Articles | Volume 9, issue 2
https://doi.org/10.5194/se-9-323-2018
© Author(s) 2018. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
https://doi.org/10.5194/se-9-323-2018
© Author(s) 2018. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Time-variable gravity fields and ocean mass change from 37 months of kinematic Swarm orbits
Christina Lück
CORRESPONDING AUTHOR
Institute of Geodesy and Geoinformation, University of Bonn, Bonn, Germany
Jürgen Kusche
Institute of Geodesy and Geoinformation, University of Bonn, Bonn, Germany
Roelof Rietbroek
Institute of Geodesy and Geoinformation, University of Bonn, Bonn, Germany
Anno Löcher
Institute of Geodesy and Geoinformation, University of Bonn, Bonn, Germany
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Kristin Vielberg, Ehsan Forootan, Christina Lück, Anno Löcher, Jürgen Kusche, and Klaus Börger
Ann. Geophys., 36, 761–779, https://doi.org/10.5194/angeo-36-761-2018, https://doi.org/10.5194/angeo-36-761-2018, 2018
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To predict the satellite's motion or its re-entry, the density surrounding the satellite needs to be known as precisely as possible. Usually empirical models are used to estimate the neutral density of the thermosphere, which is the region of the neutrally charged atmosphere. Here, based on calibrated accelerations measured by instruments on board satellites, we compute daily global maps to correct modeled densities. During times of high solar activity, corrections of up to 28 % are necessary.
Benedikt Aschenneller, Roelof Rietbroek, and Daphne van der Wal
Nat. Hazards Earth Syst. Sci., 24, 4145–4177, https://doi.org/10.5194/nhess-24-4145-2024, https://doi.org/10.5194/nhess-24-4145-2024, 2024
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Shorelines retreat or advance in response to sea level changes, subsidence or uplift of the ground, and morphological processes (sedimentation and erosion). We show that the geometrical influence of each of these drivers on shoreline movements can be quantified by combining different remote sensing observations, including radar altimetry, lidar and optical satellite images. The focus here is to illustrate the uncertainties of these observations by comparing datasets that cover similar processes.
Lara Börger, Michael Schindelegger, Mengnan Zhao, Rui M. Ponte, Anno Löcher, Bernd Uebbing, Jean-Marc Molines, and Thierry Penduff
Earth Syst. Dynam. Discuss., https://doi.org/10.5194/esd-2024-21, https://doi.org/10.5194/esd-2024-21, 2024
Revised manuscript accepted for ESD
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Flows in the ocean are driven either by atmospheric forces or by small-scale internal disturbances that are inherently chaotic. We use computer simulation results to show that these chaotic oceanic disturbances can attain spatial scales large enough to alter the motion of Earth’s pole of rotation. Given their size and unpredictable nature, the chaotic signals are a source of uncertainty when interpreting observed year-to-year polar motion changes in terms of other processes in the Earth system.
Petra Döll, Howlader Mohammad Mehedi Hasan, Kerstin Schulze, Helena Gerdener, Lara Börger, Somayeh Shadkam, Sebastian Ackermann, Seyed-Mohammad Hosseini-Moghari, Hannes Müller Schmied, Andreas Güntner, and Jürgen Kusche
Hydrol. Earth Syst. Sci., 28, 2259–2295, https://doi.org/10.5194/hess-28-2259-2024, https://doi.org/10.5194/hess-28-2259-2024, 2024
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Currently, global hydrological models do not benefit from observations of model output variables to reduce and quantify model output uncertainty. For the Mississippi River basin, we explored three approaches for using both streamflow and total water storage anomaly observations to adjust the parameter sets in a global hydrological model. We developed a method for considering the observation uncertainties to quantify the uncertainty of model output and provide recommendations.
Torsten Kanzow, Angelika Humbert, Thomas Mölg, Mirko Scheinert, Matthias Braun, Hans Burchard, Francesca Doglioni, Philipp Hochreuther, Martin Horwath, Oliver Huhn, Jürgen Kusche, Erik Loebel, Katrina Lutz, Ben Marzeion, Rebecca McPherson, Mahdi Mohammadi-Aragh, Marco Möller, Carolyne Pickler, Markus Reinert, Monika Rhein, Martin Rückamp, Janin Schaffer, Muhammad Shafeeque, Sophie Stolzenberger, Ralph Timmermann, Jenny Turton, Claudia Wekerle, and Ole Zeising
EGUsphere, https://doi.org/10.5194/egusphere-2024-757, https://doi.org/10.5194/egusphere-2024-757, 2024
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The Greenland Ice Sheet represents the second-largest contributor to global sea-level rise. We quantify atmosphere, ice and ocean-based processes related to the mass balance of glaciers in Northeast Greenland, focusing on Greenland’s largest floating ice tongue, the 79N Glacier. We find that together, the different in situ and remote sensing observations and model simulations to reveal a consistent picture of a coupled atmosphere-ice sheet-ocean system, that has entered a phase of major change.
Matthias O. Willen, Martin Horwath, Eric Buchta, Mirko Scheinert, Veit Helm, Bernd Uebbing, and Jürgen Kusche
The Cryosphere, 18, 775–790, https://doi.org/10.5194/tc-18-775-2024, https://doi.org/10.5194/tc-18-775-2024, 2024
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Shrinkage of the Antarctic ice sheet (AIS) leads to sea level rise. Satellite gravimetry measures AIS mass changes. We apply a new method that overcomes two limitations: low spatial resolution and large uncertainties due to the Earth's interior mass changes. To do so, we additionally include data from satellite altimetry and climate and firn modelling, which are evaluated in a globally consistent way with thoroughly characterized errors. The results are in better agreement with independent data.
Simon Deggim, Annette Eicker, Lennart Schawohl, Helena Gerdener, Kerstin Schulze, Olga Engels, Jürgen Kusche, Anita T. Saraswati, Tonie van Dam, Laura Ellenbeck, Denise Dettmering, Christian Schwatke, Stefan Mayr, Igor Klein, and Laurent Longuevergne
Earth Syst. Sci. Data, 13, 2227–2244, https://doi.org/10.5194/essd-13-2227-2021, https://doi.org/10.5194/essd-13-2227-2021, 2021
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GRACE provides us with global changes of terrestrial water storage. However, the data have a low spatial resolution, and localized storage changes in lakes/reservoirs or mass change due to earthquakes causes leakage effects. The correction product RECOG RL01 presented in this paper accounts for these effects. Its application allows for improving calibration/assimilation of GRACE into hydrological models and better drought detection in earthquake-affected areas.
L. Drees, J. Kusche, and R. Roscher
ISPRS Ann. Photogramm. Remote Sens. Spatial Inf. Sci., V-2-2020, 813–820, https://doi.org/10.5194/isprs-annals-V-2-2020-813-2020, https://doi.org/10.5194/isprs-annals-V-2-2020-813-2020, 2020
Isabel Meza, Stefan Siebert, Petra Döll, Jürgen Kusche, Claudia Herbert, Ehsan Eyshi Rezaei, Hamideh Nouri, Helena Gerdener, Eklavyya Popat, Janna Frischen, Gustavo Naumann, Jürgen V. Vogt, Yvonne Walz, Zita Sebesvari, and Michael Hagenlocher
Nat. Hazards Earth Syst. Sci., 20, 695–712, https://doi.org/10.5194/nhess-20-695-2020, https://doi.org/10.5194/nhess-20-695-2020, 2020
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The paper presents, for the first time, a global-scale drought risk assessment for both irrigated and rainfed agricultural systems while considering drought hazard indicators, exposure and expert-weighted vulnerability indicators. We identify global patterns of drought risk and, by disaggregating risk into its underlying components and factors, provide entry points for risk reduction.
Helena Gerdener, Olga Engels, and Jürgen Kusche
Hydrol. Earth Syst. Sci., 24, 227–248, https://doi.org/10.5194/hess-24-227-2020, https://doi.org/10.5194/hess-24-227-2020, 2020
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GRACE-derived drought indicators enable us to detect hydrological droughts based on changes observed in all storages. By performing synthetic experiments, we find that droughts identified by existing and modified indicators are biased by trends and GRACE-based spatial noise. A modified version of the Zhao et al. (2017) indicator is found to be particularly robust against spatial noise and is therefore applied to real GRACE data over South Africa.
Stefan Schröder, Anne Springer, Jürgen Kusche, Bernd Uebbing, Luciana Fenoglio-Marc, Bernd Diekkrüger, and Thomas Poméon
Hydrol. Earth Syst. Sci., 23, 4113–4128, https://doi.org/10.5194/hess-23-4113-2019, https://doi.org/10.5194/hess-23-4113-2019, 2019
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We propose deriving altimetric rating curves by
bridginggaps between time series from gauge and altimeter measurements using hydrological model simulations. We investigate several stations at the Niger River, which is a challenging region. We show that altimetry reproduces discharge well and enables continuing the gauge time series, albeit at a lower temporal resolution.
Kristin Vielberg, Ehsan Forootan, Christina Lück, Anno Löcher, Jürgen Kusche, and Klaus Börger
Ann. Geophys., 36, 761–779, https://doi.org/10.5194/angeo-36-761-2018, https://doi.org/10.5194/angeo-36-761-2018, 2018
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To predict the satellite's motion or its re-entry, the density surrounding the satellite needs to be known as precisely as possible. Usually empirical models are used to estimate the neutral density of the thermosphere, which is the region of the neutrally charged atmosphere. Here, based on calibrated accelerations measured by instruments on board satellites, we compute daily global maps to correct modeled densities. During times of high solar activity, corrections of up to 28 % are necessary.
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Mesay Geletu Gebre and Elias Lewi
Solid Earth, 14, 101–117, https://doi.org/10.5194/se-14-101-2023, https://doi.org/10.5194/se-14-101-2023, 2023
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In this work, a gravity inversion method that can produce compact and sharp images is presented. An auto-adaptive regularization parameter estimation method, improved error-weighting function and combined stopping rule are the contributions incorporated into the presented inversion method. The method is tested by synthetic and real gravity data, and the obtained results confirmed the potential practicality of the method.
Francesco Pintori, Enrico Serpelloni, and Adriano Gualandi
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We study time-varying vertical deformation signals in the European
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displacements. Recognizing and filtering out non-tectonic signals allows us
to improve the accuracy and precision of the vertical velocities.
Barend Cornelis Root, Josef Sebera, Wolfgang Szwillus, Cedric Thieulot, Zdeněk Martinec, and Javier Fullea
Solid Earth, 13, 849–873, https://doi.org/10.5194/se-13-849-2022, https://doi.org/10.5194/se-13-849-2022, 2022
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Several alternative gravity modelling techniques and associated numerical codes with their own advantages and limitations are available for the solid Earth community. With upcoming state-of-the-art lithosphere density models and accurate global gravity field data sets, it is vital to understand the differences of the various approaches. In this paper, we discuss the four widely used techniques: spherical harmonics, tesseroid integration, triangle integration, and hexahedral integration.
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Solid Earth, 12, 2523–2537, https://doi.org/10.5194/se-12-2523-2021, https://doi.org/10.5194/se-12-2523-2021, 2021
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After an earthquake, the fault continues to slip for days to months. Yet, little is know about the very early part of this phase (i.e., minutes to hours). We have looked at what happens just after an earthquake in Chile from 2015. We find that the fault responds in two ways: south of the rupture zone it slips seismically in the form of aftershocks, while north of the rupture zone it slips slowly. Early inference of such bimodal behavior could prove to be useful for forecasting aftershocks.
Séverine Liora Furst, Samuel Doucet, Philippe Vernant, Cédric Champollion, and Jean-Louis Carme
Solid Earth, 12, 15–34, https://doi.org/10.5194/se-12-15-2021, https://doi.org/10.5194/se-12-15-2021, 2021
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We develop a two-step methodology combining multiple surface deformation measurements above a salt extraction site (Vauvert, France) in order to overcome the difference in resolution and accuracy. Using this 3-D velocity field, we develop a model to determine the kinematics of the salt layer. The model shows a collapse of the salt layer beneath the exploitation. It also identifies a salt flow from the deepest and most external part of the salt layer towards the center of the exploitation.
Bogdan Matviichuk, Matt King, and Christopher Watson
Solid Earth, 11, 1849–1863, https://doi.org/10.5194/se-11-1849-2020, https://doi.org/10.5194/se-11-1849-2020, 2020
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The Earth deforms as the weight of ocean mass changes with the tides. GPS has been used to estimate displacements of the Earth at tidal periods and then used to understand the properties of the Earth or to test models of ocean tides. However, there are important inaccuracies in these GPS measurements at major tidal periods. We find that combining GPS and GLONASS gives more accurate results for constituents other than K2 and K1; for these, GLONASS or ambiguity resolved GPS are preferred.
Letizia Anderlini, Enrico Serpelloni, Cristiano Tolomei, Paolo Marco De Martini, Giuseppe Pezzo, Adriano Gualandi, and Giorgio Spada
Solid Earth, 11, 1681–1698, https://doi.org/10.5194/se-11-1681-2020, https://doi.org/10.5194/se-11-1681-2020, 2020
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The Venetian Southern Alps (Italy) are located in a slowly deforming plate-boundary region where strong earthquakes occurred in the past even if seismological and geomorphological evidence is not conclusive about the specific thrust faults involved. In this study, we integrate and model different geodetic datasets of ground velocity to constrain the seismogenic potential of the studied faults, giving an example of the importance of using vertical geodetic data for seismic hazard estimates.
Wolfgang Szwillus, Jörg Ebbing, and Bernhard Steinberger
Solid Earth, 11, 1551–1569, https://doi.org/10.5194/se-11-1551-2020, https://doi.org/10.5194/se-11-1551-2020, 2020
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At the bottom of the mantle (2850 km depth) two large volumes of reduced seismic velocity exist underneath Africa and the Pacific. Their reduced velocity can be explained by an increased temperature or a different chemical composition. We use the gravity field to determine the density distribution inside the Earth's mantle and find that it favors a distinct chemical composition over a purely thermal cause.
Yu Tian and Yong Wang
Solid Earth, 11, 1121–1144, https://doi.org/10.5194/se-11-1121-2020, https://doi.org/10.5194/se-11-1121-2020, 2020
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Given the inconsistency of the plane height and also the effects of the initial density model on the inversion results, the sequential inversion of on-orbit GOCE satellite gravity gradient and terrestrial gravity are divided into two integrated processes. Some new findings are discovered through the reliable and effective inversion results in the North China Craton.
Jérémie Giraud, Mark Lindsay, Mark Jessell, and Vitaliy Ogarko
Solid Earth, 11, 419–436, https://doi.org/10.5194/se-11-419-2020, https://doi.org/10.5194/se-11-419-2020, 2020
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We propose a methodology for the identification of rock types using geophysical and geological information. It relies on an algorithm used in machine learning called
self-organizing maps, to which we add plausibility filters to ensure that the results respect base geological rules and geophysical measurements. Application in the Yerrida Basin (Western Australia) reveals that the thinning of prospective greenstone belts at depth could be due to deep structures not seen from surface.
Marc Rovira-Navarro, Wouter van der Wal, Valentina R. Barletta, Bart C. Root, and Louise Sandberg Sørensen
Solid Earth, 11, 379–395, https://doi.org/10.5194/se-11-379-2020, https://doi.org/10.5194/se-11-379-2020, 2020
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The Barents Sea and Fennoscandia were home to large ice sheets around 20 000 years ago. After the melting of these ice sheets, the land slowly rebounded. The rebound speed is determined by the viscosity of the deep Earth. The rebound is ongoing and causes small changes in the Earth’s gravity field, which can be measured by the GRACE satellite mission. We use these measurements to obtain the viscosity of the upper mantle and find that it is 2 times higher in Fennoscandia than in the Barents Sea.
Junjie Wang, Nigel T. Penna, Peter J. Clarke, and Machiel S. Bos
Solid Earth, 11, 185–197, https://doi.org/10.5194/se-11-185-2020, https://doi.org/10.5194/se-11-185-2020, 2020
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Changes in the Earth's elastic strength at increasing timescales of deformation affect predictions of its response to the shifting weight of the oceans caused by tides. We show that these changes are detectable using GPS and must be accounted for but that 3-D or locally-tuned models of the Earth's behaviour around the East China Sea provide only slightly better predictions than a simpler model which varies only with depth. Use of this model worldwide will improve precise positioning by GPS.
Thomas Frederikse, Felix W. Landerer, and Lambert Caron
Solid Earth, 10, 1971–1987, https://doi.org/10.5194/se-10-1971-2019, https://doi.org/10.5194/se-10-1971-2019, 2019
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Due to ice sheets and glaciers losing mass, and because continents get wetter and drier, a lot of water is redistributed over the Earth's surface. The Earth is not completely rigid but deforms under these changes in the load on top. This deformation affects sea-level observations. With the GRACE satellite mission, we can measure this redistribution of water, and we compute the resulting deformation. We use this computed deformation to improve the accuracy of sea-level observations.
Christine Masson, Stephane Mazzotti, Philippe Vernant, and Erik Doerflinger
Solid Earth, 10, 1905–1920, https://doi.org/10.5194/se-10-1905-2019, https://doi.org/10.5194/se-10-1905-2019, 2019
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In using dense geodetic networks and large GPS datasets, we are able to extract regionally coherent velocities and deformation rates in France and neighboring western European countries. This analysis is combined with statistical tests on synthetic data to quantify the deformation detection thresholds and significance levels.
Evren Pakyuz-Charrier, Mark Jessell, Jérémie Giraud, Mark Lindsay, and Vitaliy Ogarko
Solid Earth, 10, 1663–1684, https://doi.org/10.5194/se-10-1663-2019, https://doi.org/10.5194/se-10-1663-2019, 2019
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This paper improves the Monte Carlo simulation for uncertainty propagation (MCUP) method for 3-D geological modeling. Topological heterogeneity is observed in the model suite. The study demonstrates that such heterogeneity arises from piecewise nonlinearity inherent to 3-D geological models and contraindicates use of global uncertainty estimation methods. Topological-clustering-driven uncertainty estimation is proposed as a demonstrated alternative to address plausible model heterogeneity.
Maurizio Milano, Maurizio Fedi, and J. Derek Fairhead
Solid Earth, 10, 697–712, https://doi.org/10.5194/se-10-697-2019, https://doi.org/10.5194/se-10-697-2019, 2019
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In this work we aim to interpret the extended magnetic low visible at satellite altitudes above central Europe by performing a joint analysis of magnetic field and total gradient intensity maps at low and high altitudes. Here we demonstrate that such a magnetic anomaly is mainly a result of the contrast between two crustal platforms differing strongly in geological and magnetic properties. Synthetic model tests have been created to support our modeling.
Martin Kobe, Gerald Gabriel, Adelheid Weise, and Detlef Vogel
Solid Earth, 10, 599–619, https://doi.org/10.5194/se-10-599-2019, https://doi.org/10.5194/se-10-599-2019, 2019
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Subrosion, i.e. the underground leaching of soluble rocks, causes disastrous sinkhole events worldwide. We investigate the accompanying mass transfer using quarter-yearly time-lapse gravity campaigns over 4 years in the town of Bad Frankenhausen, Germany. After correcting for seasonal soil water content, we find evidence of underground mass loss and attempt to quantify its amount. This is the first study of its kind to prove the feasibility of this approach in an urban area.
Christine Masson, Stephane Mazzotti, and Philippe Vernant
Solid Earth, 10, 329–342, https://doi.org/10.5194/se-10-329-2019, https://doi.org/10.5194/se-10-329-2019, 2019
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We use statistical analyses of synthetic position time series to estimate the potential precision of GPS velocities. Regression tree analyses show that the main factors controlling the velocity precision are the duration of the series, the presence of offsets, and the noise. Our analysis allows us to propose guidelines which can be applied to actual GPS data that constrain the velocity accuracies.
Sergei Rudenko, Saskia Esselborn, Tilo Schöne, and Denise Dettmering
Solid Earth, 10, 293–305, https://doi.org/10.5194/se-10-293-2019, https://doi.org/10.5194/se-10-293-2019, 2019
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A terrestrial reference frame (TRF) realization is a basis for precise orbit determination of Earth-orbiting artificial satellites and sea level studies. We investigate the impact of a switch from an older TRF realization (ITRF2008) to a new one (ITRF2014) on the quality of orbits of three altimetry satellites (TOPEX/Poseidon, Jason-1, and Jason-2) for 1992–2015, but especially from 2009 onwards, and on altimetry products computed using the satellite orbits derived using ITRF2014.
Jeremie Giraud, Mark Lindsay, Vitaliy Ogarko, Mark Jessell, Roland Martin, and Evren Pakyuz-Charrier
Solid Earth, 10, 193–210, https://doi.org/10.5194/se-10-193-2019, https://doi.org/10.5194/se-10-193-2019, 2019
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We propose the quantitative integration of geology and geophysics in an algorithm integrating the probability of observation of rocks with gravity data to improve subsurface imaging. This allows geophysical modelling to adjust models preferentially in the least certain areas while honouring geological information and geophysical data. We validate our algorithm using an idealized case and apply it to the Yerrida Basin (Australia), where we can recover the geometry of buried greenstone belts.
Karen M. Simon, Riccardo E. M. Riva, Marcel Kleinherenbrink, and Thomas Frederikse
Solid Earth, 9, 777–795, https://doi.org/10.5194/se-9-777-2018, https://doi.org/10.5194/se-9-777-2018, 2018
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This study constrains the post-glacial rebound signal in Scandinavia and northern Europe via the combined inversion of prior forward model information with GPS-measured vertical land motion data and GRACE gravity data. The best-fit model for vertical motion rates has a χ2 value of ~ 1 and a maximum uncertainty of 0.3–0.4 mm yr−1. An advantage of inverse models relative to forward models is their ability to estimate formal uncertainties associated with the post-glacial rebound process.
Hai Ninh Nguyen, Philippe Vernant, Stephane Mazzotti, Giorgi Khazaradze, and Eva Asensio
Solid Earth, 7, 1349–1363, https://doi.org/10.5194/se-7-1349-2016, https://doi.org/10.5194/se-7-1349-2016, 2016
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We present a new 3-D GPS velocity solution for 182 sites for the region encompassing the Western Alps, Pyrenees. The only significant horizontal deformation (0.2 mm/yr over a distance of 50 km) is a NNE–SSW extension in the western Pyrenees. In contrast, significant uplift rates up to 2 mm/yr occur in the Western Alps but not in the Pyrenees. A correlation between site elevations and fast uplift rates in the Western Alps suggests that part of this uplift is induced by postglacial rebound.
Gang Chen, Anmin Zeng, Feng Ming, and Yifan Jing
Solid Earth, 7, 817–825, https://doi.org/10.5194/se-7-817-2016, https://doi.org/10.5194/se-7-817-2016, 2016
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In this paper, we presented a new method on the basis of a collocation and multi-quadric equation interpolation. We introduce a multi-quadric kernel function to determine the covariance of local deformation and use the method to be a simple approximation of the covariance function. We established a horizontal velocity field model for the Chinese mainland by using a set of observed velocity data of GPS stations. The result is simple and reasonable and has a significant reference value.
M. Nordman, M. Poutanen, A. Kairus, and J. Virtanen
Solid Earth, 5, 673–681, https://doi.org/10.5194/se-5-673-2014, https://doi.org/10.5194/se-5-673-2014, 2014
A. Zlinszky, G. Timár, R. Weber, B. Székely, C. Briese, C. Ressl, and N. Pfeifer
Solid Earth, 5, 355–369, https://doi.org/10.5194/se-5-355-2014, https://doi.org/10.5194/se-5-355-2014, 2014
S. Rudenko, N. Schön, M. Uhlemann, and G. Gendt
Solid Earth, 4, 23–41, https://doi.org/10.5194/se-4-23-2013, https://doi.org/10.5194/se-4-23-2013, 2013
T. Frontera, A. Concha, P. Blanco, A. Echeverria, X. Goula, R. Arbiol, G. Khazaradze, F. Pérez, and E. Suriñach
Solid Earth, 3, 111–119, https://doi.org/10.5194/se-3-111-2012, https://doi.org/10.5194/se-3-111-2012, 2012
J. Klokočník, J. Kostelecký, I. Pešek, P. Novák, C. A. Wagner, and J. Sebera
Solid Earth, 1, 71–83, https://doi.org/10.5194/se-1-71-2010, https://doi.org/10.5194/se-1-71-2010, 2010
Cited articles
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Short summary
Since 2002, the GRACE mission provides estimates of the Earth's time-variable gravity field, from which one can derive ocean mass variability. Now that the GRACE mission has come to an end, it is especially important to find alternative ways for deriving ocean mass changes. For the first time, we use kinematic orbits of Swarm for computing ocean mass time series. We compute monthly solutions, but also show an alternative way of directly estimating time-variable spherical harmonic coefficients.
Since 2002, the GRACE mission provides estimates of the Earth's time-variable gravity field,...