  Microseismic source imaging plays a significant role in passive seismic
monitoring. However, such a process is prone to failure due to aliasing when
dealing with sparsely 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 inverse Fourier
transform to extract the source image. To be more specific, we modify the
representation of the frequency-domain wavefield to inherently satisfy the
boundary conditions (the measured data on the surface) by means of a 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. Compared with the time-reversal method, the results of the proposed
method are consistent with numerical methods but less noisy. Then, we further
apply our method to hydraulic fracturing monitoring field data, and demonstrate
that our method can correctly image the source with fewer artifacts.
