A regional scale-invariant extreme value model of rainfall Intensity-Duration- Area-Frequency relationships 

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Frequentist framework

The GEV simple scaling model in the frequentist framework (Section 2.2.1) is estimated at each station by maximizing the likelihood (II.6). Optimization is based on the gradient projection method of Byrd et al. (1995) allowing box constraints for the variables. Constraints are set on the scale parameter, which is restricted to strictly positive values, the shape parameter, which is restricted in the range (􀀀0:75; 0:75) and the scaling parameter H, which is constrained in the range (0; 1). Optimization is initialized by 1 = (Dref;1 ; Dref;1 ; 1;H1), which can be considered as a smart initialization in that it is built from that data of each station : 1 is set to 0, corresponding to a Gumbel distribution. Dref;1 and Dref;1 are estimated using the method of moments under the Gumbel assumption. Following (II.2) with q = 1, H1 is set to the opposite of the regression slope of the log average maxima on the log duration (i.e. case q = 1 in Fig. II.2). Starting from 1, the gradient projection algorithm stops in ~, the maximum likelihood estimate, if it is unable to reduce the log likelihood (II.6) by a factor of 10􀀀8 jl(~)j. Density estimates of the associated random variable are obtained i) from the theorem of asymptotic normality of the maximum likelihood estimator, and ii) by bootstrap resampling technique using 1000 bootstrap samples. Return level estimates and associated densities are derived from these estimations as detailed in Section 2.2.3.

Bayesian framework

The same starting points 1 is used to initialize Metropolis-Hastings algorithm in the Bayesian framework (Section 2.3.2). Convergence of the MCMC is monitored using the ^R convergence criteria of Gelman et al. (2014) chapter 6, based on ve runs of the Metropolis-Hastings algorithm. Convergence is considered to be reached if ^R < 1:06, which is obtained after 20; 000 iterations. The burn-in period is set to the rst half iterations and every 10th iteration of the remaining 10,000 iterations is considered for the estimation of the posterior density, in order to reduce dependence within the sample. So, the posterior density estimation is based on 1000 samples. Posterior density estimates of return levels are obtained from (II.10), using these 1000 samples. To summarize any posterior density with one single value and, in particular, compare estimations with the frequentist framework, we decide to consider the posterior mean, i.e. the mean of the posterior density. Another common choice is to consider the mode of the posterior density (maximum a posteriori) but this is slightly less stable than the posterior mean.

The example of Montpellier

Before comparing the density estimates obtained with the dierent frameworks over the whole region, we start illustrating results on the station of Montpellier. This station is chosen because i) it shows among the largest values of 3h-rainfall intensity (84 mm/h at 3h duration, in autumn 2014), ii) Montpellier is a good illustration of the temporal variability of extreme rainfall : the median value of annual maximum 3h-rainfall intensity (15mm/h at 3h duration) is 50% bigger than the median value over the region (10mm/h at 3h duration), whereas at 72h duration it equals the regional median (1.25 mm/h at 72h duration), and iii) its population is among the biggest in the region (more than 250; 000 inhabitants in 2010), which make it a sensible case of risk analysis. Fig. II.8 compares the density estimates of the parameters and 50-year return levels at 3h and 72h durations. In the frequentist framework, densities are obtained with either the theorem of asymptotic normality -in which case densities are Gaussian-, or the bootstrap resampling method. For the Bayesian framework, the posterior density is depicted. Fig. II.8 illustrates that the posterior and bootstrap densities are able to better adjust to the data by being able to produce asymmetric densities with several modes.
The posterior density of H departs particularly from the bell-like shape of a Gaussian with a attened peak between 0:83 and 0:87, which cannot be seen by application of the asymptotic normality theorem. The bootstrap method, on the opposite, produces similar density of H to the posterior density. Some asymmetry with respect to the mode is also found for in the posterior density and even more in the bootstrap density. This produces asymmetry in return levels with a heavier right tails for the bootstrap and posterior densities than for the Gaussian density, whereas the left tails of the posterior and Gaussian densities are similar. Therefore the bootstrap and Bayesian methods are able to tell there is a greater likelihood for the 50-year return level to be over than under the estimated value, which is not possible when considering symmetric Gaussian densities.
The return level plot of Fig. II.9 illustrates this asymmetry in the uncertainty of return levels for the bootstrap and posterior densities, particularly for large return periods. Whatever the return period, the flower bound of the posterior and Gaussian condence intervals are equal, whereas the upper bound diers signicantly. We can thus postulate that, by imposing symmetry, the asymptotic normality theorem tends to underestimate the upper bound of the condence interval. The bootstrap method allows asymmetry, however it gives much wider condence intervals than the two other methods, even for the lower bound. Comparing the bootstrap and posterior densities in Fig. II.8 shows that dierence in the width of the condence intervals is mainly due to dierences in the scale Dref and shape parameters.

Generic IDAF relationships

The framework of the IDAF relationship considered in this article relies on two assumptions. This rst one is a version of space-time dynamic scaling setting that : pr 􀀀HM(D;A) < x  = pr   M(D; bA) < x o ; (III.1).
where M(D;A) is the random variable representing the annual maximum rainfall intensity for the duration D and the area A, H and b are non-negative real numbers called the scaling exponents and is a non-negative real number. The second assumption relies on the denition of a duration of reference D0, taken in this article equal to the nest temporal resolution of the data, D0 = 3 h. It assumes spatial scaling of rainfall for the duration of reference D0 : for any areas A and A?, pr fM(D0;A) < xg = pr fKD0(A;A?)M(D0;A?) < xg ; (III.2) where KD0 is a non-negative deterministic function verifying KD0(A;A) = 1 for any A. The subscript « D0 » intends to remind that spatial scaling is assumed for the duration D0. We use the subscript « * » to distinguish D0 {which is xed{ from A? {which can by any. Eq. (III.2) is used in Nhat et al. (2007) with KD0(A;A?) = A A? . The model of De Michele et al. (2001) is equivalent to consider KD0(A;A?) = 1+A 1+A? . In this article, we consider a more exible version : KD0(A;A?) =  1 + PRi=1 iAi 1 + PR i=1 iAi ? ! (III.3).
where i, i are scalars, is a non-negative scalar and R is a strictly positive integer. The choice of R is discussed in Section 4.1 for an illustrative example. Combining Eqs. (III.1) and (III.2) leads to the following IDAF relationships deriving the statistical distribution of annual maximum rainfall intensity for any pair of duration and area, (D;A), from the statistical distribution of annual maximum for the duration of reference and any other area, (D0;A?) relationships (see Appendix 6.1) : pr fM(D;A) < xg = pr frD0(D; A;A?)M (D0;A?) < xg ; (III.4).

Physical meaning of the dierent components of the model

The Gumbel parameters 0 and 0 > 0 model the statistical distribution of rainfall intensity maxima at the nest spatio-temporal scales. The parameter 0 is the mode of the distribution. The larger 0, the larger the most likely intensity maxima at D0 = 3 h and A0 = 1 km2. The parameter 0 is the standard deviation multiplied by p 6=. The larger 0, the larger the variability in intensity maxima at D0 = 3 h and A0 = 1 km2.
When A = A? 0 km2, Eqs. (III.5) and (III.8) dene IDF simple-scaling relationships between M(D0; 0) and M(D; 0). Following Melese et al. (2018), this means that the parameter H is related to the temporal variability of extreme rainfall at the point scale, i.e. for A 0 km2. The closer H to one, the more extreme rainfall at local scale tends to concentrate in few hours. On the contrary, the smaller H, the more extreme rainfall is homogeneous in time.
The ARF parameters in Eq. (III.10) are dicult to interpret but the ARF relationships do have a physical meaning. Noting that ARFD0(D; A;A?) = ARFD0(D; A;A0)=ARFD0(D;A?;A0).

Extreme areal rainfall structure

In order to summarize the spatial structure of extreme rainfall in the region, we group together the pixels having similar ARF shapes. This can be seen as the dual approach to Ceresetti et al. (2012) : whereas Ceresetti et al. (2012) assume a priori that the region can be subdivided into two subregions with homogeneous extreme rainfall regimes and estimate the ARFs over these two areas based on this assumption, we estimate the ARFs everywhere across the region and we look for subregions with homogeneous ARFs. Following the interpretation of the ARF of Section 3.3, we group together the pixels featuring ARFs with similar shapes rather than similar absolute values, using the k-means algorithm proposed by Hartigan and Wong (1979). We build an ARF matrix whose rows represent the 2149 selected pixels and whose columns represent the pairs (D;A) with D from 3 to 48 h with a step of 1 h and A1=2 from 1 km to 45 km with a step of 1 km. This matrix contains then 2149 rows and 2070 columns, each row representing the values of ARFD0(D; A;A0) when A and D vary, for a given pixel. Since we are interested in the shape of the ARF curves, we normalize each row between 0 and 1. We perform a principal component analysis to reduce the matrix to 9 columns, which represents 99.9% of the explained variance. Then we apply the k-means algorithm to this matrix with a number of classes varying from 2 to 5. Below we consider the classication into 3 classes as it allows a more manageable description of the extreme rainfall structure at regional scale. of the pixel classication is shown in Fig. III.7. Below we interpret the rainfall spatial features leading to these classes, focusing on the ARF shapes of the center classes. By doing so, we purposely omit the variability of the ARF curves among the classes -two pixels of the same class being interpreted has having the same ARF curves. Our goal is to interpret the broad rainfall scaling properties in space through « idealized »ARFs curves. At this point, we note that the case with ARF lower or larger than 1 depending on the duration like in Ales (Fig. III.2) is not frequent enough to appear as a class in itself, since there are only 54 such pixels (2:5% of all pixels).

Table of contents :

I. Introduction 
1. La region des Cevennes : une region au risque hydro-meteorologique eleve
2. Sur l’etude de l’alea pluviometrique
3. Relation Intensite-Duree-Aire-Frequence sous hypothese d’invariance par changement d’echelle
4. Modelisation statistique des extr^emes pluviometriques
5. Donnees pluviometriques utilisees dans cette etude
6. Cadres de developpement des modeles
7. Vue d’ensemble
II. Uncertainty estimation of Intensity-Duration-Frequency relationships : a regional analysis 
1. Introduction
2. Two frameworks of IDF relationships
2.1. Introduction
2.2. Frequentist framework
2.2.1. Model
2.2.2. Inference
2.2.3. Uncertainty computation
2.3. Bayesian framework
2.3.1. Model and priors
2.3.2. Inference
3. Data
4. Evidence of simple scaling
5. Workow
5.1. Frequentist framework
5.2. Bayesian framework
6. Results and discussion
6.1. IDF curves
6.1.1. Estimation and goodness-of-t
6.1.2. Spatial variability of return level across durations
6.1.3. Temporal variability of extreme rainfall
6.2. IDF uncertainty
6.2.1. The example of Montpellier
6.2.2. Regional study
7. Conclusion
III. A regional scale-invariant extreme value model of rainfall Intensity-Duration- Area-Frequency relationships 
1. Introduction
2. Data
3. A
exible Gumbel-IDAF model
3.1. Generic IDAF relationships
3.2. Gumbel-IDAF relationships
3.3. Physical meaning of the dierent components of the model
3.4. Model estimation
4. Results
4.1. Empirical validation of the ARF relationships
4.2. Goodness-of-t of the Gumbel-IDAF model
4.3. Characterization of the extreme rainfall regime of the region
4.3.1. Extreme local rainfall features
4.3.2. Extreme areal rainfall structure
4.4. Areal rainfall risk
5. Conclusions
6. Appendix
6.1. Details leading to Eq. (III.4)
6.2. Details leading to Eq. (III.7)
IV. A Bayesian framework for multi-scale assessment of storm severity and related uncertainties 
1. Introduction
2. Data
3. A framework of multi-scale severity modeling
3.1. Extreme value IDAF relationships : frequentist framework
3.2. Extreme value IDAF relationships : Bayesian framework
3.3. Bayesian inference
4. Results
4.1. MCMC monitoring
4.2. Model validation
4.3. Multi-scale severity of the 2011 event
4.4. Generalization to other events
5. Conclusion
6. Appendix
6.1. Adjusted likelihood
6.2. Technical details on the MCMC algorithm
6.2.1. Choice of the priors
6.2.2. DRAM algorithm
V. Rappel des principaux resultats et perspectives 
1. Rappel des principaux resultats
2. Perspectives
2.1. Detection des regions sous echantillonnees (6 mois) et augmentation des donnees (-)
2.2. Non stationnarite de la severite (18 mois)
2.3. Vers une meilleure estimation de l’alea pluviometrique et des incertitudes associees en rel^achant
l’hypothese d’independance entre les maxima de dierentes echelles spatio-temporelles (36 mois)
2.4. Alea pluviometrique sur des bassins versants (24 mois)

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