On statistical approaches to generate Level 3 products from satellite remote sensing retrievals

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.Comment: 28 pages, 10 figures, 4 table

A technique for improving conflict alerting performance in the context of runway incursions

An effective solution to the problem of runway incursions is long overdue. To date, an average of a thousand incursions are registered yearly in the United States alone, with similar figures registed in Europe. Installing a system on-board aircraft capable of providing an alert in the case of a runway incursion has the potential of significantly reducing this. As with any conflict detection and alerting system, the difficulty lies in the fine-tuning of the parameters which define a conflict, in effect resulting in finding the right trade-off between false and missed detections and associated alerts. This is an important consideration in the design of any conflict detection system and is key in the context of runway incursion alerting where aircraft would be travelling at high speed and in close proximity of eachother. This paper addresses this problem by providing an assessement on the effects of false and missed detections in the event of a runway incursion and provides mathematical tools for tuning the conflict detection boundaries.peer-reviewe

Non-Gaussian bivariate modelling with application to atmospheric trace-gas inversion

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.Comment: 45 pages, 7 figure

Multi-Scale Process Modelling and Distributed Computation for Spatial Data

Recent years have seen a huge development in spatial modelling and prediction methodology, driven by the increased availability of remote-sensing data and the reduced cost of distributed-processing technology. It is well known that modelling and prediction using infinite-dimensional process models is not possible with large data sets, and that both approximate models and, often, approximate-inference methods, are needed. The problem of fitting simple global spatial models to large data sets has been solved through the likes of multi-resolution approximations and nearest-neighbour techniques. Here we tackle the next challenge, that of fitting complex, nonstationary, multi-scale models to large data sets. We propose doing this through the use of superpositions of spatial processes with increasing spatial scale and increasing degrees of nonstationarity. Computation is facilitated through the use of Gaussian Markov random fields and parallel Markov chain Monte Carlo based on graph colouring. The resulting model allows for both distributed computing and distributed data. Importantly, it provides opportunities for genuine model and data scaleability and yet is still able to borrow strength across large spatial scales. We illustrate a two-scale version on a data set of sea-surface temperature containing on the order of one million observations, and compare our approach to state-of-the-art spatial modelling and prediction methods.Comment: 33 pages, 10 figures, 1 tabl

Modeling Nonstationary and Asymmetric Multivariate Spatial Covariances via Deformations

Multivariate spatial-statistical models are often used when modeling environmental and socio-demographic processes. The most commonly used models for multivariate spatial covariances assume both stationarity and symmetry for the cross-covariances, but these assumptions are rarely tenable in practice. In this article we introduce a new and highly flexible class of nonstationary and asymmetric multivariate spatial covariance models that are constructed by modeling the simpler and more familiar stationary and symmetric multivariate covariances on a warped domain. Inspired by recent developments in the univariate case, we propose modeling the warping function as a composition of a number of simple injective warping functions in a deep-learning framework. Importantly, covariance-model validity is guaranteed by construction. We establish the types of warpings that allow for cross-covariance symmetry and asymmetry, and we use likelihood-based methods for inference that are computationally efficient. The utility of this new class of models is shown through two data illustrations: a simulation study on nonstationary data and an application on ocean temperatures at two different depths

From Many to One: Consensus Inference in a MIP

A Model Intercomparison Project (MIP) consists of teams who each estimate the same underlying quantity (e.g., temperature projections to the year 2070), and the spread of the estimates indicates their uncertainty. It recognizes that a community of scientists will not agree completely but that there is value in looking for a consensus and information in the range of disagreement. A simple average of the teams' outputs gives a consensus estimate, but it does not recognize that some outputs are more variable than others. Statistical analysis of variance (ANOVA) models offer a way to obtain a weighted consensus estimate of outputs with a variance that is the smallest possible and hence the tightest possible 'one-sigma' and 'two-sigma' intervals. Modulo dependence between MIP outputs, the ANOVA approach weights a team's output inversely proportional to its variation. When external verification data are available for evaluating the fidelity of each MIP output, ANOVA weights can also provide a prior distribution for Bayesian Model Averaging to yield a consensus estimate. We use a MIP of carbon dioxide flux inversions to illustrate the ANOVA-based weighting and subsequent consensus inferences

Neural Point Estimation for Fast Optimal Likelihood-Free Inference

Neural point estimators are neural networks that map data to parameter point estimates. They are fast, likelihood free and, due to their amortised nature, amenable to fast bootstrap-based uncertainty quantification. In this paper, we aim to increase the awareness of statisticians to this relatively new inferential tool, and to facilitate its adoption by providing user-friendly open-source software. We also give attention to the ubiquitous problem of making inference from replicated data, which we address in the neural setting using permutation-invariant neural networks. Through extensive simulation studies we show that these neural point estimators can quickly and optimally (in a Bayes sense) estimate parameters in weakly-identified and highly-parameterised models with relative ease. We demonstrate their applicability through an analysis of extreme sea-surface temperature in the Red Sea where, after training, we obtain parameter estimates and bootstrap-based confidence intervals from hundreds of spatial fields in a fraction of a second