Abstract: Data assimilation has always been a particularly active area of research in glaciology. While many properties at the surface of glaciers and ice sheets can be directly measured from remote sensing or in situ observations (surface velocity, surface elevation, thinning rates, etc.), many important characteristics, such as englacial and basal properties, as well as past climate conditions, remain difficult or impossible to observe. Data assimilation has been used for decades in glaciology in order to infer unknown properties and boundary conditions that have important impact on numerical models and their projections. The basic idea is to use observed properties, in conjunction with ice flow models, to infer these poorly known ice properties or boundary conditions. There is, however, a great deal of variability among approaches. Constraining data can be of a snapshot in time, or can represent evolution over time. The complexity of the flow model can vary, from simple descriptions of lubrication flow or mass continuity to complex, continent-wide Stokes flow models encompassing multiple flow regimes. Methods can be deterministic, where only a best fit is sought, or probabilistic in nature. We present in this chapter some of the most common applications of data assimilation in glaciology, and some of the new directions that are currently being developed.

This is the first central Chapter of the book that describes Riemannian geometry using Cartan's notion of soldering. Gravity first appears in this Chapter as a dynamical theory of a collection of differential forms rather than a metric. We describe thegeneral notion of geometric structures and then specialise to the case of a geometric structure corresponding to a metric. We describe the notion of a spin connection, its torsion, and then present examples of caclulations of Riemann curvature in the tetrad formalism. We then describe the Einstein-Cartan formulation of GR in terms of differential forms, and present its teleparallel version. We introduce the idea of the pure connection formulation, and compute the corresponding actino perturbatively. We then describe theso-called MacDowell-Mansouri formulation. We briefly describe the computations necessary to carry out the dimensional reduction from 5D to 4D. We then describe the so-called BF formulation of 4D GR, which in particular allows to determine the pure connection action in a closed form. We then describe the field redefinitions that are available when one works in BF formalism, and the associated formulation of BF-type plus potential for the B field.