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.