A parametric study of robustness of kriging variance as a function of range and relative nugget effect for a spherical semivariogram

In geostatistics, an estimation of blocks of a deposit is reported along with the variance of error made in their estimation. This calculation is based on the model chosen for the semivariogram of the deposit so that mistakes in its estimation can manifest themselves in the perception of accuracy with which blocks are known. Changes in kriging variance resulting from various amounts of error in modeling the relative nugget effect and range of the semivariogram are investigated for an extensive set of spherical semivariograms.     

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.     

The Impact of Biased Sampling on the Estimation of the Semivariogram Within Fractured Media Containing Multiple Fracture Sets

Monte Carlo simulation was used to examine the error (statistical bias) introduced in estimating a sample semivariogram through application of oriented sampling patterns to variables which are correlated with fracture orientation. Sample semivariograms of the directional components of the water velocity were used to illustrate that oriented sampling schemes can provide biased data sets which result in error in the estimation of the semivariogram, particularly in the estimation of the sill (or variance). Three sampling patterns were used to analyze directional semivariograms of the components of the fluid velocity: sampling along lines parallel to the mean regional hydraulic gradient, sampling among lines perpendicular to the mean regional hydraulic gradient, and sampling along fracture segments. The first two sampling patterns were shown to introduce substantial error in the sills of the velocity variograms. It is argued that this error is due to the combination of unequal sampling of fractures with different orientations (i.e., sampling bias) and the systematic variation in the magnitude of the velocity components with orientation of the fracture. As a consequence, it is suggested that correction factors developed to correct fracture frequency statistics need to be extended to improve estimation of spatial moments of variables which are correlated with fracture orientation.     

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.     

Estimation of semivariograms by the maximum entropy method

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

Determination of support in soil sampling

Short-range variation of measurements of a soil characteristic is a function of the support (i.e., size, shape, and orientation of the physical sample taken at a sample point) of the soil samples. This short-range variance often shows up as a substantial component of the nugget effect in experimental semivariograms or estimated generalized covariance functions. Proper choice of support may substantially reduce short-range variance (or nugget) of measurements made at sample points and thereby reduce estimation variance in point or block kriging based on these measurements. Appropriate variance formulas and an example are given for determination of support when support is an array of vertical cores (or bucket auger samples) of equal depth and diameter.     

Analytical studies on some statistical properties of mean values and their applications to sampling in one dimension

This paper documents the results of investigations on the variance and covariance properties of mean values within the theory of stationary random functions under a variety of conditions. Analytical expressions have been developed in each case, to facilitate direct and ready applications of the results, using the exponential function as a working model for the covariance function. These properties, in turn, have been utilized in developing mathematical expressions for estimation accuracy (error variance) of the mean estimates for one-dimensional sampling plans. Two distinctly different sampling plans have been discussed: punctual sampling with no significant linear extensions of the samples and linear sampling with significant linear extensions of samples in the direction of the section being sampled.     

Accounting for Estimation Optimality Criteria in Simulated Annealing

This paper presents both estimation and simulation as optimization problems that differ in the optimization criteria, minimization of a local expected loss for estimation and reproduction of global statistics (semivariogram, histogram) for simulation. An intermediate approach is proposed whereby an initial random image is gradually modified using simulated annealing so as to better match both local and global constraints. The relative weights of the different constraints in the objective function allow the user to strike a balance between smoothness of the estimated map and reproduction of spatial variability by simulated maps. The procedure is illustrated using a synthetic dataset. The proposed approach is shown to enhance the influence of observations on neighboring simulated values, hence the final realizations appear to be better conditioned to the sample information. It also produces maps that are more accurate (smaller prediction error) than stochastic simulation ignoring local constraints, but not as accurate as E-type estimation. Flow simulation results show that accounting for local constraints yields, on average, smaller errors in production forecast than a smooth estimated map or a simulated map that reproduces only the histogram and semivariogram. The approach thus reduces the risk associated with the use of a single realization for forecasting and planning.     

The Second-Order Stationary Universal Kriging Model Revisited

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.     

On counting the number of data pairs for semivariogram estimation

In planning spatial sampling studies for the purpose of estimating the semivariogram, the number of data pairs separated by a given distance is sometimes used as a comparative index of the precision which can be expected from a given sampling design. Because spatial data are correlated, this index can be unreliable. An alternative index which partially corrects for this correlation, themaximum equivalent uncorrelated pairs, is proposed for comparing spatial designs. The index is developed under the assumption that the underlying stochastic process is Gaussian and is appropriate when the (population) semivariogram is to be estimated by the sample semivariogram.     

Summary statistics in geochemistry: A study of the performance of robust estimates

Numerical data summaries in many geochemical papers rely on arithmetic means, with or without standard deviations. Yet the mean is the worst average (estimate of location) for those extremely common geochemical data sets which are non-normally distributed or include outliers. The widely used geometric mean, although allowing for skewed distributions, is equally susceptible to outliers. The superior performance of 19 robust estimates of location (simple median, plus various combined, adaptive, trimmed, and skipped,L, M, andW estimates) is illustrated using real geochemical data sets varying in sources of error (pure analytical error to multicomponent geological variability), modality (unimodal to polymodal), size (20 to >2000 data values), and continuity (continuous to truncated in either or both tails). The arithmetic mean tends to overestimate location of many geochemical data sets because of positive skew and large outliers; robust estimates yield consistent smaller averages, although some (e.g., Hampel's and Andrew's) do perform better than others (e.g., Shorth mean, dominant cluster mode). Recommended values for international standard rocks, and for such important geochemical concepts as average chondrite, can be reproduced far more simply via robust estimation on complete interlaboratory data sets than via the rather complicated and subjective methods (e.g., laboratory ratings) so far used in the literature. Robust estimates also seem generally less affected by truncation than the mean; for example, if values below machine detection limits are alternatively treated as missing values or as real values of zero, similar averages are obtained. The standard (and mean) deviations yield consistently larger values of scale for many geochemical data sets than the hinge width (interquartile range) or median absolute deviation from the median. Therefore, summaries of geochemical data should always include at least the simple median and hinge width, to complement the often misleading mean and standard deviation.     

Kriging in terms of projections

In the last few years, an increasing number of practical studies using so-called kriging estimation procedures have been published. Various terms, such as universal kriging, lognormal kriging, ordinary kriging, etc., are used to define different estimation procedures, leaving a certain confusion about what kriging really is. The object of this paper is to show what is the common backbone of all these estimation procedures, thus justifying the common name of kriging procedures. The word kriging (in French krigeage) is a concise and convenient term to designate the classical procedure of selecting, within agiven class of possible estimators, the estimator with a minimum estimation variance (i.e., the estimator which leads to a minimum variance of the resulting estimation error). This estimation variance can be seen as a squared distance between the unknown value and its estimator; the process of minimization of this distance can then be seen as the projection of the unknown value onto the space within which the search for an estimator is carried out.     

On the condition number of covariance matrices in kriging, estimation, and simulation of random fields

The numerical stability of linear systems arising in kriging, estimation, and simulation of random fields, is studied analytically and numerically. In the state-space formulation of kriging, as developed here, the stability of the kriging system depends on the condition number of the prior, stationary covariance matrix. The same is true for conditional random field generation by the superposition method, which is based on kriging, and the multivariate Gaussian method, which requires factoring a covariance matrix. A large condition number corresponds to an ill-conditioned, numerically unstable system. In the case of stationary covariance matrices and uniform grids, as occurs in kriging of uniformly sampled data, the degree of ill-conditioning generally increases indefinitely with sampling density and, to a limit, with domain size. The precise behavior is, however, highly sensitive to the underlying covariance model. Detailed analytical and numerical results are given for five one-dimensional covariance models: (1) hole-exponential, (2) exponential, (3) linear-exponential, (4) hole-Gaussian, and (5) Gaussian. This list reflects an approximate ranking of the models, from best to worst conditioned. The methods developed in this work can be used to analyze other covariance models. Examples of such representative analyses, conducted in this work, include the spherical and periodic hole-effect (hole-sinusoidal) covariance models. The effect of small-scale variability (nugget) is addressed and extensions to irregular sampling schemes and higher dimensional spaces are discussed.     

Kriging made easy or assessing the linear unbiased minimum variance estimate by adapting the projection theorem

A new approach has been arrived at for determining the linear unbiased minimum variance estimate of ore reserves, or the Kriging. The approach rests heavily on geometric intuition, which reduces the estimation to the simple task of determining an orthogonal projection. First, the problem is presented and solved in ordinary three-dimensional space. The result is stated in matrix formulation, lending itself to a simple generalization in higher dimensions. Then, the general problem is stated in random-variable notation and solved via the results derived. Finally, an illustrative example is given to demonstrate the new approach.     

Fitting the Linear Model of Coregionalization by Generalized Least Squares

In geostatistical studies, the fitting of the linear model of coregionalization (LMC) to direct and cross experimental semivariograms is usually performed with a weighted least-squares (WLS) procedure based on the number of pairs of observations at each lag. So far, no study has investigated the efficiency of other least-squares procedures, such as ordinary least squares (OLS), generalized least squares (GLS), and WLS with other weighing functions, in the context of the LMC. In this article, we compare the statistical properties of the sill estimators obtained with eight least-squares procedures for fitting the LMC: OLS, four WLS, and three GLS. The WLS procedures are based on approximations of the variance of semivariogram estimates at each distance lag. The GLS procedures use a variance–covariance matrix of semivariogram estimates that is (i) estimated using the fourth-order moments with sill estimates (GLS₁), (ii) calculated using the fourth-order moments with the theoretical sills (GLS₂), and (iii) based on an approximation using the correlation between semivariogram estimates in the case of spatial independence of the observations (GLS₃). The current algorithm for fitting the LMC by WLS while ensuring the positive semidefiniteness of sill matrix estimates is modified to include any least-squares procedure. A Monte Carlo study is performed for 16 scenarios corresponding to different combinations of the number of variables, number of spatial structures, values of ranges, and scale dependence of the correlations among variables. Simulation results show that the mean square error is accounted for mostly by the variance of the sill estimators instead of their squared bias. Overall, the estimated GLS₁ and theoretical GLS₂ are the most efficient, followed by the WLS procedure that is based on the number of pairs of observations and the average distance at each lag. On that basis, GLS₁ can be recommended for future studies using the LMC.     

Inner product matrices, kriging, and nonparametric estimation of variogram

Two important problems in the practical implementation of kriging are: (1) estimation of the variogram, and (2) estimation of the prediction error. In this paper, a nonparametric estimator of the variogram to circumvent the problem of the precise choice of a variogram model is proposed. Using orthogonal decomposition of the kriging predictor and the prediction error, a method for selecting, what may be considered, a statistical neighborhood is suggested. The prediction error estimates based on this scheme, in fact, reflects the true prediction error, thus leading to proper coverage for the corresponding prediction interval. By simulations and a reanalysis of published data, it is shown that the proposals made in this paper are useful in practice.     

When are relative variograms useful in geostatistics?

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 {Z_t; t D}is strongly stationary and normal over a domain D,then clearly {exp (Z_t); 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.     

Spatial hydrological drought characteristics in Karkheh River basin,southwest Iran using copulas

Investigation on drought characteristics such as severity, duration, and frequency is crucial for water resources planning and management in a river basin. While the methodology for multivariate drought frequency analysis is well established by applying the copulas, the estimation on the associated parameters by various parameter estimation methods and the effects on the obtained results have not yet been investigated. This research aims at conducting a comparative analysis between the maximum likelihood parametric and non-parametric method of the Kendall \(\tau \) estimation method for copulas parameter estimation. The methods were employed to study joint severity–duration probability and recurrence intervals in Karkheh River basin (southwest Iran) which is facing severe water-deficit problems. Daily streamflow data at three hydrological gauging stations (Tang Sazbon, Huleilan and Polchehr) near the Karkheh dam were used to draw flow duration curves (FDC) of these three stations. The \(Q_{75}\) index extracted from the FDC were set as threshold level to abstract drought characteristics such as drought duration and severity on the basis of the run theory. Drought duration and severity were separately modeled using the univariate probabilistic distributions and gamma–GEV, LN2–exponential, and LN2–gamma were selected as the best paired drought severity–duration inputs for copulas according to the Akaike Information Criteria (AIC), Kolmogorov–Smirnov and chi-square tests. Archimedean Clayton, Frank, and extreme value Gumbel copulas were employed to construct joint cumulative distribution functions (JCDF) of droughts for each station. Frank copula at Tang Sazbon and Gumbel at Huleilan and Polchehr stations were identified as the best copulas based on the performance evaluation criteria including AIC, BIC, log-likelihood and root mean square error (RMSE) values. Based on the RMSE values, nonparametric Kendall-\(\tau \) is preferred to the parametric maximum likelihood estimation method. The results showed greater drought return periods by the parametric ML method in comparison to the nonparametric Kendall \(\tau \) estimation method. The results also showed that stations located in tributaries (Huleilan and Polchehr) have close return periods, while the station along the main river (Tang Sazbon) has the smaller return periods for the drought events with identical drought duration and severity.     

Risk assessment and prediction of safety factor for circular failure slope using rock engineering systems

Circular failure is generally observed in the slope of soil, highly jointed rock mass, mine dump and weak rock. Accurate estimation of the safety factor (SF) of slopes and their performance is not an easy task. In this research, based on rock engineering systems (RES), a new approach for the estimation of the SF is presented. The introduced model involves six effective parameters on SF [unit weight (γ), pore pressure ratio (r _u), height (H), angle of internal friction (φ), cohesion (C) and slope angle (\(\beta\))], while retaining simplicity as well. In the case of SF prediction, all the datasets were divided randomly to training and testing datasets for proposing the RES model. For comparison purposes, nonlinear multiple regression models were also employed for estimating SF. The performances of the proposed predictive models were examined according to two performance indices, i.e., coefficient of determination (R ²) and mean square error. The obtained results of this study indicated that the RES is a reliable method to predict SF with a higher degree of accuracy in comparison with nonlinear multiple regression models.