Full waveform inversion is a least squares technique that estimates rocks parameters by finding the model that reduces the difference between synthetic and acquired seismic data. This project is focused on acoustic inversion. The main goal is to understand and develop an FWI algorithm that is cheap and provides a higher quality inversion. The steepest-descent method is shown to be robust with some level of guarantee to converge, when the geology is simple, and when the starting model is close to the global minimum. Understanding that the gradient can be estimated by any pre-stack depth migration of the residuals, the RTM (reverse-time migration) is replaced by a PSPI (phase shift plus interpolation) migration, to reduce cost. Even though the method was shown to be expensive, mostly by the number of synthetic shots required and the pre-stack migration. By using a monochromatic averaged gradient combined with a conjugate gradient algorithm, inversion is possible for more complex geology, like a simulation on a 2D acoustic Marmousi model. However, the cost to run the process increased significantly, as each frequency on a selected band is migrated separately to form a pseudo-gradient, and are weighted averaged for the update. Inversion is simplified by applying an impedance inversion on the reflection coefficients based gradient using a band-limited impedance inversion (BLIMP) algorithm. Cost is reduced, being com- parable to the standard steepest-descent, and the inverted model resolution is kept similar to the one using the monochromatic averaged gradient. Later, we came with a new inter- pretation of the gradient, understood to be the difference between current model and the impedance inversion of the migrated acquired data. No forward modeling or source estima- tion are required to compute the gradient. The cost per iteration is the same of the PSPI migration. A post-stack method is shown to be even cheaper and promising. Using a well sonic log to calibrate the gradient reduces method cost and improves its resolution, and is our most impressive solution.
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