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

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

The estimation of historical radiative forcing, a measure of the net change in the energy balance of the climate system in response to an external perturbation, is a matter of strong scientific interest, as evidenced by the dedication of a whole chapter to this topic in the most recent assessment report from the Intergovernmental Panel on Climate Change (IPCC)

The term radiative forcing refers to a change in Earth's energy balance relative to some predefined baseline value, usually chosen to represent pre-industrial conditions. In this study we use the effective radiative forcing (ERF), whose definition is given in

It is infeasible to calculate historical radiative forcing from observational data using the raw definition of ERF, as relevant climate variables, in particular top-of-the-atmosphere (TOA) net downward radiative flux, were unobserved for most of the historical period. Techniques have therefore been developed for diagnosing radiative forcing from general circulation model (GCM) experiments.

Alternative approaches, based on simple climate models, have been used to estimate historical radiative forcing from the observational record

The present study is motivated by the potential application of simple climate models and forcing estimation techniques to the detection and attribution problem. Simple climate models have previously been used for detection and attribution of changes in GMST

In the context of detection and attribution, surface temperature is an observable proxy for radiative forcing, which is itself not directly observable. The suitability of this proxy for regression analysis is reduced by two artefacts of the climate system's thermal inertia: (i) a delayed temperature response to changes in radiative forcing and (ii) strong temporal autocorrelation in natural temperature variability. Within the framework of linear impulse-response models, instantaneous surface temperature is simply a convolution of previous changes in radiative forcing

This paper is a proof-of-concept study in which we develop a method for performing the proposed deconvolution of temperature time series using

Here we propose a novel application of the three-box EBM filter in its inverted form.

An energy-balance model (EBM) is a simple climate model where the time evolution of global temperatures is explained by changes in Earth's radiative imbalance. The simplest class of EBM, known as the

Firstly, a linear relation between GMST anomaly

Secondly, a system of

To account for temporal variation in the relationship between

Vertical layout of the boxes in the

For a time-invariant linear system (such as the

Note that the above result for

Three-box model fitted values. Panel

Table

Maximum-likelihood parameter estimates. For descriptions of all model parameters, see Table 1 of

Note that ARMA models can alternatively be estimated using Bayesian inference

Numerical estimates of the ARMA coefficients enable, in theory, conversion of time series of surface temperatures into corresponding series of radiative forcings. The properties of this proposed temperature-forcing conversion were investigated using CMIP6 historical runs from the HadGEM3-GC3.1-LL climate model.

Model surface temperatures from 1850 to 2014 were averaged annually (January–December), globally and over four ensemble members. Anomalies were calculated by subtracting from the whole series the mean absolute temperature in the first 50 years (1850–1899). The final temperature series was fed into the fitted HadGEM3-GC3.1-LL ARMA filter using the digital filter implementation in the “signal” package in R

The resulting filtered forcing series was compared (see Fig.

Only one run was available at the time of submission.

using the ERF_trans method described inComparison of forcing series estimated from HadGEM3-GC3.1-LL simulations of the historical period. In panel

Results from the two methods agree strongly, and the filtered forcing series explains a majority of the variance in the ERF_trans series: the coefficient of determination is

The observed discrepancy in inferred volcanic forcing is not entirely unexpected. The ARMA filter is derived from a three-box EBM which is known to be unable to resolve temperature responses on timescales significantly shorter than 1 year. Because the data used here are annual averages, there is also scope for error due to discretization, as a volcanic eruption might occur earlier or later in a given year. Finally, it may also be the case that the GCM's temperature response to volcanic forcing deviates from the linearity assumption of the ARMA filter.

By applying the box-model ARMA filters to time series of historical surface temperatures, we can obtain series of estimated historical forcings. The Cowtan and Way 2.0 (CW2.0) historical temperature series is an updated version of the dataset described in

The version used was the latest available at the time of submission.

, yielding estimated historical forcing series for the period 1850–2018 (see Fig.HadGEM3-GC3.1-LL three-box ARMA filter reconstruction of historical radiative forcing. Panel

It can be seen from panel (d) of Fig.

The reconstructed forcing series are very noisy, since the natural variability which contaminates historical surface temperatures is amplified by the ARMA filter. However, unlike noise in the temperature series, the filtered noise is essentially uncorrelated in time. This follows from the fact that the three-box EBM successfully accounts for the thermal inertia (memory) of the system. The white-noise-like properties of natural variability in the filtered forcing series mean that the long-term trend can be extracted using regression techniques. A generalized additive model (GAM) was fitted to the estimated forcing series using the “mgcv” package in R

The filtered forcing series and its subsequent decomposition into signal and noise are subject to multiple sources of uncertainty which must be taken into consideration. As well as the (substantial) internal climate variability, there is observational uncertainty in the historical temperature series and parameter uncertainty in the fitted impulse responses. Given a standard assumption of zero-mean errors, the aggregate noise from observational error and internal climate variability should be well accounted for by GAM regression smoothing. Uncertainty in the fitted impulse responses is also of limited concern: since the EBMs were fitted to abrupt

While we argue that the use of a post hoc GAM regression is reasonable for the reasons given above, a more integrated approach to uncertainty quantification in future analyses might be achieved using Bayesian methods. A Bayesian alternative to GAM smoothing of the filtered forcing series is the “latent-force model” approach

A framework has been developed for estimating radiative forcing from time series of surface temperatures using GCM-calibrated

This study has developed a method which has the potential to improve the detection and attribution of past temperature changes by comparing patterns of forcings rather than temperatures. Future work will examine how performance of the method generalizes to the other GCMs in CMIP6 and will address in more detail the question of uncertainty quantification. By combining ARMA filter estimates of historical radiative forcing with known observational constraints, it may be possible to further constrain climate sensitivity metrics, such as ECS and TCR, and hence constrain projections of future warming.

The step response of the

Note that, in the statistics literature, the term “ARMA process” generally refers to an ARMA filter driven by a Gaussian white-noise input. For more information about ARMA models and the backshift operator, see

The HadGEM2-ES and HadGEM3-GC3.1-LL surface temperature and TOA radiative flux datasets are available for download from Earth System Grid Federation (ESGF) portals, e.g.

All authors contributed to the development of the statistical methodology and the interpretation of the results. DPC performed the data analysis and was responsible for writing the paper. DBS and PAS proofread and edited the paper.

The authors declare that they have no conflict of interest.

We are grateful to
Chris E. Forest and the anonymous referees for their useful comments. We thank Timothy Andrews of the Met Office Hadley Centre for providing forcing data for the HadGEM3-GC3.1-LL model, originally calculated in

This paper was edited by Chris Forest and reviewed by four anonymous referees.