Mapping optimization based on sampling size in earth related and environmental phenomena

A critical sampling grid can be defined for an earth related natural variable distributed in space, according to established theoretical results and under certain mathematical conditions. Sampling above this critical limit does not substantially improve mapping results, while based on this limit the ideal process of reproducing the original phenomenon is theoretically defined. The aim of the present paper is, by using an innovative approach; to investigate the validity of commonly used interpolation algorithms, both stochastic and deterministic, below and above this critical sampling limit. When sampling is dense, application to a simulated spatial random field shows that the results are equally accurate with those derived with more sophisticated stochastic methods. On the other hand, when the sampling grid is sparse, deterministic methods produce less accurate results, therefore stochastic algorithms with minimum estimation error are a much better option. To further demonstrate these points, the interpolation algorithms were applied in three different sampling grid densities in a contaminated waste disposal site in Russia.     

New advances in methodology for statistical tests useful in geostatistical studies

Methodology for statistical procedures to perform tests of hypothesis pertaining to various aspects of geostatistical investigations has been slow in developing. The correlated nature of the data precludes most classical tests and makes the design of new tests difficult. Recent studies have led to modifications of the classical t test which allow for the intercorrelation. In addition, results for certain nonparametric tests have been obtained. The conclusions of these studies provide a variety of new tools for the geostatistician in deciding questions on significant differences and magnitudes.This paper was presented at MGUS 87 Conference, Redwood City, California, 14 April 1987.     

Image fusion by spatially adaptive filtering using downscaling cokriging

The aim of this paper was to extend the method of downscaling cokriging for image fusion by making the method spatially adaptive in that the filter parameters (cokriging weights) can change across the image. The method can adapt itself to the usual statistical non-homogeneity (spatially variable mean, variance and correlation length) of a satellite sensor image that covers an area with different spatial patterns of geographical objects or different terrain types. The solution adopted was to estimate the models of covariances and cross-covariances (or semivariograms and cross-semivariograms) by the same procedure as described in Pardo-Iguzquiza et al. (2006) but with the method applied locally instead of globally. The correct implementation of this local estimation is the key for computational feasibility and prediction efficiency. Two parameters to be taken into account are the grid of locations on which a moving window is centred (local modelling is performed inside this window) and the size of this moving window. With respect to the latter parameter, there is a trade-off between a size small enough to make the procedure locally adaptive and large enough to produce reliable statistical estimates. The computational burden will impose limits to the distance between grid points on which the local moving window is centred. A case study with Landsat ETM+ images was used to show the implementation of the method and the result was evaluated using several statistics widely used for assessing the quality of a fused image, apart from its visual appearance.     

Increasing the accuracy of nitrogen dioxide (NO2) pollution mapping using geographically weighted regression (GWR) and geostatistics

Nitrogen dioxide (NO₂) is known to act as an environmental trigger for many respiratory illnesses. As a pollutant it is difficult to map accurately, as concentrations can vary greatly over small distances. In this study three geostatistical techniques were compared, producing maps of NO₂ concentrations in the United Kingdom (UK). The primary data source for each technique was NO₂ point data, generated from background automatic monitoring and background diffusion tubes, which are analysed by different laboratories on behalf of local councils and authorities in the UK. The techniques used were simple kriging (SK), ordinary kriging (OK) and simple kriging with a locally varying mean (SKlm). SK and OK make use of the primary variable only. SKlm differs in that it utilises additional data to inform prediction, and hence potentially reduces uncertainty. The secondary data source was oxides of nitrogen (NO_x) derived from dispersion modelling outputs, at 1 km × 1 km resolution for the UK. These data were used to define the locally varying mean in SKlm, using two regression approaches: (i) global regression (GR) and (ii) geographically weighted regression (GWR). Based upon summary statistics and cross-validation prediction errors, SKlm using GWR derived local means produced the most accurate predictions. Therefore, using GWR to inform SKlm was beneficial in this study.     

On the prediction error variance of three common spatial interpolation schemes

Three forms of linear interpolation are routinely implemented in geographical information science, by interpolating between measurements made at the endpoints of a line, the vertices of a triangle, and the vertices of a rectangle (bilinear interpolation). Assuming the linear form of interpolation to be correct, we study the propagation of error when measurement error variances and covariances are known for the samples at the vertices of these geometric objects. We derive prediction error variances associated with interpolated values at generic points in the above objects, as well as expected (average) prediction error variances over random locations in these objects. We also place all the three variants of linear interpolation mentioned above within a geostatistical framework, and illustrate that they can be seen as particular cases of Universal Kriging (UK). We demonstrate that different definitions of measurement error in UK lead to different UK variants that, for particular expected profiles or surfaces (drift models), yield weights and predictions identical with the interpolation methods considered above, but produce fundamentally different (yet equally plausible from a pure data standpoint) prediction error variances.     

Geostatistical modeling of clay spatial distribution in siliciclastic rock samples using the plurigaussian simulation method

In order to implement secondary and enhanced oil recovery processes in complex terrigenous formations as is usual in turbidite deposits, a precise knowledge of the spatial distribution of shale grains is a crucial element for the fluid flow prediction. The reason of this is that the interaction of water with shale grains can significantly modify their size and/or shape, which in turn would cause porous space sealing with the subsequent impact in the flow. In this work, a methodology for stochastic simulations of spatial grains distributions obtained from scanning electron microscopy images of siliciclastic rock samples is proposed. The aim of the methodology is to obtain stochastic models would let us investigate the shale grain behavior under various physico-chemical interactions and flux regimes, which in turn, will help us get effective petrophysical properties (porosity and permeability) at core scale. For stochastic spatial grains simulations a plurigaussian method is applied, which is based on the truncation of several standard Gaussian random functions. This approach is very flexible, since it allows to simultaneously manage the proportions of each grain category in a very general manner and to rigorously handle their spatial dependency relationships in the case of two or more grain categories. The obtained results show that the stochastically simulated porous media using the plurigaussian method adequately reproduces the proportions, basic statistics and sizes of the pore structures present in the studied reference images.     

Geotechnical modeling at the city scale using statistical and geostatistical tools: The Pessac case (France)

In the context of the RIVIERA project, the building of a 3D geotechnical model at the city scale (Pessac, France) has been undertaken, from several hundreds of boreholes and geotechnical tests. It is first shown how the combination of the lithological information and of geotechnical results can improve thanks to Bayesian statistics the knowledge of mechanical characteristics in the various alluvial terraces which can be encountered in this area. Secondly the upper and lower limits of the 3D model at the city scale are computed by improving an initial digital elevation model for the upper limit and by kriging under inequality constraints for the lower limit. These limits border Quaternary formations which are of interest for geotechnical applications. In a third stage, it is focused on the spatial modelling of the pressuremeter modulus. The sequential indicator simulation method enables to obtain the spatial probability of occurrence of a given pressiometer modulus class. Coupled with other information, the analysis of these statistical and geostatistical models makes possible to develop decision support tools such as to localise, for instance, areas more prone to the clay shrinkage–swelling hazard.     

Hierarchical Geostatistical Analysis of an Experimental Stratigraphy

A hierarchical geostatistical analysis is conducted on a high-resolution, multiscale hydraulic conductivity (ln K) map, created by scaling up an experimental stratigraphy. Unlike a previous study which evaluates ln K variograms within individual depositional environments, this study analyzes deposits (or samples) that incorporate multiple depositional environments. Based on conductivity cutoffs selected from a global ln K histogram, an indicator map is created to divide the deposits into 4 categories: sand, silty sand, clayey silt, and clay (Hierarchy-I). Based on facies and facies assemblage types selected using geological criteria, two more indicator maps are created at a higher hierarchy (Hierarchy-II) to divide the deposits into 14 units and 2 units, respectively. For each sample, its experimental ln K variogram is decomposed into 4 auto- and cross-transition component variograms. The decomposition characteristics are then evaluated against the underlying heterogeneity and specific division rule. The analysis reveals that: (1) ln K cutoffs (sand contents of the physical stratigraphy) can be used to distinguish the shifts in dominant deposition mode; (2) sample univariate modes depend on the choice of hierarchical division; (3) sample variograms exhibit smooth-varying correlation structures (exponential-like variograms are observed in samples with a large variance in mean facies length); (4) the decomposition characteristics are sensitive to the division based on conductivity cutoffs, but not sensitive to the division based on depositional environment (For all samples, with appropriate division, the sample variogram is closely approximated by the sum of the cross-transition component variograms.); and (5) at the Hierarchy-II level, the 2-unit division gives similar decomposition characteristics as the 14-unit division. For the select samples, parsimony in hierarchical division is achieved at the facies assemblage scale.     

Estimating the effects of spatial variability of infiltration on the output of a distributed runoff and soil erosion model using Monte Carlo methods

Monte Carlo procedures were used to evaluate the effects of spatial variations in the values of the infiltration parameter on the results of the ANSWERS distributed runoff and erosion model. Simulation results obtained were compared with measured values. Field infiltration measurements indicated spatial correlation at much smaller distances than the size of an element. Therefore, at first only the error of the mean had to be taken into consideration for block infiltration rates. Consequently, not only single hydrographs were produced, but also error bands. Secondly, nine other hypothetical spatial correlation structures were also evaluated using Monte Carlo methods. in particular at low nugget variances, increasing spatial correlation of infiltration resulted in increasing coefficients of variation in model outputs. In general, rainstorms with low rainfall intensities were more difficult to simulate accurately than extreme events with high rainfall intensities. This is explained by the greater influence of the infiltration uncertainties at low rainfall intensities.