Imaging wave-penetrable objects in a finite depth ocean Keji Liu ∗ Yongzhi Xu † Jun Zou ‡ Abstract. We extend the direct sampling method proposed in [13] to image a wave- penetrable inhomogeneous medium in a 3D waveguide. Incidences and receivers are available only on part of the surface of a cylinder. The proposed method is basically direct and does not involve any matrix inversions or optimizations, thus computationally very cheap and efficient. Numerical simulations are presented to show the feasibility and effectiveness of the method for acoustic detection in a 3D waveguide. The method is applicable with a few scattered fields corresponding to only one or two incident waves, and is very robust against noise. Key words. Finite depth ocean, inhomogeneous media, direct sampling method. MSC classifications. 35R30, 41A27, 76Q05. 1 Introduction The direct and inverse scattering problems of underwater acoustics have received much at- tention in recent years, see, e.g., [1, 2, 4, 5, 7, 14, 15, 17] and the references therein. One of the popular models used for acoustic waves in a finite depth ocean is the waveguide bounded by two parallel planes. Because of the geometric structure, the inverse scattering problems in a parallel waveguide are much harder than similar problems in a homogeneous space. Due to the presence of two boundaries of the waveguide, only a finite number of wave modes can propagate in long distance, while the other modes decay exponentially as a function of distance. This phenomenon increases the ill-posedness of the inverse problem considerably. Assume that the ocean has a pressure released surface and a rigid bottom, we can pose a Dirichlet condition on one of the plane and a Neumann condition on the other. Based on this model, the exact and asymptotic representations of the sound field in a stratified shallow ocean was obtained in [3]. Adding a scatterer to the stratified model, a series of studies have been carried out for the direct and inverse scattering of acoustic waves by obstacles in a waveguide with plane boundaries as well as in an ocean under different sediment settings. We refer readers to [9, 10, 11, 6, 8, 12, 16, 19, 20] for more details. In this paper we extend the direct sampling method proposed in [13] for imaging a wave- penetrable inhomogeneous medium in a 3D finite depth ocean. The method is based on a scat- tering analysis and involves only computing the inner product of the measured scattered field u s ∗ Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong. (kjliu@math.cuhk.edu.hk) † Department of Mathematics, University of Louisville, Louisville, KY 40245, USA. (ysxu0001@louisville.edu) ‡ Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong. The work of this author was substantially supported by Hong Kong RGC grant (Project 404611). (zou@math.cuhk.edu.hk) 1