Mining Goes Digital – Mueller et al. (Eds)
© 2019 Taylor & Francis Group, London, ISBN 978-0-367-33604-2
106
Covariance table and PPMT: Spatial continuity mapping
of multiple variables
J. Kloeckner, C.Z. da Silva & J.F.C.L. Costa
Universidade Federal do Rio Grande do Sul, Porto Alegre, Brazil
ABSTRACT: In most mining projects often multiple variables, which may be correlated,
are required to build models. The relationship among variables has a significant impact on
processing plant performance. Therefore, such relationships should be accounted for so that
the final model is representative of the reality in the deposit. The projection pursuit multivariate transform is a modern approach that simplifies multivariate geostatistical modeling
while assuring multigaussianity. Even though, it simplifies significantly multiple variables
modeling, it remains a laborious and crucial step: calculate and fit a semivariogram model
to each and every variable for geostatistical estimation and simulation methods. We propose
a methodology using a covariance table to map automatically spatial continuity to replace
the traditional covariance explicit defined model. A three dimensional case study of an iron
ore deposit illustrates the practical applicability for the work flow proposed. The results are
satisfactory, validated by the standard simulations verification.
Keywords: covariance table, PPMT, multivariate data
1 INTRODUCTION
The geostatistical modeling is crucial for a complete understanding of the geological phenomenon. A key feature for the modeling process is the spatial continuity within the deposit,
which affects all the planning and exploration stages of the mining endeavour. Traditionally,
the spatial continuity model is obtained through the semivariogram, which is valid within a
domain along all spatial directions. Although widely used, this approach demands human
effort that can be highly laborious and subjective for a multiple variables data set. Numerous
methods propose an automatic semivariogram model fitting. However, such methodologies
are based on the parameters that are subject to the analyst adjustment. This way, the main
goal of this study is to reduce the human laborious tasks to as little as possible if desired.
This is achievable through the use of covariance table (CT) instead of the explicit covariance model. The covariance table proposed is obtained through a three-steps work flow:
interpolating the data set to fill it up a regular grid; auto convolute the gridded data via a fast
Fourier transform algorithm; back transform the model to spatial domain ensuring positive
definiteness condition. The projection pursuit multivariate transform (PPMT) to deal with
multivariate relations amongst variables. The projection pursuit multivariate transform is a
modern approach to transform multivariate data of any distribution shape, size and dimension to an uncorrelated multivariate Gaussian distribution. Thereby, geostatistical simulation
can be performed independently. After, PPMT back-transform restores the original multivariate correlation of individual independent simulations. In doing so, the PPMT simultaneously simplifies multivariate geostatistical modeling while assuring multigaussianity.
This paper is organized in four sections. The first one explains briefly the PPMT method
and how it was applied in this study. In the second section, we propose the workflow to automatically obtain the CT for each variable and its essential step named base model to extract
the covariance (BMEC). The third section presents the methodology. In the fourth section,
© 2019 Taylor & Francis Group, London, ISBN 978-0-367-33604-2
106
Covariance table and PPMT: Spatial continuity mapping
of multiple variables
J. Kloeckner, C.Z. da Silva & J.F.C.L. Costa
Universidade Federal do Rio Grande do Sul, Porto Alegre, Brazil
ABSTRACT: In most mining projects often multiple variables, which may be correlated,
are required to build models. The relationship among variables has a significant impact on
processing plant performance. Therefore, such relationships should be accounted for so that
the final model is representative of the reality in the deposit. The projection pursuit multivariate transform is a modern approach that simplifies multivariate geostatistical modeling
while assuring multigaussianity. Even though, it simplifies significantly multiple variables
modeling, it remains a laborious and crucial step: calculate and fit a semivariogram model
to each and every variable for geostatistical estimation and simulation methods. We propose
a methodology using a covariance table to map automatically spatial continuity to replace
the traditional covariance explicit defined model. A three dimensional case study of an iron
ore deposit illustrates the practical applicability for the work flow proposed. The results are
satisfactory, validated by the standard simulations verification.
Keywords: covariance table, PPMT, multivariate data
1 INTRODUCTION
The geostatistical modeling is crucial for a complete understanding of the geological phenomenon. A key feature for the modeling process is the spatial continuity within the deposit,
which affects all the planning and exploration stages of the mining endeavour. Traditionally,
the spatial continuity model is obtained through the semivariogram, which is valid within a
domain along all spatial directions. Although widely used, this approach demands human
effort that can be highly laborious and subjective for a multiple variables data set. Numerous
methods propose an automatic semivariogram model fitting. However, such methodologies
are based on the parameters that are subject to the analyst adjustment. This way, the main
goal of this study is to reduce the human laborious tasks to as little as possible if desired.
This is achievable through the use of covariance table (CT) instead of the explicit covariance model. The covariance table proposed is obtained through a three-steps work flow:
interpolating the data set to fill it up a regular grid; auto convolute the gridded data via a fast
Fourier transform algorithm; back transform the model to spatial domain ensuring positive
definiteness condition. The projection pursuit multivariate transform (PPMT) to deal with
multivariate relations amongst variables. The projection pursuit multivariate transform is a
modern approach to transform multivariate data of any distribution shape, size and dimension to an uncorrelated multivariate Gaussian distribution. Thereby, geostatistical simulation
can be performed independently. After, PPMT back-transform restores the original multivariate correlation of individual independent simulations. In doing so, the PPMT simultaneously simplifies multivariate geostatistical modeling while assuring multigaussianity.
This paper is organized in four sections. The first one explains briefly the PPMT method
and how it was applied in this study. In the second section, we propose the workflow to automatically obtain the CT for each variable and its essential step named base model to extract
the covariance (BMEC). The third section presents the methodology. In the fourth section,