顾及模型误差的震源参数InSAR反演
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摘要
合成孔径雷达干涉测量(InSAR)技术的迅速发展,极大丰富了大地形变观测数据,使得地球科学家们能够以一个全新的角度研究与地震断层相关的各种地球物理学现象。基于大地测量形变观测数据研究震源参数可以有效弥补地表破裂数据、地震记录等提取震源参数的不足。震源参数不仅可以用来分析发震断层机制及区域构造应力状况,而且也是研究活动断层破裂及扩展、特征断层演化、震后形变机制、大陆岩石圈内应变的吸收与调整、应力变化及未来地震危险性评估的重要基础。为此,震源参数的精准度越来越受到地球科学家们的重视。以确定更加精准的震源参数为核心目标,本文创新性地对顾及模型误差的震源参数InSAR反演开展了相关研究。
本文建立了震源参数InSAR反演的数学模型并对其特征进行了分析。基于矩形位错理论、拉普拉斯平滑约束方法构建了震源参数InSAR反演的函数模型、随机模型及附等式约束的数学模型。将广义反演分析法引入震源滑动分布的反演分析中,给出了反演函数模型的数据分辨率、参数分辨率及方差的计算方法,并以走滑、逆冲断层及当雄地震为例分析了增加观测数据、附加约束条件对系数矩阵的数学特征项的影响。结果表明,增加观测数据可以在一定程度上增加系数矩阵的秩但不能改善其病态性,附加约束条件可以显著增加系数矩阵的秩并能改善其病态性,在一定程度上降低了数据分辨率,但较明显地增加了参数分辨率,并将其方差大小由几百米降低至厘米级。该结果为基于InSAR形变观测数据反演震源滑动分布提供了理论研究基础。在震源滑动分布反演中,附加一定的约束条件不可或缺。
本文对数学模型误差对震源参数InSAR反演估值的影响进行了理论和模拟反演分析,并对函数模型误差和随机模型误差的区分性进行了探讨研究。在系统总结并分析震源参数InSAR反演中数学模型误差来源的基础上,引入了测量数据统计分析中线性反演的模型误差理论和非线性反演的蒙特卡罗误差估计方法。以走滑、斜滑、逆冲三种主要震源类型为例,通过非线性反演震源参数和线性反演震源滑动分布模拟分析了函数模型和随机模型存在误差对震源参数估值的影响。结果表明,函数模型和随机模型存在误差使得震源参数估值产生偏差、精度降低,与理论分析结果相一致。讨论了震源参数InSAR反演中模型误差的估计和识别方法,并对函数模型误差和随机模型误差的区分性进行了探讨分析,指出震源参数InSAR反演系统中对二者进行有效分离具有较大的挑战性。
本文给出了震源参数InSAR反演的模型误差补偿方法并进行了反演计算分析,进一步地,给出了建议的反演方法。在系统总结并对比分析测量数据处理中模型误差补偿方法的基础上,提出了震源参数InSAR反演中通过调整随机模型对模型误差进行补偿的思路。通过引入测量数据处理中的方差分量估计和抗差估计理论和方法,设计了震源参数线性和非线性反演的方差分量估计类算法、抗差估计类算法和抗差方差分量估计类算法,给出了具体的反演计算步骤,通过模拟反演计算对算法的补偿效果进行了检测,其中,采用等价方差一协方差函数的抗差估计思想构建相应的权函数。基于虚拟观测原理将光滑约束条件方程转化为虚拟观测方程,将光滑因子表达为单位权方差与虚拟观测方差之商的形式,采用方差分量估计原理同时确定观测数据集权值和光滑因子的大小。在理论上对三类算法进行了比较分析,并对它们的补偿效果进行了分析。若观测数据集含有粗差,抗差估计算法能够较好地减免粗差对震源参数非线性反演估值的不良影响,但在滑动分布反演时具有一定的局限性,需要采用抗差方差分量估计算法；若两个或两个以上的观测数据集含有粗差,抗差方差分量估计算法的模型误差补偿效果优于方差分量估计算法。进一步地,给出了实际震源参数InSAR反演研究中建议的反演方法。
本文以2008年10月6日当雄Mw6.3级地震和2008年11月10日大柴旦Mw6.3级地震为实际震例进行了系统深入的反演研究。就当雄地震而言,利用不同轨道、不同波长的Envisat和ALOS影像数据提取该地震的高质量InSAR同震形变场,采用本文设计的抗差方差分量估计类算法减免数学模型误差对震源参数估值的影响。结果表明,震源滑动分布主要发生在4.5-11km范围内,平均滑动角为-112.58°,平均滑动量为0.50m,最大滑动量为1.53m,深度位于6.1-7.1km范围内,依据该滑动分布模型得到的地震矩为4.22×1018Nm(Mw6.38)。方差分量估计确定的震源参数估值存在一定的偏差,(抗差)方差分量估计前确定的震源参数估值偏差整体上大于方差分量估计的结果,滑动角的偏差达-4.195°,精度水平也显著差于(抗差)方差分量估计的结果。就大柴旦地震而言,利用Envisat影像数据提取该地震的高质量InSAR同震形变场,采用非线性抗差估计反演确定震源破裂的几何参数,进一步地,采用线性抗差方差分量估计反演确定精细的震源滑动分布。结果表明,震源滑动分布主要发生在10-20km范围内,平均滑动角为104.2°,平均滑动量为0.2m,最大滑动量为0.64m,深度位于13.8-14.6km范围内,依据该滑动分布模型得到的地震矩为3.74×1018N m(Mw6.35)。由于InSAR形变场中包含的粗差观测值较少且量级较小,震源参数非线性反演时,是否进行抗差估计对参数估值的精度水平影响不大,但震源滑动分布线性反演时,需进行(抗差)方差分量估计以确定合理的光滑因子。
最后,讨论并分析了同震形变观测数据在震源参数反演中的权值大小。方差分量估计法可以较好的确定观测数据集间的权比,其既不等权,也不等于基于验前方差确定的权比。非线性反演震源参数和线性反演震源滑动分布时确定的权比并不相等,且差异较大,数据集间的权比需要分别通过方差分量估计原理予以确定。观测数据点的权值既不随着远离断层而增大,也不随着靠近断层而增大,按距离断层远近为准则对观测数据点定权并非合理。考虑数学模型误差,抗差估计原理可以为观测数据点的定权提供一种合理的方法。
Rapid development of Interferometric Synthetic Aperture Rarar (InSAR), greatly enriching crustal deformation observation data, enables the geoscience research community to probe into the various geophysical phenomena related to earthquake fault in a fully new perspective. When extracting hypocenter parameter, geodetic data can effectively compensate for intrinsic limitations of other observations such as surface rupture data and seismic record. Hypocenter parameter can not only be used to analyze the seismogenic fault mechanism and regional tectonic stress, but can also provide a basis for studying rupture and expansion of the active fault, evolution of the characteristic fault, mechanism of the postseismic deformation, strain absorption and adjustment of the continental lithosphere, stress changes and seismic hazard assessment in the future. Consequently, the accuracy of hypocenter parameter holds a more and more attention of geoscience scientist. To determine a more precise hypocenter parameter, this dissertation creatively carries out the work of InSAR inversion considering model error for hypocenter parameter.
This article first establishes the mathematical model of InSAR inversion for hypocenter parameter, and studies its intrinsic features. According to the rectangular dislocation theory and Laplace smoothing constraint method, functional model, stochastic model and mathematical model with equality constraint appropriate for InSAR inversion for hypocenter parameter are proposed. Method of the generalized inverse analysis is introduced into focal slip distribution inversion to give the calculation formulas of data resolution, parameter resolution and variance for the coefficient matrix of the functional model. By taking strike, thrust faults and Dangxiong earthquake as examples, the effects of adding observation data and adding constraint condition on the mathematical feature items are studied. The results show that adding observation data can to a certain extent increase the rank of the coefficient matrix, but cannot improve its morbidity; adding constraint condition can significantly increase the rank of the coefficient matrix, can improve its morbidity, can to a certain extent reduce the data resolution, but can significantly increase the parameter resolution, and can reduce its variance from hundreds of meters down to centimeter level. These results can provide a theoretical basis for inverting InSAR deformation observations for focal slip distribution. During focal slip distribution inversion, adding constraint condition is indispensable.
This article then explores the effects of mathematical model error on InSAR inversion solutions of hypocenter parameter theoretically, performs the synthetic inversion tests, and investigates the distinction of functional model error and stochastic model error. Sources of model error in the InSAR inversion for hypocenter parameter is summarized, and model error theory of linear inversion from surveying data statistical analysis and method of Monte Carlo error estimation in the nonlinear inversion are introduced. By taking strike, oblique and thrust faults as examples, the effects of the functional model error and stochastic model error on inversion solutions of hypocenter parameter are analyzed by calculations of nonlinear inversion for hypocenter parameter and linear inversion for slip distribution, respectively. The results show that the functional model and the stochastic model containing errors can bias the hypocenter parameter solutions, and reduce the corresponding precision, consistent with results from theoretical prediction. Model error estimation and identification methods in the InSAR inversion for hypocenter parameter are discussed, and the distinction of functional model error and stochastic model error is investigated. The result indicates that effective separation of these two errors in InSAR inversion system of hypocenter parameter still faces larger challenge.
This article then presents the model error compensation method in the InSAR inversion for hypocenter parameter, implements the synthetic inversion experiments, and gives recommended inversion strategy. Model error compensation method from surveying data processing is summarized in detail, and strategy of adjusting stochastic model to compensate for model error in the InSAR inversion for hypocenter parameter is proposed. By introducing variance component estimation (VCE) and robust estimation theory and methods from surveying data processing, VCE algorithm, robust estimation algorithm and robust VCE algorithm suitable for linear and nonlinear inversion for hypocenter parameter are designed, and the corresponding concrete implementing procedures are presented respectively, where the equivalent variance-covariance function is used to construct the corresponding weight function. At the same time, compensation abilities of these methods are measured by exhaustively synthetic inversion experiments. Smoothing constraint condition equation is translated into virtual observation equation by relying on the virtual observation principle, smoothing factor is then expressed in the form of unit weight variance divided by virtual observation variance, and VCE method is used to work out the weight of observing data sets and smoothing factor, simultaneously. Three types of algorithms are elaborately compared, their compensation strengthes are also analyzed. If the observing data set contains gross error, robust estimation algorithm can better reduce or eliminate the negative effects of gross error on source parameter solutions when nonlinearly inverting for hypocenter parameter, but has certain limitations when linearly invering for slip distribution, where robust VCE algorithm is required. If two or more data sets contain gross errors, robust VCE algorithm performs better than VCE algorithm on ability of compensation for model error. And then the recommended inversion strategy in the actual InSAR inversion for hypocenter parameter is proposed.
This article then studies the October6,2008Dangxiong Mw6.3earthquake and November10,2008Dachaidan Mw6.3earthquake extensively. For Dangxiong earthquake, Envisat and ALOS image data with different tracks and different wavelengths are processed to extract high-quality InSAR coseismic deformation, robust VCE method designed by this study is used to reduce or eliminate the effects of model error on source parameter solutions. The results show that the focal slip mainly occurs in the depth range4.5-11km, the average rake angle and slip are-112.58°and0.50m, the maximum slip of1.53m is located in the depth range6.1-7.1km, and the corresponding seismic moment is4.22×1018N m (Mw6.38). Hypocenter parameter solutions from VCE have a certain bias. Biases without (robust) VCE are larger than those with VCE, and deviation of rake angle is-4.195°. Precisions without (robust) VCE are also significantly worse than those with (robust) VCE. For Dachaidan earthquake, Envisat image data is processed to extract high-quality InSAR coseismic deformation, nonlinear robust estimation is used to determine the focal fault geometry, and further linear robust VCE method is adopted to derive the fine focal slip distribution. The results show that the focal slip mainly occurs in the depth range10～20km, the average rake angle and slip are104.2°and0.2m, the maximum slip of0.64m is located in the depth range13.8-14.6km, and the corresponding seismic moment is3.74×1018N m (Mw6.35). Due to that the drived InSAR deformation field contains fewer gross errors with smaller magnitude, whether do robust estimation has little impact on the precision level of the parameter estimation when nonlinearly inverting for hypocenter parameter, but (robust) VCE is needed to fix the reasonable smoothing factor when linearly inverting for fine focal slip model.
This article finally explores the weight of coseismic deformation observation data when inverting for hypocenter parameter. VCE method can better determine the weight ratio among different observation data sets, value of which is neither equal, nor equal to value calculated according to prior variances. Weight ratios from nonlinear inversion for hypocenter parameter and linear inversion for focal slip distribution are not equal and differ widely in magnitude, and should be estimated by VCE principle, respectively. The weights of the observing data points neither increase as far from the fault, nor increase as close to the fault. The criterion of assessing weight by distance from the fault trace is not reasonable. Considering mathematical model error, robust estimation principle can provide a fittable way to determine weight for each observing data point.
本文建立了震源参数InSAR反演的数学模型并对其特征进行了分析。基于矩形位错理论、拉普拉斯平滑约束方法构建了震源参数InSAR反演的函数模型、随机模型及附等式约束的数学模型。将广义反演分析法引入震源滑动分布的反演分析中,给出了反演函数模型的数据分辨率、参数分辨率及方差的计算方法,并以走滑、逆冲断层及当雄地震为例分析了增加观测数据、附加约束条件对系数矩阵的数学特征项的影响。结果表明,增加观测数据可以在一定程度上增加系数矩阵的秩但不能改善其病态性,附加约束条件可以显著增加系数矩阵的秩并能改善其病态性,在一定程度上降低了数据分辨率,但较明显地增加了参数分辨率,并将其方差大小由几百米降低至厘米级。该结果为基于InSAR形变观测数据反演震源滑动分布提供了理论研究基础。在震源滑动分布反演中,附加一定的约束条件不可或缺。
本文对数学模型误差对震源参数InSAR反演估值的影响进行了理论和模拟反演分析,并对函数模型误差和随机模型误差的区分性进行了探讨研究。在系统总结并分析震源参数InSAR反演中数学模型误差来源的基础上,引入了测量数据统计分析中线性反演的模型误差理论和非线性反演的蒙特卡罗误差估计方法。以走滑、斜滑、逆冲三种主要震源类型为例,通过非线性反演震源参数和线性反演震源滑动分布模拟分析了函数模型和随机模型存在误差对震源参数估值的影响。结果表明,函数模型和随机模型存在误差使得震源参数估值产生偏差、精度降低,与理论分析结果相一致。讨论了震源参数InSAR反演中模型误差的估计和识别方法,并对函数模型误差和随机模型误差的区分性进行了探讨分析,指出震源参数InSAR反演系统中对二者进行有效分离具有较大的挑战性。
本文给出了震源参数InSAR反演的模型误差补偿方法并进行了反演计算分析,进一步地,给出了建议的反演方法。在系统总结并对比分析测量数据处理中模型误差补偿方法的基础上,提出了震源参数InSAR反演中通过调整随机模型对模型误差进行补偿的思路。通过引入测量数据处理中的方差分量估计和抗差估计理论和方法,设计了震源参数线性和非线性反演的方差分量估计类算法、抗差估计类算法和抗差方差分量估计类算法,给出了具体的反演计算步骤,通过模拟反演计算对算法的补偿效果进行了检测,其中,采用等价方差一协方差函数的抗差估计思想构建相应的权函数。基于虚拟观测原理将光滑约束条件方程转化为虚拟观测方程,将光滑因子表达为单位权方差与虚拟观测方差之商的形式,采用方差分量估计原理同时确定观测数据集权值和光滑因子的大小。在理论上对三类算法进行了比较分析,并对它们的补偿效果进行了分析。若观测数据集含有粗差,抗差估计算法能够较好地减免粗差对震源参数非线性反演估值的不良影响,但在滑动分布反演时具有一定的局限性,需要采用抗差方差分量估计算法；若两个或两个以上的观测数据集含有粗差,抗差方差分量估计算法的模型误差补偿效果优于方差分量估计算法。进一步地,给出了实际震源参数InSAR反演研究中建议的反演方法。
本文以2008年10月6日当雄Mw6.3级地震和2008年11月10日大柴旦Mw6.3级地震为实际震例进行了系统深入的反演研究。就当雄地震而言,利用不同轨道、不同波长的Envisat和ALOS影像数据提取该地震的高质量InSAR同震形变场,采用本文设计的抗差方差分量估计类算法减免数学模型误差对震源参数估值的影响。结果表明,震源滑动分布主要发生在4.5-11km范围内,平均滑动角为-112.58°,平均滑动量为0.50m,最大滑动量为1.53m,深度位于6.1-7.1km范围内,依据该滑动分布模型得到的地震矩为4.22×1018Nm(Mw6.38)。方差分量估计确定的震源参数估值存在一定的偏差,(抗差)方差分量估计前确定的震源参数估值偏差整体上大于方差分量估计的结果,滑动角的偏差达-4.195°,精度水平也显著差于(抗差)方差分量估计的结果。就大柴旦地震而言,利用Envisat影像数据提取该地震的高质量InSAR同震形变场,采用非线性抗差估计反演确定震源破裂的几何参数,进一步地,采用线性抗差方差分量估计反演确定精细的震源滑动分布。结果表明,震源滑动分布主要发生在10-20km范围内,平均滑动角为104.2°,平均滑动量为0.2m,最大滑动量为0.64m,深度位于13.8-14.6km范围内,依据该滑动分布模型得到的地震矩为3.74×1018N m(Mw6.35)。由于InSAR形变场中包含的粗差观测值较少且量级较小,震源参数非线性反演时,是否进行抗差估计对参数估值的精度水平影响不大,但震源滑动分布线性反演时,需进行(抗差)方差分量估计以确定合理的光滑因子。
最后,讨论并分析了同震形变观测数据在震源参数反演中的权值大小。方差分量估计法可以较好的确定观测数据集间的权比,其既不等权,也不等于基于验前方差确定的权比。非线性反演震源参数和线性反演震源滑动分布时确定的权比并不相等,且差异较大,数据集间的权比需要分别通过方差分量估计原理予以确定。观测数据点的权值既不随着远离断层而增大,也不随着靠近断层而增大,按距离断层远近为准则对观测数据点定权并非合理。考虑数学模型误差,抗差估计原理可以为观测数据点的定权提供一种合理的方法。
Rapid development of Interferometric Synthetic Aperture Rarar (InSAR), greatly enriching crustal deformation observation data, enables the geoscience research community to probe into the various geophysical phenomena related to earthquake fault in a fully new perspective. When extracting hypocenter parameter, geodetic data can effectively compensate for intrinsic limitations of other observations such as surface rupture data and seismic record. Hypocenter parameter can not only be used to analyze the seismogenic fault mechanism and regional tectonic stress, but can also provide a basis for studying rupture and expansion of the active fault, evolution of the characteristic fault, mechanism of the postseismic deformation, strain absorption and adjustment of the continental lithosphere, stress changes and seismic hazard assessment in the future. Consequently, the accuracy of hypocenter parameter holds a more and more attention of geoscience scientist. To determine a more precise hypocenter parameter, this dissertation creatively carries out the work of InSAR inversion considering model error for hypocenter parameter.
This article first establishes the mathematical model of InSAR inversion for hypocenter parameter, and studies its intrinsic features. According to the rectangular dislocation theory and Laplace smoothing constraint method, functional model, stochastic model and mathematical model with equality constraint appropriate for InSAR inversion for hypocenter parameter are proposed. Method of the generalized inverse analysis is introduced into focal slip distribution inversion to give the calculation formulas of data resolution, parameter resolution and variance for the coefficient matrix of the functional model. By taking strike, thrust faults and Dangxiong earthquake as examples, the effects of adding observation data and adding constraint condition on the mathematical feature items are studied. The results show that adding observation data can to a certain extent increase the rank of the coefficient matrix, but cannot improve its morbidity; adding constraint condition can significantly increase the rank of the coefficient matrix, can improve its morbidity, can to a certain extent reduce the data resolution, but can significantly increase the parameter resolution, and can reduce its variance from hundreds of meters down to centimeter level. These results can provide a theoretical basis for inverting InSAR deformation observations for focal slip distribution. During focal slip distribution inversion, adding constraint condition is indispensable.
This article then explores the effects of mathematical model error on InSAR inversion solutions of hypocenter parameter theoretically, performs the synthetic inversion tests, and investigates the distinction of functional model error and stochastic model error. Sources of model error in the InSAR inversion for hypocenter parameter is summarized, and model error theory of linear inversion from surveying data statistical analysis and method of Monte Carlo error estimation in the nonlinear inversion are introduced. By taking strike, oblique and thrust faults as examples, the effects of the functional model error and stochastic model error on inversion solutions of hypocenter parameter are analyzed by calculations of nonlinear inversion for hypocenter parameter and linear inversion for slip distribution, respectively. The results show that the functional model and the stochastic model containing errors can bias the hypocenter parameter solutions, and reduce the corresponding precision, consistent with results from theoretical prediction. Model error estimation and identification methods in the InSAR inversion for hypocenter parameter are discussed, and the distinction of functional model error and stochastic model error is investigated. The result indicates that effective separation of these two errors in InSAR inversion system of hypocenter parameter still faces larger challenge.
This article then presents the model error compensation method in the InSAR inversion for hypocenter parameter, implements the synthetic inversion experiments, and gives recommended inversion strategy. Model error compensation method from surveying data processing is summarized in detail, and strategy of adjusting stochastic model to compensate for model error in the InSAR inversion for hypocenter parameter is proposed. By introducing variance component estimation (VCE) and robust estimation theory and methods from surveying data processing, VCE algorithm, robust estimation algorithm and robust VCE algorithm suitable for linear and nonlinear inversion for hypocenter parameter are designed, and the corresponding concrete implementing procedures are presented respectively, where the equivalent variance-covariance function is used to construct the corresponding weight function. At the same time, compensation abilities of these methods are measured by exhaustively synthetic inversion experiments. Smoothing constraint condition equation is translated into virtual observation equation by relying on the virtual observation principle, smoothing factor is then expressed in the form of unit weight variance divided by virtual observation variance, and VCE method is used to work out the weight of observing data sets and smoothing factor, simultaneously. Three types of algorithms are elaborately compared, their compensation strengthes are also analyzed. If the observing data set contains gross error, robust estimation algorithm can better reduce or eliminate the negative effects of gross error on source parameter solutions when nonlinearly inverting for hypocenter parameter, but has certain limitations when linearly invering for slip distribution, where robust VCE algorithm is required. If two or more data sets contain gross errors, robust VCE algorithm performs better than VCE algorithm on ability of compensation for model error. And then the recommended inversion strategy in the actual InSAR inversion for hypocenter parameter is proposed.
This article then studies the October6,2008Dangxiong Mw6.3earthquake and November10,2008Dachaidan Mw6.3earthquake extensively. For Dangxiong earthquake, Envisat and ALOS image data with different tracks and different wavelengths are processed to extract high-quality InSAR coseismic deformation, robust VCE method designed by this study is used to reduce or eliminate the effects of model error on source parameter solutions. The results show that the focal slip mainly occurs in the depth range4.5-11km, the average rake angle and slip are-112.58°and0.50m, the maximum slip of1.53m is located in the depth range6.1-7.1km, and the corresponding seismic moment is4.22×1018N m (Mw6.38). Hypocenter parameter solutions from VCE have a certain bias. Biases without (robust) VCE are larger than those with VCE, and deviation of rake angle is-4.195°. Precisions without (robust) VCE are also significantly worse than those with (robust) VCE. For Dachaidan earthquake, Envisat image data is processed to extract high-quality InSAR coseismic deformation, nonlinear robust estimation is used to determine the focal fault geometry, and further linear robust VCE method is adopted to derive the fine focal slip distribution. The results show that the focal slip mainly occurs in the depth range10～20km, the average rake angle and slip are104.2°and0.2m, the maximum slip of0.64m is located in the depth range13.8-14.6km, and the corresponding seismic moment is3.74×1018N m (Mw6.35). Due to that the drived InSAR deformation field contains fewer gross errors with smaller magnitude, whether do robust estimation has little impact on the precision level of the parameter estimation when nonlinearly inverting for hypocenter parameter, but (robust) VCE is needed to fix the reasonable smoothing factor when linearly inverting for fine focal slip model.
This article finally explores the weight of coseismic deformation observation data when inverting for hypocenter parameter. VCE method can better determine the weight ratio among different observation data sets, value of which is neither equal, nor equal to value calculated according to prior variances. Weight ratios from nonlinear inversion for hypocenter parameter and linear inversion for focal slip distribution are not equal and differ widely in magnitude, and should be estimated by VCE principle, respectively. The weights of the observing data points neither increase as far from the fault, nor increase as close to the fault. The criterion of assessing weight by distance from the fault trace is not reasonable. Considering mathematical model error, robust estimation principle can provide a fittable way to determine weight for each observing data point.
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