地震波形反演的稀疏约束正则化方法
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摘要
本文考虑地震波形反演问题.为了克服传统的Tikhonov正则化方法过度光滑的弊端,引入了非线性稀疏约束正则化方法,并采用对偶方法求解稀疏约束泛函的极小点.基于二维声波方程波形反演问题进行了数值模拟,针对不同模型对稀疏约束正则化方法进行了测试.结果表明,稀疏约束正则化方法对不连续介质模型的介质边缘具有良好的识别能力.
This paper studies the problems of seismic waveform inversion.In order to overcome the oversmoothness of the traditional Tikhonov regularization,we introduce the nonlinear regularization with sparsity constraints,and adopt the dual method to obtain the minimizer of the functional with sparsity constraints.Numerical simulations are carried out based on waveform inversion of the 2-D acoustic wave equation,and lots of tests are conducted for various models.The numerical results show that the regularization method with sparsity constraints has the ability to recover the boundary of discontinuous media.
This paper studies the problems of seismic waveform inversion.In order to overcome the oversmoothness of the traditional Tikhonov regularization,we introduce the nonlinear regularization with sparsity constraints,and adopt the dual method to obtain the minimizer of the functional with sparsity constraints.Numerical simulations are carried out based on waveform inversion of the 2-D acoustic wave equation,and lots of tests are conducted for various models.The numerical results show that the regularization method with sparsity constraints has the ability to recover the boundary of discontinuous media.
引文
[1]Tarantola A.Linearized inversion of seismic reflection data.Geophysical Prosp.,1984,32(6):998-1015.
[2]Berkhout A J.Multidimensional linearized inversion andseismic migration.Geophysics,1984,49(11):1881-1895.
[3]Tarantola A.A strategy for nonlinear elastic inversion ofseismic reflection data.Geophysics,1986,51(10):1983-1903.
[4]Mora P.Nonlinear two-dimensional elastic inversion ofmultioffset seismic data.Geophysics,1987,52(9):1211-1228.
[5]Zhou C X,Cai W Y,Luo Y,et al.Acoustic wave equationtraveltime and waveform inversion of cross hole seismic data.Geophysics,1995,60(3):765-773.
[6]Pratt R G,Shin C,Hicks G J.Gauss-Newton and fullNewton methods in frequency-space seismic waveform inversion.Geophysical Journal International,1998,133(2):341-362.图5算例3的小波系数结果(c)The wavelet coefficient of inversion with p=1.2;(d)The wavelet coefficient of inversion with p=1.1.
[7]Shin C,Min D J.Waveform inversion using a logarithmic wavefield.Geophysics,2006,71(3):R31-R42.
[8]Shin C,Cha Y H.Waveform inversion in the Laplace domain.Geophysical Journal International,2008,173(3):922-931.
[9]Xu T,McMechan G A,Sun R.3-D prestack full-wavefield inversion.Geophysics,1995,60(6):1805-1818.
[10]苑书金.叠前地震反演技术的进展及其在岩性油气藏勘探中的应用.地球物理学进展,2007,22(3):879-886.Yuan S J.Progress of pre-stack inversion and application in exploration of the lithological reservoirs.Progress in Geophysics(in Chinese),2007,22(3):879-886.
[11]Sen M K,Stoffa P L.Nonlinear one-dimensional seismic waveform inversion using simulated annealing.Geophysics,1991,56(10):1624-1638.
[12]Stoffa P L,Sen M K.Nonlinear multi-parameter optimization using genetic algorithms inversion of plane-wave seismograms.Geophysics,1991,56(11):1794-1810.
[13]Bunks C,Saleck F M,Zaleski S,et al.Multiscale seismic waveform inversion.Geophysics,1995,60(5):1457-1473.
[14]傅红笋,韩波.二维波动方程速度的正则化-同伦-测井约束反演.地球物理学报,2005,48(6):1441-1448.Fu H S,Han B.A regularization homotopy method for the inverse problem of2-D wave equation and well log constraint inversion.Chinese J.Geophys.(in Chinese),2005,48(6):1441-1448.
[15]Han B,Fu H S,Liu H.A homotopy method for well-log constraint waveform inversion.Geophysics,2007,72(2):R1-R7.
[16]孟鸿鹰,刘贵忠.小波变换多尺度地震波形反演.地球物理学报,1999,42(2):241-248.Meng H Y,Liu G Z.Multiscale seismic waveform inversion by wavelet transform.Chinese J.Geophys.(in Chinese),1999,42(2):241-248.
[17]陈勇,韩波,肖龙等.多尺度全变分法及其在时移地震中的应用.地球物理学报,2010,53(8):1883-1892.Chen Y,Han B,Xiao L,et al.Multiscale total variation method and its application on time-lapse seismic.Chinese J.Geophys.(in Chinese),2010,53(8):1883-1892.
[18]Donoho D L.Superresolution via sparsity constraints.SIAM J.Math.Anal.,1992,23(5):1309-1331.
[19]Candes E,Romberg J,Tao T.Stable signal recovery from incomplete and inaccurate measurements.Comm.Pure Appl.Math.,2006,59(8):1207-1223.
[20]Loris I,Douma H,Nolet G,et al.Nonlinear regularization techniques for seismic tomography.J.Comput.Physics.,2010,229(3):890-905.
[21]Jafarpour B,Goyal V K,Laughlin D B,et al.Transform-domain sparsity regularization for inverse problems in geosciences.Geophysics,2009,74(5):R69-R83.
[22]Wang Y F.Seismic impedance inversion using l1norm regularization and gradient descent methods.J.Inv.Ill-Posed Problems,2011,18:823-838.
[23]Daubechies I,Defrise M,Mol C D.An iterative thresholding algorithm for linear inverse problems with a sparsity constraint.Comm.Pure Appl.Math.,2004,57(11):1413-1457.
[24]Ramlau R,Teschke G.A projection iteration for nonlinear operator equations with sparsity constraints.Numer.Math.,2006,104(2):177-203.
[25]Kaltenbacher B,Schopfer F,Schuster T.Iterative methods for nonlinear ill-posed problems in Banach spaces:convergence and applications to parameter identification problems.Inverse Problems,2009,25(6):065003.
[2]Berkhout A J.Multidimensional linearized inversion andseismic migration.Geophysics,1984,49(11):1881-1895.
[3]Tarantola A.A strategy for nonlinear elastic inversion ofseismic reflection data.Geophysics,1986,51(10):1983-1903.
[4]Mora P.Nonlinear two-dimensional elastic inversion ofmultioffset seismic data.Geophysics,1987,52(9):1211-1228.
[5]Zhou C X,Cai W Y,Luo Y,et al.Acoustic wave equationtraveltime and waveform inversion of cross hole seismic data.Geophysics,1995,60(3):765-773.
[6]Pratt R G,Shin C,Hicks G J.Gauss-Newton and fullNewton methods in frequency-space seismic waveform inversion.Geophysical Journal International,1998,133(2):341-362.图5算例3的小波系数结果(c)The wavelet coefficient of inversion with p=1.2;(d)The wavelet coefficient of inversion with p=1.1.
[7]Shin C,Min D J.Waveform inversion using a logarithmic wavefield.Geophysics,2006,71(3):R31-R42.
[8]Shin C,Cha Y H.Waveform inversion in the Laplace domain.Geophysical Journal International,2008,173(3):922-931.
[9]Xu T,McMechan G A,Sun R.3-D prestack full-wavefield inversion.Geophysics,1995,60(6):1805-1818.
[10]苑书金.叠前地震反演技术的进展及其在岩性油气藏勘探中的应用.地球物理学进展,2007,22(3):879-886.Yuan S J.Progress of pre-stack inversion and application in exploration of the lithological reservoirs.Progress in Geophysics(in Chinese),2007,22(3):879-886.
[11]Sen M K,Stoffa P L.Nonlinear one-dimensional seismic waveform inversion using simulated annealing.Geophysics,1991,56(10):1624-1638.
[12]Stoffa P L,Sen M K.Nonlinear multi-parameter optimization using genetic algorithms inversion of plane-wave seismograms.Geophysics,1991,56(11):1794-1810.
[13]Bunks C,Saleck F M,Zaleski S,et al.Multiscale seismic waveform inversion.Geophysics,1995,60(5):1457-1473.
[14]傅红笋,韩波.二维波动方程速度的正则化-同伦-测井约束反演.地球物理学报,2005,48(6):1441-1448.Fu H S,Han B.A regularization homotopy method for the inverse problem of2-D wave equation and well log constraint inversion.Chinese J.Geophys.(in Chinese),2005,48(6):1441-1448.
[15]Han B,Fu H S,Liu H.A homotopy method for well-log constraint waveform inversion.Geophysics,2007,72(2):R1-R7.
[16]孟鸿鹰,刘贵忠.小波变换多尺度地震波形反演.地球物理学报,1999,42(2):241-248.Meng H Y,Liu G Z.Multiscale seismic waveform inversion by wavelet transform.Chinese J.Geophys.(in Chinese),1999,42(2):241-248.
[17]陈勇,韩波,肖龙等.多尺度全变分法及其在时移地震中的应用.地球物理学报,2010,53(8):1883-1892.Chen Y,Han B,Xiao L,et al.Multiscale total variation method and its application on time-lapse seismic.Chinese J.Geophys.(in Chinese),2010,53(8):1883-1892.
[18]Donoho D L.Superresolution via sparsity constraints.SIAM J.Math.Anal.,1992,23(5):1309-1331.
[19]Candes E,Romberg J,Tao T.Stable signal recovery from incomplete and inaccurate measurements.Comm.Pure Appl.Math.,2006,59(8):1207-1223.
[20]Loris I,Douma H,Nolet G,et al.Nonlinear regularization techniques for seismic tomography.J.Comput.Physics.,2010,229(3):890-905.
[21]Jafarpour B,Goyal V K,Laughlin D B,et al.Transform-domain sparsity regularization for inverse problems in geosciences.Geophysics,2009,74(5):R69-R83.
[22]Wang Y F.Seismic impedance inversion using l1norm regularization and gradient descent methods.J.Inv.Ill-Posed Problems,2011,18:823-838.
[23]Daubechies I,Defrise M,Mol C D.An iterative thresholding algorithm for linear inverse problems with a sparsity constraint.Comm.Pure Appl.Math.,2004,57(11):1413-1457.
[24]Ramlau R,Teschke G.A projection iteration for nonlinear operator equations with sparsity constraints.Numer.Math.,2006,104(2):177-203.
[25]Kaltenbacher B,Schopfer F,Schuster T.Iterative methods for nonlinear ill-posed problems in Banach spaces:convergence and applications to parameter identification problems.Inverse Problems,2009,25(6):065003.
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