Space Deformation Non-stationary Geostatistical Approach for Prediction of Geological Objects: Case Study at El Teniente Mine (Chile)

This article addresses the problem of the prediction of the breccia pipe elevation named Braden at the El Teniente mine in Chile. This mine is one of the world’s largest known porphyry-copper ore bodies. Knowing the exact location of the pipe surface is important, as it constitutes the internal limit of the deposit. The problem is tackled by applying a non-stationary geostatistical method based on space deformation, which involves transforming the study domain into a new domain where a standard stationary geostatistical approach is more appropriate. Data from the study domain is mapped into the deformed domain, and classical stationary geostatistical techniques for prediction can then be applied. The predicted results are then mapped back into the original domain. According to the results, this non-stationary geostatistical method outperforms the conventional stationary one in terms of prediction accuracy and conveys a more informative uncertainty model of the predictions.     

地统计方法学研究进展

地统计方法学已经成为空间预测与不确定性分析的关键性工具。本研究从文献计量学和方法学演变过程两个角度开展了1967~2005年的地统计方法学研究综述研究。首先从宏观角度分析其发展趋势、应用情况与模式,然后总结其演化规律、适宜性和选择原则。研究表明:地统计方法学的演化规律表现为稳态向非稳态演变、单变量向多变量(含二次信息)演变、参数与非参数方法相互补充、线性向非线性方法演变和空间静态向时空动态演变;此外,未来研究发展方向集中在半变异函数估计新方法、不确定性地统计学、时空地统计学与多点地统计学、机理模型与地统计学耦合研究和基于地统计学模拟的不确定性决策等。     

基于XGBoost的多源降水数据融合方法研究

气象站点观测降水难以精确反映降水时空分布与变化,而雷达降水存在复杂地形区域精度不高等问题。为了最大限度发挥两者的优势,文章以广东省北部山区为研究区域,选择2018-08-26—30一次暴雨过程为研究对象,结合地形、与海岸线距离、植被指数、经纬度等地表辅助参量,分析地面站点降水与地表辅助参量、雷达降水的相关关系,利用XGBoost算法与克里金插值方法,构建地面-雷达日降水数据融合模型,得到了空间分辨率为1 km的日降水融合数据集。此外,采用多元线性回归（LM）与克里金插值方法,实现了地面-雷达日降水数据的融合,并利用地面降水数据分别对XGBoost与LM日降水融合性能进行精度验证。结果表明：1）地面降水与雷达降水存在显著的正相关,地面降水与地表辅助参量之间的相关性随时间变化;2）XGBoost预测精度整体上高于LM预测结果;经模型残差校正后,XGBoost融合模型的精度整体上优于LM融合模型,这是因为XGBoost方法在捕捉地面降水与地表辅助参量、雷达降水之间关系性能上优于LM方法。     

Geostatistical Simulation with a Trend Using Gaussian Mixture Models

Geostatistics applies statistics to quantitatively describe geological sites and assess the uncertainty due to incomplete sampling. Strong assumptions are required regarding the location independence of statistical parameters to construct numerical models with geostatistical tools. Most geological data exhibit large-scale deterministic trends together with short-scale variations. Such location dependence violates the common geostatistical assumption of stationarity. The trend-like deterministic features should be modeled prior to conventional geostatistical prediction and accounted for in subsequent geostatistical calculations. The challenge of using a trend in geostatistical simulation algorithms for the continuous variable is the subject of this paper. A stepwise conditional transformation with a Gaussian mixture model is considered to provide a stable and artifact-free numerical model. The complex features of the regionalized variable in the presence of a trend are removed in the forward transformation and restored in the back transformation. The Gaussian mixture model provides a seamless bin-free approach to transformation. Data from a copper deposit were used as an example. These data show an apparent trend unsuitable for conventional geostatistical algorithms. The result shows that the proposed algorithm leads to improved geostatistical models.     

Spatial Prediction of Lateral Variability of a Laterite-Type Bauxite Horizon Using Ancillary Ground-Penetrating Radar Data

Tropical laterite-type bauxite deposits often pose a unique challenge for resource modelling and mine planning due to the extreme lateral variability at the base of the bauxite ore unit within the regolith profile. An economically viable drilling grid is often rather sparse for traditional prediction techniques to precisely account for the lateral variability in the lower contact of a bauxite ore unit. However, ground-penetrating radar (GPR) offers an inexpensive and rapid method for delineating laterite profiles by acquiring fine-scale data from the ground. These numerous data (secondary variable) can be merged with sparsely spaced borehole data (primary variable) through various statistical and geostatistical techniques, provided that there is a linear relation between the primary and secondary variables. Four prediction techniques, including standard linear regression, simple kriging with varying local means, co-located cokriging and kriging with an external drift, were used in this study to incorporate exhaustive GPR data in predictive estimation the base of a bauxite ore unit within a lateritic bauxite deposit in Australia. Cross-validation was used to assess the performance of each technique. The most robust estimates are produced using ordinary co-located cokriging in accordance with the cross-validation analysis. Comparison of the estimates against the actual mine floor indicates that the inclusion of ancillary GPR data substantially improves the quality of the estimates representing the bauxite base surface.     

Accounting for Parameter Uncertainty in Reservoir Uncertainty Assessment: The Conditional Finite-Domain Approach

An important aim of modern geostatistical modeling is to quantify uncertainty in geological systems. Geostatistical modeling requires many input parameters. The input univariate distribution or histogram is perhaps the most important. A new method for assessing uncertainty in the histogram, particularly uncertainty in the mean, is presented. This method, referred to as the conditional finite-domain (CFD) approach, accounts for the size of the domain and the local conditioning data. It is a stochastic approach based on a multivariate Gaussian distribution. The CFD approach is shown to be convergent, design independent, and parameterization invariant. The performance of the CFD approach is illustrated in a case study focusing on the impact of the number of data and the range of correlation on the limiting uncertainty in the parameters. The spatial bootstrap method and CFD approach are compared. As the number of data increases, uncertainty in the sample mean decreases in both the spatial bootstrap and the CFD. Contrary to spatial bootstrap, uncertainty in the sample mean in the CFD approach decreases as the range of correlation increases. This is a direct result of the conditioning data being more correlated to unsampled locations in the finite domain. The sensitivity of the limiting uncertainty relative to the variogram and the variable limits are also discussed.     

Additive models in mining and exploration

In this paper we present the use of additive models (AMs) for geostatistical applications. AMs are generalizations of linear regression models which hold the central place in the toolbox of applied statisticians. Generally speaking, the linear relationship between response and predictors is replaced with a general functional form. Recently such models were introduced in geostatistics. Especially, we give an approach for binary data. In this case we get generalized additive models (GAMs). Logistic regression is quite popular in medical and biological research. Using logit links also in GAMs we get so called additive logistic models. An application for geostatistical data is introduced. In a second approach we use AMs for spatial prediction and surface modelling. In both cases an advantage of multivariate data can be taken. The proposed applications can be used in the development of exploration strategies, especially in the early stage of exploration     

Surficial and Deep Earth Material Prediction from Geochemical Compositions

Prediction of true classes of surficial and deep earth materials using multivariate spatial data is a common challenge for geoscience modelers. Most geological processes leave a footprint that can be explored by geochemical data analysis. These footprints are normally complex statistical and spatial patterns buried deep in the high-dimensional compositional space. This paper proposes a spatial predictive model for classification of surficial and deep earth materials derived from the geochemical composition of surface regolith. The model is based on a combination of geostatistical simulation and machine learning approaches. A random forest predictive model is trained, and features are ranked based on their contribution to the predictive model. To generate potential and uncertainty maps, compositional data are simulated at unsampled locations via a chain of transformations (isometric log-ratio transformation followed by the flow anamorphosis) and geostatistical simulation. The simulated results are subsequently back-transformed to the original compositional space. The trained predictive model is used to estimate the probability of classes for simulated compositions. The proposed approach is illustrated through two case studies. In the first case study, the major crustal blocks of the Australian continent are predicted from the surface regolith geochemistry of the National Geochemical Survey of Australia project. The aim of the second case study is to discover the superficial deposits (peat) from the regional-scale soil geochemical data of the Tellus Project. The accuracy of the results in these two case studies confirms the usefulness of the proposed method for geological class prediction and geological process discovery.

     

Mapping transit‐based access: integrating GIS,routes and schedules

Seabed sediment textural parameters such as mud, sand and gravel content can be useful surrogates for predicting patterns of benthic biodiversity. Multibeam sonar mapping can provide near-complete spatial coverage of high-resolution bathymetry and backscatter data that are useful in predicting sediment parameters. Multibeam acoustic data collected across a ～1000 km² area of the Carnarvon Shelf, Western Australia, were used in a predictive modelling approach to map eight seabed sediment parameters. Four machine learning models were used for the predictive modelling: boosted decision tree, random forest decision tree, support vector machine and generalised regression neural network. The results indicate overall satisfactory statistical performance, especially for %Mud, %Sand, Sorting, Skewness and Mean Grain Size. The study also demonstrates that predictive modelling using the combination of machine learning models has provided the ability to generate prediction uncertainty maps. However, the single models were shown to have overall better prediction performance than the combined models. Another important finding was that choosing an appropriate set of explanatory variables, through a manual feature selection process, was a critical step for optimising model performance. In addition, machine learning models were able to identify important explanatory variables, which are useful in identifying underlying environmental processes and checking predictions against the existing knowledge of the study area. The sediment prediction maps obtained in this study provide reliable coverage of key physical variables that will be incorporated into the analysis of covariance of physical and biological data for this area.     

基于站点观测数据的气温空间化方法评述

基于统计学的插值方法是地理学、生态学领域研究气温空间化的主要方法之一,对获取精细化气温数据进行生态模拟具有重要意义。结合国内外气温空间插值的主要研究成果,对常用气温空间化方法进行了归纳、对比,探讨各种方法的适用性和不足之处,从而为涉及气温空间化的具体研究提供一定的参考,并探讨了各类方法优化的方向。不同方法的对比分析结果表明:各种气温空间化方法各有所长,在具体的应用中都取得过较好的效果,但并不存在普适性的方法,在实际应用时必须针对研究区域具体的地理特征进行方法适用性验证或对各类方法中的具体参数进行改进,才能实现区域气温的空间最优化模拟。根据气温场的物理分布特征,结合GIS技术,考虑地形等更多的相关因子以提高气温分布微观细节的模拟精度是未来重要的发展趋势。     

Spatial interpolation using conditional generative adversarial neural networks

ABSTRACT

Spatial interpolation is a traditional geostatistical operation that aims at predicting the attribute values of unobserved locations given a sample of data defined on point supports. However, the continuity and heterogeneity underlying spatial data are too complex to be approximated by classic statistical models. Deep learning models, especially the idea of conditional generative adversarial networks (CGANs), provide us with a perspective for formalizing spatial interpolation as a conditional generative task. In this article, we design a novel deep learning architecture named conditional encoder-decoder generative adversarial neural networks (CEDGANs) for spatial interpolation, therein combining the encoder-decoder structure with adversarial learning to capture deep representations of sampled spatial data and their interactions with local structural patterns. A case study on elevations in China demonstrates the ability of our model to achieve outstanding interpolation results compared to benchmark methods. Further experiments uncover the learned spatial knowledge in the model’s hidden layers and test the potential to generalize our adversarial interpolation idea across domains. This work is an endeavor to investigate deep spatial knowledge using artificial intelligence. The proposed model can benefit practical scenarios and enlighten future research in various geographical applications related to spatial prediction.     

A Bayesian approach to inverse modelling of stratigraphy, part 2: Validation tests

A novel inverse modelling method is applied to the problem of constraining the environmental parameters (e.g. relative sea level, sediment supply) that control stratigraphic architecture. This technique links forward modelling of shallow-marine wave/storm-dominated stratigraphy to a combination of inverse methods formulated in a Bayesian framework. We present a number of examples in which relative sea-level and sediment-supply curves were inferred from synthetic vertical successions of grain size (e.g. wells) and synthetic thickness curves (e.g. seismically derived isopachs) extracted from a forward model simulation. These examples represent different scenarios that are designed to test the impact of data distribution, quantity and quality on the uncertainty of the inferred parameters. The inverse modelling approach successfully reproduces the gross stratigraphic architectures and relative sea level and sediment-supply histories of the synthetic forward model simulation, within the constraints of the modelled data quality. The relative importance of the forcing parameters can be evaluated by their sensitivity and impact on the inverted data. Of equal importance, the inverse results allow complete characterisation of the uncertainties inherent to the stratigraphic modelling tool and to the data quality, quantity and distribution. The numerical scheme also successfully deals with the problem of non-uniqueness of the solution of the inverse problem. These preliminary results suggest that the inverse method is a powerful tool in constraining stratigraphic architecture for hydrocarbon reservoir characterisation and modelling, and it may ultimately provide a process-based geological complement to standard geostatistical tools.     

Additional Samples: Where They Should Be Located

Information for mine planning requires to be close spaced, if compared to the grid used for exploration and resource assessment. The additional samples collected during quasimining usually are located in the same pattern of the original diamond drillholes net but closer spaced. This procedure is not the best in mathematical sense for selecting a location. The impact of an additional information to reduce the uncertainty about the parameter been modeled is not the same everywhere within the deposit. Some locations are more sensitive in reducing the local and global uncertainty than others. This study introduces a methodology to select additional sample locations based on stochastic simulation. The procedure takes into account data variability and their spatial location. Multiple equally probable models representing a geological attribute are generated via geostatistical simulation. These models share basically the same histogram and the same variogram obtained from the original data set. At each block belonging to the model a value is obtained from the n simulations and their combination allows one to access local variability. Variability is measured using an uncertainty index proposed. This index was used to map zones of high variability. A value extracted from a given simulation is added to the original data set from a zone identified as erratic in the previous maps. The process of adding samples and simulation is repeated and the benefit of the additional sample is evaluated. The benefit in terms of uncertainty reduction is measure locally and globally. The procedure showed to be robust and theoretically sound, mapping zones where the additional information is most beneficial. A case study in a coal mine using coal seam thickness illustrates the method.     

Non-stationary Geostatistical Modeling: A Case Study Comparing LVA Estimation Frameworks

Incorporating locally varying anisotropy (LVA) in geostatistical modeling improves estimates for structurally complex domains where a single set of anisotropic parameters modeled globally do not account for all geological features. In this work, the properties of two LVA-geostatistical modeling frameworks are explored through application to a complexly folded gold deposit in Ghana. The inference of necessary parameters is a significant requirement of geostatistical modeling with LVA; this work focuses on the case where LVA orientations, derived from expert geological interpretation, are used to improve the grade estimates. The different methodologies for inferring the required parameters in this context are explored. The results of considering different estimation frameworks and alternate methods of parameterization are evaluated with a cross-validation study, as well as visual inspection of grade continuity along select cross sections. Results show that stationary methodologies are outperformed by all LVA techniques, even when the LVA framework has minimal guidance on parameterization. Findings also show that additional improvements are gained by considering parameter inference where the LVA orientations and point data are used to infer the local range of anisotropy. Considering LVA for geostatistical modeling of the deposit considered in this work results in better reproduction of curvilinear geological features.

     

基于水文随机方法的大尺度径流空间插值图化研究——淮河流域蚌埠以上区间为例

跨越不同时空尺度水文径流时空分异性及其图化问题的研究是当前国际地理水文科学研究的热点前沿。本文针对中国湿润流域年平均径流特点,探讨了基于地理相关关系的水文随机方法(Hydro-Stochastic) 在大尺度流域上应用的可行性。在淮河流域蚌埠以上区间开展径流空间插值实验,通过对流域出口流量在空间随机场内的分解生成研究区内10 km×10 km分辨率产流分布图,以及以1 km长度为基本单元的河道内径流量图。该方法在描述径流空间分异性的过程中,综合了水文学、地貌学、地理学的特点,以地统计空间插值方法为基础,结合了水量平衡约束以及数据不确定性分析理论,最终输出不同形式的径流空间分异结果。分析表明,径流插值结果能够很好的反映流域内产流的空间分布特征,并满足河道上下游的水量平衡关系。该方法避免了水文模型的复杂性,描述径流时不依赖气象资料,可以方便的在广大地区推广使用。     

On the a priori estimation of collocation error covariance functions: a feasibility study

Error covariance estimates are necessary information for the combination of solutions resulting from different kinds of data or methods, or for the assimilation of new results in already existing solutions. Such a combination or assimilation process demands proper weighting of the data, in order for the combination to be optimal and the error estimates of the results realistic. One flexible method for the gravity field approximation is least-squares collocation leading to optimal solutions for the predicted quantities and their error covariance estimates. The drawback of this method is related to the current ability of computers in handling very large systems of linear equations produced by an equally large amount of available input data. This problem becomes more serious when error covariance estimates have to be simultaneously computed. Using numerical experiments aiming at revealing dependencies between error covariance estimates and given features of the input data we investigate the possibility of a straightforward estimation of error covariance functions exploiting known characteristics of the observations. The experiments using gravity anomalies for the computation of geoid heights and the associated error covariance functions were conducted in the Arctic region north of 64° latitude. The correlation between the known features of the data and the parameters variance and correlation length of the computed error covariance functions was estimated using multiple regression analysis. The results showed that a satisfactory a priori estimation of these parameters was not possible, at least in the area considered.     

Sparse Data Division Using Data Segmentation and Kohonen Network for Neural Network and Geostatistical Ore Grade Modeling in Nome Offshore Placer Deposit

In this paper, sparse data problem in neural network and geostatistical modeling for ore-grade estimation was addressed in the Nome offshore placer gold deposit. The problem of sparse data arises because of the random data division into training, validation, and test subsets during ore-grade modeling. In this regard, the possibility of generating statistically dissimilar data subsets by random data division was also explored through a simulation exercise. A combined approach of data segmentation and application of a Kohonen network then was used to solve the data division problem. Two neural networks and five kriging models were applied for grade modeling. The neural network was trained using an early stopping method. Performance evaluation of the models was carried out on the test data set. The study results indicated that all the models that were investigated in this study performed almost equally. It was also revealed that by using the secondary variable watertable depth the neural network and the kriging models slightly improved their prediction precision. Further, the overall R ² of the models was poor as a result of high nugget (noisy) component in ore-grade variation.     

The comap as a diagnostic tool for non-stationary kriging models

In this study, we demonstrate a novel use of comaps to explore spatially the performance, specification and parameterisation of a non-stationary geostatistical predictor. The comap allows the spatial investigation of the relationship between two geographically referenced variables via conditional distributions. Rather than investigating bivariate relationships in the study data, we use comaps to investigate bivariate relationships in the key outputs of a spatial predictor. In particular, we calibrate moving window kriging (MWK) models, where a local variogram is found at every target location. This predictor has often proved worthy for processes that are heterogeneous, and most standard (global variogram) kriging algorithms can be adapted in this manner. We show that the use of comaps enables a better understanding of our chosen MWK models, which in turn allows a more informed choice when selecting one MWK specification over another. As case studies, we apply four variants of MWK to two heterogeneous example data sets: (i) freshwater acidification critical load data for Great Britain and (ii) London house price data. As both of these data sets are strewn with local anomalies, three of our chosen models are robust (and novel) extensions of MWK, where at least one of which is shown to perform better than a non-robust counterpart.     

Deriving ground surface digital elevation models from LiDAR data with geostatistics

This paper focuses on two common problems encountered when using Light Detection And Ranging (LiDAR) data to derive digital elevation models (DEMs). Firstly, LiDAR measurements are obtained in an irregular configuration and on a point, rather than a pixel, basis. There is usually a need to interpolate from these point data to a regular grid so it is necessary to identify the approaches that make best use of the sample data to derive the most accurate DEM possible. Secondly, raw LiDAR data contain information on above‐surface features such as vegetation and buildings. It is often the desire to (digitally) remove these features and predict the surface elevations beneath them, thereby obtaining a DEM that does not contain any above‐surface features. This paper explores the use of geostatistical approaches for prediction in this situation. The approaches used are inverse distance weighting (IDW), ordinary kriging (OK) and kriging with a trend model (KT). It is concluded that, for the case studies presented, OK offers greater accuracy of prediction than IDW while KT demonstrates benefits over OK. The absolute differences are not large, but to make the most of the high quality LiDAR data KT seems the most appropriate technique in this case.     

A case study of model selection and parameter inference by maximum likelihood with application to uncertainty analysis

One of the uses of geostatistical conditional simulation is as a tool in assessing the spatial uncertainty of inputs to the Monte Carlo method of system uncertainty analysis. Because the number of experimental data in practical applications is limited, the geostatistical parameters used in the simulation are themselves uncertain. The inference of these parameters by maximum likelihood allows for an easy assessment of this estimation uncertainty which, in turn, may be included in the conditional simulation procedure. A case study based on transmissivity data is presented to show the methodology whereby both model selection and parameter inference are solved by maximum likelihood.