基于L-BFGS反演算法的ΔT精确计算磁异常分量T_(ap)方法
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摘要
磁法勘探理论中,将ΔT磁异常看作磁异常矢量Ta在地磁场方向的分量T_(ap),是ΔT异常处理与解释的物理基础,然而这种近似存在误差,理论计算及实验已经证明这种近似所产生的误差将随着Ta异常强度的增大而迅速增加。当磁异常Ta远小于地磁场T0时,误差影响小,可忽略,在强磁异常情况下,误差大,ΔT异常的处理解释精度会受到很大的影响。对于高精度磁法勘探而言,必须将ΔT转换成磁异常分量T_(ap)进行处理解释。笔者提出了基于有限储存BFGS(L-BFGS)反演算法的ΔT精确计算磁异常分量方法,首先推导了T_(ap)计算ΔT的正演公式,利用ΔT与T_(ap)的差值构建反演T_(ap)的目标函数,采用L-BFGS算法由ΔT解算T_(ap)。模型实验表明该方法计算得到的T_(ap)十分接近理论值,即可将误差降低两个数量级。在存在噪声与背景场情况下该方法也都能得到很好的结果。将本方法应用于福建阳山铁矿ΔT磁测资料的处理,得到了与实际更加符合的处理解释结果。
In the magnetic exploration theory,total-field anomaly ΔT is regarded as the component T_(ap)of the magnetic anomaly vector Taon the main field( T0) direction and thus constitutes the theoretical basis. However,there is an error in this approximation. Theoretical calculations and experiments have proved that this approximation error will increase rapidly as the Taincreases. When the magnetic anomaly Tais much smaller than T0,the influence of the error is small and negligible. In the case of a strong magnetic anomaly,the error is large,and the processing interpretation accuracy of the ΔT anomaly is greatly affected. For high-precision magnetic exploration,ΔT must be converted to a magnetic anomaly component T_(ap) for processing and interpretation. In this paper,the method of accurately calculating the magnetic anomaly component using T_(ap)based on the Limited-memory Broyden-Fletcher-Goldfarb-Shanno( L-BFGS) algorithm is proposed. Firstly,the authors derived the forward formula for ΔT from T_(ap),and then constructed the objective function of T_(ap) inversion by the difference function between ΔT and T_(ap). L-BFGS algorithm was used to solve the T_(ap) from ΔT. Model experiments show that the T_(ap) calculated by this method is very close to the real value,which can reduce the error by two orders of magnitude. This method also yields good results in the presence of noise and background fields. The method was applied to the processing of ΔT magnetic survey data of the Yangshan iron mine in Fujian Province,and the results of processing and interpretation which are more consistent with the actual results were obtained.
In the magnetic exploration theory,total-field anomaly ΔT is regarded as the component T_(ap)of the magnetic anomaly vector Taon the main field( T0) direction and thus constitutes the theoretical basis. However,there is an error in this approximation. Theoretical calculations and experiments have proved that this approximation error will increase rapidly as the Taincreases. When the magnetic anomaly Tais much smaller than T0,the influence of the error is small and negligible. In the case of a strong magnetic anomaly,the error is large,and the processing interpretation accuracy of the ΔT anomaly is greatly affected. For high-precision magnetic exploration,ΔT must be converted to a magnetic anomaly component T_(ap) for processing and interpretation. In this paper,the method of accurately calculating the magnetic anomaly component using T_(ap)based on the Limited-memory Broyden-Fletcher-Goldfarb-Shanno( L-BFGS) algorithm is proposed. Firstly,the authors derived the forward formula for ΔT from T_(ap),and then constructed the objective function of T_(ap) inversion by the difference function between ΔT and T_(ap). L-BFGS algorithm was used to solve the T_(ap) from ΔT. Model experiments show that the T_(ap) calculated by this method is very close to the real value,which can reduce the error by two orders of magnitude. This method also yields good results in the presence of noise and background fields. The method was applied to the processing of ΔT magnetic survey data of the Yangshan iron mine in Fujian Province,and the results of processing and interpretation which are more consistent with the actual results were obtained.
引文
[1] Parasnis D S. Principles of Applied Geophysics[M]. Berlin:Springer Netherlands,1979.
[2]管志宁.我国磁法勘探的研究与进展[J].地球物理学报,1997,40(1):299-307.Guan Z N. Research and development of magnetic law exploration in China[J]. Chinese Journal of Geophysics,1997,40(1):299-307.
[3] Blakely R J. Potential theory in gravity and magnetic applications[M]. Cambridge:Cambridge University Press,1995.
[4]管志宁.地磁场与磁力勘探[M].北京:地质出版社,2005.Guan Z N. Geomagnetic field and magnetic exploration[M]. Beijing:Geological Press,2005.
[5]刘天佑.磁法勘探[M].北京:地质出版社,2013.Liu T Y. Magnetic exploration[M]. Beijing:Geological Press,2013.
[6] Hinze P W J,Von Frese R R B,Saad A H. Gravity and magnetic exploration:principles,practices,and applications[M]. Cambridge:Cambridge University Press,2013.
[7]塔费耶夫.金属矿床的地球物理勘探文选[M].北京:地质出版社. 1954.Tafeev. A geophysical exploration of metal deposits[M]. Beijing:Geological Publishing House. 1954.
[8]袁晓雨,姚长利,郑元满,等.强磁性体ΔT异常计算的误差分析研究[J].地球物理学报,2015,58(12):4756-4765.Yuan X Y,Yao C L,Zheng Y M,et al. Error analysis ofΔT anomaly calculation of ferromagnetic body[J]. Chinese Journal of Geophysics,2015,58(12):4756-4765.
[9]袁晓雨.强磁异常ΔT的计算误差及高精度处理转换分析研究[D].中国地质大学(北京),2016.Yuan X Y. Research on calculation error and high-precision processing conversion analysis of strong magnetic anomalyΔT[D].China University of Geosciences(Beijing),2016.
[10] Baranov W A. New method for interpretation of aeromagnetic maps:pseudo-gravimetric anomalies[J]. Geophysics,1957,22(2):359-383.
[11] Bhattacharyya B K. Two-dimensional harmonic analysis as a tool for magnetic interpretation[J]. Geophysics,1965,30(5):829-857.
[12]袁亚湘,孙文瑜.最优化理论与方法[M].北京:科学出版社,1997.Yuan Y X,Sun W Y. Optimization theory and method[M]. Beijing:Science Press,1997.
[13] Sun W,Yuan Y X. Optimization theory and methods[M]. New York:Springer US,2006:175-200.
[14]姚姚.地球物理反演[M].北京:中国地质大学出版社,2002.Yao Y. Geophysical inversion[M]. Beijing:China University of Geosciences Press,2002.
[15]王家映.地球物理反演理论[M].北京:高等教育出版社,2002.Wang J Y,Geophysical inversion theory[M]. Beijing:Higher Education Press,2002.
[16] Avdeeva A,Avdeev D. A limited-memory quasi-Newton inversion for 1D magnetotellurics[J]. Geophysics,2006,71(5):G191-G196.
[17]张生强,刘春成,韩立国,等.基于L-BFGS算法和同时激发震源的频率多尺度全波形反演[J].吉林大学学报:地球科学版,2013,43(3):1004.Zhang S Q,Liu C C,Han L G,et al. Frequency multi-scale full waveform inversion based on L-BFGS algorithm and simultaneous excitation source[J]. Journal of Jilin University:Earth Science Edition,2013,43(3):1004.
[18] Broyden C G. The convergence of a class of double-rank minimization algorithms[J]. Journal of the Institute of Mathematics and Its Applications,1970(6):76-90.
[19] Fletcher R. A new approach to variable metric algorithms[J].Computer Journal,1970,13(3):317-322.
[20] Goldfarb D. A family of variable metric updates derived by variational means[J]. Mathematics of Computation,1970,24(109):23-26.
[21] Shanno D F. Conditioning of quasi-Newton methods for function minimization[J]. Mathematics of Computation,1970,24(111):647-656.
[22] Zheng Y K,Wang Y,Chang X. Wave-equation traveltime inversion:comparison of three numerical optimization methods[J].Computers&Geosciences,2013,60(10):88-97.
[23] Fabien-Ouellet G,Gloaguen E,Giroux B. A stochastic L-BFGS approach for full-waveform inversion[J]. Seg Technical Program Expanded,2017:1622-1627.
[24] Dong C,Liu J N. On the limited memory BFGS method for large scale optimization[J]. Mathematicial Programming,1989,45(1-3):503-528.
[25] Nocedal J. Upgrading quasi-newton matrices with limited storage[J]. Mathematics of Computation,1980,35(151):773-782.
[26] Nocedal J,Wright S J. Numerical optimization[M]. New York:Springer,2006.
[27] Pan W,Innanen K A,Liao W. Accelerating hessian-free gaussnewton full-waveform inversion via L-BFGS preconditioned conjugate-gradient algorithm[J]. Geophysics,2017,82(2):R49-R64.
[2]管志宁.我国磁法勘探的研究与进展[J].地球物理学报,1997,40(1):299-307.Guan Z N. Research and development of magnetic law exploration in China[J]. Chinese Journal of Geophysics,1997,40(1):299-307.
[3] Blakely R J. Potential theory in gravity and magnetic applications[M]. Cambridge:Cambridge University Press,1995.
[4]管志宁.地磁场与磁力勘探[M].北京:地质出版社,2005.Guan Z N. Geomagnetic field and magnetic exploration[M]. Beijing:Geological Press,2005.
[5]刘天佑.磁法勘探[M].北京:地质出版社,2013.Liu T Y. Magnetic exploration[M]. Beijing:Geological Press,2013.
[6] Hinze P W J,Von Frese R R B,Saad A H. Gravity and magnetic exploration:principles,practices,and applications[M]. Cambridge:Cambridge University Press,2013.
[7]塔费耶夫.金属矿床的地球物理勘探文选[M].北京:地质出版社. 1954.Tafeev. A geophysical exploration of metal deposits[M]. Beijing:Geological Publishing House. 1954.
[8]袁晓雨,姚长利,郑元满,等.强磁性体ΔT异常计算的误差分析研究[J].地球物理学报,2015,58(12):4756-4765.Yuan X Y,Yao C L,Zheng Y M,et al. Error analysis ofΔT anomaly calculation of ferromagnetic body[J]. Chinese Journal of Geophysics,2015,58(12):4756-4765.
[9]袁晓雨.强磁异常ΔT的计算误差及高精度处理转换分析研究[D].中国地质大学(北京),2016.Yuan X Y. Research on calculation error and high-precision processing conversion analysis of strong magnetic anomalyΔT[D].China University of Geosciences(Beijing),2016.
[10] Baranov W A. New method for interpretation of aeromagnetic maps:pseudo-gravimetric anomalies[J]. Geophysics,1957,22(2):359-383.
[11] Bhattacharyya B K. Two-dimensional harmonic analysis as a tool for magnetic interpretation[J]. Geophysics,1965,30(5):829-857.
[12]袁亚湘,孙文瑜.最优化理论与方法[M].北京:科学出版社,1997.Yuan Y X,Sun W Y. Optimization theory and method[M]. Beijing:Science Press,1997.
[13] Sun W,Yuan Y X. Optimization theory and methods[M]. New York:Springer US,2006:175-200.
[14]姚姚.地球物理反演[M].北京:中国地质大学出版社,2002.Yao Y. Geophysical inversion[M]. Beijing:China University of Geosciences Press,2002.
[15]王家映.地球物理反演理论[M].北京:高等教育出版社,2002.Wang J Y,Geophysical inversion theory[M]. Beijing:Higher Education Press,2002.
[16] Avdeeva A,Avdeev D. A limited-memory quasi-Newton inversion for 1D magnetotellurics[J]. Geophysics,2006,71(5):G191-G196.
[17]张生强,刘春成,韩立国,等.基于L-BFGS算法和同时激发震源的频率多尺度全波形反演[J].吉林大学学报:地球科学版,2013,43(3):1004.Zhang S Q,Liu C C,Han L G,et al. Frequency multi-scale full waveform inversion based on L-BFGS algorithm and simultaneous excitation source[J]. Journal of Jilin University:Earth Science Edition,2013,43(3):1004.
[18] Broyden C G. The convergence of a class of double-rank minimization algorithms[J]. Journal of the Institute of Mathematics and Its Applications,1970(6):76-90.
[19] Fletcher R. A new approach to variable metric algorithms[J].Computer Journal,1970,13(3):317-322.
[20] Goldfarb D. A family of variable metric updates derived by variational means[J]. Mathematics of Computation,1970,24(109):23-26.
[21] Shanno D F. Conditioning of quasi-Newton methods for function minimization[J]. Mathematics of Computation,1970,24(111):647-656.
[22] Zheng Y K,Wang Y,Chang X. Wave-equation traveltime inversion:comparison of three numerical optimization methods[J].Computers&Geosciences,2013,60(10):88-97.
[23] Fabien-Ouellet G,Gloaguen E,Giroux B. A stochastic L-BFGS approach for full-waveform inversion[J]. Seg Technical Program Expanded,2017:1622-1627.
[24] Dong C,Liu J N. On the limited memory BFGS method for large scale optimization[J]. Mathematicial Programming,1989,45(1-3):503-528.
[25] Nocedal J. Upgrading quasi-newton matrices with limited storage[J]. Mathematics of Computation,1980,35(151):773-782.
[26] Nocedal J,Wright S J. Numerical optimization[M]. New York:Springer,2006.
[27] Pan W,Innanen K A,Liao W. Accelerating hessian-free gaussnewton full-waveform inversion via L-BFGS preconditioned conjugate-gradient algorithm[J]. Geophysics,2017,82(2):R49-R64.