Inverse problem when the source composition is unknown

3 Inverse problem when the source composition is unknown 

Therefore, we want to decide which direction, among all possible choices, is common to all sample subspaces, or, at least, which direction represents the best zone of the sample subspaces in a least-square sense. Since a direction can be completely described by its unit vector, we can restrict the solution set to the surface of the unit sphere centered at the origin. Let us call y the solution of unitary modulus and its projection onto the fcth sample subspace (k = 1. s) represented by the matrix Ak. It is a simple matter to show that 

Finding the least-square solution reduces to minimizing the sum S of squared deviations yk — j) between the estimated source solution and its projection onto each sample subspace. Thus, finding the minimum of 

This solution has very attractive stability properties  

Assumed source concentrations C0 for four arbitrary elements (column 2), mineral 1-liquid and mineral 2-liquid partition coefficients (columns 3 and 4), residual solid-liquid bulk partition coefficients calculated from mineral abundances listed in Table 9.2. Concentration units are arbitrary. 

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