Balance-aware covariance localisation for atmospheric and oceanic ensemble Kalman filters

Covariance localisation is used in many implementations of the ensemble Kalman filter (EnKF) but has been shown by Lorenc and by Kepert to significantly degrade the main balances in the atmosphere and ocean. Kepert recently introduced an improved form of localisation that reduced or eliminated this problem. This paper presents an extension to that approach, in which the background state is decomposed into balanced and unbalanced parts as part of the localisation. This new balance-aware localisation is shown to be a slight improvement on the earlier work of Kepert and a substantial improvement on the standard Schur-product localisation. Balance-aware localisation also enables the use of some sets of alternative analysis variables that do not work well with conventional localisation in the EnKF. It is shown using identical-twin experiments with a global spectral shallow-water model and no separate initialisation step that analysis to geopotential, streamfunction and velocity potential is slightly more accurate than is analysis to geopotential and the wind components. Analysis to unbalanced (instead of total) geopotential, streamfunction and velocity potential leads to slightly less accurate but significantly better balanced analyses than the other choices of analysis variables. If nonlinear normal modes initialisation is incorporated in the analysis cycling, then the conventional localisation becomes the most accurate method. However, initialisation may be undesirable or unavailable, and the comparison of system performance without localisation is useful since it helps improve understanding of the balance issues in EnKF-based assimilation systems.     

Combining sensitivities and prior information for covariance localization in the ensemble Kalman filter for petroleum reservoir applications

Sampling errors can severely degrade the reliability of estimates of conditional means and uncertainty quantification obtained by the application of the ensemble Kalman filter (EnKF) for data assimilation. A standard recommendation for reducing the spurious correlations and loss of variance due to sampling errors is to use covariance localization. In distance-based localization, the prior (forecast) covariance matrix at each data assimilation step is replaced with the Schur product of a correlation matrix with compact support and the forecast covariance matrix. The most important decision to be made in this localization procedure is the choice of the critical length(s) used to generate this correlation matrix. Here, we give a simple argument that the appropriate choice of critical length(s) should be based both on the underlying principal correlation length(s) of the geological model and the range of the sensitivity matrices. Based on this result, we implement a procedure for covariance localization and demonstrate with a set of distinctive reservoir history-matching examples that this procedure yields improved results over the standard EnKF implementation and over covariance localization with other choices of critical length.     

A comparative error analysis of manual versus automated methods of data acquisition for algebraic strain estimation

This paper presents a statistical analysis of the algebraic strain estimation algorithm of Shimamoto and Ikeda [Shimamoto, T., Ikeda, Y., 1976. A simple algebraic method for strain estimation from deformed eillipsoidal objects: 1. Basic theory. Tectonophysics 36, 315–337]. It is argued that the error in their strain estimation procedure can be quantified using an expected discrepancy measure. An analysis of this measure demonstrates that the error is inversely proportional to the number of clasts used. The paper also examines the role of measurement error, in particular that incurred under (i) a moment based and (ii) manual data acquisition methods. Detailed analysis of these two acquisition methods shows that in both cases, the effect of measurement error on the expected discrepancy is small relative to the effect of the sample size (number of objects). Given their relative speed advantage, this result favours the use of automated measurement methods even if they incur more measurement error on individual objects. Validation of these results is carried out by means of a simulation study, as well as by reference to studies appearing in previous literature. The results are also applied to obtain an upper bound on the error of strain estimation for various studies published in the literature on strain analysis.     

Canonical variate analysis with unequal covariance matrices: Generalizations of the usual solution

Canonical variate analysis is extended for use when the covariance matrices are not equal. Linear combinations of variates are derived by generalizing either a weighted between-groups approach or the likelihood-ratio test and the associated noncentrality matrix. The usual solution and the two generalizations are compared via generated data for a few typical configurations of means in a situation in which the covariance matrices are in fact equal. The MSE of the canonical variate coefficients and group means for the generalizations are approximately three times those for the usual solution, due to corresponding changes in the variances. Two examples are discussed.     

An enhanced ensemble Kalman filter scheme incorporating model error in sequential coupling between flow and geomechanics

In this work, we construct a new methodology for enhancing the predictive accuracy of sequential methods for coupling flow and geomechanics while preserving low computational cost. The new computational approach is developed within the framework of the fixed-stress split algorithm procedure in conjunction with data assimilation based on the ensemble Kalman filter (EnKF). In this context, we identify the high-fidelity model with the two-way formulation where additional source term appears in the flow equation containing the time derivative of total mean stress. The iterative scheme is then interlaced with data assimilation steps, which also incorporate the modeling error inherent to the EnKF framework. Such a procedure gives rise to an “enhanced one-way formulation,” exhibiting substantial improvement in accuracy compared with the classical one-way method. The governing equations are discretized by mixed finite elements, and numerical simulation of a 2D slab problem between injection and production wells illustrate the tremendous achievement of the method proposed herein.     

Counting error in petrographic point-count analysis: A theoretical and experimental study

The binomial model, commonly used to estimate counting error in point-count analysis, misestimates this error when the observation points on a grid are positively or negatively correlated. A model, called the cell model, is proposed as an alternative to the binomial model for use in studies, especially with coarse-grained rocks, in which such correlation is known or thought to exist. In the new model the thin section is conceptually partitioned into a number of cells (six is recommended), and the assumption is made that the proportions in the individual cells are statistically independent and that their variance does not differ from cell to cell. Empirical relations obtained from a suite of 200 thin sections of limestones are in reasonable support of the prediction that large particle size adversely affects counting error estimates based on the binomial model.     

Operator error in petrographic point-count analysis: A theoretical approach

Operator error in petrographic point-count analysis introduces bias into the estimates of proportion in a thin section. A correction for this bias, leading to an unbiased estimator of the true proportion in that thin section, is here proposed. Operator error also affects the confidence interval, and in this situation, too, an adjustment is possible. The approach proposed requires that the probabilities associated with operator error, categorized into A-type and B-type errors, are known or assumed. The A-type operator error tends to underestimate the true proportion in a thin section, whereas the B-type operator error tends to overestimate it.     

浅谈影响GPS测量的误差因素及分析

本文简述的GPS在测量中产生的误差源,并进行了分析及提出消除和削弱对测量影响的措施。     

GPS在测量中的应用及误差分析

简述了全球定位系统（GPS）的基本结构和特点,总结了GPS用于工程测量所具有的特点,介绍了GPS在工程测量中的应用实例及体会。简述了GPS系统的定位误差直接影响着GPS定位精度,按其产生的来源、性质及对系统的影响等进行了介绍和初步分析,提出了相应的措施以便消除或削弱它们对测量结果的影响。     

Validation and error analysis of a strata modelling technique

Summary Sources of error are investigated for a two-dimensional finite difference computer program designed to model strata deformation. The program calculates the displacements of a mesh of mass points, by the iterative solution of equations of equilibrium for the stresses acting on each mass point. The effect of errors on both displacement estimates and stress estimates is considered.Round-off errors are discussed analytically, while the effect of choosing too coarse a mesh density is demonstrated by comparison of two runs of the program with identical material properties, but different mesh densities. The influence of boundary conditions and the result of incomplete relaxation of the finite difference equations is estimated by comparison with Kirsch's analytical solution for a thin plate of finite width with a circular hole under unidimensional load.As a result of the analysis, estimators for stresses and displacements are derived, which make allowance for some of the sources of error; suitable boundary conditions for first and subsequent runs of the program are proposed; and a convergence criterion for the iterative process is suggested. These results are then applied to simulations of mining situations, together with various refinements of the basic model, such as separation and slip between adjacent strata, and an allowance for failure of material.     

Generalized numerical error analysis with applications to geochronology and thermodynamics

A general numerical error propagation procedure is developed to calculate the error in a quantity derived from measurements which are subject to errors. It is an alternative to computer intensive techniques such as Monte Carlo and can be applied to quite complex analytical problems where correlation among the measurement errors and among the final errors in results are present. Previous formulations have usually ignored correlated errors. Some of the assumptions involved in error propagation can also be checked numerically. This is an advantage over analytical formulations which cannot assess the validity of the calculated errors. Formulation as computer subroutines permits the analysis to be added to existing programs. Examples from the fields of geochronology and thermodynamics are used to highlight the advantages and the flexibility of the method.     

Biot consolidation analysis with automatic time stepping and error control Part 1: theory and implementation

A new finite element algorithm for solving elastic and elastoplastic coupled consolidation problems is described. The procedure treats the governing consolidation relations as a system of first-order differential equations and is based on the backward Euler and Thomas and Gladwell schemes with automatic subincrementation of a prescribed series of time increments. The prescribed time increments, which are called coarse time steps, serve to start the procedure and are chosen by the user. The automatic consolidation algorithm attempts to select the time subincrements such that, for a given mesh, the time-stepping (or temporal discretisation) error in the displacements lies close to a specified tolerance. Unlike existing solution techniques, the new algorithm computes not only the displacements and pore pressures, but also their derivatives with respect to time. These extra variables permit a family of unconditionally stable integration algorithms to be constructed which automatically provide an estimate of the local truncation error for each time step. This error estimate is inexpensive to compute and may be used to develop a simple and efficient automatic time stepping mechanism. For the elastic case, the displacements and pore pressures at the end of each subincrement may be solved directly without the need for iteration. For elastoplastic behaviour, however, the governing relationships are non-linear and a system of non-linear equations must be solved to compute the updates. Copyright © 1999 John Wiley & Sons, Ltd.     

锚桩法单桩静载试验中群桩相互作用及误差分析

利用Poulos弹性理论法,对锚桩法单桩静载承载力试验中试桩和锚桩之间的相互作用进行了研究,提出了考虑锚桩上拔量对试桩p-s曲线影响的方法。利用该方法对某工程实测资料进行了分析,发现采用原始测试数据分析所得出的试桩极限承载力偏于危险,并为该工程提供了更加合理的单桩极限承载力值。     

基于断层滑动数据古应力反演的影响因素及其误差分析

古构造应力场是构造动力学研究中的一个重要内容,且断层滑动数据古应力反演已经成为古构造应力场恢复研究中比较常用的重要方法之一。近年来,断层滑动数据古应力反演方法研究和应用取得了一系列重要进展,但有关反演结果的解释仍存歧义,反演结果的影响因素及其误差范围等并未得到深入研究与定量分析。本文总结指出,影响断层滑动数据古应力反演结果的主要因素包括变形体制、剪切破裂面类型、断层面的形态以及地质体内薄弱面的存在等。在此基础上,分别对新生断层和先存薄弱面滑动数据的古应力反演综合误差进行了定量分析。研究指出,在可以大致厘定变形体制或误差在允许范围内的前提下,将断层滑动数据反演结果解释为应力状态是合理可行的。各种因素导致的反演误差定量分析表明,同一期构造应力场形成的破裂面滑动数据的古应力方位反演误差最大不超过35°。换言之,在没有证据表明存在不同期次的应力作用情况下,主应力方位变化小于35° 的应力状态,可以划归同一期应力场。     

Suspended sediment fluxes in a tidal wetland: Measurement, controlling factors, and error analysis

Suspended sediment fluxes to and from tidal wetlands are of increasing concern because of habitat restoration efforts, wetland sustainability as sea level rises, and potential contaminant accumulation. We measured water and sediment fluxes through two channels on Browns Island, at the landward end of San Francisco Bay, United States, to determine the factors that control sediment fluxes on and off the island. In situ instrumentation was deployed between October 10 and November 13, 2003. Acoustic Doppler current profilers and the index velocity method were employed to calculate water fluxes. Suspended sediment concentrations (SSC) were determined with optical sensors and cross-sectional water sampling. All procedures were analyzed for their contribution to total error in the flux measurement. The inability to close the water balance and determination of constituent concentration were identified as the main sources of error; total error was 27% for net sediment flux. The water budget for the island was computed, with an unaccounted input of 0.20 m³s⁻¹ (22% of mean inflow), after considering channel flow, change in water storage, evapotranspiration, and precipitation. The net imbalance may be a combination of groundwater seepage, overland flow, and flow through minor channels. Change of island water storage, caused by local variations in water surface elevation, dominated the tidally averaged water flux. These variations were mainly caused by wind and barometric pressure change, which alter regional water levels throughout the Sacramento-San Joaquin River Delta. Peak instantaneous ebb flow was 35% greater than peak flood flow, indicating an ebbdominant system, though dominance varied with the spring-neap cycle. SSC were controlled by wind-wave resuspension adjacent to the island and local tidal currents that mobilized sediment from the channel bed. During neap tides sediment was imported onto the island but during spring tides sediment was exported because the main channel became ebb dominant. Over the 34-d monitoring period 14,000 kg of suspended sediment were imported through the two channels. The water imbalance may affect the sediment balance if the unmeasured water transport pathways are capable of transporting large amounts of sediment. We estimate a maximum of 2,800 kg of sediment may have been exported through unmeasured pathways, giving a minimum ent import of 11,200 kg. Sediment flux measurements provide insight on tidal to fortnightly marsh sedimentation processes, especially in complex systems where sedimentation is spatially and temporally variable.     

考虑松动圈的卡尔曼滤波与有限元耦合反分析法及其在围岩稳定性分析中的应用

考虑洞室开挖后围岩的力学状态和岩石力学众多问题的不确定性本质,通过弹性模量折减的方法,建立了考虑松动圈的卡尔曼滤波与有限元耦合反分析法,并编写了相应算法程序。利用基于文件操作的Fortran与VC++交互通信技术,一方面,为隧道施工量测管理与围岩稳定智能分析系统增加了非确定性反分析模块;另一方面,也使反分析能够利用该系统的图形模块进行前后处理,使该方法的工程应用更加简单,为该法的应用推广创造了很好的条件。用反分析法求出围岩初始参数后,通过有限元正分析可求出全域最大剪应变,并与试验和理论推演获得的临界剪应变相比较,进而进行围岩稳定性判定。由于该围岩稳定性分析法摒弃了一般有限元正分析在围岩初始应力场上的不合理假设,因而结果更贴切实际,更具合理性。对白云隧道YK48+737断面的现场监测数据进行了跟踪反演,获得了该断面围岩塑性区历时变化图,从而及时地了解了围岩的稳定状况,指导了设计和施工。     

Estimation of GPS strain rate and its error analysis in the Chinese continent

In this paper we first estimate the strain rate field with 1202 GPS vectors from 1999 to 2005 in the Chinese continent. Then we propose a method to make error analysis of the strain rates computed from GPS vectors based on the Monte Carlo technique. In general, the orientations of compressive principal strain rates are in agreement with those of present tectonic stresses. The strain rates in the paper confirm the tectonic features of Tibetan Plateau such as NE–W compression and shortening, E–W extension, and normal faulting along near N–S faults. At the same time, the eastward extrusion in the east part of Tibet can also be seen from the strain rate field. In the error analysis, independent computation of strain rates is repeated for a large number of times, both the absolute and relative errors as well as the mean value of strain rates are obtained through statistical theory. The error result shows that the errors of strain rates may be associated with the uncertainties of GPS measurement and the distribution of GPS stations in space. The magnitude of the strain rate error is in the range of 3–8 × 10^?9/yr across China in general. However, we found that the relative errors are much larger in East China than those in the west of China. In addition, the relative errors of maximum shear strain rates are smaller than those of surface dilation rates. Also, we found that there may be some correlations between maximum shear strain rates and seismic activities. Where the maximum shear strain rates are higher, there the earthquakes occur more frequently. The computed GPS strain rates reveal the distributed nature of deformation across the whole Chinese continent, and can represent the main present-day tectonic features. The strain rate and its error in the paper may provide important constraints to geodynamical modeling.     

轴平移技术在基质吸力测控中的局限性和误差分析

为了揭示非饱和土在自然环境和轴平移环境两种条件下,基质吸力测试数据存在差异的原因,分别研究了表面张力系数和难充水微孔隙在相同吸力作用下对含水率的影响。一方面,表面张力系数随压力的增大而小幅减小,同一基质吸力条件下采用轴平移技术时对应的含水率较自然状态有偏小的趋势;另一方面,由于一端封闭微孔隙的存在,较高的孔隙气压力必然促使土中水进入部分难充水微孔隙,从而同一基质吸力条件下轴平移技术对应的含水率较自然状态又有升高的趋势。因此,在特定基质吸力条件下,轴平移方法得到的含水率较自然状态偏大还是偏小取决于上述两方面的综合效应。毛细上升试验和针对一端封闭微孔隙模型的研究表明,一端封闭微孔隙的存在对土-水特征曲线的影响远远大于表面张力系数变化的影响。针对石英砂、砂土和黏土的两种SWCC测试结果表明,随土颗粒逐渐变细,轴平移技术对SWCC的影响越来越大。从而进一步印证,张力计法和轴平移方法在测试黏土土-水特征曲线方面存在差异的原因在于一端封闭微孔隙的存在。     

智能开采工作面三维地质模型构建及误差分析

地质条件的复杂性是影响当前智能开采进一步发展的关键问题之一,亟需构建高精度回采工作面三维地质模型。通过分析智能开采地质模型的构建方法,并以黄陵一号矿某智能工作面为例,结合工作面所有的地质勘探资料,利用TIM-3D建模软件分别构建了工作面初始静态模型和回采工作面动态模型,搭载透明工作面数字孪生系统对智能开采地质模型进行展示;通过对比回采揭露真实煤厚值与地质模型预测煤厚值,分析静态地质模型与动态地质模型的误差,探讨模型误差产生的原因。分析认为：静态地质模型精度不能达到智能化开采的地质要求;更新后的动态地质模型可显著缩小煤厚预测误差,基本能达到智能化开采的地质需求;模型的误差是测量误差、采样数据量及其分布、插值算法选取共同造成的。综合认为模型的建立要充分融合工作面所有地质信息,模型建立巷道标志点的间隔应该小于10 m,模型动态更新的推采距离应该小于15 m。研究结果对于充分认识当前智能开采地质模型精度水平有重要意义,为下一步智能开采地质保障技术的发展具有借鉴意义。     

压汞法测量孔隙时减小误差的方法及分析

低−中阶煤煤质软、在高压条件下煤基质压缩效应明显,应用压汞实验能否得到可靠的孔隙结构存在争议。选取4种低−中阶煤样品,通过对比不同粒度样品的进汞曲线、压汞前后的宏观与显微特征,建立高压段进汞量的校正方法,研究了压汞实验对低−中阶煤中孔隙测试结果的可靠性。结果表明：(1) 柱状样品能保留更多的原生裂隙,有效避免低压阶段的粒间孔与麻皮效应,更适用于低−中阶煤的压汞测试;(2) 宏观与微观尺度下均未发现汞在高压下对煤基质与孔隙的破坏作用,压汞孔容偏高主要是煤基质压缩效应的结果,压缩性校正后压汞测孔隙率与氦气测孔隙率基本一致,6~100 nm的孔容校正后与低温N₂吸附结果的差值降低了29.87%~55.49%,表明压汞实验可以应用于低−中阶煤中孔隙结构的测定;(3) 针对高压阶段的煤基质压缩效应,建立了压缩性累积校正方法,对大于20 MPa高压段数据进行校正后,进汞量降低了0.017 0~0.032 3 mL/g,纳米孔隙与低温N₂吸附实验结果更为接近;(4) 低−中阶煤中孔隙的孔容主要源于大孔,褐煤与长焰煤孔隙发育差异显著,褐煤压汞孔容为0.168 7 cm³/g,长焰煤压汞孔容介于0.027 2~0.072 0 cm³/g,孔隙特别发育的样品,植物组织胞腔孔是孔容的主要来源。