Orthonormal residuals in geostatistics: Model criticism and parameter estimation

In linear geostatistics, models for the mean function (drift) and the variogram or generalized covariance function are selected on the basis of the modeler's understanding of the phenomenon studied as well as data. One can seldom be assured that the most appropriate model has been selected; however, analysis of residuals is helpful in diagnosing whether some important characteristic of the data has been neglected and, ultimately, in providing a reasonable degree of assurance that the selected model is consistent with the available information. The orthonormal residuals presented in this work are kriging errors constructed so that, when the correct model is used, they are uncorrelated and have zero mean and unit variance. It is suggested that testing of orthonormal residuals is a practical way for evaluating the agreement of the model with the data and for diagnosing model deficiencies. Their advantages over the usually employed standardized residuals are discussed. A set of tests are presented. Orthonormal residuals can also be useful in the estimation of the covariance (or variogram) parameters for a model that is considered correct.     

Recursive parameter estimation with applications in earth sciences

Parameter estimation has become increasingly interesting the last few decades for a variety of engineering topics. In such situations, one may face problems like (a) how to estimate parameters for which erroneous measurements are available (direct estimation), or (b) how to estimate coefficients of some process model governing a geological phenomenon when these coefficients are inaccessible or difficult to access by direct investigation (inverse estimation or identification). Both these problems are examined in this presentation from a modern stochastic viewpoint, where parameters sought are interpretated mathematically as random functions, generated and estimated in space or time with the aid of recursive models. Advantages of this methodology are remarkable, from both theoretical and physical points of view, as compared to conventional statistics or nonrecursive estimators. Particularly it may offer more accurate estimators, better representation of spatial variation, and a means of overcoming difficulties such as excessive computational time or computer storage. To test effectiveness of this type of estimation, a series of representative case studies from geotechnical practice have been computed in detail.     

丹麦法稳健估计在隧道椭圆拟合中的应用

隧道断面在设计时为一标准圆形,受到施工及外部环境的影响变形为偏心率很小的椭圆。减小粗差影响、精确确定椭圆的几何参数是其研究的重点。通过对上海某一隧道断面的测量试验,得出用丹麦法稳健估计可以有效减小粗差的影响,而且拟合值可以很好地逼近无粗差情况下的最小二乘拟合值。采用Matlab 语言实现算法,并详细介绍了用二项式变换求解椭圆参数的方法。     

A multiscale method for distributed parameter estimation with application to reservoir history matching

A method for multiscale parameter estimation with application to reservoir history matching is presented. Starting from a given fine-scale model, coarser models are generated using a global upscaling technique where the coarse models are tuned to match the solution of the fine model. Conditioning to dynamic data is done by history-matching the coarse model. Using consistently the same resolution both for the forward and inverse problems, this model is successively refined using a combination of downscaling and history matching until model-matching dynamic data are obtained at the finest scale. Large-scale corrections are obtained using fast models, which, combined with a downscaling procedure, provide a better initial model for the final adjustment on the fine scale. The result is thus a series of models with different resolution, all matching history as good as possible with this grid. Numerical examples show that this method may significantly reduce the computational effort and/or improve the quality of the solution when achieving a fine-scale match as compared to history-matching directly on the fine scale.     

Multiscale sources of spatial variation in soil. III. Improved methods for fitting the nested model to one-dimensional semivariograms

The previous paper in this series presented a one-dimensional stochastic nested model to account for superimposed sources of soil variation at various scales. This paper shows how the nested model can be fitted to experimental data using weighted or generalized least-squares methods that account for correlations between consecutive terms that had previously been neglected. This paper also presents a method of estimating effective degrees of freedom for each sampling interval and thus for estimating 90% confidence limits for the semivariogram of the nested model.     

水文频率分布模型的异同性与参数估计问题

回顾了20世纪50年代对水文频率分布模型异同性的认识,分析了分布模型在统计特征上的相异性与共通性。在分布的几何形态相同的条件下,目前所用的分布模型中,仅3个参数的模型基本上是相异的。通过观察与实例分析,初步发现,取4个参数,有可能使模型有统一的求解结果。用数学方法能得到参数的唯一解,但多数方法是先计算出不确定性较大的C_S值,继而据此算得其它一个或两个参数,故这种结果是一种初值,有待通过合理性分析,才能得到最终取用值。     

Automatic parameter estimation applied on a groundwater model: The problem of structure identification

The properties of a Pleistocene aquifer located in NW Germany were evaluated by means of an inverse two-dimensional numerical model in the vertical plane. The steady-state drawdown recorded at the end of a pumping test was used to calibrate the model with a Gauss-Newton parameter estimation method. The results were analyzed within a formal statistical procedure by means of eigenvalues and eigenvectors. Although a mathematically meaningful value for the standard error can be obtained, the resulting set of permeabilities depends strongly upon the choice of the zonation pattern.     

General and local estimation of local average and their application in geotechnical parameter estimations

Two kinds of estimation variance functions for estimating a local average (LA) of a stationary (homogeneous) random field (RF) are derived. One is local estimation (LE) and the other is general estimation (GE) of LA. The former is for estimating LA at the observation location, and the latter is for obtaining LA at any arbitrary location within the RF. The geotechnical implications of these two estimations are that LE is for estimating LA of geotechnical parameters at the spot where the investigations are made, whereas GE is for estimating LA at any arbitrary location within the same layer. The behavior of the two estimation variance functions differs greatly, controlled by the number of observations (i.e. sample size) and the normalized layer thickness (layer thickness divided by autocorrelation distance of RF). Based on the derived estimation variance functions, methods for determining reliable characteristic values of geotechnical parameters and necessary sample size are proposed. The methods are based on the same framework as that of the traditional statistical theory, i.e. confidence interval of estimated parameters. However, the assumption of independently and identically distributed (i.i.d.) samples in the traditional statistical theory is replaced by the assumption of correlated samples from a stationary RF. The results obtained from the proposed methods for LE and GE differ from each other as well as from the traditional results, which has significant implications for geotechnical parameter estimation in geotechnical engineering practice.     

非饱和土壤溶质迁移转化模型参数优化估算

在分析非饱和土壤中水分和溶质迁移转化规律的基础上,采用与坐标网格相适应的分块连续作用的水动力函数,研究模型参数最优化估算技术.运用Gauss-Newton最小化计算的Levenberg-Marquardt修正法来实现反求模型的迭代问题.该方法被应用于估算一维土柱和二维土槽试验研究模型的参数,探讨了参数给定与估算数量对参数的置信区间和预测效果的影响.经验证表明,参数优化估算技术是可行的,以此参数进行模型预测,其结果是可信的.     

Comparison of structural and least-squares lines for estimating geologic relations

Two different goals in fitting straight lines to data are to estimate a true linear relation (physical law) and to predict values of the dependent variable with the smallest possible error. Regarding the first goal, a Monte Carlo study indicated that the structural-analysis (SA) method of fitting straight lines to data is superior to the ordinary least-squares (OLS) method for estimating true straight-line relations. Number of data points, slope and intercept of the true relation, and variances of the errors associated with the independent (X) and dependent (Y) variables influence the degree of agreement. For example, differences between the two line-fitting methods decrease as error in X becomes small relative to error in Y. Regarding the second goal—predicting the dependent variable—OLS is better than SA. Again, the difference diminishes as X takes on less error relative to Y. With respect to estimation of slope and intercept and prediction of Y, agreement between Monte Carlo results and large-sample theory was very good for sample sizes of 100, and fair to good for sample sizes of 20. The procedures and error measures are illustrated with two geologic examples.     

新安江模型中河网汇流参数Cs的一种计算方法

新安江模型河网汇流参数Cs主要取决于河网形态及地貌特征规律,针对PUB（Prediction in Ungauged Basin）问题,从物理机理分析的角度,构建描述河网汇流动态变化的微分方程并给出适当的定解条件,用解析法得出微分方程的解,从而探索出在无实测水文资料地区适用的Cs计算方法。选取中国湿润地区和半湿润地区25个流域,统计并分析了各流域次洪模型的Cs计算值和率定值。结果表明,从总体来看计算值和率定值基本上相等;不论流域是否分块,提出的Cs计算方法均适用;当其他参数保持不变,虽然Cs取计算值时所得到的次洪模拟结果略逊于取率定值的模拟结果,但误差在可接受的范围之内,也能取得良好的模拟效果;如果令Cs取计算值,再重新率定其余的各项参数,能够使模拟精度得到进一步提高。采用该法可推求无实测资料地区的汇流参数Cs值,使模型在无资料地区的应用成为一种可能。     

Cole-Cole模型参数在时间域激电找矿中的应用

常规的时间域激电法获得参数较少,仅有电阻率和极化率两个参数,这使得激电异常往往难以准确解译,具体体现在矿体解译信息量较少和探测分辨率差上,特别是在极化率或电阻率数值接近的一些共生及伴生的矿产勘探中,如炭质灰岩、黄铁矿和褐铁矿共生的一些矿山.为了丰富矿体地质信息,提高对矿体的认识,减少误判和漏判,文中利用激电效应的时间响应可用Cole-Cole模型来描述的特征,通过阻尼最小二乘法实现时间响应与时间域Cole-Cole模型的最佳拟合,从而获得多种Cole-Cole模型参数.实际应用证明,该方法经济、有效,可通过多种物理参数综合解释,实现对激电异常更全面的认识和正确的推断解释,同时提高了时间域激电在深部找矿中的分辨率.     

线性回归与最小一乘法

论述了用最小一乘法估计两变数线性回归模型中的参数。计算方法为：通过定点的解法，一般解法，两点法和单纯形法。对于最小一乘解，在实际分析时会出现一些问题，如解的不灵敏性、解的多重性以及突出点的多少对回归线有不同影响等。     

一种新的本构模型及参数标定

在现行本构模型研究的基础上,建立了一种全新本构模型,它可以描述4种行为特征：弹性-弹塑性-软化、弹性-弹塑性-硬化、理想弹塑性、弹脆性;其模型参数只有4个,现行模型只是该模型的特例（如：邓肯-张模型）,介绍了模型参数的决定方法,其决定的依据是：模型所描述的行为特征与实测数据之间相关度最优;分析了岩土体破坏后区特征,提出岩土体破坏后区行为可以定义为视应力-视应变特征。并以实例对该模型进行了说明,结果表明,该模型可以描述岩土体的行为特征,并对虚内键模型、弹簧模型和离散单元法等具有借鉴作用。     

一个可考虑球应力和剪应力共同作用的结构性参数

土的结构性是岩土材料的基本特性,基于综合结构势理论建立的结构性参数,以表述土的结构性,但已有的表述存在缺陷,为此针对应变和应力结构性参数的缺点提出了一个可同时考虑球应力和剪应力作用的结构性参数--应力比结构性参数。通过理论分析和三轴试验研究发现,应力比结构性参数与应力或应变结构性参数相比,可同时考虑剪应力和球应力对土结构性的影响,并且剪切过程中结构性参数不再出现峰值,更符合人们以往对结构性的认识。该参数从定量上更加准确地反映了土的结构性。     

Quadratic covariance estimation and equivalence of predictions

This article illustrates the use of linear and nonlinear regression models to obtain quadratic estimates of covariance parameters. These models lead to new insights into the motivation behind estimation methods, the relationships between different methods, and the relationship of covariance estimation to prediction. In particular, we derive the standard estimating equations for minimum norm quadratic unbiased translation invariant estimates (MINQUEs) from an appropriate linear model. Connections between the linear model, minimum variance quadratic unbiased translation invariant estimates (MIVQUEs), and MINQUEs are examined and we provide a minimum norm justification for the use of one-step normal theory maximum likelihood estimates. A nonlinear regression model is used to define MINQUEs for nonlinear covariance structures and obtain REML estimates. Finally, the equivalence of predictions under various models is examined when covariance parameters are estimated. In particular, we establish that when using MINQUE, iterative MINQUE, or restricted maximum likelihood (REML) estimates, the choice between a stationary covariance function and an intrinsically stationary semivariogram is irrelevant to predictions and estimated prediction variances.     

Wu-Bauer亚塑性模型参数确定方法的改进及其验证

研究了Wu-Bauer亚塑性模型参数试验确定方法及其存在的问题。根据单形调优法,对模型参数确定方法进行了改进,并通过三轴固结排水试验、侧限压缩试验和加卸载试验进行了验证。分析表明,该方法可有效地确定模型参数。最后分析了Wu-Bauer亚塑性模型的优点和不足之处。     

基于主成分和半参数的大坝变形监测回归模型

主成分分析在一定程度上可以解决大坝变形监测回归模型因子间的复共线性,然而当提取的主成分信息不充分时,主成分回归用于大坝安全预测可能失效。提出以主成分分析提取的主成分作为半参数回归的参数分量,剩余成分和模型误差作为未知的非参数分量对主成分回归进行补偿,建立一种基于主成分和半参数的大坝变形监测混合回归模型。利用某大坝实测资料进行建模分析,结果表明该混合模型能克服回归因子间的复共线性,避免半参数回归补偿最小二乘估计中法矩阵的病态性,比传统的主成分回归和逐步回归模型具有更好的拟合和预报精度。     

深埋脆性岩石力学参数评估与变形特性探讨

隧道工程开挖深度有逐渐增加的趋势,深部岩石具高强度及高应力环境。虽然脆性岩石在高围压应力下,将转而呈现延性特性,但隧道开挖所产生的径向解压,使隧道周边一定范围内的岩体处在低围压状态,故这部分围岩的变形仍由脆性性质所掌控。对于脆性岩石呈现峰后强度弱化特性,文中汇整各种岩石峰后强度弱化参数评估方法后,提出结合Hoek-Brown准则与岩心试验的强度损失试验评估法。以实际工程案例分析结果显示,600 m埋深时岩石峰后强度弱化与否,对隧道围岩变形量影响不大,但隧道埋深超过2 000 m后,变形量的差异将达到3～4倍左右,显示脆性岩石峰后强度弱化特性,对深埋隧道施工变形影响不可忽视。