The climate response of nonlinear atmosphere-ocean systems to constant external forcing is often dominated by a linear component, which is characterized by the steady-state linear response matrix L. In this study, we use the Lorenz-63 system, a simple chaotic system, as a unified testbed to assess the errors of a few unforced-data-based algorithms for estimating the L of the Lorenz-63 system and to explore potential strategies for their improvement.
We find that the linear inverse modeling (LIM) and dynamic mode decomposition (DMD) exhibit relatively large errors at small lag times, compared with direct simulations, possibly because the linear Markov assumption embedded in both methods performs poorly. We also find that fluctuation-dissipation theorem (FDT), which does not rely on the linear Markov assumption, gives a negative (3,3) element of the response matrix L—a result that agrees with direct forced simulations—implying that a weak constant z-forcing counterintuitively drives an opposite response. By systematically assessing these errors, we aim to identify their sources and develop strategies to improve these algorithms for applications in real-world atmosphere-ocean systems. Accurately computing L from unforced data will ultimately offer a cost-effective pathway to diagnose climate sensitivity without expensive forced simulations. Such matrix L also helps us effectively conduct experiments about climate variability and extremes under climate change.