Abstract
<p>The article describes an implementation of the negative log-likelihood function in the determination of uncorrelated noise standard deviation together with the parameters of spherical signal covariance model in least squares collocation (LSC) of gravity anomalies. The correctness and effectiveness of restricted maximum likelihood (REML) estimates are fully validated by leave-one-out validation (LOO). These two complementary methods give an opportunity to inspect the parametrization of the signal and uncorrelated noise in details and can provide some guidance related to the estimation of individual parameters. The study provides the practical proof that noise variance is related with the data resolution, which is often neglected and the information on a priori noise variance is based on the measurement error. The data have been downloaded from U.S. terrestrial gravity database and resampled to enable an analysis with four different horizontal resolutions. These data are intentionally the same, as in the previous study of the same author, with the application of the planar covariance model. The aim is to compare the results from two different covariance models, which have different covariance approximation at larger distances. The most interesting outputs from this study confirm previous observations on the relations of the data resolution, a priori noise variance, signal spectrum and LSC accuracy.</p>
Highlights
Introduction and objectiveThe spatial correlation of the signal in least squares collocation (LSC) can be described by the analytical covariance model selected from a large group of models available today
The LSC based on the covariance model is widely applied in geodesy [Arabelos and Tscherning 2003, Albertella et al 2004, Kotsakis 2007, Sansò et al 2008, Reguzzoni and Tselfes 2009, Pavlis et al 2012] and has some common assumptions with the group of techniques originating from other fields of geosciences, called kriging [Reguzzoni et al 2005]
The coloured curves are based on the parameters found in restricted maximum likelihood (REML) and leave-one-out validation (LOO) analyses, realized and explained
Summary
Where cn are degree variances, RB is the radius of Bjerhammar sphere [Arabelos and Tscherning 2003] and Pn are Legendre polynomials with spherical distance }. This is the model of global gravity anomalies, since we start from the lowest harmonics, representing the most global characteristics of the gravity field. Assuming that our observations are gravity anomalies, the necessary step is to remove lower frequencies of the harmonic expansion from the signal, because: Dg = Xb + Dgr. In the presented investigations the long-wavelength part comes from the geopotential global model EGM2008 [Pavlis et al 2012] and replaces deterministic part Xb with the harmonic expansion. The probability density function (PDF) of the multivariate normal distribution based on the residual gravity Dgr and covariance C(i) reads [Kusche 2003, Koch 2007, van Loon 2008]:
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