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UBC Theses and Dissertations

A new approach to mass balance modeling: applications to igneous petrology Ke, Peiwan


An overdetermined linear system of equations is proposed to investigate mass balance relationships in igneous systems. A chi square (x²) minimization technique is used to solve the set of linear equations. The x² technique is superior to R² techniques because it considers the analytical errors on measurement data. Singular Value Decomposition (SVD) is used to search for the x² optimal solution. The attributes of the x² technique are discussed. The x² minimization fitting technique, combined with SVD methods allows the confidence regions to be calculated and expressed by a geometric shape centered on the x² optimal solution. The geometric shape varies with the dimension of the linear system: one-dimensional cases yield a line fragment; two-dimensional cases yield an ellipse and the three dimensions require an ellipsoid. The size of the confidence regions is determined by the confidence limits and is related to the coefficient matrix of the linear problem. Model with smaller confidence regions are considered to be better than models associated with larger confidence regions where the same confidence limit is used. Therefore, the confidence region can be used to select the model that best explains chemical differentiation in igneous system. Synthetic data sets have been analyzed using the new program MASSBAL.0 which solves the mass balance problems using the principles of the x² technique and SVD. The x² technique presents a solution closer to the true one than does the traditional residual minimization fitting (R² technique). It overcomes the deficiencies that the R² technique has and appears to be a more robust and sensitive fitting technique. The x² technique has been applied to the apparent clinopyroxene paradox formed in lavas from Diamond Craters, Oregon. Three models are proposed to explain the chemical difference: a three phase model (01+P1+Px), a two phase model (O1+Pl) and a one phase model (01). The x² mass balance modeling results show that the three phase model (O1+Pl+Px) has the smallest minimum x² value and the most compact confidence region; thus it is preferred and clinopyroxene did participate in the fractionation process. The x² technique has also been applied to study the assimilation problem in Paricutin volcano and the fractionation problem in PLA group of Mount Meager volcanic complex. The mass balance calculations show that assimilation has play a major role in Paricutin volcano and about 15% xenolith has contaminated the magma. In PLA fractionation processes, plagioclase and amphibole are shown to have caused the chemical difference between lavas, although other phenocrysts ( biotite and pyroxene ) did not participate.

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