Generalized covariance functions in estimation

I discuss the role of generalized covariance functions in best linear unbiased estimation and methods for their selection. It is shown that the experimental variogram (or covariance function) of the detrended data can be used to obtain a preliminary estimate of the generalized covariance function without iterations and I discuss the advantages of other parameter estimation methods.     

The equivalence of predictions from universal kriging and intrinsic random-function kriging

A proof is provided that the predictions obtained from kriging based on intrinsic random functions of orderk are identical to those obtained from anappropriate universal kriging model. This is a theoretical result based on known variability measures. It does not imply that people performing traditional universal kriging will get the same predictions as those using intrinsic random functions, because traditionally these methods differ in how variability is modeled. For intrinsic random functions, the same proof shows that predictions do not depend on the specific choice of the generalized covariance function. It is argued that the choice between these methods is really one of modeling and estimating the variability in the data.     

Distribution of kriging error and stationarity of the variogram in a coal property

If a particular distribution for kriging error may be assumed, confidence intervals can be estimated and contract risk can be assessed. Contract risk is defined as the probability that a block grade will exceed some specified limit. In coal mining, this specified limit will be set in a coal sales agreement. A key assumption necessary to implement the geostatistical model is that of local stationarity in the variogram. In a typical project, data limitations prevent a detailed examination of the stationarity assumption. In this paper, the distribution of kriging error and scale of variogram stationarity are examined for a coal property in northern West Virginia.     

A comparison between cokriging and ordinary kriging: Case study with a polymetallic deposit

Cokriging is applied to the estimation of mineral resources in a polymetallic deposit. Several major steps, which should be taken in using cokriging, are highlighted as necessary practical considerations. The case study is related to an ultramafic copper-nickel deposit. Six elements, Cu, Ni, Au, Ag, Pt, and Pd, occurring in the deposit, are partitioned into three subgroups and the elements within each group are simultaneously estimated on the basis of over 4000 drill assays. A comparison was made between ordinary kriging and cokriging methods through cross-validation. The results show that cokriging has significantly improved the estimates of resources by reducing the overall estimation error by over 15% and the variance of error by over 20%.     

Geostatistical case studies of the advantages of lognormal-de Wijsian kriging with mean for a base metal mine and a gold mine

Parallel variogram analyses, block kriging, and follow-up studies were effected for the lead content of part of the Prieska copper-zinc ore body and for the gold content of the highly variable Breef in a section of the Loraine gold mine, based first on untransformed values and second on logarithmically transformed values using the lognormal-de Wijsian model. For both models the effect was also analyzed of using the population mean or ignoring it. Practical follow-up comparisons confirm theoretical considerations and show that on these mines conditional biases can be eliminated conveniently by kriging with mean; also that the lognormal-de Wijsian model with mean gives the best results.     

A modification of minimum norm quadratic estimation of a generalized covariance function for use with large data sets

Marshall and Mardia (1985) and Kitanidis (1985) have suggested using minimum norm quadratic estimation as a method to estimate parameters of a generalized covariance function. Unfortunately, this method is difficult to use with large data sets as it requires inversion of an n × n matrix, where n is number of observations. These authors suggest replacing the matrix to be inverted by the identity matrix, which eliminates the computational burden, although with a considerable loss of efficiency. As an alternative, the data set can be broken into subsets, and minimum norm quadratic estimates of parameters of the generalized covariance function can be obtained within each subset. These local estimates can be averaged to obtain global estimates. This procedure also avoids large matrix inversions, but with less loss in efficiency.     

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.     

A new methodology for building local climate change scenarios: A case study of monthly temperature projections for Mexico City

This paper proposes a new methodology for generating climate change scenarios at the local scale based on multivariate time series models and restricted forecasting techniques. This methodology offers considerable advantages over the current statistical downscaling techniques such as: (i) it provides a better representation of climate at the local scale; (ii) it avoids the occurrence of spurious relationships between the large and local scale variables; (iii) it offers a more appropriate representation of variability in the downscaled scenarios; and (iv) it allows for compatibility assessment and combination of the information contained in both observed and simulated climate variables. Furthermore, this methodology is useful for integrating scenarios of local scale factors that affect local climate. As such, the convenience of different public policies regarding, for example, land use change or atmospheric pollution control can be evaluated in terms of their effects for amplifying or reducing climate change impacts.     

Estimation of ultimate bond strength for soil nails in clayey soils using maximum likelihood method

This paper presents a maximum likelihood estimation of the ultimate bond strength for soil nails in clays. Both uncensored and censored ultimate bond strength data for soil nails are collected from the literature. Based on the concept of maximum likelihood, a log-likelihood function is constructed for estimating the mean and coefficient of variation (COV) of the ultimate bond strength jointly using the two types of data. The mean and COV are determined as the pair that maximises the log-likelihood function. Two distribution models (normal and lognormal) are used for the estimation. A comparison of the relative competence between the two candidate distribution models that are adopted for describing the collected uncensored and censored data is performed using the Bayesian Information Criterion. Example designs of soil nail walls against internal pullout limit state of nails and overall stability limit state are provided to demonstrate the benefit of taking censored data into account for estimation of the ultimate bond strength of soil nails.     

Validation of a three‐dimensional constitutive model for nonlinear site response and soil‐structure interaction analyses using centrifuge test data

The capability of a bounding surface plasticity model with a vanishing elastic region to capture the multiaxial dynamic hysteretic responses of soil deposits under broadband (eg, earthquake) excitations is explored by using data from centrifuge tests. The said model was proposed by Borja and Amies in 1994 (J. Geotech. Eng., 120, 6, 1051‐1070), which is theoretically capable of representing nonlinear soil behavior in a multiaxial setting. This is an important capability that is required for exploring and quantifying site topography, soil stratigraphy, and kinematic effects in ground motion and soil‐structure interaction analyses. Results obtained herein indicate that the model can accurately predict key response data recorded during centrifuge tests on embedded specimens—including soil pressures and bending strains for structural walls, structures' racking displacements, and surface settlements—under both low‐ and high‐amplitude seismic input motions, which was achieved after performing only a basic material parameter calibration procedure. Comparisons are also made with results obtained using equivalent linear models and a well‐known pressure‐dependent multisurface plasticity model, which suggested that the present model is generally more accurate. The numerical convergence behavior of the model in nonlinear equilibrium iterations is also explored for a variety of numerical implementation and model parameter options. To facilitate broader use by researchers and practicing engineers alike, the model is implemented as a “user material” in ABAQUS Standard for implicit time stepping.     

白鹤滩水电站坝址区玄武岩水一岩作用模拟及应用研究

采用水一岩作用线型规划的模拟方法,以白鹤滩水电工程为实例,从水化学方面对坝区复杂水文地质条件作定量分析评价。建立由目标函数和约束条件方程组构成的数学模型,利用Excel规划求解的功能对其进行求解,为评价坝区水文地质条件提供了依据。     

白鹤滩水电站坝址区玄武岩水-岩作用模拟及应用研究

采用水-岩作用线型规划的模拟方法,以白鹤滩水电工程为实例,从水化学方面对坝区复杂水文地质条件作定量分析评价.建立由目标函数和约束条件方程组构成的数学模型,利用Excel规划求解的功能对其进行求解,为评价坝区水文地质条件提供了依据.     

A theoretical method for determining sample mass in a sieving test

Obtaining a representative particle size distribution for soil is a common challenge in earth works and is affected by the size of sieving samples. While the current standards usually specify a minimum sample mass, there is no scientific justification for the specification. Treating the preparation of sieving samples as a process of random sampling, a new method is proposed for determining the minimum mass of a sieving test sample. The proposed method is validated via laboratory tests and will improve standard specification and engineering practice.     

A complex variable solution for a deforming buoyant tunnel in a heavy elastic half‐plane

An analytical solution is presented for a buoyant tunnel in an elastic half‐plane. The tunnel undergoes a prescribed displacement along its boundary, and the surface of the half‐plane is stress‐free. The solution uses the complex variable method and consists of conformally mapping the hole and half‐plane to an annular region. The complex potentials each contain two logarithmic terms due to the resultant buoyancy force acting on the tunnel, and as a result the displacements at infinity are unbounded. An example is presented for the case of a rigid buoyant tunnel in a half‐plane with gravity loading. Even though the tunnel does not deform, the buoyancy effect can be clearly seen in the contours of the displacements and stresses. Copyright © 2002 John Wiley & Sons, Ltd.     

A technique to test finite difference schemes to model some geophysical processes in a geological structure

A preliminary problem to solve in the set-up of a mathematical model simulating a geophysical process is the choice of a suitable discrete scheme to approximate the governing differential equations. This paper presents a simple technique to test finite difference schemes used in the modeling of geophysical processes occurring in a geological structure. This technique consists in generating analytical solutions similar to the ones characterizing a geophysical process, given general information on some relevant parameters. Useful information for the choice of the discrete scheme to employ in the mathematical model simulating the original geophysical process can be obtained from the comparison between the analytical solution and the approximated numerical solutions generated by means of different discrete schemes. Two classes of numerical examples approximating the differential equation that governs the steady state earth's heat flow have been treated using three different finite differences schemes. The first class of examples deals with media whose phenomenological parameters vary as continuous space functions; the second class, instead, deals with media whose phenomenological parameters vary as discontinuous space functions. The finite difference schemes that have been utilized are: Centered Finite Difference Scheme (CDS), Arithmetic Mean Scheme (AMS), and Harmonic Mean Scheme (HMS).The numerical simulations showed that: the CDS may yield physically inconsistent solutions if the lattice internodal distance is too large, but in case of phenomenological parameters varying as a continuous function, this pitfall can be avoided increasing the lattice node refinement. In case of phenomenological parameters varying as a discontinuous function, instead, the CDS may yield physically inconsistent solutions for any lattice-node refinement. The HMS produced good results for both classes of examples showing to be a scheme suitable to model situations like these.     

A framework for economic loss estimation due to seismic transportation network disruption: a spatial computable general equilibrium approach

This paper presents a framework for assessing the economic impact of disruption in transportation that can relate the physical damage to transportation networks to economic losses. A spatial computable general equilibrium (SCGE) model is formulated and then integrated with a transportation model that can estimate the traffic volumes of freight and passengers. Economic equilibrium under a disruption in the transportation network is computed subject to the condition that the adjustment of labor and capital inputs is restricted; the model reflects slow adjustment of these linked to the state of recovery. As a case study, the model reviews the large Niigata-Chuetsu earthquake of 2004. Considering the damage to the transportation infrastructure, the model indicates the extent of the economic losses arising from the earthquake distributed over regions as a consequence of the intra- and interregional trade in a regional economy. The results show that 20% of the indirect losses occur in the Niigata region directly affected by the earthquake, whereas 40% of the total losses are experienced in the Kanto region and non-negligible losses reach rather remote zones of the country such as Okinawa.     

A novel analytical solution for constant‐head test in a patchy aquifer

A mathematical model describing the hydraulic head distribution for a constant‐head test performed in a well situated at the centre of a patchy aquifer is presented. The analytical solution for the mathematical model is derived by the Laplace transforms and the Bromwich integral method. The solution for the hydraulic head has been shown to satisfy the governing equations, related boundary conditions, and continuity requirements for the hydraulic head and flow rate at the interface of the patch and outer regions. An efficient numerical approach is proposed to evaluate the solution, which has an integral covering an integration range from zero to infinity and an integrand consisting the product and square of the Bessel functions. This solution can be used to produce the curves of dimensionless hydraulic head against dimensionless time for investigating the effect of the contrast of formation properties on the dimensionless hydraulic head distribution. Define the ratio of outer‐region transmissivity to patch‐region transmissivity as α. The dimensionless hydraulic head for α=0.1 case is about 2.72 times to that for α=10 case at dimensionless large time (e.g. τ?10⁶) when the dimensionless distance (ρ) equals 10. The results indicate that the hydraulic head distribution highly depends on the hydraulic properties of two‐zone formations. Copyright © 2006 John Wiley & Sons, Ltd.     

A generalized model for evaluating area-potential in a mineral exploration program

A generalized model for predicting the potential of geographic areas for mineral exploration is developed using computers and mathematical techniques. A cellular approach is adopted and each area is divided into cells; the data base is transformed into a computer processable form by digitizing data over each cell. Control cells are selected from a control area by two a priori subjective models using multiple linear regression and filtering techniques. These control cells are used to develop weighting factors for computer-transformed variables. The evaluation and prediction of cells are made using an evaluation model, wherein products of weighting factors and corresponding transformed variables are added to give a probability score for each cell. In the example analysis, 7.76% of cells are selected as predicted cells and checked for mining in the cell areas by comparing them with a mining data base. Of the total predicted cells, 38.13% are classified as first-order prospects and the remaining predicted cells are classified into second- and third-order prospects. The success of the prediction and the open structure of the model implies a successful, generalized model with capabilities of evaluating large areas and predicting the potential in any exploration program.Presented at the Third Decennial International Conference on Geophysical and Geochemical Exploration for Minerals and Groundwater, Sept. 27–Oct 1, 1987, Toronto, Canada.     

A spatial perspective for predicting enrollment in a regional pharmacy school

Having the ability to predict enrollment is an important task for any school’s recruiting team. The purpose of this study was to identify significant factors that can be used to predict the spatial distribution of enrollments. As a case study, we used East Tennessee State University (ETSU) pharmacy school, a regional pharmacy school located in the Appalachian Mountains. Through the application of a negative binomial regression model, we found that the most important indicators of enrollment volume for the ETSU pharmacy school were Euclidean distance, probability (based on competing pharmacy schools’ prestige, driving distance between schools and home and tuition costs), and the natural barrier of the Appalachian Mountains. Using these factors, together with other control variables, we successfully predicted the spatial distribution of enrollments for ETSU pharmacy school. Interestingly, gender also surfaced as a variable for predicting the pharmacy school’s enrollment. We found female students are more sensitive to the geographic proximity of home to school.     

A mathematical programming approach for scheduling equipment in a surface coal mining operation

Summary A mathematical programming model for scheduling open pit mining was developed and validated using data from a surface mining operation. A two-phase solution procedure was used involving repeated evaluations of an integer scheduling model and a simple transportation model.