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.

Abstract: There is a fundamental need to understand and improve the errors and uncertainties associated with estimates of seasonal snow analysis and prediction. Over the past few decades, snow cover remote sensing techniques have increased in accuracy, but the retrieval of spatially and temporally continuous estimates of snow depth or snow water equivalent remains challenging tasks. Model-based snow estimates often bear significant uncertainties due to model structure and error-prone forcing data and parameter estimates. A potential method to overcome model and observational shortcomings is data assimilation. Data assimilation leverages the information content in both observations and models while minimising inherent limitations that result from uncertainty. This chapter reviews current snow models, snow remote sensing methods, and data assimilation techniques that can reduce uncertainties in the characterisation of seasonal snow.

Abstract: Geomagnetic data assimilation aims at constraining the state of the geodynamo working at the Earth’s deep interior by sparse magnetic observations at and above the Earth’s surface. Due to difficulty separating the different magnetic field sources in the observations, spectral models of the geomagnetic field are generally used as inputs for data assimilation. However, the assimilation of raw pointwise observations can be relevant within certain configurations, specifically with paleomagnetic and historical geomagnetic data. Covariance localisation, which is a key ingredient to the assimilation performance in an ensemble framework, is relatively unexplored, and differs with respect to spectral and pointwise observations. This chapter introduces the main characteristics of geomagnetic data and magnetic field models, and explores the role of model and observation covariances and localisation in typical assimilation set-ups, focusing on the use of 3D dynamo simulations as the background model.

Abstract: Operational forecasts of volcanic clouds are a key decision-making component for civil protection agencies and aviation authorities during the occurrence of volcanic crises. Quantitative operational forecasts are challenging due to the large uncertainties that typically exist on characterising volcanic emissions in real time. Data assimilation, including source term inversion, has long been recognised by the scientific community as a mechanism to reduce quantitative forecast errors. In terms of research, substantial progress has occurred during the last decade following the recommendations from the ash dispersal forecast workshops organised by the International Union of Geodesy and Geophysics (IUGG) and the World Meteorological Organization (WMO). The meetings held in Geneva in 2010–11 in the aftermath of the 2010 Eyjafjallajökull eruption identified data assimilation as a research priority. This Chapter reviews the scientific progress and its transfer into operations, which is leveraging a new generation of operational forecast products.

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.