Earth System Mass Transport Mission (e.motion): A Concept for Future Earth Gravity Field Measurements from Space

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.

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.

Table 2 Geoid effect of water loads with a spatial distribution as a Gaussian bell of varying radii

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:

$$ \bar{h} = \frac{1}{\Upomega }\int {h^{L} \left( {\theta ,\phi } \right) \cdot W^{L} \left( {\theta ,\phi } \right)} \cdot \sin \theta {\text{d}}\theta {\text{d}}\phi $$

(1)

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:

$$ \bar{h} = \frac{1}{\Upomega }\sum\limits_{\ell = 0}^{L} {\sum\limits_{m = 0}^{\ell } {\left( {W_{\ell m}^{c} \cdot h_{\ell m}^{c} + W_{\ell m}^{s} \cdot h_{\ell m}^{s} } \right)} } $$

(2)

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:

$$ \text{var} \left( {\bar{h}} \right) = \frac{1}{{\Upomega^{2} }}\sum\limits_{\ell = 0}^{L} {\frac{{\sigma_{\ell }^{2} }}{2\ell + 1}\sum\limits_{m = 0}^{\ell } {\left( {\left( {W_{\ell m}^{c} } \right)^{2} + \left( {W_{\ell m}^{s} } \right)^{2} } \right)} } $$

(3)

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:

$$ \text{var} \left( {\bar{h}} \right) = \frac{1}{{\Upomega^{2} }}\sum\limits_{\ell = 0}^{L} {\frac{{\sigma_{\ell }^{2} }}{2\ell + 1}W_{\ell }^{2} } $$

(4)

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. Panet^18,6, 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

• ¹University of Bristol, Bristol Glaciology Centre, Bristol, United Kingdom

• ²Observatoire Midi-Pyrénées, Groupe de Recherche de Géodésie Spatiale, Toulouse, France

• ³Utrecht University, Institute for Marine and Atmospheric Research, Utrecht, The Netherlands

• ⁴Université du Luxembourg, Faculté des Sciences, de la Technologie et de la Communication, Luxembourg

• ⁵Max-Planck Institut für Gravitationsphysik, Albert-Einstein Institut, Hannover, Germany

• ⁶Institut de Physique du Globe de Paris, Paris, France

• ⁷German Research Centre for Geosciences (GFZ), Potsdam, Germany

• ⁸Leibnitz Universität Hannover, Centre for Quantum Engineering and Space-Time Research, Hannover, Germany

• ⁹Technische Universität München, Institut für Astronomische und Physikalische Geodäsie, Münich, Germany

• ¹⁰Laboratoire d’Études en Géophysique et Océanographie Spatiales, Toulouse, France

• ¹¹National Resources Canada, Ottawa, Canada

• ¹²National Oceanography Centre, Liverpool, United Kingdom

• ¹³Universität Bern, Astronomisches Institut, Bern, Switzerland

• ¹⁴Nansen Environmental and Remote Sensing Center, Bergen, Norway

• ¹⁵Technical University of Denmark, National Space Institute, Copenhagen, Denmark

• ¹⁶Universität Bonn, Institut für Geodäsie und Geoinformation, Bonn, Germany

• ¹⁷Politecnico di Milano, Dipartimento di Ingegneria Idraulica, Ambientale, Infrastrutture Viarie, Rilevamento, Milano, Italy

• ¹⁸Institut National de l’Information Géographique et Forestière, Marne-la-Vallée, France

• ¹⁹Collecte Localisation Satellites, Ramonville-Saint-Agne, France

• ²⁰Università degli Studi di Milano, Milano, Italy

• ²¹Delft University of Technology, Water Resources Section, Delft, The Netherlands

• ²²University of Calgary, Department of Geomatics Engineering, Calgary, Canada

• ²³Universität Stuttgart, Geodätisches Institut, Stuttgart, Germany

• ²⁴Universität Hamburg, Zentrum für Marine und Atmosphärische Wissenschaften, Hamburg, Germany

• ²⁵Delft University of Technology, Astrodynamics and Satellite Systems, Delft, The Netherlands

• ²⁶University of Trento, Department of Physics, Trento, Italy