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 analysis of daily variations, observed during 24 months at 94 magnetic observatories, 18 of them in the southern hemisphere. Observatories in polar or equatorial jet regions are not included. For global induction research this is 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 series analysis is a harmonic analysis of single Greenwich days, with a subsequent phase shift to zero time at local midnight. The selection of days is guided by the degree of magnetic activity, and the emphasis is on the analysis of quiet‐time daily variations. To take full advantage of extended periods 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 before 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 harmonic expansions is reversed. For each time harmonic, sums are formed over spherical 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 p th time harmonic, the remaining sums with m = p ± 1, m = p ± 2, … for its part not moving with the speed of the Sun westwards. Up to the fourth harmonic, the spectrum of spherical harmonic coefficients is dominated by the second local‐time term with m = p and n = p +  1, except during solstices. For the fifth and sixth harmonics, this dominance is lost in all seasons.  The choice of spherical terms to be included has been guided by an eigenvalue decomposition of the normal equation matrix to ensure a numerically stable 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 instability sets in with a choice of more than 12 terms, notably in the expansion for the vertical field. With this number of terms, not more than one‐half of the vertical field and about two‐thirds of the horizontal field can be accounted for by spherical harmonics, in the global average. With a hypothetical network of comparable size, but with evenly spaced observing sites, all 36 terms could have been included.