Scattering-based decomposition of sensitivity kernels for full-waveform inversion

Abstract

With the increasing demand in complexity for subsurface models in environments such as subsalt, sub-basalt and pre-salt, full-waveform inversion (FWI) is quickly becoming one of the model-building methods of choice. While in principle capable of handling all of the nonlinearity in the data, in practice nonlinear gradient-based FWI is limited due to its notorious sensitivity to the choice of starting models. To help addressing model convergence issues in FWI, in this paper we analyze the role of nonlinearity in the so-called sensitivity kernels, which are the centerpiece of gradient-based FWI algorithms. Using a scattering-based approach and assuming acousticonly data, we start by reparameterizing the subsurface model in terms of smooth and singular components for both compressibility and density. This leads to a decomposition of the data into a reference field that is sensitive only to the smooth model, and a scattered field sensitive to both smooth and sharp model components. Focussing on the model backprojections from the scattered data only, we provide expressions for the Frechet-derivative sensitivity kernels of all four model parameters. Our results provide for the decomposition of current FWI kernels into no less than nine different sub-kernels which have explicitly different levels of nonlinearity with respect to both data and model parameters. This capability to discern levels of nonlinearity within FWI kernels is key to understanding model convergence in gradient-based, iterative FWI. We illustrate this by analyzing some of the sub-kernel terms in detail. The scattering-based FWI kernel decomposition we provide could have broad potential applications, such devising multiscale FWI algorithms, and improving velocity model building in the image domain using extended image gathers. Introduction For many years, the most common imaging techniques were based on ray theory, such as, Kirchhoff migration. But lately, as the industry have been facing geologically more complex areas where these techniques were not successful, new methods based on wave-equation migration came into play – in the begining, one-way wave-equation based, and more recently, two-way waveequation. All this became possible due to new aquisition techniques which give better illumination of the subsurface, and more powerful computational capacities. But those new methods require more and more refined Earth models. on the other, hand, even if migration has advanced quickily with computer power, constructing these models is still ray-based. Recently, one tool, based on the two-way wave-equation, have been studied and developed for Earth modeling: The full waveform inversion (FWI) (Vigh et al., 2009). The basic idea behind the FWI is the minimization of a objective function that ”mesures” the difference between observed seismic data and synthetic data from a earth model. In the last decades, many studies on FWI were made. On Virieux and Operto (2009) and Vigh et al. (2009) one can find the state-of-art on the subject. A series of paper publish on the eighties (Lailly, 1983; Tarantola, 1984, 1986) brought to light the gradient-based FWI in the applied geophysics field. In a few words, this methods says that a model can be updated iteratively with the help of the sensitivity kernels (SK). The SK’s are operators that give the change in the wavefield due to changes in the model parameters. With the adjoint of the SK, one can evaluate the change in the earth model due to wavefield residuals. While in principle capable of handling all of the nonlinearity in the data, in practice nonlinear gradient-based FWI is limited due to its notorious sensitivity to the choice of starting models. To help addressing model convergence issues in FWI, in this paper we analyze the role of nonlinearity in the sensitivity kernels. To do so, we use a scattering-based approach (Vasconcelos, 2008). Assuming a acoustic medium, we reparameterize the subsurface model in terms of smooth and singular components for both compressibility and density. This leads to a decomposition of the data into a reference field that is sensitive only to the smooth model, and a scattered field sensitive to both smooth and sharp model components. Focussing on the model backprojections from the scattered data only, we provide expressions for the Frechet-derivative sensitivity kernels of all four model parameters. The scattering-based FWI kernel decomposition we provide could have broad potential applications, such devising multiscale FWI algorithms, and improving velocity model building in the image domain using extended image (EI) gathers (Rickett and Sava, 2002; Sava and Fomel, 2003; Symes, 2008; Sava and Vasconcelos, 2009, 2010). As shown by Vasconcelos et al. (2009, 2010), there’s a connection between the extended image conditions and the interferometry formalism: the EI’s behave like locally scattered wavefields in the image domain. Twelfth International Congress of The Brazilian Geophysical Society SCATTERING-BASED DECOMPOSITION OF SENSITIVITY KERNELS 2