地震同震滑动分布反演的总体最小二乘方法

同震滑动分布参数与地表形变间的线性关系依赖于格林函数矩阵的构造,格林函数矩阵元素与破裂面位置、几何参数、破裂方式及位错模型假设等因素有关。本文尝试考虑格林函数矩阵元素的误差来补偿上述原因在一定程度上对反演参数的影响,采用同时顾及系数矩阵(格林函数矩阵)和观测向量两者误差的总体最小二乘方法反演同震滑动分布。首先确定了系数矩阵元素和观测向量的协因数矩阵,考虑到格林函数矩阵的病态性(秩亏),借助拉普拉斯二阶平滑得到正则化矩阵,采用总体最小二乘正则化法反演同震滑动分布。并对2009年意大利中部拉奎拉(L’Aquila)Mw6.3级地震实例进行同震滑动分布反演研究。结果表明,拉奎拉地震的走向为144.37°,倾角为59.06°,滑动分布的最大滑动量为0.95m,平均滑动角为-96.4°,主要滑动深度为4~15km的范围,地震矩为3.63×10~(18)N·m,对应的矩震级为Mw6.34。总体最小二乘与最小二乘法的滑动分布解存在一定差别,但差别的量级在10-4以内。

Total Least Squares Method Inversion for Coseismic Slip Distribution

The coefficient matrix (Green matrix) is composed of surface point offset caused by unit slip of sub-fault patches.The elements of the coefficient matrix are related to the location, geometry of rupture surface, assumption of model and other factors.In this paper, we attempted to consider the Green'function matrix (coefficient matrix) errors in order to compensate for the effects of above-mentioned factors to some extent.The total least squares (TLS) method, which both errors of coefficient matrix and observation vector are considered, is proposed for fault slip inversion.So we dealt with the errors in both of coefficient matrix and observation at same time.And by analysis of the relations between observation vector and coefficient matrix elements, we obtained the covariance matrix of coefficient matrix elements and observation vector.Considering the coefficient matrix was ill-posed, we used the second-order Laplace smoothing to constrain the slip parameters each other, then we used the regularized total least squares method to estimate slip distribution.the total least squares (TLS)slip inversion method was applied to simulate oblique fault event and Mw6.3 earthquake occurred in L'Aquila (central Italy) on April 6, 2009, respectively.To L'Aquila earthquake, the results by total least squares method indicate that the inverted geodetic moment is 3.63×1018N·m (Mw6.34).With a maximum slip of 0.95m, and a average rake of-96.4°, the main slip occurred at depth of 4km-15km.The difference of slip distribution solutions between total least squares and least squares method is less than 10-4 order.