# Accounting for historical data uncertainty in flood frequency analysis: the Upper Rhine River

Michel Lang, Jérôme Le Coz, Benjamin Renard, Matteo Darienzo

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Michel Lang, Jérôme Le Coz, Benjamin Renard, Matteo Darienzo. Accounting for historical data uncertainty in flood frequency analysis: the Upper Rhine River. Journal of Hydrology, 2025, 660, pp.133480. 10.1016/j.jhydrol.2025.133480. hal-05078660

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Research papers

# Accounting for historical data uncertainty in flood frequency analysis: the Upper Rhine River

Michel Lang<sup>a, *</sup>, Jérôme Le Coz<sup>a</sup>, Benjamin Renard<sup>b</sup>, Matteo Darienzo<sup>a,c</sup> ◆

## ARTICLE INFO

This manuscript was handled by A. Bardossy, Editor-in-Chief, with the assistance of Felix Frances, Associate Editor

### Keywords:

Flood

Rhine

Uncertainty

Historical data

Frequency analysis

Research limitations

## ABSTRACT

This paper is based on an international expertise on the safety of Rhine dikes, downstream Strasbourg (France). Two long hydrometric series at Basel (1225–2017) and Maxau (1815–2018) were examined, with an analysis of discharge uncertainties using the BaRatin model. The width of the 95 % credibility interval for flood discharges at Basel and at Maxau varies from ±7 % over the recent period (1994–2017) to ±48 % over the oldest period (1225–1713), and from ±5 % over the recent period (1977–2018) to ±45 % for the oldest period (1815–1839), respectively. The application of statistical tests to the flood samples reveals heterogeneity of the two series, which stems in part from hydraulic developments of the Rhine over the past two centuries. The series corrected for these effects are of better quality. However, there is still some heterogeneity around 1966–1976 at both Basel and Maxau, which cannot be explained by errors on rating curves or by hydraulic engineering on the Rhine or in the catchment area. It might be due to a change in the climate circulation regime in Central Europe since the 1970 s. A Bayesian Flood Frequency Analysis model with multiplicative errors on rating curves is used to process a flood sample with variable discharge uncertainty. It shows the advantages and limitations of introducing historical information predating systematic discharge records.

## 1. Introduction

The purpose of the Flood Directive (European Union, 2007) is “to establish a framework for the assessment and management of flood risks, aiming at the reduction of their adverse consequences”. Flood risk maps are a key tool for flood risk management plans, displaying the areas sub- merged according to several flood scenarios, defined as extreme (low probability), intermediate (rarer than the 100-year flood, typically), and frequent (high probability) events. The flood peak discharge for a given occurrence probability must be estimated prior to the computation of flooded areas and risk levels, through hydrodynamical modelling, typically.

The statistical estimation of extreme floods is challenging. As stressed by Merz and Blöschl (2008a), there is a need “to expand the information beyond the flood sample at the site of interest”. The standard length of flood records is usually of a few decades, which has a strong impact on sampling uncertainty. When using a Generalized Extreme Value (GEV) distribution, or a Generalized Pareto (GP) distribution, the

estimation of the shape parameter ξ, which governs the extrapolation towards extreme quantiles, is very sensitive to the sample size. Merz and Blöschl (2008b) recommend expanding flood information with three main groups of data: 1/ historical or paleoflood data, predating continuous discharge series, which we call “systematic records”; 2/ regional data, by pooling flood information from neighbouring and similar catchments; 3/ rainfall-runoff data by coupling a rainfall generator with a hydrological model to simulate long discharge series of several thousands of years, typically.

We focus on the first group of data, which can be derived from a variety of information sources: testimonies from documentary sources (Brázdil et al., 2010), flood marks (Bösmeier et al., 2022; Renard, 2023), or paleofloods from sedimentary deposits (Benito et al., 2004; Wilhelm et al., 2022). It allows going back in time several centuries to several millennia (with historical and paleoflood data respectively), and to increase the length of the flood discharge series. Although the inclusion of historical data in Flood Frequency Analysis (FFA) is possible by using suitable likelihood functions (Stedinger and Cohn, 1986; Kuczera,

<footnote><sup>a</sup> INRAE, Research Unit RIVERLY, Lyon-Villeurbanne, France</footnote>

<footnote><sup>b</sup> INRAE, Research Unit RECOVER, Aix-Marseille University, France</footnote>

<footnote><sup>c</sup> Department of Civil, Environmental, and Architectural Engineering, University of Padova, Padova, Italy</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail addresses: michel.lang@inrae.fr (M. Lang), jerome.lecoz@inrae.fr (J. Le Coz), benjamin.renard@inrae.fr (B. Renard), matteo.darienzo@unipd.it (M. Darienzo).</footnote>
1999), it remains an uncommon practice for engineering offices (Kjeldsen et al., 2014). Several reasons explain such limitation: 1/ national discharge databases provide quasi-continuous discharge series since the establishment of hydrometric networks without a compilation of “old” historical floods; 2/ reconstruction of the discharge of old events is time consuming (data collection and review); 3/ there is a lack of dedicated software for including uncertain and censored data in FFA. Flood of record is often used as a possible scenario (e.g., for flood hazard mapping, de Moel et al., 2009) with an approximate link to an exceedance probability. There is a need to disseminate operational tools following a sound probabilistic method able to include historical floods, accounting for their uncertainty.

This paper presents a case study on the Upper Rhine River, where the assessment of extreme flood differed greatly between the two banks of the Upper Rhine. On the right bank, the Germans considered that it was not possible to include floods prior to 1903, as it was too difficult to correct past floods to be consistent with the current flow context. On the left bank, the French used the annual maximum values since 1933 and a set of six historical floods at Basel since 1808, without any flood discharge correction. The return period of the safety flood of the levees, which protect several thousand people, is estimated to be larger than 10 000 years for the Germans (IWG, 2010) and less than 1000 years for the French (EDF, 2002). On one side, the elevation of the levee crest was hence considered as acceptable, while the other side asked for additional investigation on flood risk.

The paper demonstrates how to quantify and account for discharge data uncertainties in historical FFA through the representative case of the Rhine River. Its specific objectives are: i/ to assess the Rhine flood discharge over several centuries, with the corresponding uncertainties; ii/ to check the homogeneity of flood discharge series, by comparing “raw” series with “corrected” series (taking into account river works on the High and the Upper Rhine River); iii/ to compare the estimate of extreme flood quantiles, using several data sets of varying length (from 1 to 8 centuries), with or without historical floods, and adding some regional consistency on the shape parameter of the GEV distribution. Fig. 1 shows the flow chart used to answer the question of the safety of flood protection levees on the Rhine River.

This paper is organized as follows. Section 2 discusses advantages and limitations of inclusion of historical data in FFA. Section 3 presents the Bayesian frameworks used for the estimation of rating curves with associated uncertainties, and the FFA with historical data. Section 4

presents the case study of the Upper Rhine River, with two long hydrometric series established at stations in Switzerland (Basel) and Germany (Maxau). In Section 5 flood discharge uncertainties are estimated, and in Section 6 the flood homogeneity is checked. In Section 7, FFA is applied to the combined set of systematic series and historical data, illustrating how historical flood data can be useful to improve extreme flood analysis.

## 2. Advantages and limitations of historical data

### 2.1. Statistical requirements on historical data

An implicit requirement is that the flood sample is exhaustive, without any missing value. It is clearly not the case when sampling annual maximum (AMAX) values from the period before the establishment of hydrometry networks, where only the largest floods are known. However, it remains relevant to use historical data with a censored statistical model, assuming that during the historical period all the floods beyond a perception threshold are known (Stedinger and Cohn, 1986). An expert analysis allows choosing a perception threshold high enough so that no historical flood is missing, and low enough to capture all the relevant information for further FFA. A specific attention must be payed to flood event collection based on flood damages. As the consequences of flood damages depend on both the flood magnitude and the human activity in and around the river, it may be possible to have a variable perception threshold over different periods (Barriendos et al., 2003). Another important issue is the completeness of archives or sedimentary sources, which may be partial in case of loss or destruction-alteration of documentation or flood marks (Bösmeier et al., 2022), or removal-remobilisation of flood sediment deposits (Benito and Thorn-dycraft, 2005). Cross-checking information at the catchment scale is essential to ensure that the sub-sample of values exceeding the perception threshold is nearly complete.

Extreme Value Theory (Coles, 2001) provides guidance on the limiting distribution of AMAX or Peak-Over-Threshold (POT) values. Assuming that sampled values are issued from independent and identically distributed (i.i.d) random variables, AMAX values have a GEV (Generalized Extreme Value) distribution, and POT values a GP (Generalized Pareto) distribution.

A number of challenges are associated with the i.i.d assumption when dealing with historical or paleoflood data. It is easy to meet the

<figure><img src="image_2.png" /><figcaption>Fig. 1. Flow chart used for Flood Frequency Analysis accounting for historical data uncertainty.</figcaption></figure>
requirement of independence, when sampling systematic discharge series. With AMAX values, the use of a hydrologic year, starting during the low flow period instead of an arbitrary date like the 1st of January, avoids the case of two successive AMAX values related to the same hydrometeorological event. With POT values, some inter-flood duration criteria on two consecutive peaks allows discarding correlated peaks (Lang et al., 1999).

The assumption that all sample values follow the same distribution is more challenging, due to possible changes of the flood regime (Hall et al., 2014). River training or large changes in the floodplain or the catchment during the last centuries could have influenced the flood regime. In such a case, it is better to work with a “corrected” flood series, related to a same state of the river/catchment, with either a “natural” or a “present” configuration. Using a mixed distribution is also possible when several flood types occur with different hydrometeorological drivers, for example snowmelt or rainfall floods, or slow or flash floods. A compound distribution based on weather pattern (Garavaglia et al., 2010) or flood type (Fischer, 2024) sub-sampling allows complying with the i.i.d assumption for each sub-sample. The stationarity hypothesis may also be challenged by climate variability over the last few centuries and by climate change since the Anthropocene era. In this case, it is possible to include climatic co-variables in the probabilistic model (Salas et al., 2018).

## 2.2. Tradeoff between long series and accurate series

Since the beginning of organized hydrometry networks (in the second part of the 19th century at best), hydrometric technologies have made a lot of progress in relation with advances of mechanical, electrical and electronic techniques (Le Coz, 2017). In the 19th century, flow measurements used simplified technical devices for velocity measurements, such as floats, Woltman meters or Pitot tubes. Stage was read directly on a staff gauge or recorded on paper from the position of a float in a stilling well. Estimation of past floods requires the calibration of a stage-discharge curve (or “rating curve”), which is affected by non-negligible uncertainty as no direct discharge measurement (or “gauging”) is available (or with poor accuracy) and as the description of the riverbed topography and the flow conditions is usually approximate.

There is a trade-off between the length and the accuracy of the discharge series used in FFA. The sampling uncertainty of extreme flood quantiles decreases when the series is longer, whereas “old” flood discharge estimates are more uncertain than “modern” flood estimates. Several authors showed how the final uncertainty on extreme flood quantiles evolves with the inclusion of imprecise historical data, either with synthetic data or with case studies (Frances et al., 1994; Parent and Bernier, 2003; Payrastre et al., 2011; Macdonald et al., 2014; Parkes and Demeritt, 2016). A key question is the correct estimation of discharge uncertainties (Westerberg and Karlsen, 2024) with particular emphasis on the reconstruction of old flood discharges (Lucas et al., 2023; Sabeti et al., 2024). In their study of the Rhône River at Beaucaire, Lucas et al. (2024) used both a long series (1816–2020) of AMAX discharges and a collection of 13 historical floods over three centuries (1500–1845). The results indicate that the inclusion of historical floods reduces the uncertainty of 100-or 1000-year flood quantiles, even though “old” reconstructed flood discharges have greater uncertainties than estimates of “modern” floods (e.g., 30–50 % vs 5–15 %).

# 3. Methods

## 3.1. BaRatin model for the estimation of rating curves

The BaRatin model (Le Coz et al., 2014) uses a Bayesian approach to combine expertise on the hydraulic controls that physically determine the stage-discharge relation with uncertain information from the available gaugings. The method produces an estimate of the rating curve and the associated uncertainties (95 % credibility interval, for example). The

prior elicitation of the stage-discharge relation is based on the identification of the section and channel controls of the hydrometric station and the associated parameters (geometry of the control cross-section or channel, channel slope and friction coefficient). When the flow is controlled by a channel, first the main channel and then the floodplain typically, the relationship $Q(H|\psi)$ between stage H and discharge Q can be expressed using power laws:

$$k_1 < H \le k_2 \text{ (main channel): } Q(H) = a_1(H - b_1)^{c_1} \quad (1a)$$

$$H \ge k_2 \text{ (floodplain): } Q(H) = a_1(H - b_1)^{c_1} + a_2(H - b_2)^{c_2} \quad (1b)$$

where $k_1 < k_2$ are the activation stages for each control and $\psi = (a_1, a_2, c_1, c_2, k_1, k_2)$ is the vector of parameters of the rating curve (the parameters $b_1$ and $b_2$ being deduced from the other parameters by continuity of the stage-discharge relationship at $H = k_2$). The gauging uncertainty is estimated (GDH, 2017; table 4.2) according to the type of gauging (complete exploration of the velocity field throughout the cross-section or only at the free surface, tracer dilution) and the methods and instrumentation used to measure the velocity (acoustic Doppler current profiler, current meter, radar or video-based velocimetry).

For the period prior to the existence of gaugings, the BaRatin model is applied retrospectively, going back in time from the first rating curve based on gaugings. The information available on the morphological evolution of the riverbed is used to define a series of homogeneous sub-periods, based on hypotheses on the hydraulic parameters.

## 3.2. Flood frequency framework accounting for discharge uncertainties

We use a Bayesian approach for FFA with the following assumptions: i/AMAX values are GEV distributed; ii/ the set of historical data is exhaustive (years without historical flood are supposed to have an AMAX value below a perception threshold); iii/ rating curve errors are modelled through a multiplicative error on flood discharge. No informative prior is assumed on the GEV parameters (except in Section 7.2.3).

We assume that the AMAX random variable has a GEV distribution, with a set $\theta = (\mu, \sigma, \xi)$ of location, scale and shape parameters. The cumulative distribution function (cdf) $F$ and the corresponding probability density function (pdf) $f$ are:

$$\text{Prob}(Q < q) = F(q; \theta) = \exp \left[ - \left(1 - \xi \frac{q - \mu}{\sigma}\right)^{1/\xi} \right] \quad (2)$$

and

$$f(q; \theta) = \frac{\partial F(q; \theta)}{\partial q} = \frac{1}{\sigma} (1 - \xi \frac{q - \mu}{\sigma})^{1/\xi - 1} F(q; \theta).$$

With a mixed sample of AMAX discharges $q$ from a period of $j$ years with systematic records and a set of $k$ historical floods exceeding a perception threshold $S$ from another period of $n$ years, the likelihood writes:

$$L(\theta; q, k) = \prod_{i=1}^{j} f(q_i; \theta) \underbrace{\left[ \binom{n}{k} (F(S; \theta))^n (1 - F(S; \theta))^k \right]}_{(b)} \quad (3)$$

Terms (a) and (b) in Equation (3) respectively represent the likelihood for systematic and historical data. The posterior distribution of the parameters $\theta$ given floods from systematic records and historical data is:

$$p(\theta|q, k) \propto L(\theta; q, k)p(\theta) \quad (4)$$

with $p(\theta)$ the prior distribution of the parameters.

We now consider that it is not only possible to state that historical floods exceed a perception threshold $S$, but also to reconstruct their discharges $y = (y_1, ..., y_k)$. A first way to account for the uncertainties is to define an uncertainty interval $[y_i^{inf}; y_i^{sup}]$ for each historical flood discharge $y_i$. The corresponding likelihood writes:
$$L(\theta; \mathbf{q}, \mathbf{y}, k) = \prod_{i=1}^{j} f(q_i; \theta) \prod_{i=1}^{k} [F(y_i^{sup}; \theta) - F(y_i^{inf}; \theta)] (F(S; \theta))^{n-k} \quad (5)$$

A second way is to consider a multiplicative error $\gamma_i$ on each rating curve used to reconstruct discharge from systematic records and historical data (Neppel et al., 2010). Denoting $\tilde{Y}$ the real discharge, and Y the reconstructed discharge, we have: $\tilde{Y} = \gamma_i Y$.

$$As \quad \tilde{Y} \sim F(\mathbf{y}; \theta), \quad Y \sim F(\mathbf{y}; \theta'), \quad \text{with } \theta' = \left( \frac{\mu}{\gamma_i}, \frac{\sigma}{\gamma_i}, \xi \right) \qquad (6)$$

Assuming a triangle prior distribution $p(\gamma_i)$ for each rating curve error parameter $\gamma_i$, the posterior distribution is:

$$p(\theta, \gamma | q) \propto L(\theta', \gamma; q, \mathbf{y}) p(\gamma) p(\theta) \qquad (7)$$

The multiplicative error model is justified by the common observation that absolute rating curve errors increase with the discharge value (e.g., Reitan and Petersen-Overleir, 2009; Thyer et al., 2009).

We use non-informative flat priors for the GEV parameters $\theta$. A Markov Chain Monte Carlo (MCMC) simulation method is performed with the STooDs code (https://github.com/STooDs-tools/RSTooDs/; Renard et al., 2021). It generates values that converge in probability to the posterior distribution of the parameters (Equation 4). We consider the modal parameter estimate $\theta^*$, called “max post”, which maximizes the posterior distribution, and the posterior 95 % credibility interval of flood quantile $Y_p$, which corresponds to the 2.5 % and 97.5 % quantiles of its posterior distribution.

# 4. Case study on the Upper Rhine River

## 4.1. French-German agreement on flood embankment

A French-German agreement on the development of the Rhine between Kehl-Strasbourg and Neuburgweier-Lauterbourg (IWLP, 1969) defines the level of protection against flooding along this section of the Upper Rhine River (Fig. A1). The height of the levees should have a safety margin against a reference flood $Q^* = 7500 \text{ m}^3/\text{s}$. Unfortunately, this contractual freeboard appears not to be respected, particularly in the upstream part of the Iffezheim reach, downstream of the city of Strasbourg.

In 2016, the German authority of Bade-Wurtemberg requested to carry out engineering works to comply with these contractual levee levels, following which, in 2017, the French government asked for additional investigation on the assessment of extreme floods. The return period $T(Q^*)$ was estimated to exceed 10 000 years according to University of Karlsruhe, Germany (IWG, 2010), while it was found to be less than 1000 years in some French studies (EDF, 2002). The German experts used discharge series from the 20th century only, discarding old floods as too imprecise. The French experts used a mixed sample, combining annual maximum floods recorded since 1933 and a set of six historical flood discharge estimates at Basel (1808–1933). Using the flood frequency framework accounting for discharge uncertainties that we introduced in Section 3.2, here we present an additional study (Lang, 2020; Lang et al., 2021), using data from 1808 to 2017. Both German and French sides have accepted it. The aim was to expand the available hydrological information as far as possible in the past, with a critical review of the quality of flood discharges. We now have 6 additional years (2018–2023, with 2024 not yet published) that are not used in this study. We expect that these additional years have limited impact on the results of this paper for the following reasons: 1/ no significant flood occurred during the 2018–2023 period (AMAX discharge at Basel between 2426 and 3738 m³/s, below the 10-year flood); 2/ the weight of 6 additional years in a 210-year series is low.

## 4.2. Flood information at Basel – Switzerland

The Basel gauge has one of the longest discharge series in Europe,

starting from 1808. The location of the gauging station changed twice (Fig. 2). It was initially located on the left bank of the river, at the old town bridge at Schifflände (kilometer 166.7). It moved upstream on the left bank at Rheinfelden (km 148.05), due to the commissioning of Kembs Dam in 1933. A rating curve could no longer be used at Schifflände because of the backwater created by the dam. To ensure the continuity of the discharge time series, the Rhine discharge over the 1933–1994 period is the sum of the Rhine River at Rheinfelden and two tributaries (Birs and Ergolz) joining the Rhine between the old and the new stations. The present gauge has been located at Rheinhalle (km 164.39) on the right bank since 1995.

It is therefore possible to have a continuous AMAX flood series from 1808 to 2017. A set of 53 gaugings is available for the period 1867–1933 (Fig. A2) and 88 gaugings for the period 1994–2017. We have a set of 9 rating curves for the period 1916–1932, and 25 rating curves for the period 1986–2018.

Wetter et al. (2011) provide additional information on floods since 1268 from two sources. On the right bank of the Rhine River on the Schönbeinhaus at Oberer Rheinweg 93 in Lower Basel, about 500 m upstream from Schifflände, there remain 12 flood marks covering the 1641–1881 period. The first historical flood recorded by Wetter et al. (2011) dates back to 1268, based on an inventory of 38 flooding events in the town’s historical centre near the old bridge of the city, built in 1225. We assume that there was no significant flooding in the period 1225–1268, otherwise there would have been accounts of it in the documentary archive in relation with the bridge and the Lower Basel. Based on a 1D hydraulic model, Wetter et al. (2011) identified two perception thresholds: $S_1 = 4911 \text{ m}^3/\text{s}$ over the 1268–1713 period (38 floods) and $S_2 = 4322 \text{ m}^3/\text{s}$ over the 1714–1807 period (11 floods).

The Swiss data manager (Federal Office for the Environment, FOEN) of the Basel station has a series of 5 old cross-sectional surveys (in 1914, 1916, 1918, 1922 and 1930) of the Rhine’s main channel. The comparison of these cross-sectional profiles shows a relative stability of the cross-section over the period 1914–1922 and a river incision between 1922 and 1930. A German document (BWSK, 1930, appendix 9) shows the longitudinal profile of the Rhine from Basel to Mannheim (over a length of about 250 km) for various dates: 1830, 1852, 1895, 1902 and 1926. Table A1 shows that the slope of the Rhine at Basel remained stable (around $10^{-3}$, with a slight continuous increase from 0.8 to 1.1 $10^{-3}$), but that the main channel was eroded.

## 4.3. Flood information at Maxau – Germany

The first systematic readings of the water level at Maxau (zero $Z_0 = 97.79 \text{ m}$; catchment area $50 \text{ 343 km}^2$) started in 1815. According to the German data manager (WSV, 1975), between 1815 and 1881 the frequency of readings varied between 1 and 3 times a day, depending on the period. After 1881, it was set to 3 times a day (05:00, 13:00 and 21:00), with increased frequency for high water levels, so as not to miss the flood peak. The first water level gauge was commissioned in 1888, but due to metrological problems, the first continuous and validated recordings of water levels did not begin until 1897. During the World War II, measurements were discontinued several times between May 1942 and November 1943, and the gauge was destroyed in November 1943. A new gauge started to produce data at the end of February 1946.

Continuous flood discharges have been estimated since 1815, based on 22 rating curves reconstructed over the period 1821–1973 (HSK, 1978) and 15 rating curves established using 850 gaugings taken since 1947, including 42 gaugings of flood discharges greater than 3000 m³/s.

The Upper Rhine underwent major rectification in the 19th century and major work for navigation in the 20th century. Based on the monitoring of changes in Rhine water low levels in Maxau over the period 1810–1927 (BWSK, 1930) and their interpretation (BAW, 2015), assuming a similar change between low water level and riverbed, we have plotted the changes in the Rhine bed elevation from 1815 to 1946 in Fig. 3. There was an incision of more than one meter from 1815 to
<figure><img src="image_3.png" /><figcaption>Fig. 2. Location of various hydrometric stations in Basel-Switzerland since 1808.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3. Evolution of the Rhine mean bed elevation at Maxau from 1815 to 1946 low flow level records (BWSK, 1930; BAW, 2015) and priors (means and 95% intervals) used to calculate uncertainties with the BaRatin model (see Section 5.2.2 for details).</figcaption></figure>

1904, followed by a rise of 30 to 40 cm from 1905 to 1946.

4.4. River works on the High Rhine and the Upper Rhine River

**The High Rhine** (Fig. A1), upstream of Basel, was modified by major hydraulic works in the 18th and 19th centuries, which successfully altered flood propagation.

Firstly, a diversion of the Kander River (a tributary of the Aare River) into Lake Thun in 1714 induced a reduction of the peak flow during floods. Using a hydraulic model, Wetter et al. (2011) consider that before 1714 all peak floods above 4911 m³/s at Basel would have

reduced by 270 m³/s, had the floods occurred in the current situation.

Secondly, “Jura water correction works” between 1868 and 1891 induced a larger flood mitigation effect. These consisted of diverting the course of the Aare to take it through Lake Biel (Fig. A3, Section 1), increasing the exchange capacity between Lake Biel and Lakes Neuchâtel (Neuenburgersee) and Murten (Murtensee) (Fig. A3, Sections 2 and 3), and improving the flow capacity downstream of Lake Biel (Fig. A3, Section 4). The flood mitigation effect was assessed based on hydraulic studies, with corrections N°1 (Wetter et al., 2011) and N°2 (HSK, 1978) respectively. In both cases, the mitigation effect appears when discharge Q exceeds 2600 m³/s. It increases linearly with
discharge up to a peak discharge reduction of 600 m³/s (correction N°1) or 1300 m³/s (correction N°2), when discharge is greater than or equal to 3900 m³/s (correction N°1) or 5400 m³/s (correction N°2).

During the 19th century, the shape of the upper Rhine (Fig. A1) changed from a braided river with multiple channels to a single channel river (Fig. A4): cut-off of the meanders (1815–1831) and Tulla’s river engineering works (1843–1860). During the 20th century, additional hydraulic engineering including regulation works (1906–1924 and 1930–1939), the construction of 10 dams and channelling works (1928–1977) and new retention zones (1982–2018) changed the propagation of the floods. Based on hydraulic simulations, all peak floods were corrected to the same reference state in 1977 (HSK, 1978; LUBW, 2019).

Works carried out in the 19th century significantly altered the morphology of the Rhine bed, increasing the slope of the main channel and incising the bed, which may have reduced propagation times. As the development of vegetation in the floodplain slowed flood propagation, there is finally less difference between the periods 1820–1850 and 1875–1950, than between the periods 1875–1950 and 1950–1977 (Fig. A5). It appears that the channelling of the 1948–1977 period (9 dams and locks, and construction of dikes along the main channel) had the greatest impact on flood propagation, with greatly reduced propagation times.

# 5. Estimation of flood discharge and uncertainties

## 5.1. Flood discharges of the Rhine River at Basel

### 5.1.1. Periods with gaugings

**Recent period 1994–2017.** Using a cross-section surveyed in 2011 (Fig. A6), we determine the equivalent rectangular cross-section of the main channel, which has a width $B_1 = 165 \pm 20$ m (95 % credibility interval). We estimate the width of the floodplain as $B_2 = 130 \pm 20$ m (95 % credibility interval), based on the extent of the flood area in the old centre of Basel (Wetter et al., 2011). We consider a hydraulic control with a wide rectangular channel, modeled using the Manning-Strickler equation: $Q = BKS^{1/2}(H-b)^{5/3}$. Table 1 summarizes the priors retained for the physical parameters (B: cross-section width; K: Strickler friction coefficient; S: longitudinal slope) along with the activation stages $k_1$ and $k_2$ of the two additive channel controls.

Combining these prior parameters with 57 ADCP gaugings (stage H > 5.5 m; measurement uncertainty ±5 %) from the period 1994–2018 gives a posterior BaRatin rating curve with an uncertainty of about ±4 % (Fig. 4a). For verification purpose, we plot the two most distant rating curves (FOEN, N°10 in 1995 and N° 16 in 2002) of the period, which respectively give the lowest and highest discharge Q for a stage H. We also plot the discharge estimated by FOEN for the two highest floods of the period (09/08/2007 and 12/05/1999). If we assign an uncertainty of ±7 % to the 2007 and 1999 flood discharges, the upper limit corresponds to the BaRatin estimate. In the end, we will retain this uncertainty of ±7 % for the discharge of the AMAX floods over the period 1994–2017.

**Period 1867–1933.** As no gaugings for the period 1934–1993 are available in the FOEN archives, we continue the analysis of the rating

curves with the period 1867–1933 (Basel-Schifflände), for which a series of cross-sections (1914–1930) and longitudinal profiles (1830–1926) gives an idea of the morphology of the Rhine’s main channel. Based on the 70 available gaugings, we distinguish three sub-periods: 1867–1909, 1910–1925 and 1926–1933. The rating curve uncertainties obtained with the BaRatin model are [−12 %; +26 %], [−5 %; +9 %] and [−7 %; +10 %] respectively. Conservatively, and considering that uncertainty increases as we go back in time, the rating curve uncertainty for the period 1910–1925 is taken to be identical to that for 1926–1933, i.e. [−7 %; +10 %].

### 5.1.2. Periods without gaugings

**Period 1808–1866.** For the main channel, we use the posterior values of the hydraulic parameters of the 1867–1908 rating curve as prior values for the period 1808–1866. We update some parameters when information is available (see Table 2). From Table A1 we have an incision of 0.5 m of the riverbed and an increasing of the slope of the main channel. We keep the same slope $S_2$ in the floodplain as $S_1$ in the main channel. As the urbanised area was less extensive in the past, we reduce the friction effect (increasing the Strickler coefficient $K_1$). Old maps show a narrower main channel (reducing the width $B_1$). As we have limited information on all these parameters, we enlarge their uncertainty accordingly.

Fig. 4b shows the BaRatin rating curve for the period 1808–1866. We add three pairs (H, Q) obtained by hydraulic modelling (Table 2 from Wetter et al., 2011). The discharges correspond to the flooding of various reference locations in the historic centre of Basel:

* Flooded “Blumenplatz” quay: altitude Z = 249.6 m, H = 5.70 m, Q = 4300 m³/s;

* First flooding of “Fischmarkt”: altitude Z = 250.1 m, H = 6.20 m, Q = 4900 m³/s;

* Severe flooding of “Fischmarkt”: altitude Z = 251.1 m, H = 7.20 m, Q = 6400 m³/s.

There is good consistency between the BaRatin rating curve, very approximately estimated from prior values of hydraulic parameters, and information obtained by hydraulic modelling. The discharge uncertainty intervals around the BaRatin rating curves are [−18 %; +22 %]. As a conservative measure, in order to have greater uncertainties going back into the past, as the uncertainty interval for the period 1867–1909 was [−12 %; +26 %]. We retain a final discharge uncertainty for the period 1808–1866 within the interval [−18 %; +26 %].

**Period 1268–1807.** Wetter et al. (2011) identified 38 floods that inundated the old city of Basel. We make a number of assumptions about the value of the rating curve parameters (see Table 2), using higher uncertainties (activation stage $k_1$, Strickler friction coefficients $K_1$ and $K_2$, slopes $S_1$ and $S_2$), which lead to an uncertainty of [−35 %; +44 %].

**Intermediate period 1934–1993.** This period, without any record of gauging or rating curves (that must have existed), is framed by two periods 1910–1933 and 1994–2017 where the uncertainties are low, ±9 % and ±7 % respectively, due to the availability of gaugings. As a conservative measure, we retain an uncertainty of ±20 %, close to that of the periods 1808–1866 (±22 %) and 1867–1909 (±19 %).

<caption>**Table 1**</caption>

Prior parameters used for estimating the rating curve of the Rhine River at Basel-Rheinhalle (recent period).

<table><thead><tr><th></th><th colspan="5">Main channel</th><th colspan="5">Floodplain</th></tr><tr><th></th><th>k<sub>1</sub><br>(m)</th><th>K<sub>1</sub><br>(m/s)</th><th>B<sub>1</sub><br>(m)</th><th>S<sub>1</sub></th><th>c<sub>1</sub></th><th>k<sub>2</sub><br>(m)</th><th>K<sub>2</sub><br>(m/s)</th><th>B<sub>2</sub><br>(m)</th><th>S<sub>2</sub></th><th>c<sub>2</sub></th></tr></thead><tbody><tr><th>Physical parameters</th><td>1.3 ± 0.5</td><td>35 ± 5</td><td>165 ± 20</td><td>10<sup>-3</sup> ± 5</td><td>10<sup>-4</sup></td><td>1.67 ± 0.05</td><td>9 ± 0.5</td><td>15 ± 5</td><td>130 ± 20</td><td>10<sup>-3</sup> ± 5</td><td>10<sup>-4</sup></td></tr><tr><th>Rating curve parameters</th><td>k<sub>1</sub><br>1.3 ± 0.5</td><td>a<sub>1</sub> = K<sub>1</sub>B<sub>1</sub>√S<sub>1</sub></td><td></td><td></td><td></td><td>c<sub>1</sub><br>1.67 ± 0.05</td><td>k<sub>2</sub><br>9 ± 0.5</td><td>a<sub>2</sub> = K<sub>2</sub>B<sub>2</sub>√S<sub>2</sub></td><td></td><td>c<sub>2</sub><br>1.67 ± 0.05</td><td></td></tr></tbody></table>
<figure><img src="image_5.png" /><figcaption>Fig. 4. Rating curve for the Rhine River: a/ at Basel-Rheinhalle, recent period 1994–2017, posterior BaRatin estimate with 95% credibility interval, plotting of the two extreme rating curves (N°10 and N°16) and the two highest floods (in 1999 and 2002); b/ at Basel-Schifflände, period 1808–1866, prior BaRatin with the plotting of 3 pairs (H, Q) obtained by hydraulic modelling (Wetter et al., 2011).</figcaption></figure>

<caption>**Table 2**</caption>

Prior parameters of the rating curve of the Rhine at Basel-Schifflände: periods 1808–1866 and 1225–1807.

<table><thead><tr><th rowspan="3">Period</th><th rowspan="3"></th><th colspan="5">Main channel</th><th colspan="5">Floodplain</th></tr><tr><th>k<sub>1</sub></th><th>K<sub>1</sub></th><th>B<sub>1</sub></th><th>S<sub>1</sub></th><th>c<sub>1</sub></th><th>k<sub>2</sub></th><th>K<sub>2</sub></th><th>B<sub>2</sub></th><th>S<sub>2</sub></th><th>c<sub>2</sub></th></tr><tr><th>(m)</th><th>(m)<sup>1/3</sup></th><th>(m)</th><th></th><th></th><th>(m)</th><th>(m)<sup>1/3</sup></th><th>(m)</th><th></th><th></th></tr></thead><tbody><tr><td rowspan="2">1867–1909</td><td rowspan="2">prior</td><td>k<sub>1</sub></td><td>35 ± 5</td><td>173 ± 20</td><td>10<sup>-3</sup> ± 5 10<sup>-4</sup></td><td>1.67 ± 0.05</td><td>9 ± 0.5</td><td>15 ± 5</td><td>130 ± 20</td><td>10<sup>-3</sup> ± 5 10<sup>-4</sup></td><td>1.67 ± 0.05</td></tr><tr><td>(m)</td><td>0.5 ± 0.5</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="2">1808–1866</td><td rowspan="2">posterior</td><td>k<sub>1</sub></td><td>-1.4 ± 0.1</td><td></td><td></td><td></td><td>1.66 ± 0.02</td><td>5 ± 0.3</td><td></td><td></td><td>1.67 ± 0.02</td></tr><tr><td>(m)</td><td>-1.4 ± 0.1</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="2">1225–1807</td><td rowspan="2">prior</td><td>k<sub>1</sub></td><td>38.5 ± 1</td><td>170 ± 2</td><td>8 10<sup>-4</sup> ± 1.5 10<sup>-4</sup></td><td>1.67 ± 0.05</td><td>5 ± 0.5</td><td>17 ± 3</td><td>130 ± 10</td><td>9 10<sup>-4</sup> ± 1.5 10<sup>-4</sup></td><td>1.67 ± 0.05</td></tr><tr><td>(m)</td><td>-0.9 ± 0.5</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="2">1225–1807</td><td rowspan="2">prior</td><td>k<sub>1</sub></td><td>35 ± 5</td><td>170 ± 5</td><td>8 10<sup>-4</sup> ± 5 10<sup>-4</sup></td><td>1.67 ± 0.05</td><td>5 ± 0.5</td><td>20 ± 5</td><td>130 ± 20</td><td>8 10<sup>-4</sup> ± 5 10<sup>-4</sup></td><td>1.67 ± 0.05</td></tr><tr><td>(m)</td><td>-1 ± 1</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

## 5.2. Flood discharges of the Rhine River at Maxau

### 5.2.1. Period 1947–2017, with gaugings

For the period 1947–2017, detailed topography of the river cross-section and a series of 850 gaugings are available. Without any precise information on the discharge measurement procedure used for each gauging, we assume that the operators have used a current meter until the end of 1999 and an acoustic profiler ADCP since 2000, with an uncertainty of ±10 % and ±5 % respectively on the gauged discharges.

This uncertainty estimate is increased for significant floods (discharge greater than 3000 m³/s, i.e. $H > 7.6$ m) by a multiplication factor equal to 2.

A recursive segmentation procedure (Darienzo et al., 2021), which accounts for the uncertainty of the measured discharges, determines breaks in the stage-discharge relationship as change dates. Its application to the Maxau’s 850 gaugings (period 1947–2017) yields 11 homogeneous sub-periods (Fig. 5), instead of 21 from the official rating curves. The set of the parameters ($a_i, b_i, c_i$) ($i = 1, 11$) of the 11 sub-

<figure><img src="image_6.png" /><figcaption>Fig. 5. Segmentation procedure applied to 850 gaugings of the Rhine discharge at Maxau (period 1947–2017): (upper panel) stages of the gaugings pooled (colours) over the time; (lower panel) pdf of shift times – in blue – and the most probable date of shift – in red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>
period rating curves is estimated with a single iteration, using the BaRatin-SPD model (Mansanarez et al., 2019). It gives the opportunity to use constant and varying parameters for the 11 sub-periods and to transfer gauging information across periods. We obtain a very low discharge uncertainty of [-5%; +4%] over the 1947–1968 period, and [-3%; +2%] over the 1969–2017 period.

The SPD model gives an estimate of the $b_1$ parameter (mean river bottom elevation) for all the 11 sub-periods. We observe a rise of around 50 cm from 1947 to 2017 (Fig. 6), which is consistent with morphological monitoring of the Rhine bed (BAW, 2015).

### 5.2.2. Period 1815–1946, without gaugings

For the period 1815–1946, we distinguish six homogenous sub-periods (Fig. 3): 1815–1820, 1821–1839, 1840–1874, 1875–1904, 1905–1929 and 1930–1946. In the absence of gaugings, we follow the same procedure as for the Basel station. For the 1930–1946 curve, we adopt the posterior values of the oldest BaRatin curve for the 1947–2017 period (i.e. 1947–1968), enlarging the uncertainty on the $a_1$ et $a_2$ parameters of the rating curve (Equation 1), and integrating the available information on the morphological evolution of the Rhine bed (Fig. 3). We estimate earlier curves in a similar way, by difference between the river bed elevation of two successive sub-periods (Fig. 3). As the uncertainties obtained with the BaRatin curve for certain sub-periods are similar, we retain an uncertainty of [-37%; +48%] for the period 1815–1839 and [-27%; +37%] for the period 1840–1946.

The BaRatin curves yield substantially lower discharge than the official rating curves over the period 1840–1946 (-13% to -42% for stages between 7 and 8.8 m; Fig. A7). The BaRatin curves are based on a number of assumptions about the physical parameters of the main channel and the floodplain, using information about the past morphology of the Rhine, and going back in time since the last BaRatin rating curve with gauging (1947–1968). The past official rating curves were reconstructed from an indirect interpolation (HSK, 1978; appendix 3.4) between Basel and Kaub, respectively 200 km upstream and 150 km downstream of Maxau. For a stage $H_M$ at Maxau, the probability $p(H_M) = \text{Prob}(H_{\text{Maxau}} < H_M)$ is estimated from the classified stage curve at

Maxau. Using the classified discharge curves at Basel and Kaub respectively gives an estimate of the discharges $Q_B$ and $Q_K$, such as $\text{Prob}(Q_{\text{Basel}} < Q_B) = \text{Prob}(Q_{\text{Kaub}} < Q_K) = p(H_M)$. A multiple linear regression between the discharges $Q_{\text{Basel}}$ and $Q_{\text{Kaub}}$ and the discharge $Q_{\text{Maxau}}$ at Maxau yields the corresponding discharge $Q_M$ (from $Q_B$ and $Q_K$) related to the stage $H_M$.

As the official curves are just below the upper limit of the 95% credibility interval of the BaRatin curves, we will still use the official flood discharges, but taking into account large uncertainties on these discharges for flood frequency analysis.

### 5.3. Uncertainty of the flood discharges of the Rhine River (Basel and Maxau)

The “raw” discharge $Q$ is corrected by a multiplicative factor $c$ to take into account the changes in flood regime induced by the Rhine developments (Section 4.4), for a reference condition in 1977. It corresponds to the end of the channelling period of the Upper Rhine. The corrected discharge $Q_c$ writes:

$$Q_c = (1 + c)Q \quad (8)$$

The final uncertainty $u_Q$ of the corrected discharge $Q_c$ is obtained (Appendix 1) by assuming that the variables $Q$ and $c$ are independent and follow a Normal distribution:

$$ \left( \frac{u_{Q_c}}{E(Q_c)} \right)^2 = \left( \frac{u_Q}{E(Q)} \right)^2 \left[ 1 + \left( \frac{u_c}{1 + E(c)} \right)^2 \right] + \left( \frac{u_c}{1 + E(c)} \right)^2 \quad (9) $$
(where $E$ is the expected value)

Table 3 summarizes the relative uncertainties $u_{Q_c}/E(Q_c)$ in corrected flood discharge $Q_c$, using the uncertainty $u_Q$ on flood discharge with the BaRatin model (Sections 5.1 and 5.2) and a correction factor $c$ from information of hydraulic modelling (Section 4.4).

At Basel, a sample of 10 floods (period 1808–1889, discharge between 3750 and 5700 m³/s), for which the mitigation effect of the Jura correction works was evaluated using a hydraulic model (HSK, 1978), gives a correction factor $c$ that varies between -11% and -15%.

<figure><img src="image_7.png" /><figcaption>Fig. 6. Variations in parameter $b_1$ (river bottom elevation) on the Rhine River at Maxau, for the 11 sub-periods from 1947 to 2017 (estimated value and 95% credibility interval).</figcaption></figure>
<table><caption>**Table 3**
Relative Uncertainty (RU) on Rhine flood discharges.</caption><thead><tr><th colspan="8">Flood series of the Rhine River at Basel (1225–2017)</th></tr><tr><th>Period</th><th>1225–1713</th><th>1714–1807</th><th>1808–1866(*)</th><th>1867–1889</th><th>1890–1909</th><th>1910–1933</th><th>1934–1993</th><th>1994–2017</th></tr></thead><tbody><tr><td>RU u<sub>Q</sub></td><td>±40 %</td><td></td><td>±22 %</td><td>±19 %</td><td></td><td>±9 %</td><td>±20 %</td><td>±7 %</td></tr><tr><td>Correction c</td><td>-20 %</td><td>-15 %</td><td></td><td></td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>RU u<sub>c</sub>/E(c)</td><td>±100 %</td><td>±50 %</td><td>±50 %</td><td>±50 %</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>RU u<sub>Q</sub>/E(Q<sub>c</sub>)</td><td>±48 %</td><td>±41 %</td><td>±24 %</td><td>±21 %</td><td>±19 %</td><td>±9 %</td><td>±20 %</td><td>±7 %</td></tr></tbody><thead><tr><th colspan="8">Flood series of the Rhine River at Maxau (1815–2017)</th></tr><tr><th>Period</th><th>1815–1839</th><th>1840–1890</th><th>1891–1946</th><th>1947–1968</th><th>1969–1976</th><th>1977–2018</th><th></th></tr></thead><tbody><tr><td>RU u<sub>Q</sub></td><td>±42 %</td><td>±32 %</td><td></td><td>±5 %</td><td>±3 %</td><td></td><td></td></tr><tr><td>Correction c</td><td>16 %</td><td></td><td>10 %</td><td></td><td></td><td>2 to 13 %</td><td></td></tr><tr><td>RU u<sub>c</sub>/E(c)</td><td>±100 %</td><td>±100 %</td><td>±70 %</td><td>±70 %</td><td>±70 %</td><td>±50 %</td><td></td></tr><tr><td>RU u<sub>Q</sub>/E(Q<sub>c</sub>)</td><td>±45 %</td><td>±35 %</td><td>±33 %</td><td>±9 %</td><td>±8 %</td><td>±5 %</td><td></td></tr></tbody></table>

Considering that this correction factor *c* is partially estimated by the hydraulic model, we assume that *c* varies between 1.5c and 0.5c (c < 0). This gives a semi-interval width $ΔQ_c/Q_c = (Q_{0.5c} - Q_{1.5c})/(2Q_c) = -c/[2(1+c)]$. It varies between 6 and 9 % (with respectively c = -0.11 and c = -0.15), which seems possible using a hydraulic model after calibration. We assume a relative standard deviation $u_c/E(u_c) = 50$ % for this period 1880–1889. The earliest period 1225–1713 is more uncertain (before the diversion of the Kander river in 1714), and we assume a higher value of the correction factor $c = -20$ % and its relative uncertainty $u_c/E(u_c) = 100$ %.

At Maxau, the impact of new retention zones on a sample of 7 floods (period 1988–2018, discharge between 4060 and 4440 m³/s) was evaluated using a hydraulic model (LUBW, 2019), with a correction factor *c* that varies between 2 % and 13 %. Peak discharge would have been higher in 1977, in the absence of the new retention zones. With a relative uncertainty $u_c/E(u_c) = 50$ %, the semi interval $ΔQ_c/Q_c$ varies between 1 % and 6 %, which is possible with a hydraulic model calibrated with recent floods. During the period 1891–1968, hydraulic modelling by HSK (1978) gives a correction factor $c = 10$ %, with a relative uncertainty $u_c/E(u_c) = 70$ %. The former period 1815–1890 is more uncertain (Tulla's river engineering work), and we assume a higher value of the correction factor $c = 16$ % and its relative uncertainty

$$u_c/E(u_c) = 100 \%$$

The relative final uncertainty of the corrected flood discharges (Fig. 7) varies from 5 %–7 % for the recent period and increases when going back in time (except for 1910–1933 at Basel) up to 45–48 % for the oldest floods (respectively periods 1225–1713 and 1815–1839 at Basel and Maxau).

## 6. Homogeneity tests on flood discharge series

This section evaluates the homogeneity of the long series at Basel (1225–2017) and Maxau (1815–2018). As the flood discharges have been corrected to account for the same “1977 reference” state of the Rhine and the developments that have taken place over time, the homogeneity tests will be applied to the “raw” series (without correction) and the “corrected” series (“1977 reference” state). First, we apply a homogeneity test to POT floods at Basel since 1225. Second, we check the homogeneity of the series of AMAX flood values at Basel and Maxau since 1808 and 1815, using the uncertainties in the flood discharge that have just been calculated (Table 3).

<figure><img src="image_8.png" /><figcaption>**Fig. 7.** Floods at Basel (1225–2017) and Maxau (1815–2018): a/ relative uncertainty (%) of the corrected flood discharges; b/ AMAX flood discharge series and historical floods (Basel) (m³/s).</figcaption></figure>

<footnote>(*) with correction N°1 (maximum mitigation effect of 600 m³/s).</footnote>
## 6.1. Poisson test on POT flood values at Basel since 1225

We apply a homogeneity test to the historical flood series at Basel (Fig. 8). The Poisson test (Lang et al., 1999) plots the experimental curve of the cumulative number $m_t$ of floods over an interval [0; t] as a function of date t, and checks its position in relation to the 90 % tolerance interval. We consider three series start dates: 1225, 1714 (Kander diversion) or 1808 (start of systematic recordings); and three types of series with raw data or corrected data (with correction N°1 or N°2) to make flood discharges prior to the 1714 and 1890 developments in line with the reference situation in 1977.

During the period 1225–1713, we have 27 floods with a raw discharge $Q > 4911 \text{ m}^3\text{/s}$. Over the period 1714–1807, we have 11 floods exceeding $4322 \text{ m}^3\text{/s}$, of which 8 exceeded $4911 \text{ m}^3\text{/s}$. Since 1808, we have 8 floods with a discharge $Q > 4322 \text{ m}^3\text{/s}$, of which 4 exceeded $4911 \text{ m}^3\text{/s}$. As a reminder, the thresholds of $4911$ and $4322 \text{ m}^3\text{/s}$ correspond to the extreme floods recorded by Wetter et al. (2011), which flooded Basel's old city center (see Section 5.1.2: Blumenplatz and Fischmarkt).

In Fig. 8a (left), with 39 floods larger than $4911 \text{ m}^3\text{/s}$ over 1225–2017, most floods are in the period 1651–1764 and the experimental curve is outside the tolerance interval. Using correction N°1, the $4911 \text{ m}^3\text{/s}$ threshold decreases to $4041 \text{ m}^3\text{/s}$ (Jura correction and Kander diversion). There are 45 floods exceeding $4041 \text{ m}^3\text{/s}$, and the series appears homogeneous (Fig. 8b, left). With correction N°2 (Fig. 8c, left), the series remains heterogeneous.

We then apply the Poisson test to the periods 1714–2017 and 1808–2017 (Fig. 8, middle and right). The same behavior is observed, with the raw series (a) considered as heterogeneous, as with the series modified with correction N°2 (c) (HSK, 1978). The series resulting from correction N°1 (b) produced by Wetter et al. (2011) can be considered homogeneous. It should be noted that the diagnosis carried out on the raw series 1808–2017 (Fig. 8a, right) is not necessarily very robust, given the low number of flood values with a discharge greater than $4322 \text{ m}^3\text{/s}$ (only 8 floods). For this reason, the next section focusses on

the 1808–2017 period, using the AMAX values.

## 6.2. Homogeneity tests on AMAX discharges of the Rhone River at Basel and Maxau

We apply the Pettitt test (1979) and a segmentation procedure (Darienzo et al., 2021) to the AMAX flood series at Basel (1808–2017) and Maxau (1815–2018). The former tests a shift in the central tendency of a time series and indicates the sample rank associated with this shift. The latter iteratively searches for N breaks in a series and provides the probability density of the break date for each break detected.

Table 4 shows that no break is detected in the raw series at Basel in

<caption>Table 4</caption>

Homogeneity tests on AMAX discharges on the Rhine River (p-value of the Pettitt test).

<table><thead><tr><th>Station</th><th>Period</th><th>Series</th><th>Pettitt test</th><th>Darienzo procedure</th></tr></thead><tbody><tr><td rowspan="3">Basel</td><td>1890–2017</td><td>Raw data</td><td>p = 2 10<sup>-3</sup>; shift in 1933</td><td>Break in 1929–1930</td></tr><tr><td>1808–2017</td><td>Raw data</td><td>p = 2 10<sup>-4</sup>; shift in 1933</td><td>Break in 1930–1933</td></tr><tr><td></td><td>Corrected data(1714 et 1890 river works)</td><td>p = 10<sup>-3</sup>; shift in 1964</td><td>Break in 1966 or 1976</td></tr><tr><td rowspan="2">Maxau</td><td rowspan="2">1815–2018</td><td>Raw data</td><td>p = 2 10<sup>-3</sup>; shift in 1976</td><td>Break in 1976</td></tr><tr><td>Corrected data (correction N°1)</td><td>No shift</td><td>No break (*)<br>Break in 1976 (**)</td></tr></tbody></table>

<figure><img src="image_9.png" /><figcaption>Fig. 8. Poisson test on POT flood values at Basel since 1225, 1714 and 1808: a/ with raw data; b/ and c/ with corrected data (with corrections N°1 or N°2, respectively).</figcaption></figure>

<footnote>(*) Darienzo procedure without uncertainty of flood discharges.</footnote>

<footnote>(**) Darienzo procedure with uncertainty of flood discharges (Table 3).</footnote>
connection with the Jura water correction works of 1890. However, a break in the series is detected around 1930–1933, which may correspond to the relocation of the Schifflände station to Rheinfelden and a new method of calculating discharges. The Basel corrected series is considered heterogeneous, with a break date around 1964–1966 or 1976 (Fig. 9a), with no known physical explanation. The raw Maxau series is considered heterogeneous by both statistical approaches. Pettitt's test leads to acceptance of the homogeneity hypothesis for the corrected series, as does Darienzo's procedure (Fig. 9b) if we neglect the fact that the discharges have an uncertainty. On the other hand, including the uncertainties calculated above (Table 3), a break is detected around 1976 with the Darienzo procedure (Fig. 9c).

In the end, the corrections made to the Basel and Maxau series improve their homogeneity. However, there remains a break in both series, around 1966 or 1976, which is not explained by a metrological change. We are not aware of any large-scale land development that could explain this break. Brönnimann et al. (2019) point to a more intense period of flooding in Central Europe in the 19th century, and a less active period 1943–1970 (“flood-poor period”, Brönnimann’s Fig. 11b), which suggests that the period after 1970 was indeed more active than before 1970. Schmocker-Fackel and Naef (2010a,b; Fig. 5) also mention an increase in flood frequency in Switzerland from the 1970s onwards. A major challenge lies in understanding climate variability. Blöschl et al. (2020) showed, using historical data over the last 500 years, that Europe has experienced flood-rich and flood-poor periods, usually associated with respectively cooler and warmer than usual phases. Brönnimann et al. (2022) studied the drivers of flood variability in Europe, based on changes in atmospheric circulation and water vapour. They concluded that the first driver was dominant during the 19th and early 20th centuries, whereas today it is the second driver that explains long-lasting precipitation events. Fischer and Schumann (2024) observed temporal changes in flood process in Central Europe during the second part of the 20th century, with more frequent heavy-rainfall floods and less snowmelt-induced floods. Further investigation is required to attribute a cause to the break point observed around 1966–1976, but the climatic origin seems the most likely, although this point would deserve further investigation.

# 7. Flood frequency analysis with uncertain data

## 7.1. How sensitive is the flood distribution to the length of record and the relative uncertainty of flood discharges?

At Basel, we consider four periods with variable discharge uncertainty (Table 3):

* 1910–2017 period. Annual maximum floods have small relative uncertainties, less than 10 %, except for the 1934–1993 period;

* 1808–2017 period, with the full continuous record. Additional annual maximum floods have larger relative uncertainties, about 20 %;

* 1714–2017 period, adding 11 historical floods larger than 4322 m³/s. Historical floods have large relative uncertainties, about 40 %;

* 1225–2017 period, adding 27 historical floods before the Kander diversion in 1714 and larger than 4911 m³/s. Historical floods have large relative uncertainties, about 50 %.

Fig. 10a shows an increase in the estimate of the 100-year flood as the length of the series increases (respectively +3, 6 and 9 % compared to the period 1910–2017). The half-width of the 95 % credibility interval for the 100-year flood ($Q100_{0.975} - Q100_{0.025}$) / (2 $Q100$) remains stable, between 9 and 12 %. This possibly surprising result may be due to the fact that two opposite effects may interfere. On the one hand, large series have the potential to reduce sampling uncertainty. On the other hand, adding historical floods with larger uncertainty may increase the quantile uncertainty.

At Maxau, we consider three data sets (Table 3):

* 1947–2018 period. Annual maximum floods have small relative uncertainties, less than 10 %, due to 850 gaugings;

* 1891–2018 period. Additional annual maximum floods have large relative uncertainties, about 30 %, due to the lack of gaugings;

* 1815–2018 period, with the full continuous record. Additional annual maximum floods have large relative uncertainties, about 35 to 45 %, due to the lack of gaugings and morphological changes of the river.

<figure><img src="image_10.png" /></figure>

**Fig. 9.** Results of the segmentation procedure of Darienzo et al. (2021) on the corrected AMAX flood series: a/ at Basel (1808–2017); b/ and c/ at Maxau (1815–2018) respectively without/discharge uncertainties; with the most probable break date (maxpost, dotted vertical line in blue), the probability density of the break date (below in blue), the mean value per period (horizontal line in red), and the parametric and total uncertainties (red and pink bands). In the following section, we examine the Basel and Maxau series corrected to the “1977 reference” condition, plus an additional correction of +150 m³/s before 1977 to account for the break detected at both stations. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
<figure><img src="image_11.png" /><figcaption>**Fig. 10.** Q100 flood quantile (and 95% credibility interval) estimated using increasingly long sets of flood data: a/ at Basel since 1903, 1808, 1714 and 1225; b/ at Maxau since 1947, 1891 and 1815.</figcaption></figure>

Fig. 10b shows that the 100-year flood increases with the length of the series (respectively +3 and 11 % compared to the period 1947–2018). The half-width of the 95 % credibility interval for the 100-year flood uncertainty with the 1947–2018 period is ±12 %. An interesting result is that it decreases with the addition of pre-1947 flood information (± 9–10 %), despite the greater flood discharge uncertainties during the 19th century, with significant morphological changes of the Rhine bed.

## 7.2. How does the probabilistic model deal with uncertain data?

### 7.2.1. Systematic series with raw data and corrected data

As river works on the High Rhine (18th and 19th centuries) and the Upper Rhine (19th and 20th centuries) have had an impact on riverbed morphology and flood propagation, homogeneity tests (Section 6) clearly indicate breaks in the raw discharge series at Basel and Maxau. We therefore introduced corrections on flood series based on hydraulic calculations, with the associated uncertainties (Table 3). The use of a probabilistic model with $\gamma_i$ multiplicative errors on the rating curves (Equations 6 and 7) provides another opportunity for data review, from the point of view of the homogeneity of the flood sample with respect to a GEV distribution. It enables us to “weight” uncertain data with large uncertainties. Assuming a Triangle distribution on $\gamma_i \sim \text{Triangle}(1, 1-u_{Q_c}, 1+u_{Q_c})$, Table 5 gives the prior and posterior estimates on $\gamma_i$, with raw or corrected Basel discharge series over the period 1910–2017. The break in the raw series in 1977 (Fig. 9a) is detected by the multiplicative error model, which introduces a posterior correction on flood discharge of around +5 % to +7 % before 1976, and -2 % to -3 % after 1976. The use of the corrected series yields a smaller posterior correction (+ 2 %; -1 % to -3 %).

Fig. 11 shows that although the two systematic series are different (the raw series is heterogeneous with a break of +150 m³/s in 1977), the two final distributions are similar as the maxpost estimates of $\gamma_i$ give similar maxpost values of discharge. The same pattern is observed with

the Maxau discharge series over the 1947–2017 period, with lower posterior correction before and after 1976 with the corrected series (+1%; -1 %) than with the raw series (+3%; -1%).

#### 7.2.1.1. Mixed series with systematic and historical data.
In this section, we use the AMAX Basel series for the 1808–2017 period as a reference. We consider two sub-series using: 1/ a continuous series of 1910–2017 AMAX values; 2/ a mixed series with 1910–2017 AMAX values and 5 historical floods over the 1808–1909 period (Figs. A8 and A9).

We compare the mixed distribution with a GEV distribution fitted from AMAX values (1910–2017 and 1808–2017). Fig. 12 shows a significant change in the flood distribution when using an additional century of AMAX values, from 1808 instead of 1910. An interesting result is that the inclusion of five historical floods over the 1808–1909 period gives similar results to those obtained with 110 additional AMAX values, both in terms of the maxpost estimate of the flood distribution and the width of the 95 % credibility interval.

#### 7.2.1.2. Consistency between upstream (Basel) and downstream (Maxau) flood distributions.
We now compare Basel and Maxau distributions, using AMAX values from the two last centuries (respectively 1808–2017 and 1815–2018). We expect a higher flood distribution at Maxau (50 343 km²), with a larger catchment area than Basel (35 925 km²). Some small and medium-sized floods have a similar discharge at both stations, when there is no significant flow contribution between Basel and Maxau (for example, when a snowmelt flood propagates from the Upper Rhine, with no precipitation downstream of Basel). But for large floods, e.g. with AMAX(Maxau) > 4300 m³/s (10-year flood), the mean ratio AMAX (Maxau)/AMAX(Basel) is significantly greater than 1 (equal to 1.30).

Fig. 13a shows that the two distributions will intersect beyond the 1000-year return period. This is due to the asymptotic behavior of the two distributions, conditioned by the value of the GEV shape $\xi$ coefficient (Equation 2). A value close to zero ($\xi$ = 0.066) at Basel induces a Gumbel distribution, whereas a significant positive value ($\xi$ = 0.219) at

<caption>**Table 5**</caption>

Prior and posterior estimates of the multiplicative errors on the rating curves with the probabilistic model: raw series (RS) and corrected series (CS) at Basel 1910–2017.

<table><thead><tr><th rowspan="2">Rating curve</th><th rowspan="2">Prior discharge uncertainty</th><th rowspan="2">Prior on γ<sub>i</sub>, with 95% credibility interval</th><th rowspan="2">Series</th><th colspan="3">Posterior estimates</th></tr><tr><th>Max post on γ<sub>i</sub></th><th>95% credibility interval on γ<sub>i</sub></th><th>Correction factor on Q</th></tr></thead><tbody><tr><td>1994–2017</td><td>± 7 %</td><td>1.0[0.95; 1.05]</td><td>RS</td><td>0.97</td><td>[0.94; 1.02]</td><td>-3%</td></tr><tr><td>1977–1993</td><td>±20 %</td><td>1.0[0.85; 1.15]</td><td>CS</td><td>0.97</td><td>[0.94; 1.03]</td><td>-3%</td></tr><tr><td>1934–1976</td><td>±20 %</td><td>1.0[0.85; 1.15]</td><td>RS</td><td>0.98</td><td>[0.88; 1.08]</td><td>-2%</td></tr><tr><td>1910–1933</td><td>±9 %</td><td>1.0[0.93; 1.07]</td><td>CS</td><td>0.99</td><td>[0.90; 1.11]</td><td>-1%</td></tr><tr><td></td><td></td><td></td><td>RS</td><td>1.07</td><td>[0.98; 1.15]</td><td>+7%</td></tr><tr><td></td><td></td><td></td><td>CS</td><td>1.02</td><td>[0.96; 1.12]</td><td>+2%</td></tr><tr><td></td><td></td><td></td><td>RS</td><td>1.05</td><td>[0.97; 1.08]</td><td>+5%</td></tr><tr><td></td><td></td><td></td><td>CS</td><td>1.02</td><td>[0.97; 1.08]</td><td>+2%</td></tr></tbody></table>
<figure><img src="image_12.png" /><figcaption>Fig. 11. GEV distribution and 95% confidence interval of the AMAX flood at Basel (1910–2017) with raw or corrected series.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 12. Comparison of the GEV distribution and its 95% credibility interval at Basel, with AMAX values (1910–2017 and 1808–2017) or a mixed series (AMAX 1910–2017 and five historical floods 1808–1909).</figcaption></figure>

Maxau induces a Weibull distribution with an upper limit.

These results were obtained without making any prior assumption about the set θ of parameters of the GEV distribution. We perform a new computation of the two GEV distributions, assuming a regional prior distribution on ξ, within the interval [-0.1; 0.4]. It corresponds to the range of ξ values at Basel and Maxau. The posterior distributions of ξ are less different (Fig. 13b).

Fig. 14 shows a good consistency from upstream (Basel) to downstream (Maxau). As the return period of the design discharge Q* = 7500 m³/s should be estimated at Iffezheim (46 400 km²), between Basel (35 925 km²) and Maxau (50 343 km²), we interpolate by assuming a power law relationship between flood quantile and catchment area. The design discharge Q* has a very low exceedance probability, less than 10⁻⁴. As the required level of protection of the Rhine’s riverbanks is 0.7
<figure><img src="image_14.png" /><figcaption>**Fig. 13.** AMAX values at Basel and Maxau, since 1808 or 1815: a/ GEV distribution and its 95% credibility interval; b/ ξ parameter, with local distributions (solid lines), regional prior (grey line) and posterior distributions (dashed lines).</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Fig. 14.** GEV distribution at Basel and Maxau, with AMAX values (since 1808 or 1815), using a regional prior on the ξ parameter. Power law interpolation at Iffezheim.</figcaption></figure>

$10^{-3}$ to $10^{-3}$ (France; Germany), Lang (2020) concluded that the present situation is acceptable. The 1000-year flood at Basel (5520 m³/s) is similar to the value estimated by Salinas et al. (2016), using a fuzzy Bayesian approach with historical data since 1256 (5550 m³/s: middle part of the 90 % credibility interval).

## 8. Conclusion

This paper presents a case study on the Upper Rhine, with two long hydrometric series at Basel (1225–2017) and Maxau (1815–2018). Using the available information on rating curves, gaugings, surveying for the Rhine channel and the floodplain, and hydraulic modelling, we summarize our main findings as follows:

* The width of the 95 % credibility interval of Rhine flood discharges is around ±7 % at Basel and ±5 % at Maxau. The relative uncertainty is greater for floods that occurred during the 19th century (± 20 % at Basel and ±30 to 45 % at Maxau). Historical floods at Basel during the period 1225–1807 are more imprecise: ± 41 to 48 % for the periods 1225–1773 and 1774–1807.

* It is important to check the quality of the flood series, as the main hypothesis of the probabilistic models is that the flood sample has independent and identically distributed values. All the previous studies have taken 1977 as the reference condition (the end of channelling works, before modern restoration of flood expansion zones), and flood discharge were corrected with reference to this hydraulic context. The application of statistical tests shows that the raw series are heterogeneous, with the corrected series being of better quality. The Basel and Maxau series show a degree of heterogeneity around 1966–1976, which is not related to metrological issues or engineering works, but rather to climate variability (section 6.2).

* Although the discharge of historical floods is less well known than that of modern floods, the uncertainty analysis shows that it does not affect the quality of the assessment of extreme floods, even with large uncertainties in historical floods (around ±40 to 50 %). As an example, we have similar flood distributions in Basel, using AMAX values for two centuries (period 1808–2017) or a mixed series with AMAX values for one century (period 1910–2017) and the 5 largest floods during the 19th century (period 1808–1909).
* The extrapolation of the flood distribution remains uncertain, as the estimation of the shape parameter ξ of the GEV distribution is very sensitive to the flood sample (and the discharge uncertainties). We found that the local distributions at Basel and Maxau become inconsistent for extreme floods, with a crossover between the two distributions. The assumption of a regional prior on the parameter ξ, corresponding to the range of variation of ξ at Basel and Maxau, gives better results. A stronger hypothesis would be to use the same shape parameter for both stations, but the upstream-downstream deformation of the shape of the flood hydrographs does not guarantee that this is justified.

Finally, we would like to stress the operational interest of such Bayesian approach for Flood Frequency Analysis, including historical data and taking into account discharge uncertainty. Instead of a binary decision, to include or exclude historical data, as followed by the French and German authorities respectively on the Upper Rhine River, we proposed to assess the uncertainty of flood discharges and “weight” these data in the probabilistic model. By taking into account uncertain historical floods prior to the 20th century, it was possible to answer the question of the safety of flood protection levees (Lang, 2020).

Beyond this approach using information on historical floods, simulation approaches with a hydrometeorological chain are able to model the main processes of flood generation and routing. An application by Viviroli et al. (2022) gives the opportunity to estimate extreme floods to the Aare River at Stilli (17 700 km²), one of the main tributary of the High Rhine. It will be interesting to compare future results with such approach over the entire catchment area of the upper Rhine River at Basel. The challenges are to model the concomitance between the Aare

River and the Alpine Rhine, and future climate change on rainfall intensity, atmospheric circulation and snowmelt flood reduction. Bayesian model averaging may be used, with a set of possible simulations (Raftery et al., 2005; Duan et al., 2007; Alinezhad et al., 2021).

### CRediT authorship contribution statement

**Michel Lang:** Writing – review & editing, Writing – original draft, Project administration, Methodology, Funding acquisition, Formal analysis, Data curation. **Jérôme Le Coz:** Writing – review & editing, Methodology. **Benjamin Renard:** Writing – review & editing, Software, Methodology. **Matteo Darienzo:** Writing – review & editing, Software, Methodology.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgments

This work was funded by the French Ministry of Ecology, at the request of DREAL Grand Est, Rhine Center: “Hydrological expertise on Rhine River at Gambsheim and Iffezheim” (Lang, 2020). Anne Landriere (DREAL Grand Est, France), Rüdiger Beiser (WSV, Germany), Michèle Oberhänsli (BAFU, Swiss) and Oliver Wetter (University of Bern, Swiss) are thanked for providing hydro-meteorological data.

## Appendix

### Appendix 1.: Uncertainty of the corrected discharge

If X and Y are two independent variables, we have:

$$E(XY) = E(X) E(Y)$$

$$Var(XY) = Var(X) Var(Y) + [E(Y)]^2 Var(X) + [E(X)]^2 Var(Y)$$

Assuming that the multiplicative correction factor c is independent of the raw discharge Q, the expectation and variance of the corrected discharge

$Q_c = (1 + c) Q$ write:

$$E(Q_c) = E(1 + c) E(Q)$$

$$Var(Q_c) = Var(1 + c) Var(Q) + [E(Q)]^2 Var(1 + c) + [E(1 + c)]^2 Var(Q) = Var(Q) [[E(1 + c)]^2 + Var(c)] + Var(c) [E(Q)]^2$$

Denoting $Var(Q_c) = (u_{Q_c})^2$, $Var(Q) = (u_Q)^2$ and $Var(c) = (u_c)^2$, we have:

$$(u_{Q_c})^2 = (u_Q)^2 [E(1 + c)]^2 + (u_c)^2] + (u_c)^2 [E(Q)]^2$$

$$\left(\frac{u_{Q_c}}{E(Q_c)}\right)^2 = \left(\frac{u_Q}{E(Q)}\right)^2 \left[1 + \left(\frac{u_c}{1 + E(c)}\right)^2\right] + \left(\frac{u_c}{1 + E(c)}\right)^2$$

### Appendix 2.: Auxiliary material

<table><caption>**Table A1**
Changes in the slope and elevation of the Rhine bed in Basel (period 1830–1926). Zero scale $Z_0 = 243.90$ m.</caption><thead><tr><th>Year</th><th>1830</th><th>1852</th><th>1895</th><th>1902</th><th>1926</th></tr></thead><tbody><tr><td>Riverbed slope</td><td>0.8 10<sup>-3</sup></td><td>0.8 10<sup>-3</sup></td><td>1.0 10<sup>-3</sup></td><td>1.1 10<sup>-3</sup></td><td>1.1 10<sup>-3</sup></td></tr><tr><td>Bottom of the riverbed (altitude in m)</td><td>244.5</td><td>244.5</td><td>244</td><td>244</td><td>243</td></tr></tbody></table>
<figure><img src="image_16.png" /><figcaption>Fig. A1. Plan view of the Upper Rhine, Kembs-Iffezheim sector, and location of the Basel and Maxau hydrometric stations (adapted from https://als.m.wikipedia.org/wiki/Datei:Rhine_SluicesandHydropower.jpg and https://www.iksr.org/en/topics/rhine/sub-basins).</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Fig. A2. Gaugings on the Rhine at Basel/Schifflände (float gauging from 1867 to 1908, surface current meters from 1910 to 1933).</figcaption></figure>
<figure><img src="image_18.png" /></figure>

**Fig. A3.** First "water correction works in the Jura" over the period 1868–1891: 1. Diversion of the River Aare into Lake Bienne; 2–3-4. Upstream/downstream lake correction works; 5. Marshland drainage (source: https://www.bvd.be.ch/fr/start/themen/wasser/gewaesserregulierung/juragewaesserkorrektion.html).

<figure><img src="image_19.png" /></figure>

**Fig. A4.** Plan view of the Rhine River in the Maxau section: top in 1817–1819, bottom in 1882. The white star indicates the position of the hydrometric station (source: OberDirektion des Wasser und Strassenbaues, 1885).

<figure><img src="image_20.png" /></figure>

**Fig. A5.** Rhine flood propagation times between Basel and Marlen (5 km upstream of Strasbourg) (HSK, 1978, appendix 5.7).
<figure><img src="image_21.png" /><figcaption>Fig. A6. Cross-section of the main channel of the Rhine at Basel/Rheinhalle in 2011 (from Swiss data manager FOEN) with an equivalent rectangular cross-section, and the approximate heights HX(100), HX(10), HA and HN(10) associated with the 100 and 10-year return period flood discharges, the mean annual discharge, and the ten-year low flow respectively.</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>Fig. A7. Maxau rating curves for the period 1815-1946: BaRatin curves with 95% credibility intervals and official reconstructed curves.</figcaption></figure>
<figure><img src="image_23.png" /><figcaption>Fig. A8. AMAX flood series at Basel (1808-2017) with two sub-periods and the set of values greater than 3872 m³/s.</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>Fig. A9. GEV distribution and 95 % credibility interval of the mixed series at Basel, with AMAX values (1910–2017) and five historical floods (1808–1909). Experimental distribution with the formula of Hirsch (1987).</figcaption></figure>

## Data availability

Data will be made available on request.

## References

Alinezhad, A., Gohari, A., Eslamian, S., Saberi, Z., 2021. A probabilistic Bayesian framework to deal with the uncertainty in hydro-climate projection of Zayandeh-Rud River Basin. Theor. Appl. Climatol. 144 (3–4), 847–860. https://doi.org/10.1007/s00704-021-03575-3.

Barriendos, M., Cœur, D., Lang, M., Llasat, M.C., Naulet, R., Lemaitre, F., Barrera, T., 2003. Stationarity analysis of historical flood series in France and Spain (14th–20th

centuries). Nat. Hazards Earth Sys., 3, 583–592. https://doi.org/10.5194/nhess-3-583-2003.

BAW, 2015. Veränderungen am Pegel Maxau. Bundersanstalat für Wasserbau, 8 June, 16 slides.

Benito, G., Lang, M., Barriendos, M., Llasat, M.C., Francés, F., Ouarda, T., Thorndycraft, V., Enzel, Y., Bardossy, A., Cœur, D., Bobée, B., 2004. Use of systematic, palaeoflood and historical data for the improvement of flood risk estimation. Review of scientific methods. Nat. Hazards 31, 623–643. https://doi.org/10.1023/B:NHAZ.0000024895.48463 eb.

Benito, G., Thorndycraft, V., 2005. Palaeoflood hydrology and its role in applied hydrological sciences. J. Hydrol. 313, 3–15. https://doi.org/10.1051/lhb:2006092.

Blöschl, G., Kiss, A., Viglione, A., et al., 2020. Current European flood-rich period exceptional compared with past 500 years. Nature 583, 560–566. https://doi.org/10.1038/s41586-020-2478-3.
Bösmeyer, A.S., Himmelsbach, I., Seeger, S., 2022. Reliability of flood marks and practical relevance for flood hazard assessment in southwestern Germany. Nat. Hazards Earth Sys. 22, 2963-2979. https://doi.org/10.5194/nhess-22-2963-2022.

Bráždil, R., Demarée, G.R., Deutsch, M., et al., 2010. European floods during the winter 1783/1784: scenarios of an extreme event during the 'Little Ice Age'. Theor. Appl. Climatol., 100, 163-189. https://doi.org/10.1007/s00704-009-0170-5.

Brönnimann, S., Frigerio, L., Schwander, M., Rohrer, M., Stucki, P., Franke, J., 2019. Causes for increased flood frequency in central Europe in the 19th century. Clim. Past 15 (1395-1409), 2019. https://doi.org/10.5194/cp-15-1395-2019.

Brönnimann, S., Stucki, P., Franke, J., Valler, V., Brugnara, Y., Hand, R., Slivinski, L.C., Compo, G.P., Sardeshmukh, P.D., Lang, M., Schaeffi, B., 2022. Influence of warming and atmospheric circulation changes on multidecadal European flood variability. Clim. Past 18, 919-933. https://doi.org/10.5194/cp-18-919-2022.

BWSK, 1930. Die Veränderungen der Höhenlage der Rheinsohle von Basel bis Mannheim in den Jahren 1810 bis 1927. Bad Wasser und Strassenbaudirektion in Karlsruhe, 111p + 9 annexes.

Coles, S., 2001. An introduction to statistical modelling of extreme values. In: Springer Series in Statistics. Springer-Verlag, London, p. 209.

Darienzo, M., Renard, B., Le Coz, J., Lang, M., 2021. Detection of stage-discharge rating shifts using gaugings: A recursive segmentation procedure accounting for observational and model uncertainties. Water Resour. Res., 57, e2020WR028607. https://doi.org/10.1029/2020WR028607.

de Moel, H., van Alphen, J., Aerts, J.C.J.H., 2009. Flood maps in Europe - methods, availability and use. Nat. Hazards Earth Sys., 9, 289-301. https://doi.org/10.5194/nhess-9-289-2009.

Duan, Q., Ajami, N.K., Gao, X., Sorooshian, S., 2007. Multi-model ensemble hydrologic prediction using Bayesian model averaging. Adv Water Resour 30 (5), 1371-1386. https://doi.org/10.1016/j.adwatwres.2006.11.014.

EDF, 2002. Calcul du débit milléral à Fessenheim par la méthode du renouvellement dans le cadre de la réactualisation de la CMS du site (in French). In: Design Flood Estimate by Renewal Method in Fessenheim, p. 31.

Union, E., 2007. Directive 2007/60/EC of the European Parliament and of the council of 23 October 2007 on the assessment and management of flood risks. Off. J. Eur. Union. http://data.europa.eu/eli/dir/2007/60/oi.

Fischer, S., 2024. Incorporating historical flood events in type-based statistics. J. Hydrol, 636. https://doi.org/10.1016/j.jhydrol.2024.131289.

Fischer, S., Schumann, A.H., 2024. Temporal changes in the frequency of flood types and their impact on flood statistics. J. Hydrol. X 100171. https://doi.org/10.1016/j.hydroa.2024.100171.

Frances, F., Salas, J.D., Boes, D.C., 1994. Flood frequency analysis with systematic and historical or paleoflood data based on the two-parameter general extreme value models. Water Resour Res 30 (6), 1653-1664. https://doi.org/10.1029/94WR00154.

Garavaglia, F., Gailhard, J., Paquet, E., Lang, M., Garcon, R., Bernardara, P., 2010. Introducing a rainfall compound distribution model based on weather patterns sub-sampling. Hydrol. Earth Syst. Sci. 14, 951-964. https://doi.org/10.5194/hess-14-951-2010.

GDH, 2017. Charte qualité de l'hydrométrie. Guide de bonnes pratiques. Groupe Doppler Hydrométrie 82.

Hall, J., Arheimer, B., Borga, M., Bráždil, R., Claps, P., Kiss, A., Kjeldsen, T.R., Kriauciunienė, J., Kundzewicz, Z.W., Lang, M., Llasat, M.C., Macdonald, N., McIntyre, N., Mediero, L., Merz, B., Merz, R., Molnar, P., Montanari, A., Neuhold, C., Parajka, J., Perdigão, R.A.P., Plavcová, L., Rogger, M., Salinas, J.L., Sauquet, E., Schär, C., Szolgay, J., Viglione, A., Blöschl, G., 2014. Understanding flood regime changes in Europe: A state of the art assessment. Hydrol. Earth Syst. Sci. 18, 2735-2772. https://doi.org/10.5194/hess-18-2735-2014.

Hirsch, R.M., 1987. Probability plotting position formulas for flood records with historical information. J. Hydrol. 96 (1-4), 185-199. https://doi.org/10.1016/0022-1694(87)90152-1.

HSK (Hochwasser-Studienkommission für den Rhein), 1978. Schlussbericht (Teil 1 Text, Teile 2/3 Anhang).

IWG, 2010. Ermittlung von extremen Abflüssen für die Staustiften Gambèsheim und Iffezheim. Extreme value analysis in Gambèsheim and Iffezheim (in German). Univ. of Karlsruhe 49, p.

IWLP, 1969. Convention Between the French Republic and the Federal Republic of Germany Concerning Development of the Rhine Between Strasbourg/Kehl and Lauterbourg/Neuburweier, https://www.internationalwaterlaw.org/documents/regionaldocs/rhine_river1969.html.

Kjeldsen, T.R., Macdonald, N., Lang, M., Mediero, L., Albuquerque, T., Bogdanowicz, E., Bráždil, R., Castellarin, A., David, V., Fleig, A., Gül, G.O., Kriauciunienė, J., Kohnova, S., Merz, B., Nicholson, O., Roald, L.A., Salinas, J.L., Sarauskiemien, D., Sraj, M., Strupczewski, W., Szolgay, J., Toumazis, A., Vanneuville, W., Veijalainen, N., Wilson, D., 2014. Documentary evidence of past floods in Europe and their utility in flood frequency estimation. J. Hydrol., 517, 963-973. https://doi.org/10.1016/j.jhydrol.2014.06.038.

Kuczera, G., 1999. Comprehensive at-site flood frequency analysis using Monte Carlo Bayesian inference. Water Resour. Res., 35, 1551-1557. https://doi.org/10.1029/1999WR00012.

Lang, M., 2020. Expertise sur l'hydrologie du Rhin sur les biefs de Gambèsheim et Iffezheim. In: Hydrological Expertise on Rhine River at Gambèsheim and Iffezheim. INRAE (bilingual French and German), p. 101. https://hal.science/hal-03295501.

Lang, M., Ouarda, T., Bobée, B., 1999. Towards operational guidelines for over-threshold modelling. J. Hydrol., 225, 103-117. https://doi.org/10.1016/S0022-1694(99)00167-5.

Lang M., Le Coz J., Renard B., Darienzo M., 2021. Design flood assessment on the upper Rhine using historical data and accounting for discharge uncertainty. FLOODrisk

2020 - 4th European Conference on Flood Risk Management, June, 10.3311/FloodRisk2020.3.6, hal-03295467.

Le Coz, J., 2017. Quantifying discharges and fluxes of matters in rivers. Habilitation à diriger des recherches, Univ. Grenoble Alpes, 85 p., https://hal.inrae.fr/tel-02606946.

Le Coz, J., Renard, B., Bonnifait, L., Branger, F., Le Boursicaud, R., 2014. Combining hydraulic knowledge and uncertain gaugings in the estimation of hydrometric rating curves: A Bayesian approach. J. Hydrol., 509, 573-587. https://doi.org/10.1016/j.jhydrol.2013.11.016.

Lucas, M., Renard, B., Le Coz, J., Lang, M., Bard, A., Pierrefeu, G., 2023. Are historical stage records useful to decrease the uncertainty of flood frequency analysis? A 200-year long case study. J. Hydrol., 624, https://doi.org/10.1016/j.jhydrol.2023.129840.

Lucas, M., Lang, M., Renard, B., Le Coz, J., 2024. A comprehensive uncertainty framework for historical flood frequency analysis: a 500-year-long case study. HESS 28, 5031-5047. https://doi.org/10.5194/hess-28-5031-2024.

LUBW, 2019. Bisherige Retentionseinsätze der Rückhaltemaßnahmen am Oberrhein zwischen Basel und Worms, 1p.

Macdonald, N., Kjeldsen, T.R., Prosdocimi, I., Sangster, H., 2014. Reassessing flood frequency for the Sussex Ouse, Lewes: the inclusion of historical flood information since AD 1650. Nat. Hazards Earth Syst. Sci., 14, 2817-2828. https://doi.org/10.5194/nhess-14-2817-2014.

Mansanarez, V., Renard, B., Le Coz, J., Lang, M., Darienzo, M., 2019. Shift happens! Adjusting stage-discharge rating curves to morphological changes at known times. Water Resour. Res., 55, https://doi.org/10.1029/2020WR023389.

Merz, R., Blöschl, G., 2008a. Flood frequency hydrology: 1. Temporal, spatial, and causal expansion of information. Water Resour. Res., 44, W08432. https://doi.org/10.1029/2007WR006744.

Merz, R., Blöschl, G., 2008b. Flood frequency hydrology: 2. Combining Data Evidence. Water Resour. Res., 44, W08433. https://doi.org/10.1029/2007WR006745.

Neppel, L., Renard, B., Lang, M., Ayalr, P.A., Coeur, D., Gaume, E., Jacob, N., Payrastre, O., Pobanz, K., Vinet, F., 2010. Flood frequency analysis using historical data: accounting for random and systematic errors. Hydrolog. Sci. J., 55 (2), 192-208. https://doi.org/10.1080/02626660903546092.

OberDirektion des Wasser und Strassenbaues, 1885. Beitrage zur Hydrographie des Grossherzogthums Baden. Atlas zum dritten Heft, Karlsruhe https://www.deutsche-digital-bibliothek.de/item/HMBOYFCGR15PZNESH7BDRW7EJFVRLFNO.

Parent, E., Bernier, J., 2003. Bayesian POT modeling for historical data. J Hydrol 274 (2003), 95-108. https://doi.org/10.1016/S0022-1694(02)00396-7.

Pettitt, A.N., 1979. A non-parametric approach to the change-point problem. Appl. Stat. 28 (2), 126-135. https://doi.org/10.2307/2346729.

Parkes, B., Demeritt, D., 2016. Defining the hundred-year flood: A Bayesian approach for using historic data to reduce uncertainty in flood frequency estimates. J. Hydrol., 540, 1189-1208. https://doi.org/10.1016/j.jhydrol.2016.07.025.

Payrastre, O., Gaume, E., Andrieu, H., 2011. Usefulness of historical information for flood frequency analyses: Developments based on a case study. Water Resour. Res., 47, W08511. https://doi.org/10.1029/2010WR009812.

Raftery, A.E., Gneiting, T., Balabdaoui, F., Polakowski, M., 2005. Using Bayesian model averaging to calibrate forecast ensembles. Mon Weather Rev 133 (5), 1155-1174. https://doi.org/10.1175/MWR2906.1.

Reitan, T., Petersen-Overleir, A., 2009. Bayesian methods for estimating multi-segment discharge rating curves. Stoch. Env. Res. Risk a., 23 (5), 627-642. https://doi.org/10.1007/s00477-008-0248-0.

Renard, B., 2023. Use of a national flood mark database to estimate flood hazard in the distant past. Hydrolog. Sci. J., 68 (8), 1078-1094. https://doi.org/10.1080/02626667.2023.2212165.

Renard, B., Thyer, M., McInerney, D., Kavetski, D., Leonard, M., Westra, S., 2021. A hidden climate indices modeling framework for multivariable space-time data. Water Resour. Res., 57, e2021WR030007. https://doi.org/10.1029/2021WR030007.

Sabeti, R., Stamatakis, I., Kjeldsen, T.R., 2024. Reconstructing the 1968 River Chew flash flood: merging a HEC-RAS 2D hydraulic modelling approach with historical evidence. Geomat Nat Haz Risk 15 (1), 2377655. https://doi.org/10.1080/1947505.2024.2377655.

Salas, J.D., Obeysekera, J., Vogel, R.M., 2018. Techniques for assessing water infrastructure for nonstationary extreme events: a review. Hydrolog. Sci. J., 63 (3), 325-352. https://doi.org/10.1080/02626667.2018.1426858.

Salinas, J.L., Kiss, A., Viglione, A., Viertl, R., Blöschl, G., 2016. A fuzzy Bayesian approach to flood frequency estimation with imprecise historical information. Water Resour. Res., 52, 6730-6750. https://doi.org/10.1002/2016WR019177.

Schmocker-Fackel, P., Naef, F., 2010a. Changes in flood frequencies in Switzerland since 1500. Hydr. Earth Sys. Sc., 14, 1581-1594. https://hess.copernicus.org/articles/14/1581/2010/.

Schmocker-Fackel, P., Naef, F., 2010b. More frequent flooding? Changes in flood frequency in Switzerland since 1850. J. Hydrol., 381, 1-8. https://doi.org/10.1016/j.jhydrol.2009.09.022.

Stedinger, J.R., Cohn, T.A., 1986. Flood frequency analysis with historical and paleoflood information. Water Resour. Res., 22 (5), 785-793. https://doi.org/10.1029/WR022i005p00785.

Thyer, M., Renard, B., Kavetski, D., Kuczera, G., Franks, S.W., Srikanthan, S.W., 2009. Critical evaluation of parameter consistency and predictive uncertainty in hydrological modelling: a case study using Bayesian total error analysis. Water Resour. Res., 45, W00B14. https://doi.org/10.1029/2008WR006825.

Viviroli, D., Sikorska-Senoner, A.E., Evin, G., Staudinger, M., Kauzlaric, M., Chardon, J., Favre, A.-C., Hingray, B., Nicolet, G., Raynaud, D., Seibert, J., Weingartner, R., Wheatal, C., 2022. Comprehensive space-time hydrometeorological simulations for
estimating very rare floods at multiple sites in a large river basin. Nat. Hazards Earth Sys., 22, 2891–2920. https://doi.org/10.5194/nhess-22-2891-2022.

Westerberg, I.K., Karlsen, R.H., 2024. Sharing perceptual models of uncertainty: On the use of soft information about discharge data. Hydrol Process 38, e15145. https://doi.org/10.1002/hyp.15145.

Wetter, O., Pfister, C., Weingartner, R., Luterbacher, J., Reist, T., Trösch, J., 2011. The largest floods in the High Rhine basin since 1268 assessed from documentary and

instrumental evidence. Hydrolog. Sci. J., 56 (5), 733–758. https://doi.org/10.1080/02626667.2011.583613.

Wilhelm, B., Rapuc, W., Amann, B., Anselmetti, F.S., Arnaud, F., Blanchet, J., Brauer, A., Czymzik, M., Giguet-Covex, C., Gilli, A., Glur, L., Grosjean, M., Irmler, R., Nicolle, M., Sabatier, P., Swierczynski, T., Wirth, S.B., 2022. Impact of warmer climate periods on flood hazard in the European Alps. Nat. Geosci 15, 118–123. https://doi.org/10.1038/s41561-021-00878-y.

WSV, 1975. Pegelstammbuch Maxau. p. 20.