An introduction to full waveform inversion

Full waveform inversion (FWI) is a highresolution seismic imaging technique that is based on using the entire content of seismic traces for extracting physical parameters of the medium sampled by seismic waves. The widespread strategy of seismic imaging, the single scattering formulation, at the core of FWI, assumes no prior scale in the model description. Each unexplained residual data sample at receivers for one source is assumed to come from any point of the medium, and only the summation over sources and receivers helps in locating medium property anomalies, regardless of what type of phase is involved. This pixel-oriented perturbation leads to the local optimization approach, which is a linearized differential approach based on the Newton equation. For a least-squares misfit function, there are both the gradient vector and Hessian matrix, in addition to approximations that can be considered for the related Newton equation. The forward problem of the wave propagation, used thousands of times during optimization, should be efficient, and these equations are expressed either as a firstorder hyperbolic system of velocity-stress or as a second-order hyperbolic system of displacement (or velocity) only, by using a self-adjoint formulation in both cases. Gradient vectors are built as a zero-lag crosscorrelation in time between incident and adjoint wavefields with forward and backward patterns and also could be used for obtaining Hessian-matrix approximations. Resolution and uncertainties are relevant, although the actual state of the art does not provide meaningful estimation of these quantities: the FWI remains a deterministic approach at this time. An examplary North Sea data set from the Valhall reservoir illustrates the successful story for high-resolution imaging based on datadriven components, with paleorivers stored in sediments, imprints of glaciers in the bedrock, and gas clouds at different scales in the image. Alternative sources of information on the medium, such as sonic logs and geologic interpretation, are illustrated through a model-driven component of the misfit function. Although methods can be used to increase the speed of the workflow, they are quite costly. The multiparameter reconstruction, which is mandatory for elastic FWI, starts to be feasible if one improves the Hessian-matrix influence. Thus, FWI is becoming a mature strategy for high-resolution seismic imaging.