We examine the key concepts in full waveform inversion (FWI) and relate them to processes familiar to practicing geophysicists. For clarity, we present the central theoretical result behind FWI as a mathematical statement that a linear update to a migration velocity model is proportional to a prestack reverse-time migration of the data residual (the difference between the actual data and data predicted by the model) where the proportionality factor must be estimated. We argue that in many cases this factor will be complex-valued and frequency dependent, or in the time domain, it will be a convolutional wavelet. We find an analogy between estimation of the velocity update from the migrated section and the common process of impedance inversion, and we suggest that FWI can be viewed as a practical cycle of data modeling, migration of the data residual, and “calibration” of this migration to deduce the velocity update. The calibration step can be accomplished like a conventional impedance inversion where the migrated data residual is tied to the velocity residual (the difference between actual velocity and migration velocity) at a well. As there are a great many established algorithms for impedance inversion, so there are a plethora of possibilities for calibration. We present an extended example using the Marmousi model in which we use wave-equation migration (e.g. depth stepping) of the data residual and a simple least-squares amplitude scaling and constant phase rotation, determined at a simulated well, to calibrate the migration. We find that our approach produces a much improved velocity model in only a few iterations.

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