Jorge Kazuo Yamamoto

An Alternative Measure of the Reliability of Ordinary Kriging Estimates

This paper presents an interpolation variance as an alternative to the measure of the reliability of ordinary kriging estimates. Contrary to the traditional kriging variance, the interpolation variance is data-values dependent, variogram dependent, and a measure of local accuracy. Natural phenomena are not homogeneous; therefore, local variability as expressed through data values must be recognized for a correct assessment of uncertainty. The interpolation variance is simply the weighted average of the squared differences between data values and the retained estimate. Ordinary kriging or simple kriging variances are the expected values of interpolation variances; therefore, these traditional homoscedastic estimation variances cannot properly measure local data dispersion. More precisely, the interpolation variance is an estimate of the local conditional variance, when the ordinary kriging weights are interpreted as conditional probabilities associated to the n neighboring data. This interpretation is valid if, and only if, all ordinary kriging weights are positive or constrained to be such. Extensive tests illustrate that the interpolation variance is a useful alternative to the traditional kriging variance.

Absolute Dating of Permian Ash-Fall in the Rio Bonito Formation, Paraná Basin, Brazil

Abstract The Rio Bonito Formation in southern Parana basin contains a set of tonsteins interbedded with coal-seams. These tonsteins are composed mainly of kaolinite with zircon, apatite and beta-quartz paramorphs as accessory minerals, and were interpreted as volcanic ashes deposited by ash falls over pits protected by barrier islands in a barrier-lagoon system. A U-Pb dating of zircons in the tonstein A, which furnished an age of 267.1 ± 3.4 Ma (Early Permian) confirming previous age-dates based on palynology and correlating them with one of the main periods of volcanic activity in the Gondwana. The source of the pyroclastic material was attributed to the early Permian Choiyoi magmatic arc in Argentina, developed during the Sanrafaelic orogeny, and with a main peak of volcanic activity between 260 and 272 Ma.

Quantification of Uncertainty in Ore-Reserve Estimation: Applications to Chapada Copper Deposit, State of Goiás, Brazil

Ordinary kriging (OK) has been used widely for ore-reserve estimation because of its superior characteristics in relation to other methods. One of these characteristics is related to the quantification of uncertainty by the kriging variance. However, the kriging variance does not recognize local data variability, which is an important issue in the process of ore-reserve estimation, when heterogeneous mineral deposits with richer and poorer parts are being evaluated. This paper proposes the use of interpolation variance as a reliable measure of local data variability and, therefore, adequate for ore-reserve classification. With a reliable measurement of data variability, local confidence can be calculated using the classical confidence interval around an estimate. Errors derived from local confidence then are used to assign classes according to a degree of certainty within some confidence level. Comparative tests using both OK variance and interpolation variance are carried out using exploration data from Chapada Copper Deposit, State of Goiás, Brazil. Results show that the interpolation variance provides a better way to measure uncertainty and consequently to classify reserves.

Mapping an uncertainty zone between interpolated types of a categorical variable

Categorical data cannot be interpolated directly because they are outcomes of discrete random variables. Thus, types of categorical variables are transformed into indicator functions that can be handled by interpolation methods. Interpolated indicator values are then backtransformed to the original types of categorical variables. However, aspects such as variability and uncertainty of interpolated values of categorical data have never been considered. In this paper we show that the interpolation variance can be used to map an uncertainty zone around boundaries between types of categorical variables. Moreover, it is shown that the interpolation variance is a component of the total variance of the categorical variables, as measured by the coefficient of unalikeability.

Comparison Between Kriging Variance and Interpolation Variance as Uncertainty Measurements in the Capanema Iron Mine, State of Minas Gerais—Brazil

The Capanema Mine, an iron ore deposit, is located in the central portion of the Quadrilátero Ferrífero, State of Minas Gerais, southeastern Brazil. Mine development data from approximately 7000 drillholes were used for a comparative study between kriging variance and interpolation variance as uncertainty measurements associated with ordinary kriging estimates. As known, the traditional kriging variance does not depend on local data and, therefore, does not measure the actual dispersion of data. On the other hand, the interpolation variance measures adequately the local dispersion of data used for an ordinary kriging estimate. This paper presents an application of the concept of interpolation variance for measuring uncertainties associated with ordinary kriging estimates of Fe and silica grades. These data were selected for their distinct statistical characteristics with Fe presenting a negatively skewed distribution and, consequently, a low dispersion, and silica a positively skewed distribution and, therefore, a high variability. Comparative studies between the two uncertainty measurements associated with ordinary kriging estimates of Fe and silica proved the superiority of the interpolation variance as a reliable and precise alternative to the kriging variance.

A record of Permian subaqueous vent activity in southeastern Brazil

The remarkable occurrence of more than 4,500 conical siliceous mounds in an area of less than 1.5 square kilometres has been reported in the Paraná basin, near Anhembi, São Paulo, in southeastern Brazil. These structures, which are up to two metres high, are thought to have been formed at the margin of a very shallow, broad but waning internal sea, and it was originally suggested that they are stromatolites. Yet their restricted occurrence, unusual abundance and nearly pure siliceous composition have never been satisfactorily explained by this hypothesis. Here we report field and laboratory observations on their shape, construction, composition and mineralogy. On the basis of our data we suggest that the conical mounds are the result of subaqueous Late Permian vent activity in southwestern Gondwana. The present siliceous cone field differs considerably from other Palaeozoic siliceous hot spring deposits, such as those at Rhynie, Scotland, and the Drummond basin, Australia, and therefore represents an unusual occurrence of vent activity.

CONVEX_HULL—a Pascal program for determining the convex hull for planar sets

Computer aided graphical display of geological data is usually based on a regular grid, interpolated from a scattered data set. However, the interpolation function is valid only inside the domain of sampling points, or a closed boundary which limits all the sampling points. This closed boundary, named convex hull, can be determined with the aid of an algorithm. The convex hull of a planar set of points is defined as the minimum area convex polygon containing all the points. This paper presents a review of current methods for determining the convex hull, and the computer program CONVEX_HULL, written in Pascal language and based on a new algorithm.

Remoção de Pb2+ e Cr3+ em solução por zeólitas naturais associadas a rochas eruptivas da formação serra geral, bacia sedimentar do Paraná

The capacity of natural zeolites and its host rock (dacite) to remove Pb2+ and Cr3+ from aqueous solutions has been investigated. Results showed that both samples prefer to remove Pb2+ instead of Cr3+. Almost 100% of Pb2+ was removed from solutions with concentration until 50 mg L-1 and 100 mg L-1 of this metal, respectively by dacite and zeolite. The equilibrium of metals adsorption process was reached during the first 30 min by both materials. Na+ can be used to recover Pb2+, but not to remove Cr3+ from the treated samples. The Sips model showed a good fit for experimental data of this study.

Post-processing of sequential indicator simulation realizations for modeling geologic bodies

Sequential indicator simulation realizations contain unavoidable artifacts that are geologically unrealistic. This happens because unlikely types can be drawn randomly from the cumulative distribution and be assigned to a cell in the simulated model. This cell may then be used as previously simulated data when a cell in its neighborhood is visited during a random walk. The sequential process sometimes results in geologically unrealistic realizations. However, different realizations can reveal hidden features. Each realization contains both reliable geological information and noise that is displayed as unlikely types. This paper proposes applying the averaging filter that is commonly used in seismic reflection data to improve the signal to noise ratio. After applying this filter, all L realizations will be condensed into a single geological model that contains certain and uncertain cells. This average model is then exhaustively sampled for the certain cells, and this new sample is used to post-process the uncertain cells to reduce the uncertainty. This resampling and post-processing procedure can be repeated until the final model is considered to be good enough. The proposed method is tested with a model of a dike that crosscuts two sedimentary units. The synthetic geologic model was sampled with 24 drill holes. The results show that the final geological model with reduced uncertainty reproduces very well the sedimentary units and the orientation of the dike as well. The dike shape is not fully reproduced and still presents uncertainties because of lack of neighbor data.

Backtransforming Rank Order Kriging Estimates

Kriging of raw data presenting distributions with positive skewness must be avoided because the strong influence of a few high values in the resulting estimates. The solution is to apply data transformation, which changes the shape of original distribution into a symmetric distribution. Kriging of transformed data is performed and then back-transformed to the original scale of measurement. In this paper, we examine the uniform score transform that results in a uniform distribution. Ordinary kriging estimates of uniform score data results in a bell-shaped distribution, since the tails of the distribution are lost in the estimation process because of the smoothing effect. The back-transformation of this bell-shaped distribution result in biased estimates. Therefore, the solution proposed in this paper is to correct the smoothing effect of the rank order kriging estimates before transforming them back to the scale of raw data. Results showed this algorithm is reliable and back-transformed estimates are unbiased in relation to the sample data.