Under these conditions of (uncertain) sea-level rise and raising of the asset, the overall (or effective) expected number, NovNov, of exceedances (>zP+a)(>zP+a) during the period T, becomes equation(3) Nov=∫−∞∞P(z′)Nμ−zP+Δz+z′−aλdz

The function, NN, is often well-fitted by a generalised extreme-value distribution   (GEV  ). The simplest of these, the Gumbel   distribution, fits most sea-level extremes quite well (e.g. van den Brink and Können, 2011). The Gumbel distribution may be expressed as (e.g. Coles, 2001, p. 47) equation(4) F=exp−expμ−zPλwhere F   is the probability that there will be no exceedances >zP>zP during the prescribed learn more period, T. From Eqs. (1), (2) and (4) equation(5) N=Nμ−zPλ=expμ−zPλμμ is therefore the value of z  P for which N  =1 during the period T  , and λλ, the ‘scale parameter’, is an e-folding distance in the vertical. Globally, the scale parameter has a quite narrow range; for the sea-level records described in Section 4, the 5-percentile, median and 95-percentile values of the scale parameter are 0.05 m, 0.12 m and 0.19 m, respectively.

Again, as noted in Section 1, it is assumed that the scale parameter, λλ, does not change with a rise in selleck products sea level. It will also be noted later (Section 6) that Eq. (5) is only valid over the restricted range of zP that encompasses the high extreme values. Eq. (3) therefore becomes (Hunter, 2012): equation(6) Nov=∫−∞∞P(z′)expμ−zP+Δz+z′−aλdz′=NexpΔz+λln∫−∞∞P(z′)expz′λdz′−a/λ In order to preserve the expected number of exceedances (or flooding events), we require that Nov=NNov=N. Therefore, the allowance, a  , is equal to the term Δz+λln(⋯) in the last part of Eq. (6). This C59 chemical structure allowance is composed of two parts: the mean sea-level rise, ΔzΔz, and the term λln(⋯), which arises from the uncertainty in future sea-level rise. Hunter (2012) evaluated the allowance for three types of uncertainty distribution for future sea-level rise: a normal distribution, a boxcar (uniform) distribution

and a raised cosine distribution. The resulting allowances may all be expressed as simple analytical expressions, involving the Gumbel scale parameter, λλ, the central value of the estimated rise, ΔzΔz, and its standard deviation, σσ. We here estimate the allowances using normal and raised cosine distributions, the former having fatter tails and therefore yielding higher allowances (the raised-cosine distribution falls to zero at a finite distance from the central value, the total range of the distribution being about 1.7 times the 5- to 95-percentile range). Both distributions were fitted to the 5- and 95-percentile range of the IPCC AR4 projections of sea-level rise, with the central value, ΔzΔz, being the mean of the 5- and 95-percentile values. For a normal uncertainty distribution of future sea-level rise, the allowance is given by Δz+σ2/(2λ)Δz+σ2/(2λ) (Hunter, 2012). A typical sea-level rise projection for 2100 relative to 1990 for the A1FI emission scenario is 0.5±0.