Computer Methods for Geochemical and Petrologic Mixing Problems
Mixing problems arise frequently in examinations of compositional variations in rock bodies, particularly in studies of magmatic differentiation. At one time the problems were examined graphically, but ten years ago geochemists and petrologists were shown the matrix algebra that could be used to obtain least-squares solutions. Computer programs based on the fundamental matrix operations, and variations of them, have been circulated broadly and used widely by petrologists.
More recently, it has been determined that the methods of Q-mode factor analysis, extended for treating compositional data, are well suited for all types of chemical and mineralogic mixing problems. The methods can be used not only to estimate the mixing proportions, but also to determine the number of end members required in a given problem and to aid in determination of endmember compositions. The first step is to derive a matrix of recomputed data. The recomputed data matrix can approximate closely the original data matrix even though it may be of lower rank. The rank equals the number of end members required in the mixing model. Possible end-member compositions are represented by vectors in the same space as the row vectors in the matrix of recomputed data. Various methods can be used to select end-member vectors that might represent the compositions of materials involved in the mixing process. Also, selected compositions can be tested individually for mathematical suitability and modified accordingly. Interactive computer programs allow one to test geochemical hypotheses by trying various sets of end-member compositions until the derived mixing proportions are compatible with all that is known about the samples and the geologic environment from which they were collected.
KeywordsSample Vector Original Data Matrix Initial Scaling Varimax Factor Geologic Ground
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