Reconstruction of Initial Wave Field of a Nonsteady-State Wave Propagation from Limited Measurements at a Specific Spatial Point Based on Stochastic Inversion

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1. Introduction

Water waves are one of important subjects in ocean-related fields. They are generated on the free surface, between air and water, from various sources such as a storm at sea, moving ship in calm water, and impact from a falling object. Once they are generated, wave propagation occurs on the surface of water away from the source. The propagation of water wave is a little more complex than many other kinds of waves in nature due to its dispersion.

In this paper, we are interested in a nonsteady-state water wave propagation generated by an initial surface disturbance. The classical problem in water waves is the computation of the time and spatial evolution in a whole body of water with this given initial wave field. That is, the aim of such problems is to predict the propagation of water waves as effects of the initial free surface disturbance, which is the cause of this phenomenon. This kind of problem has been widely studied because of its importance [1–6]. We name it forward or direct problem. Although much progress has been achieved in the analysis of resulting wave flow, there are only few studies [7–11] on inverse problems, whose aim is to find the cause of the wave propagation.

In this study, our attention is focused on the inverse problem of reconstructing initial wave field from the limited measurement on the specific boundary of the fluid body for two- and three-dimensional problems. Based on the assumption of linear dispersive wave theory, the reconstructing problem can be formulated as an inverse problem. The problem requires solving the Fredholm integral equation of the first kind, which causes severe ill posedness when numerically approximating. As a result, the inverse problem (reconstruction of initial wave field) is mathematically and numerically much more challenging than the forward problem (simulation of future wave field given...