Linear Response and Optimal Fingerprinting for Nonautonomous Systems

TL;DR

This paper develops a unified framework linking linear response theory, pullback measures, and optimal fingerprinting for nonautonomous systems, providing explicit linear response formulas for time-dependent Markov chains and diffusions via Green-Kubo-type constructions. It extends optimal fingerprinting to time-dependent backgrounds, enabling attribution of observed anomalies to acting forcings across multiple time slices, with solutions expressed in a nonautonomous, causality-respecting form. The authors validate the theory using a time-modulated Ghil-Sellers energy balance model, implementing a Markov-state approach via Ulam’s method and demonstrating accurate prediction of CO$_2$- and aerosol-forcing responses despite coarse graining. The work broadens climate-change attribution to nonstationary baselines and suggests broader applicability to neuroscience, finance, and other complex systems where time-dependent reference dynamics are critical. Key results include explicit expressions for $ u^{(1)}(t)$ and $ ho^{(1)}(t)$, a nonstationary OFM framework, and robust numerical demonstrations of pullback convergence and fingerprinting performance under time-varying forcings.

Abstract

We provide a link between response theory, pullback measures, and optimal fingerprinting method that paves the way for a) predicting the impact of acting forcings on time-dependent systems and b) attributing observed anomalies to acting forcings when the reference state in not time-independent. We first derive formulas for linear response theory for time-dependent Markov chains and diffusions processes. We discuss existence, uniqueness, and differentiability of the pullback measure under general (not necessarily slow or periodic) perturbations of the transition kernels. An explicit Green-Kubo-type formula for the linear response is derived. We analyze in detail the case of periodic reference dynamics, where the unperturbed pullback attractor is periodic but the response is generally not. Our formulas reduce to those of classic linear response if one considers a reference autonomous state. Finally, we show that our results allow for extending the theory of optimal fingerprinting for detection and attribution of climate change (or change in any complex system) for the case of time-dependent background state and for the case where the optimal solution is sought for multiple time slices at the same time. We provide strong numerical support for the findings by applying our theory to a modified version of the Ghil-Sellers energy balance model where we include explicit time dependence in the reference state as a result of natural forcings. We verify the accuracy of response theory in predicting the impact of increases of $CO_2$ in the temperature field even when we discretize the system using Markov state modelling approach. Additionally, we consider a more complex modelling scenario where a localized aerosol forcing is also included in the system and show that the optimal fingerprinting method developed here is able to attribute the climate change signal to the acting forcings.

Figures (3)

• Figure 1: a) Ensemble average of simulations performed considering exclusively the sun spots cycle and volcanic eruptions as forcings to the system. b) Logarithm of the probability distribution of the yearly values of $T_{AVE}$ and $\Delta T$ for the reference time-dependent system. We have used 10000 ensemble members. The Voronoi tessellation used for constructing the reduced Markov chain is shown. The arrows provide a qualitative indication of the response of the system to the occurrence of stronger (thick arrows) or weaker (thin arrows) volcanic eruptions. c) Ensemble average of $T_{AVE}$ (light blue) and $\Delta T$ (orange) under the anthropogenic forcing scenario where $\bar{m}$ perturbed as according to the modulation $n(t)$ and corresponding predictions (blue and red, respectvely) obtained using the reduced Markov model constructed with the tessellation shown in b).
• Figure 2: Optimal fingerprinting for detection and attribution of climate change for a nonautonomous reference state. a) Average value $\pm$ 2 standard deviations computed across the ensemble simulations of the weighting factor the $CO_2$ ($\beta_{CO_2}$) and aerosolos ($\beta_A$) fingerprints. The corresponding temporal modulations of the forcing $n(t)$ (cor CO$_2$) and $\gamma(t)$ (for aersolos) are shown in the insets. b) Scatter plot of the $\beta$ factors with indication of the 1 and 2 $\sigma$'s confidence region at the time horizon T=50 $y$. c) Same as b), for for T=100 $y$. d) Same as b), for for T=150 $y$. e): Same as b), for T=200 $y$.
• Figure 3: Optimal fingerprinting for detection and attribution of climate change for a nonautonomous reference state. a) Average value $\pm$ 2 standard deviations computed across the ensemble simulations of the weighting factor the $CO_2$ ($\beta_{CO_2}$) and aerosolos ($\beta_A$) fingerprints. The corresponding temporal modulations of the forcing $n(t)$ (cor CO$_2$) and $\gamma(t)$ (for aersolos) are shown in the insets. b) Scatter plot of the $\beta$ factors with indication of the 1 and 2 $\sigma$'s confidence region at the time horizon T=50 $y$. c) Same as b), for for T=100 $y$. d) Same as b), for for T=150 $y$. e): Same as b), for T=200 $y$.