基于贝叶斯一稀疏约束正则化方法的地震波形反演
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摘要
将稀疏约束正则化方法应用于地震波形反演问题.为了减弱对稀疏约束项的光滑性要求,引入贝叶斯推断,产生一组收敛于后验分布的采样点.通过数值算例记录了采样点的条件期望、方差、置信区间等具有统计意义的结果.数值结果表明,在没有光滑性的要求下,稀疏约束正则化方法对孔洞模型和分层模型中的介质边缘有良好的识别能力.特别地,当减少观测数据时,稀疏约束正则化方法仍能获得较好的反演结果.
The regularization method is applied with sparsity constraints to seismic waveform inversion in this paper.To weaken the smoothness requirement of the sparsity constraints,the Bayesian inference is introduced and a series of samplings which satisfies the posterior distribution are generated.In numerical examples,statistically significant results of samplings such as conditional expectation,variance and confidence interval are recorded.Numerical results are presented to illustrate that,without requirement of smoothness,the regularization method with sparsity constraints has a good ability to identify the edge of the media with cavity and layered models.Especially,when the observation data are reduced,the regularization method with sparsity constraints can still provide reasonable inversion results.
The regularization method is applied with sparsity constraints to seismic waveform inversion in this paper.To weaken the smoothness requirement of the sparsity constraints,the Bayesian inference is introduced and a series of samplings which satisfies the posterior distribution are generated.In numerical examples,statistically significant results of samplings such as conditional expectation,variance and confidence interval are recorded.Numerical results are presented to illustrate that,without requirement of smoothness,the regularization method with sparsity constraints has a good ability to identify the edge of the media with cavity and layered models.Especially,when the observation data are reduced,the regularization method with sparsity constraints can still provide reasonable inversion results.
引文
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[10]Jin B,Khan T,Maass P.A reconstruction algorithm for electrical impedance tomography based on sparsity regularization[J].Int.J.Numer.Meth.Engng.,2012,89(3):337-353.
[11]Cao J J,Wang Y F,Zhao J T,Yang C C.A review on restoration of seismic wavefields based on regularization and compressive sensing[J].Inverse Problems in Science and Engineering, 2011,19(5):679-704.
[12]Wang Y F.Seismic impedance inversion using Zi-norm regularization and gradient descent methods[J].Journal of Inverse and Ill-Posed Problems,2011,18(7):823-838.
[13]K(u|¨)gler P,Gaubitzer E,Miiller S.Parameter identication for chemical reaction systems using sparsity enforcing regularization:a case study for the Chlorite-Iodide reaction[J].The Journal of Physical Chemistry A,2009,113:2775-2785.
[14]Kaipio J,Somersalo E.Statistical and Computational Inverse Problems[M].New York:Springer-Verlag, 2004.
[15]Lassas M,Siltanen S.Can one use total variation prior for edge-preserving Bayesian inversion [J].Inverse Problems,2004,20(5):1537-1563.
[16]Stuart A M.Inverse problems:a Bayesian perspective[J],Acta Numerica,2010,19:451-559.
[17]Kaipio J P,Kolehmainen V,Somersalo E,Vauhkonen M.Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography[J].Inverse Problems,2000,16(5): 1487-1522.
[2]刘继军.不适定问题的正则化方法及应用[M].北京:科学出版社,2005.
[3]Ramlau R.Regularization properties of Tikhonov regularization with sparsity constraints[J]. Electron.Trans.Numer.Anal.,2008,30:54-74.
[4]Acar R,Vogel C R.Analysis of bounded variation penalty methods for ill-posed problems[J]. Inverse Problems,1994,10(6):1217-1229.
[5]陈勇,韩波,肖龙,陈小宏.多尺度全变分法及其在时移地震中的应用[J].地球物理学报,2010, 53(8):1883-1892.
[6]Candes E,Romberg J,Tao T.Stable signal recovery from incomplete and inaccurate measurements [J].Pure Appl.Math.,2006,59:1207-1223.
[7]Donoho D L.Compressed sensing[J].IEEE Transactions on Information Theory,2006,52(4): 1289-1306.
[8]Loris I,Nolet G,Daubechies I,Dahlen F A.Tomographic inversion using Zi-norm regularization of wavelet coefficients[J].Geophysical Journal International,2007,1:359-370.
[9]Loris I,Douma H,Nolet G,Daubechies I,Regone C.Nonlinear regularization techniques for seismic tomography[J].Comput.Physics,2010,229:890-905.
[10]Jin B,Khan T,Maass P.A reconstruction algorithm for electrical impedance tomography based on sparsity regularization[J].Int.J.Numer.Meth.Engng.,2012,89(3):337-353.
[11]Cao J J,Wang Y F,Zhao J T,Yang C C.A review on restoration of seismic wavefields based on regularization and compressive sensing[J].Inverse Problems in Science and Engineering, 2011,19(5):679-704.
[12]Wang Y F.Seismic impedance inversion using Zi-norm regularization and gradient descent methods[J].Journal of Inverse and Ill-Posed Problems,2011,18(7):823-838.
[13]K(u|¨)gler P,Gaubitzer E,Miiller S.Parameter identication for chemical reaction systems using sparsity enforcing regularization:a case study for the Chlorite-Iodide reaction[J].The Journal of Physical Chemistry A,2009,113:2775-2785.
[14]Kaipio J,Somersalo E.Statistical and Computational Inverse Problems[M].New York:Springer-Verlag, 2004.
[15]Lassas M,Siltanen S.Can one use total variation prior for edge-preserving Bayesian inversion [J].Inverse Problems,2004,20(5):1537-1563.
[16]Stuart A M.Inverse problems:a Bayesian perspective[J],Acta Numerica,2010,19:451-559.
[17]Kaipio J P,Kolehmainen V,Somersalo E,Vauhkonen M.Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography[J].Inverse Problems,2000,16(5): 1487-1522.
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