Estimating proportions in geochemical mixing equations by Chebyshevʹs method

A computer program was written for estimating components in a mixture by solving an overdetermined system of linear equations by Chebyshevʹs method. An overdetermined system of linear equations contains more equations than unknowns. If the equations are uncertain, then, in general, no exact solution exists. A feasible solution can nevertheless be found by minimising the total error of the equations. In Chebyshevʹs method the problem is transformed to a linear programming problem, which is solved. The program was tested on two artificial mixtures containing 4 and 10 rock types, respectively. The computation was based on geochemical XRF analyses of the components and mixtures. Best results were obtained by removing weakly fitting elements from the model in 2 or 3 steps, so that the number of elements then became 2 or 3 times the number of components. In the best solution for the mixture of 4 components the error, defined as the largest difference between the computed and true compositions, was 4%. For 10 components the error was 8.5%. Investigation of the influence of missing components on the residuals of the equations showed the largest absolute residual to increase drastically when more than 40% of the components were missing from the model. The suitability of the program was also tested on fine fractions of till, for which the proportions of glacially ground rock types and sediments were estimated.