METHODS TO REDUCE ALIASING IN SPHERICAL
HARMONIC ANALYSIS
ABSTRACT
Christopher J ekeli
Department of Geodetic Science and Surveying
The Ohio State University, Columbus, OH 43210 USA
The effect of aliasing is rigorously formulated for various currently practiced methods of
harmonic analysis of gravity anomalies on the sphere. This shows clearly not only the
suboptimality, in terms of aliasing error, of some methods, but also the inappropriateness
of uniformly weighted averaging of gravity anomalies in latitudellongitude grid cells. The
following results are obtained: 1) The simple quadratures method and related methods of
analysis are biased even with band-limited functions. 2) A modification of Colombo's
method of least squares, requiring only a slight increase in number of computations, further
reduces the aliasing error. 3) The essential elimination of aliasing can only be effected with
weighted, spherical cap averages, not with the often used, unweighted, constant angular
block averages.
INTRODUCTION
The harmonic (or spectral) representation of functions on the sphere has proved useful in
modeling the Earth's gravity field, and today's models include over 130 thousand terms
(Rapp and Pavlis, 1990), representing details of the field with surface resolution of 50 km.
In theory, the spherical spectrum of the gravity field is infinite (and discrete) because the
function it represents is continuous (and periodic; e.g., the period in longitude is 21t). But
in practice, only a finite number of function values, usually on some regular grid, is
available. This means not only that merely a finite number of spectral coefficients can be
determined, but that these are biased by the spectral content not directly observable. This is
a well known phenomenon in spectral analysis, where even though one applies orthogonal
operators to the data, they cannot fIlter out harmonics fmer in detail than that dictated by the
sampling interval - the orthogonality goes only so far. This is known as "aliasing."
Spectral aliasing in functions defined on the line or on the Cartesian plane is well
understood; it is less obvious on a non-Euclidean surface such as the sphere.
The spherical harmonic analysis is examined here with the object of clearly understanding
the effect of aliasing in conventional techniques, as well as in the modern techniques
developed by Colombo (1981) that are used to compute today's models. The investigation
is accomplished by developing a one-to-one correspondence between the spectral analysis
on the sphere and the spectral analysis on the plane, where aliasing is more easily
formulated. This leads to straightforward understanding of how well various smoothing
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HARMONIC ANALYSIS
ABSTRACT
Christopher J ekeli
Department of Geodetic Science and Surveying
The Ohio State University, Columbus, OH 43210 USA
The effect of aliasing is rigorously formulated for various currently practiced methods of
harmonic analysis of gravity anomalies on the sphere. This shows clearly not only the
suboptimality, in terms of aliasing error, of some methods, but also the inappropriateness
of uniformly weighted averaging of gravity anomalies in latitudellongitude grid cells. The
following results are obtained: 1) The simple quadratures method and related methods of
analysis are biased even with band-limited functions. 2) A modification of Colombo's
method of least squares, requiring only a slight increase in number of computations, further
reduces the aliasing error. 3) The essential elimination of aliasing can only be effected with
weighted, spherical cap averages, not with the often used, unweighted, constant angular
block averages.
INTRODUCTION
The harmonic (or spectral) representation of functions on the sphere has proved useful in
modeling the Earth's gravity field, and today's models include over 130 thousand terms
(Rapp and Pavlis, 1990), representing details of the field with surface resolution of 50 km.
In theory, the spherical spectrum of the gravity field is infinite (and discrete) because the
function it represents is continuous (and periodic; e.g., the period in longitude is 21t). But
in practice, only a finite number of function values, usually on some regular grid, is
available. This means not only that merely a finite number of spectral coefficients can be
determined, but that these are biased by the spectral content not directly observable. This is
a well known phenomenon in spectral analysis, where even though one applies orthogonal
operators to the data, they cannot fIlter out harmonics fmer in detail than that dictated by the
sampling interval - the orthogonality goes only so far. This is known as "aliasing."
Spectral aliasing in functions defined on the line or on the Cartesian plane is well
understood; it is less obvious on a non-Euclidean surface such as the sphere.
The spherical harmonic analysis is examined here with the object of clearly understanding
the effect of aliasing in conventional techniques, as well as in the modern techniques
developed by Colombo (1981) that are used to compute today's models. The investigation
is accomplished by developing a one-to-one correspondence between the spectral analysis
on the sphere and the spectral analysis on the plane, where aliasing is more easily
formulated. This leads to straightforward understanding of how well various smoothing
121