Construction of Artificial Point Sources for a Linear Wave Equation in Unknown Medium

We study the wave equation on a bounded domain of Rm and on a compact Riemannian manifold M with boundary. We assume that the coefficients of the wave equation are unknown but that we are given the hyperbolic Neumann-to-Dirichlet map. that corresponds to the physical measurements on the boundary. Using the knowledge of. we construct a sequence of Neumann boundary values so that at a time T the corresponding waves converge to zero while the time derivative of the waves converge to a delta distribution. The limit of such waves can be considered as a wave produced by an artificial point source. The convergence of the wave takes place in the function spaces naturally related to the energy of the wave. We apply the results for inverse problems and demonstrate the focusing of the waves numerically in the one-dimensional case.Peer reviewe

Micro-seismic Elastic Reflection Full Waveform Inversion with An Equivalent Source

In micro-seismic event measurements, pinpointing the passive source's exact spatial and temporal location is paramount. This research advocates for the combined use of both P- and S-wave data, captured by geophone monitoring systems, to improve source inversion accuracy. Drawing inspiration from the secondary source concept in Elastic Reflection Full Waveform Inversion (ERFWI), we introduce an equivalent source term. This term combines source functions and source images. Our optimization strategy iteratively refines the spatial locations of the source, its temporal functions, and associated velocities using a full waveform inversion framework. Under the premise of an isotropic medium with consistent density, the source is defined by two spatial and three temporal components. This offers a nuanced source representation in contrast to the conventional seismic moment tensor. To address gradient computation, we employ the adjoint-state method. However, we encountered pronounced non-linearity in waveform inversion of micro-seismic events, primarily due to the unknown source origin time, resulting in cycle skipping challenges. To counteract this, we devised an objective function that is decoupled from the source origin time. This function is formulated by convolving reference traces with both observed and predicted data. Through the concurrent inversion of the source image, source time function, and velocity model, our method offers precise estimations of these parameters, as validated by a synthetic 2D example based on a modified Marmousi model. This nested inversion approach promises enhanced accuracy in determining the source image, time function, and velocity model