C. B. Agee, Petrology of the Mantle Transition Zone, Annual Review of Earth and Planetary Sciences, vol.21, issue.1, pp.19-41, 1993.
DOI : 10.1146/annurev.ea.21.050193.000315

M. Albers and U. R. Christensen, The excess temperature of plumes rising from the core-mantle boundary, Geophysical Research Letters, vol.14, issue.24, pp.3567-3570, 1996.
DOI : 10.1029/96GL03311

A. K. Alekseev and I. M. Navon, The analysis of an ill-posed problem using multi-scale resolution and second-order adjoint techniques, Computer Methods in Applied Mechanics and Engineering, vol.190, issue.15-17, pp.1937-1953, 2001.
DOI : 10.1016/S0045-7825(00)00299-1

J. Badro, J. Rueff, G. Vanko, G. Monaco, G. Figuet et al., Electronic Transitions in Perovskite: Possible Nonconvecting Layers in the Lower Mantle, Electronic transitions in perovskite: Possible nonconvecting layers in the lower mantle, pp.383-386, 2004.
DOI : 10.1126/science.1098840
URL : https://hal.archives-ouvertes.fr/hal-00087150

J. D. Bass, Elasticity of minerals, glasses, and melts, AGU Ref. Shelf, vol.20, pp.45-63, 1995.
DOI : 10.1029/RF002p0045

A. F. Bennett, Inverse Methods in Physical Oceanography, 1992.
DOI : 10.1017/CBO9780511600807

H. Bunge, M. A. Richards, C. Lithgow-bertelloni, J. R. Baumgardner, S. P. Grand et al., Time Scales and Heterogeneous Structure in Geodynamic Earth Models, Time scales and heterogeneous structure in geodynamic Earth models, pp.91-95, 1998.
DOI : 10.1126/science.280.5360.91

H. Bunge, M. A. Richards, and J. R. Baumgardner, Mantle-circulation models with sequential data assimilation: inferring present-day mantle structure from plate-motion histories, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.360, issue.1800, pp.2545-2567, 2002.
DOI : 10.1098/rsta.2002.1080

H. Bunge, C. R. Hagelberg, and B. J. Travis, Mantle circulation models with variational data assimilation: inferring past mantle flow and structure from plate motion histories and seismic tomography, Geophysical Journal International, vol.152, issue.2, pp.280-301, 2003.
DOI : 10.1046/j.1365-246X.2003.01823.x

F. H. Busse, 3D convection at infinite Prandtl number in Cartesian geometry ??? a benchmark comparison, Geophysical & Astrophysical Fluid Dynamics, vol.75, issue.1, pp.39-59, 1993.
DOI : 10.1017/S0022112088001582
URL : https://hal.archives-ouvertes.fr/hal-00562400

D. G. Cacuci, M. Ionescu-bujor, and I. M. Navon, Sensitivity and Uncertainty Analysis. Volume II: Applications to Large-Scale Systems, 2005.
DOI : 10.1201/9780203483572

A. Chopelas and R. Boehler, Thermal expansion measurements at very high pressure, systematics, and a case for a chemically homogeneous mantle, Geophysical Research Letters, vol.243, issue.11, pp.1347-1350, 1989.
DOI : 10.1029/GL016i011p01347

K. C. Condie, Mantle Plumes and Their Record in Earth History, 2001.
DOI : 10.1017/CBO9780511810589

L. Csontos, A. Nagymarosy, F. Horvath, and M. Kovac, Tertiary evolution of the intra-Carpathian area: A model, pp.221-241, 1992.

A. Davaille, Simultaneous generation of hotspots and superswells by convection in a heterogeneous planetary mantle, Nature, vol.402, issue.6763, pp.756-760, 1999.
DOI : 10.1038/45461

A. Davaille and J. Vatteville, On the transient nature of mantle plumes, Geophysical Research Letters, vol.503, issue.1, pp.10-1029, 2005.
DOI : 10.1029/2005GL023029

C. Demetrescu, S. B. Nielsen, M. Ene, D. Z. Serban, G. Polonic et al., Lithosphere thermal structure and evolution of the Transylvanian Depression ??? insights from new geothermal measurements and modelling results, Physics of the Earth and Planetary Interiors, vol.126, issue.3-4, pp.249-267, 2001.
DOI : 10.1016/S0031-9201(01)00259-X

R. Fletcher and C. M. Reeves, Function minimization by conjugate gradients, The Computer Journal, vol.7, issue.2, pp.149-154, 1964.
DOI : 10.1093/comjnl/7.2.149

A. M. Forte and J. X. Mitrovica, Deep-mantle high-viscosity flow and thermochemical structure inferred from seismic and geodynamic data, Nature, pp.410-1049, 2001.

M. Ghil and P. Malanotte-rizzoli, Data Assimilation in Meteorology and Oceanography, Adv. Geophys, vol.33, pp.141-266, 1991.
DOI : 10.1016/S0065-2687(08)60442-2

D. H. Green and T. J. Falloon, Pyrolite, pp.311-378, 1998.
DOI : 10.1007/0-387-30845-8_205

U. Hansen, D. A. Yuen, and S. E. Kroening, Transition to hard turbulence in thermal convection at infinite Prandtl number, Physics of Fluids A: Fluid Dynamics, vol.2, issue.12, pp.2157-2163, 1990.
DOI : 10.1063/1.857802

U. Hansen, D. A. Yuen, and S. E. Kroening, Effects of depth-dependent thermal expansivity on mantle circulations and lateral thermal anomalies, Effects of depthdependent thermal expansivity on mantle circulations and lateral thermal anomalies, pp.1261-1264, 1991.
DOI : 10.1029/91GL01288

U. Hansen, D. A. Yuen, S. E. Kroening, and T. B. Larsen, Dynamical consequences of depth-dependent thermal expansivity and viscosity on mantle circulations and thermal structure, Dynamical consequences of depth-dependent thermal expansivity and viscosity on mantle circulations and thermal structure, pp.205-223, 1993.
DOI : 10.1016/0031-9201(93)90099-U

H. Harder and U. R. Christensen, A one-plume model of martian mantle convection, Nature, vol.380, issue.6574, pp.507-509, 1996.
DOI : 10.1038/380507a0

C. A. Hier-majumder, B. J. Travis, E. Belanger, G. Richard, A. P. Vincent et al., Efficient sensitivity analysis for flow and transport in the Earth's crust and mantle, Geophys, J. Int, 2006.

A. M. Hofmeister, Mantle Values of Thermal Conductivity and the Geotherm from Phonon Lifetimes, Science, vol.283, issue.5408, pp.1699-1706, 1999.
DOI : 10.1126/science.283.5408.1699

S. Honda, S. Balachandar, D. A. Yuen, and D. Reuteler, Three-dimensional mantle dynamics with an endothermic phase transition, Geophysical Research Letters, vol.72, issue.3, pp.221-224, 1993.
DOI : 10.1029/92GL02976

S. Honda, D. A. Yuen, S. Balachandar, and D. Reuteler, Three-Dimensional Instabilities of Mantle Convection with Multiple Phase Transitions, Threedimensional instabilities of mantle convection with multiple phase transitions, pp.1308-1311, 1993.
DOI : 10.1126/science.259.5099.1308

L. N. Howard, Convection at high Rayleigh number, Applied Mechanics, Proc. of the 11th Intl Congress of Applied Mechanics, pp.1109-1115, 1964.
DOI : 10.1007/978-3-662-29364-5_147

A. T. Ismail-zadeh, A. I. Korotkii, B. M. Naimark, and I. A. Tsepelev, Numerical simulation of three-dimensional viscous flows with gravitational and thermal effects, Comput. Math. Math. Phys, vol.41, pp.1399-1415, 2001.

A. T. Ismail-zadeh, A. I. Korotkii, B. M. Naimark, and I. A. Tsepelev, Three-dimensional numerical simulation of the inverse problem of thermal convection, Comput. Math. Math. Phys, vol.43, pp.587-599, 2003.

A. T. Ismail-zadeh, A. I. Korotkii, and I. A. Tsepelev, Numerical approach to solving problems of slow viscous flow backwards in time, Computational Fluid and Solid Mechanics, pp.938-941, 2003.
DOI : 10.1016/B978-008044046-0.50230-X

A. Ismail-zadeh, G. Schubert, I. Tsepelev, and A. Korotkii, Inverse problem of thermal convection: numerical approach and application to mantle plume restoration, Physics of the Earth and Planetary Interiors, vol.145, issue.1-4, pp.99-114, 2004.
DOI : 10.1016/j.pepi.2004.03.006

A. Ismail-zadeh, B. Mueller, and G. Schubert, Three-dimensional numerical modeling of contemporary mantle flow and tectonic stress beneath the earthquake-prone southeastern Carpathians based on integrated analysis of seismic, heat flow, and gravity data, Physics of the Earth and Planetary Interiors, vol.149, issue.1-2, pp.81-98, 2005.
DOI : 10.1016/j.pepi.2004.08.012

M. A. Jellinek and M. Manga, The influence of a chemical boundary layer on the fixity, spacing and lifetime of mantle plumes, Nature, vol.132, issue.6899, pp.418-760, 2002.
DOI : 10.1029/2001GL014624

E. Kalnay, Atmospheric Modeling, Data Assimilation and Predictability , 341 pp, 2003.

S. Karato, Importance of anelasticity in the interpretation of seismic tomography, Geophysical Research Letters, vol.74, issue.15, pp.1623-1626, 1993.
DOI : 10.1029/93GL01767

A. I. Korotkii and I. A. Tsepelev, Solution of a retrospective inverse problem for one nonlinear evolutionary model, Proceedings of the Steklov Institute of Mathematics, pp.80-94, 2003.

R. Lattes and J. L. Lions, The Method of Quasi-Reversibility: Applications to Partial Differential Equations, 1969.

L. Dimet, F. , I. M. Navon, and D. N. Daescu, Second-Order Information in Data Assimilation*, Monthly Weather Review, vol.130, issue.3, pp.629-648, 2002.
DOI : 10.1175/1520-0493(2002)130<0629:SOIIDA>2.0.CO;2
URL : https://hal.archives-ouvertes.fr/hal-00787272

A. M. Leitch, V. Steinbach, and D. A. Yuen, Centerline temperature of mantle plumes in various geometries: Incompressible flow, Journal of Geophysical Research: Solid Earth, vol.76, issue.46, pp.829-850, 1996.
DOI : 10.1029/96JB01784

D. C. Liu and J. Nocedal, On the limited memory BFGS method for large scale optimization, Mathematical Programming, vol.32, issue.2, pp.503-528, 1989.
DOI : 10.1007/BF01589116

M. Liu, D. A. Yuen, W. Zhao, and S. Honda, Development of Diapiric Structures in the Upper Mantle Due to Phase Transitions, Science, vol.252, issue.5014, pp.1836-1839, 1991.
DOI : 10.1126/science.252.5014.1836

A. V. Malevsky and D. A. Yuen, Plume structures in the hard-turbulent regime of three-dimensional infinite Prandtl number convection, Geophysical Research Letters, vol.216, issue.9l, pp.383-386, 1993.
DOI : 10.1029/93GL00293

A. V. Malevsky, D. A. Yuen, and L. M. Weyer, Viscosity and thermal fields associated with strongly chaotic non-Newtonian thermal convection, Geophysical Research Letters, vol.36, issue.5, pp.127-130, 1992.
DOI : 10.1029/91GL03181

M. Martin, J. R. Ritter, and . Group, High-resolution teleseismic body-wave tomography beneath SE Romania - I. Implications for three-dimensional versus one-dimensional crustal correction strategies with a new crustal velocity model, Geophysical Journal International, vol.162, issue.2, pp.448-460, 2005.
DOI : 10.1111/j.1365-246X.2005.02661.x

A. Mcdonald, Accuracy of Multiply-Upstream, Semi-Lagrangian Advective Schemes, Monthly Weather Review, vol.112, issue.6, pp.1267-1279, 1984.
DOI : 10.1175/1520-0493(1984)112<1267:AOMUSL>2.0.CO;2

D. Mclaughlin, An integrated approach to hydrologic data assimilation: interpolation, smoothing, and filtering, Advances in Water Resources, vol.25, issue.8-12, pp.1275-1286, 2002.
DOI : 10.1016/S0309-1708(02)00055-6

J. X. Mitrovica, Haskell [1935] revisited, Journal of Geophysical Research: Solid Earth, vol.87, issue.B1, pp.555-569, 1996.
DOI : 10.1029/95JB03208

J. X. Mitrovica and A. M. Forte, A new inference of mantle viscosity based upon joint inversion of convection and glacial isostatic adjustment data, Earth and Planetary Science Letters, vol.225, issue.1-2, pp.177-189, 2004.
DOI : 10.1016/j.epsl.2004.06.005

R. Montelli, G. Nolet, F. A. Dahlen, G. Masters, E. R. Engdahl et al., Finite-Frequency Tomography Reveals a Variety of Plumes in the Mantle, Science, vol.303, issue.5656, pp.338-343, 2004.
DOI : 10.1126/science.1092485

W. B. Moore, G. Schubert, and P. Tackley, Three-Dimensional Simulations of Plume-Lithosphere Interaction at the Hawaiian Swell, Science, vol.279, issue.5353, pp.1008-1011, 1998.
DOI : 10.1126/science.279.5353.1008

W. J. Morgan, Plate Motions and Deep Mantle Convection, Geol. Soc. Am. Mem, vol.132, pp.7-22, 1972.
DOI : 10.1130/MEM132-p7

I. M. Navon, X. Zou, J. Derber, and J. Sela, Variational Data Assimilation with an Adiabatic Version of the NMC Spectral Model, Monthly Weather Review, vol.120, issue.7, pp.1433-1446, 1992.
DOI : 10.1175/1520-0493(1992)120<1433:VDAWAA>2.0.CO;2

G. Nolet and F. A. Dahlen, Wave front healing and the evolution of seismic delay times, Journal of Geophysical Research: Solid Earth, vol.141, issue.B8, pp.43-19054, 2000.
DOI : 10.1029/2000JB900161

P. Olson and H. Singer, Creeping plumes, Creeping plumes, pp.511-531, 1985.
DOI : 10.1016/0012-821X(80)90165-X

P. Olson, G. Schubert, and C. Anderson, Structure of axisymmetric mantle plumes, Structure of axisymmetric mantle plumes, pp.6829-6844, 1993.
DOI : 10.1029/92JB01013

N. M. Ribe and U. Christensen, Three-dimensional modeling of plume-lithosphere interaction, Journal of Geophysical Research: Solid Earth, vol.18, issue.B1, pp.669-682, 1994.
DOI : 10.1029/93JB02386

Y. Ricard, M. A. Richards, C. Lithgow-bertelloni, and Y. L. Stunff, A geodynamic model of mantle density heterogeneity, Journal of Geophysical Research: Solid Earth, vol.66, issue.14, pp.895-21909, 1993.
DOI : 10.1029/93JB02216

M. A. Richards and R. W. Griffiths, Deflection of plumes by mantle shear flow: experimental results and a simple theory, Geophysical Journal International, vol.94, issue.3, pp.367-376, 1988.
DOI : 10.1111/j.1365-246X.1988.tb02260.x

M. A. Richards, R. A. Duncan, and V. Courtillot, Flood Basalts and Hot-Spot Tracks: Plume Heads and Tails, Science, vol.246, issue.4926, pp.103-107, 1989.
DOI : 10.1126/science.246.4926.103

J. Ritsema, H. J. Van-heijst, and J. H. Woodhouse, Complex Shear Wave Velocity Structure Imaged Beneath Africa and Iceland, Complex shear wave velocity structure imaged beneath Africa and Iceland, pp.1925-1928, 1999.
DOI : 10.1126/science.286.5446.1925

B. Romanowicz and Y. Gung, Superplumes from the Core-Mantle Boundary to the Lithosphere: Implications for Heat Flux, Science, vol.296, issue.5567, pp.513-516, 2002.
DOI : 10.1126/science.1069404

A. A. Samarskii, P. N. Vabishchevich, and V. I. Vasil-'ev, Iterative solution of a retrospective inverse problem of heat conduction, Math. Model, vol.9, issue.5, pp.119-127, 1997.

G. Silveira, E. Stutzmann, A. Davaille, J. Montagner, L. Mendes-victor et al., Azores hotspot signature in the upper mantle, Journal of Volcanology and Geothermal Research, vol.156, issue.1-2, 2006.
DOI : 10.1016/j.jvolgeores.2006.03.022
URL : https://hal.archives-ouvertes.fr/hal-00271098

N. H. Sleep, Hotspots and mantle plumes: Some phenomenology, Journal of Geophysical Research, vol.93, issue.B5, pp.6715-6736, 1990.
DOI : 10.1029/JB095iB05p06715

B. Sperner, and The CRC 461 Team Monitoring of slab detachment in the Carpathians, in Perspectives in Modern Seismology, Lect. Notes Earth Sci, vol.105, pp.187-202, 2005.

B. Steinberger, Plumes in a convecting mantle: Models and observations for individual hotspots, Journal of Geophysical Research: Solid Earth, vol.41, issue.45, pp.127-138, 2000.
DOI : 10.1029/1999JB900398

B. Steinberger and R. J. , Advection of plumes in mantle flow: implications for hotspot motion, mantle viscosity and plume distribution, Geophysical Journal International, vol.132, issue.2, pp.412-434, 1998.
DOI : 10.1046/j.1365-246x.1998.00447.x

O. Talagrand, Assimilation of observations: An introduction, J. Meteorol. Soc. Jpn, vol.75, pp.191-209, 1997.

A. N. Tikhonov and V. Y. Arsenin, Solution of Ill-Posed Problems, 1977.

R. A. Trompert and U. Hansen, On the Rayleigh number dependence of convection with a strongly temperature-dependent viscosity, Physics of Fluids, vol.10, issue.2, pp.351-360, 1998.
DOI : 10.1063/1.869527

C. Wunsch, The Ocean Circulation Inverse Problem, 1996.
DOI : 10.1017/CBO9780511629570

D. Zhao, Global tomographic images of mantle plumes and subducting slabs: insight into deep Earth dynamics, Physics of the Earth and Planetary Interiors, vol.146, issue.1-2, pp.3-34, 2004.
DOI : 10.1016/j.pepi.2003.07.032

S. Zhong, Dynamics of thermal plumes in three-dimensional isoviscous thermal convection, Geophysical Journal International, vol.162, issue.1, pp.289-300, 2005.
DOI : 10.1111/j.1365-246X.2005.02633.x

X. Zou, I. M. Navon, M. Berger, K. H. Phua, T. Schlick et al., Numerical Experience with Limited-Memory Quasi-Newton and Truncated Newton Methods, SIAM Journal on Optimization, vol.3, issue.3, pp.582-608, 1993.
DOI : 10.1137/0803029

À. , A. Ismail-zadeh, and G. Institut, Karlsruhe D-76187, Germany. (alik.ismail-zadeh@gpi.uka.de) A. Korotkii and I. Tsepelev, Institute of Mathematics and Mechanics, Russian Academy of Sciences, S. Kovalevskoy ul. Geology Building, vol.16, issue.3806, pp.90095-1567