Denis Marcotte

Multivariable variogram and its application to the linear model of coregionalization

In this article, we present the multivariable variogram, which is defined in a way similar to that of the traditional variogram, by the expected value of a distance, squared, in a space withp dimensions. Combined with the linear model of coregionalization, this tool provides a way for finding the elementary variograms that characterize the different spatial scales contained in a set of data withp variables. In the case in which the number of elementary components is less than or equal to the number of variables, it is possible, by means of nonlinear regression of variograms and cross-variograms, to estimate the coregionalization parameters directly in order to obtain the elementary variables themselves, either by cokriging or by direct matrix inversion. This new tool greatly simplifies the procedure proposed by Matheron (1982) and Wackernagel (1985). The search for the elementary variograms is carried out using only one variogram (multivariable), as opposed to thep(p + 1)/2 required by the Matheron approach. Direct estimation of the linear coregionalization model parameters involves the creation of semipositive definite coregionalization matrices of rank 1.

Estimation of hydraulic conductivity of an unconfined aquifer using cokriging of GPR and hydrostratigraphic data

Abstract Densely sampled geophysical data can supplement hydrogeological data for estimating the spatial distribution of porosity and hydraulic conductivity over an aquifer. A 3D Ground Penetrating Radar (GPR) survey was performed over a shallow unconfined aquifer consisting of a coarse to medium sand sequence overlying an impermeable clay layer. The site is instrumented with piezometers and water levels are frequently monitored. Vertical determination of moisture and granulometry at a resolution of 10 cm were made at a few locations. The GPR reflection times were correlated with piezometric and stratigraphic information; cokriging of both data yields the spatial distribution of the radar velocities within the layers. Porosity and hydraulic conductivities are estimated using the Complex Refractive Index Method (CRIM) and Kozeny–Carman formulations, respectively. A pumping test and a tracer test, both done using a well in the center of the survey zone, provide a measure of the average hydraulic conductivity and its anisotropy. The results from cokriging in the saturated zone show that the estimated parameters agree very well with the measured hydrogeological data. The geometric mean of the porosity is close to the laboratory measurements. The geometric mean of the GPR-derived hydraulic conductivities fits the values obtained from the pumping and tracer tests. The range of estimated hydraulic conductivities is quite large and indicates that flow could be faster or slower than the one predicted from the pumping test in some places. Radar attenuation is also found to be a good indicator of porosity distribution. From the observed (high) GPR attenuations and electrical conductivities of water sampled in the piezometers, porosity is determined using Archies formula. In the vadose zone, moisture content estimated from the GPR velocities using either CRIM or Topp formulations agree well with the ones from the laboratory measurements. Cokriging of the radar reflection times and of the hydrogeological/stratigraphic data leads to an accurate estimate of the radar velocities with a precision and a spatial resolution much higher than the CDP technique. Within the limits of the interpretative models, porosity, saturation and hydraulic conductivities can accurately be estimated with a high spatial resolution over the survey zone.

Cokriging with matlab

A program termed COKRI is presented, which will perform point or block kriging or cokriging in any number of dimensions, with any number of variables and basic structures. Different forms of cokriging are offered by COKRI: simple, ordinary with one nonbias condition or with the usual p (number of variables) nonbias conditions, and universal cokriging with drift of order 1 or 2. Factorial kriging or cokriging also can be performed. All the basic structures can have different geometric anisotropies thus allowing great modeling flexibility. The addition of a new basic model to the five currently offered by COKRI requires only one line of code. The program operates within the user-friendly Matlab environment.

Theoretical and Experimental Performance of Spatial Interpolation Methods for Soil Salinity Analysis

Interpolation methods are required for analysis of soil salinity data by geographic information systems. This study was conducted to determine interpolation methods that are best suited to map soil salinity. Methods of closest neighbor, kriging, inverse-distance moving average, and thin plate smoothing splines were compared by cross-validation for precision and smoothing, using 341 measured values of electrical conductivity of saturated paste extract in a 16 000 ha area in southwest Iran. Interpolation precision of all methods were low, with a mean absolute difference between measured and predicted values ranging from 42 to 76% of the mean measured soil salinity. This was due to the large ratio of nugget effect to the sill of the variogram and to the high variability of data. Thin plate smoothing splines and ordinary kriging were the most precise methods, whereas closest neighbor was the least precise. The smoothing of the methods was assessed by comparing the dispersion standard deviation of interpolated values with that of observed values. The most precise methods were also those that performed an important smoothing. Ordinary kriging and thin plate smoothing splines produced contour maps that were much easier to interpret. A theoretical analysis of the performance of the methods (precision and smoothing) led to conclusions similar to those based on the cross-validation study. Such a theoretical analysis can be used to select an appropriate interpolation method without the need for time consuming cross-validation.

Variance and spatial scales in a tropical rain forest: changing the size of sampling units

The size of a sampling unit has a critical effect on our perception of ecological phenomena; it influences the variance and correlation structure estimates of the data. Classical statistical theory works well to predict the changes in variance when there is no autocorrelation structure, but it is not applicable when the data are spatially autocorrelated. Geostatistical theory, on the other hand, uses analytical relationships to predict the variance and autocorrelation structure that would be observed if a survey was conducted using sampling units of a different size. To test the geostatistical predictions, we used information about individual tree locations in the tropical rain forest of the Pasoh Reserve, Malaysia. This allowed us to simulate and compare various sampling designs. The original data were reorganised into three artificial data sets, computing tree densities (number of trees per square meter in each quadrat) corresponding to three quadrat sizes (5×5, 10×10 and 20×20 m(2)). Based upon the 5×5 m(2) data set, the spatial structure was modelled using a random component (nugget effect) plus an exponential model for the spatially structured component. Using the within-quadrat variances inferred from the variogram model, the change of support relationships predicted the spatial autocorrelation structure and new variances corresponding to 10×10 m(2) and 20×20 m(2) quadrats. The theoretical and empirical results agreed closely, while the classical approach would have largely underestimated the variance. As quadrat size increases, the range of the autocorrelation model increases, while the variance and proportion of noise in the data decrease. Large quadrats filter out the spatial variation occurring at scales smaller than the size of their sampling units, thus increasing the proportion of spatially structured component with range larger than the size of the sampling units.

Fast variogram computation with FFT

Abstract Two programs are presented to compute direct- and cross-variograms, direct and cross-covariograms, and pseudo-cross-variograms. The programs are written in MATLAB and are based on the Fast Fourier Transform algorithm (FFT). The programs accept complete, or incomplete, regular grid data. The FFT appoach is shown to be faster than the spatial approach for this type of data. It gives exactly the same numerical variogram values as programs operating in the spatial domain. These programs could be most useful in image analysis, where images are usually 256 × 256 pixels, 512 × 512 pixels, or larger. For such large images, FFT is many orders of magnitude faster than the spatial approach.

Comparison of approaches to spatial estimation in a bivariate context

The problem of estimating a regionalized variable in the presence of other secondary variables is encountered in spatial investigations. Given a context in which the secondary variable is known everywhere (or can be estimated with great precision), different estimation methods are compared: regression, regression with residual simple kriging, kriging, simple kriging with a mean obtained by regression, kriging with an external drift, and cokriging. The study focuses on 19 pairs of regionalized variables from five different datasets representing different domains (geochemical, environmental, geotechnical). The methods are compared by cross-validation using the mean absolute error as criterion. For correlations between the principal and secondary variable under 0.4, similar results are obtained using kriging and cokriging, and these methods are superior slightly to the other approaches in terms of minimizing estimation error. For correlations greater than 0.4, cokriging generally performs better than other methods, with a reduction in mean absolute errors that can reach 46% when there is a high degree of correlation between the variables. Kriging with an external drift or kriging the residuals of a regression (SKR) are almost as precise as cokriging.

3D stochastic inversion of gravity data using cokriging and cosimulation

A new application has been developed, based on geostatistical techniques of cokriging and conditional simulation, for the 3D inversion of gravity data including geologic constraints. The necessary gravity, density, and gravity-density covariance matrices are estimated using the observed gravity data. Then the densities are cokriged or simulated using the gravity data as the secondary variable. The model allows noise to be included in the observations. The method is applied to two synthetic models: a short dipping dike and a stochastic distribution of densities. Then some geologic information is added as constraints to the cokriging system. The results show the ability of the method to integrate complex a priori information. The survey data of the Matagami mining camp are considered as a case study. The inversion method based on cokriging is applied to the residual anomaly to map the geology through the estimation of the density distribution in this region. The results of the inversion and simulation methods are in good agreement with the surface geology of the survey region.

The multivariate (co)variogram as a spatial weighting function in classification methods

The multivariate variogram and the multivariate covariogram are used as spatial weighting functions for forming spatially homogeneous groups automatically. The groups are created after either deflating similarities between distant samples with the multivariate covariogram or by inflating dissimilarities between distant samples with the multivariate variogram. These approaches can be seen as generalization of the Oliver and Webster proposal. Two data sets show the efficiency of the two weighting functions when compared to the classical approach which does not take spatial information into account. In one case study, the weighting of similarities by the multivariate covariogram showed more interpretable results than the weighting of dissimilarities by the multivariate variogram.

Mapping clay content for subsurface drainage in the Nile delta

Abstract Drain envelope requirements for large-scale subsurface drainage projects are often estimated based on the surface area with clay percentages less than a given threshold value. Contouring clay percentage data can be done by computer using spatial interpolation methods. Cross-validation was used to evaluate the precision of estimation for clay percentage using five interpolation techniques (closest neighbor, moving average, weighted moving average, kriging, and cokriging) for a study area of 33,500 ha in the Nile Delta. The clay content was measured at 485 sites. Cokriging used the correlation between clay percentage and saturated hydraulic conductivity, which had been measured at 3488 sites. The lowest mean absolute difference (MAD) between observed and interpolated clay percentage values was obtained for cokriging (12.01). MAD values were very close to that of cokriging in the case of moving average (12.13), weighted moving average (12.23), and kriging (12.21), whereas for the closest neighbor method a MAD value of 16.44 was obtained. Surface areas under 30 and 40 percent clay were over-estimated for the closest neighbor method, and under-estimated for the moving average method. Surface areas and contour maps based on weighted moving average, kriging, and cokriging were very similar. Use of moving average resulted in a loss of precision due to the smoothing effect.