Restricted kriging for mixture of grade models

A modified type of kriging, referred to as restricted kriging (RK), is proposed in this study. The method incorporates constraints on different grade classes to restrict the influence of the samples having different likelihoods in estimation. RK is motivated by the estimation of mineral reserves when grades have highly skewed distributions. Ordinary kriging tends to produce an overly smoothed interpolated surface by underestimating high grades and overestimating low grades. The fact that ordinary kriging gives a uniform prior treatment to all samples independent of their values is a major factor associated with this smoothing effect. The new approach differentiates each grade portion by preselected cutoffs. RK is developed for a single cutoff and then extended into a general form for any finite number of cutoffs. Restricted cokriging (RCK) is also formulated to simultaneously estimate a set of random functions with restriction conditions. Methods are suggested for determination of the probabilities of occurrence of different grade portions. Finally, the new approach is demonstrated on a case study of an epithermal gold deposit.     

Probabilistic analysis of collapsing soil by indicator kriging

Collapsing soils, which undergo a large decrease in bulk volume virtually instantaneously upon saturation and/or load application, are found in arid and semi-arid regions of the world. In the western and midwestern U.S., problems resulting from collapsing soils are being recognized due to rapid industrial and urban developments. A probabilistic analysis of the distribution of such soils would be a rational approach for quantifying risk involved for a project in an area where such soils are found. Indicator kriging was applied to seven sets of collapse and collapse-related soil parameters to obtain the probability that a certain parameter is more or less than a predefined critical value for low, medium, and high collapse susceptibility. Results are presented in the form of probability contour plots with known variance of estimation of the probability. The ability to predict the probability of occurrence of collapse and collapse-related soil parameters for different critical values with a known degree of certainty is invaluable to planners, developers, and geotechnical engineers.     

Nonparametric estimation of multivariate joint and conditional spatial distributions

To estimate the recoverable reserves of deposits containing more than one metal, linear combinations of sample grades based on the economic importance of each metal have been employed. These linear combinations, called equivalent grades, have inherent problems at least one of which is that the contribution of an individual metal is confounded with all others. As an alternative to equivalent grades, multivariable joint and conditional spatial distributions may be estimated using indicator variable methods. These spatial distributions may then be used to determine the joint or conditional amounts of the metals in the estimated recoverable reserve.This paper was presented at MGUS 87 Conference, Redwood City, California, 14 April 1987.     

Notes on the robustness of the kriging system

The robustness of the kriging system with respect to uncertainty of the theoretical variogram is investigated. Inequalities for possible changes of the kriging estimator and the estimation variance are derived. Results of a numerical study show that changes of kriging weights can be predicted partly with the help of the maximal kriging weight.     

The Role Of The Geographer On Multidisciplinary Resource Surveys

Turning and turning in the widening gyre The falcon cannot hear the falconer; Things fall apart; the centre cannot hold; Mere anarchy is loosed upon the world ….

“The Second Coming,” W. B. Yeats 1921     

SRTM resample with short distance‐low nugget kriging

The shuttle radar topography mission (SRTM), was flow on the space shuttle Endeavour in February 2000, with the objective of acquiring a digital elevation model of all land between 60° north latitude and 56° south latitude, using interferometric synthetic aperture radar (InSAR) techniques. The SRTM data are distributed at horizontal resolution of 1 arc‐second (～30 m) for areas within the USA and at 3 arc‐second (～90 m) resolution for the rest of the world. A resolution of 90 m can be considered suitable for the small or medium‐scale analysis, but it is too coarse for more detailed purposes. One alternative is to interpolate the SRTM data at a finer resolution; it will not increase the level of detail of the original digital elevation model (DEM), but it will lead to a surface where there is the coherence of angular properties (i.e. slope, aspect) between neighbouring pixels, which is an important characteristic when dealing with terrain analysis. This work intents to show how the proper adjustment of variogram and kriging parameters, namely the nugget effect and the maximum distance within which values are used in interpolation, can be set to achieve quality results on resampling SRTM data from 3” to 1”. We present for a test area in western USA, which includes different adjustment schemes (changes in nugget effect value and in the interpolation radius) and comparisons with the original 1” model of the area, with the national elevation dataset (NED) DEMs, and with other interpolation methods (splines and inverse distance weighted (IDW)). The basic concepts for using kriging to resample terrain data are: (i) working only with the immediate neighbourhood of the predicted point, due to the high spatial correlation of the topographic surface and omnidirectional behaviour of variogram in short distances; (ii) adding a very small random variation to the coordinates of the points prior to interpolation, to avoid punctual artifacts generated by predicted points with the same location than original data points and; (iii) using a small value of nugget effect, to avoid smoothing that can obliterate terrain features. Drainages derived from the surfaces interpolated by kriging and by splines have a good agreement with streams derived from the 1” NED, with correct identification of watersheds, even though a few differences occur in the positions of some rivers in flat areas. Although the 1” surfaces resampled by kriging and splines are very similar, we consider the results produced by kriging as superior, since the spline‐interpolated surface still presented some noise and linear artifacts, which were removed by kriging.     

Using landscape characteristics to define an adjusted distance metric for improving kriging interpolations

Interpolation of point measurements using geostatistical techniques such as kriging can be used to estimate values at non-sampled locations in space. Traditional geostatistics are based on the spatial autocorrelation concept that nearby things are more related than distant things. In this study, additional information was used to modify the traditional Euclidean concept of distance into an adjusted distance metric that incorporates similarity in terms of quantifiable landscape characteristics such as topography or land use. This new approach was tested by interpolating soil moisture content, pH and carbon-to-nitrogen (C:N) ratio measured in both the mineral and the organic soil layers at a field site in central Sweden. Semivariograms were created using both the traditional distance metrics and the proposed adjusted distance metrics to carry out ordinary kriging (OK) interpolations between sampling points. In addition, kriging with external drift (KED) was used to interpolate soil properties to evaluate the ability of the adjusted distance metric to incorporate secondary data into interpolations. The new adjusted distance metric typically lowered the nugget associated with the semivariogram, thereby better representing small-scale variability in the measured data compared to semivariograms based on the traditional distance metric. The pattern of the resulting kriging interpolations using KED and OK based on the adjusted distance metric were similar because they represented secondary data and, thus, enhanced small-scale variability compared to traditional distance OK. This created interpolations that agreed better with what is expected for the real-world spatial variation of the measured properties. Based on cross-validation error, OK interpolations using the adjusted distance metric better fit observed data than either OK interpolations using traditional distance or KED.     

Soil depth estimation through soil-landscape modelling using regression kriging in a Himalayan terrain

Soil formation depends upon several factors such as parent material, soil biota, topography and climate. It is difficult to use conventional soil survey methods for mapping the depth of soil in complex mountainous terrains. In this context, the present study aimed to estimate the soil depth for a large area (330.35 km²) using different geo-environmental factors through a soil-landscape regression kriging (RK) model in the Darjeeling Himalayas. RK with seven predictor variables such as elevation, slope, aspect, general curvature, topographic wetness index, distance from the streams and land use, was used to estimate the soil depth. While topographic parameters were derived from an 8-m resolution digital elevation model, the ortho-rectified Cartosat-1 satellite image was used to prepare the land use map. Soil depth measured at 148 sites within the study area was used to calibrate and validate the RK model. The result showed that the RK model with the seven predictors could explain 67% spatial variability of soil depth with a prediction variance between 0.23 and 0.42 m at the test site. In the regression analysis, land use (0.133) and slope (–0.016) were identified as significant determinants of soil depth. The prediction map showed higher soil depth in south-facing slopes and near valleys in comparison to other areas. Mean, mean absolute and root mean-square errors were used to access the reliability of the prediction, which indicated a goodness-of-fit of the RK model.     

结合信息熵改进的证据权法与滑坡危险度区划

滑坡是一种常见的地质灾害,其危害性极大,科学地划分滑坡危险度是防灾减灾工作中主动预测预警的重要环节.该文结合GIS技术与信息熵理论,充分利用熵值法特别适合于确定同一个指标下不同因素相对权重的特点,开展了滑坡危险度区划模型中证据权法的改进研究,并以台湾地区为实验区,在综合考虑滑坡的形成要素并提取滑坡危险度评价因子后,实现了研究区滑坡危险度区划.通过定性和定量等精度检验和对比分析,证明了结合信息熵改进的证据权法所获得的滑坡危险度区划结果更准确且更符合实际.     

1998～2001年国家授时中心(NTSC)铯原子钟运转情况

主要介绍了用于保持中国科学院国家授时中心 (原陕西天文台 )时间尺度的 6台铯原子钟 (代号分别为 :Cs0 7、Cs0 8、Cs1 1、Cs1 6、Cs1 7、Cs1 8)的运行情况。 6台铯原子钟自 1 997年下半年正式给BIPM (国际权度局 )提供数据 ,参加国际原子时(TAI)计算至今已达 5年之久 ,在此期间 ,除Cs1 8因铯束管故障在 1 999年返回美国检修外 ,其它 5台钟运行情况良好。详细介绍、分析每台钟在 1 998～ 2 0 0 1年期间的整体特性、不同采样间隔下的稳定度比较、速率漂移以及它们在TAI系统中的取权情况。通过分析比较 ,使我们对每台钟的运行情况作出恰如其分的评价 ,以便更好地利用国家授时中心的钟资源 ,做好守时和授时工作。