Source location using physics-informed neural networks with hard constraints

Locating subsurface seismic sources is crucial to both seismic monitoring and seismology. The exploding reflector assumption provides a direct imaging approach for focusing energy at microseismic source locations under the premise of time-reversal imaging. However, the imaging process is prone to aliasing problems when the observed data are sparsely sampled. Physics-informed neural networks (PINNs) provide a feasible solution to obtain aliased free images of the sources by representing the frequency-domain wavefield by as a neural network function of spatial coordinates and angular frequency. Specifically, we use a modified representation of the Helmholtz equation, which incorporates the recorded data in the partial differential equation, as a physical loss for PINNs to avoid the challenge that PINNs face in dealing with boundary conditions. The additional frequency dimension of the neural network function allows for direct image extraction of the subsurface using inverse Fourier transform. Numerical tests on the Overthrust model demonstrate that the proposed method could admit reliable source locations in multiple scenarios with coarsely sampled data in a label-free manner.