Research on rational Krylov subspace model order reduction algorithm for three-dimensional multi-frequency CSEM modelling
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摘要:
本文采用有理函数Krylov子空间模型降阶算法实现了同时求解多频可控源电磁法三维正演响应的快速计算.首先采用基于Yee氏交错网格的拟态有限体积法实现控制方程的空间离散，将任意频率的电场响应表示为关于频率参数的传递函数.采用有理函数Krylov子空间算法求解该传递函数.针对构建m维有理函数Krylov子空间需要求解m次（几十到上百）关于有理函数极点和离散控制方程系数矩阵的线性方程组的问题，本文提出采用单个重复极点的有理函数Krylov子空间模型降阶算法，结合直接法求解器PARDISO，采用Gram-Schmidt方法，只需要1次系数矩阵分解和m次矩阵回代即可实现有理函数Krylov子空间的构建，极大地减少了计算量.针对最优化有理函数极点选取问题，本文根据传递函数的有理函数Krylov子空间投影算法的误差分析理论，引入关于单个重复极点的收敛率函数，通过求解有理函数的最大收敛率直接给出最优化的单个重复极点公式.最终实现了不同发射频率的可控源电磁法三维正演响应的快速计算.分别计算了典型层状模型多发射频率的CSAMT和海洋CSEM的正演响应，通过与解析解的对比验证了本文算法在多发射频率正演的计算精度和计算效率；并通过一个三维海洋CSEM勘探设计最优化发射频率和接收区域选取的例子进一步说明本文算法的优点.
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关键词:
- 可控源电磁法 /
- 三维正演 /
- 多频 /
- 有理函数Krylov子空间 /
- 模型降阶
Abstract:In this paper, a 3D forward modeling of multi-frequency controlled-source electromagnetic (CSEM) response is proposed by using the rational Krylov subspace model order reduction algorithm. Firstly, the staggered grid is used to simulate the mimetic finite volume space discretization of the governing equations. Then we obtain a system for electric field alone. The explicit solution of this system is given in terms of the transfer function form of the frequency parameter. Rational Krylov subspace model order reduction algorithm is used to solve the transfer function. It uses the Gram-Schmidt orthogonalization algorithm to construct the orthonormal basis vectors. Through the analysis of main time-consuming part of the rational Krylov subspace algorithm, we choose to adopt a repeat pole. A simple and fast algorithm for selecting optimization repeat pole is given according to the error analysis theory of rational function approximation. Further the basis vectors and projection matrix of rational Krylov subspace is constructed by only a matrix decomposition with direct method solver PARDISO. Then 3D response of multi-frequency CSEM is obtained. Finally, the computational accuracy and computational efficiency of the proposed algorithm are verified by comparing the results of the forward modeling with CSAMT and marine CSEM layered model. Then we use the proposed algorithm to calculate a typical 3D marine CSEM model to obtain the optimal transmission frequency and receiving area.
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Key words:
- CSEM /
- 3D forward /
- Multi-frequency /
- Rational Krylov subspace /
- Model order reduction
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图 2
CSAMT一维模型及发射-接收系统
Figure 2.
The 1D conductivity model and transmitter-receiver configuration for CSAMT
图 3
不同频率有理函数Krylov子空间模型降阶解的相对误差随m的变化情况
Figure 3.
Error curves of the rational Krylov method for different frequency with m
图 4
1D解析算法和3D有理函数Krylov子空间模型降阶解算法求得的CSAMT视电阻率和相位响应
Figure 4.
CSAMT apparent resistivity and phase response of the 1D model by 1D analytic method and 3D rational Krylov method
图 6
频率范围[0.1, 0.25, 0.5, 0.75, 1.0]Hz时0.1 Hz的有理函数Krylov子空间模型降阶解的相对误差随m的变化情况
Figure 6.
Error curves of the rational Krylov method for 0.1 Hz with different m when frequency interval is [0.1, 0.25, 0.5, 0.75, 1.0]Hz
图 7
1D解析算法和3D有理函数Krylov子空间模型降阶解求得的CSEM模型沿inline方向的电场响应
Figure 7.
Inline electric field of the CSEM model by 1D analytic method and 3D rational Krylov method
图 9
发射频率为0.1 Hz和1 Hz时含有和不含目标体的3D CSEM模型沿inline方向的电场响应
Figure 9.
Inline electric field of the 3D model with and without target
图 10
频率范围[0.1~2.0]Hz时0.1 Hz的有理函数Krylov子空间模型降阶解的相对误差随m的变化情况
Figure 10.
Error curves of the rational Krylov method for 0.1 Hz with different m when frequency interval is [0.1~2.0]Hz
图 11
不同发射频率的3D CSEM模型沿inline方向的电场响应
Figure 11.
Inline electric field of the 3D model with different frequencies
表 1
计算传递函数gτ(A)X的有理函数Krylov子空间模型降阶算法
Table 1.
Rational Krylov subspace model order reduction algorithm for transfer function gτ(A)X
算法1计算gτ(A)X的有理函数Krylov子空间模型降阶算法 输入：空间离散系数矩阵A，源项X，极点ξj, j=1, …, m 输出：不同频率的电场响应的有理函数Krylov子空间模型降阶解e(τ) 1.设置v1=X/‖X‖ 2.循环j=1, 2, …, m 3.求解(A-ξjI)x=vj 4.循环i=1, 2, …, j 5.求解x=x-(x, vi)vi 6.结束循环 7.设置vj+1=x/‖x‖ 8.结束循环 9.计算子空间正交基Vm+1=[v1, v2, …, vm+1] 10.计算投影矩阵Tm+1=Vm+1AVTm+1 11.计算有理函数Krylov子空间模型降阶解e(τ)=Vm+1gτ(Tm+1)(‖X‖e1) 
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Börner R U. 2010. Numerical modelling in geo-electromagnetics:advances and challenges. Surveys in Geophysics, 31(2):225-245. doi: 10.1007/s10712-009-9087-x
Börner R U, Ernst O G, Güttel S. 2015. Three-dimensional transient electromagnetic modelling using Rational Krylov methods. Geophysical Journal International, 202(3):2025-2043. doi: 10.1093/gji/ggv224
Cai H Z, Hu X Y, Li J H, et al. 2017. Parallelized 3D CSEM modeling using edge-based finite element with total field formulation and unstructured mesh. Computers & Geosciences, 99:125-134. http://www.sciencedirect.com/science/article/pii/S009830041630694X
Constable S, Weiss C J. 2006. Mapping thin resistors and hydrocarbons with marine EM methods:Insights from 1D modeling. Geophysics, 71(2):G43-G51. doi: 10.1190/1.2187748
Constable S. 2010. Ten years of marine CSEM for hydrocarbon exploration. Geophysics, 75(5):75A67-75A81. doi: 10.1190/1.3483451
Druskin V, Knizhnerman L. 1994. Spectral approach to solving three-dimensional Maxwell's diffusion equations in the time and frequency domains. Radio Science, 29(4):937-953. doi: 10.1029/94RS00747
Druskin V L, Knizhnerman L A, Lee P. 1999. New spectral Lanczos decomposition method for induction modeling in arbitrary 3-D geometry. Geophysics, 64(3):701-706. doi: 10.1190/1.1444579
Güttel S. 2010. Rational Krylov methods for operator functions. [Ph. D. thesis] Technische Universität Bergakademie Freiberg.
Haber E, Ruthotto L. 2014. A multiscale finite volume method for Maxwell's equations at low frequencies. Geophysical Journal International, 199(2):1268-1277. doi: 10.1093/gji/ggu268
Jiang Y L. 2010. Model Reduction Method (in Chinese). Beijing:Science Press.
Key K. 2009. 1D inversion of multicomponent, multifrequency marine CSEM data:Methodology and synthetic studies for resolving thin resistive layers. Geophysics, 74(2):F9-F20. doi: 10.1190/1.3058434
Knizhnerman L, Druskin V, Zaslavsky M. 2009. On optimal convergence rate of the rational Krylov subspace reduction for electromagnetic problems in unbounded domains. SIAM Journal on Numerical Analysis, 47(2):953-971. doi: 10.1137/080715159
Kong F N, Johnstad S E, Røsten T, et al. 2008. A 2.5 D finite-element-modeling difference method for marine CSEM modeling in stratified anisotropic media. Geophysics, 73(1):F9-F19. doi: 10.1190/1.2819691
Li J H, Farquharson C G, Hu X Y, et al. 2016. A vector finite element solver of three-dimensional modelling for a long grounded wire source based on total electric field. Chinese Journal of Geophysics (in Chinese), 59(4):1521-1534, doi:10.6038/cjg20160432.
Li J M. 2005. Geoelectric Field and Electrical Exploration (in Chinese). Beijing:Geological Press.
Mittet R, Morten J P. 2013. The marine controlled-source electromagnetic method in shallow water. Geophysics, 78(2):E67-E77. http://adsabs.harvard.edu/abs/2013Geop...78E..67M
Newman G A, Alumbaugh D L. 1995. Frequency-domain modelling of airborne electromagnetic responses using staggered finite differences. Geophysical Prospecting, 43(8):1021-1042. doi: 10.1111/gpr.1995.43.issue-8
Peng R H, Hu X Y, Han B, et al. 2016. 3D frequency-domain CSEM forward modeling based on the mimetic finite-volume method. Chinese Journal of Geophysics (in Chinese), 59(10):3927-3939, doi:10.6038/cjg20161036.
Plessix R E, Darnet M, Mulder W A. 2007. An approach for 3D multisource, multifrequency CSEM modeling. Geophysics, 72(5):SM177-SM184. doi: 10.1190/1.2744234
Routh P S, Oldenburg D W. 1999. Inversion of controlled source audio-frequency magnetotellurics data for a horizontally layered earth. Geophysics, 64(6):1689-1697. doi: 10.1190/1.1444673
Ruhe A. 1994. Rational Krylov algorithms for nonsymmetric eigenvalue problems. //Golub G, Luskin M, Greenbaum A, eds. Recent Advances in Iterative Methods. The IMA Volumes in Mathematics and its Applications. New York, NY: Springer, 149-164.
Schenk O, Gärtner K. 2004. Solving unsymmetric sparse systems of linear equations with PARDISO. Future Generation Computer Systems, 20(3):475-487. doi: 10.1016/j.future.2003.07.011
Streich R. 2009. 3D finite-difference frequency-domain modeling of controlled-source electromagnetic data:Direct solution and optimization for high accuracy. Geophysics, 74(5):F95-F105. doi: 10.1190/1.3196241
Streich R. 2016. Controlled-source electromagnetic approaches for hydrocarbon exploration and monitoring on land. Surveys in Geophysics, 37(1):47-80. doi: 10.1007/s10712-015-9336-0
Tang J T, He J S. 2005. Controlled-Source Audio Magnetotelluric Method and Its Application. Changsha:Central South University Press.
Um E S, Commer M, Newman G A. 2013. Efficient pre-conditioned iterative solution strategies for the electromagnetic diffusion in the Earth:finite-element frequency-domain approach. Geophysical Journal International, 193(3):1460-1473. doi: 10.1093/gji/ggt071
Weiss C J. 2013. Project APhiD:A Lorenz-gauged A-Φ decomposition for parallelized computation of ultra-broadband electromagnetic induction in a fully heterogeneous Earth. Computers & Geosciences, 58:40-52. http://chianti.ucsd.edu/cyto_web/plugins/pluginjardownload.php?id=424
Xu Z F, Wu X P. 2010. Controlled source electromagnetic 3-D modeling in frequency domain by finite element method. Chinese Journal of Geophysics (in Chinese), 53(8):1931-1939, doi:10.3969/j.issn.0001-5733.2010.08.019.
Yang B, Xu Y X, He Z X, et al. 2012. 3D frequency-domain modeling of marine controlled source electromagnetic responses with topography using finite volume method. Chinese Journal of Geophysics (in Chinese), 55(4):1390-1399, doi:10.6038/j.issn.0001-5733.2012.04.035.
Yin C C, Ben F, Liu Y H, et al. 2014. MCSEM 3D modeling for arbitrarily anisotropic media. Chinese Journal of Geophysics (in Chinese), 57(12):4110-4122, doi:10.6038/cjg20141222.
Zhang Y, Wang H N, Tao H G, et al. 2012. Finite volume algorithm to simulate 3D responses of multi-component induction tools in inhomogeneous anisotropic formation based on coupled scalar-vector potentials. Chinese Journal of Geophysics (in Chinese), 55(6):2141-2152, doi:10.6038/j.issn.0001-5733.2012.06.036.
蒋耀林. 2010.模型降阶方法.北京:科学出版社.
李建慧, Farquharson C G, 胡祥云等. 2016.基于电场总场矢量有限元法的接地长导线源三维正演.地球物理学报, 59(4):1521-1534, doi:10.6038/cjg20160432. http://manu39.magtech.com.cn/Geophy/CN/abstract/abstract12740.shtml
李金铭. 2005.地电场与电法勘探.北京:地质出版社.
彭荣华, 胡祥云, 韩波等. 2016.基于拟态有限体积法的频率域可控源三维正演计算.地球物理学报, 59(10):3927-3939, doi:10.6038/cjg20161036. http://manu39.magtech.com.cn/Geophy/CN/abstract/abstract13165.shtml
汤井田, 何继善. 2005.可控源音频大地电磁法及其应用.长沙:中南大学出版社.
徐志锋, 吴小平. 2010.可控源电磁三维频率域有限元模拟.地球物理学报, 53(8):1931-1939, doi:10.3969/j.issn.0001-5733.2010.08.019. http://manu39.magtech.com.cn/Geophy/CN/abstract/abstract3240.shtml
杨波, 徐义贤, 何展翔等. 2012.考虑海底地形的三维频率域可控源电磁响应有限体积法模拟.地球物理学报, 55(4):1390-1399, doi:10.6038/j.issn.0001-5733.2012.04.035. http://manu39.magtech.com.cn/Geophy/CN/abstract/abstract8634.shtml
殷长春, 贲放, 刘云鹤等. 2014.三维任意各向异性介质中海洋可控源电磁法正演研究.地球物理学报, 57(12):4110-4122, doi:10.6038/cjg20141222. http://manu39.magtech.com.cn/Geophy/CN/abstract/abstract11039.shtml
张烨, 汪宏年, 陶宏根等. 2012.基于耦合标势与矢势的有限体积法模拟非均匀各向异性地层中多分量感应测井三维响应.地球物理学报, 55(6):2141-2152, doi:10.6038/j.issn.0001-5733.2012.06.036. http://manu39.magtech.com.cn/Geophy/CN/abstract/abstract8723.shtml
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