The most predictable component of a linear stochastic model

It is known that special combinations of variables can have more predictability than any variable in the combination. While such combinations can be obtained numerically in specific cases, this leaves broader questions unanswered. For example, given the dynamics of a linear stochastic model, what is the maximum predictability? What is its structure? What stochastic forcing maximizes predictability? This paper answers these questions. Specifically, this paper derives an upper bound on the predictability of linear combinations of variables. This bound is achieved when stochastic forcing correlates perfectly between eigenmodes. In a certain limit, the structure of the most predictable combination can be derived analytically. This structure is called the Pascal Mode due to its relation to Pascal's Triangle in a special case. These results provide a new perspective on observation-based climate predictability estimates. For instance, the most predictable component of monthly North Atlantic sea surface temperature is only modestly more predictable than the least-damped eigenmode. This aligns with the theoretical findings, as neither the stochastic forcing nor the dynamical eigenvalues are tailored to enhance predictability. The Pacific shows more predictability relative to the least-damped mode, but this increase is an order of magnitude smaller than the theoretical limit given the dynamics.