最小二乘估计残差的统计性质和应用

最小二乘估计求得的观测值残差,有多种形式的定义,它们具有不同的分布。本文论述它们的概率分布和性质并讨论其用途。编算了多余观测分量r_i和标准化残差e_i的临界值表。此外,应用线性变换求得了不相关残差W=U~TV和S=D_W~(-1/2)W 。建议用S作粗差检验,本文编制了相应于这种检验的max|S_i|的临界值表,最后计算了一个实测的测边网,用求得的各种残差作了统计检验。     

Magnetic interpretation of long horizontal cylinders using correlation factors between successive least-squares residual anomaly profiles

Procedures are formulated using the correlation factors between successive least-squares residual magnetic anomaly profiles due to long horizontal cylinders for interpreting the three principal anomalies (vertical, horizontal, and total). It is demonstrated that correlation values can be used to determine the depth to the center of the buried structure and the index parameter. Procedures are also formulated to estimate the amplitude coefficient. Two worked examples using theoretical data show the effectiveness of the present method.     

Systematic variations of mean travel-time residuals in space and time

Summary Mean travel-time residuals of P waves for the period 1964–1970 at stations in North America and Europe are well separated into large domains of positive and negative values. The spatial distribution of residuals in North America is in good agreement with the structure of the upper mantle obtained from Rayleigh wave dispersion and is in accord with magnitude anomalies and heat flow data. A systematic variation of residuals in time is found to be sensitive to major changes in the nature of the earthquake source distribution, as for example aftershocks, as well as to changes in methods of detection of first arrivals. Residuals at neighboring stations are correlated up to distances of the order of 2°. Half year mean residuals at stations within 2° distance from the epicenters of strong earthquakes have a sudden jump 1.5 years before the occurrence of an earthquake. A formal algorithm based on this pattern permits prediction of the times of all 16 earthquakes with magnitudeM7.5 occurring during the interval 1966–1972. Although the average duration of the alarm periods is about half the total time interval of the catalog, so that the algorithm has no practical importance, these results are statistically significant on a 99% level of confidence.     

Finite element solutions for mass oscillations in surge tanks

This paper describes a finite element technique using the method of weighted residuals for the solution of mass oscillations in surge tanks. Three weighting functions, uniform, linear and Galerkin, are applied and the results are compared with those from alternative techniques. The relatively simple case of surge analysis with flow rate change in the penstock, but neglecting tunnel friction, is first considered as a direct analytical solution is available. Finally friction is included for comparison with a graphical and analogue solution.     

The crust and upper mantle structure beneath southern Italy by earthquakes and DSS data

This paper describes a travel—time analysis performed for the Italian seismic stations, in particular those operating in southern Italy, in order to study the crust and upper mantle properties in the region. Average P-wave residuals of teleseisms in the distance range 30°–95° with respect to Jeffreys-Bullen tables, at thirteen permanent and temporary stations of southern Italy, are coherent with a high velocity zone beneath Calabria and northern Sicily and low velocity material in the mantle beneath the Eolian Islands. Travel—time residuals from Tyrrhenian intermediate earthquakes show a high velocity structure which extends in a NW direction from a depth of at least 200 km down to 450 km.A damped least-squares inversion applied to DSS data confirms the existence of low velocity zones in the crust beneath the Eolian Islands, at 8–12 km depth, that agrees with previous results and with the lack of S waves from local earthquakes.Publication No. 193, Progetto Finalizzato Geodinamica, CNR-Roma.     

Pg回折波的上地壳三维层析成象

提出利用人工地震Ｐｇ回折波走时重建区域上地壳三维速度分布的方法。正问题利用经典最小二乘迭代求解垂向速度梯度模型的参数值及相应走时残差。反问题采用泛函空间的最小二乘准则来实现。对速度不需要预先进行参数化处理，可以计算反演区内任意点的速度值。数值计算结果表明了该方法的可靠性。处理了滦县地震区人工地震Ｐｇ走时资料，获得了该区上地壳顶部的三维速度分布。     

A test sensitive to extreme hidden periodicities

A modified version of the widely used Kolmogorov-Smirnov (K-S) test of null hypothesis is constructed, that a given time series is Gaussian white noise, against the alternative hypothesis that the time series contains an added or multiplicative deterministic-periodic component of unspecified frequency. The usual KS test is treated as a special case. The proposed test is more powerful than the ordinary K-S test in detecting extreme (low or high) hidden periodicities. Computational procedure necessary for implementation are also given.     

Statistical regression applied to borehole strain measurements data analysis

Summary In mining and geotechnical engineering, it is usually necessary to carry out field measurements in order to obtain information. Parameters are often measured indirectly and calculated based on certain relationships to the measured quantities. More often, the number of measurements taken is greater than the minimum required, in order to increase the reliability of results. However, some data points are less reliable than others for reasons such as measurement errors; a solution which best fits the measurement data is obtained accordingly. As a result, there is a residual or a difference between the individual quantities measured and those predicted from the best-fit solution. This brings about a question of how big a residual is acceptable for a solution to be reliable. It is also important to know whether the data point with the largest residual is the most erroneous, whether those data points with large residuals should be deleted and how many of them should be deleted. Standard deviation may provide a measure of the data divergence but it is questionable if this parameter can be used as a measure of the reliability of solution. In order to solve these problems, the author has done extensive study in this area, especially as part of geotechnical data analysis. In this paper, the statistical multiple regression method is introduced to analyse the measurement data. The method is applied to the analysis ofin situ stress measurement and can be easily adopted to analyse data from other field measurements and laboratory tests. An example is included which illustrates the analysis procedure and shows the advantages of the method.     

北斗系统精密卫星钟差精度评价

针对目前北斗卫星导航系统尚未形成统一的事后轨道和钟差产品的现状,该文提出了一种较为全面的北斗卫星导航系统精密卫星钟差精度评价方法,从钟差拟合残差标准差和天跳变等方面对北斗系统事后精密卫星钟差的精度进行了评价。使用相同方法对2013年全年、2014年上半年的北斗系统精密卫星钟差产品进行了评价,以此来分析定轨策略调整对卫星钟差的影响。结果表明,武汉大学分析中心解算的北斗系统精密卫星钟差产品的精度为分米级,大部分卫星的钟差精度优于0.2m,且定轨策略调整后倾斜地球同步轨道/中地球轨道卫星钟差产品的精度有小幅度提升;定轨策略调整前卫星钟差天跳变在0.2~0.5m间,但是调整后钟差天跳变显著增大,变化范围为2~3m,建议在实际应用中对其加以慎重处理。     

Robust methods for assessing the accuracy of linear interpolated DEM

Methods for assessing the accuracy of a digital elevation model (DEM) with emphasis on robust methods have been studied in this paper. Based on the squared DEM residual population generated by the bi-linear interpolation method, three average-error statistics including (a) mean, (b) median, and (c) M-estimator are thoroughly investigated for measuring the interpolated DEM accuracy. Correspondingly, their confidence intervals are also constructed for each average error statistic to further evaluate the DEM quality. The first method mainly utilizes the student distribution while the second and third are derived from the robust theories. These innovative robust methods possess the capability of counteracting the outlier effects or even the skew distributed residuals in DEM accuracy assessment. Experimental studies using Monte Carlo simulation have commendably investigated the asymptotic convergence behavior of confidence intervals constructed by these three methods with the increase of sample size. It is demonstrated that the robust methods can produce more reliable DEM accuracy assessment results compared with those by the classical t-distribution-based method. Consequently, these proposed robust methods are strongly recommended for assessing DEM accuracy, particularly for those cases where the DEM residual population is evidently non-normal or heavily contaminated with outliers.