This paper addresses a factorization method for imaging the support of a wave-number-dependent source function from multi-frequency data measured at a finite pair of symmetric receivers in opposite directions. The source function is given by the inverse Fourier transform of a compactly supported time-dependent source whose initial moment or terminal moment for radiating is unknown. Using the multi-frequency far-field data at two opposite observation directions, we provide a computational criterion for characterizing the smallest strip containing the support and perpendicular to the directions. A new parameter is incorporated into the design of test functions for indicating the unknown moment. The data from a finite pair of opposite directions can be used to recover the $\Theta$-convex polygon of the support. Uniqueness in recovering the convex hull of the support is obtained as a by-product of our analysis using all observation directions. Similar results are also discussed with the multi-frequency near-field data from a finite pair of observation positions in three dimensions. We further comment on possible extensions to source functions with two disconnected supports. Extensive numerical tests in both two and three dimensions are implemented to show effectiveness and feasibility of the approach. The theoretical framework explored here should be seen as the frequency-domain analysis for inverse source problems in the time domain.

翻译：本文针对波数依赖源函数支撑成像问题，提出一种利用有限对对称接收器在相反方向测量的多频数据进行因式分解的方法。源函数由具有紧支撑的时变源的傅里叶逆变换定义，其初始辐射时刻或终止辐射时刻未知。利用两个相反观测方向的多频远场数据，我们建立了表征包含支撑且垂直于方向的最小条带的计算准则。在测试函数设计中引入新参数以指示未知时刻。有限对相反方向数据可用于恢复支撑的Θ-凸多边形。作为分析副产品，利用全方向观测可唯一恢复支撑的凸包。本文还对三维空间中有限对观测位置的多频近场数据进行了类似讨论，并进一步探讨了支撑不连通源函数的扩展可能性。二维与三维空间的数值实验验证了该方法的有效性与可行性。本理论框架应视为时域逆源问题的频域分析方法。