Adjoint tomography with full envelope for scattering and intrinsic attenuation - resolution and trade-off

A rigorous framework exists for deterministic imaging the subsurface seismic velocity structure. Full-waveform inversion (FWI) that combines the forward simulation of waveforms with an adjoint (backward) simulation of the data misfit provides the gradient of the model misfit with respect to the changes in the model parameters. This gradient is used for iterative improvements of the model to minimize the data misfit. To investigate the small scale heterogeneity of the medium below the resolution limits the waveform tomography the envelopes of high-frequency seismic waves have been used to derive a statistical description of the small scale structure. Such studies employed a variety of misfit measures or empirical parameters and various assumptions about the spatial sensitivity of the measurements to derive some information about the spatial distribution of the high-frequency attenuation and scattering properties. A rigorous framework for the inversion of seismogram envelopes for the spatial imaging of heterogeneity and attenuation has been missing so far. Here we present a mathematical framework for the full envelope inversion that is based on a forward simulation of seismogram envelopes and an adjoint (backward) simulation of the envelope misfit, in full analogy to FWI. 

Different from FWI that works with the wave equation, our approach is based on the Radiative Transfer Equation. In this study, the forward problem is solved by modelling the 2-D multiple nonisotropic scattering in a random elastic medium with spatially variable heterogeneity and attenuation using the Monte-Carlo method. The fluctuation strength ε and intrinsic quality factors Q_P^-1 and Q_S^-1 in the random medium are used to describe the spatial variability of attenuation and scattering. The misfit function is defined as the differences between the full observed and modelled envelopes.

We derived the sensitivity kernels corresponding to this misfit function that is minimized during the iterative adjoint inversion with the L-BFGS method. We have applied this algorithm in some numerical tests in the acoustic approximation. We show that it is possible in a rigorous way to image the spatial distribution of small scale heterogeneity and attenuation separately using seismogram envelopes. The resolution and the trade-off between scattering and intrinsic attenuation are discussed. Our analysis shows that relative importance of scattering and attenuation anomalies need 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.