Cumulative semivariogram models of regionalized variables

The cumulative semivariogram approach is proposed for modeling regionalized variables in the geological sciences. This semivariogram is defined as the successive summation of half-squared differences which are ranked according to the ascending order of distances extracted from all possible pairs of sample locations within a region. This procedure is useful especially when sampling points are irregularly distributed within the study area. Cumulative semivariograms possess all of the objective properties of classical semivariograms. Classical semivariogram models are evaluated on the basis of the cumulative semivariogram methodology. Model parameter estimation procedures are simplified with the use of arithmetic, semilogarithmic, or double-logarithmic papers. Plots of cumulative semivariogram values vs. corresponding distances may scatter along a straight line on one of these papers, which facilitates model identification as well as parameter estimation. Straight lines are fitted to the cumulative semivariogram scatter diagram by classical linear regression analysis. Finally, applications of the methodology are presented for some groundwater data recorded in the sedimentary basins of the Kingdom of Saudi Arabia.     

The problem of the spatial variation of a spectral band ratio: a first order approach based on probability theory

Numerous efforts have been made to study how the spatial distribution of ground surface objects controls the image semivariogram. The present paper is centered on how the histograms and semivariograms of the individual bands x and y influence the spatial variation of a simple spectral ratio u = arctan(x/y). The image histogram of each separate band is described by a proper distribution. The exponential model is used to describe the semivariograms of x and y. Taking the first derivatives of the spectral ratio u for x and y and taking into account the mathematical behavior of the histograms of bands x and y, an approximate expression for the semivariogram γ _u of the spectral ratio is derived. This mathematical expression shows how the spatial variation of the spectral ratio depends on the standard deviations of the histograms, as well as the ranges of the semivariograms of x and y. Experimentation with multispectral images is then carried out and it shows that theoretical predictions agree, in qualitative terms, with real data. The results and conclusions of this paper may be useful in assessing the efficiency of various spectral band ratios and vegetation indices, which are often used in geological and environmental research (mapping of hydrothermal zones and land cover types).     

Permeability tensors for sedimentary structures

Accurate modeling of fluid flow through sedimentary units is of great importance in assessing the performance of both hydrocarbon reservoirs and aquifers. Most sedimentary rocks display structure from the mm or cm scale upwards. Flow simulation should therefore begin with grid blocks of this size in order to calculate effective permeabilities for larger structures. In this paper, we investigate several flow models for sandstones, and examine their impact on the calculation of effective permeability for single phase flow. Crossflow arises in some structures, in which case it may be necessary to use a tensor representation of the effective permeability. We establish conditions under which tensors are required, e.g., in crossbedded structures with a high bedding angle, high permeability contrast, and laminae of comparable thickness. Cases where the off-diagonal terms can be neglected, such as in symmetrical systems, are also illustrated. We indicate how the method of calculating tensor permeabilities may be extended to model multiphase flow in sedimentary structures.     

Fractal properties of simulated bed profiles in coarse-grained channels

Bed roughness characteristics in coarse-grained channels are fairly complex. A hierarchy of roughness elements can be observed, ranging from variable particle sizes and shapes and small-scale sedimentary structures, to large-scale bedforms such as riffle-pool sequences. The effects of these scales of roughness on the flow geometry still remain to be thoroughly investigated. The semivariogram has been suggested in the past as a means of quantifying bed roughness effects on streamflow, as well as for distinguishing between scales of roughness. However, field measurements are rather time-consuming. The low number of bed profiles measured in the field precludes the identification of generally applicable relationships between the statistical properties derived from the semivariograms (such as the Hausdorff dimensions and the scale of autocorrelation corresponding to each fractal band) and the bed configuration itself (geometrical and sedimentological properties). Simulation results of gravel-bed profiles are, therefore, presented in order to complement the original investigation of Robert (1988a). The simulation experiments, based on grain characteristics of sizes and shapes and on morphological properties of small-scale bedforms, yield very significant information on boundary roughness at the microscale and give insight into the interpretation of empirical semivariograms (derived from field measurements). Bed-material sorting, variable grain shapes, and height and spacing of cluster bedforms control the fractal dimensions obtained from the semivariograms, as well as the location of the break of slope and the range of the process.     

Spectral Corrected Semivariogram Models

Fitting semivariograms with analytical models can be tedious and restrictive. There are many smooth functions that could be used for the semivariogram; however, arbitrary interpolation of the semivariogram will almost certainly create an invalid function. A spectral correction, that is, taking the Fourier transform of the corresponding covariance values, resetting all negative terms to zero, standardizing the spectrum to sum to the sill, and inverse transforming is a valuable method for constructing valid discrete semivariogram models. This paper addresses some important implementation details and provides a methodology to working with spectrally corrected semivariograms.     

Semivariogram Models Based on Geometric Offsets

Kriging-based geostatistical models require a semivariogram model. Next to the initial decision of stationarity, the choice of an appropriate semivariogram model is the most important decision in a geostatistical study. Common practice consists of fitting experimental semivariograms with a nested combination of proven models such as the spherical, exponential, and Gaussian models. These models work well in most cases; however, there are some shapes found in practice that are difficult to fit. We introduce a family of semivariogram models that are based on geometric shapes, analogous to the spherical semivariogram, that are known to be conditional negative definite and provide additional flexibility to fit semivariograms encountered in practice. A methodology to calculate the associated geometric shapes to match semivariograms defined in any number of directions is presented. Greater flexibility is available through the application of these geometric semivariogram models.     

Statistical properties of sediment bed profiles in alluvial channels

The objective of this study is to investigate in detail the statistical properties of series of bed elevations measured on gravel-bed and sand-bed alluvial channels, in order to identify means of quantifying bed roughness effects on streamflow. The semivariogram is used as the basic statistical method for investigating roughness properties of bed profiles obtained from field work and laboratory experiments. For sand bedforms, the semivariograms include exponential and periodic components from which can be obtained reliable measures of bedform spacing and height, as well as information on the degree of regularity of bedform arrangement. Because of the irregular nature of gravel-bed profiles, the approach in this case uses the semivariogram to investigate fractal properties of series of bed elevations to determine scales of bed roughness associated with grain sizes and small-scale bedforms and to estimate the Hausdorff dimension corresponding to each scale. These superimposed scales of roughness may be responsible for the greater flow resistance generally observed in gravel-bed rivers rather than predicted from the theoretical friction equation.     

Characterizing Fracture Spatial Patterns by Using Semivariograms

Semivariogram is applied to fracture data obtained from detailed scanline surveys of nine field sites in western New York, USA in order to investigate the spatial patterns of natural fractures. The length of the scanline is up to 36 m. How both fracture spacing and fracture length vary with distance is determined through semivariogram calculations. In this study, the authors developed a FORTRAN program to resample the fracture data from the scanline survey. By calculating experimental semivariogram, the authors found five different types of spatial patterns that can be described by linear, spherical, reversed spherical, polynomial I (for a<0) and polynomial II (for a>0) models, of which the last three are newly proposed in this study. The well-structured semivariograms of fracture spacing and length indicate that both the location of the fractures and the length distribution within their structure domains are not random. The results of this study also suggest that semivariograms can provide useful infor     

A parametric study of robustness of kriging variance as a function of range and relative nugget effect for a spherical semivariogram

In geostatistics, an estimation of blocks of a deposit is reported along with the variance of error made in their estimation. This calculation is based on the model chosen for the semivariogram of the deposit so that mistakes in its estimation can manifest themselves in the perception of accuracy with which blocks are known. Changes in kriging variance resulting from various amounts of error in modeling the relative nugget effect and range of the semivariogram are investigated for an extensive set of spherical semivariograms.     

Generalized Bootstrap Method for Assessment of?Uncertainty in Semivariogram Inference

The semivariogram and its related function, the covariance, play a central role in classical geostatistics for modeling the average continuity of spatially correlated attributes. Whereas all methods are formulated in terms of the true semivariogram, in practice what can be used are estimated semivariograms and models based on samples. A generalized form of the bootstrap method to properly model spatially correlated data is used to advance knowledge about the reliability of empirical semivariograms and semivariogram models based on a single sample. Among several methods available to generate spatially correlated resamples, we selected a method based on the LU decomposition and used several examples to illustrate the approach. The first one is a synthetic, isotropic, exhaustive sample following a normal distribution, the second example is also a synthetic but following a non-Gaussian random field, and a third empirical sample consists of actual raingauge measurements. Results show wider confidence intervals than those found previously by others with inadequate application of the bootstrap. Also, even for the Gaussian example, distributions for estimated semivariogram values and model parameters are positively skewed. In this sense, bootstrap percentile confidence intervals, which are not centered around the empirical semivariogram and do not require distributional assumptions for its construction, provide an achieved coverage similar to the nominal coverage. The latter cannot be achieved by symmetrical confidence intervals based on the standard error, regardless if the standard error is estimated from a parametric equation or from bootstrap.     

Another look at anisotropy in geostatistics

A thorough geostatistical data analysis includes a careful study of how the data's second-order variation, as characterized by the semivariogram, depends on the relative orientation of data locations. If the semivariogram depends on only the (Euclidean) distance between locations, then the semivariogram is isotropic; otherwise, it is anisotropic. In this article, I take another look at the modeling of anisotropy in geostatistics. A new, more specific classification of types of anisotropy is proposed. More importantly, some heretofore inadequately understood implications of the dependence of various semivariogram attributes on direction are discussed, and the wisdom of some current practices for modeling the direction-dependence of these attributes is questioned.     

Estimation of semivariograms by the maximum entropy method

A wide variety of semivariograms may be represented in terms of a first- or second-order autoregressive (AR) process, and the nugget effect may be included by use of a moving average (MA) process. The weighting parameters for these models have a simple functional dependence on the value of the sill and the semivariance at the first and second lag. These may be estimated either graphically from the semivariogram or directly from the computed values. Improved spectral estimates of geophysical data have been obtained by the use of the maximum entropy method, and the necessary equations were adapted here for the estimation of the weighting parameters of the AR and the MA processes. Comparison among the semivariograms obtained for the ideal case, the observed case, and the estimated case for artificial series show excellent correspondence between the ideal and estimated while the observed semivariogram may show marked divergence.     

Bayesian Modeling and Inference for Geometrically Anisotropic Spatial Data

A geometrically anisotropic spatial process can be viewed as being a linear transformation of an isotropic spatial process. Customary semivariogram estimation techniques often involve ad hoc selection of the linear transformation to reduce the region to isotropy and then fitting a valid parametric semivariogram to the data under the transformed coordinates. We propose a Bayesian methodology which simultaneously estimates the linear transformation and the other semivariogram parameters. In addition, the Bayesian paradigm allows full inference for any characteristic of the geometrically anisotropic model rather than merely providing a point estimate. Our work is motivated by a dataset of scallop catches in the Atlantic Ocean in 1990 and also in 1993. The 1990 data provide useful prior information about the nature of the anisotropy of the process. Exploratory data analysis (EDA) techniques such as directional empirical semivariograms and the rose diagram are widely used by practitioners. We recommend a suitable contour plot to detect departures from isotropy. We then present a fully Bayesian analysis of the 1993 scallop data, demonstrating the range of inferential possibilities.     

The Impact of Biased Sampling on the Estimation of the Semivariogram Within Fractured Media Containing Multiple Fracture Sets

Monte Carlo simulation was used to examine the error (statistical bias) introduced in estimating a sample semivariogram through application of oriented sampling patterns to variables which are correlated with fracture orientation. Sample semivariograms of the directional components of the water velocity were used to illustrate that oriented sampling schemes can provide biased data sets which result in error in the estimation of the semivariogram, particularly in the estimation of the sill (or variance). Three sampling patterns were used to analyze directional semivariograms of the components of the fluid velocity: sampling along lines parallel to the mean regional hydraulic gradient, sampling among lines perpendicular to the mean regional hydraulic gradient, and sampling along fracture segments. The first two sampling patterns were shown to introduce substantial error in the sills of the velocity variograms. It is argued that this error is due to the combination of unequal sampling of fractures with different orientations (i.e., sampling bias) and the systematic variation in the magnitude of the velocity components with orientation of the fracture. As a consequence, it is suggested that correction factors developed to correct fracture frequency statistics need to be extended to improve estimation of spatial moments of variables which are correlated with fracture orientation.     

Extending Multivariate Space-Time Geostatistics for Environmental Data Analysis

Environmental studies require multivariate data such as chemical concentrations with space-time coordinates. There are two general conditions related to such data: the existence of correlations among the coregionalized variables and the differences in numbers of data which occur because of insufficient data caused by measurement error or bad weather conditions. This study proposes geostatistical techniques for space-time multivariate modeling that take into consideration these correlations and data absences. These techniques consist of suitable modeling of semivariograms and cross-semivariograms for quantifying correlation structures among multivariables and of extending standardized ordinary cokriging. The tensor product cubic smoothing surface method is used for space-time semivariogram modeling. These methods are applied to the chemical component data of the Ariake Sea, a typical closed sea in southwest Japan. In order to clarify environmental changes in the Ariake Sea, the concentration data of four nutritive salts (NO₂–N, NO₃–N, NH₄–N, and PO₄–P) at 38 stations over 25 years are used as environmental indicators. For each of the kinds of data, there are spaces and times for which there is no data available. The effectiveness of the modeling of space-time semivariograms and the high estimation capability of the extended cokriging are demonstrated by cross-validation. Compared with ordinary kriging for a single variable, multivariate space-time standardized ordinary cokriging can provide a more detailed concentration map of nutritive salts and while elucidating their temporal changes over sparsely spaced data areas. In the space-time models by ordinary kriging, on the other hand, smooth trends are obvious.     

Impact-related clastic injections in the marine Ordovician Lockne impact structure, Central Sweden

Clastic injections generated in connection with the formation of impact craters show many similarities to injections created by other geological processes. However, circumstances such as their position relative to the impact structure and the evidence of forceful processes indicate an impact origin. The Ordovician Lockne impact structure was formed in a marine environment with both sedimentary (Cambrian and Ordovician) and underlying crystalline (Proterozoic) target rocks. Sea water played a substantial part in the cratering process, especially in the modification of the newly formed crater as the water surged back into the structure. In the Lockne area clastic dykes and sills have long been known and have earlier been interpreted as neptunian dykes and conglomerates. So far seven cases of dykes and sills are known in the area. In this work these are interpreted as clastic injections formed in connection with the Lockne impact. The clastic injections occur in the crystalline basement and the sedimentary sequence. The material in the injections comes from all local lithologies (both sedimentary and crystalline) but the sedimentary sequence dominates as a source. The dykes and sills were injected simultaneously with the fracturing and dilation of the host rock in the cratering process, and occur at different stratigraphic levels. In some dykes, clasts from the host rock wall can be fitted back to their original position; the clasts are slightly rotated and surrounded by exotic material. Quartz grains with planar deformation features were observed in the injected material. Most of the sills within the bedded Ordovician limestone are restricted to marly beds, except for the feeder dykes which cut through the overlying beds. This circumstance demonstrates how the decompression has opened the strata along weaker layers and that the underpressure created subsequently sucked the material down. Laminar flow is a conspicuous internal structure in the dykes and sills and indicates viscous flow of injected material. The lamination in the injected material is parallel to the walls in each case. The material was lithified prior to the event and was crushed, mobilized in a water/sediment slurry and injected as dykes and sills.     

基于地质统计学方法的孔压静力触探锥尖阻力固有空间变异性研究

地质统计学是用于模拟土体固有空间变异性的方法之一,以变差函数为工具,采用Kriging插值提供未采样点处土工参数值的最优线性无偏估计。将地质统计学方法应用于宿-新（宿迁至新沂）高速公路某试验段内孔压静力触探（piezocone penetration test,CPTU）锥尖阻力qt空间变异性研究中,采用回归分析移除数据中的趋势项,从而获得具有弱平稳性的残差数据。指数型理论变差函数能够准确描述试验段内土体的连续空间变异性特征。根据估计结果,试验段内锥尖阻力qt残差的变程具有显著各向异性,在水平方向和竖直方向分别为4.05 m和1.2 m。采用普通Kriging插值结合趋势分析,绘制了qt在试验段的空间分布图和平面投影图,用于指导工程实践。结果表明,普通Kriging插值的估计结果能够与试验段内实测资料形成较好的对比,仅仅在部分极值变化和远离采样点的位置处估计值可靠性会降低。     

鄂尔多斯盆地合水地区三叠系延长组长63段 三维储集层建模

通过取心井的岩心资料,综合运用岩性组合特征、沉积结构、构造特征等相标志,将延长组长63段辫状河三角洲前缘亚相划分为水下分流河道、河口坝、席状砂、分流间湾4种沉积微相;在此基础上,利用随机建模技术,建立了研究区构造模型和沉积微相模型,并以此为约束条件,采用序贯高斯模拟方法建立三维储层物性模型,精细的表征孔隙度和渗透率的空间展布特征.结果表明,三维储集层模型能够与实际砂体展布和物性特征较好吻合,为油藏模拟、剩余油预测及挖潜调整提供了可靠的依据.     

扇形沉积体生长过程的动力学机制及分形模拟

扇形沉积体是我国内陆及近海中、新生代含油气盆地的重要油气聚集体, 如何定量地表征沉积体的分布状况及内部结构参数的空间变化规律, 对指导油气勘探具有重要意义.扇形沉积体的生长过程是一个复杂的非线性动力学过程, 通过对其生长的沉积动力学、混沌动力学和分形动力学机制的分析, 认识到扇形沉积体的生长具有分形特征, 决口过程是造成其外部形态复杂多变的重要原因; 分形几何学理论在沉积体定量建模和模拟方面具有较好的应用前景.采用分形-地质统计学相结合的方法, 对扇形沉积体的外部形态和内部结构特征实施了计算机模拟, 并对模拟结果进行了讨论.