$m$ and $z$]{} at a lag as long as 30 days (Figure 8b). One can expect that at longer lag timescales, $\tilde{m}$ and $z$ eventually become decorrelated and thus Equation 8 will be valid, but it can also be expected that with such long lag time, the margin of error will be large so that the estimation is uninformative. Inspired by the observation that the strength of the mean-state-independent eddy forcing vanishes at the low-frequency limit (Figure 9), here we propose a new method to bypass this issue. Multiplied by ${Z^{*}}/{(ZZ^{*})}$ on both sides, where $Z^{*}$ denotes the conjugate of $Z$, Equation 3 becomes:

$$\renewcommand{\theequation}{9}
\frac{MZ^*}{ZZ^*} = \frac{\tilde{M}Z^*}{ZZ^*} + b$$

Using the LRF, the real component of the first term on the right hand side can be explicitly calculated and is found to be negligible at the low-frequency limit (Figure 12). As a result, the feedback strength equals the real component of the left hand side of Equation 9 at the lowest frequencies, which can be calculated as the regression coefficient of low-pass filtered $m$ on low-pass filtered $z$. [In practice, Lanczos filtering is applied with the number of weights covering the length of four times of the cut-off periods.]{} 