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\begin{document}

\title{The impact of measurement scale on the univariate statistics of K, Th,
and U in the Earth crust}



\author[1]{David Baratoux}%
\author[2]{Makhoudia Fall}%
\author[3]{Pierre-Yves Meslin}%
\author[4]{Mark Jessell}%
\author[5]{Olivier Vanderheaghe}%
\author[6]{Jean-François Moyen}%
\author[2]{Papa Moussa Ndiaye}%
\author[7]{Kwame Boamah}%
\author[8]{Lenka Baratoux}%
\author[9]{Anne-Sylvie André-Mayer}%
\affil[1]{Université Félix Houphouët-Boigny}%
\affil[2]{University Cheikh Anta Diop}%
\affil[3]{IRAP-OMP}%
\affil[4]{University of Western Australia}%
\affil[5]{Géosciences Environnement Toulouse, University of Toulouse}%
\affil[6]{Univ. St-Etienne}%
\affil[7]{Geological Survey Department of Ghana}%
\affil[8]{Institut de Recherche pour le Développement}%
\affil[9]{Université de Lorraine}%


\vspace{-1em}



  
  \date{May 24, 2024}


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\selectlanguage{english}
\begin{abstract}
The univariate statistics of Potassium (K), Thorium (Th) and Uranium (U)
concentrations, in the Earth's oceanic and continental crust are
examined by different techniques. The frequency distributions of the
concentrations of these elements in the oceanic crust are derived from a
global catalog of mid-ocean ridge basalts (MORB). Their frequency
distributions of concentrations in the continental crust are illustrated
by the North Pilbara Craton, and the West Africa Craton. For these two
cratons, the distributions of K, Th and U derived from geochemical
analyses of several thousand whole rock samples differ significantly
from those derived from airborne radiometric surveys. The distributions
from airborne surveys tends to be more symmetric with smaller standard
deviations than the right-skewed distributions inferred from whole rock
geochemical analyses. Hypothetic causes of these differences include (i)
bias in rock sampling or in airborne surveys, (ii) the differences in
the chemistry of superficial material, and (iii) the differences in
scales of measurements. The scale factor, viewed as consequence of the
central limit theorem applied to K, Th and U concentrations, appears to
account for most of the observed differences in the distributions of K,
Th and U. It suggests that the three scales of auto-correlation of K, Th
and U concentrations are of the same order of magnitude as the
resolution of the airborne radiometric surveys (50 -- 200 m) and
concentrations of K, Th, U are therefore generally heterogenous at
smaller scales.%
\end{abstract}%



\sloppy


\textbf{Supplementary Information}\selectlanguage{english}
\begin{figure}[H]
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\includegraphics[width=0.70\columnwidth]{figures/image1/image1}

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\end{figure}

Fig. 1 -- Simulated distribution for a geochemical data base with
different proportions of igneous rocks, (meta)sediments and regolith, in
the case of the Northern Pilbara Craton. Note that the effect of varying
the proportion of the different lithologies produce marginal
modifications of the distributions, and do not resolve the differences
in peak position, width and skewness between the airborne distributions
and rock samples distributions. This reflects the fact that,
distributions of U, and Th, in the various lithologies are, on the first
order, similar. Distributions for K are not estimated as the North
Pilbara database does not include regolith samples.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image2/image2}

\end{center}
\end{figure}

Fig. 2 -- Simulated distribution for a geochemical data base with
different proportions of igneous rocks, (meta)sediments and regolith, in
the case of the West African Craton. As in the Northern Pilbara Craton,
Note that the effect of varying the proportion of the different
lithologies produce marginal modifications of the distributions, and do
not resolve the differences in peak position, width and skewness between
the airborne distributions and rock samples distributions. As for the
Northern Pilbara Craton, the distributions in the various lithologies
are, on the first order, similar, except for K in the regolith. Note
that a dominant proportion of regolith (not shown here) does not resolve
the differences either, considering the fact that the low K in regolith
would not match with the distribution of K in the airborne dataset.

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