Variogram Function Research Articles

Variogram time-series analysis of wind speed

Fluctuations of wind-power production are a significant hindrance to its high penetration in power systems. System operators have to provide complementary power and relevant control strategies to smooth out the fluctuations when large-scale wind power ones is injected into the grid. To better smooth the fluctuations, the change rate of the wind speed is a critical piece of information. In this study, the variogram function is introduced to measure the change rate of the wind speed. Based on the variogram time-series, some statistical analyses are conducted. These results contribute to a better understanding of the characteristics of the change rate of the wind speed, such as the chronological variation pattern of the change rate on a day, whether the future change rate can be forecasted, and whether there is a relationship between the change rate and wind speed.

Dual self-image technique for beam collimation

We propose an accurate technique for obtaining highly collimated beams, which also allows testing the collimation degree of a beam. It is based on comparing the period of two different self-images produced by a single diffraction grating. In this way, variations in the period of the diffraction grating do not affect to the measuring procedure. Self-images are acquired by two CMOS cameras and their periods are determined by fitting the variogram function of the self-images to a cosine function with polynomial envelopes. This way, loss of accuracy caused by imperfections of the measured self-images is avoided. As usual, collimation is obtained by displacing the collimation element with respect to the source along the optical axis. When the period of both self-images coincides, collimation is achieved. With this method neither a strict control of the period of the diffraction grating nor a transverse displacement, required in other techniques, are necessary. As an example, a LED considering paraxial approximation and point source illumination is collimated resulting a resolution in the divergence of the beam of .

Stability analysis in nonstationary spatial covariance estimation

Space deformation modelling and estimation techniques based on Multidimensional Scaling (MDS) methods play an important role in nonparametric approaches to the covariance structure analysis of the spatiotemporal processes underlying environmental studies. Since any related procedure depends on the planar MDS representation, the stability of the estimated dispersion, together with the determination of the most influential stations in the estimation of the dispersion space, are important issues that must be analysed before performing the final mapping. In this paper, stability analysis, both in terms of the MDS model and of the variogram function, as well as concerning the derivation of kriging interpolation estimates, is addressed using a special analytical jackknife procedure. Furthermore, the influence of each station in the solution given is assessed, thus providing relevant information regarding not only the MDS procedure but also the interpolation process and the variogram estimation of the spatial dispersion.

Prediction of the Collapsing Risk of Mining Slopes Based on Geostatistical Interpretation of Geotechnical Parameters

Abstract: Almost all collapses of rock slopes especially in open pit mines are related to discontinuities such as bedding planes, faults and major joints. Geostatistical assessments can be used for understanding the distribution of regionalized variables in any spatial study. In this paper3D spatial dispersion of the fault planes in the Gole Gohar open pit iron mine, located in Kerman province, south east of Iran, is modeled. Then, regionalized variable theory is used to analyze and interpret spatial distribution of the following geotechnical parameters: Geological strength index (GSI), Rock quality designation (RQD), Cohesion (C) and angle of internal friction (ϕ). In order to define regionalized variable distribution, variogram functions were determined for identifying the regional behavior. Structural analysis of variograms showed that the mentioned parameters have spatial structures that make it possible to set up a geostatistical model to predict their values for each non-sampled block on the pit wall. Results showed that there is a relation between the low values of geotechnical parameters and the existence of discontinuities around the pit area. The role of discontinuities in the occurrence of collapses in the area was clearly demonstrated by comparing the estimated parameters models and the model of discontinuities dispersion.

水平井阵列持水率测井资料成像插值算法分析

随着科学技术的进步，成像观测技术越来越先进。流动成像测井方面，国外各大测井公司均推出了全井眼测量阵列持率测井仪，测量在井中相同深度上套管内壁附近或分布在不同半径的圆周上的局部持水率。基于电容阵列持水率测井仪CAT的测井数据特征，分析了简单距离反比加权插值、高斯径向基函数插值、普通克里金插值和克里金中的变差函数距离反比插值算法的原理，然后对4种插值算法在理论和实际应用方面进行了比较，试验数据成像处理效果图和误差综合分析结果表明，普通克里金插值算法的成像效果和误差估计总体比其他3种插值算法好，能较直观反映井内油水介质的分布和井筒截面上总的持水率值。 With the progress of science and technology, imaging observation technology was more and more advanced. The large well logging companies abroad pushed out the full well logging tool of array holdup applied in flow imaging logging; this tool could measure the local water holdup at the circular ring inner wall of the casing at the same depth. Based on data feature of logging data of CAT, the principles of simple inverse distance weighting value, Gauss basis function interpolation, ordinary Kriging interpolation and variogram function inverse weighting method were analyzed in this paper. The comparison of these interpolation methods both in theory and in practical application shows that the result of Kriging interpolation method is better than that of the other three methods. The result can better show the distribution of the medium of oil-water in the well and total water holdup clearly and directly.

Characterization of Anisotropy of Joint Surface Roughness and Aperture by Variogram Approach Based on Digital Image Processing Technique

The mechanical and hydraulic anisotropy of rock joints are strongly dependent on the surface roughness and aperture. To date, accurate quantification of the anisotropic characteristics of joint surfaces remains a key issue. For this purpose, the digital image processing (DIP) technique was used to retrieve the joint surface topography, and a variogram function was used to characterize the anisotropy of the joint surface roughness and estimate the joint aperture. A new index, SRV, related to both the sill and the range of the variogram is proposed to describe the anisotropy of the joint surface roughness, and a new aperture index, b, is derived to quantify the joint aperture. These newly proposed indexes, SRV and b, were validated by characterizing three artificial triangular joint surfaces, then the values of both SRV and b were calculated along 42 directions on an artificial joint surface. The range of SRV was between 0.058622 and 0.331283, while that of b was from 0.270433 to 0.397715 mm. The results show that the newly proposed indexes SRV and b are effective for quantifying the anisotropic roughness and aperture of joint surfaces, respectively. In addition, based on the hypothesis that there exists a smooth upper wall for the artificial joint, a relationship between the indexes SRV and b was obtained based on the data analysis. It indicates that the trends of the indexes SRV and b tend to coincide, although some of their individual values differ. In this respect, the hydraulic aperture of rock joints is related to not only surface roughness but also the distribution of asperities on the surface. In addition, this method can also be used to characterize the roughness of real rock joints when the joint surface is treated by dying with ink before taking digital photos. This study provides a new method for properly quantifying the directional variability of joint surface roughness and estimating the mechanical and hydraulic properties of rock joints based on the DIP technique.

Built-up Areas Extraction in High Resolution SAR Imagery based on the method of Multiple Feature Weighted Fusion

Abstract. Synthetic aperture radar in the application of remote sensing technology is becoming more and more widely because of its all-time and all-weather operation, feature extraction research in high resolution SAR image has become a hot topic of concern. In particular, with the continuous improvement of airborne SAR image resolution, image texture information become more abundant. It’s of great significance to classification and extraction. In this paper, a novel method for built-up areas extraction using both statistical and structural features is proposed according to the built-up texture features. First of all, statistical texture features and structural features are respectively extracted by classical method of gray level co-occurrence matrix and method of variogram function, and the direction information is considered in this process. Next, feature weights are calculated innovatively according to the Bhattacharyya distance. Then, all features are weighted fusion. At last, the fused image is classified with K-means classification method and the built-up areas are extracted after post classification process. The proposed method has been tested by domestic airborne P band polarization SAR images, at the same time, two groups of experiments based on the method of statistical texture and the method of structural texture were carried out respectively. On the basis of qualitative analysis, quantitative analysis based on the built-up area selected artificially is enforced, in the relatively simple experimentation area, detection rate is more than 90%, in the relatively complex experimentation area, detection rate is also higher than the other two methods. In the study-area, the results show that this method can effectively and accurately extract built-up areas in high resolution airborne SAR imagery.

Self-imaging technique for beam collimation.

A simple collimation technique based on measuring the period of one self-image produced by a diffraction grating is proposed. Transversal displacement of the grating is not required, and then automatic single-frame processing can be performed. The self-image is acquired with a CMOS camera, and the period is computed using the variogram function. Analytical and experimental results are obtained, which show the simplicity and accuracy of the proposed technique.

The Application of Reservoir Geo-Modeling Technology in Section07, Ticleni Oilfield

In the study of reservoir geo-modeling in Ticleni oilfield, structural model, the spatial framework of the geo-model was firstly constructed based on seismic and petrophysical data and integrated geological background analysis. Then the variogram function was analyzed based on petrophysical data and sedimentary setting analysis, and the litholofacies model was established. Finally, the property models were constructed with reservoir physical property controlled by lithofacies model. Therefore, the quantification of geological study achievements and the distribution of reservoir physical property are effectively combined, which provides the foundation for the sequent reservoir development of Section07.

Generating random fields with a truncated power-law variogram: A comparison of several numerical methods

In this study we describe and compare four numerical methods for the generation of random fields with a truncated power-law variogram; the Fourier method, the Randomization method, the Hybrid method as well as the Fourier–Wavelet method. We evaluate these methods with respect to their ability to represent the variogram function over a number of spatial scales as well as the Gaussianity of the generated fields. We furthermore compare these methods with respect to computational costs and investigate structural features.Results show that the Randomization method performs well if only a few number of spatial scales (4–6 orders of magnitude) need to be represented. Due to its simpler implementation it can be preferred over the Fourier–Wavelet method. For a larger interval of spatial scales (9–12 orders of magnitude) however, the Randomization method fails to represent the variogram. Under such circumstances the Hybrid method or the Fourier–Wavelet method should be used.The Matlab code, used for the simulations can be accessed on our institution website at http://www.ufz.de/index.php?en=32179.

English

The role of interpolation methods in creation of the Digital elevation models (DEM) is an actual problem from theoretical and application aspect. Geo-statistical interpolation methods respect autocorrelation of input data, which is expressed by variogram function. The aim of this paper is to describe creation of variogram model functions and estimation of their parameters and also implementation of variogram to software Surfer for using Kriging interpolation method. The main contribution is creation of variogram function by parametric and nonparametric regression methods and their comparison. Kriging geo-statistical interpolation method, some types of variogram models and their properties are presented. The Least square method is used in estimation of regression functions by parametric regression method. The statistically analyzed results of four noted model functions were presented and used for implementation variogram model to software Surfer. Comparison of variograms created by parametric and nonparametric method demonstrated that nonparametric method respects input data more than model created by parametric regression, however in prediction of data the parametric regression methods are preferable. Key words: Interpolation, Kriging, variogram, geostatistics, parametric and nonparametric regression methods.

Experiments on Image Processing for Micro-Assembly

This paper introduces a micro-assembly system based on image processing and focuses on some key technologies. The system employed some orthogonal wavelet functions to filter images and the suitible filtering effects were obtained by Symlets wavelet basis. The auto-focusing method adopted in the micro-assembly system was the combination of coarse adjustment and fine adjustment. The resolution evaluation function for coarse adjustment was based on Krisch operator and the resolution evaluation function for fine one was based on high frequency component. Experimental results show that the focusing accuracy can reach ±4.0μm. A wavelet multiscale method based on variogram function was used in the edge detection, which can eliminate noise with more details retained. Generally the system can meet the requirements of micro-assembly.

Variogram-based approach for selection of grid size in groundwater modeling

In this paper a geostatistical approach based on the variogram concept is used for decisions about the size of a grid network applicable to a groundwater model. One of the important properties of the variogram function is the range of influence, which is interpreted as a measure of similarity and correlation distance between spatial phenomena. Taking the concept of range into account, several available spatial variables of the under study aquifer were used to plot the variogram in different directions. The study area is an unconfined aquifer (Boushkan Plain in southwest Iran) with an average thickness of 50 m and area of 100 km2. The variables used for computing the variogram include aquifer thickness (D), groundwater pumping rate (Q) and electrical resistivity (R) of the aquifer material. The range of influence was estimated to be 2,000, 2,500, and 2,000 m for D, Q and R, respectively. Comparisons of statistical parameters of spatial variables over a grid size ranging from 1,000 to 5,000 m were done to confirm the proposed grid size according to variogram plots. The bounded area of each cell could be considered as homogenous media according to the spatial variation of the variables used. The optimum size of the grid network was selected according to the minimum of the variance and coefficient of variation over each cell size. Results suggest a 2,500 × 2,500 m grid size for the modeling process in the studied aquifer. The results emphasize the role of variogram function in selection of grid size for the groundwater model.

Detection of masses based on asymmetric regions of digital bilateral mammograms using spatial description with variogram and cross-variogram functions

A mammogram is an examination of the breast intended to prevent and diagnose breast cancer. In this work we propose a methodology for detecting masses by determining certain asymmetric regions between pairs of mammograms of the left and the right breast. The asymmetric regions are detected by means of structural variations between corresponding regions, defined by a spatial descriptor called cross-variogram function. After determining the asymmetric regions of a pair of images, the variogram function is applied to each asymmetric region separately, for classification as either mass or non-mass. The first stage of the methodology consists in preprocessing the images to make them adequate for registration. The following step performs the bilateral registration of pairs of left and right breasts. Pairs of corresponding regions are listed and their variations are measured by means of the cross-variogram spatial descriptor. Next, a model is created to train a Support Vector Machine (SVM) using the values of the cross-variogram function of each pair of windows as features. The pairs of breasts containing lesions are classified as asymmetric regions; the remaining ones are classified as symmetric regions. From the asymmetric regions, features are extracted from the variogram function to be used as tissue texture descriptors. The regions containing masses are classified as mass regions, and the other ones as non-mass regions. Stepwise linear discriminant analysis is used to select the most statistically significant features. Tests are performed with new cases for the final classification as either mass or non-mass by the trained SVM. The best results presented in the final classification were 96.38% of accuracy, 100% of sensitivity and 95.34% of specificity. The worst case presented 70.21% of accuracy, 100% of sensitivity and 67.56% of specificity. The average values for all tests were 90.26% of accuracy, 100% of sensitivity and 85.37% of specificity.

Mapping complex soil patterns with multiple‐point geostatistics

SummaryThe commonly used variogram function is incapable of modelling complex spatial patterns associated with repetitive, connected or curvilinear features, because it is a two‐point statistic. Because this was strongly limiting to petroleum‐ and hydrogeologists, they developed multiple‐point geostatistics (MPG), an approach that replaces the variogram by a training image (TI). However, soil scientists also face complex spatial patterns andMPGmight be of use to them as well. Therefore, this paper aims to introduceMPGto soil science and demonstrate its potential with a case study of polygonal subsoil patterns caused byWeichselian periglacial frost cracks inBelgium. A high‐resolution proximal soil sensing survey provided a reference image from which a continuous (655 sensor data) and a categorical (100 point observations) dataset were extracted. As a continuousTI, we used the geophysical data of another part of the field, and as categoricalTIwe used a classified photograph of an ice‐wedge network inAlaska. The resultingMPGmaps reconstructed the polygonal patterns very well and corresponded closely to the reference image. Consequently, we identifyMPGas a promising technique to map complex soil patterns and suggest that it should be added to the pedometrician's toolbox.

Sample weighted variograms on the sequential indicator simulation of coal deposits

The variogram is an important function in geostatistical applications such as kriging and various conditional simulation methods. In general, the actual variogram function is unknown for a particular application, and inferred from the experimental data. In the same way as first order statistics, the experimental variogram is also affected by preferentially clustered data. The irregularity of sampling has a well-known destructuring effect on the sample variograms, but it is rarely considered in their experimental calculation and modeling. This paper presents a study where a declustering method based on the data polygon of influence was applied in order to weight data pairs in experimental variogram calculations. The sample weighted variograms were appropriately modeled and used in sequential indicator simulations of coal thickness in a large Brazilian deposit.

ESVC-based extraction and segmentation of texture features

Inspired by Krige' variogram and the multi-channel filtering theory for human vision information processing, this paper proposes a novel algorithm for segmenting the textures based on experimental semi-variogram function (ESVF), which can simultaneously describe structural property and statistical property of textures. The single variogram function value (SVFV) and the variance distance obtained by ESVF are used as texture feature description for segmenting textures. The feasibility and effectiveness of the proposed method are demonstrated by testing on some texture images. The computational complexity of the proposed approach depends neither on the number of the textures nor on the number of the gray levels, and only on the size of the image blocks.We have proved theoretically that the algorithm has the advantages of direction invariability and a higher sensitivity to different textures and can detect almost all kinds of the boundaries of the shape textures. Experimental results on the Brodatz texture databases show that the performance of this algorithm is superior to the traditional techniques such as texture spectrum, SIFT, k-mean method, and Gabor filters. The proposed approach is found to be robust, efficient, and satisfactory.

Reliability of Carbon Stock Estimates in Imperata Grassland (East Kalimantan, Indonesia), Using Georeferenced Information

Knowledge of the spatial distribution of total carbon is important for understanding the impact of regional land use change on the global carbon cycle. We studied spatial total carbon variability using transect sampling in an Imperata grassland area. Spatial variability was modeled following an isotropic stationary process with spherical and exponential variogram functions. Range and sill were estimated at 100 m and 82.29 ton2 ha-2, respectively. For nugget, sill ratio was estimated at 24%, implying a rather strong spatial dependence. In a subsequent total carbon stock inventory based on the sampling design mentioned above, we applied three types of estimators, namely, “naive average procedure,” “spatial average procedure,” and “spatial optimal procedure.” Estimation of total carbon stock (in ton ha-1) following naive average procedure (which erroneously ignores the spatial dependence) resulted in a considerably too narrow 95% confidence interval of 37.52 to 39.75, whereas the outcomes using spatial average procedure and spatial optimal procedure were 36.54 to 40.73 and 37.14 to 40.78), respectively, using the spherical model, and 36.63 to 40.64 and 37.07 to 40.64, respectively, using the exponential model. Our research indicated that, when total carbon stock estimation is the main goal, random sampling is optimal, whereas wide design sampling (i.e., shortest distance between sampling locations not less than the range) can be preferred in some cases

Prediction by Regression and kriging for Spatial Data with Application

This study deals with the prediction of the non-stationary spatial stochastic process. The prediction is done by two techniques which are regression technique (generalized least square estimation) and universal kriging technique. As it is familiar, that the non-stationary stochastic process has a trend (mean) as a linear or non-linear model. By this process we can find covariance function from knowing the variogram function and the latter is attributed to a spherical variogram model, also in order to estimate parameters of spherical model by minimum norm quadratic unbiased estimator which requires that the covariance function must be linear in the parameters, then changing the spherical model into an approximated linear model by Taylor series in the linear approximation The prediction in these two techniques is applied to real data which represent height levels of ground water of 47 wells with their regional coordinates in Sinjar district in Ninevah Governorate in Iraq. The results were so encouraging where we show the approximation between the predictive values and the real values as well as computing the variance of prediction in these two techniques. It is shown that the prediction variance of universal kringing is less than that of regression.

On the Validity of Commonly Used Covariance and Variogram Functions on the Sphere

Covariance and variogram functions have been extensively studied in Euclidean space. In this article, we investigate the validity of commonly used covariance and variogram functions on the sphere. In particular, we show that the spherical and exponential models, as well as power variograms with 0<α≤1, are valid on the sphere. However, two Radon transforms of the exponential model, Cauchy model, the hole-effect model and power variograms with 1<α≤2 are not valid on the sphere. A table that summarizes the validity of commonly used covariance and variogram functions on the sphere is provided.