Multifractal analysis and simulation techniques and turbulent fields

In turbulent and other nonlinear geophysical processes, extreme variability is built up multiplicatively from large to small scales via cascade mechanisms generally leading to (canonical) universal multifractal fields. This thesis clarifies the distinction between "bare" and "dressed" cascade processes, the classification of multifractal singularities and the related nontrivial problem of sampling and estimation of their scale properties. It then describes the first data analysis techniques applicable to these highly variable multifractals: previously existing techniques are limited to much calmer geometric or microcanonical multifractals. The two techniques developed here (Probability Distribution Multiple Scaling and Double Trace Moment) are analyzed theoretically and numerically and are tested on turbulent and topographic data, providing the first estimates of their universal multifractal parameters, the Levy index $ alpha$ which characterizes the degree of multifractality, and C$ sb1$ (the codimension of the mean singularity) which characterizes the mean inhomogeneity of the process