It is now generally appreciated in the weather and climate research communities that under suitable "coarse-graining", a high-order chaotic nonlinear dynamical system can sometimes be consistently and accurately approximated as a lower-order stochastically forced damped linear system. The model parameters of this linear system, and also the statistical parameters of the stochastic forcing, can be estimated directly from the zero-lag and time-lag statistics of the full system using Fluctuation-Dissipation relationships. We have shown that the statistics of the full system need not be Gaussian for these relationships to hold. Such "Linear Inverse Models" (LIMs) have been demonstrated to be competitive with GCMs in modeling and predicting subseasonal to interdecadal scale variations in the tropics and the northern hemisphere. As such they represent attractive tools for assessing and diagnosing predictability on these time scales.

In this approximation, the predictable signal is associated with the deterministic linear dynamics of the LIM, and the forecast error with the unpredictable stochastic noise. For any forecast lead time, an average signal to noise ratio can be estimated directly at each grid point from the LIM's parameters, and converted into a potential average forecast anomaly correlation skill score at the point. Such a map of potential average skill can then be compared with a map of actual average skill to assess the potential for further skill improvements. We have generated and compared such pairs of maps for seasonal Tropical SST forecasts, weekly tropical diabatic heating forecasts, and weekly northern hemispheric 750 mb streamfunction forecasts for the past 50+ years. The single most revealing result from these comparisons is that the actual skill maps, though somewhat weaker in magnitude, are strikingly similar in pattern to the maps of the potential skill. It will be argued that this result places important constraints on further skill improvements on these time scales even using comprehensive NWP and climate models. The value of LIMs in identifying, through a singular vector analysis, special initial structures from which relatively high skill forecasts are possible will also be discussed.