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Multi-exponential inversion algorithm of nuclear magnetic resonance (NMR) T2 spectrum is an important mathematical tool for the NMR relaxation study of complicated samples. The popular algorithm usually obtains the T2 spectrum by linear fitting under the prescribed distribution of T2. When the T2 spectrum is dispersed, such a procedure is inaccurate because of the lack of adaptive prescription and the limit of linear method. Nonlinear fitting method does not fix the T2 distribution, and it provides the positions and the weights of T2 simultaneously via the nonlinear fitting of multi-exponential function. In this case, the problem of multi-exponential inversion is transformed into a nonlinear optimization problem with non-negative constraints. The optimization objective function is the residual sum of squares (or residual sum of squares with regularization). The nonlinear optimization problem can usually be solved by Levenberg-Marquardt algorithm and evolutionary algorithm. But the results of Levenberg-Marquardt algorithm are dependent on initial values, and the calculation of evolutionary algorithm is complicated. We provide an optimal model for the nonlinear fitting in the inversion of dispersed T2 spectrum based on the linear regression and least-squares. The key idea is that the optimal weights of T2 can be calculated by least square when the positions of T2 are fixed, although the positions of T2 are adjusted adaptively. So we can relate the positions to weights appropriately to improve the popular nonlinear fitting algorithms. Such an improvement can reduce the searching inversion parameters, speed up its convergence and reduce the dependence on initial value. Incorporating it into the Levenberg-Marquardt algorithm or evolutionary algorithm can improve the inversion accuracy and make the algorithm more robust. The validity of our improvement is demonstrated by the inversions of simulation data and practical NMR data by combining Levenberg- Marquardt algorithm and differential evolution algorithm with our improvement. The inversion results of simulation data show that for dispersed T2 spectrum, the algorithm using this improvement can obtain more accurate T2 spectrum than previous ones, especially in the case of low signal-to-noise ratio (SNR) cases. The inversion results also indicate that the improvement can reduce the dependence on initial value of Levenberg-Marquardt algorithm, and can accelerate the convergence of differential evolution algorithm. The inversion results of practical NMR data show that the algorithm using the improvement can obtain more accurate T2 spectrum than the widely used CONTIN program in the case of low signal-to-noise ratio (SNR). The inversion results of oil-water mixture sample NMR data also demonstrate that the relaxation time T2 is independent of dispersion degree of immiscible system components.
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Keywords:
- nuclear magnetic resonance /
- multi-exponential inversion /
- nonlinear fitting /
- differential evolution
[1] Wang W M, Li P, Ye C H 2001 Sci. China A 31 730 (in Chinese) [王为民, 李培, 叶朝辉 2001 中国科学A 31 730]
[2] Xu F, Huang Y R 2002 Acta Phys. Sin. 51 415 (in Chinese) [许峰, 黄永仁 2002 51 415]
[3] Zheng S K, Chen Z, Chen Z W, Zhong J H 2001 Chin. Phys. 10 558
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[7] Dunn K J, LaTorraca G A, Warner J L, Bergman D J 1994 SPE 69th Annual Techoical Conference and Exhibition New Orleans, Louisiana September25-28, 1994 SPE28367 45
[8] Wang Z D, Xiao L Z, Liu T Y 2003 Sci. China G 33 323 (in Chinese) [王忠东, 肖立志, 刘堂宴 2003 中国科学G 33 323]
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[10] Bro R, De Jong S 1997 J. Chemom. 11 393
[11] Liao G Z, Xiao L Z, Xie R H, Fu J J 2007 Chinese J. Geophys. 50 932 (in Chinese) [廖广志, 肖立志, 谢然红, 付娟娟 2007 地球 50 932]
[12] Berman P, Levi O, Parmet Y, Saunders M, Wiesman Z 2013 Concepts in Magnetic Resonance Part A 42 72
[13] Tikhonov A N 1963 Soviet Mathematics 4 1035
[14] Provencher S W 1982 Comput. Phys. Commun. 27 229
[15] Moody J B, Xia Y 2004 J. Magn. Res. 167 36
[16] Prange M, Song Y Q 2009 J. Magn. Res. 196 54
[17] Prange M, Song Y Q 2010 J. Magn. Res. 204 118
[18] Lin F, Wang Z W, Li J Y, Zhang X A, Jiang Y L 2011 Appl. Geophys. 8 233
[19] Wang H, Li G Y 2005 Acta Phys. Sin. 54 1431 (in Chinese) [王鹤, 李鲠颖 2005 54 1431]
[20] Pan K J, Chen H, Tan Y J 2008 Acta Phys. Sin. 57 5956 (in Chinese) [潘克家, 陈华, 谭永基 2008 57 5956]
[21] Chen H, Pan K J, Tan Y J 2009 Well Logging Technol. 33 37 (in Chinese) [陈华, 潘克家, 谭永基 2009 测井技术 33 37]
[22] Tan M J, Shi Y L, Xie G B 2007 Well Logging Technol. 31 413 (in Chinese) [谭茂金, 石耀霖, 谢关宝 2007 测井技术 31 413]
[23] Hastie T, Tibshirani R, Friedman J 2001 The Elements of Statistical Learning: Data Mining, Inference, and Prediction (New York: Springer) p11
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[1] Wang W M, Li P, Ye C H 2001 Sci. China A 31 730 (in Chinese) [王为民, 李培, 叶朝辉 2001 中国科学A 31 730]
[2] Xu F, Huang Y R 2002 Acta Phys. Sin. 51 415 (in Chinese) [许峰, 黄永仁 2002 51 415]
[3] Zheng S K, Chen Z, Chen Z W, Zhong J H 2001 Chin. Phys. 10 558
[4] Borgia G C, Brown R J S, Fantazzini P 1998 J. Magn. Res. 132 65
[5] Borgia G C, Brown R J S, Fantazzini P 2000 J. Magn. Res. 147 273
[6] Butler J P, Reeds J A, Dawson S V 1981 SIAM J. Numer. Anal. 18 381
[7] Dunn K J, LaTorraca G A, Warner J L, Bergman D J 1994 SPE 69th Annual Techoical Conference and Exhibition New Orleans, Louisiana September25-28, 1994 SPE28367 45
[8] Wang Z D, Xiao L Z, Liu T Y 2003 Sci. China G 33 323 (in Chinese) [王忠东, 肖立志, 刘堂宴 2003 中国科学G 33 323]
[9] Lawson C L, Hanson R J 1974 Solving Least Square Problems (Englewood Cliffs, New Jersey: Prentice-Hall) p158
[10] Bro R, De Jong S 1997 J. Chemom. 11 393
[11] Liao G Z, Xiao L Z, Xie R H, Fu J J 2007 Chinese J. Geophys. 50 932 (in Chinese) [廖广志, 肖立志, 谢然红, 付娟娟 2007 地球 50 932]
[12] Berman P, Levi O, Parmet Y, Saunders M, Wiesman Z 2013 Concepts in Magnetic Resonance Part A 42 72
[13] Tikhonov A N 1963 Soviet Mathematics 4 1035
[14] Provencher S W 1982 Comput. Phys. Commun. 27 229
[15] Moody J B, Xia Y 2004 J. Magn. Res. 167 36
[16] Prange M, Song Y Q 2009 J. Magn. Res. 196 54
[17] Prange M, Song Y Q 2010 J. Magn. Res. 204 118
[18] Lin F, Wang Z W, Li J Y, Zhang X A, Jiang Y L 2011 Appl. Geophys. 8 233
[19] Wang H, Li G Y 2005 Acta Phys. Sin. 54 1431 (in Chinese) [王鹤, 李鲠颖 2005 54 1431]
[20] Pan K J, Chen H, Tan Y J 2008 Acta Phys. Sin. 57 5956 (in Chinese) [潘克家, 陈华, 谭永基 2008 57 5956]
[21] Chen H, Pan K J, Tan Y J 2009 Well Logging Technol. 33 37 (in Chinese) [陈华, 潘克家, 谭永基 2009 测井技术 33 37]
[22] Tan M J, Shi Y L, Xie G B 2007 Well Logging Technol. 31 413 (in Chinese) [谭茂金, 石耀霖, 谢关宝 2007 测井技术 31 413]
[23] Hastie T, Tibshirani R, Friedman J 2001 The Elements of Statistical Learning: Data Mining, Inference, and Prediction (New York: Springer) p11
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