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Simulation-based methods for making statistical inference have evolved dramatically over the past 50 years, keeping pace with technological advancements. The field is undergoing a new revolution as it embraces the representational capacity of neural networks, optimisation libraries, and graphics processing units for learning complex mappings between data and inferential targets. The resulting tools are amortised, in the sense that they allow inference to be made quickly through fast feedforward operations. In this article we review recent progress made in the context of point estimation, approximate Bayesian inference, the automatic construction of summary statistics, and likelihood approximation. The review also covers available software, and includes a simple illustration to showcase the wide array of tools available for amortised inference and the benefits they offer over state-of-the-art Markov chain Monte Carlo methods. The article concludes with an overview of relevant topics and an outlook on future research directions., Comment: 48 pages, 5 figures, 2 tables

Tropical lands play an important role in the global carbon cycle yet their contribution remains uncertain owing to sparse observations. Satellite observations of atmospheric carbon dioxide (CO2) have greatly increased spatial coverage over tropical regions, providing the potential for improved estimates of terrestrial fluxes. Despite this advancement, the spread among satellite-based and in-situ atmospheric CO2 flux inversions over northern tropical Africa (NTA), spanning 0–24°N, remains large. Satellite-based estimates of an annual source of 0.8–1.45 PgC yr−1 challenge our understanding of tropical and global carbon cycling. Here, we compare posterior mole fractions from the suite of inversions participating in the Orbiting Carbon Observatory 2 (OCO-2) Version 10 Model Intercomparison Project (v10 MIP) with independent in-situ airborne observations made over the tropical Atlantic Ocean by the National Aeronautics and Space Administration (NASA) Atmospheric Tomography (ATom) mission during four seasons. We develop emergent constraints on tropical African CO2 fluxes using flux-concentration relationships defined by the model suite. We find an annual flux of 0.14 ± 0.39 PgC yr−1 (mean and standard deviation) for NTA, 2016–2018. The satellite-based flux bias suggests a potential positive concentration bias in OCO-2 B10 and earlier version retrievals over land in NTA during the dry season. Nevertheless, the OCO-2 observations provide improved flux estimates relative to the in situ observing network at other times of year, indicating stronger uptake in NTA during the wet season than the in-situ inversion estimates.

Abstract. Nitrous oxide is a potent greenhouse gas (GHG) and ozone-depleting substance, whose atmospheric abundance has risen throughout the contemporary record. In this work, we carry out the first global hierarchical Bayesian inversion to solve for nitrous oxide emissions, which includes prior emissions with truncated Gaussian distributions and Gaussian model errors, in order to examine the drivers of the atmospheric surface growth rate. We show that both emissions and climatic variability are key drivers of variations in the surface nitrous oxide growth rate between 2011 and 2020. We derive increasing global nitrous oxide emissions, which are mainly driven by emissions between 0 and 30∘ N, with the highest emissions recorded in 2020. Our mean global total emissions for 2011–2020 of 17.2 (16.7–17.7 at the 95 % credible intervals) Tg N yr−1, comprising of 12.0 (11.2–12.8) Tg N yr−1 from land and 5.2 (4.5–5.9) Tg N yr−1 from ocean, agrees well with previous studies, but we find that emissions are poorly constrained for some regions of the world, particularly for the oceans. The prior emissions used in this and other previous work exhibit a seasonal cycle in the extra-tropical Northern Hemisphere that is out of phase with the posterior solution, and there is a substantial zonal redistribution of emissions from the prior to the posterior. Correctly characterizing the uncertainties in the system, for example in the prior emission fields, is crucial for deriving posterior fluxes that are consistent with observations. In this hierarchical inversion, the model-measurement discrepancy and the prior flux uncertainty are informed by the data, rather than solely through “expert judgement”. We show cases where this framework provides different plausible adjustments to the prior fluxes compared to inversions using widely adopted, fixed uncertainty constraints.

interpretable, and well understood models that are routinely employed even though, as is revealed through prior and posterior predictive checks, these can poorly characterise the spatial heterogeneity in the underlying process of interest. Here, we propose a new, flexible class of spatial-process models, which we refer to as spatial Bayesian neural networks (SBNNs). An SBNN leverages the representational capacity of a Bayesian neural network; it is tailored to a spatial setting by incorporating a spatial ``embedding layer'' into the network and, possibly, spatially-varying network parameters. An SBNN is calibrated by matching its finite-dimensional distribution at locations on a fine gridding of space to that of a target process of interest. That process could be easy to simulate from or we may have many realisations from it. We propose several variants of SBNNs, most of which are able to match the finite-dimensional distribution of the target process at the selected grid better than conventional BNNs of similar complexity. We also show that an SBNN can be used to represent a variety of spatial processes often used in practice, such as Gaussian processes, lognormal processes, and max-stable processes. We briefly discuss the tools that could be used to make inference with SBNNs, and we conclude with a discussion of their advantages and limitations., Comment: 35 pages, 21 figures

Nitrous oxide is a potent greenhouse gas (GHG) and ozone-depleting substance, whose atmospheric abundance has risen throughout the contemporary record. In this work, we carry out the first global hierarchical Bayesian inversion to solve for nitrous oxide emissions, which includes prior emissions with truncated Gaussian distributions and Gaussian model errors, in order to examine the drivers of the atmospheric surface growth rate. We show that both emissions and climatic variability are key drivers of variations in the surface nitrous oxide growth rate between 2011 and 2020. We derive increasing global nitrous oxide emissions, which are mainly driven by emissions between 0 and 30∘ N, with the highest emissions recorded in 2020. Our mean global total emissions for 2011–2020 of 17.2 (16.7–17.7 at the 95 % credible intervals) Tg N yr−1, comprising of 12.0 (11.2–12.8) Tg N yr−1 from land and 5.2 (4.5–5.9) Tg N yr−1 from ocean, agrees well with previous studies, but we find that emissions are poorly constrained for some regions of the world, particularly for the oceans. The prior emissions used in this and other previous work exhibit a seasonal cycle in the extra-tropical Northern Hemisphere that is out of phase with the posterior solution, and there is a substantial zonal redistribution of emissions from the prior to the posterior. Correctly characterizing the uncertainties in the system, for example in the prior emission fields, is crucial for deriving posterior fluxes that are consistent with observations. In this hierarchical inversion, the model-measurement discrepancy and the prior flux uncertainty are informed by the data, rather than solely through “expert judgement”. We show cases where this framework provides different plausible adjustments to the prior fluxes compared to inversions using widely adopted, fixed uncertainty constraints.

The natural cycles of the surface-to-atmosphere fluxes of carbon dioxide (CO$_2$) and other important greenhouse gases are changing in response to human influences. These changes need to be quantified to understand climate change and its impacts, but this is difficult to do because natural fluxes occur over large spatial and temporal scales. To infer trends in fluxes and identify phase shifts and amplitude changes in flux seasonal cycles, we construct a flux-inversion system that uses a novel spatially varying time-series decomposition of the fluxes, while also accommodating physical constraints on the fluxes. We incorporate these features into the Wollongong Methodology for Bayesian Assimilation of Trace-gases (WOMBAT, Zammit-Mangion et al., Geosci. Model Dev., 15, 2022), a hierarchical flux-inversion framework that yields posterior distributions for all unknowns in the underlying model. We apply the new method, which we call WOMBAT v2.0, to a mix of satellite observations of CO$_2$ mole fraction from the Orbiting Carbon Observatory-2 (OCO-2) satellite and direct measurements of CO$_2$ mole fraction from a variety of sources. We estimate the changes to CO$_2$ fluxes that occurred from January 2015 to December 2020, and compare our posterior estimates to those from an alternative method based on a bottom-up understanding of the physical processes involved. We find substantial trends in the fluxes, including that tropical ecosystems trended from being a net source to a net sink of CO$_2$ over the study period. We also find that the amplitude of the global seasonal cycle of ecosystem CO$_2$ fluxes increased over the study period by 0.11 PgC/month (an increase of 8%), and that the seasonal cycle of ecosystem CO$_2$ fluxes in the northern temperate and northern boreal regions shifted earlier in the year by 0.4-0.7 and 0.4-0.9 days, respectively (2.5th to 97.5th posterior percentiles).

Deep neural network models have become ubiquitous in recent years, and have been applied to nearly all areas of science, engineering, and industry. These models are particularly useful for data that have strong dependencies in space (e.g., images) and time (e.g., sequences). Indeed, deep models have also been extensively used by the statistical community to model spatial and spatio-temporal data through, for example, the use of multi-level Bayesian hierarchical models and deep Gaussian processes. In this review, we first present an overview of traditional statistical and machine learning perspectives for modeling spatial and spatio-temporal data, and then focus on a variety of hybrid models that have recently been developed for latent process, data, and parameter specifications. These hybrid models integrate statistical modeling ideas with deep neural network models in order to take advantage of the strengths of each modeling paradigm. We conclude by giving an overview of computational technologies that have proven useful for these hybrid models, and with a brief discussion on future research directions., Comment: 27 pages, 1 figure

Flux inversion is the process by which sources and sinks of a gas are identified from observations of gas mole fraction. The inversion often involves running a Lagrangian particle dispersion model (LPDM) to generate sensitivities between observations and fluxes over a spatial domain of interest. The LPDM must be run backward in time for every gas measurement, and this can be computationally prohibitive. To address this problem, here we develop a novel spatio-temporal emulator for LPDM sensitivities that is built using a convolutional variational autoencoder (CVAE). With the encoder segment of the CVAE, we obtain approximate (variational) posterior distributions over latent variables in a low-dimensional space. We then use a spatio-temporal Gaussian process emulator on the low-dimensional space to emulate new variables at prediction locations and time points. Emulated variables are then passed through the decoder segment of the CVAE to yield emulated sensitivities. We show that our CVAE-based emulator outperforms the more traditional emulator built using empirical orthogonal functions and that it can be used with different LPDMs. We conclude that our emulation-based approach can be used to reliably reduce the computing time needed to generate LPDM outputs for use in high-resolution flux inversions., Comment: 25 pages, 8 figures, 2 tables, data & code available

WOMBAT (the WOllongong Methodology for Bayesian Assimilation of Trace-gases) is a fully Bayesian hierarchical statistical framework for flux inversion of trace gases from flask, in situ, and remotely sensed data. WOMBAT extends the conventional Bayesian-synthesis framework through the consideration of a correlated error term, the capacity for online bias correction, and the provision of uncertainty quantification on all unknowns that appear in the Bayesian statistical model. We show, in an observing system simulation experiment (OSSE), that these extensions are crucial when the data are indeed biased and have errors that are correlated. Using the GEOS-Chem atmospheric transport model, we show that WOMBAT is able to obtain posterior means and uncertainties on non-fossil-fuel CO$_2$ fluxes from Orbiting Carbon Observatory-2 (OCO-2) data that are comparable to those from the Model Intercomparison Project (MIP) reported in Crowell et al. (2019, Atmos. Chem. Phys., vol. 19). We also find that our predictions of out-of-sample retrievals from the Total Column Carbon Observing Network are, for the most part, more accurate than those made by the MIP participants. Subsequent versions of the OCO-2 datasets will be ingested into WOMBAT as they become available., Comment: 46 pages, 13 figures

The last decade has seen an explosion in data sources available for monitoring and prediction of environmental phenomena. While several inferential methods have been developed to make predictions on the underlying process by combining these data, an optimal sampling design for additional data collection in the presence of multiple heterogeneous sources has not yet been developed. Here, we provide an adaptive spatial design strategy based on a utility function that combines both prediction uncertainty and risk-factor criteria. Prediction uncertainty is obtained through a spatial data fusion approach based on fixed rank kriging that can tackle data with differing spatial supports and signal-to-noise ratios. We focus on the application of low-cost portable sensors, which tend to be relatively noisy, for air pollution monitoring, where data from regulatory stations as well as numeric modeling systems are also available. Although we find that spatial adaptive sampling designs can help to improve predictions and reduce prediction uncertainty, low-cost portable sensors are only likely to be beneficial if they are sufficient in number and quality. Our conclusions are based on a multi-factorial simulation experiment, and on a realistic simulation of pollutants in the Erie and Niagara counties in Western New York.

Visualizing both conditional and marginal associations in contingency tables by using simple diagrams is discussed

The Gaussian process is an indispensable tool for spatial data analysts. The onset of the “big data” era, however, has lead to the traditional Gaussian process being computationally infeasible for modern spatial data. As such, various alternatives to the full Gaussian process that are more amenable to handling big spatial data have been proposed. These modern methods often exploit low-rank structures and/or multi-core and multi-threaded computing environments to facilitate computation. This study provides, first, an introductory overview of several methods for analyzing large spatial data. Second, this study describes the results of a predictive competition among the described methods as implemented by different groups with strong expertise in the methodology. Specifically, each research group was provided with two training datasets (one simulated and one observed) along with a set of prediction locations. Each group then wrote their own implementation of their method to produce predictions at the given location and each was subsequently run on a common computing environment. The methods were then compared in terms of various predictive diagnostics. Supplementary materials regarding implementation details of the methods and code are available for this article online.

Integro-difference equation (IDE) models describe the conditional dependence between the spatial process at a future time point and the process at the present time point through an integral operator. Nonlinearity or temporal dependence in the dynamics is often captured by allowing the operator parameters to vary temporally, or by re-fitting a model with a temporally-invariant linear operator in a sliding window. Both procedures tend to be excellent for prediction purposes over small time horizons, but are generally time-consuming and, crucially, do not provide a global prior model for the temporally-varying dynamics that is realistic. Here, we tackle these two issues by using a deep convolution neural network (CNN) in a hierarchical statistical IDE framework, where the CNN is designed to extract process dynamics from the process' most recent behaviour. Once the CNN is fitted, probabilistic forecasting can be done extremely quickly online using an ensemble Kalman filter with no requirement for repeated parameter estimation. We conduct an experiment where we train the model using 13 years of daily sea-surface temperature data in the North Atlantic Ocean. Forecasts are seen to be accurate and calibrated. A key advantage of our approach is that the CNN provides a global prior model for the dynamics that is realistic, interpretable, and computationally efficient. We show the versatility of the approach by successfully producing 10-minute nowcasts of weather radar reflectivities in Sydney using the same model that was trained on daily sea-surface temperature data in the North Atlantic Ocean., Comment: 22 pages, 10 figures

The world is becoming increasingly complex, with larger quantities of data available to be analyzed. It so happens that much of these "big data" that are available are spatio-temporal in nature, meaning that they can be indexed by their spatial locations and time stamps. Spatio-Temporal Statistics with R provides an accessible introduction to statistical analysis of spatio-temporal data, with hands-on applications of the statistical methods using R Labs found at the end of each chapter.

Maximum likelihood (ML) estimation of spatial autocorrelation models is well established for the case where each node in the graph is directly observed. When one or more nodes are not observed, the user has a variety of computational tools at her or his disposal ranging from the expectation-maximization algorithm, which has become a standard for missing‐data problems, to marginal likelihood estimation methods and to fully Bayesian approaches. In this article, we give a comprehensive overview of likelihood‐based computational frameworks for parameter estimation of the conditional autoregressive model, and we establish connections with several algorithms in the literature that are iterative and often computationally suboptimal. We show that a vanilla marginal ML approach, which we provide computational details for, is still generally orders of magnitude faster than the iterative approaches, even on large data sets and especially so when the number of unobserved units is relatively large.

Detection and quantification of greenhouse-gas emissions is important for both compliance and environment conservation. However, despite several decades of active research, it remains predominantly an open problem, largely due to model errors and assumptions that appear at each stage of the inversion processing chain. In 2015, a controlled-release experiment headed by Geoscience Australia was carried out at the Ginninderra Controlled Release Facility, and a variety of instruments and methods were employed for quantifying the release rates of methane and carbon dioxide from a point source. This paper proposes a fully Bayesian approach to atmospheric tomography for inferring the methane emission rate of this point source using data collected during the experiment from both point-and path-sampling instruments. The Bayesian framework is designed to account for uncertainty in the parameterisations of measurements, the meteorological data, and the atmospheric model itself when performing inversion using Markov chain Monte Carlo (MCMC). We apply our framework to all instrument groups using measurements from two release-rate periods. We show that the inversion framework is robust to instrument type and meteorological conditions. From all the inversions we conducted across the different instrument groups and release-rate periods, our worst-case median emission rate estimate was within 36 % of the true emission rate. Further, in the worst case, the closest limit of the 95 % credible interval to the true emission rate was within 11 % of this true value.

Satellite remote sensing of trace gases such as carbon dioxide (CO 2 ) has increased our ability to observe and understand Earth's climate. However, these remote sensing data, specifically Level 2 retrievals, tend to be irregular in space and time, and hence, spatio-temporal prediction is required to infer values at any location and time point. Such inferences are not only required to answer important questions about our climate, but they are also needed for validating the satellite instrument, since Level 2 retrievals are generally not co-located with ground-based remote sensing instruments. Here, we discuss statistical approaches to construct Level 3 products from Level 2 retrievals, placing particular emphasis on the strengths and potential pitfalls when using statistical prediction in this context. Following this discussion, we use a spatio-temporal statistical modelling framework known as fixed rank kriging (FRK) to obtain global predictions and prediction standard errors of column-averaged carbon dioxide based on Version 7r and Version 8r retrievals from the Orbiting Carbon Observatory-2 (OCO-2) satellite. The FRK predictions allow us to validate statistically the Level 2 retrievals globally even though the data are at locations and at time points that do not coincide with validation data. Importantly, the validation takes into account the prediction uncertainty, which is dependent both on the temporally-varying density of observations around the ground-based measurement sites and on the spatio-temporal high-frequency components of the trace gas field that are not explicitly modelled. Here, for validation of remotely-sensed CO 2 data, we use observations from the Total Carbon Column Observing Network. We demonstrate that the resulting FRK product based on Version 8r compares better with TCCON data than that based on Version 7r, in terms of both prediction accuracy and uncertainty quantification.

The Authors. We present spatiotemporal mass balance trends for the Antarctic Ice Sheet from a statistical inversion of satellite altimetry, gravimetry, and elastic-corrected GPS data for the period 2003-2013. Our method simultaneously determines annual trends in ice dynamics, surface mass balance anomalies, and a time-invariant solution for glacio-isostatic adjustment while remaining largely independent of forward models. We establish that over the period 2003-2013, Antarctica has been losing mass at a rate of -84 ± 22 Gt yr-1, with a sustained negative mean trend of dynamic imbalance of -111 ± 13 Gt yr-1. West Antarctica is the largest contributor with -112 ± 10 Gt yr-1, mainly triggered by high thinning rates of glaciers draining into the Amundsen Sea Embayment. The Antarctic Peninsula has experienced a dramatic increase in mass loss in the last decade, with a mean rate of -28 ± 7 Gt yr-1 and significantly higher values for the most recent years following the destabilization of the Southern Antarctic Peninsula around 2010. The total mass loss is partly compensated by a significant mass gain of 56 ± 18 Gt yr-1 in East Antarctica due to a positive trend of surface mass balance anomalies.

The Authors. We present spatiotemporal mass balance trends for the Antarctic Ice Sheet from a statistical inversion of satellite altimetry, gravimetry, and elastic-corrected GPS data for the period 2003-2013. Our method simultaneously determines annual trends in ice dynamics, surface mass balance anomalies, and a time-invariant solution for glacio-isostatic adjustment while remaining largely independent of forward models. We establish that over the period 2003-2013, Antarctica has been losing mass at a rate of -84 ± 22 Gt yr-1, with a sustained negative mean trend of dynamic imbalance of -111 ± 13 Gt yr-1. West Antarctica is the largest contributor with -112 ± 10 Gt yr-1, mainly triggered by high thinning rates of glaciers draining into the Amundsen Sea Embayment. The Antarctic Peninsula has experienced a dramatic increase in mass loss in the last decade, with a mean rate of -28 ± 7 Gt yr-1 and significantly higher values for the most recent years following the destabilization of the Southern Antarctic Peninsula around 2010. The total mass loss is partly compensated by a significant mass gain of 56 ± 18 Gt yr-1 in East Antarctica due to a positive trend of surface mass balance anomalies.

The Authors. We present spatiotemporal mass balance trends for the Antarctic Ice Sheet from a statistical inversion of satellite altimetry, gravimetry, and elastic-corrected GPS data for the period 2003-2013. Our method simultaneously determines annual trends in ice dynamics, surface mass balance anomalies, and a time-invariant solution for glacio-isostatic adjustment while remaining largely independent of forward models. We establish that over the period 2003-2013, Antarctica has been losing mass at a rate of -84 ± 22 Gt yr-1, with a sustained negative mean trend of dynamic imbalance of -111 ± 13 Gt yr-1. West Antarctica is the largest contributor with -112 ± 10 Gt yr-1, mainly triggered by high thinning rates of glaciers draining into the Amundsen Sea Embayment. The Antarctic Peninsula has experienced a dramatic increase in mass loss in the last decade, with a mean rate of -28 ± 7 Gt yr-1 and significantly higher values for the most recent years following the destabilization of the Southern Antarctic Peninsula around 2010. The total mass loss is partly compensated by a significant mass gain of 56 ± 18 Gt yr-1 in East Antarctica due to a positive trend of surface mass balance anomalies.

The Authors. We present spatiotemporal mass balance trends for the Antarctic Ice Sheet from a statistical inversion of satellite altimetry, gravimetry, and elastic-corrected GPS data for the period 2003-2013. Our method simultaneously determines annual trends in ice dynamics, surface mass balance anomalies, and a time-invariant solution for glacio-isostatic adjustment while remaining largely independent of forward models. We establish that over the period 2003-2013, Antarctica has been losing mass at a rate of -84 ± 22 Gt yr-1, with a sustained negative mean trend of dynamic imbalance of -111 ± 13 Gt yr-1. West Antarctica is the largest contributor with -112 ± 10 Gt yr-1, mainly triggered by high thinning rates of glaciers draining into the Amundsen Sea Embayment. The Antarctic Peninsula has experienced a dramatic increase in mass loss in the last decade, with a mean rate of -28 ± 7 Gt yr-1 and significantly higher values for the most recent years following the destabilization of the Southern Antarctic Peninsula around 2010. The total mass loss is partly compensated by a significant mass gain of 56 ± 18 Gt yr-1 in East Antarctica due to a positive trend of surface mass balance anomalies.

We present spatiotemporal mass balance trends for the Antarctic Ice Sheet from a statistical inversion of satellite altimetry, gravimetry, and elastic-corrected GPS data for the period 2003–2013. Our method simultaneously determines annual trends in ice dynamics, surface mass balance anomalies, and a time-invariant solution for glacio-isostatic adjustment while remaining largely independent of forward models. We establish that over the period 2003–2013, Antarctica has been losing mass at a rate of −84 ± 22 Gt yr−1, with a sustained negative mean trend of dynamic imbalance of −111 ± 13 Gt yr−1. West Antarctica is the largest contributor with −112 ± 10 Gt yr−1, mainly triggered by high thinning rates of glaciers draining into the Amundsen Sea Embayment. The Antarctic Peninsula has experienced a dramatic increase in mass loss in the last decade, with a mean rate of −28 ± 7 Gt yr−1 and significantly higher values for the most recent years following the destabilization of the Southern Antarctic Peninsula around 2010. The total mass loss is partly compensated by a significant mass gain of 56 ± 18 Gt yr−1 in East Antarctica due to a positive trend of surface mass balance anomalies.

We present spatiotemporal mass balance trends for the Antarctic Ice Sheet from a statistical inversion of satellite altimetry, gravimetry, and elastic-corrected GPS data for the period 2003–2013. Our method simultaneously determines annual trends in ice dynamics, surface mass balance anomalies, and a time-invariant solution for glacio-isostatic adjustment while remaining largely independent of forward models. We establish that over the period 2003–2013, Antarctica has been losing mass at a rate of −84 ± 22 Gt yr−1, with a sustained negative mean trend of dynamic imbalance of −111 ± 13 Gt yr−1. West Antarctica is the largest contributor with −112 ± 10 Gt yr−1, mainly triggered by high thinning rates of glaciers draining into the Amundsen Sea Embayment. The Antarctic Peninsula has experienced a dramatic increase in mass loss in the last decade, with a mean rate of −28 ± 7 Gt yr−1 and significantly higher values for the most recent years following the destabilization of the Southern Antarctic Peninsula around 2010. The total mass loss is partly compensated by a significant mass gain of 56 ± 18 Gt yr−1 in East Antarctica due to a positive trend of surface mass balance anomalies.

Multivariate geostatistics is based on modelling all covariances between all possible combinations of two or more variables at any sets of locations in a continuously indexed domain. Multivariate spatial covariance models need to be built with care, since any covariance matrix that is derived from such a model must be nonnegative-definite. In this article, we develop a conditional approach for spatial-model construction whose validity conditions are easy to check. We start with bivariate spatial covariance models and go on to demonstrate the approach's connection to multivariate models defined by networks of spatial variables. In some circumstances, such as modelling respiratory illness conditional on air pollution, the direction of conditional dependence is clear. When it is not, the two directional models can be compared. More generally, the graph structure of the network reduces the number of possible models to compare. Model selection then amounts to finding possible causative links in the network. We demonstrate our conditional approach on surface temperature and pressure data, where the role of the two variables is seen to be asymmetric.

Atmospheric trace-gas inversion is the procedure by which the sources and sinks of a trace gas are identified from observations of its mole fraction at isolated locations in space and time. This is inherently a spatio-temporal bivariate inversion problem, since the mole-fraction field evolves in space and time and the flux is also spatio-temporally distributed. Further, the bivariate model is likely to be non-Gaussian since the flux field is rarely Gaussian. Here, we use conditioning to construct a non-Gaussian bivariate model, and we describe some of its properties through auto- and cross-cumulant functions. A bivariate non-Gaussian, specifically trans-Gaussian, model is then achieved through the use of Box-Cox transformations, and we facilitate Bayesian inference by approximating the likelihood in a hierarchical framework. Trace-gas inversion, especially at high spatial resolution, is frequently highly sensitive to prior specification. Therefore, unlike conventional approaches, we assimilate trace-gas inventory information with the observational data at the parameter layer, thus shifting prior sensitivity from the inventory itself to its spatial characteristics (e.g., its spatial length scale). We demonstrate the approach in controlled-experiment studies of methane inversion, using fluxes extracted from inventories of the UK and Ireland and of Northern Australia.

In this work we assess the most recent estimates of glacial isostatic adjustment (GIA) for Antarctica, including those from both forward and inverse methods. The assessment is based on a comparison of the estimated uplift rates with a set of elastic-corrected GPS vertical velocities. These have been observed from an extensive GPS network and computed using data over the period 2009-2014. We find systematic underestimations of the observed uplift rates in both inverse and forward methods over specific regions of Antarctica characterized by low mantle viscosities and thin lithosphere, such as the northern Antarctic Peninsula and the Amundsen Sea Embayment, where its recent ice discharge history is likely to be playing a role in current GIA. Uplift estimates for regions where many GIA models have traditionally placed their uplift maxima, such as the margins of Filchner-Ronne and Ross ice shelves, are found to be overestimated. GIA estimates show large variability over the interior of East Antarc tica which results in increased uncertainties on the ice-sheet mass balance derived from gravimetry methods.

Multivariate geostatistics is based on modelling all covariances between all possible combinations of two or more variables at any sets of locations in a continuously indexed domain. Multivariate spatial covariance models need to be built with care, since any covariance matrix that is derived from such a model must be nonnegative-definite. In this article, we develop a conditional approach for spatial-model construction whose validity conditions are easy to check. We start with bivariate spatial covariance models and go on to demonstrate the approach's connection to multivariate models defined by networks of spatial variables. In some circumstances, such as modelling respiratory illness conditional on air pollution, the direction of conditional dependence is clear. When it is not, the two directional models can be compared. More generally, the graph structure of the network reduces the number of possible models to compare. Model selection then amounts to finding possible causative links in the network. We demonstrate our conditional approach on surface temperature and pressure data, where the role of the two variables is seen to be asymmetric.

Atmospheric trace-gas inversion is the procedure by which the sources and sinks of a trace gas are identified from observations of its mole fraction at isolated locations in space and time. This is inherently a spatio-temporal bivariate inversion problem, since the mole-fraction field evolves in space and time and the flux is also spatio-temporally distributed. Further, the bivariate model is likely to be non-Gaussian since the flux field is rarely Gaussian. Here, we use conditioning to construct a non-Gaussian bivariate model, and we describe some of its properties through auto- and cross-cumulant functions. A bivariate non-Gaussian, specifically trans-Gaussian, model is then achieved through the use of Box-Cox transformations, and we facilitate Bayesian inference by approximating the likelihood in a hierarchical framework. Trace-gas inversion, especially at high spatial resolution, is frequently highly sensitive to prior specification. Therefore, unlike conventional approaches, we assimilate trace-gas inventory information with the observational data at the parameter layer, thus shifting prior sensitivity from the inventory itself to its spatial characteristics (e.g., its spatial length scale). We demonstrate the approach in controlled-experiment studies of methane inversion, using fluxes extracted from inventories of the UK and Ireland and of Northern Australia.

Atmospheric trace-gas inversion is the procedure by which the sources and sinks of a trace gas are identified from observations of its mole fraction at isolated locations in space and time. This is inherently a spatio-temporal bivariate inversion problem, since the mole-fraction field evolves in space and time and the flux is also spatio-temporally distributed. Further, the bivariate model is likely to be non-Gaussian since the flux field is rarely Gaussian. Here, we use conditioning to construct a non-Gaussian bivariate model, and we describe some of its properties through auto- and cross-cumulant functions. A bivariate non-Gaussian, specically trans-Gaussian, model is then achieved through the use of Box-Cox transformations, and we facilitate Bayesian inference by approximating the likelihood in a hierarchical framework. Trace-gas inversion, especially at high spatial resolution, is frequently highly sensitive to prior specification. Therefore, unlike conventional approaches, we assimilate trace-gas inventory information with the observational data at the parameter layer, thus shifting prior sensitivity from the inventory itself to its spatial characteristics (e.g., its spatial length scale). We demonstrate the approach in controlled- experiment studies of methane inversion, using fluxes extracted from inventories of the UK and Ireland and of Northern Australia.

Environmental Informatics uses a panoply of tools from the Statistics, Mathematics, Computing, and Visualisation disciplines. It uses these tools to reveal, quantify, and validate scientific hypotheses in the environmental sciences, with the quantification of uncertainty central to its approach. There is now a strong recognition that scientific models need to incorporate stochastic components throughout: While it has always been recognised that data have a component of measurement error, attention is now being given to the quantification of model error, and it is becoming accepted by environmental scientists that probability models for the latter allows for a coherent way to make scientific inference. In Environmental Informatics, uncertainty may be assigned not only to datasets of measurements, but also to computer-generated climate-model output. Methodological advances, in the form of hierarchical statistical models and the accompanying computational developments, have expanded the scope of statistical analyses into very large spatial domains. This has led to studies of the dynamical evolution of entire spatial fields of geophysical variables, where results are given in terms of predictive distributions. Environmental Informatics is not only involved in characterising the environment, it can also be used to make decisions about mitigation and adaptation strategies. The steps taken by environmental scientists, from data to information, from information to knowledge, and from knowledge to decisions, are all taken in the presence of uncertainty. Environmental Informatics encompasses all these aspects.

Spatio-temporal log-Gaussian Cox process models play a central role in the analysis of spatially distributed systems in several disciplines. Yet, scalable inference remains computationally challenging both due to the high resolution modelling generally required and the analytically intractable likelihood function. Here, we exploit the sparsity structure typical of (spatially) discretised log-Gaussian Cox process models by using approximate message-passing algorithms. The proposed algorithms scale well with the state dimension and the length of the temporal horizon with moderate loss in distributional accuracy. They hence provide a flexible and faster alternative to both non-linear filtering-smoothing type algorithms and to approaches that implement the Laplace method or expectation propagation on (block) sparse latent Gaussian models. We infer the parameters of the latent Gaussian model using a structured variational Bayes approach. We demonstrate the proposed framework on simulation studies with both Gaussian and point-process observations and use it to reconstruct the conflict intensity and dynamics in Afghanistan from the WikiLeaks Afghan War Diary.

Antarctica is the world’s largest fresh-water reservoir, with the potential to raise sea levels by about 60 m. An ice sheet contributes to sea-level rise (SLR) when its rate of ice discharge and/or surface melting exceeds accumulation through snowfall. Constraining the contribution of the ice sheets to present-day SLR is vital both for coastal development and planning, and climate projections. Information on various ice sheet processes is available from several remote sensing data sets, as well as in situ data such as global positioning system data. These data have differing coverage, spatial support, temporal sampling and sensing characteristics, and thus, it is advantageous to combine them all in a single framework for estimation of the SLR contribution and the assessment of processes controlling mass exchange with the ocean. In this paper, we predict the rate of height change due to salient geophysical processes in Antarctica and use these to provide estimates of SLR contribution with associated uncertainties. We employ a multivariate spatio-temporal model, approximated as a Gaussian Markov random field, to take advantage of differing spatio-temporal properties of the processes to separate the causes of the observed change. The process parameters are estimated from geophysical models, while the remaining parameters are estimated using a Markov chain Monte Carlo scheme, designed to operate in a highperformance computing environment across multiple nodes. We validate our methods against a separate data set and compare the results to those from studies that invariably employ numerical model outputs directly. We conclude that it is possible, and insightful, to assess Antarctica’s contribution without explicit use of numerical models. Further, the results obtained here can be used to test the geophysical numerical models for which in situ data are hard to obtain.

Spatio-temporal log-Gaussian Cox process models play a central role in the analysis of spatially distributed systems in several disciplines. Yet, scalable inference remains computationally challenging both due to the high resolution modelling generally required and the analytically intractable likelihood function. Here, we exploit the sparsity structure typical of (spatially) discretised log-Gaussian Cox process models by using approximate message-passing algorithms. The proposed algorithms scale well with the state dimension and the length of the temporal horizon with moderate loss in distributional accuracy. They hence provide a flexible and faster alternative to both non-linear filtering-smoothing type algorithms and to approaches that implement the Laplace method or expectation propagation on (block) sparse latent Gaussian models. We infer the parameters of the latent Gaussian model using a structured variational Bayes approach. We demonstrate the proposed framework on simulation studies with both Gaussian and point-process observations and use it to reconstruct the conflict intensity and dynamics in Afghanistan from the WikiLeaks Afghan War Diary.

Antarctica is the world’s largest fresh-water reservoir, with the potential to raise sea levels by about 60 m. An ice sheet contributes to sea-level rise (SLR) when its rate of ice discharge and/or surface melting exceeds accumulation through snowfall. Constraining the contribution of the ice sheets to present-day SLR is vital both for coastal development and planning, and climate projections. Information on various ice sheet processes is available from several remote sensing data sets, as well as in situ data such as global positioning system data. These data have differing coverage, spatial support, temporal sampling and sensing characteristics, and thus, it is advantageous to combine them all in a single framework for estimation of the SLR contribution and the assessment of processes controlling mass exchange with the ocean. In this paper, we predict the rate of height change due to salient geophysical processes in Antarctica and use these to provide estimates of SLR contribution with associated uncertainties. We employ a multivariate spatio-temporal model, approximated as a Gaussian Markov random field, to take advantage of differing spatio-temporal properties of the processes to separate the causes of the observed change. The process parameters are estimated from geophysical models, while the remaining parameters are estimated using a Markov chain Monte Carlo scheme, designed to operate in a highperformance computing environment across multiple nodes. We validate our methods against a separate data set and compare the results to those from studies that invariably employ numerical model outputs directly. We conclude that it is possible, and insightful, to assess Antarctica’s contribution without explicit use of numerical models. Further, the results obtained here can be used to test the geophysical numerical models for which in situ data are hard to obtain.

Spatio-temporal log-Gaussian Cox process models play a central role in the analysis of spatially distributed systems in several disciplines. Yet, scalable inference remains computationally challenging both due to the high resolution modelling generally required and the analytically intractable likelihood function. Here, we exploit the sparsity structure typical of (spatially) discretised log-Gaussian Cox process models by using approximate message-passing algorithms. The proposed algorithms scale well with the state dimension and the length of the temporal horizon with moderate loss in distributional accuracy. They hence provide a flexible and faster alternative to both non-linear filtering-smoothing type algorithms and to approaches that implement the Laplace method or expectation propagation on (block) sparse latent Gaussian models. We infer the parameters of the latent Gaussian model using a structured variational Bayes approach. We demonstrate the proposed framework on simulation studies with both Gaussian and point-process observations and use it to reconstruct the conflict intensity and dynamics in Afghanistan from the WikiLeaks Afghan War Diary.

Antarctica is the world’s largest fresh-water reservoir, with the potential to raise sea levels by about 60 m. An ice sheet contributes to sea-level rise (SLR) when its rate of ice discharge and/or surface melting exceeds accumulation through snowfall. Constraining the contribution of the ice sheets to present-day SLR is vital both for coastal development and planning, and climate projections. Information on various ice sheet processes is available from several remote sensing data sets, as well as in situ data such as global positioning system data. These data have differing coverage, spatial support, temporal sampling and sensing characteristics, and thus, it is advantageous to combine them all in a single framework for estimation of the SLR contribution and the assessment of processes controlling mass exchange with the ocean. In this paper, we predict the rate of height change due to salient geophysical processes in Antarctica and use these to provide estimates of SLR contribution with associated uncertainties. We employ a multivariate spatio-temporal model, approximated as a Gaussian Markov random field, to take advantage of differing spatio-temporal properties of the processes to separate the causes of the observed change. The process parameters are estimated from geophysical models, while the remaining parameters are estimated using a Markov chain Monte Carlo scheme, designed to operate in a highperformance computing environment across multiple nodes. We validate our methods against a separate data set and compare the results to those from studies that invariably employ numerical model outputs directly. We conclude that it is possible, and insightful, to assess Antarctica’s contribution without explicit use of numerical models. Further, the results obtained here can be used to test the geophysical numerical models for which in situ data are hard to obtain.

Spatio-temporal log-Gaussian Cox process models play a central role in the analysis of spatially distributed systems in several disciplines. Yet, scalable inference remains computationally challenging both due to the high resolution modelling generally required and the analytically intractable likelihood function. Here, we exploit the sparsity structure typical of (spatially) discretised log-Gaussian Cox process models by using approximate message-passing algorithms. The proposed algorithms scale well with the state dimension and the length of the temporal horizon with moderate loss in distributional accuracy. They hence provide a flexible and faster alternative to both non-linear filtering-smoothing type algorithms and to approaches that implement the Laplace method or expectation propagation on (block) sparse latent Gaussian models. We infer the parameters of the latent Gaussian model using a structured variational Bayes approach. We demonstrate the proposed framework on simulation studies with both Gaussian and point-process observations and use it to reconstruct the conflict intensity and dynamics in Afghanistan from the WikiLeaks Afghan War Diary.

Antarctica is the world’s largest fresh-water reservoir, with the potential to raise sea levels by about 60 m. An ice sheet contributes to sea-level rise (SLR) when its rate of ice discharge and/or surface melting exceeds accumulation through snowfall. Constraining the contribution of the ice sheets to present-day SLR is vital both for coastal development and planning, and climate projections. Information on various ice sheet processes is available from several remote sensing data sets, as well as in situ data such as global positioning system data. These data have differing coverage, spatial support, temporal sampling and sensing characteristics, and thus, it is advantageous to combine them all in a single framework for estimation of the SLR contribution and the assessment of processes controlling mass exchange with the ocean. In this paper, we predict the rate of height change due to salient geophysical processes in Antarctica and use these to provide estimates of SLR contribution with associated uncertainties. We employ a multivariate spatio-temporal model, approximated as a Gaussian Markov random field, to take advantage of differing spatio-temporal properties of the processes to separate the causes of the observed change. The process parameters are estimated from geophysical models, while the remaining parameters are estimated using a Markov chain Monte Carlo scheme, designed to operate in a highperformance computing environment across multiple nodes. We validate our methods against a separate data set and compare the results to those from studies that invariably employ numerical model outputs directly. We conclude that it is possible, and insightful, to assess Antarctica’s contribution without explicit use of numerical models. Further, the results obtained here can be used to test the geophysical numerical models for which in situ data are hard to obtain.

Physical processes rarely occur in isolation, rather they influence and interact with one another. Thus, there is great benefit in modeling potential dependence between both spatial locations and different processes. It is the interaction between these two dependencies that is the focus of Genton and Kleiber's paper under discussion. We see the problem of ensuring that any multivariate spatial covariance matrix is nonnegative definite as important, but we also see it as a means to an end. That "end" is solving the scientific problem of predicting a multivariate field.

Combinations of various numerical models and datasets with diverse observation characteristics have been used to assess the mass evolution of ice sheets. As a consequence, a wide range of estimates have been produced using markedly different methodologies, data, approximation methods and model assumptions. Current attempts to reconcile these estimates using simple combination methods are unsatisfactory, as common sources of errors across different methodologies may not be accurately quantified (e.g. systematic biases in models). Here we provide a general approach which deals with this issue by considering all data sources simultaneously, and, crucially, by reducing the dependence on numerical models. The methodology is based on exploiting the different space-time characteristics of the relevant ice-sheet processes, and using statistical smoothing methods to establish the causes of the observed change. In omitting direct dependence on numerical models, the methodology provides a novel means for assessing glacio-isostatic adjustment and climate models alike, using remote-sensing datasets. This is particularly advantageous in Antarctica, where in situ measurements are difficult to obtain. We illustrate the methodology by using it to infer Antarctica's mass trend from 2003 to 2009 and produce surface mass-balance anomaly estimates to validate the RACMO2.1 regional climate model.

Atmospheric trace-gas inversion refers to any technique used to predict spatial and temporal fluxes using mole-fraction measurements and atmospheric simulations obtained from computer models. Studies to date are most often of a data-assimilation flavour, which implicitly consider univariate statistical models with the flux as the variate of interest. This univariate approach typically assumes that the flux field is either a spatially correlated Gaussian process or a spatially uncorrelated non-Gaussian process with prior expectation fixed using flux inventories (e.g., NAEI or EDGAR). Here, we extend this approach in three ways. First, we develop a bivariate model for the mole-fraction field and the flux field. The bivariate approach allows optimal prediction of both the flux field and the mole-fraction field, and it leads to significant computational savings over the univariate approach. Second, we employ a lognormal spatial process for the flux field that captures both the lognormal characteristics of the flux field (when appropriate) and its spatial dependence. Third, we propose a new, geostatistical approach to incorporate the flux inventories in our updates, such that the posterior spatial distribution of the flux field is predominantly data-driven. The approach is illustrated on a case study of methane (CH4) emissions in the United Kingdom and Ireland.

We would like to thank Marc Genton and William Kleiber (hereafter, GK) for their informative review, "Cross-covariance functions for multivariate geostatistics" (forthcoming in Statistical Science), and the editor of Statistical Science for the opportunity to contribute to the discussion. Physical processes rarely occur in isolation, rather they influence and interact with one another. Thus, there is great benefit in modelling potential dependence between both spatial locations and different processes. It is the interaction between these two dependencies that is the focus of GK. We see the problem of ensuring that the matrix given in GK-(2) is nonnegative definite (nnd) as important, but we also see it as a means to an end. That "end" is solving the scientific problem of predicting a multivariate eld of, say, temperature and rainfall, based on noisy and spatially incomplete data from weather stations in a region of interest. There is also scientific interest in the behaviour of the measures of cross-spatial dependence (e.g., cross-covariance functions), but usually spatial prediction is the ultimate goal.

Atmospheric trace-gas inversion refers to any technique used to predict spatial and temporal fluxes using mole-fraction measurements and atmospheric simulations obtained from deterministic computer models. Studies to date are most often of a data-assimilation flavour, which implicitly consider univariate statistical models with the flux as the variate of interest. This univariate approach typically assumes that the flux field is either a spatially correlated Gaussian process or a spatially uncorrelated non-Gaussian process with prior expectation fixed using flux inventories (e.g., NAEI or EDGAR in Europe). Here, we extend this approach in three ways. First, we develop a bivariate model for the mole-fraction field and the flux eld. The bivariate approach allows optimal prediction of both the flux field and the mole-fraction field, and it leads to significant computational savings over the univariate approach. Second, we employ a lognormal spatial process for the flux field that captures both the lognormal characteristics of the flux field (when appropriate) and its spatial dependence. Third, we propose a new, geostatistical approach to incorporate the flux inventories in our updates, such that the posterior spatial distribution of the flux field is predominantly data-driven. The approach is illustrated on a case study of methane (CH 4) emissions in the United Kingdom and Ireland.

Multivariate geostatistics is based on modelling all covariances between all possible combinations of two or more variables and their locations in a continuously indexed domain. Multivariate spatial covariance models need to be built with care, since any covariance matrix that is derived from such a model has to be nonnegative-definite. In this article, we develop a conditional approach for model construction. Starting with bivariate spatial covariance models, we demonstrate the approach’s generality, including its connection to regression and to multi variate models defined by spatial networks. We demonstrate the fitting of such models on a minimum-maximum temperature dataset.