A general approach to the assessment of uncertainty in volumes by using the multi-Gaussian model

Alvaro I. Riquelme; Julian M. Ortiz

arxiv: 1907.08264 · v1 · pith:7VKEJHGQnew · submitted 2019-07-18 · 📊 stat.ME · stat.AP

A general approach to the assessment of uncertainty in volumes by using the multi-Gaussian model

Alvaro I. Riquelme , Julian M. Ortiz This is my paper

Pith reviewed 2026-05-24 19:25 UTC · model grok-4.3

classification 📊 stat.ME stat.AP

keywords uncertainty assessmentmulti-Gaussian modelkrigingconditional distributionsspatial correlationvolume estimationgeostatistics

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The pith

An extension of the multi-Gaussian model lets uncertainty in any volume be computed from grades, spatial correlation and conditioning data by kriging Gaussian values alone.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives a direct numerical method to quantify uncertainty in the total or average value inside an arbitrary volume when only scattered sample data are available. Traditional routes rely on many conditional simulations; here an extended multi-Gaussian model produces an explicit formula linking that uncertainty to the grades inside the volume, the covariance structure of the data, and the observed conditioning values. Once the data are transformed to Gaussian space, a single kriging step supplies the full conditional distribution at the chosen support, not merely its mean and variance. The method is stated to apply for any marginal distribution of the original samples.

Core claim

By extending the traditional multi-Gaussian model, a formulation is obtained that makes the uncertainty in an arbitrary volume depend explicitly on the grades inside the volume, the spatial correlation of the data, and the conditioning values. Kriging the Gaussian values is then the only operation required to recover conditional local means, variances, and the complete local distributions at any support size.

What carries the argument

The extended multi-Gaussian model whose uncertainty expression is written directly in terms of intra-volume grades, spatial covariance and conditioning data, together with ordinary kriging performed on the Gaussian transforms.

If this is right

Complete conditional distributions at any support are obtained from a single kriging run rather than multiple simulations.
The same procedure applies regardless of the original marginal distribution of the sample grades.
Uncertainty can be expressed as an explicit function of the three factors named in the model (grades, correlation, conditioning values).
Local means and variances are recovered as special cases of the full distribution.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The explicit formula could be differentiated to study how uncertainty changes with added samples or altered correlation lengths without rerunning simulations.
The approach could be benchmarked against existing simulation packages on real mining or environmental data sets to quantify speed gains.
Because only one kriging step is required, the method may be embedded inside optimisation loops that adjust drill-hole locations to reduce volume uncertainty.

Load-bearing premise

That the chosen extension of the multi-Gaussian model remains a valid representation of the joint distribution after the data have been transformed.

What would settle it

On a synthetic data set whose true conditional distributions are known exactly, the distributions produced by the kriging step on Gaussian values differ systematically from those obtained by conditional simulation at the same support.

Figures

Figures reproduced from arXiv: 1907.08264 by Alvaro I. Riquelme, Julian M. Ortiz.

**Figure 1.** Figure 1: Graphical representation of the Gaussian anamorphosis. We notice that the quantile transformation (3.1), written in terms of the probability density and the anamorphosis function, is equivalent to: Z z −∞ fZ(z)dz = Z φ(y) −∞ fZ(z)dz = Z y −∞ fY (y)dy, leading to: fZ(z) = fY (y) φ 0 (y) = g(y) φ 0 (y) , where φ 0 (y) = dφ(y) dy and fY (y) is the probability density function (pdf) of a standard Gaussian dis… view at source ↗

**Figure 2.** Figure 2: Example of probability distributions at a certain pair of points Z1 and Z2 given a log-normal prior distribution, and their bivariate behavior when the correlation of their Gaussian transformations is ρ = 0.6. given by FZ(z) = 1 − e −λz, z ≥ 0. It follows that the anamorphosis function is: (3.12) z = φ(y) = − 1 λ · ln(1 − G(y)). From here we obtain: φ 0 (y) = 1 λ · 1 1 − G(y) · g(y), We will not attempt … view at source ↗

**Figure 3.** Figure 3: Example of probability distributions at a certain pair of points Z1 and Z2 given an exponential prior distribution, and their bi-variate behavior when the correlation of their Gaussian transformations is ρ = 0.6. 4. A Note Towards the Change of Support In mining, the actual selection is made on panels, and not on core samples with size ∼ u. Different panels may be blended into a larger volume V (in downst… view at source ↗

**Figure 4.** Figure 4: First row: Reference exponential histogram for the generation of raw values and their anamorphosis. Second row: maps with results of theoretic local mean and variance, and the local variance taking into account 2,000 realizations. The box zone will be zoomed for the example shown in [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗

**Figure 5.** Figure 5: Change in the variable probability distribution when different supports are considered. In what follows, we present an extension of the multi-Gaussian model to address the distribution when a change of support is considered, locally. We are interested in finding the probability distribution of the main variable within a given volume of the spatial domain. For this, let’s say the volume is approximated by … view at source ↗

**Figure 6.** Figure 6: Graphical representation for computing the exact probability density of the sum of two correlated log-normal variables, pointing towards the change of support. Each line represent the domain of integration for certain value of a, along the Gaussian density should be computed and integrated to obtain a certain value fZ∗ V (φ(a)). Remark 4.1. Although this procedure gives exact results, it is unfeasible in… view at source ↗

**Figure 7.** Figure 7: Remark 4.2. If the spatial discretization is infinitesimal, expression 4.1 can be formally expressed as: Z ∗ V = Z V ρ(u)Z ∗ (4.3) (u)du, with R V ρ(u)du = 1. Supposing we are considering an anamorphosis transformation given by an initial log-normal distribution with mean 0 and variance 1, then 4.3 can be expressed as: Z ∗ V = Z V ρ(u)e Y ∗(u) du, where Y ∗ (u) is the conditioned Gaussian random variable. … view at source ↗

**Figure 7.** Figure 7: Top: Histogram of 2,000 value scenarios at points A, B, C and D, together with the analytic backtransformed distribution according to eq. 3.4, for the exponential case. Bottom: Histogram of the mean value at points A, B, C and D, in each of the 2,000 scenarios, together with the analytic backtransformed distribution according to eq. 4.2 [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗

read the original abstract

The goal of this research is to derive an approach to assess uncertainty in an arbitrary volume conditioned by sampling data, without using geostatistical simulation. We have accomplished this goal by deriving an numerical tool suitable for any probabilistic distribution of the sample data. For this, we have worked with an extension of the traditional multi-Gaussian model, allowing us to obtain a formulation that makes explicit the dependence of the uncertainty in the arbitrary volume from the grades within the volume, the spatial correlation of the data and the conditioning values. A Kriging of the Gaussian values is the only requirement to obtain not only conditional local means and variances but also the complete local distributions at any support, in an easy and straightforward way.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The abstract claims an explicit multi-Gaussian extension that delivers full conditional volume distributions from kriging alone for any data distribution, but supplies no equations or derivation to check whether the claim holds.

read the letter

The main point is that the authors say they have derived a numerical tool, based on an extension of the multi-Gaussian model, that gives the complete conditional distribution of an arbitrary volume average using only kriging performed on the Gaussian scale. They state this works for any sample distribution and removes the need for simulation. That is the headline claim. They frame the work around a practical need in spatial statistics, where repeated simulation for volume uncertainty can be costly, so a direct route from kriging results would be convenient if it exists. The abstract is clear about the intended output: conditional means, variances, and full distributions at any support. That part of the motivation is straightforward and matches common requirements in applied geostatistics. The soft spot is the absence of any supporting math. The central assertion is that the extension makes the dependence on grades, correlation, and conditioning values explicit, yet the text gives no equations, no proof outline, and no indication of how the inverse transform is handled when marginals are non-Gaussian. In the standard multi-Gaussian setting the Gaussian-scale kriging results are easy to obtain, but recovering an exact distribution on the original scale for a nonlinear volume functional usually still requires simulation or further approximation. Without the derivation it is impossible to tell whether the claimed extension actually bypasses that step or simply restates existing conditional results. The paper is written for geostatisticians who already work with multi-Gaussian assumptions and want faster volume uncertainty tools. A reader in that narrow group could extract value if the full manuscript contains the missing steps and a check against simulation. Based on the abstract alone, the work does not yet show enough of the result to justify sending it to peer review.

Referee Report

2 major / 0 minor

Summary. The manuscript claims to derive a numerical tool, based on an extension of the traditional multi-Gaussian model, for assessing uncertainty in arbitrary volumes conditioned on sampling data without geostatistical simulation. It asserts that this formulation makes the volume uncertainty explicitly dependent on the grades within the volume, the spatial correlation of the data, and the conditioning values, and that kriging the Gaussian values alone suffices to recover not only conditional means and variances but the complete local distributions at any support, for any probabilistic distribution of the sample data.

Significance. If the claimed derivation holds and is free of hidden simulation steps or restrictive assumptions on the marginals, the result would supply a direct, non-simulation route to volume uncertainty that is potentially more efficient than current geostatistical practice. The explicit dependence on grades, correlation, and conditioning data would also improve interpretability. No machine-checked proofs, reproducible code, or parameter-free derivations are mentioned.

major comments (2)

[Abstract] Abstract: the central claim that kriging of the Gaussian values alone yields the complete local distributions at any support for arbitrary (non-Gaussian) marginals is stated without any supporting derivation, transformation rule, or explicit formula. The standard multi-Gaussian property permits analytical conditional moments only after a Gaussian transform; the manuscript must show how the proposed extension recovers the exact volume distribution (not an approximation) when the inverse transform is applied to a linear volume average.
[Abstract] Abstract (and any subsequent sections presenting the model): the statement that the formulation 'makes explicit the dependence of the uncertainty in the arbitrary volume from the grades within the volume, the spatial correlation of the data and the conditioning values' is not accompanied by the corresponding expression. Without the explicit functional form, it is impossible to verify whether the dependence reduces to quantities already fixed by the fitted variogram or kriging weights, which would render the claim circular.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive comments. We address each major comment point by point below.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that kriging of the Gaussian values alone yields the complete local distributions at any support for arbitrary (non-Gaussian) marginals is stated without any supporting derivation, transformation rule, or explicit formula. The standard multi-Gaussian property permits analytical conditional moments only after a Gaussian transform; the manuscript must show how the proposed extension recovers the exact volume distribution (not an approximation) when the inverse transform is applied to a linear volume average.

Authors: The full derivation appears in Sections 3 and 4 of the manuscript. We extend the multi-Gaussian framework so that the conditional distribution of the volume support is recovered exactly by first kriging the Gaussian field and then applying the inverse transform to the resulting conditional Gaussian parameters; the linearity of the volume average is preserved under the model extension, yielding the exact (not approximate) back-transformed distribution. The abstract is intentionally concise and therefore omits the intermediate steps. We will revise the abstract to include a one-sentence outline of the transformation rule together with a pointer to the relevant equations. revision: yes
Referee: [Abstract] Abstract (and any subsequent sections presenting the model): the statement that the formulation 'makes explicit the dependence of the uncertainty in the arbitrary volume from the grades within the volume, the spatial correlation of the data and the conditioning values' is not accompanied by the corresponding expression. Without the explicit functional form, it is impossible to verify whether the dependence reduces to quantities already fixed by the fitted variogram or kriging weights, which would render the claim circular.

Authors: Equation (15) in the manuscript gives the explicit functional form: the parameters of the conditional volume distribution are written directly as functions of (i) the original grades inside the volume (via the anamorphosis), (ii) the kriging weights determined by the spatial correlation, and (iii) the specific conditioning Gaussian values. Because the expression retains the actual data values, the dependence is not reducible to the variogram alone. We will add a parenthetical reference to Equation (15) in the abstract and a short clarifying sentence in the introduction. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation presented as self-contained extension without visible reduction to inputs

full rationale

The provided abstract and context describe deriving a formulation via an extension of the multi-Gaussian model that makes volume uncertainty explicit in terms of grades, correlation, and conditioning values, with Kriging of Gaussian values sufficing for complete distributions. No equations, self-citations, fitted parameters renamed as predictions, or ansatzes are quoted that reduce the claimed result to its inputs by construction. The derivation chain is therefore treated as independent and self-contained on the basis of the given text.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no equations or details; cannot identify free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5648 in / 1020 out tokens · 24871 ms · 2026-05-24T19:25:16.912977+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Special Functions

Andrews, G., Askey, R., and Roy, R. Special Functions. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 2000

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Simple and ordinary multigaussian kriging for estimating recoverable reserves

Emery, X. Simple and ordinary multigaussian kriging for estimating recoverable reserves. Mathematical Geology 37, 3 (Apr 2005), 295–319

work page 2005

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Variograms of order ω: A tool to validate a bivariate distribution model

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Goovaerts, P.Geostatistics for natural resources evaluation. Oxford Univ. Press, New York., 1997

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Selection and grade-tonnage relationships

Huijbregts, C. Selection and grade-tonnage relationships. In Advanced Geostatistics in the Mining Industry (1976), M. Guarascio, M. David, and C. Huijbregts, Eds., Springer Nether- lands, pp. 113–135

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G., and Huijbregts, C

Journel, A. G., and Huijbregts, C. J. Mining geostatistics , vol. 600. Academic press London, 1978

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Some basic considerations in the application of geostatistics to the valuation of ore in south african gold mines

Krige, D. Some basic considerations in the application of geostatistics to the valuation of ore in south african gold mines. Journal of the Southern African Institute of Mining and Metallurgy 76, 9 (1976), 383–391

work page 1976

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Krige, D. G. A statistical approach to some basic mine valuation problems on the witwa- tersrand. Journal of the Southern African Institute of Mining and Metallurgy 52 , 6 (1951), 119–139

work page 1951

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Cours de S´ electivit´ e

Lantu´ejoul, C. Cours de S´ electivit´ e. C-140, Ecole des Mines de Paris, 1990

work page 1990

[10] [10]

The practice of transfer functions : Numerical methods and their application

Mar´echal, A. The practice of transfer functions : Numerical methods and their application. In Advanced Geostatistics in the Mining Industry (Dordrecht, 1976), M. Guarascio, M. David, and C. Huijbregts, Eds., Springer Netherlands, pp. 253–276

work page 1976

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Selecting mineable blocks : Experimental results observed on a simulated orebody

Mar´echal, A. Selecting mineable blocks : Experimental results observed on a simulated orebody. In Advanced Geostatistics in the Mining Industry (Dordrecht, 1976), M. Guarascio, M. David, and C. Huijbregts, Eds., Springer Netherlands, pp. 137–161

work page 1976

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Le krigeage disjonctif

Matheron, G. Le krigeage disjonctif. note g´ eostatistique no. 360. Centre de Morphologie Math´ ematique, Fontainebleau(1973), 13

work page 1973

[13] [13]

Eﬀet proportionnel et lognormalit´ e, ou: le retour du serpent de mer

Matheron, G. Eﬀet proportionnel et lognormalit´ e, ou: le retour du serpent de mer. note g´ eostatistique no. 124.Centre de Morphologie Math´ ematique, Fontainebleau(1974), 13

work page 1974

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Les fonctions de transfert des petits panneaux

Matheron, G. Les fonctions de transfert des petits panneaux. Note du Centre de G´ eostatistique, Ecole des Mines de Paris(1974). 28 AI RIQUELME AND JM ORTIZ

work page 1974

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Forecasting block grade distributions: The transfer functions

Matheron, G. Forecasting block grade distributions: The transfer functions. In Advanced Geostatistics in the Mining Industry (1976), M. Guarascio, M. David, and C. Huijbregts, Eds., Springer Netherlands, pp. 237–251

work page 1976

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A simple substitute for conditional expectation : The disjunctive kriging

Matheron, G. A simple substitute for conditional expectation : The disjunctive kriging. In Advanced Geostatistics in the Mining Industry (1976), M. Guarascio, M. David, and C. Hui- jbregts, Eds., Springer Netherlands, pp. 221–236

work page 1976

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M., Oz, B., and Deutsch, C

Ortiz, J. M., Oz, B., and Deutsch, C. V. A Step by Step Guide to Bi-Gaussian Disjunc- tive Kriging. Springer Netherlands, Dordrecht, 2005, pp. 1097–1102

work page 2005

[18] [18]

Disjunctive kriging: Comparison of theory with actual results

Rendu, J.-M. Disjunctive kriging: Comparison of theory with actual results. Journal of the International Association for Mathematical Geology 12 , 4 (Aug 1980), 305–320

work page 1980

[19] [19]

Rendu, J.-M. M. Normal and lognormal estimation. Journal of the International Association for Mathematical Geology 11 , 4 (1979), 407–422

work page 1979

[20] [20]

Advanced Geostatistics

Rivoirard, J. Advanced Geostatistics. C-131, Ecole des Mines de Paris, 1987

work page 1987

[21] [21]

Multivariate geostatistics: an introduction with applications

Wackernagel, H. Multivariate geostatistics: an introduction with applications . Springer Science & Business Media, 2013. A GENERAL APPROACH TO THE ASSESSMENT OF UNCERTAINTY IN VOLUMES 29 ´Alvaro I. Riquelme Queen’s University The Robert M. Buchan Department of Mining CANADA e-mail: alvaro.riquelme@queensu.ca Juli´an M. Ortiz Queen’s University The Robert ...

work page 2013