Generalised scale invariance and multiplicative processes in the atmosphere

Many geophysical fields show highly intermittent fractal structures spanning wide ranges of scale. However, few are isotropic: texture, stratification, as well as variable (scale dependent) orientation of structures is far more common. To deal with such fractals, we must generalise the idea of scale invariance beyond the familiar self-similar (or even self-affine) notions. Taking the atmosphere as our primary example (however, we also model galaxies), we outline the necessary formalism (generalised scale invariance), and show how it can be used to deal with the strongly intermittent structures which result from multiplicative (cascade type) processes concentrating matter or energy into smaller and smaller scales.We illustrate these ideas with rain data from blotting paper and radar, showing first how to directly estimate the elliptical dimension characterising the stratification, and second, how to determine universal scale-independent (invariant) codimension functions that characterise the distribution of the intense rain regions.