A spherical harmonic analysis of solar daily variations in the years 1964-1965: response estimates and source fields for global induction - I. Methods

This work is based on a time harmonic and subsequent spherical harmonic ana lysis of daily variations, observed during 24 months at 94 magnetic observa tories, 18 of them in the southern hemisphere. Observatories in polar or eq uatorial jet regions are not included. For global induction research this i s a return to the classical potential method. It requires a dual spherical harmonic analysis of horizontal and vertical components, and thereby allows their separation into internal and external parts. The preceding time seri es analysis is a harmonic analysis of single Greenwich days, with a subsequ ent phase shift to zero time at local midnight. The selection of days is gu ided by the degree of magnetic activity, and the emphasis is on the analysi s of quiet-time daily variations. To take full advantage of extended period s of quietness, a new measure is introduced, based on a fixed threshold for the sum of ap indices on the respective day and the adjoining half-days be fore and after. The spherical harmonic analysis is carried out with time harmonics from all observatories except two, but weights are assigned to them to reduce their hemispherical imbalance. Time harmonics refer either to mean monthly daily variations or to those on single days. Noting that daily variations depend primarily on local time, the usual order of summations in spherical harmon ic expansions is reversed. For each time harmonic, sums are formed over sph erical terms of the same order m and ascending degree from n = \m\ onwards. The first partial sum is with m = p for the local-time part of the pth tim e harmonic, the remaining sums with m = p +/- 1, m = p +/- 2,... for its pa rt not moving with the speed of the Sun westwards. Up to the fourth harmoni c, the spectrum of spherical harmonic coefficients is dominated by the seco nd local-time term with m = p and n = p + 1, except during solstices. For t he fifth and sixth harmonics, this dominance is lost in all seasons. The choice of spherical terms to be included has been guided by an eigenval ue decomposition of the normal equation matrix to ensure a numerically stab le least-squares solution. No generalized inverse is used in order to allow a term-by-term determination of the expansion coefficients. In tests, the total number of terms has been Varied between one and 36. Numerical instabi lity sets in with a choice of more than 12 terms, notably in the expansion for the vertical held. With this number of terms, not more than one-half of the vertical field and about two-thirds of the horizontal held can be acco unted for by spherical harmonics, in the global average. With a hypothetica l network of comparable size, but with evenly spaced observing sites, all 3 6 terms could have been included.