Element To Mineral Conversion

Description

This article describes a non-negative least-squares method for converting elemental assays into mineral compositions.

Model theory

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Excel

The Element to Mineral Conversion may be invoked from the Excel formula bar with the following function call:

=mdMineralogy_EMC(Assay as Range, C as Range, Optional returnResidual as Boolean = false)

Invoking the function with no arguments will print Help text associated with the model, including a link to this page.

The input parameters and calculation results are defined below in matrix notation, along with an example image showing the selection of the same cells and arrays in the Excel interface:

[math]\displaystyle{ \begin{align} \mathit{Assay} & = \begin{bmatrix} a_1\text{ (}%\text{ w/w)}\\ \vdots\\ a_n\text{ (}%\text{ w/w)}\\ \end{bmatrix}\\ \\ \mathit{C} & = \begin{bmatrix} c_{11}\text{ (w/w)} & \cdots & c_{1p}\text{ (w/w)}\\ \vdots & & \vdots\\ c_{n1}\text{ (w/w)} & \cdots & c_{np}\text{ (w/w)}\\ \end{bmatrix}\\ \\ \mathit{returnResidual} & = \big [ \text{(True/False)} \big ]\\ \end{align} }[/math] [math]\displaystyle{ \begin{align} \mathit{mdMineralogy\_EMC} & = \begin{bmatrix} m_1\text{ (\% w/w)}\\ \vdots\\ m_p\text{ (\% w/w)}\\ \text{Residual}\\ \end{bmatrix} \end{align} }[/math] where: [math]\displaystyle{ n }[/math] is the number of assays (elements) [math]\displaystyle{ p }[/math] is the number of minerals [math]\displaystyle{ a }[/math] is elemental assays (% w/w) [math]\displaystyle{ c }[/math] is the conversion ratio, the mass proportion of an element in a mineral (w/w) [math]\displaystyle{ m }[/math] is the mineral composition (% w/w) [math]\displaystyle{ Residual }[/math] is the residual of the least squares regression, only returned if [math]\displaystyle{ returnResidual=True }[/math]

Figure 1. Example showing the selection of the Assay (blue frame), C (red frame), and Results (light blue frame) arrays in Excel. [math]\displaystyle{ returnResidual=False }[/math] in thid example.