We commonly refer to state estimation theory in geosciences as data assimilation (DA). This term encompasses the entire sequence of operations that, starting from the observations of a system, and from additional statistical and dynamical information (such as a dynamical evolution model), provides an estimate of its state. DA is standard practice in numerical weather prediction, but its application is becoming widespread in many other areas of climate, atmosphere, ocean, and environment modeling; in all circumstances where one intends to estimate the state of a large dynamical system based on limited information. While the complexity of DA, and of the methods thereof, stands on its interdisciplinary nature across statistics, dynamical systems, and numerical optimization, when applied to geosciences, an additional difficulty arises by the continually increasing sophistication of the environmental models. Thus, in spite of DA being nowadays ubiquitous in geosciences, it has so far remained a topic mostly reserved to experts. We aim this overview article at geoscientists with a background in mathematical and physical modeling, who are interested in the rapid development of DA and its growing domains of application in environmental science, but so far have not delved into its conceptual and methodological complexities. This article is categorized under: Climate Models and Modeling > Knowledge Generation with Models

Representation, representativity, representativeness error, forward interpolation error, forward model error, observation-operator error, aggregation error and sampling error are all terms used to refer to components of observation error in the context of data assimilation. This article is an attempt to consolidate the terminology that has been used in the earth sciences literature and was suggested at a European Space Agency workshop held in Reading in April 2014. We review the state of the art and, through examples, motivate the terminology. In addition to a theoretical framework, examples from application areas of satellite data assimilation, ocean reanalysis and atmospheric chemistry data assimilation are provided. Diagnosing representation-error statistics as well as their use in state-of-the-art data assimilation systems is discussed within a consistent framework.

Disordered noninteracting quasiparticles that are governed by a Majorana-type Hamiltonian -prominent examples are dirty superconductors with broken time-reversal and spin-rotation symmetry, or the fermionic representation of the 2d Ising model with fluctuating bond strengths -are called class D. In two dimensions, weakly disordered systems of this kind may possess a metallic phase beyond the insulating phases expected for strong disorder. We show that the 2d metal phase emanates from the free Majorana fermion point, in the direction of the RG trajectory of a perturbed WZW model. To establish this result, we develop a supersymmetric extension of the method of nonabelian bosonization. On the metallic side of the metal-insulator transition, the density of states becomes nonvanishing at zero energy, by a mechanism akin to dynamical mass generation. This feature is explored in a model of N species of disordered Dirac fermions, via the mapping on a nonlinear sigma model, which encapsulates a Z 2 spin degree of freedom. We compute the density of states in a finite system, and obtain agreement with the random-matrix prediction for class D, in the ergodic limit. Vortex disorder, which is a relevant perturbation at the free-fermion point, changes the density of states at low energy and suppresses the local Z 2 degree of freedom, thereby leading to a different symmetry class, BD.1 The equation (2) fixes an orthogonal Lie algebra in even dimension, so(2N ), which is denoted by D N in Cartan's table. Hence the name "D".2 Throughout this paper, supergroups such as GL(n|n) and OSp(2n|2n) are understood to be defined over the complex number field, C, unless specified otherwise. In places where this fact is of particular importance, we will switch to the notation GL C (n|n) and OSp C (2n|2n).