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, R.; Slobbe, Cornelis; Farahani, H. Hashemi

doi:10.1007/s00190-018-1136-0

Cited by 14 publications

(9 citation statements)

References 31 publications

(68 reference statements)

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“…In this way, long‐wavelength systematic and random errors in these data sets could be reduced significantly. Though many of these data sets cover only small areas (often smaller than the spatial resolution of GOCO05s), they introduce long‐wavelengths systematic errors in the computed QG model as shown in Klees et al (). Without these bias parameters, the quality of the computed QG models would be much lower, though the combination with GOCO05s would partially compensate for the biases in the largest data sets.…”

Section: Discussionmentioning

confidence: 99%

“…The motivation for including solution V is that most methods and software used in local QG modeling cannot deal with an ill‐conditioned noise VC matrix. As shown in Klees et al (), the noise VC matrix of the GOCO05s GGM height anomaly data set has a condition number of about 10 16 . Dealing with this ill‐conditioned matrix in weighted least squares requires the use of one of the methods suggested in Klees et al ().…”

Section: Numerical Experimentsmentioning

confidence: 92%

“…As shown in Klees et al (), the noise VC matrix of the GOCO05s GGM height anomaly data set has a condition number of about 10 16 . Dealing with this ill‐conditioned matrix in weighted least squares requires the use of one of the methods suggested in Klees et al (). For information about the method used in this study, we refer to Appendix .…”

Section: Numerical Experimentsmentioning

confidence: 92%

“…Therefore, it does not make sense to include GOCO05s up to the maximum degree 280. On the other hand, using a value p 2 < 280 reduces the condition number of the noise VC matrix of the GGM height anomaly data set, which makes the computation of the weight matrix easier(Klees et al, ).…”

Section: Data Sets and Stochastic Modelsmentioning

confidence: 99%

“…The second addresses the fact that many methods and software packages currently in use cannot deal with full, highly ill‐conditioned noise VC matrices. For instance, Klees et al () have shown that the noise covariance matrix of the GOCO05s GGM, which represents the state of the art of satellite‐only GGMs and is used here, is highly ill conditioned making the computation of the weight matrix unstable.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

The Impact of Noise in a GRACE/GOCE Global Gravity Model on a Local Quasi‐Geoid

Slobbe

Klees

Farahani³

et al. 2019

JGR Solid Earth

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We present a local quasi‐geoid (QG) model which combines a satellite‐only global gravity model with local data sets using weighted least squares. The QG is computed for an area comprising the Netherlands, Belgium, and the southern North Sea. It uses a two‐scale spherical radial basis function model complemented by bias parameters to account for systematic errors in the local gravity data sets. Variance factors are estimated for the noise covariance matrices of all involved data sets using variance component estimation. The standard deviation (SD) of the differences between the computed QG and GPS/leveling data is 0.95 and 1.52 cm for the Netherlands and Belgium, respectively. The fact that the SD of the control data is about 0.60 and 1.20 cm for the Netherlands and Belgium, respectively, points to a lower mean SD of the computed QG model of about 0.7 cm for the Netherlands and 1.0 cm for Belgium. The differences to a QG model computed with the remove‐compute‐restore technique range from −5.2 to 2.6 cm over the whole model domain and from −1.5 to 1.5 cm over the Netherlands and Belgium. A variogram analysis of the differences with respect to GPS/leveling data reveals a better performance of the computed QG model compared to a remove‐compute‐restore‐based QG model for wavelengths >100 km for Belgium but not for the Netherlands. The latter is due to the fact that at the spatial scales resolved by the global gravity model, variance component estimation assigns significantly lower weights to the local data set in favor of the global gravity model.

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Section: Discussionmentioning

confidence: 99%

Section: Numerical Experimentsmentioning

confidence: 92%

Section: Numerical Experimentsmentioning

confidence: 92%

Section: Data Sets and Stochastic Modelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The Impact of Noise in a GRACE/GOCE Global Gravity Model on a Local Quasi‐Geoid

Slobbe

Klees

Farahani³

et al. 2019

JGR Solid Earth

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Error propagation in regional geoid computation using spherical splines, least-squares collocation, and Stokes’s formula

Ophaug

Gerlach

2020

J Geod

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Current International Association of Geodesy efforts within regional geoid determination include the comparison of different computation methods in the quest for the “1-cm geoid.” Internal (formal) and external (empirical) approaches to evaluate geoid errors exist, and ideally they should agree. Spherical radial base functions using the spline kernel (SK), least-squares collocation (LSC), and Stokes’s formula are three commonly used methods for regional geoid computation. The three methods have been shown to be theoretically equivalent, as well as to numerically agree on the millimeter level in a closed-loop environment using synthetic noise-free data (Ophaug and Gerlach in J Geod 91:1367–1382, 2017. 10.1007/s00190-017-1030-1). This companion paper extends the closed-loop method comparison using synthetic data, in that we investigate and compare the formal error propagation using the three methods. We use synthetic uncorrelated and correlated noise regimes, both on the 1-mGal ($$=10^{-5}~{\mathrm {m s}}^{-2}$$ = 10 - 5 m s - 2 ) level, applied to the input data. The estimated formal errors are validated by comparison with empirical errors, as determined from differences of the noisy geoid solutions to the noise-free solutions. We find that the error propagations of the methods are realistic in both uncorrelated and correlated noise regimes, albeit only when subjected to careful tuning, such as spectral band limitation and signal covariance adaptation. For the SKs, different implementations of the L-curve and generalized cross-validation methods did not provide an optimal regularization parameter. Although the obtained values led to a stabilized numerical system, this was not necessarily equivalent to obtaining the best solution. Using a regularization parameter governed by the agreement between formal and empirical error fields provided a solution of similar quality to the other methods. The errors in the uncorrelated regime are on the level of $$\sim $$ ∼ 5 mm and the method agreement within 1 mm, while the errors in the correlated regime are on the level of $$\sim $$ ∼ 10 mm, and the method agreement within 5 mm. Stokes’s formula generally gives the smallest error, closely followed by LSC and the SKs. To this effect, we note that error estimates from integration and estimation techniques must be interpreted differently, because the latter also take the signal characteristics into account. The high level of agreement gives us confidence in the applicability and comparability of formal errors resulting from the three methods. Finally, we present the error characteristics of geoid height differences derived from the three methods and discuss them qualitatively in relation to GNSS leveling. If applied to real data, this would permit identification of spatial scales for which height information is preferably derived by spirit leveling or GNSS leveling.

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Characterization and stabilization of the downward continuation problem for airborne gravity data

Huang

Klees

et al. 2022

J Geod

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In this study, we compare six commonly used methods for the downward continuation of airborne gravity data. We consider exact and noisy simulated data on grids and along flight trajectories and real data from the GRAV-D airborne campaign. We use simulated and real surface gravity data for validation. The methods comprise spherical harmonic analysis, least-squares collocation, residual least-squares collocation, least-squares radial basis functions, the inverse Poisson method and Moritz's analytical downward continuation method. We show that all the methods perform similar in terms of surface gravity values. For real data, the downward continued airborne gravity values are used to compute a geoid model using a Stokes-integral-based approach. The quality of the computed geoid model is validated using high-quality GSVS17 GPS-levelling data. We show that the geoid model quality is similar for all the methods. However, the least-squares collocation approach appears to be more flexible and easier to use than the other methods provided that the optimal covariance function is found. We recommend it for the downward continuation of GRAV-D data, and other methods for second check.

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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?

Cited by 14 publications

References 31 publications

The Impact of Noise in a GRACE/GOCE Global Gravity Model on a Local Quasi‐Geoid

The Impact of Noise in a GRACE/GOCE Global Gravity Model on a Local Quasi‐Geoid

Error propagation in regional geoid computation using spherical splines, least-squares collocation, and Stokes’s formula

Characterization and stabilization of the downward continuation problem for airborne gravity data

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