Estimating proportions in geochemical mixing equations by Chebyshev's method

A computer program was written for estimating components in a mixture by so lving an overdetermined system of linear equations by Chebyshev's method. A n overdetermined system of linear equations contains more equations than un knowns. If the equations are uncertain, then, in general, no exact solution exists. A feasible solution can nevertheless be found by minimising the to tal error of the equations. In Chebyshev's method the problem is transforme d to a linear programming problem, which is solved. The program was tested on two artificial mixtures containing 4 and 10 rock types, respectively. Th e computation was based on geochemical XRF analyses of the components and m ixtures. Best results were obtained by removing weakly fitting elements fro m the model in 2 or 3 steps, so that the number of elements then became 2 o r 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 comp uted and true compositions, was 4%. For 10 components the error was 8.5%. I nvestigation of the influence of missing components on the residuals of the equations showed the largest absolute residual to increase drastically whe n more than 40% of the components were missing from the model. The suitabil ity of the program was also tested on fine fractions of till, for which the proportions of glacially ground rock types and sediments were estimated. ( C) 1998 Elsevier Science Ltd. All rights reserved.