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The only situation for which there is orthogonality is for the vector gradients when integrated over the annular shell. In the other three cases potential over surface or annulus, and field over surface orthogonality can be restored by using an appropriate geometrical weighting factor applied to the integrand; it is therefore still possible to perform the equivalent of a classical spherical harmonic analysis.
In the special case of the sphere, there is real orthogonality in all four cases; in effect the weighting factors are all unity. In geodesy, spheroidal harmonic analysis is done using a method that relies on a particular result valid only for potential; it cannot be extended to the corresponding vector field, or to ellipsoidal geometry. We illustrate some of the problems by comparing different versions of the power spectrum for a spheroidal analysis of the global lithospheric magnetic field.
Similarly, our results do not depend on the normalization used in the basis functions. Geopotential theory , Magnetic field 1 Introduction Because the Earth is roughly spherical, it is customary to express the scalar geomagnetic or gravitational potential as a series of spherical harmonics.
The spherical harmonics have the advantage that the different basis harmonic functions are orthogonal over the sphere, making the determination of the associated numerical coefficients easier in theoretical problems.
And when using numerical data the arithmetic computation of the coefficients by quadrature or least squares is better behaved if data are or are weighted to look like roughly equally spaced over the sphere. However, the real Earth is not spherical. For a long time the only magnetic data available were on the surface of the Earth, which is more nearly spheroidal than spherical; also the directions of the magnetic field components were based on the geodetic coordinate system rather than a geocentric system.
It was very soon suggested that it might be desirable to allow for these differences before making a spherical harmonic analysis. Many authors suggested a variety of methods for approximately allowing for the difference, either by correcting the data before analysis, or correcting the Gauss coefficients after analysis. Similarly, workers producing gravity models used increasingly complicated approximate methods to extrapolate data from the spheroidal surface to a spherical surface, so that conventional spherical boundary-value methods could be used to high degree.
In geomagnetism, the first use of spheroidal harmonics, as opposed to spherical harmonics, was by Schmidt , Then availability of electronic computers meant that the full 3-D situation up to the currently available degree could be handled using spherical harmonics.
So in geomagnetism, for a long time there was no interest in using spheroidal harmonics. However, recently in some situations it is now seen to be beneficial to make the analysis itself using spheroidal harmonics.
Maus has produced a high degree spheroidal harmonic analysis of the near-surface magnetic field produced by the magnetization of crustal rocks. Similarly, in terrestrial gravity the high-degree components are better fitted by using a high degree spheroidal analysis, for example, Jekeli There are also many other smaller-scale situations where spheroidal and ellipsoidal harmonics are more appropriate.
Historically there has been a different approach to the application of spheroidal harmonics and of ellipsoidal harmonics. Although spheroidal harmonics are well described in many texts, there is very little discussion of whether the spheroidal harmonics are analogous to spherical harmonics in being orthogonal over the spheroidal surface; we show that they are not!
However, we also show that there exists a simple geometrical weighting factor that restores effective orthogonality. In contrast, most of the texts on ellipsoidal harmonics implicitly accept that the harmonics themselves are not orthogonal over the surface, and that a weighting factor is needed to restore orthogonality. And in both geometries, none of the texts discuss the situation, for example in geomagnetism, where what is of interest are the vector fields that are the gradient of the potentials, rather than the potentials themselves.
This paper aims to fills these gaps. Similarly, the reader does not need to know what normalization is being used. It turns out there is not orthogonality of the potentials over the surface of the ellipsoid including the spheroid , except for the special case of the sphere. Nor is there orthogonality when the integration is extended over the volume between two confocal ellipsoids. For the corresponding vector fields again there is not orthogonality over the surface except for the sphere , but there is orthogonality over the ellipsoidal shell.
However, it is always possible to use a geometrical weighting factor that restores orthogonality. Workers in geodesy have for some time used a spheroidal harmonic approach that, in effect, uses this weighting factor, but without making this explicit. On the other hand, users of ellipsoidal harmonics in smaller-scale problems in physics have applied the weighting function explicitly.
After explaining some conventions, in Section 2 we summarize the situation for the sphere, and in Section 3 we consider in detail the case of the oblate spheroid, because of its comparative simplicity and its good approximation to the shape of the Earth. Then in Section 4 we show that it does not matter if the spheroid is oblate or prolate, and in Section 5 that the orthogonality results we derived for the spheroid hold also for the general ellipsoid.
In Section 6 we discuss the transformation between the spherical and spheroidal harmonic representations, and the problem of convergence, and in Section 7 we discuss the implications of our results to the meaning of geodetic degree variances and geomagnetic field power spectra.
We end with a general discussion and summary. However, in this paper we need to be clear that the spheroid and sphere are special cases of the ellipsoid, so we retain the distinction between ellipsoid, spheroid and sphere throughout. Italic font is used for real, arbitrary, dimensioned, potentials V, and their corresponding vector fields B. In this paper S is the surface of a sphere, spheroid or ellipsoid. Workers in geomagnetism and in geodesy use different normalizations of the Legendre functions.
In this paper, except in Section 7 , none of the equations depends on the use of a particular normalization of the basis functions, provided the same normalization is used throughout; in Section 7 we say explicitly when we assume the conventional geomagnetic Schmidt semi-normalization as used with real functions. We use the geomagnetic convention in which these Vnm are non-dimensional, and the gnm are numerical coefficients appropriate to the reference radius, and having the dimensions of magnetic field.
The theory of spherical and ellipsoidal harmonics,
The theory of spherical and ellipsoidal harmonics,
The theory of spherical and ellipsoidal harmonics