On the modelling of sand bedforms using the semivariogram

This study shows the usefulness of the semivariogram for modelling sand ripples created by water flows of varied flow intensity. A combination of two mathematical functions is fitted to each sample semivariogram, that is an exponential (or stochastic) component and a periodic component. The parameters of each of these components have direct physical meaning. A non-dimensional ratio combining the two parameters of the exponential model is interpreted as a regularity index (which increases with the degree of regularity of bedform arrangement). This regularity index is inversely related to the Froude number of the flow. The non-dimensional wavelength, estimated from the dominant periodic function, is also inversely and closely related to the Froude number. The wave height, accurately estimated from properties of the two fitted components, is a direct function of flow velocity and is also proportional to the standard deviation of bed elevations. The bedform shape introduces a considerable discrepancy between the generally assumed normal frequency distribution and the empirical distributions of bed height. The series of bed elevations are generally characterized by a mixture of normal distributions having the same variance but different means. The calculation of a covariance assuming a constant and single mean (as in spectral analysis) can therefore be misleading and the problem may be avoided by using the semivariogram.     

A six-step practical approach to semivariogram modeling

Geostatistical prediction and simulation are being increasingly used in the earth sciences and engineering to address the imperfect knowledge of attributes that fluctuate over large areas or volumes—pollutant concentration, electromagnetic fields, porosity, thickness of a geological formation. Central to the application of such techniques is the need to know the spatial continuity, knowledge that is commonly condensed in the form of covariance or semivariogram models. Their preparation is subdivided here into the following steps: (1) Data editing, (2) Exploratory data analysis, (3) Semivariogram estimation, (4) Directional investigation, (5) Simple modeling, (6) Nested modeling. I illustrate these stages practically with a real data set from a geophysical survey from Elk County, Kansas, USA. The applicability of the approach is not limited by the physical nature of the attribute of interest.     

High-resolution grain-size characterisation of gravel bars using imagery analysis and geo-statistics

The scarcity of grain-size data from gravel-bed rivers has traditionally hindered hydraulic, sediment transport and river habitat studies. A new remote sensing methodology to estimate grain-size distribution is presented. It combines textural digital images of the riverbed at 1 : 1000 and 1 : 40 scales with grain-size sampling. It was applied to a 12-km reach of the Isábena River (Central Pyrenees NE Spain). First, textural patterns for each grain-size range were obtained, selecting the most closely related texture variables, including the use of semivariograms. Second, multiple linear regression equations were derived from the textural variables to estimate each value of the grain-size distribution. The highest correlation values (r²) were obtained from the central part of the distribution (D₅₀ with a RMS error of 12.7%). Finally, new multiple linear regression equations to estimate the D₅₀ and D₈₄ were obtained from 1 : 1000 images and four textural variables. These were used to derive D₅₀ and D₈₄ maps of the riverbed, re-sampled at a resolution of 1.5 m pixels, with RMS estimation errors of 26% and 32%, respectively. Downstream change in grain-size is also well reproduced by the method. The mean D₅₀ of 72 and 32 mm were estimated in the upper and the lower reaches of the river, respectively. The methodology shows great potential for application, the relation between the spatial resolution of the images and the mean grain-size of the riverbed sediment being the main issue for future development.     

基于Kriging估计的GPS高程转换

随着GPS测量精度的不断提高,其平面测量已经在多种领域得到应用,成为测量工作的一个全新的手段与工具,但在GPS测高方面需要改善和研究。Kriging方法是地统计学的基本方法,其理论核心是变异函数,它是一种最好的线性无偏估计法,在地理学、生态学、环境科学等领域都得到了广泛的应用,取得了很好效果。本文在详细介绍kriging法的基础上,建立了Kriging法的GPS高程拟合模型,利用该模型对两个实例的GPS高程数据拟合,得到了正常高,结果表明Kriging法在GPS高程拟合中的应用精度达到厘米级,且得出了一些有益的结论,为以后GPS高程的应用提供一些参考。     

Declustering of Clustered Preferential Sampling for?Histogram and Semivariogram Inference

Measurements of attributes obtained more as a consequence of business ventures than sampling design frequently result in samplings that are preferential both in location and value, typically in the form of clusters along the pay. Preferential sampling requires preprocessing for the purpose of properly inferring characteristics of the parent population, such as the cumulative distribution and the semivariogram. Consideration of the distance to the nearest neighbor allows preparation of resampled sets that produce comparable results to those from previously proposed methods. A clustered sampling of size 140, taken from an exhaustive sampling, is employed to illustrate this approach.     

Geostatistical modelling of spatial and depth variability of SPT data for Bangalore

The purpose of this study is to develop a geostatistical model to evaluate the spatial and depth variability of Standard Penetration Test (SPT) data from Bangalore, India. The database consists of 766 boreholes spread over a 220 km² area, with several SPT values (N) in each of them. The geostatistical analysis is done for N corrected (N _corrected) values. The N _corrected value has been corrected for different parameters such as overburden stress, size of the bore hole, type of sampler, hammer energy and length of the connecting rod. The knowledge of the semivariogram of the SPT data is used with kriging theory to estimate the values at points in the subsurface of Bangalore where field measurements are not available. The model is used to compute the variance of estimated data. The model predicts reasonably well the SPT data. The geostatistical model provides valuable results that can be used for seismic hazard analysis, site response and liquefaction studies for the development of microzonation maps. The predicted N values from the developed model can also be used to estimate the subsurface information, allowable bearing pressure of soils and elastic modulus of soils.     

模糊类别制图的空间统计学方法

类别地图是地理信息系统（ＧＩＳ）应用中所利用的重要数据类别。这类数据可以从摄影测量和遥感技术得到。用摄影测量方法（影像判读）制作的类别地图常以点、线和多边形的离散目标形式描述,而遥感图像分类方法输出的类别地图以连通光栅块形式表达。不论哪一种情况,在每一个多边形或者光栅块（即制图单元）中仅允许单一类别,边界内部非均匀性和模糊形已经被“过滤”了。这样的类别地图沿用了古曲脆集合论,因为每个制图单元只允许     

“Krige”空间内插技术在地理学中的应用

称为“Krige”技术的内插稀疏观测资料的随机方法是Matheron(1970年)提出的,D.R.Krige首先将这一方法应用于找矿上,因而命名于“Krige”技术。本文首先定义和说明了空间协方差曲线,基于无偏估计和最优原理导出了“Krige”内插权重系数的代数方程组,最后给出实例说明该方法如何应用到地理学和水文学中。     

Spatial prediction model and its application to chemistry of atmospheric precipitation in Jordan

This study is concerned with the spatial variability of some wet atmospheric precipitation parameters such as; pH, conductivity (EC). The study also depicts the spatial variability of some ions (cations and anions) of atmospheric precipitation in Jordan such as, Ca²⁺, Mg²⁺, Na⁺ and K⁺, HCO₃⁻, Cl⁻, NO₃⁻ and SO₄²⁻. The basis of the work is to establish a relationship through the cumulative semivariogram technique between the distance ratios and the spatial dependence structure of the chemical composition of atmospheric precipitation. All semivariogram models are constructed in this study in order to understand the behavior of the spatial distribution. The spatial distributions of rainwater parameters show differences from station to station which is expressed in terms of angle, where the larger the angle the weaker the correlation. The semivariogram (SV) models are constructed to show the variation of the rainfall chemistry in Jordan. The SV models show weak correlation between mountain and leeside mountain stations, i.e. mountain and desert stations. On the other hand, good correlations are observed when transferring from south to north of the country. The larger is the found angle, the weaker is the correlation. For most of the SV model the correlation is found to be very weak between desert and mountainous locality. The Standard Regional Dependence Factor (SRDF) is used for prediction of the distribution of rain fall parameters. It shows the relative error between observed and predicted values of rainwater parameters. The overall regional relative error between the observed and estimated concentrations remains less than 15%.