地震震级—频度关系b值参数的稳健估计

文中依据古登堡－里克特震级－频度公式，结合实例说明了参数ｂ值的稳健估计方法，并将其与最小二乘法及最大似然法比较，结果表明，一般情况下，稳健估算结果与实际数据分布较吻合，并且可以在不剔除低震级段和大震级的偏离分布值的情况下，取得与选取合理震级上下限最小二乘法一致的结果，此外，稳健估计结果的余差分析同样也得到地震震级－频度关系不是简单的对数线性关系的结果，而原始Ｇ－Ｒ公式需加二以上的高次项才能较好地押     

b值的稳健估计及其在地震预报中的应用

利用Ｍ估计的稳健回归方法,选取ψ函数为正态密度型函数,计算震级－频度关系中的ｂ值,并会计了稳健估计方法与最小二乘估计所得结果的差异。认为由稳健估计得到的ｂ值更加符合震级－频度关系中的大小比例,可作为可靠的背景值使用。作为应用实例对新疆地区ｂ值进行空间及时间扫描,并定义最小二乘估计ｂ值与稳健估计ｂ值的偏差,以此为中强震的前兆判定指标,对喀什－巴楚地区８０年代以来６级以上地震前的异常进行讨论,认为该指     

稳健估计b值及中强震异常特征研究

利用M估计的稳健回归方法，选取ψ函数为正态密度型函数，计算震级-频度关系中的b值，对基于随机泊松分布的模拟地震样本及天然地震样本，用稳健估计方法与最小二乘方法计算b值后进行比较得出，对于天然地震样本，稳健估计得到的b值更加符合震级-频度关系，作为应用实例，将最小二第六估计法b值与稳健估计b值更加符合震级-频度关系，作为应用实例，将最小二乘估计法b值与稳健估计b值的偏离度作为中强震的前兆判定指标，对新疆地区b值进行时间扫描，在对北天山及南天山东段Ms≥5、南天山西段Ms≥6级地震前异常特征进行讨论后，认为该指标作为中甜地震预报指标具有一定的实用性。     

地震震级—频度G—R关系的偏离度

假设震级－频度分布服从对应于Ｇ－Ｒ式母体分布污染的情况，文中引进震级分布偏离度来描述实际资料分布与Ｇ－Ｒ式的偏离程度。在Ｇ－Ｒ式分参数稳健估计的基础上给出了偏离度的计算方法。华北地区１９８０－１９９３年取资料时间窗长１年，滑动步长１个月的ζ值计算结果表明，６级左右强地震前的半年至一个半左右ζ值明显为相对稳定的低值，震后呈高值变化。     

震级—频度关系的非线性特征

在假设地震断裂系统具有分形特征前提下，推导出非线性震级－频度关系公式：ＬｏｇＮ＝ａ＋ｂ△Ｍ－Ｃ／△Ｍ，作为示例，用此公式对一些实际观测资料进行了最小二乘拟合，效果明显好于线性公式。     

怎样正确计算b值

在我们研究震级一频度客观分布的基础上,根据数理统计理论和大量的实际地震活动资料,对b值的正确计算问题进行了详细的讨论,结果认为最小二乘法是计算b值的最好方法。此外,还认真地研究了与b值正确计算有关的下限震级、上限震级、样本数以及震级分档区间等具体问题,并给出了它们的合理选择方法。     

地震活动期、幕最大地震M1的估计及中-长期地震趋势预测

提出一种地震活动期、幕最大地震M1的估计方法：利用中低震级累计频度∑N与震级近似的直线关系得到间隔地震频度N与震级的线性关系。用间隔度N与震级的线性关系估计最大地震M1。结果表明，以震级－间隔频率的线性关系估计的M1比震级－累计频度关系得到的M1更接近实际发生的最大地震震级，震级－累计频率关系得到M1通常被高估。基于震级－累计频度关系曲线的特征和震级－间隔频度的线性关系估计的最大地震M1，可为中－长期地震趋势预测提供新的信息。研究表明，中国大陆自1979年开始的地震幕即将结束，而帕米尔东北侧自1915年以来的地震平静期也将结束。     

地震前兆数据的稳健回归与建模

基于正态密度ψ函数的M 估计,建立了含有趋势和周期项组合的稳健回归数学模型,对含有离群值的仿真数据进行了最小二乘估计、稳健估计和修正离群值后的最小二乘估计.结果表明稳健估计可以克服最小二乘估计受离群值影响较大的弊病,模型参数更接近实际.对地倾斜和地下气体等前兆观测数据的实际算例表明,用稳健回归方法建立的数学模型避免了少数离群值的干扰影响,更加真实地反映了前兆观测数据的变化趋势,是前兆数据趋势变化分析的强有力的数学工具.     

地震活动性的层次模型及推导的地震活动经验公式和分维性质

文章从一个简单的地震活动层次模型出发，推导地震活动性中几个经验公式以及地震活动的时、空和震源大小分维性质。研究结果表明：（１）震级和频度关系式即Ｇｕｔｅｎｂｅｒｇ－Ｒｉｃｈｔｅｒ公式中的比例系数ｂ值等于震级和平均震中距关系式中的比例系数ｂ１的２倍，并且ｂ值也等于震级和平均时间间隔关系式中的比例系数ｂ２。（２）平均震中距分维值为２，平均时间间隔分维值为１，震源大小分布的分维值的２ｂ。     

地壳形变分析的抗差最小二乘配置迭代解法

利用最小二乘配置进行地壳形变分析,其结果的合理性关键在于经验协方差函数的拟合.考虑到观测数据存在粗差的情况,提出基于观测值中位数初值的抗差最小二乘配置方法和基于中位参数法的抗差最小二乘配置方法.两种方法首先分别利用观测值中位数给出观测值初始权阵以及利用中位参数法给出最小二乘配置初始解,然后均在给定协方差函数参数初始值的情况下,应用合适的等价权进行抗差估计并通过迭代计算,最终获得稳健的协方差函数参数估值及最小二乘配置解.利用本文提出的两种方法以及传统方法分别对庐山地震的GPS垂直位移数据和意大利L'Aquila地震的InSAR同震位移数据进行处理分析.结果表明:相对传统方法,基于观测值中位数初值的抗差最小二乘配置方法效果更好,更具稳健性.     

基于MGR模型修正我国大陆中强以上地震的震级-频度关系和确定震级极限值div

从理论上推导了修正的古登堡-里克特模型的震级上限的确定方法, 应用常用的两种模型对我国分区域进行了震级-频度关系的修正和震级极限值的确定. 结果表明, 我国大陆存在非线性的震级-频度关系, 传统的NGR模型不能反映非线性的震级-频度关系, 而MGR模型能反映适合非线性和线性的震级-频度关系, 因此MGR模型适合我国各研究区中强以上地震的震级-频度关系. 当震级 频度关系为非线性时, 用MGR模型确定的震级极限值较准确．      

A robust weighted total least squares algorithm and its geodetic applications

Total least squares (TLS) can solve the issue of parameter estimation in the errors-invariables (EIV) model, however, the estimated parameters are affected or even severely distorted when the observation vector and coefficient matrix are contaminated by gross errors. Currently, the use of existing robust TLS (RTLS) methods for the EIV model is unreasonable. Original residuals are directly used in most studies to construct the weight factor function, thus the robustness for the structure space is not considered. In this study, a robust weighted total least squares (RWTLS) algorithm for the partial EIV model is proposed based on Newton-Gauss method and the equivalent weight principle of general robust estimation. The algorithm utilizes the standardized residuals to construct the weight factor function and employs the median method to obtain a robust estimator of the variance component. Therefore, the algorithm possesses good robustness in both the observation and structure spaces. To obtain standardized residuals, we use the linearly approximate cofactor propagation law for deriving the expression of the cofactor matrix of WTLS residuals. The iterative procedure and precision assessment approach for RWTLS are presented. Finally, the robustness of RWTLS method is verified by two experiments involving line fitting and plane coordinate transformation. The results show that RWTLS algorithm possesses better robustness than the general robust estimation and the robust total least squares algorithm directly constructed with original residuals.     

THE MAXIMUM LIKELIHOOD ESTIMATION OF b-VALUE IN MAGNITUDE-FREQUENCY RELATION AND ANALYSIS OF ITS INFLUENCING FACTORS

b-value in the magnitude-frequency(G-R)relationship plays a vital role in seismicity research and seismic hazard analysis, and the most commonly used techniques to simulate it are least square approach and maximum likelihood method. Least square method is simple and easy to apply, therefore widely used in China. However, many researches show that there exist some limits in least square estimation of b-value. Earthquakes with different magnitudes are not equally weighted in this method, and larger events have higher weights, so b-value is vulnerable to the fluctuation of several big earthquakes; meanwhile, least square method needs to divide magnitude intervals artificially. With a small sample size, data points could be not enough if the magnitude interval is too wide, and events in a magnitude interval may be lacking if it is divided to be too narrow. Especially for incremental G-R relationship, it is possible that N(M_i)equals 0 in an interval with large magnitude, so log(N(M_i))loses meaning and has to be ignored, resulting in a low b-value. Therefore, under certain conditions, maximum likelihood method is recommended as an effective substitution or supplementary for least square estimation of b-value. Among numerous previous researches on maximum likelihood estimation of b-value, lots of equations have been provided, based on varied implicit assumptions and different ways of solution. A brief overview is first presented for these equations, and classification and summary are provided based on whether taking account of the effect of binned magnitude, with finite maximum magnitude, using unequal observation periods for different magnitude intervals, and with analytic solution or not. Following this, a total of 6 influential factors are analyzed, such as binning magnitude, measurement errors of magnitude, sample size, magnitude span, minimum completeness magnitude and fore- and aftershocks. At last, reasonable suggestions are provided for using those equations properly. The equations of Aki(1965), Utsu(1965), Page(1968)and Kijko and Smit(2012)are based on assumption that magnitudes are continuous random variables, and have no corrections for this, so these equations are not recommended here. For simplicity, the equations of Utsu(1966)or Tinti and Mulargia(1987)can be used, but magnitude span should be greater than 2.5 due to without finite maximum magnitude in the formulas. For researchers having capability to write code and calculate numerically, Weichert(1980)or Bender(1983)'s algorithm could be utilized. Especially when it is required to apply data with different observation periods for varied magnitudes, the formula of Weichert(1980)is recommended. This study contributes to more accurately understand and use different formulas of estimating b-value by maximum likelihood technique, which can be used as reference for peers.     

用烈度资料反演Qssub>、扩散因子和震源S波加速度.

本文采用发生在中国的35个地震的烈度资料,用阻尼最小二乘法解超定方程组的方法,反演了中国7个区的1秒周期S波品质因子Q_S的平均值、扩散因子和震源S波加速度.并探讨了震源S波加速度与震级的关系.作为一个例子,反演了台湾地区Qs值的二维平面分布,发现Qs值分布与布格重力异常、地震活动性存在某种对应关系. 对反演结果进行了初步分析和讨论,并估计了Qs的误差.      

围压下岩石破坏声发射测试系统及震级频度关系的实验研究

震级频度关系的Gutenburg-Richter公式近年来受到了挑战,一些学者在不同条件下都发现了b值的非线性表现。本文利用自己装配的实验系统对围压下岩石破裂声发射活动的震级频度关系和时间序列特征进行了研究。实验表明AE事件的震级频度关系在某个中强震级K′之前基本符合b值规律,其b值的大小与破裂方式有关。在摩擦滑动中AE序列具有低b值的特征。从K′起震级频度关系出现明显的非线性现象。根据非线性表现的不同,可将其分为第一型和第二型两类。在AE序列中,也存在着“特征震级”的现象     

A Robust Method for Relative Gravity Data Estimation with High Efficiency

When gravimetric data observations have outliers, using standard least squares (LS) estimation will likely give poor accuracies and unreliable parameter estimates. One of the typical approaches to overcome this problem consists of using the robust estimation techniques. In this paper, we modified the robust estimator of Gervini and Yohai (2002) called REWLSE (Robust and Efficient Weighted Least Squares Estimator), which combines simultaneously high statistical efficiency and high breakdown point by replacing the weight function by a new weight function. This method allows reducing the outlier impacts and makes more use of the information provided by the data. In order to adapt this technique to the relative gravity data, weights are computed using the empirical distribution of the residuals obtained initially by the LTS (Least Trimmed Squares) estimator and by minimizing the mean distances relatively to the LS-estimator without outliers. The robustness of the initial estimator is maintained by adapted cut-off values as suggested by the REWLSE method which allows also a reasonable statistical efficiency. Hereafter we give the advantage and the pertinence of REWLSE procedure on real and semi-simulated gravity data by comparing it with conventional LS and other robust approaches like M- and MM-estimators.     

Determination ofQ S,spreading factor and S-wave acceleration by inversion of seismic intensity data

In this paper, on the basis of intensity data from 85 earthquakes occurred in China, the method of resolving overdetermined equations by using the damped least squares method is applied to inversing for the mean value of 1 s. S wave quality factor, the spreading factor and the S-wave acceleration at the foci of earthquakes in 7 regions of China. The relation between S-wave acceleration at the foci of earthquakes and magnitude is discussed. As an example, 2-DQ _S distribution in Taiwan Province is obtained by inversion. It is found that there exists some corresponding relation between this distribution and Bouguer gravitational anomaly and seismicity. Preliminary analysis and discussion on results of the inversion, and estimation of error inQ _S are made.     

震级-频度关系的修改及其对部分地震区带的应用

在本文中作者根据对实际资料的计算,提出将传统的震级-频度关系修改成logN=b_1+b_2M+b_3M~2的形式。用该公式对我国部分地震区带的地震资料进行了计算,结果表明,修改后的震级-频度关系的拟合精度显著提高,其残差比原关系式缩小。用修改后的公式可方便地求得地震区带的震级上限,只要有足够时间长度的观测资料,所求得的震级上限是比较合理的。     

地球物理抗差估计和广义逆方法

为进行观测数据误差较大的地球物理资料的反演,引人抗差估计,称作地球物理抗差估计;为使病态方程组的反演解更可靠,又结合广义逆方法进行了算法改进．首先介绍抗差最小二乘（ＲＬＳ）的基本原理,然后推寻出适于广义逆反演方法的改进算式,最后举例加以讨论．分析表明,抗差估计可以有效地抑制地球物理观测异常的影响,得出正常模式下的较好估计值;而用改进后的算式和广义逆反演可以使反演解更加改善,不仅如此,改进后的方法还能直接给出解估计的可靠性评价．