This paper is concerned with the inverse problem of reconstructing the location and shape of buried obstacles in the lower half-space of an unbounded two-layered medium in two dimensions from phaseless far-field data. Similarly to the homogenous background medium case, for this problem it is also true that the modulus of the far-field pattern is invariant under translations of the scattering obstacle if only one plane wave is used as the incident field, and thus it is impossible to determine the location of the obstacle from such phaseless far-field data. Based on the idea of using superpositions of two plane waves with different directions as the incident fields, a direct imaging algorithm is developed in this paper to locate the position of small anomalies with the intensity of the far-field pattern measured in the upper half-space. This is a nontrivial extension of our previous work (2018 Inverse Problems 34 104005) from the homogenous background medium case to the two-layered background medium case. Both the limited aperture measurement data and the presence of the two-layered background medium lead to difficulties in the theoretical analysis of the proposed imaging algorithm. To overcome the difficulties we employ the theory of oscillatory integrals. Further, with the aid of the imaging algorithm, a recursive Newton-type iteration algorithm in frequencies is proposed to reconstruct both the location and shape of extended obstacles. Finally, numerical experiments are presented to illustrate the feasibility of our algorithms.

本文研究的是从无相远场数据在二维无界两层介质下半空间重建埋藏障碍物位置和形状的逆问题。与均匀背景介质的情况类似，对于这个问题，如果仅使用一个平面波作为入射场，远场模式的模量在散射障碍物的平移下也是不变的，因此不可能从这种无相远场数据中确定障碍物的位置。本文基于将两个不同方向的平面波叠加作为入射场的思想，提出了一种直接成像算法，利用上半部测量的远场图的强度来定位小异常点的位置。空间。逆问题 34 104005) 从同质背景介质情况到两层背景介质情况。有限的孔径测量数据和两层背景介质的存在导致所提出的成像算法的理论分析困难。为了克服这些困难，我们采用了振荡积分理论。此外，在成像算法的帮助下，提出了一种递归牛顿型频率迭代算法来重建扩展障碍物的位置和形状。最后，给出了数值实验来说明我们算法的可行性。