# Accelerating High-Resolution Seismic Imaging by Using Deep Learning
## 2 Methods  
### High-Resolution Imaging Using QPSTM  
The QPSTM can be expressed as [1; 2],  
\[\begin{array}{c}I(x,y,T)=\left(\frac{\tau_{g}}{\tau_{g}}\right)^{2}\int_{ \omega}F(\omega)e^{-j\frac{\pi}{2}}\epsilon^{j\omega\left(\tau_{g}+\tau_{g} \right)\left(1-\frac{ln\left(\omega/\omega_{0}\right)}{\pi Q_{eff}}\right)}\\ e^{\left(\frac{\omega}{2Q_{eff}}\left(\tau_{g}+\tau_{g}\right)\right)}\,d \omega,\end{array} \tag{1}\]  
where \(I(x,y,T)\) is the image at the horizontal location \((x,y)\) and vertical time-depth \(T\). \(\tau_{s}\) and \(\tau_{g}\) denote the travel time from the source and the receiver to the imaging point, respectively, as shown in Figure 1. \(F(\omega)\) denotes the seismic traces in the frequency domain. \(\omega_{0}\) represents the dominant frequency of the seismic data. \(Q_{eff}\) is a factor that represents the intensity of absorption and dispersion of the seismic waves during propagation. This approach is a frequency-domain imaging method as there exists an integral over \(\omega\). The frequency components for each imaging point are accumulated, which makes this approach computationally intensive.  
To verify the effectiveness of the QPSTM, we tested it on a synthetic marine model. Figure 2 shows the P-wave velocity structures. The upper-shallow layer is a water layer with a velocity of 1500 m/s. The velocity increases with an increase in the depth. In addition, the model includes several faults, and overthrust structures. A \(Q\) model with the same size as that of the velocity model was used. We generated the synthetic shot-gathers, taking into account the attenuation and dispersion, by using the modeling method proposed by [PERSON] [33]. Next, the modeled data was migrated by using the conventional PSTM and QPSTM method. The corresponding imaging results are shown in Figures 3 and 4. Sedimentary layers, faults and overthrusts can be clearly observed, which can help analyze the sedimentary relationship better. It is thus concluded that the QPSTM can significantly improve the resolution of the migration image.  
Figure 1: Imaging principle of the deabsorption prestack time migration (QPSTM). The shot point is the position where the seismic signal is generated and the geophone point is the position where the signal is received, and the image point is the underground position where the structure is studied. By calculating the travel time between the source-and-receiver pair, \(\tau_{s}\) and \(\tau_{g}\) can be obtained. Next, the compensation effect resulting from the Q factor along the wave-path in the frequency domain with an effective Q is determined to achieve the energy-compensated imaging result.  
Figure 3: Conventional pre-stack time migration (PSTM) image. The migration image has a relatively lower resolution owing to the energy attenuation and phase distortion.  
Figure 2: Synthetic marine velocity model. The velocity model consists of a water layer and several sedimentary layers, with values varying from 1500 to 4000 m/s. The viscous acoustic seismic record was modeled to test the effectiveness of the QPSTM with a same size Q model (not shown here).  
However, the computation cost of the QPSTM is larger than that of the conventional PSTM. For example, it takes 115 and 2218 s to calculate the simple 2D profile in Figures 3 and 4, respectively. Even when using GPUs, several months may be required to obtain the high-resolution results for a real 3D field dataset [3]. This prevents the application of this approach to large-scale problems. Furthermore, the low efficiency makes the parameter update more difficult and energy intensive.