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1.
This paper presents statistical aspects related to the calibration process and a comparison of different regression approaches of relevance to almost all analytical techniques. The models for ordinary least-squares (OLS), weighted least-squares (WLS), and maximum likelihood fitting (MLF) were evaluated and, as a case study, X-ray fluorescence (XRF) calibration curves for major elements in geochemical reference materials were used. The results showed that WLS and MLF models were statistically more consistent in comparison with the usually applied OLS approach. The use of uncertainty on independent and dependent variables during the calibration process and the calculation of final uncertainty on individual results using error propagation equations are the novel aspects of our work.  相似文献   

2.
Semivariogram parameters are estimated by a weighted least-squares method and a jackknife kriging method. The weighted least-squares method is investigated by differing the lag increment and maximum lag used in the fit. The jackknife kriging method minimizes the variance of the jackknifing error as a function of semivariogram parameters. The effects of data sparsity and the presence of a trend are investigated by using 400-, 200-, and 100-point synthetic data sets. When the two methods yield significantly different results, more data may be needed to determine reliably the semivariogram parameters, or a trend may be present in the data.  相似文献   

3.
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.  相似文献   

4.
A wide variety of semivariograms may be represented in terms of a first- or second-order autoregressive (AR) process, and the nugget effect may be included by use of a moving average (MA) process. The weighting parameters for these models have a simple functional dependence on the value of the sill and the semivariance at the first and second lag. These may be estimated either graphically from the semivariogram or directly from the computed values. Improved spectral estimates of geophysical data have been obtained by the use of the maximum entropy method, and the necessary equations were adapted here for the estimation of the weighting parameters of the AR and the MA processes. Comparison among the semivariograms obtained for the ideal case, the observed case, and the estimated case for artificial series show excellent correspondence between the ideal and estimated while the observed semivariogram may show marked divergence.  相似文献   

5.
Assessment of the sampling variance of the experimental variogram is an important topic in geostatistics as it gives the uncertainty of the variogram estimates. This assessment, however, is repeatedly overlooked in most applications mainly, perhaps, because a general approach has not been implemented in the most commonly used software packages for variogram analysis. In this paper the authors propose a solution that can be implemented easily in a computer program, and which, subject to certain assumptions, is exact. These assumptions are not very restrictive: second-order stationarity (the process has a finite variance and the variogram has a sill) and, solely for the purpose of evaluating fourth-order moments, a Gaussian distribution for the random function. The approach described here gives the variance–covariance matrix of the experimental variogram, which takes into account not only the correlation among the experiemental values but also the multiple use of data in the variogram computation. Among other applications, standard errors may be attached to the variogram estimates and the variance–covariance matrix may be used for fitting a theoretical model by weighted, or by generalized, least squares. Confidence regions that hold a given confidence level for all the variogram lag estimates simultaneously have been calculated using the Bonferroni method for rectangular intervals, and using the multivariate Gaussian assumption for K-dimensional elliptical intervals (where K is the number of experimental variogram estimates). A general approach for incorporating the uncertainty of the experimental variogram into the uncertainty of the variogram model parameters is also shown. A case study with rainfall data is used to illustrate the proposed approach.  相似文献   

6.
Hybrid Estimation of Semivariogram Parameters   总被引：1，自引：0，他引：1  
Two widely used methods of semivariogram estimation are weighted least squares estimation and maximum likelihood estimation. The former have certain computational advantages, whereas the latter are more statistically efficient. We introduce and study a “hybrid” semivariogram estimation procedure that combines weighted least squares estimation of the range parameter with maximum likelihood estimation of the sill (and nugget) assuming known range, in such a way that the sill-to-range ratio in an exponential semivariogram is estimated consistently under an infill asymptotic regime. We show empirically that such a procedure is nearly as efficient computationally, and more efficient statistically for some parameters, than weighted least squares estimation of all of the semivariogram’s parameters. Furthermore, we demonstrate that standard plug-in (or empirical) spatial predictors and prediction error variances, obtained by replacing the unknown semivariogram parameters with estimates in expressions for the ordinary kriging predictor and kriging variance, respectively, perform better when hybrid estimates are plugged in than when weighted least squares estimates are plugged in. In view of these results and the simplicity of computing the hybrid estimates from weighted least squares estimates, we suggest that software that currently estimates the semivariogram by weighted least squares methods be amended to include hybrid estimation as an option.  相似文献   

7.
The linear model of coregionalization (LMC) is generally fit to multivariate geostatistical data by minimizing a least-squares criterion. It is commonly believed that weighting the criterion by inverse variances will reduce the influence of those variables with large variance. We point out that this need not be so, and that in some cases the weights will have no effect whatsoever on the estimated sill matrices. When there is an effect, it is due not to a reduction of these variables’ influence, but rather due to a lack of invariance of the minimization problem; moreover, sometimes the influence may actually increase. The correct way to reduce influence is to fit the LMC after standardizing the variables to have unit variance.  相似文献   

8.
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.  相似文献   

9.
Large volumes of greenhouse gases such as CH4 and CO2 form by contact metamorphism of organic-rich sediments in aureoles around sill intrusions in sedimentary basins. Thermogenic gas generation and dehydration reactions in shale are treated numerically in order to quantify basin-scale devolatilization. We show that aureole thicknesses, defined as the zone of elevated metamorphism relative to the background level, vary within 30-250% of the sill thickness, depending on the temperature of the host-rock and intrusion, besides the sill thickness. In shales with total organic carbon content of >5 wt.%, CH4 is the dominant volatile (85-135 kg/m3) generated through organic cracking, relative to H2O-generation from dehydration reactions (30-110 kg/m3). Even using conservative estimates of melt volumes, extrapolation of our results to the scale of sill complexes in a sedimentary basin indicates that devolatilization can have generated ∼2700-16200 Gt CH4 in the Karoo Basin (South Africa), and ∼600-3500 Gt CH4 in the Vøring and Møre basins (offshore Norway). The generation of volatiles is occurring on a time-scale of 10-1000 years within an aureole of a single sill, which makes the rate of sill emplacement the time-constraining factor on a basin-scale. This study demonstrates that thousands of gigatons of potent greenhouse gases like methane can be generated during emplacement of Large Igneous Provinces in sedimentary basins.  相似文献   

10.
Bayesian Modeling and Inference for Geometrically Anisotropic Spatial Data   总被引：3，自引：0，他引：3  
A geometrically anisotropic spatial process can be viewed as being a linear transformation of an isotropic spatial process. Customary semivariogram estimation techniques often involve ad hoc selection of the linear transformation to reduce the region to isotropy and then fitting a valid parametric semivariogram to the data under the transformed coordinates. We propose a Bayesian methodology which simultaneously estimates the linear transformation and the other semivariogram parameters. In addition, the Bayesian paradigm allows full inference for any characteristic of the geometrically anisotropic model rather than merely providing a point estimate. Our work is motivated by a dataset of scallop catches in the Atlantic Ocean in 1990 and also in 1993. The 1990 data provide useful prior information about the nature of the anisotropy of the process. Exploratory data analysis (EDA) techniques such as directional empirical semivariograms and the rose diagram are widely used by practitioners. We recommend a suitable contour plot to detect departures from isotropy. We then present a fully Bayesian analysis of the 1993 scallop data, demonstrating the range of inferential possibilities.  相似文献   

11.
In the linear model of coregionalization (LMC), when applicable to the experimental direct variograms and the experimental cross variogram computed for two random functions, the variability of and relationships between the random functions are modeled with the same basis functions. In particular, structural correlations can be defined from entries of sill matrices (coregionalization matrices) under second-order stationarity. In this article, modified t-tests are proposed for assessing the statistical significance of estimated structural correlations. Their specific aspects and fundamental differences, compared with an existing modified t-test for global correlation analysis with spatial data, are discussed via estimated effective sample sizes, in relation to the superimposition of random structural components, the range of autocorrelation, the presence of correlation at another structure, and the sampling scheme. Accordingly, simulation results are presented for one structure versus two structures (one without and the other with autocorrelation). The performance of tests is shown to be related to the uncertainty associated with the estimation of variogram model parameters (range, sill matrix entries), because these are involved in the test statistic and the degrees of freedom of the associated t-distribution through the estimated effective sample size. Under the second-order stationarity and LMC assumptions, the proposed tests are generally valid.  相似文献   

12.
Variograms of hydrologic characteristics are usually obtained by estimating the experimental variogram for distinct lag classes by commonly used estimators and fitting a suitable function to these estimates. However, these estimators may fail the conditionally positive-definite property and the better results for the statistics of cross-validation, which are two essential conditions for choosing a valid variogram model. To satisfy these two conditions, a multi-objective bilevel programming estimator (MOBLP) which is based on the process of cross-validation has been developed for better estimate of variogram parameters. This model is illustrated with some rainfall data from Luan River Basin in China. The case study demonstrated that MOBLP is an effective way to achieve a valid variogram model.  相似文献   

13.
There is a confusing situation in geostatistical literature: Some authors write variogram, and some authors write semivariogram. Based on a formula for the empirical variance that relates to pairwise differences, it is shown that the values depicted in a variogram are entire variances of observations at a given spatial separation (lag). Therefore, they should not be called semivariances, and the term semivariogram should also be avoided. To name a variogram value, we suggest the use of the term gammavariance instead of the misleading semivariance.  相似文献   

14.
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.  相似文献   

15.
A distribution-free estimator of the slope of a regression line is introduced. This estimator is designated Sm and is given by the median of the set of n(n – 1)/2 slope estimators, which may be calculated by inserting pairs of points (X i, Yi)and (X j, Yj)into the slope formula S i = (Y i – Yj)/(X i – Xj),1 i < j n Once S m is determined, outliers may be detected by calculating the residuals given by Ri = Yi – SmXi where 1 i n, and chosing the median Rm. Outliers are defined as points for which |Ri – Rm| > k (median {|R i – Rm|}). If no outliers are found, the Y-intercept is given by Rm. Confidence limits on Rm and Sm can be found from the sets of Ri and Si, respectively. The distribution-free estimators are compared with the least-squares estimators now in use by utilizing published data. Differences between the least-squares and distribution-free estimates are discussed, as are the drawbacks of the distribution-free techniques.  相似文献   

16.
Fitting semivariograms with analytical models can be tedious and restrictive. There are many smooth functions that could be used for the semivariogram; however, arbitrary interpolation of the semivariogram will almost certainly create an invalid function. A spectral correction, that is, taking the Fourier transform of the corresponding covariance values, resetting all negative terms to zero, standardizing the spectrum to sum to the sill, and inverse transforming is a valuable method for constructing valid discrete semivariogram models. This paper addresses some important implementation details and provides a methodology to working with spectrally corrected semivariograms.  相似文献   

17.
A dynamical estimate of the mass of the black hole in the LMC X-1 binary system is obtained in the framework of a Roche model for the optical star, based on fitting of the He I 4471 Å and He II 4200 Å absorption lines assuming LTE. The mass of the black hole derived from the radial-velocity curve for the He II 4200 Å line is mx = 10.55 M⊙, close to the value found earlier based on a model with two point bodies [1].  相似文献   

18.
The U.S. Geological Survey is conducting a national assessment of coal resources. As part of that assessment, a geostatistical procedure has been developed to estimate the uncertainty of coal resources for the historical categories of geological assurance: measured, indicated, inferred, and hypothetical coal. Data consist of spatially clustered coal thickness measurements from coal beds and/or zones that cover, in some cases, several thousand square kilometers. Our procedure involved trend removal, an examination of spatial correlation, computation of a sample semivariogram, and fitting a semivariogram model. This model provided standard deviations for the uncertainty estimates. The number of sample points (drill holes) in each historical category also was estimated. Measurement error in the thickness of the coal bed/zone was obtained from the fitted model or supplied exogenously. From this information approximate estimates of uncertainty on the historical categories were computed. We illustrate the methodology using drill hole data from the Harmon coal bed located in southwestern North Dakota. The methodology will be applied to approximately 50 coal data sets.  相似文献   

19.
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.  相似文献   

20.
Observations of current velocity profiles and hydrography over and near a tall sill in a Chilean glacial fjord are used to illustrate the interactions between barotropic and baroclinic tides. The character of the barotropic tide in the glacial fjord is mixed with semidiurnal dominance. The ratio of sill height to water column depth at the study site is ca. 0.95. Water column stratification appeared only in the upper 5 m of the water column. Current velocity variations in the stratified surface layer were quite different to those underneath. Below the pycnocline, nonlinear interactions between semidiurnal M2 and diurnal K1 oscillations yielded a third-diurnal distortion MK3. Most interesting, surface layer currents were distortedby the superposition of semidiurnal M2 and sixthdiurnal M6 oscillations. The oscillations with M6 variability were identified, through wave superposition approaches, as reflected internal tides linked to M2 tidal variations. This was confirmed by theoretical results of stratified barotropic tidal flows interacting with abrupt bathymetry. Under the predominantly tidally mixed regime of the study area, the distortion to surface currents caused by the reflected wave was nearly symmetric during the large tidal ranges of the diurnal cycle. Nearly symmetric distortions resulted as the phase lag between incident and reflected wave-inducted currents was small (reflected currents developing a few minutes after maximum tidal flows). During the small ranges of the diurnal cycle, distortions were asymmetrical because of the relatively larger phase lags of the reflected signal (reflected currents developing tens of minutes after maximum tidal flows).  相似文献   

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