Uncertainty as the overlap of alternate conditional distributions

An important task in modern geostatistics is the assessment and quantification of resource and reserve uncertainty. This uncertainty is valuable support information for many management decisions. Uncertainty at specific locations and uncertainty in the global resource is of interest. There are many different methods to build models of uncertainty, including Kriging, Cokriging, and Inverse Distance. Each method leads to different results. A method is proposed to combine local uncertainties predicted by different models to obtain a combined measure of uncertainty that combines good features of each alternative. The new estimator is the overlap of alternate conditional distributions.     

Size Scaling of Cross Correlation Between Multiple Variables

Reservoir models have large uncertainty because of spatial variability and limited sample data. The ultimate aim is to use simultaneously all available data sources to reduce uncertainty and provide reliable reservoir models for resource assessment and flow simulation. Seismic impedance or some other attribute provides a key source of data for reservoir modeling. These seismic data are at a coarser scale than the hard well data and it not an exact measurement of facies proportions or porosity. A requirement for data integration is the cross-covariance between the well and seismic data.The size-scaling behavior of the cross correlation for different measurement scales was nvestigated. The size-scaling relationship is derived theoretically and validated by numerical studies (including an example with real data). The limit properties of the cross-correlation coefficient when the averaging volume becomes large is shown. After some averaging volume, the volume-dependent cross-correlation coefficient reaches a limit value. This plateau value is controlled mainly by the large-scale behavior of the cross and direct variograms.The cross correlation can increase or decrease with volume support depending on the relative importance of long- and short-scale covariance structures. If the direct and cross variograms are proportional, there is no change in the cross correlation as the averaging volume changes. Our study shows that the volume-dependent cross-correlation coefficient is sensitive to the shape of the cross variogram and differences between the direct variograms of the well data and seismic data.     

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.     

Indicator Simulation Accounting for Multiple-Point Statistics

Geostatistical simulation aims at reproducing the variability of the real underlying phenomena. When nonlinear features or large-range connectivity is present, the traditional variogram-based simulation approaches do not provide good reproduction of those features. Connectivity of high and low values is often critical for grades in a mineral deposit. Multiple-point statistics can help to characterize these features. The use of multiple-point statistics in geostatistical simulation was proposed more than 10 years ago, on the basis of the use of training images to extract the statistics. This paper proposes the use of multiple-point statistics extracted from actual data. A method is developed to simulate continuous variables. The indicator kriging probabilities used in sequential indicator simulation are modified by probabilities extracted from multiple-point configurations. The correction is done under the assumption of conditional independence. The practical implementation of the method is illustrated with data from a porphyry copper mine.     

Transformation of Residuals to Avoid Artifacts in Geostatistical Modelling with a Trend

Trend modelling is an important part of natural resource characterization. A common approach to account for a variable with a trend is to decompose it into a relatively smoothly varying trend and a more variable residual component. Then, the residuals are stochastically modelled independent of the trend. This decomposition can result in values outside the plausible range of variability, such as grades below zero or ratios that exceed 1.0. We transform the residuals conditional to the trend component to explicitly remove these complex features prior to geostatistical modelling. Back transformation of the modelled residual values allows the complex relations to be reproduced. A petroleum-related application shows the robustness of the proposed transformation. Furthermore, a mining application shows that when this conditional transformation is applied to the original variable, instead of the residual, simulated values are assured to be nonnegative.     

The domain structure of synthetic stoichiometric TM10–TM75 and Al-, Mg-, Mn- and V-doped TM62 titanomagnetites

The existence of domain structure has been questioned for titanomagnetites of typical oceanic basalt composition owing to the unusual temperature dependence of their susceptibility, resembling that of spin glasses. In order to make a direct test of domain structure, a series of stoichiometric titanomagnetites between magnetite (TM0) and 75% ulvöspinel content (TM75) as well as a titanomagnetite of typical oceanic basalt composition have been synthesised using the double-sintering technique at 1300°C, in controlled atmospheres. The purity, stoichiometry and homogeneity of these materials were tested by optical, X-ray and microprobe studies as well as by magnetic measurements.Domain structures were observed using the Bitter-pattern technique after ionic polishing to produce stress-free surface of the bulk material. The optimum time required for ionic polishing was found to increase with the ulvöspinel content and to be correlated with the magnetostrictive constant θ. Magnetite showed a domain configuration which is also typical for nickel (mostly lamella-shaped domains, pine-tree-shaped closure domains, high domain wall mobility in small external fields, straight domain walls). The tendency to form lamella-shaped domains is present up to TM75 (which has a Curie temperature of only 40°C), but with an increasing tendency to form curved domain walls and to have fewer and also differently shaped closure domains. This is demonstrated in a series of photographs. The results constitute unequivocal evidence for the existence of a domain structure in the classical sense in a broad range of stoichiometric pure and doped (Al, Mg, Mn, V) titanomagnetites.     

Estimating urban above ground biomass with multi-scale LiDAR

Background

Urban trees have long been valued for providing ecosystem services (mitigation of the “heat island” effect, suppression of air pollution, etc.); more recently the potential of urban forests to store significant above ground biomass (AGB) has also be recognised. However, urban areas pose particular challenges when assessing AGB due to plasticity of tree form, high species diversity as well as heterogeneous and complex land cover. Remote sensing, in particular light detection and ranging (LiDAR), provide a unique opportunity to assess urban AGB by directly measuring tree structure. In this study, terrestrial LiDAR measurements were used to derive new allometry for the London Borough of Camden, that incorporates the wide range of tree structures typical of an urban setting. Using a wall-to-wall airborne LiDAR dataset, individual trees were then identified across the Borough with a new individual tree detection (ITD) method. The new allometry was subsequently applied to the identified trees, generating a Borough-wide estimate of AGB.

Results

Camden has an estimated median AGB density of 51.6 Mg ha^–1 where maximum AGB density is found in pockets of woodland; terrestrial LiDAR-derived AGB estimates suggest these areas are comparable to temperate and tropical forest. Multiple linear regression of terrestrial LiDAR-derived maximum height and projected crown area explained 93% of variance in tree volume, highlighting the utility of these metrics to characterise diverse tree structure. Locally derived allometry provided accurate estimates of tree volume whereas a Borough-wide allometry tended to overestimate AGB in woodland areas. The new ITD method successfully identified individual trees; however, AGB was underestimated by ≤?25% when compared to terrestrial LiDAR, owing to the inability of ITD to resolve crown overlap. A Monte Carlo uncertainty analysis identified assigning wood density values as the largest source of uncertainty when estimating AGB.

Conclusion

Over the coming century global populations are predicted to become increasingly urbanised, leading to an unprecedented expansion of urban land cover. Urban areas will become more important as carbon sinks and effective tools to assess carbon densities in these areas are therefore required. Using multi-scale LiDAR presents an opportunity to achieve this, providing a spatially explicit map of urban forest structure and AGB.

     

Practical Incorporation of Multivariate Parameter Uncertainty in Geostatistical Resource Modeling

There is a need to bridge theory and practice for incorporating parameter uncertainty in geostatistical simulation modeling workflows. Simulation workflows are a standard practice in natural resource and recovery modeling, but the incorporation of multivariate parameter uncertainty into those workflows is challenging. However, the objectives can be met without considerable extra effort and programming. The sampling distributions of statistics comprise the core theoretical notion with the addition of the spatial degrees of freedom to account for the redundancy in the spatially correlated data. Prior parameter uncertainty is estimated from multivariate spatial resampling. Simulation-based transfer of prior parameter uncertainty results in posterior distributions which are updated by data conditioning and the model domain extents and configuration. The results are theoretically tractable and practical to achieve, providing realistic assessments of uncertainty by accounting for large-scale parameter uncertainty, which is often the most important component impacting a project. A simulation-based multivariate workflow demonstrates joint modeling of intrinsic shale properties and uncertainty in estimated ultimate recovery in a shale gas project. The multivariate workflow accounts for joint prior parameter uncertainty given the current well locations and results in posterior estimates on global distributions of all modeled properties. This is achieved by transferring the joint prior parameter uncertainty through conditional simulations.