Adjoint envelope tomography for scattering and absorption using radiative transfer theory

Abstract

SUMMARY To investigate the small-scale elastic structure of the subsurface at length scales below the resolution limits of waveform tomography, envelopes of high-frequency scattered seismic waveforms have been used with a variety of approaches. However, a rigorous framework for the iterative inversion of seismogram envelopes to image heterogeneity and high-frequency attenuation comparable to full waveform inversion (FWI) is missing. We present the mathematical framework for an iterative full envelope inversion using forward and adjoint simulations of the radiative transfer equations, in full analogy to FWI that is based on the wave equation. The forward and adjoint problems are solved by modelling 2-D multiple non-isotropic scattering in a random elastic medium with spatially variable heterogeneity and attenuation using the Monte Carlo method. Sensitivity kernels are derived for the squared difference between the full observed and modelled envelopes which is iteratively minimized with the L-BFGS method. We apply this algorithm in numerical tests in the acoustic approximation and show that it is possible to image the spatial distribution of small-scale heterogeneity and attenuation in iterative inversions. Our analysis shows that the relative importance of scattering and attenuation anomalies needs to be considered when the model resolution is assessed. The inversions confirm that the early coda is important for imaging the distribution of heterogeneity while later coda waves are more sensitive to intrinsic attenuation and we show that this dependency can be used to cope with the trade-off that exists between both material properties.