Application of an optimized transmitting boundary with multiple artificial wave velocities in spectral-element simulation of seismic wave propagation
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摘要: 将作者最近发展的多人工波速优化透射边界（记为ca j-MTF）应用于高精度谱元法的地震波动模拟中，并与经典的廖氏透射（MTF）边界、完美匹配层（PML）边界、黏弹性边界以及一阶旁轴近似边界进行了比较分析。结果显示：① ca j-MTF边界与MTF边界在形式上非常接近，它继承了后者公式简单、易于实现、精度可控、计算量低以及通用性好的优点；② 不同于传统MTF边界的单一人工波速参数配置，ca j-MTF边界所具有的多个人工波速参数可以分别取为P波和S波波速，此时边界计算波速与介质物理波速相匹配，能够大幅度提高复杂波动情形下的边界精度；③ ca j-MTF边界的精度略低于PML边界，不过要明显优于MTF边界、黏弹性边界以及一阶旁轴近似边界；④ ca j-MTF边界相比于PML边界的优势在于形式简洁且普遍适用。本研究为谱元法的地震波动模拟提供了一种便捷、高效的人工边界条件（即吸收边界条件）实现方法。Abstract: This paper applied an optimized transmitting boundary with multiple artificial velocities （denoted as caj-MTF） that is recently proposed by the authors to the high-accuracy spectral-element simulation of seismic wave propagation, and made a comparison study with several other classical artificial （or absorbing） boundary conditions including Liao’s multi-transmitting formula （MTF） boundary, perfectly matched layer （PML） boundary, viscous-spring boundary and the first-order Clayton-Engquist paraxial-approximation boundary. The results obtained from theoretical analysis and numerical tests are as follows: ① The formulation of ca j-MTF is very similar to that of MTF, so it has most of the advantages of the latter, i.e., very simple expressions, easy to be implemented, adjustable accuracy, minimal computation cost, and general applicability. ② Unlike the traditional MTF boundary that has only a single artificial wave velocity （i.e., computational wave velocity）, ca j-MTF has multiple artificial wave velocities. In the simulation of elastic waves, the computational wave velocity parameters of ca j-MTF can be set to be P- and S-wave velocities, respectively. On this situation, the consistency between computational and physical wave velocities makes a significant improvement in the boundary accuracy. ③ ca j-MTF boundary has an slightly lower accuracy than that of PML boundary, whereas it is significantly superior to MTF, viscous-spring boundary and the first-order paraxial-approximation boundary. ④ ca j-MTF is superior to PML as it has much simpler formulations and better versatility. This work provides a convenient and high-efficient artificial boundary （or absorbing boundary） for spectral-element simulation of seismic wave propagation.
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图 1 多人工波速优化透射边界（
$c_{{\rm{a}}j}$ -MTF）示意图 （s0t为定义人工边界条件的局部坐标系）Figure 1. Schematic diagram of optimized transmitting boundary with multiple artificial wave velocities （
$c_{{\rm{a}}j}$ -MTF） （s0t is a local coordinate system that is used to define the artificial boundary condition）![]()
图 2 优化透射边界的谱元计算格式
Figure 2. Spectral-element computational scheme of the optimized transmitting boundary
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图 3 人工边界性能分析模型
（a） 体波大角度透射问题；（b） 面波透射问题
Figure 3. Analytical models of artificial boundary performance
（a） The problem of body wave transmission over large incident angles；（b） Surface wave transmission problem
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图 5 不同边界条件模拟体波大角度透射问题的波场快照（时刻依次为0.3 s，0.4 s，0.5 s，0.7 s）
（a） $c_{{\rm{a}}j} $-MTF边界；（b） MTF边界；（c） PML边界；（d） 黏弹性边界；（e） 一阶旁轴近似边界；（f） 自由边界（物理边界）
Figure 5. Wave field snapshots of body wave propagating over large incident angles obtained from different boundary conditions （time instants are 0.3 s，0.4 s，0.5 s，0.7 s）
（a） $c_{{\rm{a}}j} $-MTF boundary；（b） MTF boundary；（c） PML boundary；（d） Viscous-spring boundary；（e） The first-order paraxial-approximation boundary；（f） Free boundary （physical boundary）
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图 6 观察点A （a），B （b）和C （c）处z方向位移
$u_z $ 的时程及误差曲线Figure 6. Time histories and error curves of the displacement
$u_z $ at the observation points A （a），B （b） and C （c）![]()
图 7 不同边界条件模拟面波问题的波场快照（时刻依次为0.5 s，0.9 s，1.2 s）
（a） $ c_{{\rm{a}}j} $-MTF边界；（b） MTF边界；（c） PML边界；（d） 黏弹性边界；（e） 一阶旁轴近似边界；（f） 自由边界（物理边界）
Figure 7. Wave field snapshots of surface waves obtained by different boundary conditions （time instants are 0.5 s，0.9 s，1.2 s）
（a） $ c_{{\rm{a}}j} $-MTF boundary；（b） MTF boundary；（c） PML boundary；（d） Viscous-spring boundary；（e） The first-order paraxial-approximation boundary；（f） Free boundary （physical boundary）
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图 8 观察点D （a），E （b）和F （c）处z方向位移
$u_z $ 的时程及误差曲线Figure 8. Time history and error curves of the displacement
$u_z $ at the observation points D （a），E （b） and F （c）表 1 体波情形下A，B和C点误差曲线峰值（绝对值）以及每种边界条件下误差峰值与参考解峰值的百分比
Table 1 The peaks （absolute value） of error curves of the points A，B，C，and the percentages between those error peaks and reference solution peaks on different boundary conditions：Body wave case
边界条件 ux误差峰值/m uz误差峰值/m 误差峰值与参考
解峰值的百分比A点 B点 C点 A点 B点 C点 ca j-MTF边界 0.006 5 0.002 1 0.005 1 0.013 9 0.004 4 0.003 9 15.3% MTF边界 0.021 9 0.013 1 0.008 0 0.033 0 0.008 5 0.009 8 37.0% PML边界 0.001 0 0.000 9 0.001 5 0.001 5 0.001 1 0.000 6 3.2% 黏弹性边界 0.019 3 0.028 7 0.016 9 0.038 3 0.025 3 0.012 4 62.1% 一阶旁轴近似边界 0.022 3 0.004 0 0.010 0 0.008 3 0.011 0 0.012 9 33.7% 自由边界 0.058 5 0.036 4 0.016 2 0.097 6 0.076 4 0.013 7 119.2% 参考解峰值 0.062 6 0.045 8 0.021 1 0.075 3 0.028 2 0.021 3 − 
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表 2 面波情形下D，E和F点误差曲线峰值（绝对值）以及每种边界条件下误差峰值与参考解峰值的百分比
Table 2 The peaks （absolute value） of error curves of the points D， E，F， and the percentages between those error peaks and reference solution peaks under different boundary conditions：Surface wave case
边界条件 ux 误差峰值/m uz误差峰值/m 误差峰值与参考
解峰值的百分比D点 E点 F点 D点 E点 F点 ca j-MTF边界 0.019 6 0.012 2 0.002 5 0.018 1 0.012 5 0.002 8 11.7% MTF边界 0.102 0 0.049 7 0.017 8 0.103 2 0.061 0 0.009 9 59.7% PML边界 0.005 0 0.002 5 0.002 5 0.002 8 0.002 3 0.001 4 4.8% 黏弹性边界 0.086 4 0.014 0 0.004 5 0.168 5 0.026 8 0.011 0 30.4% 一阶旁轴近似边界 0.075 4 0.034 8 0.009 8 0.069 9 0.059 6 0.009 6 41.0% 自由边界 0.552 3 0.196 5 0.204 9 0.447 7 0.153 4 0.154 9 425.4% 参考解峰值 0.233 3 0.039 8 0.015 3 0.336 7 0.252 2 0.053 1 −
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