Band-limited functions on a bounded spherical domain: the Slepian problem on the sphere

Albertella, A.; Sansò, F.; Sneeuw, Nico

doi:10.1007/pl00003999

Cited by 80 publications

(87 citation statements)

References 11 publications

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“…Since timelimited functions cannot be simultaneously bandlimited in the frequency domain, nor vice versa, the optimally concentrated signal is considered to be the one with the least energy outside the interval of interest. The concentration problem has been extended and generalized for the purpose of signal estimation, representation and analysis on geographical domains by Albertella et al (1999), Pail et al (2001) and in geodesy, and by Wieczorek and Simons (2007) and Dahlen and Simons (2008) in more general settings. The quadratic maximization of the spatial energy of bandlimited functions is one way to achieve localization in one domain while curbing leakage in the other.…”

Section: The Spherical Slepian Basismentioning

confidence: 99%

The spherical Slepian basis as a means to obtain spectral consistency between mean sea level and the geoid

Slobbe

Simons

Klees

2012

J Geod

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The mean dynamic topography (MDT) can be computed as the difference between the mean sea level (MSL) and a gravimetric geoid. This requires that both data sets are spectrally consistent. In practice, it is quite common that the resolution of the geoid data is less than the resolution of the MSL data, hence, the latter need to be low-pass filtered before the MDT is computed. For this purpose conventional low-pass filters are inadequate, failing in coastal regions where they run into the undefined MSL signal on the continents. In this paper, we consider the use of a bandlimited, spatially concentrated Slepian basis to obtain a low-resolution approximation of the MSL signal. We compute Slepian functions for the oceans and parts of the oceans and compare the performance of calculating the MDT via this approach with other methods, in particular the iterative spherical harmonic approach in combination with Gaussian low-pass filtering, and various modifications. Based on the numerical experiments, we conclude that none of these methods provide a lowresolution MSL approximation at the sub-decimetre level. In particular, we show that Slepian functions are not appropriate basis functions for this problem, and a Slepian representation of the low-resolution MSL signal suffers from broadband leakage. We also show that a meaningful definition of a lowresolution MSL over incomplete spherical domains involves orthogonal basis functions with additional properties that Slepian functions do not possess. A low-resolution MSL signal, spectrally consistent with a given geoid model, is obtained by a suitable truncation of the expansions of the MSL signal in terms of these orthogonal basis functions. We compute one of these sets of orthogonal basis functions using the Gram-Schmidt orthogonalization for spherical harmonics. For the oceans, we could construct an orthogonal basis only for resolutions equivalent to a spherical harmonic degree 36. The computation of a basis with a higher resolution fails due to inherent instabilities. Regularization reduces the instabilities but destroys the orthogonality and, therefore, provides unrealistic low-resolution MSL approximations. More research is needed to solve the instability problem, perhaps by finding a different orthogonal basis that avoids it altogether.

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Section: The Spherical Slepian Basismentioning

confidence: 99%

The spherical Slepian basis as a means to obtain spectral consistency between mean sea level and the geoid

Slobbe

Simons

Klees

2012

J Geod

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“…The problem is due to the polar gaps, that is to the fact that the nonpolar orbit of GOCE results in subsatellite tracks never crossing the two polar caps (with latitude above 83 • .5 and below −83 • .5). This is related to the non-orthogonality of the spherical harmonics over a latitude band (Albertella et al, 1999;Pail et al, 2001). …”

Section: Consequences Of Polar Gapsmentioning

confidence: 99%

A timewise kinematic method for satellite gradiometry: GOCE simulations

Milani

Rossi

Villani³

2005

Earth Moon Planet

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Abstract. We have defined new algorithms for the data processing of a satellite geodesy mission with gradiometer (such as the next European mission GOCE) to extract the information on the gravity field coefficients with a realistic estimate of their accuracy. The large scale data processing can be managed by a multistage decomposition. First the spacecraft position is determined, i.e., a kinematic method is normally used. Second we use a new method to perform the necessary digital calibration of the gradiometer. Third we use a multiarc approach to separately solve for the global gravity field parameters. Fourth we use an approximate resonant decomposition, that is we partition in a new way the harmonic coefficients of the gravity field. Thus the normal system is reduced to blocks of manageable size without neglecting significant correlations. Still the normal system is badly conditioned because of the polar gaps in the spatial distribution of the data. We have shown that the principal components of the uncertainty correspond to harmonic anomalies with very small signal in the region where GOCE is flying; these uncertainties cannot be removed by any data processing method. This allows a complete simulation of the GOCE mission with affordable computer resources. We show that it is possible to solve for the harmonic coefficients up to degree 200 ÷ 220 with error to signal ratio ≥ 1, taking into account systematic measurement errors. Errors in the spacecraft orbit, as expected from state of the art satellite navigation, do not degrade the solution. Gradiometer calibration is the main problem. By including a systematic error model, we have shown that the results are sensitive to spurious gradiometer signals at frequencies close to the lower limit of the measurement band. If these spurious effects grow as the inverse of the frequency, then the actual error is larger than the formal error only by a factor 2, that is the results are not compromised.

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“…For the polar gap problem we refer to the literature, e.g. Sneeuw and van Gelderen (1997) or Albertella et al (1999).…”

Section: Introductionmentioning

confidence: 99%

An improved sampling rule for mapping geopotential functions of a planet from a near polar orbit

Weigelt

Sneeuw

Schrama

et al. 2012

J Geod

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One of the limiting factors in the determination of gravity field solutions is the spatial sampling. Especially during phases, when the satellite repeats its own track after a short time, the spatial resolution will be limited. The Nyquist rule-of-thumb for mapping geopotential functions of a planet, also referred to as the Colombo-Nyquist rule-ofthumb, provides a limit for the maximum achievable degree of a spherical harmonic development for repeat orbits. We show in this paper that this rule is too conservative and solutions with better spatial resolutions are possible. A new rule is introduced which limits the maximum achievable order (not degree!) to be smaller than the number of revolutions if the difference between the number of revolutions and the number of nodal days is of odd parity and to be smaller than half the number of revolutions if the difference is of even parity. The dependence on the parity is reflected in the eigenvalue spectrum of the normal matrix and becomes especially important in the presence of noise. The rule is based on applying the Nyquist sampling theorem separately in North-South and East-West direction. This is only possible for satellites in highly inclined orbits like e.g. CHAMP and GRACE. Tables for these two satellite missions are also provided which indicate the passed and (in case of GRACE) expected repeat cycles and possible degradations in the quality of the gravity field solutions.

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Band-limited functions on a bounded spherical domain: the Slepian problem on the sphere

Cited by 80 publications

References 11 publications

The spherical Slepian basis as a means to obtain spectral consistency between mean sea level and the geoid

The spherical Slepian basis as a means to obtain spectral consistency between mean sea level and the geoid

A timewise kinematic method for satellite gradiometry: GOCE simulations

An improved sampling rule for mapping geopotential functions of a planet from a near polar orbit

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