• ISSN 2097-1893
  • CN 10-1855/P

地球物理反演问题中的贝叶斯方法研究

蒋星达 张伟 杨辉

引用本文: 蒋星达,张伟,杨辉. 2022. 地球物理反演问题中的贝叶斯方法研究. 地球与行星物理论评,53(2):159-171
Jiang X D, Zhang W, Yang H. 2022. The research on Bayesian inference for geophysical inversion. Reviews of Geophysics and Planetary Physics, 53(2): 159-171

地球物理反演问题中的贝叶斯方法研究

doi: 10.19975/j.dqyxx.2021-042
基金项目: 国家自然科学基金资助项目(U1901602);深圳市深远海油气勘探技术重点实验室资助项目(ZDSYS20190902093007855);深圳市科技计划资助项目(KQTD20170810111725321)
详细信息
    通讯作者:

    蒋星达(1989-),男,博士研究生,主要从事微地震监测、正反演方法的研究及应用. E-mail:11749289@mail.sustech.edu.cn

  • 中图分类号: P315

The research on Bayesian inference for geophysical inversion

Funds: Supported by the National Natural Science Foundation of China (Grant No. U1901602), the Shenzhen Key Laboratory of Deep Offshore Oil and Gas Exploration Technology (Grant No. ZDSYS20190902093007855) and the Shenzhen Science and Technology Program (Grant No. KQTD20170810111725321)
  • 摘要: 基于统计理论的贝叶斯反演方法在先验信息和观测数据的约束下,以后验概率分布的形式表征模型参数在不同区间的可能性大小. 相对于确定性反演理论,贝叶斯反演通过提取模型参数边缘概率分布、最大后验解、平均解、相关系数等定量评价反演结果的不确定性以及模型参数之间的相互关系,通过模型参数后验概率分布反映观测数据和先验信息对模型参数的约束能力. 本文基于贝叶斯方法在地球物理反演中的应用,总结了贝叶斯反演的基本流程,详细介绍了不同背景条件下的先验信息概率分布选择、似然函数建立、后验概率公式求解. 在优化参数方面,介绍了模型参数的固定维和变维反演概念,以及超参数的优化方法;在反演方法方面,着重介绍了固定维和变维反演马尔科夫链蒙特卡罗采样方法;在模型参数评价方面,介绍了不同情况下贝叶斯统计参数的求取. 然后讨论了贝叶斯反演方法采样效率提升的具体措施. 最后对贝叶斯方法在地球物理反演中的应用作出总结.

     

  • 图  1  贝叶斯反演一般流程

    Figure  1.  Flowchart of Bayesian inversion

    图  2  贝叶斯原理示意图

    Figure  2.  Schematic diagram of Bayesian principle

    图  3  模型参数化示意图.(a)一维模型Voronoi 参数化(修改自Sambridge et al., 2013);(b)二维模型Voronoi 网格参数化(修改自Bodin et al., 2012a). 方块表示网格节点,根据相邻网格节点垂直平分线确定网格边. m为模型参数,z为界面位置

    Figure  3.  The schematic diagram of model parameterization. (a) 1D Voronoi model parameterization (modified from Sambridge et al., 2013); (b) 2D Voronoi model parameterization (modified from Bodin et al., 2012a). The squares represent grid nodes, the grid boundaries are determined by the perpendicular bisector of adjacent grid nodes. m is model parameter. z is interface position

    图  4  根据后验概率分布确定S波速度结构(修改自Bodin et al., 2012c

    Figure  4.  S-wave velocity structure determined by posterior probability density distribution (modified from Bodin et al., 2012c)

  • [1] Akaike H. 1980. Likelihood and the Bayes procedure[J]. Trabajos de estadística y de investigación operativa, 31(1): 143-166.
    [2] Alemie W, Sacchi M D. 2011. High-resolution three-term AVO inversion by means of a Trivariate Cauchy probability distribution[J]. Geophysics, 76(3): R43-R55. doi: 10.1190/1.3554627
    [3] Amey R M J, Hooper A, Walters R J. 2018. A Bayesian method for incorporating self-similarity into earthquake slip inversions[J]. Journal of Geophysical Research: Solid Earth, 123(7): 6052-6071. doi: 10.1029/2017JB015316
    [4] Aster R C, Borchers B, Thurber C H. 2013. Parameter Estimation and Inverse Problems[M]. Elsevier Academic Press.
    [5] Avseth P, Mukerji T, Jørstad A, et al. 2001. Seismic reservoir mapping from 3-D AVO in a North Sea turbidite system[J]. Geophysics, 66(4): 1157-1176. doi: 10.1190/1.1487063
    [6] Bagnardi M, Hooper A. 2018. Inversion of surface deformation data for rapid estimates of source parameters and uncertainties: A Bayesian approach[J]. Geochemistry, Geophysics, Geosystems, 19(7): 2194-2211. doi: 10.1029/2018GC007585
    [7] Bayes T. 1763. An essay towards solving a problem in the doctrine of chances[J]. MD Computing: Computers in Medical Practice, 8(3):157-171(1991).
    [8] Bodin T, Sambridge M. 2009. Seismic tomography with the reversible jump algorithm[J]. Geophysical Journal International, 178(3), 1411-1436. doi: 10.1111/j.1365-246X.2009.04226.x
    [9] Bodin T, Sambridge M, Gallagher K. 2009. A self-parametrizing partition model approach to tomographic inverse problems[J]. Inverse Problems, 25(5): 055009.
    [10] Bodin T, Salmon M, Kennett B, et al. 2012a. Probabilistic surface reconstruction from multiple data sets: An example for the Australian Moho[J]. Journal of Geophysical Research: Solid Earth, 117(B10307): 1-13.
    [11] Bodin T, Sambridge M, Rawlinson N, et al. 2012b. Transdimensional tomography with unknown data noise[J]. Geophysical Journal International, 189(3): 1536-1556. doi: 10.1111/j.1365-246X.2012.05414.x
    [12] Bodin T, Sambridge M, Tkalčić H, et al. 2012c. Transdimensional inversion of receiver functions and surface wave dispersion[J]. Journal of Geophysical Research: Solid Earth, 117(B02301): 1-24.
    [13] Buland A, Omre H. 2003. Bayesian linearized AVO inversion[J]. Geophysics, 68(1): 185-198. doi: 10.1190/1.1543206
    [14] Buland A, Kolbjørnsen O. 2012. Bayesian inversion of CSEM and magnetotelluric data[J]. Geophysics, 77(1): E33-E42. doi: 10.1190/geo2010-0298.1
    [15] Chen J, Kemna A, Hubbard S S. 2008. A comparison between Gauss-Newton and Markov-chain Monte Carlo–based methods for inverting spectral induced-polarization data for Cole-Cole parameters[J]. Geophysics, 73(6): F247-F259. doi: 10.1190/1.2976115
    [16] de Figueiredo L P, Grana D, Roisenberg M, et al. 2019. Multimodal Markov chain Monte Carlo method for nonlinear petrophysical seismic inversion[J]. Geophysics, 84(5): M1-M13. doi: 10.1190/geo2018-0839.1
    [17] Dosso S E, Dettmer J, Steininger G, et al. 2014. Efficient trans-dimensional Bayesian inversion for geoacoustic profile estimation[J]. Inverse Problems, 30(11): 114018. doi: 10.1088/0266-5611/30/11/114018
    [18] Downton J E, Lines L R. 2001. Constrained three parameter AVO inversion and uncertainty analysis[J]. SEG Technical Program Expanded Abstracts. Society of Exploration Geophysicists, 251-254.
    [19] Duijndam A J W. 1988. Bayesian estimation in seismic inversion. Part I: princiles[J]. Geophysical Prospecting, 1988, 36(8): 878-898. doi: 10.1111/j.1365-2478.1988.tb02198.x
    [20] Eidsvik J, Avseth P, Omre H, et al. 2004. Stochastic reservoir characterization using prestack seismic data[J]. Geophysics, 69(4): 978-993. doi: 10.1190/1.1778241
    [21] Fukahata Y, Yagi Y, Matsu'ura M. 2003. Waveform inversion for seismic source processes using ABIC with two sorts of prior constraints: Comparison between proper and improper formulations[J]. Geophysical Research Letters, 30(6):38-1-38-4.
    [22] Geman S, Geman D. 1984. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images[J]. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6(PAMI-6): 721-741.
    [23] Gesret A, Desassis N, Noble M, et al. 2015. Propagation of the velocity model uncertainties to the seismic event location[J]. Geophysical Journal International, 200(1): 52-66. doi: 10.1093/gji/ggu374
    [24] Gouveia W P, Scales J A. 1998. Bayesian seismic waveform inversion: Parameter estimation and uncertainty analysis[J]. Journal of Geophysical Research: Solid Earth, 103(B2): 2759-2779. doi: 10.1029/97JB02933
    [25] Grana D. 2016. Bayesian linearized rock-physics inversion[J]. Geophysics, 81(6): D625-D641. doi: 10.1190/geo2016-0161.1
    [26] Grana D, Passos de Figueiredo L, Azevedo L. 2019. Uncertainty quantification in Bayesian inverse problems with model and data dimension reduction[J]. Geophysics, 84(6): M15-M24. doi: 10.1190/geo2019-0222.1
    [27] Grana D. 2020. Bayesian petroelastic inversion with multiple prior models[J]. Geophysics, 85(5): M57-M71. doi: 10.1190/geo2019-0625.1
    [28] Grandis H, Menvielle M, Roussignol M. 1999. Bayesian inversion with Markov chains—I. The magnetotelluric one-dimensional case[J]. Geophysical Journal International, 138(3): 757-768. doi: 10.1046/j.1365-246x.1999.00904.x
    [29] Green P J. 1995. Reversible jump Markov chain Monte Carlo computation and Bayesian model determination[J]. Biometrika, 82(4): 711-732. doi: 10.1093/biomet/82.4.711
    [30] Guitton A, Symes W W. 2003. Robust inversion of seismic data using the Huber norm[J]. Geophysics, 68(4): 1310-1319. doi: 10.1190/1.1598124
    [31] Guo R, Dosso S E, Liu J, et al. 2011. Non-linearity in Bayesian 1-D magnetotelluric inversion[J]. Geophysical Journal International, 185(2): 663-675. doi: 10.1111/j.1365-246X.2011.04996.x
    [32] 郭荣文, 柳建新. 2017. 大地电磁贝叶斯反演方法与理论[M]. 长沙: 中南大学出版社.

    Guo R W, Liu J X. 2017. Bayesian Inversion for Magnetotelluric Problems [M].Changsha: Central South University Press (in Chinese).
    [33] Haario H, Laine M, Lehtinen M, et al. 2004. Markov chain Monte Carlo methods for high dimensional inversion in remote sensing[J]. Journal of the Royal Statistical Society: series B (statistical methodology), 66(3): 591-607. doi: 10.1111/j.1467-9868.2004.02053.x
    [34] Haario H, Laine M, Mira A, et al. 2006. DRAM: efficient adaptive MCMC[J]. Statistics and Computing, 16(4): 339-354.
    [35] Hastings W K. 1970. Monte Carlo sampling methods using Markov chains and their applications[J]. Biometrika, 57(1), 97-109. doi: 10.1093/biomet/57.1.97
    [36] 何沛阳, 卢建旗, 李山有, 马云漪. 2020. 地震预警震级估算方法的不确定性评估模型——以τ~p_(max)法为例[J]. 内陆地震, 34(4): 317-329.

    He P Y, Lu Z Q, Li S Y, Ma Y Y. 2020. Uncertainty evaluation model for earthquake magnitude estimation method taking τ~p_(max) as an example[J]. Inland Earthquake, 34(4): 317-329 (in Chinese).
    [37] Hopcroft P O, Gallagher K, Pain C C. 2007. Inference of past climate from borehole temperature data using Bayesian Reversible Jump Markov chain Monte Carlo[J]. Geophysical Journal International, 171(3): 1430-1439. doi: 10.1111/j.1365-246X.2007.03596.x
    [38] 胡华锋, 印兴耀, 吴国忱. 2012. 基于贝叶斯分类的储层物性参数联合反演方法[J]. 石油物探, 51(3): 225-232. doi: 10.3969/j.issn.1000-1441.2012.03.003

    Hu H F, Yin X Y, Wu G C. 2012. Joint inversion of petrophysical parameters based on Bayesian classification[J]. Geophysical Prospecting for Petroleum, 51(3):225-232 (in Chinese). doi: 10.3969/j.issn.1000-1441.2012.03.003
    [39] 黄捍东, 赵迪, 任敦占, 王玉梅. 2011. 基于贝叶斯理论的薄层反演方法[J]. 石油地球物理勘探, 46(6): 919-924.

    Huang H D, Zhao D, Ren D Z, et al. 2011. A thin bed inversion method based on Bayes theory[J]. Oil Geophysical Prospecting, 46(6): 919-924 (in Chinese).
    [40] Jiang X, Zhang W and Yang H. 2021. Transdimensional downhole velocity optimization by incremental pseudo-master method[J]. SEG Technical Program Expanded Abstracts. Society of Exploration Geophysicists(in Press).
    [41] Kjønsberg H, Hauge R, Kolbjørnsen O, et al. 2010. Bayesian Monte Carlo method for seismic predrill prospect assessment[J]. Geophysics, 75(2): O9-O19. doi: 10.1190/1.3339678
    [42] Kolbjørnsen O, Buland A, Hauge R, et al. 2020. Bayesian seismic inversion for stratigraphic horizon, lithology, and fluid prediction[J]. Geophysics, 85(3): R207-R221. doi: 10.1190/geo2019-0170.1
    [43] Laloy E, Vrugt J A. 2012. High-dimensional posterior exploration of hydrologic models using multiple-try DREAM (ZS) and high-performance computing[J]. Water Resources Research, 48(1): W01526.
    [44] Laplace P S. 1814. A Philosophical Essay on Probabilities[M]. Translated by Bell E T. An Introductory Note. New York: Dover Publications Inc.
    [45] 李承瑾, 郭荣文, 柳建新, 刘黎明. 2018. 跨维贝叶斯反演在地球物理中的研究进展[J]. 工程地球物理学报, 15(4): 501-508. doi: 10.3969/j.issn.1672-7940.2018.04.015

    Li C J, Guo R W, Liu J X, Liu L M. 2018. Research progress of trans-dimensional Bayesian inversion in geophysics[J]. Chinese Journal of Engineering Geophysics, 15(4): 501-508 (in Chinese). doi: 10.3969/j.issn.1672-7940.2018.04.015
    [46] 刘彦, 吕庆田, 李晓斌, 等. 2015. 基于模型降阶的贝叶斯方法在三维重力反演中的实践[J]. 地球物理学报, 58(12): 4727-4739. doi: 10.6038/cjg20151233

    Liu Y, Lü Q T, Li X B, et al. 2015. 3D gravity inversion based on Bayesian method with model order reduction. Chinese Journal of Geophysics, 58(12): 4727-4739 (in Chinese). doi: 10.6038/cjg20151233
    [47] 刘艳杰. 2020. 参数反演的贝叶斯方法及其应用研究[D]. 淄博: 山东理工大学.

    Liu Y J. 2020. Bayesian method of parameter inversion and its application [D]. Zibo: Shandong University of Technology (in Chinese).
    [48] Malinverno A. 2002. Parsimonious Bayesian Markov chain Monte Carlo inversion in a nonlinear geophysical problem[J]. Geophysical Journal International, 151(3): 675-688. doi: 10.1046/j.1365-246X.2002.01847.x
    [49] Malinverno A, Briggs V A. 2004. Expanded uncertainty quantification in inverse problems: Hierarchical Bayes and empirical Bayes[J]. Geophysics, 69(4): 1005-1016. doi: 10.1190/1.1778243
    [50] Mallick S. 1995. Model-based inversion of amplitude-variations-with-offset data using a genetic algorithm[J]. Geophysics, 60(4): 939-954. doi: 10.1190/1.1443860
    [51] Metropolis N, Ulam S. 1949. The monte carlo method[J]. Journal of the American statistical association, 44(247): 335-341. doi: 10.1080/01621459.1949.10483310
    [52] Metropolis N, Rosenbluth A W, Rosenbluth M N, et al. 1953. Equation of state calculations by fast computing machines[J]. The Journal of Chemical Physics, 21(6): 1087-1092. doi: 10.1063/1.1699114
    [53] Minson S E, Simons M, Beck J L, et al. 2014. Bayesian inversion for finite fault earthquake source models–II: the 2011 great Tohoku-oki, Japan earthquake[J]. Geophysical Journal International, 198(2): 922-940. doi: 10.1093/gji/ggu170
    [54] Mosegaard K, Tarantola A. 1995. Monte Carlo sampling of solutions to inverse problems[J]. Journal of Geophysical Research: Solid Earth, 100(B7): 12431-12447. doi: 10.1029/94JB03097
    [55] Mosegaard K, Sambridge M. 2002. Monte Carlo analysis of inverse problems[J]. Inverse Problems, 18(3): R29-R54. doi: 10.1088/0266-5611/18/3/201
    [56] Parker R L. 1977. Understanding inverse theory[J]. Annual Review of Earth & Planetary Sciences, 5(1): 35-64.
    [57] Pugh D J, White R S. 2018. MTfit: A Bayesian approach to seismic moment tensor inversion[J]. Seismological Research Letters, 89(4): 1507-1513. doi: 10.1785/0220170273
    [58] Ray A, Key K. 2012. Bayesian inversion of marine CSEM data with a trans-dimensional self parametrizing algorithm[J]. Geophysical Journal International, 191(3): 1135-1151.
    [59] Ray A, Alumbaugh D L, Hoversten G M, et al. 2013. Robust and accelerated Bayesian inversion of marine controlled-source electromagnetic data using parallel tempering[J]. Geophysics, 78(6): E271-E280. doi: 10.1190/geo2013-0128.1
    [60] Robert C P, Chopin N, Rousseau J. 2009. Harold Jeffreys’s theory of probability revisited[J]. Statistical Science, 24(2): 141-172.
    [61] Sacchi M D, Ulrych T J. 1995. High-resolution velocity gathers and offset space reconstruction[J]. Geophysics, 60(4): 1169-1177. doi: 10.1190/1.1443845
    [62] Sambridge M, Gallagher K, Jackson A, et al. 2006. Trans-dimensional inverse problems, model comparison and the evidence[J]. Geophysical Journal International, 167(2): 528-542. doi: 10.1111/j.1365-246X.2006.03155.x
    [63] Sambridge M, Bodin T, Gallagher K, et al. 2013. Transdimensional inference in the geosciences[J]. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 371(1984): 20110547. doi: 10.1098/rsta.2011.0547
    [64] Sambridge M. 2014. A parallel tempering algorithm for probabilistic sampling and multimodal optimization[J]. Geophysical Journal International, 196(1): 357-374. doi: 10.1093/gji/ggt342
    [65] Scales J A, Snieder R. 1998. What is noise?[J]. Geophysics, 63(4): 1122-1124. doi: 10.1190/1.1444411
    [66] Schott J J, Roussignol M, Menvielle M, et al. 1999. Bayesian inversion with Markov chains—II. The one-dimensional DC multilayer case[J]. Geophysical Journal International, 138(3): 769-783. doi: 10.1046/j.1365-246x.1999.00905.x
    [67] Schwarz G. 1978. Estimating the dimension of a model[J]. Annals of statistics, 6(2):461-464.
    [68] Sen M K, Stoffa P L. 1996. Bayesian inference, Gibbs' sampler and uncertainty estimation in geophysical inversion[J]. Geophysical Prospecting, 44(2): 313-350. doi: 10.1111/j.1365-2478.1996.tb00152.x
    [69] Tarantola A, Valette B. 1982. Inverse problems= Quest for information[J]. Journal of Geophysics, 50(1): 159-170.
    [70] Tarantola A. 1987. Inverse Problem Theory and Methods for Model Parameter Estimation[M]. Society for Industrial and Applied Mathematics.
    [71] Theune U, Jensås I Ø, Eidsvik J. 2010. Analysis of prior models for a blocky inversion of seismic AVA data[J]. Geophysics, 75(3): C25-C35. doi: 10.1190/1.3427538
    [72] 田军, 吴国忱, 宗兆云. 2013. 鲁棒性AVO三参数反演方法及不确定性分析[J]. 石油地球物理勘探, 48(3): 443-449.

    Tian J, Wu G C, Zong Z Y. 2013. Robust three-term AVO inversion and uncertainty analysis[J]. Oil Geophysical Prospecting, 48(3): 443-449 (in Chinese).
    [73] Ulrych T J, Sacchi M D, Woodbury A. 2001. A Bayes tour of inversion: A tutorial[J]. Geophysics, 66(1): 55-69. doi: 10.1190/1.1444923
    [74] Ulvmoen M, Omre H. 2010. Improved resolution in Bayesian lithology/fluid inversion from prestack seismic data and well observations: Part 1—Methodology[J]. Geophysics, 75(2): R21-R35. doi: 10.1190/1.3294570
    [75] Visser G, Guo P, Saygin E. 2019. Bayesian transdimensional seismic full-waveform inversion with a dipping layer parameterization[J]. Geophysics, 84(6): R845-R858. doi: 10.1190/geo2018-0785.1
    [76] Vrugt J A, Ter Braak C J F, Clark M P, et al. 2008. Treatment of input uncertainty in hydrologic modeling: Doing hydrology backward with Markov chain Monte Carlo simulation[J]. Water Resources Research, 44(12): W00B09.
    [77] 王家映. 2002. 地球物理反演理论[M]. 北京: 高等教育出版社.

    Wang J Y. 2002. Inverse Theory in Geophysics[M]. Beijing: Higher Education Press (in Chinese).
    [78] 王家映. 2007. 地球物理资料非线性反演方法讲座(一)地球物理反演问题概述[J]. 工程地球物理学报, 4(1): 1-3. doi: 10.3969/j.issn.1672-7940.2007.01.001

    Wang J Y. 2007. Introduction to Geophysical Inverse Problems[J]. Chinese Journal of Engineering Geophysics, 4(1): 1-3 (in Chinese). doi: 10.3969/j.issn.1672-7940.2007.01.001
    [79] 王朋岩, 李耀华, 赵荣. 2015. 叠后MCMC法岩性反演算法研究[J]. 地球物理学进展, 30(4): 1918-1925.

    Wang P Y, Li Y H, Zhao R. 2015. Algorithm research of post-stack MCMC lithology inversion method[J]. Progress in Geophysics, 30(04): 1918-1925 (in Chinese).
    [80] 肖爽, 巴晶, 符力耘, 等. 2020. 基于高斯先验和马尔科夫随机场约束的非线性叠前地震反演研究及应用[J]. 地球物理学进展, 35(6): 2250-2258. doi: 10.6038/pg2020EE0034

    Xiao S, Ba J, Fu L Y, et al. 2020. Research and application of nonlinear pre-stack seismic inversion based on Gaussian prior and Markov random field constraints[J]. Progress in Geophysics, 35(6): 2250-2258 (in Chinese). doi: 10.6038/pg2020EE0034
    [81] Xiong Z, Zhuang J, Zhou S, et al. 2021. Crustal strain-rate fields estimated from GNSS data with a Bayesian approach and its correlation to seismic activity in Mainland China[J]. Tectonophysics, 815: 229003. doi: 10.1016/j.tecto.2021.229003
    [82] Yabuki T, Matsu'Ura M. 1992. Geodetic data inversion using a Bayesian information criterion for spatial distribution of fault slip[J]. Geophysical Journal International, 109(2): 363-375. doi: 10.1111/j.1365-246X.1992.tb00102.x
    [83] 杨培杰, 印兴耀. 2008. 非线性二次规划贝叶斯叠前反演[J]. 地球物理学报, 51(6): 1876-1882. doi: 10.3321/j.issn:0001-5733.2008.06.030

    Yang P J, Yin X Y. 2008. Non-linear quadratic programming Bayesian prestack inversion[J]. Chinese Journal of Geophysics, 51(6): 1876-1882 (in Chinese). doi: 10.3321/j.issn:0001-5733.2008.06.030
    [84] 姚铭, 高刚, 胡瑞卿, 等. 2020. 一种改进的贝叶斯反演算法[J]. 地球物理学进展, 35(5): 1911-1918. doi: 10.6038/pg2020DD0379

    Yao M, Gao G, Hu R Q, et al. 2020. Improved Bayesian inversion algorithm[J]. Progress in Geophysics, 35(5): 1911-1918 (in Chinese). doi: 10.6038/pg2020DD0379
    [85] 尹彬, 胡祥云. 2016. 非线性反演的贝叶斯方法研究综述[J]. 地球物理学进展, 31(3): 1027-1032. doi: 10.6038/pg20160313

    Yin B, Hu X Y. 2016. Overview of nonlinear inversion using Bayesian method[J]. Progress in Geophysics, 31(3): 1027-1032 (in Chinese). doi: 10.6038/pg20160313
    [86] 印海燕. 2008. AVO叠前反演方法研究[D]. 青岛: 中国石油大学(华东).

    Yin H Y. 2008. The study on methods of AVO prestack inversion[D].Qingdao: China University of Petroleum (East China) (in Chinese).
    [87] Yin X, Zhang S. 2014. Bayesian inversion for effective pore-fluid bulk modulus based on fluid-matrix decoupled amplitude variation with offset approximation[J]. Geophysics, 79(5): R221-R232. doi: 10.1190/geo2013-0372.1
    [88] 印兴耀, 周琪超, 宗兆云, 刘汉卿. 2014. 基于t分布为先验约束的叠前AVO反演[J]. 石油物探, 53(1): 84-92. doi: 10.3969/j.issn.1000-1441.2014.01.012

    Yin X Y, Zhou Q C, Zong Z Y, Liu H Q. 2014. AVO inversion with t-distribution as priori constraint[J]. Geophysical Prospecting for Petroleum, 53(1): 84-92 (in Chinese). doi: 10.3969/j.issn.1000-1441.2014.01.012
    [89] 余小东, 陆从德, 王绪本. 2020. 时间域航空电磁数据的自适应变维贝叶斯反演研究[J]. 地球物理学进展, 35(05): 2023-2032. doi: 10.6038/pg2020DD0455

    Yu X D, Lu C D, Wang X B. 2020. Adaptive trans-dimensional Bayesian inversion of airborne time-domain electromagnetic data[J]. Progress in Geophysics, 35(5): 2023-2032 (in Chinese). doi: 10.6038/pg2020DD0455
    [90] 袁成, 李景叶, 陈小宏. 2016. 地震岩相识别概率表征方法[J]. 地球物理学报, 59(1): 287-298. doi: 10.6038/cjg20160124

    Yuan C, Li J Y, Chen X H. 2016. A probabilistic approach for seismic facies classification[J]. Chinese Journal of Geophysics, 59(1): 287-298 (in Chinese). doi: 10.6038/cjg20160124
    [91] 苑闻京. 2012. 叠前反演和地震吸收技术在复杂天然气藏地震预测中的应用[J]. 地球物理学进展, 27(3): 1107-1115. doi: 10.6038/j.issn.1004-2903.2012.03.035

    Yuan W J. 2012. Application of pre-stack seismic inversion and seismic absorption technology for forecasting of complex gas reservoir[J]. Progress in Geophysics, 27(3), 1107-1115 (in Chinese). doi: 10.6038/j.issn.1004-2903.2012.03.035
    [92] 张繁昌, 肖张波, 印兴耀. 2014. 地震数据约束下的贝叶斯随机反演[J]. 石油地球物理勘探, 49(1): 176-182.

    Zhang F C, Xiao Z B, Yin X Y. 2014. Bayesian stochastic inversion constrained by seismic data[J]. Oil Geophysical Prospecting, 49(1): 176-182 (in Chinese).
    [93] 张广智, 王丹阳, 印兴耀, 李宁. 2011. 基于MCMC的叠前地震反演方法研究[J]. 地球物理学报, 54(11): 2926-2932. doi: 10.3969/j.issn.0001-5733.2011.11.022

    Zhang G Z, Wang D Y, Yin X Y, Li N. 2011. Study on prestack seismic inversion using Markov Chain monte carlo[J]. Chinese Journal of Geophysics, 54(11): 2926-2932 (in Chinese). doi: 10.3969/j.issn.0001-5733.2011.11.022
    [94] 张世鑫, 印兴耀, 张繁昌. 2011. 基于三变量柯西分布先验约束的叠前三参数反演方法[J]. 石油地球物理勘探, 46(5): 737-743.

    Zhang S X, Yin X Y, Zhang F C. 2011. Prestack three term inversion method based on Trivariate Cauchy distribution prior constraint[J]. Oil Geophysical Prospecting, 46(5): 737-743 (in Chinese).
    [95] Zhang Z, Rector J W, Nava M J. 2017. Simultaneous inversion of multiple microseismic data for event locations and velocity model with Bayesian inference[J]. Geophysics, 82(3): KS27-KS39. doi: 10.1190/geo2016-0158.1
    [96] Zhang Z, Du J, Gao F. 2018. Simultaneous inversion for microseismic event location and velocity model in Vaca Muerta Formation [J]. Geophysics, 83(3): KS23-KS34. doi: 10.1190/geo2017-0010.1
    [97] 赵小龙, 吴国忱, 曹丹平. 2016. 多尺度地震资料稀疏贝叶斯联合反演方法[J]. 石油地球物理勘探, 51(6): 1156-1163.

    Zhao X L, Wu G C, Cao D P. 2016. A sparse Bayesian joint inversion of multi-scale seismic data[J]. Oil Geophysical Prospecting, 51(6):1156-1163 (in Chinese).
    [98] Zhu D, Gibson R. 2018. Seismic inversion and uncertainty quantification using transdimensional Markov chain Monte Carlo method[J]. Geophysics, 83(4): R321-R334. doi: 10.1190/geo2016-0594.1
    [99] Zhu H, Li S, Fomel S, et al. 2016. A Bayesian approach to estimate uncertainty for full-waveform inversion using a priori information from depth migration[J]. Geophysics, 81(5): R307-R323. doi: 10.1190/geo2015-0641.1
  • 加载中
图(4)
计量
  • 文章访问数:  708
  • HTML全文浏览量:  429
  • PDF下载量:  269
  • 被引次数: 0
出版历程
  • 收稿日期:  2021-08-04
  • 录用日期:  2021-08-27
  • 网络出版日期:  2021-09-06
  • 刊出日期:  2022-03-05

目录

    /

    返回文章
    返回