Downscaling and generating the missing half of geophysical fields with blunt extension of discrete Universal Multifractal cascades
Résumé
Scale issues are ubiquitous in geosciences and particularly in atmospheric sciences, notably for rainfall. This has been often illustrated with the help of discrete multiplicative cascades, whose tree structure is rather intuitive and pedagogic, but has problematic consequences in particular with respect to translation invariance. In spite of these drastic limitations, discrete cascades have also been used for engineering applications. Fortunately, continuous in scale cascades had enabled to straightforwardly overcome these difficulties and link them to pushback transforms of fields and pull forward transforms of measures, i.e. their transforms under change of space-time scale that could be not only continuous, but also infinitesimal. This is essential for generating fractional differential equations.
In this talk, we discuss a recently introduced approach that is based on the parsimonious framework of Universal Multifractals (UM) and enables to tackle the scale invariance issue while preserving the simple structure of discrete cascades. It basically consists in smoothing at each cascade step the random multiplicative increments with the help of a geometric interpolation over a moving window whose size is tailored to the resolution to preserve scale invariance. Theoretically expected multifractal behaviour is theoretically derived and numerically confirmed in 1D, 2D and 2D + time.
Finally, straightforward applications of this framework to two very common issues in geosciences are discussed through illustrations with rainfall fields. The first one is downscaling to generate ensemble of realistic realizations of rainfall fields at resolutions higher than observed or simulated. The second one is to reconstruct the missing half of a field. Considering applications in 2D+time, it is a needed step towards nowcasting of rainfall fields.