Efficient Estimation of Climate State and Its Uncertainty Using Kalman Filtering with Application to Policy Thresholds and Volcanism

Abstract We present the energy-balance model–Kalman filter (EBM–KF), a hybrid model projecting and assimilating the global mean surface temperature (GMST) and ocean heat content anomaly (OHCA). It combines an annual energy-balance model (difference equations) with 17 parameters drawn from the literature and a statistical extended Kalman filter assimilating GMST and OHCA, either observed time series or simulated by Earth system models. Our motivation is to create an efficient and natural estimator of the climate state and its uncertainty, which we believe to be Gaussian at a global scale. We illustrate four applications: 1) EBM–KF generates a similar estimate to the 30-yr time-averaged climate state 15 years sooner, or a model-simulated hindcasts’ annual ensemble average, depending on whether volcanic forcing is filtered or not; 2) EBM–KF conveniently assesses annually likelihoods of crossing a policy threshold. For example, based on temperature records up to the end of 2023, p = 0.0017 indicates that the climate state was 1.5°C over preindustrial, but there is a 16% likelihood that the GMST in 2023 itself could have been over that threshold; 3) a variant of the EBM–KF also approximates the spread of an entire climate model large ensemble using only one or a few ensemble members; and 4) all variants of the EBM–KF are sufficiently fast to allow thorough sampling from non-Gaussian probabilistic futures, e.g., the impact of rare but significant volcanic eruptions. This sampling with the EBM–KF better determines how future volcanism may affect when policy thresholds will be crossed and what an ensemble with thousands of members exploring future intermittent volcanism reveals. Significance Statement The global average of Earth’s historical climate over the past 150 years can be explained by a thermal/radiation physical equation involving a small number of constants, atmospheric carbon dioxide (CO 2 ) concentration, anthropogenic clouds, and volcanic emissions. Global mean surface temperature measurements vary around this climate state within a consistent normal distribution. This physical equation and statistical depiction allowed us to construct a simple model that can rapidly estimate the uncertainty in Earth’s current climate, aid in policy discussions, and provide an alternative for some applications to expensive ensemble modeling.