﻿ 用U曲线法确定地震同震滑动分布反演正则化参数
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 大地测量与地球动力学  2018, Vol. 38 Issue (11): 1196-1201  DOI: 10.14075/j.jgg.2018.11.019

### 引用本文

WANG Leyang, ZHAO Xiong. Using U Curve Method to Determine the Regularization Parameters of Coseismic Earthquake Slip Distribution Inversion[J]. Journal of Geodesy and Geodynamics, 2018, 38(11): 1196-1201.

### Foundation support

National Natural Science Foundation of China, No. 41874001, 41664001;Support Program for Outstanding Youth Talents in Jiangxi Province, No. 20162BCB23050; National Key Research and Development Program of China, No. 2016YFB0501405; Innovation Fund Designated for Graduate Students of Jiangxi Province, No. YC2017-S279.

### 第一作者简介

WANG Leyang, PhD, associate professor, majors in geodetic inversion and geodetic data processing, E-mail:wleyang@163.com.

### 文章历史

1. 东华理工大学测绘工程学院，南昌市广兰大道418号, 330013;
2. 流域生态与地理环境监测国家测绘地理信息局重点实验室，南昌广兰大道418号, 33001;
3. 江西省数字国土重点实验室，南昌广兰大道418号, 330013

1 U曲线法确定正则化参数 1.1 地震滑动分布反演基本方程

 $\mathit{\boldsymbol{d}} = \mathit{\boldsymbol{Gm}} + \mathit{\boldsymbol{\varepsilon }}$ (1)

 $\mathit{\boldsymbol{Hm}} = {\bf{0}}$ (2)

 $\begin{array}{l} \;\;{\left\| {\mathit{\boldsymbol{d}} - \mathit{\boldsymbol{Gm}}} \right\|^2} + \alpha \Omega \left( \mathit{\boldsymbol{m}} \right) = \\ {\left\| {\mathit{\boldsymbol{d}} - \mathit{\boldsymbol{Gm}}} \right\|^2} + \alpha {\mathit{\boldsymbol{m}}^{\rm{T}}}\mathit{\boldsymbol{Rm}} = {\rm{min}} \end{array}$ (3)

1.2 U曲线法基本原理

 $\mathit{\boldsymbol{m}} = {({\mathit{\boldsymbol{G}}^{\rm{T}}}\mathit{\boldsymbol{pG}} + \alpha \mathit{\boldsymbol{R}})^{-1}}{\mathit{\boldsymbol{G}}^{\rm{T}}}\mathit{\boldsymbol{pd}}$ (4)

U曲线法确定正则化参数与L曲线法类似。L曲线法是根据不同的α值分别得到一组‖ d-Gm2、‖Hm2值，以‖ d-Gm2为横坐标、‖Hm2为纵坐标拟合成一条类似“L”形状的曲线，取L曲线上拐点附近对应的α值作为最优正则化参数[10]。可以看出，L曲线法的精度依赖于‖d-Gm2与‖Hm2数据之间的拟合程度。U曲线法则是通过定义U(α)函数，根据α的取值得到U(α)-α曲线，曲线左侧近似垂直部分曲率最大点所确定的α值即为最优正则化参数[11-12]。U曲线法定义如下：

 $U\left( \alpha \right) = \frac{1}{{{{\left\| {\mathit{\boldsymbol{d}} - \mathit{\boldsymbol{Gm}}} \right\|}^2}}} + \frac{1}{{{{\left\| {\mathit{\boldsymbol{Hm}}} \right\|}^2}}}$ (5)

1.3 两种方法比较

 ${\rm{RMS}} = \sqrt {\frac{{\sum\limits_{i = 1}^n {{P_i}{{({d_i} - {c_i})}^2}} }}{n}}$ (6)

2 模拟实验

 图 1 模拟InSAR数据形变观测点 Fig. 1 Simulating the deformation observation points of InSAR data

 图 2 用L曲线法、U曲线法确定正则化参数 Fig. 2 The regularization parameters obtained by using L-curve and U-curve method

 图 3 模拟实验两种方法反演地震滑动分布结果及解的残差 Fig. 3 Slip distribution results of simulation experiment and the residuals of two methods

3 芦山实际震例反演

2013-04-20 08：02四川芦山发生M7.0地震，震源深度13 km。地震发生后，不同研究机构对此次地震进行快速反演分析，研究地震震源机制和反演地震震源参数，但基于不同的手段所反演出的震源参数结果具有一定的差异。本文采用文献[14]获取的GPS三维形变数据来约束地表形变场，利用文献[1]通过多峰值颗粒群算法获得的芦山地震断层参数作为反演芦山地震的断层参数。在此基础上将断层破裂面沿断层走向、倾向把断层长度、宽度均扩展至51 km，并将破裂面延伸至地表，将断层面均匀剖分成1.5 km×1.5 km大小的矩形单元，共得到了1 156个矩形断层单元。

 图 4 用L曲线法、U曲线法确定正则化参数 Fig. 4 The regularization parameters obtained by using L-curve and U-curve method

 图 5 芦山地震滑动分布反演结果及两种方法反演残差分布 Fig. 5 Slip distribution results of Lushan earthquake and the residuals of two methods

4 结语

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Using U Curve Method to Determine the Regularization Parameters of Coseismic Earthquake Slip Distribution Inversion
WANG Leyang1,2,3     ZHAO Xiong1,2
1. Faculty of Geomatics, East China University of Technology, 418 Guanglan Road, Nanchang 330013, China;
2. Key Laboratory of Watershed Ecology and Geographical Environment Monitoring, NASMG, 418 Guanglan Road, Nanchang 330013, China;
3. Key Laboratory for Digital Land and Resources of Jiangxi Province, 418 Guanglan Road, Nanchang 330013, China
Abstract: The determination of regularization parameters is the key to the inversion of coseismic earthquake slip distribution. In view of the selection of regularization parameters in seismic slip distribution inversion, the U curve method is proposed in this paper. Using the U curve method and L curve method to design the simulation experiment. Moreover, the two methods are applied to the inversion of Lushan earthquake. The inversion results of simulation experiment and the Lushan earthquake show that the regularization parameters obtained by U curve method has the advantage of high accuracy and no need to rely on data fitting accuracy than L curve method in coseismic slip distribution inversion.
Key words: coseismic slip distribution; regularization parameters; U curve method; L curve method; Lushan earthquake