Trapezoid-Grid Finite-Difference Time-Domain Method for 3D Seismic Wavefield Modeling Using CPML Absorbing Boundary Condition

The large computational memory requirement is an important issue in 3D large-scale wave modeling, especially for GPU calculation. Based on the observation that wave propagation velocity tends to gradually increase with depth, we propose a 3D trapezoid-grid finite-difference time-domain (FDTD) method to achieve the reduction of memory usage without a significant increase of computational time or a decrease of modeling accuracy. It adopts the size-increasing trapezoid-grid mesh to fit the increasing trend of seismic wave velocity in depth, which can significantly reduce the oversampling in the high-velocity region. The trapezoid coordinate transformation is used to alleviate the difficulty of processing ununiform grids. We derive the 3D acoustic equation in the new trapezoid coordinate system and adopt the corresponding trapezoid-grid convolutional perfectly matched layer (CPML) absorbing boundary condition to eliminate the artificial boundary reflection. Stability analysis is given to generate stable modeling results. Numerical tests on the 3D homogenous model verify the effectiveness of our method and the trapezoid-grid CPML absorbing boundary condition, while numerical tests on the SEG/EAGE overthrust model indicate that for comparable computational time and accuracy, our method can achieve about 50% reduction on memory usage compared with those on the uniform-grid FDTD method.

A stable implementation of multiple Higdon’s absorbing boundary condition for Maxwell’s equations

<p></p><p>Highlights</p> <p>1) This letter presents a stable implementation of Higdon’s ABC at multiple layers without attenuation factor.</p> <p>2) The multiple Higdon’s ABC works well with wide-band frequency and ultra-thin layers, such as three.</p> <p>3) Optimal parameters of the multiple Higdon’s ABC are presented in this letter.</p> <p>4) The optimal angels of the multiple Higdon’s ABC satisfy a circle equation.</p><br><p></p>

A stable implementation of multiple Higdon’s absorbing boundary condition for Maxwell’s equations

<p></p><p>Highlights</p> <p>1) This letter presents a stable implementation of Higdon’s ABC at multiple layers without attenuation factor.</p> <p>2) The multiple Higdon’s ABC works well with wide-band frequency and ultra-thin layers, such as three.</p> <p>3) Optimal parameters of the multiple Higdon’s ABC are presented in this letter.</p> <p>4) The optimal angels of the multiple Higdon’s ABC satisfy a circle equation.</p><br><p></p>

Time-Inhomogeneous Feller-type Diffusion Process with Absorbing Boundary Condition

A time-inhomogeneous Feller-type diffusion process with linear infinitesimal drift $$\alpha (t)x+\beta (t)$$ α ( t ) x + β ( t ) and linear infinitesimal variance 2r(t)x is considered. For this process, the transition density in the presence of an absorbing boundary in the zero-state and the first-passage time density through the zero-state are obtained. Special attention is dedicated to the proportional case, in which the immigration intensity function $$\beta (t)$$ β ( t ) and the noise intensity function r(t) are connected via the relation $$\beta (t)=\xi \,r(t)$$ β ( t ) = ξ r ( t ) , with $$0\le \xi <1$$ 0 ≤ ξ < 1 . Various numerical computations are performed to illustrate the effect of the parameters on the first-passage time density, by assuming that $$\alpha (t)$$ α ( t ) , $$\beta (t)$$ β ( t ) or both of these functions exhibit some kind of periodicity.

Semi-exact local absorbing boundary condition for seismic wave simulation

An absorbing boundary condition is necessary in seismic wave simulation for eliminating the unwanted artificial reflections from model boundaries. Existing boundary condition methods often have a trade-off between numerical accuracy and computational efficiency. We proposed a local absorbing boundary condition for frequency-domain finite-difference modelling. The proposed method benefits from exact local plane-wave solution of the acoustic wave equation along predefined directions that effectively reduces the dispersion in other directions. This method has three features: simplicity, accuracy and efficiency. Numerical simulation demonstrated that the proposed method has higher efficiency than the conventional methods such as the second-order absorbing boundary condition and the perfectly matched layer (PML) method. Meanwhile, the proposed method shared the same low-cost feature as the first-order absorbing boundary condition method.

Passive-seismic Inversion of SH-Wave Input Motions in a Domain Truncated by Wave Absorbing Boundary Condition

This paper introduces a new inversion method for the reconstruction of complex, incoherent SH incident wavefield in a domain that is truncated by a wave-absorbing boundary condition (WABC), using a partial differential equation (PDE)-constrained optimization method. In numerical examples, dynamic traction at the WABC mimics seismic incident wavefield. Estimated traction is discretized over space and time, and the discretized values are reconstructed by using seismic motion data that are sparsely made by sensors on the top surface of a domain and a vertical array. The discretize-then-optimize (DTO) approach is used in the mathematical modeling and numerical implementation, and the finite element method (FEM) is applied to solve state and adjoint problems.The numerical results show that incident, inclined plane waves, cannot be fully reconstructed if using only the top surface sensors. In order to improve the inversion performance, a vertical array of sensors on the side boundary of a domain should be included. Second, a sufficiently large number of sensors must be employed to improve the algorithm's inversion performance. Third, the minimizer suffers less from solution multiplicity when it identifies lower frequency traction (e.g., a realistic seismic signal). Fourth, the larger value of the inversion error in the reconstructed traction does not necessarily translate to an error of the same order of magnitude in the corresponding reconstructed wave responses in the computational domain. Fifth, our presented inversion algorithm's accuracy is not compromised by the material complexity of a background domain. Lastly, the error in the reconstructed traction and the error in the corresponding wave responses grow when the noise of a larger level is added to the measurement data, but not in the same proportion. By extending the presented method into realistic 3D settings, this algorithm can indicate where large amplitudes of stress waves (i.e., weak points) occur in built environments and soils in a domain of interest during seismic events.