Abstract
In the last decade, satellite gravimetry has been revealed as a pioneering technique for mapping mass redistributions within the Earth system. This fact has allowed us to have an improved understanding of the dynamic processes that take place within and between the Earth’s various constituents. Results from the Gravity Recovery And Climate Experiment (GRACE) mission have revolutionized Earth system research and have established the necessity for future satellite gravity missions. In 2010, a comprehensive team of European and Canadian scientists and industrial partners proposed the e.motion (Earth system mass transport mission) concept to the European Space Agency. The proposal is based on two tandem satellites in a pendulum orbit configuration at an altitude of about 370 km, carrying a laser interferometer inter-satellite ranging instrument and improved accelerometers. In this paper, we review and discuss a wide range of mass signals related to the global water cycle and to solid Earth deformations that were outlined in the e.motion proposal. The technological and mission challenges that need to be addressed in order to detect these signals are emphasized within the context of the scientific return. This analysis presents a broad perspective on the value and need for future satellite gravimetry missions.
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Acknowledgments
This paper is based on the comprehensive work and analysis realized to prepare the e.motion proposal, in response to the European Space Agency Call for proposals Earth Explorer Opportunity Mission EE-8. As such, the results presented here greatly benefited from numerous inputs and discussions with the members of the e.motion science team, listed in Appendix 3. We gratefully thank them for their contributions. Industrial support was provided from SpaceTech GmbH Immenstaad and from the Office National d’Études et de Recherches Aérospatiales. We thank Michel Diament for helping us to improve this manuscript. We are grateful to the Editor, Anny Cazenave, and two anonymous reviewers, for their suggestions that contributed to improve this manuscript. Work by Isabelle Panet, Richard Biancale, Pascal Gegout, and Guillaume Ramillien was supported by CNES (Centre National d’Etudes Spatiales) through the TOSCA committee. This is IPGP contribution number 3344.
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Appendices
Appendix 1: Amplitude of Geoid Variations Associated with Local Water Loads
The amplitude of geoid variations associated with a water mass load depends on the spatial scale of this load, as shown in Wahr et al. (1998). Here, we computed the geoid effect of a water load spatially distributed as a Gaussian bell, with amplitude of 1 cm of equivalent water thickness at the center of the Gaussian. We considered the direct Newtonian attraction of the load and the Earth’s elastic deformation as given by the Love numbers. Results are provided in Table 2 for a Gaussian bell of varying radii r. The radius corresponds to the distance to the centre for which the water thickness has decreased by a factor of two. These results allow for easier conversions between EWH signals and geoid variations.
Appendix 2: Local Error Estimates
Earth system mass signals associated with various physical processes are most often local. This means that the estimates of the precision required to recover those signals should be local. Here we recall, following Dickey et al. (1997) and Wahr et al. (1998), how they are related to global spherical harmonics error spectra.
Let h(\( \theta ,\varphi , \)) denote the water height at the point of colatitude \( \theta \) and longitude \( \varphi \), and the average error over an area of interest. A localizing filter W(\( \theta ,\varphi \)) is associated with this area, describing its shape. However, because of the limited resolution of satellite gravity data, it is not possible to perfectly localize the target area, since an infinite number of spherical harmonics would be needed to build a perfect localizing filter, with a value of 1 inside the study area and 0 outside. Consequently, water height estimates from a truncated spherical harmonic spectrum will never be perfectly localized. Contrarily, a filter with a perfect spectral localization, such as that realized by a cumulative spherical harmonic spectrum, is associated with a highly oscillating spatial window. The construction of local basin filters is extensively discussed in the literature (e.g., Swenson and Wahr 2002; Seo and Wilson 2005).
Let us denote \( W_{lm}^{c} \), \( W_{lm}^{s} \) and \( h_{lm}^{c} \), \( h_{lm}^{s} \) the coefficients of the spherical harmonic expansion of the filter and the water height, respectively. The average water height within the approximate shape of the basin, resulting from the truncation at degree L of the spherical harmonic expansion of W, is given by:
where \( \Upomega \) is the solid angle subtended by the basin, equal to \( 4\pi \frac{{S_{\text{Area}} }}{{S_{\text{Earth}} }} \), where S Area is the surface of the area and S Earth the surface of the Earth. From the orthogonality of the spherical harmonics, one obtains the following equation:
Now, let us denote \( \delta h_{lm}^{c} \) and \( \delta h_{lm}^{s} \) the errors on the spherical harmonics coefficients of the water height measured by a satellite gravity mission, and \( \sigma_{\ell }^{2} = \sum\limits_{m = 0}^{\ell } {\left( {\left( {\delta h_{\ell m}^{c} } \right)^{2} + \left( {\delta h_{\ell m}^{s} } \right)^{2} } \right)} \) the degree variance. To simplify the expressions, we suppose that \( \delta h_{lm}^{c} \) and \( \delta h_{{l^{'} m^{'} }}^{s} \) are uncorrelated for any degrees and orders, and \( \delta h_{lm}^{c} \) and \( \delta h_{{l^{'} m^{'} }}^{c} \) (\( \delta h_{lm}^{s} \) and \( \delta h_{{l^{'} m}}^{s} \), respectively) are uncorrelated if \( \ell \ne \ell^{'} \) or \( m \ne m^{'} \). We make the approximation that \( \left( {\delta h_{\ell m}^{c} } \right)^{2} = \left( {\delta h_{\ell m}^{s} } \right)^{2} = \frac{{\sigma_{\ell }^{2} }}{2\ell + 1} \). The variance on the average water height resulting from the errors \( \delta h_{lm}^{c} \) and \( \delta h_{lm}^{s} \) can then be written as follows:
Introducing the degree spectrum of the localizing filter \( W_{\ell }^{{}} = \sqrt {\sum\limits_{m = 0}^{\ell } {\left( {\left( {W_{\ell m}^{c} } \right)^{2} + \left( {W_{\ell m}^{s} } \right)^{2} } \right)} } \), we end up with:
If we use a localizing window shaped as an axisymmetric Gaussian bell, then the spectrum W l is given in Wahr et al. (1998), based on Jekeli (1981). However, this Gaussian is normalized so that its global integral is equal to unity. Thus, to be used in Eq. (4), the W l coefficients given in equation (34) of Wahr et al. (1998) should be divided by the normalization factor \( \frac{b}{2\pi }\frac{1}{{1 - e^{ - 2b} }} \), with \( b = \frac{\ln \left( 2 \right)}{{1 - \cos \left( {r/a} \right)}} \), a the semi-major axis of the Earth’s ellipsoid (a = 6,378,136.46 m), and r the distance at the Earth’s surface for which the value of the Gaussian window is equal to half its maximum.
Appendix 3: The e.motion Science Team
J. Bamber1, R. Biancale2, M. van den Broeke3, T. van Dam4, K. Danzmann5, M. Diament6, H. Dobslaw7, F. Flechtner7, J. Flury8, P. Gegout2, T. Gruber9, A. Güntner7, G. Heinzel5, M. Horwath10, J. Huang11, C.W. Hughes12, A. Jäggi13, J. Johannessen14, P. Knudsen15, J. Kusche16, B. Legresy10, F. Migliaccio17, R. Pail9, I. Panet18,6, G. Ramillien2, M-H. Rio19, R. Sabadini20, I. Sasgen7, H. Savenije21, L. Seoane2, B. Sheard5, M. Sideris22, N. Sneeuw23, D. Stammer24, M. Thomas7, B. Vermeersen25, P. Visser25, S. Vitale26, P. Woodworth12
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1University of Bristol, Bristol Glaciology Centre, Bristol, United Kingdom
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2Observatoire Midi-Pyrénées, Groupe de Recherche de Géodésie Spatiale, Toulouse, France
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3Utrecht University, Institute for Marine and Atmospheric Research, Utrecht, The Netherlands
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4Université du Luxembourg, Faculté des Sciences, de la Technologie et de la Communication, Luxembourg
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5Max-Planck Institut für Gravitationsphysik, Albert-Einstein Institut, Hannover, Germany
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6Institut de Physique du Globe de Paris, Paris, France
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7German Research Centre for Geosciences (GFZ), Potsdam, Germany
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8Leibnitz Universität Hannover, Centre for Quantum Engineering and Space-Time Research, Hannover, Germany
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9Technische Universität München, Institut für Astronomische und Physikalische Geodäsie, Münich, Germany
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10Laboratoire d’Études en Géophysique et Océanographie Spatiales, Toulouse, France
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11National Resources Canada, Ottawa, Canada
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12National Oceanography Centre, Liverpool, United Kingdom
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13Universität Bern, Astronomisches Institut, Bern, Switzerland
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14Nansen Environmental and Remote Sensing Center, Bergen, Norway
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15Technical University of Denmark, National Space Institute, Copenhagen, Denmark
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16Universität Bonn, Institut für Geodäsie und Geoinformation, Bonn, Germany
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17Politecnico di Milano, Dipartimento di Ingegneria Idraulica, Ambientale, Infrastrutture Viarie, Rilevamento, Milano, Italy
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18Institut National de l’Information Géographique et Forestière, Marne-la-Vallée, France
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19Collecte Localisation Satellites, Ramonville-Saint-Agne, France
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20Università degli Studi di Milano, Milano, Italy
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21Delft University of Technology, Water Resources Section, Delft, The Netherlands
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22University of Calgary, Department of Geomatics Engineering, Calgary, Canada
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23Universität Stuttgart, Geodätisches Institut, Stuttgart, Germany
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24Universität Hamburg, Zentrum für Marine und Atmosphärische Wissenschaften, Hamburg, Germany
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25Delft University of Technology, Astrodynamics and Satellite Systems, Delft, The Netherlands
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26University of Trento, Department of Physics, Trento, Italy
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Panet, I., Flury, J., Biancale, R. et al. Earth System Mass Transport Mission (e.motion): A Concept for Future Earth Gravity Field Measurements from Space. Surv Geophys 34, 141–163 (2013). https://doi.org/10.1007/s10712-012-9209-8
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DOI: https://doi.org/10.1007/s10712-012-9209-8