Variogram Estimation Research Articles

The textures of sarcoidosis: quantifying lung disease through variograms.

Sarcoidosis is a granulomatous disease affecting the lungs in over 90% of patients. Qualitative assessment of chest CT by radiologists is standard clinical practice and reliable quantification of disease from CT would support ongoing efforts to identify sarcoidosis phenotypes. Standard imaging feature engineering techniques such as radiomics suffer from extreme sensitivity to image acquisition and processing, potentially impeding generalizability of research to clinical populations. In this work, we instead investigate approaches to engineering variogram-based features with the intent to identify a robust, generalizable pipeline for image quantification in the study of sarcoidosis. For a cohort of more than 300 individuals with sarcoidosis, we investigated 24 feature engineering pipelines differing by decisions for image registration to a template lung, empirical and model variogram estimation methods, and feature harmonization for CT scanner model, and subsequently 48 sets of phenotypes produced through unsupervised clustering. We then assessed sensitivity of engineered features, phenotypes produced through unsupervised clustering, and sarcoidosis disease signal strength to pipeline. We found that variogram features had low to mild association with scanner model and associations were reduced by image registration. For each feature type, features were also typically robust to all pipeline decisions except image registration. Strength of disease signal as measured by association with pulmonary function testing and some radiologist visual assessments was strong (optimistic AUC ≈ 0.9, p ≪ 0.0001 in models for architectural distortion, conglomerate mass, fibrotic abnormality, and traction bronchiectasis) and fairly consistent across engineering approaches regardless of registration and harmonization for CT scanner. Variogram-based features appear to be a suitable approach to image quantification in support of generalizable research in pulmonary sarcoidosis.

Uncertainty estimation of regionalised depth–duration–frequency curves in Germany

Abstract. The estimation of rainfall depth–duration–frequency (DDF) curves is necessary for the design of several water systems and protection works. These curves are typically estimated from observed locations, but due to different sources of uncertainties, the risk may be underestimated. Therefore, it becomes crucial to quantify the uncertainty ranges of such curves. For this purpose, the propagation of different uncertainty sources in the regionalisation of the DDF curves for Germany is investigated. Annual extremes are extracted at each location for different durations (from 5 min up to 7 d), and local extreme value analysis is performed according to Koutsoyiannis et al. (1998). Following this analysis, five parameters are obtained for each station, from which four are interpolated using external drift kriging, while one is kept constant over the whole region. Finally, quantiles are derived for each location, duration and given return period. Through a non-parametric bootstrap and geostatistical spatial simulations, the uncertainty is estimated in terms of precision (width of 95 % confidence interval) and accuracy (expected error) for three different components of the regionalisation: (i) local estimation of parameters, (ii) variogram estimation and (iii) spatial estimation of parameters. First, two methods were tested for their suitability in generating multiple equiprobable spatial simulations: sequential Gaussian simulations (SGSs) and simulated annealing (SA) simulations. Between the two, SGS proved to be more accurate and was chosen for the uncertainty estimation from spatial simulations. Next, 100 realisations were run at each component of the regionalisation procedure to investigate their impact on the final regionalisation of parameters and DDF curves, and later combined simulations were performed to propagate the uncertainty from the main components to the final DDF curves. It was found that spatial estimation is the major uncertainty component in the chosen regionalisation procedure, followed by the local estimation of rainfall extremes. In particular, the variogram uncertainty had very little effect on the overall estimation of DDF curves. We conclude that the best way to estimate the total uncertainty consisted of a combination between local resampling and spatial simulations, which resulted in more precise estimation at long observation locations and a decline in precision at unobserved locations according to the distance and density of the observations in the vicinity. Through this combination, the total uncertainty was simulated by 10 000 runs in Germany, and it indicated that, depending on the location and duration level, tolerance ranges from ± 10 %–30 % for low-return periods (lower than 10 years) and from ± 15 %–60 % for high-return periods (higher than 10 years) should be expected, with the very short durations (5 min) being more uncertain than long durations.

Spatial Characteristics of the Fungus Powdery Mildew (Erysiphe neolycopersici) on Tomatoes and its Spread in Industrial Greenhouses

In regions with cool temperate climates, tomatoes are grown on an industrial scale in large greenhouses. There the crops are susceptible to infection by powdery mildew, the fungus Erysiphe neolycopersici, which is introduced largely as fungal spores from outside the greenhouses and spread by wind within them. We have monitored the spread of the disease and mapped its distribution in four commercial greenhouses throughout the growing season to understand its aetiology. We modelled the patterns of infection geostatistically, each comprising a deterministic long-range trend plus a short-range spatially correlated random residual. We identified three main kinds of pattern; one consisted of a constant plus a spatially correlated residual, second comprised a linear trend throughout the greenhouse plus a correlated random residual, and third, the trend had the form of a bell akin to a Gaussian surface plus, again, a correlated random residual. Here, we show three examples of these distributions and the detail of their geostatistical analysis using both the traditional method of moments (MoM) estimation of variograms and residual maximum likelihood (REML) to separate the deterministic and random components. The analytical modelling is followed by ordinary punctual kriging in the first case, by universal kriging in the second, and by regression kriging in the third case to display the infection as isarithmic ("contour") maps. We interpret the first form of distribution as arising from numerous foci as spores landed on the leaves from various sources spread by air currents and the movement of workers along the paths through the greenhouse. In the second case, the disease seemed to have spread from an infection introduced through the main door in one corner of the greenhouse and spread from there by the workers and air currents. The third infection arose near the centre of the greenhouse by the main path and spread outwards from there. In all three examples, the main pathways seemed important routes along which the fungus spread.

Interpolation of rainfall observations during extreme rainfall events in complex mountainous terrain

AbstractThe representation of rainfall in space is important for hydrological modelling. Accurate estimation of rainfall is particularly challenging in mountainous regions where observations are often sparse relative to the spatial variability of rainfall. In these regions, orographic processes lead to complex patterns of rainfall enhancement and rain shadow depletion. This study testsNatural Neighbour Interpolation (NNI),ordinary kriging (OK)andordinary cokriging (CK), to determine ifCKimproves rainfall interpolation during three extreme rainfall events. Three different elevation indices were considered as secondary variables forCK. Preliminary analysis using long‐term annual average rainfall totals, including additional high elevation rainfall observations, showed thatCKwith an effective elevation index (a directionally smoothed elevation, corrected for degree of ‘orographic processing’ and shifted to account for ‘wind‐drift’ of rainfall) as a secondary variable performed better thanNNIandOKwith an overall improvement of around 40%. Using rainfall totals for long‐term wind direction and wind speed rainfall classes,CKperformance was variable but provided an improvement of approximately 15% for wind direction classeswithoutan easterly wind component. For 15‐min timesteps during extreme rainfall events, there were comparatively small differences (cross‐validation usingRMSE) between interpolation methods, partly attributed to having only relatively low elevation rainfall observations, providing weak constraint. Using cross‐validation andmean biasdid, however, show an improvement for both high and low elevation observation classes. Importantly, cross‐variogram estimation provided differing cross‐validation results when estimated for different rainfall accumulation periods: 15‐min, hourly, daily and long‐term. Variograms and cross variograms estimated at a 15‐min timestep frequency were robust for many timesteps, but were difficult to fit automatically for others. Variograms estimated from longer periods were more reliably estimated, but tended to have lower variance and cross‐variance and longer correlation ranges producing a smoother interpolated rainfall field. Given the weak cross‐validation constraint, care must be taken in identifying the most appropriate method and variogram estimation period.

On Robustness for Spatio-Temporal Data

The spatio-temporal variogram is an important factor in spatio-temporal prediction through kriging, especially in fields such as environmental sustainability or climate change, where spatio-temporal data analysis is based on this concept. However, the traditional spatio-temporal variogram estimator, which is commonly employed for these purposes, is extremely sensitive to outliers. We approach this problem in two ways in the paper. First, new robust spatio-temporal variogram estimators are introduced, which are defined as M-estimators of an original data transformation. Second, we compare the classical estimate against a robust one, identifying spatio-temporal outliers in this way. To accomplish this, we use a multivariate scale-contaminated normal model to produce reliable approximations for the sample distribution of these new estimators. In addition, we define and study a new class of M-estimators in this paper, including real-world applications, in order to determine whether there are any significant differences in the spatio-temporal variogram between two temporal lags and, if so, whether we can reduce the number of lags considered in the spatio-temporal analysis.

GSTools v1.3: a toolbox for geostatistical modelling in Python

Abstract. Geostatistics as a subfield of statistics accounts for the spatial correlations encountered in many applications of, for example, earth sciences. Valuable information can be extracted from these correlations, also helping to address the often encountered burden of data scarcity. Despite the value of additional data, the use of geostatistics still falls short of its potential. This problem is often connected to the lack of user-friendly software hampering the use and application of geostatistics. We therefore present GSTools, a Python-based software suite for solving a wide range of geostatistical problems. We chose Python due to its unique balance between usability, flexibility, and efficiency and due to its adoption in the scientific community. GSTools provides methods for generating random fields; it can perform kriging, variogram estimation and much more. We demonstrate its abilities by virtue of a series of example applications detailing their use.

Facilitating mapping and understanding of within‐vineyard variation in fruit composition using data pooled from multiple vineyards

Background and Aims A large number of fruit samples is required for adequate variogram estimation, making the development of prescriptive maps for vineyard management cost prohibitive for most growers. The project assessed the efficacy of aggregating samples from multiple vineyards, over multiple years, to estimate a ‘common’ variogram that could be generated and applied more efficiently. Methods and Results Fifteen hundred berry samples were collected over 3 years (2017–2019) in four vineyards in California for analysis of fruit composition and spatial variability. Maps were produced for anthocyanins, malic acid and β-damascenone in each vineyard using four separate aggregations of samples and showed only subtle changes in patterns of spatial variability in any of the three analytes assessed. A common variogram generated without points from the vineyard to be mapped indicated lower kriging variances over 100 simulations and was able to correctly classify up to 70% of sample values. Conclusions The use of a common variogram in describing spatial variability in vineyards adds important statistical support to the generation of robust maps that could be used for targeted vineyard management. Grower collaboration across multiple regional vineyards could therefore improve mapping support for all involved. Though high-density sampling may still be required in some cases, once stable zones of fruit quality have been characterised, the sample size could potentially be reduced in subsequent years. Significance of the Study Maps produced from combined datasets collected from multiple vineyards and years could provide growers and wineries more confidence in zonal management by showing the temporal stability of the spatial variability of several aspects of fruit quality.

SciKit-GStat 1.0: a SciPy-flavored geostatistical variogram estimation toolbox written in Python

Abstract. Geostatistical methods are widely used in almost all geoscientific disciplines, i.e., for interpolation, rescaling, data assimilation or modeling. At its core, geostatistics aims to detect, quantify, describe, analyze and model spatial covariance of observations. The variogram, a tool to describe this spatial covariance in a formalized way, is at the heart of every such method. Unfortunately, many applications of geostatistics focus on the interpolation method or the result rather than the quality of the estimated variogram. Not least because estimating a variogram is commonly left as a task for computers, and some software implementations do not even show a variogram to the user. This is a miss, because the quality of the variogram largely determines whether the application of geostatistics makes sense at all. Furthermore, the Python programming language was missing a mature, well-established and tested package for variogram estimation a couple of years ago. Here I present SciKit-GStat, an open-source Python package for variogram estimation that fits well into established frameworks for scientific computing and puts the focus on the variogram before more sophisticated methods are about to be applied. SciKit-GStat is written in a mutable, object-oriented way that mimics the typical geostatistical analysis workflow. Its main strength is the ease of use and interactivity, and it is therefore usable with only a little or even no knowledge of Python. During the last few years, other libraries covering geostatistics for Python developed along with SciKit-GStat. Today, the most important ones can be interfaced by SciKit-GStat. Additionally, established data structures for scientific computing are reused internally, to keep the user from learning complex data models, just for using SciKit-GStat. Common data structures along with powerful interfaces enable the user to use SciKit-GStat along with other packages in established workflows rather than forcing the user to stick to the author's programming paradigms. SciKit-GStat ships with a large number of predefined procedures, algorithms and models, such as variogram estimators, theoretical spatial models or binning algorithms. Common approaches to estimate variograms are covered and can be used out of the box. At the same time, the base class is very flexible and can be adjusted to less common problems, as well. Last but not least, it was made sure that a user is aided in implementing new procedures or even extending the core functionality as much as possible, to extend SciKit-GStat to uncovered use cases. With broad documentation, a user guide, tutorials and good unit-test coverage, SciKit-GStat enables the user to focus on variogram estimation rather than implementation details.

Problems of Estimating the Resources of Accompanying Elements: A Case Study from the Cu-Ag Rudna Deposit (Legnica-Głogów Copper District, Poland)

As a result of the exploitation of ore deposits, in addition to the main elements, the accompanying elements are also partially recovered. Some of them increase the profitability of exploitation, while others reduce it because they hinder the recovery of the main elements and thus increase the costs of the recovery process. A comprehensive economic calculation to assess the profitability of ore mining depends on an appropriately accurate estimation of the resources of both the main and associated elements. This issue was analyzed with the example of the Cu-Ag Rudna ore deposit (LGCD, Poland). The subject of the assessment was the resources prediction accuracy of the main element (Cu) and four (4) accompanying elements (Co, Ni, Pb, and V) using geostatistical estimation method, in particular the ordinary kriging after the estimation of the relative variograms for describing the spatial variability structures of elements abundance. It was found that the standard kriging errors (deviations) in accompanying elements resources that are scheduled for exploitation within a one-year period in some parts of deposits are drastically greater (2 to 5 times) than the estimation errors of the main element resources. This is due to the sparse sampling pattern for their determinations and/or the high variability (among others nugget effect) of their abundance. In this situation, without additional sampling and a denser sampling pattern, the possibilities of a reliable assessment of the influence of accompanying elements on the economic consequences of exploitation are very limited.

A computational validation for nonparametric assessment of spatial trends

The analysis of continuously spatially varying processes usually considers two sources of variation, namely, the large-scale variation collected by the trend of the process, and the small-scale variation. Parametric trend models on latitude and longitude are easy to fit and to interpret. However, the use of parametric models for characterizing spatially varying processes may lead to misspecification problems if the model is not appropriate. Recently, Meilan-Vila et al. (TEST 29:728–749, 2020) proposed a goodness-of-fit test based on an $$L_2$$ -distance for assessing a parametric trend model with correlated errors, under random design, comparing parametric and nonparametric trend estimates. The present work aims to provide a detailed computational analysis of the behavior of this approach using different bootstrap algorithms for calibration, one of them including a procedure that corrects the bias introduced by the direct use of the residuals in the variogram estimation, under a fixed design geostatistical framework. Asymptotic results for the test are provided and an extensive simulation study, considering complexities that usually arise in geostatistics, is carried out to illustrate the performance of the proposal. Specifically, we analyze the impact of the sample size, the spatial dependence range and the nugget effect on the empirical calibration and power of the test.

A Robust Gaussian variogram estimator for cartography of hydrological extreme events

Kriging is used for spatial interpolation of geophysical and hydrological variables. This technique requires the definition of a mathematical function of spatial correlation known as variogram. Extreme value mapping is critical for the analysis of hydrological problems such as multiple risk and disaster damage valuation. However, the formulation of a variogram is complex and the selection of the variogram model is often crucial in the final representation of extreme values. There is some evidence that a variogram estimator may reduce the negative impact of using data samples with outliers. This is an approach that eliminates the term that takes into consideration the squared differences of empirical values in the calculation of a variogram. Known as Cressie–Hawkins Estimator (CH), it is based on considering a Gaussian distribution for the calculation of the variogram. The CH satisfies the main statistical considerations of normality; however, the formulation of kurtosis is not used to guarantee total normality. The present paper modifies the CH, adding the term corresponding to the fourth statistical moment. The numerical example proposed in the literature to evaluate the efficiency of the CH is reproduced, and it is verified that the new variogram (CH-GLo) decreases further the effect of the extreme values in the cartography. CH-GLo is applied to cartography an extreme storm that happened in August 2014 in the city of Queretaro, Mexico. It is concluded that this new variogram formulation is a suitable alternative for mapping extreme values monitored in real-time in urban areas.

Stochastic Simulation of the Spatial Heterogeneity of Deltaic Hydrofacies Accounting for the Uncertainty of Facies Proportions

The spatial geological heterogeneity of an aquifer significantly affects groundwater storage, flow and the transport of solutes. In the particular case of coastal aquifers, spatial geological heterogeneity is also a major determining factor of the spatio-temporal patterns of water quality (salinity) due to seawater intrusion. Whist the hydraulics of coastal hydrogeology can be modelled effectively by various density flow equations, the aquifer geology is highly uncertain. A stochastic solution to the problem is to generate numerical realisations of the geology using sequential stratigraphy, geophysical models or geostatistical approaches. The geostatistical methods (two-point geostatistics, Markov chain models and multiple-point geostatistics) have the advantage of minimal data requirements, e.g., when the only data available are from cores from a few sparsely located boreholes. We provide an extension of sequential indicator simulation by including the uncertainty of the hydrofacies proportions in the simulation approach. We also deal with the problem of variogram estimation from sparse boreholes and we discuss the implicit transition probabilities and the connectivity of simulated realisations of a number of categorical variables. The variogram model used in the simulation of hydrofacies significantly influences the degree of connectivity of the hydrofacies in the simulated model. The choice of model is critical as connectivity determines the amount and extent of seawater intrusion and hence the environmental risk. The methodology is illustrated with a case study of the Andarax river delta, a coastal aquifer in south-eastern Spain. This is a semi-arid Mediterranean region in which the increasing use of, and demand for, groundwater is exacerbated by a transient tourist population that reaches its peak in the summer when the demand for the permanent population is at its highest. The work reported here provides a sound basis for designing flow simulation models for the optimal management of groundwater resources. This paper is an extended version of a presentation given at the 2012 GeoENV Conference held in Valencia, Spain.

Disaggregating a regional-extent digital soil map using Bayesian area-to-point regression kriging for farm-scale soil carbon assessment

Abstract. Most soil management activities are implemented at farm scale, yet digital soil maps are commonly available at regional or national scale. Disaggregating these regional and/or national maps is applicable for farm-scale tasks, particularly in data-poor or limited situations. Although disaggregation is a frequently discussed topic in recent digital soil mapping literature, the uncertainty of the disaggregation process is not often discussed. Underestimation of inferential or predictive uncertainty in statistical modelling leads to inaccurate statistical summaries and overconfident decisions. The use of Bayesian inference allows for quantifying the uncertainty associated with the disaggregation process. In this study, a framework of Bayesian area-to-point regression kriging (ATPRK) is proposed for downscaling soil attributes, in particular, maps of soil organic carbon. An estimation of point support variograms from block-supported data was carried out using the Monte Carlo integration via the Metropolis–Hastings algorithm. A regional soil carbon map with a resolution of 100 m (block support) was disaggregated to 10 m (point support) information for a farm in northern New South Wales (NSW), Australia. The derived point support variogram has a higher partial sill and nugget, while the range and parameters do not deviate much from the block support data. The disaggregated fine-scale map (point support with a grid spacing of 10 m) using Bayesian ATPRK had an 87 % concordance correlation with the original coarse-scale map. The uncertainty estimates of the disaggregation process were given by a 95 % confidence interval (CI) limit. Narrow CI limits indicate that the disaggregation process gives a fair approximation of the mean soil organic carbon (SOC) content of the study site. The Bayesian ATPRK approach was compared with dissever, which is a regression-based disaggregation algorithm. The disaggregated maps generated by dissever had 96 % concordance correlation with the coarse-scale map. Dissever achieves this higher concordance correlation through an iteration process, while Bayesian ATPRK is a one-step process. The two disaggregated products were validated with 127 independent topsoil carbon observations. The validation concordance correlation coefficient for Bayesian ATPRK disaggregation was 23 %, while downscaled maps generated from dissever had 18 % concordance correlation coefficient (CCC). The advantages and limitations of both disaggregation algorithms are discussed.

Space-time variogram for daily rainfall estimates using rain gauges and satellite data in mountainous tropical Island of Bali, Indonesia (Preliminary Study)

High spatio-temporal variability of daily rainfall in Bali Island can create the absence of structure in the daily variogram in certain days. This research proposes a new technique applying space–time variogram for 3-successive daily rainfall to detect structure in variogram estimation by merging rain gauges and satellite data applying ordinary kriging (OK), regression kriging with CMORPH (CM), regression kriging with TRMM (TR), and blended monthly rainfall (MONT) to obtain daily blended gridded rainfall estimates. Original retrieval of CMORPH (CM_OR) and TRMM (TR_OR) also used as control points to assess the proposed method. Uncertainty and sensitivity analysis including cross validation were carried out to validate the proposed method.The result shows that a new technique, adapted from CMORPH specific character, can be applied to detect the existence of structure in variogram. Blended gridded daily rainfall estimates of CM has highest probability detection of rainy events, while OK the lowest. Generally, the four interpolation methods (OK, CM, MONT, TR) have low accuracy at leeward and in a coastal area. All of them have almost similar performance indicated by no clear distinction of RMSE value from cross validation. CM, MONT, and TR have a good sensitivity with maximum and minimum temperature, indicating that satellite data can improve the gridded rainfall estimates. However, it is required an improvement of daily gridded rainfall estimates since there are large RMSE values and low coefficient correlation in certain days because of strong and erratic behavior of rainy events in a mountainous tropical island of Bali.

Estimation of causal continuous‐time autoregressive moving average random fields

AbstractWe estimate model parameters of Lévy‐driven causal continuous‐time autoregressive moving average random fields by fitting the empirical variogram to the theoretical counterpart using a weighted least squares (WLS) approach. Subsequent to deriving asymptotic results for the variogram estimator, we show strong consistency and asymptotic normality of the parameter estimator. Furthermore, we conduct a simulation study to assess the quality of the WLS estimator for finite samples. For the simulation, we utilize numerical approximation schemes based on truncation and discretization of stochastic integrals and we analyze the associated simulation errors in detail. Finally, we apply our results to real data of the cosmic microwave background.

Application of Dueling DQN and DECGA for Parameter Estimation in Variogram Models

Parameter estimation of variogram models is an important problem in geostatistics and environmental engineering. Most of existing works aim to estimate parameters of variogram single models, while neglecting the parameter estimation of variogram nested model. Most recently, the evolutionary algorithms(EA), including genetic algorithm(GA), are exploited to calculate the parameters of variogram model, which can obtain a more accurate solution. These methods have some hyper-parameters to set and suffer from the well-recognized premature convergence and slow global convergence problem of EA. In this paper, a double elite co-evolutionary genetic algorithm(DECGA) and deep reinforcement learning(dueling DQN) was introduced to estimate the parameters of variogram single or nested models so as to achieve better generalization performance. The DECGA can get the global optimal solution faster than GA with the help of dueling DQN, which can set the hyper-parameters according to the state of DECGA. To verify the effectiveness of the proposed method(DDQNGA), we conduct experiments on the agricultural heavy metal database. Experimental results demonstrate that our method can obtain parameter estimation more accurately. The method proposed in this paper have a certain practical value in the field of geostatistics and environmental engineering.

Ergodic and non-ergodic variogram uncertainty in Agbabu bitumen field porosity data

A recent investigation estimated and modeled the variogram of the porosity attribute of the bitumen field located at Agbabu, south-west Nigeria. In spite of methodological interventions made during the estimation, the variogram estimates obtained are still uncertain due to preferential and limited sample data. This current research therefore assesses the uncertainty in the ergodic and non-ergodic variogram estimates. Multiple samples required for ergodic and non-ergodic variogram estimates have been drawn from simulated realizations of the porosity random field. Variogram estimates have been obtained (using a recently-formulated lag-cluster technique and conventional technique) for each sample. The sampling distributions of these estimates have been examined with histograms and Normal Q-Q plots. The estimates exhibit approximately Normal distributions. The uncertainty in the ergodic variogram increases from 1.4×10−7 at lag 16.7 m to 4.6×10−7 at lag 4601.4 m; thereafter, it appears to flatten out. For most lag distances, the uncertainty in the non-ergodic variogram is constant at about 1.8×10−7; it fluctuates at large lag distances. The uncertainty in non-ergodic estimates is observed to be lower than the uncertainty in ergodic estimates. Consequently, that the current well spacing and distribution in Agbabu field is deemed adequate in the context of variogram estimation.

Saddlepoint approximations for the distribution of some robust estimators of the variogram

In this paper, we obtain a saddlepoint approximation for the small sample distribution of several variogram estimators such as the classical Matheron’s estimator, some M-estimators like the robust Huber’s variogram estimator, and also the $$\alpha $$-trimmed variogram estimator. The tail probability approximation obtained is very accurate even for small sample sizes. In the approximations we consider that the observations follow a distribution close to the normal, specifically, a scale contaminated normal model. To obtain the approximations we transform the original observations into a new ones, which can be considered independent if a linearized variogram can be accepted as model for them. To check this, a goodness of fit test for a variogram model is defined in the last part of the paper.

Data generation for axially symmetric processes on the sphere

In this article, we propose an algorithm that generates data on the entire sphere following the given axially symmetric covariance. We demonstrate that the axially symmetric data can be decomposed as Fourier series on circles, where the random Fourier coefficients can be expressed as circularly-symmetric complex random vectors. Our algorithm has the advantages of efficiency and scalability comparing with the classical method using the given covariance function directly. The data validation through the classical variogram estimation method demonstrates that our algorithm produces the data with smaller biases in recovering the given covariance function while maintaining the comparable mean squared errors.

Central limit theorem for mean and variogram estimators in Lévy–based models

AbstractWe consider an infinitely divisible random field in ℝd given as an integral of a kernel function with respect to a Lévy basis. Under mild regularity conditions, we derive central limit theorems for the moment estimators of the mean and the variogram of the field.