## 1 Introduction
### Theory and Algorithm
\begin{array}{c}\sin^{2}\!\theta\frac{\delta v}{v_{0}}+\cos^{2}\!\theta\frac {\delta(\rho v)}{\rho v_{0}v_{0}}-\iint\iint\frac{32\cos^{2}\!\theta^{\prime} \;\cos\!\beta_{r}}{|\omega|}\frac{\cos\!\beta_{r}}{v(\mathbf{x})}A(\mathbf{x} ;\mathbf{x}_{s})A(\mathbf{x}_{r};\mathbf{x})\\ \times(\theta^{\prime}-\theta)\delta d(\mathbf{x}_{r};\omega;\mathbf{x}_{s})e^ {-i\omega t\mathcal{I}(\mathbf{x}_{s}\omega;\mathbf{x}_{s})}d\mathbf{x}_{d}d \mathbf{x}_{d}\omega d\theta^{\prime}\end{array}  \tag{5}\]  
where \(\beta_{s}\) and \(\beta_{r}\) represent the takeoff angles at the source and receivers, respectively. Within the framework of amplitude-preserving RTM, the asymptotic forms of \(p_{F}(\mathbf{x};\omega;\mathbf{x}_{s})\) and \(p_{B}(\mathbf{x};\mathbf{x}_{s};\omega)\) for 2D acoustic case are given as follows [10; 12]:  
\[p_{F}^{*}(\mathbf{x};\omega;\mathbf{x}_{s})=-2\frac{\cos\beta_{s}}{v(\mathbf{ x}_{s})}\sqrt{|\omega|}A(\mathbf{x};\mathbf{x}_{s})e^{-i\omega\tau(\mathbf{x} \mathbf{x}_{s})-i(\frac{\pi}{4})sgn(\omega)} \tag{6}\]  
and  
\[p_{B}(\mathbf{x};\omega;\mathbf{x}_{s})=2\!\int\frac{\cos\beta_{r}}{v(\mathbf{ x}_{r})}\sqrt{|\omega|}A(\mathbf{x}_{r};\mathbf{x})e^{-i\omega\tau(\mathbf{x} _{r};\mathbf{x})-i(\frac{\pi}{4})sgn(\omega)}\delta d(\mathbf{x}_{r};\omega; \mathbf{x}_{s})d\mathbf{x}_{r} \tag{7}\]  
Substituting Equations (6) and (7) for the terms on the right-hand side of Equation (5), we can obtain:  
\[\sin^{2}\!\theta\frac{\delta v}{v_{0}}+\cos^{2}\!\theta\frac{\delta(\rho v)}{ \rho_{0}v_{0}}=\iiint\frac{8\cos^{2}\!\theta^{\prime}}{\omega^{2}}\delta( \theta^{\prime}-\theta)p_{F}^{*}(\mathbf{x};\omega;\mathbf{x}_{s})p_{B}( \mathbf{x};\omega;\mathbf{x}_{s})d\omega d\theta^{\prime}d\mathbf{x}_{s} \tag{8}\]  
The left-hand side of Equation (8) is the composite form of relative velocity perturbation and impedance perturbation. The right-hand side has the same form as RTM. This shows that the composite parameter on the left-hand side can be estimated using the RTM framework. In order to invert each individual parameter, [PERSON] et al. [10] pointed out that the near-angle stacked image can output impedance perturbation, \(\frac{\delta(\rho v)}{\rho v_{0}v_{0}}\), while the far-angle stacked image can be used to estimate the velocity perturbation, \(\frac{\delta v}{v_{0}}\). They separated the effects of impedance and velocity by first generating RTM angle-domain common-image gathers, and then using stacked images within different angle sections for velocity and impedance estimations. However, it could be computationally expensive to compute RTM angle gathers, especially in 3D. Note that the right-hand side of Equation (8) shares a similar form as the true amplitude imaging principle proposed by [PERSON] et al. [13], where the \(\cos^{2}\!\theta\) term is approximated by the ray theoretical slowness vectors in the Fourier domain and computed by applying the time derivatives and spatial gradients of wavefields. The time derivatives and spatial gradients of wavefields were also utilized in the inverse scattering imaging condition proposed by [PERSON] and [PERSON] [11]. Next, we investigated the inverse scattering imaging condition, and further proposed a modified inverse scattering imaging condition to output the near-angle stacked images without computing the angles. Our method can reduce the computational costs compared with the method using angle gathers, and still utilize the imaging capabilities of RTM for the relative impedance estimation.