MINSQ – a least squares spreadsheet method for calculating mineral proportions from whole rock major element analyses Walter Herrmann & Ron F. Berry Centre for Ore Deposit Research, University of Tasmania, GPO Box 252-79, Hobart, Tasmania 7001, Australia (e-mail:
[email protected]) ABSTRACT: MINSQ is a spreadsheet adaptation of the least squares method that utilizes the Solver tool in Microsoft Excele to quantitatively estimate the proportions of constituent minerals in rocks from whole rock lithogeochemical data. It is simple to use and easily and interactively adaptable to observed mineral or normative assemblages. The structure facilitates input of actual mineral analyses for individual rock samples. The spreadsheet format and the numerical Solver tool in Microsoft Excele put the user in a powerful position to adapt this method to a wide range of numerical problems in geochemistry. KEYWORDS: MINSQ, mineral proportion, modal mineralogy, normative, lithogeochemistry, least squares, Solver
INTRODUCTION This paper describes MINSQ – a personal computer spreadsheet method for calculating the proportions of minerals, of any specified compositions in either observed or theoretical assemblages, from major element analyses of rocks. Le Maitre (1981) published a least squares petrological mixing model program of this type, called GENMIX. It was written in FORTRAN and requires a specific data format, which obscures some of the computation procedure. The process of calculation described here is a modification of the method of least squares, which utilizes the Solver analysis tool in the widely used spreadsheet Microsoft Excele. It is adaptable to virtually any mineral assemblage. In terms of assemblage versatility, MINSQ has some similarities to the metasomatic normative approach of Cheng & Sinclair (1995) and the SEDNORM program of Cohen & Ward (1991). However, MINSQ has the advantage that it does not incrementally assign components in a strict mineral hierarchy. MINSQ iteratively adjusts the proportions of all selected phases to provide a best-fit solution to the composition data. The development of the MINSQ spreadsheet was prompted by the need for quantitative mineralogical data in a research project testing the accuracy of portable short wavelength infrared mineral spectrometry to estimate the proportions of phyllosilicates in hydrothermally altered volcanic rocks. It is a potent and versatile means of relating lithogeochemical data to hydrothermal mineral assemblages, with greater rigour than alteration indices (e.g. Large et al. 2001). It also has applications in petrology, as a flexible method of calculating mineralogical proportions for use in igneous mixing models, or in whole rock oxygen isotope geo-thermometry to determine mineral proportions for fractionation corrections. Where lithogeochemical data are available, it is an efficient alternative to tedious microscopic point counting and expensive quantitative X-ray diffraction (XRD) techniques. Geochemistry: Exploration, Environment, Analysis, Vol. 2 2002, pp. 361–368
SOLVER IN EXCELe The Solver, in Microsoft Excele is a spreadsheet tool that uses a non-linear optimization code (Anonymous 1998). It minimizes, maximizes or sets to a certain value, the value in a specified target cell in a spreadsheet, by iteratively changing the values in up to 200 other cells related by formulae to the target cell, subject to up to 500 specified numerical constraints. In this geochemical application of the least squares method, Solver minimizes the sum of squared residuals by changing its estimates of the proportions of minerals in a specified assemblage. The residuals represent the differences between the actual (analysed) and estimated (calculated) weight percent proportions of each major chemical component of the rock analysis. The spreadsheet uses a ‘constrained minimization’ approach to ensure realistic estimates of mineral proportions. That is, the Solver is constrained to keep the changing cells (which contain the estimated proportions of specified minerals) at values between 0 and 100%, which sum to less than or equal to 100%. This prevents it from offering solutions involving negative proportions of minerals and unrealistic totals of greater than 100%. DESCRIPTION OF THE MINSQ SPREADSHEET Mineral compositions The upper part of the MINSQ worksheet (Fig. 1), under the heading ‘Mineral Compositions’, contains a table of mineral analyses with the 10 major components of whole rock analyses listed left to right in the customary order. In this example CO2, S, Ba, Cu, Pb and Zn are additional components included to allow for carbonate, baryte- and sulphide-bearing assemblages. All components are in weight percent units. The components and number of components can be modified to suit different types of analyses. Routine XRF analysis usually includes a component called loss on ignition (LOI) which comprises H2O, H2O+ and other volatiles such as S. Unless H2O+ is accurately determined, it is best to leave it out of the calculation 1467-7873/02/$15.00 2002 AEG/Geological Society, London
Fig. 1. Example of the MINSQ worksheet. The analyses tabulated under ‘Wholerock Analyses’ are of altered felsic volcanics associated with ore and in the footwall of the Rosebery massive sulphide deposit, western Tasmania.
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MINSQ – a least squares spreadsheet method and treat the compositions of hydrous minerals as partial analyses (i.e. totalling
