## 4 Inversion on a 3D Field Dataset  
After extracting the depth-domain generalized seismic wavelet, it can be applied to the depth-domain AI inversion. Basis pursuit (BP) is an optimization method with an \(L_{1}\) norm. Considering the sparsity of subsurface stratigraphy, the BP has been gradually applied to seismic inversion in recent years, which improves the detectability and resolution of thin layers [23; 24]. Here, we use the BP inversion approach to obtain the reflectivity, and the inverted AI should be recovered by the following:  
\[I_{H}=I_{0}\exp\left(2\sum_{h=h_{0}}^{H}r_{h}\right), \tag{28}\]  
where \(I_{H}\) is the AI at depth \(H\) and \(I_{0}\) is the AI at the initial depth \(h_{0}\). The initial AI model can be constructed by interpolating well-log measured AI. We also use the inversion result of the pseudo-depth-domain SPE method for comparison, which converts depth-domain seismic data to the pseudo-depth domain and performs inversion using the same BP inversion technique before converting the result back to the depth domain. We use the same depth-domain seismic data, the same initial AI model, and the same regularization parameters for the two methods. To quantify the accuracy of different inversion results, we use the mean relative error (MRE), written as follows:  
\[\text{MRE}=\frac{100\%}{MN}\sum_{i=1}^{M}\sum_{j=1}^{N}\left|\frac{\hat{I}_{ i,j}-I_{i,j}}{I_{i,j}}\right|, \tag{29}\]  
where \(\hat{I}\) is the inverted AI, and \(I\) is the actual AI. In this study, we perform inversion on a 3D depth-domain field seismic dataset from East China, as depicted in Figure 13a. The seismic data comprises 246 survey lines with a line interval of 0.025 km. Each line has 261 traces or 261 CDP, and the trace spacing is also 0.025 km. The depth ranges from 2.0 km to 4.5 km with a depth sample interval of 0.005 km. The red arrows shown in Figure 13a,b indicate the direction. In this paper, the black dotted lines indicate the actual positions of the slices in 3D figures. Take Figure 13a as an example; the slice shown at the top is a horizontal slice with a depth of 3.3 km in the seismic dataset; the situation is the same for the below. Figure 13b shows the four seismic horizons of this seismic dataset. It can be seen that the terrain of this region is high in the southwest and low in the northeast, and layers 3 and 4 are gradually separated along the southwest to  
Figure 12: Test of the depth-domain strong attenuation model. (**a**) \(Q\); (**b**) depth-domain attenuated wavelets; (**c**) depth-domain attenuated seismogram; (**d**) extracted depth-domain generalized seismic wavelets; (**e**) comparison between the real seismogram (black) and those obtained by the pseudo-depth-domain SPE method (blue) and the proposed method (red); (**f**) pseudo-depth-domain constant wavelet. northeast direction. There are 19 wells in this region, as shown in Figure 13c, named wells A to Q. And the distribution of wells is relatively uniform. Figure 13d presents the initial AI model established by interpolating the well-log data along the seismic horizons. First, our method extracts the generalized seismic wavelet parameters from the 3D depth domain seismic data, as shown in Figure 6. Figure 14a,b show the extracted fractional derivative and reference wavenumber, respectively. The fractional derivative ranges from 1.9 to 2.04, indicating that the shape of the extracted depth-domain generalized seismic wavelet, is close to the Ricker wavelet. The reference wavenumber tends to be lower with increased depth. Next, the pseudo-depth-domain SPE method is used for comparison. The pseudo-depth-domain SPE method can estimate a constant wavelet at each well for this 3D depth-domain seismic dataset. However, only one wavelet should be used during the inversion. Therefore, after extracting the pseudo-depth domain constant wavelet from each well, it is applied to all wells to obtain the pseudo-depth domain synthetic records. The PCCs between the synthetic records and the well bypass seismograms are computed in the pseudo-depth domain.