The manuscript presents the applicability of the Gumbel distribution in the frequency analysis of extreme events in hydrology. The advantages and disadvantages of using the distribution are highlighted, as well as recommendations regarding its proper use. A literature review was also carried out regarding the methods of estimating the parameters of the Gumbel distribution. Thus, for the verification of the methods, case studies are presented regarding the determination of the maximum annual flows and precipitations using nine methods for parameter estimation. The influence of the observed data lengths on the estimation of the statistical indicators, the parameters and the quantiles corresponding to the field of low annual exceedance probabilities (p <1%) is also highlighted. The results are compared to those obtained with the Generalized Extreme Value distribution, respectively the Burr and the Wakeby distributions with parameters estimated using the L-moments. The results highlight and reaffirm the statistical, mathematical and hydrological recommendation regarding the avoidance of using the Gumbel distribution in Flood Frequency Analysis and its use with reservations in the case of maximum precipitation, especially when the statistical indicators of the analyzed data are not close to the characteristic ones and unique to the distribution.

AbstractLocal estimates of coastal flood risk are required for coastal planning and development, including the location and design of sea-defences and coastal buildings, such as harbours and associated infrastructure. This paper discusses the use of three parameters associated with estimating such risks; the flood return period, the instantaneous flood probability and the flood design risk, and it describes the mathematical background for their derivation. The discussion is extended to include the effects of sea level rise and how it can be incorporated into the calculations. Flood height can vary quite rapidly with distance along the coast, being affected by coastal topology, which may magnify or diminish the tidal and surge effects. Similarly land heave influences the local effects of sea level rise and can be influenced by water extraction, tectonic movements and melting ice. Tide gauge measurements provide a local historical record from which the various parameters can be retrieved. This paper discusses the algorithms used to derive these measures from tide-gauge records. The figures have been derived for four tide gauges located on the UK east coast.

This work is divided into two parts. In the first one the combinatorics of a new class of randomly generated objects, exhibiting the same properties as the distribution of prime numbers, is solved and the probability distribution of the combinatorial counterpart of the n-th prime number is derived, together with an estimate of the prime-counting function π(x). A proposition equivalent to the Prime Number Theorem (PNT) is proved to hold, while the equivalent of the Riemann Hypothesis (RH) is proved to be false with probability 1 (w.p. 1) for this model. Many identities involving Stirling numbers of the second kind and harmonic numbers are found, some of which appear to be new. The second part is dedicated to generalizing the model to investigate the conditions enabling both PNT and RH. A model representing a general class of random integer sequences is found, for which RH holds w.p. 1. The prediction of the number of consecutive prime pairs, as a function of the gap d, is derived from this class of models and the results are in agreement with empirical data for large gaps. A heuristic version of the model, directly related to the sequence of primes, is discussed and new integral lower and upper bounds of π(x) are found.