On the solution of certain overdetermined systems of linear equations that arise in geophysics

Recent results in numerical analysis on the solution of overdetermined systems of linear equations by the least-squares method are used to shed more light on a series of papers that have appeared in this Journal, which discussed the best procedure for generating a spherical harmonic analysis of a magnetic disturbance. This discussion was initially stimulated by a suggestion of P. F. Fougere (1963). Fougere's original suggestion has certain computational advantages over other suggestions, but its implementation should be reformulated. As a result of the reformulation, the method of solution for an overdetermined system of linear equations (such as arises in the generation of a spherical harmonic analysis) must be based on the use of modified Gram-Schmidt orthonormalization. Though the generation of a spherical harmonic analysis is the only geophysical application discussed in any detail, the conclusions of this paper apply to any geophysical analysis that reduces to the solution of an overdetermined system of linear equations.