One of the most widely used method for the time-series analysis of continuous Global Navigation Satellite System (GNSS) observations is Maximum Likelihood Estimation (MLE) which in most implementations requires O(n 3 ) operations for n observations. Previous research by the authors has shown that this amount of operations can be reduced to O(n 2 ) for observations without missing data. In the current research we present a reformulation of the equations that preserves this low amount of operations, even in the common situation of having some missing data.Our reformulation assumes that the noise is stationary to ensure a Toeplitz covariance matrix. However, most GNSS time-series exhibit power-law noise which is weakly non-stationary. To overcome this problem, we present an Toeplitz covariance matrix that provides an approximation for power-law noise that is accurate for most GNSS time-series. * mbos@ciimar.up.pt † rmanuel@di.ubi.pt ‡ sdwil@noc.ac.uk § lcbastos@fc.up.pt 1 Numerical results are given for a set of synthetic data and a set of International GNSS Service (IGS) stations, demonstrating a reduction in computation time of a factor of 10-100 compared to the standard MLE method, depending on the length of the time-series and the amount of missing data.

It has been generally accepted that the noise in continuous GPS observations can be well described by a power-law plus white noise model. Using maximum likelihood estimation (MLE) the numerical values of the noise model can be estimated. Current methods require calculating the data covariance matrix and inverting it, which is a significant computational burden. Analysing 10 years of daily GPS solutions of a single station can take around 2 h on a regular computer such as a PC with an AMD Athlon™ 64 X2 dual core processor. When one analyses large networks with hundreds of stations or when one analyses hourly instead of daily solutions, the long computation times becomes a problem. In case the signal only contains power-law noise, the MLE computations can be simplified to a O(N log N ) Electronic supplementary material The online version of this article (doi:10.1007/s00190-007-0165-x) contains supplementary material, which is available to authorized users. process where N is the number of observations. For the general case of power-law plus white noise, we present a modification of the MLE equations that allows us to reduce the number of computations within the algorithm from a cubic to a quadratic function of the number of observations when there are no data gaps. For time-series of three and eight years, this means in practise a reduction factor of around 35 and 84 in computation time without loss of accuracy. In addition, this modification removes the implicit assumption that there is no environment noise before the first observation. Finally, we present an analytical expression for the uncertainty of the estimated trend if the data only contains powerlaw noise. M. S. Bos (B)·

[1] Studies of intra-and inter-plate deformation typically need a model describing the motions of the stable part of the tectonic plates for reference purposes. We have developed DEOS2k, a model for the current motion of seven major tectonic plates derived from space-geodetic observations. This paper focuses on relative motion between Africa and Eurasia. In the past, this motion has been poorly established because of poor data coverage for Africa. DEOS2k is based on ITRF2000 [Altamimi et al., 2002] and new African GPS observations. It is an improvement over the NUVEL-1A model for predicting the present-day relative motions of these two plates. DEOS2k predicts in northeastern Africa that Africa-Eurasia relative motion is about 40% smaller in magnitude than NUVEL-1A and trends more to the northwest. This is consistent with independent local geodetic observations. A similar shift in orientation, clockwise, is observed at the western tip of the plate boundary.

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