一阶Rytov近似有限频走时层析
|
摘要
传统的波动方程走时核函数(或走时Fréchet导数)多基于互相关时差测量方式及地震波场的一阶Born近似导出,其成立条件非常苛刻.然而,地震波走时与大尺度的速度结构具有良好的线性关系,对于小角度的前向散射波场,Rytov近似优于Born近似.因此,本文基于Rytov近似和互相关时差测量方式,导出了基于Rytov近似的有限频走时敏感度核函数的两种等价形式:频率积分和时间积分表达式.在此基础之上,本文提出了一种隐式矩阵向量乘方法,可以直接计算Hessian矩阵或者核函数与向量的乘积,而无需显式计算和存储核函数及Hessian矩阵.基于隐式矩阵向量乘方法,本文利用共轭梯度法求解法方程实现了一种高效的Gauss-Newton反演算法求解走时层析反问题.与传统的敏感度核函数反演方法相比,本文方法在每次迭代过程中,无需显式计算和存储核函数,极大降低了存储需求.与基于Born近似的伴随状态方法走时层析相比,本文方法具有准二阶的收敛速度,且适用范围更广.数值试验证明了本文方法的有效性.
The conventional wave-equation traveltime sensitivity kernel(TSK)or traveltime Fréchet derivative is derived from the Born approximation and cross-correlation measurement,which has a very narrow valid condition.In fact,the seismic traveltime has a more linear relationship with the large-scale velocity structure.For small-angle forward scattered wavefield,Rytov approximation is proved to be superior to Born approximation.Based on the Rytov approximation and cross-correlation measurement,a new wave-equation traveltime sensitivity kernel is derived.Meanwhile,an implicit matrix-vector product method is proposed,which can directly calculate the product of a matrix(TSK)and a model-space vector as well as the product of a matrix transpose and a data-space vector,eliminating the need of calculating TSK explicitly.Based on the proposed implicit matrix-vector product method,traveltime tomography using the Gauss-Newton inversion algorithm is implemented efficiently by solving the normal equation iteratively using a conjugate gradient method.Compared with the conventional TSK method,the proposed inversion strategy is free of TSK calculation and storage,making it more practical for large-scale problem.Compared with the adjoint traveltime tomography,the proposed method hasaquasi-second-order convergent rate and a broader valid condition.Numerical examples demonstrate the effectiveness of the proposed method.
The conventional wave-equation traveltime sensitivity kernel(TSK)or traveltime Fréchet derivative is derived from the Born approximation and cross-correlation measurement,which has a very narrow valid condition.In fact,the seismic traveltime has a more linear relationship with the large-scale velocity structure.For small-angle forward scattered wavefield,Rytov approximation is proved to be superior to Born approximation.Based on the Rytov approximation and cross-correlation measurement,a new wave-equation traveltime sensitivity kernel is derived.Meanwhile,an implicit matrix-vector product method is proposed,which can directly calculate the product of a matrix(TSK)and a model-space vector as well as the product of a matrix transpose and a data-space vector,eliminating the need of calculating TSK explicitly.Based on the proposed implicit matrix-vector product method,traveltime tomography using the Gauss-Newton inversion algorithm is implemented efficiently by solving the normal equation iteratively using a conjugate gradient method.Compared with the conventional TSK method,the proposed inversion strategy is free of TSK calculation and storage,making it more practical for large-scale problem.Compared with the adjoint traveltime tomography,the proposed method hasaquasi-second-order convergent rate and a broader valid condition.Numerical examples demonstrate the effectiveness of the proposed method.
引文
Beydoun W B,Tarantola A.1988.First Born and Rytov approximations:modeling and inversion conditions in a canonical example.The Journal of the Acoustical Society of America,83(3):1045-1055.
Brown W P.1967.Validity of the Rytov approximation.Journal of the Optical Society of America,57(12):1539-1542.
Chen P,Jordan T,Zhao L.2007.Full three-dimensional tomography:Acomparison between the scattering-integral and adjoint-wavefield methods.Geophysical Journal International,170(1):175-181.
Cheng Q S.2003.Digital Signal Processing(in Chinese).Beijing:Peking University Press.
Dahlen F A,Hung S H,Nolet G.2000.Fréchet kernels for finitefrequency traveltimes-I.Theory.Geophysical Journal International,141(1):157-174.
Das R,Rai S S.2016.Seismic interferometry and ambient noise tomography:theoretical background and application in south India.Journal of Physics:Conference Series,759(1):012006,doi:10.1088/1742-6596/759/1/012006.
DeWolf D A.1967.Validity of Rytov′s approximation.Journal of the Optical Society of America,57(8):1057-1058.
Feng B,Wang H Z.2015.Data-domain wave equation reflection traveltime tomography.Journal of Earth Science,26(4):487-494.
Hung S H,Dahlen F A,Nolet G.2000.Fréchet kernels for finitefrequency traveltime-II.Examples.Geophysical Journal International,141(1):175-203.
Hung S H,Shen Y,Chiao L Y.2004.Imaging seismic velocity structure beneath the Iceland hot spot:A finite frequency approach.Journal of Geophysical Research:Solid Earth,109(B8):B08305,doi:10.1029/2003JB002889.
Jocker J,Spetzler J,Smeulders D,et al.2006.Validation of firstorder diffraction theory for the traveltimes and amplitudes of propagating waves.Geophysics,71(6):T167-T177.
Keller J B.1969.Accuracy and validity of the Born and Rytov approximations.Journal of the Optical Society of America,59(8):1003-1004.
Li X X,Li Q C.2016.Near-surface ambient noise tomography in the Baogutu copper deposit area.Journal of Geophysics and Engineering,13(6):868-874.
Liu Q Y,Tromp J.2006.Finite-frequency kernels based on adjoint methods.Bulletin of the Seismological Society of America,96(6):2383-2397.
Liu Q Y,Tromp J.2008.Finite-frequency sensitivity kernels for global seismic wave propagation based upon adjoint methods.Geophysical Journal International,2008,174(1):265-286.
Liu Y Z,Dong L G,Li P M,et al.2009.Fresnel volume tomography based on the first arrival of the seismic wave.Chinese Journal of Geophysics(in Chinese),52(9):2310-2320,doi:10.3969/j.issn.0001-5733.2009.09.015.
Liu Y Z,Wu Z,Geng Z C.2019.First-arrival phase-traveltime tomography.Chinese Journal of Geophysics(in Chinese),62(2):619-633,doi:10.6038/cjg2019L0524.
Luo Y,Schuster G T.1991.Wave-equation traveltime inversion.Geophysics,56(5):645-653.
Marquering H,Dahlen F A,Nolet G.1999.Three-dimensional sensitivity kernels for finite-frequency traveltimes:The bananadoughnut paradox.Geophysical Journal International,137(3):805-815.
Mercerat E D,Nolet G.2013.On the linearity of cross-correlation delay times in finite-frequency tomography.Geophysical Journal International,192(2):681-687.
Montelli R,Nolet G,Dahlen F A,et al.2004.Finite-frequency tomography reveals a variety of plumes in the mantle.Science,303(5656):338-343.
Nocedal J,Wright S J.1999.Numerical Optimization.New York:Springer.
Rao Y,Wang Y,Zhang Z D,et al.2016.Reflection seismic waveform tomography of physical modelling data.Journal of Geophysics and Engineering,13(2):146-151.
Snieder R,Lomax A.1996.Wavefield smoothing and the effect of rough velocity perturbations on arrival times and amplitudes.Geophysical Journal International,125(3):796-812.
Spetzler J,Snieder R.2001.The effect of small scale heterogeneity on the arrival time of waves.Geophysical Journal International,145(3):786-796.
Spetzler J,Snieder R.2004.The Fresnel volume and transmitted waves.Geophysics,69(3):653-663.
Tape C,Liu Q Y,Tromp J.2007.Finite-frequency tomography using adjoint methods-methodology and examples using membrane surface waves.Geophysical Journal International,168(3):1105-1129.
Tape C,Liu Q Y,Maggi A,et al.2009.Adjoint tomography of the Southern California crust.Science,325(5943):988-992.
Tarantola A.1984.Linearized inversion of seismic reflection data.Geophysical Prospecting,32(6):998-1015.
Tromp J,Tape C,Liu Q Y.2005.Seismic tomography,adjoint methods,time reversal and banana-doughnut kernels.Geophysical Journal International,160(1):195-216.
Woodward M J.1992.Wave-equation tomography.Geophysics,57(1):15-26.
Wu R S.2003.Wave propagation,scattering and imaging using dual-domain one-way and one-return propagators.Pure and Applied Geophysics,160(3-4):509-539.
Xie X B,Yang H.2008.The finite-frequency sensitivity kernel for migration residual moveout and its applications in migration velocity analysis.Geophysics,73(6):S241-S249.
Xu W J,Xie X B,Geng J H.2015.Validity of the Rytov approximation in the form of finite-frequency sensitivity kernels.Pure and Applied Geophysics,172(6):1409-1427.
Yuan Y O,Simons F J,Tromp J.2016.Double-difference adjoint seismic tomography.Geophysical Journal International,206(3):1599-1618.
Zhang J X,Yang Q,Meng X H,et al.2016.Reflection tomography based on a velocity model with implicitly described structure information.Journal of Geophysics and Engineering,13(5):721-732.
Zhao L,Jordan T H,Chapman C H.2000.Three-dimensional Fréchet differential kernels for seismic delay times.Geophysical Journal International,141(3):558-576.
Zhao L,Jordan T H,Olsen K B,et al.2005.Fréchet kernels for imaging regional Earth structure based on three-dimensional reference models.Bulletin of the Seismological Society of America,95(6):2066-2080.
程乾生.2003.数字信号处理.2版.北京:北京大学出版社.
刘玉柱,董良国,李培明等.2009.初至波菲涅尔体地震层析成像.地球物理学报,52(9):2310-2320,doi:10.3969/j.issn.0001-5733.2009.09.015.
刘玉柱,伍正,耿志成.2019.初至波相位走时层析.地球物理学报,62(2):619-633,doi:10.6038/cjg2019L0524.
Brown W P.1967.Validity of the Rytov approximation.Journal of the Optical Society of America,57(12):1539-1542.
Chen P,Jordan T,Zhao L.2007.Full three-dimensional tomography:Acomparison between the scattering-integral and adjoint-wavefield methods.Geophysical Journal International,170(1):175-181.
Cheng Q S.2003.Digital Signal Processing(in Chinese).Beijing:Peking University Press.
Dahlen F A,Hung S H,Nolet G.2000.Fréchet kernels for finitefrequency traveltimes-I.Theory.Geophysical Journal International,141(1):157-174.
Das R,Rai S S.2016.Seismic interferometry and ambient noise tomography:theoretical background and application in south India.Journal of Physics:Conference Series,759(1):012006,doi:10.1088/1742-6596/759/1/012006.
DeWolf D A.1967.Validity of Rytov′s approximation.Journal of the Optical Society of America,57(8):1057-1058.
Feng B,Wang H Z.2015.Data-domain wave equation reflection traveltime tomography.Journal of Earth Science,26(4):487-494.
Hung S H,Dahlen F A,Nolet G.2000.Fréchet kernels for finitefrequency traveltime-II.Examples.Geophysical Journal International,141(1):175-203.
Hung S H,Shen Y,Chiao L Y.2004.Imaging seismic velocity structure beneath the Iceland hot spot:A finite frequency approach.Journal of Geophysical Research:Solid Earth,109(B8):B08305,doi:10.1029/2003JB002889.
Jocker J,Spetzler J,Smeulders D,et al.2006.Validation of firstorder diffraction theory for the traveltimes and amplitudes of propagating waves.Geophysics,71(6):T167-T177.
Keller J B.1969.Accuracy and validity of the Born and Rytov approximations.Journal of the Optical Society of America,59(8):1003-1004.
Li X X,Li Q C.2016.Near-surface ambient noise tomography in the Baogutu copper deposit area.Journal of Geophysics and Engineering,13(6):868-874.
Liu Q Y,Tromp J.2006.Finite-frequency kernels based on adjoint methods.Bulletin of the Seismological Society of America,96(6):2383-2397.
Liu Q Y,Tromp J.2008.Finite-frequency sensitivity kernels for global seismic wave propagation based upon adjoint methods.Geophysical Journal International,2008,174(1):265-286.
Liu Y Z,Dong L G,Li P M,et al.2009.Fresnel volume tomography based on the first arrival of the seismic wave.Chinese Journal of Geophysics(in Chinese),52(9):2310-2320,doi:10.3969/j.issn.0001-5733.2009.09.015.
Liu Y Z,Wu Z,Geng Z C.2019.First-arrival phase-traveltime tomography.Chinese Journal of Geophysics(in Chinese),62(2):619-633,doi:10.6038/cjg2019L0524.
Luo Y,Schuster G T.1991.Wave-equation traveltime inversion.Geophysics,56(5):645-653.
Marquering H,Dahlen F A,Nolet G.1999.Three-dimensional sensitivity kernels for finite-frequency traveltimes:The bananadoughnut paradox.Geophysical Journal International,137(3):805-815.
Mercerat E D,Nolet G.2013.On the linearity of cross-correlation delay times in finite-frequency tomography.Geophysical Journal International,192(2):681-687.
Montelli R,Nolet G,Dahlen F A,et al.2004.Finite-frequency tomography reveals a variety of plumes in the mantle.Science,303(5656):338-343.
Nocedal J,Wright S J.1999.Numerical Optimization.New York:Springer.
Rao Y,Wang Y,Zhang Z D,et al.2016.Reflection seismic waveform tomography of physical modelling data.Journal of Geophysics and Engineering,13(2):146-151.
Snieder R,Lomax A.1996.Wavefield smoothing and the effect of rough velocity perturbations on arrival times and amplitudes.Geophysical Journal International,125(3):796-812.
Spetzler J,Snieder R.2001.The effect of small scale heterogeneity on the arrival time of waves.Geophysical Journal International,145(3):786-796.
Spetzler J,Snieder R.2004.The Fresnel volume and transmitted waves.Geophysics,69(3):653-663.
Tape C,Liu Q Y,Tromp J.2007.Finite-frequency tomography using adjoint methods-methodology and examples using membrane surface waves.Geophysical Journal International,168(3):1105-1129.
Tape C,Liu Q Y,Maggi A,et al.2009.Adjoint tomography of the Southern California crust.Science,325(5943):988-992.
Tarantola A.1984.Linearized inversion of seismic reflection data.Geophysical Prospecting,32(6):998-1015.
Tromp J,Tape C,Liu Q Y.2005.Seismic tomography,adjoint methods,time reversal and banana-doughnut kernels.Geophysical Journal International,160(1):195-216.
Woodward M J.1992.Wave-equation tomography.Geophysics,57(1):15-26.
Wu R S.2003.Wave propagation,scattering and imaging using dual-domain one-way and one-return propagators.Pure and Applied Geophysics,160(3-4):509-539.
Xie X B,Yang H.2008.The finite-frequency sensitivity kernel for migration residual moveout and its applications in migration velocity analysis.Geophysics,73(6):S241-S249.
Xu W J,Xie X B,Geng J H.2015.Validity of the Rytov approximation in the form of finite-frequency sensitivity kernels.Pure and Applied Geophysics,172(6):1409-1427.
Yuan Y O,Simons F J,Tromp J.2016.Double-difference adjoint seismic tomography.Geophysical Journal International,206(3):1599-1618.
Zhang J X,Yang Q,Meng X H,et al.2016.Reflection tomography based on a velocity model with implicitly described structure information.Journal of Geophysics and Engineering,13(5):721-732.
Zhao L,Jordan T H,Chapman C H.2000.Three-dimensional Fréchet differential kernels for seismic delay times.Geophysical Journal International,141(3):558-576.
Zhao L,Jordan T H,Olsen K B,et al.2005.Fréchet kernels for imaging regional Earth structure based on three-dimensional reference models.Bulletin of the Seismological Society of America,95(6):2066-2080.
程乾生.2003.数字信号处理.2版.北京:北京大学出版社.
刘玉柱,董良国,李培明等.2009.初至波菲涅尔体地震层析成像.地球物理学报,52(9):2310-2320,doi:10.3969/j.issn.0001-5733.2009.09.015.
刘玉柱,伍正,耿志成.2019.初至波相位走时层析.地球物理学报,62(2):619-633,doi:10.6038/cjg2019L0524.