地震数据曲网格模拟与叠后深度偏移
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摘要
在直角坐标系下用传统伪谱法模拟地下界面时，弯曲起伏的连续界面会因离散化而成为阶梯状界面，这种阶梯状界面在模拟波的传播时，将产生人为绕射，降低合成记录的精度。本论文实现了一种精确有效的方法来模拟曲界面波场。
该方法结合了多块映射和超限插值技术，这项技术出自计算流体力学，它可使产生的网格网线沿着地下所有界面变化。这一映射技术将直角坐标系下的曲界面变换成曲坐标系下的规则界面，在曲坐标系下用伪谱法模拟波场，最后利用插值得到实际波场的合成记录。这种方法能有效地消除在直角坐标系下因离散化曲界面产生的不必要的人为绕射。曲网格伪谱法用较少的网格数可以获得与直网格伪谱法相同的精度，从而节省了计算机内存，这一点对于三维地震模型的计算具有重要意义。通过声波模型和弹性波模型在直网格下的合成记录与在曲网格下的合成记录的比较验证了方法的有效性与精确性。
傅立叶有限差分(简称FFD)偏移算法是一种叠后深度偏移算法，其向下延拓算子是一种混合算子，包括三项：一项是处理常速的相移算子，一项是一阶相移修正算子，最后一项是类似45度方程的有限差分算子，用来处理剧烈横向变速。FFD算法既保持了相移法的精确性，又可以处理剧烈横向变速问题，在一定程度上克服了相移法偏移和有限差分法偏移的缺点，是一种比较有效的偏移方法。本论文用连分式近似单程波波动方程中的平方根导出FFD算法的基本公式，并对FFD算法中的有限差分算子进行了系数优化，进一步提高了计算的有效性。
When applying the conventional Fourier pseudospectral method on Cartesian grids, curved interfaces are represented in a 'staircase fashion' causing spurious diffractions. It is demonstrated that these non-physical diffractions can be eliminated by using curved grids that generally follow all curved interfaces.
The curved grids are generated by using the transfinite interpolation and the so-called multiblock techniques that originally developed for computation fluid dynamics. The curved grid is taken to constitute a generalized curvilinear system. Thus, the wave equations have to be written in curvilinear coordinates before applying the numerical scheme. Because the grid is Cartesian in the curvilinear domain, standard pseudospectral technique can be applied. At last the synthetic record of physical configuration can be evaluated. Compared to similar methods on Cartesian grids, the same accuracy is obtained with a lot fewer grid points, which means that considerable savings in computer memory can be obtained. This fact has an important implication for extension to 3D configurations. Comparing the synthetic seismogram by using Cartesian grids with the synthetic seismogram by using the curved grid, it can prove the approach is effective.
A hybrid migration method, named "Fourier finite-difference migration", is a post-stack depth migration scheme. The downward extrapolation operator is split into three operators: one operator is a phase-shift operator for a chosen constant background velocity, another operator is the well-known first-order correction term, and the third operator is a finite-difference operator for the varying of the velocity function. Phase-shift downward extrapolation and finite-difference downward extrapolation preserves the advantage of phase-shift method and finite-difference method. So FFD migration method is an effective scheme. In the thesis, the fundamental formula of FFD method derives from the square root that is approximated by a continued fraction expansion in the one-way wave equation. Optimizations of the parameters of the finite-difference operator improve the validity of the method.
该方法结合了多块映射和超限插值技术，这项技术出自计算流体力学，它可使产生的网格网线沿着地下所有界面变化。这一映射技术将直角坐标系下的曲界面变换成曲坐标系下的规则界面，在曲坐标系下用伪谱法模拟波场，最后利用插值得到实际波场的合成记录。这种方法能有效地消除在直角坐标系下因离散化曲界面产生的不必要的人为绕射。曲网格伪谱法用较少的网格数可以获得与直网格伪谱法相同的精度，从而节省了计算机内存，这一点对于三维地震模型的计算具有重要意义。通过声波模型和弹性波模型在直网格下的合成记录与在曲网格下的合成记录的比较验证了方法的有效性与精确性。
傅立叶有限差分(简称FFD)偏移算法是一种叠后深度偏移算法，其向下延拓算子是一种混合算子，包括三项：一项是处理常速的相移算子，一项是一阶相移修正算子，最后一项是类似45度方程的有限差分算子，用来处理剧烈横向变速。FFD算法既保持了相移法的精确性，又可以处理剧烈横向变速问题，在一定程度上克服了相移法偏移和有限差分法偏移的缺点，是一种比较有效的偏移方法。本论文用连分式近似单程波波动方程中的平方根导出FFD算法的基本公式，并对FFD算法中的有限差分算子进行了系数优化，进一步提高了计算的有效性。
When applying the conventional Fourier pseudospectral method on Cartesian grids, curved interfaces are represented in a 'staircase fashion' causing spurious diffractions. It is demonstrated that these non-physical diffractions can be eliminated by using curved grids that generally follow all curved interfaces.
The curved grids are generated by using the transfinite interpolation and the so-called multiblock techniques that originally developed for computation fluid dynamics. The curved grid is taken to constitute a generalized curvilinear system. Thus, the wave equations have to be written in curvilinear coordinates before applying the numerical scheme. Because the grid is Cartesian in the curvilinear domain, standard pseudospectral technique can be applied. At last the synthetic record of physical configuration can be evaluated. Compared to similar methods on Cartesian grids, the same accuracy is obtained with a lot fewer grid points, which means that considerable savings in computer memory can be obtained. This fact has an important implication for extension to 3D configurations. Comparing the synthetic seismogram by using Cartesian grids with the synthetic seismogram by using the curved grid, it can prove the approach is effective.
A hybrid migration method, named "Fourier finite-difference migration", is a post-stack depth migration scheme. The downward extrapolation operator is split into three operators: one operator is a phase-shift operator for a chosen constant background velocity, another operator is the well-known first-order correction term, and the third operator is a finite-difference operator for the varying of the velocity function. Phase-shift downward extrapolation and finite-difference downward extrapolation preserves the advantage of phase-shift method and finite-difference method. So FFD migration method is an effective scheme. In the thesis, the fundamental formula of FFD method derives from the square root that is approximated by a continued fraction expansion in the one-way wave equation. Optimizations of the parameters of the finite-difference operator improve the validity of the method.
引文
1 J．F．克莱鲍特，地震成像理论及方法 石油工业出版社，1991
2 Gazdag J., Wave equation migration with the phase-shift method: Geophysics, 1978, 73, 1342-1351
3 马在田，高阶逼近的有限差分法：中国国家石油与天然气开发与发展会议的技术报告，1981
4 Stolt R., Migration by Fourier transform: Geophysics, 1978, 43, 23-48
5 Gazdag J., and Sguazzero P., Migration of seismic data by phase shift plus interpolation: Geophysics, 1984, 49, 124-131
6 Staffa P.L., Fokkema J. T., de Luna Freire R. M., and Kessinter W. P., Split-step Fourier migration: Geophysics, 1990, 56, 410-421
7 Ristow D., and Ruhl T., Fourier finite-difference migration: Geophysics, 1994, 59, 1882-1893
8 范祯祥，郑仙种，地震波数值模拟与偏移成像 河南科学技术出版社 1994
9 Trorey A.W., A simple theory for seismic diffractions: Geophysics, 1970, 35, 762-784
10 Kelly K. R., Ward R.W., Treitel S., and Alford R. M., Synthetic seismograms: A finite-difference approach: Geophysics, 1976, 41, 2-27
11 Gazdag J., Modeling of the acoustic wave equation with transform methods: Geophysics, 1981, 46, 854-859
12 Seron F. J., Sanz F. J. and Kindelan M., Elastic Wave Propagation with the Finite-Element Method: Comparison of Different Numerical Techniques: SEG Annual Meeting, 1989, 1283-1287
13 Vafidis A., Abramovic F., and Kanasewich E. R., Elastic wave propagation using fully vectorized high order finite-difference algorithms: Geophysics, 1992, 57, 218-232
14 Fornberg B., High-order finite differences and the pseudospectral method on staggered grids: SIAM J. NUMER. ANAL, 1990, 27, 904-918
15 Fornberg B., The pseudospectral method: Accurate representation of interfaces in elastic wave calculations: Geophysics, 1988, 53, 625-637
16 Nielsen P., If F., Berg P. and Skovgarrd O., Using the pseudospectral technique on curved grids for 2D acoustic forward modelling: Geophysical Prospecting, 1994, 42, 321-341
17 Kosloff D. D., and Baysal E., Forward model by a Fourier method: Geophysics, 1982, 47, 1402-1412
18 程乾生，信号数字处理的数学原理 石油工业出版社，第一版 1979
19 陆金甫，关治，偏微分方程数值解法 清华大学出版社 1987
20 Thompson J. F., Warsi Z. U. A. and Mastin C. W., Numerical Grid Generation-Foundations and Application North Holland Publishing Co, 1985
21 李岳生，黄友谦，数值逼近 高等教育出版社，第二版 1987，第二章
22 Berg P., If F. and Skovgaard O., A spectral method for seismic wave propagation in elastic media: Wave Motion, 1990, 12, 415-427
23 Nielsen P., If F., Berg P. and Skovgarrd O., Using the pseudospectral method on curved grids for 2D elastic forward modeling: Geophysical Prospecting, 1995, 43, 369-395
24 Talezer H., Kosloff D. and Koren Z., An accurate scheme for seismic forward modeling: Geophysics Prespecting, 1987, 35, 479-490
25 张叔伦，孙沛勇，基于平面波合成的傅立叶有限差分叠前深度偏移，石油地球物理勘探，1999，1，1-7
2 Gazdag J., Wave equation migration with the phase-shift method: Geophysics, 1978, 73, 1342-1351
3 马在田，高阶逼近的有限差分法：中国国家石油与天然气开发与发展会议的技术报告，1981
4 Stolt R., Migration by Fourier transform: Geophysics, 1978, 43, 23-48
5 Gazdag J., and Sguazzero P., Migration of seismic data by phase shift plus interpolation: Geophysics, 1984, 49, 124-131
6 Staffa P.L., Fokkema J. T., de Luna Freire R. M., and Kessinter W. P., Split-step Fourier migration: Geophysics, 1990, 56, 410-421
7 Ristow D., and Ruhl T., Fourier finite-difference migration: Geophysics, 1994, 59, 1882-1893
8 范祯祥，郑仙种，地震波数值模拟与偏移成像 河南科学技术出版社 1994
9 Trorey A.W., A simple theory for seismic diffractions: Geophysics, 1970, 35, 762-784
10 Kelly K. R., Ward R.W., Treitel S., and Alford R. M., Synthetic seismograms: A finite-difference approach: Geophysics, 1976, 41, 2-27
11 Gazdag J., Modeling of the acoustic wave equation with transform methods: Geophysics, 1981, 46, 854-859
12 Seron F. J., Sanz F. J. and Kindelan M., Elastic Wave Propagation with the Finite-Element Method: Comparison of Different Numerical Techniques: SEG Annual Meeting, 1989, 1283-1287
13 Vafidis A., Abramovic F., and Kanasewich E. R., Elastic wave propagation using fully vectorized high order finite-difference algorithms: Geophysics, 1992, 57, 218-232
14 Fornberg B., High-order finite differences and the pseudospectral method on staggered grids: SIAM J. NUMER. ANAL, 1990, 27, 904-918
15 Fornberg B., The pseudospectral method: Accurate representation of interfaces in elastic wave calculations: Geophysics, 1988, 53, 625-637
16 Nielsen P., If F., Berg P. and Skovgarrd O., Using the pseudospectral technique on curved grids for 2D acoustic forward modelling: Geophysical Prospecting, 1994, 42, 321-341
17 Kosloff D. D., and Baysal E., Forward model by a Fourier method: Geophysics, 1982, 47, 1402-1412
18 程乾生，信号数字处理的数学原理 石油工业出版社，第一版 1979
19 陆金甫，关治，偏微分方程数值解法 清华大学出版社 1987
20 Thompson J. F., Warsi Z. U. A. and Mastin C. W., Numerical Grid Generation-Foundations and Application North Holland Publishing Co, 1985
21 李岳生，黄友谦，数值逼近 高等教育出版社，第二版 1987，第二章
22 Berg P., If F. and Skovgaard O., A spectral method for seismic wave propagation in elastic media: Wave Motion, 1990, 12, 415-427
23 Nielsen P., If F., Berg P. and Skovgarrd O., Using the pseudospectral method on curved grids for 2D elastic forward modeling: Geophysical Prospecting, 1995, 43, 369-395
24 Talezer H., Kosloff D. and Koren Z., An accurate scheme for seismic forward modeling: Geophysics Prespecting, 1987, 35, 479-490
25 张叔伦，孙沛勇，基于平面波合成的傅立叶有限差分叠前深度偏移，石油地球物理勘探，1999，1，1-7