Conditional estimation variances: An empirical approach

One of the tasks routinely carried out by geostatisticians is the evaluation of global mining reserves corresponding to a given cutoff grade and size of selective mining units. A long with these recovery figures, the geostatistician generally provides an assessment of the global estimation variance, which represents the precision of the overall average grade estimate, when no cutoff is applied. Such a global estimation variance is of limited interest for evaluating mining projects; what is required is the reliability of the estimate of recovered reserves or, in other words, the conditional estimation variance. Unfortunately, classical linear geostatistical methods fail to provide an easy way to estimate this variance. Through the use of simulated deposits (representing various types of regionalization)the present paper reviews and discusses the effects of changes in cutoff grade and selective mining unit size on the conditional estimation variance. It is shown that, when the cutoff grade is applied to a pointsupport (sample-size)distribution, the conditional estimation variance appears to be readily accessible by classical formulas, once the conditional semivariogram is known. However, the evaluation of the conditional estimation variance seems to be less straightforward for the general case when a cutoff is applied to the average grade distribution of selective mining units. Empirical approximation formulas for the conditional estimation variance are tentatively proposed, and their performance in the case of the simulated deposits is shown. The limitations of these approximations are discussed, and possible ways of formalizing the problem suggested.     

Analysis of the Variability of a Two-Dimensional Finite Strain Estimate

Finite strain estimation is a widely used technique for the study of rock deformation in structural geology. One particular algorithm proposed by Shimamoto and Ikeda uses the ‘average shape matrix’ of deformed markers. This paper provides a detailed error analysis for resulting strain estimates in two dimensions. When the number of markers exceeds 100, estimators of components of the strain tensor are shown to have an approximately Gaussian distribution with variances that increase with their mean. Equal variance estimators are obtained by applying a log transform for the elongation and an arcsin transformation for the orientation estimates. Confidence interval formulae for strain tensor components are proposed. Lithology specific constants arising in these formulae are estimated from undeformed samples. The results are validated by application to simulated data as well as observational data from thin sections of sandstone sampled from SE Ireland.     

Notes on the robustness of the kriging system

The robustness of the kriging system with respect to uncertainty of the theoretical variogram is investigated. Inequalities for possible changes of the kriging estimator and the estimation variance are derived. Results of a numerical study show that changes of kriging weights can be predicted partly with the help of the maximal kriging weight.     

An application of the deterministic variogram to design-based variance estimation

When estimating the mean value of a variable, or the total amount of a resource, within a specified region it is desirable to report an estimated standard error for the resulting estimate. If the sample sites are selected according to a probability sampling design, it usually is possible to construct an appropriate design-based standard error estimate. One exception is systematic sampling for which no such standard error estimator exists. However, a slight modification of systematic sampling, termed 2-step tessellation stratified (2TS) sampling, does permit the estimation of design-based standard errors. This paper develops a design-based standard error estimator for 2TS sampling. It is shown that the Taylor series approximation to the variance of the sample mean under 2TS sampling may be expressed in terms of either a deterministic variogram or a deterministic covariance function. Variance estimation then can be approached through the estimation of a variogram or a covariance function. The resulting standard error estimators are compared to some more traditional variance estimators through a simulation study. The simulation results show that estimators based on the new approach may perform better than traditional variance estimators.     

Estimator Stability in Geological Applications

Classic mathematical statistics recommends maximum likelihood estimators of parameters of a model because they have minimal variance in the model. The theory of robustness showed that these estimators were unstable to small deviations of probability density. The estimator stability is necessary for applications, where reality is always more complex than any model, especially in geology, where objects are unique. Methods of calculus of variations give a measure of the estimator stability, and the maximum likelihood estimators have little stability. Simultaneous maximization of efficiency and stability gives new estimators more suitable for applications. The estimator instability is especially harmful in the estimation of the multivariate normal distribution. To avoid instability, multivariate problems are reduced to sequences of bivariate problems. An example of the solution of a geological problem shows that methods of classic statistics are not good and the reductive method is much better.     

General and local estimation of local average and their application in geotechnical parameter estimations

Two kinds of estimation variance functions for estimating a local average (LA) of a stationary (homogeneous) random field (RF) are derived. One is local estimation (LE) and the other is general estimation (GE) of LA. The former is for estimating LA at the observation location, and the latter is for obtaining LA at any arbitrary location within the RF. The geotechnical implications of these two estimations are that LE is for estimating LA of geotechnical parameters at the spot where the investigations are made, whereas GE is for estimating LA at any arbitrary location within the same layer. The behavior of the two estimation variance functions differs greatly, controlled by the number of observations (i.e. sample size) and the normalized layer thickness (layer thickness divided by autocorrelation distance of RF). Based on the derived estimation variance functions, methods for determining reliable characteristic values of geotechnical parameters and necessary sample size are proposed. The methods are based on the same framework as that of the traditional statistical theory, i.e. confidence interval of estimated parameters. However, the assumption of independently and identically distributed (i.i.d.) samples in the traditional statistical theory is replaced by the assumption of correlated samples from a stationary RF. The results obtained from the proposed methods for LE and GE differ from each other as well as from the traditional results, which has significant implications for geotechnical parameter estimation in geotechnical engineering practice.     

The updated kriging variance and optimal sample design

A number of criteria based on kriging variance calculations may be used for infill sampling design in geologic site characterization. Searching for the best new sample locations from a set of candidate locations can result in excessive computation time if these criteria and the naive rekriging are used. The relative updated kriging estimate and variance for universal kriging estimation are demonstrated as a simple kriging estimate and variance, respectively. The updated kriging variance is demonstrated as the multiplication of two kriging variances. Using these updated kriging variance equations can increase the computational speed for selecting the best new sample locations. The application results for oil rock thickness in an oilfield indicate that minimizing the average relative updated kriging variance is a useful alternative to the other criteria based on kriging variance in optimal infill sampling design for geologic site characterization.     

On the equivalence of kriging and maximum entropy estimators

This study compares kriging and maximum entropy estimators for spatial estimation and monitoring network design. For second-order stationary random fields (a subset of Gaussian fields) the estimators and their associated interpolation error variances are identical. Simple lognormal kriging differs from the lognormal maximum entropy estimator, however, in both mathematical formulation and estimation error variances. Two numerical examples are described that compare the two estimators. Simple lognormal kriging yields systematically higher estimates and smoother interpolation surfaces compared to those produced by the lognormal maximum entropy estimator. The second empirical comparison applies kriging and entropy-based models to the problem of optimizing groundwater monitoring network design, using six alternative objective functions. The maximum entropy-based sampling design approach is shown to be the more computationally efficient of the two.     

Adaptive state estimation of groundwater contaminant boundary input flux in a 2-dimensional aquifer

In many circumstances involving heat and mass transfer issues,it is considered impractical to measure the input flux and the resulting state distribution in the domain.Therefore,the need to develop techniques to provide solutions for such problems and estimate the inverse mass flux becomes imperative.Adaptive state estimator(ASE)is increasingly becoming a popular inverse estimation technique which resolves inverse problems by incorporating the semi-Markovian concept into a Bayesian estimation technique,thereby developing an inverse input and state estimator consisting of a bank of parallel adaptively weighted Kalman filters.The ASE is particularly designed for a system that encompasses independent unknowns and/or random switching of input and measurement biases.The present study describes the scheme to estimate the groundwater input contaminant flux and its transient distribution in a conjectural two-dimensional aquifer by means of ASE,which in particular is because of its unique ability to efficiently handle the process noise giving an estimation of keeping the relative error range within 10%in 2-dimensional problems.Numerical simulation results show that the proposed estimator presents decent estimation performance for both smoothly and abruptly varying input flux scenarios.Results also show that ASE enjoys a better estimation performance than its competitor,Recursive Least Square Estimator(RLSE)due to its larger error tolerance in greater process noise regimes.ASE's inherent deficiency of being slower than the RLSE,resulting from the complexity of algorithm,was also noticed.The chosen input scenarios are tested to calculate the effect of input area and both estimators show improved results with an increase in input flux area especially as sensors are moved closer to the assumed input location.     

Adaptive state estimation of groundwater contaminant boundary input flux in a 2-dimensional aquifer

In many circumstances involving heat and mass transfer issues, it is considered impractical to measure the input flux and the resulting state distribution in the domain. Therefore, the need to develop techniques to provide solutions for such problems and estimate the inverse mass flux becomes imperative. Adaptive state estimator (ASE) is increasingly becoming a popular inverse estimation technique which resolves inverse problems by incorporating the semi-Markovian concept into a Bayesian estimation technique, thereby developing an inverse input and state estimator consisting of a bank of parallel adaptively weighted Kalman filters. The ASE is particularly designed for a system that encompasses independent unknowns and /or random switching of input and measurement biases. The present study describes the scheme to estimate the groundwater input contaminant flux and its transient distribution in a conjectural two-dimensional aquifer by means of ASE, which in particular is because of its unique ability to efficiently handle the process noise giving an estimation of keeping the relative error range within 10% in 2-dimensional problems. Numerical simulation results show that the proposed estimator presents decent estimation performance for both smoothly and abruptly varying input flux scenarios. Results also show that ASE enjoys a better estimation performance than its competitor, Recursive Least Square Estimator (RLSE) due to its larger error tolerance in greater process noise regimes. ASE’s inherent deficiency of being slower than the RLSE, resulting from the complexity of algorithm, was also noticed. The chosen input scenarios are tested to calculate the effect of input area and both estimators show improved results with an increase in input flux area especially as sensors are moved closer to the assumed input location.     

Recursive parameter estimation with applications in earth sciences

Parameter estimation has become increasingly interesting the last few decades for a variety of engineering topics. In such situations, one may face problems like (a) how to estimate parameters for which erroneous measurements are available (direct estimation), or (b) how to estimate coefficients of some process model governing a geological phenomenon when these coefficients are inaccessible or difficult to access by direct investigation (inverse estimation or identification). Both these problems are examined in this presentation from a modern stochastic viewpoint, where parameters sought are interpretated mathematically as random functions, generated and estimated in space or time with the aid of recursive models. Advantages of this methodology are remarkable, from both theoretical and physical points of view, as compared to conventional statistics or nonrecursive estimators. Particularly it may offer more accurate estimators, better representation of spatial variation, and a means of overcoming difficulties such as excessive computational time or computer storage. To test effectiveness of this type of estimation, a series of representative case studies from geotechnical practice have been computed in detail.     

On-line estimation of nonlinear physical systems

Recursive algorithms for estimating states of nonlinear physical systems are presented. Orthogonality properties are rediscovered and the associated polynomials are used to linearize state and observation models of the underlying random processes. This requires some key hypotheses regarding the structure of these processes, which may then take account of a wide range of applications. The latter include streamflow forecasting, flood estimation, environmental protection, earthquake engineering, and mine planning. The proposed estimation algorithm may be compared favorably to Taylor series-type filters, nonlinear filters which approximate the probability density by Edgeworth or Gram-Charlier series, as well as to conventional statistical linearization-type estimators. Moreover, the method has several advantages over nonrecursive estimators like disjunctive kriging. To link theory with practice, some numerical results for a simulated system are presented, in which responses from the proposed and extended Kalman algorithms are compared.     

Determining the best search neighbourhood in reserve estimation, using geostatistical method: A case study anomaly No 12A iron deposit in central Iran

Ordinary kriging and non-linear geostatistical estimators are now well accepted methods in mining grade control and mine reserve estimation. In kriging, the search volume or ‘kriging neighbourhood’ is defined by the user. The definition of the search space can have a significant impact on the outcome of the kriging estimate. In particular, too restrictive neighbourhood, can result in serious conditional bias. Kriging is commonly described as a ‘minimum variance estimator’ but this is only true when the neighbourhood is properly selected. Arbitrary decisions about search space are highly risky. The criteria to consider when evaluating a particular kriging neighbourhood are the slope of the regression of the ‘true’ and ‘estimated’ block grades, the number of kriging negative weights and the kriging variance. Search radius is one of the most important parameters of search volume which often is determined on the basis of influence of the variogram. In this paper the above-mentioned parameters are used to determine optimal search radius.     

Hybrid Estimation of Semivariogram Parameters

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.     

Random vectors and spatial analysis by geostatistics for geotechnical applications

Geostatistics is extended to the spatial analysis of vector variables by defining the estimation variance and vector variogram in terms of the magnitude of difference vectors. Many random variables in geotechnology are in vectorial terms rather than scalars, and its structural analysis requires those sample variable interpolations to construct and characterize structural models. A better local estimator will result in greater quality of input models; geostatistics can provide such estimators: kriging estimators. The efficiency of geostatistics for vector variables is demonstrated in a case study of rock joint orientations in geological formations. The positive cross-validation encourages application of geostatistics to spatial analysis of random vectors in geoscience as well as various geotechnical fields including optimum site characterization, rock mechanics for mining and civil structures, cavability analysis of block cavings, petroleum engineering, and hydrologic and hydraulic modelings.     

Seismic hazard assessment under complex source size distribution of mining-induced seismicity

It is well-documented that a variety of factors controlling the rockmass fracturing process in mines often results in a complexity of mining event size distribution. In such cases, the estimation of the probability functions of source size parameterizations, with the use of presently known distribution models, brings about an unacceptable and systematic over- or underestimation of the seismic hazard parameters. It is, therefore, recommended that the non-parametric, kernel estimators of the event size distribution functions, be applied to stationary hazard studies in mining seismicity.These data-driven estimators, adapted to seismic source size characterization, accurately fit all kinds of data underlying distributions, regardless of their complexity. Recently, the non-parametric approach to size characterization was supported by a special method of uncertainty analysis based on resampling techniques. At present, it is a fully developed method, which provides point and interval estimates of size distribution functions and related hazard parameters. Two examples of its use in studying mining seismic data are presented and discussed in this paper. The analyzed data sets were recorded in two different copper mines in Poland. The smoothed bootstrap test for multimodality, which is a specialized tool for investigating the shapes of probability densities, provided highly significant proof that in both cases the probability densities of source size parameterization were complex thus implied the superiority of the non-parametric estimation to the classic, model-based approach in the studied cases. The data were then used to construct non-parametric, kernel estimates of the source size cumulative distribution function (CDF), the exceedance probability and the mean return period. Furthermore, confidence intervals for these quantities were also estimated. The intervals for CDF were narrow, showing that the procedures of non-parametric estimation and resampling based uncertainty analysis were precise. Due to the fact that the mean return period is very sensitive to values of the CDF, in particular for larger events sizes, the uncertainty of the return period estimates was not insignificant but remained manageable. The point and interval estimates of source size CDF and hazard parameters so obtained were compared with the respective point estimates achieved from the inappropriate in the case of complex magnitude distributions, model-based approach.     

Evaluation and comparison of spatial interpolators II

The performance of several variations on ordinary kriging and inverse distance estimators is evaluated. Mean squared errors (MSE) were calculated for estimates made on multiple resamplings from five exhaustive data bases representing two distinctly different types of estimation problem. Ordinary kriging, when performed with variograms estimated from the sample data, was more robust than inverse-distance methods to the type of estimation problem, and to the choice of estimation parameters such as number of neighbors.Notice: Although the research described in this article has been funded in part by the United States Environmental Protection Agency through Cooperative Agreement CR818526 to the Harry Reid Center for Environmental Studies, University of Nevada-Las Vegas, it has not been subjected to Agency review. Therefore it does not necessarily reflect the views of the Agency. Mention of trade names or commercial products does not constitute endorsement or recommendation for use.     

On least squares estimation of generalized covariance functions

Least squares estimation (LSE) is theoretically related to quadratic unbiased estimation of variance components. It is argued that these methods of estimation of variance components essentially generalize LSE though they are not formally equivalent.     

An Alternative Measure of the Reliability of Ordinary Kriging Estimates

This paper presents an interpolation variance as an alternative to the measure of the reliability of ordinary kriging estimates. Contrary to the traditional kriging variance, the interpolation variance is data-values dependent, variogram dependent, and a measure of local accuracy. Natural phenomena are not homogeneous; therefore, local variability as expressed through data values must be recognized for a correct assessment of uncertainty. The interpolation variance is simply the weighted average of the squared differences between data values and the retained estimate. Ordinary kriging or simple kriging variances are the expected values of interpolation variances; therefore, these traditional homoscedastic estimation variances cannot properly measure local data dispersion. More precisely, the interpolation variance is an estimate of the local conditional variance, when the ordinary kriging weights are interpreted as conditional probabilities associated to the n neighboring data. This interpretation is valid if, and only if, all ordinary kriging weights are positive or constrained to be such. Extensive tests illustrate that the interpolation variance is a useful alternative to the traditional kriging variance.     

Estimation of the Continuous Ranked Probability Score with Limited Information and Applications to Ensemble Weather Forecasts

The continuous ranked probability score (CRPS) is a much used measure of performance for probabilistic forecasts of a scalar observation. It is a quadratic measure of the difference between the forecast cumulative distribution function (CDF) and the empirical CDF of the observation. Analytic formulations of the CRPS can be derived for most classical parametric distributions, and be used to assess the efficiency of different CRPS estimators. When the true forecast CDF is not fully known, but represented as an ensemble of values, the CRPS is estimated with some error. Thus, using the CRPS to compare parametric probabilistic forecasts with ensemble forecasts may be misleading due to the unknown error of the estimated CRPS for the ensemble. With simulated data, the impact of the type of the verified ensemble (a random sample or a set of quantiles) on the CRPS estimation is studied. Based on these simulations, recommendations are issued to choose the most accurate CRPS estimator according to the type of ensemble. The interest of these recommendations is illustrated with real ensemble weather forecasts. Also, relationships between several estimators of the CRPS are demonstrated and used to explain the differences of accuracy between the estimators.