Notes on the meaning of adjoint wave equation derived via Lagrange multiplier approach

Full-wave adjoint inversion method can be formulated via the Lagrange multiplier approach, taking the forward seismic wave equation as an additional independent constraint on synthetic wavefield. The adjoint wave equation is explained as the Lagrange multiplier where the local minimum existed, in case the model parameters are unperturbed. Such an explanation is unsatisfactory, due to the fact that the variation of synthetic wavefield equals to zero in case the model parameters are unperturbed. To avoid such a contradiction, we give a more clear explanation about the meaning of the adjoint wave equation, that it serves for converting the gradient of misfit function with respect to synthetic wavefield to the gradient of misfit function with respect to the model parameter.