Singular measures versus nondifferentiability: from the solid earth to the atmosphere and their interface (Invited)

In the 1980’s, the paradigm of Fractal Geometry popularized the fact that the ubiquitous geostatistical power laws imply nondifferentiabilty of the corresponding fractal sets, fractal functions. The prototypical examples of such scaling have flucutations Δf which change with scale Δx accordng to laws of the form Δf ≈ φΔx**H where H is the scaling exponent and φ is flux. The famous Kolmorogov turbulence law is the special case where Δf = Δv for velocity fluctuations across a distance Δx with H = 1/3 and φ = ɛ**1/3 where ɛ is the turbulent energy flux. Similarly, there is much evidence that topographic altitude fluctuations Δh are of the same form with Δf = Δh , H ≈1/2 and with φ a fundamental flux field governing topography dynamics. In both cases the basic laws are quite classical going back to 1941 (Kolmogorov; Δv) and to 1951 (Venig-Meinsz; Δh) respectively. From the above form we see that Δf/Δx ≈ φΔx**(H-1) which (when H<1) diverges as Δx tends to zero implying the divergence of the first derivative. Indeed, all (fractional) derivatives of order >H diverge. The focus on the geometry / differentiability properties puts the spotlight on the old H parameter. This has unfortunately drawn attention away from the more important consequences of our new understanding of the nonclassical flux φ as a singular multifractal measure. Over the last 25 years it has become clear that nonlinear processes that are scale invariant over wide ranges generically give rise to singular measures with statistics satisfying the relation <φ(λ)**q> ≈ λ**K(q) where λ is the resolution (defined as the ratio of the largest scale of the variability to the averaging scale), q is an arbitrary order of moment “<>” means statistical averaging, and K(q) is a scaling exponent function which characterizes the statistics of the flux at all the scales λ. Since φ(λ) is the ratio of the (Lebesque) integral of φ over a λ-resolution line, square or cube to the corrsponding length, area or volume of the integration set, such fluxes are singular with respect to Lebesgue measures (the scaling exponent K(q) - which is a cumulant generating function - diverges in the small scale limit i.e. as λ -> infinity). We give examples of such singular geomeasures ranging from ore concentrations, geopotential fields, topography, to surface and atmospheric radiances and to the state variables showing the ubiquity of singular measures throughout the geosciences. Classical geostatistics is based on point process random functions; it can easily handle nondifferentiability. However, it implicitly assumes that the relevant geomeasures are on the contrary regular with respect to Lebesgue measures. It would thus seem that real world geodata are outside the domain of application of classical geostatistics. We discuss the consequences.