> What do you mean by QUANTIZED VERSION? and CONTINUOUS INDEX, as in a continuous random variable made up from an interpolation function?

The continuous index is just a time series of a real variable. The quantized version is where you apply some thresholding function to the time series (e.g., 1 if the index is above some value for some time, 0 otherwise).

Anyway, the point is that it's probably most useful to be working with some simple aggregate function of the observational data (sea surface temperatures), not a discretized variable, or a correlation function, or something like that. It's the temperature itself that matters to people.