Nonlinear LS-RTM based on seismic scale separation and full wavefield multi-parameter FWI Øystein Korsmo Yang Yang Nizar Chemingui PGS PGS PGS oystein.korsmo@pgs.com yang.yang@pgs.com nizar.chemingui@pgs.com Andrey Pankov Antonio Castiello Andrew Long PGS PGS PGS andrey.pankov@pgs.com antonio.castiello@pgs.com andrew.long@pgs.com SUMMARY Least-squares migration (LSM) images the subsurface through an inversion process and seeks the reflectivity model that best explains the measured data. This data-fitting process will implicitly compensate for wavefield propagation effects and limited resolution seen with conventional imaging. These favourable properties have made it the method of choice in complex geological settings. In this paper, we will discuss the various properties and implementations of LSM and Full Waveform Inversion (FWI) and suggest a new nonlinear least-squares reverse time migration (LS-RTM) implemented in the framework of FWI. The new inversion estimates the earth reflectivity while updating the velocity model at every iteration. We demonstrate the new solution on a field data example from the Norwegian Sea. Key words: FWI, RTM, Least Squares Migration, Multi-Parameter. INTRODUCTION LSM is a linear inversion process with a convex objective function. It updates the reflectivity by matching the high wavenumber perturbations in the observed and synthetic data (Nemeth et al., 1999). The linear property of the inversion relates to the fixed background model, meaning that the velocity field is assumed known and kept unchanged during the inversion, making it feasible to estimate with a linear solver (Gauthier et al., 1986; Mora, 1989). The synthetic data are generated based on the Born approximation, where only the first order scattering term is utilized in the forward modelling step (Woodhouse, 1980; Tanimoto, 1984). In practice, this means that only the near- to mid-reflection angles can be utilized in the inversion process. On the other hand, FWI is a highly nonlinear inversion with a complex objective function consisting of many local minima (Tarantola, 1987). The aim of FWI is to minimize the misfit between observed and modelled data. To linearize the inversion, the model parameters can be updated with a gradient descent approach that utilizes smaller steps in combination with a multi-stage process. In practice, this means that the lowest frequencies are first used and matched before higher frequencies are included in the inversion to reveal more details in the model. FWI is commonly implemented with the adjoint-state method (Chavent, 1974), where the adjoint states refer to the backpropagated data-residuals and the gradient of the least-squares misfit is a migration operator (Lailly, 1983; Tarantola, 1984). Refractions and diving waves have proven to be robust and straightforward to utilize in FWI and have become the basis of most velocity model building projects. FWI can also incorporate reflections, which allow velocity estimation beyond the penetration depth of diving waves. For this to be feasible, the modelling engine must initiate the complete wavefield; diving waves, refractions, and reflections (beyond Born approximations). In Figure 1, we show a comparison between Born modelling (a) and the full wavefield modelling (b). Notice that only the near to mid offsets/angles are generated with the Born approximation, while the full wavefield modelling generates the head-wave and diving waves/refractions in addition to low and high angle reflections. To make use of the complete wavefield, not limited by Born modelling, we propose a new nonlinear seismic inversion process built on the FWI framework. The method can be considered as a multi-parameter FWI where each model will be updated simultaneously without parameter-leakage. This can be achieved with a unique imaging condition, that naturally honours the seismic scale separation, and solve for the background model (velocities) and the perturbations (reflectivity) separately. The background model can be interpreted as the FWI long-wavelength velocities and the reflectivity can be described as a nonlinear LS-RTM, implemented with the adjoint state method. To handle the nonlinearity of this inversion, we follow the multi-stage approach until the background model reach convergence. From this point, we can perform large steps in frequency with the focus primarily on the reflectivity model.