The Born series utilizes a model perturbation framework to explain the difference between the seismic modeled data corresponding to a background model and those measured in the field corresponding to the real Earth. These perturbations include short wavelength features like those predicted by full waveform inversion (FWI) gradients, and long wavelength features often constrained by migration velocity analysis (MVA) objectives. The Born series, however, is not a convergent series. If the perturbations are large, we probably will not be able to explain the data difference. Thus, using the leading terms of the Born in an iterative process, in which they are scaled properly, allows us to avoid such limitations and update the short and long wavelength components of the velocity model. In fact, the FWI gradient is manifested in the first term of the Born series, and the MVA update resides in the transmission (first Fresnel zone) part of the second term. In optimizing such terms where we find the perturbations that fit the residuals in the data, we are effectively applying a Hessian preconditioner and more to linear, quadratic, and higher-order terms of the series that will help with convergence, and normalizes the limited series contributions. In this case, FWI and MVA are code names for dividing the optimized update to reflectivity-based portions and those adequate for the background, respectively. Examples on synthetic and real data demonstrate this logic.
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