The climate is highly variable over wide ranges of scale in both space and time so that the amplitude of changes systematically depends on the scale of observations. As a consequence, climate variations recorded in time series or spatial distributions, which are produced through modelling or empirical analyses are inextricably linked to their space-time scales and is a significant part of the uncertainties in the proxy approaches. Rather than treating the variability as a limitation to our knowledge, as a distraction from mechanistic explanations and theories, in this course the variability is treated as an important, fundamental aspect of the climate dynamics that must be understood and modelled in its own right. Long considered as no more than an uninteresting spectral “background”, modern data shows that in fact it contains most of the variance.

We review techniques that make it possible to systematically analyse and model the variability of instrumental and proxy data, the inferred climate variables and the outputs of GCM’s. These analyses enable us to cover wide ranges of scale in both space and in time - and jointly in space-time - without trivializing the links between the measurements, proxies and the state variables (temperature, precipitation etc.). They promise to systematically allow us to compare model outputs with data, to understand the climate processes from small to large and from fast to slow. Specific tools that will be covered include spectral analysis, scaling fluctuation analysis, wavelets, fractals, multifractals, and stochastic modeling; we discuss corresponding software. We also include new developments in the Fractional Energy Balance Equation approach that combines energy and scale symmetries.