We consider numerical boundary conditions for high order finite difference schemes for solving convection-diffusion equations on arbitrary geometry. The two main difficulties for numerical boundary conditions in such situations are: (1) the wide stencil of the high order finite difference operator requires special treatment for a few ghost points near the boundary; (2) the physical boundary may not coincide with grid points in a Cartesian mesh and may intersect with the mesh in an arbitrary fashion. For purely convection equations, the so-called inverse Lax-Wendroff procedure [28], in which we convert the normal derivatives into the time derivatives and tangential derivatives along the physical boundary by using the equations, have been quite successful. In this paper, we extend this methodology to convection-diffusion equations. It turns out that this extension is non-trivial, because totally different boundary treatments are needed for the diffusion-dominated and the convection-dominated regimes. We design a careful combination of the boundary treatments for the two regimes and obtain a stable and accurate boundary condition for general convection-diffusion equations. We provide extensive numerical tests for one-and two-dimensional problems involving both scalar equations and systems, including the compressible Navier-Stokes equations, to demonstrate the good performance of our numerical boundary conditions.
Microseismic monitoring is an indispensable technique in characterizing the physical processes that are caused by extraction or injection of fluids during the hydraulic fracturing process. Microseismic data, however, are often contaminated with strong random noise and have a low signal-to-noise ratio (S/N). The low S/N in most microseismic data severely affects the accuracy and reliability of the source localization and source-mechanism inversion results. We have developed a new denoising framework to enhance the quality of microseismic data. We use the method of adaptive sparse dictionaries to learn the waveform features of the microseismic data by iteratively updating the dictionary atoms and sparse coefficients in an unsupervised way. Unlike most existing dictionary learning applications in the seismic community, we learn the features from 1D microseismic data, thereby to learn 1D features of the waveforms. We develop a sparse dictionary learning framework and then prepare the training patches and implement the algorithm to obtain favorable denoising performance. We use extensive numerical examples and real microseismic data examples to demonstrate the validity of our method. Results show that the features of microseismic waveforms can be learned to distinguish signal patches and noise patches even from a single channel of microseismic data. However, more training data can make the learned features smoother and better at representing useful signal components.
Efficient modeling schemes currently exist to handle the spatially variable-order fractional Laplacians in the fractional Laplacian viscoacoustic wave equation. The simplest approach is to change the spatially variable-order fractional Laplacians into a linear combination of several constant fractional-order Laplacians. We generalize the constant fractional-order scheme to a spatially variable fractional-order viscoelastic wave equation and develop an almost-equivalent constant fractional-order viscoelastic wave equation. Our constant fractional-order scheme avoids the simulation error introduced by directly averaging the spatially varying fractional order; thus, our scheme simulates seismic wave propagation in viscoelastic media with sharp [Formula: see text] contrasts well. The fast Fourier transform is used in the approximation of the fractional Laplacians, which improves the spectral accuracy in space. Several simulation examples are performed to verify that the numerical solution of a homogeneous [Formula: see text] model obtained by solving our constant fractional-order viscoelastic wave equation agrees well with that obtained by solving the original viscoelastic wave equation. The numerical simulations for spatially varying [Formula: see text] models obtained by the new wave equation are more straightforward than those currently in use and match the reference solutions obtained by accurate, but inefficient, methods. This match of simulation results verifies the accuracy of our viscoelastic wave equation.
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 propose a data-driven low-frequency recovery method based on deep learning from the high-frequency signals. In the proposed method, we introduce the idea of employing a basic data patch of seismic data to build a local data-driven mapping in low-frequency recovery. Energy balancing and data patches are employed 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 can then be utilized to predict low-frequency data from high-frequency data. The proposed 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 both 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 the proposed approach can reliably recover the low-frequency data.
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