S U M M A R YThe lithospheric contribution to the geomagnetic field arises from magnetized rocks in a thin shell at the Earth's surface. The lithospheric field can be calculated as an integral of the distribution of magnetization using standard results from potential theory. Inversion of the magnetic field for the magnetization suffers from a fundamental non-uniqueness: many important distributions of magnetization yield no potential magnetic field outside the shell. We represent the vertically integrated magnetization (VIM) in terms of vector spherical harmonics that are new to geomagnetism. These vector functions are orthogonal and complete over the sphere: one subset (I) represents the part of the magnetization that produces a potential field outside the shell, the observed field; another subset (E) produces a potential field exclusively inside the shell; and a third, toroidal, subset (T ) produces no potential field at all. E and T together span the null space of the inverse problem for magnetization with perfect, complete data. We apply the theory to a recent global model of VIM, give an efficient algorithm for finding the lithospheric field, and show that our model of magnetization is dominated by E, the part producing a potential field inside the shell. This is largely because, to a first approximation, the model was formed by magnetizing a shell with a substantial uniform component by an potential field originating inside the shell. The null space for inversion of lithospheric magnetic anomaly data for VIM is therefore huge. It can be reduced if the magnetization is assumed to be induced by a known inducing field, but the null space for susceptibility is not so easily recovered.

SUMMARY Spherical harmonic analysis of the main magnetic field of the Earth and its daily variations is the numerical determination of coefficients of solid spherical harmonics in the mathematical expressions used for the magnetic scalar potential of fields of internal and external origin. The coefficients are determined from vector components of the field and their purpose is to represent the vector field, not to reconstruct the magnetic scalar potential. An alternative interpretation of the spherical harmonic analysis is presented: namely the determination of the coefficients of a series representation of the magnetic vector field on a spherical surface in orthonormal real vector spherical harmonics, which correspond to the internal and external fields, and an additional non‐potential toroidal field. The numerical values of the coefficients of an orthonormal vector spherical harmonic series have a direct physical significance, which is not obscured by some arbitrary normalization of the vector spherical harmonics. Therefore, we propose a Schmidt vector normalization to be used in conjunction with the Schmidt quasi‐normalization of associated Legendre functions. A property of orthonormalized functions is that the standard deviations of the coefficients determined by the method of least squares from ideal data, which are uniformly accurate and uniformly globally distributed, are constant for all coefficients. The real vector spherical harmonic analysis of the geomagnetic field is extended to a spherical shell and conditions that restrict the radial dependence of the vector spherical harmonic coefficients are examined. In particular, two hypotheses for the current systems deriving from the non‐potential toroidal component of the magnetic field over the surface of a sphere are presented, namely, Earth–air currents and field‐aligned currents.

[1] Previous studies of the longitudinal variation of the local noon electrojet have yielded doubtful results either because of the poor data quality or because the local times of equatorial crossings occurred in the early morning or late afternoon. The recent launch of the Ørsted satellite in a near-circular orbit with slow drift in local time of equatorial crossing has provided the opportunity for researchers to study the electrojet more accurately. Most studies remove the main field using a spherical harmonic model and then search the daytime equatorial passes for the distinctive electrojet trough in total intensity. The present study examines the electrojet for two consecutive 6-month periods and consequently two local time ranges. Pure signal processing is used to remove the main field directly. The residuals are binned separately for night and day passes on a 1°by 1°grid to enhance the signal to noise ratio and are bin centered by a least squares fitted linear model to compensate for the variations in satellite altitude. Thereafter, for each period the compensated night and day binned values are subtracted from each other to produce a difference set. Global plots of the subsequently spatially filtered difference sets reveal an almost constant electrojet 1/e half width of 3°, as seen at satellite altitude apart from a region in the western Pacific. There are four maxima in the electrojet amplitude at 0°-30°E, 90°-120°E, 180°-220°E, and 260°-290°E in each local time range.

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