The wavefield is often reconstructed through solving a wave equation corresponding to an active source and using our best knowledge of the medium to reproduce the data that we seek to fit the observed data as part of a process we call full-waveform inversion (FWI). Alternatively, the wavefield can be inverted and used to invert for the velocity model, in an extended optimization problem, in which we can relax the requirement that the wavefield satisfies the wave equation. In this case, the wavefield calculation, as in FWI, requires a matrix inversion (which depends on the velocity) at practically every iteration. Thus, we formulate a bilinear optimization problem with respect to the wavefield and a modified source function, as independent variables. We specifically recast the wave equation so that velocity perturbations are included in this modified source function (which includes secondary or contrast sources); thus, it represents the velocity perturbations implicitly. The optimization includes a measure of the wavefield’s fit to the data at the sensor locations and the wavefield, as well as the modified source function, compliance with a wave equation corresponding to the background model. This problem is, however, complex with an extended model space that is ill-posed, so we use an alternating-direction method to reduce the inversion to two subproblems for inverting each of the wavefields and extended source. On the other hand, the velocity perturbations can be extracted in a separate step via direct division (deconvolution). Because we avoid using gradient methods in extracting the velocity perturbations, we are less prone to crosstalk artifacts when we use simultaneous sources. We evaluate these features on a simple two-anomalies model and the modified Marmousi model.

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