Second order up-scaling : theory and an exercise with a complex photosynthesis model

Model objectives related to higher levels of description in general require the application of aggregation methods which shorten computation time but preserve the effect of fine scale variability. It is shown that in particular temporal up-scaling problems can be solved through a second order approximation of a statistical expectation operator. For the application of the second order up-scaling method one has to evaluate estimates of second order derivatives of the model functions as well as variances of fluctuating boundary conditions. Here, we propose to calculate second order derivatives on the base of look-up tables, which significantly reduces computational effort. This technique, however, should be used with care since round-offs of the model output largely affect second order terms. In order to obtain values of fine scale variances in external conditions one can simply correlate these with longer-term averages. The correctness of the correlation method is demonstrated for different irradiance and temperature time-series. After a more general derivation, the combination of methods is tested by aggregating a complex photosynthesis model in time. This exercise shows that computation time can be reduced by many orders of magnitude using the combined second order up-scaling technique. Other potentials as well as limitations of look-up tables, second order moment approximations and the statistical representation of daily climate fluctuations are discussed.