Conversional formulation of the gradient based on the cross-correlation of the derivatives of forward and backward particle displacement wavefield is derived from the second-order wave equations. During the process of back-propagation of the data residuals, the adjoint wave equations are just the same as the forward ones. Without preprocessing the data residuals before back-propagation, one cannot obtain a properly scaled gradient when applying the first-order velocity-stress differential equations. In this paper, based on the first-order elastic system in time domain, we propose a new form of adjoint wave equations, meanwhile corresponding formulation of gradient is described as well. Without integrating particle velocity in time to convert it to displacement and preprocessing the data residuals before back-propagation, the new scheme is tested to be more efficient. In addition by using dimensional analysis, it is obvious that the new formula of gradient is perfectly correct.
Arbitrarily wide-angle wave equations (AWWEs) are capable of imaging steep dips in heterogeneous media and convenient in numerical calculations, which enable them to be powerful tools for migration. The seismic wave modeling and migration are always carried out in a limited space, so an effective absorbing boundary condition (ABC) is required to avoid spurious edge reflections. In spite of an extensive utilization of perfectly matched layer (PML) in full wave equation, applications of PML for one-way wave equation (OWWE) are rare. In this abstract we derive a PML formulation for 3D scalar AWWEs to provide a good approach to suppress the undesired edge reflections. We finally formulate the PML in terms of a split field in the time domain and give out the discrete scheme using finite-difference method. Several numerical examples are given to show the effectiveness of the derived PML condition.