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.

We present a hierarchical Bayesian method for atmospheric trace gas inversions. This method is used to estimate emissions of trace gases as well as "hyper-parameters" that characterize the probability density functions (PDFs) of the a priori emissions and model-measurement covariances. By exploring the space of "uncertainties in uncertainties", we show that the hierarchical method results in a more complete estimation of emissions and their uncertainties than traditional Bayesian inversions, which rely heavily on expert judgment. We present an analysis that shows the effect of including hyper-parameters, which are themselves informed by the data, and show that this method can serve to reduce the effect of errors in assumptions made about the a priori emissions and model-measurement uncertainties. We then apply this method to the estimation of sulfur hexafluoride (SF₆) emissions over 2012 for the regions surrounding four Advanced Global Atmospheric Gases Experiment (AGAGE) stations. We find that improper accounting of model representation uncertainties, in particular, can lead to the derivation of emissions and associated uncertainties that are unrealistic and show that those derived using the hierarchical method are likely to be more representative of the true uncertainties in the system. We demonstrate through this SF₆ case study that this method is less sensitive to outliers in the data and to subjective assumptions about a priori emissions and model-measurement uncertainties than traditional methods.

Accurate accounting of emissions and removals of CO₂ is critical for the planning and verification of emission reduction targets in support of the Paris Agreement. Here, we present a pilot dataset of country-specific net carbon exchange (NCE; fossil plus terrestrial ecosystem fluxes) and terrestrial carbon stock changes aimed at informing countries' carbon budgets. These estimates are based on “top-down” NCE outputs from the v10 Orbiting Carbon Observatory (OCO-2) modeling intercomparison project (MIP), wherein an ensemble of inverse modeling groups conducted standardized experiments assimilating OCO-2 column-averaged dry-air mole fraction ($X_{{\mathrm{CO}}_{\mathrm{2}}}$ ) retrievals (ACOS v10), in situ CO₂ measurements or combinations of these data. The v10 OCO-2 MIP NCE estimates are combined with “bottom-up” estimates of fossil fuel emissions and lateral carbon fluxes to estimate changes in terrestrial carbon stocks, which are impacted by anthropogenic and natural drivers. These flux and stock change estimates are reported annually (2015–2020) as both a global 1^∘ × 1^∘ gridded dataset and a country-level dataset and are available for download from the Committee on Earth Observation Satellites' (CEOS) website: https://doi.org/10.48588/npf6-sw92 (Byrne et al., 2022). Across the v10 OCO-2 MIP experiments, we obtain increases in the ensemble median terrestrial carbon stocks of 3.29–4.58 Pg CO₂ yr⁻¹ (0.90–1.25 Pg C yr⁻¹). This is a result of broad increases in terrestrial carbon stocks across the northern extratropics, while the tropics generally have stock losses but with considerable regional variability and differences between v10 OCO-2 MIP experiments. We discuss the state of the science for tracking emissions and removals using top-down methods, including current limitations and future developments towards top-down monitoring and verification systems.

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.

Modern conflicts are characterized by an ever increasing use of information and sensing technology, resulting in vast amounts of high resolution data. Modelling and prediction of conflict, however, remain challenging tasks due to the heterogeneous and dynamic nature of the data typically available. Here we propose the use of dynamic spatiotemporal modelling tools for the identification of complex underlying processes in conflict, such as diffusion, relocation, heterogeneous escalation, and volatility. Using ideas from statistics, signal processing, and ecology, we provide a predictive framework able to assimilate data and give confidence estimates on the predictions. We demonstrate our methods on the WikiLeaks Afghan War Diary. Our results show that the approach allows deeper insights into conflict dynamics and allows a strikingly statistically accurate forward prediction of armed opposition group activity in 2010, based solely on data from previous years.