We investigate the identification of the time-dependent source term in the diffusion equation using boundary measurements. This facilitates tracing back the origins of environmental pollutants. Employing the concept of dynamic complex geometrical optics (CGO) solutions, a variational formulation of the inverse source problem is analyzed, leading to a proof of uniqueness result. Our proposed two-step reconstruction algorithm first determines the point source locations and subsequently reconstructs the Fourier components of the emission concentration functions. Numerical experiments on simulated data are conducted. The results demonstrate that the proposed two-step reconstruction algorithm can reliably reconstruct multiple point sources and accurately reconstruct the emission concentration functions. Additionally, by partitioning the algorithm into online and offline computations, and concentrating computational demand offline, real-time pollutant traceability becomes feasible. This method, applicable in various fields - especially those related to water pollution, can identify the source of a contaminant in the environment, thus serving as a valuable tool in environmental protection.

翻译：我们研究利用边界测量识别扩散方程中与时间相关的源项，从而追溯环境污染物来源。通过引入动态复几何光学解的概念，分析了逆源问题的变分形式，并证明了唯一性结果。我们提出的两步重建算法首先确定点源位置，随后重建排放浓度函数的傅里叶分量。基于模拟数据的数值实验表明，该两步重建算法能够可靠地重建多个点源并准确恢复排放浓度函数。此外，通过将算法划分为在线计算与离线计算两个阶段，并将计算需求集中于离线部分，实时污染物溯源成为可能。该方法可应用于多个领域（尤其与水污染相关领域），能够识别环境中污染物的来源，从而为环境保护提供重要工具。