A Determination of an Abrupt Motion of the Sea Bottom by Using Snapshot Data of Water Waves

This paper presents an inverse problem and its solution procedure, which are aimed at identifying a sudden underwater movement of the sea bottom. The identification is mathematically shown to work with a known snapshot data of generated water wave configurations. It is also proved that the problem has a unique solution. However, the inverse problem is involved in an integral equation of the first kind, resulting in an ill-posed problem in the sense of stability. That is, the problem lacks solution stability properties. To overcome the difficulty of solution instability, in this paper, a stabilization technique, called regularization, is incorporated in the present solution procedure for the identification of the sea bottommovement. A numerical experiment is presented to demonstrate that the proposed numerical solution procedure operates.


Introduction
In the field of natural science and ocean engineering, it is not only of interest but important to examine how waves are generated in the ocean surface by the underwater abrupt movement of the sea bottom.If we knew the information of the underwater abrupt movement in advance, it would enable us to determine how the waves propagate in space and time.In practice, this can be extremely crucial, for example, for a Tsunami Warning System TWS , which is used to detect tsunamis and issue warnings to prevent loss of life and property.
The problem of finding the resulting wave flow field has been usually solved based on the potential wave model.For example, excellent research has been made on the subject of wave generation and propagation 1-4 .Even if much progress has been achieved in determining the resulting wave flow or forward problem , only few attempts have been

Transient Waves by Abrupt Movement of Sea Bottom
We consider an inviscid incompressible water of finite depth and a system of coordinates in which the y-axis is vertical and the x-axis horizontal in mutually perpendicular directions, as shown in Figure 1.The water is assumed to have a constant density and negligible surface tension.The surface water waves are induced in the body of the water initially at rest for time t < 0 by an abrupt movement of the sea bottom.

Boundary Conditions
We assume that the flow is irrotational in a simply connected fluid domain such that there exists a single-valued velocity potential function Φ x, y, t .Then, by the continuity equation, in the system of Cartesian coordinates x, y , the free-surface wave motion is governed by the elliptic type of Laplace equation with regard to the velocity potential Φ x, y, t because water is assumed to be incompressible.

Wave Spectrum
We suppose that the sea bottom changes suddenly at t 0 such that its movement can be mathematically expressed as Applying Fourier-Laplace transform to the velocity potential in the governing equation 2.1 and the boundary conditions of 2.2 and 2.4 , we can derive the wave elevation of the freesurface 1 A k e i kx ωt e i kx−ωt dk 2.6 by using the method of inverse Fourier-Laplace transforms.The resulting wave system of 2.6 is of dispersive waves, whose dispersion relation between the wave-number k and the frequency ω and the spectrum A k are Here, the notation F 0 k stands for the Fourier transform F of F 0 x and is expressed as follows:

Integral Equation
As mentioned before, the abrupt bottom motion is assumed to arise at t 0. The resulting wave system propagates in space with time, according to the dispersion relation equation 2.7 .In this study, we will measure the spatial wave distribution of the resulting wave at an instant of time t T > 0, which is symbolized as 3.1 for some real positive constant > v g • T : v g is the group velocity, v g gh 1/2 1 .We then are able to find a relationship between the spectrum F 0 k in 2.8 and the η T from 2.6 and 2.7 3.2 Equation 3.2 can be regarded as an integral equation of the first kind, in which the kernel function K is expressed as cos T gk tanh kh 2π cosh kh .

3.3
The integral equation 3.2 is rewritten with a symbolic notation as The meaning of the integral equation 3.2 is as follows.If we can measure snapshot data of the surface-wave elevation η T at t T > 0, it then enables us to identify the spectrum F 0 k in 2.6 .Physically, this implies that we are able to know completely the whole information of the dispersive wave system in 2.7 if we know a partial information about the wave system, for example, the snapshot data of the surface-wave elevation η T .We finally discover the abrupt displacement F 0 x in 2.5 by employing the inverse Fourier transform

Uniqueness
Before the detailed discussion of solving the integral equation of 3.2 , we need to examine whether the integral equation has a unique solution.
Physically, this is crucial and essential to recover the real movement of the sea bottom.We note that it suffices to prove that the null space of 3.2 is trivial because 3.2 is linear; that is, we want to show that η T x ≡ 0 identically for − < x < means F 0 k ≡ 0 identically.
We first rewrite 3.2 as

4.1
where Mathematical Problems in Engineering Thus, the following holds for From the injectivity of the Fourier transform, we have from 4.3 or 4.4 that the quantity Because the function cos T gk tanh kh / cosh kh in 4.2 has isolated zeros, it follows that F 0 k is zero almost everywhere from 4.2 .That is, we show that η T x ≡ 0 identically for This completes the proof.

Construction of the Wave Spectrum
Although the uniqueness of the solution of the wave spectrum has been established, we have still a question of its stability, that is, the solution F 0 k depends continuously on the snapshot data of the wave configuration η T in 3.2 .

The Discontinuous Operator L c
Because the computer memory is limited in practice, in this study, we replace the integration limit of ∞ in 3.3 with a large but finite real number κ 5 as follows: or, in operator notation, Equation 5.1 is classified as an integral equation of the first kind rather than the second kind.
It is thus necessary to check the stability of the solution, that is, whether it depends on the snapshot wave data in a continuous manner.According to the theory of integral equations, the solution lacks stability properties 5, 16 , as the kernel K in 5.1 is regular.This means that a small amount of noisy data in a snapshot wave configuration may be amplified and cause an unreliable solution.In other words, mathematically, the operator L c in 5.2 is discontinuous with respect the usual topology.

Tikhonov's Regularization
To overcome the difficulty encountered in Section 5.1, we suppress the lack of stability through a stabilization technique.To this end, we suggest the use of the following regularization: for a real positive constant α, known as the regularization parameter, where the symbol I is the identity operator and L * c the adjoint operator of L 16, 22-29 , for a square integrable function g x .According to the regularization theory, it is proven that the inverse operator αI L * c L c −1 in 5.3 is bounded and always exists.Moreover, the q α converges to the solution to 5.1 as α → 0 .The stabilization process characterized as in 5.3 is called Tikhonov's regularization 16, 22, 23 .

Numerical Examples
We will examine a numerical example, where we follow the procedure proposed in this paper to measure an impulsive movement of the sea bottom.For that, we first start with the following specification for the underwater displacement F 0 x in 2.5 : Note that the Fourier transform 30 of 6.1 is known as follows: We normalize the water depth h and the constant c as the unit.The water waves that result from this movement are briefly sketched in Figure 1.Graphical illustrations of F 0 x and its Fourier transform are depicted in Figures 2 and 3.In this paper, for numerical calculation, the interval for the physical variable x is taken as −50 < x < 50, and the interval for the frequency k as −12 < k < 12.
The impulsive movement of the sea bottom equation 6.1 leads to the generation of a wave system, the spatial distribution of which is

Noise Level
Our aim is to inversely recover F 0 x in 6.1 by using the data η T in 6.3 .However, in practice, the measured or calculated data are always deteriorated somewhat due to noise.Thus, we assume that we know measured data, denoted by η T, δ .Thereby, we define a noise level δ > 0, which satisfies the following norm inequality: That is, the noise level is a quantity measuring an error intensity concerning the data η T, δ .
Here, the notation • 2 refers to the L 2 norm 23 .We have, in this study, the data η T, δ randomly generated but with the two different noise levels δ of 10 −4 and 10 −6 , respectively.The noise is assumed to have the normal distribution with zero mean.The numerical values for 6.3 are plotted in Figure 4 a noise-free condition , which shows the spatial wave distribution when T 10.The results in Figures 4 b and 4 c correspond to the noise levels 10 −4 and 10 −6 .

Optimal Regularization Parameter
To achieve an accurate solution during the regularization, it is important to decide optimal regularization parameter in the Tikhonov regularization 16, 22, 23 .Based on the L-curve criterion 31 , we depict the curves of log-log plot as shown in Figures 5 and 8: log L c q α − η T,δ 2 , log q α 2 , 6.5  where q α denotes the regularized solution, calculated by 5.3 , depending on the regularization parameter α.Here, we discretize 5.3 directly; the Simpson's numerical integration rule is applied to the direct discretization.The number of intervals used for the Simpson's rule is chosen as 400.We obtain the optimal regularization parameter of α 10 −8 when the noise level is 10 −4 .This is because the optimal regularization parameter corresponds to the corner of the L-curve in Figure 5 31 .A brief explanation for this reason is as follows.When the regularization parameter α decreases, the size of error, L c q α − η T,δ 2 , reduces, because q α approaches the true solution.However, subsequently q α potentially deviates far away; that is, the function norm, q α 2 , begins to increase, when 1/α exceeds a certain threshold.This means that there exists an appropriate regularization parameter α such that the q α is an optimal solution for the present problem.This optimal solution is shown to correspond to the corner of the L-curve by Hansen 31 .In a similar way, we can obtain the optimal regularization parameter α 10 −9 corresponding to the noise level 10 −6 , as depicted in Figure 8.

Determining the Wave Spectrum
The graphs shown in Figures 6 a -6 d , concerning the noise level 10 −4 , compare the exact wave spectrum in 6.2 with the regularized wave spectrum q α in 5.3 .In addition,  Figure 6 e shows a residual, which is the L 2 norm of difference between the exact and regularized spectra divided by the L 2 norm of the exact.In fact, we did calculate many of various regularized wave spectra q α with various values for α.Among them, only four selected regularization parameters α 10 −2 , 10 −4 , 10 −8 , and 10 −11 are depicted in Figures 6 a -6 d .This immediately shows that the best approximation for the wave spectrum is found, when the regularization parameter α equals 10 −8 ; as mentioned above, this value of α is an optimal regularization parameter corresponding to the corner of the L-curve.Similarly, when the noise level is equal to 10 −6 , we did carry out a lot of calculations of regularized wave spectra q α for various regularization parameters α.We also select only four α 10 −2 , 10 −4 , 10 −9 , and 10 −15 among them, as shown in Figures 9 a -9 d , and the residual is seen in Figure 9 e .Here, the exact wave spectra are compared with the regularized ones q α .It is clear from Figure 9 that the best approximation to the exact one occurs when α 10 −9 , which was anticipated from the L-curve in Figure 8.

Discussions
We proposed a new method to find sudden movements of the sea bottom using just a local snapshot data of wave configurations, − < x < l.However, it is interesting to observe also how the method proposed is suitable to retrieve whole wave configurations, that is, we can identify η T η x, T , −∞ < x < ∞, just using a local data η x, T , − < x < l.This is realized by simply estimating the integration where q α k is a Tikhonov's regularized solution in 5.3 .In the present inverse study, we assumed that the sea bottom movement is instantaneous.In fact, this may be a usual assumption in these kinds of wave generation problems, especially studying tsunamigenic earthquakes.However, sometimes the sea bottom movement can be relatively slow, for example in case of a tsunami earthquake 32 .In this case, the sea bottom movement is not considered to be instantaneous, which means that the inverse method proposed in this study does not work.

Conclusion
Sudden underwater movements of the sea bottom result in the free-surface flow of ocean waves.The problem of finding the resulting wave flow is well studied, and is known as the forward problem.We examine whether an inverse problem can be defined as an alternative approach.We explore whether it is possible to indirectly measure sudden underwater movements using a local snapshot data of the resulting wave motion.We propose an indirect measurement procedure that successfully confirms the viability of the inverse problem approach.A numerical example is presented that verifies the proposed procedure and confirms its workability.As application, it is interesting and important to know that if we have a local snapshot data of wave configurations by remote control airplane, we can recover a wider range of wave configuration, of course, including the local data, by using the method proposed in this study.

FFigure 1 :
Figure 1: Impulsive movement of the sea-bottom and the resulting wave flow.

cos T gk tanh kh 2π cosh kh e ikx F 0 k dk 6. 3 from 3 .Figure 2 :Figure 3 :
Figure 2: Graphical illustration of the impulsive movement F 0 x of the sea bottom in 6.1 .

Figure 4 : 2 log 2 Figure 5 :
Figure 4: The spatial distribution of the generated waves when T 10 for a unit water depth at −50 < x < 50.

eFigure 6 :
Figure 6:Comparison of the regularized wave spectrum with the exact one for four cases of regularization parameters α a -d : noise level 10 −4 .Legend: the dotted and solid lines stand for the exact wave spectrum F 0 k in 6.2 and the Tikhonov regularized solution q α in 5.3 , respectively.

e
Residual the L 2 norm of difference between the exact and regularized displacement divided by the L 2 norm of the exact

Figure 7 :
Figure 7: Comparison of the recovered sudden displacement of the sea bottom with the exact displacement for four cases of regularization parameters α a -d : noise level 10 −4 .Legend: the dotted and solid lines stand for the exact F 0 x in 6.1 and the recovered sudden displacement of the sea bottom, respectively.The lines show good agreement with each other when the regularization parameter α 10 −8 .

2 log 2 Figure 8 :
Figure 8: Illustration of the L-curve noise level 10 −6 .Note: The optimal regularization parameter α 10 −9 occurs at the corner of the L-curve.

Following 3 . 5 ,
we can recover the sudden movement of the sea bottom F 0 x , which are depicted in Figures 7 a -7 d and in Figures 10 a -10 d .Here, the residual is also calculated and depicted in Figures 7 e and 10 e , respectively.There are fairly good agreements between the exact and recovered results.The most accurate results are found when the regularization parameters are optimal.

eFigure 9 :
Figure 9:Comparison of the regularized wave spectrum with the exact wave spectrum for four cases of regularization parameters α a -d : noise level 10 −6 .Legend: the dotted and solid lines stand for the exact wave spectrum F 0 k in 6.2 and the Tikhonov regularized solution q α in 5.3 , respectively.

eFigure 10 :
Figure10: Comparison of the recovered, sudden displacement of the sea bottom with the exact displacement for four cases of regularization parameters α a -d : noise level 10 −6 .Legend: the dotted and solid lines stand for the exact F 0 x in 6.1 and the recovered sudden displacement of the sea bottom, respectively.The lines show good agreement with each other when the regularization parameter α 10 −9 .