In spatial analysis, the VARIOGRAM function that determines the spatial correlation structure of the data is usually unknown and it should be estimated by observations. Although there are several estimation methods for VARIOGRAM PARAMETERS, limited number of observations produces in much uncertainty in VARIOGRAM parameter estimates and finally non-precision of the spatial prediction. In this paper, precision measures of the weighted least squares estimates of VARIOGRAM PARAMETERS are determined by separate block bootstrap method. Next, these estimates are corrected using precision measures. Then, it is shown by cross-validation method that using a VARIOGRAM with corrected parameter estimates with result in increasing spatial prediction precision.

Lahiri (2003) proposed the moving block bootstrap method for spatial data, in which observations are divided into several moving blocks and resampling is done from them. Since, in this method, the presence of boundary observations in the resampling blocks have less selection chance than the other observations, therefore, the estimator of the precision measures would be biased. In this paper, revising the moving block bootstrap method, the separate block bootstrap method was presented for estimating the precision measures of the VARIOGRAM PARAMETERS estimator and spatial prediction. Then its usage was illustrated in an applied example.

Calculation of VARIOGRAMs and spatial continuity is one of the first and most important processes in geostatistical modeling, which is a long and experience-oriented process. Due to the complexities of calculating experimental VARIOGRAMs, interpretation and fitting the appropriate model are always the main challenges in this field. This article presents an intelligent VARIOGRAM modeling method using deep learning that can increase the speed of VARIOGRAM modeling and also prevent common errors in manual VARIOGRAM model fitting. In this method, two convolutional neural networks are used. The first CNN network converts the initial data into a 2D simulated map based on various VARIOGRAM models. For this purpose, it is necessary to train the first network with initial data and their corresponding simulations. The output of this model is entered into the second convolutional neural network as input, and the VARIOGRAM PARAMETERS (including range, azimuth, ratio, and nugget effect) are predicted. In this article, the proposed algorithm is implemented on synthetic 2D data and the PARAMETERS of the CNN models are optimized. The accuracy of the proposed model was 97 %, and then the proposed algorithm was used for VARIOGRAM modeling of Nouchon area geochemical data, which included the elements Cu, Zn, and Pb. the accuracy of the obtained model compared to manual fitting was 90%.

Nowadays, kriging has been accepted as the most common method of grade estimation in a mineral resource evaluation stage. Access to the crisp assay data and a VARIOGRAM model are necessary tools of utilizing this method. Since fitting a crisp VARIOGRAM model is generally difficult, if not impossible, the fitted theoretical model is usually tainted with uncertainty due to various reasons especially limitation in the number of drill holes. Although the geostatistical kriging model is incapable of taking into account the uncertainties, the fuzzy kriging method (presented based on the fuzzy concept) is capable of calculating the effects of uncertainties on the fitted model (and even on the assay data). To evaluate the Zu II Jajarm mineral resource, effort was made to use Bardossy’ s fuzzy kriging method (proposed based on the extension principle) instead of ordinary kriging because of high uncertainties tainted with the fitted VARIOGRAM model. Since no comprehensive software existed to be used for this method, the “ FuzzyKrig” was developed for the required calculations. A key advantage of the fuzzy kriging method compared with the general, simple, ordinary, and log-kriging is that it presents, as a parameter, the width of the fuzzy number of every block as a criterion for the evaluation of uncertainties in the estimation process. The advantage of this parameter is that, unlike the estimation variance, it depends not only on the data arrangement, but also on the grade data, and therefore, can play a key role in risk management studies.

An important problem in spatial data analysis is VARIOGRAM modeling to specify the correlation structure of data. Usually the values of a VARIOGRAM estimator at different lags are used for VARIOGRAM modeling. Since the number of lags affects the accuracy of the model fitted to the VARIOGRAM estimates, finding an optimal value for the number of lags plays an important role in VARIOGRAM modeling. In this paper a simulation study is carried out to find the optimal value of the number of lags for data generated from a Gaussian random field. The result has been used in a real practical example to fit an exponential VARIOGRAM model to the VARIOGRAM estimates. Then, the VARIOGRAM is used in kriging to predict tuberculosis disease incidence rates in some cities of Iran.    

This research aimed to study and compare accuracy of three geo-statistical methods (kriging, co-kriging, and inverse distance weighting) in order to estimate some soil properties, and determine the effect of sampling density on VARIOGRAM PARAMETERS. This study was conducted in Laghar plain in south of Fars province with an area of 12986 hectares. Soil particle size percentage, calcium carbonate equivalent, gypsum percentage, mottling distribution, and soil saturated hydraulic conductivity were measured and studied and zoned based on the most appropriate model. Then, suitable interpolation and evaluation methods as well as suitable interpolator were selected. The results showed that the kriging estimator was better and had less error than the inverse distance weighting and co-kriging methods for interpolating saturated hydrolytic conductivity, mottling, and gypsum percentage. To interpolate sand, silt and clay percentage PARAMETERS, the inverse distance weighting method had better results than the other two methods and was preferred. To interpolate calcium carbonate percentage, the co-kriging method presented better results than the kriging and inverse distance weighting methods. Sampling density effect on VARIOGRAM PARAMETERS was studied too. In the first level, all samples (80 samples) and, in the second level, 40 samples were randomly selected and considered for determining VARIOGRAM and interpolation. According to the results, there was no regular change in values of VARIOGRAM PARAMETERS, but increase in nugget effect could be observed in most of the studied properties.

The evaluation of potential human and economic losses arising from earthquakes, which may affect urban infrastructures that are spatially extended over an area, is important for national authorities, local municipalities, and the insurance and reinsurance industries. However, seismic-risk analysis of distributed systems and infrastructures need to apply a different approach with respect to the classical site-specific hazard and risk analysis. Ground motion intensity measures (IMs) and resulting structural responses are correlated in neighborhood sites. The correlation value depends on the distance between the adjacent sites and the natural vibration period of structures. In particular, when a lifeline system is of concern, classical site-specific hazard tools, which consider IMs at different locations independently, may not be accurate enough to assess the seismic risk. In fact, modeling of ground motion as a random field, which consists of assigning a spatial correlation to the IM of interest, is required. It is very common in the seismic design of spatially distributed structures and lifelines to include the correlation of the nearby earthquake records, through empirical semi-VARIOGRAM functions. In this study, the semi-VARIOGRAM of vertical components as a function of inter-site separation distance with respect to the ground motion prediction equations for the Iranian acceleration data (vertical peak ground acceleration (PGA) and vertical pseudo spectral acceleration (PSA)) are presented for the first time using acceleration data from 220 earthquakes. The calculations were carried out for five natural vibration periods in the range of 0 to 3 seconds and using ground motion prediction equations for vertical component. The selected ground motion prediction equation is the local model proposed by Soghrat & Ziaeifar (2017). For estimation of empirical semi-VARIOGRAM, two classical and robust estimators, and to fit the data, the exponential and Goda models are used. For the ground motion prediction equation by Soghrat & Zyiaeifar (2017), the values of the range (b) in the exponential model and the values of a and b in the model of Goda (i.e. a continuous function fitted to experimental values in order to deduce semiVARIOGRAM values for any possible site separation distance, Goda & Hong, 2008) are estimated. It is observed that the correlation trend range generally increases with period.

The spatial analysis became very important in various fields of landscape archeology and statistical analysis. Spatial relationships of archaeological data, patterns created by human activities, their implications for the interior space of archaeological sites as well as their surroundings are studied by spatial archeology. The main purpose of the study is the geometric analysis and clustering of three archaeological periods of Khonj County. The central question of this research is,“, What patterns do the spatial distribution of the archaeological sites of the Khonj plain follow based on periodic clustering (prehistoric, historical-Islamic, Islamic) and what factors have played a role in locating these sites? ”,The present study is a descriptive-analytical. Through field studies, initially, 192 archaeological sites were recorded in three periods of classification and their location (latitude and longitude). By studying the research literature, then, to extract 8 effective indicators (distance from the river, altitude, slope, direction of slope, climatic conditions, vegetation, precipitation, distance from villages) in the distribution of centers and ancient sites in the Khonj County were extracted at the level of 4 identified villages using Delphi technique. The final status of the points was analyzed using the SemiVARIOGRAM tool in the Geostatistical analyst section with ArcGIS software. Research results show,more than 45% of the areas are scattered at an altitude of 1100-900 meters and on a slope of 5 to 10%, their climate with an average precipitation of less than 100 to 150 mm, temperate climate, and pastures around them can be irrigated. The average area is more than 1 hectare and the average distance from the villages is 3000 meters. On average, the distance from water sources is more than 3 km,the areas are concentrated in the eastern half and partly in the southeastern part of the county.

To enhance the certainty of the grade block model, it is necessary to increase the number of exploratory drillholes and collect more data from the deposit. The inputs of the process of locating these additional drillholes include the VARIOGRAM model PARAMETERS, locations of the samples taken from the initial drillholes, and the geological block model. The uncertainties of these inputs will lead to uncertainties in the optimal locations of additional drillholes. Meanwhile, the locations of the initial data are crisp, but the VARIOGRAM model PARAMETERS and the geological model have uncertainties due to the limitation of the number of initial data. In this paper, effort has been made to consider the effects of VARIOGRAM uncertainties on the optimal location of additional drillholes using the fuzzy kriging and solve the locating problem with the genetic algorithm (GA) optimization method. A bauxite deposit case study has shown the efficiency of the proposed model.