Abstract: This chapter provides a broad introduction to Bayesian data assimilation that will be useful to practitioners in interpreting algorithms and results, and for theoretical studies developing novel schemes with an understanding of the rich history of geophysical data assimilation and its current directions. The simple case of data assimilation in a ‘perfect’ model is primarily discussed for pedagogical purposes. Some mathematical results are derived at a high level in order to illustrate key ideas about different estimators. However, the focus of this chapter is on the intuition behind these methods, where more formal and detailed treatments of the data assimilation problem can be found in the various references. In surveying a variety of widely used data assimilation schemes, the key message of this chapter is how the Bayesian analysis provides a consistent framework for the estimation problem and how this allows one to formulate its solution in a variety of ways to exploit the operational challenges in the geosciences.

Abstract: During the past two decades, there have been significant efforts to better quantify emissions of environmentally important trace gases along with their trends. In particular, there has been a clear need for robust estimates of emissions on policy-relevant scales of trace gases that impact air quality and climate. This need has driven the expansion of the observing network to better monitor the changing composition of the atmosphere. This chapter will discuss the use of various data assimilation and inverse modelling approaches to quantify these emissions, with a focus on the use of satellite observations. It will discuss the inverse problem of retrieving the atmospheric trace gas information from the satellite measurements, and the subsequent use of these satellite data for quantifying sources and sinks of the trace gases.

Abstract: The primary observables of the Global Positioning System (GPS) ground tracking sites for geodynamics are the Earth’s surface motions, and their geophysical interpretation is based on the numerical models of various tectonic processes. The key issues for geophysical interpretation of the GPS observations are adequate mechanical models of brittle and ductile rock behaviour used to predict surface motions related to various tectonic processes, and the corresponding inversion techniques which allow separation of the processes, and evaluation of their parameters. For large-scale heterogeneous processes, the inversion of the GPS observations requires regularisation because it implies evaluation of some complicated distributed underground motions from their discrete manifestation at the surface. One of the fastest growing applications of the satellite geodetic observations is investigation of the seismotectonic deformation associated with great earthquakes worldwide at all stages of the seismic cycle – inter-seismic, co-seismic, post-seismic. The inversion techniques based on dislocation models in elastic or viscoelastic medium is one of the approaches that may be widely used for GPS-based studies of various seismotectonic deformations.

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: In this chapter, we survey some recent developments in the field of geophysical inversion. We aim to provide an accessible general introduction to the breadth of current research, rather than focusing in depth on particular topics. We hope to give the reader an appreciation for the similarities and connections between different approaches, and their relative strengths and weaknesses.

Abstract: Variational data assimilation through the adjoint method is a powerful emerging technique in geodynamics. It allows one to retrodict past states of the Earth’s mantle as optimal flow histories relative to the current state, so that poorly known mantle flow parameters such as rheology and composition can be tested explicitly against observations gleaned from the geologic record. By yielding testable time dependent Earth models, the technique links observations from seismology, geology, mineral physics, and paleomagnetism in a dynamically consistent way, greatly enhancing our understanding of the solid Earth system. It motivates three research fronts. The first is computational, because the iterative nature of the technique combined with the need of Earth models for high spatial and temporal resolution classifies the task as a grand challenge problem at the level of exa-scale computing. The second is seismological, because the seismic mantle state estimate provides key input information for retrodictions, but entails substantial uncertainties. This calls for efforts to construct 3D reference and collaborative seismic models, and to account for seismic data uncertainties. The third is geological, because retrodictions necessarily use simplified Earth models and noisy input data. Synthetic tests show that retrodictions always reduce the final state misfit, regardless of model and data error. So the quality of any retrodiction must be assessed by geological constraints on past mantle flow. Horizontal surface velocities are an input rather than an output of the retrodiction problem; but viable retrodiction tests can be linked to estimates of vertical lithosphere motion induced by mantle convective stresses.

Abstract: Earthquake early warning (EEW) systems aim to provide advance warning of impending strong ground shaking, in which earthquake ground shaking is predicted in real-time or near real-time. Many EEW systems are based on a strategy which first quickly determines the earthquake hypocentre and magnitude, and then predicts the strength of ground shaking at various locations using the hypocentre distance and magnitude. Recently, however, a new strategy was proposed in which the current seismic wavefield is rapidly estimated by using data assimilation, and then the future wavefield is predicted on the basis of the physics of wave propagation. This technique for real-time prediction of ground shaking in EEW does not necessarily require the earthquake hypocentre and magnitude. In this paper, I review real-time shake-mapping and data assimilation for precise estimation of ongoing ground shaking, and prediction of future shaking in EEW.

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.

Abstract: Energetic charged particles trapped by the Earth’s magnetic field present a significant hazard for Earth-orbiting satellites and humans in space. Application of the data assimilation tools allows us to reconstruct the global state of the radiation particle environment from sparse single-point observations. The measurements from different satellites with different observational errors can be blended in an optimal way with physics-based models. The mathematical formulation on the diffusion and diffusion-advection equations for the Earth’s Van Allen radiation belts and ring current is described. We further describe several recent studies that successfully applied the data assimilation tools to the near-Earth space radiation environment. The applications to the reanalysis of the radiation belts and ring current, real-time predictions, and analysis of the missing physical processes are described and motivation for these studies is provided. We further discuss various assimilation techniques and potential topics for future research.

Abstract: This chapter presents a third-order predictive modelling methodology which aims at obtaining best-estimate results with reduced uncertainties (acronym: 3rd-BERRU-PM) for applications to large-scale models comprising many parameters. The building blocks of the 3rd-BERRU-PM methodology include quantification of third-order moments of the response distribution in the parameter space using third-order adjoint sensitivity analysis (which overcomes the curse of dimensionality), assimilation of experimental data, model calibration, and posterior prediction of best-estimate model responses and parameters with reduced best-estimate variances/covariances for the predicted responses and parameters. Applications of these concepts to an inverse radiation transmission problem, to an oscillatory dynamical model, and to a large-scale computational model involving 21,976 uncertain parameters, respectively, are also presented, thus illustrating the actual computation and impacts of the first-, second-, and third-order response sensitivities to parameters on the expectation, variance, and skewness of the respective model responses.

Abstract: Trans-dimensional Markov chain Monte Carlo (MCMC) treats the number of model parameters as an unknown, and provides a natural approach to assess models of variable complexity. We demonstrate the application of these methods to geochronology and thermochronology. The first is mixture modelling, physically a finite dimension problem, which aims to extract the number and characteristics of component age distributions from an overall distribution of radiometric age data. We demonstrate the MCMC method with Gaussian and skew-t component distributions, the latter containing the former as a special case, applied to a suit of U-Pb zircon data from a sediment in northern France. When considering the posterior distributions obtained from the MCMC samplers, the asymmetrical skew distribution models imply fewer components than the symmetrical Gaussian distribution models. We present some heuristic criteria based on different ways to look the results and aid in model choice in the mixture modelling problem. The second application is a thermal history model, physically a continuous time-temperature function but here parametrised in terms of a finite number of time temperature nodes. We consider a suite of synthetic data from a vertical profile to demonstrate the variable resolution in models constrained from single and multiple samples. Provided the implicit assumptions made when grouping multiple samples are valid, the multi-sample approach is preferable as we exploit the variable information on the model (thermal history) contained in different samples.

Abstract: The continuously increasing quantity and quality of seismic waveform data carry the potential to provide images of the Earth’s internal structure with unprecedented detail. Harnessing this rapidly growing wealth of information, however, constitutes a formidable challenge. While the emergence of faster supercomputers helps to accelerate existing algorithms, the daunting scaling properties of seismic inverse problems still demand the development of more efficient solutions. The diversity of seismic inverse problems – in terms of scientific scope, spatial scale, nature of the data, and available resources – precludes the existence of a silver bullet. Instead, efficiency derives from problem adaptation. Within this context, this chapter describes a collection of methods that are smart in the sense of exploiting specific properties of seismic inverse problems, thereby increasing computational efficiency and usable data volumes, sometimes by orders of magnitude. These methods improve different aspects of a seismic inverse problem, for instance, by harnessing data redundancies, adapting numerical simulation meshes to prior knowledge of wavefield geometry, or permitting long-distance moves through model space for Monte Carlo sampling.

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: In this chapter, I discuss an alternative perspective on interpreting the results of joint and constrained inversions of geophysical data. Typically such inversions are performed based on inductive reasoning (i.e. we fit a limited set of observations and conclude that the resulting model is representative of the Earth). While this has seen many successes, it is less useful when, for example, the specified relationship between different physical parameters is violated in parts of the inversion domain. I argue that in these cases a hypothesis testing perspective can help to learn more about the properties of the Earth. I present joint and constrained inversion examples that show how we can use violations of the assumptions specified in the inversion to study the subsurface. In particular I focus on the combination of gravity and magnetic data with seismic constraints in the western United States. There I see that high velocity structures in the crust are associated with relatively low density anomalies, a possible indication of the presence of melt in a strong rock matrix. The concepts, however, can be applied to other types of data and other regions and offer an extra dimension of analysis to interpret the results of geophysical inversion algorithms.

Abstract: Geomagnetic data assimilation is a recently established research discipline in geomagnetism. It aims to optimally combine geomagnetic observations and numerical geodynamo models to better estimate the dynamic state of the Earth’s outer core, and to predict geomagnetic secular variation. Over the past decade, rapid advances have been made in geomagnetic data assimilation on various fronts by several research groups around the globe, such as using geomagnetic data assimilation to understand and interpret the observed geomagnetic secular variation, estimating part of the core state that is not observable on the Earth’s surface, and making geomagnetic forecasts on multi-year time scales. In parallel, efforts have also been made on proxy systems for understanding fundamental statistical properties of geomagnetic data assimilation, and for developing algorithms tailored specifically for geomagnetic data assimilation. In this chapter, we provide a comprehensive overview of these advances, as well as some of the immediate challenges of geomagnetic data assimilation, and possible solutions and pathways to move forward.