T}
is the transpose of matrix
A
. The inverse matrix of
A^{T}A
exists, Eq. (10) can be expressed by its inverse. However, if the matrix is illconditioned, 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
).
where
I
denotes the identity matrix. By selecting an appropriate value of
α
, the matrix can be stabilized. (
Kwon et al., 2007
;
Tikhonov, 1963
)
 Lcurve
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 Lcurve 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 Lcurve. 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
)
Also,
X
in Eq. (11) can be written as follows:
Substituting
X
in Eqs. (12) and (13),
ξ
(
α
) and
ζ
(
α
) can be expressed as:
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 tradingoff 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
)
By using the Lcurve method, the desired regularization parameter is obtained from the point of maximum curvature in the Lcurve and this point is marked in red circle in
Fig. 4
.
The Lcurve.
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 85
m
× 10
m
× 3.5
m
(L × B × D). The wave tank is shown in
Fig. 5
. The wave maker is multiplunger 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 tank.
Wave maker.
Specifications of wave maker.
Specifications of wave maker.
Wave gauge and amplifier (KENEK, TK650).
Specifications of wave gauge.
Specifications of wave gauge.
 Wave condition and experimental setup
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 3.5
m
, 3.26
rad/m
, and 1.2
rad/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
.
Experimental condition of the Gaussian wave packet.
Experimental condition of the Gaussian wave packet.
Time series of wave maker displacement.
The installed position of the wave gauges is shown in
Fig. 9
. The measuring point and target point are located at 18.0
m
and 30.5
m
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.
Installation position of the wave gauges.
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.
Experimental and predicted results : Case 1.
Experimental and predicted results : Case 2.
Experimental and predicted results : Case 3.
Experimental and predicted results : Case 4.
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 illconditioned matrix. Thus the solution becomes unstable. To remedy this difficulty, the groundbreaking 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 Lcurve method where the optimum value of the parameter is taken at the maximumcurvature point in the Lcurve. 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 Lcurve 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 illposed problem.
Acknowledgements
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) through GCRCSOP (No. 20110030013).
Calvettia D.
,
Morigib S.
,
Reichelc L.
,
Sgallarid F.
2000
Tikhonov regularization and the Lcurve for large discrete illposed problems
Journal of Computational and Applied Mathematics
123
(12)
423 
446
DOI : 10.1016/S03770427(00)004143
Fridman V.
1956
A Method of successive approximations for fredholm integral equations of the first kind
Uspekhi Matematicheskikh Nauk
11
(1(67))
233 
234
Groetsch C.W.
1993
Inverse problems in the mathematical sciences
Springer Vieweg
USA
Hanke M.
1996
Limitations of the Lcurve method in illposed problems
BIT Numerical Mathematics
36
(2)
287 
301
DOI : 10.1007/BF01731984
Isakov V.
1998
Inverse problems for partial differential equations
Springer
New York
Johnston P.R.
,
Gulrajani R.M.
2000
Selecting the corner in the LCurve approach to tikhonov regularization
IEEE Transactions on Biomedical Engineering
47
(9)
1293 
1296
DOI : 10.1109/10.867966
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
40
(3)
547 
573
DOI : 10.1016/0022247X(72)900029
Kwon S.H.
,
Kim C.H.
,
Jang T.S.
2007
An identification of wave propagation based on a singlepoint measurement
Ocean Engineering
34
(10)
1405 
1412
DOI : 10.1016/j.oceaneng.2006.10.008
Lawson C.L.
,
Hanson L.J.
1995
Solving least squares problems
SIAM
Philadelphia, PA
Strang G.
1980
Linear algebra and its applications
Academic Press
New York
Tikhonov A.N.
1963
Solution of incorrectly formulated problems and the regularization method
Soviet Doklady
4
1035 
1038
Vogel C.R.
2002
Computational Methods for Inverse Problems
SIAM
Philadelphia, PA