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.
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