A 4-D spatio-temporal misfit function for waveform inversion of Distributed Acoustic Sensing data

Summary Standard Full-Waveform Inversion (FWI) objective functions often struggle to resolve deep structures when data are dominated by localised near-surface scattering. To address this, we develop a 4-D time-frequency-space-wavenumber (TFXK) objective function that explicitly enforces spatial coherence, leveraging the dense sampling of Distributed Acoustic Sensing (DAS). We derive the corresponding adjoint sources for both phase and envelope misfits. By suppressing localised scattering from shallow heterogeneities, the TFXK misfit shifts kernel sensitivity toward deeper, larger-scale structural targets. Furthermore, this 4-D mapping smooths the objective function topology, mitigating local minima while improving inversion convergence, as demonstrated in a 2D synthetic FWI example with strong shallow heterogeneity. The proposed idea of a spatio-temporal objective function is generalisable and can be adapted to other objective functions. We also show how it can accommodate curved fibre geometries via arc-length parametrization, Local Polynomial Fourier Transforms, or a full vectorial wavenumber approach, depending on the complexity of the fibre layout. These mathematical extensions are necessary for applying the method to varied fibre geometries and low-frequency inversions. The foundational principles of our approach extend beyond optical fibres and can also be applied to structural inversions using traditional dense nodal arrays.