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.

翻译：微震源成像在被动地震监测中具有重要意义。然而，当处理稀疏测量数据时，由于混叠问题，该过程容易失效。为此，我们提出了一种基于物理约束神经网络（PINNs）的微震源直接成像框架，即使是在极稀疏的记录条件下，也能生成聚焦的震源图像。我们利用PINNs表示多频率波场，并通过逆傅里叶变换提取震源图像。特别地，我们对频域波场的表示进行了修改，通过硬约束手段使其天然满足边界条件（地表测量数据），这有助于避免PINNs中数据损失与偏微分方程损失平衡的难题。此外，我们提出了基于深度方向的因果损失实现方法，以增强PINNs的收敛性。在Overthrust模型上的数值实验表明，该方法能够对单源或多源情况，甚至在被动监测环境下，提供可靠且精确的震源成像。随后，我们进一步将该方法应用于水力压裂现场数据，并证明该方法能够正确地对震源进行成像。