Docteur De L'

Matteo Darienzo

Thèse Dirigée Par

Michel Michel Lang

Jérôme Le Coz

Benjamin Renard

Thèse

Anne-Catherine Mme

Favre

M Gil

M Dirceu

Reis Professeur

M Asgeir Petersen-Øverleir

M Jérôme

L E Coz

Invité M Benjamin

Invité M Arnaud Belleville

Karine Mme

Delamarre

THÈSE Pour obtenir le grade de

large experience in Hydrology and in supervising PhD students. He always tried to valorise my work. Jérôme Le Coz, for his ability to keep clear the overall picture of the scientic problems of this thesis, to manage the planning in order to respect the deadlines and to guide me over the operational aspects. Benjamin Renard, for his undeniable support during the statistical developments of this thesis work. I learnt so much from him. Moreover, they all showed great pedagogical skills, in particular during the redaction period, giving me advices, remarks and tips with patience and kindness.

Then, I would like to thank the members of the jury for their interest in this thesis project, and the members of the three thesis committees (Anne-Catherine Favre (UGA), Arnaud Belleville (EDF), Sara Puijalon (CNRS), Alexandre Hauet (EDF), Damien Sevrez (EDF), Elodie Dufeu (SCHAPI), David Besson (DREAL), Gilles Pierrefeu (CNR), Karine Delamarre (CNR), Benoit Camenen (INRAE)) for their undeniable contribution. Their criticisms and suggestions, especially from operational perspectives, have greatly contributed to the completion of this thesis work.

I also thank the funders of this thesis (Electricité de France -EDF, Compagnie Nationale du Rhône -CNR, and Service central d'hydrométéorologie et d'appui à la prévision des inondations -SCHAPI) to have demonstrated from the very beginning a true interest in the objectives of this project. This has always motivated my work, even during the most challenging periods. Moreover vii CONTENTS the substantial fundings allowed me to participate in enriching international conferences and formations.

Data and expert knowledge used in this thesis were provided by the French National Hydrological Services (Unité d'Hydrométrie et de Prévision des Crues Grand Delta, Guillaume Fourquet as regards the Ardèche River at Meyras station), the New Zealand Regional Council (Mike Ede, for the Wairau River at Barnetts Bank), the CVH Ile de la Réunion and Oce de l'Eau de la Réunion (Valérie Payet, for the Mat River at Escalier).

Other stations have been processed using data and knowledge from USGS (Thomas Over, Julie Kiang, Robert Mason). This thesis work has also made use of a few existing codes: BaM, Bayesian

Modelling (developed by Benjamin Renard and which also uses the DMSL library of Dmitri Kavetski), BaRatinAGE (https://forge.irstea.fr/projects/baratinage_v2), BaRatin-SPD (https: //forge.irstea.fr/projects/bam/les) developed by Valentin Mansanarez during his thesis.

I thank the holding institute (Institut national de recherche pour l'agriculture, l'alimentation et l'environnement -INRAE, ex Irstea), especially the Research Unit RiverLy based in Lyon-Villeurbanne for these years of perfect working conditions: the comfortable oces, the convivial atmosphere with colleagues, the breaks and pots at coin café". I want to thank Etienne for the RCEM conference in Auckland. And then, all other PhD students at INRAE of the UR RiverLy. In particular, Sheng who welcomed me with extreme kindness at my arrival, Yassine who started the PhD at the same period as me and helped me on numerous occasions, Ivan for our interesting discussions on programming and R, Alexandre for our discussions on statistics but more generally about life, Peng (who was bearing me during the last year of my thesis work, xièxiè), Miguel (El mexicano, hasta luego amigo!), but also Mathieu, Emilie, Guillaume B., Mohammad, Tarek, Musaab, Shashank, Junjian, Clarisse, Juliette. Finally, I thank all postdocs and cdd (among them Emeline, Aurélien, Léa, Jules, Quentin), all "stagiaires", all other researchers and Guillaume D for his technical tips for managing the defense in video-conferencing. Merci beaucoup. I also thank all other doctorants/postdocs met during the international conferences Matteo Darienzo CONTENTS (among them, my compatriots Marco DM and Andrea B).

My scientic skills are certainly also the results of the patience and professionalism of the supervisors of my previous research experiences: Olivier Boutron, Marion Vittecoq (La Tour du Valat) and Laurent Oxarango (IGE, ex LTHE). Moreover, I got some important skills in hydrometry from my experience at SEGI Ingénierie. Merci beaucoup.

Finally, but not for importance, I deeply thank Olga for her constant support and ability to ease the stress during these years in Lyon ("spasibo bolshoye"). Je remercie tous mes amis à

Lyon et tous ceux que j'ai rencontrés en France, ma ovviamente anche tutti i miei amici d'infanzia in Italia con i quali ho cresciuto, e la mia famiglia sempre presente nel momento del bisogno.

Grazie innite.

RÉSUMÉ

Les séries temporelles de débit des rivières sont établies à l'aide de "courbes de tarage", qui sont des modèles avec les hauteurs d'eau en entrée et les débits en sortie. Malheureusement, de nombreuses stations hydrométriques ont une relation hauteur-débit instable, notamment à cause de l'évolution du lit de la rivière lors des crues. Ces "détarages" posent problème à la fois pour l'établissement des séries hydrologiques de long-terme (analyse rétrospective) et pour la fourniture de données en temps réel, par exemple pour la prévision des inondations, avec des incertitudes quantiées et ables. Les méthodes existantes pour la mise à jour de la courbe de tarage sont basées sur une analyse statistique des données de calibration (jaugeages) passées ou sur des règles empiriques. Cette thèse a permis de développer des méthodes originales pour la détection automatique des détarages et l'estimation de leur amplitude en rétrospectif et en temps réel : une méthode de segmentation des résidus entre les jaugeages et une courbe de référence, une analyse des récessions du limnigramme et une détection de détarages potentiels à partir d'un indicateur disponible en temps réel (par exemple, transport sédimentaire cumulé). L'approche probabiliste permet d'une part de prendre en compte l'incertitude des informations sur les détarages et d'autre part de quantier les incertitudes sur les débits calculés. La combinaison des trois méthodes a été appliquée à la station de l'Ardèche à Meyras, en France, qui présente des détarages nets après chaque crue importante. Une bonne détection et estimation des détarages a été observée en rétrospectif et en temps réel. La méthode est générique et, après davantage de validation, applicable en opérationnel à d'autres sites.

xi ABSTRACT River discharge time series are established using "rating curves", which are models with stage as input and discharge as output. Unfortunately, many hydrometric stations have an unstable stage-discharge relation, particularly because of the change in the river bed during oods. These "rating shifts" pose a problem both for the establishment of long-term hydrological series (retrospective analysis) and for the delivery of real-time data, for example for ood forecasting, with quantied and reliable uncertainties. The existing methods for updating the rating curve are based on a statistical analysis of past calibration data (the gaugings) or on empirical rules. This thesis aims at developing some original methods for the automatic detection of rating shifts and the estimation of their magnitude in both retrospective and real time: a method of segmentation of the residuals between the gaugings and a base rating curve, an analysis of the stage recessions and a method for detecting potential shifts from an indicator available in real time (e.g. cumulative sediment transport). The probabilistic approach allows on the one hand to take into account the uncertainty of the information on the shift and on the other hand to quantify the uncertainties of the calculated streamow. The combination of the three methods has been applied to the Ardèche at Meyras, France, which shows net shifts after each major ood. Good detection and estimation of the rating shift has been observed retrospectively and in real time.

The method is generic and, after further validation, operationally applicable to other sites.

xiii LIST OF FIGURES

1.1

The Ardèche River at Meyras station (France), view downstream of Barutel bridge (taken from [START_REF] Mansanarez | Shift Happens! Adjusting Stage-Discharge Rating Curves to Morphological Changes at Known Times[END_REF]): a very common hydraulic conguration. . 1.11 Top panels: segmentation procedure applied to the time series of the stage residuals computed between the gaugings and a baseline RC [START_REF] Morlot | Dynamic rating curve assessment for hydrometric stations and computation of the associated uncertainties: Quality and station management indicators[END_REF].

Bottom panels: associated stage-discharge gaugings. . . . . . . . . . . . . . . . . 17

2.6

Examples of synthetic data sets characterised by known rating shift times using parameters of Classes 1-3-4-6-9-10 dened in Table 2.1. . . . . . . . . . . . . . . .

2.7

Schematic example of the gaugings classication into true positive (T P ), false negative (F N ), true negative (T N ), false positive (F P ) for the performance evaluation of the segmentation results. s i represent the known shift times, while τi are the change point estimates. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.8

Results of the rst experiment with Strategy D, comparing four criteria for the optimal choice of K and using all 100 simulated data sets. . . . . . . . . . . . . .

2.9

Results of the second experiment: comparison between the four strategies for the

3.3

Results of the proposed method applied to the Ardèche River at Meyras by using = 50 cm and models M1, M2, M4 of Table 3. for each period delimited by the detected shift times (a) and of estimated shift ∆ β (b) for nine dierent stage-recession models and χ = 50 cm. Also the estimated b 1 (MAP) and shifts ∆ b1 obtained from the gaugings [Darienzo et al., 2021; Mansanarez et al., 2019]

4.5

Results of the estimation of the cumulative volume of transported sediments after each potential morphogenic event for the Ardèche River at Meyras, France. . . .

4.6

Relation ∆b 1 ↔ V (a) and ∆b 2 ↔ V (b) for the Ardèche River at Meyras, France.

Calibration data are the estimated ∆b and V for each reference morphogenic event.

The number above the calibration data represent the index of the reference event.

The blue line represents the maximum a posteriori and the blue ribbon illustrates Streamow time series are therefore the baseline data for most hydrological studies. They support decision-making for the management of water resources and aquatic environments and for the prevention of ood risk, erosion and river pollution.

However, for many natural rivers, streamow time series are not direct observations: indeed, streamow cannot be measured continuously. Instead, at some specic locations along the river easily accessible and equipped for data logging (called hydrometric stations) the water level (hereafter called "stage") is monitored with a constant frequency (e.g., every 15 minutes) or a frequency increasing with stage variation.

Consequently, streamow time series are derived from the transformation of continuous measurement of water level via a stage-discharge relation, called the "rating curve" [WMO, 2010;[START_REF] Rantz | wesurement nd omputtion of strem)owF olume PX gomputtion of dishrge[END_REF], hereafter "RC".

1.1.2 Hydraulic principles behind the rating curve

The hydraulic relation between discharge and stage is determined by hydraulic controls.

They are classied into two main categories: the section controls, characterised by critical ow conditions induced by obstacles or change in the cross-section, e.g., natural rie, articial weir, and the channel controls, mainly inuenced by the bed slope and roughness and characterised by fairly uniform or friction-dominated ow. For both types, elementary controls have been extensively studied in the literature with typical formulas of the power-law form:

Q(h) = a(h -b) c (1.1)
linking the discharge Q to stage h, where a is the coecient related to the physical and geometrical properties of the control (e.g., the channel width, the longitudinal slope, the roughness), b is the oset (with respect to the instrument measuring stage) below which the ow is zero, and c is an exponent related to the type and shape of the hydraulic control [START_REF] Le Coz | Combining hydraulic knowledge and uncertain gaugings in the estimation of hydrometric rating curves: A Bayesian approach[END_REF]. The term h -b represents the water depth y.

In general several controls add or succeed to each other as ow increases. As an example of very common situations, Figure 1.1 illustrates the hydraulic conguration proposed by [START_REF] Mansanarez | Shift Happens! Adjusting Stage-Discharge Rating Curves to Morphological Changes at Known Times[END_REF] and [START_REF] Sikorska | Calibrating a hydrological model in stage space to account for rating curve uncertainties: general framework and key challenges[END_REF] for the Ardèche River at Meyras station located in a relatively small catchment of Mediterranean France. At this gravel bed stretch of the river the low ows are controlled by a natural rie (located almost 50 m downstream of the bridge, where stage is measured) which can be modelled as a rectangular weir section control. The medium-high ows are controlled by the characteristics of the main channel which can be modelled as a wide rectangular channel control. At very high ows, water also ows in the lateral oodplain which can be modelled as a wide rectangular channel control added to the main channel control.

Once the hydraulic conguration has been dened, the RC parameters are calibrated using some occasional stage-discharge measurements (also called "gaugings"). Unfortunately, the gaugings are aected by measurement uncertainty. While the uncertainty of gauged stage is often neglected for the RC estimation, the uncertainty of gauged discharges cannot be neglected.

Standard

Rating curve uncertainty

Estimating the parameters of the RC based on a limited number of uncertain gaugings leads to substantial RC uncertainty. Two sources of uncertainty can be distinguished [START_REF] Le Coz | Combining hydraulic knowledge and uncertain gaugings in the estimation of hydrometric rating curves: A Bayesian approach[END_REF][START_REF] Kiang | A comparison of methods for streamow uncertainty estimation[END_REF]:

-the parametric uncertainty resulting from the imperfect identication of RC parameters;

-the structural (also called "remnant" or "epistemic") uncertainty related to the imperfection of the considered RC model.

1 Unless specied otherwise, uncertainties expressed as ± u denote the width of a 95 % probability interval.

For instance a value x aected by a probability ± 10 % means that a 95 % probability interval is [0.9 x; 1.1 x]. As an order of magnitude, the total uncertainty is typically of ± 50 -100 % for low ows, ± 10 -20% for medium ows, and ±40% for out of bank ows [START_REF] Mcmillan | Benchmarking observational uncertainties for Matteo Darienzo hydrology: rainfall, river discharge and water quality[END_REF].

The various methods for estimating the RC with quantitative uncertainty are in general based on the regression of piecewise power functions accounting for gaugings uncertainty [START_REF] Kiang | A comparison of methods for streamow uncertainty estimation[END_REF]. They basically dier in:

-the assumption they made about the main sources of errors (e.g., structural error not formally accounted for, or accounted for with a standard deviation, constant or linearly increasing with discharge);

-the statistical tools they use (e.g. least square regression, Bayesian inference);

-the amount of hydraulic information used in addition to the gaugings (e.g., the Bayesian approaches allow specifying informative priors on the hydraulic knowledge of the RC parameters);

-the way they estimate the low and high ows outside the gaugings range (e.g., by extrapolation, or by prior specication on the very low ow control and on the ood type (modied from [START_REF] Herschy | trem)ow mesurement[END_REF] and [START_REF] Coxon | A novel framework for discharge uncertainty quantication applied to 500 UK gauging stations[END_REF]).

plain).

As an example, Figure 1.2 illustrates the results of the RC estimation with uncertainty for the Meyras station by using the BaRatin method [START_REF] Le Coz | Combining hydraulic knowledge and uncertain gaugings in the estimation of hydrometric rating curves: A Bayesian approach[END_REF], which introduces in the Bayesian inference the preliminary hydraulic analysis through the prior specication. The gure does not only illustrate the most probable (maximum a posteriori) RC and the two types of RC uncertainty, but also displays the activation stages, i.e. the stage values corresponding to the transitions between two subsequent controls.

Rating changes

Unfortunately, one of the major issues aecting the RCs is that the stage-discharge relation is not only uncertain but can be unstable and aected by rating changes over time. The rating changes impose episodic or continuous RC updates.

Various causes lead to the RC changing in various ways, including simple translations of the Modied from [START_REF] Mansanarez | Shift Happens! Adjusting Stage-Discharge Rating Curves to Morphological Changes at Known Times[END_REF].

whole curve to more complex modications. The causes of this instability have been extensively studied in the literature (e.g. [START_REF] Herschy | trem)ow mesurement[END_REF]) and some examples are illustrated in Figure 1.3.

Rating changes can be classied into two main categories: transient and sudden changes.

Transient changes are caused by progressive phenomena that lead to a wide variety of RC changes, such as sediment dynamics, hysteresis during oods, aquatic vegetation, variable backwater eect and ice sheets covering.

On the other hand, sudden changes (hereafter called rating shifts) are related to specic and short events inducing net rating changes between before and after the event, such as morphogenic oods, relocation or modication of the gauging station, operation of hydraulic structures such as the change in the opening of sluice gates, dike/levee break, dams built by swimmers or beavers, gravel or sand extraction from the river bed, etc.

In the case of Meyras station (Figure 1.4) the RC changes are most likely caused by sudden vertical shifts of the gravel bed elevation. These shifts are induced by some episodic and intense After having characterised the processes aecting the RC stability, the next step is to identify the RC parameters that are more likely to vary over time. For instance processes aecting width, slope and roughness of the channel induce a change in the parameter a of the control equation (Equation 1.1), and processes aecting the channel elevation induce a change in the parameter b. Instead, parameter c, which denes the type and the shape of the hydraulic control, is more rarely aected by rating changes. Moreover, identifying the unstable RC parameters may be very challenging also because some processes may occur simultaneously.

However, determining the nature of the change is not sucient. The hydrologist also needs to identify among the hydraulic controls of the station which ones are more likely aected. For example in the case of morphological changes induced by oods the low ow section control (e.g. natural rie) is in general more frequently and severely aected by net shifts than the high ow channel control. This is because the section controls are sensitive to local adjustments in the bed micro-topography whereas the channel controls are not. Thus, minor events may suce to induce net shifts at the low-ow controls. To induce net shifts at high ow controls, much greater events are required.

In the Meyras example, assuming that the channel banks have a rigid rocky composition (Figure 1.1), the channel bottom mostly degrades or raises vertically while the channel width remains stable. This induces mainly shifts of the oset parameters of the low ow controls: b 1 (mean elevation of the natural rie crest) and b 2 (mean elevation of the main channel bed).

Managing rating changes in real time

Tracking and estimating these rating changes in order to update the RC is of primary importance for many operational applications, for instance for ood forecasting, hydroelectricity, compliance with environmental ows and nutrient/pollutant ux limits, administrative decisions related to low ows, restrictions or prohibitions on water diversion (for irrigation, etc.), shutdown of nuclear reactors, etc. [START_REF] Osorio | A Bayesian Approach for the Evaluation of Rating Curve Uncertainties in Flood Frequency Analyses[END_REF][START_REF] Mcmillan | How uncertainty analysis of streamow data can reduce costs and promote robust decisions in water management applications[END_REF].

There is a strong interest in obtaining and communicating values of streamow in real time, accompanied by quantied uncertainties. This is particularly challenging at unstable hydrometric stations, aected by rating changes.

In the operational practice, the main source of information to detect and estimate RC changes is represented by the gaugings. When a gauging is far away from the last stable rating curve then the practitioner is aware that a potential shift may have occurred. Unfortunately, gauging campaigns are relatively costly and time consuming and can be problematic in particular site and hydraulic conditions. Thus, in general detecting a rating shift may take several months. During this period the ocial RC is obsolete and the discharge values estimated in real time by using this model might be biased.

This leads to the following questions: how to detect rating shifts as quickly as possible?

Which information other then a gauging can be used to this aim? How to manage the uncertainty around the released streamow data? In real time, when rating shifts cannot be Thus, a poor streamow forecasting may have two types of consequences: a) it may fail to issue a warning for a ood event leading to potential loss of life and infrastructure; or b) it may issue a warning for an event that does not occur, which may erode people's trusting in the forecast and lead them to not respond to the next warning.

Two distinct sources of streamow information are used for real-time ood risk management:

-the real-time observed streamow (directly measured at some stations or estimated through the RC).

Chapter 1. Introduction -the forecasted streamow for the actual and next time steps, usually obtained from hydrological rainfall-runo models describing the water balance in the river catchement.

These models use meteorological data (rainfall, wind, temperature, etc) as input and historic streamow data for the model calibration.

To reduce ood forecasting uncertainty sources a standard practice is to assimilate the real-time streamow observations into the forecasting process in order to correct the registered deviations between the uncertain forecasted ows and the uncertain observed ows.

In the case of unstable rating curves the real-time analysis of streamow data uncertainty is very challenging but is still necessary since it aects the data assimilation process and hence ood forecasting [START_REF] Ocio | The role of rating curve uncertainty in real-time ood forecasting[END_REF]. During oods in June 2016 on the Cher and Seine catchments in France hydrometric services (the DREAL Centre-Val-de-Loire and the DRIEE Ile-de-France)

had to extrapolate and re-estimate their rating curves in emergency (the same day), especially due to the dierence in oodplain vegetation between summer and winter.

Hydroelectric power plants

Hydroelectric power plants can be mainly split in impoundment facilities (where the water is stored in a reservoir controlled by a dam and then released from the reservoir to the penstocks containing the turbines located at a lower elevation) and run-of-river plants (with little or no water storage, composed of a diversion structure, not necessarily a dam, that derives the ow destined to the turbines, example in Figure 1.7a). Both types need to carefully monitor streamow in order to:

-optimise the energy production, by combining the energy price market, the energy demand cycles and the available streamow diverted to the turbines (for example Q 2 in Figure 1.7b).

-control the oods through dam operations and to defend structures and workers from dangerous oods. This requires the monitoring of the incoming main streamow (for example Q 1 in Figure 1.7b).

-guarantee that the instream ow released in the main channel (for example Q 3 in A poor real-time streamow estimation may lead to economical losses for the hydropower company, to material and human losses in case of oods, or to environmental issues during low ows followed by sanctions implemented after Water Authorities investigations.

(a) As for hydropower plants all the phases of the siting, design, construction and realtime operation of an NPP (Figure 1.8a) require accurate streamow estimation in order to ensure the necessary water supply (in this case water is used for cooling the condenser and the spent-fuel pool), to protect structures and workers against oods, and to ensure a minimum environmental ow.

Q 3 Q 1 Q 4 POWER PLANT Q 2 (b)
As an example, in 2011, the NPP of Fort Calhoun in Nebraska along the Missouri River was ooded (Figure 1.8b) and shut down (source: https://en.wikipedia.org/wiki/Fort_Calhoun_Nuclear_Generating_Station).

In addition to these issues the NPP needs to avoid or limit accidental pollution, that can be thermal (the hot water released by the NPP to the main stream may In conclusion both water quantity and quality need to be monitored both upstream and downstream of an NPP. The stability of the rating curve at the hydrometric station of the NPP needs to be frequently monitored, in particular if rating changes are suspected to occur often. In order to describe the transport and the dispersion of the contaminant by water, reliable streamow data is essential. It is particularly important in real time for the accident forecasting and prevention and/or the estimation of the short and long-term environmental eects. As an example a recent (2020) contamination of the Ambarnaya River in Russia (Figure 1.9) by 20,000 tons of diesel oil was constantly and carefully monitored by state Authorities in order to take actions to limit the environmental consequences.

Water-use restrictions

Real-time streamow data is also of primary importance for those complex systems governed by one or more reservoirs where water resources need to be optimised for multiple uses at the same time: e.g. irrigation (Figure 1.10a), water consumption, hydroelectricity production. Administrative decisions related to low ows may lead to restrictions or prohibition of withdrawals.

In France, hydrological situation bulletins have to be released every three weeks or so by water authorities to allow for decisions on water use and allocation during drought seasons.

A severe drought in 2019 has impacted the Mekong River (Figure 1.10b), causing record low water levels in Thailand, Laos, Cambodia and Vietnam. Lack of water had devastating consequences for sh, as well as the tens of millions of people living and working along the river. Poor real-time streamow estimation in this case may possibly cause political conicts on the water management since the Mekong River ows through several countries characterised by dierent water regulation laws.

1.2 State-of-the art for the detection and estimation of rating changes Several methods (manual or automated) have been proposed in the literature to formally or empirically track and estimate the magnitude of rating changes over time, as reviewed by [START_REF] Mansanarez | Shift Happens! Adjusting Stage-Discharge Rating Curves to Morphological Changes at Known Times[END_REF]. However their applicability to real-time applications is challenging and quite limited.

The transient and sudden changes (mentioned in Section 1.1.4) require dierent approaches.

While transient changes require dynamic modelling or continuous updates of the RC, sudden changes require, rstly, the detection of the shift times with the denition of stationarity periods of the RC, secondly, the RC estimation for each period.

Dynamic modelling of transient changes

In the past, dynamic approaches have existed in the operational practice with gradual modication of the RC (called "correction curve"). These methods are time-intensive and the applied corrections are done without considering the underlying hydraulic controls. Moreover, the calibration of the RCs and the review of the results remain very manual, without quantifying the uncertainties, and unsuitable for real-time management.

Bayesian methods have recently been developed to introduce some physical knowledge about the rating changes. [START_REF] Reitan | Dynamic rating curve assessment in unstable rivers using Ornstein-Uhlenbeck processes[END_REF] developed a dynamic model based on time-varying RC parameters within a hierarchical Bayesian framework. [START_REF] Mansanarez | xonEunique stgeEdishrge reltionsX fyesin nlysis of omplex rting urves nd their unertinties[END_REF] proposed a method for complex ratings, including stage-fall-discharge models for twin gauge stations aected by variable backwater (introducing an additional stage input variable, h 2 ), and stage-gradient-discharge (SGD) models to address hysteresis due to transient ows and the eect of the ood wave propagation [START_REF] Mansanarez | Stage-gradient-discharge models: Bayesian analysis of hysteresis due to unsteady ows. ter esoures eserh[END_REF]. [START_REF] Perret | A Rating Curve Model Accounting for Cyclic Stage-Discharge Shifts due to Seasonal Aquatic Vegetation[END_REF] developed physically-based models to account for the aquatic vegetation dynamics (through the Strickler roughness coecient, which involves a modication of parameter a of the RC). The models are calibrated using the gaugings and some qualitative information on vegetation density through a Bayesian approach.

Detecting and estimating sudden changes

Several methods have also been proposed to detect the rating shifts in order to identify the periods characterised by RC stability.

The most common practice to detect a shift is to use gaugings. Some empirical approaches have been proposed, such as the detection of a shift when the gauging departs from the previous RC for more than some predened threshold. However these approaches do not consider the uncertainty of both the gaugings and the RC. A more formal way to analyse the set of gaugings is to apply a statistical analysis (segmentation) to the residuals between the gaugings and a baseline RC, as proposed by Morlot et al. [2014] (Figure 1.11). A more detailed review on the segmentation issue will be proposed in Chapter 2.

Another way to detect rating shift is to use the stage record, which, contrarily to gaugings, is available in continuous and much cheaper. apuszek [2003] demonstrated that analysing the annual minimum stages (called H in Figure 1.12) may be of interest in order to detect anomalies or a trend in river bed evolution. However this method is not suited to the real time application because of its annual time step. A shift may be detected after several years.

Once the periods of validity have been dened, dierent methods have been proposed in the literature to update the corresponding RC. The "shift correction" techniques (used by USGS and Water Survey of Canada [START_REF] Rainville | Hydrometric manual: Data computations, stage-discharge model development and maintenance[END_REF]) apply a shift s(h) to the RC oset:

Q(h) = a(h -b -s(h)) c (1.2)
These corrections allow the hydrologist to account for the hydraulic conguration of the river and to select (based on the expertise) the specic controls assumed to be aected by the shift.

Alternatively, the RC can be estimated by using, in a sequential way, one of the methods mentioned in Section 1.1.3 for RC estimation with uncertainty [START_REF] Kiang | A comparison of methods for streamow uncertainty estimation[END_REF]. However, this becomes challenging for those periods characterised by few (or no) available gaugings, which may cause a large RC uncertainty. Thus a current practice is to re-use some gaugings from other periods to estimate the RC of the current period, therefore assuming that the corresponding controls are stable [START_REF] Mansanarez | Shift Happens! Adjusting Stage-Discharge Rating Curves to Morphological Changes at Known Times[END_REF]]: e.g., this is typically applied to high-ow gaugings [START_REF] Mcmillan | Impacts of uncertain river ow data on rainfall-runo model calibration and discharge predictions[END_REF][START_REF] Puechberty | Charte qualité de l'hydrométrie: Guide de bonnes pratiques. qroupe hoppler rydrométrieF winistère de l9invironnementD de l9inergie et de l werD prne[END_REF].

A solution to this issue has been proposed by [START_REF] Mansanarez | Shift Happens! Adjusting Stage-Discharge Rating Curves to Morphological Changes at Known Times[END_REF] who developed stageperiod-discharge models for estimating successive RCs and their uncertainty with a transfer of information across periods through a Bayesian approach. In these models, RC parameters that are suspected to shift between periods are specic to each period, while other parameters remain unchanged across all periods. As an example, the application of this method to the Meyras case study is illustrated in Figure 1.4.

Real-time challenges

All the traditional and emerging methods introduced in the previous subsections are usually performed in retrospective, thus when the hydrometric data sets are fully available and when a rating shift has already occurred. This is acceptable when the interest of the station managers 1.2. State-of-the art for the detection and estimation of rating changes is to perform a re-analysis of the historical hydrograph for purposes such as ood frequency analysis or to dene the instability level of a station for planning future gauging campaigns.

On the contrary, the use of the traditional methods is quite limited for real-time purposes such as the operational applications described in Section 1.1.5, because of their dependence on gaugings (seldom performed and very uncertain if performed during oods) or on the stage record but with a too long time step (year).

Detecting a rating shift (real or suspected) may be required in real time, e.g., during oods or soon after, without waiting for the next gaugings. To this aim, station managers use visual control (more seldom bathymetry surveys) and all kind of information on, e.g., river works, gravel mining operations, vegetation growth or ice jams. These methods are, in the opinion of the operational sta themselves, unsatisfactory, in particular because they are not formalized and hardly reproducible but it is the best that can be done with the tools currently available.

Moreover, each type of process causing rating change is associated with a dierent time step. During oods, the rating change induced by morphological change or due to hysteresis may occur in a period of a few hours or a few days. During a period of growth of aquatic vegetation the rating change may occur for several months with observable changes at a daily/weekly/monthly time step. A dierent time step may be used also depending on the operational application. For ood forecasting purposes or in the case of accidental pollution the hydrometric services and water authorities may want to update the RC or at least its uncertainty, for example, every 15 minutes or every hour during/after the event. Instead for water restrictions during droughts a weekly/monthly time step may suce.

The real-time management of the RC was the initial motivation for the development of the GesDyn method [START_REF] Morlot | Dynamic rating curve assessment for hydrometric stations and computation of the associated uncertainties: Quality and station management indicators[END_REF] by EDF and LTHE through the PhD thesis of Morlot [2014]. However, the real-time application could not be achieved because a new gauging is required to trigger an RC update, with no specic mechanism to update the curve or its uncertainty in etween gaugings.

Chapter 1. Introduction

Alternatively, the operational services often analyse the spatial and temporal behavior of several stations to detect potential shifts by examining the hydrographs from correlated stations (upstream-downstream station, station located in a tributary river or in a neighboring catchment, etc.). A change in the regression between the stations (e.g., after a ood) leads to question the rating curves of the two stations [START_REF] Puechberty | Charte qualité de l'hydrométrie: Guide de bonnes pratiques. qroupe hoppler rydrométrieF winistère de l9invironnementD de l9inergie et de l werD prne[END_REF]. This method is usually based on a monthly time step, thus its applicability during oods is limited. Moreover, it requires the existence of a hydrologically comparable and stable station, which is never granted.

In addition, rainfall-runo modelling is widely available and is used for real-time purposes by operational services [START_REF] Puechberty | Charte qualité de l'hydrométrie: Guide de bonnes pratiques. qroupe hoppler rydrométrieF winistère de l9invironnementD de l9inergie et de l werD prne[END_REF][START_REF] Lucas | Développement d'une méthode de détection en temps-réel des ruptures dans la relation hauteur/débit. pport de stge de (n d9étudesD endré pr ierre frigodeD olyteh xieEophiD et hmien evrezD ihpEhq[END_REF]. A rating shift may be detected by comparing the output of such models with the streamow obtained from the RC. 1. tools for the detection of eective rating shifts using essentially the gaugings and the stage record.

2. tools based on proxy models for the detection of potential rating shifts using the only information always available in real time: the stage record.

3. a general real-time framework for the streamow quantication based on the sequential re-estimation of the RC accounting for potential shifts.

As regards the rst objective two tools for detecting eective rating shifts are presented:

An original segmentation procedure applied to the time series of the residuals computed between the gaugings and a reference rating curve (Chapter 2). [START_REF] Darienzo | Detection of Stage-Discharge Rating Shifts Using Gaugings: A Recursive Segmentation Procedure Accounting for Observational and Model Uncertainties[END_REF].

Chapter 1. Introduction

An original stage-recession analysis to detect changes in the recession shape over time (Chapter 3). The method aims at detecting and estimating step changes in the river bed elevation using the stage record as an input. It is based on: the extraction of all available recessions; the estimation of a recession model through a Bayesian approach; the detection of shifts in specic parameters of the recession model over time. The text in Chapter 3 is also written as a journal article that will be submitted shortly.

As regards the second objective, Chapter 4 describes a tool for detecting potential rating shifts caused by morphological changes during oods. The tool is based on a sediment transport proxy analysis and is of particular interest for real-time applications. Using the stage record, it computes a cumulative bed load estimate for each ood event in order to identify the events more likely to cause a net river bed change.

The third objective of this PhD work will be treated in Chapter 5 by discussing the results of a rst proof-of-concept application of a real-time streamow estimation framework. In particular, the real-time framework requires:

One tool for estimating the RC estimation and its uncertainties. In this manuscript the Bayesian approach (and in particular the BaRatin method [START_REF] Le Coz | Combining hydraulic knowledge and uncertain gaugings in the estimation of hydrometric rating curves: A Bayesian approach[END_REF]) will be preferred because of its ability to include in the statistical inference the physical knowledge of the rating shift through the prior specication of RC parameters).

At least one tool for detecting and estimating rating shifts in a real-time context, and thus capable of providing a continuous information about the RC stability. This tool needs to be calibrated and validated through a retrospective analysis that consists in detecting and studying all past rating shifts.

The general perspectives for future work including possible improvement, potential operational application of the tools proposed in this manuscript, and the nal conclusions will be presented in Chapter 6.

CHAPTER 2

SEGMENTATION OF GAUGINGS

Abstract

The stage-discharge rating curve is subject at many hydrometric stations to sudden changes (shifts) typically caused by morphogenic oods. We propose an original method for estimating shift times using the stage-discharge observations, also known as gaugings. This method is based on a recursive segmentation procedure that accounts for both gaugings and rating curve uncertainties through a Bayesian framework. It starts with the estimation of a baseline rating curve using all available gaugings. Then it computes the residuals between the gaugings and this rating curve with uncertainties. It proceeds with the segmentation of the time series of residuals through a multi change point Bayesian estimation accounting for residuals uncertainties. Once a rst set of shift times is identied, the same procedure is recursively applied to each subperiod through a "top-down" approach searching for all eective shifts. The proposed method is illustrated using the Ardèche River at Meyras in France (a typical hydrometric site subject Chapter 2. Segmentation of gaugings to river bed degradation) and evaluated using several synthetic data sets for which the true shift times are known. The applications conrm the added value of the recursive segmentation compared with a "single-pass" approach and highlight the importance of properly accounting for uncertainties in the segmented data. The recursive procedure eectively disentangles rating changes from observational and rating curve uncertainties.

Plain Language Summary

For many hydrological and hydraulic issues, such as ood forecasting, a reliable river discharge estimate is needed. In general discharge is derived from the recorded water level (stage) through a stage-discharge relation (rating curve). This relation is calibrated using direct observations (gaugings). Unfortunately the rating curve is not only uncertain but it can also be subject to sudden changes or shifts due for example to intense oods that modify the river bed geometry.

One solution to identify periods of rating curve stability is to apply a segmentation procedure to the gaugings. We propose in this paper an original recursive segmentation procedure that accounts for both gaugings and rating curve uncertainties.

Introduction

Rating curves

River discharge, or streamow, is one of the most important variables in hydrology and hydraulics. Hydrometric data are essential for the calibration of hydrological models, ood forecasting and warning, engineering design (of dam and bridges for example) and policy decisions related to water resources and environmental management. However, streamow time series are not direct observations as streamow cannot generally be continuously measured in natural rivers.

Instead, the water level (also called "stage") is continuously monitored.

Streamow time series is hence derived from rating curves [WMO, 2010;[START_REF] Rantz | wesurement nd omputtion of strem)owF olume PX gomputtion of dishrge[END_REF], hereafter called "RCs", which are models transforming an input stage into an output discharge. These models are estimated using occasional stage-discharge measurements (also known as gaugings)

and hydraulic constraints. The physical relation between discharge and stage is determined by hydraulic controls, that is, physical characteristics of the river section or channel inuencing the ow: geometry, friction, longitudinal slope, head losses, etc. A hydraulic analysis of the site, through eld expertise or modelling, allows identifying the succession or addition of hydraulic controls as ow increases [START_REF] Le Coz | Combining hydraulic knowledge and uncertain gaugings in the estimation of hydrometric rating curves: A Bayesian approach[END_REF].

RCs are aected by many sources of uncertainty, including structural uncertainty (imperfection of the RC model), gaugings measurement uncertainty, and parametric uncertainty (estimation of RC parameters). Several methods for quantifying RC uncertainty have been developed as recently reviewed by [START_REF] Kiang | A comparison of methods for streamow uncertainty estimation[END_REF].

Unfortunately, the stage-discharge relation can be unstable and aected by rating changes. When ignored, these changes may be the main source of RC uncertainty [START_REF] Ibbitt | Gauging frequency and detection of rating changes[END_REF]. Concerning the causes of this instability, it is useful to distinguish between transient and sudden rating changes. Transient changes are caused by progressive phenomena [START_REF] Herschy | trem)ow mesurement[END_REF] such as hysteresis in unsteady ow during oods (rising limb and falling limb have dierent discharge values for the same stage), variable backwater due to unsteady downstream boundary conditions (e.g., stage controlled by a reservoir, sea tidal eect), growth and decay of aquatic vegetation, evolution of ice sheets covering cold-water rivers, river bed evolution due to sediment dynamics with progressive scour and ll. On the other hand, sudden changes (shifts) are related to specic and occasional events such as morphogenic oods, dike breaks, dams built by swimmers or beavers, etc.

The next sections review the methods proposed in the literature to estimate the magnitude of rating changes and therefore to track the variability of RCs in time. The aforementioned transient and sudden changes require dierent approaches.

Detecting and modelling transient changes

Modelling transient changes requires dynamic approaches. Such methods have actually existed in the operational practice since the beginning of the 20th century at least, in the form of time-varying RCs accounting for gradual rating changes. [START_REF] Schmidt | enlysis of tgeEhishrge eltions for ypenEghnnel plows nd heir essoited nertinties[END_REF] described the Stout's method (circa 1900) based on gradual daily shifts estimated from the gaugings. The same author also described the similar Bolster's method (circa 1910) which interpolates gaugings every day and develops parallel rating curves. Recently, [START_REF] Westerberg | Stage-discharge uncertainty derived with a non-stationary rating curve in the Choluteca River, Honduras[END_REF] and [START_REF] Guerrero | Temporal variability in stagedischarge relationships[END_REF] proposed to estimate RCs on moving temporal windows containing 30 successive gaugings. [START_REF] Morlot | Dynamic rating curve assessment for hydrometric stations and computation of the associated uncertainties: Quality and station management indicators[END_REF] proposed to compute as many RCs as there are gaugings and introduced the concept of RC aging: following an RC update, uncertainty increases with time according to a variographic analysis [START_REF] Jalbert | Temporal uncertainty estimation of discharges from rating curves using a variographic analysis[END_REF]. [START_REF] Reitan | Dynamic rating curve assessment in unstable rivers using Ornstein-Uhlenbeck processes[END_REF] developed a dynamic model based on time-varying RC parameters within a hierarchical Bayesian framework. Finally, in the specic context of sites aected by aquatic vegetation, [START_REF] Puechberty | Charte qualité de l'hydrométrie: Guide de bonnes pratiques. qroupe hoppler rydrométrieF winistère de l9invironnementD de l9inergie et de l werD prne[END_REF] proposed time-varying stage corrections and [START_REF] Perret | A Rating Curve Model Accounting for Cyclic Stage-Discharge Shifts due to Seasonal Aquatic Vegetation[END_REF] introduced the Bayesian estimation of a time-dependent rating curve model accounting for vegetation growth and decay, with year-specic parameters.

Detecting sudden changes

As rating changes often result from episodic morphogenic oods, models assuming sudden rating changes between stability periods are more widespread than dynamic models in the operational practice. This approach requires solving two main issues: detecting changes (which includes estimating shift times), and estimating the successive stable RCs with their associated uncertainties. In this paper we will focus on the rst issue only. We refer the reader to [START_REF] Mansanarez | Shift Happens! Adjusting Stage-Discharge Rating Curves to Morphological Changes at Known Times[END_REF] for a review on the second issue.

Several methods exist for sudden rating change detection. The most common approach is arguably to use gaugings. For instance, an empirical rule [WMO, 2010] prescribes to start a new period (and hence a new RC) when a gauged discharge departs from the current RC by more than some predened threshold, or when successive gauged discharges are systematically above or below the current RC. This rule varies across agencies and site conditions, for instance:

±5 % of discharge or ±0.6 cm (±0.02 ft) in stage for low-ow controls for the USGS [START_REF] Rantz | wesurement nd omputtion of strem)owF olume PX gomputtion of dishrge[END_REF]; ±5 % of discharge for the Water Survey of Canada [START_REF] Rainville | Hydrometric manual: Data computations, stage-discharge model development and maintenance[END_REF]; ±10 % of discharge and/or ±2 -3 cm in stage in France [START_REF] Puechberty | Charte qualité de l'hydrométrie: Guide de bonnes pratiques. qroupe hoppler rydrométrieF winistère de l9invironnementD de l9inergie et de l werD prne[END_REF]. While easy to apply, this approach is based on empirical rules and ignores both gaugings and RC uncertainties.

In addition, operational services monitor the evolution of the river bed elevation to detect a change in the corresponding RC parameter. They use eld observations such as information about river works, gravel mining operations and bathymetry surveys. It is also possible to install submersible pressure transducers at various locations along the reach [START_REF] Phillips | Detecting the Timing of Morphologic Change Using Stage-Discharge Regressions: A Case Study at Fishtrap Creek, British Columbia, Canada[END_REF]. An observed drastic drop or raise in the stage record may indicate a sudden river bed change. Moreover, apuszek and Lenar-Matyas [2015] evaluated whether changes in the annual minimum stages may indicate changes of the river bed level. This method is useful to provide a trend of the river bed evolution, but due to its annual resolution it cannot precisely identify the dates of rating shifts. Alternatively, [START_REF] Mcmillan | Impacts of uncertain river ow data on rainfall-runo model calibration and discharge predictions[END_REF] proposed to arbitrarily select the 0.5-year return period discharge as a threshold triggering a new RC period.

Furthermore, some operational services perform correlation analyses with reference stable discharge time series (e.g., the output of a hydrological model, or a discharge time series from a stable and well gauged neighboring station). Changes in the correlation structure may be indicative of rating shifts.

Finally, a formal way to detect changes by using gaugings is to apply a segmentation procedure to the time series of residuals between the gaugings and a time-invariant RC in order to identify homogeneous families of gaugings, as done for instance by [START_REF] Morlot | Dynamic rating curve assessment for hydrometric stations and computation of the associated uncertainties: Quality and station management indicators[END_REF]. This

Chapter 2. Segmentation of gaugings paper focuses on this approach.

Change point detection methods

The problem of nding abrupt changes in a time series is known in the literature as Change

Point Detection (CPD). Reviews of the most representative methods have been proposed by many authors, such as [START_REF] Basseville | hetetion of erupt ghnge E heory nd epplition[END_REF]; [START_REF] Ducré-Robitaille | Comparison of techniques for detection of discontinuities in temperature series[END_REF]; [START_REF] Keogh | Segmenting time series: A survey and novel approach[END_REF]; [START_REF] Jandhyala | Inference for single and multiple change-points in time series[END_REF]; [START_REF] Aminikhanghahi | A survey of methods for time series change point detection[END_REF].

A distinction can be made between single (sCPD) and multiple (mCPD) change point detection methods. sCPD methods are based on statistical tests questioning the existence of one single change aecting typically the mean or the median of the series, sometimes higher moments as well. The literature proposes several methods, e.g. likelihood-ratio tests [START_REF] Hinkley | Inference about the change-point in a sequence of random variables[END_REF][START_REF] Chen | Information criterion and change point problem for regular models. nkhy© X he sndin tournl of ttistis @[END_REF][START_REF] Bibliography Chen | rmetri sttistil hnge point nlysisX ith pplitions to genetisD mediineD nd[END_REF], non-parametric tests [START_REF] Wilcoxon | Individual comparisons by ranking methods[END_REF][START_REF] Pettitt | A Non-Parametric Approach to the Change-Point Problem. tournl of the oyl ttistil oietyF eries g @epplied ttistisA[END_REF][START_REF] Kruskal | Use of ranks in one-criterion variance analysis[END_REF][START_REF] Mcgilchrist | Note on a distribution-free cusum technique[END_REF], Standard Normal Homogeneity tests [START_REF] Hawkins | Testing a sequence of observations for a shift in location. tournl of the emerin ttistil essoition[END_REF]; [START_REF] Alexandersson | A homogeneity test applied to precipitation data[END_REF], and Bayesian procedures [START_REF] Cherno | Estimating the current mean of a normal distribution which is subjected to changes in time[END_REF][START_REF] Lee | A shift of the mean level in a sequence of independent normal random variables: A bayesian approach[END_REF][START_REF] Booth | A Bayesian approach to retrospective identication of changepoints[END_REF][START_REF] Perreault | Detection of changes in precipitation and runo over eastern canada and u.s. using a bayesian approach[END_REF]Perreault et al., , 2000a,b],b].

Alternatively, mCPD methods look for multiple change points in the series. Unlike sCPD, mCPD is aected by a combinatorial challenge induced by the large number of possible change point positions. The Binary Segmentation or BS [START_REF] Scott | A cluster analysis method for grouping means in the analysis of variance[END_REF] recursively performs sCPD until no more changes are detected in any of the obtained segments. However this approach is prone to known issues such as premature termination (schematized in Figure 2.1a) and mislocated splits (Figure 2.1b). To overcome these issues, several options are available: e.g. the Circular BS method proposed by [START_REF] Olshen | Circular binary segmentation for the analysis of array-based DNA copy number data[END_REF], sequential methods [START_REF] Page | Continuous Inspection Schemes[END_REF][START_REF] Hubert | Approche statistique de l'aridication de l'Afrique de l'Ouest[END_REF][START_REF] Hubert | Segmentation des séries hydrométéorologiques. Application à des séries de précipitations et de débits de l'Afrique de l'Ouest[END_REF], Dynamic Programming [START_REF] Auger | Algorithms for the optimal identication of segment neighborhoods[END_REF][START_REF] Killick | Optimal detection of changepoints with a linear computational cost[END_REF], Bayesian inference [START_REF] Green | Reversible jump Markov chain Monte Carlo computation and Bayesian model determination[END_REF][START_REF] Chib | Estimation and comparison of multiple change-point models[END_REF][START_REF] Lavielle | An application of mcmc methods for the multiple changepoints problem[END_REF] or Hidden Markov Models [START_REF] Cappé | snferene in ridden wrkov wodels @pringer eries in ttistisA[END_REF][START_REF] Luong | Hidden markov model applications in change-point analysis[END_REF].

However, these methods lack exibility in the treatment of uncertainties aecting the segmented data. Indeed, the total uncertainty aecting RC residuals is induced by both gaugings and RC uncertainties. This uncertainty is not only potentially large, but it may Chapter 2. Segmentation of gaugings also strongly vary from point to point. This should be accounted for in the segmentation procedure, as illustrated in the conceptual example of Figure 2.1c: ignoring data uncertainty clearly suggests three periods, while recognizing that two points have a much larger uncertainty than the others suggests a single period may suce. An option is to consider rescaled residuals instead of absolute residuals (e.g. dividing the absolute residuals by the standard deviation representing their uncertainty). However in some cases this rescaling might create spurious periods as illustrated in Figure 2.1d.

Furthermore, mCPD methods attempt to detect all changes in a given data set. In the context of RC shifts, this data set is derived using a baseline RC tted to all gaugings, which may be a very poor representation of the stage-discharge relation.

The large scatter and uncertainty of residuals may hide smaller changes that may be missed by such single-pass" procedure.

A recursive procedure, re-estimating the RC on each sub-period and deriving updated data sets of RC residuals, may hence be of interest.

Finally, the estimated change points provided by mCPD methods are not well-adapted to the context of RC shifts for the following reasons: (i) they are expressed in terms of position (i.e. observation index) rather than time, which is not ideal for irregular data such as gaugings; (ii) the uncertainty around the change point positions is rarely quantied. [START_REF] Nam | Quantifying the uncertainty in change points[END_REF] underlined the importance of accounting for the uncertainty of change point estimates. Estimating change points in terms of uncertain shift times would be useful to look for specic events that may have caused the change -e.g. a large ood that would typically occur in etween gaugings.

Objectives of the paper

The main objective of this paper is to propose an original method for the detection of rating shifts through the segmentation of residuals between the gaugings and a baseline RC. The method must:

1. account for uncertainties in segmented data.

2. recursively re-estimate the baseline RC and apply the segmentation on each sub-period to reveal minor changes.

Chapter 2. Segmentation of gaugings 2.2 The proposed method for rating shift detection

Overview

Figure 2.2 illustrates in a schematic way the algorithm of the proposed method. The main steps are listed below and detailed in the next subsections.

1. Estimation of the baseline RC and its uncertainty using all available gaugings (Section 2.2.2).

2. Computation of the residuals between the gaugings and the baseline RC, and their uncertainties. (Section 2.2.3).

3. Multiple change point detection applied to the residuals time series (Section 2.2.4) and choice of the optimal number of change points (Section 2.2.5).

4. Shift times adjustment (Section 2.2.6).

5. Top-down recursion: re-apply steps 1-4 to each period until no more changes are detected (Section 2.2.7).

Estimation of the baseline rating curve

The rst step of the proposed method is to estimate the baseline RC and its uncertainty using all gaugings. Since one of the basic objectives of this paper is to account for both gaugings and RC uncertainties, it is necessary to select an RC estimation method that provides quantitative uncertainties (see [START_REF] Kiang | A comparison of methods for streamow uncertainty estimation[END_REF] for a review of such methods). In this paper the BaRatin method [START_REF] Le Coz | Combining hydraulic knowledge and uncertain gaugings in the estimation of hydrometric rating curves: A Bayesian approach[END_REF] is used for convenience. We refer to the aforementioned paper for a more detailed description and we stress that any other method could be used, as long as it provides the uncertainty around the RC.

Computation of residuals and their uncertainty

The residuals between the gaugings and the RC are dened as follows:

Iteration 2.2.2 Iteration 2.2.1 Time t 0 Residual r Stage h Q r h t 1
Multi change point detection applied to the residuals between the gaugings and the RC End Iteration 0

Iteration 1.1

r h h Iteration 1.2 End End End Iteration 1.3 h h Q Q Q Q t t t t
r i = Q i -Q i i = 1, ..., N (2.1)
where Q i is the gauged discharge, Q i is the RC-estimated discharge and N is the number of gaugings. Each residual is aected by two sources of uncertainty. The rst one is the measurement uncertainty aecting the observed discharge Q i , which does not depend on the RC method and should ideally be determined by an uncertainty analysis of the measurement process. Assuming zero-mean Gaussian measurement errors, this uncertainty can be quantied by the standard deviation u Q i . By contrast, the uncertainty aecting the RC discharge Q i is obviously dependent on the RC method being used. The BaRatin method used in this paper assumes zero-mean Gaussian RC errors with standard deviation u

Q i = γ 1 + γ 2 Q i ,
where γ 1 and γ 2 are estimated as part of the RC estimation process. We reiterate that any other RC method could be used as long as it provides an uncertainty u Q i .

Further assuming that measurement and RC errors are independent, the combined standard uncertainty aecting residuals r i is equal to:

u r i = u 2 Q i + u 2 Q i i = 1, . . . , N (2.2)

Segmentation model and Bayesian inference

The third step of the proposed method is the mCPD of the time series of residuals (t i , r i ) i=1...N through the Bayesian estimation of a segmentation model.

General segmentation model

Generally speaking, a segmentation model can be viewed as a piece-wise constant model of the form:

r i = r i + ε i (2.3) r i =                        µ 1 , t 1 ≤ t i < τ 1 µ 2 , τ 1 ≤ t i < τ 2 . . . µ K , τ K-1 ≤ t i ≤ t N (2.4)
In Equation 2.4, K is the known number of segments; it will be selected based on a modelselection procedure described in the following section 2.2.5. The means µ j of each segment j and the change point τ j that separate segment j from segment j + 1 are unknown and are grouped into the vector of inferred parameters θ = (µ 1 , ..., µ K , τ 1 , ..., τ K-1 ). The treatment of segmentation errors ε i depends on how these errors are interpreted: it is a key focus of this paper, and two distinct approaches will be presented in the next section. Finally, it is noted that many segmentation models in the literature use the observation index i rather than the time t i . We favor the latter option because it will allow expressing uncertainties on the change point in terms of time rather than position, which is particularly useful in the RC context where gaugings are performed irregularly.

Two approaches for describing segmentation errors

Segmentation errors ε i are generally assumed to be realisations from a zero-mean Gaussian distribution. The two approaches considered here dier in the way they treat their standard deviation σ i :

1. Type-1 approach: σ i is assumed to be unknown but identical for all segmented data, i.e.

σ i = σ 2.
Type-2 approach: σ i is assumed to be known but to vary between segmented data, i.e.

σ i = u r i
Type-2 approach is particularly suitable for cases where the segmented data r i are RC residuals (or more generally, residuals between a model and observations): indeed, Equation 2.2 provides the known standard deviation u r i to be used in this case. Type-1 approach is arguably the most standard procedure, since it corresponds to the assumption made in standard regression models with homoscedastic errors. However, it ignores the uncertainty aecting the segmented data, despite the fact that it is known before applying the segmentation procedure.

Bayesian estimation

Assuming that segmentation errors ε i are independent, the likelihood associated with the segmentation model can be written as follows:

Type-1 approach: p (r|θ, σ)

= N i=1 φ (r i | r i (θ), σ) Type-2 approach: p (r|θ) = N i=1 φ (r i | r i (θ), u r i ) (2.5)
where φ (z|m, s) is the probability density function (pdf ) of a Gaussian distribution with mean m and standard deviation s, evaluated at value z.

Bayesian inference requires specifying the prior distribution of parameters (θ, σ).

Independent priors are specied for each inferred parameter. By default a uniform prior distribution is specied for each change point, τ j ∼ U(t 1 , t N ). Note that on top of this prior distribution, change points are also constrained by the relation τ

1 < • • • < τ K-1 . An
order-of-magnitude Gaussian distribution is specied for each segment mean, µ j ∼ N (0, 10 m );

the value of m is case-specic and should reect the expected order of magnitude of RC residuals, which in turn is specic to the studied catchment.

Bayes' theorem allows combining the information brought by the data through the likelihood with the prior information on the inferred parameters into a posterior distribution of the parameters, whose pdf is dened by: Type-1 approach: p (θ, σ|r) ∝ p (r|θ, σ) p (θ, σ)

Type-2 approach: p (θ|r) ∝ p (r|θ) p (θ)

(2.6)

A MCMC approach based on a multi-block Metropolis algorithm is used to explore this multidimensional posterior distribution. The variance of each parameter jump distribution is also adapted during iterations in order to reach an user-dened acceptance rate. The implemented algorithm is detailed in [START_REF] Renard | An application of Bayesian analysis and Markov chain Monte Carlo methods to the estimation of a regional trend in annual maxima[END_REF]. By default the rst half of the simulations is ignored (burned) and only the second half of simulated values is used for inference. MCMC convergence is assessed by visually inspecting trace plots and density plots (except in the synthetic case studies of Section 2.4 where this is not feasible given the large number of replications). In addition, the Potential Scale Reduction Factor [START_REF] Brooks | General methods for monitoring convergence of iterative simulations[END_REF] is computed for each parameter, verifying that it is smaller than 1.2.

Finally, to avoid short segments containing no observations, which may lead to an ill-posed inference, additional constraints can be enforced: a minimum number of points N min ≥ 1 for each segment and a minimum duration d min ≥ 0 between two consecutive change points. These constraints are case-specic and user-dened.

Choice of the optimal number of segments

The number of segments K is selected by minimizing the Deviance Information Criterion (DIC, [START_REF] Spiegelhalter | Bayesian measures of model complexity and t (with discussion)[END_REF]), in the formulation suggested by [START_REF] Pooley | Bayesian model evidence as a practical alternative to deviance information criterion[END_REF].

The DIC is selected because it is adapted to the Bayesian estimation described in Section 2.2.4.3.

We note that a maximum-likelihood estimation could also be used and possibly favored by some users. In this case, the DIC should be replaced by an alternative model-selection criterion such as the Akaike Information Criterion (AIC, [START_REF] Akaike | A new look at the statistical model identication[END_REF]), the Bayesian Information Criterion (BIC, [START_REF] Schwarz | Estimating the dimension of a model[END_REF]), or the Hannan-Quinn information Criterion (HQC, [START_REF] Hannan | The determination of the order of an autoregression. tournl of the oyl ttistil oietyF eries f @wethodologilA[END_REF]).

Let Θ denote the vector of all inferred parameters, i.e. Θ = (θ, σ) for the type-1 approach (N par = 2K) and Θ = θ for the type-2 approach (N par = 2K -1). Moreover, let D(Θ) denote the deviance dened as D(Θ) = -2 ln (p (r|Θ)). The four criteria discussed above are computed as follows:

AIC = D( Θ) + 2N par HQC = D( Θ) + 2N par ln(ln(N )) BIC = D( Θ) + N par ln(N ) DIC = E [D(Θ)] + 1 2 Var [D(Θ)]
(2.7)

In the rst three criteria, Θ is the maximum-likelihood parameter estimate. In the fourth criterion DIC, E [.] and Var [.] represent the posterior mean and variance, and the corresponding quantities can easily be computed using MCMC samples.

Adjustment of shift times

Bayesian estimation results provide the marginal posterior distribution of each inferred parameter.

Rating shift times s = (s 1 , . . . , s K-1 ) can be obtained from the posterior distributions of parameters τ .

As illustrated in Figure 2.3, each posterior distribution provides a point estimate τ j and a credibility interval CI j . Typically, τ j is the Maximum A Posteriori (MAP) estimate maximizing the posterior density. The interval CI j can be explored to nd a physically-justied shift time.

The following three options can be considered for instance:

1. s j = τ j can be used as a default option.

2. If the stage record is available, s j can be set to the time of the largest stage value within CI j (cf. Figure 2.3), reasoning that a large ood is a likely cause of the shift.

3. s j can also be set manually within CI j using any other information on possible causes of the shift (e.g., gravel mining operations, beavers/swimmers dams, works in the river bed, earthquake).

Recursive segmentation

Once a rst set of shift times has been identied, a recursive top-down" procedure is performed (Figure 2.2). The segmentation procedure described in the previous sections (2.2.2 to 2.2.6) is recursively performed within each sub-period. At each iteration only the gaugings of the current sub-period are considered. The iterations stop when within all periods no more shift times are detected.

It may happen that at some iteration of the recursive segmentation only few gaugings are available. This may lead to challenging RC estimation and, if the BaRatin method is used, to challenging quantication of u Q i . When moving from one period to its sub-period, a better t and hence a smaller standard deviation u Q i is expected; thus a prior uniform distribution between zero and the posterior mean of the parent period is specied for u Q i .

2

Segmentation strategies

Several segmentation strategies are used in order to compare single-pass" vs. recursive procedures and type-1 vs. type-2 treatment of segmentation errors:

A Single-pass" mCPD method from the literature.

The R function ptFmen of the hngepoint package [START_REF] Killick | Changepoint: An R Package for Changepoint Analysis[END_REF] For each iteration Figure 2.4b shows the RC estimated using gaugings from the current period. At iteration 0 the baseline RC has a large uncertainty, conrming the presence of multiple stage-discharge relations for this data set. This uncertainty decreases in subsequent iterations, reecting the fact that the RC is estimated using more homogeneous gaugings.

Figure 2.4c shows the evolution of four criteria (AIC, BIC, HQC and DIC) for the choice of the optimal segmentation (see Section 2.2.5). A similar behavior is observed between BIC, DIC and HQC, in particular between BIC and DIC. On the contrary AIC tends to favor a higher number of periods for the iterations shown in the gure and for other iterations (not shown).

Figure 2.4d shows the segmented residuals. These three iterations illustrate the added value of the top-down" recursion: the large RC uncertainty at iteration 0 leads to the detection of two major shifts only; then iterations 1.2 and 1.3 lead to the detection of other minor shifts based on rened RCs. Iteration 1.3 also illustrates that, because of the uncertainty in the estimated change point location, the adjusted shift time (based on ood occurrence) may be far from the optimal time identied using gaugings only.

Finally, by the last iterations, segmentation errors (ε i in Equation 2.3) do not generally exhibit any signicant autocorrelation (not shown).

Comparison of Strategies A-D

The results of the four strategies are then summarized in period (red gaugings) the detected shifts correspond to the largest gaugings. This might suggest that the shifts result from the much larger uncertainty aecting these residuals (which cannot be accommodated with the constant-sigma type-1 approach), rather than from a genuine change in the underlying RC. This evidences the problem illustrated in Figure 2.1c-d.

However the objective choice of the most ecient segmentation strategy is challenging without knowing the true shift times. In the next section, the model selection criteria and the segmentation strategies are compared based on synthetic data with known shift times and magnitudes, thus enabling a more objective evaluation of their performance.

Chapter 2. Segmentation of gaugings 2.4 Performance evaluation from simulated rating shifts

Generation of synthetic data

The generation of synthetic gauging data is based on the following steps (the corresponding R code is also provided as online material):

1. Select the length of the studied period [0; T ] (in years).

2. Shift times: generate inter-shift durations from an exponential distribution with rate λ s (e.g. λ s = 1/5 corresponds to 1 shift every 5 years on average); shift times are then derived as the cumulative sum of the inter-shift durations. The generation stops when the shift time exceeds T , leading to N s shifts.

3. Shift magnitudes: it is assumed that RC shifts only aect the oset of the lower control

(i.e. the b in equation Q = a(h -b) c ). Each shift magnitude δb (i) is generated from
a Gaussian distribution with mean 0 and standard deviation σ b . If there exists at least one shift, for each stable period j (j ≥ 2), the oset parameter b (j) is hence equal to b (j) = b (1) + j i=2 δb (i) .

4. Gauging times: use the same approach as for shift times, using a rate λ g leading to N g gaugings.

5. Gauging true discharge: for each gauging, the true discharge Q i is generated by rst sampling a non-exceedance probability p between 0 and 1, then transforming it into discharge using the quantile function of a LogNormal LN (ln(50), 0.5) distribution.

Probability p is sampled from a beta distribution B(0.1, 0.9) which is strongly skewed toward zero, mimicking the typical situation where gaugings are mostly performed during low ows.

6. Gauged stage: for each gauging, the stage h i is computed by applying the inverse RC function to the true discharge Q i .

7. Gauged discharge: for each gauging, the gauged discharge is obtained by adding a measurement error ξ i to the true discharge Q i . ξ i is sampled from a Gaussian distribution with mean 0 and standard deviation ρ i × Q i .

Design of experiments

In order to assess how the properties of the data set impact the performances of the segmentation approaches, several classes of simulation are dened as described in Table 2.1.

Each class is characterised by xed values of the parameters described in Section 2.4.1.

Comparing these classes allows assessing the impact of the following Properties:

P1: number of hydraulic controls N c (classes 8 and 10).

P2: mean number of gaugings per period, which is equal to the ratio between the gaugings frequency λ g and the shifts frequency λ s (classes 1, 6, 2, 7, 3, 8), as suggested by [START_REF] Ibbitt | Gauging frequency and detection of rating changes[END_REF].

P3: uncertainty in gauged discharges as controlled by ρ LF and ρ HF (classes 8 and 9).

P4: shift magnitude as controlled by σ b (classes 4, 5 and 8).

For each class, 10 replications are generated, for a total of 100 simulations (some data sets are reported in Figure 2.6). To minimize computational cost all data sets are generated with a maximum number of 150 gaugings and a maximum number of 15 true shift times within a

Metrics for performance evaluation

The performance evaluation uses some of the metrics proposed by [START_REF] Aminikhanghahi | A survey of methods for time series change point detection[END_REF]. At the end of the segmentation procedure each gauging is classied into one of T P , F N , F P , T N (see example in Figure 2.7), where:

A gauging is classied T P (true positive) if it is the nearest neighbor of a true shift and this true shift is within the 95 % CI of an estimated shift.

A gauging is classied F N (false negative) if it is the nearest neighbor of a true shift but this true shift is outside all 95 % CI of estimated shifts.

A gauging is classied F P (false positive) if it is the nearest neighbor of an estimated shift but the 95 % CI of this estimated shift does not contain any true shift.

Otherwise a gauging is classied T N (true negative). The Accuracy A is dened as the rate of correctly classied gaugings:

N g = n T P + n F N + n F P + n T N
A = n T P + n T N N g (2.8)
The Sensitivity S is maximal when no shift has been missed; low values hence correspond to under-segmentation:

S = n T P n T P + n F N (2.9)
The Precision P is maximal when all detected shifts are real; low values hence correspond to over-segmentation:

P = n T P n T P + n F P (2.10)
The RM SE between the times of correctly-detected shifts s T P i and the times of corresponding true shifts s true k i is also computed: and HQC lead to a higher degree of over-segmentation with, on average, 125 % and 67 %, respectively, more detected shift times than there really are. On the contrary BIC and DIC over-estimate the number of shifts by only 13% and 40%, respectively.

RM SE = n T P i=1 (s T P i -s true k i ) 2 n T P (2.
BIC and DIC sometimes miss a few shifts leading to lower values of Sensitivity than AIC and HQC.

These results indicate that AIC and HQC have a marked tendency to over-segmentation and should therefore be avoided. BIC and DIC have similar performances, and it is therefore sensible to select the one that is conceptually more adapted to the chosen inference paradigm.

More specically, BIC is solely based on the likelihood and is hence more adapted to maximumlikelihood estimation (despite what its name confusingly suggests). By contrast, DIC makes use of the whole posterior distribution and should therefore be favored in a Bayesian context such as the one adopted in this paper. Chapter 2. Segmentation of gaugings 2.5 Discussion 2.5.1 Contributions to the operational practice and the scientic literature

Comparison of segmentation strategies

The proposed method represents a more formal way to detect rating shifts using gaugings, compared to empirical rules commonly used in the operational practice [WMO, 2010]. A similar formalization objective was pursued by [START_REF] Morlot | Dynamic rating curve assessment for hydrometric stations and computation of the associated uncertainties: Quality and station management indicators[END_REF] who applied the Hubert segmentation method [START_REF] Hubert | Segmentation des séries hydrométéorologiques. Application à des séries de précipitations et de débits de l'Afrique de l'Ouest[END_REF] for segmenting the residuals between the gaugings and a baseline RC. Their method yielded satisfying results, even though it neglected the residuals' uncertainties. However, the synthetic simulations and the real case study proposed in this paper suggest that neglecting residuals' uncertainties may lead to over-segmentation.

The proposed method diers from the mCPD literature in its handling of shift times in two aspects: considering time rather than position and providing shift time uncertainty. Existing performance evaluation metrics proposed in the literature [START_REF] Aminikhanghahi | A survey of methods for time series change point detection[END_REF] were also adapted to account for shift times uncertainty while comparing the true shift times and the detected ones.

In addition to the segmentation procedure, this paper proposes a protocol for the generation of synthetic data sets of gaugings and shift times. This protocol is very useful to evaluate the performance of the segmentation method.

The case studies proposed in this paper indicate that the mean number of gaugings per period remarkably aects the performance of the segmentation: this is consistent with the observation of [START_REF] Ibbitt | Gauging frequency and detection of rating changes[END_REF]. It is therefore important to consider this indicator when planning gauging campaigns or deriving gauging strategies. According to the simulations, the availability of 20 gaugings per period on average leads to an acceptable identication of rating shifts. On the contrary, less than 10 gaugings per period may lead to a poor segmentation. However, these numbers should be considered as rough orders of magnitudes rather than precise gures since not all existing hydraulic congurations, shift magnitudes and gaugings uncertainties have been Matteo Darienzo 56/174 2.5. Discussion tested.

Current limitations

The method proposed in this paper is built on the main assumption that changes correspond to sudden shifts (as opposed to slower transient changes), which may be inadequate for phenomena such as vegetation growth and decay. The segmentation model proposed here is not designed to analyse a trend in the residuals. Dynamic approaches such as those discussed in the introduction Section 2.1.2 should instead be favored.

The case studies have illustrated the added value of expressing change points in terms of time (rather than position) and of adjusting the shift time by looking for some causative events within the uncertainty bound of the estimated change point, as described in Section 2.2.6.

Nevertheless, this adjustment must be done with precaution. Determining the cause of a shift is a complex decision since several potential processes might be suspected (e.g., ood, gravel mining). This necessarily comes with a degree of expertise and subjectivity. For instance, how to separate the sudden shift created by a morphogenic ood from apparent shifts induced by transient phenomena such as vegetation, backwater, etc.? How to choose when a single shift may be attributed to several oods? Introducing some degree of expertise and subjectivity is not problematic in our opinion. It may even well improve an otherwise fully automated procedure.

Avenues for future work

The method proposed in this paper can, in principle, be used with any RC method that provides RC uncertainty. Results from various RC methods could be compared in future work to assess the extent to which the detected changes are robust with respect to the RC method.

The method is also based on the analysis of residuals computed with respect to discharge.

They may be computed with respect to stage too, as suggested by [START_REF] Morlot | Dynamic rating curve assessment for hydrometric stations and computation of the associated uncertainties: Quality and station management indicators[END_REF]. It would therefore be of interest to modify the method to enable the use of stage residuals, and to evaluate whether it has an important impact on the detected changes. We note however that the treatment of stage residuals, and in particular of their uncertainty, is not straightforward.

This is because the equivalent of Equation 2.2 for stage is not immediately available and may require some additional error propagation.

More work could also be carried out to rene operational gauging strategies in the presence of shifty RCs. Many factors may aect the eciency of the rating shift detection, in particular, the number of gaugings, their uncertainty and their location along the RC. A single gauging may be sucient if it is precise and far away from the base RC. But in the case of minor shifts or very uncertain gaugings several gaugings may be required. Notice also that the gaugings for Meyras station (Section 2.3) have been performed for mostly all ow conditions. This leads to the estimation of relatively precise RCs. More uncertain RCs are likely to require more gaugings to detect a shift of a given magnitude.

The case studies have also evidenced that, among the proposed criteria for the choice of the optimal number of segments, the BIC and the DIC obtained the highest performance.

However the segmentation appears sensitive to these criteria. Future work includes a more exhaustive comparison, as discussed by [START_REF] Buckland | Model selection: An integral part of inference[END_REF] and [START_REF] Burnham | Multimodel Inference: Understanding AIC and BIC in Model Selection[END_REF], by analysing the weight of each penalty term related to data t and number of parameters.

The proposed method is also inherently limited by gaugings availability -no change can be detected in their absence. However, we stress that gaugings are not the only information available at hydrometric stations. The proposed tool may be complemented with other sources of information such as the stage record (e.g., by detecting a change in recession shapes or by deriving sediment transport estimates), other independent hydrologic data (e.g., correlation

analysis with neighboring stations or with the output of a rainfall-runo model) and direct observations (e.g., bathymetric surveys).

Finally, as a general perspective, the proposed method could probably be extended to other elds where a relationship between two variables, calibrated with uncertain data, is subject to sudden changes (e.g. relation turbidity vs. total suspended sediment concentration).

Matteo Darienzo 58/174 2.6. Conclusion

Conclusion

We propose a method for the detection of rating shifts using gaugings. The method applies a recursive segmentation procedure to the time series of residuals between the gaugings and a time-invariant baseline RC. Unlike other classical methods for the segmentation of residuals, the proposed method formally accounts for both gaugings and RC uncertainties through a Bayesian approach. It also expresses change points and their uncertainties in terms of time rather than position, which is of interest to search for specic events that may have caused the shift. It performs a "top-down" recursive procedure, progressively rening the RC estimation on homogeneous sub-periods and leading to the detection of minor shifts.

The method yielded encouraging results for the Ardèche River at Meyras, France, with the detection of eective rating shifts, in good agreement with the ocial dates of RC update.

Accounting for the uncertainty in the change point times allowed identifying ood events as likely causes of the shifts. Furthermore a performance evaluation procedure based on synthetic gauging data sets for which the true shift times are known highlighted the added value of the recursive segmentation procedure and the importance of accounting for both gaugings and RC uncertainties. This approach yielded more accurate results than a "single-pass" strategy or a strategy assuming homoscedastic residuals.

Introduction

3.1.1 Stage-discharge rating shifts at hydrometric stations Some hydrometric stations especially located in natural rivers are aected by changes in the stage-discharge rating curve (hereafter called RC), i.e. the relation used to estimate streamow, Q, from the recorded water level, h (stage). These rating changes may undermine the accuracy of the streamow estimates.

One of the causes of rating changes [START_REF] Herschy | trem)ow mesurement[END_REF][START_REF] Mansanarez | Shift Happens! Adjusting Stage-Discharge Rating Curves to Morphological Changes at Known Times[END_REF] is related to river morphodynamics. The geometry of the river cross-section is subject to changes governed by streamow intensity and sediment transport processes that cause erosion and deposition [START_REF] Coleman | Fluvial sediment-transport processes and morphology. tournl of rydrology @xew elndA[END_REF]. The intensity of these processes depends on many factors:

geological and geotechnical properties (bed soil type, grain-size distribution, soil mechanical properties), catchment hydrology, channel sinuosity and geometry, vegetation cover, etc.

Some rivers are subject to sudden changes of river bed elevation during morphogenic oods, but are characterized by periods of stability between these events. Other rivers are characterized by a continuous evolution of the river bed geometry, especially sandy bed rivers [START_REF] Jia | Impacts of the large amount of sand mining on riverbed morphology and tidal dynamics in lower reaches and delta of the dongjiang river[END_REF][START_REF] Wang | Long-term geomorphic response to ow regulation in a 10km reach downstream of the mississippiatchafalaya river diversion[END_REF] and rivers with dunes or alternated bars [START_REF] Rodrigues | Alternate bars in a sandy gravel bed river: Generation, migration and interactions with superimposed dunes[END_REF].

The issue of sudden morphogenic changes is well known by the operational services and the decision makers in charge of an "unstable" station. Indeed, their main interest is to detect and estimate river bed changes after a ood event with the least possible delay.

Methods for estimating river bed evolution

Currently methods for monitoring the river bed evolution include bathymetry measurements, stage-discharge rating curves approaches and analysis of the stage record.

Bathymetry surveys provide a direct and relatively precise estimation of the river bed geometry [START_REF] Zhao | Research on precise monitoring method of riverbed evolution[END_REF].

Nevertheless, these measurements are performed sporadically, usually during gaugings campaigns, and require important mobilization of sta and equipment. New techniques such as camera time lapse [START_REF] Leduc | Technical note: Stage and water width measurement of a mountain stream using a simple time-lapse camera[END_REF] or Satellite-Derived Bathymetry [START_REF] Legleiter | Mapping gravel bed river bathymetry from space. tournl of qeophysil eserhX irth urfe[END_REF] can also provide information on the river bed bathymetry. However, these techniques are restricted to large rivers in the absence of vegetation and under low turbidity conditions [START_REF] Legleiter | Evaluating the potential for remote bathymetric mapping of a turbid, sand-bed river: 2. application to hyperspectral image data from the platte river[END_REF].

A standard method to estimate the evolution of the river bed elevation at hydrometric stations is based on gaugings and RC estimation [START_REF] Rantz | wesurement nd omputtion of strem)owF olume PX gomputtion of dishrge[END_REF]WMO, 2010;[START_REF] Le Coz | Estimating the long-term evolution of river bed levels using hydrometric data[END_REF]. If the RC equation is a power law of type Equation 1.1, then the oset b is the parameter reecting the river bed elevation. River bed changes can therefore be deduced by changes of parameter b over time. Note that the exact meaning of the expression "river bed elevation" depends on the type of control represented by Equation 1.1. For a channel control the oset b corresponds to the mean elevation of the controlling reach; alternatively, if low ows are controlled by a weir/natural rie, the oset b denotes the mean crest elevation.

We propose in this paper an alternative method based on the stage record, which is continuously available at hydrometric stations and may represent a simple and useful information to detect the morphological changes and estimate the river bed elevation.

Surprisingly, this option has not been thoroughly studied in the research literature, even though it is performed by many hydrometry operators who monitor visually the anomalies in the stage record.

apuszek [2003] suggested a method based on the assumption that the stage at low ows tends towards the river bed elevation as streamow tends towards zero. Assuming that annual minimum stages are close to the river bed elevation, the author could thus study the long-term trends in river bed elevation. A limitation of this approach is that focusing on annual minimum stages restricts the detection of changes at an annual resolution. Consequently, the dates of the ood events responsible for morphological changes are dicult to identify. In addition, annual minimum stages might be far from the river bed elevation during wet years with no signicant droughts. To overcome this issue, a recession analysis on the stage record may be of interest.

Q(t) = Q 0 e -Λ t (3.1)
where:

-Q 0 is the initial ow of the recession period (at t = 0); -Q(t) is the ow after a recession time t;

-Λ > 0 is the recession rate.

Sometimes several exponential terms are distinctly visible [START_REF] Barnes | The structure of discharge-recession curves. iosD rnstions emerin qeophysil nion[END_REF][START_REF] Larue | L'assèchement de cours d'eau dans le bassin de la Maine entre 1989 et[END_REF], denoting dierent streamow origins: the rapid runo due to fast ow after oods, the sub-surface ow and the very slow emptying of aquifers. By adding N exp exponential terms, streamow-recession can be seen as a superposed exponential function [START_REF] Barnes | The structure of discharge-recession curves. iosD rnstions emerin qeophysil nion[END_REF]:

Q(t) = Nexp i=1 Q i e -Λ i t (3.2)
An alternative to the use of several exponential terms for modeling complex recessions is to use models based on non-linear reservoirs [START_REF] Wittenberg | Nonlinear analysis of ow recession curves[END_REF][START_REF] Toebes | On recession curves: 1 Recession Equations. tournl of rydrology @xew elndA[END_REF]]: e.g. the double-exponential model [START_REF] Horton | The rôle of inltration in the hydrologic cycle. iosD rnstions emerin qeophysil nion[END_REF] or the hyperbolic model [START_REF] Werner | On the groundwater recession curve for large watersheds[END_REF][START_REF] Drogue | Analyse statistique des hydrogrammes de decrues des sources karstiques. Statistical analysis of hydrographs of karstic springs[END_REF].

Objectives and structure of the paper

The objective of this manuscript is to propose a method for estimating the river bed evolution at hydrometric stations through the analysis of stage-recessions. In particular, the proposed method pursues the following specic objectives:

overcome the annual resolution of the method proposed by apuszek [2003] by analysing all available recessions rather than annual minimum stages only;

evaluate whether models proposed in the literature for streamow-recessions can be applied to stage-recessions;

build recession models that consider some parameters as common to all recessions (in the spirit of a "master curve" concept) and other parameters as specic to each recession; The structure of the paper can be summarised as follows. Section 3.2 describes the proposed method for the recessions extraction, estimation and segmentation. Then Sections 3.3 applies the method to a well-documented case-study by evaluating dierent recession models. Section 3.4

proposes a discussion on the results and the future perspectives for this work, such as the realtime application. Finally, Section 3.5 summarises the main ndings and conclusions.

3.2 The proposed method for river bed estimation using stage recessions

The proposed procedure for the stage-recession analysis is composed of three main steps, detailed in the following subsections:

Step 1: Extraction of the stage-recessions from the stage record.

Step 2: Estimation of the stage-recession curves in a unique model through a Bayesian approach.

Step 3: Segmentation of the time series of the recession-specic parameters estimates.

Step 1: Extraction of the stage-recessions

Let h = (h(t 1 ), ..., h(t n )) dene the values of the stage record. Note that times t i denote the absolute times at which stage is recorded. This is a slight abuse of notation since in recession equations, such as Equation 3.1-3.2, t denotes the recession time, i.e. the time since the beginning of the recession. The proposed algorithm for the extraction of the recessions from the recorded stage time series h is based on the following steps:

1. Selection of all decreasing stage values h d among h such that h(t i ) < h(t i-1 ) (black and empty dots in Figure 3.1).

Denition of the continuous sequence h rec from the values h d such that every value h(t i )

of the sequence is smaller than all the previous elements, h(t i-1 ), h(t i-2 ), ... (black dots in Figure 3.1).

3. Separation of the recessions: a threshold parameter, χ, is used to separate one recession period (k) from the next one (k + 1). If:

[h rec (t i ) -h rec (t i-1 )] > χ (3.3)
then the recession k ends at time t i-1 and a new recession k + 1 starts at time t i (see Figure 3.1).

4. Selection of the recessions that fulll the following user-dened conditions: 1.1) gives the following stage-recession model:

frnesD IWQW M 1. h(t, k) = α (k) e -λ t + β (k) M 2. h(t, k) = α (k) 1 e -λ1 t + α 2 e -λ2 t + β (k) M 3. h(t, k) = α (k) 1 e -λ1 t + α (k) 2 e -λ2 t + β (k) M 4. h(t, k) = α (k) 1 e -λ1 t + α 2 e -λ2 t + α 3 e -λ3 t + β (k) M 5. h(t, k) = α (k) 1 e -λ1 t + α (k) 2 e -λ2 t + α 3 e -λ3 t + β (k) λ λ 1 , α 2 , λ 2 λ 1 , λ 2 α 2 , α 3 , λ 1 , λ 2 , λ 3 α 3 , λ 1 , λ 2 , λ 3 α, β α 1 , β α 1 , α 2 , β α 1 , β α 1 , α 2 , β Double- exponential rortonD IWQQ M 6. h(t, k) = α (k) e -λ t η + β (k) M 7. h(t, k) = α (k) e -λ (k) t η + β (k) λ, η η α, β α, λ, β Hyperbola hrogueD IWUP M 8. h(t, k) = α (k) (1+λ t) η + β (k) M 9. h(t, k) = α (k) (1+λ (k) t) η + β (k) λ, η η α, β α, λ, β (Equation
h(t) = Q 0 a 1 c e -Λ c t + b (3.4)
Therefore, as for the streamow-recession, the stage-recession can still be modelled with an exponential function with the unique dierence that it does not tend to zero, but rather to the oset b.

Unfortunately, Equation 3.4 does not hold for more complex recession models and/or for a multi-control piecewise RC. Despite this, we postulate that models proposed in the literature for streamow-recessions can reasonably be applied to stage-recessions by adding a parameter β representing the asymptotic stage.

In particular, nine dierent stage-recession models are proposed in this paper and reported in Table 3.1. They all use one or more parameters representing the initial stage (α i ), one or more parameters representing the recession rate (λ i ) and one parameter representing the asymptotic stage (β).

Model M1, representing a simple exponential behavior, is often used for studying the lowest part of the recession and may not be capable of describing the entire recession behavior. Instead, all other models aim at describing the entire recession from the fast runo to the slow emptying of the aquifer.

For this purpose models M2-M5 add to the simple exponential function few exponential terms based on the superposed-exponential concept (where λ i > λ i+1 ). In this paper we restrict to three exponential terms, but more terms may be added, keeping in mind that the additional exibility comes at the cost of additional parameters.

On the other hand, models M6-M9 aim at describing the entire recession through the introduction of an additional parameter (η) acting on the shape of the recession.

Models M1-9 also dier in the choice of the common and recession-specic parameters. All models M1-5 assume a static λ i , implying that the recession rates do not vary over recessions, and a recession-specic α 1 , implying that the initial stage of the rst exponential contribution (which describes the fast runo ) is specic to each recession, thus dependent on the high variability of ood peaks. Regarding the two-superposed-exponential models, M2-3, while M3 assumes a recession-specic initial stage α 2 , M2 assumes the second exponential contribution is constant across recessions and the variability between recessions comes from the rst exponential term only. Models M4-M5 add a static third exponential contribution to models M2-3, assuming that the aquifer emptying contribution is constant across recessions.

On the other hand, Models M6-9, which consider one single recession contribution, all assume a recession-specic initial stage α and a static shape parameter η. Finally, while models M6, M8 consider a static λ, implying that the recession rate does not vary over recessions, for models M7, M9 parameter λ is considered recession-specic.

For a given recession model, let θ R denote the inferred parameters of the recession model, comprising static θ static and recession-specic θ recession parameters.

Likelihood computation

Let t i and k i denote the time and the recession index, respectively, associated to the iEth recession stage value. Then, the observed stage value hi is written as the stage value predicted by the regression model ĥi = f (t i , k i |θ R ) plus a structural error and the observed stage error.

Both types of error are assumed independent and Gaussian distributed with mean equals to zero and standard deviation equals to σ R,i = γ R1 + γ R2 ĥi -min(h rec ) and to a given u h,i , respectively. The vector of all inferred parameters is θ = (θ R , γ R1 , γ R2 ). The likelihood L of the N tot observed stage recession values h is:

L h|θ, t, k = Ntot i=1 p norm hi |f (t i , k i |θ R ), σ 2 R,i + u 2 h,i (3.5)
where p norm hi |m, s is the pdf of a gaussian distribution with mean m and standard deviation s at the observed stage value hi .

The static recession parameters θ static appear in all N tot terms of the product in Equation 3.5.

Consequently, the information contained in all the recession observations is used to infer these static parameters. On the contrary, the recession-specic parameters θ (k) recession only appear in the terms of the product involving observations from the k -th recession.

Prior specication

Bayesian inference requires prior specication on parameters θ. The joint prior distribution is:

p(θ) = p(γ R1 ) p(γ R2 ) p(θ static ) p(θ recession ) (3.6)
Positive uniform distributions are assigned to the initial stage parameters: α ∼ U(0, 10 n ),

where n is the order of magnitude of the stage values. A uniform distribution is assigned to the asymptotic stage parameter: β ∼ U(-10 n , +10 n ). Log-normal priors are used for the recession rate parameters λ. It is convenient to specify this prior in terms of the more intuitive half-life τ (the time required for decreasing the initial stage by a factor of two), which is related to the rate parameter by: λ = ln 2 τ

(3.7)

The log-normal prior on τ can be translated into a log-normal prior on λ whose logarithm has mean equal to ln (ln 2) -ln τ and standard deviation equal to σ log :

λ ∼ LN (ln (ln 2) -ln τ, σ log )

(3.8)

A log-normal prior is also specied to the shape parameter η ∼ LN (ln 1, 1). For the structural error parameters a uniform prior is specied: γ R1 ∼ U(0, 10 n ) and γ R2 ∼ U(0, 100),

Posterior distribution

Bayes' theorem combines likelihood with priors through:

posterior p θ| h, t, k ∝ likelihood L h|θ, t, k prior p (θ)
(3.9)

The multi-dimensional posterior distribution is explored with an adaptive block Metropolis sampler described in [START_REF] Renard | An application of Bayesian analysis and Markov chain Monte Carlo methods to the estimation of a regional trend in annual maxima[END_REF]. The MCMC samples provide marginal and joint properties of parameters θ (e.g. posterior mean, standard deviation, credibility interval) but also the most probable parameters values θ (maximum a posteriori) having the largest posterior pdf. MCMC convergence is visually checked through trace and density plots and by ensuring the Gelman factor [START_REF] Brooks | General methods for monitoring convergence of iterative simulations[END_REF]] is smaller than 1.2. In this paper, 150,000 iterations are performed, but the rst half of MCMC is discarded (burned). Finally, statistics are done considering one iteration every 100.

Third step: recessions segmentation

Once the recession model has been estimated, the temporal evolution of the recessionspecic parameters θ recession is analysed. Particular attention is given to the evolution of the asymptotic stage parameter β, which corresponds to the elevation b of the lowest control.

A segmentation procedure is therefore applied to the series of the estimated asymptotic stages β = (β (1) , ..., β (Nrec) ) searching for net shifts. Any segmentation method proposed in the Chapter 3. Stage-recession analysis literature can be applied. In this paper the segmentation procedure developed by Darienzo et al.

[2021] (Chapter 2) is used. It is a multi change point detection method based on a Bayesian approach and on a model selection criterion to choose the optimal number of change points, e.g.

DIC [START_REF] Gelman | fyesin ht enlysis[END_REF][START_REF] Pooley | Bayesian model evidence as a practical alternative to deviance information criterion[END_REF]. This method is selected because it has the following properties:

it accounts for the uncertainty aecting each estimated parameters (potentially variable from recession to recession); it expresses change points in terms of time (rather than position), which is convenient since the recessions are irregularly located along the time series. To this aim the time associated to each parameter β (k) is the time at which the corresponding recession begins.

it provides an uncertainty on the detected shift times.

Finally, the method also provides the segments mean with uncertainty. They correspond to the estimation of the river bed elevation during each period of stability delimited by the shift times.

In the next section, the proposed stage recession analysis is applied to the Ardèche River at Meyras in France, a gravel bed river characterised by river bed degradation after episodic oods.

3.3 Application: Ardèche River at Meyras, France

Description of the station site

The Ardèche River at Meyras is located in a relatively small catchment (98 km

2 ). It is a gravel bed river degrading during each important ood resulting in vertical shifts in the stage record (as evidenced in Figure 3.2). [START_REF] Sikorska | Calibrating a hydrological model in stage space to account for rating curve uncertainties: general framework and key challenges[END_REF] and [START_REF] Mansanarez | Shift Happens! Adjusting Stage-Discharge Rating Curves to Morphological Changes at Known Times[END_REF] have exhaustively studied this station, proposing a three-controls hydraulic conguration: one rectangular weir section control activated at very low ows by a natural rie, one main wide rectangular channel control activated at medium-high ows, and one wide rectangular oodway channel control added to the main channel control at very high ows. For this station the asymptotic stage of the stage-recession model, β, corresponds to the oset of the rst control, b 1 , i.e. the mean crest elevation of the natural rie. The stage record is studied for the period between 07/11/2001 and 29/10/2018. Gaugings and comments about rating shift times have been provided by the hydrometric service, UHPC Grand Delta. of this work is the detection and estimation of shifts of the asymptotic stage, longer recessions are more suitable to the analysis, thus a reasonably high χ is here preferred.

Step 1: Recessions extraction

Step 2: Recessions estimation

The second step consists of estimating the stage-recession models. For comparison purposes, all the recession models proposed in Table 3.1 are used for the recession data sets obtained with χ = 10, 30, 50 cm. A standard deviation u h,i = 0.5 cm is assigned to represent the uncertainty in all stage observations. The priors for the inferred parameters for each model are specied as described in Section 3.2.2.3. Parameter λ is specied according to Equation 3.8. For all models having a single parameter λ we assume τ = 0.5 days describing the fast runo and σ log = 1. For models M2-M5 we assume τ = 80 days and σ log = 0.5 for the slowest rate parameter (λ 2 for M2-M3, λ 3 for M4-M5) to describe the slow emptying of the aquifer. Finally, for models M4 and M5 we assume τ = 50 days and σ log = 1 for parameter λ 2 to describe the subsurface ow.

As an example, Figure 3.3a illustrates the results of the recession estimation using χ = 50 cm and using models M1, M2, M4. For simplicity, only the MAP (maximum a posteriori) curves are plotted. As expected, results evidence that model M1 (one simple exponential term) does not t well to the entire recession behavior. Instead, both model M2 and M4 (with two and three exponential terms, respectively) lead to quite similar curves and seem to better represent the fast runo component as well as the slower components associated with low ow levels. As the true dates of river bed shift and the true values of river bed elevation are unknown, the choice of the most adapted model and χ value is challenging. This issue is addressed in the next section.

Step 3: Recessions segmentation

Sensitivity to the selected recession model

In order to select the most adapted model we propose two approaches:

One general method applicable to all case studies and based on a model selection criterion.

One method applicable to stations where information about past rating shifts is available, and based on the comparison between the shift times and magnitudes detected by the proposed method and those obtained by other methods (e.g. using the gaugings).

Using a model selection criterion

The recession models can be compared by means of the Deviance Information Criterion, which measures the trade-o between the t and the number of parameters required to achieve it and which is well suited to the Bayesian framework adopted here (see Section 2.4.4.1). We use the version proposed by [START_REF] Pooley | Bayesian model evidence as a practical alternative to deviance information criterion[END_REF] from the formulation of [START_REF] Gelman | fyesin ht enlysis[END_REF] and based on the Deviance D(θ) = -2 ln (p (h|θ)), dened as:

DIC = E [D(θ)] + 1 2 Var [D(θ)]
(3.10)

where E [.] and Var [.] represent the posterior mean and variance, computed using the MCMC samples. According to this criterion, the model to be chosen is the one with the lowest DIC. Another remarkable result from using models M2-M5 is that considering a recession-specic α 2 lowers the DIC (compare M3 vs. M2 and M5 vs. M4) with the exception for M2-M3 and χ = 10. In a similar way, for models M6-M9 considering a recession-specic λ leads to a lower DIC.

Thus, if long recessions are available (as in this case study) the superposed exponential model with recession specic α 2 is preferred. On the contrary, for those case studies characterised by frequent oods thus with short recessions, the hyperbolic model with recession-specic λ, M9, may be preferred because not signicantly depending on χ, thus on the length of the recessions.

Comparing the shift times and their magnitudes

In addition to using the DIC, comparing the segmentations obtained with dierent models and dierent values of χ can also be insightful.

Figure 3.5 evidences that increasing the value of χ (longer recessions) leads to slightly dierent segmentation for a given model. However it also shows that one shift time is detected by all models and by all χ values, and other two shift times by several models, which means that these three shifts are most likely three genuine shifts.

If additional information is available about the rating shift times from other methods (e.g. segmentation of the gaugings, or ocial dates of RC update) then the comparison can be even more exhaustive. In general we observe that models M1, M2, M4, M5 are the most consistent with the gauging segmentation (provided in Chapter 2), independently on the χ value, despite some of these models having a pretty high DIC. On the other hand, the models that yielded the less consistent segmentation independently on the χ values are models M6, M7, M8. Comparing the corresponding shift magnitudes is also insightful. Figure 3.6b shows the values of ∆ β (the dierences between the consecutive β estimates (MAP)) and their uncertainty (the quadratic sum of the uncertainties of the two consecutive β) for the nine models and χ = 50 cm, which are compared to the shift estimates ∆ b1 obtained from the gaugings.

Results reveal that all models lead to shift estimates similar to those obtained by using the gaugings, with the exception of models M6, M7 which lead to some unrealistic (and highly uncertain) shift estimates. This interesting result suggests that reasonably estimating shift magnitudes is possible even when the estimation of the absolute river bed elevation is inaccurate. Moreover, the two shifts (the rst and the last one) detected by the gaugings and missed by almost all models appear to be very small (of the order of 2 cm).

Finally, an interesting observation from this comparison with gauging-based segmentation results is that recession models ranking quite poorly in terms of DIC (Figure 3.4) can still estimate shift times and amplitudes quite acceptably (e.g. M2 and even M1). The only models that lead to consistent results both in terms of DIC and comparison with gaugings are models M5 (with χ = 10 -30 cm) and M9 (with χ =50 cm).

Matteo Darienzo 84/174 3.4. Discussion

Discussion

Limitations

The recession extraction results have shown for some models a high sensitivity of the threshold parameter χ. A reasonably high value of the threshold parameter χ for the recession extraction is in general recommended, to ensure the availability of quite long recessions.

However, even with a high value of χ, obtaining recessions long enough may be challenging for those catchments characterised by frequent oods. For such catchments the shift detection and estimation may be challenging.

The recession models considered in this paper are adapted to undisturbed natural systems but may not be adapted when low ows are aected by other processes: e.g. gravel mining operations, beavers/swimmers dams.

This paper is limited to river bed vertical instability. However, the shape of the river crosssection can change horizontally or vertically [apuszek, 2003] and the horizontal instability implies change in the width. Moreover, the method is geared toward change in low ow controls, and it's unlikely that changes in e.g. oodways of high-ow channels can be detected from the stage record.

Finally, the proposed stage-recession analysis needs to be tested for stations subject to frequent oods, or for stations located at sandy bed rivers subject to a continuous evolution of the river bed. For sandy bed rivers the segmentation step should be replaced by some form of trend analysis.

Perspective: real-time stage-recession analysis

A stage-recession analysis may be able to detect and estimate a RC change even before any gauging is performed, which is of interest to manage the RC in "real time". Indeed, the real-time application of the proposed recession analysis is a promising perspective of this work.

Chapter 3. Stage-recession analysis Note that the expression "real time" refers to the attempt to use the continuously incoming stage data to make decision about the RC, but it does not imply a notion of quickness. The estimation of the river bed already a few days after a ood may be very useful to provide fast information about river bed shifts and update the RC. While the retrospective analysis of the stage-recessions is characterised by the availability of all entire recession data sets, the real-time recession analysis sequentially adds the incoming stage values to the current recession.

The real-time application rstly requires the retrospective analysis of all past recessions in order to estimate the static parameters of the considered recession model. This can be done through the method described in Section 3.2. Then, the real-time estimation of the current recession can be performed in a sequential way by adding at each time step the incoming stage and updating the recession estimation. The recession estimation can be done through the same Bayesian regression approach but using the data of the current recession only and by considering as priors for parameters θ static the corresponding posteriors obtained from the retrospective analysis. Chapter 3. Stage-recession analysis

Conclusion

The proposed method for detecting river bed changes and estimating their evolution using the stage record is based on a stage-recession analysis and on the assumption that stage tends to the river bed as streamow tends to zero.

The method consists of three main steps. It starts with the extraction of all available recessions from the stage record.

It proceeds with the estimation of a unique model for all recessions through a Bayesian approach distinguishing recession-specic parameters and common parameters. Among the recession-specic parameters the asymptotic stage corresponds to the river bed elevation. Finally, the method applies a segmentation procedure to the series of the asymptotic stage parameter of all recessions. If some step changes are detected it provides the river bed change estimates with quantitative uncertainty.

The method yielded encouraging results with the detection and estimation of major river bed changes for the Ardèche River at Meyras in France. These changes can be detected quite accurately, even when the absolute river bed elevations cannot. Some recession models ranked poorly in terms of DIC but were still capable of detecting and estimating changes. Future perspectives include the application of the proposed recession analysis to the real-time context.

The prediction of changes in the river bed elevation may be possible a few days after the ood. One possible way of using the stage data for the fast detection of a potential change is based on the fact that a minimum water depth is needed to trigger bedload, hence a potential morphological change (c.f. critical threshold for incipient motion [START_REF] Meyer-Peter | Formulas for bed load transport. ser Pnd meetingD tokholmD ppendix PD rydruli ingineering eports[END_REF][START_REF] Bungton | A systematic analysis of eight decades of incipient motion studies, with special reference to gravel-bedded rivers[END_REF][START_REF] Perret | rnsport of modertely sorted grvels t low ed sher stress X impt of ed rrngement nd (ne sediment in(ltrtion[END_REF]). Thus, the morphogenic ood can be dened as the period of the stage record characterised by values larger than a triggering stage.

Moreover, the volume of transported sediments, and therefore the potential for morphological change, increase with the duration and the intensity of the ood.

The change detected by simply considering the exceedance of a critical stage value merely represents a potential change. Indeed, it may happen that both scour and ll processes occur during oods [START_REF] Laronne | Scour chain employment in gravel bed rivers[END_REF] or that bedload transport is at equilibrium so that no net changes (signicant dierence before and after the ood) really occur. Furthermore the consolidation degree of bed material may change in time. Thus, two oods with the same intensity and duration may cause dierent morphological changes, depending on upstream sediment sources and local conditions.

From an operational point of view a potential change requires increasing the uncertainty around the parameters of the RC, with longer and more intense oods leading to a larger uncertainty increase. By contrast, methods using gaugings or stage-recessions (see previous chapters) aim at detecting eective changes and re-estimating these RC parameters, thus modifying not only their variance but also their mean value.

There is the need to develop a tool for detecting potential changes and to calibrate it against the historic morphological changes of the station assuming that the dynamics of the causal processes do not vary over time. The proposed approach is developed by borrowing ideas from river sediment transport modelling.

Sediment transport modelling

River sediments are heterogeneous aggregates of minerals, organic matter and biological matter. Their density and grain-size distribution strongly inuence the processes of erosion and deposition. The sediment transport is initiated when the bed shear stress τ b exerted by the water exceeds a critical value τ c [START_REF] Meyer-Peter | Formulas for bed load transport. ser Pnd meetingD tokholmD ppendix PD rydruli ingineering eports[END_REF]. Under the assumptions of wide rectangular channel and uniform ow conditions the shear stress condition can be re-expressed in terms of water depth y, with the triggering depth y c being a function of the characteristic sediment diameter, the water and the sediment densities, the river bed longitudinal slope and the critical shear stress [START_REF] Shields | Application of similarity principles and turbulence research to bed-load movement[END_REF].

Once the critical shear stress is dened, there exist many semi-empirical formulas (e.g. [START_REF] Meyer-Peter | Formulas for bed load transport. ser Pnd meetingD tokholmD ppendix PD rydruli ingineering eports[END_REF]; [START_REF] Engelund | A monograph on sediment transport in alluvial streams (3rd edition). rydruli ingineering eportsD ehnil niversity of henmrk ystervoldgde IHD gopenhgen uF[END_REF]; [START_REF] Camenen | Simple and general formula for the settling velocity of particles[END_REF]; van [START_REF] Van Rijn | Sediment Transport, Part I: Bed Load Transport[END_REF]) to estimate the sediments ux as reviewed and compared by [START_REF] Davies | Comparisons between sediment transport models and observations made in wave and current ows above plane beds[END_REF]; [START_REF] Camenen | Comparison of sediment transport formulae for the coastal environment[END_REF] for instrumented rivers. These formulas make various assumptions on where this ux comes from (e.g. bed load ux for well-sorted ne gravel).

Sediment transport models as proxys for potential changes

Sediment transport models need to be calibrated using measurements of transported uxes which, unfortunately, are not available in general at hydrometric stations (especially during oods). Therefore it is not possible to use these models to precisely predict the volume of transported sediment.

However, we postulate that these models can still provide a valid information on the dynamics of transport (not the uxes themselves), which we consider as a proxy for potential change. We also postulate that this proxy model, to be related to morphological changes, should compute the cumulative volume of transported sediments, rather than the instantaneous ux. This proxy model rst has to be calibrated in retrospective mode, using previously detected changes identied by other methods: e.g. the segmentation of gauging residuals proposed in Chapter 2 or the stage-recession analysis proposed in Chapter 3. This calibration can be based on the principle that all past ood-induced changes should be detected by the proxy model.

Objectives and structure of the chapter

The main goal of this work is to propose a sediment transport proxy analysis for the fast detection of potential morphological changes at hydrometric stations during oods. The proposed method must be: calibrated without the use of sediment transport measurements, but by using only information widely available at hydrometric stations: past rating changes and the stage record;

suitable for a real-time application, when the managers of a station may want to quickly detect a potential morphological change.

The structure of this chapter can be summarised as follows. Section 4.2 describes the steps of the proposed method for the sediment transport proxy analysis. Section 4.3 then presents and discusses the results of its application to the Ardèche River at Meyras in France. Then limitations and possible improvements are discussed in Section 4.4. Finally Section 4.5 summarises the main ndings of the study.

Matteo Darienzo 92/174 4.2. The proposed sediment transport proxy analysis 4.2 The proposed sediment transport proxy analysis

Overview

The method is based on a retrospective analysis of the stage record and on documented rating shifts. It starts with the selection of the reference morphogenic events from the information available from the station history. Then it applies a sediment transport model to the stage record in order to reproduce the reference morphogenic events and to identify all other potential morphogenic events. It nally establishes a relation between the cumulative sediment transport and the associated potential rating shift to be used in real time. All steps are described in the following subsections.

Information available from the station history

In order to calibrate the proposed method, some information from the station history is required, and more specically the following two sets of data:

1. the set of the eective ood-induced shifts, leading to the reference shift times t ref .

2. the set of the associated shift estimates. Usually the shifts aect the oset of the lowest control. We therefore use the notation ∆b.

As regards the set of t ref , one can use the methods proposed in the literature (see Chapter 2)

or the methods proposed in the previous chapters to detect eective rating shifts. However, only the morphological changes caused by sediment dynamics should be included, and hence there may be more ∆b's related to other processes (e.g. gravel mining). Since the causes of the rating shifts are in general challenging to determine, if no additional information is available, then we suggest to apply the simple following rule: if the reference shift time refers to a ood then the rating shift is kept because most probably due to the sediment dynamics. On the contrary if it refers to a period of low ows it is discarded because probably caused by other processes.

As regards the shift estimates ∆b, they can be obtained from the RCs established for each period delimited by the times t ref .

To this aim, several methods are proposed in the literature Chapter 4. Fast detection of potential rating shifts based on the stage record and bedload assessment (as reviewed by [START_REF] Kiang | A comparison of methods for streamow uncertainty estimation[END_REF]). We use in this manuscript the BaRatin-SPD method proposed by [START_REF] Mansanarez | Shift Happens! Adjusting Stage-Discharge Rating Curves to Morphological Changes at Known Times[END_REF] because of its ability to estimate RCs also for periods characterised by few or no gaugings. Alternatively, in the absence of gaugings the recession analysis proposed in Chapter 3 can be used too. Then, if we consider a single-control hydraulic conguration a h -b (k) c valid for the period k before a ood peak t

∆b (k) = b (k+1) -b (k) ∼ N µ (k+1) b -µ (k) b ; σ (k+1) b 2 + σ (k) b 2 (4.1)
where µ b indicates the estimated value of b and σ b its uncertainty. In the Bayesian context they correspond to the mean and the standard deviation of the posterior distribution of parameter b, respectively. Notice that Equation 4.1 can be generalised to more complex hydraulic congurations.

Estimation of the triggering stage and detection of all potential morphogenic events

The basic principle of the proposed method is that a minimum water depth is needed to trigger a potential morphological change. More precisely, the sediment transport is initiated when the bed shear stress at time t, τ b (t), exerted by the water exceeds a critical value τ c . In a wide rectangular channel in uniform ow conditions this relation can be written as:

τ b (t) = ρ g y(t) S 0 > τ c (4.2)
where:

ρ = 1000 kg/m3 is the water density; g = 9.81 m/s 2 is the gravity acceleration;

S 0 is the river bed longitudinal slope (whose approximated value can be obtained from a hydraulic modelling or topographical survey, e.g. 0.005); y(t) is the water depth at time t, which is not known directly but can be obtained from the stage record through the hydrometric relation:

y(t) = h(t) -b(t).
Matteo Darienzo 94/174 4.2. The proposed sediment transport proxy analysis For non-cohesive sediments this condition is in general made dimensionless [START_REF] Shields | Application of similarity principles and turbulence research to bed-load movement[END_REF]:

τ * b (t) = τ b (t) (ρ s -ρ) g d > τ * c (4.3)
where:

ρ s = 2650 kg/m 3 is the usual quartz-rich sediment density;

d is the median or other characteristic diameter of sediments. Even though its value is dicult to estimate without specic grain size measurements, its order of magnitude (e.g. d = 0.05 m) can be evaluated from prior knowledge on the site and/or photos;

τ * c is the dimensionless critical bed shear stress with typical value within the range [0.03; 0.09] [START_REF] Bungton | A systematic analysis of eight decades of incipient motion studies, with special reference to gravel-bedded rivers[END_REF][START_REF] Soulsby | Dynamics of marine sands: a manual for practical applications[END_REF] according to grain size, slope, friction, hiding/exposure etc. It can be set to the traditional value for gravels, 0.047 [START_REF] Meyer-Peter | Formulas for bed load transport. ser Pnd meetingD tokholmD ppendix PD rydruli ingineering eports[END_REF].

[
The triggering stage h c (t) can be obtained by combining Equation 4.2 and Equation 4.3:

h c (t) = τ * c ρ s -ρ ρ d S 0 + b(t) (4.4)
All parameters of Equation 4.4 can be set to their predened values, except for parameters d and S 0 , which are strongly site-specic and may be aected by much larger uncertainty (if no specic surveys are performed). Therefore, the method needs to compute the triggering stage h c (t) by calibrating the value of the fraction d/S 0 (hereafter called φ) such that all the stage peaks corresponding to the reference times t ref are above h c (t). To this aim, we propose two main steps:

1. Class the N reference events (with reference shift times t ref ) with respect to the maximum stage: h

(1)

max < h (2) max < ... < h (N ) max .
2. Fix a value of φ such that the obtained h c < h

(1) max .

Unfortunately, while parameter b is known for each period delimited by the reference shift times t ref (see previous step), the evolution of b(t) during the oods is unknown and challenging to determine. If no precise information is available, a linear interpolation can be applied to solve this discontinuity (see example in The application of Equation 4.4 to the stage record determines the set of all potential morphogenic events. Then, the peak of these detected events dene the times of the potential ood-induced shifts, t pot . Notice that the impossibility of verifying and estimating the corresponding shifts makes them potential shifts only.

Computation of the sediment transport

In addition to the potential shift times detection, the analysis may also provide an estimate on the variance of the associated shifts ∆b by establishing a relation between the cumulative volume of sediment V at the end of each event and the associated shift magnitude ∆b.

Several models have been proposed in the literature to estimate the sediment discharge per unit of width q s (t) expressed in m 2 /s.

q s (t) =        8 ρs-ρ ρ g d 3 [τ * b (t) -τ * c ] 1.5 , τ * b (t) > τ * c 0 , τ * b (t) ≤ τ * c (4.5)
By substituting Equations 4.3-4.4 in Equation 4.5, the bed load equation becomes:

q s (t) =        ψ [h(t) -h c (t)] 1.5 , h(t) > h c (t) 0 , h(t) ≤ h c (t) (4.6)
where coecient ψ is equal to ψ = 8 ρs-ρ ρ g d 3 S 0 d ρ ρs-ρ 1.5

. All parameters included in ψ are xed to generic values, except for the ratio S 0 /d which has been calibrated as described in Section 4.2.3.

The cumulative volume of mobilised material V (k) expressed in m 3 computed from the beginning t (k) in to the end t (k) f of the morphogenic event k is:

V (k) = t (k) f t=t (k) in q s (t) B ∆t (4.7)
where B is the average active channel width at the hydrometric station.

Estimation of the uncertainty on the potential shifts

The principle behind the relation ∆b ↔ V is that the larger the volume of transported sediment during a ood is, the larger the associated potential shift is or at least larger is its uncertainty. When the volume V is small, the change cannot be large, but when V increases, it has the potential for being larger. The variance of ∆b is expected to increase with V . Moreover, the shift can be positive (sediment deposition, hence river bed raise) or negative (erosion, hence river bed lowering). Therefore to t the set of (V (k) , ∆b (k) ) pairs we propose the following probabilistic assumption:

∆b (k) ∼ N (µ (k) , σ (k) ) (4.8)
Chapter 4. Fast detection of potential rating shifts based on the stage record and bedload assessment with:

       µ (k) = 0 σ (k) = σ (k) ∆b 2 + ξ V (k) 2 (4.9)
where σ

(k) ∆b is the standard deviation representing the uncertainty in the shift ∆b (k) (Equation 4.1) and ξ is a positive parameter that is estimated though a Bayesian-MCMC approach described in [START_REF] Renard | An application of Bayesian analysis and Markov chain Monte Carlo methods to the estimation of a regional trend in annual maxima[END_REF] and in Chapters 2-3. Poorly-informative prior is specied for ξ (e.g.

ξ ∼ U(0, 1)).

In real time, starting from the beginning of the morphogenic event, the uncertainty of RC parameter b (k) can be updated by accounting for the standard deviation ξ(V ) of the potential shift. In a Bayesian context the prior distribution of parameter b (k) can be updated (e.g.

assuming a Gaussian distribution) as follows:

b(t) ∼ N µ (k) b ; σ (k) b 2 + ξ V (k) (t) 2 (4.10)
To illustrate the method in the next section the proposed sediment transport analysis is applied to the case study of the Ardèche River at Meyras, France (also treated in Chapters 1 and 2). Stage-Period-Discharge analysis [START_REF] Mansanarez | Shift Happens! Adjusting Stage-Discharge Rating Curves to Morphological Changes at Known Times[END_REF]. The choice of the proper value of φ can be done through the two steps described in Section 4.2.3. In particular, Figure 4.4 shows that, while the set of shift times t pot obtained with φ = 30 misses one of the reference shift times (the last one), both φ = 10 and φ = 20 lead to set t pot including all t ref . The choice between φ = 10 and φ = 20 is subjective, though.

h c (ϕ=10) h c (ϕ=20) h c (ϕ=30) (ϕ=10) (ϕ=20) (ϕ=30)
However we may want that the cumulative volume of bed load for the period when h(t) > h c is large enough. If for example we take φ such that h c = h

(1) max , the bed load would occur for a too short time (equal to zero) to induce a river bed shift. With φ = 20, h c is still too much close to h

(1) max . However, if we take φ = 10 too many potential shifts are detected. Instead, using φ = 15 can be considered as a good trade-o between detecting all reference events and detecting other potentials shifts. Thus, φ = 15 is selected for this application.

Relation between shift ∆b and sediments volume V

The cumulative volumes of sediment V for each potential morphogenic event have been computed by using Equation 4.7 and by xing φ = 15 and B = 15 m. They are plotted with blue bars in Figure 4.5. Several potential events have a large volume but are not associated with any reference events (whose peaks are identied with circles in the gure), e.g. the volume at t = 5500 is larger than the volumes of events 1, 3, 6. This suggests that no change occurred despite the large transported volume. The sediment dynamics can be very dierent during each event leading to dierent responses: the alternation of scour and ll processes may cause net shift very close to zero at the end of the ood.

Then, the volumes V of the reference events are used as calibration data for the relations ∆b 1 ↔ V and ∆b 2 ↔ V . Each reference event has been selected as the largest ood within the uncertainty interval of the shift time obtained by the segmentation of gaugings (see Chapter 2).

It may happen that several oods are included within this interval and they are probably contributing to the rating shift. Therefore we propose that some very close events (separated by maximum 100 days, the order of magnitude of the shift times uncertainty) are merged in unique events (see ribbons in pink in Figure 4.5). Notice in particular that reference events 5 and 6 have been merged in a unique event. The eective shifts ∆b 1 and ∆b 2 associated to each

Discussion

Main limitations

The applicability of the approach may be restricted to gravel bed rivers. Indeed, for sandy rivers the estimated triggering water depth may be extremely small which may lead to the continuous detection of potential shifts. The proposed method should still be applicable but using sediment transport models more suitable to sand granulometry [e.g. [START_REF] Engelund | A monograph on sediment transport in alluvial streams (3rd edition). rydruli ingineering eportsD ehnil niversity of henmrk ystervoldgde IHD gopenhgen uF[END_REF]] instead of the one proposed by Meyer-Peter and Müller [1948]. However the formulation proposed by [START_REF] Engelund | A monograph on sediment transport in alluvial streams (3rd edition). rydruli ingineering eportsD ehnil niversity of henmrk ystervoldgde IHD gopenhgen uF[END_REF] (frequently used for ne sediments) does not include any thresholds for incipient motion. A dynamic modelling of the stage-discharge relation may be necessary in this case.

Moreover, the calibration of the triggering stage is based on the assumption that all reference shift times are correctly identied. However this assumption may be wrong, if a shift is missed or attributed to the incorrect ood, or if a shift that doesn't exist is detected. The impact that this issue may have on the results needs to be studied.

Another issue is that the method excludes the breathing phenomena of scour and ll that may occur during oods. To properly study these events a solution may be to apply theoretical sediment transport modelling under unsteady ow and unsteady sediment transport conditions [START_REF] Davies | Physical model study of stage-discharge relationships in a braided river gorge[END_REF]. However this requires experimental surveys on the sediment size distribution and on the time-scale of bed deformation. Notice also that during oods the variation of the river bed may be complex and may aect not only parameter b of the low controls but also the width of the control or the channel slope, hence parameter a (see the example in Figure 4.7 of the braided Wairau river at Barnetts Bank in New zealand). However, detecting in real time changes in the parameter a, based on the stage record only, may be challenging. On the other hand, photos or videos might be enough for this purpose, since a change in the width should be easier to observe visually than a bed change.

Another possible issue is related to the merging of multiple morphogenic events. The choice of the maximum inter-event time-lag for merging the events (e.g. 100 days for Meyras) is quite subjective and case-specic. Further work is necessary to improve this step. A solution may be to merge the events there are located within the uncertainty interval (e.g. at 95 %) of the shift times obtained from the gaugings segmentation (see Chapter 2).

Moreover, the relations ∆b ↔ V , by using the model described by Equations 4.8-4.9 with mean equals to zero and standard deviation increasing proportionally with V , successfully yielded the estimation of the uncertainty on the potential rating shift during oods. However, we could also use a linear or proportional model which would provide the trend and the direction of the morphological evolution (e.g. raising or lowering of the bed elevation).

Finally, the assumption of constant channel longitudinal slope S 0 and of constant characteristic diameter d (hence constant φ) may be too strong in some cases. A sensitiviy analysis could be done in the future to assess whether this assumption has an impact on the results or not.

Use of the method for retrospective purposes

Notice that the objective of this chapter is the fast detection and estimation of potential rating shifts in the real time context. However, the station managers may be also interested in the retrospective detection of all past potential rating shifts in order to re-calibrate the RC for each stable period and reconstruct the historic streamow time series. Working with many near-zero potential changes is not useful and creates practical problems. Filtering out such minor potential changes would therefore be valuable.

In fact, the condition h(t) > h c (t) does not imply that a net morphological change occurred but only that the bed sediment transport is initiated. A minimum volume of transported sediments (V min ) from the beginning of the event may be necessary to result in a net morphological change, justifying an update of the RC. Moreover an armouring phenomenon of the river bed may occur. Sediment transport can start well beyond the triggering stage and suddenly mobilise a lot of material.

The minimum volume of transported sediments V min (necessary for a morphological change) can be estimated as the minimum volume among the reference events. Then, the manager may decide to select among all the potential events only those ones characterised by V ≥ V min .

Other perspectives

Perspectives of the proposed sediment transport proxy analysis include the tests on more challenging stations: in particular rivers with sandy bed and continuous bed evolution, and rivers characterised by frequent oods. The method to be validated should also be tested on a station where sediment transport measurements are available.

Finally, the calibration of the triggering stage and the sediment transport, thus the detection and the estimation of potential shifts, may be performed through a Bayesian method, using a binary detection index I, where I = 0 if the potential shift is not part of the reference shifts and I = 1 if it is. This approach may allow a more formal inclusion of the prior knowledge on d/S 0 and may provide uncertainty on the estimates of the cumulative volumes V that is otherwise ignored. Further work is necessary to test its feasibility and its added value.

Matteo Darienzo 108/174 4.5. Conclusion

Conclusion

The method proposed in this chapter for the fast detection of potential shifts uses the stage record and a proxy sediment transport analysis.

The method is based on the exceedance of a triggering stage. Computing this threshold requires calibrating a parameter in retrospective such that the triggering stage is exceeded by all the reference morphogenic oods detected by other methods (e.g. segmentation of gaugings and stage-recession analysis). The method yielded encouraging results for the Ardèche River at Meyras conrming all reference eective shifts. Several other potential morphogenic events have been identied. As expressed in the general introduction, it is sometimes necessary to detect and estimate rating shifts with the least possible delay in order to obtain more reliable streamow data which may lead to more informed decisions, for example for the following applications:

during a ood event for forecasters, a poorly-estimated streamow assimilated into a ood forecasting modeling chain may compromise its reliability;

during drought the decision-makers (Water Authorities) must know the streamow each day in order to be able to take orders regulating the use of water resources;

the minimum environmental ow downstream of a hydroelectric facility or a nuclear power plant must be respected continuously, hence known at all times. Upstream streamow is also required in real time for optimizing the production and the safety of the facility.

It is important to draw a clear distinction between real-time and retrospective analyses.

The retrospective analysis is the study of the past hydrometric data (e.g. gaugings, stage record) which are fully available. In this context there is no need to detect a rating shift right after its occurrence. A shift can be detected with a delay of one year for example and then it is possible to search back for the event most probably associated with the shift (e.g. the most intense ood). The detection of rating shifts and the periods of RC stability delimited by the shift times may then lead to the re-construction of the past hydrograph, which is essential for several purposes such as ood frequency analysis and design ood maps, or to simply classify the stations into "stable" and "unstable" categories [START_REF] Morlot | Dynamic rating curve assessment for hydrometric stations and computation of the associated uncertainties: Quality and station management indicators[END_REF] in order to give priority to the unstable stations when planning gauging campaigns.

On the contrary, the real-time analysis is based on the analysis of any data or information available in real time to detect a shift while it is occurring or at least as soon as possible after its occurrence. Real-time analysis should be applicable before any eld intervention has been performed. Therefore the real-time analysis cannot only rely on the segmentation of gaugings (Chapter 2) and the stage-recession analysis proposed in Chapter 3 can be applied only when the stage is located in a recession period. During a morphogenic ood the real-time application should use all kind of information, however uncertain it may be, to suspect a potential shift.

The sediment transport proxy analysis proposed in Chapter 4 is a solution to achieve this, but other tools may be explored.

Solutions proposed in the literature and main diculties

Unfortunately, while for the retrospective analysis several methods have been proposed in the literature (see the introduction of the previous chapters), less information is available on the management of rating curves in real time, and as far as we know, no formalised and automated methods have been proposed for this purpose.

The necessity of solving this issue is mentioned by [START_REF] Kiang | A comparison of methods for streamow uncertainty estimation[END_REF], Mansanarez et al.

[2019], Le Coz et al. [2017], [START_REF] Puechberty | Charte qualité de l'hydrométrie: Guide de bonnes pratiques. qroupe hoppler rydrométrieF winistère de l9invironnementD de l9inergie et de l werD prne[END_REF], but they do not propose any general protocol. In the operational practice, each hydrometric service makes use of expertise and available shift detection tools but especially of good sense for taking decisions and actions about the suspicion of a rating shift. A real-time quality check of gaugings and RC is performed but this remains very empirical, with very little automated procedure to help the operator to detect and estimate a potential shift during its occurrence.

The national (French) hydrometry quality plan [START_REF] Puechberty | Charte qualité de l'hydrométrie: Guide de bonnes pratiques. qroupe hoppler rydrométrieF winistère de l9invironnementD de l9inergie et de l werD prne[END_REF] proposes to analyze the streamow consistency with another neighboring station to detect a discrepancy either with upstream-downstream linear regression (on stage or on streamow). Alternatively a correlation analysis can be performed between the estimated streamow and the output of a hydrological model [START_REF] Garçon | Prévision opérationnelle des apports de la Durance à Serre-Ponçon à l'aide du modèle MORDOR. Bilan de l'année 1994-1995[END_REF], looking for discrepancies in the correlation [START_REF] Puechberty | Charte qualité de l'hydrométrie: Guide de bonnes pratiques. qroupe hoppler rydrométrieF winistère de l9invironnementD de l9inergie et de l werD prne[END_REF][START_REF] Lucas | Développement d'une méthode de détection en temps-réel des ruptures dans la relation hauteur/débit. pport de stge de (n d9étudesD endré pr ierre frigodeD olyteh xieEophiD et hmien evrezD ihpEhq[END_REF].

However, the basic principle discussed in this chapter is that if we succeed in providing a warning about a potential rating shift to the operator and to quantify its uncertainty in real time this would be already an important step forward. However, these estimates will necessary come with large uncertainties, and they do not intend to replace a new eld measurement (e.g gauging, bathymetry) to decrease this uncertainty.

Outline of a real-time procedure

The functioning of such a real-time application should basically be structured around one or more methods for the detection and estimation of rating shifts and one method for the RC estimation with uncertainty. All of the traditional and emerging methods for detecting/estimating rating shifts (including the three methods proposed in the previous chapters) can be used. These tools should rst be tested retrospectively by detecting all past shifts before their application in real time.

The time step of the real-time application depends on the incoming stage frequency.

However many time steps may be needed before being able to detect and estimate an eective rating shift, i.e. a shift for which the magnitude can be estimated. In the meantime, we may want to use some tools (e.g. the sediment transport proxy analysis proposed in Chapter 4) to detect the shift as quickly as possible, thus a potential shift, for which the objective is to estimate its uncertainty, as opposed to its precise magnitude. summarises the main ndings of the proposed real-time application.

Incoming stage data

While for the retrospective analysis the entire data sets of past gaugings and stage record are available, in the real time iterations the input data are unending sequence of high-frequency stage observations and low-frequency gaugings. Since the detection tools developed and used in this manuscript for fast detection of potential rating shifts are mostly based on the stage record, the real-time step ∆t is adapted to the frequency of the incoming stage (e.g. 15 minutes). In this manuscript the stage data are assumed not aected by any instrumental error, but this assumption is discussed in Section 5.4.2.

Shift detection

Once the new stage data have been received, the application proceeds by investigating the RC stability at time t. To this aim the proposed application makes use of all available tools for rating shift detection and estimation. By default these tools are kept on stand-by until some predened threshold is exceeded or some conditions are met. In this manuscript three tools are used:

During oods, when the stage exceeds the triggering stage for incipient motion (see Chapter 4) the tool based on the sediment transport proxy analysis is activated and a potential rating shift is detected.

During stage recessions, after a few consecutive days of recession, the tool based on the stage-recession analysis is activated.

As described in Chapter 3, this analysis allows detecting a shift but also estimating its magnitude.

If a new gauging is performed between two time steps then it is added to the gaugings data set of the base RC and used as input for the detection of eective rating shifts, as described in Chapter 2.

We stress that any rating shift detection tool and on-site information on the shift (photos, witnesses) can be used at this step.

parameter (e.g. b(t)) needs to be updated as:

b(t) ∼ N µ * b + µ ∆b (t); σ * b 2 + (σ ∆b (t)) 2 (5.3)
where µ * b and σ * b are the posterior mean and posterior standard deviation, respectively, of parameter b from the base RC right before the beginning of the provisional period, and µ ∆b (t) and σ ∆b (t) are the mean and the standard deviation, respectively, of the shift estimate ∆b(t) during the provisional period at time t. If there are not new gaugings then the RC estimation is done through prior propagation, otherwise it is performed by using the BaRatin method.

Discharge computation

The new RC is used to compute the discharge Q(t) (the main output of each time iteration).

This is a standard RC procedure in hydrometry and leads to the most probable value of Q(t)

(MAP) and its uncertainty [START_REF] Le Coz | Combining hydraulic knowledge and uncertain gaugings in the estimation of hydrometric rating curves: A Bayesian approach[END_REF].

Start of a new stable period

At the end of the real-time iteration, after having provided the provisional RC and the value of discharge at time t, we need to decide whether or not a new stable period starts after this iteration. Considering the situations described in the previous step:

For situations 1 nd 2, the stable period is still ongoing.

For situations 3 and 4, the provisional RC becomes the new base RC only if there are new gaugings. In this case a new retrospective analysis is performed with the addition of the newly detected shift information to update the calibration of the tools for rating shift detection. When there are no new gaugings the provisional period proceeds.

The next section illustrates a prototype of the proposed real-time method applied to the case study of the Ardèche River at Meyras in France during one ood event.

5.3 Application to the Ardèche River at Meyras: a demo

Overview of the application

A real-time re-analysis is applied to one morphogenic ood event at the Meyras station on the Ardèche River (France). The hydraulic analysis and the retrospective analysis are performed at time t 0 . A truly real-time context is studied, by fully ignoring any data that would not be available at time t 0 . The retrospective analysis detects and estimates all past rating shifts in order to calibrate the shift detection tools and estimate the stable RC at t 0 . Next, the crucial real-time conditions for the studied event are:

A rst period where the RC is stable and equal to the base RC (no shifts are detected).

The occurrence of a morphogenic ood inducing a potential rating shift A recession period after the ood peak. The stage-recession analysis proposed in Chapter 3 is used. The longer the recession period, the more precise the estimation of the asymptotic stage becomes, and consequently the RC uncertainty may be rened with time.

Finally the arrival of new gaugings may conrm or dismiss the rating shift previously detected. The segmentation procedure applied to gaugings and proposed in Chapter 2 is used to this aim. If the shift is conrmed, then a new RC stable period begins.

Figure 5.2 illustrates the selected period for the analysis, subdivided in two parts by the initialization time t 0 : the retrospective analysis and the real-time analysis. t 0 is chosen such that the RC stability is ensured: no oods located nearby, no new gaugings since the previous stage data and a very low stage value.

The retrospective analysis

The retrospective analysis has been performed on the stage record and the gaugings preceding t 0 . The results are summarised in Figure 5.3. The stage-recession analysis is here performed using the model and the parameter χ that lead to the most accurate estimation of shift times and magnitudes for the application of Chapter 3: M2 (two superposed exponential terms and one asymptotic stage parameter), χ = 50 cm and all other options for the stage-recession extraction used in the application of Chapter 3. Then, the posterior distributions of the common parameters of the stage-recession model will be used as priors for the recession estimation during the real-time iterations, as described in Section 3.4.2 of Chapter 3.

Retrospective

The combined results of the segmentation of gaugings and of the stage-recession analysis are used to dene the four RC stable periods of the retrospective period. A BaRatin-SPD analysis [START_REF] Mansanarez | Shift Happens! Adjusting Stage-Discharge Rating Curves to Morphological Changes at Known Times[END_REF] is performed to estimate the RC for each period (Figure 5.4a). These combined results are then used to calibrate the sediment transport proxy analysis.

Firstly, the triggering stage h c is computed and reported in green in Figure 5.3. This leads to the detection of several other potential rating shifts. Secondly the sediment transport, and in particular its cumulative volume, is computed. A relation between the cumulative volume of sediments transported during the morphogenic event and the associated potential rating shifts ∆b 1 and ∆b 2 is then established (as described in Chapter 4) and shown in Figure 5.5. At this stage, only three calibration data (V, ∆b) are available.

Finally, the BaRatin-SPD analysis [START_REF] Mansanarez | Shift Happens! Adjusting Stage-Discharge Rating Curves to Morphological Changes at Known Times[END_REF]] also provides the estimate of the stable rating curve (base RC) preceding the initialisation and still valid at time t 0 . The initial conditions of the real-time application, including the data and the RC of the last stable green: the tool is on stand-by;

yellow: the tool is activated but no rating shift is detected red: the tool is activated and a rating shift is detected

The proposed application makes use of (from top to bottom of Figure 5.6C):

the segmentation of the residuals between the gaugings and the base RC (procedure described in Chapter 2). If no new gauging is performed then no rating shift warning is provided by this tool keeping the light on "green". For the next time iterations every time that a new gauging is performed, its residual with respect to the base RC (with uncertainty) is then computed and plotted here and the light is turned on "yellow". The Chapter 5. The real-time application segmentation procedure is then applied to the time series of the residuals looking for one shift. The mean of the residuals is reported by the horizontal red segment.

The sediment transport proxy analysis (see details in Chapter 4). If the current stage is below the triggering stage h c no potential rating shifts are detected by this tool and the light is kept on "green". The volume of transported sediments V in this case is equal to zero. Instead, when the actual stage exceeds the triggering stage h c , the light is turned on "red" warning for a potential rating shift. The sediment transport volume is in this case larger than zero and the relation in Figure 5.5 is used to estimate the uncertainty of the potential shifts ∆b1 and ∆b2, thus to increase the uncertainty on corresponding RC parameters.

the stage-recession analysis (Chapter 3). If the current stage is not in a recession period then no warning for rating shift is provided by this tool, keeping the light on green. Instead, when the current stage is located in a recession period then the new stage data is plotted and the light is turned on "yellow". If the number of the stage-recession data and the duration of the recession period are larger then the minimum number N min and minimum duration t min , respectively, then the stage-recession analysis is performed.

Subsequently, each new incoming stage data dene the following iterations of the real-time application. The following subsections will focus on the most crucial iterations. At this iteration the stage-recession analysis cannot be performed because the stage is increasing, thus the tool is kept on standby.

Matteo

Moreover, no new gaugings have been performed thus also the tool based on the segmentation of gaugings is kept on standby.

Instead, the stage h(t) slightly exceeds the triggering stage h c (t) for sediment incipient motion. Hence, a potential ood-induced rating shift is detected and the tool based on the sediment transport analysis is activated (red light). This iteration denes the beginning of a morphogenic ood event. The volume of transported sediments cumulated from the beginning of the event V (t) is computed through the proxy model described in Chapter 4. This tool provides also an estimate on the uncertainty of the rating shift through the relation V -∆b ST established in retrospective (Figure 5.5), so that for each value of V corresponds an estimate of the shift magnitude ∆b and in particular of its uncertainty.

The priors of parameters b 1 (t) and b 2 (t) are thus updated accounting for the rating shift using Equation 5.3 with σ ∆b 1 (t) and σ ∆b 2 (t) around 0.0003 m and V = 28 m 3 . At such an early stage of the ood, the potential shift cannot be too large. However, since the available gaugings are now potentially not valid anymore, the base RC is obsolete and the new provisional RC is now estimated through prior propagation only. As for iteration 82 the RC is merely estimated through a prior estimation by updating the priors of parameters b 1 (t) and b 2 (t) of the base RC using Equation 5.3. However, at this iteration σ ∆b 1 (t) and σ ∆b 2 (t) assume values around 0.17 the stage is located in a recession period after the ood peak, with a length larger than 5 days.

Matteo

Thus a stage-recession analysis can be performed. The estimated recession is illustrated in green.

Then, the segmentation of the estimated asymptotic level parameters β does not lead to the detection of any shift. Even though the recession analysis does not detect at this iteration a rating shift, it does provide a valuable uncertainty ∆b rec (t) on it as dened by Equation 5.1. If this uncertainty is inferior to the one obtained through the sediment transport analysis ∆b ST (t)

it can be used to rene the prior on b 1 (t) and b 2 (t) as explained in Section 5.2.5. Thus also the new provisional RC shows a smaller uncertainty compared with the previous iterations. 5.3.8 Iteration 311: new gauging and rating shift conrmation Figure 5.12 illustrates the results of iteration 311. At this time step a new gauging has been performed. Thus, the tool based on the segmentation of gaugings is activated. A shift is detected. Thus the ligth is turned on "red". This corresponds to an eective rating shift, which means that the provisional period is ended and that a new base RC can be estimated using the new gauging.

Matteo

The stage is still located in a recession period, thus the estimation of the rating shift ∆b rec is used to update the prior before the RC estimation with the new gauging. The new gauging conrms the potential rating shift previously detected by the sediment transport analysis and renes the estimation of parameters b 1 and b 2 . The estimated shifts ∆b 1 and ∆b 2 are quite small, around -0.066 m and -0.092 m, respectively. This possibly explains why the change wasn't detected by the stage-recession analysis.

This time step determines the beginning of a new stable period.

Summary of the application

In conclusion, during this application we have tested the tools for rating shift detection in real time during a morphogenic ood event known to have caused a minor shift of the RC. The tool based on the segmentation of gaugings does not detect any shifts when there are no shifts (before the ood) and instead detects the minor shift when the rst gauging is performed after the ood. The sediment transport analysis properly detects a potential rating shift which is conrmed subsequently. Moreover the estimation of the shift uncertainty through the relation

V -∆b provides a useful information on the RC uncertainty during the ood. On the other hand the tool based on the stage-recession analysis shows some diculties in detecting a minor shift. However, it provides a very useful information on the rating shift uncertainty which may reduce the one provided by the sediment transport analysis, leading to more reliable streamow uncertainty. Chapter 5. The real-time application 5.4 Discussion

Matteo

Main limitations

While the scheme of the proposed application for the real-time management of rating shifts is conceived to account for general real-time situations, it still have some limitations. Some of these limitations are related to the tools used for the shift detection and are already discussed in the previous chapters. More specically related to the real-time framework itself, the structure of the proposed method is conceived for the management of rating shifts only. Instead, managing transient shifts, such as those induced by the growth/decline of aquatic vegetation, may require a dynamic RC modelling [START_REF] Perret | A Rating Curve Model Accounting for Cyclic Stage-Discharge Shifts due to Seasonal Aquatic Vegetation[END_REF] which, in turn, may require a specic adaptation to the real-time context.

Stage pre-treatment

In this application the uncertainty in the stage input data is ignored. However, this uncertainty may sometimes be not negligible [START_REF] Horner | Impact of stage measurement errors on streamow uncertainty[END_REF], especially in a real-time context. The stage observation may be aected by a general noise (due to instrument sensitivity or to the water surface oscillations induced by high ow conditions) that need to be ltered out, or by instrumental bugs (due to data logger issues or to icing conditions) that need to be corrected before the stage can be exploited (Figure 5.13).

Future perspectives of the proposed method include some pre-treatments of the raw stage measurements. Figure 5.13 schematizes three possible situations discussed in Puechberty et al.

[2017]:

1. The instrument breaks down or is frozen, then the stage raw signal appears frozen too. As long as the instrument is out of order, no correction of the signal is possible. A solution needs to be found to provide streamow estimate during this latent period.

2. Sudden instrumental bugs may also occur causing invalid aberrant stage values. These bugs can easily be corrected retrospectively by interpolation (on short periods at least), but this issue is more challenging in real time. It may happen at some iterations that multiple sources of information are available about the rating shift. Therefore we may want to aggregate the dierent probability estimates. The issue of combining dierent "opinions" has been studied in the literature: e.g. by [START_REF] Genest | Combining probability distributions: A critique and an annotated bibliography[END_REF] or more recently by [START_REF] Albert | Combining expert opinions in prior elicitation[END_REF] and [START_REF] Dietrich | Probabilistic opinion pooling. yxford rndook of roility nd hilosophy[END_REF]. However no unique and general solutions have been provided yet. 1. considering the product of the two pdfs (consensus prior) which in this case is also Gaussian.

2. averaging the two pdfs (mixture prior).

Further work is needed to apply the approaches proposed in the literature to the real-time operational management of RCs.

Matteo Darienzo 142/174 5.4. Discussion

Adding other tools for detecting rating shifts

Other tools for rating shift detection and estimation are planned to be developed in future work and are further discussed in Chapter 6, such as:

-correlation analysis with the output of a rainfall-runo hydrological model or with the discharge record of the hydropower plants [START_REF] Puechberty | Charte qualité de l'hydrométrie: Guide de bonnes pratiques. qroupe hoppler rydrométrieF winistère de l9invironnementD de l9inergie et de l werD prne[END_REF];

-correlation analysis with the surface velocity measure by means of non-contact techniques (e.g. radar, [START_REF] Thollet | Mesure de débit en rivière par station radar hauteur / vitesse selon la méthode de la vitesse témoin [Streamow monitoring at stage / velocity radar stations using the index velocity method[END_REF]).

-on-site bathymetric surveys;

-other eld observations (e.g. presence of dams built by beavers or swimmers; data logger bugs);

-information about mining operations in the river bed or civil works aecting the ow.

Conclusion

The proposed method for the real-time management of unstable rating curves is based on the application to the real-time context of the available tools for rating shift detection and estimation, and of the available tools for RC estimation with uncertainty.

Each iteration is based on the incoming new stage data, then it applies the aforementioned tools searching for a rating shift (potential or eective) and it eventually updates the RC with the estimation of a provisional RC, which in the Bayesian context is done by updating the prior specication. When an eective shift is estimated then a new base RC is dened and a new stable period begins.

The method has been applied to the case study of the Ardèche River at Meyras during one morphogenic ood event. The application yielded promising results with the detection of the potential shift by the sediment transport proxy model and the rening of the RC uncertainty by the stage-recession analysis and nally the conrmation of the shift by the rst gauging. However, these results cannot be generalized yet. Further work is needed to test the performance of the method on several other case studies with dierent characteristics and dene the applicability limitations.

The second tool has been developed to detect and estimate eective shifts using a stagerecession analysis. Its originality is mainly to apply for the rst time in hydrometry the recession analysis concepts, broadly studied in the literature, to the stage record instead of the streamow record. The main assumption is that the stage-recession curve tends toward the elevation of the lowest control as streamow tends toward zero and that a morphological change of the river bed induces a parallel change of this asymptotic stage.

After having estimated all the available recession curves, the same segmentation procedure that was applied to the gaugings residuals is applied to the time series of asymptotic stage estimates.

The method leads not only to the detection of a rating change but also to estimate its amplitude.

The rst and second tools are limited for real-time applications since they need information (gauging, recession) that is only occasionally available in real time. For the same reason, these tools may miss rating shifts in the retrospective analysis. Thus a third tool has been proposed

in this manuscript to detect potential rating shifts. While for eective shifts the magnitude can be estimated precisely enough, for potential shifts the primary objective is to estimate its uncertainty. This tool is based on a sediment transport proxy analysis using as input the stage record available in real time. The shift detection is issued when stage exceeds a triggering stage for incipient motion which is calibrated in retrospective by analysing all past rating shifts and the corresponding morphogenic oods. Then, a relation established between the cumulative volume of transported sediments (computed using classical models) and the past shift estimates can be used in real time to provide an uncertainty estimate on the detected potential shift.

The advantages of using this approach are that it detects a potential shift while it is occurring, without delay, and that it requires only the stage record and some knowldege on the past rating shifts.

The three tools have been applied initially for the retrospective detection of rating shifts and identication of the stable periods. This may allow the managers of a station to review the historical series of hydrometic data and to reduce the number of stable periods when a conservative analysis led to over-segmentation of the periods (e.g. after each ood).

Finally the proposed methods are implemented together into an original framework for the Matteo Darienzo 146/174 6.1. Summary real-time management of shifts aecting the rating curves and the streamow estimation. Such real-time scheme is an original outcome of this thesis work since it was never formally developed before in the hydrometry eld. As a proof-of-concept, the method has been applied to the station of the Ardèche River at Meyras in France during a morphogenic ood. The combination of the three tools for shift detection and estimation yielded promising results and showed that they can be successfully used at least for gravel bed rivers aected by morphogenic oods. However further work is needed to validate the method and demonstrate its operational applicability, in particular by testing it on several other stations. The scheme of the proposed framework is also conceived to be general, thus other potential shift detection criteria can be included as detailed in the following perspectives sections. the real-time application. The most suitable recession model might be dierent in retrospective and in real time. Further work is necessary in order to test other models in real time.

Other interesting perspectives include applying the stage-recession analysis to several case studies in order to provide some statistics on the stable parameters of the recession model (such as the recession rate λ), searching for hydrological signatures (as studied in the PhD thesis of [START_REF] Horner | hesign nd evlution of hydrologil signtures for the dignosti nd improvement of proessEsed distriuted hydrologil model[END_REF]).

Sediment transport proxy analysis

According to the method proposed in Chapter 4, the criterion used for detecting potential shifts is based on the exceedance of the triggering stage h c for incipient motion. This criterion leads to the detection of many events characterized by little sediment transport, thus by near-zero potential changes. This may certainly create practical problems for retrospective applications but also in real time when the manager of the station may not want to question the validity of the base RC too often to account for near-zero changes. An alternative to this may be to condition the detection of a rating shift to a minimum volume of transported sediments likely to cause a signicant shift.

Finally another possible perspective, but not formalised yet, is the calibration of the triggering stage through a Bayesian framework. This may allow a more formal inclusion of the prior knowledge on the unknown parameters and considering some additional uncertainty that is otherwise ignored.

Performance evaluation using a wide range of hydrometric stations

Another important perspective of this work is the evaluation of the performance of the proposed methods on a range of dierent stations. While not described in this manuscript, we tested a few stations and the tests on a few others are planned in collaboration with several French and international services (e.g., CNR, EDF, DREAL, SCHAPI, NIWA, USGS, DEAL-Réunion). The stations to study are characterised by: Matteo Darienzo 150/174 6.2. Perspectives Dierent river bed composition: gravel bed or sand beds. We focus in this manuscript on gravel bed rivers. We suppose that the application of the proposed methods to sand bed rivers (e.g. the Loire River at Montjean in France) may be more challenging due to the progressive river bed modication induced by continuous sediment transport and the great contribution of the suspended sediments to the erosion/deposition processes.

Dierent frequency and intensity of shifts (e.g., the case of Séveraissette River at Vilar-Loubière, France which is highly unstable and the Illinois River at Tinley Creek, USA which has very small shifts) and input data (such as the available gaugings and recessions between the shift times). The ability of the segmentation of gaugings to detect rating shifts depends on the gaugings frequency with respect to the shift frequency, the gaugings uncertainty and their location along the RC. Two stations have been already processed:

the results of the Wairau River at Barnetts Bank in New Zealand (Figure 6.3b) show that shifts occur very frequently with respect to the gaugings, thus many periods have few gaugings only. The results of the Mat River at Escalier, Reunion Island (Figure 6.3a)

show that gaugings are mostly located at low ows leading to very uncertain base RCs and challenging segmentation. Moreover the stage-recession analysis depends on the number of long recessions that are available. Its application to stations with very frequent oods (e.g., the Wairau River at Barnetts Bank in New Zealand) may be very challenging and some improvements may be necessary.

More complex controls and shifts which may aect other RC parameters (e.g., parameter a due to shifts of the channel width or the roughness coecient or the longitudinal slope), such as braided or meandering rivers (e.g., the Drôme River at Loriol in France, the Rakaia River at Fighting Hill, in New zealand) or twin-gauge stations (e.g., the Isère River at Beaumont-Monteux). The correlation analysis between the discharge time series produced at the station and the output of a hydrological rainfall-runo model might be the most promising perspective.

Many watersheds have models of this type. A rating shift may be detected by applying a segmentation procedure to the series of residuals derived from a linear regression between the two sets of discharge data. This approach is mentioned in the national (French) hydrometry quality plan [START_REF] Puechberty | Charte qualité de l'hydrométrie: Guide de bonnes pratiques. qroupe hoppler rydrométrieF winistère de l9invironnementD de l9inergie et de l werD prne[END_REF]. Moreover, [START_REF] Lucas | Développement d'une méthode de détection en temps-réel des ruptures dans la relation hauteur/débit. pport de stge de (n d9étudesD endré pr ierre frigodeD olyteh xieEophiD et hmien evrezD ihpEhq[END_REF] has tested this approach by using the hydrological model MORDOR [START_REF] Garçon | Prévision opérationnelle des apports de la Durance à Serre-Ponçon à l'aide du modèle MORDOR. Bilan de l'année 1994-1995[END_REF]).

Such residuals are characterized by large and strongly varying uncertainties, like the gaugings residuals (Chapter 2). Thus the segmentation procedure proposed in Chapter 2 would be well suited to them. However, a big dierence with gaugings is that the number of data is much larger and their errors not independent (model bias). The risk may be a bias towards overcondence in model outputs (cf. [START_REF] Sikorska | Calibrating a hydrological model in stage space to account for rating curve uncertainties: general framework and key challenges[END_REF]). Moreover this tool depends on the availability of the input climatic data such as the precipitation and of the calibration streamow data, and one problem to solve is how to quantify the model output uncertainty.

Correlation with neighboring stations

Also the spatial and temporal analysis of neighboring stations can be a powerful tool.

It consists of comparing the discharge time series of the studied station with the discharge time series of another station with comparable hydrological behavior (e.g., located upstream or downstream along the same river, located in a tributary river, located in a neighboring catchment, etc.). Again, a segmentation procedure can be applied to the residuals. This approach is discussed in [START_REF] Puechberty | Charte qualité de l'hydrométrie: Guide de bonnes pratiques. qroupe hoppler rydrométrieF winistère de l9invironnementD de l9inergie et de l werD prne[END_REF] and the issue of characterizing the spatial correlation of daily streamows has been studied by several authors, such as [START_REF] Betterle | Characterizing the spatial correlation of daily streamows[END_REF].

However, it is necessary to consider a suciently long time step (e.g., monthly) to ensure some reasonable degree of correlation between the two stations. Moreover this analysis cannot 153/174 Matteo Darienzo Chapter 6. Conclusions and perspectives be done without the existence of a hydrologically comparable and stable station, which is never granted.

This correlation approach could also be used for sites where independent discharge measurement from a structure (e.g. dam gates / turbines) is available. Usually it is much less precise than the RC but insensitive to bed evolution, hence "stable" (e.g. the Isère River at Beaumont-Monteux and the stations of the Vieux-Rhône by-passed channels, in France).

Analysis of residuals of the stage-surface velocity relation

Another tool possibly useful for detecting potential shifts in real time during oods is the analysis of the residuals between the stage and the surface velocity measured by non contact techniques such as radar or image velocimetry. This approach has been studied by Thollet et al.

[2017] and [START_REF] Jacob | Débits et incertitudes issus d'enregistrements de niveau et de vitesse par radar sur des cours d'eau de montagne à lit instable. pport de stge endré pr pien holletD térôme ve goz et fenoît gmenenD sxei vyonEilleurnne[END_REF] with encouraging results: Figure 6.4 shows the stage -velocity relation at a station in France evidencing the existence of at least two distinct periods.

More and more hydrometric/hydrological services start to equip their stations with this type of instrumentation. Thus the analysis of the stage-surface velocity relation may become a valuable alternative to the sediment transport proxy analysis since it is based on measurements and available in real time during oods. The framework proposed in Chapter 5 for the real-time management of rating shifts is conceived to be complemented by any kind of information available in real time: e.g., the notice of dams built by beavers or swimmers; data logger bugs or instrumental failure reports, bathymetric surveys; information about sediment mining operations in the river bed or about works.

Before starting the real-time application shift detection tools must be selected. This choice may depend on the available data (e.g., gaugings, stage record, bathymetry) but also on the ow conditions (ood, drought), the tool-specic limitations and the shift causes. Table 6.1

summarises the conditions of application of the main shift detection tools from the literature and for those proposed in this manuscript.

The delay for precisely identifying and estimating a rating shift depends on the type of process causing the shifts: e.g., during oods the rating change induced by morphological change or due to hysteresis may occur in a period of a few hours or a few days; during a period of growth of aquatic vegetation the rating change may occur for several months in a progressive way.

Implementation into operational applications

The transfer of the methods proposed in this manuscript to their operational use may require additional improvements and adjustments. First of all a pre-treatment or visual checking of the input data is necessary in order to detect anomalies (e.g., errors due to wrong selection of rating shift date, sensor fault) and this must be done at dierent time scales (e.g., month, year) [START_REF] Puechberty | Charte qualité de l'hydrométrie: Guide de bonnes pratiques. qroupe hoppler rydrométrieF winistère de l9invironnementD de l9inergie et de l werD prne[END_REF].

Regarding the computational costs, the tools proposed in this manuscript may take some time (from days to a week) when applied in retrospective to a long series and when several tests need to be carried out (e.g., to nd the most adequate stage-recession model). Instead in real time the required computations for detecting a shift, updating the RC and providing the streamow estimate are performed within 5 minutes for each time iteration, thus within the typical time step of the stage records (e.g., 15 minutes). This makes the method applicable operationally. The graphical interface can certainly be improved and adjusted according to the specic purposes of the services. Moreover, while the method is based on the automatic detection and estimation of the shifts, their validation and the beginning of a new stable period may remain partly manual and rely on the hydrologist expertise.

Another possible use of the proposed methods is their integration into existing operational real-time ood forecasting systems to provide streamow data with more realistically quantied uncertainty to be assimilated in real time [START_REF] Ocio | The role of rating curve uncertainty in real-time ood forecasting[END_REF][START_REF] Barbetta | Real-time ood forecasting downstream river conuences using a Bayesian approach[END_REF]. The proposed methods may also help improving and optimising the future strategies for gauging campaigns.

Finally, future perspectives can clearly extend the objectives of this manuscript to study the processes inducing transient rating changes, such as: growth and decline of the aquatic vegetation, hysteresis, sea tidal eects, scour and ll of the river bed. Some of these processes are an ongoing study in the literature. A dynamic modelling may be more adequate than the shift-oriented approach taken in this manuscript, but this requires a better understanding of the physics behind the processes and of what precisely makes the RC changing over time.

  Leblois and Michel Lang who gave me the opportunity to teach at the National School of State Public Works -ENTPE in Vaulx-en-Valin for the courses of Hydrology/Hydraulics, and Christine Poulard for her patience and technical support during the teaching experience. Among the other colleagues at INRAE I want to thank Benoit, Laura, Emeline, Jérôme for sharing with me the amazing trip to New Zealand in the end of 2019 (just a few months before Covid chaos)

  photo: http://media.bom.gov.au/social/blog/2156/explainer-what-is-a-ood/. Credit: Australian Defence Force. . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1. 12

 12 Variability of annual minimum stages (H) and estimated linear trends at the Brzegi cross-section on the Nida River [apuszek and Lenar-Matyas, 2015]. . . . . typical segmentation pitfalls: (a) premature termination in Binary Segmentation: no single change point model provides a reasonable t to the data and BS therefore stops at the rst iteration, with no change detected; (b) mislocated split in Binary Segmentation: optimal single change point is in the middle of a stable period; (c) neglecting versus accounting for data uncertainties; (d) artifact induced by residual rescaling: while raw residuals with standard deviation close to zero lead to rescaled residuals with high absolute values, highly uncertain residuals are sent to zero by the rescaling, thus creating a spurious period. 29 2.2 a) Conceptual owchart of the proposed algorithm. b) Schematic representation of the iterative procedure. Each iteration consists in the succession of Steps 1-4 described in Section 2.2. Colored ribbons and error bars represent 95% uncertainty intervals for RCs (pink), shift times (blue), gaugings (black dots) and residuals (empty dots). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3 Example of shift time adjustment options. Instead of setting the shift time to the MAP estimate τ 1 , a better option may be to choose the time of the maximum stage t f lood,1 within CI 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.4 Strategy D applied to the gaugings of the Ardèche River at Meyras. Subgure (a): structure of the recursion. Subgures (b): baseline RC using the gaugings of the current period. Subgures (c): evolution of four criteria for the selection the optimal segmentation applied to the residuals. Subgures (d): results of the segmentation considering the lowest DIC. . . . . . . . . . . . . . . . . . . . . . . 42 2.5 Results of the segmentation procedure applied to gaugings of the Ardèche River at Meyras, France. Gaugings are plotted against the stage record with a dierent color for each period of RC stability identied by Strategy D. For each strategy, results are presented as posterior pdf of τ j (blue ribbons) and adjusted shift times s j (red segments). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

  segmentation of residuals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Results of the second experiment: impact of the four properties P1 to P4 on the performance metrics (Strategy D). . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Schematic illustration of the recessions extraction method. . . . . . . . . . . . . . 3.2 Results of the proposed recession extraction method applied to the Ardèche River at Meyras: the time series (on the left) and the overlapped recessions (on the right) extracted from the stage record with dierent values of the threshold parameter χ: a) χ = 10 cm; b) χ = 30 cm; c) χ = 50 cm. A specic color is assigned to each extracted recession according to a rainbow gradient, going from the oldest recessions (in red) to the most recent ones (in violet). . . . . . . . . . . . . . . . .

  1: a) Estimation of the recession curves. b) Results of the segmentation applied to the time series of recessionspecic parameter β. The horizontal red lines and the vertical blue lines are the most probable values of the segments mean and of the shift times, respectively. c) Gaugings, detected shifts and pdfs of the shift times plotted against the stage record. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Values of DIC computed by using nine dierent recession models and three different values of extraction parameter χ. . . . . . . . . . . . . . . . . . . . . . . . segmentation in terms of shift times (only the most probable time values -maximum a posteriori -are plotted) yielded by using nine dierent recession models and three dierent values of extraction parameter χ. Also the shift times obtained from the gaugings [Darienzo et al., 2021] (Chapter 2) are plotted for comparison. Notice that the models have been ordered according to increasing values of DIC obtained with χ = 50 cm. . . . . . . . . . . . . . . . . . . . . . . . 82 3.6 Results of the segmentation in terms of estimated river bed elevation β (MAP)

  are plotted for comparison. Error bars represent the 95% uncertainty interval of the shift estimates. . . . . . . . . . . . . . . . . . 83 3.7 Results of the real time application by using model M2 and χ = 50 cm and for three dierent real-time iterations: t = 5.9 -17.8 -81.3 days after the ood peak. The current recession estimation is illustrated in blue in the top panels and the asymptotic stage estimation β in the bottom panels. Also the past recessions are plotted in gray for comparison. The gray horizontal line and the gray ribbon represent the mean and the uncertainty interval (at 95 %), respectively, of the β estimates of all past recessions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.1 Linear interpolation of the river bed elevation b during a morphogenic ood event. 96 4.2 Stage record of the Ardèche River at Meyras (France) for the period 07/11/2001 -29/10/2018 illustrating the combined results of the detection of eective rating shifts through the segmentation of gaugings and the recession analysis on the stage record. Moreover the RC parameters b 1 and b 2 are estimated through the BaRatin-Stage-Period-Discharge analysis [Mansanarez et al., 2019] and are illustrated (posterior mean value and 95% uncertainty) with blue and red segments and ribbons, respectively, for each reference stable period (whose index is indicated below the segments). . . . . . . . . . . . . . . . . . . . . . . . . . . . the RC parameters b 1 and b 2 estimated for each reference stable periods of the Ardèche River at Meyras (France) for the period 07/11/2001 -29/10/2018 through the BaRatin-Stage-Period-Discharge analysis [Mansanarez et al., 2019]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Results of the detection of all potential rating shifts t pot caused by sediment dynamics for the Ardèche River at Meyras station. Three dierent values of φ = d/S 0 are used and compared against the reference shift times t ref (shown on stage record by red open dots). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  the total uncertainty at 95 %. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.7Aerial view of the Wairau River at Barnetts Bank in New zealand (taken from[START_REF] Mansanarez | Shift Happens! Adjusting Stage-Discharge Rating Curves to Morphological Changes at Known Times[END_REF]): a braided gravel bed river subject to very frequent oods causing frequent sudden modication of elevation and width of the lowest control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 General workow of the proposed real-time application. The area in green denotes the rating curve; the area in red denotes the shift detection tools; the area in blue denotes the updating of the provisional RC. . . . . . . . . . . . . . . . . . . . . . 5.2 Selection of the period for the real-time analysis (in red) as well as for the retrospective analysis (in blue) applied to the Ardèche River at Meyras. The real-time initialisation of the application is indicated by the time t 0 . Points on the stage record indicate the gaugings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Results of the retrospective analysis applied to the Ardèche River at Meyras in terms of the detected shift times (and their pdfs). Three tools have been used: the segmentation of gaugings, the stage-recession analysis for detecting eective rating shifts and the sediment transport proxy analysis for detecting potential rating shifts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . of the RC for each of the four past stable periods using BaRatin-SPD [Mansanarez et al., 2019]. Colors correspond to Figure 5.3. b) Boxplots of RC parameters b 1 and b 2 for each period. . . . . . . . . . . . . . . . . . . . . . . 124 5.5 Results of the retrospective sediment transport analysis applied to the Ardèche River at Meyras: relation between the cumulative sediments volume V and the rating shifts ∆b 1 and ∆b 2 . The blue ribbon illustrates the total 95 % uncertainty interval for the shift. The number above the points indicate the indexes of the reference morphogenic events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.6 Results of the proposed real-time application to the Ardèche River at Meyras from the initialization time 3600. A) discharge and stage time series B) RC with uncertainty and gaugings, C) toolbox with three tools for rating shift detection, D) schematic representation of the river cross-section geometry. Iteration 0. . . . 127 5.7 Same as Figure 5.6 for iteration 15. . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.8 Same as Figure 5.6 for iteration 16. . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.9 Same as Figure 5.6 for iteration 82. . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.10 Same as Figure 5.6 for iteration 191. . . . . . . . . . . . . . . . . . . . . . . . . . 135 5.11 Same as Figure 5.6 for iteration 287. . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.12 Same as Figure 5.6 for iteration 311. . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.13 Stage errors examples. Adapted from Puechberty et al. [2017]. . . . . . . . . . . . 141 5.14 Example of two approaches for combining two distinct individual priors on the same rating shift ∆b. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 6.1 Segmentation procedure proposed in Chapter 2 applied to the relation Turbidity vs Total Suspended Sediment concentration (MES in french) for the Arc River at Pontamafrey in France for the period 2011-2019. . . . . . . . . . . . . . . . . . . 149 6.2 The proposed segmentation procedure applied to the series of maximum annual discharge for the Rhine River at Maxau (Karlsruhe, Germany) during the period 1947-2017 [Lang, 2020]. Red and pink ribbons show the parametric and total uncertainty, respectively, of the segments mean. The shift time is represented by a dashed line and by its pdf in the graph below. . . . . . . . . . . . . . . . . . . applied to the gaugings of the Mat River at Escalier in Reunion Island (a) and the Wairau River at Barnetts Bank in New Zealand (b). Dashed vertical lines indicate the most probable values of the oset of the lowest control, b 1 , for each stable period. Solid vertical lines in panel a) indicate the estimates of b 2 . Ribbons around each rating curve represent the 95% uncertainty intervals obtained through the Bayesian approach of Mansanarez et al. [2019]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 6.4 Stage-velocity radar-based measurements for the Arvan River at Saint-Jean de Maurienne in France, for the period March-June 2013 (modied from Jacob [2014]).155 INTRODUCTION 1.1 Context and challenges related to discharge quantication 1.1.1 Monitoring streamow: the rating curve River discharge, or streamow, is the volume of water passing through a river cross-section per unit of time (in m 3 .s -1 ).

  Figure 1.1: The Ardèche River at Meyras station (France), view downstream of Barutel bridge (taken from Mansanarez et al. [2019]): a very common hydraulic conguration.

Figure 1

 1 Figure 1.2: RC estimation with quantitative uncertainty for the Ardèche River at Meyras using the BaRatin approach [Le Coz et al., 2014] for the period 07/11/2001 -08/11/2006.

Figure 1

 1 Figure 1.3: Schematic representation of the dierent types of rating changes, of both transient and sudden

Figure 1

 1 Figure 1.5: Stage record at the Ardèche River at Meyras (France) for the period 07/11/2001 16:19 -29/10/2018 17:45 (measured by piezometer probe and a bubbler system); 151 gaugings provided by the hydrometric service UHPC Grand Delta for the period 07/11/2001 16:30 -25/09/2018 10:55 are also plotted (dots). Stars denote morphogenic oods that are suspected to have caused a rating shift.

Figure 1

 1 Figure 1.6: Flood of the Ross River in Townsville, Australia, on February 2019. Source of the photo: http: // media.bom.gov.au/ social/ blog/ 2156/ explainer-what-is-a-ood/ . Credit: Australian Defence Force.

Figure 1 .

 1 Figure 1.7b) respects the minimum environmental ow (MEF) xed by the law, to ensure

Figure 1

 1 Figure 1.7: a) Run-of-River hydropower plant along the Rhin River, at Kembs in France, operated by EDF. Source image: https:// congress.hydropower.org/ 2019-paris/ programme/ study-tours/ study-tour-kembs-france/ . b) Schematics of a generic Run-of-River hydropower plant with upstream main ow Q 1 , derived ow Q 2 , instream reserved ow Q 3 , downstream ow Q 4 .

  Figure 1.8: a) NPP of Tricastin, France (EDF). Source photo: https:// en.wikipedia.org/ wiki/ Tricastin_ Nuclear_Power_Plant. Credit: Marianne Casamance. b) Flood of Missouri River at the Calhoun NPP in Nebraska, 2011. Source photo: https:// en.wikipedia.org/ wiki/ Fort_Calhoun_Nuclear_Generating_ Station. Credit: U.S. Army Corps of Engineer.

  Figure 1.9: May 2020, a tank containing diesel oil at the Nadezhda plant on the Taymyr peninsula in the Russian Arctic, accidentally released 20 thousand tonnes of oil contaminating more than 20 km of Ambarnaya River. Source photo: https:// www.greenpeace.org/ international/ story/ 43553/ oil-spill-russian-arctic/ . Credit: Greenpeace.

  Figure 1.10: a) Water uptake from the river for irrigation purposes. Source photo: https: // wmanorthamerica.weebly.com/ colorado-river.html. b) Mekong River in Thailand suering from a terrible drought in the end of 2019. Source photo: https:// vietnamtimes.org.vn/ mekong-river-facing-severe-drought-amidst-serious-ooding-in-china-21905.html.

Figure 1 .

 1 Figure 1.11: Top panels: segmentation procedure applied to the time series of the stage residuals computed between the gaugings and a baseline RC [Morlot et al., 2014]. Bottom panels: associated stage-discharge gaugings.

Figure 1 .

 1 Figure 1.12: Variability of annual minimum stages (H) and estimated linear trends at the Brzegi crosssection on the Nida River [apuszek and Lenar-Matyas, 2015].

Figure 2

 2 Figure 2.1: Schematic illustration of typical segmentation pitfalls: (a) premature termination in Binary Segmentation: no single change point model provides a reasonable t to the data and BS therefore stops at the rst iteration, with no change detected; (b) mislocated split in Binary Segmentation: optimal single change point is in the middle of a stable period; (c) neglecting versus accounting for data uncertainties; (d) artifact induced by residual rescaling: while raw residuals with standard deviation close to zero lead to rescaled residuals with high absolute values, highly uncertain residuals are sent to zero by the rescaling, thus creating a spurious period.

Figure 2

 2 Figure 2.2: a) Conceptual owchart of the proposed algorithm. b) Schematic representation of the iterative procedure. Each iteration consists in the succession of Steps 1-4 described in Section 2.2. Colored ribbons and error bars represent 95% uncertainty intervals for RCs (pink), shift times (blue), gaugings (black dots) and residuals (empty dots).

Figure 2 .

 2 Figure 2.4 shows the results of some signicant steps of Strategy D. The structure and the enumeration of the iterations are schematised in Subgure 2.4a. Seven shift times are detected in thirteen iterations. Panels (2.4b-2.4d) zoom to the intermediate results of iterations 0, 1.2, 1.3.

Figure 2

 2 Figure 2.4: Strategy D applied to the gaugings of the Ardèche River at Meyras. Subgure (a): structure of the recursion. Subgures (b): baseline RC using the gaugings of the current period. Subgures (c): evolution of four criteria for the selection the optimal segmentation applied to the residuals. Subgures (d): results of the segmentation considering the lowest DIC.

  Figure 2.5 in terms of detected shift times against the stage record. The ocial dates of RC updates are provided by the hydrometric Service SPC Grand Delta and are conrmed by the analysis proposed by Mansanarez et al. [2019]. Only Strategy D shows results similar to the ocial segmentation. However the ocial dates cannot be considered as a truth" against which the performance of competing methods can be judged. Comparing the results of Strategies A-B-C-D may still be insightful. The rst striking observation is that the four Strategies lead to markedly dierent segmentations, in particular between Strategy D and Strategies A, B, C. While both Strategies C and D are recursive, they lead to very dierent numbers of shifts (42 detected in 56 iterations vs. 7 detected in 13 iterations). This indicates that the treatment of segmentation errors (type-1 vs. type 2) is of prime importance. Strategies A and B are both non-recursive, and dier in the following two aspects: Strategy A is index-based while B is time-based; they use a dierent criterion for selecting the number of shifts. Both strategies lead to very similar results in terms of number of shifts (27 vs. 29) and their location. The slight dierences may be due to the stronger penalty term of the BIC or to Binary Segmentation issues as illustrated in Figure 2.1a-b. Strategies A, B and C lead to many more change points than Strategy D. During the rst

Figure 2 . 5 :

 25 Figure 2.5: Results of the segmentation procedure applied to gaugings of the Ardèche River at Meyras, France. Gaugings are plotted against the stage record with a dierent color for each period of RC stability identied by Strategy D. For each strategy, results are presented as posterior pdf of τ j (blue ribbons) and adjusted shift times s j (red segments).

Figure 2 . 6 :

 26 Figure 2.6: Examples of synthetic data sets characterised by known rating shift times using parameters of Classes 1-3-4-6-9-10 dened inTable 2.1.

Figure 2 . 7 :

 27 Figure 2.7: Schematic example of the gaugings classication into true positive (T P ), false negative (F N ), true negative (T N ), false positive (F P ) for the performance evaluation of the segmentation results. s i represent the known shift times, while τi are the change point estimates.

  of criteria for choosing the number of change points

Figure 2 .Figure 2

 22 Figure 2.8 summarizes the results of the rst experiment. Results reveal that:

Figure 2 .Figure 2 Figure 2

 222 Figure 2.9 summarizes the results of the second experiment. Strategies A, B, C, D are compared considering all simulations.

  in the time series of the recession-specic parameters can indicate morphological changes.

Figure 3 .

 3 Figure3.2 illustrates the results of the rst step of the proposed recession analysis by using, for comparison, three dierent reasonable values of the threshold parameter χ: 10, 30, 50 cm. All other options have been xed to: n min = 10 (to have enough points to estimate recession-specic parameters), t min = 10 days (consistently with the literature). We also sub-sampled the stage time series to one value per day to reduce computational cost.

Figure 3 .Figure 3

 33 Figure 3.2 evidences the shifting of the low-stage recession levels, probably due to river bed erosion, conrming the interest of the proposed analysis to this station.

Figure 3 .

 3 Figure 3.3b illustrates the results of the segmentation applied to the time series of the asymptotic stage parameter β (corresponding to the mean elevation of the low-ow control) for models M1, M2, M4. The minimum segment length has been xed to 1 and the maximum number of change points has been xed to 7. The results of the segmentation of β time series are also plotted against the stage record and the gaugings in Figure 3.3c.

Figure 3 . 3 :

 33 Figure 3.3: Results of the proposed method applied to the Ardèche River at Meyras by using = cm and models M1, M2, M4 of Table 3.1: a) Estimation of the recession curves. b) Results of the segmentation applied to the time series of recession-specic parameter β. The horizontal red lines and the vertical blue lines are the most probable values of the segments mean and of the shift times, respectively. c) Gaugings, detected shifts and pdfs of the shift times plotted against the stage record.

Figure 3 .

 3 Figure 3.4 illustrates the results for the Meyras case study by considering the dierent proposed models and the three dierent proposed values of parameter χ. Results reveal that while the ranking of the exponential models M1-M5 is sensitive to parameter χ (dierent χ values lead to dierent model rankings), models M6-M9 lead to similar DIC for all values of χ. Models that yielded the lowest DIC are M5, M9, independently on the values of χ, and M3 with χ = 30 -50 cm.

Figure 3

 3 Figure 3.4: Values of DIC computed by using nine dierent recession models and three dierent values of extraction parameter χ.

Figure 3 Figure 3 . 6 :

 336 Figure 3.5: Results of the segmentation in terms of shift times (only the most probable time valuesmaximum a posteriori -are plotted) yielded by using nine dierent recession models and three dierent values of extraction parameter χ.Also the shift times obtained from the gaugings[START_REF] Darienzo | Detection of Stage-Discharge Rating Shifts Using Gaugings: A Recursive Segmentation Procedure Accounting for Observational and Model Uncertainties[END_REF] (Chapter 2) are plotted for comparison. Notice that the models have been ordered according to increasing values of DIC obtained with χ = 50 cm.

Figure 3 .

 3 Figure 3.7 shows an example of real-time re-analysis for the Meyras case study by using model M2 and χ = 50 cm. The considered real-time window comprises the recession period that follows the morphogenic ood at day 2550 (see the stage record of Figure 3.3) for which a net shift was observed. Model M2 detects a shift after only nearly 6 days and the shift is conrmed at all subsequent time steps.

  in previous chapters, one frequent cause of rating changes at the hydrometric stations is river bed morphological changes. Erosion and deposition are mainly due to the sediment transport induced by intense oods. During these events the RC may become obsolete and the estimation of streamow inaccurate. For this reason, station managers and ood forecasting services may want to quickly detect potential morphological changes during and after the oods.However, gaugings and bathymetry surveys are rarely performed during oods because of the diculty and danger of accessing the site. Thus gauging-based methods such as the one proposed in Chapter 2 are not suitable for the fast detection of rating changes. Also the recession analysis proposed in Chapter 3 is not adequate during oods since it applies to recession periods only. Sometimes non-contact techniques, such as radar or imagery techniques, are deployed during oods for fast detection but they cannot be generalized, thus we cannot only rely on them. The stage record h(t) is in fact the only information always available, including during oods, to detect and estimate morphological changes (as long as the recording instrument is not damaged by the ood).

  ref where the oset b represents the mean elevation of the channel bed, the control becomes a h -b (k+1) c after the ood, with the shift of parameter b:

Figure 4

 4 Figure 4.1: Linear interpolation of the river bed elevation b during a morphogenic ood event.

Figure 4

 4 Figure 4.2: Stage record of the Ardèche River at Meyras (France) for the period 07/11/2001 -29/10/2018 illustrating the combined results of the detection of eective rating shifts through the segmentation of gaugings and the recession analysis on the stage record. Moreover the RC parameters b 1 and b 2 are estimated through the BaRatin-Stage-Period-Discharge analysis[START_REF] Mansanarez | Shift Happens! Adjusting Stage-Discharge Rating Curves to Morphological Changes at Known Times[END_REF] and are illustrated (posterior mean value and 95% uncertainty) with blue and red segments and ribbons, respectively, for each reference stable period (whose index is indicated below the segments).

Figure 4 . 3 :

 43 Figure 4.3: Posterior boxplots of the RC parameters b 1 and b 2 estimated for each reference stable periods of the Ardèche River at Meyras (France) for the period 07/11/2001 -29/10/2018 through the BaRatin-

Figure 4 . 4 :

 44 Figure 4.4: Results of the detection of all potential rating shifts t pot caused by sediment dynamics for the Ardèche River at Meyras station. Three dierent values of φ = d/S 0 are used and compared against the reference shift times t ref (shown on stage record by red open dots).

t

  ref are then estimated through Equation 4.1 and used as calibration data for the relations ∆b 1 ↔ V and ∆b 2 ↔ V illustrated in Figure 4.6.

Figure 4 . 6 :

 46 Figure 4.6: Relation ∆b 1 ↔ V (a) and ∆b 2 ↔ V (b) for the Ardèche River at Meyras, France. Calibration data are the estimated ∆b and V for each reference morphogenic event. The number above the calibration data represent the index of the reference event. The blue line represents the maximum a posteriori and the blue ribbon illustrates the total uncertainty at 95 %.

Figure 4

 4 Figure 4.7: Aerial view of the Wairau River at Barnetts Bank in New zealand (taken from Mansanarez et al. [2019]): a braided gravel bed river subject to very frequent oods causing frequent sudden modication of elevation and width of the lowest control.

  The analysis then proceeds by establishing a relation between the cumulative volume of transported sediments during the ood and the associated potential rating shift. This relation provides quantitative information on the uncertainty of the shift, which can be used in a real time context. The application on the Meyras case study shows that the method provides a realistic relation between transported volume and potential change.

Figure 5 . 1 :

 51 Figure 5.1: General workow of the proposed real-time application. The area in green denotes the rating curve; the area in red denotes the shift detection tools; the area in blue denotes the updating of the provisional RC.

  . The sediment transport proxy model proposed in Chapter 4 is used to provide an estimate on the potential shift uncertainty. Consequently the RC uncertainty increases as the cumulative volume of transported sediments increases.

Figure 5

 5 Figure 5.2: Selection of the period for the real-time analysis (in red) as well as for the retrospective analysis (in blue) applied to the Ardèche River at Meyras. The real-time initialisation of the application is indicated by the time t 0 . Points on the stage record indicate the gaugings.

Figure 5 .Figure 5 . 3 :

 553 Figure 5.4b also shows the boxplots of the estimation of RC parameters b 1 (mean elevation of the natural rie) and b 2 (mean elevation of the main channel bed), assumed to be the only two unstable parameters for this case study.

  Figure 5.4: a) Estimation of the RC for each of the four past stable periods using BaRatin-SPD [Mansanarez et al., 2019]. Colors correspond to Figure 5.3. b) Boxplots of RC parameters b 1 and b 2 for each period.

Figure 5 . 5 :

 55 Figure 5.5: Results of the retrospective sediment transport analysis applied to the Ardèche River at Meyras: relation between the cumulative sediments volume V and the rating shifts ∆b 1 and ∆b 2 . The blue ribbon illustrates the total 95 % uncertainty interval for the shift. The number above the points indicate the indexes of the reference morphogenic events.

Figure 5 .

 5 Figure 5.6B shows the current base RC with total uncertainty valid at time t 0 . The intersection of the two dashed black segments indicates the current position along the RC.

Figure 5 .

 5 Figure 5.6D shows a very schematic representation of the river cross-section in order to visualize the current hydraulic conditions.

  Scheme of the river cross-section

Figure 5 .

 5 Scheme of the river cross-section

  Scheme of the river cross-section

Figure 5

 5 Scheme of the river cross-section

Figure 5

 5 Scheme of the river cross-section

Figure 5 .

 5 Figure 5.13: Stage errors examples. Adapted from Puechberty et al. [2017].

  Figure 5.14: Example of two approaches for combining two distinct individual priors on the same rating shift ∆b.

Figure 5 .

 5 Figure 5.14 proposes an example of two possible approaches for combining the prior on the rating shift obtained from the stage-recession analysis ∆b rec ∼ N (µ ∆brec , σ ∆brec ) with the one obtained by the sediment transport proxy analysis ∆b ST ∼ N (µ ∆b ST , σ ∆b ST ) into an unique prior by:

Figure 6

 6 Figure 6.1: Segmentation procedure proposed in Chapter 2 applied to the relation Turbidity vs Total Suspended Sediment concentration (MES in french) for the Arc River at Pontamafrey in France for the period 2011-2019.

Figure 6 . 2 :

 62 Figure 6.2: The proposed segmentation procedure applied to the series of maximum annual discharge for the Rhine River at Maxau (Karlsruhe, Germany) during the period 1947-2017 [Lang, 2020]. Red and pink ribbons show the parametric and total uncertainty, respectively, of the segments mean. The shift time is represented by a dashed line and by its pdf in the graph below.

  Figure 6.3: Proposed segmentation procedure applied to the gaugings of the Mat River at Escalier in Reunion Island (a) and the Wairau River at Barnetts Bank in New Zealand (b). Dashed vertical lines indicate the most probable values of the oset of the lowest control, b 1 , for each stable period. Solid vertical lines in panel a) indicate the estimates of b 2 . Ribbons around each rating curve represent the 95% uncertainty intervals obtained through the Bayesian approach of Mansanarez et al. [2019].

Figure 6

 6 Figure 6.4: Stage-velocity radar-based measurements for the Arvan River at Saint-Jean de Maurienne in France, for the period March-June 2013 (modied from Jacob [2014]).

  Figure 2.3: Example of shift time adjustment options. Instead of setting the shift time to the MAP estimate τ 1 , a better option may be to choose the time of the maximum stage t f lood,1 within CI 1 .

				2.2. The proposed method for rating shift detection
		4
	Stage h [m]	0 1 2 3
		-1	2000	2250	2500
		1.00
	Scaled pdf	0.25 0.50 0.75
		0.00	2000	2250	2500
				Time [days]
	Legend
		Stage record	Gaugings	τ 1 ^tflood,1	Posterior pdf of change point time τ 1
	Matteo Darienzo		38/174

  .3 Application to a real case study: the Ardèche River at Presentation of the stationThe Ardèche River at Meyras station is located in Mediterranean France, with a catchment area of 98 km 2 . This station is characterized by a gravel bed degrading during each important ood. It has been already studied by Le Coz et al.[2014],[START_REF] Sikorska | Calibrating a hydrological model in stage space to account for rating curve uncertainties: general framework and key challenges[END_REF] and[START_REF] Mansanarez | Shift Happens! Adjusting Stage-Discharge Rating Curves to Morphological Changes at Known Times[END_REF]. They all proposed a three-control hydraulic conguration: a section control governed by a natural rie for low ows, a main channel control for medium ows and one oodway channel control added to the main channel for very high ows. The stage record is available for the period between 07/11/2001 and 29/10/2018. Gaugings and comments about shift times have been provided by the hydrometric Service in charge of the station (UHPC Grand

	Meyras, France
	2.3.1 Delta).

  Note that in this method, the segmentation model of Equation2.4 is expressed in terms of observation index i rather than time t i , and does not provide uncertainty on the change point. Consequently, each change point τ k is assumed to be uniformly distributed between times t i k (the time associated with the k -th detected change point) and t i (ii) DIC selection of K; (iii) minimum number of data in a segment N min = 1; (iv) minimum duration of a segment d min = 0 days. This approach is very similar to the previous Strategy A, except that the segmentation model of Equation2.4 is expressed in terms of time t i . Shift time adjustment is therefore applied by looking for the largest ood in the 95% credibility interval of the change point, as described in Section 2.2.6.C Recursive mCPD method proposed in this paper with a type-1 treatment of segmentation errors (i.e., unknown but constant uncertainty). The maximum number of segments is now set to 5 because by using a recursive procedure, there is no requirement to nd all changes during the rst pass. All other options are identical to approach B.

is used with the following options: (i) maximum number of segments K = 30; (ii) change in the mean only; (iii) Binary Segmentation method Scott and Knott [1974]; (iv) Normal statistic test; (v) BIC selection of K; (vi) minimum number of data in a segment N min = 1. k -1 . The shift times are then adjusted (Section 2.2.6) on the largest stage value within this interval.

B Single-pass" mCPD method proposed in this paper with a type-1 treatment of segmentation errors (i.e., unknown but constant uncertainty).

The following options are chosen: (i) maximum number of segments K = 30;

D Recursive mCPD method proposed in this paper with a type-2 treatment of segmentation errors (i.e., known residuals uncertainties). All options are identical to approach C.

2.3.3 Results with Strategy D

Table 2 .

 2 1: Classes of simulation for the performance evaluation.

		Frequency	Frequency	Mean number of	Shift	Gauging error	Number of
	Class
		of gaugings	of shifts	gaugings/period	st.dev.	Low ows	High ows	controls
		λ g (year -1 ) λ s (year -1 ) λ g /λ s	σ b (m) ρ LF (%)	ρ HF (%)	N c
	1	2	1/5	10	0.5	2.5	5	1
	2	4	1/5	20	0.5	2.5	5	1
	3	7	1/5	35	0.5	2.5	5	1
	4	10	1/5	50	0.1	2.5	5	1
	5	10	1/5	50	0.3	2.5	5	1
	6	10	1	10	0.5	2.5	5	1
	7	10	1/2	20	0.5	2.5	5	1
	8	10	1/5	50	0.5	2.5	5	1
	9	10	1/5	50	0.5	10	15	1
	10	10	1/5	50	0.5	2.5	5	3

  Table 2.1.period of 15 years. As an order of magnitude, the CPU time to apply approach C to one data set with 54 gaugings is around 50 minutes. This is to be multiplied by the number of data sets (100) times the number of approaches (4) or the number of criteria (4), which amounts to several days of eective running time.The rst experiment aims at comparing criteria AIC, BIC, DIC and HQC (see Section 2.2.5). To this aim, Strategy D is applied to all classes above.The second experiment aims at comparing Strategies A, B, C and D. To this aim, all four Strategies are applied to the same 100 data sets of the rst experiment. The stage record is not available for synthetic data sets, thus in Strategies B-C-D, the shift times are taken as the estimated parameters τ j (i.e., option 1 in Section 2.2.6). Since Strategy A provides the estimated change point as an observation index k (rather than a time), the shift time is taken as the middle of the interval [t k-1 ; t k ].

Table 3 .

 3 1: Stage-recession models h(t, k) = f (t, k|θ R ) used in the paper, where t is the recession time, k is the recession index and θ R is the vector of model parameters.

	Stable	Recession-specic
	Stage-recession model
	parameters	parameters
	Superposed-
	exponential

  We selected the widely used model for bed load proposed

		4.2. The proposed sediment transport proxy analysis
	by Meyer-Peter and Müller [1948]:
	Matteo Darienzo	96/174

Matteo Darienzo

1.2. State-of-the art for the detection and estimation of rating changes

Matteo Darienzo100/174

Matteo Darienzo158/174

ACKNOWLEDGEMENTS

So many people helped me during these years of PhD and it is impossible for me to thank all of them. I apologize in advance to those I unfairly forget.

First of all, I am deeply grateful to the perfect trio of supervisors: Michel Lang, for his

LIST OF TABLES

2.1

Classes of simulation for the performance evaluation. . . . . . . . . . . . . . . . . 47

3.1

Stage-recession models h(t, k) = f (t, k|θ R ) used in the paper, where t is the recession time, k is the recession index and θ R is the vector of model parameters. 70 The structure of the paper can be summarized as follows. Section 2.2 describes the proposed method. In Section 2.3, this method is applied to a typical hydrometric station with the aims of illustrating the main properties of the method and introducing several possible variants.

Section 2.4 then describes a more thorough evaluation of the method's performance based on synthetic data sets where change points are known. Section 2.5 discusses results and proposes future perspectives. Finally, Section 2.6 summarizes the main ndings.

CHAPTER 3

STAGE-RECESSION ANALYSIS

This chapter is written as an article to be submitted to a scientic journal with the title "Estimation of river bed evolution at hydrometric stations using stage-recessions".

Abstract

Tracking and estimating the evolution of the river geometry is particularly important at hydrometric stations where streamow estimation is based on a stage-discharge relation. The evolution of the river bed (e.g. due to intense oods) may induce rating changes, which may undermine streamow accuracy. We propose an original method for the detection and estimation of net variations in the river bed elevation after oods using the stage record only. The method is based on the fact that when streamow tends towards zero, stage tends towards the mean elevation of the main channel bottom or, if it exists, the mean crest elevation of the low ow section control. The method comprises three main steps. Firstly, the stage-recessions are extracted from the stage record. Then all extracted recessions are estimated together in a unique regression model through a Bayesian pooling approach. Finally a segmentation procedure is applied to detect multiple step changes in specic parameters of this model. The method is applied to the Ardèche River at Meyras in France, a gravel bed river subject to intense oods causing episodic river bed shifts. The method yields encouraging results with the detection and

Recession analysis

The recession analysis is usually performed on streamow. Two main issues are addressed in the literature: the extraction of the individual recessions and the estimation of the corresponding recession curves.

Many methods (both manual and automated) exist in the literature to separate streamow recessions, as reviewed by [START_REF] Chapman | A comparison of algorithms for stream ow recession and baseow separation[END_REF]; [START_REF] Tallaksen | A review of baseow recession analysis[END_REF]; [START_REF] Hall | Base-ow recessionsa review[END_REF]; [START_REF] Sujono | A comparison of techniques for hydrograph recession analysis[END_REF].

In general, a recession period lasts as long as the streamow does not rise. [START_REF] Vogel | Estimation of baseow recession constants[END_REF] proposed to start a recession period when a 3-day moving average begins to decrease and ends when it starts to increase. Other algorithms aim at separating the "storm runo" caused by the ood event from the hydrograph [START_REF] Chapman | A comparison of algorithms for stream ow recession and baseow separation[END_REF], in order to isolate the baseow, i.e.

the result of groundwater discharging into the stream. Some authors proposed to simply remove the initial portion of the recession period, e.g. the initial 30 % [START_REF] Vogel | Estimation of baseow recession constants[END_REF]. In addition to streamow, other approaches use precipitation data to dene periods not inuenced by precipitation [START_REF] Lang | Une méthode d'analyse du tarissement des cours d'eau pour la prévision des débits d'étiage[END_REF][START_REF] Tallaksen | A review of baseow recession analysis[END_REF]. Moreover, the recessions are also selected according to a minimum duration for the recession period, usually chosen between 4 and 10 days [START_REF] Tallaksen | A review of baseow recession analysis[END_REF].

The literature also proposes several methods for estimating the streamow recession curves, as reviewed by [START_REF] Johnson | Outline for Compiling Precipitation, Runo, and Ground Water Data from Small Watersheds. he porest ervieD outhestern porest ixperiment ttionD yld ttion per[END_REF]; [START_REF] Tallaksen | A review of baseow recession analysis[END_REF]; [START_REF] Langbein | Some channel-storage studies and their application to the determination of inltration[END_REF]; [START_REF] Lang | Une méthode d'analyse du tarissement des cours d'eau pour la prévision des débits d'étiage[END_REF]; [START_REF] Dewandel | Evaluation of aquifer thickness by analysing recession hydrographs. Application to the Oman ophiolite hard-rock aquifer[END_REF]. Some methods aim at estimating the individual recession curves separately [START_REF] Barnes | The structure of discharge-recession curves. iosD rnstions emerin qeophysil nion[END_REF]. Other methods aim at overcoming the high variability of the recession behavior through the estimation of a master recession curve [START_REF] Toebes | On recession curves: 1 Recession Equations. tournl of rydrology @xew elndA[END_REF][START_REF] Nathan | Evaluation of automated techniques for base ow and recession analyses[END_REF], obtained from various individual recessions. Moreover, [START_REF] Morlet | Wave propagation and sampling theorypart i: Complex signal and scattering in multilayered media[END_REF] and [START_REF] Sujono | Hydrograph recession analysis using wavelet transforms[END_REF] proposed a method for analysing the recession characteristics based on the wavelet transform.

The recession analysis is conceptually based on storageoutow models in linear and nonlinear forms [START_REF] Brutsaert | Regionalized drought ow hydrographs from a mature glaciated plateau[END_REF]. The single linear reservoir is commonly used in engineering practice, in particular using the simple exponential Maillet's law [START_REF] Tallaksen | A review of baseow recession analysis[END_REF]:

3.2.2

Step 2: Bayesian estimation of the stage-recessions

The second step of the proposed method is based on the estimation of a unique model for all recessions through a Bayesian pooling approach. We refer the reader to a similar approach described in [START_REF] Mansanarez | Shift Happens! Adjusting Stage-Discharge Rating Curves to Morphological Changes at Known Times[END_REF].

The stage-recession model

As mentioned in the introduction, the simplest model for streamow-recession is the single exponential function (Equation 3.1). Combining Equation 3.1 with the RC power law 4.3 Application to the Ardèche River at Meyras, France

Information from the station history

For the Meyras station, as described in the previous chapters, a documented knowledge on the hydraulic conguration, the stage record, the gaugings and the ocial dates of RC update is available for the period between 07/11/2001 and 29/10/2018 [START_REF] Sikorska | Calibrating a hydrological model in stage space to account for rating curve uncertainties: general framework and key challenges[END_REF][START_REF] Mansanarez | Shift Happens! Adjusting Stage-Discharge Rating Curves to Morphological Changes at Known Times[END_REF].

Moreover, some eective rating shifts for the studied period are proposed in the previous chapters of this manuscript by the analysis of the gaugings (Chapter 2) and by the analysis of the stage-recessions (Chapter 3). The combined results are illustrated by vertical dotted lines in Figure 4.2. All detected rating shifts refer to ooding events and can thus be related to sediment transport dynamics, leading to the set of reference morphogenic events with peak times t ref .

By analysing the gaugings for this station, [START_REF] Mansanarez | Shift Happens! Adjusting Stage-Discharge Rating Curves to Morphological Changes at Known Times[END_REF] found that the rating shifts aect in particular the elevation of the low ow rie control (b 1 ) and the mean elevation of the main channel (b 2 ). The rie width and channel width and slope (and consequently, parameters a 1 and a 2 ) as well as the oodplain control are assumed stable.

The BaRatin-SPD method [START_REF] Mansanarez | Shift Happens! Adjusting Stage-Discharge Rating Curves to Morphological Changes at Known Times[END_REF]

Estimation of the triggering stage and detection of all potential shift times

The detection of all potential morphological shifts is based on the selection of the events exceeding the triggering stage h c , which is estimated through Equation 4.4. As explained in Section 4.2.3 the parameter that needs to be "calibrated" is φ = d/S 0 . By observing the

Time [days]

h c (ϕ = 15)

The proposed real-time application

The steps of the proposed general method for the real-time management of unstable RCs are schematized in Figure 5.1 and detailed in the following subsections.

Initialisation: hydraulic analysis

The application starts at time t 0 by performing the hydraulic analysis of the station to get some knowledge on the hydraulic and geometric properties of the river stretch: the average geometry of the cross and longitudinal sections; the roughness and composition of the bed material, the number, the type and the succession of the elementary hydraulic controls dening the RC model.

Retrospective analysis

After the hydraulic analysis and before starting the real-time iterations, a retrospective analysis is performed on the entire past period searching for all past rating shifts (and hence identifying all periods of RC stability), with three main objectives:

Identify the type and the causes of the past rating shifts and the identication of the most unstable RC parameters. This is important for the selection of adequate tools for the rating shift detection and estimation in real time.

Calibrate the tools for the shift detection and estimation to be used in real time.

Estimate the "base RC" at the initial time t 0 of the real time application. To this aim we could use one of the methods proposed in the literature for the estimation of RC with uncertainty [START_REF] Kiang | A comparison of methods for streamow uncertainty estimation[END_REF] applied to the last stable period before t 0 . However the "stage-period-discharge" method proposed by [START_REF] Mansanarez | Shift Happens! Adjusting Stage-Discharge Rating Curves to Morphological Changes at Known Times[END_REF] is preferred since it allows the transfer of information across the periods through the common RC parameters, which is particularly important for periods with very few gaugings.

As will be explained in Section 5.2.8, this retrospective analysis is also performed every time that the beginning of a new stable period is declared.

Shift estimation

If no rating shifts have been detected using the available tools at time t then the priors for the RC parameters are not updated and the base RC is kept invariant. On the contrary, if a rating shift has been detected the validity of the current base RC is ended and a provisional period begins. In this case the real-time application proceeds with the estimation of the shift magnitude, when it is possible.

As regards the stage-recession analysis, if a shift is detected, the time series of the asymptotic stages β is segmented by applying the segmentation model with 2 predened segments: one segment for the past recessions (which belong to the stable period before the shift) and one segment for the ongoing recession, thus with only one point (which belongs to the provisional period during/after the shift), with uncertainties. The dierence between the two segment means is used to estimate the shift of the RC parameter b (oset of the low ow control) at time t obtained from the recession analysis:

where µ β (t) and µ * β indicates the mean of the estimated parameter β for the current provisional period at time t and for the past stable period, respectively, while σ β (t) and σ * β are their standard deviation.

The sediment transport proxy analysis leads to the estimation of potential shift of parameter b at time t, by using the relation between the cumulative sediments volume and the magnitude of the potential shift (as described in Chapter 4):

where the mean of the shift is zero (corresponding to a potential change), unlike in Equation 5.1, and the standard deviation is proportional to the cumulative sediment volume V (t) computed from the beginning of the ood event t in to time t. Finally, it may happen that at time t more than one tool lead to the estimation of the rating shift (e.g. ∆b ST (t) from the sediment transport proxy analysis, ∆b rec (t) from the stage-recession analysis). In this manuscript the combined shift estimate ∆b(t) is equal to the shift estimate having the smallest standard deviation, but alternative choices are discussed in Section 5.4.3.2.

Update of RC priors and RC estimation

The RC estimation is here performed through a Bayesian approach, which requires the gaugings and the prior distributions of the RC parameters and provides their posterior distributions. In the real-time context at time t we could have the four main following situations:

1. No detected shift and no new gaugings: the RC priors are taken equal to the posteriors of the current base RC, thus in fact the estimated RC is equal to the base RC. 3. Shift detected by the gauging segmentation: in this case we enter in a provisional period and the validity of the base RC is ended. All the past gaugings now belong to the past base RC. Only the new gauging is used to estimate the provisional RC at time t. The BaRatin-SPD developed by [START_REF] Mansanarez | Shift Happens! Adjusting Stage-Discharge Rating Curves to Morphological Changes at Known Times[END_REF] can be used to allow the transfer of the information on the parameters in common between the past base RC and the current provisional period. Assumptions on the stable and unstable RC parameters are necessary: e.g. all parameters remain unchanged except for parameter b of the low ow control. Since no estimate on ∆b is available a weakly informative prior is specied for b.

4. Shift detected and estimated (e.g. using the tool based on the stage-recession analysis or on the sediment transport proxy analysis): the prior on the corresponding unstable RC CHAPTER 6

CONCLUSIONS AND PERSPECTIVES

Summary

In this dissertation the issue of unstable rating curves is investigated.

In particular three original tools for the detection and estimation of rating shifts are proposed. They are a continuation of the PhD work of Valentin Mansanarez [START_REF] Mansanarez | xonEunique stgeEdishrge reltionsX fyesin nlysis of omplex rting urves nd their unertinties[END_REF] who developed the Stage-Period-Discharge method leading to the estimation of the multiple RCs at known times.

The rst tool has been developed for the detection and estimation of eective rating shifts using the gaugings. It is based on an original recursive segmentation procedure applied to the time series of the residuals computed between the gaugings and a reference RC. In this segmentation method there are no assumptions on the number of homogeneous segments.

Moreover its originality is to account for the uncertainty of the data (here, the gaugings residuals) which may have large and variable uncertainty. This avoids detecting false shifts, as demonstrated through the application to several synthetic data sets. Finally it also provides useful information about the uncertainty on the shift times, which allows searching for the causal event, e.g. the largest ood. Finally, the proposed segmentation procedure may be applied beyond the eld of hydrometry to a wide range of problems. Some examples are reported in Section 6.2.1.1.

Perspectives

The proposed tools for rating shift detection and estimation could be further improved in the future. Some ideas are discussed in the next subsections.

6.2.1 Improvement of the proposed tools for rating shift detection

Segmentation of gaugings

The tool based on the segmentation of the gaugings residuals could be improved by dening the most suitable criterion used for determining the optimal number of segments, among several proposed in the literature. The simulations using synthetic data in Chapter 2 have shown that the segmentation is sensitive to the chosen criterion. The criteria that yield the highest performance are the DIC and the BIC. A better understanding of the dierences between BIC and DIC is needed in future work.

Finally, as a general perspective, the proposed segmentation procedure could probably be extended to other elds where the relation between two observed variables, estimated using uncertain calibration data, is subject to sudden changes. As an example, Figure 6.1 shows the results of the proposed segmentation method applied to the relation turbidity vs Total Suspended Sediment concentration. More generally, the proposed method could be used for segmenting uncertain time series. It can be applied to test the homogeneity of the series used for the analysis of the distributions of oods and droughts. It has been used in a recent report [START_REF] Lang | Expertise sur l'hydrologie du Rhin sur les biefs de Gambsheim et Iezheim[END_REF] to study the oods on the Rhine River and shows its utility in the case of data with uncertainty strongly varying in time (cf. historical oods, see Figure 6.2). Détection et estimation de détarages dans les modèles hauteur-débit pour la quantification du débit des cours d'eau en rétrospectif et en temps réel Les séries temporelles de débit des rivières sont établies à l'aide de "courbes de tarage", qui sont des modèles avec les hauteurs d'eau en entrée et les débits en sortie. Malheureusement, de nombreuses stations hydrométriques ont une relation hauteur-débit instable, notamment à cause de l'évolution du lit de la rivière lors des crues. Ces "détarages" posent problème à la fois pour l'établissement des séries hydrologiques de long-terme (analyse rétrospective) et pour la fourniture de données en temps réel, par exemple pour la prévision des inondations, avec des incertitudes quantifiées et fiables. Les méthodes existantes pour la mise à jour de la courbe de tarage sont basées sur une analyse statistique des données de calibration (jaugeages) passées ou sur des règles empiriques. Cette thèse a permis de développer des méthodes originales pour la détection automatique des détarages et l'estimation de leur amplitude en rétrospectif et en temps réel : une méthode de segmentation des résidus entre les jaugeages et une courbe de référence, une analyse des récessions du limnigramme et une détection de détarages potentiels à partir d'un indicateur disponible en temps réel (par exemple, transport sédimentaire cumulé). L'approche probabiliste permet d'une part de prendre en compte l'incertitude des informations sur les détarages et d'autre part de quantifier les incertitudes sur les débits calculés. La combinaison des trois méthodes a été appliquée à la station de l'Ardèche à Meyras, en France, qui présente des détarages nets après chaque crue importante. Une bonne détection et estimation des détarages a été observée en rétrospectif et en temps réel. La méthode est générique et, après davantage de validation, applicable en opérationnel à d'autres sites.

Stage-recession analysis

Mots clés : courbes de tarage, analyse Bayésienne, temps réel, détarages, jaugeages, analyse rétrospective Detection and estimation of stage-discharge rating shifts for retrospective and real-time streamflow quantification River discharge time series are established using "rating curves", which are models with stage as input and discharge as output. Unfortunately, many hydrometric stations have an unstable stage-discharge relation, particularly because of the change in the river bed during floods. These "rating shifts" pose a problem both for the establishment of long-term hydrological series (retrospective analysis) and for the delivery of real-time data, for example for flood forecasting, with quantified and reliable uncertainties. The existing methods for updating the rating curve are based on a statistical analysis of past calibration data (the gaugings) or on empirical rules. This thesis aims at developing some original methods for the automatic detection of rating shifts and the estimation of their magnitude in both retrospective and real time: a method of segmentation of the residuals between the gaugings and a base rating curve, an analysis of the stage recessions and a method for detecting potential shifts from an indicator available in real time (e.g. cumulative sediment transport). The probabilistic approach allows on the one hand to take into account the uncertainty of the information on the shift and on the other hand to quantify the uncertainties of the calculated streamflow. The combination of the three methods has been applied to the Ardèche at Meyras, France, which shows net shifts after each major flood. Good detection and estimation of the rating shift has been observed retrospectively and in real time.

The method is generic and, after further validation, operationally applicable to other sites. Key words: rating curves, Bayesian analysis, real time, rating shifts, gaugings, retrospective analysis