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| Titel |
Annual Rainfall Maxima: Practical Estimation Based on Large-Deviation Results |
| VerfasserIn |
C. Lepore, D. Veneziano, A. Langousis |
| Konferenz |
EGU General Assembly 2009
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| Medientyp |
Artikel
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| Sprache |
Englisch
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| Digitales Dokument |
PDF |
| Erschienen |
In: GRA - Volume 11 (2009) |
| Datensatznummer |
250024683
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| Zusammenfassung |
| In a separate communication (Veneziano et al., “Annual Rainfall Maxima: Large-deviation
Alternative to Extreme-Value and Extreme-Excess Methods,” EGU 2009), we show that, at
least for scale-invariant rainfall models, classical extreme value analysis based on Gumbel’s
extreme value (EV) theory and peak-over-threshold (PoT) analysis based on Pickands’
extreme excess (EE) theory do not apply to annual rainfall maxima (AM). A more
appropriate theoretical setting is provided by large-deviation (LD) theory. This paper delves
with some practical implications of these findings.
All above theories predict that, as the averaging durations d - 0, (1) the annual
maximum rainfall intensity in d, Iyear(d), has generalized extreme value (GEV) distribution,
(2) the excess of the average intensity in d, I(d), above a level u on the order of the annual
maximum has generalized Pareto (GP) distribution, and (3) the GEV and GP distributions
have the same shape parameter k. The value of k depends on the theory used. According to
EV and EE theories, k is determined by the upper tail of I(d), whereas LD theory shows that
k is determined by less extreme regions of the distribution of I(d). The LD parameter kLD is
always in the EV2 range and is larger than the value kEV-EE predicted by EV and EE
theories.
Since all theories predict that the annual maxima have GEV distribution and the
corresponding excesses have GP distribution, methods that directly fit GEV and GP
distributions to the data without reference to its asymptotic value should not be affected by
which theory is correct. However, the theoretical results have other significant practical
implications:
Accurate estimation of k from at-site data is difficult. For this reason, k is often
estimated regionally. The estimate of k from LD theory is much more robust than
that from EV and EE theories and relies on the scaling of the moments of rainfall
of order 2.5-3.5. This scaling is nearly universal for rainfall, providing a good
“prior” value of k (around 0.3-0.4), which can be used also at un-gauged sites.
The shift of focus to regions of the marginal distribution of I(d) below the
extreme upper tail, and the recognition that in practice one needs extreme rainfall
estimates over a range of finite durations dfor which Iyear(d) does not have
GEV distribution make non-asymptotic methods more attractive. These methods
fit marginal distributions to the order statistics of I(d) or to PoT values above
thresholds not much below the level of the annual maxima and estimate the
distribution of Iyear(d) as
P [Iyear(d) > x] - {P [I(d) > x]}n(d) (1)
P [Iyear(d) > x] - e-λd,uP[IPoT(d;u)>x-u] (2)
where n(d) is a parameter that gives the effective number of independent I(d) variables in
one year, λd,u is the annual rate at which I(d) up-crosses level u, and IPoT(d;u) is the PoT
intensity for averaging duration d and threshold u.
We have implemented procedures based on these non-asymptotic approaches, with the
following specific characteristics:
The distributions of I(d) (in the upper region) and IPoT(d;u) are taken to have
scaled lognormal shape, with 3 parameters (the mean value m, the variance Ï2,
and a scaling factor c >0 on the probability density). This choice of distribution
is based on both empirical evidence and asymptotic multifractal results;
The unknown parameters {m, Ï2, c, n(d)} or {m, Ï2, c, λd,u} are estimated
simultaneously from marginal or PoT and AM data (the latter data mainly
constrain n(d) and λd,u) using maximum likelihood.
The upper region for I(d) is chosen such that the predicted AM distribution from
Eq. 1 closely matches the empirical AM distribution.
Application to several actual and simulated rainfall records shows that this approach is
superior in accuracy and robustness to conventional AM and PoT methods.
This work is funded by project RISK (High Speed Rail) of the M.I.T.-Portugal Program. |
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