Scalar and vector Slepian functions, spherical signal estimation and spectral analysis

It is a well-known fact that mathematical functions that are timelimited (or spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the finite precision of measurement and computation unavoidably bandlimits our observation and modeling scientific data, and we often only have access to, or are only interested in, a study area that is temporally or spatially bounded. In the geosciences we may be interested in spectrally modeling a time series defined only on a certain interval, or we may want to characterize a specific geographical area observed using an effectively bandlimited measurement device. It is clear that analyzing and representing scientific data of this kind will be facilitated if a basis of functions can be found that are "spatiospectrally" concentrated, i.e. "localized" in both domains at the same time. Here, we give a theoretical overview of one particular approach to this "concentration" problem, as originally proposed for time series by Slepian and coworkers, in the 1960s. We show how this framework leads to practical algorithms and statistically performant methods for the analysis of signals and their power spectra in one and two dimensions, and, particularly for applications in the geosciences, for scalar and vectorial signals defined on the surface of a unit sphere.Comment: Submitted to the 2nd Edition of the Handbook of Geomathematics, edited by Willi Freeden, Zuhair M. Nashed and Thomas Sonar, and to be published by Springer Verlag. This is a slightly modified but expanded version of the paper arxiv:0909.5368 that appeared in the 1st Edition of the Handbook, when it was called: Slepian functions and their use in signal estimation and spectral analysi

Impact of accounting for coloured noise in radar altimetry data on a regional quasi-geoid model

We study the impact of an accurate computation and incorporation of coloured noise in radar altimeter data when computing a regional quasi-geoid model using least-squares techniques. Our test area comprises the Southern North Sea including the Netherlands, Belgium, and parts of France, Germany, and the UK. We perform the study by modelling the disturbing potential with spherical radial base functions. To that end, we use the traditional remove-compute-restore procedure with a recent GRACE/GOCE static gravity field model. Apart from radar altimeter data, we use terrestrial, airborne, and shipboard gravity data. Radar altimeter sea surface heights are corrected for the instantaneous dynamic topography and used in the form of along-track quasi-geoid height differences. Noise in these data are estimated using repeat-track and post-fit residual analysis techniques and then modelled as an auto regressive moving average process. Quasi-geoid models are computed with and without taking the modelled coloured noise into account. The difference between them is used as a measure of the impact of coloured noise in radar altimeter along-track quasi-geoid height differences on the estimated quasi-geoid model. The impact strongly depends on the availability of shipboard gravity data. If no such data are available, the impact may attain values exceeding 10 centimetres in particular areas. In case shipboard gravity data are used, the impact is reduced, though it still attains values of several centimetres. We use geometric quasi-geoid heights from GPS/levelling data at height markers as control data to analyse the quality of the quasi-geoid models. The quasi-geoid model computed using a model of the coloured noise in radar altimeter along-track quasi-geoid height differences shows in some areas a significant improvement over a model that assumes white noise in these data. However, the interpretation in other areas remains a challenge due to the limited quality of the control data.Novel Aerospace MaterialsPhysical and Space Geodes

Estimation of mass change trends in the Earth’s system on the basis of GRACE satellite data, with application to Greenland

The Gravity Recovery and Climate Experiment (GRACE) satellite mission measures the Earth’s gravity field since March 2002. We propose a new filtering procedure for post-processing GRACE-based monthly gravity field solutions provided in the form of spherical harmonic coefficients. The procedure is tuned for the optimal estimation of linear trends and other signal components that show a systematic behavior over long time intervals. The key element of the developed methodology is the statistically optimal Wiener-type filter which makes use of the full covariance matrices of noise and signal. The developed methodology is applied to determine the mass balance of the Greenland ice sheet, both per drainage system and integrated, as well as the mass balance of the ice caps on the islands surrounding Greenland. The estimations are performed for three 2-year time intervals (2003–2004, 2005–2006, and 2007–2008), as well as for the 6-year time interval (2003–2008). The study confirms a significant difference in the behavior of the drainage systems over time. The average 6-year rate of mass loss in Greenland is estimated as 165 ± 15 Gt/year. The rate of mass loss of the ice caps on Ellesmere Island (together with Devon Island), Baffin Island, Iceland, and Svalbard is found to be 22 ± 4, 21 ± 6, 17 ± 9, and 6 ± 2 Gt/year, respectively. All these estimates are corrected for the effect of glacial isostatic adjustment.Geoscience and Remote SensingCivil Engineering and Geoscience

Sea-Level Change in the Dutch Wadden Sea

Rising sea levels due to climate change can have severe consequences for coastal populations and ecosystems all around the world. Understanding and projecting sea-level rise is especially important for low-lying countries such as the Netherlands. It is of specific interest for vulnerable ecological and morphodynamic regions, such as the Wadden Sea UNESCO World Heritage region. Here we provide an overview of sea-level projections for the 21st century for the Wadden Sea region and a condensed review of the scientific data, understanding and uncertainties underpinning the projections. The sea-level projections are formulated in the framework of the geological history of the Wadden Sea region and are based on the regional sea-level projections published in the Fifth Assessment Report of the Intergovernmental Panel on Climate Change (IPCC AR5). These IPCC AR5 projections are compared against updates derived from more recent literature and evaluated for the Wadden Sea region. The projections are further put into perspective by including interannual variability based on long-Term tide-gauge records from observing stations at Den Helder and Delfzijl. We consider three climate scenarios, following the Representative Concentration Pathways (RCPs), as defined in IPCC AR5: The RCP2.6 scenario assumes that greenhouse gas (GHG) emissions decline after 2020; the RCP4.5 scenario assumes that GHG emissions peak at 2040 and decline thereafter; and the RCP8.5 scenario represents a continued rise of GHG emissions throughout the 21st century. For RCP8.5, we also evaluate several scenarios from recent literature where the mass loss in Antarctica accelerates at rates exceeding those presented in IPCC AR5. For the Dutch Wadden Sea, the IPCC AR5-based projected sea-level rise is 0.07±0.06m for the RCP4.5 scenario for the period 2018-30 (uncertainties representing 5-95%), with the RCP2.6 and RCP8.5 scenarios projecting 0.01m less and more, respectively. The projected rates of sea-level change in 2030 range between 2.6mma-1 for the 5th percentile of the RCP2.6 scenario to 9.1mma-1 for the 95th percentile of the RCP8.5 scenario. For the period 2018-50, the differences between the scenarios increase, with projected changes of 0.16±0.12m for RCP2.6, 0.19±0.11m for RCP4.5 and 0.23±0.12m for RCP8.5. The accompanying rates of change range between 2.3 and 12.4mma-1 in 2050. The differences between the scenarios amplify for the 2018-2100 period, with projected total changes of 0.41±0.25m for RCP2.6, 0.52±0.27m for RCP4.5 and 0.76±0.36m for RCP8.5. The projections for the RCP8.5 scenario are larger than the high-end projections presented in the 2008 Delta Commission Report (0.74m for 1990-2100) when the differences in time period are considered. The sea-level change rates range from 2.2 to 18.3mma-1 for the year 2100. We also assess the effect of accelerated ice mass loss on the sea-level projections under the RCP8.5 scenario, as recent literature suggests that there may be a larger contribution from Antarctica than presented in IPCC AR5 (potentially exceeding 1m in 2100). Changes in episodic extreme events, such as storm surges, and periodic (tidal) contributions on (sub-)daily timescales, have not been included in these sea-level projections. However, the potential impacts of these processes on sea-level change rates have been assessed in the report