Optimizing interpolation of shoot density data from a Posidonia oceanica seagrass bed

A case study on the optimization of Posidonia oceanica density interpolation, using a data set from a large meadow at Porto Conte Bay (NW Sardinia, Italy), is presented. Ordinary point kriging, cokriging and a weighted average based on inverse square distance were used to interpolate density data measured in 36 sampling stations. The results obtained from different methods were then compared by means of a leave‐one‐out cross‐validation procedure. The scale at which interpolation was carried out was defined on the basis of the Hausdorff dimension of the variogram. Optimizing spatial scale and data points search strategy allowed obtaining more accurate density estimates independently of the interpolation method.     

On the geologic structure of the Explora Escarpment (Weddell Sea,Antarctica) revealed by satellite and Shipboard data evaluation

Bathymetric, gravity, and magnetic data from Antarctic expeditions with RV POLARSTERN and satellite altimeter data from the Geosat Geodetic Mission are analysed using methods from geostatistics and geophysical inverse theory.The Explora Escarpment represents the edge between the Antarctic Continental Shelf and the Weddell Abyssal Plain. It is an important link in the reconstruction of Gondwana breakup, but a feature as large as the 2000 m deep Wegener Canyon was only discovered in 1984, when extensive bathymetric, gravimetric, and magnetic surveys with RV POLARSTERN began.Geostatistics, the theory of regionalized variables, is applied to integrate dense surveys of Wegener Canyon and sparse observations in adjacent areas into maps with full coverage of the 230 km by 330 km area at 10°–20° W/70°–72° S. The resultant highresolution bathymetric and gravity maps reveal detailed structures of the Explora Escarpment. Using geophysical inversion, the gravity terrain effect is calculated. Satellite data are used for their better coverage, but have much lower resolution. Nevertheless, the structures of Wegener Canyon and other more prominent features appear with surprisingly good correlation also in the Geosat altimeter data. While it was initially supposed that Wegener Canyon is purely an erosional structure, the magnetic map now provides evidence of the canyon's tectonic origin.     

Modelling Positional Uncertainty of Line Features by Accounting for Stochastic Deviations from Straight Line Segments

The assessment of positional uncertainty in line and area features is often based on uncertainty in the coordinates of their elementary vertices which are assumed to be connected by straight lines. Such an approach disregards uncertainty caused by sampling and approximation of a curvilinear feature by a sequence of straight line segments. In this article, a method is proposed that also allows for the latter type of uncertainty by modelling random rectangular deviations from the conventional straight line segments. Using the model on a dense network of sub‐vertices, the contribution of uncertainty due to approximation is emphasised; the sampling effect can be assessed by applying it on a small set of randomly inserted sub‐vertices. A case study demonstrates a feasible way of parameterisation based on assumptions of joint normal distributions for positional errors of the vertices and the rectangular deviations and a uniform distribution of missed sub‐vertices along line segments. Depending on the magnitudes of the different sources of uncertainty, not accounting for potential deviations from straight line segments may drastically underestimate the positional uncertainty of line features.     

Hierarchical decomposition of variance with applications in environmental mapping based on satellite images

A quadtree-based image segmentation procedure (HQ) is presented to map complex environmental conditions. It applies a hierarchical nested analysis of variance within the framework of multiresolution wavelet approximation. The procedure leads to an optimal solution for determining mapping units based on spatial variability with constraints on the arrangement and shape of the units. Linkages to geostatisiics are pointed out, but the HQ decomposition algorithm does not require any homogeneity criteria. The computer implementation can be parameterized by either the number of required mapping units or the maximum within-unit variance, or it can provide a spectrum of significances of nested ANOVA. The detailed mathematical background and methodology is illustrated by a salt-affected grassland mapping study (Hortobágy, Hungary), where heterogeneous environmental characteristics have been sampled and predicted based on remotely sensed images using these principles.     

Efficient generation of conditional simulations by chebyshev matrix polynomial approximations to the symmetric square root of the covariance matrix

Consider the problem of generating a realization y₁ of a Gaussian random field on a dense grid of points ₁ conditioned on field observations y₂ collected on a sparse grid of points ₂. An approach to this is to generate first an unconditional realization y over the grid =_{1 2}, and then to produce y₁ by conditioning y on the data y₂. As standard methods for generating y, such as the turning bands, spectral or Cholesky approaches can have various limitations, it has been proposed by M. W. Davis to generate realizations from a matrix polynomial approximations to the square root of the covariance matrix. In this paper we describe how to generate a direct approximation to the conditional realization y₁, on ₁ using a variant of Davis' approach based on approximation by Chebyshev polynomials. The resulting algorithm is simple to implement, numerically stable, and bounds on the approximation error are readily available. Furthermore we show that the conditional realization y₁ can be generated directly with a lower order polynomial than the unconditional realization y, and that further reductions can be achieved by exploiting a nugget effect if one is present. A pseudocode version of the algorithm is provided that can be implemented using the fast Fourier transform if the field is stationary and the grid ₁ is rectangular. Finally, numerical illustrations are given of the algorithm's performance in generating various 2-D realizations of conditional processes on large sampling grids.     

Spatial analysis of elements in soils from Chengdu City

Three kinds of spatial analysis methods （geostatistics, concentration-area fractal model and the multifractal analysis called the moment method） were used for almost 50 elements, including heavy metals, disperse elements, rare elements and even others, in 6586 top soil （0-20 cm） samples and 1833 deep soil （150-200 cm） samples from Chengdu metropolitan area of 12400 km^2, southwestern China. The ranges of spatial correlation revealed by variograms are quite different for different kinds of elements in the top and deep soils. The most interest is the fact that the multifractal spectra of environmentally important elements such as Pb, Cr, Cd and Ni in top soils in the metropolitan area show systematic change from those in the deep soils, revealing a strong anthropogenic addition, while Hg, Zn, As, Cu and all common elements show no such kind of addition. In terms of multifractal properties based on the multifractal spectrum curves, those disperse and rare elements show great deviation from other major and trace elements, which is also of great interest.     

Optimization of SAGD Well Elevation

The application of steam-assisted gravity drainage (SAGD) to recover heavy oil sands is becoming increasingly important in the northern Alberta McMurray Formation because of the vast resources/reserves accessible with this mechanism. Selecting the stratigraphic elevations of SAGD well pairs is a vital decision for reservoir evaluation and planning. The inherent uncertainty in the distribution of geological variables significantly influences this decision. Geostatistical simulation is used to capture geological uncertainty, which is used can be transformed into a distribution of the best possible well pair elevations. A simple exhaustive calculation scheme is used to determine the optimum stratigraphic location of a SAGD well pair where the recovery R is maximized. There are three basic steps to the methodology: (1) model the uncertainty in the top continuous bitumen (TCB) and bottom continuous bitumen (BCB) surfaces, (2) calculate the recovery at all possible elevation increments within the TCB and BCB interval, and (3) identify the elevation that maximizes R. This is repeated for multiple TCB/BCB pairs of surfaces to assess uncertainty. The methodology is described and implemented on a subset of data from the Athabasca Oilsands in Fort McMurray, Alberta.     

Carbon dioxide and radon gas hazard in the Alban Hills area (central Italy)

The sudden and catastrophic, or slow and continuous, release at surface of naturally occurring toxic gases like CO₂, H₂S and Rn poses a serious health risk to people living in geologically active regions. In general this problem receives little attention from local governments, although public concern is raised periodically when anomalous toxic-gas concentrations suddenly kill humans or livestock. For example, elevated CO₂ concentrations have been linked to the death of at least 10 people in the central Italian region of Lazio over the last 20 years, while it was the CO₂ asphyxiation of 30 cows in a heavily populated area near Rome in 1999 which prompted the present soil-gas study into the distribution of the local health risk. A detailed geochemical survey was carried out in an area of about 4 km² in the Ciampino and Marino districts, whereby a total of 274 soil-gas samples were collected and analysed for more than 10 major and trace gas species. Data were then processed using both statistical and geostatistical methods, and the resulting maps were examined in order to highlight areas of elevated risk. General trends of elevated CO₂ and Rn concentrations imply the presence of preferential pathways (i.e. faults and fractures) along which deep gases are able to migrate towards the surface. The CO₂ and Rn anomalous trends often correspond to and are usually elongated parallel to the Apennine mountain range, the controlling structural feature in central Italy. Because of this fundamental anisotropy in the factors controlling the soil-gas distribution, it was found that a geostatistical approach using variogram analysis allowed for a better interpretation of the data. With regard to the health risk to local inhabitants, it was found that although some high risk areas had been zoned as parkland, others had been heavily developed for residential purposes. For example, many new houses were found to have been built on ground which has soil-gas CO₂ concentrations of more than 70% and radon values of more than 250 kBq m⁻³. It is recommended that land-use planners incorporate soil-gas and/or gas flux measurements in environmental assessments in areas of possible risk (i.e. volcanic or structurally active areas).     

Stepwise Conditional Transformation for Simulation of Multiple Variables

Most geostatistical studies consider multiple-related variables. These relationships often show complex features such as nonlinearity, heteroscedasticity, and mineralogical or other constraints. These features are not handled by the well-established Gaussian simulation techniques. Earth science variables are rarely Gaussian. Transformation or anamorphosis techniques make each variable univariate Gaussian, but do not enforce bivariate or higher order Gaussianity. The stepwise conditional transformation technique is proposed to transform multiple variables to be univariate Gaussian and multivariate Gaussian with no cross correlation. This makes it remarkably easy to simulate multiple variables with arbitrarily complex relationships: (1) transform the multiple variables, (2) perform independent Gaussian simulation on the transformed variables, and (3) back transform to the original variables. The back transformation enforces reproduction of the original complex features. The methodology and underlying assumptions are explained. Several petroleum and mining examples are used to show features of the transformation and implementation details.