I happened across what I consider to be an amazing slide while “reading around” the work of Deser and colleagues. It is reproduced below, taken from Dagg and Wills:
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They are actually just conveying a definition from Hawkins and Sutton (2009), specifically from their Appendix A, Equation (1), and the paragraph following it. Specifically, internal variability is defined as the residual random variable after fitting a fourth order polynomial to a time series using ordinary least squares. The data were single members of prediction ensembles from three scenarios for each of 15 models from an IPCC study. What the source of the time series was is really not all that important to my point here. Now, quoting Hawkins and Sutton:
Each individual prediction was fit, using ordinary least squares, with a fourth-order polynomial over the years 1950–2099. The raw predictions for each model , scenario and year can be written as
where the reference temperature is denoted by , the smooth fit is represented by , and the residual (internal variability) is . The reference temperatures used were the year 2000 (Fig. 3) and the mean of the years 1971 to 2000 (for all other analyses), both of which were estimated from the smooth fits.
Okay, forget about how the smooth fit was obtained. Hawkins and Sutton used an OLS-derived polynomial, but it could have been a Bayesian smoother or some other more elaborate model-based or model-free device (e.g., splines) which delivers a curve with vanishing high order time derivatives. What’s astonishing to me is that they explicitly identify the residual as internal variability. I thought internal variability was more complicated than that. I thought interval variability was something like the time integral of variance in climate if suddenly all forcings (boundary conditions) were held at constant values, integrating out to infinity. (The transition to constancy in the conditions can be smoothed, if you like, to impede “ringing” effects.) Apparently not.
Now, if that were a model I had, I would be keenly interested in seeing what made up In particular, I’d like to further break it down into variance terms which were statistically independent of one another.
There’s nothing at all wrong with modeling things this way, even using a polynomial. But lumping things into an unexplored internal variability and leaving it at that seems, well, incomplete. Maybe there’s a good reason, like it’s already known what’s in and it’s been decided by the field to be uninteresting or not useful. Or maybe for some good reason really is the time derivative of the time integral of variance in climate if suddenly all forcings (boundary conditions) were held at constant values.
But I sure would like to know.