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