Least-squares reverse time migration in frequency domain using the adjoint-state method

A new scheme is presented to implement a least-squares frequency domain reverse time migration (LS-FRTM). This scheme expresses the gradient of the misfit function with respect to the model as the product of the conjugated Green's functions and the data residuals in the frequency domain based on the adjoint state method. In the 2D case, for each frequency all the Green's functions from the shots to the reflectors and from the reflectors to the receivers which depend on the background velocity can be calculated once using the lower/upper decomposition. The pseudo-Hessian matrix which is also expressed as a function of the Green's function is used as a substitute for the approximate Hessian to amplitude compensation for the gradient. Since the linearized inversion does not update the background velocity, the Green's function needs to be calculated only once. An iteration based LS-FRTM can be implemented with high efficiency. As examples supporting our assertion, we present the results obtained by applying our method to the 2D Marmousi model.