The lack of low-frequency signals in seismic data makes the full-waveform inversion (FWI) procedure easily fall into local minima leading to unreliable results. To reconstruct the missing low-frequency signals more accurately and effectively, we have developed a data-driven low-frequency recovery method based on deep learning from high-frequency signals. In our method, we develop the idea of using a basic data patch of seismic data to build a local data-driven mapping in low-frequency recovery. Energy balancing and data patches are used to prepare high- and low-frequency data for training a convolutional neural network (CNN) to establish the relationship between the high- and low-frequency data pairs. The trained CNN then can be used to predict low-frequency data from high-frequency data. Our CNN was trained on the Marmousi model and tested on the overthrust model, as well as field data. The synthetic experimental results reveal that the predicted low-frequency data match the true low-frequency data very well in the time and frequency domains, and the field results show the successfully extended low-frequency spectra. Furthermore, two FWI tests using the predicted data demonstrate that our approach can reliably recover the low-frequency data.

Prestack inversion has become a common approach in reservoir prediction. At present, the critical issue in the application of seismic inversion is the estimation of elastic parameters in the thin layers and weak reflectors. To improve the resolution and the accuracy of the inversion results, we introduced the difference of L1 and L2 norms as a nearly unbiased approximation of the sparsity of a vector, denoted as the L1−2 norm, to the prestack inversion. The nonconvex penalty function of the L1−2 norm can be decomposed into two convex subproblems via the difference of convex algorithm, and each subproblem can be solved efficiently by the alternating direction method of multipliers. Compared with the L1 norm regularization, the L1−2 minimization can reconstruct reflectivities more accurately. In addition, the f-x predictive filtering was introduced to guarantee the lateral continuity of the location and the amplitude of the reflectivity series. The generalized linear inversion and f-x predictive filtering are combined for stable elastic impedance inversion results, and three parameters of P-wave velocity, S-wave velocity, and density can be inverted with the Bayesian linearized amplitude variation with offset inversion. The inversion results of synthetic and real seismic data demonstrate that the proposed method can effectively improve the resolution and accuracy of the inversion results.

Seismic acoustic impedance inversion plays an important role in reservoir prediction. However, single-trace inversion methods often suffer from spatial discontinuities and instability due to poor-quality seismic records with spatially variable signal-to-noise ratios or missing traces. The specified hyperparameters for seismic inversion cannot be suitable to all seismic traces and subsurface structures. In addition, conventional multichannel inversion imposes lateral continuity with a prespecified mathematical model. However, the inversion results constrained with specified lateral regularization are inferior when the subsurface situations violate the hypothesis. A data-driven multichannel acoustic impedance inversion method with patch-ordering regularization is introduced, in which the spatial correlation of seismic reflection is used. The method decomposes the seismic profile into patches and constructs the patch-ordering matrix based on the similarity among seismic patches to record the impedance structural extension. So the patch-ordering matrix can record the spatial extension of the acoustic impedance. Then, a simple regularization with difference operators of varying weights can reduce the random noise presented in the inverted impedance profile, stabilize the inversion result, and enhance the spatial continuity of the layer extension. The objective function for multichannel poststack seismic impedance inversion can be constructed by integrating the observed seismic record and the spatial continuity in the form of patch-ordering regularization, and it can be solved effectively with the limited-memory BFGS algorithm. The synthetic and field data tests illustrate the improvement of accuracy and lateral continuity of inverted results with our method, compared to conventional model-based inversion results.

Seismic acoustic impedance inversion plays an important role in subsurface quantitative interpretation. Due to the band-limited property of the seismic record and the discretization of the continuous elastic parameters with a limited sampling interval, the inverse problem suffers from serious ill-posedness. Various regularization methods are introduced into the seismic inversion to make the inversion results comply with the prespecified characteristics. However, conventional seismic inversion methods can only reflect fixed distribution characteristics and do not take into account discretization challenges. We have adopted a new poststack seismic impedance inversion method with upsampling and adaptive regularization. The adaptive regularization is constructed with two trained dictionaries from the true model and upsampled model-based inversion result to capture the features of high- and low-resolution details, and a sparsity-based statistical model is proposed to build the relationship between their sparse representations. The high-resolution components can be recovered based on the prediction model and low-resolution sparse representations, and the parameters of the statistical prediction model can be obtained effectively with conventional optimization algorithms. The synthetic and field data tests show that the model-based inversion is dependent on the sample interval, and our method can reveal more thin layers and enhance the extension of the strata compared with conventional inversion methods. Moreover, the inverted impedance variance of our method matches well with the borehole observations. The tests demonstrate that the interpolated model-based inversion result combined with the sparsity-based prediction model can effectively improve the resolution and accuracy of the inversion results.

Bayesian full-waveform inversion (FWI) addresses the problem of the nonuniqueness of solutions in traditional deterministic FWI by quantifying the model uncertainties, which can be realized by a variational inference (VI) approach. As an efficient VI algorithm, the Stein variational gradient descent (SVGD) has been used to develop a VI-based FWI method, which approximates the posterior probability density function using the distribution of a particle set. However, the SVGD-based FWI method reported in the existing literature usually uses some weak priors, such as a uniform distribution, to generate prior particles (or models, a term commonly used in the geophysics community). The particles generated from such priors exhibit random structures. Although this maximizes the retention of all possible solutions, it usually requires a large number of iterations to ensure convergence to the results with clear geologic implications. To address this, a geostatistical method was introduced to extract geologic structure information from seismic images and this information was used to generate prior particles. Specifically, the particles were generated by perturbing a smooth model with products of a pattern-feature correlation (PFC) matrix and random vectors. The elements of the PFC matrix, quantitatively determined as correlation coefficients of the pattern score vectors at each point, represent similarities of the geologic patterns at different positions. To reduce the storage amount of the PFC matrix and eliminate the spurious spatial correlations, which typically occur between two spatially distant points, the variogram function in geostatistics was adopted to determine the maximum correlation radius and the PFC matrix was sparsified according to this radius. The sparsified PFC matrix was then used to generate informed prior particles for SVGD-based FWI. Numerical examples demonstrate clearly that using the geostatistical prior particles as initial particles enhances the convergence of SVGD-based FWI visibly and yields an accurate characterization of the posterior distribution of the velocity model.

As an irreplaceable quantitative interpretation method, prestack seismic inversion enables the effective estimation of subsurface elastic parameters for reservoir prediction. However, for the model-driven prestack seismic inversion, the band-limited characteristics and noise interference of observed seismic data result in its high dependence on the initial models. This suggests that reasonable initial models act as a supplement to reliable variation trends in formation and can reduce the non-uniqueness of inversion results. In this article, we introduce a well-log interpolation method with a feature map-guided non-local means algorithm, which is for establishing high-fidelity initial models used for prestack seismic inversion. First, we briefly review the basic theory of general model-driven prestack seismic inversion. Then, we use dictionary learning to split the poststack seismic record into patches, and represent them with sparse vectors, instead of directly using seismic record. The advantage of dictionary learning is that it can adaptively extract useful signals from noisy observed data and provide fine structures by sparse reconstruction. Therefore, the proposed feature extraction method can improve the noise immunity and reliability of the well-log interpolation. More accurate initial models are pre-constructed efficiently by our feature extraction method, which improves the reliability of prestack seismic inversion results. Two kinds of observed seismic data are used, including the poststack seismic record for well-log interpolation and prestack seismic data used for inversion. Synthetic and field data tests both demonstrate the favorable performance of the proposed well-log interpolation method. In summary, a novel and convenient initial model building approach is provided, which contributes to seismic exploration and geologic modeling.