An Application of Oblate Spheroidal Harmonic Functions to the Determination of Geomagnetic potential

Winch, D. E.

doi:10.5636/jgg.19.49

Cited by 11 publications

(17 citation statements)

References 7 publications

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“…Following and evaluating the gradient in ellipsoidal coordinates, the vector components of the magnetic field are related to the magnetic potential as [ Winch , 1967] where X is the local north component, Y is east and Z is the down component of the magnetic field. Inserting the magnetic potential V given by leads to …”

Section: Ellipsoidal Harmonicsmentioning

confidence: 99%

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An ellipsoidal harmonic representation of Earth's lithospheric magnetic field to degree and order 720

Maus

2010

Geochem Geophys Geosyst

108

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[1] While high-degree models of the Earth's gravity potential have been inferred from measurements for more than a decade, corresponding geomagnetic models are difficult to produce. The primary challenge lies in the estimation of the magnetic potential, which is not completely determined by available field intensity measurements and cannot be computed by direct integration. Described here is the methodology behind the third generation of the National Geophysical Data Center's degree 720 magnetic model. Key issues are (1) the ellipsoidal harmonic representation of the magnetic potential, (2) the reduction of ambiguities by a suitable penalty function, and (3) the use of an iterative method to estimate the model coefficients. The NGDC-720 model provides the lithospheric magnetic field vector at any desired location and altitude close to and above the Earth's surface. Anticipated uses are in geological and tectonic studies of the lithosphere, as a local correction for magnetic navigation and heading systems, and the calibration of ground, marine, airborne, and spaceborne magnetometers. The NGDC-720 model is available at http://geomag.org/ models/ngdc720.html and for long-term archive at

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Section: Ellipsoidal Harmonicsmentioning

confidence: 99%

“…[17] Following (8) and evaluating the gradient in ellipsoidal coordinates, the vector components of the magnetic field are related to the magnetic potential as [Winch, 1967]…”

Section: Representation Of Magnetic Vector Componentsmentioning

confidence: 99%

An ellipsoidal harmonic representation of Earth's lithospheric magnetic field to degree and order 720

Maus

2010

Geochem Geophys Geosyst

108

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“…(e.g. Winch 1967), where the coefficients κ l m are real and Q l m is an associated Legendre function of the second kind. We need to calculate (6) in this coordinate system.…”

Section: A Magnetized Oblate Spheroidal Shellmentioning

confidence: 99%

Geomagnetic effects of the Earth’s ellipticity

Jackson¹,

Winch

Lesur³

1999

Geophys. J. Int.

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We analyse the external field generated by a uniform distribution of magnetic susceptibility contained in an oblate spheroidal shell when it is magnetized by an internal magnetic field of arbitrary complexity. The situation is more relevant to the Earth than that of a spherical shell considered by Runcorn (1975a) (in the context of lunar magnetism), because of the larger flattening of the Earth than that of the Moon. We find that, to first order in the susceptibility, each internal harmonic in a spheroidal harmonic expansion of the magnetic potential generates just one non‐vanishing external field coefficient, unlike in the spherical case when all harmonics vanish identically. The field generated is proportional to the susceptibility, thickness of the shell and square of the Earth’s eccentricity, and hence it appears that this field amplification mechanism will be very ineffective for the Earth.

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“…In geomagnetism these spheroidal harmonics were first introduced by Schmidt (1889 theory, 1895 application), who performed a numerical analysis based on data scaled from charts. Winch (1967) introduced the topic again, clarifying and simplifying Schmidt's approach, and reworking the Schmidt data, mainly in the context of improving the separation of the fields having their origin internal and external to the Earth's spheroidal surface. [Because they were working with data at the Earth's surface, these authors were able to use an approximation, valid to first order in ( E / u ) 2 to calculate dU/d u .]…”

Section: The Oblate Spheroidmentioning

confidence: 99%

“…Yu & Cao (1996), eq. (2.1), or, in a slightly different notation, Winch (1967)] where r =√( u 2 + E 2 ) is the radius of the auxiliary sphere, the (equatorial) semi‐major axis of the spheroid.…”

Section: The Oblate Spheroidmentioning

confidence: 99%

Orthogonality of harmonic potentials and fields in spheroidal and ellipsoidal coordinates: application to geomagnetism and geodesy

Lowes¹,

Winch²

2012

Geophysical Journal International

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SUMMARY We investigate the orthogonality of the potential distributions that are the basis solutions of Laplace's equation appropriate to 3‐D ellipsoidal (including spheroidal) coordinate systems, and also the orthogonality of the corresponding vector gradient fields, both over the surface of the ellipsoid, and for integration over the volume of the annular shell between two confocal ellipsoids. 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. The lack of orthogonality over the surface means that care must be taken when interpreting conventional geomagnetic ‘power spectra’, and geodetic ‘degree variance’, as these no longer correspond exactly to the mean‐square values over the actual ellipsoidal surface. We illustrate some of the problems by comparing different versions of the power spectrum for a spheroidal analysis of the global lithospheric magnetic field. We use only simple vector algebra, and do not need to know the details of the actual basis solutions, only that they are the product of three functions, one for each coordinate and involving only that coordinate, and that they satisfy Laplace's equation. Similarly, our results do not depend on the normalization used in the basis functions.

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An Application of Oblate Spheroidal Harmonic Functions to the Determination of Geomagnetic potential

Abstract: The general theory of harmonic

Cited by 11 publications

References 7 publications

An ellipsoidal harmonic representation of Earth's lithospheric magnetic field to degree and order 720

An ellipsoidal harmonic representation of Earth's lithospheric magnetic field to degree and order 720

Geomagnetic effects of the Earth’s ellipticity

Orthogonality of harmonic potentials and fields in spheroidal and ellipsoidal coordinates: application to geomagnetism and geodesy

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