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 [Le Coz et al., 2014]. 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 Mansanarez et al.  and Sikorska and Renard  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.
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. [Osorio and Reis, 2016; McMillan et al., 2017].
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.
Flood forecasting and ood risk management
River oods (Figure 1.6) are still nowadays one of the major natural disasters all around the world [UNDRR, 2020]. In real time the decision makers supported by the services in charge of the hydrometric stations use forecasted streamow to provide reliable and timely ood alert.
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).
– 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 [Ocio et al., 2017]. 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.
Sudden accidental pollution of a river may occur for several reasons: release of toxic industrial waste water into the river, pipelines failure, rain and snowmelt run-o from contaminated watershed, etc.
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.
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 Mansanarez et al. . 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. Reitan and Petersen-Øverleir  developed a dynamic model based on time-varying RC parameters within a hierarchical Bayesian framework. Mansanarez  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, h2), and stage-gradient-discharge (SGD) models to address hysteresis due to transient ows and the eect of the ood wave propagation [Mansanarez et al., 2020].
Perret et al.  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.
Table of contents :
1.1 Context and challenges related to discharge quantication
1.1.1 Monitoring streamow: the rating curve
1.1.2 Hydraulic principles behind the rating curve
1.1.3 Rating curve uncertainty
1.1.4 Rating changes
1.1.5 Managing rating changes in real time
1.2 State-of-the art for the detection and estimation of rating changes
1.2.1 Dynamic modelling of transient changes
1.2.2 Detecting and estimating sudden changes
1.2.3 Real-time challenges
1.3 Objectives and outline of the manuscript
2 Segmentation of gaugings
2.1.1 Rating curves
2.1.2 Detecting and modelling transient changes
2.1.3 Detecting sudden changes
2.1.4 Change point detection methods
2.1.5 Objectives of the paper
2.2 The proposed method for rating shift detection
2.2.2 Estimation of the baseline rating curve
2.2.3 Computation of residuals and their uncertainty
2.2.4 Segmentation model and Bayesian inference
2.2.5 Choice of the optimal number of segments
2.2.6 Adjustment of shift times
2.2.7 Recursive segmentation
2.3 Application to a real case study: the Ardèche River at Meyras, France
2.3.1 Presentation of the station
2.3.2 Segmentation strategies
2.3.3 Results with Strategy D
2.3.4 Comparison of Strategies A-D
2.4 Performance evaluation from simulated rating shifts
2.4.1 Generation of synthetic data
2.4.2 Design of experiments
2.4.3 Metrics for performance evaluation
2.4.4 Results of the experiments
2.5.1 Contributions to the operational practice and the scientic literature
2.5.2 Current limitations
2.5.3 Avenues for future work
3 Stage-recession analysis
3.1.1 Stage-discharge rating shifts at hydrometric stations
3.1.2 Methods for estimating river bed evolution
3.1.3 Recession analysis
3.1.4 Objectives and structure of the paper
3.2 The proposed method for river bed estimation using stage recessions
3.2.1 Step 1: Extraction of the stage-recessions
3.2.2 Step 2: Bayesian estimation of the stage-recessions
3.2.3 Third step: recessions segmentation
3.3 Application: Ardèche River at Meyras, France
3.3.1 Description of the station site
3.3.2 Step 1: Recessions extraction
3.3.3 Step 2: Recessions estimation
3.3.4 Step 3: Recessions segmentation
3.3.5 Sensitivity to the selected recession model
3.4.2 Perspective: real-time stage-recession analysis
4 Fast detection of potential rating shifts based on the stage record and bedload assessment
4.1.1 General principle
4.1.2 Sediment transport modelling
4.1.3 Sediment transport models as proxys for potential changes
4.1.4 Objectives and structure of the chapter
4.2 The proposed sediment transport proxy analysis
4.2.2 Information available from the station history
4.2.3 Estimation of the triggering stage and detection of all potential morphogenic events
4.2.4 Computation of the sediment transport
4.2.5 Estimation of the uncertainty on the potential shifts
4.3 Application to the Ardèche River at Meyras, France
4.3.1 Information from the station history
4.3.2 Estimation of the triggering stage and detection of all potential shift times
4.3.3 Relation between shift b and sediments volume V
4.4.1 Main limitations
4.4.2 Use of the method for retrospective purposes
4.4.3 Other perspectives
5 The real-time application
5.1.1 Retrospective vs Real-time analysis
5.1.2 Solutions proposed in the literature and main diculties
5.1.3 Outline of a real-time procedure
5.1.4 Objectives and structure of the chapter
5.2 The proposed real-time application
5.2.1 Initialisation: hydraulic analysis
5.2.2 Retrospective analysis
5.2.3 Incoming stage data
5.2.4 Shift detection
5.2.5 Shift estimation
5.2.6 Update of RC priors and RC estimation
5.2.7 Discharge computation
5.2.8 Start of a new stable period
5.3 Application to the Ardèche River at Meyras: a demo
5.3.1 Overview of the application
5.3.2 The retrospective analysis
5.3.3 Iteration 15: recession analysis but no shift
5.3.4 Iteration 16: recession analysis and new gauging but no shift
5.3.5 Iteration 82: exceedance of the triggering stage and detection of a potential shift
5.3.6 Iteration 191: ood peak
5.3.7 Iteration 287: application of the stage-recession analysis after the ood
5.3.8 Iteration 311: new gauging and rating shift conrmation
5.3.9 Summary of the application
5.4.1 Main limitations
5.4.2 Stage pre-treatment
5.4.3 Future perspectives
6 Conclusions and perspectives
6.2.1 Improvement of the proposed tools for rating shift detection
6.2.2 Performance evaluation using a wide range of hydrometric stations
6.2.3 Development of other tools for potential rating shift detection
6.2.4 Choice of the tools for shift detection/estimation
6.3 Implementation into operational applications