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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

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 [

Recently, Jang et al. [

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 [

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 [

We consider an inviscid incompressible water of finite depth and a system of coordinates in which the

Impulsive movement of the sea-bottom and the resulting wave flow.

We assume that the flow is irrotational in a simply connected fluid domain such that there exists a single-valued velocity potential function

We suppose that the sea bottom changes suddenly at

As mentioned before, the abrupt bottom motion is assumed to arise at

Before the detailed discussion of solving the integral equation of (

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 (

We first rewrite (

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

Because the computer memory is limited in practice, in this study, we replace the integration limit of

To overcome the difficulty encountered in Section

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

We normalize the water depth

Graphical illustration of the impulsive movement

Fourier transform of

The impulsive movement of the sea bottom equation (

Our aim is to inversely recover

The numerical values for (

The spatial distribution of the generated waves when

Noise-free condition

Noise level of

Noise level of

To achieve an accurate solution during the regularization, it is important to decide optimal regularization parameter in the Tikhonov regularization [

Illustration of the

In a similar way, we can obtain the optimal regularization parameter

The graphs shown in Figures

Comparison of the regularized wave spectrum with the exact one for four cases of regularization parameters

Residual (the

Similarly, when the noise level is equal to

Following (

Comparison of the recovered sudden displacement of the sea bottom with the exact displacement for four cases of regularization parameters

Residual (the

Illustration of the

Comparison of the regularized wave spectrum with the exact wave spectrum for four cases of regularization parameters

Residual (the

Comparison of the recovered, sudden displacement of the sea bottom with the exact displacement for four cases of regularization parameters

Residual (the

We proposed a new method to find sudden movements of the sea bottom using just a local snapshot data of wave configurations,

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 [

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).