The integral of the semivariogram: A powerful method for adjusting the semivariogram in geostatistics

A good fining of the structural junction that describes the variability of a spatial phenomenon is an essential stage in the building of an accurate estimator by kriging. The technique of the integral of the semivariogram (ISV) makes it possible to find this structural function while overcoming the problem of grouping together the pairs of experimental points into classes of distances when the data are not sampled on a regular grid. The ISV is particularly useful when the dispersion of the values of the classical Semivariogram (SV) makes it difficult to fit a model. Since the ISV is composed of a large number of values, it is more continuous than a SV and therefore easier to fit analytically. In fact, when the general shape of the SV is known, the ISV method proves its worth in finding the parameters that best fit a given variogram model. The analytical models of ISV which will be used, are the integral expressions of the traditional analytical SV. In this paper and on the basis of hydrogeological examples, we propose a method to adjust all the parameters of each model. The first derivative of a filled ISV, used in the kriging equations, appears to be systematically the best SV for a cross-validation on the data. This is why we think that the ISV technique should be used when the strong spatial variability of a parameter spreads out the values of the experimental SV.     

Estimating Regional Hydraulic Conductivity Fields—A Comparative Study of Geostatistical Methods

Geostatistical estimations of the hydraulic conductivity field (K) in the Carrizo aquifer, Texas, are performed over three regional domains of increasing extent: 1) the domain corresponding to a three-dimensional groundwater flow model previously built (model domain); 2) the area corresponding to the 10 counties encompassing the model domain (County domain), and; 3) the full extension of the Carrizo aquifer within Texas (Texas domain). Two different approaches are used: 1) an indirect approach where transmissivity (T) is estimated first and K is retrieved through division of the T estimate by the screen length of the wells, and; 2) a direct approach where K data are kriged directly. Due to preferential well screen emplacement, and scarcity of sampling in the deeper portions of the formation (> 1 km), the available data set is biased toward high values of hydraulic conductivities. Kriging combined with linear regression, simple kriging with varying local means, kriging with an external drift, and cokriging allow the incorporation of specific capacity as secondary information. Prediction performances (assessed through cross-validation) differ according to the chosen approach, the considered variable (log-transformed or back-transformed), and the scale of interest. For the indirect approach, kriging of log T with varying local means yields the best estimates for both log-transformed and back-transformed variables in the model domain. For larger regional scales (County and Texas domains), cokriging performs generally better than other kriging procedures when estimating both (log T)^∗ and T^∗. Among procedures using the direct approach, the best prediction performances are obtained using kriging of log K with an external drift. Overall, geostatistical estimation of the hydraulic conductivity field at regional scales is rendered difficult by both preferential well location and preferential emplacement of well screens in the most productive portions of the aquifer. Such bias creates unrealistic hydraulic conductivity values, in particular, in sparsely sampled areas.     

Instability in Principal Component Analysis and the Quantification of Polyphenism in Palaeontological Data

The occurrence of cryptic polyphenism (variation in morphological properties within a single species) in ammonites is used to exemplify the application of the multivariate set of techniques known in analytical chemistry as cross-validation to quantify and isolate deviating specimens (ecomorphs) in a genetically homogeneous sample. A byproduct of the analysis bears on a method of identifying redundant variables. A species of Nigerian Turonian (Cretaceous) ammonites of the genus Thomasites is used in the exemplification.     

Application of universal kriging for estimation of earthquake ground motion: Statistical significance of results

Universal kriging is compared with ordinary kriging for estimation of earthquake ground motion. Ordinary kriging is based on a stationary random function model; universal kriging is based on a nonstationary random function model representing first-order drift. Accuracy of universal kriging is compared with that for ordinary kriging; cross-validation is used as the basis for comparison. Hypothesis testing on these results shows that accuracy obtained using universal kriging is not significantly different from accuracy obtained using ordinary kriging. Tests based on normal distribution assumptions are applied to errors measured in the cross-validation procedure;t andF tests reveal no evidence to suggest universal and ordinary kriging are different for estimation of earthquake ground motion. Nonparametric hypothesis tests applied to these errors and jackknife statistics yield the same conclusion: universal and ordinary kriging are not significantly different for this application as determined by a cross-validation procedure. These results are based on application to four independent data sets (four different seismic events).     

hv-block交叉验证法在率定水文模型参数中的比较研究

本文指出了应用分离样本验证法率定水文模型参数的不足,引入并探讨了hv-block交叉验证法的适用性,并与分离样本验证法进行比较。不同流域、不同系列长度和不同时段资料的计算结果表明,hv-block交叉验证法的模拟预报精度要明显优于分离样本验证法,该法能够充分利用资料信息,特别是当资料系列较短的情况下,其优越性更加突出。     

Geotechnical characterization of the upper Pleistocene–Holocene alluvial deposits of Roma (Italy) by means of multivariate geostatistics: Cross-validation results

We are presenting an attempt to evaluate the spatial variability of geotechnical parameters in the upper Pleistocene–Holocene alluvial deposits of Roma (Italy) by means of multivariate geostatistics.The upper Pleistocene–Holocene alluvial deposits of Roma are sensitive to high levels of geohazard. They occupy a sizable and significant part of the city, being the foundation for many monuments, historical neighborhoods, and archaeological areas, and the main host of the present and future subway lines. We have stored information from more than 2000 geotechnical boreholes crossing the alluvial deposits into a relational database. For the present study, only the boreholes with lithologic/textural interpretation and geotechnical information were selected. The set includes 283 boreholes and 719 samples, which have a set of geotechnical information comprising physical properties and mechanical parameters.Techniques of multivariate statistics and geostatistics were combined and compared to evaluate the estimation methods of the mechanical parameters, with special reference to the drained friction angle from direct shear test (φ′). Principal Component Analysis was applied to the dataset to highlight the relationships between the geotechnical parameters. Through cross-validation analysis, multiple linear regression, kriging, and cokriging were tested as estimators of φ′. Cross-validation demonstrates that the cokriging with granulometries as auxiliary variables is the most suitable method to estimate φ′. In addition to proving that cokriging is a good estimator of φ′, cross-validation demonstrates that input data are coherent and this allows us to use them for estimation of geotechnical parameters, although they come from different laboratories and different vintages.Nevertheless, to get the same good results of cross-validation in estimation, it is necessary for granulometries to be available at grid points. Since this information being not available at all grid points, it is expected that, in the future, textural information can be derived in an indirect way, i.e., from lithologic/textural spatial reconstructions.     

Comparison of kriging techniques in a space-time context

Space-time processes constitute a particular class, requiring suitable tools in order to predict values in time and space, such as a space-time variogram or covariance function. The space-time co-variance function is defined and linked to the Linear Model of Coregionalization under second-order space-time stationarity. Simple and ordinary space-time kriging systems are compared to simple and ordinary cokriging and their differences for unbiasedness conditions are underlined. The ordinary space-time kriging estimation then is applied to simulated data. Prediction variances and prediction errors are compared with those for ordinary kriging and cokriging under different unbiasedness conditions using a cross-validation. The results show that space-time kriging tend to produce lower prediction variances and prediction errors that kriging and cokriging.     

Intelligent modelling of clay compressibility using hybrid meta-heuristic and machine learning algorithms

Compression index Ccis an essential parameter in geotechnical design for which the effectiveness of correlation is still a challenge.This paper suggests a novel modelling approach using machine learning(ML)technique.The performance of five commonly used machine learning(ML)algorithms,i.e.back-propagation neural network(BPNN),extreme learning machine(ELM),support vector machine(SVM),random forest(RF)and evolutionary polynomial regression(EPR)in predicting Cc is comprehensively investigated.A database with a total number of 311 datasets including three input variables,i.e.initial void ratio e0,liquid limit water content wL,plasticity index Ip,and one output variable Cc is first established.Genetic algorithm(GA)is used to optimize the hyper-parameters in five ML algorithms,and the average prediction error for the 10-fold cross-validation(CV)sets is set as thefitness function in the GA for enhancing the robustness of ML models.The results indicate that ML models outperform empirical prediction formulations with lower prediction error.RF yields the lowest error followed by BPNN,ELM,EPR and SVM.If the ranges of input variables in the database are large enough,BPNN and RF models are recommended to predict Cc.Furthermore,if the distribution of input variables is continuous,RF model is the best one.Otherwise,EPR model is recommended if the ranges of input variables are small.The predicted correlations between input and output variables using five ML models show great agreement with the physical explanation.     

改进 GＲNN 网络预测致密砂岩气层压裂产能

致密砂岩储层孔隙度小、渗透率低、含气饱和度低,基本上没有自然产能,需要进行压裂,所以压裂产能的预测很重要。广义回归神经网络 ( GＲNN) 稳定,对样本数量的要求低。产能预测关键是样本的选取以及扩展因子的选取。在原有的 GＲNN 预测产能的基础上,利用交叉验证法改进 GＲNN 网络,选取最优的样本确定最优的 GＲNN 网络结构,利用循环判断法,选取最优的扩展因子。改进的 GＲNN 神经网络可以避免确定 GＲNN 网络结构和扩展因子过程中过多的人为影响。笔者利用灰色关联分析法分析压裂产能的影响因素,利用改进的 GＲNN 网络有针对性地建立适合苏里格地区致密砂岩气层的压裂产能预测模型。结果表明该方法在苏里格地区气层压裂产能预测中有较好的应用效果。     

T -distributed Random Fields: A Parametric Model for Heavy-tailedWell-log Data1

Histograms of observations from spatial phenomena are often found to be more heavy-tailed than Gaussian distributions, which makes the Gaussian random field model unsuited. A T-distributed random field model with heavy-tailed marginal probability density functions is defined. The model is a generalization of the familiar Student-T distribution, and it may be given a Bayesian interpretation. The increased variability appears cross-realizations, contrary to in-realizations, since all realizations are Gaussian-like with varying variance between realizations. The T-distributed random field model is analytically tractable and the conditional model is developed, which provides algorithms for conditional simulation and prediction, so-called T-kriging. The model compares favourably with most previously defined random field models. The Gaussian random field model appears as a special, limiting case of the T-distributed random field model. The model is particularly useful whenever multiple, sparsely sampled realizations of the random field are available, and is clearly favourable to the Gaussian model in this case. The properties of the T-distributed random field model is demonstrated on well log observations from the Gullfaks field in the North Sea. The predictions correspond to traditional kriging predictions, while the associated prediction variances are more representative, as they are layer specific and include uncertainty caused by using variance estimates.     

Spatial Variability of Rock Depth in Bangalore Using Geostatistical, Neural Network and Support Vector Machine Models

Geospatial technology is increasing in demand for many applications in geosciences. Spatial variability of the bed/hard rock is vital for many applications in geotechnical and earthquake engineering problems such as design of deep foundations, site amplification, ground response studies, liquefaction, microzonation etc. In this paper, reduced level of rock at Bangalore, India is arrived from the 652 boreholes data in the area covering 220 km². In the context of prediction of reduced level of rock in the subsurface of Bangalore and to study the spatial variability of the rock depth, Geostatistical model based on Ordinary Kriging technique, Artificial Neural Network (ANN) and Support Vector Machine (SVM) models have been developed. In Ordinary Kriging, the knowledge of the semi-variogram of the reduced level of rock from 652 points in Bangalore is used to predict the reduced level of rock at any point in the subsurface of the Bangalore, where field measurements are not available. A new type of cross-validation analysis developed proves the robustness of the Ordinary Kriging model. ANN model based on multi layer perceptrons (MLPs) that are trained with Levenberg–Marquardt backpropagation algorithm has been adopted to train the model with 90% of the data available. The SVM is a novel type of learning machine based on statistical learning theory, uses regression technique by introducing loss function has been used to predict the reduced level of rock from a large set of data. In this study, a comparative study of three numerical models to predict reduced level of rock has been presented and discussed.     

A correction model for conditional bias in selective mining operations

A nonlinear correction functionK(Z*) is proposed to transform any initial linear grade estimatorZ* into a conditional unbiased estimatorZ**=K(Z*) with reduced conditional estimation variance. Such a corrected estimator allows more accurate prediction of ore reserves at any level of selection performed during the mine lifetime. The correction is based upon an analytical or isofactorial representation of a bivariate distribution model of true gradeZ and its estimatorZ*. This correction model allows derivation of conditional estimation variances for both estimatorsZ* andZ** and provides a solution to the problem of change of support. A case study is presented and performance of the proposed correction model is evaluated in terms of actual conditional bias and mean squared errors. Results obtained stress the practical importance of the correction model in selective mining operations.     

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.     

Support Vector Machines for Landslide Susceptibility Mapping: The Staffora River Basin Case Study, Italy

The aim of this study is the application of support vector machines (SVM) to landslide susceptibility mapping. SVM are a set of machine learning methods in which model capacity matches data complexity. The research is based on a conceptual framework targeted to apply and test all the procedural steps for landslide susceptibility modeling from model selection, to investigation of predictive variables, from empirical cross-validation of results, to analysis of predicted patterns. SVM were successfully applied and the final susceptibility map was interpreted via success and prediction rate curves and receiver operating characteristic (ROC) curves, to support the modeling results and assess the robustness of the model. SVM appeared to be very specific learners, able to discriminate between the informative input and random noise. About 78% of occurrences was identified within the 20% of the most susceptible study area for the cross-validation set. Then the final susceptibility map was compared with other maps, addressed by different statistical approaches, commonly used in susceptibility mapping, such as logistic regression, linear discriminant analysis, and naive Bayes classifier. The SVM procedure was found feasible and able to outperform other techniques in terms of accuracy and generalization capacity. The over-performance of SVM against the other techniques was around 18% for the cross-validation set, considering the 20% of the most susceptible area. Moreover, by analyzing receiver operating characteristic (ROC) curves, SVM appeared to be less prone to false positives than the other models. The study was applied in the Staffora river basin (Lombardy, Northern Italy), an area of about 275 km² characterized by a very high density of landslides, mainly superficial slope failures triggered by intense rainfall events.     

Sensitivity of Spatial Analysis Neural Network Training and Interpolation to Structural Parameters

A Spatial Analysis Neural Network (SANN) algorithm was applied for the analysis of geospatial data, on the basis of nonparametric statistical analysis and the concepts of traditional Artificial Neural Networks. SANN consists of a number of layers in which the neurons or nodes between layers are interconnected successively in a feed-forward direction. The Gaussian Kernel Function layer has several nodes, and each node has a transfer or an activation function that only responds (or activates) when the input pattern falls within its receptive field, which is defined by its smoothing parameter or width. The activation widths are functions of the model structural parameters, including the number of the nearest neighbor points P and a control factor F. The estimation method is based on two operational modes, namely, a training-validation mode in which the model structure is constructed and validated, and an interpolation mode. In this paper we discuss the effect of varying F and P upon the accuracy of the estimation in a two-dimensional domain for different input field sizes, using spatial data of wheat crop yield from Eastern Colorado. Crop yield is estimated as a function of the two-dimensional Cartesian coordinates (easting and northing). The results of the research led to the conclusion that optimal values of F and P depend on the sample size, i.e., for small data sets F=1.5 and P=7 while for large data sets F=2.5 and P=9. In addition, the accuracy of the interpolated field varies with the sample size. As expected for small sample sizes, the interpolated field and its variability may be significantly underestimated.     

Probabilistic assessment of soil liquefaction considering spatial variability of CPT measurements

The conventional liquefaction potential assessment methods (also known as simplified methods) profoundly rely on empirical correlations based on observations from case histories. A probabilistic framework is developed to incorporate uncertainties in the earthquake ground motion prediction, the cyclic resistance prediction, and the cyclic demand prediction. The results of a probabilistic seismic hazard assessment, site response analyses, and liquefaction potential analyses are convolved to derive a relationship for the annual probability and return period of liquefaction. The random field spatial model is employed to quantify the spatial uncertainty associated with the in-situ measurements of geotechnical material.     

天山山区降水量的空间分布及其估算方法

利用天山山区及周边31个国家基本站1998～2008年期间的逐月降水资料结合TRMM卫星月平均降水资料,使用卫星结合雨量计的降水估算方法,得到天山山区逐月降水空间分布,并运用交叉检验方法对降水估算精度评估,Nash-Sutcliffe效率系数在0.5以上,相关系数高达0.9以上。结果表明,TRMM卫星对西北山区的降水活动有一定的探测能力,能够较好的反映天山山区降水时空变化特征,为山区降水数据稀缺条件下的降水空间分布估算提供方法,为相关的水文、气象等研究提供数据支持。     

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%.     

普朗铜矿床铜品位分布地质统计学研究

运用地质统计学的方法对普朗矿区铜品位分布情况进行了研究,建立了铜品位变异规律的数学模型,分别得到了矿体厚度、倾向和走向3个方向的变异函数,该函数呈几何异向性,比值为1:1.612:3.869,反映出矿体铜品位在3个方向上的相关性较好,说明铜品位分布总体变化系数不大,有利于矿山的开采.统计得出铜品位属于正态分布,表明下一步用普通克立格法进行估值效果最好,为矿山规划设计和生产提供了理论依据.     

Back-analysis for initial ground stress field at a diamond mine using machine learning approaches

Exact knowledge for ground stress field guarantees the construction of various underground engineering projects as well as prediction of some geological hazards such as the rock burst. Limited by costs, field measurement for initial ground stresses can be only conducted on several measure points, which necessitates back-analysis for initial stresses from limited field measurement data. This paper employed a multioutput decision tree regressor (DTR) to model the relationship between initial ground stress field and its impact factor. A full-scale finite element model was built and computed to gain 400 training samples for DTR using a submodeling strategy. The results showed that correlation coefficient r between field measurement values and back-analysis values reached 0.92, which proved the success of DTR. A neural network was employed to store the global initial ground stress field. More than 600,000 node data extracted from the full-scale finite element model were used to train this neural network. After training, the stresses on any location can be investigated by inputting corresponding coordinates into this neural network.