[1] A total of 954 continuous GPS position time series from 414 individual sites in nine different GPS solutions were analyzed for noise content using maximum likelihood estimation (MLE). The lengths of the series varied from around 16 months to over 10 years. MLE was used to analyze the data in two ways. In the first analysis the noise was assumed to be white noise only, a combination of white noise plus flicker noise, or a combination of white noise plus random walk noise. For the second analysis the spectral index and amplitude of the power law noise were estimated simultaneously with the white noise. In solutions where the sites were globally distributed, the noise can be best described by a combination of white noise plus flicker noise. Both noise components show latitude dependence in their amplitudes (higher at equatorial sites) together with a bias to larger values in the Southern Hemisphere. In the regional solutions, where a spatially correlated (common mode) signal has been removed, the noise is significantly lower. The spectral index of the power law in regional solutions is more varied than in the global solutions and probably reflects a mixture of local effects. A significant reduction in noise can be seen since the first continuous GPS networks began recording in the early 1990s. A comparison of the noise amplitudes to the different monument types in the Southern California Integrated GPS Network suggests that the deep drill braced monument is preferred for maximum stability.

1) model also fits the data and suggests that the velocity uncertainties should be 3-6 times larger than for the white noise model. We cannot adequately distinguish between these two noise models, nor can we rule out the possibility of a random walk signal at the lowest frequencies; these questions await the analysis of longer time series. In any case, reducing the magnitude of low-frequency colored noise is critical and appears to be best accomplished by building sites with deeply anchored and braced monuments. Otherwise, rate uncertainties estimated from continuous GPS measurements may not be improved significantly compared to those estimated from infrequent campaign-mode measurements.

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[1] We use 2.5 to 14 years long position time series from >800 continuous Global Positioning System (GPS) stations to study vertical deformation rates in the Euro-Mediterranean region. We estimate and remove common mode errors in position time series using a principal component analysis, obtaining a significant gain in the signal-to-noise ratio of the displacements data. Following the results of a maximum likelihood estimation analysis, which gives a mean spectral index~À0.7, we adopt a power law + white noise stochastic model in estimating the final vertical rates and find 95% of the velocities within ±2 mm/yr, with uncertainties from filtered time series~40% smaller than from the unfiltered ones. We highlight the presence of statistically significant velocity gradients where the stations density is higher. We find undulations of the vertical velocity field at different spatial scales both in tectonically active regions, like eastern Alps, Apennines, and eastern Mediterranean, and in regions characterized by a low or negligible tectonic activity, like central Iberia and western Alps. A correlation between smooth vertical velocities and topographic features is apparent in many sectors of the study area. Glacial isostatic adjustment and weathering processes do not completely explain the measured rates, and a combination of active tectonics and deep-seated geodynamic processes must be invoked. Excluding areas where localized processes are likely, or where subduction processes may be active, mantle dynamics is the most likely process, but regional mantle modeling is required for a better understanding.

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.

Over the last 10 years, several papers have established that daily estimates of GPS coordinates are temporally correlated and it is therefore incorrect to assume that the observations are independent when estimating parameters from them. A direct consequence of this assumption is the over-optimistic estimation of the parameter uncertainties. Perhaps the perceived computational burden or the lack of suitable software for time series analysis has resulted in many heuristic methods being proposed in the scientific literature for estimating these uncertainties. We present a standalone C program, CATS, developed to study and compare stochastic noise processes in continuous GPS coordinate time series and, as a consequence, assign realistic uncertainties to parameters derived from them. The name originally stood for Create and Analyze Time Series. Although the name has survived, the creation aspect of the software has, after several versions, been abandoned. The implementation of the method is briefly described to aid understanding and an example of typical input, usage, output and the available stochastic noise models are given.

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Bedrock uplift in Antarctica is dominated by a combination of glacial isostatic adjustment (GIA) and elastic response to contemporary mass change. Here, we present spatially extensive GPS observations of Antarctic bedrock uplift, using 52% more stations than previous studies, giving enhanced coverage, and with improved precision. We observe rapid elastic uplift in the northern Antarctic Peninsula. After considering elastic rebound, the GPS data suggests that modeled or empirical GIA uplift signals are often over‐estimated, particularly the magnitudes of the signal maxima. Our observation that GIA uplift is misrepresented by modeling (weighted root‐mean‐squares of observation‐model differences: 4.9–5.0 mm/yr) suggests that, apart from a few regions where large ice mass loss is occurring, the spatial pattern of secular ice mass change derived from Gravity Recovery and Climate Experiment (GRACE) data and GIA models may be unreliable, and that several recent secular Antarctic ice mass loss estimates are systematically biased, mainly too high.