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We present a new method to obtain dynamic body force at virtual interfaces to reconstruct shear wave motions induced by a source outside a truncated computational domain. Specifically, a partial differential equation (PDE)-constrained optimization method is used to minimize the misfit between measured motions at a limited number of sensors on the ground surface and their counterparts reconstructed from optimized forces. Numerical results show that the optimized forces accurately reconstruct the targeted ground motions in the surface and the interior of the domain. The proposed optimization framework yields a particular force vector among other valid solutions allowed by the domain reduction method (DRM). Per this optimized or inverted force vector, the reconstructed wave field is identical to its reference counterpart in the domain of interest but may differ in the exterior domain from the reference one. However, we remark that the inverted solution is valid and introduce a simple post-process that can modify the solution to achieve an alternative force vector corresponding to the reference wave field. We also study the desired sensor spacing to accurately reconstruct the wave responses for a given dominant frequency of interest. We remark that the presented method is omnidirectionally applicable in terms of the incident angle of an incoming wave and is effective for any given material heterogeneity and geometry of layering of a reduced domain. The presented inversion method requires information on the wave speeds and dimensions of only a reduced domain. Namely, it does not need any informa- tion on the geophysical profile of an enlarged domain or a seismic source profile outside a reduced domain. Thus, the computational cost of the method is compact even though it leads to the high-fidelity reconstruction of wave re- sponse in the reduced domain, allowing for studying and predicting ground and structural responses using real seismic measurements. 

This work studies the feasibility of imaging a coupled fluid-solid system by using the elastodynamic and acoustic waves initiated from the top surface of a computational domain. A one-dimensional system, where a fluid layer is surrounded by two solid layers, is considered. The bottom solid layer is truncated by using a wave-absorbing boundary condition (WABC). The wave responses are measured on a sensor located on the top surface, and the measured signal contains information about the underlying physical system. By using the measured wave responses, the elastic moduli of the solid layers and the depths of the interfaces between the solid and fluid layers are identified. To this end, a multi-level Genetic Algorithm (GA) combined with a frequency- continuation scheme to invert for the values of sought-for parameters is employed. The numerical results show the following findings. First, the depths of solid-fluid interfaces and elastic moduli can be reconstructed by the presented method. Second, the frequency-continuation scheme improves the convergence of the estimated values of parameters toward their targeted values. Lastly, a preliminary inversion, using an all- solid model, can be employed to identify if a fluid layer is presented in the model by showing one layer with a very large value of Young's modulus (with a similar value to that of the bulk modulus of water) and the value of mass density being similar to that of water. Then, the primary GA inversion method, based on a fluid-solid model, can be utilized to adjust the soil characteristics and fine-tune the locations of the fluid layer. If this work is extended to a 3D setting, it can be instrumental to finding unknown locations of fluid-filled voids in geological formations that can lead to ground instability and/or collapse (e.g., natural/anthropogenic sinkhole, urban cave-in subsidence, etc.).