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摘要:
有限差分方法(Finite-difference Method, FD)广泛用于地震波场数值模拟, 但其存在固有的数值频散问题, 影响模拟的计算效率和数值精度.本文主要研究了有限差分方法的空间数值频散误差和网格划分精度以及差分算子的关系, 基于计算量最小准则, 提出了最优化有限差分参数选取流程, 为有限差分数值模拟参数选取提供理论指导.本文主要工作包括: (1) 提出了空间数值频散正变换过程(Forward Space Dispersion Transform, FSDT)方法, 该方法可以高效模拟出不同网格划分精度(波长采样点数)的带有空间数值频散的波场; (2) 提出了波场空间数值频散误差衡量准则, 可以定量地判断出数值模拟导致的波形频散程度, 选取合适的频散误差阈值; (3) 研究了给定空间数值频散误差阈值下, 差分算子系数、差分算子阶数、网格划分精度与计算量之间的关系.文中基于雷米兹交换方法(Remez Exchange Method, RE)和泰勒级数展开方法(Taylor-series Expansion Method, TE)的差分系数, 在空间数值频散误差阈值0.01时, 数值模拟了不同差分算子阶数、网格划分精度与计算量的关系, 并给出了有限差分参数选取的参考值.
Abstract:The finite-difference (FD) method is widely used for simulating complex wavefield propagation because of its high accuracy and efficiency. However, it suffers from numerical dispersions caused by spatial discretization with coarse grid sizes. The dispersions affect the computational efficiency and modeling accuracy. This study investigates the relationship among numerical dispersion error, FD operators, and the number of samples per shortest wavelength. Based on the criterion of minimum computational cost, we propose a scheme to estimate the optimal FD parameters, i.e., the number of samples per shortest wavelength and the spatial order of FD operators. Firstly, we introduce a method called Forward Space Dispersion Transform (FSDT) to add spatial dispersion to the reference wavefield for various scenarios of grid spacing and FD orders; Secondly, we measure the normalized L2 norm of the error between the reference wavefield and dispersed wavefield, so we can estimate the dispersion error directly to set a proper error threshold; Finally, we study the relationship among FD coefficients, FD operator length, grid spacing, and computational cost to find out the optimal grid spacing and FD order for giving the minimum computational cost under a preset error threshold. We also show some numerical tests of the relationship among the FD operator length, grid points per shortest wavelength, and computational cost under an error threshold of 0.01 with the finite-difference coefficients generated using the Remez Exchange (RE) method and Taylor-series Expansion (TE) method. Based on the tests, some FD parameters are recommended.
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图 1
FSDT方法模拟的2阶精度有限差分空间数值频散示意图
Figure 1.
Simulating the spatial dispersion of the second-order accuracy finite-difference by using the FSDT method
图 2
FSDT方法模拟的空间数值频散波场示意图
Figure 2.
The spatially dispersed wavefield simulated by using the FSDT method
图 3
FSDT方法模拟的2阶精度有限差分在不同网格大小下的空间数值频散波场示意图
Figure 3.
The spatially dispersed wavefield simulated by using the FSDT method of 2nd order finite-difference with different grid size
图 4
雷米兹交换算法(RE)和泰勒级数展开方法(TE)的16阶精度空间数值频散曲线示意图
Figure 4.
The numerical dispersion curves of the 16th-order accuracy RE and TE methods
图 5
设定误差阈值下，差分算子阶数、网格大小及计算量的关系图
Figure 5.
The relationship between the orders of FD operators, the grid size, and the computational cost with a fixed error threshold
图 6
计算量最小参数对应的波场快照
Figure 6.
The wavefield snapshot with the parameters of minimum computational cost
图 8
设定误差阈值下，差分算子阶数、网格大小及计算量的关系图
Figure 8.
The relationship between the orders of FD operators, the grid size, and the computational cost with a fixed error threshold
图 10
误差阈值0.01，常规Ricker子波在不同传播距离对应的最小计算量的差分算子阶数和最小波长采样点数
Figure 10.
The orders of FD operators and grid points per shortest wavelength for the minimum computational cost at different propagation distances with an original Ricker wavelet. The error threshold is 0.01
图 11
低通滤波的Ricker子波在不同传播距离对应的最小计算量的差分算子阶数和最小波长采样点数，滤波截止频率为2.5倍主频(即50 Hz)，误差阈值0.01
Figure 11.
The orders of FD operators and grid points per shortest wavelength for the minimum computational cost at different propagation distances with a low-pass-filtered Ricker wavelet. The stop frequency is 2.5 times the dominate frequency (i.e., 50 Hz). The error threshold is set to 0.01
图 12
(a) Marmousi速度模型及(b)传播时间1 s的波场快照
Figure 12.
(a) Marmousi velocity model and (b) wavefield snapshot at 1 second
图 13
单道波场记录对比(TE, G=3.18, 2N=16)
Figure 13.
The single trace waveform comparison (TE, G=3.18, 2N=16)
图 14
单道波场记录对比(RE, G=2.7, 2N=16)
Figure 14.
The single trace waveform comparison (RE, G=2.7, 2N=16)
表 1
运算量统计
Table 1.
Number of arithmetic operations
不同维度方程 加法运算量 乘法运算量 一维 3×N+3 N+3 二维 7×N+3 N+3 三维 11×N+3 N+3 
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表 2
2阶偏导数的中心差分算子的雷米兹交换方法差分系数(2N =12~24阶精度)
Table 2.
Remez exchange method FD coefficients of central-difference operator for the second partial derivative (2N=12~24)
差分系数 精度阶数 2N=12 2N=14 2N=16 2N=18 2N=20 2N=22 2N=24 c0 -3.08567404 -3.13546354 -3.17049888 -3.19557676 -3.21402708 -3.22789034 -3.23841242 c1 1.80490080 1.85080445 1.88374360 1.90764916 1.92541257 1.93885792 1.94911937 c2 -0.32938272 -0.36521045 -0.39251710 -0.41319258 -0.42902907 -0.44128665 -0.45079990 c3 0.08448092 0.10784253 0.12764668 0.14379201 0.15682499 0.16730741 0.17567973 c4 -0.02064818 -0.03304081 -0.04541707 -0.05669729 -0.06654180 -0.07491935 -0.08189626 c5 0.00389460 0.00897909 0.01546686 0.02241071 0.02917261 0.03539281 0.04087661 c6 -0.00040840 -0.00185764 -0.00457368 -0.00824342 -0.01240476 -0.01665681 -0.02069872 c7 0.00021460 0.00103068 0.00262046 0.00485956 0.00749850 0.01026837 c8 -0.00013053 -0.00063864 -0.00164834 -0.00310188 -0.00484390 c9 0.00008797 0.00043206 0.00111639 0.00210097 c10 -0.00006428 -0.00031314 -0.00079756 c11 0.00004997 0.00023791 c12 -0.00004041
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