Variogram Parameters Research Articles

Research on variogram analysis method for 3D modeling of sandstone reservoir

Analysis of variogram is a necessary step in 3D stochastic modeling, and different settings of variograms can directly affect the final distribution of model attribute. How to optimize the setting of variogram parameters by unit and facies type has become a key step in geological modeling. This article focuses on the reservoir within sedimentary background of fluvial-deltas in the layer SII7+8 which develops multiple sedimentary microfacies, such as fluvial channels, abandoned fluvial channels, flood plains, and natural levee with strong heterogeneity, so as to study the impact of changes of the variogram parameters on the simulation results of reservoir properties, in order to explore the analysis method of the variogram. On this basis, the reservoir is divided into 3 types of sand bodies and 11 types of sedimentary microfacies based on net pay thickness, sedimentary environment, and main sand scale and morphology. The distribution characteristics and extension scale of different facies types on the plane are classified and studied, in order to determine the characteristic parameters of the variogram of different types of sand bodies, accurately simulate the distribution characteristics of reservoir physical property, and further improve the accuracy of the 3D attribute model.

Empirical assessment of the physico-chemical determinants of soil spatial variability in Sub-Saharan Africa

An appropriate understanding of soil properties’ spatial variability could help to perform sustainable soil nutrient management. The study aims to identify the most important soil characteristics driving spatial variability in tropical soil. A total of 5000 sample locations were randomly generated from the Sub-Saharan Africa map and the sample values were obtained from www.soilgrid.org. Various variogram models were tested and the best fitted variogram parameters were used to simulate 10000 replications of each attributes and the spatial dependence indices were computed. Results suggested that soil N, pH and organic carbon are the most driving spatial variability to better control experimental error.

Accounting for analytical and proximal soil sensing errors in digital soil mapping

AbstractDigital soil mapping (DSM) approaches provide soil information by utilising the relationship between soil properties and environmental variables. Calibration of DSM models requires measurements that may often have substantial measurement errors which propagate to the DSM outputs and need to be accounted for. This study applied a geostatistical‐based DSM approach that incorporates measurement error variances in the covariance structure of the spatial model, weights measurements in accordance with their measurement accuracies and assesses the effects of measurement errors on the accuracies of DSM outputs. The method was applied in the Western Cameroon, where soil samples from 480 locations were collected and analysed for pH, clay and soil organic carbon (SOC) using conventional and mid‐infrared spectroscopy methods. Variogram parameters and regression coefficients were estimated using residual maximum likelihood under two scenarios: with and without taking measurement errors into account. Performance of the spatial models in the two scenarios was compared using validation metrics obtained with three types of cross‐validation. Acknowledging measurement errors impacted the regression coefficients and influenced the variogram parameters by reducing the nugget and sill variance for the three soil properties. Validation metrics including mean error, root mean square error and model efficiency coefficient were quite similar in both scenarios, but the prediction uncertainties were more realistically quantified by the models that account for measurement errors, as indicated by accuracy plots. There were relatively small absolute differences in predicted values of soil properties of up to 0.1 for pH, 1.6% for clay and 2 g/kg for SOC between the two scenarios. We emphasised the need of incorporating measurement errors in DSM approaches to improve uncertainty quantification, particularly when applying spectroscopy for estimating soil properties. Further development of the approach is the extension to non‐linear machine learning regression methods.Highlights Errors in soil measurements are usually not accounted for and may affect DSM results. Measurement error variances were incorporated in the geostatistical models of three soil properties. Quantifying measurement errors in DSM allows to weigh measurements in accordance with their accuracy. Accounting for measurement errors in DSM better assesses prediction accuracy.

Spatial statistics and soil mapping: A blossoming partnership under pressure

For the better part of the 20th century pedologists mapped soil by drawing boundaries between different classes of soil which they identified from survey on foot or by vehicle, supplemented by air-photo interpretation, and backed by an understanding of landscape and the processes by which soil is formed. Its limitations for representing gradual spatial variation and predicting conditions at unvisited sites became evident, and in the 1980s the introduction of geostatistics and specifically ordinary kriging revolutionized thinking and to a large extent practice. Ordinary kriging is based solely on sample data of the variable of interest—it takes no account of related covariates. The latter were incorporated from the 1990s onward as fixed effects and incorporated as regression predictors, giving rise to kriging with external drift and regression kriging. Simultaneous estimation of regression coefficients and variogram parameters is best done by residual maximum likelihood estimation. In recent years machine learning has become feasible for predicting soil conditions from huge sets of environmental data obtained from sensors aboard satellites and other sources to produce digital soil maps. The techniques are based on classification and regression, but they take no account of spatial correlations. Further, they are effectively ‘black boxes’; they lack transparency, and their output needs to be validated if they are to be trusted. They undoubtedly have merit; they are here to stay. They too, however, have their shortcomings when applied to spatial data, which spatial statisticians can help overcome. Spatial statisticians and pedometricians still have much to do to incorporate uncertainty into digital predictions, spatial averages and totals over regions, and to take into account errors in measurement and spatial positions of sample data. They must also communicate their understanding of these uncertainties to end users of soil maps, by whatever means they are made.

Land price polarization and dispersion in Tokyo: a spatial model approach

Intensifying spatial inequality in housing markets is of great concern for urban planners and policymakers. This study employed a year-on-year analysis using universal kriging to fit the official land price data in Tokyo over the period 1997–2021. By comparing estimated regression coefficients with the estimated spatial variogram parameters, we found evidence of rising spatial polarization and dispersion in land prices of Tokyo. This was confirmed via price differences between western and central Tokyo that are becoming larger with each passing year. The determinants of the land price have different impacts on prices in western and central Tokyo. As for the variogram parameter, the nugget effects of the residuals in spatial models are increasing for the non-central region and decreasing in central Tokyo, which indicates that the degree of spatial dispersion is increasing.

Optimisation of geotechnical surveys using a BIM-based geostatistical analysis

PurposeA principle prerequisite for designing and constructing an underground structure is to estimate the subsurface's properties and obtain a realistic picture of stratigraphy. Obtaining direct measure of these values in any location of the built environment is not affordable. Therefore, any evaluation is afflicted with uncertainty, and we need to combine all available measurements, observations and previous knowledge to achieve an informed estimate and quantify the involved uncertainties. This study aims to enhance the geotechnical surveys based on a spatial estimation of subsoil to customised data structures and integrating the ground models into digital design environments.Design/methodology/approachThe present study's objective is to enhance the geotechnical surveys based on a spatial estimation of subsoil to customised data structures and integrating the ground models into digital design environments. A ground model consisting of voxels is developed via Revit-Dynamo to represent spatial uncertainties employing the kriging interpolation method. The local arrangement of new surveys are evaluated to be optimised.FindingsThe visualisation model's computational performance is modified by using an octree structure. The results show that it adapts the structure to be modelled more efficiently. The proposed concept can identify the geological models' risky locations for further geological investigations and reveal an optimised experimental design. The modifications criteria are defined in global and local considerations.Originality/valueIt provides a transparent and repeatable approach to construct a spatial ground model for subsequent experimental or numerical analysis. In the first attempt, the ground model was discretised by a grid of voxels. In general, the required computing time primarily depends on the size of the voxels. This issue is addressed by implementing octree voxels to reduce the computational efforts. This applies especially to the cases that a higher resolution is required. The investigations using a synthetic soil model showed that the developed methodology fulfilled the kriging method's requirements. The effects of variogram parameters, such as the range and the covariance function, were investigated based on some parameter studies. Moreover, a synthetic model is used to demonstrate the optimal experimental design concept. Through the implementation, alternative locations for new boreholes are generated, and their uncertainties are quantified. The impact of the new borehole on the uncertainty measures are quantified based on local and global approaches. For further research to identify the geological models' risky spots, the development of this approach with additional criteria regarding the search neighbourhood and consideration of barriers and trends in real cases (by employing different interpolation methodologies) should be considered.

Automatic variogram model fitting of a variogram map based on the Fourier integral method

One of the cornerstones of geostatistics is the variogram. Fitting a variogram model to observed data is certainly one of the routine tasks in daily geostatistics practice. The central role of the variogram and the difficulty of performing variogram modeling manually call for an automatic, or at least a computer-assisted, method. Fitting a parametric curve to observational points in an XY chart is far from new and the GSLib software package has included a module (called VARFIT) to perform automatic variogram fitting in 1D since its debut many years ago. Nevertheless, extending its objective function to fitting to variogram map (2D) resulted in a method with poor performance. Hence, this work proposes a computational method to fit variogram models to variogram maps with a different objective function based on the principles of the Fourier Integral Method (FIM). A map is synthesized from the theoretical variogram model using the principles of FIM. The metric of the objective function is the sum of squares of differences between each pixel value of the synthetic map and the input map. The genetic algorithm uses this metric to find the variogram parameters of a model with good fitness. Since the genetic algorithm has a stochastic component, the best fit is not certain. Hence, several runs with different random number generator seeds are performed. The parameters of all fitted models are collected and cluster analysis in parameter space is done with the DBSCAN algorithm. The centers of the clusters are the parameters of the sought variogram model. Open-source software was written and made available, in which the proposed objective function was tested alongside the 2D version of the one in VARFIT. Results showed better effectiveness of the proposed method in fitting variogram models for the variogram map case.

An enhanced dual IDW method for high-quality geospatial interpolation

Many geoscience problems involve predicting attributes of interest at un-sampled locations. Inverse distance weighting (IDW) is a standard solution to such problems. However, IDW is generally not able to produce favorable results in the presence of clustered data, which is commonly used in the geospatial data process. To address this concern, this paper presents a novel interpolation approach (DIDW) that integrates data-to-data correlation with the conventional IDW and reformulates it within the geostatistical framework considering locally varying exponents. Traditional IDW, DIDW, and ordinary kriging are employed to evaluate the interpolation performance of the proposed method. This evaluation is based on a case study using the public Walker Lake dataset, and the associated interpolations are performed in various contexts, such as different sample data sizes and variogram parameters. The results demonstrate that DIDW with locally varying exponents stably produces more accurate and reliable estimates than the conventional IDW and DIDW. Besides, it yields more robust estimates than ordinary kriging in the face of varying variogram parameters. Thus, the proposed method can be applied as a preferred spatial interpolation method for most applications regarding its stability and accuracy.

Spatial variability of epi- and mesopelagic 38 kHz backscatter from nekton and macrozooplankton across the southeastern US shelf break

Acoustic echosounders collect detailed information on the location of patchily distributed pelagic organisms over varying spatial scales. This study measured the spatial variability of epi- and mesopelagic 38 kHz backscatter along the US Mid- and South Atlantic continental shelf and slope. We used variogram analysis to estimate the horizontal spatial structure of backscatter measurements, examined whether environmental variables might affect these estimates, and assessed potential impacts of acoustic survey design. Backscatter data were collected during ship-based surveys (50-100 km2) at 7 sites during 4 cruises between November 2017 and November 2019. Average patch size estimates were consistently between 2 and 4 km among locations. Modeled variogram range varied significantly with the depth of the backscatter layer, but linear effect sizes were negligible (<1 m). Chlorophyll a (chl a) concentration had a significant positive effect on range (95 m), suggesting that patch sizes are slightly larger in the epipelagic where chl a concentration is higher. Incorporating variogram parameters of range, sill, and nugget produced some clustering of spatial correlation parameters with scattering layer depth, particularly for the deepest sites assessed (700-900 m depth). Spatial characteristics of a given location were not significantly different between surveys of the same size, but sometimes differed with smaller (25% of area) survey sizes. These results offer insight into nekton and macrozooplankton backscatter patterns in important shelf break and slope systems across horizontal and vertical dimensions, and provide needed information for monitoring fine- to mesoscale offshore marine habitat areas.

Estimation the Variogram Function Indicator which represent the Transmissivity Coefficient in the groundwater

The problem tackled in this paper is the estimation of variogram function Indicator of spatial stochastic process for the Levels of groundwater, by the method of weighted Least squares. This methods is well known in regression analysis in estimating the coefficient of ression model. After defining the indicator variable the parameters of Indicator variogram estimated based on mean squares error. The final formula of weighted least squares estimator can be not be solved exactly, then through the use of iterative Newten - Raphson algorithm and for some iterations the convergence of solution is obtained with certain termination criterion or number of repeats (that used in this paper).

How many sampling points are needed to estimate the mean nitrate-N content of agricultural fields? A geostatistical simulation approach with uncertain variograms

Knowledge of how many sampling points are needed to estimate the mean content of soil nutrients in agricultural fields, given a precision requirement on the estimated mean, is limited. This paper describes a versatile geostatistical simulation approach for predicting the variance of the mean nitrate-N (NO3-N) content within an agricultural field estimated by random sampling. In fall of 2016 sixteen agricultural fields were sampled on a square grid to model the spatial variation of NO3-N. On twelve out of sixteen fields NO3-N showed a lognormal distribution rather than a normal distribution. Variograms for (log-transformed) NO3-N are estimated using a Bayesian approach, resulting in 100 vectors with possible variogram parameters per field, obtained by MCMC sampling from the posterior distribution. Each of these variograms is used to simulate 100 maps of NO3-N, resulting in 100 × 100 maps of NO3-N per field. Each map is used to compute the variance of the estimated mean with stratified simple random sampling of 5,10,…,50 points, with one point per compact geographical stratum. For each sample size (number of sampling points) the mean, median and P90 of the uncertainty distribution are computed. Based on the medians, the sample size required for a maximum expanded measurement uncertainty of 50% varies from <5 to >50. This large variation in required sample size shows the large variation among the sixteen fields in variance of NO3-N within a field.

Parameter fitting of variogram based on hybrid algorithm of particle swarm and artificial fish swarm

Variation function is an important tool for describing the spatial correlation characteristics of regionalized variables in geostatistical methods. Variation function modeling is an important part of kriging interpolation and will directly affect the accuracy of the final interpolation result. The purpose of this work is to address the shortcomings of traditional variogram fitting methods, introduce particle swarm algorithm and artificial fish swarm algorithm under swarm intelligence framework, and design a variogram parameter fitting based on the hybrid algorithm of particle swarm and artificial fish swarm method. With this method, the minimum difference between the variation function fitting model and the given experimental variation value is utilized as the optimization goal. An appropriate objective function is set to convert it into a minimum problem. The hybrid algorithm has a strong search ability and convergence, as well as the ability to obtain the satisfactory fitness values. By comparing the results of the VARFIT fitting and the results of the optimization algorithm, it can be concluded that the absolute deviation of the fitting results of the optimization algorithm is 3.39 lower than the results of the VARFIT fitting. Compared with the traditional variogram modeling approach, this method has a strong optimization ability and high precision, and can effectively realize the automatic fitting of variogram parameters.

Improved Kriging model based on weighted self-adaptive differential evolution and application for structural monitoring data

The integrity of structural health monitoring data plays an important role in extracting data features. An improved Kriging model based on weighted self-adaptive differential evolution algorithm (KWSDE) is proposed to repair the spatial missing data. In the KWSDE model, the control parameters of variogram in Kriging are optimized by a novel differential evolution (DE) with both scale factor and crossover rate adaptation. Besides, a weighted least-square method is presented for the optimization process to restrain the variable scale of the fitness function, which makes it more suitable for the lack of monitoring data. Finally, a distributed optical fiber monitoring experiment is designed to verify the reliability and functionality of the proposed model. Compared with the Kriging based on DE model (KDE), the interpolation precision of the KWSDE model can be improved by more than 10%, which has great significance for its engineering application.

Transferability of a soil variogram for sampling design: A case study of three grasslands in Ireland

AbstractIt is commonly accepted that an estimated soil variogram can be transferred to another similar area for deriving the tolerable spacing of a sampling grid or, more generally, the sample size, given a requirement on the quality of the soil property map of the recipient area. The quality of the derived tolerable grid spacing depends on how similar the population variograms of the donor area and recipient area are. In practice we are uncertain about the variograms of both areas due to sampling errors. Ideally, the uncertainty about the variogram of the donor area is accounted for in deriving the tolerable grid spacing. To assess the transferability, we should also account for uncertainty in the estimated variogram of the recipient area. In this study the transferability of variograms of soil pH, P, Mg and K is analysed for three grassland fields in Ireland, which are similar in soil‐forming factors. One field served as donor area, the other two as recipient area. For all three fields and for each soil property, 500 variograms were sampled from the posterior distribution of the variogram parameters. Results showed that the estimated variogram parameters of the recipient fields differed largely from those of the transferred variograms. The ranges of estimated mean kriging variance values for the various grid spacings, as obtained with the two sets of variograms (one set of the donor field, one set of the recipient field), did not overlap. Even after scaling the transferred variogram with an estimate of the variance of the recipient field, the transferred variogram was of no use for determining the tolerable grid spacing. The difference in the variograms can possibly be explained by the difference in historical land use.Highlights Transferability of variograms to derive tolerable grid spacing for mapping grassland fields is assessed Transferability should be based on the uncertainty distributions of the tolerable grid spacings Due to difference in historical land use, local and transferred variograms differed largely Transferability of a variogram is very poor, even after scaling the transferred variogram

Probabilistic estimation of variogram parameters of geotechnical properties with a trend based on Bayesian inference using Markov chain Monte Carlo simulation

ABSTRACT When geotechnical properties show a trend with spatial coordinates, estimation of a variogram model is a challenging task. In the previous studies, the trend-removal method based on ordinary least-squares approach has been commonly used. However, the obtained variogram is biased because the residuals are assumed to be statistically independent. In this study, the ability of Bayesian inference using Markov chain Monte Carlo (MCMC) simulation to estimate the variogram of geotechnical properties with a trend is explored using cone penetration resistance (q c) data of piezocone penetration tests (CPTU). The results show that the Bayesian inference method can estimate variogram parameters and the coefficients of trend function accurately. Based on the posterior variogram models, the predictive uncertainty and total uncertainty bounds are presented using kriging and sequential Gaussian simulation (SGS) methods. 96% validation points lie within the 95% confidence intervals of the total uncertainty based on 200 measurements of q c at NS31. The median and mean SSD of the prediction are 0.34 and 0.87, which is closer to the SSD criterion than the trend-removal method. The model uncertainty of the variograms, the predictive and total uncertainty of prediction all decrease as the sampling density increases at NS31 and NS12.

Quantifying the criteria for classification of mineral resources and reserves through the estimation of block model uncertainty using geostatistical methods: a case study of Khoshoumi Uranium deposit in Yazd, Iran

ABSTRACT Investments and progress of mineral projects depend on the quantity (tonnage) and quality (grade) of mineral resources and reserves. This study examines the impact of various criteria used in the classification of mineral deposits or parameters defining these criteria. The data used in this study include the uranium assay analysis from 127 exploratory boreholes, which were then subjected to a three-directional variography after statistical studies to identify regional anisotropy. A grade block model was built using the optimal parameters of variograms and with the help of kriging estimator. Then, by using different methods of estimating the block model uncertainty including kriging estimation variance, block error estimation, kriging efficiency and slope of regression, classification of mineral reserves was carried out in accordance with the JORC standard code. Based on different cut-off grades, the tonnage and average grade were calculated and plotted. An innovative quantitative method based on the distribution function of the mentioned parameters and the fractal pattern of separation of populations was used for the classification of mineral reserves. The existence of the least difference between the use of standard and fractal patterns in the slope of regression method indicated less error and was a proof of more reliable results.

Shear Behavior and Anisotropy Characteristics of the Fracture Morphology of Sandstone with Different Water Contents

Hydromechanical coupling in rock masses is an important issue for many rock mechanics and hydrogeology applications. The change of a water-bearing state will induce the fracture of the intact rocks and further accelerate the shear slip instability of the sheared surface. To investigate the weakening effect of water content on the mechanical properties of a rock mass, laboratory direct shear tests combined with three-dimensional analysis of sheared surfaces were carried out on sandstone samples with different water contents. The variogram parameters, sill and range, were applied to quantify the morphology of shear fracture surfaces, to reflect the shear failure process of the intact rock, and to provide a basis for resliding instability of jointed rock. It was determined that the sill represents the height of the fluctuation body in the fracture surface and the range represents the single fluctuation body and may reflect the frequency of fluctuations. The test results revealed that the increase in water content had a clear weakening effect on the shear strength and deformation behavior of rock, especially under saturated conditions. Moreover, the distribution of water in the samples directly affected the crack initiation and propagation and characteristics of the fracture morphology.

Assessing the accuracy of geostatistical techniques for mapping soil macronutrients on basaltic landscape of central India

A study was conducted to interpolate and to explore the analysis of spatial variability of major soil nutrients in Basaltic Terrain of Bemetara district, Chhattisgarh. A total of 182 soil samples (0-25 cm) were collected randomly using GPS. Soil chemical properties i.e. available nutrients (N, P and K) were measured in laboratory. Data were interpolated by Ordinary Kriging (Spherical, Circular, and Gaussian). The performance of methods was evaluated using Mean Absolute Error (MAE), Root Mean Square Error (RMSE) and Goodness of prediction (G) obtained from a cross-validation procedure. The results showed that Circular, spherical, and Gaussian models were found best fit for available N, P, and K, respectively. All variables showed strong spatial dependence. Cross validation of kriged map showed that spatial prediction of soil nutrients using semi variogram parameters is better than assuming mean of observed value for any unsample location. Therefore it is a suitable alternative method for accurate estimation of soil properties in unsampled positions as compared to direct measurement which has time and costs concerned.

The Effect of Using Cokriging against Ordinary Kriging When Estimating Phosphate Deposits at Elsebaiya Area, Egypt

Application of geostatistical techniques when evaluating mineral deposits could reflect some geological characteristics which help through the stage of planning and production. In the present study, an attempt has been done on two phosphate deposits at Elsebaiya area on both sides of the River Nile namely: Um Tondoba mine at Elsebaiya East area and Western River Nile mine in Elsebaiya West area. Depending on the available data, statistical analysis illustrated differences in the distribution of P2O5 % and ore thickness within the studied areas. Geostatistics used to start with constructing variograms for P2O5 % and thickness for the two phosphate deposits to be used with ordinary kriging models, also constructing cross variograms between P2O5 % and thickness to be used with cokriging models where the difference in the variogram parameters reflected a specific variation for each deposit horizontally and vertically. The ordinary kriging models and cokriging models illustrated different distribution behavior through both the two kriging techniques.

Study on the Variogram Analysis Methods of 3D Geological Modeling for Sandstone Reservoirs

Variogram analysis is a necessary step for 3D stochastic modeling. Different variogram settings will directly affect the final distribution characteristics of model attributes. Setting the variogram parameters on the optimization results of sedimentary types of sub-units has become a key step in geological modeling. In this paper, the influence of variogram parameters on the simulation results of reservoir properties is studied to improve the analysis method of the variogram. This paper is for typical river-delta sedimentary reservoirs, specifically taking the SII7+8 layers as the research object. Multiple sedimentary microfacies types, including river, abandoned river, floodplain, natural dyke, coexist and show highly heterogeneous in the SII7+8 layers. On this basis, the reservoir is divided into 3 types of sand bodies and 11 kinds of sedimentary microfacies according to the effective thickness, sedimentary conditions, scale and shape of main sand bodies to study the distribution characteristics and extension scale of different sedimentary types in plane, so as to determine the variogram parameters of different sand body types, and accurately simulate the reservoir property distributions. The process in variogram analysis methods can further improving the accuracy of 3D geological modeling.