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

TL;DR

This work proposes a direct microseismic imaging framework based on physics-informed neural networks (PINNs), which can generate focused source images, even with very sparse recordings, and proposes the causality loss implementation with respect to depth to enhance the convergence of PINNs.

Abstract

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.

Figures (23)

• Figure 1: The comparison between the NN function and the 3Hz observed data. a) shows the prediction of NN function trained with 20% to 90% of the dataset, and b) is the comparison between the corresponding second-order derivates calculated by AD or the finite-difference method (FD) for various coverage percentages.
• Figure 2: The pipeline of the proposed method. There are two branches: the data fitting branch (top) and the PDE fitting branch (bottom). The recorded data $d(x,t)$ are first transformed to the frequency domain. Then, they are used to train the data NN. Further, the data NN is combined into the PDE fitting branch to train the PDE NN. The output of the PDE NN is transformed to the time-domain snapshots using inverse FFT, where the red box denotes the source imaging where the energy focuses. The module PE denotes the positional encoding with sinusoidal functions Huang2021.
• Figure 3: The filtered (3-12 Hz) recording of one source event (located at (5.0, 3.0) km) with receivers covering the whole surface.
• Figure 4: a) shows the Overthrust velocity model, the source event location (denoted by the black star), and the random receivers (denoted by the red triangles); b) is the source image courtesy of the proposed method.
• Figure 5: The source image courtesy of a numerical finite-difference solver.