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: We introduce direct and inverse problems, which describe dynamical processes causing change in the Earth system and its space environment. A well-posedness of the problems is defined in the sense of Hadamard and in the sense of Tikhonov, and it is linked to the existence, uniqueness, and stability of the problem solution. Some examples of ill- and well-posed problems are considered. Basic knowledge and approaches in data assimilation and solving inverse problems are discussed along with errors and uncertainties in data and model parameters as well as sensitivities of model results. Finally, we briefly review the book’s chapters which present state-of-the-art knowledge in data assimilation and geophysical inversions and applications in many disciplines of the Earth sciences: from the Earth’s core to the near-Earth environment.

Abstract: Lava flow and lava dome growth are two main manifestations of effusive volcanic eruptions. Less-viscous lava tends to flow long distances depending on slope topography, heat exchange with the surroundings, eruption rate, and the erupted magma rheology. When magma is highly viscous, its eruption on the surface results in a lava dome formation, and an occasional collapse of the dome may lead to a pyroclastic flow. In this chapter, we consider two models of lava dynamics: a lava flow model to determine the internal thermal state of the flow from its surface thermal observations, and a lava dome growth model to determine magma viscosity from the observed lava dome morphological shape. Both models belong to a set of inverse problems. In the first model, the lava thermal conditions at the surface (at the interface between lava and the air) are known from observations, but its internal thermal state is unknown. A variational (adjoint) assimilation method is used to propagate the temperature and heat flow inferred from surface measurements into the interior of the lava flow. In the second model, the lava dome viscosity is estimated based on a comparison between the observed and simulated morphological shapes of lava dome shapes using computer vision techniques.