EarthRef.org Reference Database (ERR) -- Campbell 2004

I thank Maus et al. for providing this opportunity to explain spherical harmonic analysis (SHA) methods to our Eos readers. Gauss devised the SHA as a means for separating the external and internal geomagnetic field contributions at an analysis spherical surface. The SHA starts with the selection of an analysis center and axis through that center defining the study coordinates of longitude F, latitude ?, and radius r of the analysis sphere. Using Maxwell's equations, the three orthogonal geomagnetic field components at the analysis spherical surface are converted to equivalent potential function values to which two special series of terms are then fitted. One series has coefficients of increasing powers of r (the external separated field part) and the other has coefficients of increasing powers of 1/r (the internal part). The sought internal field is then represented by a double summation of order m (m=0 to n) and degree n (n=1 to 8, with n = m) containing coefficients gnm and hnm for the cosine(mF), sine(mF), and Legendre polynomials, Pnm(?). This procedure is somewhat like fitting the daily amplitude of quiet daily field change to a Fourier series of the harmonic oscillating ups and downs of the cosine and sine terms; only now we are fitting the surface variations of the potential function to the appropriate amplitudes of the bulges and depressions in the harmonic polynomials over a spherical surface, with their symmetry defined by F and ?. I thank Maus et al. for providing this opportunity to explain spherical harmonic analysis (SHA) methods to our Eos readers. Gauss devised the SHA as a means for separating the external and internal geomagnetic field contributions at an analysis spherical surface. The SHA starts with the selection of an analysis center and axis through that center defining the study coordinates of longitude F, latitude ?, and radius r of the analysis sphere. Using Maxwell's equations, the three orthogonal geomagnetic field components at the analysis spherical surface are converted to equivalent potential function values to which two special series of terms are then fitted. One series has coefficients of increasing powers of r (the external separated field part) and the other has coefficients of increasing powers of 1/r (the internal part). The sought internal field is then represented by a double summation of order m (m=0 to n) and degree n (n=1 to 8, with n = m) containing coefficients gnm and hnm for the cosine(mF), sine(mF), and Legendre polynomials, Pnm(?). This procedure is somewhat like fitting the daily amplitude of quiet daily field change to a Fourier series of the harmonic oscillating ups and downs of the cosine and sine terms; only now we are fitting the surface variations of the potential function to the appropriate amplitudes of the bulges and depressions in the harmonic polynomials