How much could be gained from ocean-data assimilation if ENSO were a simple stochastic oscillator?

Assimilation of sub-surface ocean data is one of the factors that have contributed to the increase in skill of ENSO forecasting systems, but it is not well known how much. Here this question will be addressed in the context of about the simplest ENSO model possible: the stochastic oscillator,a noise-driven two-variable system. The first variable is a NiñoSST index, the second is an indicator of the sub-surface ocean. Measuring the second variable stands for ocean-data assimilation in this system. It is found that the impact of ocean data assimilation depends critically on the noise terms. That is, different choices exist that have the same properties of the Niño index, and the same predictability if the second variable is not measured, yet very different increases in predictability if sub-surface ocean data are assimilated. This illustrates the need for a proper representation of small-scale processes if one wishes to assess the predictability of the ENSO system.