Fitting the Linear Model of Coregionalization by Generalized Least Squares

In geostatistical studies, the fitting of the linear model of coregionalization (LMC) to direct and cross experimental semivariograms is usually performed with a weighted least-squares (WLS) procedure based on the number of pairs of observations at each lag. So far, no study has investigated the efficiency of other least-squares procedures, such as ordinary least squares (OLS), generalized least squares (GLS), and WLS with other weighing functions, in the context of the LMC. In this article, we compare the statistical properties of the sill estimators obtained with eight least-squares procedures for fitting the LMC: OLS, four WLS, and three GLS. The WLS procedures are based on approximations of the variance of semivariogram estimates at each distance lag. The GLS procedures use a variance–covariance matrix of semivariogram estimates that is (i) estimated using the fourth-order moments with sill estimates (GLS₁), (ii) calculated using the fourth-order moments with the theoretical sills (GLS₂), and (iii) based on an approximation using the correlation between semivariogram estimates in the case of spatial independence of the observations (GLS₃). The current algorithm for fitting the LMC by WLS while ensuring the positive semidefiniteness of sill matrix estimates is modified to include any least-squares procedure. A Monte Carlo study is performed for 16 scenarios corresponding to different combinations of the number of variables, number of spatial structures, values of ranges, and scale dependence of the correlations among variables. Simulation results show that the mean square error is accounted for mostly by the variance of the sill estimators instead of their squared bias. Overall, the estimated GLS₁ and theoretical GLS₂ are the most efficient, followed by the WLS procedure that is based on the number of pairs of observations and the average distance at each lag. On that basis, GLS₁ can be recommended for future studies using the LMC.     

Statistical evaluation of methods for the calculation of static formation temperatures in geothermal and oil wells using an extension of the error propagation theory

The feasibility of using the more sophisticated weighted least-squares (WLS) model, as opposed to the traditional ordinary least-squares (OLS), in linear regressions of BHT data to estimate the static formation temperatures (SFT) was investigated. The most commonly used analytical methods (line-source; spherical and radial heat flow; and cylindrical heat source) were applied. Error propagation equations were derived to calculate errors in the time function of each method. These errors were combined with the BHT measurement errors to compute weighting factors for applying the WLS. Intercept uncertainties were estimated for all regressions using sets of synthetic and actual borehole logs taken from geothermal and oil applications. SFT computed with the spherical and radial heat flow method were generally greater than those from the other two methods.     

气相色谱法测定地下水中六六六结果的不确定度评定

依照《测量不确定度评定与表示》,对气相色谱法测定地下水中六六六(HCH)四种单体结果进行了不确定度评定。分析了测量过程中引入的不确定度来源,包括提取液体积的量取、样品提取溶液的定容体积、分析仪器的进样量、标准系列溶液的测量以及仪器重复测定等分量引入不确定度及其各参数的采集和计算方法,最后合成标准不确定度,通过乘以95%概率下的扩展因子2,获得测量结果的扩展不确定度。     

The influence of model structure on groundwater recharge rates in climate-change impact studies

Numerous modeling approaches are available to provide insight into the relationship between climate change and groundwater recharge. However, several aspects of how hydrological model choice and structure affect recharge predictions have not been fully explored, unlike the well-established variability of climate model chains—combination of global climate models (GCM) and regional climate models (RCM). Furthermore, the influence on predictions related to subsoil parameterization and the variability of observation data employed during calibration remain unclear. This paper compares and quantifies these different sources of uncertainty in a systematic way. The described numerical experiment is based on a heterogeneous two-dimensional reference model. Four simpler models were calibrated against the output of the reference model, and recharge predictions of both reference and simpler models were compared to evaluate the effect of model structure on climate-change impact studies. The results highlight that model simplification leads to different recharge rates under climate change, especially under extreme conditions, although the different models performed similarly under historical climate conditions. Extreme weather conditions lead to model bias in the predictions and therefore must be considered. Consequently, the chosen calibration strategy is important and, if possible, the calibration data set should include climatic extremes in order to minimise model bias introduced by the calibration. The results strongly suggest that ensembles of climate projections should be coupled with ensembles of hydrogeological models to produce credible predictions of future recharge and with the associated uncertainties.     

The effects of ionic strength and organic matter on virus inactivation at low temperatures: general likelihood uncertainty estimation (GLUE) as an alternative to least-squares parameter optimization for the fitting of virus inactivation models

This study examined how the inactivation of bacteriophage MS2 in water was affected by ionic strength (IS) and dissolved organic carbon (DOC) using static batch inactivation experiments at 4 °C conducted over a period of 2 months. Experimental conditions were characteristic of an operational managed aquifer recharge (MAR) scheme in Uppsala, Sweden. Experimental data were fit with constant and time-dependent inactivation models using two methods: (1) traditional linear and nonlinear least-squares techniques; and (2) a Monte-Carlo based parameter estimation technique called generalized likelihood uncertainty estimation (GLUE). The least-squares and GLUE methodologies gave very similar estimates of the model parameters and their uncertainty. This demonstrates that GLUE can be used as a viable alternative to traditional least-squares parameter estimation techniques for fitting of virus inactivation models. Results showed a slight increase in constant inactivation rates following an increase in the DOC concentrations, suggesting that the presence of organic carbon enhanced the inactivation of MS2. The experiment with a high IS and a low DOC was the only experiment which showed that MS2 inactivation may have been time-dependent. However, results from the GLUE methodology indicated that models of constant inactivation were able to describe all of the experiments. This suggested that inactivation time-series longer than 2 months were needed in order to provide concrete conclusions regarding the time-dependency of MS2 inactivation at 4 °C under these experimental conditions.     

Prior model identification during subsurface flow data integration with adaptive sparse representation techniques

Construction of predictive reservoir models invariably involves interpretation and interpolation between limited available data and adoption of imperfect modeling assumptions that introduce significant subjectivity and uncertainty into the modeling process. In particular, uncertainty in the geologic continuity model can significantly degrade the quality of fluid displacement patterns and predictive modeling outcomes. Here, we address a standing challenge in flow model calibration under uncertainty in geologic continuity by developing an adaptive sparse representation formulation for prior model identification (PMI) during model calibration. We develop a flow-data-driven sparsity-promoting inversion to discriminate against distinct prior geologic continuity models (e.g., variograms). Realizations of reservoir properties from each geologic continuity model are used to generate sparse geologic dictionaries that compactly represent models from each respective prior. For inversion initially the same number of elements from each prior dictionary is used to construct a diverse geologic dictionary that reflects a wide range of variability and uncertainty in the prior continuity. The inversion is formulated as a sparse reconstruction problem that inverts the flow data to identify and linearly combine the relevant elements from the large and diverse set of geologic dictionary elements to reconstruct the solution. We develop an adaptive sparse reconstruction algorithm in which, at every iteration, the contribution of each dictionary to the solution is monitored to replace irrelevant (insignificant) elements with more geologically relevant (significant) elements to improve the solution quality. Several numerical examples are used to illustrate the effectiveness of the proposed approach for identification of geologic continuity in practical model calibration problems where the uncertainty in the prior geologic continuity model can lead to biased inversion results and prediction.     

高效液相色谱法测定地下水中苯并(a)芘的不确定度评定

通过对高效液相色谱法测定地下水中苯并(a)芘含量的全过程分析,确定了测定结果不确定度的来源。采用不确定度连续传递模型,对引入的不确定度分量进行评定,并采用最小二乘法对标准曲线进行拟合,确定了地下水中苯并(a)芘含量标准不确定度由样品取样量、样品定容体积及测定体积、样品重复性测定、标准溶液浓度和标准曲线拟合误差6部分不确定度合成。通过对2个不同含量样品测定结果不确定度评定,证明苯并(a)芘含量越低,测定结果的相对标准不确定度越大;且样品重复性测定和标准曲线拟合误差是测定结果不确定度的重要来源。     

Characterization and Quantification of Uncertainty in?Coregionalization Analysis

Coregionalization analysis has been presented as a method of multi-scale analysis for multivariate spatial data. Despite an increasing use of this method in environmental and earth sciences, the uncertainty associated with the estimation of parameters in coregionalization analysis (e.g., sills and functions of sills) is potentially high and has not yet been characterized. This article aims to discuss the theory underlying coregionalization analysis and assess the robustness and limits of the method. A theoretical framework is developed to calculate the ergodic and fluctuation variance-covariance matrices of least-squares estimators of sills in the linear model of coregionalization. To adjust for the positive semidefiniteness constraint on estimated coregionalization matrices, a confidence interval estimation procedure for sills and functions of sills is presented. Thereafter, the relative importance of uncertainty measures (bias and variance) for sills and structural coefficients of correlation and determination is assessed under different scenarios to identify factors controlling their uncertainty. Our results show that the sampling grid density, the choice of the least-squares estimator of sills, the positive semidefiniteness constraint, the presence of scale dependence in the correlations, and the number and range of variogram models, all affect the level of uncertainty, sometimes through multiple interactions. The asymptotic properties of variogram model parameter estimators in a bounded sampling domain impose a theoretical limit to their accuracy and precision. Because of this limit, the uncertainty was found to be high for several scenarios, especially with three variogram models, and was often more dependent on the ratio of variogram range to domain extent than on the sampling grid density. In practice, in the coregionalization analysis of a real dataset, the circular requirement for sill estimates in the calculation of uncertainty measures makes the quantification of uncertainty very problematic, if not impossible. The use of coregionalization analysis must be made with due knowledge of the uncertainty levels and limits of the method.     

Resolving uncertainty in chemical speciation determinations

Speciation determinations involve uncertainty in system definition and experimentation. Identification of appropriate metals and ligands from basic chemical principles, analytical window considerations, types of species and checking for consistency in equilibrium calculations are considered in system definition uncertainty. A systematic approach to system definition limits uncertainty in speciation investigations. Experimental uncertainty is discussed with an example of proton interactions with Suwannee River fulvic acid (SRFA). A Monte Carlo approach was used to estimate uncertainty in experimental data, resulting from the propagation of uncertainties in electrode calibration parameters and experimental data points. Monte Carlo simulations revealed large uncertainties present at high (>9–10) and low (<4) −log[H⁺], which result from larger instabilities in the proton balance function in these regions. Uncertainties in speciation parameters were compared for uniresponse fitting (linear programming and least-squares) and multiresponse fitting. Linear programming and least-squares approaches both fit the observed data, but suggest different mixtures of monoprotic ligands. Least-squares fit the data with 21 sites, whereas linear programming fit the data equally well with 9 sites. Multiresponse fitting, involving simultaneous fluorescence and pH measurements, improved model discrimination. Deconvolution of the excitation versus emission fluorescence surface for SRFA establishes a minimum of five sites. Diprotic sites are also required for the five fluorescent sites, and one non-fluorescent monoprotic site was added to accommodate the pH data. Consistent with greater complexity, the multiresponse method had broader confidence limits than the uniresponse methods, but corresponded better with the accepted total carboxylic content for SRFA. Overall there was a 40% standard deviation in total carboxylic content for the multiresponse fitting, versus 10% and 1% for least-squares and linear programming, respectively.     

Characterization and Quantification of Uncertainty in Coregionalization Analysis

Coregionalization analysis has been presented as a method of multi-scale analysis for multivariate spatial data. Despite an increasing use of this method in environmental and earth sciences, the uncertainty associated with the estimation of parameters in coregionalization analysis (e.g., sills and functions of sills) is potentially high and has not yet been characterized. This article aims to discuss the theory underlying coregionalization analysis and assess the robustness and limits of the method. A theoretical framework is developed to calculate the ergodic and fluctuation variance-covariance matrices of least-squares estimators of sills in the linear model of coregionalization. To adjust for the positive semidefiniteness constraint on estimated coregionalization matrices, a confidence interval estimation procedure for sills and functions of sills is presented. Thereafter, the relative importance of uncertainty measures (bias and variance) for sills and structural coefficients of correlation and determination is assessed under different scenarios to identify factors controlling their uncertainty. Our results show that the sampling grid density, the choice of the least-squares estimator of sills, the positive semidefiniteness constraint, the presence of scale dependence in the correlations, and the number and range of variogram models, all affect the level of uncertainty, sometimes through multiple interactions. The asymptotic properties of variogram model parameter estimators in a bounded sampling domain impose a theoretical limit to their accuracy and precision. Because of this limit, the uncertainty was found to be high for several scenarios, especially with three variogram models, and was often more dependent on the ratio of variogram range to domain extent than on the sampling grid density. In practice, in the coregionalization analysis of a real dataset, the circular requirement for sill estimates in the calculation of uncertainty measures makes the quantification of uncertainty very problematic, if not impossible. The use of coregionalization analysis must be made with due knowledge of the uncertainty levels and limits of the method.     

岩溶碳汇效应研究中“碳”测定的不确定度分析——以溶解性无机碳为例

为建立非分散红外吸收法测定岩溶区流域水体中溶解性无机碳(DIC)分析结果不确定度的评定方法,采用不确定度连续传递模式,对两个岩溶地下水样的DIC测定结果的不确定度进行了评定。其主要做法是:先采用双误差回归方式对校准曲线进行拟合,对评定过程中各个不确定度分量进行量化,然后合成得到DIC测定结果的不确定度评定模型。通过实验结果计算表明:(1)测定结果的不确定度主要来源于标准溶液引入的不确定度、曲线拟合产生的不确定度和测定过程产生的不确定度。(2)样品的DIC含量越低,其相对不确定度越大,且校准曲线在拟合过程中所引入的不确定度对其测定结果的总不确定度具有较大的贡献率。      

基于土壤属性的VIC模型基流参数估计框架

"异参同效"现象是水文模型参数率定过程中的一个难题。为了减少需要率定的模型参数个数,从而降低模型参数的相互作用,构建了利用土壤属性直接估计VIC(Variable Infiltration Capacity)模型3个基流参数的框架,并在3个位于不同水文气候条件下的典型流域中作了实例研究。传统的VIC模型参数估计方法需要率定6个参数;而在新的参数估计框架下,需要率定的模型参数从6个减少为3个。蒙特卡罗模拟结果表明新的参数估计框架提高了参数的敏感性。同时,新的参数估计方法模拟的径流过程和之前通过率定得到的径流过程差别很小。GLUE(Generalised Likelihood Uncertainty Estimation)方法分析得到径流模拟的不确定性结果表明:新的参数估计框架计算径流的90%置信区间要明显小于传统方法,而且对于低流量的模拟效果更加明显,也就是说采用新的参数估计框架可以显著降低径流模拟的不确定性。这种基流参数估计方法可以被应用于其他相似的水文模型中。     

Modeling water resources of a highly irrigated alluvial plain (Italy): calibrating soil and groundwater models

Modern and effective water management in large alluvial plains that have intensive agricultural activity requires the integrated modeling of soil and groundwater. The models should be complex enough to properly simulate several, often non-linear, processes, but simple enough to be effectively calibrated with the available data. An operative, practical approach to calibration is proposed, based on three main aspects. First, the coupling of two models built on well-validated algorithms, to simulate (1) the irrigation system and the soil water balance in the unsaturated zone and (2) the groundwater flow. Second, the solution of the inverse problem of groundwater hydrology with the comparison model method to calibrate the model. Third, the use of appropriate criteria and cross-checks (comparison of the calibration results and of the model outputs with hydraulic and hydrogeological data) to choose the final parameter sets that warrant the physical coherence of the model. The approach has been tested by application to a large and intensively irrigated alluvial basin in northern Italy.     

Modelling the response of river systems to environmental change: Progress,problems and prospects for palaeo-environmental reconstructions

Over the last decade several computational models, and several types of model, have been developed to simulate the response of river systems to environmental change over time scales of decades to millennia: hydrological models, flood inundation models, channel morphology models, channel network models, models of river meandering and river braiding, alluvial stratigraphy models, and landscape evolution models. Each type of model simulates different aspects of a river's response to changes in environmental inputs such as climate and land-use — and to changes in these inputs. And each type of model has its abilities, advantages and limitations. We provide an overview of the different types of model that have been developed, and we evaluate their suitability for testing hypotheses about past environmental conditions, as well as for investigating the response of alluvial river systems to future environmental change. Additionally, we discuss the general issues and problems of computational modelling (e.g. scale and resolution, data availability, process representation, process parameterization, model calibration, non-linearity, and uncertainty), and the extent to which these hamper the usefulness of the models as a tool in environmental landscape studies. Finally, we identify trends in computational modelling research to outline possible future directions of the discipline.     

Comparison of Two Empirical Models for Gully-Specific Debris Flow Hazard Assessment in Xiaojiang Valley of Southwestern China

Two sets of empirical models for debris flow hazard assessment on gully-specific scale are presented herein, which may be the first step toward developing a formalized procedure for debris flow hazard assessment. Emphasis is put on comparison of the two models between Multiple Factor Composite Assessment Model (MFCAM) and Multivariate Stepwise Regression Assessment Model (MSRAM). From the aspects of data acquisition and model construct, this paper shows the advantages and limitations of two models. From the assessment results, two models are still of uncertainty. Fundamental researches of debris flow hazard, especially the relationship between debris flow magnitude and its frequency of occurrence should be given a priority in future study.     

Importance of establishing sources of uncertainty for the derivation of reliable SHRIMP ages

The ion microprobe, as exemplified by SHRIMP, has long been an invaluable resource for the derivation of geological ages. The derivation of those ages is critically dependent on the identification and individual quantification of all sources of contributing uncertainty. In recent years, it has been proposed that the only component of uncertainty arising from the instrument itself is predictable from counting statistics. The adoption of that approach has led to several conclusions including: (i) that zircon U–Pb ages are relatively easily reset, which necessitates the enhanced editing of individual analyses before a grouped age can be obtained; and (ii) that other studies have overestimated analytical uncertainties and, as a consequence, have reported incorrect and/or overly imprecise ages. We present evidence for the presence of additional sources of instrument‐related uncertainty that necessitates a different (but not new) approach for the processing of SHRIMP data. Fortunately, this complication does not represent a serious problem, provided that a high‐quality zircon‐calibration standard has been used for Pb/U calibration. SHRIMP ages obtained some time ago from the Crudine Group of the Hill End Trough (New South Wales) have recently been placed at the centre of this controversy. A significant part of the problem is that most of those ages were based on a standard (SL 13) that is now known to be heterogeneous. The more reliable parts of the original data have been re‐reprocessed on the basis of the new evidence. They fail to detect a significant age difference between the bottom and the top of the Merrions Formation, a conclusion that is contrary to earlier expressed opinions.     

Empirical evaluation of solution models for pelitic minerals and their application to thermobarometry

One of the major sources of uncertainty for thermobarometry in pelitic rocks is the inability of solution models to represent accurately the non-ideal behavior of important pelitic minerals, in particular garnet, biotite, and plagioclase. In recent years, a large number of different solution models have been proposed for these phases. The purpose here is to test the effects on the overall uncertainty of the most commonly-used solution models using the garnet-muscovite-annite-plagioclase (GMAP) barometer and the empirical calibration technique. The GMAP reaction has been empirically calibrated several times with varying results, due in large part to the choice of solution models. The method's dependence on the choice of solution model makes it an excellent was in which to address the quality of specific models. By repeating the calibration exercise for different combinations of solution models, it is possible to determine which set of solution models results in the lowest uncertainties, using a sample set of 72 published electron microprobe analyses of pelitic rocks containing the assemblage garnet+muscovite+biotite +plagioclase+aluminum silicate+quartz. As an independent test of the suitability of solution models, we compare the fit of the sample set to the kyanite-sillimanite equilibrium curve, obtaining thermobarometric estimates by simultaneous solution of the garnet-biotite exchange thermometer and GMAP calibrated from thermochemical data.     

三种水文模型不确定性分析方法比较

模型不确定性研究是水文科学的重要课题。以尼泊尔Bagmati流域为案例,采用了马尔科夫链蒙托卡罗(Markov Chain Monte Carlo)、蒙托卡罗(Monte Carlo)和拉丁超立方体(Latin Hypercube)等三种方法,分析了水箱模型输出成果的不确定性,并将三种方法所获得参数不确定性进行了比较。另外,运用Meta-Gaussian模型计算了总体不确定性,在基于所采用的似然函数基础上,对由参数导致模型输出的不确定性和模型输出的总体不确定性进行了比较。结果显示,模型的不确定性比参数不确定性更为重要,同时也表明,尽管蒙托卡罗和拉丁超立方体两种模拟方法产生几乎相同的结果,但两者都与马尔科夫链蒙托卡罗方法有很大的不同。     

水文模型参数区域化研究进展

参数区域化方法是解决资料缺乏地区水文模拟和预报的有效手段,主要包括回归法、空间邻近法和属性相似法三类方法,可将有资料流域的水文模型参数移用到资料缺乏流域。首先回顾了区域化方法的基本原理和应用方法,并分析了三类主要区域化方法的适用性。从流域特征因子、水文模型及参数、不确定性探讨三个方面综述了区域化方法的研究进展。分析发现,当前区域化方法缺乏完善的理论基础,流域特征因子选择存在主观性,水文模型及参数的适用性方面研究不足。最后展望了未来的研究重点：（1）多维度适用性比较;（2）水文过程和参数的空间分布规律;（3）参数的尺度问题;（4）参数区域化的不确定性问题。     

等离子体发射光谱法测定土壤和沉积物中6个常量元素结果不确定度评定

采用等离子体发射光谱法对测定土壤和水系沉积物中铝、铁、钙、镁、钾、钠6个常量元素进行系统的不确定度评定。不确定度的来源主要包括分析过程中所用的天平、玻璃器皿、标准曲线、标准溶液、试液定容体积及测量重复性等引入的不确定度分量。计算出各分量的不确定度,通过合成得到测量结果的合成不确定度、扩展不确定度及测试结果的报告形式。