The focus of this book is on recent developments in filtering for high-dimensional, chaotic and noisy dynamical systems. Filtering basically means guiding a dynamical model to shadow the actual state of the system and extracting predictions in real time. This is a two-stage process. In the first step, the probability distribution of the state of the system is predicted using the model and the current distribution. The result is called the “prior” distribution. In the second step, measurements are used to adjust the prior distribution to one closer to the actual state of the system, resulting in a “posterior” distribution. Both the model and the measurements are assumed to have uncertainties associated with them, taken into account as random variables.
While the importance of developing accurate, real-time prediction algorithms for complex systems is obvious in many applied sciences, the authors concentrate on geophysical fluid dynamics and the climate system in general—hence the word “turbulence” in the title. Regarding this system as highly chaotic, as well as stochastic—if only to account for physical processes not explicitly modelled or ill-understood—is very much part of the modern paradigm of climate dynamics. It is also one of the most challenging systems to which to apply filtering techniques. Difficulties include
- the huge number of contributing processes, from cloud formation to sea-ice dynamics, only some of which will be represented in any given model;
- the wide range of time scales of these processes, from hours to decades;
- the wide range of their spatial scales, from kilometres to planetary scales;
- the high dimension of the chaotic (hypothetical) chaotic attractor;
- the sparseness of observations, in space and time as well as in the kind of variables that can be measured.
The authors take up this enormous challenge by combining the classical theory of Kalman filtering with stability analysis for numerical methods for partial differential equations, and by testing their ideas on comparatively simple stochastic dynamical systems constructed to reproduce certain statistics of turbulent motion.
The first part of the book is devoted to basics and uses low-dimensional systems to illustrate Kalman filtering. It also goes over some elementary stability theory for time-stepping of partial differential equations, in particular the Courant-Friedrichs-Lewy instability and the Lax equivalence theorem. In the second part, entitled “mathematical guidelines for filtering turbulent signals,” linear models for turbulent motion are developed and used to demonstrate basic filtering techniques. The analysis is done mostly in Fourier space, so that the stochastic forcing can be chosen to produce certain energy spectra (such as Kolmogorov’s -5/3 spectrum). In the third part, nonlinearity is introduced, again in a toy model cleverly designed to yield exact statistics. In the final chapter, the focus is on filtering techniques that are feasible for nonlinear systems with many degrees of freedom.
The persistent use of models that yield exact solutions, yet include essential complications, such as time-scale separation or the random switching from damped to amplified dynamics for ranges of Fourier modes, is a very attractive feature of this work. It allows us to see the various filtering strategies at work and reveals their strengths and weaknesses, so that the reader can understand the advanced filtering techniques better than theory alone or uncontrolled experiments would allow.
The more basic theory, on the other hand, is hard to study from this book.
The introduction claims that “the book contains enough background material from filtering, turbulence theory and numerical analysis to make the presentation self-contained and is suitable for graduate courses as well as for researchers in a range of disciplines across science and engineering....” I beg to differ. Firstly, the topic of stochastic dynamics is conspicuously absent from the list, and without at least a graduate course in that area I think the book would be quite hard to read. From the very first chapter, the reader is assumed to be familiar with Fokker-Planck and Langevin equations, Bayes’ theorem and other such concepts. Secondly, there is no introduction to Kalman filtering as such. Essential aspects such as observability, controllability and filter stability are brought up whenever they come up in the example computation in chapter two, but an explanation of their significance and interrelation is lacking. Especially in the first part of the book, the authors have used very few words, often referring to research papers for concepts and computations rather than explaining them in the text. In addition, a working knowledge of geophysical fluid dynamics is tacitly assumed, which makes the examples had to appreciate for those who are not familiar with such phenomena as quasi-geostrophic flow, gravity waves and the baroclinic instability.
Rather than a book that explains the topic of filtering and brings the reader from the basics to the state of affairs in this field, this appears to be more of a compendium of recent work, directly accessible only for experts and useful to a lay audience as a guide for further reading.