Research Article | Open Access
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 bottom movement. A numerical experiment is presented to demonstrate that the proposed (numerical) solution procedure operates.
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 made so far on analysis of an inverse problem to the resulting wave flow. The present study concentrates on the cause of the resulting wave flow (or inverse problem).
Recently, Jang et al.  have considered a problem that involves the indirect measurement of the movement of the sea bottom when the time history of the resulting wave motion is observed. They proposed a procedure for finding the wave source based on time-history data acquisition. Although the procedure by Jang et al.  is robust for measuring the wave source, it requires a relatively long time to acquire the time-history data at a fixed measurement position. However, in some cases, it may be convenient to obtain a snapshot data of the resulting wave configuration, for example, using a high-speed Remote Control airplane camera, rather than the acquiring of the full-time history data.
Motivated by this, we propose, in this paper, a new systematic procedure for the indirect measurement of abrupt underwater movement of the sea bottom by analyzing “snapshot” data of a local wave configuration. That is to say, the only needed data for identifying the underwater wave source is a snapshot involving a local wave configuration such as wave photos  taken from a remote control airplane at a fixed time. It is interesting to note that the new procedure proposed is also suitable to recover whole (or global) wave configurations only by using a local data of wave configuration. This work is classified as an inverse problem which occurs in many branches of science and engineering [5–15].
As a first step, we begin with a simplified mathematical wave model. That is, the two-dimensional irrotational wave flow is modeled with a constant water depth within the framework of linear dispersive wave theory. Based on the wave model, we propose an inverse problem characterized by an integral equation. The problem proposed is shown to have a unique solution of wave source. However, the problem lacks solution stability properties. This means that a small amount of noise from the snapshot data may be amplified, eventually leading to unreliable solutions due to the lack of stability. This is an unwelcome instability phenomenon which contrasts to the usual well-posed problem arising in natural sciences. A stabilization technique is applied to overcome this difficulty .
We investigate the workability of our approach through a numerical experiment. Although this work is a fundamental first step toward the indirect measurement of underwater movement, it may be related to a problem concerning the nature of tsunami generation using photographic (or snapshot) wave configuration. This, in turn, provides the basis for a photographic identifying problem for wave sources such as submarine-landslide, earthquakes, and underwater explosions or the testing of nuclear weapons [17–21].
2. 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 -axis is vertical and the -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 ) by an abrupt movement of the sea bottom.
2.1. 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 . Then, by the continuity equation, in the system of Cartesian coordinates , the free-surface wave motion is governed by the elliptic type of Laplace equation with regard to the velocity potential because water is assumed to be incompressible. The kinematic and dynamic free-surface boundary conditions are imposed on the mean free surface, , respectively as where denotes the free-surface elevation, and is the gravity acceleration . Denoting the sea bottom displacement by the boundary condition of the sea bottom is written as 
2.2. Wave Spectrum
We suppose that the sea bottom changes suddenly at 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 free-surface  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 and the frequency ) and the spectrum are respectively [1–3]. Here, the notation stands for the Fourier transform of and is expressed as follows:
3. Integral Equation
As mentioned before, the abrupt bottom motion is assumed to arise at . 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 , which is symbolized as (3.1) for some real positive constant is the group velocity, . We then are able to find a relationship between the spectrum in (2.8) and the from (2.6) and (2.7) Equation (3.2) can be regarded as an integral equation of the first kind, in which the kernel function is expressed as 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 at , it then enables us to identify the spectrum 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 . We finally discover the abrupt displacement in (2.5) by employing the inverse Fourier transform
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 identically for means identically.
We first rewrite (3.2) as where Thus, the following holds for Alternatively, From the injectivity of the Fourier transform, we have from (4.3) or (4.4) that the quantity if for . Because the function in (4.2) has isolated zeros, it follows that is zero almost everywhere from (4.2). That is, we show that identically for means . Therefore, if identically for , . This completes the proof.
5. 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 depends continuously on the snapshot data of the wave configuration in (3.2).
5.1. The Discontinuous Operator
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  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 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 in (5.2) is discontinuous with respect the usual topology.
5.2. 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 , known as the regularization parameter, where the symbol is the identity operator and the adjoint operator of [16, 22–29], for a square integrable function . According to the regularization theory, it is proven that the inverse operator in (5.3) is bounded and always exists. Moreover, the converges to the solution to (5.1) as . The stabilization process characterized as in (5.3) is called Tikhonov’s regularization [16, 22, 23].
6. 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 in (2.5): Note that the Fourier transform  of (6.1) is known as follows:
We normalize the water depth and the constant as the unit. The water waves that result from this movement are briefly sketched in Figure 1. Graphical illustrations of and its Fourier transform are depicted in Figures 2 and 3. In this paper, for numerical calculation, the interval for the physical variable is taken as , and the interval for the frequency as .
6.1. Noise Level
Our aim is to inversely recover in (6.1) by using the data 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 . Thereby, we define a noise level , which satisfies the following norm inequality: That is, the noise level is a quantity measuring an error intensity concerning the data . Here, the notation refers to the norm . We have, in this study, the data randomly generated but with the two different noise levels of and , 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 . The results in Figures 4(b) and 4(c) correspond to the noise levels and .
(a) Noise-free condition
(b) Noise level of
(c) Noise level of
6.2. 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 -curve criterion , we depict the curves of log-log plot as shown in Figures 5 and 8: where denotes the regularized solution, calculated by (5.3), depending on the regularization parameter . 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 when the noise level is . This is because the optimal regularization parameter corresponds to the corner of the -curve in Figure 5 . A brief explanation for this reason is as follows. When the regularization parameter decreases, the size of error, , reduces, because approaches the true solution. However, subsequently potentially deviates far away; that is, the function norm, , begins to increase, when exceeds a certain threshold. This means that there exists an appropriate regularization parameter such that the is an optimal solution for the present problem. This optimal solution is shown to correspond to the corner of the -curve by Hansen .
In a similar way, we can obtain the optimal regularization parameter corresponding to the noise level , as depicted in Figure 8.
6.3. Determining the Wave Spectrum
The graphs shown in Figures 6(a)–6(d), concerning the noise level , compare the exact wave spectrum in (6.2) with the regularized wave spectrum in (5.3). In addition, Figure 6(e) shows a residual, which is the norm of difference between the exact and regularized spectra divided by the norm of the exact. In fact, we did calculate many of various regularized wave spectra with various values for . Among them, only four selected regularization parameters , , , and 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 equals ; as mentioned above, this value of is an optimal regularization parameter corresponding to the corner of the -curve.
(e) Residual (the norm of difference between the exact and regularized spectra divided by the norm of the exact)
Similarly, when the noise level is equal to , we did carry out a lot of calculations of regularized wave spectra for various regularization parameters . We also select only four , , , and 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 . It is clear from Figure 9 that the best approximation to the exact one occurs when , which was anticipated from the -curve in Figure 8.
Following (3.5), we can recover the sudden movement of the sea bottom , 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.
(e) Residual (the norm of difference between the exact and regularized displacement divided by the norm of the exact)
(e) Residual (the norm of difference between the exact and regularized spectra divided by the norm of the exact)
(e) Residual (the norm of difference between the exact and regularized displacement divided by the norm of the exact)
We proposed a new method to find sudden movements of the sea bottom using just a local snapshot data of wave configurations, . However, it is interesting to observe also how the method proposed is suitable to retrieve whole wave configurations, that is, we can identify , , just using a local data , . This is realized by simply estimating the integration where 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 . 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.
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.
This work is partially supported by the principal R&D program of KORDI: “Performance Evaluation Technologies of Offshore Operability for Transport and Installation of Deep-sea Offshore Structures” granted by Korea Research Council of Public Science and Technology. In addition, the first author and the third author are partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no.: 2011-0010090). And they were also supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no.: K20902001780-10E0100-12510).
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