Julien Ackerer
email: julien.ackerer@orange.fr

Jérôme Van Der Woerd
email: jerome.vanderwoerd@unistra.fr

Anne Sophie Meriaux

Coralie Ranchoux

Gerhard Schäfer

Frederick Delay

François Chabaux
email: fchabaux@unistra.fr

Amelie Aubert

René

Régis Braucher

Vincent Godard

K Keddadouche

Quantifying geomorphological evolution from 10Be denudation rates: Insights from high-resolution depth profiles, topsoils, and stream sediments

Keywords: 10 Be isotopes, geomorphological evolution, denudation rates, watershed 10 Be isotopes, geomorphological evolution, denudation rates, watershed

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Introduction

Denudation, which is defined as the sum of chemical weathering and physical erosion, is a key process for understanding the Earth's surface evolution. Denudation exhumes fresh bedrock, exerts a strong control on landscape structure, and carries regolith particles from watershed slopes to rivers [START_REF] Vanacker | Restoring dense vegetation can slow mountain erosion to near natural benchmark levels[END_REF][START_REF] Lupker | Predominant floodplain over mountain weathering of Himalayan sediments (Ganga basin)[END_REF][START_REF] Schoonejans | Coupling uranium series and 10Be cosmogenic radionuclides to evaluate steady-state soil thickness in the Betic Cordillera[END_REF]. Quantifying denudation rates is important for the understanding of regolith residence time [START_REF] Foster | Hillslope lowering rates and mobile-regolith residence times from in-situ and meteoric 10Be analysis[END_REF], the sediment transport in rivers [START_REF] Dongen | Cosmogenic 10 Be in river sediment: where grain size matters and why[END_REF], the evolution of mountain ranges [START_REF] Meyer | Determining the growth rate of topographic relief using in-situ-produced 10Be: A case study in the Black Forest, Germany[END_REF], or the coupling between chemical and physical processes in the critical zone (CZ; [START_REF] West | Thickness of the chemical weathering zone and implications for erosional and climatic drivers of weathering and for carbon-cycle feedbacks[END_REF][START_REF] Larsen | Rapid soil production and weathering in the Southern Alps[END_REF].

In-situ terrestrial cosmogenic 10 Be is a powerful tool to study the Earth's surface evolution and to gather quantitative estimates on denudation rates (e.g. [START_REF] Dixon | Tracing and pacing soil across slopes[END_REF]. Many studies used in-situ 10 Be to estimate watershed-scale averaged denudation rates from stream or river sediments (e.g. [START_REF] Brown | Denudation rates determined from the accumulation of in-situ-produced 10Be in the Luquillo Experimental Forest, Puerto Rico[END_REF][START_REF] Granger | Spatially averaged long-term erosion rates measured from in-situ-produced cosmogenic nuclides in alluvial sediment[END_REF]. This widely used approach is based on several important assumptions: (1) the denudation is uniform over time and 10 Be concentrations have reached a steady-state, reflecting a balance between 10 Be cosmogenic production and 10 Be loss by denudation and radioactive decay. (2) each eroding area contributes quartz to the mixed sediment sample in proportion to its erosion rate (valid with a single lithology). ( 3) if several lithologies are present, the contributing rock types contain similar grain size distributions. (4) quartz is not enriched during weathering or transport. (5) sediment storage is minimal in the watershed. And (6) the denudational time scale is smaller than the timescale for radioactive decay [START_REF] Blanckenburg | The control mechanisms of erosion and weathering at basin scale from cosmogenic nuclides in river sediment[END_REF][START_REF] Dunai | Cosmogenic nuclides: principles, concepts and applications in the earth surface sciences[END_REF].

In parallel to the watershed-scale averaged denudation rates, soil and regolith denudation rates were also locally determined from in-situ 10 Be in topsoil samples (e.g. [START_REF] Riebe | Long-term rates of chemical weathering and physical erosion from cosmogenic nuclides and geochemical mass balance[END_REF][START_REF] Meyer | Determining the growth rate of topographic relief using in-situ-produced 10Be: A case study in the Black Forest, Germany[END_REF]. However, calculations based on topsoil samples only provide estimates of the minimum exposure age under the assumption of a negligible denudation or estimates of the maximum denudation rate under the assumption of steady-state 10 Be concentrations [START_REF] Braucher | Determination of both exposure time and denudation rate from an in-situ-produced 10Be depth profile: a mathematical proof of uniqueness. Model sensitivity and applications to natural cases[END_REF]. Soil denudation rates are frequently converted into soil or regolith production rates assuming a steady-state of regolith thickness (regolith denudation at the soil level is balanced by the regolith production at depth; e.g. [START_REF] Heimsath | The soil production function and landscape equilibrium[END_REF].

One of the major issues is that the steady-state of denudation and in-situ 10 Be concentrations may not be verified for watersheds with a complex evolution including the transport of mobile regolith along slopes or complex exposure histories (assumption 1, Von [START_REF] Blanckenburg | The control mechanisms of erosion and weathering at basin scale from cosmogenic nuclides in river sediment[END_REF]. Many studies taking place in mountain regions with potentially young surfaces and/or mobile regolith do not quantify the impact of this assumption (e.g. [START_REF] Riebe | Long-term rates of chemical weathering and physical erosion from cosmogenic nuclides and geochemical mass balance[END_REF][START_REF] Meyer | Determining the growth rate of topographic relief using in-situ-produced 10Be: A case study in the Black Forest, Germany[END_REF], primarily because the analysis of in-situ 10 Be in superficial samples cannot independently determine both regolith exposure age and denudation rate [START_REF] Norton | Cosmogenic nuclide-derived rates of diffusive and episodic erosion in the glacially sculpted upper Rhone Valley, Swiss Alps[END_REF]. Another problem is the lack of a clear distinction between three different equilibrium states that do not necessary coincide in time: the steady-state of denudation and 10 Be concentrations, the steady-state of regolith thickness, and the steadystate of geomorphology and topography.

Sampling of in-situ 10 Be depth profiles raises a new research perspective (e.g. [START_REF] Schaller | Paleoerosion rates from cosmogenic 10Be in a 1.3 Ma terrace sequence: response of the River Meuse to changes in climate and rock uplift[END_REF]2018;[START_REF] Ackerer | Regolith evolution on the millennial timescale from combined U-Th-Ra isotopes and in-situ cosmogenic 10Be analysis in a weathering profile (Strengbach catchment, France)[END_REF][START_REF] Hidy | Age-erosion constraints on an Early Pleistocene paleosol in Yukon, Canada, with profiles of 10Be and 26Al: Evidence for a significant loess cover effect on cosmogenic nuclide production rates[END_REF]. By gathering regolith and bedrock samples and by combining the contributions from neutrons and muons in modeling approaches, depth profiles opened the possibility of determining both exposure ages and denudation rates [START_REF] Braucher | Determination of both exposure time and denudation rate from an in-situ-produced 10Be depth profile: a mathematical proof of uniqueness. Model sensitivity and applications to natural cases[END_REF]. Depth profiles also provided new opportunities to investigate the thickness of mobile regolith, the mixing processes near the surface, or the polyphase evolution of weathering profiles [START_REF] Ackerer | Regolith evolution on the millennial timescale from combined U-Th-Ra isotopes and in-situ cosmogenic 10Be analysis in a weathering profile (Strengbach catchment, France)[END_REF][START_REF] Schaller | Addressing the contribution of climate and vegetation cover on hillslope denudation, Chilean Coastal Cordillera (26-38 S)[END_REF].

We supplement the data obtained by [START_REF] Ackerer | Regolith evolution on the millennial timescale from combined U-Th-Ra isotopes and in-situ cosmogenic 10Be analysis in a weathering profile (Strengbach catchment, France)[END_REF] for a high-resolution depth profile located on the summit of a mountain watershed (the Strengbach watershed). Two new highresolution depth profiles in addition with topsoils, rock outcrops and stream sediments were sampled across the Strengbach watershed to gather a wide database of mineralogical, geochemical and in-situ 10 Be analyses. This rare combination allows us to investigate: (a) how the mobile regolith is generated and how it affects the critical zone (CZ) evolution, (b) how the assumption of steady denudation and 10 Be concentrations impacts the denudation estimates, (c) how denudation rates vary temporally and spatially in a mountain watershed and (d) what are the timescales to reach steady-state for 10 Be concentrations, regolith thickness and geomorphology.

The Strengbach watershed is a particularly well suited location to investigate these points as it is a small mountain watershed (0.8 km²) with a complex evolution and a first order stream [START_REF] Ackerer | Regolith evolution on the millennial timescale from combined U-Th-Ra isotopes and in-situ cosmogenic 10Be analysis in a weathering profile (Strengbach catchment, France)[END_REF]. The bedrock is relatively homogeneous [START_REF] Fichter | Mineral evolution of acid forest soils in the Strengbach catchment (Vosges mountains, NE France)[END_REF] and alluvial terraces or major zones of sediment storage were not observed in the watershed [START_REF] Cotel | Evaluation of fluxes of suspended matters and bedload in the small granitic Strengbach catchment (Vosges massif, France)[END_REF]. Quartz enrichment during weathering is not an issue and the denudational time scale is clearly smaller than the timescale for radioactive decay [START_REF] Ackerer | Regolith evolution on the millennial timescale from combined U-Th-Ra isotopes and in-situ cosmogenic 10Be analysis in a weathering profile (Strengbach catchment, France)[END_REF]. By consequence, most of the assumptions classically used to interpret 10 Be data are validated in this mountain watershed (assumptions 2 to 6; Von [START_REF] Blanckenburg | The control mechanisms of erosion and weathering at basin scale from cosmogenic nuclides in river sediment[END_REF], with the exception of the critical assumption of balance between the productions and losses of 10 Be within the watershed (assumption 1). Our study will thus allow to focus on the specific impact of this major assumption on denudation estimates in a complex watershed.

Study site

This study was conducted in the Strengbach watershed (figure 1), one of the reference sites of the French critical zone network (OZCAR), where multidisciplinary studies have been conducted since 1986 ("Observatoire Hydrogéochimique de l'Environnement", OHGE; [START_REF] Fichter | Mineral evolution of acid forest soils in the Strengbach catchment (Vosges mountains, NE France)[END_REF][START_REF] Viville | Erosion and weathering fluxes in granitic basins: the example of the Strengbach catchment (Vosges massif, eastern France)[END_REF][START_REF] Ackerer | Regolith evolution on the millennial timescale from combined U-Th-Ra isotopes and in-situ cosmogenic 10Be analysis in a weathering profile (Strengbach catchment, France)[END_REF]2020a;[START_REF] Chabaux | Geochemical tracing and modeling of surface and deep water-rock interactions in elementary granitic watersheds (Strengbach and Ringelbach CZOs, France)[END_REF]2019;[START_REF] Pierret | The Strengbach catchment: A multidisciplinary environmental sentry for 30 years[END_REF][START_REF] Wild | In-situ dissolution rates of silicate minerals and associated bacterial communities in the critical zone (Strengbach catchment, France)[END_REF][START_REF] Ranchoux | Characterization of groundwater circulations in a headwater catchment from an analysis of chemical concentrations, Sr-Nd-U isotope ratios, and CFC, SF6 gas tracers (Strengbach CZO, France)[END_REF]. The Strengbach watershed is a headwater watershed (0.8 km²) located in the Vosges Mountains of northeastern France at elevations between 883 and 1147 m (figures 1a and 1b).

The current climate is mountainous oceanic, with an annual mean temperature of 7 °C and an annual mean rainfall of approximately 1400 mm. An updated summary of the Strengbach hydrologic and climatic features can be found in [START_REF] Pierret | The Strengbach catchment: A multidisciplinary environmental sentry for 30 years[END_REF]. The bedrock is a Hercynian granite, fractured, hydrothermally overprinted, and mainly composed of quartz (35%), albite (31%), K-feldspar (22%) and biotite (6%). The granite was affected by weathering processes during the Quaternary [START_REF] Ackerer | Regolith evolution on the millennial timescale from combined U-Th-Ra isotopes and in-situ cosmogenic 10Be analysis in a weathering profile (Strengbach catchment, France)[END_REF] and the watershed is currently covered by a 50 to 100 cm-thick soil overlying a 0.5 to 8 m-thick regolith [START_REF] Weill | Variability of water transit time distributions at the Strengbach catchment (Vosges Mountains, France) inferred through integrated hydrological modeling and particle tracking algorithms[END_REF]Ackerer et al., 2020a). The regolith is defined here by the layer of weathered material that has lost a physical coherence with the bedrock in place. The granitic bedrock is outcropping at few locations, with tors located on the summit of the northern slopes and at mid-elevation on the southern slopes (figure 1).

The watershed geomorphology was likely affected by periglacial processes during the Quaternary as the Vosges massif experienced Pleistocene glaciations [START_REF] Mercier | Deglaciation of the Vosges dated using 10Be[END_REF][START_REF] Heyman | Paleo-climate of the central European uplands during the last glacial maximum based on glacier mass-balance modeling[END_REF]. Cold conditions likely persisted at the elevation of the Strengbach watershed after the Late Glacial Maximum (LGM), and the forest cover probably developed only at the beginning of the Holocene [START_REF] Leroy | Palynological analyses in the laminated sediment of Lake Holzmaar (Eifel, Germany): duration of Lateglacial and Preboreal biozones[END_REF]. The Strengbach watershed was affected by anthropogenic deforestation associated with pastoralism before the 20 th century [START_REF] Etienne | Searching for ancient forests: A 2000 year history of land use in northeastern French forests deduced from the pollen compositions of closed depressions[END_REF], and the return to a dense forest is due to the planting of spruce stands at the beginning of the 20 th century [START_REF] Fichter | Mineral evolution of acid forest soils in the Strengbach catchment (Vosges mountains, NE France)[END_REF].

Methods

Sampling methods

Samples were collected at different locations to explore the variability of denudation rates within the watershed (figure 1b). Two new high-resolution depth profiles extending from the topsoil (0 m) to the granitic bedrock (2-3 m) were realized by digging pits in the slope below the depth profile located on the summit and analyzed by [START_REF] Ackerer | Regolith evolution on the millennial timescale from combined U-Th-Ra isotopes and in-situ cosmogenic 10Be analysis in a weathering profile (Strengbach catchment, France)[END_REF]. These new depth profiles are located on a gentle slope of the southern part of the watershed where no modern perturbations are visible (figure 1b). Our high-resolution sampling includes: (1) a high number of samples (10 to 15), (2) a detailed density of sampling (one measure every 15-25 cm maximum) to identify specific zones along the vertical direction and (3) a sampling not neglecting the lower part of the depth profiles (the lower regolith and the bedrock), as this part contains important information for the long term muonic exposure and the inherited 10 Be. The three depth profiles are named P1, P2 and P3 (figures 1b and 1c). Observations were performed to study the CZ structure and to estimate soil and regolith thicknesses in the three profiles.

In addition, topsoil samples were collected at different places in the watershed, with independent topsoil samples taken on the top of depth profiles, and other samples taken at lower elevations and on the summit of the northern part of the watershed (figure 1b). Two rock samples were also collected on granitic tors outcropping nearby the summit of northern slopes and at mid elevation location on southern slopes (figure 1b). Stream sediment samples were finally collected along the Strengbach stream network before and at the watershed outlet (figure 1b).

Analytical methods

For samples collected in depth profiles, depth, major element concentrations, mineralogy, density and in-situ 10 Be concentrations were characterized (table 1, supplementary tables S1 and S2). For topsoil, rock outcrop and stream sediment samples, only density and in-situ 10 Be concentrations were characterized (table 2).

Bulk densities of regolith samples collected from depth profiles were estimated by combining measurements of fine fraction and block densities [START_REF] Ackerer | Regolith evolution on the millennial timescale from combined U-Th-Ra isotopes and in-situ cosmogenic 10Be analysis in a weathering profile (Strengbach catchment, France)[END_REF]. For each sample in the upper part of profiles (0-100 cm), the fine fraction density was obtained by weighing a known volume of regolith sampled with a steel cylinder. The proportion of blocks was estimated via macroscopic description and photos and a constant density of 2.7 g.cm -3 was used for the granitic bedrock. The bulk density of each sample was then calculated by combining the fine fraction density and the density of blocks with respect to the proportion of blocks in the horizon.

Quartering and sequential crushing provided representative subsamples of regolith bulk samples. These subsamples were used for mineralogical, major elements, and in-situ 10 Be analysis following the methodology detailed in [START_REF] Ackerer | Regolith evolution on the millennial timescale from combined U-Th-Ra isotopes and in-situ cosmogenic 10Be analysis in a weathering profile (Strengbach catchment, France)[END_REF]. Mineralogical analysis was performed by X-ray diffraction, and major element concentrations were determined by tetraborate alkaline fusion followed by ICP-MS and ICP-AES analysis at ITES laboratory (Institut Terre et Environnement de Strasbourg, France; [START_REF] Gangloff | Characterization and evolution of dissolved organic matter in acidic forest soil and its impact on the mobility of major and trace elements (case of the Strengbach watershed[END_REF][START_REF] Prunier | Monitoring of geochemical and isotopic (Sr,U) signatures in soil solutions for the evaluation of soil weathering evolution (the Strengbach case[END_REF] and BRGM (Bureau des Recherches Géologiques et Minières, Orléans, France). The analytical accuracy was checked by measuring the San Joaquin soil standard (NIST SRM 2709a), and the overall uncertainty was <5 % for major element concentrations.

For in-situ 10 Be, subsamples consist of approximately 150 g of 250-500 µm sieved grains.

Purified quartz was obtained from regolith samples by following the approach involving sequential HCl and HF leaching to remove oxides, organic matter and meteoric beryllium (e.g. [START_REF] Kohl | Chemical isolation of quartz for measurement of in-situ-produced cosmogenic nuclides[END_REF]. After addition of a 9 Be carrier, beryllium separation, purification, hydroxylation, and targeting, the 10 Be/ 9 Be ratios were measured by accelerator mass spectrometry at ASTER (CEREGE laboratory, France). The 10 Be/ 9 Be ratios of the blanks and samples were approximately 1×10 -15 and 1×10 -13 , respectively, and the overall analytical uncertainty for 10 Be concentrations was <5 % (table 1).

Quantification of chemical mobility and volumetric variations in depth profiles

To quantify the mobility of major elements within each depth profile, mass transfer coefficients (𝜏 𝑖 ) were calculated as [START_REF] Brimhall | Quantitative geochemical approach to pedogenesis: importance of parent material reduction, volumetric expansion, and eolian influx in lateritization[END_REF][START_REF] Riebe | Long-term rates of chemical weathering and physical erosion from cosmogenic nuclides and geochemical mass balance[END_REF]:

𝜏 𝑖 = ( 𝐶(𝑖) 𝑠𝑎𝑚𝑝𝑙𝑒 𝐶(𝑗) 𝑏𝑒𝑑𝑟𝑜𝑐𝑘 𝐶(𝑖) 𝑏𝑒𝑑𝑟𝑜𝑐𝑘 𝐶(𝑗) 𝑠𝑎𝑚𝑝𝑙𝑒 -1) (1)
with 𝐶(𝑖) 𝑠𝑎𝑚𝑝𝑙𝑒 and 𝐶(𝑖) 𝑏𝑒𝑑𝑟𝑜𝑐𝑘 the concentrations of the element 𝑖 for a given sample and for the deepest sample of the depth profile, and 𝐶(𝑗) 𝑠𝑎𝑚𝑝𝑙𝑒 and 𝐶(𝑗) 𝑏𝑒𝑑𝑟𝑜𝑐𝑘 the concentrations of the immobile element 𝑗 for a given sample and for the deepest sample of the depth profile. Positive or negative 𝜏 𝑖 imply the gain or loss of the element 𝑖 relative to the deepest bedrock sample of a profile, taken as the reference rock. Titanium (Ti) was used as an immobile element for the 𝜏 𝑖 calculation due to evidence of immobile behavior during weathering in the profile P1 [START_REF] Ackerer | Regolith evolution on the millennial timescale from combined U-Th-Ra isotopes and in-situ cosmogenic 10Be analysis in a weathering profile (Strengbach catchment, France)[END_REF]. In addition, volumetric strain index (𝜀) was calculated to quantify the volumetric variations within each depth profile as [START_REF] Brimhall | Quantitative geochemical approach to pedogenesis: importance of parent material reduction, volumetric expansion, and eolian influx in lateritization[END_REF]:

𝜀 = ( 𝐶 (

Analytical results

Subdivision of depth profiles

Soil thickness is variable and reaches 50 cm, 70 cm and 80 cm in depth profiles P1, P2 and P3, respectively (table 1). Soil type changes from alocrisol in P1 to ochric podzol in P2 and P3, and few weakly weathered blocks of granite are observed within the upper soil in P2 and P3

(figure 1c). Lower regolith extends between 50 and 100 cm, 70 and 190 cm, and 80 and 190 cm of depth for P1, P2 and P3, respectively (table 1, figure 1). In P1, the amount of blocks gradually increases from top to bottom of the lower regolith, while the proportion of granitic grus decreases (figure 1c). In P2 and P3, the thicker lower regolith has a different texture, with higher amount of clay minerals and some large blocks showing angular and sharp faces with signs of transport. The amount of blocks is also much more variable along the vertical direction in P2 and P3. Granitic bedrock extends below 100 cm for P1 and below 190 cm for P2 and P3, respectively (figure 1c). Bedrock is in place for the three depth profiles and shows weaker traces of weathering. As also detailed in [START_REF] Ackerer | Regolith evolution on the millennial timescale from combined U-Th-Ra isotopes and in-situ cosmogenic 10Be analysis in a weathering profile (Strengbach catchment, France)[END_REF], profile P1 has a simple and continuous vertical structure indicating a soil and a regolith developed from in-situ bedrock weathering. P2 and P3 show more complex vertical structures indicative of regolith mobility along the slope.

Mineralogy, chemical mobility and density in depth profiles

A clear enrichment of interstratified clay minerals is observed in the soil of all depth profiles (supplementary table S1). Minerals resistant to weathering tend to accumulate in the soil (quartz, K-feldspar) while illites, biotite and albite are depleted. Variations from bedrock to regolith are relatively smooth in profile P1 with enrichment of quartz and depletion of most other primary minerals (table S1). In P2 and P3, abrupt depletion of albite and enrichment of illites and micas occur at the transition between bedrock and mobile regolith. Albite is less depleted in the upper part of the mobile regolith (table S1), indicating a non-continuous weathering along the vertical direction in profiles P2 and P3.

𝜏 𝑖 values close to zero indicate limited mobility of major elements in the bedrock of the three profiles (figure 2, concentrations in table S2). In P1, 𝜏 𝑖 values point to a progressive depletion of major elements from the bottom of the regolith (𝜏 𝐶𝑎𝑂 = -0.1) to the topsoil (𝜏 𝐶𝑎𝑂 = -0.5, figure 2a). In P2 and P3, the mobile regolith shows stronger variations of 𝜏 𝑖 coefficients with marked depletion of sodium and calcium (𝜏 𝑁𝑎2𝑂 = -0.8, 130 cm depth, P2, figure 2b; 𝜏 𝑁𝑎2𝑂 = -0.7, 180 cm depth, P3, figure 2c). 𝜏 𝑖 values are less negative in the upper part of the mobile regolith and more negative in the soil of P2 and P3, supporting a non-continuous weathering along the vertical direction in these two profiles (figures 2b and 2c). These complex variations are less visible for silica, aluminum and potassium as these elements are controlled by less weatherable minerals (quartz, K-feldspar, …). Overall, the mobility of major elements follows the structure of depth profiles, with a progressive depletion of major elements in P1 and more complex variations in the mobile regolith of P2 and P3.

Bulk density is assumed constant in the bedrock of the three profiles (d=2.7 g.cm -3 ) before decreasing in the lower regolith (d=1.8-2.6 g.cm -3 ) and in the soil (d=0.9-1.8 g.cm -3 , table 1 and figure 3). Bulk density primarily follows the soil and regolith thicknesses. Volumetric variations are limited in the bedrock and the lower regolith, but expansion occurs in the soil of the three depth profiles (𝜀=0.5-1.4, figure 3b).

In-situ 10 Be variations in depth profiles

At first order, in-situ 10 Be concentrations increase towards the soil surface within the three depth profiles (table 1, figure 3a). 10 Be concentrations are similar in the granitic bedrock for all profiles with concentrations around 25000-30000 atoms. g qtz -1 . The exponential increase of 10 Be concentrations is continuous within the bedrock and the lower regolith of P1, while 10 Be concentrations show a linear trend in the soil of P1 (figure 3a). For P2 and P3 profiles, 10 Be concentrations show an exponential trend within the mobile regolith, but concentrations are systematically higher than the concentrations measured in the lower regolith of P1. 10 Be concentrations are also higher in the soil of P2 and P3 compared to P1, with scattered concentrations in the soil of P3 (figure 3a). Similarly to the profile subdivision and major elements, 10 Be concentrations point to a relatively continuous evolution in P1 and more complex variations in P2 and P3. This complexity tends to increase from the summit to the lower part of the hillslope.

In-situ 10 Be variations in topsoils, rock outcrops and stream sediments

In-situ 10 Be concentrations measured in topsoil samples tend to decrease with increasing elevation, with concentrations around 180000-190000 atoms. g qtz

Modeling methods

Mean denudation rate and cosmogenic exposure age of profile P1 were determined from a numerical inverse procedure in [START_REF] Ackerer | Regolith evolution on the millennial timescale from combined U-Th-Ra isotopes and in-situ cosmogenic 10Be analysis in a weathering profile (Strengbach catchment, France)[END_REF]. Due to the complex structure of the two new depth profiles P2 and P3, an accurate estimation of denudation rates and exposure ages is not possible from a numerical inverse procedure for these profiles. We therefore propose a direct modeling of the evolution of the depth profiles P2 and P3 integrating regolith production, denudation, and the effect of the regolith mobility on 10 Be concentrations. This direct modeling of regolith evolution at the depth profile scale is supplemented by denudation rates derived from topsoil, rock outcrop and stream sediment samples.

Our modeling approaches are based on the same general differential equation describing the evolution of 10 Be concentration (C) as a function of depth and time (e.g. Lal, 1991):

𝜕𝐶 𝜕𝑡 = 𝑃 exp ( -𝜌𝑧 𝐾 ) + 𝑑 𝜕𝐶 𝜕𝑧 -𝜆 𝐶 (3)
with 𝑃 the total surface production rate (in atoms. g qtz -1 . yr -1 ), 𝑡 the cosmic ray exposure age of the profile (in yr), 𝑧 the depth (in cm), 𝜌 the integrated density (in g. cm -3 ), 𝐾 the attenuation length of cosmic ray particles (g. cm -2 ), 𝑑 the mean denudation rate of the surface (in cm. yr -1 ) and 𝜆 the radioactive decay constant of the 10 Be (in yr -1 ). This general differential equation 3 was adapted for the interpretation of 10 Be concentrations from depth profile, topsoil, rock outcrop and stream sediment samples.

Modeling of 10 Be concentrations in depth profiles

By considering constant values over time of inherited concentration from previous exposure 𝐶 0 and of respective proportions of secondary neutrons, negative muons and fast muons in 10 Be production, the evolution of 10 Be concentration (𝐶) as a function of time (𝑡), depth (𝑧)

and integrated density (ρ) for a given denudation rate (d) along a depth profile can be determined from equation 3 as (adapted from [START_REF] Siame | Local erosion rates versus active tectonics: cosmic ray exposure modelling in Provence (south-east France)[END_REF]:

𝐶(𝑡, 𝑧, 𝜌) = 𝐶 0 exp(-𝜆𝑡) + 𝑃. 𝑃 𝑛 𝜌𝑑 𝐾 𝑛 + 𝜆 exp ( -𝜌𝑧 𝐾 𝑛 ) (1 -exp (-𝑡( 𝜌𝑑 𝐾 𝑛 + 𝜆))) + 𝑃. 𝑃 𝑛𝑚 𝜌𝑑 𝐾 𝑛𝑚 + 𝜆 exp ( -𝜌𝑧 𝐾 𝑛𝑚 ) (1 -exp (-𝑡( 𝜌𝑑 𝐾 𝑛𝑚 + 𝜆))) + 𝑃. 𝑃 𝑓𝑚 𝜌𝑑 𝐾 𝑓𝑚 + 𝜆 exp ( -𝜌𝑧 𝐾 𝑓𝑚 ) (1 -exp (-𝑡( 𝜌𝑑 𝐾 𝑓𝑚 + 𝜆))) (4)
with Pn, Pnm and Pfm the relative contributions of 97.85, 1.5 and 0.65 % to the total production rate 𝑃, and Kn, Knm and Kfm the effective attenuation lengths of 150, 1500 and 5300 g.cm -², for secondary neutrons, negative muons and fast muons, respectively [START_REF] Braucher | Determination of both exposure time and denudation rate from an in-situ-produced 10Be depth profile: a mathematical proof of uniqueness. Model sensitivity and applications to natural cases[END_REF].

-Modeling of depth profiles A direct modeling was developed to determine the variations of soil and regolith production rates, soil and regolith thicknesses, and bulk and integrated densities during depth profile evolution (flow chart in supplementary figure S1). Simulations based on a finite difference scheme were performed for each depth profile along a 1D vertical direction. The CZ vertical structure was subdivided into three parts: the soil, the lower regolith and the bedrock.

During simulations, a constant mean denudation rate 𝑑 was imposed at the soil surface of each profile by following:

𝑑𝑧 𝑠𝑜𝑖𝑙_𝑠𝑢𝑟𝑓𝑎𝑐𝑒 = 𝑑 * 𝑑𝑡 (5)
with 𝑧 𝑠𝑜𝑖𝑙_𝑠𝑢𝑟𝑓𝑎𝑐𝑒 the elevation of the soil surface. In parallel, the soil is formed from lower regolith weathering and the lower regolith is formed from bedrock weathering. Calculation of soil and regolith production rates was based on a law of exponential decline with increasing soil and regolith thicknesses (e.g. [START_REF] Heimsath | Eroding Australia: rates and processes from Bega Valley to Arnhem land[END_REF][START_REF] Ma | Regolith production rates calculated with uranium-series isotopes at Susquehanna/Shale Hills Critical Zone Observatory[END_REF]:

𝐹 𝑠𝑜𝑙 = 𝐹 0 exp(-𝛼 ℎ 𝑠𝑜𝑙 ) (6) 𝐹 𝑟𝑒𝑔 = 𝐹 0 exp(-𝛽 ℎ 𝑟𝑒𝑔 ) (7)
with ℎ 𝑠𝑜𝑙 and ℎ 𝑟𝑒𝑔 the soil and regolith thicknesses (in cm), 𝐹 𝑠𝑜𝑙 and 𝐹 𝑟𝑒𝑔 the production rates of soil and regolith (in g.cm -².yr -1 ), 𝐹 0 the maximum soil and regolith production rate when soil and regolith thicknesses equal zero (in g.cm -².yr -1 ), and 𝛼 and 𝛽 the depth scaling factors of soil and regolith production rates (in cm -1 ). ℎ 𝑠𝑜𝑙 and ℎ 𝑟𝑒𝑔 were actualized at each new time step during simulations by considering the competition between soil surface lowering due to denudation and soil and regolith production. For the parameter 𝐹 0 governing the maximum soil and regolith production rate, we used for all profiles a high production rate determined by [START_REF] Egli | Soil formation rates on silicate parent material in alpine environments: Different approaches-different results?[END_REF] on a very thin regolith from a mountain environment (table S3). Scaling factors 𝛼 and 𝛽 controlling the exponential decrease of soil and regolith production rates with soil and regolith thicknesses were adjusted for each profile to obtain the best agreement between modeled and observed thicknesses at present time.

We assumed that bulk densities of topsoil, soil base and granitic bedrock remained constant through time and we used present day measurements of density for these three interfaces (values in table S3). Bulk density variations throughout the regolith were calculated by a linear interpolation between topsoil, soil base and bedrock densities. The integrated density at a given depth was then calculated by taking the average of bulk densities along the regolith column above the given depth.

-Initial state and inherited 10 Be Previous modeling of 10 Be concentrations in the depth profile P1 located at the summit of the watershed indicated that a major erosion event removing the soil and part of the regolith occurred at approximately 19 kyr BP [START_REF] Ackerer | Regolith evolution on the millennial timescale from combined U-Th-Ra isotopes and in-situ cosmogenic 10Be analysis in a weathering profile (Strengbach catchment, France)[END_REF]. This major erosion event removed approximately 2 m of material and was followed by a much more stable exposure period during the Holocene. The inherited 10 Be concentrations are derived from the deeper bedrock samples (≈15 000 atoms. g qtz -1 ), that would correspond to a long-term muonic component equivalent to an exposure age of approximately 150 kyr.

Based on the results of [START_REF] Ackerer | Regolith evolution on the millennial timescale from combined U-Th-Ra isotopes and in-situ cosmogenic 10Be analysis in a weathering profile (Strengbach catchment, France)[END_REF], a relic regolith in place and a bedrock in place were taken as initial state for the direct modeling of P1 at 19 kyr BP (parameters in table S3).

Given the proximity of the three profiles, P2 and P3 were also probably affected by the major erosion event at 19 kyr BP. In addition, a mobile regolith within profiles P2 and P3 was highlighted by profile section descriptions and geochemical data (section 4). We therefore used an extremely thin soil after erosion, a mobile regolith stabilized on the slopes, and a bedrock in place as initial state for P2 and P3 at 19 kyr BP (table S3).

The inherited 10 Be concentration from previous exposure (𝐶 0 in equation 4) was adapted for each depth profile in agreement with their initial states. In the bedrock of all depth profiles, 𝐶 0 was set to have an inherited 10 Be concentrations from a long-term muonic pre-exposure of 150 kyr (table S3). The same 𝐶 0 was used for the lower regolith of the profile P1, as the relic regolith probably experienced the same muonic exposure. By contrast, in the mobile regolith of P2 and P3, 𝐶 0 values represent the 10 Be concentrations in the regolith transported along the slope and stabilized on P2 and P3 after the important erosive event. 𝐶 0 values in the mobile regolith of P2 and P3 are unknown and were adjusted to obtain the best agreement between modeled and measured 10 Be concentrations in these profiles.

-Final 10 Be concentrations

The surface total 10 Be production rates were determined for the three depth profiles following [START_REF] Martin | The CREp program and the ICE-D production rate calibration database: A fully parameterizable and updated online tool to compute cosmic-ray exposure ages[END_REF] (scaling LAL-ERA Musch). After this step, 10 Be production rates along depth were calculated for each time step from equation 4 and from the time evolution of the integrated density. Final 10 Be concentrations were computed by taking the sum of 10 Be produced during simulation time, and exposure age and denudation rate can be estimated by comparing modeled and observed 10 Be concentrations. Best fits were identified with the help of Kling-Gupta coefficients [START_REF] Gupta | Decomposition of the mean squared error and NSE performance criteria: Implications for improving hydrological modelling[END_REF]. Overall, the denudation rate 𝑑 is inferred from equation 4 with the help of the direct modeling of depth profile evolution that does not rely on a steady-state assumption for the 10 Be concentrations.

Modeling denudation rates from topsoil, rock outcrop and stream sediment samples

In-situ 10 Be concentrations are also widely used to estimate local or catchment-wide denudation rates from their analysis in topsoil, rock outcrop or stream sediment samples (e.g. [START_REF] Granger | Spatially averaged long-term erosion rates measured from in-situ-produced cosmogenic nuclides in alluvial sediment[END_REF][START_REF] Meyer | Determining the growth rate of topographic relief using in-situ-produced 10Be: A case study in the Black Forest, Germany[END_REF][START_REF] Portenga | Understanding Earth's eroding surface with 10 Be[END_REF]. By assuming that there is no inherited 10 Be concentration, that 10 Be concentration has reached a steady-state, and that the exposure age is much longer than the effective irradiation time (𝑡 ≫ 𝑇 𝑒𝑓𝑓 ; with

𝑇 𝑒𝑓𝑓 = 1 𝜆+ 𝜌 𝐾 𝑛
.𝑑 𝑚𝑎𝑥 ; Lal, 1991), the 10 Be concentration for a surface sample (z=0) can be expressed as [START_REF] Charreau | Basinga: A cellby-cell GIS toolbox for computing basin average scaling factors, cosmogenic production rates and denudation rates[END_REF]: (Lal, 1991;[START_REF] Stone | Air pressure and cosmogenic isotope production[END_REF] and revised 10 Be production rates [START_REF] Martin | The CREp program and the ICE-D production rate calibration database: A fully parameterizable and updated online tool to compute cosmic-ray exposure ages[END_REF]. Shielding effect is limited in the watershed and was estimated by following the method from [START_REF] Codilean | Calculation of the cosmogenic nuclide production topographic shielding scaling factor for large areas using DEMs[END_REF].

𝐶(𝑧 0 ) = 𝑃.
For topsoil and rock outcrop samples, local 10 Be production rates were calculated at each location and equation 9 gives the maximum denudation rate at the sample location (referred as local 𝑑 𝑚𝑎𝑥 ). The local 𝑑 𝑚𝑎𝑥 obtained from a single topsoil sample independently collected on the surface of a depth profile can then be compared to the denudation rate 𝑑 inferred from equation 4 to quantify the impact of the assumption of steady-state 10 Be concentrations.

For stream sediment samples, average elevation was used to calculate the average 10 Be production rate within each subcatchment. Equation 9gives the spatially-averaged maximum denudation rate of the catchment above the sampling point (referred as spatiallyaveraged 𝑑 𝑚𝑎𝑥 ). This calculation is based on steady-state 10 Be concentrations at the catchment scale and on the other assumptions stated in the introduction (assumptions 1 to 6). Finally, spatially-averaged 𝑑 𝑚𝑎𝑥 obtained from stream sediments can be used to determine subcatchment maximum denudation rates between two sampling points [START_REF] Granger | Spatially averaged long-term erosion rates measured from in-situ-produced cosmogenic nuclides in alluvial sediment[END_REF]:

𝑑 𝑚𝑎𝑥 𝑠𝑢𝑏𝑐𝑎𝑡𝑐ℎ𝑚𝑒𝑛𝑡 = 𝑑 𝑚𝑎𝑥,2 𝐴 2 -𝑑 𝑚𝑎𝑥,1 𝐴 1 𝐴 2 -𝐴 1 (10)
with 𝐴 2 and 𝐴 1 the areas of downstream and upstream catchments, respectively.

One of the important issue for the calculation of 𝑑 𝑚𝑎𝑥 from equation 9 is the choice of density of the eroding material. Apart for the case of granitic tors, for which the density of 2.7 g.cm -3 is relatively obvious, several possibilities arise for topsoil and stream sediment samples. One possibility is to use the current average densities of topsoil (≈1 g.cm -3 , as in [START_REF] Meyer | Determining the growth rate of topographic relief using in-situ-produced 10Be: A case study in the Black Forest, Germany[END_REF] and of stream sediment samples (≈2.2 g.cm -3 ) for the calculation of local and spatially-averaged 𝑑 𝑚𝑎𝑥 . Another approach is to use the current average density of regolith (≈1.7 g.cm -3 ) for both cases, arguing that this density could be a better estimate for the available erodible materials at the surface. However, it must be recalled that the assumption of steady-state 10 Be concentrations implies a long time of cosmic ray exposure (𝑡 ≫ 𝑇 𝑒𝑓𝑓 ) and this point must be kept in mind for the choice of densities. If soils are young and thin, and if most of the 10 Be accumulation occurs in the deeper regolith or in the bedrock, using lower density values measured from topsoils is biasing the denudation rates inferred from equation 9 toward higher values. A strategy more coherent with this assumption is to use the bedrock density (2.7 g.cm -3 ) for all calculations. This reflects that at longer timescales, all the surface processes (soil development during interglacial and soil scraping during glacial times) are averaged out and what is eroding is in effect the underlying bedrock. We quantitatively evaluate these three approaches bellow.

Modeling results

Evolution, exposure age and denudation rates of depth profiles

Modeled soil production rates are variable between profiles and tend to increase with decreasing elevation (from P1 to P2 and P3, adjusted parameters in table S3). By contrast, regolith production rates are much lower in profiles P2 and P3 due to the thicker regolith already present at the initial state (table S3). Modeled soil thicknesses rapidly increase during the first 5 kyr before reaching more stable values after 10 kyr of simulation time (illustration for profile P1 in figure 4). Soil thicknesses at present time reach 50 cm for P1 and 75 cm for P2 and P3 in accordance with measurements. Final simulated bulk and integrated densities are also consistent with measurements from all depth profiles (figures 4a, 4b, 5a, 5b, 6a and 6b).

The best agreement between modeled and measured 10 Be concentrations are obtained with the same cosmic ray exposure age of 19 kyr, the same muonic pre-exposure age of 150 kyr in the bedrock, and the same denudation rate 𝑑 of 22 mm.kyr -1 for the three depth profiles.

The 10 Be concentrations inherited from muons are presented in figure 4d. The best fit of data was characterized by Kling-Gupta coefficients of 0.96, 0.95 and 0.82 for P1, P2 and P3 profiles, respectively [START_REF] Gupta | Decomposition of the mean squared error and NSE performance criteria: Implications for improving hydrological modelling[END_REF]. In profile P1, 10 Be concentrations in the bedrock and the lower regolith are well explained by the direct modeling (figure 5c). The 10 Be concentrations are well explained in the whole profile P1 after considering a recent soil expansion constrained by the volumetric strain index (by expanding the soil layers towards the surface; as in [START_REF] Ackerer | Regolith evolution on the millennial timescale from combined U-Th-Ra isotopes and in-situ cosmogenic 10Be analysis in a weathering profile (Strengbach catchment, France)[END_REF]; figure 5d). In P2 and P3 bedrock, 10 Be concentrations are well explained by the direct modeling, similarly to the P1 bedrock (figures 6c and 6d).

Accurate fitting of 10 Be concentrations in P2 and P3 regolith is more challenging due to impact of mobile regolith within these profiles. Higher 10 Be concentrations in the mobile regolith of P2 and P3 are explained by the inherited 10 Be from mobile regolith pre-exposure (𝐶 0 in table S3). Best agreements between modeled and measured 10 Be concentrations in P2 lower regolith are obtained with a full mixing of mobile regolith during the transport event that occurred before 19 kyr BP, implying a constant 𝐶 0 value over depth in P2 lower regolith (table S3, figure 6c). Best results in P3 lower regolith are obtained with a limited mixing of mobile regolith and a variable 𝐶 0 value over depth (table S3, figure 6d). Scattered 10 Be concentrations, probably due to surface mixing in P3 soil, is not described by our direct modeling approach (figure 6d).

Maximum denudation rates from topsoils, rock outcrops and stream sediments

Maximum denudation rates inferred from equation 9 are shown in figure 7 for topsoils, rock outcrops and stream sediments. For this calculation, we used a constant granite density of 2.7 g.cm -3 for all samples. Calculated spatially-averaged 𝑑 𝑚𝑎𝑥 are similar for all the stream sediments (36-40 mm.kyr -1 ), apart from the sediments collected at the outlet that indicate higher denudation rate (43 mm.kyr -1 , table 2, figure 7a). The highest spatially-averaged 𝑑 𝑚𝑎𝑥 (56 mm.kyr -1 ) is calculated for the closest subcatchment to the watershed outlet (equation 10, figure 7a). Local 𝑑 𝑚𝑎𝑥 for rock outcrops are lower (30-34 mm.kyr -1 ), while local 𝑑 𝑚𝑎𝑥 for soils range between 30 and 42 mm.kyr -1 with higher denudation near the crests (figure 7a).

Local 𝑑 𝑚𝑎𝑥 calculated for topsoil samples independently collected at the surface of depth profiles (34-42 mm.kyr -1 ) are approximately two times higher than the denudation rate 𝑑 determined from the direct modeling of the high-resolution depth profiles (22 mm.kyr -1 ).

Other choices of density for topsoil and stream sediment samples (i.e. section 5.2) increase the biases observed between the local 𝑑 𝑚𝑎𝑥 determined on depth profiles and the denudation rate 𝑑 determined on the same depth profiles from the direct modeling (from a factor three to four).

We find no relation between catchment mean slope orientations and denudation rates (figure 7b, data in table S4), suggesting a limited role of differences in freezing-thaw cycles between north facing and south facing slopes. There is also no clear relation between catchment mean slopes and denudation rates (figure 7c, table S4): similar 𝑑 𝑚𝑎𝑥 are obtained for catchments with moderate mean slopes (12-18°) and only above a threshold of around 20° higher denudation is observed for the outlet subcatchment (figure 7b and table S4).

Discussion

Origin of mobile regolith

Our Study of creeping velocity for regolith particles in a warmer and drier environment was conducted by [START_REF] Heimsath | Creeping soil[END_REF]. Much slower velocity was inferred for the downward migration of regolith particles by processes analogous to diffusive erosion (6 et 12 m.kyr -1 ; [START_REF] Heimsath | Creeping soil[END_REF]. By taking a mean velocity of 10 m.kyr -1 , 19 kyr would be needed for regolith particles to cover the distance of 200 m between P1, P2 and P3 profiles. This time is too long and incompatible with the similar 10 Be concentrations observed in the upper part of all profiles (table 1) and with the modeling results pointing to a similar exposure age across the slope (table S3). Therefore, the deep mobile regolith observed in the watershed is most likely explained by rapid peri-glacial processes occurring during the LGM (episodic erosion; [START_REF] Norton | Cosmogenic nuclide-derived rates of diffusive and episodic erosion in the glacially sculpted upper Rhone Valley, Swiss Alps[END_REF], and not by continuous creeping as suggested in other mountain catchments [START_REF] West | Regolith production and transport at the Susquehanna Shale Hills Critical Zone Observatory, part 2: insights from meteoric 10Be[END_REF].

In contrast, the near surface (< 1 m of depth) regolith mobility is probably a recent mechanism (Holocene to actual). A longer cosmic ray exposure of a stable profile would inexorably produce a simple exponential decrease of 10 Be concentrations with depth, which is not observed in P2 and P3. Given the moderate slopes and the dense forest cover currently present around the depth profiles, recent erosion processes are probably dominated by diffusive erosion [START_REF] Montgomery | Topographic controls on erosion rates in tectonically active mountain ranges[END_REF] and soil layers are likely impacted by biological processes [START_REF] Wilkinson | Breaking ground: pedological, geological, and ecological implications of soil bioturbation[END_REF]. In P1, the linear trend of 10 Be concentrations observed in the soil is explained by the volumetric expansion due to the incorporation of organic matter [START_REF] Ackerer | Regolith evolution on the millennial timescale from combined U-Th-Ra isotopes and in-situ cosmogenic 10Be analysis in a weathering profile (Strengbach catchment, France)[END_REF]. In P2 and P3, the soil mixing appears incomplete, as 10 Be concentrations are not homogenized over depth as illustrated by [START_REF] Dixon | Tracing and pacing soil across slopes[END_REF]. This soil mixing is likely related to recent diffusive erosion, bioturbation by living organisms, tree throw, or anthropogenic perturbations during the last centuries [START_REF] Etienne | Searching for ancient forests: A 2000 year history of land use in northeastern French forests deduced from the pollen compositions of closed depressions[END_REF].

The differences between superficial and deep mobile regolith are rarely discussed in the literature (e.g. [START_REF] Dixon | Tracing and pacing soil across slopes[END_REF][START_REF] Schaller | Addressing the contribution of climate and vegetation cover on hillslope denudation, Chilean Coastal Cordillera (26-38 S)[END_REF]. One reason is that most of studies investigated shallower depth profiles or that sampling resolution in the deeper part of depth profiles is too low to highlight complex 10 Be concentration trends and the presence of deep mobile regolith. We therefore emphasize the importance of a high-resolution sampling in the lower part of depth profiles to identify deep mobile regolith and to better constrain the long-term evolution of the regolith along hillslopes.

Role of mobile regolith in CZ evolution

Our data and modeling results indicate that the mobile regolith has a key role in the CZ evolution as it modifies and complexifies the CZ structure (summary figure 8). Firstly, the occurrence of deep mobile regolith implies a disconnection between the bedrock in place and the regolith. This induces a greater heterogeneity, explaining the sharper variations of mineralogical and chemical composition at the bedrock/regolith interface within P2 and P3 profiles (figure 2). The deep mobile regolith also modifies the inherited 10 Be concentrations (figures 3 and 8), increasing 10 Be concentrations in the regolith of profiles located downslope (P2 and P3).

The deep mobile regolith also impacts soil and regolith production rates. Higher soil formation rates from lower regolith weathering are modeled in P2 and P3 profiles (section 5.2), and thicker and more evolved soils (ochric podzols) are observed within profiles developed on a deep mobile regolith (figures 1 and 8). These points are potentially explained by a faster weathering of the transported material found in the mobile regolith compared to the direct weathering of a lower regolith in place. By contrast, modeled values of regolith production rates from bedrock weathering are faster at the summit (P1) than at lower elevation along the hillslope (P2 and P3). [START_REF] Ma | Regolith production rates calculated with uranium-series isotopes at Susquehanna/Shale Hills Critical Zone Observatory[END_REF] focused on a watershed affected by periglacial processes and based on Useries isotopes also concluded that regolith production rates decrease with increasing regolith thickness and distance from the crests. Our results support this scheme and show that one possible explanation of this functioning is the deposition of mobile regolith along slopes. In fact, this process is explaining both the thickening of regolith at lower elevation and the protection of bedrock from weathering, which reduces regolith production rate.

The deep mobile regolith has also implications on the hydrological processes and water storage across landscape. It induces a thickening of weathered material in the lower parts of the watershed, which tends to enhance water storage capacity. Higher water storage capacity at mid-slope locations compared to crests was identified as an important feature to correctly model the hydrogeochemical functioning of the watershed [START_REF] Weill | Variability of water transit time distributions at the Strengbach catchment (Vosges Mountains, France) inferred through integrated hydrological modeling and particle tracking algorithms[END_REF]Ackerer et al., 2020a).

Many studies investigating the CZ evolution pictured that the mobile regolith is relatively superficial, affecting soils, and implying an almost complete mixing of soil layers (e.g. [START_REF] Anderson | Physical and chemical controls on the critical zone[END_REF][START_REF] Dixon | Soils as pacemakers and limiters of global silicate weathering[END_REF][START_REF] Riebe | Controls on deep critical zone architecture: A historical review and four testable hypotheses[END_REF][START_REF] Schaller | Addressing the contribution of climate and vegetation cover on hillslope denudation, Chilean Coastal Cordillera (26-38 S)[END_REF]. If a near surface mobile regolith is typically identified, the deep mobile regolith extending below the soil is too often neglected to understand the CZ structure and its evolution. As shown by the detailed modeling of depth profiles, the deep mobile regolith is not always well mixed, suggesting that a depth dependent transport law as proposed by [START_REF] Heimsath | The illusion of diffusion: Field evidence for depth-dependent sediment transport[END_REF] is also valid in the context of shallow gradient slopes of watersheds affected by periglacial processes. The mobile regolith may also change the relationship between physical erosion and chemical denudation rates across landscape, by complexifying the idealized soil mantled hillslope where soil is directly produced from bedrock weathering (e.g. Ferrier and Kirchner; 2008).

Impact of the assumption of steady-state 10 Be concentrations on denudation rates

The results show that the denudation rate 𝑑 (22 mm.kyr -1 ) determined from the direct modeling of depth profiles without any steady-state assumption on 10 Be concentrations is approximately two times lower than the local 𝑑 𝑚𝑎𝑥 (34-42 mm.kyr -1 , figure 7a) calculated on the same profiles with independent topsoil samples. This bias rises to a factor of three or even four when considering current average regolith or topsoil densities (≈1.7 g.cm -3 and ≈1 g.cm -3 respectively) instead of the bedrock density (2.7 g.cm -3 ) in the calculation of local

𝑑 𝑚𝑎𝑥 .
This bias is explained by the fact that in the Strengbach catchment, the 10 Be concentrations are currently not at steady-state, i.e. production, denudation and radioactive decay are not balanced. Additional simulations were performed on the profiles for the next 5,000 or 10,000 years, indicating a significant increase in 10 Be concentrations in the soil, regolith and bedrock for all depth profiles. Such behavior is related to the short cosmic ray exposure age of the regolith. Calculation of the effective irradiation time (𝑇 𝑒𝑓𝑓 ) indicates that the exposure age should be longer than 40-50 kyr to validate the assumption of steady-state 10 Be concentrations, which is not the case given the important erosion event that affected the profiles at around 19 kyr BP (i.e. 𝑡 < 𝑇 𝑒𝑓𝑓 ).

A key finding is that the depth profiles, although not in steady-state for the 10 Be concentrations, are close to a steady-state for their soil and regolith thicknesses (figure 4).

The simulated regolith production rates are close to the simulated denudation rates within the Holocene timeframe. This conclusion is supported by the observation that our simulated regolith production rate in P1 (approximately 14 mm.kyr -1 over the last 15 kyr, table S3) is consistent with the regolith production rate independently determined on the millennial timescale from U-Th-Ra isotopes (approximately 12 mm.kyr -1 , [START_REF] Ackerer | Regolith evolution on the millennial timescale from combined U-Th-Ra isotopes and in-situ cosmogenic 10Be analysis in a weathering profile (Strengbach catchment, France)[END_REF].

We also highlight the importance of considering soil expansion (non-isovolumetric weathering) when deciphering the evolution of depth profiles. In P1, a steady-state of regolith thickness and mass balance has been reached with a regolith production rate of 12 mm.kyr -1 and a denudation rate of 22 mm.kyr -1 : the weathering of a given thickness of bedrock produces approximately twice more thickness of soil due to the volumetric expansion and the decrease in density. When considering the differences in density and expansion processes, regolith production and denudation rates expressed in specific mass fluxes are in fact similar in P1 (30±10 and 37±15 T.km -².yr -1 , respectively; [START_REF] Ackerer | Regolith evolution on the millennial timescale from combined U-Th-Ra isotopes and in-situ cosmogenic 10Be analysis in a weathering profile (Strengbach catchment, France)[END_REF].

We therefore advocate for a clear distinction between two different equilibrium states: a steady sate of 10 Be concentrations achieved when 10 Be production is balanced by denudation and radioactive decay, and a steady-state of regolith thickness (or mass balance)

reached when regolith production is balanced by total denudation. Our results demonstrate that these two steady-states are not necessarily reached over the same timescale. In the case of the Strengbach watershed and nearby the crests, the steady-state of regolith thickness is reached before that of the 10 Be concentrations.

One way to correct the bias induced by the assumption of steady-state 10 Be concentrations on the local 𝑑 𝑚𝑎𝑥 is to rather estimate denudation rates 𝑑 from equation 4, at z=0, using the inherited 10 Be concentrations and the exposure age both deduced from the modeling of high-resolution depth profiles. The determination of these corrected denudation rates 𝑑 can be done with a numerical inverse procedure [START_REF] Ackerer | Regolith evolution on the millennial timescale from combined U-Th-Ra isotopes and in-situ cosmogenic 10Be analysis in a weathering profile (Strengbach catchment, France)[END_REF], by minimizing the difference between 10 Be concentrations measured in samples and 10 Be concentrations calculated from equation 4.

On our depth profiles, the corrected local 𝑑 from topsoil samples indicate that denudation rates range between 33 mm.kyr -1 (in P1) and 14 mm.kyr -1 (in P2), much more consistent with the value of 22 mm.kyr -1 estimated by the direct modeling of depth profiles (table 2). We used for this calculation the average regolith density of 1.7 g.cm -3 , as this density is representative of the eroded regolith during the time-window from 19 kyr BP to present.

The scattering of the revised denudations is certainly affected by the mixing processes of the shallow samples. Even though the corrected denudation values are only calculated for 3 samples, it is interesting to note that the average of these three 𝑑 values is 22 mm.kyr -1 , i.e.

identical within error to the value derived from the modelling of depth profiles. This result is consistent with the classical approach that recommends averaging the denudation rates of several samples for a given area, rather than relying on a single sample estimate.

The bias induced by the classical assumption of steady-state 10 Be concentrations (an overestimation of local denudation rates by a factor two to four) has probably a limited consequence for studies dealing with first-order estimates of denudation rates and/or comparing several watersheds with very different denudation rates. However, this bias is significant for studies attempting to better understand the geomorphological evolution of a watershed, as for example studies comparing local denudation of ridge crests with watershed-scale averaged denudation (e.g. [START_REF] Meyer | Determining the growth rate of topographic relief using in-situ-produced 10Be: A case study in the Black Forest, Germany[END_REF]. Our study shows that the assumption of steady-state 10 Be concentrations should be used very carefully in mountain environments where regoliths with short cosmic ray exposure ages are present. Combining high-resolution depth profiles with local regolith and topsoil samples is a way to quantify the impact of this assumption on denudation estimates.

Spatiotemporal variability of denudation rates and geomorphological implications

The temporal variability of denudation can firstly be estimated from the direct modeling of depth profiles. Denudation rates likely strongly decreased between the LGM and the last 19 kyr, as indicated by the cosmic ray exposure age of depth profiles pointing to important erosion events before 19 kyr BP and a period of relative stability after 19 kyr BP (figure 8). A second comparison can be made from the modern measurements of denudation. The research conducted in the Strengbach watershed shows that the current denudation determined from solute and solid fluxes at the outlet (around 15 T.km -².yr -1 , with 5 T.km -².yr - 1 of solute exports and 10 T.km -².yr -1 of solid exports over 20 yr of record; [START_REF] Viville | Erosion and weathering fluxes in granitic basins: the example of the Strengbach catchment (Vosges massif, eastern France)[END_REF][START_REF] Cotel | Evaluation of fluxes of suspended matters and bedload in the small granitic Strengbach catchment (Vosges massif, France)[END_REF][START_REF] Ackerer | Crossing hydrological and geochemical modeling to understand the spatiotemporal variability of water chemistry in a headwater catchment (Strengbach, France)[END_REF] is lower than the long-term denudation inferred from 10 Be data (around 37 T.km -².yr -1 , determined on the last 19 kyr on P1; [START_REF] Ackerer | Regolith evolution on the millennial timescale from combined U-Th-Ra isotopes and in-situ cosmogenic 10Be analysis in a weathering profile (Strengbach catchment, France)[END_REF]. A general trend of denudation decrease with time is then highlighted from the LGM to the last 19 kyr, and from this 19 kyr time-window to present. A probable explanation is the progressive development of vegetation during the Holocene at this elevation [START_REF] Leroy | Palynological analyses in the laminated sediment of Lake Holzmaar (Eifel, Germany): duration of Lateglacial and Preboreal biozones[END_REF], and the dense forest cover observed today in the watershed [START_REF] Pierret | The Strengbach catchment: A multidisciplinary environmental sentry for 30 years[END_REF].

Another explanation which cannot be excluded is that modern measurements were not affected by rare but high magnitude erosion events, as suggested by [START_REF] Kirchner | Mountain erosion over 10 yr, 10 ky, and 10 my time scales[END_REF].

The spatial variability of denudation can be discussed by the comparison of denudations rates determined from depth profile, topsoil, rock outcrop and stream sediment samples. As discussed in section 7.3, the assumption of steady-state 10 Be concentrations is not applicable for these sites. This suggests that the same assumption is probably too strong for the determination of denudation rates from other soil, outcrop, or sediment samples in the watershed and that the determined 𝑑 𝑚𝑎𝑥 values are biased. The approach to correct theses estimates developed in section 7.3 has been applied for these different samples to estimate more realistic denudation rates from 10 Be concentrations. For all the Strengbach samples, a corrected denudation rate 𝑑 has been determined by numerical inverse procedure from equation 4, with the assumption that the entire watershed is characterized by the same exposure age and the same inherited 10 Be concentration as established on depth profiles.

Comparison of corrected denudation rates inferred from topsoil, rock outcrop and stream sediment samples allows for a detailed understanding of the evolution of the watershed geomorphology (figure 9, table 2). On the upper part of the watershed, lower denudation rates on rock outcrops compared to soil and sediment denudation rates indicate an ongoing individualization and growth of bare rocks. The observation that rock outcrops have lower denudation rates than other parts of the basin is classically invoked to explain the prominence of these structures in the landscape [START_REF] Riebe | Long-term rates of chemical weathering and physical erosion from cosmogenic nuclides and geochemical mass balance[END_REF][START_REF] Heimsath | Eroding Australia: rates and processes from Bega Valley to Arnhem land[END_REF].

With denudation rates of around 22 mm.kyr -1 for the regolith and 4 to 9 mm.kyr -1 for the tors, in 19 kyr the topographic difference between the soil surface and tors would be a few tens of cm, consistent with field observations for the STR 11-1 site. However, these values are far too low to explain the morphologies observed at site STR11-5 where the bedrock outcrop surface overhangs the surrounding soils by nearly 2 m. With the estimated denudation rates presented above, such topographic differences would require nearly 150 kyr to be obtained, i.e. the time of the penultimate strong denudation event of the site based on the interpretations of [START_REF] Ackerer | Regolith evolution on the millennial timescale from combined U-Th-Ra isotopes and in-situ cosmogenic 10Be analysis in a weathering profile (Strengbach catchment, France)[END_REF]. We therefore propose that bedrock outcrops in these areas result from their exposure during stripping events of locally resistant bedrock portions, accounting for denudation at lower rates in these areas compared to other rocks. This interpretation remains to be validated by detailed studies of these tors, which is beyond the scope of this study.

It can also be noted that in the upper part of the watershed, the highest local denudation rates are observed on the crests (33-36 mm.kyr -1 ), with values significantly higher than the local rate determined above the main catchment knickpoint (i.e. STR11-7; 12 mm.kyr -1 ; figure 9). By comparing corrected spatially-averaged 𝑑 from sediments from the upper part of the stream with corrected local 𝑑 from soils on crests (as in [START_REF] Meyer | Determining the growth rate of topographic relief using in-situ-produced 10Be: A case study in the Black Forest, Germany[END_REF], the relief decrease between the crests and above the river knickpoint would be around 15 mm.kyr -1 .

However, it must be noted that this relief variation is close to uncertainties, if we consider the substantial variability of local denudation rates determined near the crest at the surface of depth profiles (33-14 mm.kyr -1 ; figure 9). Therefore, except the significant topographic growth of local granitic outcrops, the upper part of the watershed is likely affected by a limited relief variation. This is clearly in contrast with the variability of denudation rates at the scale of the entire watershed. The lower part of the watershed stands out from the rest of the watershed, as the highest denudation rate in the whole watershed is undeniably determined for the outlet subcatchment (99 mm.kyr -1 , figure 9). With the above assumptions (i.e. same exposure age and inherited 10 Be for the entire watershed), the high denudation rate of the outlet subcatchment compared to the ones determined from the sediments collected in the upper part of the stream (23-18 mm.kyr -1 ) implies that a regressive erosion from the bottom of the valley steepens the topography nearby the stream outlet. By comparing corrected spatiallyaveraged 𝑑 in the outlet subcatchment with corrected local 𝑑 from soils on crests, the total watershed relief increase would be around 65 mm.kyr -1 , following the approach given in [START_REF] Meyer | Determining the growth rate of topographic relief using in-situ-produced 10Be: A case study in the Black Forest, Germany[END_REF]. From the above discussion, it can also be proposed that the Strengbach watershed is subdivided into two different geomorphological systems, with an upper part with a nearly constant relief and a lower part characterized by a relief increase localized near the stream incision.

Different quantitative estimates would be determined for the geomorphological evolution of the watershed from denudation rates uncorrected or differently corrected. The lowest relief change between crests and outlet would be obtained with uncorrected denudation rates (≈20 mm.kyr -1 , figures 7). But from a qualitative point of view this would not change the conclusion that the Strengbach watershed is marked by a noticeable spatial variability of its denudation with an upstream subcatchment with limited relief change and a downstream one with a localized regressive denudation near the stream outlet. The Strengbach case therefore illustrates that even in a watershed where the steady-state assumption of 10 Be concentrations is not verified (𝑡 < 𝑇 𝑒𝑓𝑓 ), the spatial variations of in-situ 10 Be concentrations in sediments still carry qualitatively relevant information on the geomorphological evolution of landscapes.

Conclusion

We presented in this study a rare combination of in-situ 10 Be data from high resolution depth profiles, soils, rock outcrops and stream sediments in a small mountain watershed.

The high resolution of depth profile sampling was a key point to understand how the mobile regolith is generated and how it affects the CZ evolution (summary figure 8). Two different types of mobile regolith were identified: a deep mobile regolith explained by periglacial processes exclusively occurring during the LGM and a superficial mobile regolith generating a recent expansion or mixing of soil layers. Deep mobile regolith clearly modifies 10 Be concentrations and mineralogical, geochemical and physical evolution of the CZ. The modeling of regolith evolution and of 10 Be concentrations along depth allows to estimate the cosmic ray exposure age (19 kyr) and the mean denudation rate (22 mm.kyr -1 ) of highresolution depth profiles without any assumption of steady-state 10 Be concentrations.

Comparison with maximum denudation rates inferred from topsoil samples independently collected at the surface of depth profiles and interpreted under the assumption of steadystate 10 Be concentrations shows that this assumption leads to an overestimation of local denudation rates by a factor of approximately two. One important conclusion of our study is to show that if one wants to estimate denudation rates by following the classical approach (i.e. single topsoil sample, assumption of steady-state 10 Be concentrations), the best estimates are obtained when using the density of the bedrock. Results also indicate that maximum spatially-averaged denudation rates determined from stream sediment samples overestimate denudation rates by a factor of approximately two. These biases are significant to understand the precise geomorphological evolution of a watershed, and we propose a method to estimate corrected denudation rates. The comparison between corrected local and spatially-averaged denudation rates indicates that the watershed geomorphology is evolving and that regressive erosion increases the relief nearby the outlet. A key finding is also that a steady-state of 10 Be concentrations and a steady-state of regolith thickness (or mass balance) are two different situations that do not necessarily overlap. In the case of the Strengbach watershed crests, a steady-state of 10 Be concentrations is not achieved while regolith thickness tends to a steady-state. However, even in a watershed where the steadystate assumption of 10 Be concentrations is not verified (𝑡 < 𝑇 𝑒𝑓𝑓 ), the spatial variations of in-situ 10 Be concentrations in sediments still provide relevant information on the geomorphological evolution of landscapes. Be analysis were performed. The profile P1 was investigated by [START_REF] Ackerer | Regolith evolution on the millennial timescale from combined U-Th-Ra isotopes and in-situ cosmogenic 10Be analysis in a weathering profile (Strengbach catchment, France)[END_REF]. Mor litter is defined by a thick mat of undecomposed to partially decomposed litter that is not significantly incorporated into the soil, present in coniferous forests. Moder litter is defined by an undecomposed and partially decomposed remains of broad-leaved deciduous forest litter that is shallowly incorporated into the soil. Calculations of the muonic pre-exposure consider the erosion of regolith over time, and especially the important erosion event that removed approximately 2 m of regolith during the Last Glacial Maximum [START_REF] Ackerer | Regolith evolution on the millennial timescale from combined U-Th-Ra isotopes and in-situ cosmogenic 10Be analysis in a weathering profile (Strengbach catchment, France)[END_REF]. Figure 9: corrected denudation rates calculated from equation 4 by assuming a uniform cosmic ray exposure time across the watershed equal to the exposure time determined on depth profiles (t=19 kyr). Corrected denudation rates written in black are inferred from 10 Be concentrations measured in topsoil, rock outcrop and stream sediment samples. We used in calculations the density of 2.7 g.cm -3 for rock outcrops and the average regolith density of 1.7 g.cm -3 for topsoil and stream sediment samples. Denudation rate written in blue is determined through the modeling of high resolution depth profiles without assumption of steady state 10 Be concentrations (𝑑, equation 4). Denudation rate written in bold black for the outlet subcatchment is calculated from equation 10. Denudation rates are expressed in mm.kyr -1 and uncertainties include the uncertainties on the 10 Be concentrations, blank correction, and 10 Be production rates (uncertainties in table 2).

Figure caption

Table caption

Table 1: sample type, depth, bulk density, integrated density and 10 Be data for the three high resolution depth profiles from the southern part of the Strengbach watershed. Corrected denudation rates d were calculated without assuming a steady state of 10 Be concentrations (equation 4) by using inherited 10 Be and cosmic ray exposure age determined on depth profiles. The density of 2.7 g.cm -3 for rock outcrops and the average regolith density of 1.7 g.cm -3 for topsoil and stream sediment samples were used in corrected d calculations. Table S1: mineralogical composition of the samples collected within the three depth profiles from the southern part of the Strengbach watershed. The mineralogical compositions of all the powdered 250μm bulk samples were determined by XRD with a Siemens D5000 diffractometer (BRGM) or a X BRUKER D8 ADVANCE Eco Diffractometer (ITES). The <2 µm clay fractions were separated by centrifugation and analyzed after air drying and treatment with glycol ethylene and hydrazine to distinguish illites, smectites, kaolinites and inter-stratified clay minerals. Estimation of mineral proportion has been performed by the Rietveld method with the Siroquant software. (Uncertainty of ≈ 3 % for quartz and plagioclase mass proportion and ≈5 % for micas and clay minerals.

  study on the Strengbach watershed indicates that two different types of mobile regolith can occur along a slope: (1) a deep mobile regolith able to transport the soil and the lower regolith and (2) a near surface mobile regolith generating an expansion or a mixing of soil layers.For the deep mobile regolith, modeling results obtained for the three depth profiles indicate that regolith mobilization occurred around the last glacial maximum (LGM, around 24-19 kyr). The homogenization of mobile regolith was total (in P2) or partial (in P3) during transport, and stabilization occurred along the slope around 19 kyr BP. Given the moderate slopes around the three depth profiles (10-12°), the probable lack of vegetation and the prevalence of cold conditions at the elevation of the Strengbach watershed during the LGM[START_REF] Leroy | Palynological analyses in the laminated sediment of Lake Holzmaar (Eifel, Germany): duration of Lateglacial and Preboreal biozones[END_REF][START_REF] Heyman | Paleo-climate of the central European uplands during the last glacial maximum based on glacier mass-balance modeling[END_REF], the transport of deep mobile regolith is probably explained by periglacial solifluction and cryoturbation processes. Studies of periglacial processes in present day cold environments indicate that rock debris can be transported at velocities up to 10-50 cm.yr -1[START_REF] Kääb | Surface geometry, thickness changes and flow fields on creeping mountain permafrost: automatic extraction by digital image analysis[END_REF][START_REF] Wangensteen | Surface displacements and surface age estimates for creeping slope landforms in Northern and Eastern Iceland using digital photogrammetry[END_REF]. With a similar downward velocity during the LGM, the distance of 200 m between P1, P2 and P3 profiles would be covered by rock debris in only 400-2000 years. These estimates indicate that the transport of mobile regolith in a periglacial context can be fast compared to the 10 Be production timescale, explaining why 10 Be concentrations within the three profiles are well modeled by the same exposure age of 19 kyr.

Figure 1 :

 1 Figure 1: study watershed and sampling location. (a) Strengbach watershed location in northeastern France. (b) Sampling location of 10 Be samples including topsoil, rock outcrop, stream sediment and high-resolution depth profile samples. (c) Photography of the three high-resolution depth profiles realized on the southern part of the watershed. Yellow rectangles represent collected samples of soil, regolith or bedrock. Blue diamonds represent samples for which mineralogical, geochemical and 10 Be analysis were performed. The profile P1 was investigated by[START_REF] Ackerer | Regolith evolution on the millennial timescale from combined U-Th-Ra isotopes and in-situ cosmogenic 10Be analysis in a weathering profile (Strengbach catchment, France)[END_REF]. Mor litter is defined by a thick mat of undecomposed to partially decomposed litter that is not significantly incorporated into the soil, present in coniferous forests. Moder litter is defined by an undecomposed and partially decomposed remains of broad-leaved deciduous forest litter that is shallowly incorporated into the soil.

Figure 2 :

 2 Figure 2: chemical mobility coefficients (𝜏 𝑖 = ( 𝐶(𝑖) 𝑠𝑎𝑚𝑝𝑙𝑒 𝐶(𝑗) 𝑏𝑒𝑑𝑟𝑜𝑐𝑘 𝐶(𝑖) 𝑏𝑒𝑑𝑟𝑜𝑐𝑘 𝐶(𝑗) 𝑠𝑎𝑚𝑝𝑙𝑒 -1) ) for the three high-resolution depth profiles on the southern slope of the watershed. (a) Subplot for the depth profile P1. (b) Subplot for the depth profile P2. (c) Subplot for the depth profile P3.

Figure 3 :

 3 Figure 3: 10 Be data, volumetric variations, and bulk density from the three high-resolution depth profiles (P1, P2 and P3). (a) 10 Be concentrations measured in bulk samples. (b) Volumetric strain index ( 𝜀 = ( 𝐶(𝑗) 𝑏𝑒𝑑𝑟𝑜𝑐𝑘 𝜌 𝑏𝑒𝑑𝑟𝑜𝑐𝑘 𝐶(𝑗) 𝑠𝑎𝑚𝑝𝑙𝑒 𝜌 𝑠𝑎𝑚𝑝𝑙𝑒 ) -1). (c) Bulk density.

Figure 4 :

 4 Figure 4: direct modeling of the evolution of P1 profile. Black lines represent from top to bottom, soil surface, soil base and bedrock interfaces. (a) Bulk density evolution. (b) Integrated density evolution. (c) 10 Be production rate evolution. (d) 10 Be concentrations from muonic pre-exposure (150 kyr).Calculations of the muonic pre-exposure consider the erosion of regolith over time, and especially the important erosion event that removed approximately 2 m of regolith during the Last Glacial Maximum[START_REF] Ackerer | Regolith evolution on the millennial timescale from combined U-Th-Ra isotopes and in-situ cosmogenic 10Be analysis in a weathering profile (Strengbach catchment, France)[END_REF].

Figure 5 :

 5 Figure 5: modeling results for the P1 profile. (a) Measured and modeled bulk density. (b) Measured and modeled integrated density. (c) Comparison between measured and simulated 10 Be concentrations. (d) Comparison between measured and simulated 10 Be concentrations including a correction of soil expansion.

Figure 6 :

 6 Figure 6: modeling results for P2 and P3 profiles. (a) Measured and modeled integrated density for the profile 1. (b) Measured and modeled integrated density for the P2 profile. (c) Comparison between measured and simulated 10 Be concentrations for the P2 profile. (d) Comparison between measured and simulated 10 Be concentrations for the P3 profile.

Figure 7 :

 7 Figure 7: slopes, catchments of stream sediments and spatial variability of denudation rates in the Strengbach watershed. (a) Map of the watershed. Maximal denudation rates (𝑑 𝑚𝑎𝑥 ) written in black are inferred from 10 Be concentrations measured in topsoil, rock outcrop and stream sediment samples with the GIS toolbox Basinga (Charreau et al., 2019) using equation 9 (𝑑 𝑚𝑎𝑥 ≈ (

  Figure 1

  Maximal denudation rates (𝑑 𝑚𝑎𝑥 ) written in black are inferred from 10 Be concentrations measured in topsoil, rock outcrop and stream sediment samples with the GIS toolbox Basinga[START_REF] Charreau | Basinga: A cellby-cell GIS toolbox for computing basin average scaling factors, cosmogenic production rates and denudation rates[END_REF] using equation 9 (𝑑 𝑚𝑎𝑥 ≈ ( ) and under the assumption of steady state 10 Be concentrations. A constant bedrock density of 2.7 g.cm -3 was used in all calculations. The denudation rate written in blue is determined through the modeling of high-resolution depth profiles without the assumption of steady state 10 Be concentrations (𝑑, equation 4). The denudation rate written in bold black for the outlet subcatchment is calculated from equation 10. Denudation rates are expressed in mm.kyr -1 and uncertainties include the uncertainties on the 10 Be concentrations, blank correction, and 10 Be production rates (uncertainties in table 2). (b) Spatially-averaged 𝑑 𝑚𝑎𝑥 calculated for stream sediment catchments as a function of mean catchment aspect (° from N). (c) Spatially-averaged 𝑑 𝑚𝑎𝑥 calculated for stream sediment catchments as a function of mean catchment slope (°). Error bars represent one standard deviation of data. Figure 8: summary diagram of the four major stages of the CZ evolution along the southern slope of the Strengbach watershed. Stage (1) around 25 000 BP: presence of paleo profiles with inherited insitu 10 Be concentrations in the bedrock corresponding to the long term muonic exposure during approximately 150 kyr. Stage (2) around 24 000-20 000 BP: major erosion event removing the paleo profile soil and regolith in P2 and P3 but keeping in place part of the regolith in P1. Stage (3) around 19 000 BP: stabilization of deep mobile regolith on the bedrock of P2 and P3 profiles. P1 is still covered by part of the in-situ regolith. Stage (4) from 19 000 BP to present: period with lower denudation rates explaining the soil development and the exposure age of 19 000 yr determined for all profiles. In P1, a thin soil was developed from the weathering of in situ regolith and was affected by near surface mobile regolith and volumetric expansion. In P2 and P3, thicker soils were formed from the weathering of deep mobile regolith and were affected by near surface mobile regolith and surface mixing. Slopes and distances are purely illustrative but height variations were represented in

	(	𝑃.𝑃 𝑛𝑚 𝐶	)	𝐾 𝑛𝑚 𝜌	+ (	𝑃.𝑃 𝑓𝑚 𝐶	)	𝐾 𝑓𝑚
	accordance			with	modeling	results.
									𝑃.𝑃 𝑛 𝐶	)	𝐾 𝑛 𝜌	+

𝜌

Table 2 :

 2 sample type, location, elevation and 10 Be data for topsoil, rock outcrop and stream sediment samples collected across the Strengbach watershed. Maximum denudation rates (𝑑 𝑚𝑎𝑥 ) were inferred from the GIS toolbox Basinga[START_REF] Charreau | Basinga: A cellby-cell GIS toolbox for computing basin average scaling factors, cosmogenic production rates and denudation rates[END_REF] using equation 9 and assuming a steady state of 10 Be concentrations. A constant bedrock density of 2.7 g.cm -3 was used in 𝑑 𝑚𝑎𝑥 calculations.

48.209 °N, 7.193 °E, 1120 m)

	Sample STR11-3 STR13-43 STR13-45 STR13-47 STR13-49 STR13-51 STR13-53 STR13-55 STR13-59 STR13-61 STR13-42B STR13-43B STR13-45B STR13-47B STR13-49B STR17-4 STR17-14 STR17-20 STR17-26 STR17-28 STR17-30 STR17-32 STR17-34 STR17-36 STR17-40 STR17-43 Profile P3 (STR13-3 Type Depth (cm) Bulk density (g.cm -3 ) Integrated density (g.cm -3 ) Profile P1 (48.208 °N, 7.191 °E, 1146 m) Quartz mass (g) 9 Be (mg) Soil 7.0 1.0 0.9 25.02 0.477 Soil 22.5 1.5 1.2 17.23 0.285 Soil 32.5 1.2 1.2 19.72 0.293 Soil 42.5 1.7 1.3 14.68 0.290 Soil 52.5 1.7 1.3 22.54 0.287 Lower regolith 62.5 2.0 1.4 18.60 0.277 Lower regolith 72.5 2.1 1.5 23.79 0.281 Lower regolith 82.5 2.1 1.6 22.03 0.252 Lower regolith 102.5 2.5 1.7 22.45 0.303 Bedrock 112.5 2.6 1.8 18.32 0.274 Bedrock 135.0 2.7 1.9 29.31 0.291 Bedrock 145.0 2.7 2.0 40.35 0.248 Bedrock 165.0 2.7 2.1 38.00 0.278 Bedrock 185.0 2.7 2.2 48.84 0.268 Bedrock 205.0 2.7 2.2 57.09 0.280 Profile P2 (48.20900 °N, 7.19214 °E, 1131 m) Soil 7.5 1.1 1.1 25.49 0.519 Soil 57.5 1.9 1.5 21.67 0.518 Lower regolith 87.5 2.3 1.6 24.30 0.514 Lower regolith 117.5 2.6 1.8 30.84 0.511 Lower regolith 135.0 2.7 1.9 32.12 0.515 Lower regolith 155.0 2.7 2.0 37.16 0.516 Lower regolith 175.0 2.7 2.1 34.09 0.515 Bedrock 195.0 2.7 2.2 31.20 0.521 Bedrock 205.0 2.7 2.2 31.14 0.519 Bedrock 250.0 2.7 2.3 38.61 0.519 Bedrock 287.5 2.7 2.3 36.58 0.518 Soil 12.5 1.3 1.0 21.0 0.295 STR13-6 Soil 27.5 1.5 1.2 15.6 0.298 STR13-9 Soil 42.5 1.6 1.3 13.5 0.294 STR13-12 Soil 57.5 1.9 1.4 18.7 0.295 STR13-15 Soil 72.5 1.6 1.5 16.0 0.297 STR13-18 Lower regolith 87.5 1.9 1.6 17.6 0.297 STR13-21 Lower regolith 102.5 2.2 1.7 24.4 0.295 STR13-24 Lower regolith 117.5 2.4 1.7 14.3 0.294 STR13-27 Lower regolith 132.5 2.4 1.8 13.8 0.295 STR13-33 Lower regolith 162.5 2.5 1.9 21.6 0.297 STR13-36 Lower regolith 177.5 2.5 2.0 32.4 0.312 STR13-26B Bedrock 187.5 2.7 2.0 47.4 0.314 STR13-28B Bedrock 215.0 2.7 2.1 48.8 0.313 STR13-29B Bedrock 227.5 2.7 2.1 30.1 0.295 STR13-30B Bedrock 237.5 2.7 2.1 58.3 0.311 Type Location Elevation (m) Quartz mass (g) STR11-1 Rock outcrop 48.20951 °N -7.19587 °E 1115 25.09 STR11-5 Rock outcrop 48.21582 °N -7.19528 °E 1117 24.96 STR11-3 Soil (top profile P1) 48.20817 °N -7.19111 °E 1146 25.02 STR17-4 Soil (top profile P2) 48.20900 °N -7.19214 °E 1131 25.49 STR13-3 Soil (top profile P3) 48.20942 °N -7.19278 °E 1120 20.97 STR11-6 Soil 48.21621 °N -7.19550 °E 1110 39.95 STR11-7 Soil 48.21300 °N -7.19646 °E 1017 29.96 STR11-8 Stream sediments 48.21319 °N -7.19866 °E 986 39.91 STR11-9 Stream sediments 48.21341 °N -7.20348 °E 922 40.86 STR11-10 Stream sediments 48.21357 °N -7.19994 °E 962 39.34 STR11-11 Stream sediments 48.21377 °N -7.20012 °E 964 40.06 STR11-12 Stream sediments 48.21214 °N-7.20447 °E 891 39.35 Supplementary materials Supplementary tables Sample Type Depth (cm) Quartz (%) Plagioclase Feldspars (%) K-feldspars (%) Micas and 0.469 1.87E-13 10 Be/ 9 Be Err (%) 1.47E-13 3.1 1.55E-13 3.4 1.65E-13 4.0 1.16E-13 3.9 1.70E-13 3.9 1.32E-13 3.3 1.45E-13 3.8 1.23E-13 3.2 8.75E-14 3.9 6.39E-14 4.0 8.49E-14 3.9 1.09E-13 5.2 7.48E-14 4.5 8.31E-14 3.9 8.55E-14 3.8 1.61E-13 2.6 1.08E-13 3.1 7.79E-14 3.1 8.01E-14 3.3 7.38E-14 3.6 7.41E-14 3.1 5.87E-14 3.1 4.58E-14 3.4 3.55E-14 4.0 3.31E-14 4.9 2.45E-14 6.1 2.17E-13 3.2 1.30E-13 3.1 1.56E-13 3.3 1.88E-13 3.7 1.53E-13 3.1 1.39E-13 3.3 2.18E-13 3.4 8.81E-14 3.2 6.94E-14 3.7 8.00E-14 3.6 9.57E-14 5.3 9.50E-14 6.9 9.06E-14 7.1 3.74E-14 13 9.99E-14 9 Be (mg) 10 Be/ 9 Be 0.487 1.68E-13 0.477 1.45E-13 0.518 4.82E-13 0.295 2.17E-13 0.473 2.28E-13 0.474 2.13E-13 0.501 2.18E-13 0.508 2.28E-13 0.500 2.22E-13 0.481 2.34E-13 0.504 1.87E-13 illites (%) Kaolinite (%) Interstratified In-situ Be (atoms. g qtz -1 ) 188144 171954 163601 153162 144995 131079 114030 219429 171153 109549 196609 168477 228979 196839 194345 161022 172978 116320 Err (%) In-situ 10 Be Err (%) 3.5 3.9 4.5 4.4 4.4 3.9 4.3 3.7 4.4 4.5 4.4 5.5 4.9 4.4 4.3 3.3 3.7 3.7 3.9 4.1 3.7 3.7 3.9 4.4 5.3 6.4 3.8 3.7 3.9 4.2 3.7 3.9 4.0 3.8 4.2 4.1 5.6 7.2 7.4 13 (atoms. g qtz -1 ) 3.5 233458 3.2 218546 3.1 185587 2.6 219429 3.2 204375 3.1 180207 3.3 225541 3.1 182825 3.0 189279 3.1 188241 3.0 188057 3.1 159699 clays (%) Hematite (%) Profile P1 (48.208 °N, 7.191 °E, 1146 m) STR13-41 Soil 12.5 43 17 25 8 2 5 STR13-43 Soil 22.5 43 18 25 8 1 5 STR13-45 Soil 32.5 40 20 23 9 2 6 STR13-47 Soil 42.5 37 22 26 8 2 5 STR13-49 Soil 52.5 38 18 31 9 3 1 STR13-51 Lower regolith 62.5 46 15 21 16 2 0 STR13-53 Lower regolith 72.5 48 15 18 17 2 0 STR13-55 Lower regolith 82.5 52 16 17 11 4 0 STR13-57 Lower regolith 92.5 45 16 22 15 2 0 STR13-59 Bedrock 102.5 35 22 22 20 1 0 STR13-61 Bedrock 112.5 34 20 26 19 0 0 STR13-41B Bedrock 117.5 37 18 29 14 2 0 STR13-42B Bedrock 135 40 22 22 13 2 0 STR13-43B Bedrock 145 33 19 30 14 2 0 STR13-45B Bedrock 165 37 22 28 11 2 0 STR13-47B Bedrock 185 32 20 29 17 2 0 Profile P2 (48.20900 °N, 7.19214 °E, 1131 m) STR17-4 Soil 7.5 51 11 20 7 1 10 STR17-6 Soil 17.5 49 10 20 12 1 7 STR-17-8 Soil 27.5 48 13 23 7 1 8 STR17-12 Soil 49 48 14 22 6 1 9 STR17-14 Soil 61 47 16 25 10 1 0 STR17-18 Lower regolith 80 38 15 20 27 0 0 STR17-20 Lower regolith 88 38 18 20 24 0 0 STR17-24 Lower regolith 104 40 10 20 24 1 4 STR17-26 Lower regolith 112.5 51 4 21 16 0 6 STR17-28 Lower regolith 130 55 4 15 24 0 1 STR17-30 Lower regolith 150 43 15 18 24 0 0 STR17-32 Lower regolith 166.5 45 17 18 15 0 4 STR17-36 Bedrock 207.5 43 17 20 19 0 1 STR17-40 Bedrock 247.5 43 18 17 21 0 1 STR17-43 Bedrock 287.5 48 17 16 18 0 1 15 Sample Profile P3 (	Err (%) 4.0 3.7 3.7 3.3 3.8 3.6 3.9 3.7 3.6 3.7 3.6 3.7	dmax (mm.kyr -1 ) 30±1 34±2 42±3 34±2 37±3 42±2 30±1 40±2 36±2 38±2 38±2 43±3	Corrected d (mm.kyr -1 ) 4±1 9±2 33±4 14±3 20±4 36±4 12±2 23±2 18±2 19±2 19±2 38±5

48.209 °N, 7.193 °E, 1120 m)

	STR13-3	Soil	12.5	46	15	25	8	3	3
	STR13-6	Soil	27.5	39	19	26	9	1	6
	STR13-9	Soil	42.5	42	17	27	8	1	5
	STR13-12	Soil	57.5	43	19	25	7	2	4
	STR13-15	Soil	72.5	41	20	26	8	4	1
	STR13-18	Lower regolith	87.5	38	21	26	14	1	0
	STR13-21	Lower regolith	102.5	42	23	23	10	3	0
	STR13-24	Lower regolith	117.5	33	18	18	29	0	0
	STR13-27	Lower regolith	132.5	30	19	20	31	0	0
	STR13-30	Lower regolith	147.5	35	16	21	28	0	0
	STR13-33	Lower regolith	162.5	33	18	20	26	3	0
	STR13-36	Lower regolith	177.5	44	6	24	24	0	2
	STR13-26B	Bedrock	187.5	40	10	24	26	0	0
	STR13-27B	Bedrock	197.5	37	23	24	13	3	0
	STR13-28B	Bedrock	215	39	20	24	17	0	0
	STR13-29B	Bedrock	227.5	36	25	24	15	0	0
	STR13-30B	Bedrock	237.5	35	26	24	15	0	0

This study was financially supported by the French ANR program under grant agreement ANR-15-CE06-0014 (Project CANTARE-Alsace) and by the 2020 IPGS-LHyGeS project call (CNRS-University of Strasbourg). The study also benefited from the funding of a post-doctoral salary to Julien Ackerer by the Labex G-eau-Thermie (University of Strasbourg). for their help during chemical analysis at ITES and Martine Trautmann for the sample preparation (EOST). Daniel Viville and Eric Pelt are thanked for their involvement in the pit supported by the INSU-CNRS, the French Ministry of Research and Higher Education, IRD and CEA. We are grateful to Julien Charreau (CRPG, Nancy) for the discussions concerning the GIS Toolbox Basinga. This work also benefited from the support of the OZCAR research infrastructure.

Table S3: variables and numerical values used for the direct modeling of the three depth profiles from the southern part of the watershed. P is the total production rate of 10 Be at the surface of the soil (determined with the Basinga GIS toolbox; [START_REF] Charreau | Basinga: A cellby-cell GIS toolbox for computing basin average scaling factors, cosmogenic production rates and denudation rates[END_REF]. Pn, Pnm and Pfm are the relative contributions of 97.85, 1.5 and 0.65 % to the total production and Kn, Knm and Kfm are the effective attenuation lengths of 150, 1500 and 5300 g.cm -², for secondary neutrons, negative muons and fast muons, respectively [START_REF] Braucher | Determination of both exposure time and denudation rate from an in-situ-produced 10Be depth profile: a mathematical proof of uniqueness. Model sensitivity and applications to natural cases[END_REF]. We used λ=5E-7 yr -1 for the radioactive decay constant of 10 Be. Be concentrations (equation 4) by using inherited 10 Be and cosmic ray exposure age determined on depth profiles. The density of 1.7 g.cm -3 for stream sediment samples were used in corrected d calculations.

Supplementary figures

Figure S1: flow chart of the modeling of depth profile evolution and of 10 Be concentration.

Model flow chart