Error distribution in regional inversion of potential field data

Schachtschneider, Reyko; Holschneider, Matthias; Mandea, Mioara

doi:10.1111/j.1365-246x.2010.04598.x

Cited by 11 publications

(15 citation statements)

References 26 publications

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“…Indeed, it arises whenever different data sets and/or models with various spatial or temporal resolutions are to be merged for interpretation e.g. in seismology and geomagnetic studies (Trampert and Snieder 1996;Boschi and Dziewonski 1999;Schachtschneider et al 2010;Schott and Thébault 2011).…”

Section: Introductionmentioning

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: Introductionmentioning

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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“…In our study, we found empirically that an extension of 5 • reduces the distortions over the area of interest to a level below 2 mm. This result is independent of the size of the area of interest as shown in (Schachtschneider et al 2010). Hence, only for data areas much smaller than the one considered in the numerical experiments of Sect.…”

Section: Impact Of the Size Of The Data Areamentioning

confidence: 84%

“…These distortions, which are usually referred to as edge effects, reduce towards the centre of the data area (e.g. Schachtschneider et al 2010). Hence, to reduce the distortions, the data area has to be chosen larger than the area of interest.…”

Section: Impact Of the Size Of The Data Areamentioning

confidence: 99%

How to deal with the high condition number of the noise covariance matrix of gravity field functionals synthesised from a satellite-only global gravity field model?

Klees

Slobbe

Farahani

2018

J Geod

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The posed question arises for instance in regional gravity field modelling using weighted least-squares techniques if the gravity field functionals are synthesised from the spherical harmonic coefficients of a satellite-only global gravity model (GGM), and are used as one of the noisy datasets. The associated noise covariance matrix, appeared to be extremely ill-conditioned with a singular value spectrum that decayed gradually to zero without any noticeable gap. We analysed three methods to deal with the ill-conditioned noise covariance matrix: Tihonov regularisation of the noise covariance matrix in combination with the standard formula for the weighted least-squares estimator, a formula of the weighted least-squares estimator, which does not involve the inverse noise covariance matrix, and an estimator based on Rao's unified theory of least-squares. Our analysis was based on a numerical experiment involving a set of height anomalies synthesised from the GGM GOCO05s, which is provided with a full noise covariance matrix. We showed that the three estimators perform similar, provided that the two regularisation parameters each method knows were chosen properly. As standard regularisation parameter choice rules do not apply here, we suggested a new parameter choice rule, and demonstrated its performance. Using this rule, we found that the differences between the three least-squares estimates were within noise. For the standard formulation of the weighted least-squares estimator with regularised noise covariance matrix, this required an exceptionally strong regularisation, much larger than one expected from the condition number of the noise covariance matrix. The preferred method is the inversion-free formulation of the weighted least-squares estimator, because of its simplicity with respect to the choice of the two regularisation parameters.

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“…The error distributions obtained in case of elliptical data regions look very similar to the ones obtained in circular data region experiments ( cf. Schachtschneider et al 2010). The minimum value is reached in the centre, the error increases towards the boundary where the maximum is reached.…”

Section: Resultsmentioning

confidence: 99%

“…In such cases there are edge effects that have to be taken into account. The boundary region that is necessary and a comprehensive error analysis for circular spherical cap shaped regions is found in Schachtschneider et al (2010).…”

Section: Introductionmentioning

confidence: 99%

Error distribution in regional modelling of the geomagnetic field

Schachtschneider

Holschneider

Mandea

2012

Geophysical Journal International

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SUMMARY In this study we analyse the error distribution in regional models of the geomagnetic field. Our main focus is to investigate the distribution of errors when combining two regional patches to obtain a global field from regional ones. To simulate errors in overlapping patches we choose two different data region shapes that resemble that scenario. First, we investigate the errors in elliptical regions and secondly we choose a region obtained from two overlapping circular spherical caps. We conduct a Monte‐Carlo simulation using synthetic data to obtain the expected mean errors. For the elliptical regions the results are similar to the ones obtained for circular spherical caps: the maximum error at the boundary decreases towards the centre of the region. A new result emerges as errors at the boundary vary with azimuth, being largest in the major axis direction and minimal in the minor axis direction. Inside the region there is an error decay towards a minimum at the centre at a rate similar to the one in circular regions. In the case of two combined circular regions there is also an error decay from the boundary towards the centre. The minimum error occurs at the centre of the combined regions. The maximum error at the boundary occurs on the line containing the two cap centres, the minimum in the perpendicular direction where the two circular cap boundaries meet. The large errors at the boundary are eliminated by combining regional patches. We propose an algorithm for finding the boundary region that is applicable to irregularly shaped model regions.

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Error distribution in regional inversion of potential field data

Cited by 11 publications

References 26 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

How to deal with the high condition number of the noise covariance matrix of gravity field functionals synthesised from a satellite-only global gravity field model?

Error distribution in regional modelling of the geomagnetic field

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