伴随数据同化方法及其在地幔对流中的应用
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摘要
伴随数据同化方法是一个基于梯度法的反演技巧,尤其适用于对非线性问题的反演。最近几年里,该方法在地球物理问题中的应用取得了长足发展。文中试图从理论推导到其在地幔对流中的应用对该方法进行系统阐述,并附例图加以说明。伴随数据同化方法的基础是扰动理论,将模型输出与观测值的差别归因于模型输入中存在的误差,而该输入误差可以通过输出误差的最小二乘对输入条件的一阶倒数(梯度)来表示,这个联系就称作伴随算子。非线性问题的反演需要用到多次迭代;对输入误差的预估程度会直接影响计算和结果收敛的速度。作为描述当前地幔结构最有力证据的地震层析成像技术的不断进步,不论是在区域还是全球尺度上,都为地幔对流的反演提供了一个出发点。通过进一步同化或者比较相关的地质学证据特别是地表动力沉降观测,地幔对流反演可以克服目前仍然存在的地幔动力学各参数的不确定性的影响,从而进一步揭示壳幔系统的动力学机制。讨论的一个实际的例子是如何使用该方法反推出Farallon板块于晚白垩世时期在北美板块下的平俯冲过程以及该研究所导致的地球物理学及地质学意义。
The adjoint method for data assimilation is a gradient-based inversion technique which is especially useful for inverting nonlinear dynamic systems.Recently,its application for simulating geophysical problems has been proved promising.In this paper,we try to introduce the adjoint method in a systematic way,from the theoretical basis to its practical implementation in numerical models of mantle convection,with several examples to help the understanding.The adjoint operator of a temporally evolving system can be derived based on the perturbation theory,where a mismatch in the model output against observation is attributed to an error in the input,with their relation approximated as a first-order derivative (gradient) of the least-squared mismatch with respect to the input.For a nonlinear system,iterative processing is inevitable;the efficiency and convergence rate depend on the amount of a-priori information about the input (e.g.the initial condition).Seismic tomography,which describes the present-day mantle structures,has seen steady progress on both regional and global scales,and allows inverting mantle convection to the past.By either assimilation or comparison with geological observations,especially the dynamic topography inferred from stratigraphy,adjoint calculation of mantle convection can constrain unknown mantle dynamic properties while recovering the initial condition.Enormous new insights about the earth's dynamic mechanisms,therefore,can emerge.A specific example is the inversion of the Farallon flat subduction under North America during the Late Cretaceous time.
The adjoint method for data assimilation is a gradient-based inversion technique which is especially useful for inverting nonlinear dynamic systems.Recently,its application for simulating geophysical problems has been proved promising.In this paper,we try to introduce the adjoint method in a systematic way,from the theoretical basis to its practical implementation in numerical models of mantle convection,with several examples to help the understanding.The adjoint operator of a temporally evolving system can be derived based on the perturbation theory,where a mismatch in the model output against observation is attributed to an error in the input,with their relation approximated as a first-order derivative (gradient) of the least-squared mismatch with respect to the input.For a nonlinear system,iterative processing is inevitable;the efficiency and convergence rate depend on the amount of a-priori information about the input (e.g.the initial condition).Seismic tomography,which describes the present-day mantle structures,has seen steady progress on both regional and global scales,and allows inverting mantle convection to the past.By either assimilation or comparison with geological observations,especially the dynamic topography inferred from stratigraphy,adjoint calculation of mantle convection can constrain unknown mantle dynamic properties while recovering the initial condition.Enormous new insights about the earth's dynamic mechanisms,therefore,can emerge.A specific example is the inversion of the Farallon flat subduction under North America during the Late Cretaceous time.
引文
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[11]Ismail-Zadeh A,Schubert G,TsepelevI,et al.Inverse prob-lemof thermal convection:Numerical approach and applica-tion to mantle plume restoration[J].Physics of the Earth Planetary Interiors,2004,145:99-114.
[12]Liu L,Gurnis M.Si multaneous inversion of mantle proper-ties and initial conditions using an adjoint of mantle convec-tion[J].Journal of Geophysical Research,2008,113:B08405.
[13]Gurnis M,Turner M,DiCaprio L,et al.Global plate recon-structions with continuously closing plates[J].Geochemis-try,Geophysics,Geosystems,(in revision).
[14]Grand S P.Mantle shear-wave tomography and the fate of subducted slabs[J].Philosophical Transactions of the Royal Society of London,Series A,2002,A360:2475-2491.
[15]Liu L,Spasojevi'cS,Gurnis M.Reconstructing Farallon Plate subduction beneath North America back to the Late Cretaceous[J].Science,2008,322:934-938.
[16]Coney P J,Reynolds S J.Cordilleran Benioff zones[J].Na-ture,1977,270:403-406.
[17]Bond G.Evidence for continental subsidencein North Ameri-ca during the Late Cretaceous global submergence[J].Geolo-gy,1976,4:557-560.
[18]Miller K G,Kominz MA,Browning J V,et al.The Phaner-ozoic record of global sea-level change[J].Science,2005,310:1293-1298.
[19]Spasojevi'cS,Liu L,Gurnis M,et al.The case for dynamic subsidence of the United States east coast over the Cenozoic[J].Geophysical Research Letters,2008,35:L08305.
[20]Liu L,Gurnis M.Dynamic subsidence and uplift of the Colo-rado Plateau[J].Geology,doi:10.1130/G30624.1(in press).
[21]Liu L,Gurnis M,Seton M,et al.Predictions of oceanic plat-eau subduction beneath North America and newinsights into the Laramide Orogeny[J].Nature Geoscience,doi:10.1038/NGEO829.(Advanced Online Publication).
[2]Van der Hilst R D,Widiyantoro S,Engdahl E R.Evidence of deep mantle circulation from global tomography[J].Na-ture,1997,386:578-584.
[3]Li C,van der Hilst R D,Engdahl E R,et al.A new global model for P wave speed variationsin Earth s mantle[J].Geo-chemistry,Geophysics,Geosystems,2008,9:10.1029/2007GC001806.
[4]Lithgow-Bertelloni C,Richards MA.The dynamics of Ceno-zoic and Mesozoic plate motions[J].Reviews of Geophysics,1998,36:27-78.
[5]Bunge H P,Richards M A,Lithgow-Bertelloni C,et al.Ti me scales and heterogeneous structurein geodynamic Earth models[J].Science,1998,280:91-95.
[6]Steinberger B,O Connell R.Effects of mantle flow on hots-pot motion[M]∥Richards M A,Gordon R G,van der Hilst R D.The History of Dynamics of Global Plate Motions.Geo-physical Monograph Series,Vol.121.Washington DC:AGU,2000:377-398.
[7]Conrad C P,Gurnis M.Seismic tomography,surface uplift,and the breakup of Gonwanaland:Integrating mantle convec-tion backwards in ti me[J].Geochemistry,Geophysics,Geo-systems,2003,4(3):1031.
[8]Talagrand O,Courtier P.Variational assi milation of meteor-ological observation with the adjoint vorticity equation[J].Quaternary Journal of the Royal Meteorology Society,1987,113:1311-1328.
[9]Ismail-Zadeh A T,Talbot C J,Volozh Y A.Dynamic resto-ration of profiles across diapiric salt structures:Numerical ap-proach and its applications[J].Tectonophysics,2001,337(1/2):23-38.
[10]Bunge H P,Hagelberg C R,Travis B J.Mantle circulation models with variational data assi milation:Inferring past man-tle flowand structure fromplate motions histories and seismic tomography[J].Geophysical Journal International,2003,152:280-301.
[11]Ismail-Zadeh A,Schubert G,TsepelevI,et al.Inverse prob-lemof thermal convection:Numerical approach and applica-tion to mantle plume restoration[J].Physics of the Earth Planetary Interiors,2004,145:99-114.
[12]Liu L,Gurnis M.Si multaneous inversion of mantle proper-ties and initial conditions using an adjoint of mantle convec-tion[J].Journal of Geophysical Research,2008,113:B08405.
[13]Gurnis M,Turner M,DiCaprio L,et al.Global plate recon-structions with continuously closing plates[J].Geochemis-try,Geophysics,Geosystems,(in revision).
[14]Grand S P.Mantle shear-wave tomography and the fate of subducted slabs[J].Philosophical Transactions of the Royal Society of London,Series A,2002,A360:2475-2491.
[15]Liu L,Spasojevi'cS,Gurnis M.Reconstructing Farallon Plate subduction beneath North America back to the Late Cretaceous[J].Science,2008,322:934-938.
[16]Coney P J,Reynolds S J.Cordilleran Benioff zones[J].Na-ture,1977,270:403-406.
[17]Bond G.Evidence for continental subsidencein North Ameri-ca during the Late Cretaceous global submergence[J].Geolo-gy,1976,4:557-560.
[18]Miller K G,Kominz MA,Browning J V,et al.The Phaner-ozoic record of global sea-level change[J].Science,2005,310:1293-1298.
[19]Spasojevi'cS,Liu L,Gurnis M,et al.The case for dynamic subsidence of the United States east coast over the Cenozoic[J].Geophysical Research Letters,2008,35:L08305.
[20]Liu L,Gurnis M.Dynamic subsidence and uplift of the Colo-rado Plateau[J].Geology,doi:10.1130/G30624.1(in press).
[21]Liu L,Gurnis M,Seton M,et al.Predictions of oceanic plat-eau subduction beneath North America and newinsights into the Laramide Orogeny[J].Nature Geoscience,doi:10.1038/NGEO829.(Advanced Online Publication).
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