A criticism of the Gram-Schmidt Orthogonalization process applied to spherical harmonic analysis

In a recent series of papers on a new method of spherical harmonic analysis of the geomagnetic field using observatory data, the Gram-Schmidt process has been presented as a method with important advantages over the solution of the normal equations by compact elimination methods, such as Crout's method, Doolittle's method, or the modernized Gauss method. The main feature of the Gram-Schmidt process is the generation of a series of quantities by means of scalar products of vectors. It is proved here that these quantities are identical with those that arise in the solution of the normal equations by compact elimination methods. On the basis of this identity, the advantages claimed for the Gram-Schmidt process have been examined and are shown to be merely interpretations of well-known statistical procedures in terms of orthogonal vectors. One important procedure not discussed by Fougere has been presented in terms of his orthogonal vectors.