Experimental Variograms Research Articles

Variogram models reconstruction for damaged ERT profiles

Reconstructing signals which are embedding spatial patterns such as Electrical resistivity tomography, is a process that should require to reconstruct first the spatial correlation of the damaged signals. This paper proposes an approach that implements an Unstationary Kriging (UNK) to reconstruct the experimental variogram of a damaged synthetic pseudo section within a set of pseudo sections coming from the same survey. We used and compared 02 other simple methods which are Linear Regression (LR) and Ordinary Kriging (OK), to test the hypothesis we formulate to link the experimental variograms coming from the same ERT survey. We implemented the UNK using Discrete Fourier Transforms (DFT) for trend modeling. After an implementation of the hybrid process (UNK) on 02 sets of data which are synthetics, we observed that the LR and the UNK methods present an interest. They both reconstruct signals with a +90% rate of accuracy, but when there is no structure or spatial correlation within the data, the LR is unstable. DFT was also tested alone for reconstruction but was mainly used in this study to help in computing the trends for each set of variographic signals. In the end, we conclude on an evidence that is: the proposed hybrid process is a promising way to reconstruct variographic signals, since we can improve it after more time invested to dig deep into the modeling of each of his components.

Rainfall risk over the city of Abidjan (Côte d'Ivoire): first contribution of the joint analysis of daily rainfall from a historical record and a recent network of rain gauges

Abstract. Every year, rains cause material damage and human losses, in Abidjan (Côte d'Ivoire). The objective of this study is to contribute to the characterization of the rain hazard in the District of Abidjan. The available data are made up of daily rainfall from a historical station “Abidjan airport” (1961–2014) and an academic network of rain gauges (21) progressively implemented in Abidjan since 2015. A descriptive analysis (date of occurrence, rainfall depth, mean wet days intensity and number of rainy days) on the Highest Cumulative Rainfall Periods (HCRP: 60 d) is conducted on the long-term station. The periods of highest risk of flooding during the long and short rainy seasons are characterized. The Experimental variograms of extreme rainfalls derived from the current network, allow to evaluate their extensions according to the rainy season.

Addressing Geological Challenges in Mineral Resource Estimation: A Comparative Study of Deep Learning and Traditional Techniques

Spatial prediction of orebody characteristics can often be challenging given the commonly complex geological structure of mineral deposits. For example, a high nugget effect can strongly impact variogram modelling. Geological complexity can be caused by the presence of structural geological discontinuities combined with numerous lithotypes, which may lead to underperformance of grade estimation with traditional kriging. Deep learning algorithms can be a practical alternative in addressing these issues since, in the neural network, calculation of experimental variograms is not necessary and nonlinearity can be captured globally by learning the underlying interrelationships present in the dataset. Five different methods are used to estimate an unsampled 2D dataset. The methods include the machine learning techniques Support Vector Regression (SVR) and Multi-Layer Perceptron (MLP) neural network; the conventional geostatistical methods Simple Kriging (SK) and Nearest Neighbourhood (NN); and a deep learning technique, Convolutional Neural Network (CNN). A comparison of geologic features such as discontinuities, faults, and domain boundaries present in the results from the different methods shows that the CNN technique leads in terms of capturing the inherent geological characteristics of given data and possesses high potential to outperform other techniques for various datasets. The CNN model learns from training images and captures important features of each training image based on thousands of calculations and analyses and has good ability to define the borders of domains and to construct its discontinuities.

A power-law exponential model for variograms with quick transition and known range: Construction and application to geostatistical time series

The variogram is an important element in geostatistical studies that has been used as a tool to investigate the variability and continuity of the study phenomenon. The classical Matheron estimator for calculating the experimental variogram is used in this paper. A new variogram model is proposed, the power-law exponential model and a comparative study of three-time series datasets with the characteristic of being periodic is performed: rainfalls, particulate matter PM10, and Ozone series. Three other models were considered for comparison: spherical, exponential, and quadratic–exponential. The numerical fit and analysis are carried out using the least-squares method provided by the R statistics software, and testing the solutions by the coefficient of determination R2 and the RMSE measure. Our results provided a good adaptation of the proposed model to the different datasets, satisfying the equality of variances, and also showing a robust fit when adding noise. Moreover, our model presented several advantages over the others, regarding the mentioned characteristics. This suggests that the power-law exponential model is a variant that adds new attributes to the set of existing variogram models.

Cephy: Astand-alone, free and open-source toolkitfor hydrocarbon in place evaluation by ordinary kriging

Cephy is a software toolkit written inJava that enables the user to perform spatial interpolation of subsurface data by ordinary kriging. Key features of the software include data exploration by map plot with symbol dimension proportional to the magnitude of the property at a given location;variogram modeling and interpolation of petrophysical properties;as well as oil in placeestimationper area unit throughout the reservoir. Cephy enables the user to define the tolerance and the minimum number of pairs required to validate the experimental variogram of the corresponding lag distance when learning the relationship between geolocated data. The software is entirelymade of a graphical user interface for adjusting the variogram modeling parameters and visualizing the subsequent interpolationally-generated map.Cephy is applied on Kinkasi reservoir in Kinkasi field (Coastal basin of DR Congo) to demonstrate its functionalities. The result supports a better understanding of oil distribution throughout the reservoir and can be used as a baseline forthesuggestion of new infill drilling locations.

Spatial variability of half-cell potential data from a reinforced concrete structure—a geostatistical analysis

Corrosion in reinforced concrete structures is among the major degradation mechanisms. The quantification and description of the spatial distribution of the corrosion condition within a structure on the basis of condition assessments are important. This study considered half-cell potential mapping data as a widely used technique to detect corrosion in reinforced concrete structures. A four-step workflow was proposed to analyse half-cell potential data with geostatistical techniques, first consisting of trend identification and possible trend removal. The obtained residuals were then subjected to a quantile-quantile transformation. Subsequently, experimental variograms were calculated and fitted with variogram models to estimate the correlation lengths. A case study with data from a road tunnel confirms the applicability of the workflow. It was assumed that the identified trend is primarily a result of the heterogeneity of the exposure conditions within the structure that ranges several metres. The residuals are interpreted as the result of the heterogeneities of material resistances that give rise to spatial variability in corrosion probability on a shorter distance range. The proposed analysis may be utilised for service life modelling (e.g. based on random fields), planning maintenance works, or optimising the grid size for half-cell potential measurements.

Novel MLR-RF-Based Geospatial Techniques: A Comparison with OK

Geostatistical estimation methods rely on experimental variograms that are mostly erratic, leading to subjective model fitting and assuming normal distribution during conditional simulations. In contrast, Machine Learning Algorithms (MLA) are (1) free of such limitations, (2) can incorporate information from multiple sources and therefore emerge with increasing interest in real-time resource estimation and automation. However, MLAs need to be explored for robust learning of phenomena, better accuracy, and computational efficiency. This paper compares MLAs, i.e., Multiple Linear Regression (MLR) and Random Forest (RF), with Ordinary Kriging (OK). The techniques were applied to the publicly available Walkerlake dataset, while the exhaustive Walker Lake dataset was validated. The results of MLR were significant (p < 10 × 10−5), with correlation coefficients of 0.81 (R-square = 0.65) compared to 0.79 (R-square = 0.62) from the RF and OK methods. Additionally, MLR was automated (free from an intermediary step of variogram modelling as in OK), produced unbiased estimates, identified key samples representing different zones, and had higher computational efficiency.

Geostatistical Ore Body Modeling on Uranium Mineralization in Remaja Sector, Kalan Area, West Kalimantan

Manual ore body modeling on Remaja Sector, Kalan, West Kalimantan generally takes a long time and is subjective. On the other hand, automatic modeling (implicit modeling) is faster, objective, and equipped with uncertainty factors. This study aimed to analyze the comparison between the geostatistical Sequential Indicator Simulation (SIS) ore body model to the manual ore body model. The lithology database was used as input for variogram analysis and SIS simulation. The directional variogram was used to construct an experimental variogram for the lithology with orientation data. The orientation of the lithologies corresponds to the anisotropy of their variogram map. The SIS was carried out in Block A and Block B with block sizes of 6×6×6 m3 and 5×5×5 m3 respectively. The simulation results were processed to produce a lithology probability model. By using maximum probability as block lithology, simulation results were well validated by the composite database histogram, the lithologies along the tunnel on the geological map of level 450 masl of Eko Remaja Tunnel., and the lithologies along boreholes. The weakness of the geostatistical ore body model was the results depending on the input parameters. Meanwhile, several advantages of the geostatistical ore body model were a faster processing process, equipped with an uncertainty factor, and the block size of the model has taken into account the distance between grade data so that it can be used directly for grade estimation. Quantitatively, the geostatistical ore body model had a higher average percentage of conformity to the lithology of the mineralized zone along the borehole than the manual ore body model.

Spatial correlations of NDVI and MSAVI2 indices of green and forested areas of urban agglomeration, case study Warsaw, Poland

AimThe aim of this study was to analyze spatial correlations of NDVI and MSAVI2 indices in a large urban agglomeration. The study area, encompassing the Warsaw agglomeration, was located almost in the center of Poland and the central part of Europe. MethodsThe study was based on the Sentinel 2 images with 0% cloud coverage that were taken in the years 2017, 2018, 2019, and 2020. After the NDVI and MSAVI2 indices were calculated, resulting rasters were exported to the text files and imported to the Stanford Geostatistical Modeling Software (SGeMS) software. Experimental variograms were calculated using values of vegetation indices from each pixel, separately for areas covered with natural forests, with a predominance of grass vegetation, and for areas with only rare vegetation of various types. Next, experimental variograms were modeled using the spherical or exponential model, and its parameters: range of correlation, nugget effect, and a sill. ResultsSpatial differences in the extents of green areas determined using NDVI and MSAVI2 indices were observed, especially for areas characterized by low values of both indices. Temporal variation of the extents of the green area was minimal and could have resulted from usual variability related to weather conditions, precipitation, and other factors.The longest ranges of correlation of NDVI and MSAVI2 were observed in natural forests, probably because they represent the greatest spatial continuity and homogeneity of all the examined areas. Long ranges of correlation were observed in strictly urban areas where the vegetation was relatively sparse and scattered. Despite the NDVI and MSAVI2 values being not high, this area was characterized by a relatively good spatial continuity of urban infrastructures such as buildings and roads. The shortest correlations ranges were observed in areas with a relatively large share of grasslands.

Statistical Analysis of PM10 Concentration in the Monterrey Metropolitan Area, Mexico (2010–2018)

Air-quality monitoring and analysis are initial parts of a comprehensive strategy to prevent air pollution in cities. In such a context, statistical tools play an important role in determining the time-series trends, locating areas with high pollutant concentrations, and building predictive models. In this work, we analyzed the spatio-temporal behavior of the pollutant PM10 in the Monterrey Metropolitan Area (MMA), Mexico during the period 2010–2018 by applying statistical analysis to the time series of seven environmental stations. First, we used experimental variograms and scientific visualization to determine the general trends and variability in time. Then, fractal exponents (the Hurst rescaled range and Higuchi algorithm) were used to analyze the long-term dependence of the time series and characterize the study area by correlating that dependence with the geographical parameters of each environmental station. The results suggest a linear decrease in PM10 concentration, which showed an annual cyclicity. The autumn-winter period was the most polluted and the spring-summer period was the least. Furthermore, it was found that the highest average concentrations are located in the western and high-altitude zones of the MMA, and that average concentration is related in a quadratic way to the Hurst and Higuchi exponents, which in turn are related to some geographic parameters. Therefore, in addition to the results for the MMA, the present paper shows three practical statistical methods for analyzing the spatio-temporal behavior of air quality.

Multiscale analysis of living benthic foraminiferal heterogeneity: Ecological advances from an intertidal mudflat (Loire estuary, France)

An unprecedented sampling effort on the Loire estuary allowed a multi scale approach to identify parameters controlling density variations of benthic foraminifera. Indeed, the distances between the samples analysed for this study vary from 1 cm to hundreds of kilometres. To catch this range of distance variations, a model called Scale Variance Analysis was build describing the participation of each scale to the total observed variance. The SVA model requires, for each scale, the stability of relative variance. A comparison with the Moran's Index and experimental variogram is proposed showing coherent conclusions with the SVA analysis. The analysis shows that in order to maximize information on foraminiferal density variation, sampling campaigns should be designed with stations distant from few meters to 1 km, with a particular focus on the hectometre scale. A range of scale too rarely investigated in the community of benthic foraminifera ecology. Next, based on two intertidal mudflat stations separated of few hundred meters, the present study shows that for Ammonia tepida, the scale dependant preponderant parameters is the Chl a concentration in the top first centimetre. Contrastingly, the indicators of food quality such as the lability index and the oxygen penetration depth do not seem to affect A. tepida densities. This high quantity, low quality diet is interpreted as an opportunistic behaviour that is indirectly confirmed by a kinetic approach. This approach compares the deep infaunal microhabitat density with the shallow infaunal microhabitat density. The identical ratio indicates quick saturation of the available resources.

A Quadratic–Exponential Model of Variogram Based on Knowing the Maximal Variability: Application to a Rainfall Time Series

Variogram models are a valuable tool used to analyze the variability of a time series; such variability usually entails a spherical or exponential behavior, and so, models based on such functions are commonly used to fit and explain a time series. Variograms have a quasi-periodic structure for rainfall cases, and some extra steps are required to analyze their entire behavior. In this work, we detailed a procedure for a complete analysis of rainfall time series, from the construction of the experimental variogram to curve fitting with well-known spherical and exponential models, and finally proposed a novel model: quadratic–exponential. Our model was developed based on the analysis of 6 out of 30 rainfall stations from our case study: the Río Bravo–San Juan basin, and was constructed from the exponential model while introducing a quadratic behavior near to the origin and taking into account the fact that the maximal variability of the process is known. Considering a sample with diverse Hurst exponents, the stations were selected. The results obtained show robustness in our proposed model, reaching a good fit with and without the nugget effect for different Hurst exponents. This contrasts to previous models, which show good outcomes only without the nugget effect.

Geostatistical modeling of porosity and evaluating the local and global distribution

Porosity is one of the main variables needed for reservoir characterization. For this volumetric variable, there are many methods to simulate the spatial distribution. In this article, porosity was analyzed and modeled in the local and global distribution. For simulation, Sequential Gaussian simulation (SGS) and Gaussian Random Function (GRFS) were applied. Also, kriging was used to estimate the porosity at specific locations. The main purpose of this work was to investigate the porosity to compare geostatistical simulation and estimation methods in a sandstone reservoir as a real case study. First, the data sets were normalized by the Normal Scores Transformation (NST) and stratigraphic coordinate. The model of experimental variograms was fitted in the vertical and horizontal directions. For the simulation methods, 10 realizations were generated by each method. The Q-Q plots were calculated, and both sets of quintiles (Target Porosity Distribution versus Porosity realization) came from normal distributions with the following correlation coefficients: 0.93, 0.94 and 0.97 related to GRFS, SGS and Kriging, respectively. The extracted variograms from realizations showed that the kriging couldn’t reproduce the variograms with global distribution. For local validation, the cross-validation was evaluated and three wells were omitted. The re-estimation of porosity was considered at located well logs through the well sections window where the kriging had a better performance with minimum error to estimate porosity locally. Finally, the cross-sectional models were generated by each algorithm which showed that the simple kriging tries to produce smoother distribution, whereas conditional simulations (SGS and GRFS) try to represent more global-detailed sections.

GEOSTATISTICAL MODELLING FOR SULFUR ELEMENT IN GHORABI IRON ORE MINE AREA, BAHARIYA OASIS, EGYPT

Sulfuris a concern in Ghorabi iron oreand it is an unwantedcomponent when the ore is chargedinto a blast furnace. The goal of this paper aims to track thesulfur contentin Ghorabi iron ore by using geostatistical modelling. Geologically, the ore is splitinto fourdistinct zones. The experimental variograms were created tohighlight the spatial variability of sulfur element within the deposit. The experimental variogramswere fitted with Gaussian, linear and exponentialvariogram models. Finally,the appropriate fitted models were utilized to create a distribution mapsfor the four separated zones using the ordinary kriging and indicator kriging in 0.7% cut-offs. يمثل الکبريت مشکلة في خام حديد منطقة غرابي ويعتبر مکونا غير مرغوب فيه عند شحن الخام للفرن العالي. تهدفهذه الورقة إلى رصد تباين الکبريت الموجود في خام حديد منطقة غرابي باستخدام النمذجة الجيوإحصائية. تم تقسيمالخام جيولوجي ا إلى أربع مناطق منفصلة. لإظهار التباين المکاني لعنصر الکبريت ، تم إنشاء المتغيرات التجريبيةداخل الرواسب. تم ترکيب نماذج التباين )جاوسيان – الخطية – الأسية( للمتغيرات التجريبية. أخيرا ، تم استخدا م النماذج المناسبة لإنشاء خرائط توزيع لعنصر الکبريت باستخدام الکريجنج العادي و الکريجنج المؤشر لحد قطع معين للمناطق الأربعة المختلفة.

Assessing the effectiveness of ground truth data to capture landscape variability from an agricultural region using Gaussian simulation and geostatistical techniques

Predictive modeling with remotely sensed data requires an accurate representation of spatial variability by ground truth data. In this study, we assessed the reliability of the size and location of ground truth data in capturing the landscape spatial variability embedded in the Airborne Visible Infrared Imaging Spectrometer-Next Generation (AVIRIS-NG) hyperspectral image in an agricultural region in Anand, India. We derived simulated spectral vegetation and soil indices using Gaussian simulation from AVIRIS-NG image for two point-location datasets, (1) ground truth points from adaptive sampling and (2) points from conditional Latin Hypercube Sampling (cLHS). We compared values of the simulated image indices against the actual image indices (measured) through the analysis of mean absolute errors. Modeling the variogram of the measured indices with the hyperspectral image in high spatial resolution (4m), is an effective way to characterize the spatial heterogeneity at the landscape level. We used geostatistical techniques to analyze the shapes of experimental variograms in order to assess whether or not the ground truth points, when compared against the cLHS-derived points, captured the spatial structures and variability of the studied agricultural area using measured indices. In addition, we explored the capability of the variogram by running tests in different point sample sizes. The ground truth and cLHS datasets were able to derive equivalent values for field spatial variability from image indices, according to our findings. Furthermore, this research presents a methodology for selecting spectral indices and determining the best sample size for efficiently replicating spatial patterns in hyperspectral images.

Geostatistical analysis of spatial variability of the liquefaction potential – Case study of a site located in Algiers (Algeria)

Abstract The city of Algiers (Algeria) is a highly seismic area, and therefore, soil liquefaction poses a major concern for structures resting on sandy soil. A campaign of 62 static penetration tests or cone penetrometer tests (CPT) was carried out on a site located in the commune of Dar El Beïda in Algiers. The soil Liquefaction Potential Index (LPI) values were assessed, for each borehole, based on the simplified procedure of Seed and Idriss. On the other hand, the geographic information system and geostatistical analysis were used to quantify the risk of soil liquefaction at the studied site. It is worth mentioning that the (LPI) was taken as a regionalized variable. In addition, the experimental variogram was modeled on the basis of a spherical model. Also, the interpolation of the LPI values in the unsampled locations was performed by the Kriging technique using both isotropic and anisotropic models. Kriging standard deviation maps were produced for both cases. The cross-validation showed that the anisotropic model exhibited a better fit for the interpolation of the values of the soil liquefaction potential. The results obtained indicated that a significant part of the soil is liable to liquefy, in particular in the northwestern region of the study area. The findings suggest that there is a proportional relationship between the risk of liquefaction and the increase or decrease in seismic acceleration.

3D simulation of a giant oilfield in calcareous formations and scrutiny study of the interaction of the calculated parameters (Asmari formation in Maroon oilfield, Iran)

Simulation is an art to model the reservoir’s structure in the petroleum industry, which transfers all possible data digitally to the reservoir management system’s suitable computation software. The present study attempts to construct a 3D structural model using the geostatistical method to access in situ oil volume and the best oil production position. The geological area is the Maroon oilfields located at Dezful Embayment in southwest of Iran. In this oilfield, the Asmari reservoir is divided into ten zones. Several steps such as griding, kriging, and experimental variogram were carried out to access all available data to make up the model. The evaluation revealed that zones one and five were the main oil-bearing layers since it contains about 60% of total oil volume in this field. Petrophysical parameter distribution indicated that zones four and seven are good candidates for oil production since they have higher porosity and net to growth thickness than other zones. From the seventh to the tenth zone, the oil volume decreased due to high water saturation in these areas. The best zones for the future program are located in the western part. This study shows that 3D modeling will find elongation-type anticline and reservoir direction and the best location for future drilling programs based on the interaction of the petrographic parameters. The results can be used in various steps during the lifetime of the different reservoirs around the globe.

Geological Modeling for Nahr Umr Formation in Subba Oil Field, Southern Iraq

Geological modeling is very important in reservoir study, where it is described the framework of the structure and provided a distribution of petrophysical properties, the geological model helps to comprehend the fluid flow behavior affected by the heterogeneity of reservoir, also it is used to calculate oil initially in place and to choose the new well location in reservoir management. In this study, a geological model is built for Nahr Umr Formation based on data of 11 wells (SU-2, SU-3, SU-4, SU-5, SU-6, SU-7, SU-9, SU-10, SU-11, SU-13, and SU-14), including the location of wells, well tops, also contour map of the Nahr Umr Formation in Subba oil field is utilized. The structural model is constructed for the Nahr Umr Formation, which consists of two domes. The formation is composed of sandstone mainly interbedded with shale. The formation is divided into four units (B1, B2, C1, and C2) depending on well log data, each unit is divided into layers for getting better distribution of petrophysical properties, where unit B2 is divided to the number of layers is higher than other units due to its heterogeneity. The variogram is used as a geostatistical method to predict the petrophysical properties at unsampled locations, experimental variogram is calculated from well log data in a horizontal and vertical direction. The spherical variogram model shows a good fitting with the experimental variogram. Sequential Gaussian algorithm is used in distribution of petrophysical properties in a geological model which shows good petrophysical properties in unit B2 and unit B1, while units C1 and C2 are highly saturated with water.

Local estimates of available water capacity and effect of measurement errors on the spatial estimates and their uncertainties

The purpose of this work was to: (i) propose a methodology to infer local estimates of the available water capacity (AWC) at a plot from a few measurements in laboratory of AWC carried out on horizons of a pit on the same plot, (ii) examine the effect of measurement errors on spatial estimates of AWC and the associated uncertainties. For each horizon identified in the pit, the water content was determined at field capacity and at the permanent wilting point, and thus AWC. 47 soundings were carried out on a regular grid covering the plot. For each sounding, an AWC value was estimated by matching sounding horizons to the pit horizons. Laboratory measurements of AWC on 14 sites were used to validate the 47 local estimates. Statistics such as correlation coefficient, mean error and root mean square error are promising at 0.84, 4.0 mm and 17.5 mm, respectively. The spatialization of AWC was carried out by ordinary kriging, considering or ignoring the measurement errors on AWC. The precision of AWC estimates allowed making evident that accounting for measurement errors provided estimates that were more precise. This result, confirmed by the prediction interval coverage probability statistic, underlined that taking into account measurement errors in spatial modeling is an effective way to reduce the confidence interval of any estimate. These results suggest that it would be better to fix the nugget based on a preliminary test on the measurement error rather than to fix it based on the experimental variogram.

Geostatistical analysis of static level evolution between 1995-2005 and 2007-2017 in San Luis Potosí Aquifer, México

Knowing the piezometric levels evolution of an aquifer is essential for planning the use of water resources in the cities of central México, such as San Luis Potosí, whose population depends almost entirely on groundwater. Applying geostatistical methods from georeferenced data can reliably provide information on the spatial variability of valley piezometry. The objective of this study is to evaluate the evolution of the static levels of the aquifer in San Luis Potosí Valley, through an interpolation analysis by the kriging method. For both periods, the experimental variogram was adjusted to the Gaussian model and the method with the best results was ordinary kriging. From this analysis it was obtained that the decay of the static levels is more critical in the central zone of the valley, while the zones of the northwest and towards the south, the aquifer levels have shown an important recovery; the important values for the decrease in static levels coincide with the areas where the population density is greater.