## 1 Introduction
### Theory and Algorithm
The inverse scattering imaging condition proposed by [PERSON] and [PERSON] [11]:  
\[\mathrm{I}(\mathbf{x};\mathbf{x}_{s})=I_{\
abla}(\mathbf{x};\mathbf{x}_{s})+B (\mathbf{x};\mathbf{x}_{s})I_{dt}(\mathbf{x};\mathbf{x}_{s})\] (9a) where \[I_{\
abla}(\mathbf{x};\mathbf{x}_{s})=\int\
abla p_{F}(\mathbf{x};t;\mathbf{x} _{s})\cdot\
abla p_{B}(\mathbf{x};t;\mathbf{x}_{s})dt \tag{9b}\]and  
\[I_{dt}(\mathbf{x};\mathbf{x}_{s})=-\int\frac{1}{v^{2}(\mathbf{x})}\frac{\partial p _{F}(\mathbf{x};t;\mathbf{x}_{s})}{\partial t}\frac{\partial p_{B}(\mathbf{x};t; \mathbf{x}_{s})}{\partial t}dt \tag{9c}\]  
and where \(B(\mathbf{x};\mathbf{x}_{s})\) is a weighting function to attenuate backscattered energy. Note that here we used the backpropagated wavefields \(p_{B}(\mathbf{x};t;\mathbf{x}_{s})\) in terms of wavefield propagation time \(t\) (instead of \(T-t\), where \(T\) is the maximum extrapolation time), and used the relationship \(\frac{\partial p_{B}(\mathbf{x};T-t;\mathbf{x}_{s})}{\partial t}=-\frac{ \partial p_{B}(\mathbf{x};t;\mathbf{x}_{s})}{\partial t}\). The relationship between the product of the time derivatives of \(p_{F}\) and \(p_{B}\), and the products of their spatial gradients is given by the following equation [11; 14; 15]:  
\[-\frac{\partial p_{F}}{\partial t}\frac{\partial p_{B}}{\partial t}\frac{ \cos(2\theta)A(\mathbf{x})}{v^{2}(\mathbf{x})}=\
abla p_{F}\cdot\
abla p_{B} (\mathbf{x};t;\mathbf{x}_{s}) \tag{10}\]  
where \(A(\mathbf{x})\) is the parameter to compensate for far field approximation. We can ignore it by assuming far field approximation and obtain the following equations:  
\[\
abla p_{F}(\mathbf{x};t;\mathbf{x}_{s})\cdot\
abla p_{B}(\mathbf{x};t; \mathbf{x}_{s})\approx-\frac{\cos(2\theta)}{v^{2}(\mathbf{x})}\frac{\partial p _{F}(\mathbf{x};t;\mathbf{x}_{s})}{\partial t}\frac{\partial p_{B}(\mathbf{x}; t;\mathbf{x}_{s})}{\partial t} \tag{11a}\] \[-\frac{1}{v^{2}(\mathbf{x})}\frac{\partial p_{F}(\mathbf{x};t; \mathbf{x}_{s})}{\partial t}\frac{\partial p_{B}(\mathbf{x};t;\mathbf{x}_{s})}{ \partial t}-\
abla p_{F}(\mathbf{x};t;\mathbf{x}_{s})\cdot\
abla p_{B}( \mathbf{x};t;\mathbf{x}_{s})\approx-\frac{2\cos^{2}\theta}{v^{2}(\mathbf{x}) }\frac{\partial p_{F}(\mathbf{x};t;\mathbf{x}_{s})}{\partial t}\frac{\partial p _{B}(\mathbf{x};t;\mathbf{x}_{s})}{\partial t}\] (11b) \[-\frac{1}{v^{2}(\mathbf{x})}\frac{\partial p_{F}(\mathbf{x};t; \mathbf{x}_{s})}{\partial t}\frac{\partial p_{B}(\mathbf{x};t;\mathbf{x}_{s})}{ \partial t}-\
abla p_{F}(\mathbf{x};t;\mathbf{x}_{s})\cdot\
abla p_{B}( \mathbf{x};t;\mathbf{x}_{s})\approx-\frac{2\sin^{2}\theta}{v^{2}(\mathbf{x}) }\frac{\partial p_{F}(\mathbf{x};t;\mathbf{x}_{s})}{\partial t}\frac{\partial p _{B}(\mathbf{x};t;\mathbf{x}_{s})}{\partial t} \tag{11c}\]  
Choosing \(\mathrm{B}(\mathbf{x})=1\), Equation (9a) can be expressed in the frequency domain as:  
\[ \begin{array}{c}\mathrm{I}_{\mathbf{Y}}(\mathbf{x};\mathbf{x}_{s})+I_{dt}( \mathbf{x};\mathbf{x}_{s})\\ =\int\left(-\frac{1}{v^{2}(\mathbf{x})}\frac{\partial p_{F}(\mathbf{x};t; \mathbf{x}_{s})}{\partial t}\frac{\partial p_{B}(\mathbf{x};t;\mathbf{x}_{s}) }{\partial t}+\
abla p_{F}(\mathbf{x};t;\mathbf{x}_{s})\cdot\
abla p_{B}( \mathbf{x};t;\mathbf{x}_{s})\right)dt\\ \approx-\int\frac{2\cos^{2}\theta}{v^{2}(\mathbf{x})}\frac{\partial p_{F}( \mathbf{x};t\mathbf{x}_{s})}{\partial t}\frac{\partial p_{B}(\mathbf{x};t \mathbf{x}_{s})}{\partial t}dt\\ =-\int\frac{2\omega^{2}\cos^{2}\theta^{2}}{v^{2}(\mathbf{x})}p_{F}^{*}( \mathbf{x};\omega;\mathbf{x}_{s})p_{B}(\mathbf{x};\omega;\mathbf{x}_{s}) \mathrm{d}\omega\end{array}  \tag{12}\]  
Equation (12) is similar to the energy norm imaging condition proposed by [PERSON] et al. [16], which was used to attenuate reflections with opening angles close to 180\({}^{\circ}\).