In a nonlinear climate system, the response to a forcing can be opposite to that forcing—a result a linear model without memory effect (linear Markov model, a common approach) cannot produce.  Using the Lorenz-63 system, a 3-variable nonlinear system simplified from 2D convection, we show that a linear model with memory effect (linear state-space model, SSM) can quantitatively explain this opposite response, which arises from the system’s memory effects.  To quantitatively demonstrate this, we apply weak sinusoidal forcing, construct a 32-order SSM (explained 87% variance) and decouple it into an 11-order z-subsystem and a 21-order (x,y)-subsystem.  Using this model, we: (i) accurately recover the linear frequency response function, including the steady-state response; (ii) explain the failure of a common construction of linear Markov model; and (iii) identify inherent frequencies, 1.33, 1.60 and 2.67, of the system.  For (ii), linear Markov models are commonly constructed directly using the observable vector, which inevitably entails information loss, and therefore cannot explain the opposite response of the Lorenz-63 system under weak z-forcing.  Our SSM considers this as a partially excited and partially observed linear Markov process and finds that an exponentially decaying mode explains most of the opposite response.  While demonstrating on a simple model, the method can be applied real-world ocean-atmosphere systems and analyze their climate response and natural variability.

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