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.

This chapter presents a systematic theory of generalized (or universal) Fechnerian scaling, based on the intuition underlying Fechner’s original theory. The intuition is that subjective distances among stimuli are computed by cumulating small discriminability values between “neighboring” stimuli. A stimulus space is supposed to be endowed by a dissimilarity function, computed from a discrimination probability function for any pair of stimuli chosen in two distinct observation areas. On the most abstract level, one considers all possible chains of stimuli leading from stimulus a to stimulus b and back to a, and takes the infimum of the sums of the dissimilarities along these chains as the subjective distance between a and b. In arc-connected spaces, the cumulation of dissimilarity values along all possible chains reduces to their cumulation along continuous paths, leading to a fully fledged metric geometry. In topologically Euclidean spaces, the cumulation along paths further reduces to integration along smooth paths, and the geometry in question acquires the form of a generalized Finsler geometry. The chapter also discusses Fechner’s original derivation of his logarithmic law, observation sorites paradox, a generalized Floyd--Warshall algorithm for computing metric distances from dissimilarities, and an ultra-metric version and data-analytic application of Fechnerian scaling.

Encoding models of neuroimaging data combine assumptions about underlying neural processes with knowledge of the task and the type of neuroimaging technique being used to produce equations that predict values of the dependent variable that is measured at each recording site (e.g., the fMRI BOLD response). Voxel-based encoding models include an encoding model that predicts how every hypothesized neural population responds to each stimulus, and a measurement model that first transforms neural population responses into aggregate neural activity and then into values of the dependent variable being measured. Encoding models can be inverted to produce decoding schemes that use the observed data to make predictions about what stimulus was presented on each trial, thereby allowing unique tests of a mathematical model. Representational similarity analysis is a multivariate method that provides unique tests of a model by comparing its predicted similarity structures to similarity structures extracted from neuroimaging data. Model-based fMRI is a set of methods that were developed to test the validity of purely behavioral computational models against fMRI data. Collectively, encoding methods provide useful and powerful new tests of models – even purely cognitive models – that would have been considered fantasy just a few decades ago.

Approximate Bayesian analysis is presented as the solution for complex computational models where no explicit maximum likelihood estimation is possible. The activation-suppression racemodel (ASR), which does have a likelihood amenable to Markov chain Monte Carlo methods, is used to demonstrate the accuracy with which parameters can be estimated with the approximate Bayesian methods.