Welcome to Albert Tarantola's web page

Professor Tarantola worked at the Institut de Physique du Globe de Paris
Professor Tarantola died in December 2009
Contact: zamora@ipgp.org - Street address: IPGP, 4 place Jussieu, 75005 Paris, France. Phone: +33 1 44 27 38 98.

Logo IPG

Albert Tarantola, March 2009
Me, while teaching at Princeton University, February-May 2009.

My Institute is part of the University of Paris (widely known as La Sorbonne), founded in the XII century.
Our campus (Jussieu) is at the center of Paris (see photo).
This is a link to my lecture notes (I am presently teaching at Princeton University, in August-September I will teach at the University of Santiago de Chile, and my next teaching in Paris will start in October).
This is a link to my Curriculum Vitæ.

Current Projects:

Theoretical Physics

My new book, Elements for Physics - Quantities, Qualities, and Intrinsic Theories, has just been published (see the description of the book below). Independently of this work, with my colleague Bartolomé Coll -from Observatoire de Paris- we are trying to propose a new paradigm for properly running a GPS system, using fully relativistic concepts (see details here).

Inverse Problems in Geophysics

I wrote a book on this topic (Inverse Problem Theory), that is widely used (for a download, see below). This gives me the opportunity of collaboration with many colleagues interested in the different aspects of interpretation of geophysical data. My own interest is in understanding the structure of our Planet, using mainly seismic waves. It is my feeling that seismology will one day provide detailed images of the Earth's crust, incommensurable to the gross pictures we obtain today. This will require new technology for data recording and new methods of data interpretation that will, possibly, take a dozen of years to develop. Only then we will understand how predictable earthquakes are. I have presently two projects ongoing: I am writing a new book Mapping of Probabilities (downloadable version below), and I am trying, with my colleague Klaus Mosegaard, from the University of Copenhagen, to set the basic definitions and properties of the computational complexity attached to multidimensional sampling problems. Click here for a 10-slides tutorial on Inverse Problems.

Human Prehistory

Where Neolithic culture developed? With my colleague Kurt Lambeck (from the Australian National University) we conjecture that, before the recent sea level rise, what is today the Persian Gulf was a fertile valley, inhabited by people that moved to Mesopotamia when the waters invaded their land. We plan a geophysical survey in the Gulf to test some of the predictions of our hypothesis. Click here to view the two main figures of Lambeck's 1996 paper (PDF file, 0.6 MB). My interest in human prehistory fortifies my rejection of the hyper-technological society that we are today creating (more about this at the bottom of the page).

Principal Publications:

Elements for Physics - Quantities, Qualities, and Instrinsic Theories. Albert Tarantola, Springer, 2006.

While usual presentations of physical theories emphasize the notion of physical quantity, this book shows that there is much to gain when introducing the notion of physical quality. The usual physical quantities simply appear as coordinates over the manifolds representing the physical qualities. This allows developing physical theories that have a degree of invariance much deeper than the usual one. It is shown that properly developed physical theories contain logarithms and exponentials of tensors: their conspicuous absence in usual theories suggests, in fact, that the fundamental invariance principle stated in this book is lacking in present-day mathematical physics. The book reviews and extends the theory of Lie groups, develops differential geometry, proposing compact definitions of torsion and of curvature, and adapts the usual notion of linear tangent application to the intrinsic point of view proposed for physics. As an illustration, two simple theories are studied with some detail, the theory of heat conduction and the theory of linear elastic media. The equations found differ quantitatively and qualitatively from those usually presented. You can purchase the book via Amazon (99 $) or Springer (79.95 €). If you can not purchase the book, please download it for free.
Elements for Physics

Inverse Problem Theory and Model Parameter Estimation. Albert Tarantola, SIAM, 2005.

Prompted by recent developments in inverse theory, this book is a completely rewritten version of my 1987 book. In this version there are many algorithmic details for Monte Carlo methods, least-squares discrete problems, and least-squares problems involving functions. In addition, some notions are clarified, the role of optimization techniques is underplayed, and Monte Carlo methods are taken much more seriously. The first part of the book deals exclusively with discrete inverse problems with a finite number of parameters, while the second part of the book deals with general inverse problems. The book is directed to all scientists, including applied mathematicians, facing the problem of quantitative interpretation of experimental data in fields such as physics, chemistry, biology, image processing, and information sciences. Considerable effort has been made so that this book can serve either as a reference manual for researchers or as a textbook in a course for undergraduate or graduate students. To order the book, please use any of the following links: - SIAM (85 $) - Amazon (85 $). If you can not purchase the book, please download it for free.
Inverse Problem Theory

Mapping of Probabilities - Theory for the Interpretation of Uncertain Physical Measurements. Albert Tarantola, in preparation.

In this book (still in preparation) I try to show that there is an important development missing in traditional Probability Theory. When using only Set Theory to solve problems of assimilation of observations, the obvious conceptual tools to be used are the notions on intersection of sets, image of set (via a mapping), and reciprocal image of a set. I show here that one can introduce the same notions inside Probability Theory (intersection of two probabilities, image of a probability, and reciprocal image of a probability). These three notions are related through a nontrivial compatibility theorem that gives credibility (I hope) to the theory so obtained. In fact, this text provides a mathematical foundation to the epistemological point of view expressed in the short article below (Popper, Bayes, and...). My plan is to work on this book for two more years before I send it to the press. In the meantime, any interested reader is invited to download the manuscript (at present, only the first two chapters are readable). [Comment: I have now extracted the basic definitions and theorems, and have posted them in the arXiv, see paper Image and Reciprocal Image of a Measure; Compatibility Theorem below.]
Mapping of Probabilities

Gravimetry, Relativity, and the Global Navigation Satellite Systems, by Albert Tarantola, Ludek Klimes, José Maria Pozo, and Bartolomé Coll, arXiv:0905.3798, May 2009.

Relativity is an integral part of positioning systems, and this is taken into account in today's practice by applying many "relativistic corrections" to computations performed using concepts borrowed from Galilean physics. A different, fully relativistic paradigm can be developed for operating a positioning system. This implies some fundamental changes. For instance, the basic coordinates are four times (with a symmetric meaning, not three space coordinate and one time coordinate) and the satellites must have cross-link capabilities. Gravitation must, of course, be taken into account, but not using the Newtonian theory: the gravitation field is, and only is, the space-time metric. This implies that the positioning problem and the gravimetry problem can not be separated. An optimization theory can be developed that, because it is fully relativistic, does not contain any `relativistic correction'. We suggest that all positioning satellite systems should be operated in this way. The first benefit of doing so would be a clarification and a simplification of the theory. We also expect, at the end, to be able to run the positioning systems with increased accuracy. (PDF file: arXiv, local).

Using pulsars to define space-time coordinates, by Bartolomé Coll and Albert Tarantola, arXiv:0905.4121, May 2009.

Fully relativistic coordinates have been proposed for (relativistically) running a "GPS" system. These coordinates are the arrival times of the light signals emitted by four "satellites" (clocks). Replacing the signals emitted by four controlled clocks by the signals emitted by four pulsars defines a coordinate system with lower accuracy, but valid across the whole Solar System. We here precisely define this new coordinate system, by choosing four particular pulsars and a particular event as the origin of the coordinates. (PDF file: arXiv, local).

Popper, Bayes and the inverse problem, by Albert Tarantola, Nature Physics, Vol. 2, August 2006, p 492-494, 2006.

In this small commentary (3 pages) I argue that one should not try to infer models from observations. Rather, models should be invented (using intuition or using more formal Bayesian methods), and observations should only be used to try to falsify them (a kind of Popperian approach). (PDF file).

Image and Reciprocal Image of a Measure; Compatibility Theorem, by Albert Tarantola, arXiv:0810.4749, 27 October 2008.

While the previous note is entirely qualitative, this one provides the underground mathematics. The style of this note (strict mathematical language) is quite different from my previous ones. It is proposed that to the usual probability theory, three definitions and a new theorem are added, the resulting theory allows one to displace the central role usually given to the notion of conditional probability. When a mapping \phi is defined between two measurable spaces, to each measure \mu introduced on the first space, there corresponds an image \phi[\mu] on the second space, and, reciprocally, to each measure \nu defined on the second space the corresponds a reciprocal image \phi^{-1}[\nu] on the first space. As the intersection \cap of two measures is easy to introduce, a relation like  \phi[ \mu \cap \phi^{-1} [\nu] ] = \phi[\mu] \cap \nu  makes sense. It is, indeed, a theorem of the theory. This theorem gives mathematical consistency to inferences drawn from physical measurements. Official arXiv posting: http://arxiv.org/abs/0810.4749, local copy: here.

Stress and strain in symmetric and asymmetric elasticity, by Albert Tarantola, arXiv:0907.1833, 10 July 2009.

I profoundly dislike the usual theory of finite deformation: too lousy definition of what a "tensor" is, wrong definition of strain (its definition has to be logarithmic), unphysical stress tensors, etc. In this posting I take an old-fashioned point of view about tensors and coordinates, and try to produce the correct theory of (finite) elasticity. This theory accounts for the possibility of asymmetric stress, but -contrary to usual developments of Cosserat elasticity- I do not take as fundamental the microscopic rotations, but the rotation velocities. My relation between the deformation tensor and the deformation velocity tensor -via the matricant- seems original. It may be important. Official arXiv posting: http://arxiv.org/abs/0907.1833, local copy: here.

Probabilistic Approach to Inverse Problems, by Klaus Mosegaard and Albert Tarantola, International Handbook of Earthquake & Engineering Seismology, Part A., p 237-265, Academic Press, 2002.

You can here obtain an electronic version of a manuscript published in IASPEI's centennial Handbook of Seismology (here a complete version, with all the appendixes). Quite general formulas are proposed, and the Monte Carlo approach for the resolution of inverse problems is emphasized. The paper may also have an interest in pure probability, as an intrinsic definition of conditional probability density is proposed. (PDF file,).

Mathematical Basis for Physical Inference, by Albert Tarantola and Klaus Mosegaard, arXiv:math-ph/0009029.

While the axiomatic introduction of a probability distribution over a space is common, its use for making predictions, using physical theories and prior knowledge, suffers from a lack of formalization. We propose to introduce, in the space of all probability distributions, two operations, the OR and the AND operation, that bring to the space the necessary structure for making inferences on possible values of physical parameters. While physical theories are often asumed to be analytical, we argue that consistent inference needs to replace analytical theories by probability distributions over the parameter space, and we propose a systematic way of obtaining such "theoretical correlations", using the OR operation on the results of physical experiments. Predicting the outcome of an experiment or solving "inverse problems" are then examples of the use of the AND operation. This leads to a simple and complete mathematical basis for general physical inference. (PDF file). Published at the Los Alamos physics e-print archive (http://arXiv.org/abs/math-ph/0009029).

Monte Carlo Sampling of Solutions to Inverse Problems, by Klaus Mosegaard and Albert Tarantola. Journal of Geophysical Research , Vol. 10, No B7, p 12,431-12,447, 1995.

Realistic Inverse Problems are, generally, nonlinear. When nonlinearities are strong, deterministic, iterative methods are not useful. We present here a general approach, adapting the Metropolis algorithm to the standard inverse problem, with data, a priori information, etc. (PDF file from new LaTeX rewriting, 0.5 MB) (PDF file (scan from original paper), 3.3 MB)

Inverse Problems = Quest for Information, by Albert Tarantola and Bernard Valette. J. geophys., 50, p 159-170, 1982.

This work, made in collaboration with Bernard Valette, was my fist attempt to formulate the general inverse problem, using a probabilistic point of view. This was the first introduction of the notion of conjunction of states of information. (PDF file from new LaTeX rewriting, 0.8 MB). (PDF file (scan from original paper), 3.2 MB)

Generalized Nonlinear Inverse Problems Solved using the Least Squares Criterion, by Albert Tarantola and Bernard Valette. Reviews of Geophysics and Space Physics, Vol. 20, No. 2, p 219-232, 1982.

We attempt to give a general definition of the nonlinear least squares inverse problem. First, we examine the discrete problem (finite number of data and unknowns), setting the problem in its fully nonlinear form. Second, we examine the general case where some data and/or unknowns may be functions of a continuous variable and where the form of the theoretical relationship between data and unknowns may be general (in particular, nonlinear integro-differential equations). As particular cases of our nonlinear algorithm we find linear solutions well known in geophysics, like Jackson's (1979) solution for discrete problems or Backus and Gilbert's (1970) solution for continuous problems. (PDF file from new LaTeX rewriting, 0.4 MB) (PDF file (scan from original paper), 3.8 MB)

Nonlinear Inversion of Seismic Waveforms, by Marwan Charara, Christophe Barnes and Albert Tarantola.

I value very high this collaborative work made by my old students Marwan Charara and Christophe Barnes, as it is a very serious demonstration that complex seismic waveform fitting (i.e., waveform inversion) is possible. The price to pay, of course, is the use of a very realistic (so expensive) simulation of the propagation of the elastic waves (including attenuation), and an inversion process where there is a nontrivial use of Monte Carlo techniques (otherwise, it is not possible to discover in which region of the model space the actual Earth is). Note that even a very simple medium (here, it was not extremely far from a layered medium), the observed seismograms can be very complex. The enclosed PDF document is a composite made from The state of affairs in inversion of seismic data, by Charara, Barnes and Tarantola, and from Monte Carlo Inversion of arrival times for multiple phases, by Barnes, Charara and Tarantola. Be careful, large PDF file. (PDF file, 6.6 MB).

Monte Carlo Estimation and Resolution Analysis of Seismic Background Velocities, by Zvi Koren, Klaus Mosegaard, Evgeny Landa, Pierre Thore and Albert Tarantola, Journal of Geophysical Research, Vol. 96, No. B12, 1991.

The movie philosophy is here applied to a large scale inverse problem: the solution of the inverse problem is not one model, but a probability distribution over the model space, that is here sampled. In addition to the published paper, the actual movie is presented. (click here).

Three-Dimensional Inversion Without Blocks, by Albert Tarantola and Alexandre Nercessian. Geophys. J. R. astr. Soc.,76, p 299-306, 1984.

A method for solving non-linear inverse problems in the case where the unknown is a function of the spatial coordinates, and the data set is discrete. The method is based on a generalized least-squares criterion. It may be used, for instance, to the tomography problem, where the data are provided by rays. (PDF file).

Three-Dimensional seismic transmission prospecting of the Mont Dore volcano, France, by Al. Nercessian, Al. Hirn, Al. Tarantola (et al.). Geophys. J. R. astr. Soc., 76 , p 307-315, 1984.

An application of the theory of the previous paper to an original volcanic prospecting method. (PDF file,).

Inversion of Seismic Reflection Data in the Acoustic Approximation. Albert Tarantola, Geophysics, Vol. 49, No. 8, p 1259-1266, 1984.

The nonlinear inverse problem for seismic reflection data is solved in the acoustic approximation. The full waveform approach proposed in this article had some historical importance: a citation analysis conducted by the Society of Exploration Geophysicists showed that this paper was the most cited of all papers published by Geophysics that year. (PDF file, 0.6 MB). You can also download a minor paper: Linearized Inversion of Seismic Reflection Data, A. Tarantola, 1984, Geophysical Prospecting, 32, 998-1015. (PDF file, 8.8 MB).

A Strategy for Nonlinear Elastic Inversion of Seismic Reflection Data. Albert Tarantola, Geophysics, Vol. 51, No. 10, p 1893-1903, 1986.

Extension of the previous paper to the elastic case. Contains the diffraction patterns of some standard diffractors. The lest-squares, iterative algorithm is presented with some detail. (PDF file, 2.3 MB).

Theoretical Background for the Inversion of Seismic Waveforms, Including Elasticity and Attenuation. Albert Tarantola, PAGEOPH (Pure and Applied geophysics), Vol. 128, Nos. 1/2, p 365-399, 1988.

To account for elastic ans attenuating effects in the elastic wave equation, the stress-strain relationship can be defined through a general, anisotropic, causal relaxation function K^{ijkl}(x,t). Then, thewave equation operator is not necessarily symmetric ("self-adjoint"), bu the reciprocity property is still satisfied. The representation theorem contains a term proportional to the history of strain. The dual problem consists of solving the wave equation with final time conditions and an anti-causal relaxation function. The problem of interpretation of seismic waveforms can be set as the nonlinear inverse problem of estimating the matter density rho(x) and all the functions K^(ijkl)(x,t). This inverse problem can be solved using iterative gradient methods, each iteration consisting of the propagation of the actual source in the current medium, with causal attenuation, the propagation of the residuals - acting as if they were sources - backwards in time, with anti-causal attenuation, and the correlation of the two wavefields thus obtained. (PDF file from new LaTeX rewriting, 0.3 MB). (PDF file (scan from original paper), 4.1 MB).

Geophysical Tomography. Yves Desaubies, Albert Tarantola and Jean Zinn-Justin (editors), North-Holland, 1990.

A quite complete review of the tomographic methods for both, the Solid Earth and the Oceans, with chapters written by well known specialists: Bruce Cornuelle (Some practical aspects of ocean acoustic tomography), Yves Desaubies (Ocean acoustic tomography), George Frisk (Inverse methods in ocean bottom acoustics), Finn Jensen (Ocean seismo-acoustic modeling: Numerical methods), Dan Kosloff (Seismic numerical modeling), Theodore Madden (Inversion of low-frequency electromagnetic data), Peter Mora (A unifying view of inversion), Paul Richards (A short course on theoretical seismology), Barbara Romanowicz (Asymptotic theory of normal modes and surface waves), Albert Tarantola (Probabilistic foundation of inverse theory) and Karl Wunsch (Using data with models; Ill-posed and time-dependent ill-posed problems). These texts are expanded (and well written) versions of the lessons given by the authors during the 50-th session (Oceanographic and geophysical tomography) of the Les Houches school on theoretical physics. To download the lessons (i.e., the chapters of the book), please click here.

Geodetic Evidence for Rifting in Afar: A Brittle-Elastic Model of the Behaviour of the Lithosphere. Albert Tarantola, Jean-Claude Ruegg, and Jean-Claude Lépine, EPSL (Earth and Planetary Science Letters), Vol. 45, p 435-444, 1979.

An important episode of rifting occurred in 1978 in southwest Afar, and we developed a numerical model for interpreting the geodetic data. This nice paper was ignored by our american colleagues. Now, it has only historical interest. (PDF file, 0.5 MB)

Linear Inverse Gaussian theory and geostatistics. T.H. Hansen, A,G. Journel, A.Tarantola, and K. Mosegaard, Geophysics, Vol. 71, No. 6, p R101-R111, 2006.

The concepts and the language used in geostatistics is different from that used in inverse theory, but the algorithms are quite similar. In this paper we show how the tomography inverse problem (using rays) can be solved using the concepts from inverse theory, and the sequential random generation typical of geostatistics. (PDF file, 3.3 MB)


The Periodic Table of the Elements.

There are some well known alternatives to the (present form) of Mendeleev's Table of Elements. A quite natural one is obtained when starting from a simple arrangement of the energy levels of the atoms. I built this table while I was a young student (back in 1970), in fact, rediscovering it forty-one years after Janet (1929). Still, it is unfortunate that this form of the Periodic Table is not better known. Click here for more details.

The Musical Scale.

A theoretical reflexion on well-defined logarithmic parameters leads to the suggestion of a modification of the musical scale (PDF file, 0.2 MB).

Noninformative Probability Density for the Elastic Tensor.

There is a lack of good probabilistic (statistic) models for tensors. You can learn here something about the noninformative probability density for the elastic tensor (PDF file, 1.2 MB).

The Torino Scale.

I like the Torino Scale for expressing the potential danger of a possible meteoritic impact. You can read a very modest suggestion made to Richard P. Binzel to ameliorate the scale (PDF file, 0.05 MB).

The Mathematics of Continuity.

In Gravitation Theory, space-time curvature couples with mass. When allowing for torsion, one obtains the Einstein-Cartan theory, where space-time torsion couples with spin. The conservation equations (Bianchi identities) correspond to the dynamical equations representing the conservation of mass, linear momentum and angular momentum. A 3D formulation of these equations allows to study the different gravitational forces appearing in the theory, in particular the mass-mass (Newton) force and the spin-spin force. The nonrelativistic limit of the equations proposes a nontrivial system of equations for classical continuous media. (PDF file, 1.6 MB). This text is not a finished work: some of the discoveries I made while in the middle of the work forced me to change perspective, so I stopped working in this text, to start developing my Elements for Physics.

A Maxwellian Theory of Gravitation.

When I was a young student, I tried to develop a quite simple Theory of Gravitation by simply modifying a sign in the Maxwell equations. Trivial as it may be, the theory has, at least, a pedagogical interest (PDF file, 3.6 MB) (it is in French).

Coping with peer rejection.

One of the most frustrating experiences of a researcher is to see his best papers rejected for publication. Per Bak, the discoverer of the notion of Self Organized Criticality (SOC) explains in his book How Nature Works (OUP, 1997) how difficult it was to publish the idea that earthquakes are SOC phenomena. I was the one (while I was serving as Editor for the Journal of Geophysical Research) that, against the opinion of the referees, decided to publish the work (and I am quite proud of this). Here is the excerpt of Bak's book where he mentions this episode (PDF file, 0.04 MB).

I have opinions, too:

Animal Experiments (letter to the Editor), Albert Tarantola, Nature, Vol. 340, page 424, 10 August 1989.

This (extremely short) letter to the editor was sent to mark my desagreement with animal experimentation. It was prompted by an experiment made in a French laboratory (see Nature,337, 265-267, 1989, and Nature, 339, 248, 1989) that would possibly not have been allowed under the U.K. or the U.S.A. law. Also, I was quite irritated by the general editorial viewpoint of Nature,where animal experimentation is seen as "obviously necessary" (PDF file, 0.8 MB).

Le champignon qui cache la forêt, Albert Tarantola, Le Monde, jeudi 21 septembre 1995, page 14. In French.

When French president Jacques Chirac decided in 1995 to start a last round of nuclear tests in Mururoa, a debate was open in France. Here I say that fission or fusion bombs are fatally going to be replaced, some day, by antimatter bombs. Unless we take a strong decision on time (PDF file, 0.5 MB).

My technophobia.

In recent years, I have been following an unexpected path.

Café Beaubourg Institute of Mathematical Physics
Viewgraphs of my recent presentations
The Geophysical Tomography Group (Wikipedia article)
Link to my family's page
The analemma