The conventional correlation-based objective function for FWI has a limitation: the desired (accurate) solution is not exactly located at the global minimum (or maximum) since the auto-correlation of the observed wavefield has some energy at non-zero time-lags. We propose two optimized correlation-based objective functions incorporating the auto-correlation term. Including this term in the new objective functions as a reference for the cross-correlation to fit locates the desired solution at the global minimum (or maximum). We derive the gradient expressions of the new objective functions using the adjoint-state method, where only two modelling steps are needed to calculate the gradients. The numerical examples show that FWI based on the new objective functions generate good convergent results, whereas the conventional correlation-based FWI does not.


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