Flood frequency analysis on the Ardèche river using French documentary sources from the last two centuries

Journal of Hydrology 313 (2005) 58–78 www.elsevier.com/locate/jhydrol

Flood frequency analysis on the Arde`che river using French documentary sources from the last two centuries Robin Nauleta, Michel Langa,*, Taha B.M.J. Ouardab, Denis Coeurc, Bernard Bobe´eb, Alain Reckinga, David Moussaya a

Cemagref, Unite´ de Recherche en Hydrologie-Hydraulique, 3bis quai Chauveau, 69336 Lyon Cedex 09, France Chair in Statistical Hydrology, INRS-Ete, University of Quebec, 2800 Einstein, C.P. 7500, Sainte-Foy, Que., Canada G1V 4C7 c Acthys Diffusion, 36 rue Bizanet, 38 000 Grenoble, France

b

Received 14 August 2003; revised 9 February 2005; accepted 11 February 2005

Abstract Fitting statistical laws from a short time series does not give any guarantee of reliability on extreme flood estimation. Historical investigation through documentary sources can enlarge the record period. This paper presents a case study on the Arde`che river, based on collaboration between historians, archivists, hydraulic engineers and hydrologists for a better assessment of the flood risk. A list of historical flood levels from 1644 to the present has been drawn up and converted into discharge using hydraulic modelling. A sensitivity analysis provides error intervals on discharge estimates taking into account uncertainties on water level, roughness coefficient and channel geometry, and the impact of a nonpermanent discharge or the backward effect. A flood frequency analysis using systematic and nonsystematic data gives a practical example on how historical information can improve flood knowledge. It reduces the sampling uncertainty and shows also a good agreement with a hydrometeorological approach (Gradex model). q 2005 Elsevier B.V. All rights reserved. Keywords: Flood frequency analysis; Historical information; Arde`che river; Gradex; Discharge estimate; Error analysis

1. Introduction 1.1. Use of historical data for frequency analysis Flood risk mitigation is usually based on design floods, which have a large magnitude when the potential damage is great. But extreme flood estimation is by definition a difficult task as it usually concerns * Corresponding author. E-mail address: [email protected] (M. Lang).

0022-1694/$ - see front matter q 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jhydrol.2005.02.011

a limited number of such events during a human lifetime. Historical analysis can widen the information on the major flood events, which already occurred in the past. Some difficulties are related to its use, mainly the availability and exhaustiveness of historical information, the quality and accuracy of historical accounts, climatic variability and the morphological changes of the rivers over several centuries. Collaboration between historians, archivists and hydrologists can provide a significant improvement in the field of flood hazard studies (Naulet et al., 2001).

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Historians are able to produce an exhaustive reconstitution of accounts on large to catastrophic floods, using specific techniques for archive investigation (calendar adjustments, conversion of units of measure, palaeography, etc.) and a good knowledge of the cultural and socio-economic context in past centuries. The final quality of a testimony is then assessed through a long critical process, taking into account the intrinsic quality of each document such as type, author, date and circumstances of writing. One of the main products of the historical investigation is the General State of Sources, which provides an exhaustive document. Hydrologists are also able to define a framework which will direct the inventory made by historians and its use for hydrological purposes. Three chronological accounts will be investigated: (a) those of local authorities and municipalities who are in charge of land planning or produce technical data (topography, hydrology, etc.); (b) scientific and technical methodologies and measuring instruments used to analyse natural phenomena; (c) catchment events such as floods but also major engineering works in the basin and along the river. These information sources constitute the background of the analytical enquiry.

before joining the Rhoˆne at an altitude of 37 m (Fig. 2). Its two main tributaries, the Chassezac and the Beaume rivers, converge at nearly the same point on its course. These three sub-basins are separated by mountains (1200–1600 m altitude), with steep slopes (900 m in a few kilometres). The influence of the Mediterranean sea produces heavy rainfall each autumn when warm and wet air rises along the Cevennes hills (1500 m). Seven rainfall events exceeded a 400 mm threshold during the 1961–1996 period, amongst 66 rainfall events with more than 190 mm (Deblae`re and Fabry, 1998). Such a meteorological pattern on the upper part of the catchment, with steep slopes and granite and basalt formations, results in flash floods, where flows increase to a strong magnitude in a few hours. The average 10 year flood discharge estimate is about 3300 m3/s at the final confluence of the Arde`che river (2372 km2) with the Rhoˆne river. Amongst a list of large floods in France by Duband (1994), the specific discharge of the Arde`che twice exceeded a 2 m3/s km2 threshold at Vallon (1930 km2), during the 8 Nov. 1982 and 30 Sept. 1958 flood events (3660 and 4550 m3/s).

1.2. The Arde`che catchment

This paper presents a flood frequency analysis at two places, just before and after the Arde`che canyon, at Vallon and Saint-Martin-d’Arde`che (St Martin, 2240 km2): first documentary sources provide a list of historical flood levels from 1644 to the present (Section 2), then a hydraulic modelling with a sensitivity analysis is used to convert historical water levels into discharge with error intervals (Section 3), and finally a flood frequency analysis using systematic or nonsystematic data gives a practical example of the potential of historical data for extreme flood assessment (Section 4).

The Arde`che catchment is an appropriate case study for the historical reconstitution of flood events: first a large set of documentary sources is available from the Flood Warning Office (FWO) from the two last centuries, second the basin remains preserved from major hydraulic engineering structures which may have changed the flood distribution over time, and third the river flows through a canyon where the bedrock provides the most suitable settings for reconstructing historical discharges. The Arde`che is a tributary of the Rhoˆne river (France), located in the southeast of the Massif Central (Fig. 1). Its upper part (1420 m) flows from west to east in a deep granite valley with a slope of about 10 m/km down to Thueyts. The Arde`che then crosses basalt formations and then schist in a north– south direction with a slope of a few metres per kilometre. Finally, it returns to its initial orientation from Vallon–Pont-d’Arc (Vallon) and runs with a low gradient, through a canyon through bedded limestone,

1.3. Outline of the paper

2. Historical water level data series at Vallon and St Martin 2.1. Available data and history of data sources 2.1.1. Surveying Cassini’s map is the first general topographic cover of France produced in the 18th century, but it does not

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Fig. 1. Catchment area of the Arde`che river (hydrometric stations over period 1857–2003).

Fig. 2. Longitudinal profile of Arde`che and its two main tributaries.

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include any altitudes. From a practical point of view, the first maps with absolute values are found in first French levelling survey dating from the 1860s. Before, altitudes were expressed as a relative value compared to a bench mark. A comprehensive chronological account of the successive topographic surveys (Bourdaloue 1857–1864, Lallement 1884–1892, IGN69 1962–1969) is presented in Landon (1999), with tables allowing conversion from one reference frame to another, depending on the area in question. Specific information on the Arde`che area has been available from the present French Ordnance Survey (personal communication from P. Bonnetain about the altimetric system of the Rhoˆne area before 1857). 2.1.2. Water level authorities The extreme flood on 22 Sept. 1890 was the largest in the Arde`che for several centuries. This is a turning date which corresponds to the creation of the Arde`che FWO, with systematic manual records from 1892 to today. Some previous nonsystematic information is available for the 19th century from water level scales installed from 1857 (Road and Bridge Office of the Arde`che, RBO) or within hydraulic reports with sometimes references to the levels reached by the largest floods (Fig. 3). The first flood discharge

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measurements at Vallon and St Martin were carried out from Oct. 1897 to Nov. 1900, with also a low flow gauging on 5 March 1857. Flood gaugings are then available for short time periods (1921–1926, 1954– 1964) till systematic measurements started in the 1980s. 2.1.3. Meteorological measurements After some quantitative information from local naturalists (episodes between 1805–1830 and 1853– 1859), the first official network of rain gauges was installed by Marchegay (1861) following the great flood of 1857. Observations became more numerous from 1872 (RBO) and then 1892 (FWO). During the 20th century the network was extended from about 20 to 50 rain gauges. A documentary investigation by Me´te´o-France (1995) provided detailed information on the episodes of heavy rain in the Arde`che in the 19th and 20th centuries. Cemagref (2001) have produced a preliminary website with an inventory of rainfall and flood events from 1901 to 1950. 2.2. Review of historical levels at Vallon and St Martin Hydrometric data at Vallon and St Martin are available for three measuring periods, with nonsystematic records only during the larger historical floods (1644–1891 and 1827–1891), systematic manual records by the FWO (1892–1960 and 1892–1954) and automatic records (from 1961 and 1955).

Fig. 3. Cross section at the natural arc of Pont d’Arc with water levels of the Oct. 9–10, 1827 and Sept. 28–29, 1846 floods (Combier, 1857).

2.2.1. Nonsystematic period From a general point of view, historical information at Vallon is more complete than at St Martin, as during the 19th century a permanent officer was in charge of the maintenance of the Salavas bridge (between Salavas and Vallon villages, where the Vallon staff gauge is located) and flood observations. Previous information is available at Salavas mill, 430 m upstream (e.g. 1644, 1772 floods). The data series at St Martin is less detailed as it begins in 1827, and afterwards some floods observed at Vallon were not recorded. Table 1 gives the water level above low flow of the main historical floods (annual maximum values). Using the fact that no major tributaries alter the peak discharge between Vallon and St Martin,

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Table 1 Main historical flood levels above low flow at Vallon and St Martin (Arde`che river) Date

Water level above low flow at Vallon (m)

Date

Water level above low flow at St Martin (m)

Deaths

03/09/1644 09/09/1772 09/07/1826 10/10/1827 at 0.30 a.m. 28/10/1840 29/09/1846 at 0.00 a.m. – 10/09/1857 at 7.30 p.m. 15/10/1859 at 0.00 a.m. 15/07/1861 at 10.00 p.m. 03/10/1872 22/10/1878 at 0.00 a.m. 31/12/1888 22/09/1890 at 12.00 a.m. 21/10/1891 at 7.00 a.m.l

16a 14a !12.5b 16.1a 8–10d 13.1a – 13.52f 11.27 g 9h 10i 14.5j 9.4 k 17.3j 11.1j

– – – 10/10/1827 – 29/09/1846 at 0.30 a.m. 29/10/1853 at 1.00 p.m. 10/09/1857 at 10.30 p.m. 15/10/1859 at 0.00 a.m. – – – – 22/09/1890 at 3.00 p.m.l 21/10/1891 at 10.40 a. m.l

– – – 8.9c – 7.75c or 7.97e 6.25e 6.6f 6.2 g – – – – 8.45j 6.7j

– – – 30 – 2 –

50 12 – – 5 – 35 –

a

From Combier (1849, 1857) and de Mardigny (1860): historical levels at Salavas mill (430 m upstream) for the 1644, 1772, 1827 and 1846 floods (16.88, 14.7, 17 and 13.55 m); Water level is available both at Salavas and Vallon for the 1644 and 1772 floods. b From Combier (1857). c From Combier (1857) at usine Castagne. d From Parde´ (1936). e From Perret (1857). f From a limnigraph in Morlie`re (1857a,b), which has been preferred to a table of levels in Marchegay (1861) and Gros (in Lemoine, 1896) which gives lower values at St Martin (resp. 6.5 m at 10.20 p.m., and 6.06 m). g From Ponts et Chausse´es (1859). h From Julien (1861). i From Me´te´o-France (1995). j From Lemoine (1896); Delemer (1904). k From Vaschalde (1890). l From Parde´ (1925a), handwritten data from a FWO telegram.

Delemer (1904) proposed a relation between the water levels at the two staff gauges. Fig. 4 presents an updating of this relation (using historical levels up to 1964), which has been used to fill some gaps in the St Martin series. During the 19th century historical levels were measured in relation to water depth above low flow. These have been related to the level at the staff gauges by using the following information. In 1921, a topographic survey of the whole longitudinal profile of the Arde`che river (Morel, 1921) gave the first accurate information on the zero altitude of the two scales, which remains valid from 1890 to now. The cross-checking of several documentary sources allowed us to convert these values into altitude above sea level, taking into account old cross sections with the low flow level and historical flood levels

(e.g. Fig. 3) and historical marks on buildings which have been surveyed. Table 2 gives the reviewed list of flood levels from 1644 to 1892. Taking into account an G0.70 m error on the flood level at St Martin (Fig. 4), the rankings at both sites are in agreement (e.g. the 1846 flood, respectively, in 5th and 3rd position during the 1826– 1891 period). It will be considered that the five largest floods at Vallon (1890, 1827, 1878, 1857 and 1846) are an exhaustive list of those that occurred between 1826 and 1891. This corresponds to a threshold of perception of 12.5 m at Vallon and 6.95 m at St Martin (using Fig. 4). 2.2.2. The systematic period All data from 1892 to 1964 were entered in a computerised database during the Historisk–Arde`che

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Fig. 4. Relation between the water levels at Vallon and St Martin.

project (Cœur et al., 2001). A review of the annual minimum levels shows that the scale at Vallon remained stable (Fig. 5a) from 1875 to 1964, while the data series of St Martin has two homogeneous

sub-periods, with a 1 m lowering since 1941 (Fig. 5b). The explanation is provided by correspondence between the priest of St Martin and the geographer Maurice Parde´ (N, 1959), which points out that a weir

Table 2 Reviewed list of 14 historical flood levels at Vallon and St Martin (Arde`che river) Date

22/09/1890 10/10/1827 03/09/1644 21/10/1878 09/09/1772 10/09/1857 28/09/1846 a

Vallon low flow at 78.20 m NGF

St Martin low flow at 45.70 m NGF

Flood level (m)

Rank during 1826–1891

Flood level (m)

Rank during 1826–1891

17.3 16.1 16.0 14.5 14.0 13.5 13.1

1st 2nd

8.45 8.35c 8.15b 7.70b 7.50b 6.6e 7.75f

1st 2nd

3rd 4th 5th

4th 6th 3rd

Date

09/07/1826 15/10/1859 21/10/1891 29/10/1853 03/10/1872 31/12/1888 15/07/1861

Vallon low flow at 78.20 m NGF

St Martin low flow at 45.70 m NGF

Flood level (m)

Flood level (m)

!12.5a 11.3 11.1 10.8d 10.0 9.4 9.0

!6.95b 6.20 6.70 6.25 5.90b 5.6b 5.40b

Perception threshold that insures exhaustiveness during the 1826–1891 period. No data at St Martin, estimated value from Vallon (using Fig. 4). c 8.9 m by Combier (1857). Parde´ in N (1959) proposed a review, taking into account the observed maximum levels at St Martin church stairs during the 1890 and 1827 floods, the latter being 10–18 cm below the former. d No data at Vallon, estimated value from St Martin. e Data from Morliere (1857a,b) has been preferred to Marchegay (1861) and Gros (in Lemoine, 1896): limnigraph instead of a table of levels. f Data from Combier (1849) has been preferred to Perret (1857): more compatible with the other flood marks. b

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0

0

Water elevation (m)

(b) 0.5

Water elevation (m)

(a) 0.5

-0.5

-1

-0.5

-1

-1.5

-1.5

Disappearingof the mill weir of St Martin. Height about 1m , 200 m downstream limnimetric scale.

-2

-2 1875

1895

1915

1935

1955

Dates (years)

1900

1920

1940

1960

Dates (years)

Fig. 5. Annual minimum levels: (a) at Vallon (1875–1964); (b) at St Martin (1892–1964).

(1–1.5 m high) existed in the 19th century, 200 m downstream the St Martin scale. This weir which is present in the 1921 topographic survey disappeared in 1941. The basic data from FWO are monthly sheets with daily levels, and more detailed values when the water exceeded some fixed thresholds (e.g. at Vallon: three records per day if HO4 m; one record per 2 h if HO 7 m; one record per hour if HO10 m). Determination of annual maxima from an incomplete year (with monthly sheet missing) has been possible when a copy has been found in other documentary sources (such as annual sheets from FWO, technical reports from RBO or the Maurice Parde´ archives) or when no flood occurred during the missing periods (cross-checking with rainfall data, and Vallon or St Martin water level data). Six annual maximum levels at St Martin have been estimated from the Vallon level (using Fig. 4): 1924, 1927, 1930, 1933, 1949 and 1959. Finally, all annual maximum levels above a 8 m level at Vallon (5 m at St Martin) are available from 1892 to 1964.

(12.2 m), when only two values were recorded in the monthly sheets: 30 Sept. 7 a.m. (K0.1 m) and 1 Oct. 7 a.m. (5.5 m). Even if such failures seldom occurred it was decided to use a threshold of perception for the systematic period (1892–1954 or 1960), based on the flooding of St Martin and Vallon villages. According to N. (1959) when the square in front of the church is flooded this corresponds to a 5 m level at St Martin. After such events, if the observer failed to record the maximum, a field survey by FWO gave an indication of the maximum level at St Martin and Vallon. The latter threshold has been estimated as the 8 m level at Vallon using the correspondence between both sites (Fig. 4). This level is also related to the flooding of the major bed at Vallon.

2.2.3. The automatic period A comparison between the monthly sheets from FWO and the automatic records shows that some maximum levels have been missed by the observer. As an example, the limnigraph at Vallon recorded the peak level of the 30 Sept. 1958 flood at 7.30 p.m.

3.1.1. Model calibration As explained in Section 2.2, the Vallon gauging station (1892–2003) is located at the Salavas bridge, between the Salavas and Vallon villages, where almost all the historical information is available, from 1857 to 1890. The Salavas mill is located 430 m

3. Historical discharge series at Vallon and St Martin 3.1. Hydraulic modelling at Vallon

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Fig. 6. 3D view of cross sections used within the Vallon hydraulic model.

upstream, with historical marks. Flows at Vallon are affected by a hydraulic constriction at Pont d’Arc, 4 km downstream (Fig. 6). This is a natural arch, where the width of the section is only 58 m (Fig. 3) instead of 200 m upstream and 250 m downstream. The arch can be bypassed by an abandoned meander, which becomes active only during extreme floods. Two hydraulic models will be used, taking into account topographic surveys from 1990 to 1849. 3.1.1.1. Present situation. The hydraulic model MAGE (Giraud et al., 1997), with about 40 cross sections along 10 km, has been fitted with uniform conditions using the observed Surface Water Level (SWL) profile of the 1992 flood, which has a 5-year return period. This flood has been gauged at 2800 m3/s and a detailed topographic survey is available (1990). Optimization of the roughness coefficient provided the following Manning vector n (1/25, 1/10) for the main channel and the flood plain. As the narrowing of the geometry by the arch leads to a specific flood

behaviour, like a basin emptying through a natural orifice, the natural arch has been considered as an orifice, with a specific discharge–energy function: pffiffiffiffiffiffiffiffiffiffiffiffi Q Z mS 2gDH (1) where Q is the discharge (m3/s), m the discharge coefficient, S the hydraulic section (m2), g the gravitational acceleration (m2/s), DH the head loss through the arch (m). The value mZ0.25 provided a good fit with the observed loss of energy through the arch during the 1992 flood. 3.1.1.2. 19th Century. During the 19th century, RBO (Hydraulic Department) produced several hydraulic studies, such as scientific accounts of catastrophic floods or reports on the hydraulic impact of new concessions on the river (e g. mills). A technical report from Combier (1857) provides information on the topographic survey of the bed river (seven cross sections in 1849), SWL longitudinal profile during the low flow in 1849 (with a gauged discharge)

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Fig. 7. Bed river topography with 1849 and 1990 surveys at Vallon.

and historical flood marks. The comparison between the survey from 1849 to 1990 shows evidence that the upper part of the study area remained stable, at Vallon gauge station, while the bed river rose by 2 m at Pont d’Arc. Longitudinal profiles during the 1921 drought (Morel, 1921) show that the 1990 topographic survey remains valid during the 20th century. Then a second hydraulic model was built, using the same roughness coefficients as today and modified 1990 cross sections, taking into account both the 2 m raising at Pont d’Arc and the seven cross sections (Fig. 7). A check was made that the model fits well with the observed SWL longitudinal profile at the 1849 low flow. Comparison between observed and computed SWL longitudinal profiles. Observed SWL longitudinal profiles are available for six historical floods (1827, 1846, 1855, 1857, 1859, 1890) and the 1958 and 1992 floods. A hydraulic error 3 has been defined, as follows: 3Z

N 100 X f½ðhobs Þi K ðhcomp Þi =ðhobs Þi g2 N iZ1

(2)

where N is the number of flood marks, and (hobs)i and (hcomp)i are the observed and computed water depth of mark no i. The hydraulic error 3 is 2.3% for the 1992 flood which has been used for calibration, 6.7% for the 1958 flood, and is in the range of 2.3–8.7% for the six

historical floods (2.7; 5.1; 2.3; 5.5; 8.7; 4.0, respectively). Fig. 8 shows the computed SWL longitudinal profiles with the estimated 19th century topography. 3.1.3. Sensitivity analysis on discharge estimation 3.1.3.1. Downstream condition: backward effect. The downstream condition which will be used for this hydraulic study is a relation Q(H) between discharge and water depth, 700 m downstream from Pont d’Arc at PK47950. As Sogreah (1994, 1995) already investigated the Arde`che floods along the whole canyon, we selected the computed water levels at PK57230 (10 km downstream of Pont d’Arc) ZZ71.3 and 74.5 m, for two discharges: QZ2450 and 3850 m3/s (TZ3 and 20 years). The backward effect of a G1 m error on this downstream condition is effective up to PK49000 and PK44710 (QZ2450 and 3850 m3/s), which are located downstream of the Salavas bridge (PK43405). The backward effect is negligible at Vallon gauging station, with a hydraulic error 3 of less than 1%. 3.1.3.2. Upstream condition: nonpermanent effect. Hydraulic computation has been developed using both hypotheses: permanent conditions with various peak discharges (QZ2500, 4500, 6500, 8500 m 3 /s)

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Fig. 8. Longitudinal profiles of six historical floods during the 19th century.

and nonpermanent conditions with triangular flood hydrographs (time to peak: 6 h). As the narrowing through Pont d’Arc leads to a large storage, the transient hydraulic computation gives a SWL longitudinal profile which is 1.5 m lower than with the permanent discharge computation. For extreme floods (QZ8500 m3/s), the permanent computation leads to a 10% hydraulic error 3. The effect on discharge estimate is less great, about 2% (Fig. 9). 3.1.3.3. Roughness calibration. The impact of roughness calibration on the discharge estimate has been assessed using the following procedure: after adding an error Dn (G1/5; G1/5) to the Manning vector, an iterative search gives the discharge Q* which minimizes the hydraulic error 3. Table 3 shows that the relative error on discharge estimate DQ/Q is about 20% for medium floods (QZ2700 m3/s) and 10% for large floods (QZ5500 m3/s), the latter having large water depths which reduce the effect of roughness calibration errors. These results are similar to those from Wohl (1998), who indicates that varying n by G25% produces a maximum change of 20% in discharge. 3.1.3.4. Geometry variations and discharge coefficient at Pont d’Arc. As mentioned in Section 3.1.1, the river bed near Pont d’Arc seems to have significantly risen

in the last two centuries, with a 2 m difference between 1849 and 1990. Combination of the hydrodynamic MAGE computation of flow velocities with a solid transport formulae (such as Meyer–Peter) provides evidence that during large floods a strong degradation of the bed river under the arch may occur (Moussay, 2002). As a first approximation, the sand layer above the bed rock will be assumed to have a thickness of 4 m under the arch (Combier, 1856). Table 4 shows that the effect of bed river variations (K4; C2 m) under the arch is about 7% for medium floods (QZ2700 m3/s) and 10% for extreme floods (QZ6900 m3/s), the discharge being more sensitive to the hydraulic section when the water level rises. As explained in Section 3.1.1, the flow modelling through the arch is based on a specific discharge– energy function (see Eq. (1)). It is a kind of black box approach, which merges the head loss produced by the narrowing of the section, and two-dimensional circulations upstream of the arch. The discharge coefficient (mZ0.25) has been fitted with information from the 1992 flood, which is a medium event. For extreme floods the question of which coefficient should be used remains open. Table 4 gives the effect of a G20% variation of this coefficient (0.25G0.05), with a relative error of 4% for medium floods and 10% for extreme floods.

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Fig. 9. Impact of transient hydraulic computation on the rating curve of Vallon.

3.1.3.5. Overall uncertainty. The combination of the previous relative uncertainties will be developed with the most restrictive hypotheses (i.e. overestimation of the rating curve): less severe Manning coefficient (1/35; 1/15), larger hydraulic conveyance through the arch (mZ0.30; 4 m lowering of the bed river during floods), triangular hydrograph (instead of a permanent discharge), and lowering of the downstream condition (K0.20 m). The overall relative error on the discharge estimate DQ/Q is about 30% for a medium flood (Q*Z3900 instead of 2900 m3/s), and 40% for a large flood (Q*Z7600 instead of 5500 m3/s).

floods. Historical testimonies and documents report that the arch was bypassed during the 1890 flood, through the old meander (Combe d’Arc, 1200 m long) on the left bank, whose altitude is 17 m higher than the bed river. As the MAGE hydraulic model is able to solve the set of differential equations that may be used to describe a looped channel network, it was possible to simulate extreme floods at Pont d’Arc. The natural arch and the abandoned meander are two reaches which are connected by two junction nodes. After a specific survey of five cross sections in Combe

3.1.3.6. Diversion through the meander. An additional model has been developed for extreme

Table 4 Relative error on Vallon discharge estimate versus Pont d’Arc parameters (arch geometry, discharge coefficient)

Table 3 Relative error on Vallon discharge estimate versus roughness coefficients Roughness coefficient n (1/20, 1/5) n (1/30, 1/15)

QZ2700 m3/s

QZ5500 m3/s

Q*

DQ/Q (%)

Q*

DQ/Q (%)

2250 3000

K17 C11

5000 5750

K9 C5

Pont d’Arc

Bed river altitude under the arch Discharge coefficient

Parameters

QZ2700 m3/s

QZ6900 m3/s

Q*

DQ/Q (%)

Q*

DQ/Q (%)

ZK4 m

2900

C7

7600

C10

ZC2 m mZ0.20 mZ0.30

2500 2600 2775

K7 K4 C3

6300 6350 7700

K9 K10 C10

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Fig. 10. Longitudinal profile of the Arde`che river at St Martin.

d’Arc (April, 2002), a sensitivity analysis by Moussay (2002) provided the threshold discharge (m3/s) above which the arch can be bypassed. This varies in the range of 5700–6500 m3/s with the present river bed configuration and 6500–7500 m3/s in the 19th century. Therefore only two floods (1827 and 1890) were able to flow through Combe d’Arc in the last two centuries. The old meander takes only a minor part of the overall discharge of such floods (less than 6%), with an average velocity in the range of 1–1.5 m/s. 3.2. Hydraulic modelling at St Martin 3.2.1. Model calibration Historical and FWO data are available at St Martin bridge from 1827 to 1964, as a gauging station was installed (limnigraph and cable way) in 1955 at Sauze (2.3 km upstream). The two sites are located at the exit of the Arde`che canyon on bedrock, and about 10 km upstream from the confluence with the Rhoˆne river. The MAGE hydraulic model was routed along a 12 km length with about 70 cross sections. The fitting of the roughness coefficient produced the following Manning vector n (1/30, 1/10 and 1/15) for the main channel and the flood plain (forest and crops) with

a 5% hydraulic error 3 between the computed values and the 1992 flood levels (Fig. 10). 3.2.2. Relative error on discharge estimate A sensitivity analysis by Cœur et al. (2001) provided a ranking of the different errors for the hydraulic modelling at St Martin: (1) the backward effect from the Rhoˆne level is less than 1% when using a 1 m error on the 1890 flood downstream condition at the Rhoˆne confluence (ZZ44 m). Furthermore, historical data shows that the two rivers never have simultaneous floods. (2) The errors due to the river topographic survey and variations in river bed produce a small error of less than 1%, adding G 0.5–2.0 m error on cross sections. (3) The nonpermanent effect is restricted to the confluence area (about 8%) and does not exceed 4% at St Martin bridge. (4) The impact of an error Dn (G1/5; G1/5) in the Manning vector leads to a 35% relative error on the discharge estimate. As no SWL longitudinal profiles are available for historical data a G40% overall relative error (DQ/Q)1 will be assumed for large floods, when no gauging is available (one meter above the maximum gauged discharge, i.e. about 4000 m3/s). Discharge error on small to intermediate floods is reduced (25% before 1954, 10% from 1955)

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Fig. 11. Rating curve at St Martin. (a) Before 1941; (b) till 1941.

by gauging information. A relative error DHZG 0.70 m on historical floods should be added when using historical levels before 1890 (see Section 2.2.1: agreement between Vallon and St Martin rankings) or when transferring levels from Vallon to St Martin (using Fig. 4). The combination of both errors with a first error calculation gives an overall error (DQ/Q)2 of between 60 and 80% for the 14 largest floods at St Martin: Q Z aðH K H0 Þb and ðDQ=QÞ2 Z Da=a C bDH=ðH K H0 Þ

exists along the Arde`che canyon. The small catchments between both sites do not have a significant runoff, as the orographic effect is limited to the upper part of the Arde`che basin. Therefore, the small

(3)

ðDa=a Z ðDQ=QÞ1 et Db=b z0Þ The final uncertainty on the discharge estimate at St Martin is a function of the date, the water level and the available data (observed or calculated maximum level): G10–40, 25–40 and 60–80%. Figs. 11 and 12 give the rating curves at St Martin bridge and Sauze, with the gauged values and the hydraulic extrapolation. 3.3. Discharge data series from 1644 to 2001 Comparison between discharge estimate at Vallon and St Martin is straightforward as no major tributary

Fig. 12. Rating curve at Sauze.

R. Naulet et al. / Journal of Hydrology 313 (2005) 58–78

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Fig. 13. Correspondence between flood discharges at Vallon and St Martin or Sauze.

tributaries do not greatly affect the Arde`che floods, as they have low rainfall amounts and flood peaks which do not coincide in time with the Arde`che maximum values. A linear regression between discharge estimates Qv at Vallon (AvZ1930 km2) and Qss at St Martin or Sauze (AssZ2240 km2) explains 96% of the variance (Fig. 13): Qss Z Qv ðAss =Av Þa with a Z 0:50

(4)

The area exponent a is not very similar to the standard regressions used in France for the 10-year peak discharge estimate (aZ0.8 in Ctgref, 1980), but the physical explanation is difficult as the flood hydrograph at Vallon is first reduced by the Pont d’Arc configuration and then increased along the Arde`che canyon. A comparison between the flood volumes Vv at Vallon and Vss at St Martin or Sauze shows that the ratio (VvKVvss/Vv) is in the range [K12; C24%], with an median value equal to C10%. The Arde`che flood discharge estimates by Parde´ (1925b, 1942, 1953) and the present hydraulic studies are very close (Table 5). Figs. 14 and 15 give the final discharge data series which will be used for flood frequency analysis. Different scenarios have been considered: (1) during the 1645–1771 period no event exceeded the 1644 flood (hypothesis 1) or exceeded the 1772 and 1644 floods (hypothesis 2). (2) During

the 1773–1826 period no event exceeded the 1772 flood. (3) During the 1828–1891 period the exhaustiveness criteria is defined by a non-exceedance level (Vallon: 12.5 m, i.e. 4750 m3/s; St Martin 6.95 m, i.e. 5050 m3/s. (4) During the 1892–1980 period the quality criteria is defined by the awareness of the observer (Vallon: 8 m, i.e. 2250 m3/s; St Martin 5 m, i.e. 2900 m3/s this being decreased to 2400 m3/s in order to have the same number of events). All annual maximum values from 1980 are available with confident values. Table 5 Comparison between Arde`che flood discharge estimates by Parde´ and hydraulic modelling Date

10/10/1827 22/09/1890 28/09/1900 16/10/1907 08/10/1933 10/11/1951 30/09/1958

Vallon Parde´

Hydraulic model

6800 7500 5600 4300 3700 3600 !4600

6900 7550 5800 4450 3500 3500 4750

St Martin hydraulic model 7400 7550 5750 4800 3700a 3700 4800

a Water level is available at Vallon and has been transferred using Fig. 4.

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Fig. 14. Discharge data series at Vallon from 1644 to 2001.

4. Flood frequency analysis using systematic and nonsystematic data 4.1. Stationary analysis with homogeneity tests at St Martin The great majority of statistical tests on extreme values deal with annual maximum or minimum series. These kind of tests cannot be used with historical

series, where the requirement of continuity within the data records is not fulfilled year by year. A good alternative is to use peak over threshold series, which contains all the events larger than a threshold. Lang et al. (1999) provided some guidelines in order to choose an appropriate threshold. This includes a stationary test based on the computation of the tolerance interval of the number mt of floods within an interval [0; t]. The null hypothesis H0 is to assume

Fig. 15. Discharge data series at St Martin from 1644 to 2001.

R. Naulet et al. / Journal of Hydrology 313 (2005) 58–78

that the flood process can be described by a homogeneous Poisson process, such as: Prob½mt Z k Z expðKmtÞðmtÞk =k!

(5)

The test has been applied to the St Martin data series, using different thresholds relating to three different periods, taking into account the type of available data: gauging station (1955–2000), water level scale (1892–1954) and historical data (1708– 1891). The process remains stationary for each of the three periods (Fig. 16), when dealing with

73

over-threshold values so that the hypothesis of exhaustiveness can be accepted. 4.2. Flood distribution using systematic and nonsystematic data at Vallon The Q parameters of the GEV distribution F(Qjx) have been fitted using the maximum likelihood method, using: (1) systematic data with ðx1 ; .; xN S_ Þ annual maximum values during N S_ years and (2) nonsystematic data with ðy1 ; .; yN H_ Þ annual

Fig. 16. Poisson test on the time flood process at St Martin on the Arde`che river.

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maximum values during NH years. A threshold of perception is related to each year with an unknown value assumed to be lower than yUi . The logarithm of the likelihood function is: LLTOT Z LLS_ C LLH_ C LLH!

(6)

where LLS_ Z

N S_ X

ln½f ðQjxi Þ; LLH_ Z

iZ1

N H_ X

3900–4900 to 5800–6100 m3/s when adding nonsystematic data. The influence of the various thresholds of perception and the number NH of the historical period (Section 3.3) has not been directly considered as these parameters were assessed from data quality and exhaustiveness criteria. Nevertheless, the distributions c–d–e–f–g based on various historical data sets show a good agreement with weak differences (Fig. 17).

ln½f ðQjyi Þ

iZ1

4.3. Flood distribution using rainfall data at St Martin LLH! Z

NH! X

ln½FðQjyUi Þ

iZ1

where NH!Z NH K N H_ Initialization of the iterative search was processed using the expected moment method (Lane and Cohn, 1996; Cohn et al., 1997) and its adaptation to the GEV distribution (Naulet, 2002). Fig. 17 gives a comparison of the annual maximum flood distribution at Vallon using different starting dates: 1892, 1827, 1772 and 1645 (hypotheses 1 and 2). Above the 10-year return period, flood quantiles are significantly different using systematic data (20th century) or nonsystematic data (before 1892). As an example, the 100-year flood estimate increases from

In France, the hydraulic safety of large dams has to be checked against major floods with an annual exceedance probability of the order of 10K3 or less. Design floods are usually estimated from the Gradex method (Guillot and Duband, 1967; CFGB, 1994) which is based on the assumption of a parallelism between the upper tails of the flood volume distribution and the rainfall volume distribution, the latter being exponential. Michel (1982) introduced a progressive extrapolation, which gives an asymptotic parallelism:   aqg T K Tg Qv ðTÞ Z Qv ðTg Þ C ap Ln 1 C (7) ap Tg

Fig. 17. GEV distribution of annual maximum floods at Vallon, using systematic and nonsystematic data.

R. Naulet et al. / Journal of Hydrology 313 (2005) 58–78

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Fig. 18. Flood distribution at St Martin, using Gradex/Agregee extrapolation.

where Qv is the flood volume on a fixed duration, Tg is the starting point of extrapolation, ap is the scale parameter of the rainfall distribution, called ‘gradex’, aqg is the scale parameter of the flood volume distribution. The Agregee model (Margoum et al., 1994; Lang, 1997a) is a generalization to skewed rainfall distributions, dealing with ‘pseudo-gradex’ values defined as the slope of the rainfall distribution with a logarithmic scale on return period: ap(T)Z dP(T)/d Ln T. Eq. (7) is a particular case of the Agregee extrapolation when the rainfall distribution fits closely to a Gumbel law (ap(T)Zconst.Zap). Extrapolation of the flood distribution at St Martin has been carried out using the Agregee model from systematic data (1955–2000 daily discharge series at St Martin, 1955–1998 daily rainfall series at 11 rain gauges) with the following steps: (1) estimation of the basin rainfall gradex from the annual maximum values of daily rainfall in autumn (Sept. to Nov.). Areal rainfall has been computed by averaging daily records of rain gauges (Safege, 2000). (2) Extrapolation of the annual maximum daily discharge with the basin gradex (apZ28 mm, i.e. 726 m3/s). (3) Conversion of flood volume Qv(T) to peak discharge Q(T) using a constant peak ratio estimated from the largest hydrographs. (4) Calculation of the 95% confidence

interval from a computation of the asymptotic variance of the Agregee quantiles (Lang, 1997b). Fig. 18 presents a comparison of three extrapolations at St Martin: (1) GEV three parameter distributions using automatic records (1980–2001) or systematic data (1892–2001); (2) Gradex method using rainfall information and the 10-year flood from systematic data (1955–2000); (3) GEV three parameter distribution using systematic data, the largest historical discharges, and assuming that unknown floods are lower than a perception threshold. The different distributions c–d–e–f–g based on historical data sets lie inside the 95% confidence interval of the Gradex quantiles, contrary to the distributions a–b based on discharge data from the twentieth century. The 100year discharge estimate differs between method 1 (4600–5300 m3/s) and methods 2 and 3 (6000– 6400 m3/s). It shows a good agreement between the last two approaches, which use very different information: rainfall during the 20th century and historical discharge data during the 1644–1892 period.

5. Conclusions This case study on Arde`che river gave the opportunity to present a practical example of how

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historical information can improve flood knowledge. It has been possible to produce long discharge series at Vallon and St Martin from 1644 to 2001, taking into account the reliability of documentary sources, the uncertainties in water level, roughness coefficient and channel geometry, and the impact of a nonpermanent discharge or the backward effect. Inclusion of the two historical data series into flood frequency analysis reduces the sampling uncertainty and also shows a good agreement with a hydrometeorological approach (Gradex/Agregee model). Further historical studies will be undertaken on the Vallon area, based on former archive sources during the 1400–1800 period. Such qualitative testimonies of ancient floods will be used to enlarge the data set before the nineteenth century, as the present historical investigation focussed on the last two centuries and produced only two historical levels of the 1644 and 1772 floods. Paleoflood studies have also been developed at Vallon and were able to find flood deposits from several millennium (Sheffer et al., 2003). They indicate that the recent 19th century floods were the largest at the millennium scale. This provide question on the assumption of stationarity, which may be cautiously tested when dealing with large time scales. Barriendos et al. (2003), showed on a set of historical flood series in France and Spain that no homogeneous behaviour was observed for ordinary floods, as the assumption of stationarity could be accepted on the largest flood events from 1300 to the present day. As a conclusion of the European Sphere research project (Benito et al., 2004a,b), a methodological guide (Benito and Thorndycraft, 2004) presents recommendations on the use of nonsystematic flood data for statistical purposes, which amongst several case studies gives the main results on the Arde`che catchment, with historical and palaeofloods (Lang et al., 2004; Benito et al., 2004a,b).

Acknowledgements The financial support provided during the Historisk-Arde`che project (Contrat de Plan Etat-Region Rhoˆne-Alpes, 1998–99) and the European SPHERE project (2000–2003) is gratefully acknowledged. The PhD research of R. Naulet was funded through

a scientific collaboration between France and Canada, supported by Cemagref and INRS-ETE. The authors also thank for her support C. Gigon from the Arde`che Flood Warning Office, who made access to documentary data easier.

Historical sources A historical reference is relative to a physical set of documents which can be accessed by shelf marks in archive funds (i.e. the exact reference can change from one archive system to another one). Main references are issued from Archives De´partementales de l’Arde`che (ADA). Combier, 1849. Rivie`re d’Arde`che. Profil en long d’une partie de la rivie`re d’Arde`che, depuis le moulin de Salavas jusqu’au Pont d’Arc. Plan, Ponts et Chausse´es, Arrondissement d’Aubenas, p. 33, ADA, 1262W154. Combier, 1856. Du re´gime de l’Arde`che. De l’importance et de la dure´e des crues de cette rivie`re. Renseignements provoque´s par la circulaire du 26 juillet 1856. Ponts et Chausse´es, Arrondissement d’Aubenas, Aubenas, 19 sept., ADA, 1262W154. Combier, 1857. Etudes relatives aux inondations: rivie`re d’Arde`che. Ponts et Chausse´es, Arrondissement d’Aubenas, Aubenas, 15–16 Fee´v., ADA, 1262W154. Julien, L., 1861. Observations hydrome´triques au pont suspendu de Vallon—Feuilles mensuelles, juillet 1861, Ponts et Chausse´es, De´partement de l’Arde`che, ADA, 1262W152. Morel, 1921. Bassin du Rhoˆne—L’Arde`che—Profil en long de la ligne d’eau. Planches 1 (Sortie des Gorges-Rhoˆne) et 2 (RuomsVallon). Leve´s effectue´s par le Service du Nivellement Ge´ne´ral de la France pour le compte du Service des Forces Hydrauliques, Institut de Ge´ographie Alpine, Fonds Maurice Parde´, 1116, B2–21. Morlie`re, 1857a. Rivie`re d’Arde`che. Rapport sur la crue du 10 septembre. Ponts et Chausse´es, Arrondissement d’Aubenas, 23 octobre, p. 10, ADA, 1262W154. Morlie`re, 1857b. Etudes relatives aux inondations. Rivie`re d’Arde`che. Profils des crues du 10 septembre et du 5 octobre 1857. Plan, Ponts et Chausse´es, Arrondissement d’Aubenas, p. 8, ADA, 1262W154. N., 1959. Courrier du cure´ de St-Martin a` Maurice Parde´. 21/05/1959. Institut de Ge´ographie Alpine, Fonds Maurice Parde´. 1116, p. B2–27. Parde´, M., 1925a. Ensemble de notes manuscrites. Institut de Ge´ographie Alpine, Fonds Maurice Parde´. 1116, B2–28. Perret, 1857. Etudes relatives aux inondations. Bassin du Rhoˆne. Courbes des de´bits de l’Arde`che a` St Martin d’Arde`che. Crues du 28 au 29 septembre 1846, du 26 octobre au 3 novembre 1853 et du 10 mai au 10 juillet 1856. Plan, leve´ du 5 mars, Ponts et Chausse´es, De´partement de l’Arde`che, Arrondissement de Privas, p. 4, ADA, 1262W154. Ponts et Chausse´es, De´partement de l’Arde`che, 1859. Crues des 14 et 15 octobre 1859. Rivie`res du Doux, d’Eyrieux, d’Arde`che, de Chassezac, d’Ibie. Manuscrit, p. 2, ADA, 1270W1.

R. Naulet et al. / Journal of Hydrology 313 (2005) 58–78 Vaschalde, H., 1890. Les inondations du Vivarais depuis le XIIIe`me sie`cle, pre´diction et historique de celle du 22 septembre 1890, Imprimerie de Mme Robert, Aubenas, Bibliothe`que Nationale de France, LK2-3834.

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