Geophjs. J. Int. ( 1997) 130, 548

Book review

The Ocean Circulation Inverse Problem C. Wunsch, Cambridge Unioersity Press, Cambridge 1996, 442 pp., ISBN 0-521-48090 6, Hardback, E35.00.

The ocean is huge, not just in comparison with our everyday measures of size but also in comparison with its own natural scales. To illustrate this point it is useful first to consider the atmosphere, where the natural length scales are a few hundred kilometres. The high- and low-pressure regions are thus a thousand kilometres across and the jet streams are four hundred kilometres wide. Because of this, the regular observing network can catch quite small-scale features of the weather system, information that can be used to predict its evolution with time.

The problem of observing the ocean is more difficult. The natural scale is a few tens of kilometres. The equivalent cyclonic and anticyclonic eddies are thus only a hundred kilometres across and many major currents, such as the Gulf Stream, are only 40 km wide. Oceans such as the Pacific are 10000 km across, measurements are few and far between, ships are expensive, and good data from within the ocean are like gold dust.

The situation is somewhat better at the surface of the ocean, where the current generation of satellites is providing detailed data on surface temperatures, elevation, currents and fluxes, However, these data are really only concerned with the top few metres of the ocean. Thus a major problem facing the oceanographic community is how to make the best possible use of the surface and scarce in situ data, in order to provide an accurate overall picture of the present-day ocean’s structure and circulation.

For the last 20 years, Professor Carl Wunsch of MIT has been one of the leading investigators in the field. Through his students and collaborators his influence has become wide- spread. Thus his book is to be welcomed, not only for giving a good background to new entrants to theoretical oceanogra- phy but also for the insight it gives to investigators working on other aspects of ocean research.

The text is written primarily for graduate students, and for that reason contains extensive background sections on ocean- ography and inverse theory. Unfortunately these are also the most idiosyncratic and may lead to some confusion. Thus in the first chapter, Laplace’s equation, solving for the potential given the charge distribution and boundary conditions, is called a forward problem. Later, the related problem of solving for the flow field in the ocean, given poorly observed boundary data, is called an inverse problem. (A time-dependent parabolic equation would have been a better example of the forward problem.) Elsewhere, an equation describing conservation of density is said to be a result of the first law of thermodynamics (conservation of energy).

The background chapter on oceanography is very tightly focused, concentrating primarily on aspects of the ocean circu- lation that are important in later sections of the book. Thus there are long discussions on geostrophic balance and the

problems of estimating a level of no motion. As a teaching text, the book therefore needs to be balanced by something with a broader approach and with a better coverage of ocean physics.

Wunsch’s own excellent work on the ocean inverse problem has concentrated on methods that represent the ocean by a finite-dimension state space. The book reflects this, being concerned with state vectors and variance matrices of large but finite size. The background chapter on the standard matrix inverse definitions and methods is long, over a hundred pages, but it is comprehensive. It also contains a good discussion on the close connections between the different methods. This is useful because, given the scarcity of data and the large number of degrees of freedom of the ocean, an understanding of the connections between the different schemes, and the ability to optimize them, is essential for anyone wanting to make progress in the field.

For this reason the fourth chapter is also particularly useful, especially to the graduate student community. Starting with some simple examples, Wunsch works through a number of problems comparing the Gauss-Markov and singular value decomposition solutions, illustrating their behaviour, strengths and weaknesses, before moving on to applications with real data. The chapter also covers Munk’s estimate of vertical mixing coefficients and the beta spiral and related methods; however, these are really only of historic interest.

Chapter five is concerned with a number of problems not discussed earlier. These include inequality constraints, linear programming, empirical orthogonal functions, total inversion and non-linear functions. The chapter also discusses the prob- lem of objective functions with multiple minima and the use of simulated annealing and genetic drift methods to find the optimum solution.

Finally, chapter six moves onto the more demanding time- dependent inverse problem. Methods covered include the Kalman filter, smoothing, adjoint methods and the use of Green’s functions. As before, the chapter starts with some simple examples, primarily an oscillator subject to random forcing, and then moves on to more realistic applications. The main thing missing, as in chapter four, is a better discussion of the covariance matrix and its approximation. Given the problems in handling these matrices, with possibly a million or more elements, it appears that the development of good approximations is essential before there can be much further progress in the field.

Overall, the book is a significant contribution to the field of oceanographic data assimilation. It has its flaws, suffering from long rambling discussions and concentrating on the New England view of the ocean and its circulation. However these are really minor criticisms. It is a long book but I found it a good read and believe that it will be a useful student text for those entering the field.

DAVID WEBB Southampton Oceanography Centre,

Southampton

548 0 1991 RAS