Modélisation des surcharges induites par les fluides superficiels.

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Modélisation des surcharges induites par les fluides

superficiels.

Jean-Paul Boy

To cite this version:

Jean-Paul Boy. Modélisation des surcharges induites par les fluides superficiels.. Géophysique [physics.geo-ph]. Université Louis Pasteur, Strasbourg I, 2007. �tel-01260184�

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MEMOIRE

pr´esent´e par

Jean-Paul Boy

Pour l’obtention de l’Habilitation `a Diriger les Recherches

de l’Universit´e Louis Pasteur, Strasbourg I

Spécialité : Géophysique

Mod´elisation des surcharges induites

par les fluides superficiels

Soutenu le 7 mars 2007 devant le jury compos´e de :

Trevor Baker (Proudman Oceanographic Laboratory, Liverpool) Examinateur

Richard Biancale (CNES/GRGS, Toulouse)

Rapporteur Externe

Marianne Greff-Leftz (Institut de Physique du Globe de Paris)

Rapporteur Externe

Jacques Hinderer (Institut de Physique du Globe de Strasbourg) Garant

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R´esum´e

La dynamique des fluides superficiels (atmosphère, océans, etc.) engendre des redistributions de masse en surface de la Terre, et donc, des déformations globales et des variations du champ de gravité terrestre. La précision désormais atteinte par les différents observables géodésiques permet l’étude de ces phénomènes sur un large spectre tant fréquentiel (périodes comprises entre quelques heures et les cycles saisonniers, voire des variations séculaires) que spatial (longueurs d’onde de quelques kilomètres aux phénomènes globaux).

Nous présentons ici notre effort de modélisation de ces différentes contributions et leur com-paraison aux observables géodésiques, principalement aux variations temporelles de la gravité me-surées par les gravimètres supraconducteurs. Celles-ci sont déterminées par convolution des sorties de modèles de circulation atmosphérique, océanique ou hydrologique avec une fonction de Green, exprimant la réponse de la Terre solide.

Nous présentons tout d’abord l’apport d’une modélisation précise des surcharges atmosphériques sur différents observables géodésiques, i.e. les variations de gravité mesurées en surface, du champ de pesanteur terrestre mesuré par satellite et les déformations de surface. Leur calcul précis nécessite la prise en compte de la réponse des océans à la pression ; nous montrons l’apport de modèles dynamiques de circulation océanique par rapport à l’approximation statique (baromètre inversé) pour l’interprétation des mesures de gravité.

Les marées océaniques constituent la source la plus importante de redistribution de masses à la surface de la Terre. La précision et la stabilité des gravimètres supraconducteurs permettent l’étude des surcharges induites sur l’ensemble de leur spectre, c’est-à-dire des ondes longues périodes aux ondes non-linéaires générées sur les différents plateaux continentaux. Nous présentons ici notre apport à leur compréhension et à leur modélisation.

Une dernière source importante de variations de gravité et de déformations est l’hydrologie continentale ; même si sa modélisation n’a pas encore atteint la maturité et la précision des modèles de circulation atmosphérique et océanique, elle permet une estimation satisfaisante des variations de gravité aux basses fréquences (cycles saisonniers). Nous présentons également le cas particulier du remplissage du barrage des Trois Gorges, sur le fleuve Yangtze (Chine), comme outil de validation des missions de gravité spatiale.

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Abstract

The dynamics of the surface geophysical fluids (atmosphere, oceans, etc.) induce mass redistri-bution at the Earth’s surface, and therefore a global deformation and gravity field changes. The accuracy achieved nowadays by most of geodetic observations allows their study on a wide fre-quency band (periods between a few hours and seasonal cycles, and up to secular variations) as well as on a large wavelength domain (from a few kilometers to global ranges).

We present here our effort in modeling these contributions and their comparison to various geodetic observations, especially surface gravity changes recorded by superconducting gravimeters. They are computed through a convolution of atmospheric, oceanic or hydrological general circula-tion model outputs and Green’s funccircula-tions, describing the solid Earth’s response.

We first present the impact of the precise modeling of atmospheric loading on various geo-detic observations, i.e. surface gravity variations, Earth’s time-variable gravity field and surface displacements. An accurate computation of these effects requires a model of the ocean response to atmospheric pressure forcing ; we show the improvement in gravity interpretations using dynamic ocean models, compared to the static approximation (inverted barometer).

Ocean tides are the largest source of mass redistribution at the Earth’s surface. The accuracy and the stability of superconducting gravimeters allow studying their loading effects on their complete frequency range, i.e. from long period to non-linear tidal waves, generated on major continental shelves. We present here our contribution to their understanding and modeling.

The last major source of gravity variations and deformation is the continental hydrology ; even if its modeling is not as accurate as, for example, atmospheric or oceanic general circulation models, it allows a correct estimation of long period gravity changes (seasonal cycle). We also show the case of the water impoudment of the Three Gorges Dam, on the Yangtze River (China), as a validation tool of space gravity missions.

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Table des mati`eres

Curriculum Vitae 9

Curriculum Vitae . . . 9

Liste de publications . . . 10

Contrats de financement . . . 12

Encadrement d’´etudiants de second et troisi`eme cycle . . . 12

Introduction 13 I Surcharges atmosphériques 17 I.1 Surcharges atmosphériques et variations de gravité en surface . . . 17

I.2 Surcharges atmosph´eriques et champ de gravit´e . . . 31

I.3 Surcharges atmosph´eriques et d´eplacements en surface . . . 43

I.4 Conclusion . . . 58

II Réponse des océans à la pression atmosphérique 59 II.1 Variations de gravité en surface et modèles barotropes . . . 59

II.1.1 Calcul des surcharges atmosph´eriques et oc´eaniques . . . 61

II.1.2 Traitement des donn´ees gravim´etriques . . . 62

II.1.3 R´esultats . . . 62

II.1.4 Exemple d’un phénomène transitoire en janvier-février 2000 . . . 66

II.2 Champ de gravit´e et mod`eles barotropes . . . 69

II.3 For¸cage d’un mod`ele barocline par la pression atmosph´erique . . . 72

II.4 Conclusion . . . 84

III Marées océaniques 85 III.1 Marées longues périodes . . . 85

III.2 Mar´ees diurnes et semi-diurnes . . . 93

III.3 Mar´ees non-lin´eaires . . . 111

III.4 Conclusion . . . 130

IV Surcharges hydrologiques 133 IV.1 Surcharges hydrologiques et variations de gravit´e en surface . . . 133

IV.2 Validation de GRACE : Barrage des Trois-Gorges . . . 141

IV.3 Conclusion . . . 148

Conclusion 151

Bibliographie 153

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Curriculum Vitae

Etat-Civil Jean-Paul BOY

Né le 25 mai 1974 à Sainte Adresse (France) Nationalité Fran¸caise

C´elibataire

Adresse professionnelle

Ecole et Observatoire des Sciences de la Terre

Institut de Physique du Globe de Strasbourg (UMR 7516 CNRS-ULP) 5, rue Ren´e Descartes

67084 Strasbourg Cedex, France.

Tel : 03 90 24 01 09 Fax : 03 90 24 02 91 Email : [email protected]

Formation

– Doctorat de l’Université Louis Pasteur, Strasbourg I, soutenu le 29 novembre 2000. Effets des surcharges atmosphériques sur les variations de gravité et les déplacements de surface de la Terre, sous la direction de Jacques Hinderer.

Prix de th`ese de l’ADRERUS en 2001

– Diplôme d’Ingénieur Géophysicien de l’Ecole et Observatoire de Physique du Globe, soutenu le 8 novembre 1996.

– DEA de Physique et Chimie de la Terre, `a l’Universit´e Louis Pasteur, Strasbourg I, obtenu en juin 1996.

Exp´eriences professionnelles

– Depuis Janvier 2004, Physicien-Adjoint `a l’Ecole et Observatoire des Sciences de la Terre, au sein de l’Equipe de Dynamique Globale.

– Septembre 2001 - Décembre 2003 : stage post-doctoral au NASA Goddard Space Flight Center (USA), au sein de l’équipe de Géodésie spatiale, et sous la direction de Benjamin F. Chao. – Septembre 2000 - Août 2001 : ATER à l’Ecole et Observatoire des Sciences de la Terre.

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10 Liste de publications

Liste de publications

Revues `a comit´e de lecture

1. Nicolas, J., J.-M. Nocquet, M. Van Camp, T. van Dam, J.-P. Boy, J. Hinderer, P. Gegout, E. Calais et M. Amalvict, Seasonal effect on vertical positioning by Satellite Laser Ranging and Global Positioning System and on Absolute Gravity at the OCA geodetic station, Grasse, France, Geophysical Journal International, accept´e.

2. Boy, J.-P., M. Llubes, R. Ray, J. Hinderer et N. Florsch, Validation of long-period oceanic tidal models with superconducting gravimeters, Journal of Geodynamics, 41, 112–118, 2006. 3. Boy, J.-P., et J. Hinderer, Study of the seasonal gravity signal in superconducting gravimeter

data, Journal of Geodynamics, 41, 227–233, 2006.

4. Boy, J.-P., R. Ray et J. Hinderer, Diurnal atmospheric tide and induced gravity variations, Journal of Geodynamics, 41, 253–258, 2006.

5. Hinderer, J., O. Andersen, F. Lemoine, D. Crossley et J.-P. Boy, Seasonal changes in the European gravity field from GRACE : A comparison with superconducting gravimeters and hydrology model predictions, Journal of Geodynamics, 41, 59–68, 2006.

6. Sato, T., J.-P. Boy, Y. Tamura, K. Matsumoto, K. Asari, H.-P. Plag et O. Francis, Gravity tide and seasonal gravity variation at Ny-˚Alesund, Svalbard in Arctic, Journal of Geodyna-mics, 41, 234–241, 2006.

7. Boy, J.-P., et B. F. Chao, Precise evaluation of atmospheric loading effects on Earth’s time-variable gravity field, Journal of Geophysical Research, 110, B08412, doi:10.1029/2002JB002333, 2005.

8. Crossley, D., J. Hinderer et J.-P. Boy, Time variation of the European gravity field from superconducting gravimeters, Geophysical Journal International, 161, 257–264, 2005. 9. Boy, J.-P., M. Llubes, R. Ray, J. Hinderer, N. Florsch, S. Rosat, F. Lyard et T.

Letel-lier, Non-linear oceanic tides observed by superconducting gravimeters in Europe, Journal of Geodynamics, 38, 391–405, 2004.

10. Crossley, D., J. Hinderer et J.-P. Boy, Regional gravity variations in Europe from supercon-ducting gravimeters, Journal of Geodynamics, 38, 325–342, 2004.

11. Petrov, L., et J.-P. Boy, Study of the atmospheric pressure loading signal in VLBI observa-tions, Journal of Geophysical Research, 109, B03405, doi:10.1029/2003JB002500, 2004. 12. Rosat, S., J. Hinderer, D. Crossley et J.-P. Boy, Performance of superconducting gravimeters

from long-period seismology to tides, Journal of Geodynamics, 38, 461–476, 2004.

13. de Viron, O., J.-P. Boy et H. Goosse, Geodetic effects of the ocean response to atmospheric forcing in an ocean general circulation model, Journal of Geophysical Research, 109, B03411, doi:10.1029/2003JB002837, 2004.

14. Boy, J.-P., M. Llubes, J. Hinderer et N. Florsch, A comparison of tidal ocean loading mo-dels using superconducting gravimeter data, Journal of Geophysical Research, 108, 2193, doi:10.1029/ 2002JB002050, 2003.

15. Chao, B. F., A. Y. Au, J.-P. Boy et C. M. Cox, Time-variable gravity signal of an anoma-lous redistribution of water mass in the extratropic Pacific during 1998-2002, Geochemistry, Geophysics, Geosystems, 4, doi:10.1029/2003GC000589, 2003.

16. Boy, J.-P., et B. F. Chao, Time-variable gravity signal during the water impoundment of China’s Three-Gorges Reservoir, Geophysical Research Letters, 29, 2200, doi:10.1029/ 2002GL016457, 2002.

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17. Boy, J.-P., P. G´egout et J. Hinderer, Reduction of surface gravity data from global atmos-pheric pressure loading, Geophysical Journal International, 149, 534–545, 2002.

18. Amalvict, M., J. Hinderer, J.-P. Boy et P. G´egout, A three year comparison between a superconducting gravimeter (GWR CO26) and an absolute gravimeter (FG56) in Strasbourg (France), Journal of the Geodetic Society of Japan, 47, 334–340, 2001.

19. Boy, J.-P., P. G´egout et J. Hinderer, Gravity variations and global pressure loading, Journal of the Geodetic Society of Japan, 47, 267–272, 2001.

20. Hinderer, J., J.-P. Boy, P. G´egout, P. Defraigne, F. Roosbeek et V. Dehant, Are the free core nutation parameters variable in time ?, Physics of the Earth and Planetary Interiors, 117, 37–49, 2000.

21. Loyer, S., J. Hinderer et J.-P. Boy, Determination of the gravimetric factor at the Chandler period from Earth orientation data and superconducting gravimetry observations, Geophysical Journal International, 136, 1–7. 1999.

22. Boy, J.-P., J. Hinderer et P. G´egout, Global atmospheric loading and gravity, Physics of the Earth and Planetary Interiors, 109, 161–177. 1998.

Autres revues

1. Crossley, D., J. Hinderer, J.-P. Boy et C. de Linage, Status of the GGP Satellite Project, Bulletin d’Information des Mar´ees Terrestres, 142, 11423–11432, 2006.

2. Hinderer, J., C. de Linage et J.-P. Boy, How to validate satellite-derived gravity observations with gravimeters on the ground ?, Bulletin d’Information des Mar´ees Terrestres, 142, 11433– 11442, 2006.

3. de Linage, C., J. Hinderer et J.-P. Boy, search on the gravity/height ratio induced by surface loading : theoretical investigation and numerical applications, Bulletin d’Information des Mar´ees Terrestres, 142, 11451–11460, 2006.

4. Hinderer, J., S. Rosat, D. Crossley, M. Amalvict, J.-P. Boy et P. G´egout, Influence of different processing methods on the retrieval of gravity signals from GGP data, Bulletin d’Information des Mar´ees Terrestres, 135, 10653–10668, 2002.

5. Boy, J.-P., J. Hinderer, M. Amalvict et E. Calais, On the use of long records of supercon-ducting and absolute gravity observations with special application to the Strasbourg station, France, Cahiers du Centre Européen de Géodynamique et de Séismologie (ECGS), 17, 67–83, 2000.

6. Amalvict, M., J. Hinderer et J.-P. Boy, A comparative analysis between an absolute gravi-meter (FG5-206) and a superconducting gravigravi-meter (GWR CO26) in Strasbourg : new results on calibration and long-term gravity changes, Bollettino di Geofisica, Teorica ed Applicata, 40 (3-4), 519–525, 1999.

7. Boy, J.-P., et J. Hinderer, Atmospheric pressure effects on gravity : local versus global corrections, Bulletin d’Information des Mar´ees Terrestres, 131, 10113–10127, 1999.

8. Boy, J.-P., J. Hinderer et P. Gégout, The effect of atmospheric loading on gravity, Proceedings of the 13th International Symposium on Earth Tides, éditeurs B. Ducarme et P. Pâquet, Bruxelles, 439–446, 1998.

9. Hinderer, J., J.-P. Boy, et H. Legros, A 3000 day registration of the superconducting gravi-meter GWR T005 in Strasbourg (France), Proceedings of the 13th International Symposium on Earth Tides, ´editeurs B. Ducarme et P. Pˆaquet, Bruxelles, 439–446, 1998.

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12 Encadrement d’´etudiants

Contrats de financement

2006 CNES (Etude et Observation de la Terre : hydrologie continentale Géophysique interne, géodynamique et Géodésie)

Variation de pesanteur et déformation crustale induites par les charges superficielles hy-drologiques ; comparaison gravimétrique sol-orbite et apport des données GPS.

100 K-Euros sur la p´eriode 2006-2011 2005 Groupe Mission MERCATOR / CORIOLIS

Variations de pesanteur, de la rotation et d´eformations induites par la circulation oc´eanique.

Encadrement d’´etudiants de second et troisi`eme cycle

Th`ese de doctorat

Encadrement :

– C. de Linage : Mesures gravimétriques (sol, satellite) et positionnement géodésique : quel lien entre les variations de pesanteur et les mouvements verticaux ?, thèse de doctorat de 3ème cycle, Université Louis Pasteur, Strasbourg I, sous la direction de J. Hinderer (25 %). – S. Rosat : Variations temporelles de la gravité en relation avec la dynamique interne de la

Terre - Apport des gravimètres supraconducteurs, thèse de doctorat de 3ème cycle soutenu en mars 2004, Université Louis Pasteur, Strasbourg I, sous la direction de J. Hinderer (10 %). Rapporteur :

– D. Garc´ıa Garc´ıa : Space Geodesy techniques applied to study the steric and eustatic sea level variations, Universit´e d’Alicante (Espagne), sous la direction de I. Vigo Aguiar et B. F. Chao. Rapporteur de th`ese, soutenue le 19 juillet 2006.

Supervision de diplôme d’Ingénieur Géophysicien

– A. Deforges : Contribution à une méthodologie de contrôle qualité en traitement sismique (2006).

– D. Rocher : Modélisation hydrogéologique du site nucléaire de St Laurent des Eaux (2005). – P. Lacroix : Characterization of the Mauna Loa volcano (Hawaii) deformation using InSAR

measurements of Radarsat (2004).

Encadrement de Projets de Recherche

– I. Thollet : Pluviométrie, recharge d’aquifère et effets de gravité (2006). – R. Valois : Pluviométrie et stockage d’eau dans le sous-sol (2006).

– B. Gauthier : Effets de pression sur les enregistrements sismologiques ; analyse du bruit (2006). – A. Peyrefitte : Observation du tsunami du 26 d´ecembre 2004 par altim´etrie (2006).

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Introduction

La thématique principale de mes activités de recherche consiste en l’étude des interactions entre la Terre solide et les différents fluides géophysiques de surface, c’est-à-dire l’atmosphère, les océans et les réservoirs d’eau continentale, au moyen des différents observables géodésiques. En effet, la circulation globale de ces différents fluides induit des redistributions de masses à la surface de la Terre, et donc des variations du champ de gravité et des déformations. La précision des différents observables géophysiques (déplacements en surface, variations de gravité mesurées en surface, variations du champ de pesanteur global, etc.) permet dorénavant des estimations précises des variations de masse en surface sur un large domaine temporel (périodes de quelques heures aux cycles saisonniers et au-delà) et spatial (longueurs d’onde de quelques kilomètres aux échelles continentales).

Même si l’étude des variations de gravité, mesurées par les gravimètres supraconducteurs, consti-tue la principale thématique de mes activités de recherche, mes différentes collaborations, notam-ment avec l’équipe de géodésie spatiale du NASA Goddard Space Flight Center, m’ont permis d’étendre mes champs d’intéret aux autres observables géodésiques comme les variations tempo-relles du champ de pesanteur, dans le cadre de la mission GRACE (Gravity Recovery And Climate Experiment), ainsi qu’aux déplacements de surface.

De même, si mes travaux de thèse étaient principalement focalisés sur la contribution at-mosphérique, la stabilité et la sensibilité des gravimètres supraconducteurs permettent l’étude des variations de gravité sur un large domaine spectral : périodes de quelques heures, comme les sur-charges induites par les ondes de marées non-linéaires (Boy et al., 2004), aux cycles saisonniers et annuels comme les contributions induites par les variations d’humidité du sol (Boy et Hinderer, 2006) ou par le mouvement du pôle de rotation (Loyer et al., 1999).

Même si cette séparation peut parfois paraˆıtre assez artificielle (notamment les chapitres I et II), j’ai choisi de décomposer ce manuscrit en fonction des sources des différentes surcharges : l’atmosphère, les océans (circulation rapide forcée par la pression et les vents, ainsi que les marées océaniques) et l’hydrologie continentale. Dans chaque chapitre, peuvent donc se retrouver plusieurs techniques (gravité ou déplacements en surface, champ de pesanteur) malgré des sensibilités spa-tiales et/ou temporelles différentes.

L’ensemble de ces différents travaux ont été effectués au sein de l’équipe de Dynamique Globale de l’Institut de Physique du Globe de Strasbourg, avec Nicolas Florsch (Université Pierre et Marie Curie, Paris VI) et Muriel Llubes (Laboratoire d’Etudes en Géophysique et Océanographie Spatiale, Toulouse), en particulier pour les études des surcharges de marées, ainsi que l’équipe de Géodésie Spatiale du NASA/GSFC.

Surcharges atmosph´eriques

L’atmosphère est très certainement le fluide géophysique le mieux modélisé, hormis peut-être les phénomènes les plus rapides (de périodes inférieures à, typiquement, une heure) et de très courtes

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14 Introduction

longueurs d’onde horizontales (dizaine de kilomètres au plus). Le développement de modèles de prévision météorologique à moyen-terme, depuis la fin de la seconde guerre mondiale, nous permet de disposer aujourd’hui, en quasi-temps réel, de champs globaux atmosphériques avec des pas de quelques heures (typiquement 3 ou 6 heures) et de quelques dizaines de kilomètres, permettant une estimation très précise des variations de la forme et du champ de pesanteur terrestre.

L’étude de la contribution des surcharges atmosphériques sur les variations de gravité mesurées en surface par les différents gravimètres supraconducteurs a constitué le corps de mon travail de thèse. J’ai pu alors montré qu’un calcul rigoureux des surcharges atmosphériques à partir des champs de pression surfacique permettait, systématiquement et significativement, une meilleure estimation de leurs effets induits par rapport aux corrections empiriques classiques (Boy, 2000 ; Boy et al., 2002). Cette méthodologie a ensuite été appliquée avec succès à d’autres observables géodésiques comme les déplacements en surface mesurés par VLBI (Very Long Baseline Interfe-rometry) (Petrov et Boy, 2004) et les mesures spatiales des variations temporelles du champ de pesanteur (Boy et Chao, 2005), ainsi qu’aux autres sources de surcharge (océans et hydrologie continentale).

Réponse des océans à la pression atmosphérique

La principale limitation des calculs des surcharges atmosphériques provient de l’estimation de la réponse, aux hautes fréquences (périodes inférieures à un mois), des océans aux variations de pres-sion et des vents atmosphériques. La précision des observables géodésiques, et plus particulièrement des gravimètres supraconducteurs, et des missions de gravité spatiale, ne permet plus d’utiliser les approximations classiques :

– le baromètre non-inversé pour lequel on supposait que les variations de pression étaient intégralement transmises au fond des océans (la Terre se comportant comme s’il n’y avait pas d’océans en surface),

– le baromètre inversé pour lequel on supposait que les variations de pression étaient intégralement compensées par des variations statiques de hauteur d’eau.

Si le premier modèle ne peut être vérifé que pour des mers fermées, le second modèle, parce qu’il néglige toute dynamique, ne peut être valide aux courtes périodes (périodes inférieures à un mois). La nécessaire amélioration du traitement des données altimétriques ainsi que des missions de gravité spatiale a imposé de construire des modèles dynamiques, forcés, entre autre, par la pression atmosphérique (Carrère et Lyard, 2003 ; de Viron et al., 2004).

Même s’il s’agit, en partie, de travaux en cours, je présente l’impact de cette modélisation sur certains observables géodésiques, principalement les variations de gravité en surface, mais également le champ de pesanteur terrestre et la rotation.

Mar´ees oc´eaniques

Les marées océaniques constituent certainement le phénomène le mieux connu et le plus étudié dans les océans. Depuis le lancement des satellites altimétriques, et notamment la mission franco-américaine Topex/Poseidon, la qualité des modélisations des marées a progressé très rapidement, tant au niveau de la résolution des équations hydrodynamiques que de l’assimilation des données altimétriques et marégraphiques. De plus, les marées océaniques sont de loin, avec l’atmosphère, la plus importante source de surcharges superficielles.

Les marées océaniques présentent un spectre plus complexe que les marées solides, en raison de la dynamique de la réponse des océans et de la complexité de la bathymétrie : si les marées zonales longue-période sont presque d’équilibre, ce n’est pas le cas des ondes principales diurnes

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et semi-diurnes. En outre, sur les différents plateaux continentaux, des ondes non-linéaires sont générées par interaction des ondes diurnes et semi-diurnes principales.

L’amplitude des surcharges de marées décroissant avec la distance aux côtes, une étude précise des surcharges impose de placer des instruments proches des côtes (GPS par exemple), ou bien de disposer d’instruments précis comme les gravimètres supraconducteurs. Par exemple, l’amplitude des surcharges induites par les ondes non-linéaires n’atteint que quelques nanogals (10−11 m/s2) à quelques centaines de kilomètres des côtes ouest-européennes (Boy et al., 2004).

L’extrême sensibilité des gravimètres supraconducteurs m’a permis d’étudier les contributions induites par les marées océaniques sur l’ensemble de leur domaine spectral, c’est-à-dire les marées longue période (Boy et al., 2006), les marées diurnes et semi-diurnes (Boy et al., 2003) et les marées non-linéaires (Boy et al., 2004).

Surcharges hydrologiques

Après l’atmosphère et les océans, les variations d’hydrologie continentale constitue la dernière source importante de variations de déplacement (van Dam et al., 2001a), de gravité en surface (van Dam et al., 2001b ; Boy et Hinderer, 2006) et du champ de pesanteur. L’objectif principal de la mission GRACE est d’ailleurs la détermination des variations temporelles des différents réservoirs d’eau continentale et des calottes glaciaires (NRC, 1997).

Nous présentons ici deux études consacrées aux effets des surcharges hydrologiques sur les variations de gravité, d’une part mesurées en surface par les gravimètres supraconducteurs, et d’autre part une expérience de validation/calibration de GRACE, menée en collaboration avec le groupe de géodésie spatiale du Goddard Space Flight Center.

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Chapitre I

Surcharges atmosph´eriques

L’atmosphère constitue l’une des principales sources de perturbation des variations de gravité en surface (voir, par exemple, Boy et al., 2002), du champ de gravité terrestre (voir, par exemple, Gégout et Cazenave, 1993 ; Boy et Chao, 2005) et des déplacements en surface (voir, par exemple, Petrov et Boy, 2004). Leur modélisation a constitué une partie importante de mes travaux de thèse (Boy, 2000), et en particulier la correction des données des différents gravimètres supraconducteurs du réseau mondial GGP (Global Geodynamics Project) (Crossley et al., 1999).

Les calculs des surcharges atmosphériques ne sont pas récents ; par exemple, van Dam et al. (1994) et van Dam et Herring (1994) ont calculé les contributions atmosphériques sur les déplacements en surface, et les ont comparé à des mesures GPS (Global Positioning System) et VLBI (Very Long Baseline Interferometry). Toutefois, les progrès tant des mesures géodésiques que des modèles atmosphériques n’ont permis de corriger efficacement les différentes observables géodésiques des perturbations atmosphériques que très récemment.

Nous présentons, dans ce chapitre, la modélisation des effets induits par les surcharges at-mosphériques sur les variations de gravité en surface, mesurées à l’aide des gravimètres supracon-ducteurs (Boy et al., 2002), sur les variations temporelles du champ de pesanteur, en tenant compte notamment de la structure tri-dimensionnelle de l’atmosphère (Boy et Chao, 2005) ainsi que sur les déplacements mesurés en surface par VLBI (Petrov et Boy, 2004).

I.1 Surcharges atmosph´eriques et variations de gravit´e en surface

La modélisation précise des contributions atmosphériques sur les variations de gravité en sur-face a donc constitué la partie la plus importante de mon travail de thèse (Boy, 2000). Nous avons pu montrer que l’utilisation des champs globaux de pression de surface induisait une réduction systématique et significative des résidus de gravité, par rapport aux corrections empiriques clas-siques (Kroner et Jentzsch, 1999), ne dépendant que de la seule donnée de pression locale. Ces travaux ont fait l’objet de plusieurs publications, notamment Boy et al. (1998) et Boy et al. (2002), ci-jointe.

La méthodologie utilisée pour l’atmosphère est la même que celle adoptée pour les autres fluides géophysiques de surface (océans, hydrologie continentale) : elle consiste à décrire la réponse de la Terre dans le domaine spatial (fonctions de Green) (Farrell, 1972), plutôt que dans le domaine spec-tral (c’est-à-dire en harmoniques sphériques) dans lequel sont calculées les solutions des équations de la gravito-élasticité (nombres de Love). Les effets en gravité sont alors calculés comme la convo-lution de la fonction de Green considérée avec le champ de pression atmosphérique.

δg(θ, λ, t) = ZZ

G(ψ)p(θ0, λ0, t)ds0 (I.1)

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18 Chapitre I. Surcharges atmosph´eriques

Comme nous supposons des modèles de Terre à symétrie sphérique et isotrope, la fonction de Green G(ψ) ne dépend que de la distance angulaire ψ entre le gravimètre (θ, λ) et l’élément ´

elémentaire de surface ds0 de coordonnées (θ0, λ0). La fonction de Green est calculée à partir des nombres de Love de charge (Farrell, 1972) et des polynômes de Legendre.

Dans le cas des variations de gravité, les effets induits par l’atmosphère peuvent être séparés en un effet dit Newtonien, c’est-à-dire un effet d’attraction par les masses atmosphériques, et un effet ´

elastique correspondant à la déformation. L’épaisseur de l’atmosphère (une dizaine de kilomètres) ne peut être négligée dans le calcul des effets d’attraction ; toutefois, nous n’avons pas considéré les variations tri-dimensionnelles réelles de la densité de l’air comme l’ont fait Neumeyer et al. (2004), mais supposé qu’il pouvait être déduit des seules données de surface de pression et eventuellement de température, suivant ainsi les travaux de Merriam (1992).

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Geophys. J. Int. (2002)149, 534–545

Reduction of surface gravity data from global atmospheric

pressure loading

Jean-Paul Boy,

1,2

Pascal Gegout

1

and Jacques Hinderer

1

1_{EOST-IPGS (UMR 7516 CNRS-ULP), 5 rue Rene Descartes, 67084 Strasbourg Cedex, France} 2_{Space Geodesy Branch, Code 926, NASA’s Goddard Space Flight Center, Greenbelt, MD 20771, USA}

Accepted 2001 December 6. Received 2001 December 6; in original form 2001 March 12

S U M M A R Y

Besides solid Earth and ocean tides, atmospheric pressure variations are one of the major sources of surface gravity perturbations. As shown by previous studies (Merriam 1992; Mukai et al. 1995; Boy et al. 1998), the usual pressure correction with the help of local pressure measurements and the barometric admittance (a simple transfer function between pressure and gravity, both measured locally) does not allow an adequate estimation of global atmospheric loading. We express the response of the Earth to pressure forcing using a Green’s function formalism (Farrell 1972). The atmosphere acts on surface gravity through two effects: first, a direct gravitational attraction by air masses which is sensitive to regional (about 1000 km around the gravimeter) pressure variations; second, an elastic process induced by the Earth’s surface deformation and mass redistribution which is sensitive to large scale pressure variations (wavelengths greater than 4000 km).

We estimate atmospheric loading using Green’s functions and global pressure charts provided by meteorological centres. We introduce different hypotheses on the atmospheric thickness and atmospheric density variations with altitude for the modelling of the direct Newtonian attrac-tion. All computations are compared to gravity data provided by superconducting gravimeters of the GGP (Global Geodynamics Project) network. We show the improvement by modelling global pressure versus the local estimates in terms of reduction of the variance of gravity resid-uals. We can also validate the inverted barometer (IB) hypothesis as the oceanic response to pressure forcing for periods exceeding one week. The non-inverted barometer (NIB) hypoth-esis is shown to be definitely an inadequate assumption for describing the oceanic response to atmospheric pressure at seasonal timescales.

Key words: atmospheric loading, Green’s function, pressure, superconducting gravimeter.

1 I N T R O D U C T I O N

Superconducting gravimeters are a privileged tool to study the Earth’s global and internal dynamics (see for example Crossley et al. 1999) over a large period range (from a few minutes to several years). However the atmosphere is, after solid Earth tides, one of the major sources of perturbations of surface gravity and masks small contri-butions such as core modes or anelastic effects on tides.

Atmospheric loading effects are usually corrected using an em-pirical estimation, called barometric admittance (Warburton & Goodkind 1977; Crossley et al. 1995), which is a simple trans-fer function adjusted by least square fitting between pressure and gravity, both measured locally. In fact, however, the global at-mosphere acts on surface gravity through two effects: a direct Newtonian attraction by air masses and an elastic contribution due to the Earth’s surface displacement and mass redistribution (e.g. Spratt 1982; Mukai et al. 1995; Boy et al. 1998). The simple cor-rection using only the local pressure measurement cannot take into

account the large scale pressure structures existing over the whole surface of the Earth, and is therefore not sufficient to estimate either the induced global deformation and mass redistribution of the Earth or the Newtonian direct attraction induced by regional (1000 km around the gravimeter) mass variations.

The purpose of this paper is to estimate surface gravity varia-tions induced by the atmospheric circulation with a physical ap-proach and to propose an operational modelling using global at-mospheric data provided by meteorological centres. As a first step, we determine and describe the physics of the source (the atmosphere) and the transfer functions (Newtonian and elastic atmospheric Green’s functions). For the computation of the grav-itational attraction, we consider different models for the atmo-sphere. We then describe the superconducting gravimeter obser-vations and their processing. The next section is devoted to the determination of the optimal model for computing the atmo-spheric loading; special attention is paid to the ocean response to pressure.

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Reduction of surface gravity data from global atmospheric pressure loading 535

2 G L O B A L A T M O S P H E R I C C I R C U L A T I O N

In this section, we present the two atmospheric data sets that we will use to estimate global atmospheric loading. We show that pressure variations are characterized by large scale structures with wave-lengths greater then 4000 km. We recall that the hydrostatic equi-librium hypothesis is verified for periods larger than 12 hr (Green 1999), so the estimation of atmospheric density variations with al-titude, as a function of surface pressure and temperature conditions with the help of the hydrostatic equilibrium equation, is adequate for our atmospheric data.

2.1 Atmospheric data

We use global meteorological data sets provided by the U.S. Na-tional Center for Environmental Prediction (NCEP) and the Euro-pean Centre for Medium-range Weather Forecasts (ECMWF).

The first data set, provided by the NCEP, consists of surface pres-sure data with a sampling rate of 6 hr and a spatial resolution of 2.5◦ in latitude and longitude. These NCEP reanalysis data are provided using a state-of-the-art analysis system to perform data assimilation from 1948 to the present.

The second surface data set, provided by the ECMWF, covers the period 1985–1996. The time sampling rate is the same as the NCEP reanalysis series (6 hr). The spatial sampling is 1.125◦for latitude and longitude.

2.2 Spectral energy

The spectral energy per degree represents the energy of pressure variations as a function of the harmonic degree n,

R(n, t) = n m=0 pm n(t) 2 +p˜m n(t) 2 , (1) where pm

n(t) and ˜pmn(t) are the cosine and sine terms of degree n and order m in a spherical harmonic decomposition of the time dependent surface pressure field.

In a spherical harmonic decomposition, the wavelengthλ(n) can be associated to the harmonic degree n,

λ(n) = 2πa

n, (2)

where a is the mean radius of the Earth.

Fig. 1 shows the spectral energy of the surface pressure field provided by ECMWF for the year 1994. Pressure variations are dominated by very low harmonic degrees, typically n< 10, corre-sponding to large scale atmospheric structures (λ > 4000 km).

Degrees 1 and 3 which include S1 and Sa (diurnal and annual

thermal waves) are shown to be variable in time with a seasonal component.

For periods exceeding a few hours, the atmospheric circulation is principally governed by horizontal displacements (Green 1999). The hydrostatic equilibrium is verified and we can use the corresponding equation to estimate density variations with altitude as a function of surface parameters (temperature and pressure).

There is also a relationship between the horizontal wavelength and the period of atmospheric variations (Green 1999). High fre-quency pressure variations are coherent on small scale surfaces whereas low frequency atmospheric structures are coherent on large scale surfaces. For example, mid-latitude circulation

(anticyclonic-Figure 1. Spectral energy of surface pressure field (ECMWF) in Pa2_{as a} function of spherical harmonic degree n and for the year 1994.

depression) has typical periods of about 5–10 days and wavelengths of several thousands of kilometres.

3 A T M O S P H E R I C G R E E N ’ S F U N C T I O N S

As shown by previous studies (e.g. Spratt 1982; Mukai et al. 1995; Boy et al. 1998), the global atmosphere acts on surface gravity through two effects:

(1) a direct gravitational attraction by air masses

(2) an elastic contribution from the Earth’s surface deformation and self-potential variations due to loading boundary conditions at the Earth’s surface.

We determine hereafter the Earth response to atmospheric loading in terms of surface gravity variations in the spatial domain using a Green’s function formalism. We first introduce different models for the atmosphere used in the computation of the Newtonian attraction effect, then we discuss the modelling of the elastic contribution.

3.1 Direct Newtonian attraction

In this section devoted to the Newtonian attraction caused by atmo-spheric masses, we introduce different models for the atmosphere. The most complete one is the 3-D model where pressure, tempera-ture and humidity are changing with latitude, longitude and altitude. We keep the thin-layer model as the classical simple model used for treating atmospheric loading following Farrell’s (1972) approach. We then derive our preferred pseudo-stratified model from the 3-D model using reasonable approximations and we show that the grav-ity changes can then be obtained from the convolution of the Green’s function with surface pressure data.

3.1.1 3-D atmosphere

The Newtonian effect corresponds to a direct gravitational attraction by air masses on the gravimeter. Following Merriam (1992), we define

G S(ψ, z) = G [a− (a + z) cos ψ]

[a2+ (a + z)2− 2a(a + z) cos ψ](3/2), (3)

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536 J.-P. Boy, P. Gegout and J. Hinderer

where z is the altitude of the atmospheric elementary volume of den-sityρA and spherical coordinates (θ, λ) and G is the Newtonian constant of gravitation. ψ is the angular distance between the gravimeter of coordinates (θ, λ) and the elementary atmospheric mass. That angular distance can be expressed as a function of both coordinates,

cosψ = cos θ cos θ+ sin θ sin θcos (λ − λ) (4) The Newtonian attraction of the surface gravity variations in (θ, λ) are in the case of a stratified atmosphere, of finite thickness (20 km) and with variations of pressure, temperature and humidity with height,

δgNewtonian(θ, λ, t) =

G S(ψ, z₎_ρ

A(θ, λ, z, t) dv. (5)

In this general case, the 3-D integration over the whole atmo-sphere needs to estimate the atmospheric density everywhere to compute the direct Newtonian attraction of the atmosphere. Global atmospheric data provided by meteorological centres consist of pres-sure, temperature and specific humidity data at different vertical levels. We have to use these data to rebuild atmospheric density variations.

The atmosphere is classically treated as a mixture of dry air and water vapor. The ideal gas equation gives us relations between tem-perature T, densityρA and pressure p for both components (Gill 1982),

pd = ρdRT

p_v= ρ_vR_vT, (6)

where pdandρdare respectively the partial pressure and the density of dry air. pvandρvare the partial pressure and the density of water vapor. R_vand R are respectively the universal gas constant for water vapor and for dry air and are respectively equal to 461.50 J kg−1K−1 and 287.04 J kg−1K−1.

For a gas mixture, the total pressure p is the sum of all partial pressures,

p= pd+ pv. (7)

We note q, the specific humidity, the ratio between density of water vapor and the density of gas mixture and is given by

q= ρv ρA

= ρv

ρd+ ρv

. (8)

The total densityρAbecomes

ρA= p RT 1− q +q , (9) where is equal to  = R R_v = 0.62197. (10)

We use the classical virtual temperature, noted T_v,

T_v= T 1− q +q . (11)

The density of the atmosphere becomes

ρA= p RT_v = p RT 1− q +q . (12)

This model requires a convolution of the Green’s function G S(ψ, z) with the 3-D atmospheric density (eq. 12) for the whole atmosphere.

This computation requires large computing resources. However, as described below, there is a way to approximate the estimation of the direct attraction of air masses on the gravimeter by simplify-ing the 3-D convolution into a 2-D convolution ussimplify-ing only surface atmospheric data.

3.1.2 Thin-layer model

As a first approximation, we can classically neglect the atmospheric thickness. In this case, the surface atmospheric densityσA(θ, λ) is directly linked to the surface pressure variation p0(θ, λ),

p0(θ, λ)= σA(θ, λ)g0, (13)

where g0is the mean surface gravity. This value is taken constant

and we neglect the small changes due to the latitude dependence, which are not taken into account by the meteorological models in any case.

In this model we define the Newtonian Green’s function by (Farrell 1972) G N (ψ) = − G g0a2 +∞ n=0 n Pn(cosψ), (14)

where Pn is the Legendre polynomial of degree n. Notice that this Green’s function does not have the same dimension as GS in eq. (3). The series expansion can be written as (Farrell 1972)

G N (ψ) = G g0a2 1 4 sin (ψ/2)− 2πa 2_δ(ψ) (15) whereδ(ψ) is the Dirac function and represents the contribution of the mass just above the gravimeter and is equal to the Bouguer semi-infinite plate value (−4.27 nm s−2hPa−1).

For a more detailed derivation of these results, the reader may refer to previous studies (e.g. Spratt 1982; Boy et al. 1998).

In this case, the surface gravity variations in (θ, λ) induced by direct Newtonian attraction are equal to

surface

G N (ψ)p0(θ, λ, t) ds, (16)

where ds= a2_sin_θ_d_θ_d_λ_{is the surface element of integration.}

This approximation allows a drastic reduction of the computing time because the Newtonian Green’s function is only convolved with the surface pressure field. However, this model does not take into account the curvature of the atmosphere, in particular the relative position of the atmospheric masses being above or under the local horizon (Merriam 1992).

3.1.3 Pseudo-stratified model

The second approximation does not neglect the atmospheric thick-ness but approximates density variations with altitude as only de-pending on surface pressure and temperature.

For periods exceeding a few hours, i.e. the temporal sampling of global meteorological data, the atmospheric circulation is prin-cipally governed by horizontal displacements (e.g. Green 1999). Hydrostatic equilibrium is then a valid approximation and gives pressure variations d p as a function of altitude variations d z,

d p= −ρAg0d z. (17)

If we consider the atmosphere as a perfect gas, the state law gives the relation between pressure p, densityρAand temperature T,

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p= ρART. (18)

The decrease of temperature with altitude can be modelled as a linear trend,

T (z)= T0+ αz. (19)

α = −6.49 K km−1 _{in the case of the Standard Atmosphere}

(NOAA/NASA/USAF 1976).

We neglect in this case the influence of water vapor content. In fact, it is very small compared to the temperature influence and is impossible to model using a simple law (linear, exponential...).

By combining the last three equations, we can write the variations of density dρA(z) as a function of altitude variations d z,

dρA(z) ρA(z) = −g0 R d z T0+ αz 1+ Rα g0 . (20)

The integration of this equation leads to the expression of the density as a function of the altitude and surface conditions (temperature T0

and pressure P0) ρA(z)= P0 RT (z) 1+ α T0 z −g0/Rα . (21)

The law expressing the pressure variations is very similar,

P(z)= P0 1+ α T0 z −g0/Rα . (22)

For an isothermal atmosphere (T (z)= T0orα = 0), the variations

of density and pressure with the altitude z follow from eq. (20) and are equal to ρA(z)= P0 RT0 exp −g0z RT0 (23) P(z)= P0exp −g0z RT0 (24) Fig. 2 shows the differences in pressure, temperature and density between the US Standard Atmosphere (NOAA/NASA/USAF 1976), the hydrostatic approximation with a linear decrease of temperature with altitude (eqs 21 and 22) and the hydrostatic approximation with a constant temperature (isothermal) (eqs 23 and 24).

In Fig. 2, the pressure and temperature conditions at the surface in the hydrostatic models are the same as for the standard atmosphere model. For an altitude smaller than 12 km, the temperature profile in the hydrostatic approximation with linear temperature decrease is equivalent to the one corresponding to the standard atmosphere.

However Fig. 2 also shows that the isothermal approximation does not change significantly the density profile. We hence use these assumptions (isothermal and hydrostatic) to compute the direct Newtonian attraction of air masses and call it the pseudo-stratified model.

This simple model of atmosphere using pressure and temperature at the Earth’s surface allows us a good estimation of density varia-tions with altitude up to 20 km which is the upper limit used in this study.

In this case, we can introduce a pseudo-stratified Newtonian Green’s function G SP S(ψ), G SP S(ψ) = 20 km z=0 G S(ψ, z) 1 RT0 exp −g0z RT0 d z, (25) where G S(ψ, z) is the Newtonian Green’s function for a stratified atmospheric model.

The gravity changes in (θ, λ) are only a function of surface pres-sure conditions and become

surface

G SP S(ψ)p0(θ, λ, t) ds. (26)

We neglect the temperature variations in time and space as they lead to smaller effects than the pressure-induced ones, of the order of a few per cent as shown by Merriam (1992).

This equation is very similar to the one obtained for the thin-layer model (eq. 16) and allows us to decrease the computing time by simplifying the 3-D convolution for the real atmosphere into a 2-D convolution.

3.2 Elastic contribution

The elastic contribution in gravity originates from the surface de-formation (vertical motion in the Earth’s gravity field) and from the Earth’s mass redistribution (altering the gravitational potential) and is usually expressed in the spectral domain, i.e. using a spherical har-monic approach (Sneeuw & Bun 1996), with dimensionless Love numbers, i.e. non-dimensional factors between the source (here a loading process) and the consequences (displacement, potential).

We assume that the elastic contribution is induced here by a sur-face loading process (thin-layer approximation for the atmosphere) even though we model the gravitational attraction using a strati-fied loading process. We can use the load Love numbers h_nand k_n which are respectively the load radial and potential Love numbers (Hinderer & Legros 1989). These numbers are computed in this study from the Preliminary Reference Earth Model (PREM) (Dziewonski & Anderson 1981).

The elastic Green’s function is equal to (e.g. Farrell 1972; Spratt 1982; Boy et al. 1998), G E(ψ) = − G g0a2 +∞ n=0 [2hn− (n + 1)kn]Pn(cosψ). (27)

The elastic contribution to the gravity changes at (θ, λ) is equal to

δgElastic(θ, λ, t) =

surface

G E(ψ)p0(θ, λ, t) ds. (28)

Numerical estimates of Green’s functions require the computation of Love numbers up to a high spherical harmonic degree (n= 9000 in this study). Numerical integration of the elasto-gravitational equa-tions provide a set of independent soluequa-tions in each layer. The fluid core in hydrostatic equilibrium is governed by a set of only two dif-ferential equations. Solving the set of boundary conditions at each interface and at the surface provides the Love numbers. Love num-bers for high degrees (n> 250) are obtained by considering that sur-face loading produces no deformation within the liquid core and the solid inner core, i.e. a new set of boundary conditions on the mantle is defined with no deformations at the CMB (core-mantle boundary) and boundary conditions at the Earth’s surface. We also took into account in the spherical harmonic expansion the degree-one Love numbers (Greff-Lefftz & Legros 1997) which are expressed here in the centre-of-mass reference frame. We do not take into account the Earth’s deformation of degree n= 0. However, as we subtract the mean pressure field to the surface pressure data, the degree n= 0 contribution can be neglected.

The surface gravity variations due to a surface loading can be ex-pressed in the spectral domain, i.e. in a spherical harmonic approach as follows (Spratt 1982; Hinderer & Legros 1989):

δg(θ, λ, t) = +∞ n=0 n m=0 − 3 2n+ 1 nδ n aρ pm n(t)Y m n(θ, λ), (29) C 2002 RAS, GJI,149, 534–545

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Figure 2. Pressure, temperature and density variations with altitude for the US Standard Atmosphere (NOAA/NASA/USAF 1976), the hydrostatic (linear decrease of temperature with altitude) and the hydrostatic and isothermal approximations.

where pm

n(t) are the spherical harmonic coefficients of the time dependent surface pressure field and Ym

n (θ, λ) are the spherical har-monic functions.ρ is the mean density of the Earth. δ_n is the gravi-metric factor of degree n and is equal to (Hinderer & Legros 1989)

δ n = 1 + 2 nh n− n+ 1 n k n. (30)

Fig. 3 shows the spectral response of the Earth, in terms of sur-face gravity changes, to a thin-layer sursur-face loading for the elastic contribution (the term 2

nhn−

n+1

n kn in eq. 30), and for the direct Newtonian attraction (the term 1 in eq. 30). These effects are char-acterized by opposite physical processes. The elastic contribution appears to be induced by large scale pressure coherence (or low harmonic degrees), typically of wavelengths greater than 4000 km (n<10), whereas the direct Newtonian attraction acts with the oppo-site sign and is induced by regional (distances smaller than 1000 km) pressure variations around the gravimeter.

We finally propose two hypotheses for the estimation of the di-rect Newtonian attraction: a thin-layer surface approximation which neglects the atmospheric thickness, and a pseudo-stratified isother-mal model which approximates density variations with altitude us-ing the hydrostatic equilibrium equation. The elastic contribution, which has an amplitude roughly ten times smaller than the direct attraction, is modelled using a thin-layer surface loading hypothesis. For both cases, the gravity variations in (θ, λ) are only functions of surface pressure conditions p0(θ, λ, t) and are equal to

δg(θ, λ, t) =

Surf

(G E(ψ) + G N(ψ)) p0(θ, λ, t) ds, (31)

for the thin-layer approximation, and

δg(θ, λ, t) =

Surf

(G E(ψ) + GSP S(ψ)) p0(θ, λ, t) ds (32)

for the pseudo-stratified hypothesis.

Fig. 4 shows the elastic and the Newtonian Green’s functions, i.e. the spatial response of the Earth to a pseudo-stratified atmospheric loading model as a function of angular distanceψ. The non-regular shape (slope change) appearing in the elastic Green’s function for

Figure 3. Spectral Earth response (elastic contribution and Newtonian at-traction) to a surface pressure loading in terms of surface gravity changes as a function of spherical harmonic degree n.

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Figure 4. Spatial Earth response to pressure loading (a) elastic contribution and (b) pseudo-stratified Newtonian attraction.

specific angular distances (between 0.2 and 0.7◦) is related to the evolution of the Love numbers as a function of the harmonic degree,

n, (see e.g. Boy 2000, Fig. I.1). This is probably linked to some major

discontinuities in the elastic parameters in the Earth.

4 S U P E R C O N D U C T I N G G R A V I M E T E R O B S E R V A T I O N S A N D M E T H O D O L O G Y

The surface gravity residuals are the observed gravity corrected for the following principal effects:

(1) the instrumental drift modelled by a linear or an exponential function;

(2) the Earth’s rotation induced effects (length-of-day variations and polar motion);

(3) solid and oceanic tides adjusted by least-squares fitting using the tidal analysis software ETERNA 3.30 (Wenzel 1996); and

(4) an atmospheric correction, either the local barometric ad-mittance adjustment or the subtraction of the global atmospheric loading estimated with the help of the Green’s functions previously calculated and global pressure charts provided by ECMWF or NCEP.

Figure 5. Location of the six gravimeters of GGP network whose data are analyzed in this paper.

Our purpose is to assess whether the modelling of global atmo-spheric loading produces a significant and systematic reduction of the variance of surface gravity residuals using different supercon-ducting gravimeters from the GGP (Global Geodynamics Project) network (Crossley et al. 1999) with the global atmospheric loading estimates versus the usual local pressure correction.

We explain the preprocessing of raw minute gravity and pressure data provided by GGP and the modelling of the Earth’s rotation induced effects. We also discuss the practical computation of global atmospheric loading.

4.1 Data preprocessing

Among the twenty superconducting gravimeters of the GGP net-work, we analyze data provided by six instruments: Boulder (BO) in Colorado, Canberra (CB) in Australia, Esashi (ES) in Japan, Mem-bach (MB) in Belgium, Strasbourg (ST) in France and Vienna (VI) in Austria; their locations are shown in Fig. 5. We chose these six stations because of their locations. Canberra and Esashi are close to the Pacific Ocean, whereas Boulder is far from the Atlantic and Pacific Ocean. Membach, Strasbourg and Vienna are located at in-creasing distances from the North Sea and the North Atlantic in western Europe.

This network delivers raw gravity and pressure data at a sampling rate of one minute. Raw gravity and pressure data are first corrected for major perturbations on a few samples such as offsets, gaps and spikes by substituting a synthetic local tide (Crossley et al. 1993). Data are then filtered to a sampling of 6 hr corresponding to the sampling rate of global pressure charts.

The effects induced by the Earth’s rotation variations (polar mo-tion and length-of-day) are then subtracted, assuming a static polar tide in the oceans. The gravimetric factor ˜δ2becomes equal to 1.18

(1.16 for the solid Earth only) for the amplitude and 0◦for the phase (Loyer et al. 1999; Boy 2000). This assumption is generally a good approximation of the oceanic response to Earth’s rotation variations, except for some specific areas like the North and Baltic Seas (Xie & Dickman 1995). However notice that we do not model other en-vironmental contributions such as oceanic circulation, water table or soil moisture which can lead, in some cases, to effects reaching several tens of nm s−2(Van Dam & Wahr 1998), i.e. larger than the effect induced by the dynamics of the pole tide in the North and Baltic Seas.

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The gravity changes in (θ, λ) are equal to (e.g. Loyer et al. 1999)

δgr ot(θ, λ, t) = ˜δ22a

2m3sin2θ − sin 2θ(m1cosλ + m2sinλ)

,

(33) where the dimensionless quantities are defined as

m1= p1+ 1 d p2 dt m2= p2− 1 d p1 dt (34) m3= 1 d p3 dt

The quantities pi are related to the Earth’s orientation parameters (X and Y) and length-of-day variations provided by IERS (EOPC04 series)

p1= X

p2= −Y

p3= (UT 1 − TAI )

, (35)

where UT1 and TAI are the Universal Time and the International Atomic Time. is the Earth’s mean angular velocity.

Daily gravity changes induced by the Earth’s rotation variations are interpolated to 6 hr time span and are subtracted from filtered gravity variations.

4.2 Practical computation of global atmospheric loading

The spatial sampling of the pressure fields provided by meteorolog-ical centres are respectively 2.5 and 1.125 degrees for NCEP and ECMWF. The accuracy is about 10 Pa (0.1 millibar).

The local pressure measurements, in conjunction with surface gravity, are available with a precision of about 1 Pa (0.01 millibar). As we pointed out before, the Newtonian attraction is induced by regional (about 1000 km around the station) pressure variations. We would like to use these precise local pressure measurements to en-hance our computations. For this reason, we choose to separate the practical computation of global atmospheric loading as the convo-lution of elastic and Newtonian Green’s functions with pressure into two spatial domains.

(1) a near area, called zone 1, for which the pressure is assumed to be constant and equal to the pressure measured at the gravity station. We choose to represent this area as a spherical cap of radius

ψ1.

(2) a more distant area, called zone 2, for angular distances be-tweenψ1 andψ2(ψ2 equal to 180◦ corresponds to an integration

on the whole Earth surface) for which we use the pressure fields provided by the meteorological centres.

Before testing the different hypotheses of Newtonian attraction (sur-face or pseudo-stratified models) and oceanic response to pres-sure forcing, i.e. inverted barometer (IB) or non-inverted barometer (NIB) hypotheses, we determine the optimal value ofψ1, the

an-gular radius of zone 1, which minimizes the standard deviation of surface gravity residuals.

An implicit assumption made throughout this paper is that the models leading to the smaller gravity residuals are better than those yielding larger residuals; however one has to keep in mind that other processes than atmospheric loading are affecting the data and this might modify our conclusions from the minimum variance approach.

Fig. 6 shows the standard deviation of surface gravity residuals as a function ofψ1for the gravimeter CO26 installed in Strasbourg

Figure 6. Determination of the optimal angular radius of homogeneous spherical cap (ψ1) in the modelling of the local Newtonian attraction with the help of local pressure. Hereψ2is equal to 180◦.

(France). In the computation of atmospheric loading, we consider the local pressure measurements for angular distances between 0◦ andψ1, and the NCEP pressure field for angular distances between

ψ1and 180◦.

The value ofψ1equal to 0◦corresponds to the use of only global

pressure charts and no local pressure measurements. The optimal value of ψ1, corresponding to a minimum in the surface gravity

variance, is obtained for values between 0.25◦ and 0.5◦. We fix henceforth the value ofψ1at 0.5◦(about 50 km) even if we found

that this value depends slightly on the site (coastal or inland). In the next sections, we determine with the same criterion the optimal model of atmospheric loading in terms of Newtonian attrac-tion (thin-layer or pseudo-stratified models) and oceanic response to pressure forcing (IB or NIB).

5 D E T E R M I N A T I O N O F T H E O P T I M A L L O A D I N G M O D E L

5.1 Direct Newtonian attraction

5.1.1 Thin-layer versus pseudo-stratified model

In Section 3, we presented two different hypotheses for the op-erational computation of the direct Newtonian attraction: super-ficial thin-layer and pseudo-stratified models. We show, in this section, the differences between both hypotheses in terms of reduc-tion of the variance of gravity residuals for SG C026, installed in Strasbourg (France).

Fig. 7 shows the decrease of the standard deviation of gravity residuals as a function ofψ2, the angular radius of zone 2 for which

we consider both hypotheses of direct Newtonian attraction. The surface pressure data are provided by NCEP. Forψ2equal to 0.5◦,

the convolution domain is restricted to zone 1. Whenψ2is equal to

180◦, the convolution domain covers the whole Earth’s surface. We also show the value of the standard deviation of gravity residues using the classical and empirical pressure correction with the help of the barometric admittance (found equal to −2.71 ± 0.03 nm s−2 hPa−1). A thin-layer surface Newtonian model does not allow a reduction of gravity residual standard deviation versus

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Figure 7. Reduction of the standard deviation of gravity residuals (SG C026) as a function of the solid angle ψ2 in the integration of global pressure loading (NCEP data and IB ocean) for a surface and a pseudo-stratified processes. The value of the standard deviation of gravity residues using a local pressure correction (the admittance is equal to −2.71 ± 0.03 nm s−2hPa−1) is shown by the dot-dashed line.

the local empirical estimation. However, differences between pseudo-stratified and thin-layer surface attraction models become small for angular distances larger than 10◦. The contribution of the pseudo-stratified model to the estimation of the direct Newtonian attraction takes into account the geometry of the atmosphere with the local horizon of the gravimeter (e.g. Merriam 1992). In fact, the atmospheric thickness is not negligible for small angular distances in the estimation of the direct Newtonian attraction.

The two curves in Fig. 7 can be decomposed into two areas. First a rapid decrease of the standard deviation between 0.5◦(about 50 km) and 10.0◦(about 1000 km); second a slighter decrease from 10.0◦ to 180.0◦(i.e. by considering the pressure on the whole surface). In the first part, we model the direct Newtonian attraction increasingly better (i.e. about 90 per cent of the total signal of loading) which is induced by the regional pressure variations (i.e. about 1000 km around the gravimeter). In the second part, we estimate the elas-tic contribution increasingly better which is induced by large scale (wavelength larger than 4000 km) pressure variations and leads to about 10 per cent of the loading effects.

5.1.2 Effects induced by the topography around the gravimeter

In the case of SG C024, installed at TMGO (Table Mountain Geodetic Observatory), near the Rocky Mountains close to Boulder (Colorado), we cannot neglect topography in the computation of atmospheric loading. Fig. 8 shows the topography around.

For a gravimeter at altitude h and in (θ, λ), the Newtonian attrac-tion Green’s funcattrac-tion is after eq. 3 equal to

G Stopo(ψ, z, h)

= G (a+ h) − (a + z) cos ψ

((a+ z)2+ (a + h)2− 2(a + h) (a + z) cos ψ)3/2.

(36) The topography modifies the relative position of air masses com-pared to the local horizon. The horizon is also shifted by the value of the altitude of the gravimeter.

Figure 8. Topography (in m) around the station of Table Mountain (Boulder, Colorado) (latitude and longitude are expressed in degrees). The circle rep-resents zone 1 where the pressure is assumed to be constant.

Fig. 9 shows the standard deviation of surface gravity residuals at Boulder for two global pressure loading models. The first takes into account the topography and the second does not. For both models, the oceans are supposed to respond like an IB process and the Newtonian attraction is modelled using the pseudo-stratified model.

Fig. 9 demonstrates that the topography must be taken into ac-count in the computation of the Newtonian attraction in this case. For large distances (ψ2>10◦), the differences between both models

become small, showing once more that the Newtonian attraction is a regional (about 1000 km around the station) process.

5.2 Effects due to differences between NCEP and ECMWF surface pressure fields

For SG T005, the previous instrument in Strasbourg (1987–1996), we have corrected gravity measurements with two global surface pressure data sets provided by ECMWF and NCEP. Since the

Figure 9. Reduction of the standard deviation of gravity residuals (SG C024) as a function of the solid angleψ2in the integration of global pressure loading (NCEP data, pseudo-stratified model and IB ocean) by taking or not into account the topography around the station at Boulder (Colorado).

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