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We propose a new statistical reduced complexity climate model. The centerpiece of the model consists of a set of physical equations for the global climate system which we show how to cast in non-linear state space form. The parameters in the model are estimated using the method of maximum likelihood with the likelihood function being evaluated by the extended Kalman filter. Our statistical framework is based on well-established methodology and is computationally feasible. In an empirical analysis, we estimate the parameters for a data set comprising the period 1959-2022. A likelihood ratio test sheds light on the most appropriate equation for converting the level of atmospheric concentration of carbon dioxide into radiative forcing. Using the estimated model, and different future paths of greenhouse gas emissions, we project global mean surface temperature until the year 2100. Our results illustrate the potential of combining statistical modelling with physical insights to arrive at rigorous statistical analyses of the climate system.

This study presents a statistical time-domain approach for identifying transitions between climate states, referred to as breakpoints, using well-established econometric tools. We analyze a 67.1 million year record of the oxygen isotope ratio delta-O-18 derived from benthic foraminifera. The dataset is presented in Westerhold et al. (2020), where the authors use recurrence analysis to identify six climate states. Fixing the number of breakpoints to five, our procedure results in breakpoint estimates that closely align with those identified by Westerhold et al. (2020). By treating the number of breakpoints as a parameter to be estimated, we provide the statistical justification for more than five breakpoints in the time series. Further, our approach offers the advantage of constructing confidence intervals for the breakpoints, and it allows for testing the number of breakpoints present in the time series., Comment: Authors in alphabetical order

Time series analysis of delta-O-18 and delta-C-13 measurements from benthic foraminifera for purposes of paleoclimatology is challenging. The time series reach back tens of millions of years, they are relatively sparse in the early record and relatively dense in the later, the time stamps of the observations are not evenly spaced, and there are instances of multiple different observations at the same time stamp. The time series appear non-stationary over most of the historical record with clearly visible temporary trends of varying directions. In this paper, we propose a continuous-time state-space framework to analyze the time series. State space models are uniquely suited for this purpose, since they can accommodate all the challenging features mentioned above. We specify univariate models and joint bivariate models for the two time series of delta-O-18 and delta-C-13. The models are estimated using maximum likelihood by way of the Kalman filter recursions. The suite of models we consider has an interpretation as an application of the Butterworth filter. We propose model specifications that take the origin of the data from different studies into account and that allow for a partition of the total period into sub-periods reflecting different climate states. The models can be used, for example, to impute evenly time-stamped values by way of Kalman filtering. They can also be used, in future work, to analyze the relation to proxies for CO2 concentrations., Comment: Authors in alphabetical order

The global fraction of anthropogenically emitted carbon dioxide (CO$_2$) that stays in the atmosphere, the CO$_2$ airborne fraction, has been fluctuating around a constant value over the period 1959 to 2022. The consensus estimate of the airborne fraction is around $44\%$; the remaining $56\%$ is absorbed by the oceanic and terrestrials biospheres. In this study, we show that the conventional estimator of the airborne fraction, based on a ratio of changes in atmospheric CO$_2$ concentrations and CO$_2$ emissions, suffers from a number of statistical deficiencies, such as non-existence of moments and a non-Gaussian limiting distribution. We propose an alternative regression-based estimator of the airborne fraction that does not suffer from these deficiencies. We show that the regression-based estimator has a Gaussian limiting distribution and reduces estimation uncertainty substantially. Our empirical analysis leads to an estimate of the airborne fraction over 1959--2022 of $47.0\%$ ($\pm 1.1\%$; $1 \sigma$), implying a higher, and better constrained, estimate than the current consensus. Using climate model output, we show that a regression-based approach provides sensible estimates of the airborne fraction, also in future scenarios where emissions are at or near zero.

In a paper recently published in this journal, van Marle et al. (van Marle et al., 2022) introduce an interesting new data set for land use and land cover change CO2 emissions (LULCC) that they use to study whether a trend is present in the airborne fraction (AF), defined as the fraction of CO2 emissions remaining in the atmosphere. Testing the hypothesis of a trend in the AF has attracted much attention, with the overall consensus that no statistical evidence is found for a trend in the data (Knorr, 2009; Gloor et al., 2010; Raupach et al., 2014; Bennedsen et al., 2019). In their paper, van Marle et al. analyze the AF as implied by three different LULCC emissions time series (GCP, H&N, and their new data series). In a Monte Carlo simulation study based on their new LULCC emissions data, van Marle et al. find evidence of a declining trend in the AF. In this note, we argue that the statistical analysis presented in van Marle et al. can be improved in several respects. Specifically, the Monte Carlo study presented in van Marle et al. is not conducive to determine whether there is a trend in the AF. Further, we re-examine the evidence for a trend in the AF by using a variety of different statistical tests. The statistical evidence for an uninterrupted (positive or negative) trend in the airborne fraction remains mixed at best. When allowing for a break in the trend, there is some evidence for upward trends in both subsamples.