The Theoretical Links Between Sequential Gaussian Simulation, Gaussian Truncated Simulation, and Probability Field Simulation

Sequential Gaussian simulation (sgsim), Gaussian truncated simulation (gtsim), and probability field simulation (pfsim) are three algorithms frequently used for conditional stochastic simulation. They were developed independently and are seen as different algorithms in applications. This paper establishes that gtsim and pfsim can be bridged by a simple quantile transform between Gaussian and uniform distributions. As for the sgsim algorithm, the normal score back-transform can be seen as a series of truncations of the simulated Gaussian field. All three algorithms are shown to be applicable to both continuous and categorical variables. In practice, gtsim can be most often replaced by the more CPU-efficient pfsim algorithm.     

Production of conditional simulations via the LU triangular decomposition of the covariance matrix

This paper reviews the turning band method and fast Fourier transform method of producing a nonconditional simulation of a multinormal random function with a given covariance structure. A review of the two common methods of conditioning the simulation to honor the data shows that they are formally equivalent. Another method for directly pondering a conditional simulation based on the LU triangular decomposition of the covariance matrix is presented. Computational and implementation difficulties are discussed.     

Geostatistics: Models and tools for the earth sciences

The probability construct underlying geostatistical methodology is recalled, stressing that stationary is a property of the model rather than of the phenomenon being represented. Geostatistics is more than interpolation and kriging(s) is more than linear interpolation through ordinary kriging. A few common misconceptions are addressed.     

Spectral Corrected Semivariogram Models

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.     

Compensating for estimation smoothing in kriging

Smoothing is a characteristic inherent to all minimum mean-square-error spatial estimators such as kriging. Cross-validation can be used to detect and model such smoothing. Inversion of the model produces a new estimator—compensated kriging. A numerical comparison based on an exhaustive permeability sampling of a 4-ft² slab of Berea Sandstone shows that the estimation surface generated by compensated kriging has properties intermediate between those generated by ordinary kriging and stochastic realizations resulting from simulated annealing and sequential Gaussian simulation. The frequency distribution is well reproduced by the compensated kriging surface, which also approximates the experimental semivariogram well—better than ordinary kriging, but not as well as stochastic realizations. Compensated kriging produces surfaces that are more accurate than stochastic realizations, but not as accurate as ordinary kriging.     

现代地质统计学的新进展

结合现代地质统计学最新进展,对现代地质统计学的现状进行了研究,着重时时空域中的多元信息地质编译学和时空多元动态模拟进行了探讨,并在最后指出了现代地质统计学今后的发展方向。     

Reducing the Impact of Outliers in Ore Reserves Estimation

Mining applications commonly faces surprising high values designated as outliers. These values impact dramatically statistical analysis and interpretation. A comprehensive analysis on the causes for the presence of unexpected high values was recommended. However, if an erroneous value was accepted as a part of the solution, some form of correction is recommended. A methodology based on the robust kriging (RoK) algorithm is proposed to be used in exploratory data analysis and also to deal with problems associated with the presence of outliers in the sample data set. The efficiency of RoK method as an interpolator is tested in different types of mineralizations. Importantly, the parent population from which the data was sampled is available, thus allowing direct quantitative assessment of the effectiveness of the estimation technique. The performance of the method is tested in the context of ore reserves estimation. RoK model is compared to models generated by ordinary kriging, median indicator kriging, and lognormal kriging. RoK proved to be more accurate and more precise than those methods reducing substantially the number of misclassified blocks.     

Models for Support and Information Effects: A Comparative Study

The recoverable reserves in an ore deposit depend on several factors, in particular the size of the selective mining units (support effect) and the misclassifications when sending these units to mill or dump according to their estimated grade (information effect). Both effects imply a loss of selectivity and have to be correctly forecasted. In this work, several models are reviewed and applied to a synthetic ore deposit characterized by a highly skewed grade histogram and a spatial connectivity of high grades. The affine correction, mosaic correction, and discrete Gaussian model are compared when assessing the global recoverable reserves, whereas local estimations are performed by indicator kriging with affine correction, bigaussian disjunctive kriging, and multigaussian conditional expectation. Despite their convenience and simplicity, distribution-free methods like affine correction or indicator kriging have a poorer accuracy than the other methods. In the global framework, the discrete Gaussian model is a better alternative and is based on mild assumptions. Local estimations are not accurate and may be improved by resorting to a more suitable parametric model or to conditional simulations.     

Numerical Method for Conditional Simulation of Levy Random Fields

Stochastic simulations of subsurface heterogeneity require accurate statistical models for spatial fluctuations. Incremental values in subsurface properties were shown previously to be approximated accurately by Levy distributions in the center and in the start of the tails of the distribution. New simulation methods utilizing these observations have been developed. Multivariate Levy distributions are used to model the multipoint joint probability density. Explicit bounds on the simulated variables prevent nonphysical extreme values and introduce a cutoff in the tails of the distribution of increments. Long-range spatial dependence is introduced through off-diagonal terms in the Levy association matrix, which is decomposed to yield a maximum likelihood type estimate at unobserved locations. This procedure reduces to a known interpolation formula developed for Gaussian fractal fields in the situation of two control points. The conditional density is not univariate Levy and is not available in closed form, but can be constructed numerically. Sequential simulation algorithms utilizing the numerically constructed conditional density successfully reproduce the desired statistical properties in simulations.     

Is Lognormal Kriging Suitable for Local Estimation?

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?     

A Comparison of the Sequential Gaussian and Markov-Bayes Simulation Methods for Small Samples

We compared the performance of sequential Gaussian simulation (sGs) and Markov-Bayes simulation (MBs) using relatively small samples taken from synthetic datasets. A moderate correlation (approximately r = 0.70) existed between a continuous primary variable and a continuous secondary variable. Given the small sample sizes, our objective was to determine whether MBs, with its ability to incorporate the secondary information, would prove superior to SgS. A split-split-plot computer experiment was conducted to compare the two simulation methods over a variety of primary and secondary sample sizes as well as spatial correlations. Using average mean square prediction error as a measure of local performance, sGs and MBs were roughly equivalent for random fields with short ranges (2 m). As range increased (15 m) the average mean square prediction error for sGs was less than or equal to that for MBs, even when number of noncollocated secondary observations was twice the number of collocated observations. Median variance within nonoverlapping subregions was used as a measure of the local heterogeneity or surface texture of the image. In most situations sGs images more faithfully reflected the true local heterogeneity, while MBs was more erratic, sometimes oversmoothing and sometimes undersmoothing.     

Conditional simulation with data subject to measurement error: Post-simulation filtering with modified factorial kriging

Conditional simulation with data subject to measurement error has received little attention in the geostatistical literature. The treatment of measurement error in simulation must be different from its treatment in estimation. Two approaches are examined: pre- and post-simulation filtering of data measurement error. The pre-simulation filtering is shown to be inefficient. The post-simulation filtering performs best. It is done by factorial kriging and a modified version of factorial kriging which ensures predetermined theoretical variance for the filtered data. It also is shown that the theoretical variogram of the filtered data reproduces the underlying variogram (i.e., without noise) almost perfectly. A simulation with a high level of correlated noise is used for validation and comparison. The post-simulation filtered values show an experimental variogram in agreement with the previously identified underlying variogram. Moreover, the filtered image compares well with the true image. The theoretical variogram corresponding to the post-simulation filter can be computed beforehand. Thus, the size of the simulation grid and of the filter neighborhood can be adjusted to ensure good reproduction of the underlying variogram.     

Substitution Random Fields with Gaussian and Gamma Distributions: Theory and Application to a Pollution Data Set

This paper presents random field models with Gaussian or gamma univariate distributions and isofactorial bivariate distributions, constructed by composing two independent random fields: a directing function with stationary Gaussian increments and a stationary coding process with bivariate Gaussian or gamma distributions. Two variations are proposed, by considering a multivariate directing function and a coding process with a separable covariance, or by including drift components in the directing function. Iterative algorithms based on the Gibbs sampler allow one to condition the realizations of the substitution random fields to a set of data, while the inference of the model parameters relies on simple tools such as indicator variograms and variograms of different orders. A case study in polluted soil management is presented, for which a gamma model is used to quantify the risk that pollutant concentrations over remediation units exceed a given toxicity level. Unlike the multivariate Gaussian model, the proposed gamma model accounts for an asymmetry in the spatial correlation of the indicator functions around the median and for a spatial clustering of high pollutant concentrations.     

Semivariogram Models Based on Geometric Offsets

Kriging-based geostatistical models require a semivariogram model. Next to the initial decision of stationarity, the choice of an appropriate semivariogram model is the most important decision in a geostatistical study. Common practice consists of fitting experimental semivariograms with a nested combination of proven models such as the spherical, exponential, and Gaussian models. These models work well in most cases; however, there are some shapes found in practice that are difficult to fit. We introduce a family of semivariogram models that are based on geometric shapes, analogous to the spherical semivariogram, that are known to be conditional negative definite and provide additional flexibility to fit semivariograms encountered in practice. A methodology to calculate the associated geometric shapes to match semivariograms defined in any number of directions is presented. Greater flexibility is available through the application of these geometric semivariogram models.     

Conditional LAS stochastic simulation of regionalized variables in random fields

Ore reserves forecasts are required to aid in investment decisions, mine design and valuation, short and long term production plans and proper and efficient mill design. In random multivariable fields with limited data and high levels of uncertainty, the kriged block estimates produce a smoothing effect resulting in underestimating high values and overestimating low values. The modified conditional simulation (MCS) methodology solves these problems by simulating the random field to preserve its mean and the variance structure. The simulation model is conditioned to reproduce the data at known sample points to minimize the variability between the simulated data and the true field data. In this study, the authors develop the MCS methodology to simulate ore reserve grades using the best linear unbiased estimation (BLUE) and the local average subdivision (LAS) techniques. The MCS methodology is applied to simulate block grades in a section of the Sabi Gold Project in Zimbabwe. The results are compared with the kriged estimates for this section. Analysis of the results shows that the MCS methodology reproduces the known sample grades with minimum estimation error of 0.001 while the estimation error associated with the kriged estimates is 1.104, a 100% efficiency of the MCS method over the kriging technique.     

Conditional Simulation of Random Fields by Successive Residuals

This paper presents a new approach to the LU decomposition method for the simulation of stationary and ergodic random fields. The approach overcomes the size limitations of LU and is suitable for any size simulation. The proposed approach can facilitate fast updating of generated realizations with new data, when appropriate, without repeating the full simulation process. Based on a novel column partitioning of the L matrix, expressed in terms of successive conditional covariance matrices, the approach presented here demonstrates that LU simulation is equivalent to the successive solution of kriging residual estimates plus random terms. Consequently, it can be used for the LU decomposition of matrices of any size. The simulation approach is termed conditional simulation by successive residuals as at each step, a small set (group) of random variables is simulated with a LU decomposition of a matrix of updated conditional covariance of residuals. The simulated group is then used to estimate residuals without the need to solve large systems of equations.     

Approximate local confidence intervals under change of support

Approximate local confidence intervals are constructed from uncertainty models in the form of the conditional distribution of the random variable Z given values of variables [Z_i, i=1,...,n]. When the support of the variable Z is any support other than that of the data, the conditional distributions require a change of support correction. This paper investigates the effect of change of support on the approximate local confidence intervals constructed by cumulative indicator kriging, class indicator kriging, and probability kriging under a variety of conditions. The conditions are generated by three simulated deposits with grade distributions of successively higher degree of skewness; a point support and two different block supports are considered. The paper also compares the confidence intervals obtained from these methods using the most used measures of confidence interval effectiveness.     

Hydraulic Effects of Shales in Fluvial-Deltaic Deposits: Ground-Penetrating Radar, Outcrop Observations, Geostatistics, and Three-Dimensional Flow Modeling for the Ferron Sandstone, Utah

Ground-penetrating radar (GPR) surveys, outcrop measurements, and cores provide a high-resolution 3D geologic model to investigate the hydraulic effects of shales in marine-influenced lower delta-plain distributary channel deposits within the Cretaceous-age Ferron Sandstone at Corbula Gulch in central Utah, USA. Shale statistics are computed from outcrop observations. Although slight anisotropy was observed in mean length and variogram ranges parallel and perpendicular to pale of low , the anisotropy is not statistically significant and the estimated mean length is 5.4 m. Truncated Gaussian simulation was used to create maps of shales that are placed on variably dipping stratigraphic surfaces interpreted from high-resolution 3D GPR surveys, outcrop interpretations, and boreholes. Sandstone permeability is estimated from radar responses calibrated to permeability measurements from core samples. Experimentally designed flow simulations examine the effects of variogram range, shale coverage fraction, and trends in shale coverage on predicted upscaled permeability, breakthrough time, and sweep efficiency. Approximately 1500 flow simulations examine three different geologic models, flow in the 3 coordinate directions, 16 geostatistical parameter combinations, and 10 realizations for each model. ANOVA and response models computed from the flow simulations demonstrate that shales decrease sweep, recovery, and permeability, especially in the vertical direction. The effect on horizontal flow is smaller. Flow predictions for ideal tracer displacements at Corbula Gulch are sensitive to shale-coverage fraction, but are relatively insensitive to twofold variations in variogram range or to vertical trends in shale coverage. Although the hydraulic effects of shale are statistically significant, the changes in flow responses rarely exceed 20%. As a result, it may be reasonable to use simple models when incorporating analogous shales into models of reservoirs or aquifers.     

Conditioning Simulations of Gaussian Random Fields by Ordinary Kriging

Conditioning realizations of stationary Gaussian random fields to a set of data is traditionally based on simple kriging. In practice, this approach may be demanding as it does not account for the uncertainty in the spatial average of the random field. In this paper, an alternative model is presented, in which the Gaussian field is decomposed into a random mean, constant over space but variable over the realizations, and an independent residual. It is shown that, when the prior variance of the random mean is infinitely large (reflecting prior ignorance on the actual spatial average), the realizations of the Gaussian random field are made conditional by substituting ordinary kriging for simple kriging. The proposed approach can be extended to models with random drifts that are polynomials in the spatial coordinates, by using universal or intrinsic kriging for conditioning the realizations, and also to multivariate situations by using cokriging instead of kriging.     

A Simple and Robust Lognormal Estimator

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.