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Elastic full-waveform inversion (FWI) when successfully applied can provide accurate and high-resolution subsurface parameters. However, its high computational cost prevents the application of this method to large-scale field-data scenarios. To mitigate this limitation, we propose a target-oriented elastic FWI methodology based on a redatuming step that relies upon an extended least-squared migration process. In our approach, the surface-reflection data can be attributed to a given subsurface portion when mapped into the image space. This process allows us to reconstruct reflection data generated by a target area and recorded with a virtual acquisition geometry positioned directly above it. The redatuming step enables the application of an elastic FWI method within the target portion only. The entire workflow drastically diminishes the overall cost of the surface-data inversion and allows the retrieval of accurate elastic parameters of the area of interest. We demonstrate the effectiveness of our approach on a synthetic case based on the well-known Marmousi2 model and on a 3D ocean-bottom-node (OBN) pressure data recorded in the Gulf of Mexico. We first discuss the fundamental aspects of the methodology and apply the proposed workflow to the synthetic test case. We also employ the methodology on the field-data scenario and show its efficacy at correctly retrieving the elastic parameters and rock-physical properties of a gas-bearing sand reservoir positioned in proximity of a salt-dome flank.