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Recently, multi-trace impedance inversion has attracted great interest in seismic exploration because it improves the horizontal continuity and fidelity of the inversion results by exploiting the lateral structure information of the strata. However, computational inefficiency affects its practical application. Furthermore, in terms of vertical constraints on the model parameters, it only considers smooth features while ignoring sharp discontinuity features. This leads to yielding an over-smooth solution that does not accurately reflect the distribution of underground rock. To deal with the above situations, we first develop a low-dimensional multi-trace impedance inversion (LMII) framework. Inspired by compressed sensing, this framework utilizes low-dimensional measurements in sparse space containing the maximum information of the signal to construct the objective function for multi-trace inversion, which can significantly reduce the size of the inversion problem and improve the inverse efficiency. Then, we introduce the elastic half (EH) norm as a vertical constraint on the model parameters in the LMII framework and formulate a novel constrained LMII model for impedance inversion. Because the introduced EH norm takes into account both the smoothness and blockiness of rock impedance, the constrained LMII model can effectively raise the inversion accuracy of complex strata. Finally, an efficient alternating multiplier iteration algorithm is derived based on the variable splitting technique to optimize the constrained LMII model. The performance of the developed approaches is tested using synthetic and practical data, and the results prove their feasibility and superiority.

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The problem of recovering the complete seismic data from under-sampled field-observed data is a long-term challenge. Many recent efforts to address this problem develop model-based recovery methods. However, current model-based methods cannot accurately capture inherent priors of seismic data to obtain optimal recovery results. For this issue, we propose a novel model-based seismic data recovery method, which integrates a deep-learning-based targeted denoiser into the seismic data recovery model and leverages the deep targeted denoising priors learned by this denoiser to implement high accuracy data recovery. Specifically, we first use operator splitting technology to decouple the data-consistency term and prior constraint term in a seismic data recovery model, resulting in an alternating optimization scheme composed of a least-squares inversion subtask and a proximal minimization subtask. The deep-learning-based targeted denoiser, dual-channel deep denoising network (DcDDNet), is then plugged into the alternating optimization scheme as a modular component, according to the mathematical equivalence between the proximal operator and the targeted denoiser. This modular component replaces the formalized proximal minimization subtask and plays the role of an implicit prior in the recovery optimization problem. Finally, the least-squares inversion and deep denoising subtasks in the alternating optimization scheme are iteratively executed to achieve seismic data recovery. Due to the integration of the powerful DcDDNet, the proposed method not only enjoys high flexibility and impressive generalization capability but also has high computational efficiency and excellent performance. Synthetic and field data with regular and irregular under-sampling are used to evaluate the recovery capability of this method. The results demonstrate that, compared with the f-x adaptive interpolation and Curvelet-based recovery methods, our method has advantages in visual and quality metrics. In addition, this method is also more practical in seismic applications because it works in the time-space domain and can handle both spatial aliasing and spectral leakage.

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