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1.
Analysis and Estimation of Natural Processes with Nonhomogeneous Spatial Variation Using Secondary Information 总被引：1，自引：0，他引：1
Natural processes encountered in mining, hydrogeologic, environmental, etc. applications usually are poorly known because of scarcity of data over the area of interest. Therefore, stochastic estimation techniques are the tool of choice for a careful accounting of the heterogeneity and uncertainty involved. Within such a framework, a better utilization of all available data concerning the process of interest and all other natural processes related to it, is of primary importance. Because many natural processes show complicated spatial trends, the hypothesis of spatial homogeneity cannot be invoked always, and the more general theory of intrinsic spatial random fields should be employed. Efficient use of secondary information in terms of the intrinsic model requires that suitable permissibility criteria for the generalized covariances and cross-covariances are satisfied. A set of permissibility criteria are presented for the situation of two intrinsic random fields. These criteria are more general and comprehensive than the ones currently available in the geostatistical literature. A constrained least-square technique is implemented for the inference of the generalized covariance and cross-covariance parameters, and a synthetic example is used to illustrate the methodology. The numerical results show that the use of secondary information can lead to significant reductions in the estimation errors. 相似文献
2.
J. A. Vargas-Guzmán 《Mathematical Geology》2004,36(3):307-322
This paper introduces geostatistical approaches (i.e., kriging estimation and simulation) for a group of non-Gaussian random fields that are power algebraic transformations of Gaussian and lognormal random fields. These are power random fields (PRFs) that allow the construction of stochastic polynomial series. They were derived from the exponential random field, which is expressed as Taylor series expansion with PRF terms. The equations developed from computation of moments for conditional random variables allow the correction of Gaussian kriging estimates for the non-Gaussian space. The introduced PRF geostatistics shall provide tools for integration of data that requires simple algebraic transformations, such as regression polynomials that are commonly encountered in the practical applications of estimation. The approach also allows for simulations drawn from skewed distributions. 相似文献
3.
The geostatistical approach was applied to integrate MT (Magneto-telluric) resistivity data and borehole information for the spatial RMR (Rock Mass Rating) evaluation. Generally, resistivity of the subsurface is believed to be positively related to the RMR, thus the resistivity and borehole RMR information was combined in a geostatistical approach. To relate the two different sets of data, the MT resistivity data were used as secondary information and the RMR mean values were estimated at unsampled points by identification of the resistivity to the borehole data. Two types of approach are performed for the estimation of RMR mean values. Then the residuals of the RMR values around the borehole sites are geostatistically modeled to infer the spatial structure of difference between real RMR values and estimated mean values. Finally, this geostatistical estimation is added to the previous means. The result applied to a real situation shows prominent improvements to reflect the subsurface structure and spatial resolution of RMR information. 相似文献
4.
Cheng-Shin Jang 《Environmental Earth Sciences》2018,77(22):760
Because Taiwan is a subtropical island, many pleasure beaches are situated on its coast. However, according to long-term monitoring data, fecal contamination at Taiwanese coastal beaches frequently exceeds the U.S. Environmental Protection Agency (EPA) guidelines. To avoid public health hazards, mapping the spatial extent of this contamination is crucial. This study applied indicator kriging (IK) to probabilistically assess the water quality of bathing beaches on the Taiwanese coast. Moreover, because the discontinuity of the traditional Cartesian coordinate established on an island coastline is difficult for geostatistical estimates, this study proposed a novel kriging estimation approach to deal with this problem. First, a one-dimensional (1-D) cyclic coordinate system of the Taiwanese coast was established using primary and secondary coordinates at each beach site. Escherichia coli (E. coli) and enterococci concentrations at coastal beaches were converted into indicator variables according to the U.S. EPA guidelines. IK was then used to spatially model the occurrence probabilities that exceeded the U.S. EPA guidelines for E. coli and enterococci. Finally, the water quality of bathing beaches on the Taiwanese coast was classified on the basis of the estimated probabilities. The study results indicated that bathing on the central western, northeastern, and southeastern Taiwanese coasts poses a potential threat to human health caused by high levels of fecal contamination. Moreover, primary and secondary coordinates established at beach sites were capable of analyzing the spatial variability and kriging estimates of the 1-D cyclic coordinates along the coastline. 相似文献
5.
Estimation of regional-scale groundwater flow properties in the Bengal Basin of India and Bangladesh
Quantitative evaluation of management strategies for long-term supply of safe groundwater for drinking from the Bengal Basin aquifer (India and Bangladesh) requires estimation of the large-scale hydrogeologic properties that control flow. The Basin consists of a stratified, heterogeneous sequence of sediments with aquitards that may separate aquifers locally, but evidence does not support existence of regional confining units. Considered at a large scale, the Basin may be aptly described as a single aquifer with higher horizontal than vertical hydraulic conductivity. Though data are sparse, estimation of regional-scale aquifer properties is possible from three existing data types: hydraulic heads, 14C concentrations, and driller logs. Estimation is carried out with inverse groundwater modeling using measured heads, by model calibration using estimated water ages based on 14C, and by statistical analysis of driller logs. Similar estimates of hydraulic conductivities result from all three data types; a resulting typical value of vertical anisotropy (ratio of horizontal to vertical conductivity) is 104. The vertical anisotropy estimate is supported by simulation of flow through geostatistical fields consistent with driller log data. The high estimated value of vertical anisotropy in hydraulic conductivity indicates that even disconnected aquitards, if numerous, can strongly control the equivalent hydraulic parameters of an aquifer system. 相似文献
6.
Allan D. Woodbury 《Mathematical Geology》1989,21(3):285-308
Bayesian updating methods provide an alternate philosophy to the characterization of the input variables of a stochastic mathematical model. Here, a priori values of statistical parameters are assumed on subjective grounds or by analysis of a data base from a geologically similar area. As measurements become available during site investigations, updated estimates of parameters characterizing spatial variability are generated. However, in solving the traditional updating equations, an updated covariance matrix may be generated that is not positive-definite, particularly when observed data errors are small. In addition, measurements may indicate that initial estimates of the statistical parameters are poor. The traditional procedure does not have a facility to revise the parameter estimates before the update is carried out. alternatively, Bayesian updating can be viewed as a linear inverse problem that minimizes a weighted combination of solution simplicity and data misfit. Depending on the weight given to the a priori information, a different solution is generated. A Bayesian updating procedure for log-conductivity interpolation that uses a singular value decomposition (SVD) is presented. An efficient and stable algorithm is outlined that computes the updated log-conductivity field and the a posteriori covariance of the estimated values (estimation errors). In addition, an information density matrix is constructed that indicates how well predicted data match observations. Analysis of this matrix indicates the relative importance of the observed data. The SVD updating procedure is used to interpolate the log-conductivity fields of a series of hypothetical aquifers to demonstrate pitfalls and possibilities of the method. 相似文献
7.
Allan D. Woodbury 《Mathematical Geosciences》1989,21(3):285-308
Bayesian updating methods provide an alternate philosophy to the characterization of the input variables of a stochastic mathematical model. Here, a priori values of statistical parameters are assumed on subjective grounds or by analysis of a data base from a geologically similar area. As measurements become available during site investigations, updated estimates of parameters characterizing spatial variability are generated. However, in solving the traditional updating equations, an updated covariance matrix may be generated that is not positive-definite, particularly when observed data errors are small. In addition, measurements may indicate that initial estimates of the statistical parameters are poor. The traditional procedure does not have a facility to revise the parameter estimates before the update is carried out. alternatively, Bayesian updating can be viewed as a linear inverse problem that minimizes a weighted combination of solution simplicity and data misfit. Depending on the weight given to the a priori information, a different solution is generated. A Bayesian updating procedure for log-conductivity interpolation that uses a singular value decomposition (SVD) is presented. An efficient and stable algorithm is outlined that computes the updated log-conductivity field and the a posteriori covariance of the estimated values (estimation errors). In addition, an information density matrix is constructed that indicates how well predicted data match observations. Analysis of this matrix indicates the relative importance of the observed data. The SVD updating procedure is used to interpolate the log-conductivity fields of a series of hypothetical aquifers to demonstrate pitfalls and possibilities of the method. 相似文献
8.
Ordinary Cokriging Revisited 总被引：12，自引：0，他引：12
P. Goovaerts 《Mathematical Geology》1998,30(1):21-42
This paper sets up the relations between simple cokriging and ordinary cokriging with one or several unbiasedness constraints. Differences between cokriging variants are related to differences between models adopted for the means of primary and secondary variables. Because it is not necessary for the secondary data weights to sum to zero, ordinary cokriging with a single unbiasedness constraint gives a larger weight to the secondary information while reducing the occurrence of negative weights. Also the weights provided by such cokriging systems written in terms of covariances or correlograms are not related linearly, hence the estimates are different. The prediction performances of cokriging estimators are assessed using an environmental dataset that includes concentrations of five heavy metals at 359 locations. Analysis of reestimation scores at 100 test locations shows that kriging and cokriging perform equally when the primary and secondary variables are sampled at the same locations. When the secondary information is available at the estimated location, one gains little by retaining other distant secondary data in the estimation. 相似文献
9.
This paper sets up the relations between simple cokriging and ordinary cokriging with one or several unbiasedness constraints. Differences between cokriging variants are related to differences between models adopted for the means of primary and secondary variables. Because it is not necessary for the secondary data weights to sum to zero, ordinary cokriging with a single unbiasedness constraint gives a larger weight to the secondary information while reducing the occurrence of negative weights. Also the weights provided by such cokriging systems written in terms of covariances or correlograms are not related linearly, hence the estimates are different. The prediction performances of cokriging estimators are assessed using an environmental dataset that includes concentrations of five heavy metals at 359 locations. Analysis of reestimation scores at 100 test locations shows that kriging and cokriging perform equally when the primary and secondary variables are sampled at the same locations. When the secondary information is available at the estimated location, one gains little by retaining other distant secondary data in the estimation. 相似文献
10.
Microdiamonds offer several advantages as a resource estimation tool, such as access to deeper parts of a deposit which may be beyond the reach of large diameter drilling (LDD) techniques, the recovery of the total diamond content in the kimberlite, and a cost benefit due to the cheaper treatment cost compared to large diameter samples. In this paper we take the first step towards local estimation by showing that micro-diamond samples can be treated as a regionalised variable suitable for use in geostatistical applications and we show examples of such output. Examples of microdiamond variograms are presented, the variance-support relationship for microdiamonds is demonstrated and consistency of the diamond size frequency distribution (SFD) is shown with the aid of real datasets. The focus therefore is on why local microdiamond estimation should be possible, not how to generate such estimates. Data from our case studies and examples demonstrate a positive correlation between micro- and macrodiamond sample grades as well as block estimates. This relationship can be demonstrated repeatedly across multiple mining operations. The smaller sample support size for microdiamond samples is a key difference between micro- and macrodiamond estimates and this aspect must be taken into account during the estimation process. We discuss three methods which can be used to validate or reconcile the estimates against macrodiamond data, either as estimates or in the form of production grades: (i) reconcilliation using production data, (ii) by comparing LDD-based grade estimates against microdiamond-based estimates and (iii) using simulation techniques.
相似文献11.
In the framework of various studies to characterize the aquifer at the groundwater experimental field near Montalto Uffugo, Italy, the present work estimates the spatial distribution of hydraulic conductivity of the unconsolidated deposits that underlie the area, by applying the geostatistical technique of kriging with external drift to electrical-resistivity and hydraulic-conductivity data. The reliability of the estimation method was tested by implementing a model, based on the method of cells, that simulates groundwater flow, with the estimated values of hydraulic conductivity. The results obtained indicate that the estimation method used has an acceptable degree of reliability. 相似文献
12.
Ute C. Herzfeld 《Mathematical Geosciences》2008,40(5):587-618
The topic of this paper is the retrieval of hidden or secondary information on complex spatial variables from geophysical
data. Typical situations of obscured geological or geophysical information are the following: (1) Noise may disturb the signal
for a variable for which measurements have been collected. (2) The variable of interest may be obscured by other geophysical
processes. (3) The information of interest may formally be captured in a secondary variable, whereas data may have been collected
for a primary variable only, that is related to the geophysical process of interest. Examples discussed here include mapping
of marine-geologic provinces from bathymetric data, identification of sea-ice properties from snow-depth data, analysis of
snow surface data in an Alpine environment and association of deformation types in fast-moving glaciers from airborne video
material or satellite imagery. Data types include geophysical profile or trackline data, image data, grid or matrix-type data,
and more generally, any two-dimensional or three-dimensional discrete or discretizable data sets.
The framework for a solution is geostatistical characterization and classification, which typically involves the following
steps: (1) calculation of vario functions (which may be of higher order or residual type, or combinations of both), (2) derivation
of classification parameters from vario functions, and (3) characterization, classification or segmentation, depending on
the applied problem. In some situations, spatial surface roughness is utilized as an auxiliary variable, for instance, roughness
of the seafloor may be derived from bathymetric data and be indicative of geological provinces.
The objective of this paper is to present components of the geostatistical classification method in a summarizing and synoptical
manner, motivated by applied examples and integrating principal and generalized concepts, such as hyperparameters and parameters
that relate to the same physical processes and work for data in oversampled and undersampled situations, parameters that facilitate
comparison among different data types, data sets and across scales, variograms and vario functions of higher order, and deterministic
and connectionist classification algorithms. 相似文献
13.
Katherine Campbell 《Mathematical Geology》1988,20(6):699-715
Use of intrinsic random function stochastic models as a basis for estimation in geostatistical work requires the identification of the generalized covariance function of the underlying process. The fact that this function has to be estimated from data introduces an additional source of error into predictions based on the model. This paper develops the sample reuse procedure called the bootstrap in the context of intrinsic random functions to obtain realistic estimates of these errors. Simulation results support the conclusion that bootstrap distributions of functionals of the process, as well as their kriging variance, provide a reasonable picture of variability introduced by imperfect estimation of the generalized covariance function.This paper was presented at Emerging Concepts, MGUS-87 Conference, Redwood City, California, 13–15 April 1987. 相似文献
14.
Method of Data Reduction and Uncertainty Estimation for Platinum-Group Element Data Using Inductively Coupled Plasma-Mass Spectrometry 总被引：1，自引：0，他引：1
A data reduction method is described for determining platinum-group element (PGE) abundances by inductively coupled plasma-mass spectrometry (ICP-MS) using external calibration or the method of standard addition. Gravimetric measurement of volumes, the analysis of reference materials and the use of procedural blanks were all used to minimise systematic errors. Internal standards were used to correct for instrument drift. A linear least squares regression model was used to calculate concentrations from drift-corrected counts per second (cps). Furthermore, mathematical manipulations also contribute to the uncertainty estimates of a procedure. Typical uncertainty estimate calculations for ICP-MS data manipulations involve: (1) Carrying standard deviations from the raw cps through the data reduction or (2) calculating a standard deviation from multiple final concentration calculations. It is demonstrated that method 2 may underestimate the uncertainty estimate of the calculated data. Methods 1 and 2 do not typically include an uncertainty estimate component from a regression model. As such models contribute to the uncertainty estimates affecting the calculated data, an uncertainty estimate component from the regression must be included in any final error calculations. Confidence intervals are used to account for uncertainty estimates from the regression model. These confidence intervals are simpler to calculate than uncertainty estimates from method 1, for example. The data reduction and uncertainty estimation method described here addresses problems of reporting PGE data from an article in the literature and addresses both precision and accuracy. The method can be applied to any analytical technique where drift corrections or regression models are used. 相似文献
15.
基于地质统计学的NDVI图像估值技术 总被引：2，自引：1，他引：1
将疏采样后的NDVI图像作为未受云层影响的已知数据,分别用普通克里格、泛克里格、指示克里格和序贯指示模拟对NDVI图像进行恢复并比较其效果。研究发现,各种克里格法对NDVI图像的估值效果由高到低依次为泛克里格、普通克里格、指示克里格,通常计算方便的普通克里格法就能够满足图像恢复所要求的精度;普通克里格方差和泛克里格方差只能反映数据的构型,不能很好地衡量估值图像的不确定性,指示克里格的条件方差的分布和实际误差的分布基本一致,能够较好地衡量估值图像的不确定性,并且其大小与NDVI影像数据的不确定性大小的分布一致。序贯指示模拟得到的多个等概率实现表现出很大的空间变异性,多个实现的均值图像光滑效应明显,估值精度不高,但是多个实现的方差分布可以很好地表征空间数据的不确定性分布。 相似文献
16.
Ali Akbar Daya 《Journal of the Geological Society of India》2014,83(5):567-576
Most significant iron ore deposits in Iran are located in Central Iran Zone. These deposits belong to the Bafq mining district. The Bafq mining district is located in the Early Cambrian Kashmar-Kerman volcanic arc of Central Iran. Linear estimation of regionalized variables (for example by inverse distance weighting or ordinary Kriging) results in relatively high estimation variances, i.e. the estimates have very low precision. Assessment of project economics (or other critical decision making) based on linear estimation is therefore risky. Non-linear estimation methods like disjunctive kriging perform better and the lower estimation variance allows less risky economic decision-making. Another advantage of disjunctive kriging is that it allows estimation of functions of the primary variable, which here is the grade (Fe %) of the ore. In particular it permits estimation of indicator functions defined using thresholds on the primary variable. This paper is devoted to application of disjunctive kriging method in Choghart North Anomaly iron ore deposit in Central Iran, Yazd province, Iran. In this study, the Fe concentration of Choghart North Anomaly iron ore deposit was modelled and estimated. The exploration data consists of borehole samples measuring the Fe concentration. A Gaussian isofactorial model is fitted to these data and disjunctive kriging was used to estimate the regionalized variable (Fe %) at unsampled locations and to assess the probabilities that the actual concentrations exceed a threshold value at a given location. Consequently a three dimensional model of probability of exceeding a threshold value and the estimated value are provided by disjunctive kriging to divide the ore into an economic and uneconomic part on the basis of estimation of indicator functions using thresholds grades defined on point support. The tools and concepts are complemented by a set of computer programs that are applied to the case study. The study showed that disjunctive kriging can be applied successfully for modeling the grade of an ore deposit. Results showed that the correlation between the estimated value and real value at locations close to each other is 81.9%. 相似文献
17.
In many parts of the world sedimentary horizons with potential for hydrocarbon are located below flood basalt provinces. However,
the presence of high velocity basaltic overburden makes delineation of sediments difficult due to the low velocity layer problem.
Electrical and electromagnetic methods have been used in such scenarios because of the good electrical conductivity contrast
between basalts and underlying sediments. However, mapping of the target sediments becomes difficult when the layer is thin
as the data errors due to inherent noise lead to equivalent solutions. To tackle such difficult situations, a joint inversion
scheme incorporating seismic reflection and refraction, magnetotelluric and deep electrical resistivity datasets is presented.
Efficacy of the scheme is tested for a model comprising a thin sedimentary layer sandwiched between a thick basalt cover and
a granitic basement. The results indicate that the parameters of the target sedimentary layer are either poorly resolved or
equivalent solutions are obtained by the inversion of individual datasets. Joint inversions of seismic reflection (RFLS) and
refraction (RFRS), or DC and MT dataset pairs provide improved results and the range of equivalent solutions is narrowed down.
Combination of any three of the above datasets leads to further narrowing of this range and improvements in mean model estimates.
Joint inversion incorporating all the datasets is found to yield good estimates of the structure. Resolution analysis is carried
out to appraise estimates of various model parameters obtained by jointly inverting different combinations of datasets. 相似文献
18.
This paper describes a geostatistical technique based on conditional simulations to assess confidence intervals of local
estimates of lake pH values on the Canadian Shield. This geostatistical approach has been developed to deal with the estimation
of phenomena with a spatial autocorrelation structure among observations. It uses the autocorrelation structure to derive
minimum-variance unbiased estimates for points that have not been measured, or to estimate average values for new surfaces.
A survey for lake water chemistry has been conducted by the Ministère de l'Environnement du Québec between 1986 and 1990, to assess surface water quality and delineate the areas affected by acid precipitation on the southern
Canadian Shield in Québec. The spatial structure of lake pH was modeled using two nested spherical variogram models, with
ranges of 20 km and 250 km, accounting respectively for 20% and 55% of the spatial variation, plus a random component accounting
for 25%. The pH data have been used to construct a number of geostatistical simulations that produce plausible realizations
of a given random function model, while 'honoring' the experimental values (i.e., the real data points are among the simulated
data), and that correspond to the same underlying variogram model. Post-processing of a large number of these simulations,
that are equally likely to occur, enables the estimation of mean pH values, the proportion of affected lakes (lakes with pH≤5.5),
and the potential error of these parameters within small regions (100 km×100 km). The method provides a procedure to establish
whether acid rain control programs will succeed in reducing acidity in surface waters, allowing one to consider small areas
with particular physiographic features rather than large drainage basins with several sources of heterogeneity. This judgment
on the reduction of surface water acidity will be possible only if the amount of uncertainty in the estimation of mean pH
is properly quantified.
Received: 3 March 1997 · Accepted: 16 November 1998 相似文献
19.
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
20.
Craig Loehle 《Mathematical Geology》2005,37(2):127-140
Timeseries of estimated temperature have been combined to create global or hemispheric climate series over periods exceeding 1000 yr. The data used in these studies, however, may be subject to dating errors. It is shown that when timeseries with dating error are combined, the noise in the data smoothes periodic signals but leaves linear trends intact. This means that the effect of dating error of sample data in a timeseries reconstruction is to smooth out any signals (waves, cycles) that may be present. The purpose of this study was to develop signal extraction methods that will work for this type of historical data. The method used was nonlinear estimation of sample series where dating error has been added by Monte Carlo sampling. Several algorithms were tested for handling the dating error problem. Results were that using nonlinear model fitting, the periods of signals can be identified even from the averaged data. In a second stage of the estimation procedure, the cycle magnitudes can be estimated. Very good fits were achieved for two example cases. Temperature estimation error (white noise due to the use of proxies) was also considered and the method was extended to cover this case with quite good results. Using the new estimation methods, the information inherent in multiple series can be used to overcome the problem of dating error. 相似文献