Passive seismic inversion of SH wave input motions in a truncated domain

This paper introduces a new inversion method for the reconstruction of complex, incoherent SH incident wavefield in a domain that is truncated by a wave-absorbing boundary condition (WABC), using a partial differential equation (PDE)-constrained optimization method. In numerical examples, dynamic traction at the WABC mimics seismic incident wavefield. Estimated traction is discretized over space and time, and the discretized values are reconstructed by using seismic motion data that are sparsely made by sensors on the top surface of a domain and a vertical array. The discretize-then-optimize (DTO) approach is used in the mathematical modeling and numerical implementation, and the finite element method (FEM) is applied to solve state and adjoint problems. The numerical results show that incident, inclined plane waves can be reconstructed if sensors are located both on the top surface and at a vertical array. Without the vertical array, the accuracy to invert for the particular part of the inclined waves that pass the vertical array declines. Such effectiveness of the vertical array is observed regardless of the dominant frequencies of incident waves, the complexity of their time signals, and their angles in a homogeneous or layered background domain. Second, the optimizer undergoes less severe solution multiplicity when identifying lower-frequency traction (e.g., realistic earthquake signal). Third, a sufficiently large number of sensors must be employed to improve the algorithm's inversion performance. The desired number of sensors per unit length increases as the wavelength of the incident waves decreases. Fourth, a large value of the inversion error in the reconstructed traction of a high dominant frequency does not necessarily translate to an error of the same order of magnitude in the corresponding reconstructed wave responses in the computational domain because of the intrinsic low-pass filtering of the FEM wave solver. Fifth, our presented inversion algorithm's accuracy is not compromised by the material complexity of a background domain. Lastly, the error in the reconstructed traction and the error in the corresponding wave responses grow when the noise of a larger level is added to the measurement data, but not in the same proportion. By extending the presented method into realistic 3D settings, this algorithm can indicate where large amplitudes of stress waves (i.e., weak points) occur in built environments and soils in a domain of interest during seismic events.