Optimal multi-step collocation: application to the space-wise approach for GOCE data analysis

Abstract

Collocation is widely used in physical geodesy. Its application requires to solve systems with a dimension equal to the number of observations, causing numerical problems when many observations are available. To overcome this drawback, tailored step-wise techniques are usually applied. An example of these step-wise techniques is the space-wise approach to the GOCE mission data processing. The original idea of this approach was to implement a two-step procedure, which consists of first predicting gridded values at satellite altitude by collocation and then deriving the geo-potential spherical harmonic coefficients by numerical integration. The idea was generalized to a multi-step iterative procedure by introducing a time-wise Wiener filter to reduce the highly correlated observation noise. Recent studies have shown how to optimize the original two-step procedure, while the theoretical optimization of the full multi-step procedure is investigated in this work. An iterative operator is derived so that the final estimated spherical harmonic coefficients are optimal with respect to the Wiener–Kolmogorov principle, as if they were estimated by a direct collocation. The logical scheme used to derive this optimal operator can be applied not only in the case of the space-wise approach but, in general, for any case of step-wise collocation. Several numerical tests based on simulated realistic GOCE data are performed. The results show that adding a pre-processing time-wise filter to the two-step procedure of data gridding and spherical harmonic analysis is useful, in the sense that the accuracy of the estimated geo-potential coefficients is improved. This happens because, in its practical implementation, the gridding is made by collocation over local patches of data, while the observation noise has a time-correlation so long that it cannot be treated inside the patch size. Therefore, the multi-step operator, which is in theory equivalent to the two-step operator and to the direct collocation, is in practice superior thanks to the time-wise filter that reduces the noise correlation before the gridding. The criteria for the choice of this filter are investigated numerically

Expected geoneutrino signal at JUNO

“…The total crustal thickness of each cell and its associated uncertainty correspond, respectively, to the mean and the half range of three crustal models obtained from different approaches: the global crustal model based on reflection and refraction data 'CRUST 2.0' (Bassin et al 2000;Laske et al 2001), the global shear velocity model of the crust and upper mantle 'CUB 2.0' (Shapiro and Ritzwoller 2002), and the high-resolution map of Moho (crust-mantle boundary) depth based on the gravity field data 'GEMMA' (Reguzzoni and Tselfes 2009;Reguzzoni and Sampietro 2015). The reference model incorporates the relative proportional thickness of the crustal layers along with density and elastic properties (compressional and shear wave velocity) reported in CRUST 2.0.…”

First GOCE gravity field models derived by three different approaches

“…12 is obtained by modifying the gridding collocation operator G as follows (see Reguzzoni and Tselfes 2009):…”

Rosborough Formulation in Satellite Gravity Gradiometry

“…Sharifi, 2006) and a collocation solution. In the former technique all the observations to be interpolated must carry the same spatial information while in the latter technique, different functionals may be used as observations and any other functional can be predicted; for example, estimating the second radial derivative on grid points from a set of observed data (Reguzzoni and Tselfes, 2009). It is clear that in the case of the space-wise approach for GOCE gradiometry data processing, collocation has to be used with its numerical heaviness, while the Rosborough method can benefit from both techniques.…”

“…Many authors have developed different representations of the gravity field, and consequently different data processing strategies, to solve such a huge system of equations (see e.g. Reguzzoni and Tselfes, 2009;Pail et al, 2010;Xu et al, 2008;Sneeuw, 2000).…”