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摘要:
传统的伪谱（PS）方法，采用傅里叶变换（FT）计算空间导数具有很高的精度，每个波长仅需要两个采样点，而时间导数采用有限差分（FD）近似因而精度较低.当采用大时间步长时，由于时空精度不平衡，PS法存在不稳定性问题.原始的k-space方法可以有效地克服这些问题但是却无法适用于非均匀介质.为了提高原始k-space方法模拟非均匀介质波动方程的精度，我们提出了一种新的k-space算子族.它是用非均匀介质的变速度代替原k-space算子中的常数补偿速度构造得到，引入低秩近似可以高效求解.我们将构造的新的k-space算子应用于耦合的二阶位移波动方程，而不是交错网格一阶速度应力波动方程，使模拟弹性波的计算存储量减少.我们从数学上证明了基于二阶波动方程的k-space方法与基于一阶波动方程的k-space方法是等价的.数值模拟实验表明，与传统的PS、交错网格PS和原始的k-space方法相比，我们的新方法可以在时间和空间步长较大的均匀和非均匀介质中，为弹性波的传播提供更精确的数值解.在保持稳定性和精度的同时，采用较大的时空采样间隔，可以大大降低数值模拟的计算成本.
Abstract:The traditional Pseudo-Spectral (PS) method achieve an accurate spatial approximation with only two points required per wavelength, whereas it has low accuracy in temporal approximation because using FD approximation. This unbalanced scheme of the PS method has temporal dispersion and instability problems when a large time step is used. The original k-space method can effectively overcome these problems; however, it is not suitable for seismic modeling in heterogeneous media and requires a large memory. To solve this problem, we have developed a new family of k-space operators which are constructed by replacing the constant compensation velocity in the original k-space operators with the variable velocity of heterogeneous media. A low-rank approximation is introduced to obtain solutions at a high efficiency. To deal with the large memory requirement, the constructed k-space operators are applied to the second-order displacement wave equations instead of the first-order velocity-stress wave equations on the staggered grid, which reduces the computational memory. We mathematically demonstrate that this k-space scheme based on the second-order wave equations is equivalent to the original one based on the first-order wave equations. Numerical modeling experiments show that this new scheme can provide analytical and more accurate solutions for acoustic and elastic wave propagation in homogeneous and heterogeneous media with large time and space steps compared with the traditional PS, staggered grid PS and k-space methods. By using larger temporal sampling intervals while maintaining the stability and accuracy, this scheme can greatly reduce the computational cost for long-time modeling.
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Key words:
- k-space operator /
- Low-rank approximation /
- Wave propagation /
- Heterogeneous media
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图 1
(a) 高斯一阶导数子波-∂tiV；(b)雷克子波-∂t2iV
Figure 1.
(a) The Gaussian first-order derivative wavelet -∂tiV; (b) The Ricker wavelet -∂t2iV
图 2
解析解和不同方法的数值解之间的对比
Figure 2.
Comparisons between the analytical solution and numerical solutions calculated by different methods
图 3
新k-space方法在Δt=2.5 ms和Δt=0.5 ms下与解析解的对比(三种解十分匹配)
Figure 3.
Comparisons between the analytical solution and solutions obtained by the new k-space scheme using Δt=2.5 ms and Δt=0.5 ms (three solutions are in good agreement with each other)
图 5
新k-space方法下时间步长为1.6 ms，在1.6 s时的波场快照
Figure 5.
Snapshots calculated by the new k-space scheme at 1.6 s with the time step of 1.6ms
图 6
SGPS方法下时间步长为1.6 ms，在1.6 s时的波场快照
Figure 6.
Snapshots calculated by the SGPS scheme at 1.6 s with the time step of 1.6 ms
图 7
PA方法下时间步长为1.6 ms，在1.6 s时的波场快照
Figure 7.
Snapshots calculated by the PA scheme at 1.6 s for with the time step of 1.6 ms
图 8
与8 km偏移距对应的单道比较，时间步长为1.6 ms
Figure 8.
Single trace comparisons corresponding to 8 km offset with the time step of 1.6 ms
图 9
大时间步长2 ms下对应的8 km偏移距的单道对比，其中SGPS方法变得不稳定
Figure 9.
Single trace comparisons corresponding to 8 km offset with the large time step of 2ms, where SGPS becomes unstable
图 10
小时间步长为1 ms对应的8 km偏移距单道比较
Figure 10.
Single trace comparisons corresponding to 8 km offset with the small time step of 1 ms
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