This paper presents a method for reconstructing an acoustic source located in a two-layered medium from multi-frequency phased or phaseless far-field patterns measured on the upper hemisphere. The interface between the two media is assumed to be flat and infinite, while the source is buried in the lower half-space. In the phased case, a Fourier method is proposed to identify the source based on far-field measurements. This method assumes that the source is compactly supported and can be represented by a sum of Fourier basis functions. By utilizing the far-field patterns at different frequencies, the Fourier coefficients of the source can be determined, allowing for its reconstruction. For the case where phase information is unavailable, a phase retrieval formula is developed to retrieve the phase information. This formula exploits the fact that the far-field patterns are related to the source through a linear operator that preserves phase information. By developing a suitable phase retrieval algorithm, the phase information can be recovered. Once the phase is retrieved, the Fourier method can be adopted to recover the source function. Numerical experiments in two and three dimensions are conducted to validate the performance of the proposed methods.

翻译：本文提出了一种方法，用于根据上半球面上测量的多频相位或相位缺失远场模式，重建位于两层介质中的声源。假设两种介质之间的界面是平坦且无限的，而声源埋藏在下半空间中。在相位已知的情况下，提出了一种基于远场测量的傅里叶方法来识别声源。该方法假设声源具有紧支集，并且可以表示为傅里叶基函数的和。通过利用不同频率下的远场模式，可以确定声源的傅里叶系数，从而实现其重建。对于无法获取相位信息的情况，开发了一种相位恢复公式来检索相位信息。该公式利用了远场模式通过一个保持相位信息的线性算子与声源相关联这一事实。通过开发合适的相位恢复算法，可以恢复相位信息。一旦相位被恢复，就可以采用傅里叶方法来恢复源函数。进行了二维和三维的数值实验，以验证所提出方法的性能。