Correcting the Smoothing Effect of Ordinary Kriging Estimates

Yamamoto, Jorge Kazuo

doi:10.1007/s11004-005-8748-7

Cited by 93 publications

(60 citation statements)

References 8 publications

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“…This paper shows how we can reproduce both, the sample histogram and the sample semivariogram of rank transformed data. Actually, the main objective of this paper is to show a new application of the post-processing algorithm for correcting the smoothing effect of ordinary kriging estimates (Yamamoto, 2005(Yamamoto, , 2007. Moreover, as it is shown in this paper, back-transformed values reproduce the original sample histogram and, consequently, the mean and variance of the original data.…”

Section: Introductionmentioning

confidence: 82%

“…It has been proven the heteroscedastic interpolation variance is more reliable than the homoscedastic kriging variance (see Yamamoto, 2000Yamamoto, , 2005Yamamoto, , 2007. Although the interpolation variance has been used by this author since 1989 (Yamamoto, 1989), it was published only after Journel and Rao (1996) interpreted ordinary kriging weights as conditional probabilities.…”

Section: Reproducing the Uniform Score Histogrammentioning

confidence: 99%

“…If we want to reproduce the sample histogram, estimates have to be corrected by adding a certain amount, in order that the variance of corrected estimates be equal to the sample variance. The correcting amount can be obtained using simulation methods or a post-processing algorithm proposed by Yamamoto (2005Yamamoto ( , 2007, as follows: (5) where is the correcting amount added to ordinary kriging estimates and factor is a number that makes the variance of corrected estimates equal to the sample variance. For details on the algorithm for correcting the smoothing effect of ordinary kriging estimates, please see Yamamoto (2005Yamamoto ( , 2007.…”

Section: Reproducing the Uniform Score Histogrammentioning

confidence: 99%

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Backtransforming Rank Order Kriging Estimates

Yamamoto

2010

Geol. USP, Sér. cient.

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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.

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Section: Introductionmentioning

confidence: 82%

Section: Reproducing the Uniform Score Histogrammentioning

confidence: 99%

Section: Reproducing the Uniform Score Histogrammentioning

confidence: 99%

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Backtransforming Rank Order Kriging Estimates

Yamamoto

2010

Geol. USP, Sér. cient.

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“…Mathematically this is expressed in , and means that small values are usually overestimated and large values underestimated: where Var() is the variance and and Z ( x i ) are the estimated and observed value at x i , respectively. This so‐called smoothing effect is found among other weighted average estimators and can be corrected for when necessary (Yamamoto 2005). Also, the choice of model and inclusion of a linear drift on the mean of the observations implies that, beyond the range of observed data, estimates are asymptotically linear.…”

Section: Discussionmentioning

confidence: 99%

Estimating Material and Energy Flows in Life Cycle Inventory with Statistical Models

Moreau

Bage

Marcotte

et al. 2012

J of Industrial Ecology

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Data (un)availability and uncertainty are recurring problems in life cycle assessment, and particularly inventory analysis. Advances in life cycle inventory have focused on the propagation and management of uncertainty, but this article addresses the question of how to account for unavailable data and corresponding uncertainty. Large and complicated systems often lack complete data due to confidential practices or the efforts required in the data collection process. Electricity production with multiple processes generating a single product is a classic example. Instead of the conventional process-based models to estimate missing data, the approach developed in this article divides systems based on functionally equivalent objects. Each one of these objects is then described in terms of characteristic variables, such as power capacity. Kriging, a flexible statistical estimator, allows for the estimation of unknown material and energy flows based on the objects' characteristic variables. Both univariate and multivariate kriging are tested and compared to regression analysis. It is found that kriging performs better than linear regression, according to the mean absolute error criterion. Multivariate kriging provides an even more accurate joint estimation method to bridge data gaps scattered across inventories and when observable values of material and energy flows differ from one object to the next. Parameters of the underlying models are interpreted in terms of data uncertainty.Additional supporting information may be found in the online version of this article.Supporting Information S1: This supporting information provides mathematical details for implementing kriging.

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“…Voxel-based modelling is commonly used to represent the geological objects with continuous attribute values, such as geophysical and geochemical fields. Inverse Distance Weighted (IDW) [29,30] and Kriging [31,32] have been approved methods in interpolation which is important for constructing field models. Voxelization is a key approach to convert the surface model into a block model with some methods, such as Flood-fill and Octree-based Divisive Algorithm [8,9].…”

Section: Introductionmentioning

confidence: 99%

Quantitative 3D Association of Geological Factors and Geophysical Fields with Mineralization and Its Significance for Ore Prediction: An Example from Anqing Orefield, China

Qin

Liu

2018

Minerals

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Quantitative 3D spatial association of geological factors and geophysical fields with orebodies is critical for ore prediction. The Anqing orefield, a principal Cu-Fe orefield in China, is closely associated with the Yueshan intrusion. By compiling the data from drilling and tunnelling exploration, Controlled Source Audiofrequency Magnetotelluric (CSAMT) surveying and the computational modelling of magmatic intrusion's cooling process, we constructed models of the Yueshan intrusion, ore-favourable carbonate formation, orebodies, resistivity field and volume strain field. These models are used as evidential factors to analyse their spatial association with mineralization by the weights-of-evidence (WofE) method. The location of orebodies is closely related to the shape of the contact zone of the Yueshan intrusion. The spaces with the distance ≤200 m to the concaves that were selected by minimum principal curvature (|K min | ≥ 0.0025) from contact zones, are very favourable for localization of orebodies. Most orebodies are not located in the spaces of the lowest resistivity, suggesting that the lowest resistivity cannot be used as an indicator for mineralization. The spaces with higher positive volumetric strain have higher positive weights with orebodies, implying that the mineralization is positively related to the positive volumetric strain. The spaces of all evidential factors that had positive correlation with mineralization were integrated to create a 3D prospectivity map by calculating posterior probability. Five areas with higher posterior probability, indicating higher prospectivity potential, are selected as targets for future exploration.

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Correcting the Smoothing Effect of Ordinary Kriging Estimates

Cited by 93 publications

References 8 publications

Backtransforming Rank Order Kriging Estimates

Backtransforming Rank Order Kriging Estimates

Estimating Material and Energy Flows in Life Cycle Inventory with Statistical Models

Quantitative 3D Association of Geological Factors and Geophysical Fields with Mineralization and Its Significance for Ore Prediction: An Example from Anqing Orefield, China

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