At-site methods with identically distributed variables
The standard at-site FA uses the at-site observations to estimate parameters from a prespecified distribution. More formally, given a sample of observations 1,…, n Y Y that are assumed iid in most cases, the parameters of a given distribution D( ) are estimated using a particular estimation method (e.g. maximum likelihood, moment-based approaches, Bayesian estimation). A wealth of research has been carried out within this context during the last decades. Most studies focused on the choice of the parent distribution and of the estimation approach (e.g., Durrans and Tomic ; He and Valeo ; Hosking et al. ; Kroll and Stedinger ; Lang et al. ; Madsen et al. [1997a]; Meshgi and Khalili ; Ribatet et al. ; Sankarasubramanian and Srinivasan ), or the quantification of uncertainty (e.g. Chowdhury et al. ; Cohn et al. ; Kysely ; Stedinger ; Stedinger and Tasker ; Stedinger et al. ). Besides the basic at-site FA based on estimating a pre-specified distribution, some model-based FA methods were also developed, based on models reproducing the main characteristic of hydrological variables, such as rainfall [Arnaud and Lavabre, 1999] and flood [Boughton and Droop, 2003; Hundecha and Merz, 2012]. Moreover, since a common problem of the at-site analysis is the relatively small data length, additional documentary sources on historical flood events, or data obtained from sediment deposits can be used to extent the data period. Historical and paleoflood data analysis were developed to deal with these additional data (e.g these additional data (e.g Naulet et al.Naulet et al. [2005; ]; Neppel et al.Neppel et al. [; ]; O’Connell et al.O’Connell et al. [2002; ]; Payrastre et al.Payrastre et al. [2011; ]; Reis and StedingerReis and Stedinger [2005; ]; Stedinger and CohnStedinger and Cohn [1986]). ).
AtAt–site methods with timesite methods with time–varying variablesvarying variables In the time
In the time–varying context, varying context, Renard Renard et al.et al. [2006a[2006a]] and and Ouarda and ElOuarda and El–AdlouniAdlouni [2011] discussed some new FA models by estimating timediscussed some new FA models by estimating time–varying parameters from a prevarying parameters from a pre–specified specified distribution. With simildistribution. With similar structures, ar structures, Rust et al.Rust et al. [2009] discussed the seasonality of extreme discussed the seasonality of extreme precipitation in UK, precipitation in UK, Kysely et al.Kysely et al. [2010] described the trends on ddescribed the trends on daily temperature and aily temperature and NogNogaj aj et al.et al. [2006] analyzed the amplitude and frequency of extreme temperature. More generally, analyzed the amplitude and frequency of extreme temperature. More generally, Khaliq et al.Khaliq et al. [2006] reviewed timereviewed time–varying atvarying at–site FA methods. In asite FA methods. In addition to the analysis on ddition to the analysis on the temporal variation, climate/weather information were also integrated to the analysis: for the temporal variation, climate/weather information were also integrated to the analysis: for instance, instance, MicMicevski et al.evski et al. [2006] used the Interused the Inter–decadal Pacific Oscillation (IPO) to decadal Pacific Oscillation (IPO) to characterize the flood hazard in Australia; characterize the flood hazard in Australia; Tramblay et al.Tramblay et al. [2011] used various climate used various climate covariates to analyze the heavy rainfall in Southern France; and covariates to analyze the heavy rainfall in Southern France; and Garavaglia et al.Garavaglia et al. [2010; ]; [; ]; Paquet et al.Paquet et al. [2013] incorporated weather type information to quantify the rainfall incorporated weather type information to quantify the rainfall hazard. hazard.
While at–site FA methods enabling the inclusite FA methods enabling the inclusion of climate information or nonsion of climate information or non–stationarity are becoming common, such atstationarity are becoming common, such at–site models remain limited by two important site models remain limited by two important drawbacks: drawbacks:
(1) Local analysis cannot be applied to ungauged sites.
(1) Local analysis cannot be applied to ungauged sites.
(2) Uncertainty in parameter estimates (and hence predictive estim.
(2) Uncertainty in parameter estimates (and hence predictive estimates) tends to be very ates) tends to be very large due to the limited number of observations in a local model. In addition, if climate large due to the limited number of observations in a local model. In addition, if climate information is included and more complex models are proposed, these observations may not information is included and more complex models are proposed, these observations may not be sufficient to identify the parameters be sufficient to identify the parameters [[Thyer et al.Thyer et al., 2006, 2006]]..
This motivates the development of regional frequency analysis (RFA) models that use information from multiple sites to overcome these shortcomings.information from multiple sites to overcome these shortcomings.
RFA RFA methods with identically distributed variablesmethods with identically distributed variables
In classical RFA methods, information from multiple sites is used to perform the In classical RFA methods, information from multiple sites is used to perform the inference, which may provide more precise estimations. More precisely, the basis of most inference, which may provide more precise estimations. More precisely, the basis of most RFA methods is to assume that some paraRFA methods is to assume that some parameters are common to all sites within a given meters are common to all sites within a given homogeneous region, or that the parameters can be predicted from a regression with site homogeneous region, or that the parameters can be predicted from a regression with site characteristics such as (for precipitation) elevation, distance to sea, etc.characteristics such as (for precipitation) elevation, distance to sea, etc.
Under the strict stationarity assumption, many RFA methodologies have been developed y RFA methodologies have been developed over the years (e.g. over the years (e.g. Durrans and KirbyDurrans and Kirby [2004; ]; Overeem et al.Overeem et al. [2008; ]; Yu et al.Yu et al. [2004]; ; Cooley Cooley et al.et al. [2007; ]; Madsen and RosbjeMadsen and Rosbjergrg [1997a[1997a]; ; [1997b1997b]; ]; Madsen et al.Madsen et al. [1997b[1997b]; ; ] and and Ghosh and MallickGhosh and Mallick [2011]). A comparison between regional and). A comparison between regional and atat–site approaches for site approaches for extreme rainfall was performed by e.g. extreme rainfall was performed by e.g. Kysely et al.Kysely et al. [2011]. .
RFA methods with time-varying variables
In order to move beyond the assumption of strict stationarity, Cunderlik and Burn  and Leclerc and Ouarda  described non-stationary RFA models for flood analysis, and Hanel et al. [2009a] introduced a time-varying index-flood model for extreme precipitation. Recently, several authors (Aryal et al. ; Lima and Lall ; Maraun et al. ; Maraun et al. ; Sang and Gelfand ) started investigating spatio-temporal models.
In the same vein, Gregersen et al.  also used Poisson regression models to describe the frequency of extreme rainfall in both space and time. A common difficulty for all these approaches is the treatment of the spatial dependence existing between data. This is also one of the main topics that will be discussed in this thesis.
The notion of return period in the time-varying context
In risk analysis and assessment, the rarity of an extreme event is generally quantified through a return period, rather than in terms of a probability of exceedance. In the context of identically distributed variables, the “T year event” corresponds to the event with nonexceedance probability p 11/T . A 10-year event has therefore a probability 1/10 to be exceeded every year. However, in the time-varying context, such simple equivalence does not hold any more, because the distribution varies with time. Renard  and Cooley  discussed two definitions of the return period in the time-varying context: (i) the return period associated with a value Q is the expected waiting time between two successive exceedances of level Q, and (ii) the return period associated with a value Q is computed so that the expected number of exceedances of level Q in the next T years is equal to one. While both definition are equivalent in an identical distribution context, and indeed correspond to an annual probability of exceedance equal to 1/T, this is not the case if a temporal trend exists.
The interpretation of “T year events” is much less direct than under the strict stationarity assumption. An alternative quantification of the rarity of an event uses the concept of “failure probability”. The failure probability associated with a level Q and a duration n is defined as the probability that at least one event exceeds Q during the next n years. Typically, n can be considered as, the lifetime of a hydraulic structure. This concept is much easier to interpret than return periods outside of the stationary context. Moreover, it allows investigating questions such as “what should be the capacity of a dam to ensure that the probability of exceeding the dam capacity during its lifetime of n years is less than 0.01?” A more thorough discussion of return periods and failure probabilities can be found in Salas and Obeysekera .
The concept of return period is not easier to handle with climate-informed models. Indeed, as explained in Section 2, the stationarity of climate-informed models is generally unknown. Moreover, even if the marginal distribution turned out to be stationary, we may still be interested in computing conditional probabilities of exceedance. It may be interesting to know how different the extreme events are during different climate conditions. For instance, if the main resource of a river is snowmelt, the probability of observing a large snowmelt flood will surely be different if one knows that temperatures are particularly high or low. This refers to the return level based on the conditional probability. More precisely, if the cdf of a random variable Y is F(y | ) , then the “T year event” conditional on 0 corresponds to the event with non-exceedance probability 0 p 11/T F(y | ) . A 10-year event conditional on 0 has therefore a conditional probability 1/10 to be exceeded every year for which the climate condition satisfies 0 .
Posterior distribution of regression parameters
Figure 3.4 illustrates the posterior distribution of regression parameters in Nor-L0 and Nor-L1 models. The mean value μ0 of Nor-L0 is significantly different from the intercept μ0 of Nor-L1. This is because a significant positive slope μ1 is detected for Nor-L1 model. Moreover, the standard deviation is slightly smaller in Nor-L1, suggesting that a part of the data variability is due to the existence of a trend. Similar results are found for the negative binomial distribution.
The slope parameter μ1 in both Nor-L1 and NB-L1 models shows a significantly positive value (Figure 3.5). Moreover, the posterior distributions are similar in both models. This consistency shows that the temporal trend on the mean can be detected with two different parent distributions.
Table of contents :
PART I TIME VARYING FREQUE NCY ANALYSIS FRAMEWO RK: LOCAL MODEL
CHAPTER 2 DEVELOPMENT OF A GEN ERAL TIME VARYING MODELING FRA MEWORK AT THE LOCAL SCALE
1L OCAL MODEL CONSTRUCT ION
1.1 Parent distribution for local model
1.2 Regression with temporal covariates
1.3 An illustration of local model construction
1.4 Relationship with other modeling frameworks
2 P OSTERIOR DISTRIBUTIO N AND PARAMETER INFE RENCE
2.1 Parameter estimation methods
2.2 Posterior distribution
2.3 Quantile computation based on the posterior distribution
3 M ODEL DIAGNOSIS AND S ELECTION
3.1 Diagnostic tools
3.2 Model comparison tools
4 S YNTHETIC CASE STUDIE S
4.1 Synthetic study 1
4.2 Synthetic study 2
5 C ONCLUS ION ON THE LOCAL CLI MATE INFORMED FRAMEWORK
CHAPTER 3 CASE STUDIES WITH LO CAL TIME VARYING MODELS
1 P ROJECTED CHANGES IN T HE PRECIPITATION REG IME OF THE D URANCE CATCHMENT
1.2 Precipitation variables
1.3 Parent distribution selection
1.4 Regression models
1.5 Posterior distribution of regre ssion parameters
1.6 Goodness of fit
1.7 Model comparison
1.8 Accounting for non stationarity in GCM projections: stationary sub periods vs. continuous trend
1.9 Conclusion and discussion
2 NAO EFFECTS AND TEMPORAL TRENDS IN EXTREME PR ECIPITATION IN M EDITERRANEAN F RANCE
2.2 Regression modeRegression models under three competing hypothesesls under three competing hypotheses
2.3 Posterior distribution of regression parametersPosterior distribution of regression parameters
2.5 Conditional predictionsConditional predictions
2.6 Temporal trend and NAO impact for all 92 sitesTemporal trend and NAO impact for all 92 sites
2.7 Conditional quantiles for all 92 sites and their uncertaintyConditional quantiles for all 92 sites and their uncertainty
2.8 Model comparisonModel comparison
2.9 Discussion and conclusDiscussion and conclusionion
PART II TIMETIME–VARYING FREQUENCY ANVARYING FREQUENCY ANALYSIS FRAMEWORK: REALYSIS FRAMEWORK: REGIONAL MODELGIONAL MODEL
CHAPTER 4 DEVELOPMENT OF A GENDEVELOPMENT OF A GENERAL SPATIOERAL SPATIO–TEMPORAL RTEMPORAL REGIONAL FREQUENCY ANEGIONAL FREQUENCY ANALYSIS ALYSIS FRAMEWORKFRAMEWORK
1 RREGIONAL MODEL CONSTREGIONAL MODEL CONSTRUCTIONUCTION
1.1 Parent distributionParent distribution
1.2 SpatioSpatio–temporal regressiontemporal regression
1.3 An illustration of the regional modelAn illustration of the regional model
1.4 AccountinAccounting for spatial dependence between sitesg for spatial dependence between sites
1.5 Parameter inferenceParameter inference
1.6 Missing valuesMissing values
1.7 MCMC sampling, model diagnosis and model comparisonMCMC sampling, model diagnosis and model comparison
2 CCAN THE MODEL DETECT AN THE MODEL DETECT SPATIOSPATIO–TEMPORAL VARIATIONSTEMPORAL VARIATIONS?? SSYNTHETIC CASE STUDIEYNTHETIC CASE STUDIESS
2.1 Synthetic study 1Synthetic study 1
2.2 Synthetic study 2Synthetic study 2
3 CCONCLUSION ON THE SPAONCLUSION ON THE SPATIOTIO–TEMPORAL REGIONATEMPORAL REGIONAL MODELL MODEL
CHAPTER 5 ON THE TREATMENT OF ON THE TREATMENT OF SPATIAL DEPENDENCESPATIAL DEPENDENCE
1 DDOES IGNORING SPATIALOES IGNORING SPATIAL DEPENDENCE LEADS TO DEPENDENCE LEADS TO AN UNDERAN UNDER–ESTIMATION OESTIMATION OF UNCERTAINTIESF UNCERTAINTIES??
1.1 First simulation with a Gaussian parent distributionFirst simulation with a Gaussian parent distribution
1.2 Second simulation with a GEV parent distriSecond simulation with a GEV parent distributionbution
2 SSPATIAL DEPENDENCE FOPATIAL DEPENDENCE FOR EXTREMESR EXTREMES:: CCOPULAS VSOPULAS VS.. MAXIMUM STABLE PROCEMAXIMUM STABLE PROCESSESSSES
2.1 Basics of maximum stable processesBasics of maximum stable processes
2.2 Gaussian copula inference with various spatial dataGaussian copula inference with various spatial data
2.3 Comparison with different spatial dependence modelsComparison with different spatial dependence models
PART III GENERAL APPLICATIONSGENERAL APPLICATIONS: E: ENSO IMPACT ON PRECIPNSO IMPACT ON PRECIPITATIONSITATIONS
GENERAL ENERAL IINTRODUCTION ABOUT THNTRODUCTION ABOUT THE IMPACT OF E IMPACT OF ENSOENSO ON PRECIPITATIONON PRECIPITATION
CHAPTER 6 QUANTIFYING THE IMQUANTIFYING THE IMPACT OF ENSO ON SUMMPACT OF ENSO ON SUMMER RAINFALL IN SOUTHER RAINFALL IN SOUTHEAST QUEENSLAND, EAST QUEENSLAND, AUSTRALIAAUSTRALIA
1 QQUANTIFYING THE IMPACUANTIFYING THE IMPACT OF T OF ENSOENSO ON SUMMER RAINFALL TON SUMMER RAINFALL TOTALS USING LOCAL MOOTALS USING LOCAL MODELSDELS
1.1 Data and covariatesData and covariates
1.2 Local model for the summer rainfall totalsLocal model for the summer rainfall totals
2 QQUANTIFYING THE IMPACUANTIFYING THE IMPACT OF T OF ENSOENSO ON SUMMER MAXIMUM DAON SUMMER MAXIMUM DAILY RAINFALLS USING ILY RAINFALLS USING LOCAL AND REGIONAL MLOCAL AND REGIONAL
2.1 Data and covariatesData and covariates
2.2 Models for summer rainfall maximumModels for summer rainfall maximum
2.3 AssessAssessing competing hypotheses of ENSO impact on summer maximum daily rainfallsing competing hypotheses of ENSO impact on summer maximum daily rainfalls
3.1 Assumption of homogeneous regionsAssumption of homogeneous regions
3.2 Spatial dependence Spatial dependence modelingmodeling
3.3 Spatial regression modelingSpatial regression modeling
3.4 Practical Implications: utilizing predictions of extreme rainfall distributPractical Implications: utilizing predictions of extreme rainfall distributions from the climateions f
CHAPTER 7 A GLOBAL ANALYSIS OFA GLOBAL ANALYSIS OF THE ASYMMETRIC IMPACTHE ASYMMETRIC IMPACT OF ENSO ON ET OF ENSO ON EXTREME PRECIPITATIONXTREME PRECIPITATION
1 DDATA AND METHODATA AND METHOD
1.2 A regional extreme value modelA regional extreme value model
2.1 Regional parameter estimatesRegional parameter estimates
2.2 The impact of ENSO on precipitation quantilesThe impact of ENSO on precipitation quantiles
2.3 Asymmetry of the impact of ENSO on extreme precipitationsAsymmetry of the impact of ENSO on extreme precipitations
2.4 Seasonality of the impact of ENSO on extreme precipitationsSeasonality of the impact of ENSO on extreme precipitations
3.1 Limitation of the model aLimitation of the model and reliability of the definition of a regionnd reliability of the definition of a region
3.2 Changes in ENSO teleconnectionsChanges in ENSO teleconnections
3.3 Impact of other large scale modes oImpact of other large scale modes of climate variabilityf climate variability
5.1 Slope of the location parameter with respect to SOISlope of the location parameter with respect to SOI
5.2 Percentage change for the intensity of 1 in 10 year precipitation relative to SOI=0Percentage change for the intensity of 1 in 10 year precipitation relative to SOI=0
5.3 Difference between the slope of SOI during La Niña and El Niño phasesDifference between the slope of SOI during La Niña and El Niño phases