Microseismic source imaging using physics-informed neural networks with hard constraints

Microseismic source imaging plays a significant role in passive seismic monitoring. However, such a process is prone to failure due to the aliasing problem when dealing with sparse measured data. Thus, we propose a direct microseismic imaging framework based on physics-informed neural networks (PINNs), which can generate focused source images, even with very sparse recordings. We use the PINNs to represent a multi-frequency wavefield and then apply the inverse Fourier transform to extract the source image. Specially, we modify the representation of the frequency-domain wavefield to inherently satisfy the boundary conditions (the measured data on the surface) by means of the hard constraint, which helps to avoid the difficulty in balancing the data and PDE losses in PINNs. Furthermore, we propose the causality loss implementation with respect to depth to enhance the convergence of PINNs. The numerical experiments on the Overthrust model show that the method can admit reliable and accurate source imaging for single- or multiple- sources and even in passive monitoring settings. Then, we further apply our method on the hydraulic fracturing field data, and demonstrate that our method can correctly image the source.