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1.
Robust estimation of the variogram: I 总被引：9，自引：0，他引：9
It is a matter of common experience that ore values often do not follow the normal (or lognormal) distributions assumed for them, but, instead, follow some other heavier-tailed distribution. In this paper we discuss the robust estimation of the variogram when the distribution is normal-like in the central region but heavier than normal in the tails. It is shown that the use of a fourth-root transformation with or without the use of M-estimation yields stable robust estimates of the variogram.Visiting Scientist, NRIMS, during the period in which this work was carried out. 相似文献
2.
Undiscovered oil and gas assessments are commonly reported as aggregate estimates of hydrocarbon volumes. Potential commercial value and discovery costs are, however, determined by accumulation size, so engineers, economists, decision makers, and sometimes policy analysts are most interested in projected discovery sizes. The lognormal and Pareto distributions have been used to model exploration target sizes. This note contrasts the outcomes of applying these alternative distributions to the play level assessments of the U.S. Geological Survey's 1995 National Oil and Gas Assessment. Using the same numbers of undiscovered accumulations and the same minimum, medium, and maximum size estimates, substitution of the shifted truncated lognormal distribution for the shifted truncated Pareto distribution reduced assessed undiscovered oil by 16% and gas by 15%. Nearly all of the volume differences resulted because the lognormal had fewer larger fields relative to the Pareto. The lognormal also resulted in a smaller number of small fields relative to the Pareto. For the Permian Basin case study presented here, reserve addition costs were 20% higher with the lognormal size assumption. 相似文献
3.
It is generally agreed that particle size distributions of sediments tend ideally to approximate the form of the lognormal probability law, but there is no single widely accepted explanation of how sedimentary processes generate the form of this law. Conceptually, and in its simplest form, sediment genesis involves the transformation of a parent rock mass into a particulate end product by processes that include size reduction and selection during weathering, transportation, and deposition. The many variables that operate simultaneously during this transformation can be shown to produce a distribution of particle sizes that approaches asymptotically the lognormal form when the effect of the variables is multiplicative. This was first shown by Kolmogorov (1941). Currently available models combine breakage and selection in differing degrees, but are similar in treating the processes as having multiplicative effects on particle sizes. The present paper, based on careful specification of the initial state, the nth breakage rule and the nth selection rule, leads to two stochastic models for particle breakage, and for both models the probability distributions of particle sizes are obtained. No attempt is made to apply these models to real world sedimentary processes, although this topic is touched upon in the closing remarks. 相似文献
4.
The Second-Order Stationary Universal Kriging Model Revisited 总被引：3，自引：0，他引：3
Universal kriging originally was developed for problems of spatial interpolation if a drift seemed to be justified to model the experimental data. But its use has been questioned in relation to the bias of the estimated underlying variogram (variogram of the residuals), and furthermore universal kriging came to be considered an old-fashioned method after the theory of intrinsic random functions was developed. In this paper the model is reexamined together with methods for handling problems in the inference of parameters. The efficiency of the inference of covariance parameters is shown in terms of bias, variance, and mean square error of the sampling distribution obtained by Monte Carlo simulation for three different estimators (maximum likelihood, bias corrected maximum likelihood, and restricted maximum likelihood). It is shown that unbiased estimates for the covariance parameters may be obtained but if the number of samples is small there can be no guarantee of good estimates (estimates close to the true value) because the sampling variance usually is large. This problem is not specific to the universal kriging model but rather arises in any model where parameters are inferred from experimental data. The validity of the estimates may be evaluated statistically as a risk function as is shown in this paper. 相似文献
5.
For national or global resource estimation of frequencies of metals a lognormal distribution has sometimes been assumed but never adequately tested. Tests of frequencies of Cu, Zn, Pb, Ag, Au, Mo, Re, Ni, Co, Nb2O3, REE2O3, Cr2O3, Pt, Pd, Ir, Rh, and Ru, contents in over 3000 well-explored mineral deposits display a poor fit to the lognormal distribution. Neither a lognormal distribution nor a power law is an adequate model of the metal contents across all deposits. When these metals are grouped into 28 geologically defined deposit types, only nine of the over 100 tests fail to be fit by the lognormal distribution, and most of those failures are in two deposit types suggesting problems with those types. Significant deviations from lognormal distributions of most metals when ignoring deposit types demonstrate that there is not a global lognormal or power law equation for these metals. Mean and standard deviation estimates of each metal within deposit types provide a basis for modeling undiscovered resources. When tracts of land permissive for specific deposit types are delineated, deposit density estimates and contained metal statistics can be used in Monte Carlo simulations to estimate total amounts of undiscovered metals with associated explicit uncertainties as demonstrated for undiscovered porphyry copper deposits in the Tibetan Plateau of China. 相似文献
6.
In an open pit mine, the selection of blocks for mill feed necessitates the use of a conditionally unbiased estimator not only to maximize profits, but also to predict precisely the grades at the mill. Estimation of blocks usually is done using a series of blasthole assays on a regular grid. In many instances, the blasthole grades show a lognormal-like distribution. This study examines an estimator based on the hypothesis of bilognormality between the true block grade and the estimate obtained using the blastholes. The properties of the estimator are established and the estimator is proven to be conditionally unbiased. It is almost as precise as the lognormal kriging estimator when the points are multilognormal. However, it is more precise than lognormal krigings when only univariate lognormality is present or when the distribution is not exactly lognormal. The estimator also is shown to be robust to errors in the specifications of the variogram model or of the expectation of Z. Contrary to lognormal krigings, the estimator does only a slight correction to the original estimate obtained using the blastholes assays. 相似文献
7.
In studies that involve a finite sample size of spatial data it is often of interest to test (statistically) the assumption that the marginal (or univariate) distribution of the data is Gaussian (normal). This may be important per se because, for example, a data transformation may be desired if the normality hypothesis is rejected, or it may provide a way of testing other hypotheses, such as lognormality, by testing the normality of the logarithms of the observations. The most commonly used tests, such as the Kolmogorov–Smirnov (K–S), chi-square (2), and Shapiro–Wilks (S–W) tests, are designed on the assumption that the observations are independent and identically distributed (iid). In geostatistical applications, however, this is not usually the case unless the spatial covariance (semivariogram) function is a pure nugget variance. If the covariance structure has a (practical) range greater than the minimum distance between observations, the data are correlated and the standard tests cannot be applied to the probability density function (pdf) or cumulative probability function (cdf) estimated directly from the data. The problem with correlated data arises not from the correlation per se but from cases in which correlated data are clustered rather than being located on a regular grid. In these cases inferences requiring iid assumptions may be seriously biased because of the spatial correlation among the observations. If unbiased (i.e., de-clustered) estimates of the pdf or cdf are obtained, then normality tests, such as K-S, 2, or S–W, can be applied using the unbiased estimates and an effective number of samples equivalent to the iid case. There are three questions to be addressed in these cases: Is the distribution ergodic? 相似文献
8.
On unbiased backtransform of lognormal kriging estimates 总被引：4，自引：0，他引：4
Jorge Kazuo Yamamoto 《Computational Geosciences》2007,11(3):219-234
Lognormal kriging is an estimation technique that was devised for handling highly skewed data distributions. This technique
takes advantage of a logarithmic transformation that reduces the data variance. However, backtransformed lognormal kriging
estimates are biased because the nonbias term is totally dependent on a semivariogram model. This paper proposes a new approach
for backtransforming lognormal kriging estimates that not only presents none of the problems reported in the literature but
also reproduces the sample histogram and, consequently, the sample mean. 相似文献
9.
ZOLTÁN SYLVESTER 《Sedimentology》2007,54(4):847-870
Turbidite bed thickness distributions are often interpreted in terms of power laws, even when there are significant departures from a single straight line on a log–log exceedence probability plot. Alternatively, these distributions have been described by a lognormal mixture model. Statistical methods used to analyse and distinguish the two models (power law and lognormal mixture) are presented here. In addition, the shortcomings of some frequently applied techniques are discussed, using a new data set from the Tarcău Sandstone of the East Carpathians, Romania, and published data from the Marnoso‐Arenacea Formation of Italy. Log–log exceedence plots and least squares fitting by themselves are inappropriate tools for the analysis of bed thickness distributions; they must be accompanied by the assessment of other types of diagrams (cumulative probability, histogram of log‐transformed values, q–q plots) and the use of a measure of goodness‐of‐fit other than R2, such as the chi‐square or the Kolmogorov–Smirnov statistics. When interpreting data that do not follow a single straight line on a log–log exceedence plot, it is important to take into account that ‘segmented’ power laws are not simple mixtures of power law populations with arbitrary parameters. Although a simple model of flow confinement does result in segmented plots at the centre of a basin, the segmented shape of the exceedence curve breaks down as the sampling location moves away from the basin centre. The lognormal mixture model is a sedimentologically intuitive alternative to the power law distribution. The expectation–maximization algorithm can be used to estimate the parameters and thus to model lognormal bed thickness mixtures. Taking into account these observations, the bed thickness data from the Tarcău Sandstone are best described by a lognormal mixture model with two components. Compared with the Marnoso‐Arenacea Formation, in which bed thicknesses of thin beds have a larger variability than thicknesses of the thicker beds, the thinner‐bedded population of the Tarcău Sandstone has a lower variability than the thicker‐bedded population. Such differences might reflect contrasting depositional settings, such as the difference between channel levées and basin plains. 相似文献
10.
Empirical Maximum Likelihood Kriging: The General Case 总被引：4，自引：0，他引：4
Although linear kriging is a distribution-free spatial interpolator, its efficiency is maximal only when the experimental data follow a Gaussian distribution. Transformation of the data to normality has thus always been appealing. The idea is to transform the experimental data to normal scores, krige values in the “Gaussian domain” and then back-transform the estimates and uncertainty measures to the “original domain.” An additional advantage of the Gaussian transform is that spatial variability is easier to model from the normal scores because the transformation reduces effects of extreme values. There are, however, difficulties with this methodology, particularly, choosing the transformation to be used and back-transforming the estimates in such a way as to ensure that the estimation is conditionally unbiased. The problem has been solved for cases in which the experimental data follow some particular type of distribution. In general, however, it is not possible to verify distributional assumptions on the basis of experimental histograms calculated from relatively few data and where the uncertainty is such that several distributional models could fit equally well. For the general case, we propose an empirical maximum likelihood method in which transformation to normality is via the empirical probability distribution function. Although the Gaussian domain simple kriging estimate is identical to the maximum likelihood estimate, we propose use of the latter, in the form of a likelihood profile, to solve the problem of conditional unbiasedness in the back-transformed estimates. Conditional unbiasedness is achieved by adopting a Bayesian procedure in which the likelihood profile is the posterior distribution of the unknown value to be estimated and the mean of the posterior distribution is the conditionally unbiased estimate. The likelihood profile also provides several ways of assessing the uncertainty of the estimation. Point estimates, interval estimates, and uncertainty measures can be calculated from the posterior distribution. 相似文献
11.
Chris Roth 《Mathematical Geology》1998,30(8):999-1009
Lognormal kriging was developed early in geostatistics to take account of the often seen skewed distribution of the experimental mining data. Intuitively, taking the distribution of the data into account should lead to a better local estimate than that which would have been obtained when it is ignored. In practice however, the results obtained are sometimes disappointing. This paper tries to explain why this is so from the behavior of the lognormal kriging estimator. The estimator is shown to respect certain unbiasedness properties when considering the whole working field using the regression curve and its confidence interval for both simple or ordinary kriging. When examined locally, however, the estimator presents a behavior that is neither expected nor intuitive. These results lead to the question: is the theoretically correct lognormal kriging estimator suited to the practical problem of local estimation? 相似文献
12.
Normal and lognormal data distribution in geochemistry: death of a myth. Consequences for the statistical treatment of geochemical and environmental data 总被引：13，自引：0，他引：13
All variables of several large data sets from regional geochemical and environmental surveys were tested for a normal or
lognormal data distribution. As a general rule, almost all variables (up to more than 50 analysed chemical elements per data
set) show neither a normal or a lognormal data distribution. Even when different transformation methods are used more than
70 % of all variables in every single data set do not approach a normal distribution. Distributions are usually skewed, have
outliers and originate from more than one process. When dealing with regional geochemical or environmental data normal and/or
lognormal distributions are an exception and not the rule. This observation has serious consequences for the further statistical
treatment of geochemical and environmental data. The most widely used statistical methods are all based on the assumption
that the studied data show a normal or lognormal distribution. Neglecting that geochemcial and environmental data show neither
a normal or lognormal distribution will lead to biased or faulty results when such techniques are used.
Received: 21 June 1999 · Accepted: 14 August 1999 相似文献
13.
Kriging Prediction Intervals Based on Semiparametric Bootstrap 总被引：1，自引：0，他引：1
Kriging is a widely used method for prediction, which, given observations of a (spatial) process, yields the best linear unbiased
predictor of the process at a new location. The construction of corresponding prediction intervals typically relies on Gaussian
assumptions. Here we show that the distribution of kriging predictors for non-Gaussian processes may be far from Gaussian,
even asymptotically. This emphasizes the need for other ways to construct prediction intervals. We propose a semiparametric
bootstrap method with focus on the ordinary kriging predictor. No distributional assumptions about the data generating process
are needed. A simulation study for Gaussian as well as lognormal processes shows that the semiparametric bootstrap method
works well. For the lognormal process we see significant improvement in coverage probability compared to traditional methods
relying on Gaussian assumptions. 相似文献
14.
Extreme value analysis provides a semiparametric method for analyzing the extreme long tails of skew distributions which may be observed when handling mining data. The estimation of important tail characteristics, such as the extreme value index, allows for a discrimination between competing distribution models. It measures the thickness of such long tailed distributions, if only a limited sample is available. This paper stresses the practical implementation of extreme value theory, which is used to discriminate a lognormal from a mixed lognormal distribution in a case study of size distributions for alluvial diamonds. 相似文献
15.
Geological data frequently have a heavy-tailed normal-in-the-middle distribution, which gives rise to grade distributions that appear to be normal except for the occurrence of a few outliers. This same situation also applies to log-transformed data to which lognormal kriging is to be applied. For such data, linear kriging is nonrobust in that (1)kriged estimates tend to infinity as the outliers do, and (2)it is also not minimum mean squared error. The more general nonlinear method of disjunctive kriging is even more nonrobust, computationally more laborious, and in the end need not produce better practical answers. We propose a robust kriging method for such nearly normal data based on linear kriging of an editing of the data. It is little more laborious than conventional linear kriging and, used in conjunction with a robust estimator of the variogram, provides good protection against the effects of data outliers. The method is also applicable to time series analysis. 相似文献
16.
J. A. Vargas-Guzmán 《Mathematical Geology》2005,37(6):551-567
This paper devises an analytical solution to the classic change of support problem which is to find an upscaled probability
density function (pdf) from a non-Gaussian point support pdf. The solution considers that change of support is a transformation,
and then its expectation is not the transform of the first moment but the expectation of transformed input random variables.
If the pdf is from transformation of a Gaussian pdf, as in the case of the lognormal, the expectation of the upscaled random
variable is treated as a separate operation of the spatial expectation. This recognition allows finding the correct transform
between the point support and the upscaled or block support conditional mean estimates. This novel consideration is applied
to the change of support for the lognormal resolving the question of conservation of log-normality with an upscaled pdf that
has an extra term for balancing the center of mass after change of support. 相似文献
17.
Noel Cressie 《Mathematical Geology》1985,17(7):693-702
The relative variogram has been employed as a tool for correcting a simple kind of nonstationarity, namely that in which local variance is proportional to local mean squared. In the past, this has been linked in a vague way to the lognormal distribution, although if {Zt; t D}is strongly stationary and normal over a domain D,then clearly {exp (Zt); t D}will stillbe stationary, but lognormal. The appropriate link is made in this article through a universal transformation principle. More general situations are considered, leading to the use of a scaled variogram. 相似文献
18.
We derive the bias and sampling variation of the harmonic average. These expressions, based on the lognormal distribution,
are validated using Monte Carlo and jackknife analysis of field data. The average has a positive bias. The sampling variation
results suggest that, for moderate to large sample sizes, the harmonic average is no more variable than the arithmetic average
if the medium is appropriately sampled. 相似文献
19.
采用频率分析法计算入库设计洪水时,需要通过相关分析将坝址洪水系列插补得到对应的入库洪水系列。常用的线性回归法假设两者满足线性关系且入库洪水系列服从正态分布,可能与实际情况并不相符。引入Copula函数构建坝址洪水与入库洪水的联合概率分布和条件概率分布,计算给定坝址洪水时入库洪水的条件最可能值和置信区间,提出了一种基于Copula函数的入库洪水插补新方法。三峡水库的应用实例表明:线性回归法得到的入库洪水值在坝址洪水量级较大时明显偏小,甚至稀遇洪水时不在90%置信区间内。所提方法能较好地反映坝址洪水与入库洪水的内在关系,不仅可以计算入库洪水的各种点估计值,而且能够定量评价估计的不确定性。 相似文献
20.
M. Power 《Mathematical Geology》1992,24(8):929-945
Economic filtration has been offered as an explanation of the observed lognormality in the size distribution of discovered oil and gas deposits. The result leads to the conclusion that one cannot impute the shape of the underlying parent distribution from the observed discoveries size distribution. The fact that the largest pools tend to be discovered early in the exploration history of an area of interest suggests the existence of an inherent sampling bias in the discovery process. The bias is influenced by the levels of geologic knowledge and technological sophistication. Furthermore, the existence of the bias leads to lognormality in the observed discoveries size distribution of oil and gas pools. A discovery process model explicitly incorporating the notion of sampling bias was applied to a series of Weibull parent frequency size distributions. The selected parent distributions are of a class suggested in the literature as more reflective of nature's size distribution and have empirical support. The distribution of discoveries resulting from the application of the model to the chosen parent size distributions were tested for lognormality using a chi-squared test. Lognormality was found to be an acceptable model of the discoveries size distribution over a wide range of resource exhaustion measures. When combined with the notion of economic filtration, sampling bias leads to the conclusion that one should not expect the lognormal distribution to accurately represent the underlying parent size distribution of oil and gas deposits. 相似文献