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In micro-seismic event measurements, pinpointing the passive source's exactspatial and temporal location is paramount. This research advocates for thecombined use of both P- and S-wave data, captured by geophone monitoringsystems, to improve source inversion accuracy. Drawing inspiration from thesecondary source concept in Elastic Reflection Full Waveform Inversion (ERFWI),we introduce an equivalent source term. This term combines source functions andsource images. Our optimization strategy iteratively refines the spatiallocations of the source, its temporal functions, and associated velocitiesusing a full waveform inversion framework. Under the premise of an isotropicmedium with consistent density, the source is defined by two spatial and threetemporal components. This offers a nuanced source representation in contrast tothe conventional seismic moment tensor. To address gradient computation, weemploy the adjoint-state method. However, we encountered pronouncednon-linearity in waveform inversion of micro-seismic events, primarily due tothe unknown source origin time, resulting in cycle skipping challenges. Tocounteract this, we devised an objective function that is decoupled from thesource origin time. This function is formulated by convolving reference traceswith both observed and predicted data. Through the concurrent inversion of thesource image, source time function, and velocity model, our method offersprecise estimations of these parameters, as validated by a synthetic 2D examplebased on a modified Marmousi model. This nested inversion approach promisesenhanced accuracy in determining the source image, time function, and velocitymodel.

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