PML边界条件下二维粘弹性介质波场模拟
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摘要
在波场模拟中较多使用的有限差分法和有限元法都或多或少地存在一些缺陷,如为了提高计算精度而使运算效率降低,在泊松比变化大的界面存在稳定性问题等。伪谱法是一种计算精度高且计算效率也较高的方法,它是对空间坐标通过快速傅里叶变换在时间域作差分运算,避免了求偏导数;不存在有限差分法和有限元法对高频成分限制的问题,可以实现全频带地震波模拟;对内存容量的需求也远远低于有限差分法和有限元法。为此,采用了伪谱法进行粘弹性介质波场模拟。首先给出了开尔芬粘弹性介质中的波动方程,推导出二维粘弹性介质最佳匹配层(PML)吸收边界条件及其相应的伪谱法计算公式;然后对均匀介质模型进行了模拟,结果表明,该方法的边界吸收效果很好;最后通过数值模拟分析了具有不同粘滞系数介质对地震波的吸收和衰减,结果表明,随着粘滞系数的增大,粘弹性反射波的主频向低频方向移动,高频吸收明显,有效频带变窄,振幅降低。此外,在伪谱法中引入匹配层边界条件后,只需对空间做一维傅里叶变换,大大提高了伪谱法的计算效率。
Finite difference (FD) method and finite element (FE) method widely used in simulating the wavefield have some deficiencies. For example, the computational efficiency will decrease if the computational accuracy is to be improved and there exists a problem in stability for the interface with big change in Poisson's ratio. Pseudo-spectral (PS) method has a high computational accuracy and a high computational efficiency. Compared to the FD method and FE method, the PS method has such advantages as 1) it utilizes Fast Fourier Transform (FFT) in volume coordinate to calculate the time in time domain and avoids computing the partial derivative, 2) it can realize complete frequency band seismic wave simulation, and 3) it has lower dispersion and needs lower internal memory. So, the PS method has been used in the wavefield simulation of viscoe-lastic medium. Through the given wave equation in Kelvin viscoe-lastic medium, the PML absorbing boundary condition and its pseudo-spectral method formulae in 2-D viscoelastic medium were derived. The numerical simulation of homogeneous medium indicates that the effect of this absorbing boundary condition is apparent. The influence of different coefficient of viscosity on wavefield absorption and attenuation was analyzed through wavefield simulation, showing that the dominant frequency of the viscoelastic reflected wave was moved to low frequency, the high frequency absorbed obviously, the effectively frequency band narrowed down, and the amplitude decreased with the increase of viscosity coefficient. With the matched layer boundary conditions being introduced, only the 1-D Fourier transform in space is needed for the PS method, greatly improving the computational efficiency.
Finite difference (FD) method and finite element (FE) method widely used in simulating the wavefield have some deficiencies. For example, the computational efficiency will decrease if the computational accuracy is to be improved and there exists a problem in stability for the interface with big change in Poisson's ratio. Pseudo-spectral (PS) method has a high computational accuracy and a high computational efficiency. Compared to the FD method and FE method, the PS method has such advantages as 1) it utilizes Fast Fourier Transform (FFT) in volume coordinate to calculate the time in time domain and avoids computing the partial derivative, 2) it can realize complete frequency band seismic wave simulation, and 3) it has lower dispersion and needs lower internal memory. So, the PS method has been used in the wavefield simulation of viscoe-lastic medium. Through the given wave equation in Kelvin viscoe-lastic medium, the PML absorbing boundary condition and its pseudo-spectral method formulae in 2-D viscoelastic medium were derived. The numerical simulation of homogeneous medium indicates that the effect of this absorbing boundary condition is apparent. The influence of different coefficient of viscosity on wavefield absorption and attenuation was analyzed through wavefield simulation, showing that the dominant frequency of the viscoelastic reflected wave was moved to low frequency, the high frequency absorbed obviously, the effectively frequency band narrowed down, and the amplitude decreased with the increase of viscosity coefficient. With the matched layer boundary conditions being introduced, only the 1-D Fourier transform in space is needed for the PS method, greatly improving the computational efficiency.
引文
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2 Stekl 1, Pratt R G. Accurate viscoelastic modeling by freduency-domain finite differences using rotated oper-ators[J]. Geophysics, 1998,63(5) :811~823
3崔建军,何继善．粘弹性波动方程正演和偏移[J]．中南工业大学学报,2001,32(5):441～444
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6牛滨华,孙春岩．空间介质与地震波传播[M]．北京:石油工业出版社,2002．130～151
7崔兴福,张关泉．地震波方程人工边界的两种处理方法[J]．石油物探,2003,42(4):452～455
8 Berenger J P. A perfectly matched layer for the absorption of electromagnetics waves [J]. J Comput Phys, 1994,114(2):185~200
9 Zeng Y Q, He J Q, Liu Q H. The application of the perfectly matched layer in numerical modeling of wave propagation in poroelastic media[J]. Geophysics,2001, 66(4):1 258-1 266
10 Komatitsch D, Tromp J. A perfectly matched layer absorbing boundary condition for the second-order seismic wave equation [J]. Geophysical Journal International, 2003,154(l):146~153
2 Stekl 1, Pratt R G. Accurate viscoelastic modeling by freduency-domain finite differences using rotated oper-ators[J]. Geophysics, 1998,63(5) :811~823
3崔建军,何继善．粘弹性波动方程正演和偏移[J]．中南工业大学学报,2001,32(5):441～444
4奚先,姚姚．二维粘弹性随机介质中的波场特征分析[J]．地球物理学进展,2004,19(3):608～615
5傅旦丹,何樵登．正交各向异性介质地震弹性波场的伪谱法正演[J]．石油物探,2001,40(3):8～14
6牛滨华,孙春岩．空间介质与地震波传播[M]．北京:石油工业出版社,2002．130～151
7崔兴福,张关泉．地震波方程人工边界的两种处理方法[J]．石油物探,2003,42(4):452～455
8 Berenger J P. A perfectly matched layer for the absorption of electromagnetics waves [J]. J Comput Phys, 1994,114(2):185~200
9 Zeng Y Q, He J Q, Liu Q H. The application of the perfectly matched layer in numerical modeling of wave propagation in poroelastic media[J]. Geophysics,2001, 66(4):1 258-1 266
10 Komatitsch D, Tromp J. A perfectly matched layer absorbing boundary condition for the second-order seismic wave equation [J]. Geophysical Journal International, 2003,154(l):146~153
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