Statistical modelling of extreme rainfall in Madeira Island

Extreme rainfall events have triggered a significant number of flash floods in Madeira Island along its past and recent history. Madeira is a volcanic island where the spatial rainfall distribution is strongly affected by its rugged topography. In this thesis, annual maximum of daily rainfall data from 25 rain gauge stations located in Madeira Island were modelled by the generalised extreme value distribution. Also, the hypothesis of a Gumbel distribution was tested by two methods and the existence of a linear trend in both distributions parameters was analysed. Estimates for the 50– and 100–year return levels were also obtained. Still in an univariate context, the assumption that a distribution function belongs to the domain of attraction of an extreme value distribution for monthly maximum rainfall data was tested for the rainy season. The available data was then analysed in order to find the most suitable domain of attraction for the sampled distribution. In a different approach, a search for thresholds was also performed for daily rainfall values through a graphical analysis. In a multivariate context, a study was made on the dependence between extreme rainfall values from the considered stations based on Kendall’s τ measure. This study suggests the influence of factors such as altitude, slope orientation, distance between stations and their proximity of the sea on the spatial distribution of extreme rainfall. Groups of three pairwise associated stations were also obtained and an adjustment was made to a family of extreme value copulas involving the Marshall–Olkin family, whose parameters can be written as a function of Kendall’s τ association measures of the obtained pairs.

The possibility of primordial black hole direct detection

This thesis explores the possibility of directly detecting blackbody emission from Primordial Black Holes (PBHs). A PBH might form when a cosmological density uctuation with wavenumber k, that was once stretched to scales much larger than the Hubble radius during ination, reenters inside the Hubble radius at some later epoch. By modeling these uctuations with a running{tilt power{law spectrum (n(k) = n0 + a1(k)n1 + a2(k)n2 + a3(k)n3; n0 = 0:951; n1 = ????0:055; n2 and n3 unknown) each pair (n2,n3) gives a di erent n(k) curve with a maximum value (n+) located at some instant (t+). The (n+,t+) parameter space [(1:20,10????23 s) to (2:00,109 s)] has t+ = 10????23 s{109 s and n+ = 1:20{2:00 in order to encompass the formation of PBHs in the mass range 1015 g{1010M (from the ones exploding at present to the most massive known). It was evenly sampled: n+ every 0.02; t+ every order of magnitude. We thus have 41 33 = 1353 di erent cases. However, 820 of these ( 61%) are excluded (because they would provide a PBH population large enough to close the Universe) and we are left with 533 cases for further study. Although only sub{stellar PBHs ( 1M ) are hot enough to be detected at large distances we studied PBHs with 1015 g{1010M and determined how many might have formed and still exist in the Universe. Thus, for each of the 533 (n+,t+) pairs we determined the fraction of the Universe going into PBHs at each epoch ( ), the PBH density parameter (PBH), the PBH number density (nPBH), the total number of PBHs in the Universe (N), and the distance to the nearest one (d). As a rst result, 14% of these (72 cases) give, at least, one PBH within the observable Universe, one{third being sub{stellar and the remaining evenly spliting into stellar, intermediate mass and supermassive. Secondly, we found that the nearest stellar mass PBH might be at 32 pc, while the nearest intermediate mass and supermassive PBHs might be 100 and 1000 times farther, respectively. Finally, for 6% of the cases (four in 72) we might have substellar mass PBHs within 1 pc. One of these cases implies a population of 105 PBHs, with a mass of 1018 g(similar to Halley's comet), within the Oort cloud, which means that the nearest PBH might be as close as 103 AU. Such a PBH could be directly detected with a probability of 10????21 (cf. 10????32 for low{energy neutrinos). We speculate in this possibility.