Fitted Variogram Model Research Articles

Spatial Interpolations of Annual Rainfall in Ethiopia Using Simple and Universal Kriging Techniques

The study of spatial distribution of precipitation is an important aspect of water-related research like, hydrological modeling, disaster prediction and watershed management. But, spatial interpolations of annual average rainfall in Ethiopia using Simple and Universal Kriging type through different models has not yet investigated. This study was focused on to select the best fit semivarogram model for annual rainfall interpolation and to map the spatial distribution of rainfall of Ethiopia. In this study, long term mean annual rainfall which was collected from 931 meteorological stations were interpolated using different mathematical variogram models of Kriging to select the best fit variogram model useful to predict continuous surface values of annual average rainfall in Ethiopia. Spherical model was found to be the best fitted variogram model and most accurate model for interpolation using both Simple and Universal Kriging with the value of ME (0.005), RMSE (1.21) and MSDR (121.32) and ME (0.001), RMSE (0.88) and MSDR (144.11), respectively. Since the kriging results of spherical model indicate good model efficacy for this study, it may be useful to map spatial pattern of annual rainfall in Basin level in Ethiopia.

AN INVESTIGATION OF CRIMES RATES USING CO-KRIGING AND INVERSE DISTANCE WEIGHTS IN KATSINA STATE, NIGERIA

The aim of the study is to investigates and predict the spatial variability of crimes rates using Co-kriging and inverse Distance Weight (IDW) in ten (10) Selected Local Government in Katsina State. The reported crimes to the Katsina State Police Command Consists of grievous hurt and Wound, Cattle rustling, Rape, Kidnapping, Assaults, Stealing and Burglary. The data is collected from Criminal Investigations Department Unit in Katsina State Command from 2010 to 2019. The method applied for this study was Co- Kriging Model (CK) and Inverse Distance Weight (IDW) to test the fitted Variogram Model and Spatial dependencies of variables examined. The findings predicted that, from the study areas the crimes rates likely happen in Dantamba, Sheme, Dofar-Mato, Gunya, Dankar, Rubau, Sawai, Wurma, Birinya Tsakatsa, Dankamtsa, Gardawa, and Katsalle Villages in a long run. The results also confirmed that, high level of unemployed youth and poverty rate in the study areas have a positive impact on socio-economic factors that influenced the crimes rates. The study suggests Government should encourage the youth in such localities by providing social amenities, employment opportunities, Good education, access of Road, Nearby Police stations and police outpost should be provided in each study areas. Peaceful campaign programs on dangers of Drug abuse and Trafficking, Illegal sects and sensitization of the communities on the need for peaceful co-existence. The study suggests the Government to put more emphasis on border patrol especially Kaduna, Niger, Sokoto, and Zamfara to avoid the such occurrence of crimes in the State and Local

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.

Mapping topsoil electrical conductivity by a mixed geographically weighted regression kriging: A case study in the Heihe River Basin, northwest China

Spatial prediction is an important approach to obtain location-specific values of soil electrical conductivity (EC), which is a proxy of soil salinity and important for agricultural management in arid and semi-arid areas. Linear regression models assume that the relation between soil EC and environmental covariates is constant over the area to be predicted. This is problematic at the regional scale, at which some of the regression parameters may indeed be globally constant, whereas others may vary locally. Moreover, model residuals often exhibit spatial dependence, which invalidates the ordinary least squares linear regression. This study examined the combination of a mixed geographically weighted regression model with simple kriging of the residuals (MGWGK) for mapping soil EC in the Heihe River Basin, an inland river basin in arid northwest China. We compared the performance of MGWRK with those of multiple linear regression (MLR), regression kriging (RK), geographically weighted regression (GWR), geographically weighted regression kriging (GWRK) and mixed geographically weighted regression (MGWR). Environmental covariates were developed from spatial information on topography, climate, vegetation, and geographic position. A ten-fold cross-validation was applied to evaluate predictive accuracy of the various methods. Soil EC ranged from 0.031 to 182.100 dS m−1, exhibiting a contrasting distribution of soil EC in the upper, middle, and lower river reaches. The MGWRK method outperformed other methods. The effects of different environmental covariates on soil EC were revealed by the fixed and geographically varying parameters of MGWRK. The nugget-to-sill ratios of fitted variogram models all fell between 25 and 32%, exhibiting moderate spatial autocorrelation of models residuals. Predictive accuracy was improved by MGWRK, as the spatial dependence of model residuals were included in the prediction. When selecting an optimal linear regression model, covariates should be tested to see if they are constant (as in MLR) or spatially varying (as in GWR) or semi-varying (as in MGWR), and model residuals should be tested for spatial dependence (as in MGWRK).

Spatial interpolation of precipitation indexes in Sierra Nevada (Spain): comparing the performance of some interpolation methods

The objective of this paper is to examine the spatial distribution of several precipitation indexes in Sierra Nevada, Spain: mean annual number of wet days (R ≥ 1 mm), mean annual number of heavy rainy days (R ≥ 10 mm) and mean annual number of very heavy precipitation days (R ≥ 20 mm) and test the performance of several interpolation methods using these variables. In total, 17 univariate and multivariate methods were tested. A set of 36 metereological stations distributed in Sierra Nevada and neighbouring areas was analysed in this study. The original data did not followed the normal distribution; thus, a logarithm was applied to data meet normality purposes. Interpolator’s performance was assessed using the root mean square error generated from cross-validation. The results showed that the mean annual R ≥ 10 mm and R ≥ 20 mm have a higher variability than R ≥ 1 mm. While the elevation and longitude did not show a significant correlation with the studied indexes, the latitude (i.e. distance to the sea) showed a significant negative correlation. The regressions carried out confirmed that elevation was the covariate with higher capacity to explain the variability of the indexes. The incorporation of elevation and longitude slightly increased the explanation capacity of the models. The data of LogR ≥ 1 mm, LogR ≥ 10 mm and LogR ≥ 20 mm displayed a clustered pattern, especially the last two indexes that also showed a strong spatial dependency attributed to the effects of local topography, slope, aspect and valley orientation. The best fitted variogram model to LogR ≥ 1 mm was the linear one while for the LogR ≥ 10 mm and LogR ≥ 20 mm, the Gaussian was the most appropriate. The best interpolator for LogR ≥ 1 mm was the local polinomyal with the power of 1, whereas for LogR ≥ 10 mm and LogR ≥ 20 mm, regression kriging (ROK) using as auxiliary variables the elevation, latitude and longitude was the most accurate. ROK methods significantly improved the interpolations accuracy, especially in LogR ≥ 10 mm and LogR ≥ 20 mm. Nevertheless, the covariates, when used as auxiliary information in ordinary kriging, did not improve the precision of the interpolation.

Gradient-Based Assessment of Habitat Quality for Spectral Ecosystem Monitoring

The monitoring of ecosystems alterations has become a crucial task in order to develop valuable habitats for rare and threatened species. The information extracted from hyperspectral remote sensing data enables the generation of highly spatially resolved analyses of such species’ habitats. In our study we combine information from a species ordination with hyperspectral reflectance signatures to predict occurrence probabilities for Natura 2000 habitat types and their conservation status. We examine how accurate habitat types and habitat threat, expressed by pressure indicators, can be described in an ordination space using spatial correlation functions from the geostatistic approach. We modeled habitat quality assessment parameters using floristic gradients derived by non-metric multidimensional scaling on the basis of 58 field plots. In the resulting ordination space, the variance structure of habitat types and pressure indicators could be explained by 69% up to 95% with fitted variogram models with a correlation to terrestrial mapping of >0.8. Models could be used to predict habitat type probability, habitat transition, and pressure indicators continuously over the whole ordination space. Finally, partial least squares regression (PLSR) was used to relate spectral information from AISA DUAL imagery to floristic pattern and related habitat quality. In general, spectral transferability is supported by strong correlation to ordination axes scores (R2 = 0.79–0.85), whereas second axis of dry heaths (R2 = 0.13) and first axis for pioneer grasslands (R2 = 0.49) are more difficult to describe.

Ordinary kriging for the estimation of vein type copper deposit: A case study of the Chelkureh, Iran

Estimation of mineral resources and reserves with low values of error is essential in mineral exploration. The aim of this study is to estimate and model a vein type copper deposit using ordinary kriging method. After studying of variograms in different directions it was found out that the ore deposit has no anisotropy. The best fitted variogram model was considered for ordinary kriging estimation. Cross-validation was used to evaluate the accuracy of the variogram model for kriging. After trial and error a variogram with the best summary statistics was chosen. Model consists of a pure nugget effect with 0,30 amplitude plus a spherical scheme with sill 1.10 and range 30 m. The cross validation results showed that the correlation coefficient between estimated and real data was 0.829. The resource was classified based on calculated estimation errors by JORC code. Results showed that ordinary kriging can be used to model and estimate the vein type deposit. Consequently a three dimensional model of estimated value and error estimated value was provided by ordinary kriging to divide the ore into an economic and uneconomic part.

An Application of Geostatistics to Analysis of Water Quality Parameters in Rivers and Streams in Niger State, Nigeria

Assessment of surface water quality using multivariate statistical techniques does not incorporate the spatial locations of data into their defining computations. Information on spatial continuity of surface water concentrations can help in identifying the magnitude of contamination by runoff and anthropogenic pollutions. In the present study, spatial behavior of five (5) surface water quality parameters of some rivers/streams in Niger State of Nigeria was studied using R geostatistical package gstat, in conjunction with packages sp, rgdal, spatstat and maptools. The variograms and ordinary krigged spatial maps were generated for rainy and dry seasons. The characteristics of the best variable models; range; sill and nugget effects of each parameter were obtained. The variogram analysis indicated a high spatial coherence for E.co, Mg and TDS, whereas TCo and TH indicated a low spatial coherence. The nugget to sill ratios of experimental and linear fitted variogram models in all cases were less than 0.25 indicating that the rivers/streams water level has strong spatial coherence in both seasons. This result shows that linear model is the best for both seasons. Krigged spatial variability maps revealed that an average range of 48km variograms for dry season changes more rapidly than it does in rainy season with an average range of 4.3 km and R2 values of 0.80 to 0.92.

Spatial risk assessment related to abandoned mining activities: an environmental management tool

Mining activities and resulting wastes can be considered one of the most important sources of toxic metals and metalloids in the environment. To assess environmental risk in the surrounding areas of old abandoned W-Sn and Pb–Zn mines and resulting tailings and rejected materials, 333 samples were collected in stream sediments under the influence of abandoned mines. Samples were prepared and analyzed for Fe, Ba, P, Cu, Cr, Ag, B, Zn, Be, Y, Nb, Pb, Ni, V, Mn, Mo, As, W, Co, Cd, Sn and U. The inexistence of Portuguese legislation concerning parametric values for stream sediments led to the application of a quantitative index for progressive contamination on stream sediments, the Geoaccumulation Index (Igeo), as variables to create risk maps. A first exploratory multivariate statistical analysis, using the Principal Component Analysis (PCA), applied to the obtained Igeos, shows a first factor (F1) explaining the dependence of P and B (positive correlation with the axis) and the inverse correlation of these two elements with the cluster formed by Cr, Ni and V (negative correlation with the axis); the second factor (F2) explains Ni, Fe, Zn and As; Cd and U Igeos are not explained in the new factorial space and, therefore, are characterized individually. The variographic studies showed the existence of spatial structure for the new synthesis variables (F1, F2) as well as for Cd and U Igeos. The experimental point-support data was then interpolated using ordinary kriging within a narrow search window as shown in the fitted variogram models. The obtained maps show extremely high levels of pollution in Cd and W and strongly high levels of pollution in Cr, B, Ag, Zn and Pb. The accumulation of these elements in the studied stream sediments is higher on the abandoned mining areas and in their vicinity.

Spatiotemporal Variability of Soil Moisture as Affected by Soil Properties during Irrigation Cycles

Information on spatiotemporal soil water variability and its association with soil properties during multiple irrigation cycles is important for precisely estimating soil water storage and developing effective irrigation strategies. Temporal change in the spatial soil water distribution and its correlation with topography and soil properties (texture, bulk density, and C content) were studied in a furrow-irrigated 7.56-ha vineyard located in northwestern China. Soil water measurements were taken 17 times from May to October 2012 and at soil depths of 0 to 20 and 20 to 40 cm at 135 georeferenced locations. There were four irrigation events during the study period. The coefficient of variation (CV) of spatial soil water ranged from 14.5 to 25.8% and decreased with higher spatial mean soil water content. The relation between CV and mean soil water content was found to be similar between the two soil layers. Soil water content was significantly and consistently correlated with soil texture and bulk density, while correlations with topography were mostly insignificant. The correlation of soil water content with soil sand content were stronger during the late period of an irrigation cycle when the soil was drier than at the earlier period when the soil is wetter. The spatial structure of the standardized soil water content, characterized by variograms, was found to be similar at different periods of irrigation cycles, and nugget/sill ratios of the fitted variogram model suggested that a relatively larger portion of spatial variability in soil water content was spatially structured at the surface than in the subsurface soil.

Geostatistical analysis of fruit yield and detachment force in coffee

The aim of this study was to use geostatistical analysis to evaluate the spatial variation in the detachment force of coffee fruit and coffee yield by variograms and kriging for precision agriculture. This study was conducted at Brejao farm, Tres Pontas, Minas Gerais, Brazil. The detachment force of green and mature coffee fruit was measured with a prototype dynamometer and georeferenced. The yield data were obtained from manual harvesting and were georeferenced. The data were evaluated by variograms estimated by residual maximum likelihood (REML), which provided a satisfactory approach for modeling all the variables with a small sample size. Spherical and exponential models were fitted, the first provided the better fit to mature fruit detachment force and the latter provided the better fit to coffee yield and green fruit detachment force. They were used to describe the structure and magnitude of spatial variation in the variables studied. Kriged estimates were obtained with the best fitting variogram models and mapped. The statistical and geostatistical analyses enabled us to characterize the spatial variation of the detachment force of green and mature coffee fruit and coffee yield and to visualize the spatial relations among these variables. The precision agriculture techniques used in this paper to collect, map and analyze the variables studied will help coffee farmers to manage their fields. Maps of coffee yield will enable farmers to apply nutrients site-specifically and manage harvesting either manually or mechanically. In addition, maps of detachment force of coffee fruit can enable farmers to harvest coffee selectively by choosing the appropriate places and the right time to start. This will improve the quality of the final product and also increase profits.

Robust Resampling Confidence Intervals for Empirical Variograms

The variogram function is an important measure of the spatial dependencies of a geostatistical or other spatial dataset. It plays a central role in kriging, designing spatial studies, and in understanding the spatial properties of geological and environmental phenomena. It is therefore important to understand the variability attached to estimates of the variogram. Existing methods for constructing confidence intervals around the empirical variogram either rely on strong assumptions, such as normality or known variogram function, or are based on resampling blocks and subject to edge effect biases. This paper proposes two new procedures for addressing these concerns: a quasi-block-bootstrap and a quasi-block-jackknife. The new methods are based on transforming the data to decorrelate it based on a fitted variogram model, resampling blocks from the decorrelated data, and then recorrelating. The coverage properties of the new confidence intervals are compared by simulation to a number of existing resampling-based intervals. The proposed quasi-block-jackknife confidence interval is found to have the best properties of all of the methods considered across a range of scenarios, including normally and lognormally distributed data and misspecification of the variogram function used to decorrelate the data.

Data-Dependent Choice of Optimal Number of Lags in Variogram Estimation

Geostatistical data among spatial data is analyzed in three stages: (1) variogram estimation, (2) model fitting for the estimated variograms and (3) spatial prediction using the fitted variogram model. It is very important to estimate the variograms properly as the first stage(i.e., variogram estimation) affects the next two stages. In general, the variogram is estimated with the moment estimator. To estimate the variogram, we have to decide the 'lag increment' or the 'number of lags'. However, there is no established rule for selecting the number of lags in estimating the variogram. The present paper proposes a method of choosing the optimal number of lags based on the PRESS statistic. To show the usefulness of the proposed method, we perform a small simulation study and show an empirical example with with air pollution data from Korea.

Exploring the Geostatistical Method for Estimating the Signal-to-Noise Ratio of Images

The signal-to-noise ratio (SNR) has been estimated for remotely sensed imagery using several image-based methods such as the homogeneous area (HA) and geostatistical (GS) methods. For certain procedures such as regression, an alternative SNR (SNRvar), the ratio of the variance in the signal to the variance in the noise, is potentially more informative and useful. In this paper, the GS method was modified to estimate the SNRvar, referred to as the SNRvar(GS). Specifically, the sill variance c of the fitted variogram model was used to estimate the variance of the signal component and the nugget variance c0 of the fitted model was used to estimate the variance of the noise. The assumptions required in this estimation are presented. The SNRvar(GS) was estimated using the modified GS method for six different land-covers and a range of wavelengths to explore its properties. The SNR*var(GS) was found to vary as a function of both wavelength and land-cover. The SNR*var(GS) represents a useful statistic that should be estimated and presented for different land-cover types and even per-pixel using a local moving window kernel.

Spatial Analysis of <I>Aedes albopictus</I> (Diptera: Culicidae) Oviposition in Suburban Neighborhoods of a Piedmont Community in North Carolina

Temporal and spatial distribution of egg-laying by Aedes albopictus (Skuse) (Diptera: Culicidae) was investigated in suburban neighborhoods in Raleigh, NC, by using oviposition traps (ovitraps) at fixed sampling stations during the 2002 and 2003 mosquito seasons. Variations in the phenology of oviposition between the two mosquito seasons resulted from differences in the patterns and amounts of rainfall early in the season. Aerial images of each study neighborhood were digitized, and the proportions of specific types of land cover within buffer zones encompassing ovitraps were estimated. Retrospective analyses showed that in some neighborhoods, oviposition intensity was significantly associated with specific types of land cover. However, in general, it seemed that gravid Ae. albopictus searched throughout the landscape for water-filled containers in which to lay eggs. Peridomestic surveys were carried out concurrently with ovitrap collections to estimate production of Ae. albopictus pupae in discarded water-filled containers and the abundance of females in vegetation that made up the resting habitat. Results of linear regression analyses indicated that the mean standing crop of pupae (total and per container) per residence was not a significant predictor of mean egg densities in ovitraps. However, the mean standing crop of adult females was a significant but weak predictor variable, because the magnitude and sign of regression coefficients varied between neighborhoods. Linear spatial regression analyses revealed that oviposition intensity was not spatially dependent on pupal standing crop or the numbers of pupae-positive containers distributed peridomestically. However, a weak spatial dependence on the standing crop of adult females was found in some neighborhoods. Based on spherical variogram models, kriging was carried out to predict the spatial patterns of oviposition in suburban neighborhoods. Focal areas of high and low oviposition intensity were evident in most neighborhoods; however, the spatial patterns of oviposition changed between mosquito seasons. Kriging predictions were evaluated, using cross-validation, by systematically removing each data point from our data set and predicting the removed point by using the remaining points. The root mean square (standardized) error values of best fitting variogram models approximated 1, and plots of standardized PRESS residuals showed no distinct pattern for most neighborhoods, indicating that predictions of the spatial distribution of oviposition intensity were valid. Spherical variogram models are a satisfactory method for describing the spatial distribution of Ae. albopictus oviposition, and kriging can be a useful technique for predicting oviposition intensity at locations that have not been sampled.

Spatial Analysis ofAedes albopictus(Diptera: Culicidae) Oviposition in Suburban Neighborhoods of a Piedmont Community in North Carolina

Temporal and spatial distribution of egg-laying by Aedes albopictus (Skuse) (Diptera: Culicidae) was investigated in suburban neighborhoods in Raleigh, NC, by using oviposition traps (ovitraps) at fixed sampling stations during the 2002 and 2003 mosquito seasons. Variations in the phenology of oviposition between the two mosquito seasons resulted from differences in the patterns and amounts of rainfall early in the season. Aerial images of each study neighborhood were digitized, and the proportions of specific types of land cover within buffer zones encompassing ovitraps were estimated. Retrospective analyses showed that in some neighborhoods, oviposition intensity was significantly associated with specific types of land cover. However, in general, it seemed that gravid Ae. albopictus searched throughout the landscape for water-filled containers in which to lay eggs. Peridomestic surveys were carried out concurrently with ovitrap collections to estimate production of Ae. albopictus pupae in discarded water-filled containers and the abundance of females in vegetation that made up the resting habitat. Results of linear regression analyses indicated that the mean standing crop of pupae (total and per container) per residence was not a significant predictor of mean egg densities in ovitraps. However, the mean standing crop of adult females was a significant but weak predictor variable, because the magnitude and sign of regression coefficients varied between neighborhoods. Linear spatial regression analyses revealed that oviposition intensity was not spatially dependent on pupal standing crop or the numbers of pupae-positive containers distributed peridomestically. However, a weak spatial dependence on the standing crop of adult females was found in some neighborhoods. Based on spherical variogram models, kriging was carried out to predict the spatial patterns of oviposition in suburban neighborhoods. Focal areas of high and low oviposition intensity were evident in most neighborhoods; however, the spatial patterns of oviposition changed between mosquito seasons. Kriging predictions were evaluated, using cross-validation, by systematically removing each data point from our data set and predicting the removed point by using the remaining points. The root mean square (standardized) error values of best fitting variogram models approximated 1, and plots of standardized PRESS residuals showed no distinct pattern for most neighborhoods, indicating that predictions of the spatial distribution of oviposition intensity were valid. Spherical variogram models are a satisfactory method for describing the spatial distribution of Ae. albopictus oviposition, and kriging can be a useful technique for predicting oviposition intensity at locations that have not been sampled.

Species richness patterns and metapopulation processes – evidence from epiphyte communities in boreo‐nemoral forests

For several epiphyte species, dispersal limitation and metapopulation dynamics have been suggested. We studied the relative importance of local environmental conditions and spatial aggregation of species richness of facultative and obligate epiphytic bryophytes and lichens within two old‐growth forests in eastern Sweden. The effect of the local environment was analyzed using generalized linear models (GLM). We tested whether species richness was spatially structured by fitting variogram models to the residuals of the GLM. In addition, we analyzed the species‐area relationship (area=tree diameter). Different environmental variables explained the richness of different species groups (bryophytes vs lichens, specialists vs generalists, sexual vs asexual dispersal). In most groups, the total variation explained by environmental variables was higher than the variation explained by the spatial model. Spatial aggregation was more pronounced in asexually than in sexually dispersed species. Bryophyte species richness was only poorly predicted by area, and lichen species richness was not explained by area at all.Spatial aggregation may indicate effects of dispersal limitation and metapopulation dynamics on community species richness. Our results suggest that species groups differ in habitat requirements and dispersal abilities; there were indications that presence of species with different dispersal strategies is linked to the age of the host tree. Separate analyses of the species richness of species groups that differ in the degree of habitat specialization and dispersal ability give insights into the processes determining community species richness. The poor species‐area relationship, especially in lichens, may indicate species turnover rather than accumulation during the lifetime of the host tree. Epiphyte species extinctions may be mainly caused by deterministic processes, e.g. changes in habitat conditions as the host tree grows, ages and dies, rather than by stochastic population processes.

Estimating Variogram Uncertainty

The variogram is central to any geostatistical survey, but the precision of a variogram estimated from sample data by the method of moments is unknown. It is important to be able to quantify variogram uncertainty to ensure that the variogram estimate is sufficiently accurate for kriging. In previous studies theoretical expressions have been derived to approximate uncertainty in both estimates of the experimental variogram and fitted variogram models. These expressions rely upon various statistical assumptions about the data and are largely untested. They express variogram uncertainty as functions of the sampling positions and the underlying variogram. Thus the expressions can be used to design efficient sampling schemes for estimating a particular variogram. Extensive simulation tests show that for a Gaussian variable with a known variogram, the expression for the uncertainty of the experimental variogram estimate is accurate. In practice however, the variogram of the variable is unknown and the fitted variogram model must be used instead. For sampling schemes of 100 points or more this has only a small effect on the accuracy of the uncertainty estimate. The theoretical expressions for the uncertainty of fitted variogram models generally overestimate the precision of fitted parameters. The uncertainty of the fitted parameters can be determined more accurately by simulating multiple experimental variograms and fitting variogram models to these. The tests emphasize the importance of distinguishing between the variogram of the field being surveyed and the variogram of the random process which generated the field. These variograms are not necessarily identical. Most studies of variogram uncertainty describe the uncertainty associated with the variogram of the random process. Generally however, it is the variogram of the field being surveyed which is of interest. For intensive sampling schemes, estimates of the field variogram are significantly more precise than estimates of the random process variogram. It is important, when designing efficient sampling schemes or fitting variogram models, that the appropriate expression for variogram uncertainty is applied.

VARFIT: a fortran-77 program for fitting variogram models by weighted least squares

The inference of spatial variogram parameters by the classical non-parametric approach (method of moments) requires the fitting of a model to the experimental variogram (the variogram calculated from the observational data). The usual approach is to perform this fitting `by eye' in a trial-and-error process that continues until the fitting is considered satisfactory. This procedure allows the user more freedom to include his/her own knowledge about the geology, geophysics, etc., of the spatial variable. Nevertheless a program of automatic fitting may be of considerable help for a number of different reasons: (a) as a preliminary estimation of the parameters, (b) when a high number of variograms must be modelled on a routine basis, (c) in order to make a more objective modelling (so that different practitioners will obtain the same results) and (d) in order to perform a more statistically sound fitting (each experimental variogram point should be weighed according to its own statistical uncertainty). The VARFIT program performs a nonlinear minimization of the weighed squared differences between the experimental variogram and the model. Five different weighing functions as well as four different variogram models (spherical, exponential, Gaussian and power models) are provided as options. It is possible to fit a nugget effect and a geometric anisotropy (in 2D). The fitting of an anisotropic power variogram model to rainfall data is shown to illustrate the versatility of the program. Results from simulation experiments are provided in order to make some general conclusions about the performance of the different weighing functions. An experimental variogram data set is given in order to check the program implementation.

A geostatistical descriptor of the spatial distribution of soil classes, and its use in predicting the purity of possible soil map units

A geostatistical descriptor of the spatial distribution of soil profile classes is discussed, the Sum of Indicator Variograms (SIV). This function may be estimated for a region from observations of the soil profile class occurring at a systematic array of sample sites, and fitted by an authorised variogram model. It is shown that the fitted variogram model can be used, given some simplifying assumptions, to predict the average purity of map units for a soil map of the region with a legend of simple units corresponding to the profile classes. An indication of the mean boundary interval of alternative maps is also given by the variogram model, although this should be treated with caution. The method proposed could therefore be useful when analysing observations from a reconnaissance soil survey, as an aid to selecting appropriate methods for more detailed survey.