Comparing the robustness of ordinary kriging and lognormal kriging: Outlier resistance

Ordinary kriging is well-known to be optimal when the data have a multivariate normal distribution (and if the variogram is known), whereas lognormal kriging presupposes the multivariate lognormality of the data. But in practice, real data never entirely satisfy these assumptions. In this article, the sensitivity of these two kriging estimators to departures from these assumptions and in particular, their resistance to outliers is considered. An outlier effect index designed to assess the effect of a single outlier on both estimators is proposed, which can be extended to other types of estimators. Although lognormal kriging is sensitive to slight variations in the sill of the variogram of the logs (i.e., their variance), it is not influenced by the estimate of the mean of the logs.This paper was presented at MGUS 87 Conference, Redwood City, California, 14 April 1987.     

Spatial prediction of categorical variables: the Bayesian maximum entropy approach

 Being a non-linear method based on a rigorous formalism and an efficient processing of various information sources, the Bayesian maximum entropy (BME) approach has proven to be a very powerful method in the context of continuous spatial random fields, providing much more satisfactory estimates than those obtained from traditional linear geostatistics (i.e., the various kriging techniques). This paper aims at presenting an extension of the BME formalism in the context of categorical spatial random fields. In the first part of the paper, the indicator kriging and cokriging methods are briefly presented and discussed. A special emphasis is put on their inherent limitations, both from the theoretical and practical point of view. The second part aims at presenting the theoretical developments of the BME approach for the case of categorical variables. The three-stage procedure is explained and the formulations for obtaining prior joint distributions and computing posterior conditional distributions are given for various typical cases. The last part of the paper consists in a simulation study for assessing the performance of BME over the traditional indicator (co)kriging techniques. The results of these simulations highlight the theoretical limitations of the indicator approach (negative probability estimates, probability distributions that do not sum up to one, etc.) as well as the much better performance of the BME approach. Estimates are very close to the theoretical conditional probabilities, that can be computed according to the stated simulation hypotheses.     

Geostatistical analyses of interactions between killer whales (Orcinus orca) and recreational whale-watching boats

Johnstone Strait in coastal British Columbia, Canada, is a core habitat for seasonal concentrations of killer whales (Orcinus orca), which have attracted considerable attention from commercial whale-watching operators and recreational boaters. Within the Strait lies the Robson Bight–Michael Bigg Ecological Reserve, a marine reserve set aside as critical habitat for killer whales and closed to recreational boat traffic. The geography of encounters between killer whales and seven types of whale-watching vessels (including kayaks, charter and pleasure craft) in and near this reserve was analysed with a suite of geostatistics in a geographic information system (GIS) vector environment. Reserve boundary violation was high among most user groups, with kayakers being the most frequent offenders. Motorized vessels had significantly longer contact times with whales compared to kayaks and sailboats. Motorized vessels showed the travel characteristic of deliberate tracking of whales. The movements of killer whales also appear to be affected by boats. These results have important implications for killer whale conservation and management in areas where they are subject to intensive whale-watching activities, and possible chronic disturbance.     

A Monte Carlo-based framework for risk-return analysis in mineral prospectivity mapping

Quantification of a mineral prospectivity mapping (MPM) heavily relies on geological, geophysical and geochemical analysis, which combines various evidence layers into a single map. However, MPM is subject to considerable uncertainty due to lack of understanding of the metallogenesis and limited spatial data samples. In this paper, we provide a framework that addresses how uncertainty in the evidence layers can be quantified and how such uncertainty is propagated to the prediction of mineral potential. More specifically, we use Monte Carlo simulation to jointly quantify uncertainties on all uncertain evidence variables, categorized into geological, geochemical and geophysical. On stochastically simulated sets of the multiple input layers, logistic regression is employed to produce different quantifications of the mineral potential in terms of probability. Uncertainties we address lie in the downscaling of magnetic data to a scale that makes such data comparable with known mineral deposits. Additionally, we deal with the limited spatial sampling of geochemistry that leads to spatial uncertainty. Next, we deal with the conceptual geological uncertainty related to how the spatial extent of the influence of evidential geological features such as faults, granite intrusions and sedimentary formations. Finally, we provide a novel way to interpret the established uncertainty in a risk-return analysis to decide areas with high potential but at the same time low uncertainty on that potential. Our methods are illustrated and compared with traditional deterministic MPM on a real case study of prospecting skarn Fe deposition in southwestern Fujian, China.     

Estimation of regionalized phenomena by geostatistical methods: lake acidity on the Canadian Shield

 This paper describes a geostatistical technique based on conditional simulations to assess confidence intervals of local estimates of lake pH values on the Canadian Shield. This geostatistical approach has been developed to deal with the estimation of phenomena with a spatial autocorrelation structure among observations. It uses the autocorrelation structure to derive minimum-variance unbiased estimates for points that have not been measured, or to estimate average values for new surfaces. A survey for lake water chemistry has been conducted by the Ministère de l'Environnement du Québec between 1986 and 1990, to assess surface water quality and delineate the areas affected by acid precipitation on the southern Canadian Shield in Québec. The spatial structure of lake pH was modeled using two nested spherical variogram models, with ranges of 20 km and 250 km, accounting respectively for 20% and 55% of the spatial variation, plus a random component accounting for 25%. The pH data have been used to construct a number of geostatistical simulations that produce plausible realizations of a given random function model, while 'honoring' the experimental values (i.e., the real data points are among the simulated data), and that correspond to the same underlying variogram model. Post-processing of a large number of these simulations, that are equally likely to occur, enables the estimation of mean pH values, the proportion of affected lakes (lakes with pH≤5.5), and the potential error of these parameters within small regions (100 km×100 km). The method provides a procedure to establish whether acid rain control programs will succeed in reducing acidity in surface waters, allowing one to consider small areas with particular physiographic features rather than large drainage basins with several sources of heterogeneity. This judgment on the reduction of surface water acidity will be possible only if the amount of uncertainty in the estimation of mean pH is properly quantified. Received: 3 March 1997 · Accepted: 16 November 1998     

储层地质建模的现状与展望

储层地质建模对于科学的油藏评价、油藏开发管理以及三维油藏数值模拟具有很大的意义。目前已有的建模算法和商业软件可满足地质特征三维分布的图形要求,并可进行初步的井间预测,但预测精度有待于进一步提高。简要介绍了各种建模方法研究现状,分析了已有算法中亟需改进的问题,并从建模算法的改进、原型模型的丰富、地震信息的整合以及加强地质约束等方面论述了储层地质建模的发展前景。     

Reducing geological uncertainty by conditioning on boreholes: the coupled Markov chain approach

The CMC (coupled Markov chain) model, which is based on the extension of Markov chains in two-dimensions, is used in the reduction of uncertainty in geological structures when conditioned (i.e., honours the data and their location) on a number of boreholes. The model has been applied to an unconsolidated aquifer deposit located in the central Rhine-Meuse delta (the Gorkum study area) in the Netherlands. A comparison is also made between the CMC and the SIS (sequential indicator simulation) model, which is based on Kriging and co-Kriging theories on the same deposit. The results show the potential applicability of the CMC model in reducing the uncertainty in geological configurations when a sufficient number of boreholes is available. Reproduction of the global geological features requires relatively few boreholes (in this case study, nine boreholes with 30-m spacing over a distance of 240 m). However, reproduction of the proportion of each state requires a relatively large number of boreholes (in this case study 31 boreholes with 8-m spacing over a distance of 240 m). It has been shown that variograms can be deceptive in modeling the spatial pattern and that they reflect only part of the complete spatial structure in the field. The use of transition probabilities via the CMC model provides a better alternative approach, because it uses multiple point information. Amro M. M. Elfeki on leave from Department of Irrigation and Hydraulics, Faculty of Engineering, Mansoura University, Mansoura, Egypt     

Transmissivity estimation for highly heterogeneous aquifers: comparison of three methods applied to the Edwards Aquifer,Texas, USA

Obtaining reliable hydrological input parameters is a key challenge in groundwater modeling. Although many quantitative characterization techniques exist, experience applying these techniques to highly heterogeneous real-world aquifers is limited. Three geostatistical characterization techniques are applied to the Edwards Aquifer, a limestone aquifer in south-central Texas, USA, for the purposes of quantifying the performance in an 88,000-cell groundwater model. The first method is a simple kriging of existing hydraulic conductivity data developed primarily from single-well tests. The second method involves numerical upscaling to the grid-block scale, followed by cokriging the grid-block conductivity. In the third method, the results of the second method are used to establish the prior distribution for a Bayesian updating calculation. Results of kriging alone are biased towards low values and fail to reproduce hydraulic heads or spring flows. The upscaling/cokriging approach removes most of the systematic bias. The Bayesian update reduced the mean residual by more than a factor of 10, to 6 m, approximately 2.5% of the total head variation in the aquifer. This agreement demonstrates the utility of automatic calibration techniques based on formal statistical approaches and lends further support for using the Bayesian updating approach for highly heterogeneous aquifers.     

Diagnosis of trace Pb in domestic wells,Upper Gloucester Catchment,Maine, USA

Dissolved Pb in 32 wells associated with corroding submersible pumps is examined within a rural water district after almost 20 years (1984–2002). Groundwater Pb ranged from 0.4–24.9 μg L⁻¹ after 24 h pump flushing. Preliminary geochemistry and representative borehole lithology examinations were extrapolated by Markov chain modeling. The first-order geostatistical realizations of glacial sediments coupled with the Monte Carlo Metropolis-Hasting method suggest that elevated trace Pb persists in sand and gravel units, and continues at least to 40 m depth in the catchment. The ²⁰⁷Pb/²⁰⁶Pb and ²⁰⁸Pb/²⁰⁶Pb isotope compositions of groundwater Pb were decisive in discriminating the importance of leached Pb from submersible pump materials among geogenic sources.     

Spatial correlation of shear-wave velocity in the San Francisco Bay Area sediments

Ground motions recorded within sedimentary basins are variable over short distances. One important cause of the variability is that local soil properties are variable at all scales. Regional hazard maps developed for predicting site effects are generally derived from maps of surficial geology; however, recent studies have shown that mapped geologic units do not correlate well with the average shear-wave velocity of the upper 30 m, V_s(30). We model the horizontal variability of near-surface soil shear-wave velocity in the San Francisco Bay Area to estimate values in unsampled locations in order to account for site effects in a continuous manner. Previous geostatistical studies of soil properties have shown horizontal correlations at the scale of meters to tens of meters while the vertical correlations are on the order of centimeters. In this paper we analyze shear-wave velocity data over regional distances and find that surface shear-wave velocity is correlated at horizontal distances up to 4 km based on data from seismic cone penetration tests and the spectral analysis of surface waves. We propose a method to map site effects by using geostatistical methods based on the shear-wave velocity correlation structure within a sedimentary basin. If used in conjunction with densely spaced shear-wave velocity profiles in regions of high seismic risk, geostatistical methods can produce reliable continuous maps of site effects.