颗粒粒径分布测量反演算法的改进
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摘要
综合奇异值截断法、奇异值修正法、Tikhonov正则化思想及Chahine迭代算法,提出一种改进的病态问题求解算法来测量颗粒系的粒径分布。结合Backus-Gilbert折中准则与奇异值最小原则确定了奇异截断值,采用L曲线法确定了最优正则化参数,并利用联合迭代反演法(SIRT)实现解的非负约束。模拟及实验结果表明,该算法对单、双峰分布的测量误差均小于3%,其抗噪性能、测量准确性、时效性及粒径测量范围相较其他反演算法都有明显优势。
An improved solution algorithm of the ill-posed problem is proposed to measure the particle size distribution, which is combined with the truncated singular value decomposition(TSVD) method, the modified singular value decomposition method, the Tikhonov regularization method, and the Chahine iteration method. The singular cutoff value is determined by the Backus-Gilbert tradeoff criteria and the minimum principle of singular value. The optimal regularization parameters are determined by the L-curve method, and the simultaneous iterative reconstruction technique(SIRT) is adopted to realize the non-negative constraint of the solution. The simulation and experimental results show that the measurement errors of the single-peak and bimodal distributions are both less than 3% by the proposed algorithm. In addition, the proposed algorithm has obvious advantages superior to the other inversion algorithms in the anti-noise performance, measurement accuracy, timeliness, and measurement range of the particle size.
An improved solution algorithm of the ill-posed problem is proposed to measure the particle size distribution, which is combined with the truncated singular value decomposition(TSVD) method, the modified singular value decomposition method, the Tikhonov regularization method, and the Chahine iteration method. The singular cutoff value is determined by the Backus-Gilbert tradeoff criteria and the minimum principle of singular value. The optimal regularization parameters are determined by the L-curve method, and the simultaneous iterative reconstruction technique(SIRT) is adopted to realize the non-negative constraint of the solution. The simulation and experimental results show that the measurement errors of the single-peak and bimodal distributions are both less than 3% by the proposed algorithm. In addition, the proposed algorithm has obvious advantages superior to the other inversion algorithms in the anti-noise performance, measurement accuracy, timeliness, and measurement range of the particle size.
引文
[1] Riefler N, Wriedt T. Intercomparison of inversion algorithms for particle sizing using Mie scattering[J]. Particle & Particle Systems Characterization, 2008, 25(3): 216-230.
[2] Cai X S, Su M X, Shen J Q. Particle size measurement technology and application[M]. Beijing: Chemical Industry Press, 2010: 18-34. 蔡小舒, 苏明旭, 沈建琪. 颗粒粒度测量技术及应用[M]. 北京: 化学工业出版社, 2010: 18-34.
[3] Wang Y W. Light scattering theory and application[M]. Beijing: Science Press, 2013: 42-68. 王亚伟. 光散射理论及其应用技术[M]. 北京: 科学出版社, 2013: 42-68.
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[5] Ferri F. Use of a charge coupled device camera for low-angle elastic light scattering[J]. Review of Scientific Instruments, 1997, 68(6): 2265-2274.
[6] Wang S M, Lu Y, Ye M. Measurement of the particle mean size by the ratio of the scattering light intensified at small angles near-forward[J]. Journal of Wuhan University (Natural Science Edition), 1997, 43(5): 691-696. 王式民, 陆勇, 叶茂. 前向小角散射法测量颗粒平均尺寸[J].武汉大学学报(自然科学版), 1997, 43(5): 691-696.
[7] Magatti D, Alaimo M D, Potenza M A C, et al. Dynamic heterodyne near field scattering[J]. Applied Physics Letters, 2008, 92(24): 2547.
[8] Maleknejad K, Aghazadeh N, Mollapourasl R. Numerical solution of Fredholm integral equation of the first kind with collocation method and estimation of error bound[J]. Applied Mathematics & Computation, 2006, 179(1): 352-359.
[9] Morozov V A. Methods for solving incorrectly posed problems[M]. New York: Springer Verlag, 2006: 103-121.
[10] Carstensen C, Praetorius D. Averaging techniques for the effective numerical solution of Symm′s integral equation of the first kind[J]. SIAM Journal on Scientific Computing, 2006, 27(4): 1226-1260.
[11] Wang B E. Study on numerical methods of discrete ill-posed problem in inverse problem[D]. Xi'an: Xi'an University of Technology, 2006: 25-28. 王宝娥. 反问题中离散不适定问题的数值求解方法[D]. 西安: 西安理工大学, 2006: 25-28.
[12] Ma T, Chen L W, Wu M P, et al. The selection of regularization parameter in downward continuation of potential field based on L-curve method[J]. Progress in Geophysics, 2013, 28(5): 2485-2494. 马涛, 陈龙伟, 吴美平, 等. 基于L曲线法的位场向下延拓正则化参数选择[J]. 地球物理学进展, 2013, 28(5): 2485-2494.
[13] Wang Y J, Dou Z, Shen J, et al. Multi-scale inversion combining TSVD-Tikhonov regularization for dynamic light scattering[J]. Chinese Journal of Lasers, 2017, 44(1): 0104003. 王雅静, 窦智, 申晋, 等. TSVD-Tikhonov正则化多尺度动态光散射反演[J]. 中国激光, 2017, 44(1): 0104003.
[14] Wang T E, Shen J Q, Lin C J. Vector similarity retrieval algorithm in particle size distribution analysis of forward scattering[J]. Acta Optica Sinica, 2016, 36(6): 0629002. 王天恩, 沈建琪, 林承军. 前向散射颗粒粒径分布分析中的向量相似度反演算法[J]. 光学学报, 2016, 36(6): 0629002.
[15] Potenza M A C,Pescini D, Magatti D, et al. A new particle sizing technique based on near field scattering[J]. Nuclear Physics B (Proceedings Supplements), 2006, 150(1): 334-338.
[16] Kouzelis D, Candel S M, Esposito E, et al. Particle sizing by laser light diffraction: Improvements in optics and algorithms[J]. Particle & Particle Systems Characterization, 1987, 4(1/2/3/4): 151-156.
[17] Ferri F, Bassini A, Paganini E. Modified version of the Chahine algorithm to invert spectral extinction data for particle sizing[J]. Applied Optics, 1995, 34(25): 5829-5839.
[18] Li P, Wang C Y, Xu H Z, et al. Discussion on the application of singular value decomposition in geophysical inversion[J]. Progress in Natural Science, 2001, 11(8): 891-896. 李平, 王椿镛, 许厚泽, 等. 地球物理反演中奇异值分解应用的若干问题探讨[J]. 自然科学进展, 2001, 11(8): 891-896.
[19] Wang Z J. Research on the regularization solutions of ill-posed problems in geodesy[D]. Wuhan: Institute of measurement and Geophysics of the Chinese Academy of Sciences, 2003: 34-36. 王振杰. 大地测量中不适定问题的正则化解法研究[D]. 武汉: 中国科学院测量与地球物理研究所, 2003: 34-36.
[20] Xu X N, Li J C, Wang Z T, et al. The simulation research on the Tikhonov regularization applied in gravity field determination of GOCE satellite mission[J]. Acta Geodaetica et Cartographica Sinica, 2010, 39(5): 465-470. 徐新禹, 李建成, 王正涛, 等. Tikhonov正则化方法在GOCE重力场求解中的模拟研究[J]. 测绘学报, 2010, 39(5): 465-470.
[21] Xiao L Z, Zhang H R, Liao G Z, et al. Inversion of NMR relaxation in porous media based on Backus-Gilbert theory[J]. Chinese Journal of Geophysics, 2012, 55(11): 3821-3828. 肖立志, 张恒荣, 廖广志, 等. 基于Backus-Gilbert理论的孔隙介质核磁共振弛豫反演[J]. 地球物理学报, 2012, 55(11): 3821-3828.
[22] Wang Z D, Xiao L Z, Liu T Y. A new method for multiexponential inversion of NMR relaxation signal and its application[J]. Science in China (Series G), 2003, 33(4): 323-332. 王忠东, 肖立志, 刘堂宴. 核磁共振弛豫信号多指数反演新方法及其应用[J]. 中国科学G辑, 2003, 33(4): 323-332.
[2] Cai X S, Su M X, Shen J Q. Particle size measurement technology and application[M]. Beijing: Chemical Industry Press, 2010: 18-34. 蔡小舒, 苏明旭, 沈建琪. 颗粒粒度测量技术及应用[M]. 北京: 化学工业出版社, 2010: 18-34.
[3] Wang Y W. Light scattering theory and application[M]. Beijing: Science Press, 2013: 42-68. 王亚伟. 光散射理论及其应用技术[M]. 北京: 科学出版社, 2013: 42-68.
[4] Yang Y F, Yang H, Zheng G, et al. Progress of particle size measurement by laser diffraction and scattering[J]. Optical Technique, 2011, 37(1): 19-24. 杨依枫, 杨晖, 郑刚, 等. 衍射散射式颗粒粒度测量法的研究新进展[J]. 光学技术, 2011,37(1): 19-24.
[5] Ferri F. Use of a charge coupled device camera for low-angle elastic light scattering[J]. Review of Scientific Instruments, 1997, 68(6): 2265-2274.
[6] Wang S M, Lu Y, Ye M. Measurement of the particle mean size by the ratio of the scattering light intensified at small angles near-forward[J]. Journal of Wuhan University (Natural Science Edition), 1997, 43(5): 691-696. 王式民, 陆勇, 叶茂. 前向小角散射法测量颗粒平均尺寸[J].武汉大学学报(自然科学版), 1997, 43(5): 691-696.
[7] Magatti D, Alaimo M D, Potenza M A C, et al. Dynamic heterodyne near field scattering[J]. Applied Physics Letters, 2008, 92(24): 2547.
[8] Maleknejad K, Aghazadeh N, Mollapourasl R. Numerical solution of Fredholm integral equation of the first kind with collocation method and estimation of error bound[J]. Applied Mathematics & Computation, 2006, 179(1): 352-359.
[9] Morozov V A. Methods for solving incorrectly posed problems[M]. New York: Springer Verlag, 2006: 103-121.
[10] Carstensen C, Praetorius D. Averaging techniques for the effective numerical solution of Symm′s integral equation of the first kind[J]. SIAM Journal on Scientific Computing, 2006, 27(4): 1226-1260.
[11] Wang B E. Study on numerical methods of discrete ill-posed problem in inverse problem[D]. Xi'an: Xi'an University of Technology, 2006: 25-28. 王宝娥. 反问题中离散不适定问题的数值求解方法[D]. 西安: 西安理工大学, 2006: 25-28.
[12] Ma T, Chen L W, Wu M P, et al. The selection of regularization parameter in downward continuation of potential field based on L-curve method[J]. Progress in Geophysics, 2013, 28(5): 2485-2494. 马涛, 陈龙伟, 吴美平, 等. 基于L曲线法的位场向下延拓正则化参数选择[J]. 地球物理学进展, 2013, 28(5): 2485-2494.
[13] Wang Y J, Dou Z, Shen J, et al. Multi-scale inversion combining TSVD-Tikhonov regularization for dynamic light scattering[J]. Chinese Journal of Lasers, 2017, 44(1): 0104003. 王雅静, 窦智, 申晋, 等. TSVD-Tikhonov正则化多尺度动态光散射反演[J]. 中国激光, 2017, 44(1): 0104003.
[14] Wang T E, Shen J Q, Lin C J. Vector similarity retrieval algorithm in particle size distribution analysis of forward scattering[J]. Acta Optica Sinica, 2016, 36(6): 0629002. 王天恩, 沈建琪, 林承军. 前向散射颗粒粒径分布分析中的向量相似度反演算法[J]. 光学学报, 2016, 36(6): 0629002.
[15] Potenza M A C,Pescini D, Magatti D, et al. A new particle sizing technique based on near field scattering[J]. Nuclear Physics B (Proceedings Supplements), 2006, 150(1): 334-338.
[16] Kouzelis D, Candel S M, Esposito E, et al. Particle sizing by laser light diffraction: Improvements in optics and algorithms[J]. Particle & Particle Systems Characterization, 1987, 4(1/2/3/4): 151-156.
[17] Ferri F, Bassini A, Paganini E. Modified version of the Chahine algorithm to invert spectral extinction data for particle sizing[J]. Applied Optics, 1995, 34(25): 5829-5839.
[18] Li P, Wang C Y, Xu H Z, et al. Discussion on the application of singular value decomposition in geophysical inversion[J]. Progress in Natural Science, 2001, 11(8): 891-896. 李平, 王椿镛, 许厚泽, 等. 地球物理反演中奇异值分解应用的若干问题探讨[J]. 自然科学进展, 2001, 11(8): 891-896.
[19] Wang Z J. Research on the regularization solutions of ill-posed problems in geodesy[D]. Wuhan: Institute of measurement and Geophysics of the Chinese Academy of Sciences, 2003: 34-36. 王振杰. 大地测量中不适定问题的正则化解法研究[D]. 武汉: 中国科学院测量与地球物理研究所, 2003: 34-36.
[20] Xu X N, Li J C, Wang Z T, et al. The simulation research on the Tikhonov regularization applied in gravity field determination of GOCE satellite mission[J]. Acta Geodaetica et Cartographica Sinica, 2010, 39(5): 465-470. 徐新禹, 李建成, 王正涛, 等. Tikhonov正则化方法在GOCE重力场求解中的模拟研究[J]. 测绘学报, 2010, 39(5): 465-470.
[21] Xiao L Z, Zhang H R, Liao G Z, et al. Inversion of NMR relaxation in porous media based on Backus-Gilbert theory[J]. Chinese Journal of Geophysics, 2012, 55(11): 3821-3828. 肖立志, 张恒荣, 廖广志, 等. 基于Backus-Gilbert理论的孔隙介质核磁共振弛豫反演[J]. 地球物理学报, 2012, 55(11): 3821-3828.
[22] Wang Z D, Xiao L Z, Liu T Y. A new method for multiexponential inversion of NMR relaxation signal and its application[J]. Science in China (Series G), 2003, 33(4): 323-332. 王忠东, 肖立志, 刘堂宴. 核磁共振弛豫信号多指数反演新方法及其应用[J]. 中国科学G辑, 2003, 33(4): 323-332.