Short-term Predictions - Climate Change: Identification and Projections - page 122

Whatever the transfer-state conversion adopted for G ( s ), we decided to
estimate the state of the model by optimal statistical filtering (Kalman). To
do so, as has already been done [HAS 76], we enhance the structure of the
model with a noise of measurement v and random turbulence w , with
respective power spectra r and q :

TTGsQwv =+ ()( ++ )

G E

We accept that noises of measurement v are independent from one year to
the next, and we assess their standard deviation at around 0.05 ° C. Hence
r = 0.0025.

The noise w added to the inputs Q is a catch-all. Firstly, it models the
causes of internal variability. These are characterized by a zero power
around frequency 0, due to the very principle of energy balance models,
which cannot have long-term “missing heat”. It also models measurement
errors on forcing inputs u i , where bias and calibration errors this time
contribute to the zero frequency. Finally, for lack of anything better, we
accept that the spectra of w is constant over a broad frequency band,
determining its spectra q by using the following theoretical equality:

∞

∫

2

G Vqgt t

= () + r

0

where V G is the variance observed in the output error of the model and g ( t ) the
impulse response to the transmittance G ( s ). In the end, the triplet
[ G ( s ) , r, q ] contains all the data necessary for determining a state estimation
filter. This will be statistically optimal only if v and w are indeed white noises
of spectra r and q . In any case, whether optimal or not, it leads to a stable state
observer, according to a rational procedure where no degree of freedom is left
at the disposal of its designer: once again, the input-output data speaks for
itself, without being oriented towards any presumption whatsoever.

The observer obtained in this way operates over the whole of the
identification period. After which, from the prediction start date, the model
continues to simulate, initialized by the estimated state and driven by
anticipated input data.

All these technical operations were guided by a simple objective: linking
the current observed temperature to long-term predictions.