D. Veneziano, A. Langousis, and C. Lepore, Annual rainfall maxima: Theoretical Estimation of the GEV shape parameter k using multifractal models, Eos Trans. AGU, 90(52), San Francisco, American Geophysical Union, 2009.
The annual maximum of the average rainfall intensity in a period of duration d, Iyear(d), is typically assumed to have generalized extreme value (GEV) distribution. The shape parameter k of that distribution is especially difficult to estimate from either at-site or regional data, making it important to constraint k using theoretical arguments. In the context of multifractal representations of rainfall, we observe that standard theoretical estimates of k from extreme value (EV) and extreme excess (EE) theories do not apply, while estimates from large deviation (LD) theory hold only for very small d. We then propose a new theoretical estimator based on fitting GEV models to the numerically calculated distribution of Iyear(d). A standard result from EV and EE theories is that k depends on the tail behavior of the average rainfall in d, I(d). This result holds if Iyear(d) is the maximum of a sufficiently large number n of variables, all distributed like I(d); therefore its applicability hinges on whether n = 1yr/d is large enough and the tail of I(d) is sufficiently well known. One typically assumes that at least for small d the former condition is met, but poor knowledge of the upper tail of I(d) remains an obstacle for all d. In fact, in the case of multifractal rainfall, also the first condition is not met because, irrespective of d, 1yr/d is too small (Veneziano et al., 2009, WRR, in press). Applying large deviation (LD) theory to this multifractal case, we find that, as d -> 0, Iyear(d) approaches a GEV distribution whose shape parameter kLD depends on a region of the distribution of I(d) well below the upper tail, is always positive (in the EV2 range), is much larger than the value predicted by EV and EE theories, and can be readily found from the scaling properties of I(d). The scaling properties of rainfall can be inferred also from short records, but the limitation remains that the result holds under d -> 0 not for finite d. Therefore, for different reasons, none of the above asymptotic theories applies to Iyear(d). In practice, one is interested in the distribution of Iyear(d) over a finite range of averaging durations d and return periods T. Using multifractal representations of rainfall, we have numerically calculated the distribution of Iyear(d) and found that, although not GEV, the distribution can be accurately approximated by a GEV model. The best-fitting parameter k depends on d, but is insensitive to the scaling properties of rainfall and the range of return periods T used for fitting. We have obtained a default expression for k(d) and compared it with estimates from historical rainfall records. The theoretical function tracks well the empirical dependence on d, although it generally overestimates the empirical k values, possibly due to deviations of rainfall from perfect scaling. This issue is under investigation.