Optimal Estimation of Stochastic Energy Balance Model Parameters

Abstract

This study has developed a rigorous and efficient maximum likelihood method for estimating the parameters in stochastic energy balance models (with any k > 0 number of boxes) given time series of surface temperature and top-of-the-atmosphere net downward radiative flux. The method works by finding a state-space representation of the linear dynamic system and evaluating the likelihood recursively via the Kalman filter. Confidence intervals for estimated parameters are straightforward to construct in the maximum likelihood framework, and information criteria may be used to choose an optimal number of boxes for parsimonious k-box emulation of atmosphere-ocean general circulation models (AOGCMs). In addition to estimating model parameters the method enables hidden state estimation for the unobservable boxes corresponding to the deep ocean, and also enables noise filtering for observations of surface temperature. Feasibility, reliability and performance of the proposed method are demonstrated in a simulation study. In order to obtain a set of optimal k-box emulators, models are fitted to the 4×CO2 step responses of 16 AOGCMs in CMIP5. It is found that for all 16 AOGCMs three boxes are required for optimal k-box emulation. The number of boxes, k, is found to influence, sometimes strongly, the impulse responses of the fitted models.

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

“…where κ 1 denotes the "climate feedback" parameter, which is conventionally denoted by λ. The notation used here is consistent with Fredriksen and Rypdal (2017) and Cummins et al (2020). Secondly, a system of k vertically stacked boxes recreates the thermal inertia of the ocean mixed layer and deep ocean, determining the characteristic timescales over which the response unfolds (e.g.…”

“…Maximum-likelihood parameter estimates. For descriptions of all model parameters, see Table 1 of Cummins et al (2020). Note that parameters γ and σ η are not used in this study.…”

“…2). It should however be noted that, under forcing scenarios less drastic than 4 × CO 2 , fitting of exponential response functions can suffer from ill-conditioning (De Groen and De Moor, 1987;Kaufmann, 2003). Of arguably greater concern is uncertainty due to inter-model variation between GCMs, which can be seen as a measure of confidence in a particular GCM's ability to represent the true climate.…”

“…The step response of the k-box EBM is a linear combination of saturating exponentials (Geoffroy et al, 2013;Tsutsui, 2020;Cummins et al, 2020). The discrete-time impulse response R(v) is a linear function of the step response and is a sum of decaying exponentials:…”

“…The dashed line in the middle is an abbreviation of the intervening boxes. SeeCummins et al (2020) for more discussion.…”