Based on a data-ﬁtting procedure, full waveform inversion (FWI) aims to build high resolution subsurface structures using full waveform information. Although FWI is a highly nonlinear inverse problem, it is usually solved as a local optimization problem under a linear approximation. The gradient calculation of FWI during each iteration is usually studied with the aid of sensitivity kernel, or Fréchet derivative. Recently applications of FWI show that FWI can successfully build high resolution models in shallow regions, where long-to-intermediate wavelength structures can be reconstructed from diving waves and post-critical reﬂections. When ﬁrst order scattering is considered during the construction of sensitivity kernel, recently developed reflection waveform inversion (RWI) provides the possibility to retrieve long-to-intermediate wavelengths in deeper regions from pre-critical reﬂections. In this study, we ﬁrst present the construction of nonlinear sensitivities under the scattering theory. Extending the sensitivity kernel to higher order can help reduce the nonlinearity and improve the convergence of FWI. To construct higher order sensitivities, the model perturbation from the forthcoming iteration is needed. We then present a twoiteration approach to perform nonlinear FWI in the frequency domain. Finally, we apply this nonlinear FWI on the Marmousi model. The inverted models with different frequency ranges and different initial models show that this nonlinear FWI can build a reliable high resolution model in both shallow and deeper regions.
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