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Surface magnetic resonance sounding (MRS) has generally been considered to be an efficient tool for hydrological investigations. As is well known, the effective relaxation time
$ T_2^*$ which characterizes the decay rate of MRS free-decay-induction (FID) signal and is used to measure pore-scale properties, is particularly limited for several special cases (e.g. areas with magnetic rock subsurfaces). Recent years, the transverse relaxation time$ T_2$ obtained from spin-echo signal was adopted to implement the surface MRS, and showed great potentials for estimating the porosity and permeability. However, owning to the short period of development, the related modeling and inversion strategies are rarely introduced and summarized. Actually, the general practice for surface MRS$ T_2$ measurement fits the spin-echo by the exponential function and the fitting line was directly used as the FID signal for inversion. This scheme not only limits the precision of interpretation, but also loses part of valid information about original field data. Aiming at these problems, in this paper, we introduce the calculation of forward model and thus a two-stage framework with singular value decomposition (SVD) linear inversion involved is derived to quantify the$ T_2$ distributed with depth. Considering the fact that the inversion result of SVD is always strongly affected by the noise level, an improved method which combines the simultaneous iterative reconstruction technology (SIRT) with SVD is proposed. To be specific, we compare the measurement schemes with kernel functions between$ T_2$ and the original theory in MRS, and then provide the forward and inversion formulations. In order to substantiate the effectiveness of this method, we conduct the synthetic experiments for Carr-Purcell-Meiboom-Gill sequence and explain the dataset with the mentioned strategies. As expected, the combined approach possesses a better performance in shallow layer with an error of 1.5% and 0.02 s for water content and$ T_2$ for the contaminated data, respectively. With these advantages, it is expected to realize the adoption of the SVD with SIRT in field applications and further investigate the aquifer characterizations in the future.-
Keywords:
- surface magnetic resonance sounding /
- spin-echo /
- T2 forward and inversion /
- linear spatial inversion
[1] Di Q Y, Wang M Y 2010 Bull. Eng. Geol. Environ. 69 105
Google Scholar
[2] Ronezka M K, Hellman K, Günther T, Wisén R, Dahlin T 2017 Solid Earth 8 671
Google Scholar
[3] Xue G Q, Hou D Y, Qiu W Z 2018 J. Environ. Eng. Geoph. 23 297
Google Scholar
[4] Chen K, Xue G Q, Chen W Y, Zhou N N, Li H 2019 Mine Water Environ. 38 49
Google Scholar
[5] 林君, 段清明, 王应吉 2010 核磁共振找水仪原理与应用 (北京: 科学出版社) 第7—13页
Lin J, Duan Q M, Wang Y J 2010 Theory and Design of Magnetic Resonance Sounding Instrument for Groundwater Detection and its Applications (Beijing: Science Press) pp7−13 (in Chinese)
[6] Weichman P B, Lavely E M, Ritzwoller M H 2000 Phys. Rev. E 62 1290
Google Scholar
[7] Legchenko A, Baltassat J M, Bobachev A, Martin C, Robain H, Vouillamoz J M 2004 Groundwater 42 363
Google Scholar
[8] Mohnke O, Yaramance U 2005 Near Surf. Geophys 3 165
Google Scholar
[9] Hertrich M 2008 Prog. Nucl. Mang. Reson. Spectrosc. 53 227
Google Scholar
[10] Dlugoschet R, Günther T, Müller-Petke M, Yaramanci U 2014 Near Surf. Geophys. 12 231
Google Scholar
[11] Legchenko A, Descloitres M, Vincent C, Guyard H, Garambois S, Chalikakis K, Ezersky M 2011 New J. Phys. 13 025022
Google Scholar
[12] Grombacher D, Liu L, Kass M A, Osterman G, Fiandaca G, Auken E, Larsen J J 2020 J. Appl. Geophys. 172 103869
Google Scholar
[13] Fallahsafari M, Ghanati R, Hafizi M K, Müller-Petke M 2020 J. Appl. Geophys. 175 103985
Google Scholar
[14] Chen Q, Marble A E, Colpitts B G, Balcom B J 2005 J. Magn. Reson. 175 300
Google Scholar
[15] Grunewald E, Knight R 2011 Near Surf. Geophys. 9 169
Google Scholar
[16] Grombacher D, Behroozmand A A, Auken E 2017 Geophysics 82 JM23
Google Scholar
[17] Legchenko A, Voullamoz J M, Roy J 2010 Geophysics 75 L91
Google Scholar
[18] Grunewald E, Walsh D 2013 Geophys. Res. Lett. 40 6346
Google Scholar
[19] Grunewald E, Knight R, Walsh D 2014 Geophysics 79 EN15
Google Scholar
[20] Grombacher D, Fiandaca G, Auken E 2019 Geophys. J. Int. 218 1892
Google Scholar
[21] Dlubac K, Knight R, Song Y Q, Bachman N, Grau B, Cannia J, Williams J 2013 Water Resour. Res. 49 1871
Google Scholar
[22] Walsh D, Tumer P, Grunewald E, Zhang H, Butler J J, Reboulet E, Knobbe S, Christy T, Lane J W, Johnson C D, Munday T, Fitzpatrick A 2013 Ground Water 51 914
Google Scholar
[23] Hahn E L 1950 Phys. Rev. 80 580
Google Scholar
[24] Carr H Y, Purcell E M 1954 Phys. Rev. 94 630
Google Scholar
[25] 刘骏妍 2017 硕士学位论文 (长春: 吉林大学)
Liu J Y 2017 M. S. Thesis (Changchun: Jilin University) (in Chinese)
[26] Walsh D O 2008 J. Appl. Geophys. 66 140
Google Scholar
[27] Weichman P B, Lun D R, Ritzwoller M H, Lavely E M 2002 J Appl. Geophys. 50 129
Google Scholar
[28] Hansen P C 1994 Numer. Algorithms 6 1
Google Scholar
[29] 姚旭刚, 王忠东 2003 测井技术 27 373
Google Scholar
Yao X G, Wang Z D 2003 Well Logging Technology 27 373
Google Scholar
[30] Müller-Petke M, Yaramanci U 2010 Geophysics 75 WA199
Google Scholar
[31] Mohnke O, Yaramanci U 2008 J. Appl. Geophys. 66 73
Google Scholar
[32] Bloom A L 1955 Phys. Rev. 98 1105
Google Scholar
[33] Müller-Petke M, Yaramanci U 2008 J. Appl. Geophys. 66 165
Google Scholar
[34] Dines K A, Lyttle R J 1979 Proc. IEEE 67 1065
Google Scholar
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图 1 地面磁共振CPMG序列激发示意图(为获取包含T2衰减信息的完整SE信号, 发射激发脉冲q, 并间隔时间τ重复发射多个重聚脉冲2q, 重聚脉冲间的间隔为2τ)
Figure 1. Typical time diagram of CPMG apply in surface MRS. Measurement for excitation pulse moment q requires a set of refocusing pulse 2q repeating for a defined 2τ delay after the first τ delay and obtains a complete SE response to determine the T2 decay curve.
图 2 2 A·s激发脉冲下的地面磁共振(a) FID与(b) SE激发横向磁化强度分布剖面(实验配置为100 m方形发射/接收线圈, 单匝; 填充颜色表示激发横向磁化强度相对于总磁化强度比例)
Figure 2. Excitation profile (2 A·s) for (a) FID and (b) SE responses in a surface MRS case with 100 m square transmitting/receiving loop, 1-turn configuration. Color indicates the amplitude ratio of the excited transverse magnetization to the total magnetization.
图 3 单匝100 m方形发射/接收线圈探测配置下的地面磁共振(a) FID与(b) SE一维灵敏度核函数(填充颜色反映在对应于激发脉冲矩, 某一固定深度下存在地下水能够诱发得到的磁共振信号幅度)
Figure 3. Comparison of kernel function with 100 m transmitting/receiving square loop, 1-turn configuration for (a) FID and (b) SE excitation. Color reflects the signal amplitude induced by underground water at a given depth layer corresponding to each pulse moment.
图 4 单匝100 m方形发射/接收线圈探测配置下, 同一含水情况下的地面磁共振响应(a) FID与(b) SE信号(填充颜色反映对应激发脉冲矩, 其FID或SE信号随时间衰减的磁共振信号幅度)
Figure 4. Comparison of forward responses for the same aquifer distribution, with 100 m transmitting/receiving square loop, 1-turn configuration for (a) FID and (b) SE excitation. Color reflects the signal amplitude induced by underground water decays with receiving time corresponding to each pulse moment.
图 5 (a)模型与其在(b)无噪声, (c) 3 nV及(d) 6 nV高斯白噪声情况下的SVD线性空间反演结果
Figure 5. (a) Simulated modeling and its linear spatial inversion results employing SVD with (b) no noise, (c) 3 nV and (d) 6 nV Gaussian noise.
图 6 (a)模型与其在(b) 3 nV和(c) 6 nV高斯白噪声情况下的SVD与SIRT联合线性空间反演结果
Figure 6. (a) Simulated modeling and its linear spatial inversion results employing SVD and SIRT with (b) 3 nV and (c) 6 nV Gaussian noise.
图 7 不同噪声情况下, SIRT线性空间反演迭代次数与拟合误差间的关系
Figure 7. Relationship between iteration and fitting errors for SIRT linear spatial inversion with different noise cases.
图 8 地面磁共振SE信号线性空间反演及含水量-横向弛豫时间分布拟合示意图
Figure 8. Schematic diagram of linear spatial inversion and non-linear fitting of water content and transverse relaxation time for surface MRS spin-echo responses.
图 9 (a)地面磁共振CPMG探测实验模型与(b)正演数据(加入3 nV高斯白噪声), 假设地下存在两个含水层, 分别分布在–5 m至–15 m及–30 m至–50 m.
Figure 9. (a) Simulated modeling and (b) dataset (adding 3 nV Gaussian noise) for CPMG sequence assuming a surface MRS experiment with two aquifers, which distributed from –5 m to –15 m and –30 m to –50 m, respectively.
图 10 加入3 nV高斯白噪声的地面磁共振仿真数据SVD线性空间反演结果 (a) 地下水SE响应随深度及接收时间的变化; (b)与(c)分别为应用15 ms时间窗(单指数)拟合得到的w与T2分布结果
Figure 10. Linear spatial inversion results of SVD method for synthetic surface MRS experiment data with 3 nV Gaussian noise polluted: (a) SE responses of underground water separated as a function of depth z and decayed over time, while the amplitude scale to water content; (b) and (c) are the subsurface w and T2 distribution fitted (mono-exponential) from (a) with a time window of 15 ms.
图 11 加入3 nV高斯白噪声的地面磁共振仿真数据SVD与SIRT线性空间反演结果 (a) 地下水SE响应随深度及接收时间的变化; (b)与(c)分别为应用15 ms时间窗(单指数)拟合得到的w与T2分布结果
Figure 11. Linear spatial inversion results of SVD and SIRT method for synthetic surface MRS experiment data with 3 nV Gaussian noise polluted: (a) SE responses of underground water separated as a function of depth z and decayed over time, while the amplitude scale to water content; (b) and (c) are the subsurface w and T2 distribution fitted (mono-exponential) from (a) with a time window of 15 ms.
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[1] Di Q Y, Wang M Y 2010 Bull. Eng. Geol. Environ. 69 105
Google Scholar
[2] Ronezka M K, Hellman K, Günther T, Wisén R, Dahlin T 2017 Solid Earth 8 671
Google Scholar
[3] Xue G Q, Hou D Y, Qiu W Z 2018 J. Environ. Eng. Geoph. 23 297
Google Scholar
[4] Chen K, Xue G Q, Chen W Y, Zhou N N, Li H 2019 Mine Water Environ. 38 49
Google Scholar
[5] 林君, 段清明, 王应吉 2010 核磁共振找水仪原理与应用 (北京: 科学出版社) 第7—13页
Lin J, Duan Q M, Wang Y J 2010 Theory and Design of Magnetic Resonance Sounding Instrument for Groundwater Detection and its Applications (Beijing: Science Press) pp7−13 (in Chinese)
[6] Weichman P B, Lavely E M, Ritzwoller M H 2000 Phys. Rev. E 62 1290
Google Scholar
[7] Legchenko A, Baltassat J M, Bobachev A, Martin C, Robain H, Vouillamoz J M 2004 Groundwater 42 363
Google Scholar
[8] Mohnke O, Yaramance U 2005 Near Surf. Geophys 3 165
Google Scholar
[9] Hertrich M 2008 Prog. Nucl. Mang. Reson. Spectrosc. 53 227
Google Scholar
[10] Dlugoschet R, Günther T, Müller-Petke M, Yaramanci U 2014 Near Surf. Geophys. 12 231
Google Scholar
[11] Legchenko A, Descloitres M, Vincent C, Guyard H, Garambois S, Chalikakis K, Ezersky M 2011 New J. Phys. 13 025022
Google Scholar
[12] Grombacher D, Liu L, Kass M A, Osterman G, Fiandaca G, Auken E, Larsen J J 2020 J. Appl. Geophys. 172 103869
Google Scholar
[13] Fallahsafari M, Ghanati R, Hafizi M K, Müller-Petke M 2020 J. Appl. Geophys. 175 103985
Google Scholar
[14] Chen Q, Marble A E, Colpitts B G, Balcom B J 2005 J. Magn. Reson. 175 300
Google Scholar
[15] Grunewald E, Knight R 2011 Near Surf. Geophys. 9 169
Google Scholar
[16] Grombacher D, Behroozmand A A, Auken E 2017 Geophysics 82 JM23
Google Scholar
[17] Legchenko A, Voullamoz J M, Roy J 2010 Geophysics 75 L91
Google Scholar
[18] Grunewald E, Walsh D 2013 Geophys. Res. Lett. 40 6346
Google Scholar
[19] Grunewald E, Knight R, Walsh D 2014 Geophysics 79 EN15
Google Scholar
[20] Grombacher D, Fiandaca G, Auken E 2019 Geophys. J. Int. 218 1892
Google Scholar
[21] Dlubac K, Knight R, Song Y Q, Bachman N, Grau B, Cannia J, Williams J 2013 Water Resour. Res. 49 1871
Google Scholar
[22] Walsh D, Tumer P, Grunewald E, Zhang H, Butler J J, Reboulet E, Knobbe S, Christy T, Lane J W, Johnson C D, Munday T, Fitzpatrick A 2013 Ground Water 51 914
Google Scholar
[23] Hahn E L 1950 Phys. Rev. 80 580
Google Scholar
[24] Carr H Y, Purcell E M 1954 Phys. Rev. 94 630
Google Scholar
[25] 刘骏妍 2017 硕士学位论文 (长春: 吉林大学)
Liu J Y 2017 M. S. Thesis (Changchun: Jilin University) (in Chinese)
[26] Walsh D O 2008 J. Appl. Geophys. 66 140
Google Scholar
[27] Weichman P B, Lun D R, Ritzwoller M H, Lavely E M 2002 J Appl. Geophys. 50 129
Google Scholar
[28] Hansen P C 1994 Numer. Algorithms 6 1
Google Scholar
[29] 姚旭刚, 王忠东 2003 测井技术 27 373
Google Scholar
Yao X G, Wang Z D 2003 Well Logging Technology 27 373
Google Scholar
[30] Müller-Petke M, Yaramanci U 2010 Geophysics 75 WA199
Google Scholar
[31] Mohnke O, Yaramanci U 2008 J. Appl. Geophys. 66 73
Google Scholar
[32] Bloom A L 1955 Phys. Rev. 98 1105
Google Scholar
[33] Müller-Petke M, Yaramanci U 2008 J. Appl. Geophys. 66 165
Google Scholar
[34] Dines K A, Lyttle R J 1979 Proc. IEEE 67 1065
Google Scholar
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