###### Abstract  
The seismic image produced by pre-stack depth migration is more accurate and has clearer geological significance than the time image. However, the waveform of the depth-domain seismic image is affected not only by depth-dependent velocity variation but also by media attenuation, resulting in strong spectral variation of depth-domain seismic data. Therefore, depth-domain seismic inversion is still challenging. We propose a wavelet extraction method of attenuation media based on the generalized seismic wavelet, to address this issue. Then, the estimated depth-domain wavelets were applied to the direct acoustic impedance inversion. First, we investigated the effect of attenuation media on depth-domain source wavelets and derived an analytical formula for the depth-domain wavelets of attenuation media. Next, the time-domain generalized seismic wavelet was extended to the depth domain, which was utilized to study the feasibility of using the generalized seismic wavelet to characterize the seismic wavelet of the depth-domain attenuation media. Based on the orthogonal matching pursuit, we propose a method to extract the depth-domain generalized seismic wavelet directly from depth-domain seismic data. Finally, we applied this method to the depth-domain direct acoustic impedance inversion of a 3D field data example. Tests on the synthetic and 3D field datasets show that the proposed method can correctly extract the depth-domain seismic wavelet of attenuation media and attain direct inversion of the depth-domain acoustic impedance with high accuracy. Therefore, our method is effective and has robust potential in reservoir characterization, fluid prediction, and attribute extraction in the depth domain. d 2024 14 2478 103390 1039  
PSDM results. For consistency, we refer to non-stationarity as a feature of strong spectral variation and depth-domain seismic data as seismic images obtained by PSDM in this paper. Therefore, how to deal with this non-stationarity to achieve high-precision reservoir prediction represents an urgent geophysical problem. Geophysicists have made many efforts to achieve depth-domain seismic inversion, which can be divided into three categories according to how they deal with non-stationarity. The first category of methods does not consider non-stationarity and treats depth-domain seismic data like time-domain seismic data. Typical approaches for this category include deep learning [7; 8; 9] and seismic waveform indicator inversion [10; 11]. Deep learning trains a nonlinear mapping between well logs and well bypass seismic traces. Then the trained model is used to invert seismic traces at non-well positions. However, deep learning is data-driven and does not need any deterministic operator, so it has no explicit geophysical significance. Seismic waveform indicator inversion builds the initial geologic model based on seismic waveform similarity and the spatial distance between wells, and the direct inversion is carried out on depth-domain seismic data constrained by the Bayesian framework. In addition, [PERSON] (2021) proposed a workflow for depth-domain colored inversion implemented with commercial software [12]. Because the accuracy for the first category of methods depends on the trained model, it has poor generalization performance on other depth-domain datasets. The second category of methods, domain conversion, avoids the non-stationarity of depth-domain seismic data. It converts the depth-domain seismic data to other domains. And the matured methods in other domains have been used to carry out inversion before returning the inversion results to the depth domain. Examples of this include depth-time-depth conversion [13] and pseudo-depth conversion [14]. The depth-time-depth conversion converts seismic data between the depth and time domains using the time-depth relation [13], and the pseudo-depth conversion converts seismic data between the depth and pseudo-depth domains using both constant velocity substitution and pseudo-depth-depth relations [14].