Because deep-learning networks can “learn” the complex mapping function between labeled inputs and outputs, they have shown great potential in seismic inversion. Conventional deep-learning algorithms require a large amount of labeled data for sufficient training. However, in practice, the number of well logs is limited. To address this problem, we have adopted a closed-loop fully convolutional residual network (FCRN) combined with a transfer learning strategy for seismic inversion. This closed-loop FCRN consists of an inverse network and a forward network. The inverse network predicts the inversion target from seismic data, whereas the forward network calculates seismic data from the inversion target. The inverse network is initialized by pretraining on the Marmousi2 model and is fine-tuned with the limited labeled data around the wells through transfer learning, to suit the target seismic data. The forward network is initialized by training with the limited labeled data around the wells. In this way, the closed-loop network is well initialized to ensure relatively good convergence. Then, the misfit of the limited labeled data and the error between the true and the forward seismic data are used to regularize the training of the initialized closed-loop network. The inverse network of the optimized closed-loop network is used to obtain the final inversion results. Our workflow can be used for velocity, density, and impedance inversion from poststack seismic data. We take velocity inversion as an example to illustrate the effectiveness of the method. The experimental results show that closed-loop FCRN with transfer learning is superior to open-loop FCRN with better lateral continuity and velocity details. Closed-loop FCRN can effectively predict velocity with high accuracy on synthetic data, has good antinoise performance, and can also be effectively used for field data with spatial heterogeneity.

Because there are many similar geologic structures underground, seismic profiles have an abundance of self-repeating patterns. Thus, we can divide a seismic profile into groups of blocks with a similar seismic structure. The matrix formed by stacking together similar blocks in each group should be of low rank. Hence, we can transfer the seismic denoising problem to a series of low-rank matrix approximation (LRMA) problems. The LRMA-based model commonly adopts the nuclear norm as a convex substitute of the rank of a matrix. However, the nuclear norm minimization (NNM) shrinks the different rank components equally and may cause some biases in practice. The recently introduced truncated nuclear norm (TNN) has been proven to more accurately approximate the rank of a matrix, which is given by the sum of the set of the smallest singular values. Based on this, we have adopted a novel denoising method using truncated nuclear norm minimization (TNNM). The objective function of this method consists of two terms, the F-norm data fidelity and a TNN regularization. We develop an efficient two-step iterative algorithm to solve this objective function. Then, we apply the proposed TNNM algorithm to groups of blocks with similar seismic structure, and we aggregate all resulting denoised blocks to get the denoised seismic data. We update the denoised results during each iteration to gradually attenuate the heavy noise. Numerical experiments demonstrate that, compared with f-x deconvolution and the curvelet- and NNM-based methods, TNNM not only attenuates noise more effectively even when the signal-to-noise ratio is as low as −10 dB and seismic data have complex structures, but it also accurately preserves the seismic structures without inducing Gibbs artifacts.

Wavelet and reflectivity inversion (WRI) is an important issue in seismic data processing. To overcome the ill-posedness of WRI inversion with more efficient parameter selection and better lateral continuity of reflectivities, we propose a new WRI algorithm named compact smoothness and relative sparsity (CSRS) algorithm, where a normalized compact constraint and a normalized smooth regularization is proposed for the wavelet inversion, and a relative sparsity constraint is proposed for the reflectivity inversion. The proposed constraints and regularization make the parameters of WRI easy to be selected. The proposed relative sparsity constraint can lead to a reflectivity profile with good lateral continuity as it can be suitable for various seismic data with a fixed sparsity parameter. We also propose an efficient algorithm for solving corresponding WRI optimization problem. The whole WRI problem is divided into reflectivity inversion subproblem and wavelet inversion subproblem by using alternating iterative method, where the initial wavelet is estimated by smoothing the absolute amplitude spectrum of averaged seismic data. The proximal algorithm is applied to solve both reflectivity inversion subproblem and wavelet inversion subproblem. By replacing Toeplitz matrix multiplication with fast Fourier transform (FFT) and using compact wavelet, our algorithm can be efficient for 3D seismic data. The numerical examples on 2D synthetic data, 2D offshore field data and 3D onshore field data demonstrate that, compared to Toeplitz-sparse matrix factorization (TSMF) algorithm, the CSRS algorithm with fixed default parameters can get high-resolution reflectivities with better lateral continuity, and requires much less computational time.

Conventionally, we usually carry out Q estimation first and then perform seismic attenuation compensation (Ac). In this paper, we propose a method based on semi-supervised multi-task 2D V-Net combined with transfer learning to simultaneously conduct Q estimation and Ac for seismic data. The semi-supervised network consists of an inversion network and a forward network, both of which are 2D V-Net. And two channels are used to accomplish two tasks. The network pretrained on the synthetic data generated by the typical model is transferred to the target field data. The loss between the predictions of the inversion network and the label data, along with the loss between the seismic data obtained by the forward network and the input seismic data, is utilized to jointly guide the network training. As a result of fully exploiting the synthetic data, labeled data, and unlabeled seismic data for network training, the demand for labeled data is reduced. For the situation where the number of well logs is extremely small, a two-step strategy is developed. Firstly, a certain number of labels are generated using conventional method based on stratigraphic structure constrained molecular decomposition to train the network, and then the labels produced by well logs are used to further train the network. Both synthetic and field examples demonstrate that the two-step strategy enables our method to simultaneously achieve Q and compensated seismic data with high accuracy using attenuated seismic data, even with only one well log. Moreover, this approach exhibits good robustness against noise.

The CSIAM Transactions on Applied Mathematics (CSIAM-AM) is the flagship journal of the Chinese Society for Industrial and Applied Mathematics. It publishes original research articles of high quality on a topic in applied mathematics or computational mathematics or scientific computing. Papers should address mathematically and computationally interesting and challenging problems. Manuscripts applying known methods to previously solved problems will not be accepted.