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. Reliable estimates of historical effective radiative forcing (ERF) are important for understanding the causes of past climate change and for constraining predictions of future warming. This study proposes a new linear-filtering method for estimating historical radiative forcing from time series of global mean surface temperature (GMST), using energy-balance models (EBMs) fitted to GMST from CO2 -quadrupling general circulation model (GCM) experiments. We show that the response of any k -box EBM can be represented as an ARMA( k , k−1 ) (autoregressive moving-average) filter. We show how, by inverting an EBM's ARMA filter representation, time series of surface temperature may be converted into radiative forcing. The method is illustrated using three-box EBM fits to two recent Earth system models from CMIP5 and CMIP6 (Coupled Model Intercomparison Project). A comparison with published results obtained using the established ERF_trans method, a purely GCM-based approach, shows that our new method gives an ERF time series that closely matches the GCM-based series (correlation of 0.83). Time series of estimated historical ERF are obtained by applying the method to a dataset of historical temperature observations. The results show that there is clear evidence of a significant increase over the historical period with an estimated forcing in 2018 of 1.45±0.504 W m−2 when derived using the two Earth system models. This method could be used in the future to attribute past climate changes to anthropogenic and natural factors and to help constrain estimates of climate sensitivity.

A new energy-balance approach to linear filtering for estimating effective radiative forcing from temperature time series