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.     

Aqueous alteration of the Bali CV3 chondrite: evidence from mineralogy, mineral chemistry, and oxygen isotopic compositions

A petrographic, geochemical, and oxygen isotopic study of the Bali CV3 carbonaceous chondrite revealed that the meteorite has undergone extensive deformation and aqueous alteration on its parent body. Deformation textures are common and include flattened chondrules, a well-developed foliation, and the presence of distinctive (100) planar defects in olivine. The occurrence of alteration products associated with the planar defects indicates that the deformation features formed prior to the episode of aqueous alteration. The secondary minerals produced during the alteration event include well-crystallized Mg-rich saponite, framboidal magnetite, and Ca-phosphates. The alteration products are not homogeneously distributed throughout the meteorite, but occur in regions adjacent to relatively unaltered material, such as veins of altered material following the foliation. The alteration assemblage formed under oxidizing conditions at relatively low temperatures (<100 degrees C). Altered regions in Bali have higher Na, Ca, and P contents than unaltered regions which suggests that the fluid phase carried significant dissolved solids. Oxygen isotopic compositions for unaltered regions in Bali fall within the field for other CV3 whole-rocks, however, the oxygen isotopic compositions of the heavily altered material lie in the region for the CM and CR chondrites. The heavy-isotope enrichment of the altered regions in Bali suggest alteration conditions similar to those for the petrographic type-2 carbonaceous chondrites.     

Solar system isotopic anomalies: Supernova neighbor or presolar carriers?

I evaluate several nuclear and chemical problems related both to the recent scenario suggesting that the known isotopic anomalies in the solar system have resulted from a supernova near the protosolar nebula and to the model of extinct presolar carriers. Major features include: (1) Large quantities of extinct ²⁴⁸Cm and ³⁶Cl are predicted from the Cameron-Truran model of a minor injection about 10⁶ yr before condensation; (2) an extinct-carrier model of ²⁶Mg is set forth in detail with a solid chemistry picture of the early solar system; (3) a major thermonuclear supernova responsible for ²⁶Al, ²⁴⁴Pu, and ⁴⁰K would have to have occurred several million years (3 m.y.) before condensation and contributed a large fraction of the major stable chemical elements; (4) carbon isotope families are to be expected if the oxygen isotope families are due to a late injection of ¹⁶O; (5) the Earth and E meteorites may have condensed primarily in a carbon-rich nebula existing before admixtures of a major late ¹⁶O-rich mixture; (6) the extinct-presolar-carrier model is the single best explanation of all anomalies.     

Neotectonics, sea-level changes and biological evolution in the Fennoscandian Border Zone of the southern Kattegat Sea

Shallow seismic data and vibrocore information, sequence stratigraphic and faunal evidence have been used for documentation of Late Weichselian reactivation of faulting in the south central Kattegat, southern Scandinavia. The study area is situated on the Fennoscandian Border Zone, where tectonic activity has been recurrent since Early Palaeozoic time and still occurs, as shown by present earthquake activity. New data from the area south of the island of Anholt show that after deglaciation fast isostatic rebound resulted in reactivation of a NW-SE striking normal fault system. This tectonic episode is dated to a period starting shortly before 15.0 cal. ka BP and ending around 13.5 cal. ka BP, after regression had already reached a level of about 30 m b.s.l. The vertical displacement associated with the faulting was in the order of 20 m. More generally, the results support the previously reported late Weichselian sea-level highstand, which was followed by forced regression until the eustatic sea-level rise surpassed the rate of glacio-isostatic rebound in early Preboreal. Our findings further imply that drainage of the Baltic Ice Lake through the Øresund at c. 15 cal. ka BP (Bergsten & Nordberg 1992) may have been triggered by tectonic activity in this region.     

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.     

Oxygen isotope studies of the geothermal system at Wairakei,New Zealand

Volcanic rocks in the Wairakei geothermal field have undergone extensive oxygen isotope exchange with the thermal waters, resulting in an O¹⁸-depletion averaging about 4%. A lower limit on the ratio of the mass of water to rock in the exchange system is 4·3, at least ten times greater than the corresponding figure for the Salton Sea geothermal system. Carbonates, present as alteration products in most samples, are found to be in equilibrium with waters at present-day temperatures in some wells, and to record higher ‘fossil’ temperatures in others. Quartz phenocrysts and xenocrysts remain unexchanged, and only new hydrothermal quartz is in isotopic equilibrium with geothermal solutions.