Summary
In standard seismic full waveform inversion updates (e.g., of Gauss- Newton type) small angle, backscattered amplitudes are incorporated linearly. Making an effort to include nonlinearity in each update may be useful, however, both for estimation of difficult-to-discriminate parameters such as density, and for improvement of convergence rates. We consider, in a theoretical scalar environment, one possible approach to including nonlinearity, wherein sensitivities at iteration n are computed by varying the field associated with the n+1th, rather than nth, model iterate. This produces an extended, series form, sensitivity expression. To understand the basic character of updates based on these revised sensitivities, the expression is truncated at second order, and the resulting Gauss-Newton-like updates are analyzed to expose their use of 1st and 2nd order reflectivity information. Differences between standard and nonlinear updates suggest that the latter may hold promise for the effective incorporation of high angle and high contrast reflectivity information in FWI.
Introduction
Seismic full waveform inversion updates (Lailly, 1983; Tarantola, 1984; Virieux and Operto, 2009) can be constructed so as to respond to small-angle backscattered data, e.g., pre-critical specular reflections, in a manner consistent with linearized inverse scattering and AVO inversion (Innanen, 2014). This means a multi-parameter reflection FWI updating scheme can be protected against parameter cross-talk to the same degree as AVO and linear inverse scattering. However, it also means that linearization error will be present to the same degree as it is in those other methods, and concern registered in those domains (e.g., Weglein et al., 1986) is equally applicable to FWI.
Backscattered wave amplitudes are generally nonlinear in medium property variations. In the special case of two elastic half-spaces, for instance, the Zoeppritz equations define a highly nonlinear relationship between reflection coefficients and relative changes in elastic properties across a reflecting boundary. The relationship is often linearized; in the two half-space example, the Aki-Richards approximation, which is linear in the relative changes (Aki and Richards, 2002; Castagna and Backus, 1993; Foster et al., 2010), is commonly used in AVO inversion. Linearization error takes the form of a decrease in accuracy with an increase of incidence angle and/or magnitude of relative changes. This is one reason in AVO/AVAZ inversion why density is difficult to separate from VP (e.g., Stolt, 1989), and why certain anisotropic parameters are difficult to determine (e.g., Mahmoudian et al., 2013). These parameters are best distinguished at high angle, where the Aki- Richards formula and its extensions are insufficiently accurate.
In FWI, nonlinearity is primarily accommodated through iteration. Nevertheless, mitigating linearization error within individual FWI updates could play an important role, in principle making available to FWI both (1) the improved parameter discrimination known to be possible at large scattering angles, and (2) an uplift in convergence rate.
Linearization error has been mitigated in inverse scattering environments (Weglein et al., 2003; Zhang and Weglein, 2009a,b) and AVO environments (e.g., Stovas and Ursin, 2001; Innanen, 2011, 2013). In this paper we consider means by which nonlinearity can also be at least partially accommodated during FWI iterations—specifically, if it is possible to make changes to the basic ingredients of FWI such that an update naturally accommodates nonlinearity in the reflectivity/ step length relationship. General nonlinear sensitivity formalisms have been discussed in the literature, in a seismic context by Wu and Zheng (2014) and elsewhere (e.g., in optical tomography by Kwon and Yazici, 2010). Here we will discuss the construction of very particular, analyzable FWI update formulas that have direct expression in terms of nonlinear operations on the residuals.
