As noted in the paragraph after equation (26), the aliasing effect obtained with the
constant angular block average is much larger because, even in the ideal case, the spectrum
in region A+B is passed without attenuation. Therefore, the spectral region A+B of the
point function contributes entirely to the aliasing error in this kind of average; 16 mgal and
19 cm according to the TIR model.
SUMMARY
Two methods of harmonic analysis on the sphere were investigated from the point of view
of aliasing. The aliasing error was formulated rigorously in terms of spherical harmonic
coefficients in the case when the function is sampled on a regular grid of latitudes and
longitudes. This yielded several results, some of which are known, but perhaps more
analytically illustrated, here: 1) The simple quadratures method is biased even with bandlimited functions. 2) The proposed new method eliminates this bias. It is superior even to
Colombo's (1981) method of least squares (that also eliminates this bias) because it further
reduces the aliasing error by doubling the number of estimated coefficients with minimal
computational increase. 3) The substantial reduction of aliasing can only be effected with
spherical cap averages, not with the often used constant angular block averages.
So far, the data measurement noise has not been considered. This is an important aspect
of modem harmonic analyses. However, the computational tractability of the least-squares
analysis for high-degree models (such as K = 360) restricts the allowable variances and
covariances among the errors of the given data (Colombo, 1981). The new method
described here has no such restriction because it is not a least-squares analysis, since as
many harmonic coefficients as observations are estimated.
Acknowledgment. This work was supported by contracts FI9628-93-K-0033 and
FI9628-94-K-0005 with the Air Force Phillips Laboratory, Hanscom AFB, MA.
References
Colombo, O.L. (1981). Numerical methods for harmonic analysis on the sphere, Report
no.31O, Department of Geodetic Science, The Ohio State University . .
Gaposchkin, E.M. (1980). Averaging on the surface of a sphere, .I. Geophys. Res.,
85(B6), 3187-3193.
Jekeli, C. (1981). Alternative methods to smooth the earth's gravity field, Report no.327,
Department of Geodetic Science and Surveying, Ohio State University.
Jekeli, C. (1995). Spherical harmonic analysis, aliasing, and filtering, Manuscripta
Geodaetica, in press.
Moritz, H. (1989). Advanced Physical Geodesy, 2nd ed., Wichmann Verlag, Karlsruhe.
Rapp, R.H. and N.K. Pavlis (1990). The development and analysis of geopotential
coefficient models to spherical harmonic degree 360, J. Geophys. Res., 95(B 13),
21885-21911.
Sjoberg, L. (1980). A recurrence relation for the ~n-function, Bulletin Geodesique,
54(1), 69-72.
130
constant angular block average is much larger because, even in the ideal case, the spectrum
in region A+B is passed without attenuation. Therefore, the spectral region A+B of the
point function contributes entirely to the aliasing error in this kind of average; 16 mgal and
19 cm according to the TIR model.
SUMMARY
Two methods of harmonic analysis on the sphere were investigated from the point of view
of aliasing. The aliasing error was formulated rigorously in terms of spherical harmonic
coefficients in the case when the function is sampled on a regular grid of latitudes and
longitudes. This yielded several results, some of which are known, but perhaps more
analytically illustrated, here: 1) The simple quadratures method is biased even with bandlimited functions. 2) The proposed new method eliminates this bias. It is superior even to
Colombo's (1981) method of least squares (that also eliminates this bias) because it further
reduces the aliasing error by doubling the number of estimated coefficients with minimal
computational increase. 3) The substantial reduction of aliasing can only be effected with
spherical cap averages, not with the often used constant angular block averages.
So far, the data measurement noise has not been considered. This is an important aspect
of modem harmonic analyses. However, the computational tractability of the least-squares
analysis for high-degree models (such as K = 360) restricts the allowable variances and
covariances among the errors of the given data (Colombo, 1981). The new method
described here has no such restriction because it is not a least-squares analysis, since as
many harmonic coefficients as observations are estimated.
Acknowledgment. This work was supported by contracts FI9628-93-K-0033 and
FI9628-94-K-0005 with the Air Force Phillips Laboratory, Hanscom AFB, MA.
References
Colombo, O.L. (1981). Numerical methods for harmonic analysis on the sphere, Report
no.31O, Department of Geodetic Science, The Ohio State University . .
Gaposchkin, E.M. (1980). Averaging on the surface of a sphere, .I. Geophys. Res.,
85(B6), 3187-3193.
Jekeli, C. (1981). Alternative methods to smooth the earth's gravity field, Report no.327,
Department of Geodetic Science and Surveying, Ohio State University.
Jekeli, C. (1995). Spherical harmonic analysis, aliasing, and filtering, Manuscripta
Geodaetica, in press.
Moritz, H. (1989). Advanced Physical Geodesy, 2nd ed., Wichmann Verlag, Karlsruhe.
Rapp, R.H. and N.K. Pavlis (1990). The development and analysis of geopotential
coefficient models to spherical harmonic degree 360, J. Geophys. Res., 95(B 13),
21885-21911.
Sjoberg, L. (1980). A recurrence relation for the ~n-function, Bulletin Geodesique,
54(1), 69-72.
130