Abstract: In this chapter, we review basic methods for data assimilation used in geodynamic modelling: backward advection (BAD), variational/adjoint (VAR), and quasi-reversibility (QRV). The VAR method is based on a search for model parameters (e.g. mantle temperature and flow velocity in the past) by minimising the differences between present observations of the relevant physical parameters (e.g. temperature derived from seismic tomography, geodetic measurements) and those predicted by forward models for an initial guess temperature. The QRV method is based on introduction of the additional term involving the product of a small regularisation parameter and a higher-order temperature derivative into the backward heat equation. The data assimilation in this case is based on a search of the best fit between the forecast model state and the observations by minimising the regularisation parameter. To demonstrate the applicability of the considered data assimilation methods, a numerical model of the evolution of mantle plumes is considered. Also, we present an application of the data assimilation to dynamic restoration of the thermal state of the mantle beneath the Japanese islands and their surroundings. The geodynamic restoration for the last 40 million years is based on the assimilation of the present temperature inferred from seismic tomography, and constrained by the present plate movement derived from geodetic observations, and paleogeographic and paleomagnetic plate reconstructions. Finally, we discuss some challenges, advantages, and disadvantages of the data assimilation methods.

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.