How to deal with the ill-conditioned noise covariance matrix of GRACE/GOCE global gravity models in regional quasi-geoid modelling?
21/09/2016 | 10:30 | Session 3: Recent Development in Theory and Modelling
Author(s): Roland Klees, Hassan Hashemi Farahani and Cornelis Slobbe
Roland Klees, Hassan Hashemi Farahani and Cornelis Slobbe
The quality of recent satellite-only global gravity field models (GGMs), which are mainly based on GRACE and GOCE satellite data, has improved dramatically in terms of accuracy and spatial resolution compared to pre-mission models. Today, some of these models like GOCO05s are equipped with properly scaled, full noise covariance matrices. This paves the way to a proper combination of the GGM with terrestrial gravity and other datasets when estimating a regional quasi-geoid model. No matter whether Stokes-based approaches, least-squares collocation or least-squares techniques are used, they all require to invert the full noise covariance matrix associated with the dataset representing the GGM. In regional quasi-geoid modelling, the noise covariance matrix of the spherical harmonic coefficients if often not used directly, but propagated to some gravity field functional. Depending on the size of the target area, this propagated noise covariance matrix may be extremely ill-conditioned, and how to compute a meaningful weight matrix is an open question.
In this study we provide a detailed analysis of the structure of the full noise covariance matrix of a state-of-the-art GRACE/GOCE combined gravity model and of the propagated noise covariance matrix. We show that the noise propagation requires multi-precision arithmetic in order to prevent the loss of significant digits, which otherwise would render the noise propagation meaningless. We analyze the singular values using multi-precision arithmetic and demonstrate the extreme ill-conditioning of these matrices with condition numbers far above of what can be represented in IEEE double-precision arithmetic. Then, we investigate various techniques designed to cope with these high condition numbers including covariance function modelling, truncated singular value decomposition, and Tikhonov regularization. We show that none of them is able to provide a meaningful weight matrix, leading to a wrong weighting of the GGM dataset in combined regional quasi-geoid models. Finally, we present a method, which is based on a reduced floating point representation of the covariance matrix, which provides a proper weight matrix for the GGM dataset and reasonable combined quasi-geoid models.