Theoretical background for full-waveform inversion with distributed acoustic sensing and integrated strain sensing

Full-waveform inversion (FWI) is a powerful imaging technique that produces high-resolution subsurface models. In seismology, FWI workflows are traditionally based on seismometer recordings. The development of fibre-optic sensing presents opportunities for harnessing information from new types of measurements. With dense spatial and temporal sampling, fibre-optic sensing captures the seismic wavefield at metre-scale resolution along the cable. Applying FWI to fibre-optic measurements requires the reformulation of the forward and adjoint problems due to two fundamental differences to seismometer data: i) fibre-optic measurements are sensitive to strain rather than translational motion, and ii) they do not represent the motion at a single spatial point, but instead capture the average deformation over a pre-defined cable segment, known as the gauge length. This study introduces the theoretical and computational framework to apply FWI to distributed acoustic sensing (DAS) and integrated fibre-optic sensing (IFOS) in 3-D for complex cable geometries and subsurface structures. Our proposed forward model for fibre-optic data incorporates gauge length effects, and direction— and curvature-dependent sensitivity, as well as arbitrary cable layouts. For the numerical simulations, we use a spectral-element solver that allows us to incorporate surface topography, and coupled viscoacoustic and viscoelastic rheologies. We derive an adjoint source based on moment tensors. In illustrative examples, we present how our theoretical developments can be used in inversions of synthetic fibre-optic data generated for a realistically curved cable placed on irregular topography. As examples, we invert for source parameters, including moment tensor, location, and origin time for noise-free DAS data, noise-contaminated DAS data, and IFOS data. Further, we present FWI for the three data groups to obtain 3-D tomographic images of P- and S-wave speeds. In all example inversion, we compare how close the found model is to the ground truth. The codes to produce these results are accessible and ready to be applied to real data inversions.