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Discussion papers
https://doi.org/10.5194/se-2019-60
© Author(s) 2019. This work is distributed under
the Creative Commons Attribution 4.0 License.
https://doi.org/10.5194/se-2019-60
© Author(s) 2019. This work is distributed under
the Creative Commons Attribution 4.0 License.

Research article 05 Apr 2019

Research article | 05 Apr 2019

Review status
This discussion paper is a preprint. It is a manuscript under review for the journal Solid Earth (SE).

Regularization methods for the combination of heterogeneous observations using spherical radial basis functions

Qing Liu1, Michael Schmidt1, Roland Pail2, and Martin Willberg2 Qing Liu et al.
  • 1Deutsches Geodätisches Forschungsinstitut, Technische Universität München (DGFI-TUM), Arcisstr. 21, 80333 München, Germany
  • 2Institute for Astronomical and Physical Geodesy, Technische Universität München, Arcisstr. 21, 80333 München, Germany

Abstract. Various types of heterogeneous observations can be combined within a parameter estimation process using spherical radial basis functions (SRBF) for regional gravity field refinement. However, this process is in most cases ill-posed, and thus, regularization is indispensable. We discuss two frequently used methods for choosing the regularization parameter which are the L-curve method and variance component estimation (VCE). Based on these two methods, we propose two new approaches for the regularization parameter determination, which combine the L-curve method and VCE.

The first approach, denoted as ‘VCE + L-curve method’, starts with the calculation of the relative weights between the observation techniques by means of VCE. Based on these weights the L-curve method is applied to determine the regularization parameter. In the second approach, called ‘L-curve method + VCE’, the L-curve method determines first the regularization parameter and it is set to be fixed during the calculation of the relative weights between the observation techniques from VCE.

These methods are investigated based on two different estimation concepts for combining various observation techniques. All the methods are applied and compared in six study cases using four types of observations in Europe. The results show that the ‘VCE + L-curve method’ delivers the best results in all the six cases, no matter using SRBFs with smoothing or non-smoothing features. The ‘L-curve method + VCE’ also gives rather good results, generally outperforming the cases just using the L-curve method or VCE. Therefore, we conclude that the newly proposed methods are decent and stable for regularization parameter determination when different data sets are combined and can be recommended regardless of the type of SRBFs used.

Qing Liu et al.
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Short summary
Regularization is indispensable in regional gravity field modelling. In this paper, we propose two new approaches for the regularization parameter determination, which combine the L-curve method and variance component estimation (VCE). The performance of each method is studied for combining heterogeneous observations using spherical radial basis functions. The results show that our newly proposed methods are decent and stable for regularization parameter determination.
Regularization is indispensable in regional gravity field modelling. In this paper, we propose...
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