Variogram Parameters Research Articles

Landscape‐based geostatistics: a case study of the distribution of blue crab in Chesapeake Bay

AbstractGeostatistical techniques have gained widespread use in ecology and environmental science. Variograms are commonly used to describe and examine spatial autocorrelation, and kriging has become the method of choice for interpolating spatially‐autocorrelated variables. To date, most applications of geostatistics have defined the separation between sample points using simple Euclidean distance. In heterogeneous environments, however, certain landscape features may act as absolute or semi‐permeable barriers. This effective separation may be more accurately described by a measure of distance that accounts for the presence of barriers. Here we present an approach to geostatistics based on a lowest‐cost path (LCP) function, in which the cost of a path is a function of both the distance and the type of terrain crossed. The modified technique is applied to 13 years of survey data on blue crab abundance in Chesapeake Bay. Use of this landscape‐based distance metric significantly changed estimates of all three variogram parameters. In this case study, although local differences in kriging predictions were apparent, the use of the landscape‐based distance metric did not result in consistent improvements in kriging accuracy. Copyright © 2006 John Wiley & Sons, Ltd.

Characterizing the spatial structures of riparian plant communities in the lower reaches of the Heihe River in China using geostatistical techniques

AbstractGeostatistical techniques are used to evaluate spatial characteristics of riparian plant communities that were mapped within a portion of the lower reaches of the Heihe River. The spatial structures of different plant communities were characterized using nugget, range and sill parameters of spherical or exponential model variograms. Model variograms revealed different plant communities with distinctive spatial properties that were quantified effectively by the parameters used in these models. Model variograms were fit to experimental variograms calculated from diameter at breast height and plant cover data collected for three 2,000 m × 5 m, nine 500 m × 5 m, and three 200 m × 2 m transects. The 2,000 m × 5 m transects were oriented perpendicular to the river and the other transects were located in such a way as to sample communities dominated by different plants. Riparian plant communities in the lower reaches of the Heihe River show patterns of patchiness, and the plant communities are mainly dominated by mature Populus euphratica, young P. euphratica, Tamarix chinensis, and Sophora alopecuroides. The maximum model variogram range for all communities was 55 m, suggesting that transects must be at least this long in order to be able to classify them unambiguously into recognizable communities. Experimental variograms were calculated for all plants in each of four mapped plant communities. The results show that the T. chinensis and S. alopecuroides dominated plant communities have distinctively large ranges, and the spatial structures of P. euphratica dominated plant communities show multiscale change. In the 2,000 m × 5 m transects, the spatial structures of riparian plant communities show a distinct change at the scale of 430 m, because this scale also is the scale of vegetation patchiness turnover, and at scales of more than 430 m the spatial structure of the community increases significantly. These findings confirm the potential of using high‐resolution remote sensing data and geostatistics for determining the vegetation community structures of riparian plant communities.

The modifiable areal unit problem of spatial heterogeneity of plant community in the transitional zone between oasis and desert using semivariance analysis

The modifiable areal unit problem has significant implications for ecological research that involve investigating and analyzing the spatial heterogeneity of plant community. In this paper, semivariance analysis was used to evaluate the spatial characteristics of plant community in the␣transitional zone between oasis and desert. The spatial structures of the plant community were characterized using exponent model variogram parameters, including nugget (C0) range (A0) and sill (C0+C). Two methods were employed to determine the scale effect of spatial heterogeneity. (1) A constant grain size (10×10 m2) and variational plot areas have been used to analyze spatial heterogeneity of the plant community. (2) The grain size was only changed to analyze spatial heterogeneity. In addition, the plot of 500×500 m2 was clustered into nested units of different shapes and different directions to analyze zoning effect. Finally, using semivariance analysis, we obtained a suitable plot area and zoning approach to weaken the scale and zoning effects. The results showed that the effects of scale on different variogram parameters had significant difference. For example, C0 and C0+C were very sensitive at small scales, A0 was influenced significantly by plot area at larger scales, and C0 and A0 were relatively sensitive to different zoning approaches. In order to get more representative characteristic of spatial heterogeneity of the plant community, the sampling area should be more than 200×200 m2 for Nitraria sphaerocarpa populations, 100×100 m2 for Reaumuria soongorica populations, and a grain size from 20×20 m2 to 30×30 m2 for both populations.

Inverse stochastic moment analysis of steady state flow in randomly heterogeneous media

Nonlocal stochastic moment equations have been used successfully to analyze steady state and transient flow in randomly heterogeneous media conditional on measured values of medium properties. We present a nonlinear geostatistical inverse algorithm for steady state flow that makes it possible to further condition such analyses on measured values of state variables. Our approach accounts for all scales of spatial variability resolvable by the computational grid. It is based on recursive finite element approximations of exact nonlocal first and second conditional moment equations. Hydraulic conductivity is parameterized geostatistically on the basis of measured values at discrete locations and unknown values at discrete “pilot points.” Prior estimates of pilot point values are obtained (optionally) by generalized kriging. Posterior estimates at pilot points and (optionally) at measurement points are obtained by calibrating mean flow against measured values of head. The parameters are projected onto a computational grid via kriging. Maximum likelihood calibration allows estimating not only hydraulic but also (optionally) unknown variogram parameters with or without prior information about the former. The approach yields covariance matrices for parameter estimation as well as head and flux prediction errors, the latter being obtained a posteriori. We implement our inverse approach on highly and mildly heterogeneous media under superimposed mean uniform and convergent flows in a bounded two‐dimensional domain. Our examples illustrate how conductivity and head data act separately and jointly to reduce parameter estimation errors and to model predictive uncertainty. We also evaluate the functional form of the log conductivity variogram and its parameters using likelihood‐based model discrimination criteria.

Spatial variation features description of soil available P, K, Mg and soil pH by proportional effect

This paper investigates the proportional effect of selected soil properties – low spatial variation changes are related to their local magnitudes (here standard deviation vs. mean). Content of available P, K, and Mg, and soil pH were analysed on nine agricultural fields of the Xzech Republic. Firstly, strong direct within-field proportional effect based on Moving Window Statistics (MWS) was found for soil P and K, while Mg did not exhibit any clear proportionality. Soil pH showed indication of inverse proportional effect with high field-to-field fluctuations. The relationship strength of the effect was functionally related to the asymmetry (skewness) of distribution (r = 0.31 × skew 0.08). Secondly, between-field proportional effect of 9 surveyed fields, as a measure at different scale, showed generally parallel results with the MWS approach. Proportionality is therefore not scale dependent. However, slopes of linear relationships were different for the two scales. Finally, models for prediction of proportional variogram parameters were calculated. Correlation coefficients of relationship between semivariance parameters and mean proved that sill-nugget is more stable (r = 0.74 for P and 0.83 for K) than nugget (r = 0.30 for P and 0.53 for K).

DEVIN: A forecasting approach using stochastic methods applied to the Soufrière Hills Volcano

Time series recorded at active volcanoes are often incomplete and can consist of small data sets. Due to the complexity of volcanic processes and inherent uncertainty, a probabilistic framework is needed for forecasting. A stochastic approach, named DEVIN, was developed to perform forecasts of volcanic activity. DEVIN is a multivariate approach based on geostatistical concepts which enables: (1) detection and quantification of time correlation using variograms, (2) identification of precursors by parameter monitoring and (3) forecasting of specific volcanic events by Monte Carlo methods. The DEVIN approach was applied using seismic data monitored from the Soufrière Hills Volcano (Montserrat). Forecasts were produced for the onset of dome growth with the help of potential precursors identified by monitoring of variogram parameters. Using stochastic simulations of plausible eruptive scenarios, these forecasts were expressed in terms of probability of occurrence. They constitute valuable input data as required by probabilistic risk assessments.

Scale Effects in Estimating the Variogram and Implications for Soil Hydrology

In monitoring and modeling landscape soil processes, the sampling and modeling scales should, ideally, be commensurate with the scales of the soil characteristics of interest. Unfortunately, this is usually not possible, both because the true covariance structure of the variable of interest is unknown a priori and because of logistical constraints. We examine the biases and random errors in variogram parameters that result from the choice of a sample scale triplet (spacing, extent, and support) that is not commensurate with the scales of the underlying “true” soil variability. We generate numerous two‐dimensional random fields, from which we sample data sets randomly or on a grid. We use these samples to estimate the variogram parameters with various methods. The results indicate that both biases and random errors may be large and depend on the sampling scale triplet relative to the scale of the underlying soil variability. The maximum likelihood (ML) method of parameter estimation gives the smallest biases for largely spaced random samples, while the weighted least squares (WLS) method gives the smallest biases for largely spaced gridded samples. Nonparametric estimates exhibit smaller random errors but larger biases than estimates from the two parametric methods (ML, WLS).

Pore structure changes during decomposition of fresh residue: X-ray tomography analyses

The morphology of soil pores is a crucial factor in the understanding of the ecology of soil microorganisms at a small scale. X-ray computer tomography (CT) enables to visualize the soil pore space in three dimensions. We aimed at exploring possible changes in porosity and pore morphology during the incubation of soil to which fresh residue was added. These changes were compared to changes in porosity and pore morphology in field aggregates. A set of CT images is presented from newly formed, incubated and field aggregates from both a sandy loam and a silt loam soil. These aggregates are about 6–8 mm diameter. After incubation of the artificial aggregates, we observed the formation of micro-cracks. At the same time, the total void porosity > 27 μm increased from 3.5% to 7%. The pore size distribution indicated that this extra porosity was present in the 27–67 μm range. Furthermore, it was found that this new pore space was (partly) located near decomposing residue. To quantify the observed changes in pore morphology, we explored the use of the mass fractal dimension and variograms. After correction for differences in porosity, we found that incubation decreased the mass fractal dimension to the level of field aggregates in the sandy loam soil, while no differences were observed in the silt loam soil. In contrast, variogram parameters of the different treatments did not change consistently with the observations in both soils. In conclusion, the data suggest that microbial activity changes the morphology of the pore structure. However, the observed changes in the pore architecture were not sufficient to explain the large differences in aggregate stability among the analyzed samples.

Identification of Heterogeneous Aquifer Transmissivity Using an AE‐Based Method

Determination of hydraulic head, H, as a function of spatial coordinates and time, in ground water flow is the basis for aquifer management and for prediction of contaminant transport. Several computer codes are available for this purpose. Spatial distribution of the transmissivity, T(x,y), is a required input to these codes. In most aquifers, T varies in an erratic manner, and it can be characterized statistically in terms of a few moments: the expected value, the variance, and the variogram. Knowledge of these moments, combined with a few measurements, permits one to estimate T at any point using geostatistical methods. In a review of transmissivity data from 19 unconsolidated aquifers, Hoeksema and Kitanidis (1985) identified two types of the logtransmissivity Y= ln(T) variations: correlated variations with variance sigma2Yc and correlation scale, I(Y), on the order of kilometers, and uncorrelated variations with variance sigma2Yn. Direct identification of the logtransmissivity variogram, Gamma(Y), from measurements is difficult because T data are generally scarce. However, many head measurements are commonly available. The aim of the paper is to introduce a methodology to identify the transmissivity variogram parameters (sigma2Yc, I(Y), and sigma2Yn) using head data in formations characterized by large logtransmissivity variance. The identification methodology uses a combination of precise numerical simulations (carried out using analytic element method) and a theoretical model. The main objective is to demonstrate the application of the methodology to a regional ground water flow in Eagle Valley basin in west-central Nevada for which abundant transmissivity and head measurements are available.

Using geostatistics to quantify seasonal distribution and aggregation patterns of fishes: an example of Atlantic cod (Gadus morhua)

Geostatistical methods were used to (i) quantify fish aggregation patterns over a range of scales (100 m to 67 km) using both simulated and acoustic density data of Atlantic cod (Gadus morhua) and (ii) examine how changes in aggregation patterns influenced the precision of geostatistical density indices. Variogram parameters (range, sill, and nugget) reflected changes in distribution patterns. Variograms of dispersed and low-density aggregations had large range and small sill and nugget values. In contrast, when fish were aggregated in a small portion of the study area, the range was low and the sill and nugget large. The precision of density indices (coefficient of variation) was below 20% in all cases but at a maximum during summer when cod were broadly distributed in small, moderate to dense aggregations. Geostatistical modeling allowed us to describe and quantify distribution patterns of fish density over different scales of observation, comparisons of spatiotemporal changes in density distribution, and estimations of the precision of density indices while accounting for the effects of heterogeneous distributions, outliers and the typically large number of zero and low-density observations. Geostatistical methods have particular applicability to fishes exhibiting gregarious behaviour and seasonally variable distributions, which include many temperate and high-latitude fish species.

Statistical characterization of TEM images of silica‐filled rubber

Transmission electron microscopy is used to study the micro-dispersion of silica fillers within the polymer matrix of rubber. The resulting grey-value images are interpreted as realizations of random fields and are characterized by means of variograms. The so-called Cauchy class is a suitable model for this purpose. Statistical analysis shows that different filler dispersion properties are reflected in different variogram parameters. As a case study, the random field approach is demonstrated for four exemplary rubber compounds.

The spatial variability of mineral nitrogen content in topsoil and subsoil

Spatial variability of soil mineral nitrogen N<sub>min </sub>(N-NH<sub>4</sub><sup>+</sup>, N-NO<sub>3</sub><sup>–</sup>) in a soil profile down to 60 cm was determined in a 19-ha experimental field in Prague-Ruzyně for four years. Winter wheat was grown in the years 2000 and 2001, oats in 2002 and mustard in 2003. Root length distribution and depth of the crops were determined at four locations representing different soil conditions within the experimental field. The coefficient of variation as the measure of the variability of nitrate N in topsoil and subsoil (0–30 and 30–60 cm, respectively) in the experimental years ranged between 18–39 and 20–37%, respectively. It was mostly the same or slightly greater in subsoil than in topsoil in respective years. The variability of ammonium N in topsoil and subsoil ranged between 4–58 and 11–27%, respectively. It was similar in topsoil and subsoil, except for autumn 2000. There was a positive relation between nitrate content in top and subsoil on all sampling terms. We did not find a relationship between N<sub>min</sub> contents in experimental years. Spatial dependencies were evaluated on the basis of model variogram parameters. The nugget value expressed as a percentage value of the total variogram’s sill was used for the class of spatial dependence determination. When a spatial dependence of the observed factor was found it was within a range of medium-strong dependence. Only in two cases a strong spatial dependence was found. A considerable variability was also found out in the variogram’s range, which was between 61 and396 m. All these facts pose a problem for further actions, such as appropriate design of a sampling grid, measured data spatial interpolation and application maps design.

Residual diagnostics for variogram fitting

The variogram cloud is increasingly employed in geostatistics both as an exploratory tool as well as for variogram fitting. Unfortunately in its pure form the variogram cloud cannot be effectively used to assess the fit of a corresponding parametric variogram model. This is due to the fact that its entries are highly correlated and heteroscedastic (and possibly distorted by anisotropy). We propose fitting of a model by feasible generalized least squares, suggest some diagnostics for assessing the fit and identifying potential outliers. For illustrative purposes we apply them to an example data set of chloride concentrations in the Südliche Tullnerfeld in Austria. An R-package called vardiag for the interactive exploration of the proposed diagnostics was provided.

Determining the Spatial Representativeness of Air-Temperature Records Using Variogram-Nugget Time Series

In climatology, spatial representativeness can be measured as the degree to which an instrumental temperature record resolves the climatic variability across an area. Some station records may disproportionately resolve characteristics of their immediate surroundings and, therefore, have less utility in representing climate (or weather) over larger spatial scales. To evaluate spatial representativeness of historical air-temperature records, a modified geostatistical procedure was developed. The procedure measures the spatial representativeness of a station based on variogram parameter estimates—in this case the nugget. This application is novel for two reasons: (1) variogram models are fit to station-specific or "point-centered" semivariance and (2) variogram models are fit to semivariance computed at intervals in a time series. Fitting variograms to point-centered semivariance may be used to create time series of nuggets for each station in a network. Nugget time series may show subsets of a time series that are spatially unrepresentative. These periods could be omitted from climatological analyses without excluding the entire station record. Examples from the central-U.S. subset of the Daily Historical Climatology Network illustrate how anomalous nuggets or step changes in nuggets can be identified relative to station changes.

Three-dimensional mapping of light transmittance and foliage distribution using lidar

The horizontal and vertical distributions of light transmittance were evaluated as a function of foliage distribution using lidar (light detection and ranging) observations for a sugar maple (Acer saccharum) stand in the Turkey Lakes Watershed. Along the vertical profile of vegetation, horizontal slices of probability of light transmittance were derived from an Optech ALTM 1225 instrument's return pulses (two discrete, 15-cm diameter returns) using indicator kriging. These predictions were compared with (i) below canopy (1-cm spatial resolution) transect measurements of the fraction of photosynthetically active radiation (FPAR) and (ii) measurements of tree height. A first-order trend was initally removed from the lidar returns. The vertical distribution of vegetation height was then sliced into nine percentiles and indicator variograms were fitted to them. Variogram parameters were found to vary as a function of foliage height above ground. In this paper, we show that the relationship between ground measurements of FPAR and kriged estimates of vegetation cover becomes stronger and tighter at coarser spatial resolutions. Three-dimensional maps of foliage distribution were computed as stacks of the percentile probability surfaces. These probability surfaces showed correspondence with individual tree-based observations and provided a much more detailed characterization of quasi-continuous foliage distribution. These results suggest that discrete-return lidar provides a promising technology to capture variations of foliage characteristics in forests to support the development of functional linkages between biophysical and ecological studies.

Influence of Spatial Structure on Accuracy of Interpolation Methods

Effectiveness of precision agriculture depends on accurate and efficient mapping of soil properties. Among the factors that most affect soil property mapping are the number of soil samples, the distance between sampling locations, and the choice of interpolation procedures. The objective of this study is to evaluate the effect of data variability and the strength of spatial correlation in the data on the performance of (i) grid soil sampling of different sampling density and (ii) two interpolation procedures, ordinary point kriging and optimal inverse distance weighting (IDW). Soil properties with coefficients of variation (CV) ranging from 12 to 67% were sampled in a 20‐ha field using a regular grid with a 30‐m distance between grid points. Data sets with different spatial structures were simulated based on the soil sample data using a simulated annealing procedure. The strength of simulated spatial structures ranged from weak with nugget to sill (N/S) ratio of 0.6 to strong (N/S ratio of 0.1). The results indicated that regardless of CV values, soil properties with a strong spatial structure were mapped more accurately than those that had weak spatial structure. Kriging with known variogram parameters performed significantly better than the IDW for most of the studied cases (P < 0.01). However, when variogram parameters were determined from sample variograms, kriging was as accurate as the IDW only for sufficiently large data sets, but was less precise when a reliable sample variogram could not be obtained from the data.

Incorporating parameter uncertainty into prediction intervals for spatial data modeled via a parametric variogram

In spatial predictions, researchers usually treat the estimated theoretical variogram parameters as known without error and ignore the variability of the parameter estimators. Although the prediction is still unbiased, the prediction error is usually underestimated. Therefore, the coverage probability of the prediction interval usually is lower than the nominal probability. A simulation study is performed to show how the coverage probability for prediction relates to the true range and sill of an exponential variogram. This article proposes two parametric bootstrap methods to incorporate the variability of the corresponding parameter estimators. A simulation study is performed to evaluate the coverage probability of these proposed methods. Finally, we apply the parametric bootstrap methods to a real dataset and compare the results with those from naive (i.e., treating estimated parameters as known) and Bayesian methods.

Fitting a linear model of coregionalization for soil properties using simulated annealing

It is not simple to model cross and auto-variograms to describe the covariation of two or more soil properties, since the models that are fitted must meet certain constraints. These constraints are most readily met by fitting a linear model of coregionalization (LMCR). This presents practical problems. Not all combinations of authorized variogram functions constitute a LMCR. This paper presents a method for automated fitting of variogram functions to auto and cross-variogram estimates, subject to the constraints of the LMCR. The method uses simulated annealing to minimize a weighted sum of squares between the observed and modelled variograms. The method was applied to some data on soil. It was found to be robust to the initial choice of variogram parameters. Practical methods for setting up a good cooling schedule for the simulated annealing are discussed.

Cokriging optimization of monitoring network configuration based on fuzzy and non-fuzzy variogram evaluation.

A number of optimization approaches regarding monitoring network design and sampling optimization procedures have been reported in the literature. Cokriging Estimation Variance (CEV) is a useful optimization tool to determine the influence of the spatial configuration of monitoring networks on parameter estimations. It was used in order to derive a reduced configuration of a nitrate concentration monitoring well network. The reliability of the reduced monitoring configuration suffers from the uncertainties caused by the variographer's choices and several inherent assumptions. These uncertainties can be described considering the variogram parameters as fuzzy numbers and the uncertainties by means of membership functions. Fuzzy and non-fuzzy approaches were used to evaluate differences among well network configurations. Both approaches permitted estimates of acceptable levels of information loss for nitrate concentrations in the monitoring network of the aquifer of the Plain of Modena, Northern Italy. The fuzzy approach was found to require considerably more computational time and numbers of wells at comparable level of information loss.

Conditioning fractal (fBm/fGn) porosity and permeability fields to multiwell pressure data

Stochastic fractal (fGn and fBm) porosity and permeability fields are conditioned to given variogram, static (or hard), and multiwell pressure data within a Bayesian estimation framework. Because fGn distributions are normal/second-order stationary, it is shown that the Bayesian estimation methods based on the assumption of normal/second-order stationary distributions can be directly used to generate fGn porosity/permeability fields conditional to pressure data. However, because fBm is not second-order stationary, it is shown that such Bayesian estimation methods can be used with implementation of a pseudocovariance approach to generate fBm porosity/permeability fields conditional to multiwell pressure data. In addition, we provide methods to generate unconditional realizations of fBm/fGn fields honoring all variogram parameters. These unconditional realizations can then be conditioned to hard and pressure data observed at wells by using the randomized maximum likelihood method. Synthetic examples generated from one-, two-, and three-dimensional single-phase flow simulators are used to show the applicability of our methodology for generating realizations of fBm/fGn porosity and permeability fields conditioned to well-test pressure data and evaluating the uncertainty in reservoir performance predictions appropriately using these history-matched realizations.