Full waveform inversion with wave equation migration and well control

Gary F. Margrave, Robert James Ferguson, Chad M. Hogan

We examine the key concepts in full waveform inversion (FWI) and relate them to processes familiar to the practicing geophysicists. The central theoretical result behind FWI is presented as a mathematical theorem, the Fundamental Theorem of FWI. This theorem says that a linear update to a migration velocity model can be obtained from a reverse-time migration of the data residual (the difference between the actual data and data predicted by the model). Critically, this migration is only proportional to the required update and the proportionality must be estimated. We argue that in many cases this proportionality factor will be complex-valued and frequency dependent, or in the time domain, a wavelet. The estimation of the velocity update from the migrated section is closely related to the common process of impedance inversion. Then we argue that FWI can be viewed as a cycle of data modelling, migration of the data residual, and calibration of this migration to deduce the velocity update. We present an extended example using the Marmousi model in which we use wave-equation migration of the data residual and we calibrate the migration by matching it to the velocity residual (the difference between actual velocity and migration velocity) at a well. Our example produces an encouragingly detailed inversion but raises many questions.