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Quantifying the risk of global warming exceeding critical targets such as 2.0 K requires reliable projections of uncertainty as well as best estimates of Global Mean Surface Temperature (GMST). However, uncertainty bands on GMST projections are often calculated heuristically and have several potential shortcomings. In particular, the uncertainty bands shown in IPCC plume projections of GMST are based on the distribution of GMST anomalies from climate model runs and so are strongly determined by model characteristics with little influence from observations of the real-world. Physically motivated time-series approaches are proposed based on fitting energy balance models (EBMs) to climate model outputs and observations in order to constrain future projections. It is shown that EBMs fitted to one forcing scenario will not produce reliable projections when different forcing scenarios are applied. The errors in the EBM projections can be interpreted as arising due to a discrepancy in the effective forcing felt by the model. A simple time-series approach to correcting the projections is proposed based on learning the evolution of the forcing discrepancy so that it can be projected into the future. These approaches give reliable projections of GMST when tested in a perfect model setting, and when applied to observations lead to well constrained projections with lower mean warming and narrower projection bands than previous estimates. Despite the reduced uncertainty, the lower warming leads to a greatly reduced probability of exceeding the 2.0 K warming target., Comment: 21 pages, 7 figures, 4 tables

Existing methods for diagnosing predictability in climate indices often make a number of unjustified assumptions about the climate system that can lead to misleading conclusions. We present a flexible family of state-space models capable of separating the effects of external forcing on inter-annual time scales, from long-term trends and decadal variability, short term weather noise, observational errors and changes in autocorrelation. Standard potential predictability models only estimate the fraction of the total variance in the index attributable to external forcing. In addition, our methodology allows us to partition individual seasonal means into forced, slow, fast and error components. Changes in the predictable signal within the season can also be estimated. The model can also be used in forecast mode to assess both intra- and inter-seasonal predictability. We apply the proposed methodology to a North Atlantic Oscillation index for the years 1948-2017. Around 60% of the inter-annual variance in the December-January-February mean North Atlantic Oscillation is attributable to external forcing, and 8% to trends on longer time-scales. In some years, the external forcing remains relatively constant throughout the winter season, in others it changes during the season. Skillful statistical forecasts of the December-January-February mean North Atlantic Oscillation are possible from the end of November onward and predictability extends into March. Statistical forecasts of the December-January-February mean achieve a correlation with the observations of 0.48., Comment: 16 pages, 10 figures

We develop Bayesian state space methods for modelling changes to the mean level or temporal correlation structure of an observed time series due to intermittent coupling with an unobserved process. Novel intervention methods are proposed to model the effect of repeated coupling as a single dynamic process. Latent time-varying autoregressive components are developed to model changes in the temporal correlation structure. Efficient filtering and smoothing methods are derived for the resulting class of models. We propose methods for quantifying the component of variance attributable to an unobserved process, the effect during individual coupling events, and the potential for skilful forecasts. The proposed methodology is applied to the study of winter-time variability in the dominant pattern of climate variation in the northern hemisphere, the North Atlantic Oscillation. Around 70% of the inter-annual variance in the winter (Dec-Jan-Feb) mean level is attributable to an unobserved process. Skilful forecasts for winter (Dec-Jan-Feb) mean are possible from the beginning of December., Comment: 27 pages, 8 figures, 4 tables

Numerical climate models are used to project future climate change due to both anthropogenic and natural causes. Differences between projections from different climate models are a major source of uncertainty about future climate. Emergent relationships shared by multiple climate models have the potential to constrain our uncertainty when combined with historical observations. We combine projections from 13 climate models with observational data to quantify the impact of emergent relationships on projections of future warming in the Arctic at the end of the 21st century. We propose a hierarchical Bayesian framework based on a coexchangeable representation of the relationship between climate models and the Earth system. We show how emergent constraints fit into the coexchangeable representation, and extend it to account for internal variability simulated by the models and natural variability in the Earth system. Our analysis shows that projected warming in some regions of the Arctic may be more than 2C lower and our uncertainty reduced by up to 30% when constrained by historical observations. A detailed theoretical comparison with existing multi-model projection frameworks is also provided. In particular, we show that projections may be biased if we do not account for internal variability in climate model predictions., Comment: Revised manuscript 6 figures, 2 tables

We present a framework for inference for spatial processes that have actual values imperfectly represented by data. Environmental processes represented as spatial fields, either at fixed time points, or aggregated over fixed time periods, are studied. Data from both measurements and simulations performed by complex computer models are used to infer actual values of the spatial fields. Methods from geostatistics and statistical emulation are used to explicitly capture discrepancies between a spatial field's actual and simulated values. A geostatistical model captures spatial discrepancy: the difference in spatial structure between simulated and actual values. An emulator represents the intensity discrepancy: the bias in simulated values of given intensity. Measurement error is also represented. Gaussian process priors represent each source of error, which gives an analytical expression for the posterior distribution for the actual spatial field. Actual footprints for 50 European windstorms, which represent maximum wind gust speeds on a grid over a 72-hour period, are derived from wind gust speed measurements taken at stations across Europe and output simulated from a downscaled version of the Met Office Unified Model. The derived footprints have realistic spatial structure, and gust speeds closer to the measurements than originally simulated.

We present a hybrid continuum-atomistic scheme which combines molecular dynamics (MD) simulations with on-the-fly machine learning techniques for the accurate and efficient prediction of multiscale fluidic systems. By using a Gaussian process as a surrogate model for the computationally expensive MD simulations, we use Bayesian inference to predict the system behaviour at the atomistic scale, purely by consideration of the macroscopic inputs and outputs. Whenever the uncertainty of this prediction is greater than a predetermined acceptable threshold, a new MD simulation is performed to continually augment the database, which is never required to be complete. This provides a substantial enhancement to the current generation of hybrid methods, which often require many similar atomistic simulations to be performed, discarding information after it is used once. We apply our hybrid scheme to nano-confined unsteady flow through a high-aspect-ratio converging-diverging channel, and make comparisons between the new scheme and full MD simulations for a range of uncertainty thresholds and initial databases. For low thresholds, our hybrid solution is highly accurate\,---\,within the thermal noise of a full MD simulation. As the uncertainty threshold is raised, the accuracy of our scheme decreases and the computational speed-up increases (relative to a full MD simulation), enabling the compromise between precision and efficiency to be tuned. The speed-up of our hybrid solution ranges from an order of magnitude, with no initial database, to cases where an extensive initial database ensures no new MD simulations are required.

Predictability estimates of ensemble prediction systems are uncertain due to limited numbers of past forecasts and observations. To account for such uncertainty, this paper proposes a Bayesian inferential framework that provides a simple 6-parameter representation of ensemble forecasting systems and the corresponding observations. The framework is probabilistic, and thus allows for quantifying uncertainty in predictability measures such as correlation skill and signal-to-noise ratios. It also provides a natural way to produce recalibrated probabilistic predictions from uncalibrated ensembles forecasts. The framework is used to address important questions concerning the skill of winter hindcasts of the North Atlantic Oscillation for 1992-2011 issued by the Met Office GloSea5 climate prediction system. Although there is much uncertainty in the correlation between ensemble mean and observations, there is strong evidence of skill: the 95% credible interval of the correlation coefficient of [0.19,0.68] does not overlap zero. There is also strong evidence that the forecasts are not exchangeable with the observations: With over 99% certainty, the signal-to-noise ratio of the forecasts is smaller than the signal-to-noise ratio of the observations, which suggests that raw forecasts should not be taken as representative scenarios of the observations. Forecast recalibration is thus required, which can be coherently addressed within the proposed framework., Comment: 36 pages, 10 figures

A flexible spatio-temporal model is implemented to analyse extreme extra-tropical cyclones objectively identified over the Atlantic and Europe in 6-hourly re-analyses from 1979-2009. Spatial variation in the extremal properties of the cyclones is captured using a 150 cell spatial regularisation, latitude as a covariate, and spatial random effects. The North Atlantic Oscillation (NAO) is also used as a covariate and is found to have a significant effect on intensifying extremal storm behaviour, especially over Northern Europe and the Iberian peninsula. Estimates of lower bounds on minimum sea-level pressure are typically 10-50 hPa below the minimum values observed for historical storms with largest differences occurring when the NAO index is positive., Comment: Published in at http://dx.doi.org/10.1214/14-AOAS766 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

This study considers the application of the Ignorance Score (also known as the Logarithmic Score) in the context of ensemble verification. In particular, we consider the case where an ensemble forecast is transformed to a Normal forecast distribution, and this distribution is evaluated by the Ignorance Score. It is shown that the standard Ignorance score is biased with respect to the ensemble size, such that larger ensembles yield systematically better expected scores. A new estimator of the Ignorance score is derived which is unbiased with respect to the ensemble size. In an application to seasonal climate predictions it is shown that the standard Ignorance score assigns better expected scores to simple climatological ensembles or biased ensembles that have many members, than to physical dynamical and unbiased ensembles with fewer members. By contrast, the new bias-corrected Ignorance score ranks the physical dynamical and unbiased ensembles better than the climatological and biased ones, independent of ensemble size. It is shown that the unbiased estimator has smaller estimator variance and error than the standard estimator, and that it is a fair verification score, which is optimized if the ensemble members are statistically consistent with the observations. The finite ensemble bias of ensemble verification scores is discussed more broadly. It is argued that a bias-correction is appropriate when forecast systems with different ensemble sizes are compared, and when an evaluation of the underlying distribution of the ensemble is of interest; possible applications to unbiased parameter estimation are discussed., Comment: 31 pages, 9 figures