Roberta Farina
email: roberta.farina@crea.gov.it

Renata Sándor

Mohamed Abdalla

Jorge Álvaro-Fuentes

Luca Bechini

Martin A Bolinder

Lorenzo Brilli

Claire Chenu

Hugues Clivot

Massimiliano De

Antoni Migliorati

Bechini

Luca

Brilli

Lorenzo

Antoni De

Massimiliano Migliorati

Di Bene

Christopher D Dorich

Ehrhardt, Fiona 13

Rosa Francaviglia

Donna L Giltrap

Grant, Brian, B. 17 16

Matthew T Harrison

20

Miko U F Kirschbaum

Kulmala

Liisa

Matthew J Mcgrath

18

Elizabeth Meier

Menichetti

Nils Reibold

Anita Shepherd

Ward N Smith

Pete Smith

Jean-François Soussana

Stella Tommaso

Arezoo 28

Elena C Tsutskikh

Bellocchi

Gianni

C Di Bene

F Ferchaud

N Fitton

U Franko

B B Grant

B Guenet

K Kuka

L Kulmala

J Liski

L Menichetti

N, Reibold F Moyano

A. Taghizadeh-Toosi T Stella

L Dorich

R Menichetti

S Francaviglia

W Recous

F Smith

H Ferchaud

M A Clivot

W Bolinder

A Smith

L Taghizadeh-Toosi

R Brilli

G Farina

T Bellocchi

U Stella

Franko

Ensemble modelling, uncertainty and robust predictions of organic carbon in long-term bare-fallow soils

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

LIST OF SYMBOLS AND ABBREVIATIONS

Symbol/abbreviation Long version Explanation

System variables Heatwave frequency Number of at least seven consecutive days when the maximum air temperature is higher than the average summer (June, July and August) maximum temperature of a baseline value +3 °C

C

INTRODUCTION

The ability of soils to sequester and store large amounts of carbon (C) is well known (e.g. [START_REF] Lehmann | The contentious nature of soil organic matter[END_REF]. Soil organic carbon (SOC) stocks are crucial for maintaining soil fertility and preventing erosion and desertification, and they positively influence the provision of ecosystem services at the local as well as the global scale (e.g. [START_REF] Lal | Soil carbon sequestration impacts on global climate change and food security[END_REF][START_REF] Lal | Soil conservation and ecosystem services[END_REF]. For these reasons, farmers aim to establish and maintain high organic C stocks in agricultural soils, which have often been depleted trough historical land use practices [START_REF] Fuchs | Assessing the influence of historic net and gross land changes on the carbon fluxes of Europe[END_REF][START_REF] Gardi | High Nature Value Farmland: assessment of soil organic carbon in Europe[END_REF][START_REF] Chenu | Increasing organic stocks in agricultural soils: Knowledge gaps and potential innovations[END_REF]. The continuing studies on SOC sources and biogeochemical processes in the soil environment provide key insights into climate-C feedbacks, and help prioritizing C sequestration initiatives [START_REF] Gross | The case for digging deeper: soil organic carbon storage, dynamics, and controls in our changing world[END_REF]. In light of the climate change issue, the storage of C and additional sequestration of atmospheric C have received increasing attention recently [START_REF] Rumpel | Put more carbon in soils to meet Paris climate pledges[END_REF][START_REF] Whitehead | Management practices to reduce losses or increase soil carbon stocks in temperate grazed grasslands: New Zealand as a case study[END_REF][START_REF] Lavallee | Conceptualizing soil organic matter into particulate and mineral-associated forms to address global change in the 21 st century[END_REF], promoting land management, and agroecosystems in particular, as a key mitigation option (e.g. the '4 per mille Soils for Food Security and Climate' initiative, [START_REF] Minasny | Soil carbon 4 per mille[END_REF][START_REF] Soussana | Matching policy and science: Rationale for the '4 per 1000 -soils for food security and climate' initiative[END_REF]. However, the slow response of SOC to changes in management and environmental factors hampers our understanding of how SOC can be increased in a sustainable manner, especially under changing climatic conditions.

Long-term field experiments (LTEs), in which SOC responses have been observed over several decades, provide this information and deliver reference data on SOC content for knowledge gain and model development [START_REF] Johnston | The importance of long-term experiments in agriculture: their management to ensure continued crop production and soil fertility; the Rothamsted experience[END_REF]. However, LTEs are costly to maintain, and it is generally difficult to extrapolate experimental results across space and time (Debreczeni and Körschens, 2003;[START_REF] Mirtl | Genesis, goals and achievements of long-term ecological research at the global scale: a critical review of ILTER and future directions[END_REF]. Simulation models play a prominent role in SOC research because they provide a mathematical framework to integrate, examine and test the understanding of SOC dynamics [START_REF] Campbell | Current developments in soil organic matter modeling and the expansion of model applications: a review[END_REF]. They can also be used to extrapolate from micro-(e.g. carbohydrate production during photosynthesis) to macro-scale dynamics (e.g. global C cycling) (e.g. [START_REF] Gottschalk | How will organic carbon stocks in mineral soils evolve under future climate? Global projections using RothC for a range of climate change scenarios[END_REF][START_REF] Sitch | Evaluation of ecosystem dynamics, plant geography and terrestrial carbon cycling in the LPJ dynamic global vegetation model[END_REF]. In particular, complex agricultural and environmental models incorporate a mechanistic view of processes and system interactions, in which the soil components are often represented by different, operationally defined, pools of different sizes and with different properties (e.g. [START_REF] Parton | Modeling the dynamics of soil organic matter and nutrient cycling[END_REF]. The concept of multiple C-N pools represents C-N dynamics with an idealised description [START_REF] Hill | Generating generic response signals for scenario calculation of management effects on carbon sequestration in agriculture: approximation of main effects using CENTURY[END_REF]. The relative proportion of C and N (and sometimes lignin to N ratio) in the plant residue is the primary mode to divide plant inputs (from e.g. leaf litter and root exudates) into fresh litter pools, which then decompose into SOC (or SOM, i.e. soil organic matter) pools, each being modelled with different residence (or turnover) times, varying from months for labile products of microbial decomposition to hundreds to thousands of years for organic substances with firm organic-mineral bonds (e.g. [START_REF] Yadav | Progress in soil organic matter research: litter decomposition, modelling, monitoring and sequestration[END_REF][START_REF] Dungait | Soil organic matter turnover is governed by accessibility not recalcitrance[END_REF]. Plant material and animal manures are often modelled to enter the soil environment as either readily decomposable (carbohydrate-like) or resistant (lignin and cellulose-like) materials. A varying number of pools (often including inert and slowdecomposing organic matter, and microbial biomass) linked by first-order equations is usually simulating both C and N fluxes within and between each pool [START_REF] Falloon | Modelling soil carbon dynamics[END_REF]. However, different models vary considerably in the underlying assumptions and C processes in current models, e.g. regarding number of pools, type of decomposition kinetics used and processes regulating SOC retention [START_REF] Manzoni | Soil carbon and nitrogen mineralization: Theory and models across scales[END_REF][START_REF] Cavalli | Sensitivity analysis of C and N modules in biogeochemical crop and grassland models following manure addition to soil[END_REF].

Each model offers a distinctive synthesis of scientific knowledge [START_REF] Brilli | Review and analysis of strengths and weaknesses of agro-ecosystem models for simulating C and N fluxes[END_REF] and multi-model ensembles developed from several models may reduce uncertainties in biological and physical outputs that occur over large scales, such as regions and continents (e.g. [START_REF] Rötter | Simulation of spring barley yield in different climatic zones of Northern and Central Europe -A comparison of nine crop models[END_REF][START_REF] Asseng | Uncertainty in simulating wheat yields under climate change[END_REF][START_REF] Ehrhardt | Assessing uncertainties in crop and pasture ensemble model simulations of productivity and N 2 O emissions[END_REF]. The advantage of using ensemble estimates over individual models is that caused by compensation of errors across models, and a broader integration of model processes [START_REF] Martre | Multimodel ensembles of wheat growth: Many models are better than one[END_REF]. It has been recommended to use model ensembles for reducing uncertainties in simulations of agricultural production [START_REF] Asseng | Uncertainty in simulating wheat yields under climate change[END_REF][START_REF] Bassu | How do various maize crop models vary in their responses to climate change factors?[END_REF][START_REF] Challinor | Making the most of climate impacts ensembles[END_REF][START_REF] Li | Uncertainties in predicting rice yield by current crop models under a wide range of climatic conditions[END_REF][START_REF] Ruane | Multi-wheat-model ensemble responses to interannual climate variability[END_REF][START_REF] Maiorano | Crop model improvement reduces the uncertainty of the response to temperature of multi-model ensembles[END_REF] and other biophysical/biogeochemical outputs [START_REF] Sándor | Multimodel simulation of soil temperature, soil water content and biomass in Euro-Mediterranean grasslands: Uncertainties and ensemble performance[END_REF](Sándor et al., , 2018a;;[START_REF] Ehrhardt | Assessing uncertainties in crop and pasture ensemble model simulations of productivity and N 2 O emissions[END_REF].

However, after the pioneering study of [START_REF] Smith | A comparison of the performance of nine soil organic matter models using datasets from seven long-term experiments[END_REF], who evaluated nine SOC models using 12 datasets from seven LTEs, other modelling studies targeting SOC dynamics have often been limited in scope. [START_REF] Smith | Crop residue removal effects on soil carbon: Measured and inter-model comparisons[END_REF] used four models to assess the effect on SOC of crop residues' removal in 14 experiments in North America. [START_REF] Todd-Brown | Causes of variation in soil carbon simulations from CMIP5 Earth system models and comparison with observations[END_REF][START_REF] Lal | Soil conservation and ecosystem services[END_REF] performed global estimates of SOC changes with 11 Earth system models. [START_REF] Kirschbaum | Modelling carbon and water exchange of a grazed pasture in New Zealand constrained by eddy covariance measurements[END_REF] used one simulation model and two years of eddy covariance measurements collected over an intensively grazed dairy pasture in New Zealand to better understand the drivers of changes in SOC stocks. [START_REF] Puche | Modelling carbon and water fluxes of managed grasslands: comparing flux variability and net carbon budgets between grazed and mowed systems[END_REF] performed a similar study in France. Using multi-model ensembles in scenario studies at eight sites worldwide, [START_REF] Basso | Soil organic carbon and nitrogen feedbacks on crop yields under climate change[END_REF] highlighted the importance of soil feedback effects (C and N) on the prediction of wheat and maize yield. We are not aware of any recent model inter-comparison studies specifically assessing soil C dynamics with several models across a range of experimental sites. This is a field where there is a need for standardised guidance to estimate C stocks at various spatial scales [START_REF] Bispo | Accounting for carbon stocks in soils and measuring GHGs emission fluxes from soils: do we have the necessary standards[END_REF]. A difficulty in testing and comparing various models (and interpreting model outputs) lies in the interaction between soil and plant processes so that any of the model-data discrepancies could be due to errors in either component (e.g. [START_REF] Ehrmann | Plant: soil interactions in temperate multi-cropping production systems[END_REF]. A rigorous model testing and comparison would require different model components, e.g. plant and soil modules, to be assessed separately. Bare-fallow plots offer such an opportunity in that they are plots maintained for decades without any plant inputs. The changes in SOC stocks therefore result only from decomposition processes. To assess the function of soil-model components without interaction with plant processes, we conducted a model intercomparison using a dataset from long-term bare-fallow experiments where plant inputs were zero.

In this study, we refer to bare-fallow plots that were kept free of plants by manual and/or chemical means for several decades. We used seven bare-fallow treatments included in six long-term agricultural experiments (>25 years), all located in Europe (Denmark, France, Russia, Sweden and United Kingdom). In these plots, the soils became progressively depleted in the more labile SOM components, as they decomposed, and relatively enriched in more stable SOM [START_REF] Barré | Quantifying and isolating stable soil organic carbon using long-term bare fallow experiments[END_REF].

The soil C concentrations determined at given years in these sites represented a unique opportunity to follow the decay of SOC from a multi-model ensemble perspective, without any interference from new plant C inputs, and conduct a multi-model ensemble comparison. The model intercomparison included 26 process-based models from an international modelling community. Some models only accounted for soils and used C input from plants as an external input where others were full agro-ecosystem models that explicitly simulate plant growth and resulting C input into soils. These models all simulate interactions between the soil-atmosphere continuums in different ways, but for this comparison all models were run assuming no input of fresh plant-derived C, allowing the comparison of just the soil components of the models.

Here, we assess the models, by comparing multi-decadal simulations to experimental data from seven sites in Europe. The primary goal of this study was to assess the multi-model ensemble in simulating SOC dynamics across bare-fallow sites in Europe. To achieve this goal, model evaluation against actual measurements was performed before and after model calibration. In addition, deficient areas in models and their processes were identified, paving the road for future research directions.

MATERIALS AND METHODS

Simulation models

The ensemble of models consisted of 26 process-based models, mainly developed for crop or grassland ecosystems (or focussing just on soils) and covering a broad variety of approaches (Table 1). While they are mostly based on first-order decay kinetics of multiple C pools (where C losses are proportional to SOC stocks with additional modifiers to represent the effects of other factors),

ESOC1 simulates C fluxes with second-order kinetics equations based on concepts applied in [START_REF] Schimel | The implications of exoenzyme activity on microbial carbon and nitrogen limitation in soil: a theoretical model[END_REF] and reviewed in [START_REF] Wutzler | Colimitation of decomposition by substrate and decomposers -a comparison of model formulations[END_REF]. In this case, organic matter decomposition includes reactions between SOC and decomposers (i.e. a microbial or enzyme pool). These different approaches depend mainly on alternative ways in which the C pools are linked. For instance, MONICA is one of the most complex models, considering three types of organic matter in six conceptual pools, viz. newly added organic matter, living soil microbial biomass and native non-living soil organic matter, each sub-divided into fast and slowly decomposing sub-pools. It simulates the turnover of C pools by applying first-order degradation to each pool due to microbial growth and maintenance respiration (after [START_REF] Abrahamsen | Daisy: an open soil-crop-atmosphere system model[END_REF]. Then, like other models (e.g. CenW), MONICA also includes a coupled N-cycle and sophisticated temperature and water-balance calculations that act as modifiers of degradation and respiration rates. The decomposition rates of individual pools in such multi-pool SOC models are typically controlled by vastly different reaction coefficients that can result in highly nonlinear behaviour of the overall system (e.g. [START_REF] Caruso | Soil organic carbon dynamics matching ecological equilibrium theory[END_REF]. The initial list included 34 models, but eight of them were excluded from further analysis because they showed severe limitations to run properly either under bare-fallow soils or under the given climate conditions. For all models, estimates of SOC were compared with measured SOC data.

197

a Some models/model versions include options for varying C pools (this varying number may depend on the fact that the full 198 set of pools including fresh C can be optionally simplified in the case of bare-fallow treatments).

Experimental sites

We used data from a network of six long-term bare-fallow experimental sites (LTBF) in Europe (with two fields located in Askov, Denmark; [START_REF] Barré | Quantifying and isolating stable soil organic carbon using long-term bare fallow experiments[END_REF], to test the ability of the models to represent SOC dynamics. The sites were located at a range of latitudes between 48° to 59° North (Table 2; Fig. 1a), with experiments running for at least 28 years, which were used as a test bed for the models to represent SOC dynamics. Table 2 shows the main characteristics of each site and provides a brief description of the historical land use and management of the area (more details are given by Barré et al., 2010 and references therein). The documented history of the experimental sites referred to the presence of agricultural areas (grassland or cropland), without woodlands. Soil 1956-1985 1959-2007 1965-2001 1959-2008 1956-2007 1929-2008 1956-1987/1929-2008/1944-2003/1856-2006/1956-1999/1929-2008. b Köppen-Geiger climate classification [START_REF] Kottek | World map of the Köppen-Geiger climate classification updated[END_REF].

c Mean values over the bare-fallow period. Reference evaporation was estimated based on the [START_REF] Thornthwaite | An approach toward a rational classification of climate[END_REF] equation.

d Mean difference in temperature between the warmest and the coldest month of the year. [START_REF] Keel | Large uncertainty in soil carbon modelling related to method of calculation of plant carbon input in agricultural systems[END_REF] in order to initialise the SOC pools, which can sometimes be calculated analytically. In order to keep the legacy effect of previous land-use and past management practices, in SP models (e.g. DayCent) SOC pools are routinely initialised by running the models to achieve their own states of equilibrium, where change in C stocks is minimised (e.g. [START_REF] Lardy | A new method to determine soil organic carbon equilibrium[END_REF][START_REF] Huntzinger | The North American Carbon Program Multi-scale synthesis and Terrestrial Model Intercomparison Project-Part 1: Overview and experimental design[END_REF]. However, if soils are not at equilibrium (e.g. after a sudden disturbance), spin-up runs may not always be valid with the risk of starting simulations with biased initial values (e.g. [START_REF] Wutzler | Soils apart from equilibrium -consequences for soil carbon balance modelling[END_REF]Nemo et al., 2017) but a fuller discussion on the "spin-up problem" [START_REF] Reynolds | Sustainable forestry: from monitoring and modelling to knowledge management and policy science[END_REF] is not within the scope of this paper. Carbon inputs are usually estimated through sub-models calculating total net primary production (TNPP). As it was not possible to derive TNPP data from local sources at each study-site, TNPP estimates were obtained at each site (Table 2) based on precipitation levels according to the approach of Del Grosso et al. (2008). In this way, the creation of the TNPP database used by modellers was based on an identical methodology, which is widely used worldwide, though the uncertainty in quantifying productivity across ecosystems is highlighted (e.g. [START_REF] Wieder | Evaluating soil biogeochemistry parameterizations in Earth system models with observations[END_REF].

The distinction between SP and NS models can appear somewhat arbitrary as virtually any model with more than one C pool could be spun-up or, alternatively, a function (or analytical procedures) can be used to make an initial pool partition. We refer here to common modelling practice, as performed by users within the constraints imposed by packaged (operational) solutions of SOC models (for which spin-up procedures may be operationally more difficult) or relying on the procedure suggested by previous experience. For instance, although spin-up equilibrium runs are documented for RothC (e.g. [START_REF] Herbst | Correspondence of measured soil carbon fractions and RothC pools for equilibrium and non-equilibrium states[END_REF], it is common practice to initialise three C pools for subsequent simulations through an internal routine over 10,000 years, with limited model inputs including clay fraction and weather, and a pre-defined ratio of decomposable over recalcitrant plant material (e.g. [START_REF] Xu | Modeling the change in soil organic carbon of grassland in response to climate change: Effects of measured versus modelled carbon pools for initializing the Rothamsted Carbon model[END_REF][START_REF] Weihermüller | Simple pedotransfer functions to initialize reactive carbon pools of the RothC model[END_REF]. Modellers were left to choose one option or the other when both were available for use in their models (e.g. C-TOOL).

About 40% of the models (10 models) in the study did not use SP processes and set the initial SOC values manually (using the initial SOC observation).

For each model category (SP and NS), two main modelling approaches were identified: sitespecific versus generic (single set of parameter values for all the sites). For the site-specific approach, at each site users informed models about historical management practices and land uses such as grassland or cropland (with both SP and NS models), SOC decomposition parameters (only for SP models) or the partitioning of C among different soil pools (only for NS models). With the generic (not site-specific) approach, model calibration was not applied separately for each experimental site but simultaneously on all available multi-location datasets to find for each model parameter values that would be applicable at regional scales. In this case, multi-location calibration was used to capture generic model parameter values so that the models could still perform well across a range of climate and management conditions in Europe (Dechow et al., 2019). Sitespecific and non-site-specific approaches were variously combined with factors affecting model initialisation/parameterisation (Table 3) to create simulation scenarios Gen (generic), Mix (mixed)

and Spe (specific).

Scenario Mix uses a site-specific approach for the initialisation of C pools with both SP and NS models and, for each model, a unique calibration of decomposition parameters. Fixed decomposition rate parameters (but not rate modifiers) were maintained at a constant value throughout all sites (e.g. the maximum passive pool decomposition rate in M25 was set to 0.003 yr -1 at all sites), while site-specific climate and soil textural conditions provided supplementary factors driving the actual decomposition curve (likely in the uncalibrated blind simulations as well). In scenario Spe, decomposition rates could be changed separately at each experimental site, which constrained the modelling to a fitting exercise, but made it possible to explore the spatial variability of model parameters. Scenario Gen ignored base histories of each site: arable crops and grasslands were not distinguished, past climate conditions were disregarded, and this translated into discounting the variability in the TNPP levels among sites affecting the starting SOC level. [START_REF] Parton | A general model for soil organic matter dynamics: sensitivity to litter chemistry, texture and management[END_REF][START_REF] Moriasi | Model evaluation guidelines for systematic quantification of accuracy in watershed simulations[END_REF][START_REF] Confalonieri | Multi-metric evaluation of the models WARM, CropSyst, and WOFOST for rice[END_REF][START_REF] Bellocchi | An indicator of solar radiation model performance based on a fuzzy expert system[END_REF][START_REF] Bellocchi | Validation of biophysical models: issues and methodologies. A review[END_REF].

EF = 1 - ∑ n i = 1 (P i -O i ) 2 ∑ n i = 1 (O i -O) 2
negative infinity to 1 (optimum): the closer the values are to 1, the better the model parameters with the uncertainty in model structure [START_REF] Wallach | Estimating uncertainty in crop model predictions: Current situation and future prospects[END_REF]. While the uncertainty in model predictions could be due to parameterisation, model calibration from different users (i.e. ensemble of users within ensemble of models) cannot be regarded as the solution to estimate uncertainty due to parameterization (Confalonieri et al., 2016). As well, different calibration techniques do not seem to be primarily responsible for differences in model performance [START_REF] Wallach | How well do crop models predict phenology, with emphasis on the effect of calibration? bioRxiv[END_REF] and the contribution of the initialisation to the uncertainty in SOC changes can be negligible compared to the uncertainty related to the model itself and simulated systems characteristics [START_REF] Dimassi | The impacts of CENTURY model initialization scenarios on soil organic carbon dynamics simulation in French long-term experiments[END_REF]. As uncertainty could not be associated with any individual simulation, we focussed on the analysis of model residuals. We documented the variability of the multi-model simulation exercise across two stages (blind test and alternative calibration scenarios), while inspecting how the multi-model median (MMM) converged to the observations. We used box-plots to compare the variability of estimates by different models (with focus on multi-year averages) to the observed variability, and we represented model ensembles with MMM, which has the advantage to exclude distinctly biased model members with a disproportionate influence on the mean [START_REF] Rodríguez | Implications of crop model ensemble size and composition for estimates of adaptation effects and agreement of recommendations[END_REF]. The advantage of using MMM was established in practical studies in crop and grassland modelling but also on a theoretical basis [START_REF] Wallach | Multimodel ensembles improve predictions of crop-environment-management interactions[END_REF].

R 2 = ∑ n i = 1 (P i -O i )•(O i -O) ∑ n i = 1 (P i -P) 2 • ∑ n i = 1 (O i -O) 2 0
∑ n i = 1 (O i -P i ) 2 ∑ n i = 1 (|P i -O| + |O i -O|) 2
We also quantified the relationship among standardised model residuals of SOC, based on uncalibrated (Bln) and calibrated (Gen, Mix, Spe) simulations. Moreover, we quantified the relationship between residuals of agro-climatic metrics (annual values): temperature amplitude, mean maximum temperature and annual precipitation. Arrays of pairwise scatterplots (scatterplot matrices) were generated with the panel plot option in the R language and environment for statistical computing ('panel.smooth', https://stat.ethz.ch/R-manual/Rdevel/library/graphics/html/panel.smooth.html), which also overlaid a local non-parametric smoother curve (locally estimated scatterplot smoothing) on each plot to give some indication of trends (after [START_REF] Cleveland | Robust locally weighted regression and smoothing scatterplots[END_REF].

To explore how MMM varied with the number of models in the ensemble, we performed a calculation for each z-score transformed MMM, , which was obtained by dividing the 𝑧 = 𝑀𝑀𝑀 -𝑂 𝑠𝑑 𝑜𝑏𝑠 multi-model data deviation from the mean of observations ( ) by the standard deviation of the 𝑂 observations (sd obs ) [START_REF] Sándor | Ensemble modelling of carbon fluxes in grasslands and croplands[END_REF]. A z-score can be placed on the normal distribution curve to indicate how much it deviates from the mean of the distribution. The units of a z-score are sd units: zero equals the mean, positive z-scores exceed the mean, and negative z-scores are less than the mean. A z-score allows comparisons to be made between combinations of models with different distribution characteristics, i.e. different and sd obs (used here as practical descriptors of time-𝑂 series central tendency and spread). As illustrated in Fig. 2, different sites occupy distinct zones in the sd obs versus space. Low variability and low mean SOC observations were found at Askov 𝑂 (S1, S2), Grignon (S3) and Utuna (S6). The variability was higher at Rothamsted (S5) and Versailles (S7), while the mean was the highest at Kursk (S4). None of the site occupies the upper right quadrant, i.e. high variability and high mean.

(Fig. 2 here)

We calculated z-scores for all possible combinations of sets of k out of n=26 models (k=2, … n).

The minimum number of models providing plausible estimates at each site was that for which the z-scores lay within the ranges -1 to +1 or -2 to +2. The arbitrary choice of these thresholds was due to a conventional rule, for which values falling within 1 and 2 times the standard deviation approximate the 68% (|z|=1) and 95% (|z|=2) confidence limits of a normal distribution, respectively (after [START_REF] Ehrhardt | Assessing uncertainties in crop and pasture ensemble model simulations of productivity and N 2 O emissions[END_REF]. R software (https://cran.r-project.org) was used for statistical analysis and graphical visualization.

RESULTS

Evaluation of SOC dynamics

Fig. 3 show the range of model results (represented by the shaded area) for each scenario and the multi-model median (MMM hereinafter) together with the measured values. In general, the greatest spread of model results was found under the Bln scenario, followed by the Gen scenario.

In some cases, the multi-model median of Bln and Gen scenarios overestimate observations (e.g. at S5, S6 and S7 sites). As expected, the tightest range of model results (simulation envelope) was found with site-specific simulations. MMM simulations of Spe came closest to the observations.

All the MMM lines were remarkably close to the observations at sites S1, S2 and S3 (Fig. 3), despite the much wider spread of the individual simulations, while the MMM at other sites differed more substantially from the observations (e.g. S5, S6 and S7, Fig. 3). Overall, most of the simulations (Bln, Gen and Mix) tended to overestimate the amount of SOC (e.g. S5, S6 and S7, Fig. 3).

SOC stocks decreased under all bare-fallow sites during the investigated period. At S1, S2, S3, S4 and S6 (Fig. 3) sites, the decrease in SOC stock was from minimum to moderate whereas at S5 and S7 (Fig. 3) SOC loss in the top 0.20 m was more rapid, with initial SOC halved during ~30 years. The decay tended to be more rapid in the first years and then the rate of loss decreased (e.g. at S7 site between 1929 and 1962, Fig. 3).

(Fig. 3 here) On the other hand, the MMM of Gen scenarios showed the closest values to the observed median at S5 and S7 (Fig. 4.).

Ensemble performance by site

Overall, with some exceptions, the MMM of calibrated runs were within the range of the 25 th and 75 th percentiles of observations. The Spe scenario provided the best MMM estimation.

(Fig. 4 here)

Individual models versus multi-model ensemble

The scatterplot analysis for both each model and the MMM shows that SOC estimates were improved when moving from the Bln runs (Fig. 5) to the calibration Spe scenario (Fig. 6). Model performances for calibration Mix and Spe scenarios also showed better simulation results than the Bln simulations (see also Appendix A and Appendix B). Considering all the sites and years, the predictions of some of the models (e.g. M02, M13, M22, M24 and MMM) were close to the observations even for the blind level simulations (correlation coefficient >0.9, Fig. 5). Simulations improved even further (correlation coefficient >0.98 for half of the models, Fig. 6) under scenario Spe.

All the correlation coefficients of the simulations by other models also considerably improved with the site-specific data and got closer to the 1:1 line. For instance, for M31, the spread of simulation data in the blind simulations (Fig. 5) was mainly caused by incorrect initial SOC estimates for the different sites. When the model was re-run with correctly set initial SOC amounts (Fig. 6), the subsequent drawdown of SOC over the bare-fallow period was estimated fairly well.

Even with blind simulations, MMM gave results in agreement with the observations (R 2 =0.94).

This level of agreement was only exceeded by M22 (R 2 =0.95) and approached by M02 (R 2 =0.92) and M13 (R 2 =0.90). The MMM simulations continued to give the closest agreement with the observations even under the full site-specific calibrations (R 2 =0.99) with several other models performing equally well (i.e. M02, M05, M09, M13, M23, M26). Overall, with some specific information for model calibration, many models did remarkably well in reproducing the observed patterns of SOC loss over time.

(Fig. 5 here) (Fig. 6 here)

Analysis of model residuals

The plots of the discrepancy between MMM and observations (Fig. 7) as a function of time shows a limited scatter (within ±1) at each site. While Bln, Gen and Mix scenario overestimated the SOC decomposition rate at Kursk (where the highest SOC content was measured), the standardized residuals were around zero at Grignon and both Askov sites during the whole of experimental period. However, the departure from observations may increase over time especially with Bln and Gen scenarios at some site (e.g. at Rothamsted, Ultuna, Versailles) indicating that models underestimate decomposition rates after a few years/decades.

(Fig. 7 (Fig. 8 here) .

Minimum ensemble size

We attempted to identify the minimum number of models required to obtain reliable results for Bln and calibration scenarios Mix, Spe and Gen (Fig. 9 and Appendix C-E). We observed that there could be large differences in the z-scores obtained across sites with different ensemble sizes and scenarios. Overall, Bln is characterised by greater z-scores than the calibration scenarios. Our analysis suggests that the ensemble size could be reduced to four models (or even fewer) at S3, S6 and S7. For the other sites (e.g. S4), only ensemble sizes of at least 9-10 models reduced z-scores to within the range from -2 to +2, but this number should be raised to 20 or higher to comply with the most stringent criterion of z=|1|. A minimum ensemble size of 9-10 models was also identified with Gen at S4 (Fig. 9), while with Mix and Spe scenarios the number of models could be reduced down to 7 and 3, respectively (up to about 14 [Gen], 8 [Mix] and 4 [Spe] to comply with z=|1|)

(Appendix C-E).
(Fig. 9 here)

DISCUSSION

Scenarios of ensemble SOC estimates

For Bln, Mix, Gen and Spe scenarios, the overall differences between the simulated and the observed first-year SOC values were -0.46, +3.49, +2.40 and +1.92 Mg C ha -1 , respectively, for the NS models, and +0.58, -0.29, +0.95 and -0.12 Mg C ha -1 , respectively, for the SP models.

Despite manually setting the initial SOC values (magnitude of first SOC observation for the simulation period), the NS models mostly overestimated SOC content in the initial year of the model run. In first-year estimates of the calibrated (mainly with Spe and Mix scenarios), SP models deviated less from observations than NS models that overestimated SOC stocks for the first year with the exception of M25 (+8.4 Mg C ha -1 for Gen), M29 (+18.6, +21.1 and +23.7 Mg C ha -1 for Spe, Gen and Mix, respectively) and M31 (+25.2 Mg C ha -1 for Gen). In the case of M25, the model was run with a generic grassland spin-up (i.e. 7,000 years), which was applied to all sites.

Thus, a generic history was simulated without considering the cropping history at each site. This spin-up protocol affected the simulated SOC, showing the poor ability of Gen scenario to produce results consistent with observations, which questions the practicality of spin-up processes under generic calibration. With M31, there was a greater difference between simulated and observed SOC values in the initial simulation year and the model gave results that did not correspond to the observations at all sites (Appendix F), especially under the Bln and Gen scenarios. Though M31 used the initial SOC observation as default parameter, it failed to reproduce the LTBF dynamics between sites because of large differences in C input to the soil from the former vegetation during the spin-up period. Consequently, the starting points of the LTBF simulations differed greatly from the observations, which were overestimated at S1, S2, S3 and S6, and underestimated at S4.

Overall, Mix and Spe calibrations showed better performance indices than the Gen scenario (Appendix F). We note, however, that M13, for which the SOC pool sizes (humads and humus)

were generically calibrated across sites, produced low RRMSE for Gen (5.7%).

The improved calibration knowledge obtained with the site-specific information also improved model accuracy. Moving from Bln (with knowledge of weather and soil texture, historical land use and management, and initial SOC; section 2.3) to the Gen scenario, we reproduced SOC data in a number of European bare-fallow experimental sites with a single set of calibrated, regional-scale parameter values (regardless of the possible soil, climate and past land-use dissimilarities between different sites). According to performance indicators in Appendix F, in the Bln simulations the NS models performed better than the SP models. For instance, average RRMSE and EF were 19.44% and 0.60, and 26.94% and 0.24, for NS and SP models, respectively. Compared to the Bln scenario, the discrepancy between the measured and estimated SOC values under the Gen scenario was slightly reduced with NS models and increased with SP models. Multi-site calibration can be characterised by lower uncertainty than site-specific calibration, because more data contribute to the calibration process (e.g. [START_REF] Minunno | Integrating ecosystems measurements from multiple eddy-covariance sites to a simple model of ecosystem processare there possibilities for a uniform model calibration[END_REF][START_REF] Ma | Regional-scale analysis of carbon and water cycles on managed grassland systems[END_REF]. The availability of a variety of detailed data from multiple sites thus offers the possibility of a genuine multi-location calibration of the model, assuming that a single calibration across sites is appropriate. The limit of the Gen scenario calibration was that it did not make it possible to explore the spatial variability of model parameters. The latter was done with scenarios Mix and Spe, for which a basic requisite is that model parameters are not hard coded but configuration files are left open to the users. From Gen to Mix, parameters describing initial values of each pool were determined separately for each site.

Moving from Mix to Spe, the decomposition parameters became site-specific. Hence, modellers needed to invest increasingly more knowledge (and more time-demanding calibration effort) than in Gen. Under these conditions, the improvement of simulations in SP models was evident (up to 70% for some indicators, e.g. RRMSE and EF). On the contrary, NS models only had a slight improvement in accuracy of simulations from Bln (RRMSE=21.5%; EF=0.58) to Mix (RRMSE=18.6%, EF=0.55) or Gen (RRMSE=20.5%; EF=0.45). In our analysis, the two types of models (NS and SP) appear to be suitable for different sets of data. NS-type models, in most cases, can perform well even when data are limited to climate, initial C and historic land use, while SP models generally benefit from the availability of more detailed data. All metrics related to the performance of the SP models were improved with calibration. There were some differences in model performance among the sites, but site-specific soil or climatic conditions cannot easily explain such differences.

Overall, across the seven LTEs and using simulated and observed SOC data at the end of the experimental period we observe that the greatest and least differences from observations were approximately +14.3% with Bln and +2.2% with Spe (Fig. 10). The Gen scenario achieved almost half the error (+8.9%) of is closest competitor, i.e. the Bln scenario. More than one-third of the Bln-scenario error is achievable with the Mix scenario (+4.0%).

(Fig. 10 here)

This study has shown that it is difficult to define an a priori criterion that could be used to select a subset of models that would perform better than others would. In terms of the minimum number of models required to obtain reliable results, our study indicates that the suggested minimum ensemble size (~10 models) proposed by [START_REF] Martre | Multimodel ensembles of wheat growth: Many models are better than one[END_REF] for crop growth could be a reference also when model ensembles are implemented to blindly simulate SOC in bare-fallow soils, which can be reduced down to 3-4 models with a site-specific calibration. These sizes are lower than that found by [START_REF] Sándor | Ensemble modelling of carbon fluxes in grasslands and croplands[END_REF] to provide reliable C-flux estimates in croplands and grasslands (i.e.

~13 models). While the current study applied the same methodology as [START_REF] Sándor | Ensemble modelling of carbon fluxes in grasslands and croplands[END_REF], but

as the present study focuses on one output variable only, SOC, evaluated in simplified systems (bare-fallow soils), its relative ease of simulation offers great advantages for scenario analyses in the absence of vegetation cover and plant residues, nor farming practices (only occasional tillage operations occurred at some sites and were considered by models which can simulate this option).

This is reflected in the several z-scores within the range of -2 and +2, as obtained with a limited number of models, showing that reduced ensemble sizes can satisfactorily estimate the SOC content in bare-fallow systems, mainly when site-specific calibration is possible. However, our analysis of the Russian site (S4), which had low observed variability and high mean (sd obs =6.9, 𝑂 =91.8 Mg C ha -1 ), is challenging because it showed that model ensembles that are too small might not always guarantee sufficient accuracy in SOC estimates of C-rich soils. An application to the peatlands located on the Mid-Russian Upland (e.g. [START_REF] Shumilovskikh | Long-term dynamics of the East European forest-steppe ecotone[END_REF] should thus be considered with caution.

Possibilities for model inaccuracies

We presented an approach that uses a correlation matrix (with graphical representation) to account for possible correlations between Bln, Mix, Gen and Spe residuals and, additionally, climatic factors (mean air temperature amplitude, maximum air temperature and precipitation total). This residual analysis helps find correlations among alternative scenarios, which might indicate comparable scenarios in which error propagation within models is similar, though the way of error propagation cannot be easily retrieved from the correlation matrix. This is the case of Bln, Gen and Mix, whose residuals are highly correlated, while the weak correlations between Spe and other scenarios highlight the distinct behaviour of the latter. This analysis can also help find correlations between the SOC output and external drivers, and thus suggest additional predictors that may need to be included in the models (e.g. Medlyn et al., 2005). This need emerged especially when specific models were run under Bln, Gen and Mix scenarios, for which some correlations (r>|0.4|) were obtained between model residuals and drivers of thermal and moisture conditions. A weaker but significant correlation (r=0.26, p=0.02) was also obtained between Spe residuals and precipitation.

These correlations indicate some limitations related to the response functions of SOC decomposition to soil temperature and soil moisture, though the relative uncertainties of our model ensemble are attenuated by the presence in the models of physical and chemical processes that explain the intra-and inter-annual variability of SOC. We add that such biophysical conditions affect the microbial activity (e.g. [START_REF] Blagodatskaya | Mechanisms of real and apparent priming effects and their dependence on soil microbial biomass and community structure: critical review[END_REF][START_REF] Guenet | Is there a linear relationship between priming effect intensity and the amount of organic matter input[END_REF][START_REF] Wutzler | Priming and substrate quality interactions in soil organic matter models[END_REF], and care should be taken when extrapolating our results over long time frames (especially without locally calibrated models, Fig. 7) if no corroborating field evidence for long-term decay rates can be obtained (e.g. on how models are dealing such situations in which microbes become increasingly C limited as no new C input by plants occurs; [START_REF] Kuhry | Fossil carbon/nitrogen ratios as a measure of peat decomposition[END_REF].

CONCLUSIONS AND FUTURE DIRECTIONS

This paper on SOC modelling offers a tentative answer to the questions about: In our study, we did not aim to identify the best model(s) for simulating SOC dynamics for barefallows and no probability of success was assigned to prove the suitability of using one model rather than another. Overall, we showed that a calibration scenario with generic system knowledge was adequate for providing sufficiently reliable output, but additional site-specific knowledge can further improve results under certain circumstances. This is operationally relevant because the effort required to gather calibration data might no longer be feasible for modelling scenarios moving from single sites to increasingly larger spatial scales. Site-specific calibration could help refine model estimates. However, geographical locations have characteristics (e.g. soil and climate conditions, past history) that require specific model structures and local optimisation, and the application of models may be limited by the ability to provide representative parameter values.

Soil-C model inter-comparisons including more models and experimental data from other regions should be continued to improve our ability to simulate biogeochemical processes with acceptable accuracy. Additional assessments are also recommended to complete the analysis of model behaviour in the long term (like thousands of years) with constant inputs. While the various models evaluated here did not include all available modelling approaches used to simulate soil C dynamics, the present model inter-comparison was large compared to other studies. As such, it is a distinct improvement over previously published quantitative approaches because it represents a reasonable sub-population of common and current approaches. In this, we offer a method to allow a broad ensemble of models to be implemented using existing datasets and current modelling practices. Overall, this multi-model ensemble sets a precedent for key progress in soil C modelling because it provides essential information about SOC modelling and opens a path to a more indepth analysis of the response of individual models and their uncertainties against soil and climate drivers. Now that we have examined SOC decomposition in-depth without the difficulties of C input uncertainties, a similar modelling study should be conducted on LTEs that examine both plant derived C inputs as well as C inputs from manures and other organic materials recycled in agroecosystems. In fact, under field conditions, the amount of C input is not only an important factor driving the changes in SOC stocks (including the changes due to tillage), but the amount of C input also drives the mineralization rate of the SOC [START_REF] Mary | Soil carbon storage and mineralization rates are affected by carbon inputs rather than physical disturbance: Evidence from a 47-year tillage experiment[END_REF]. How simulation models compare under such conditions is important for improving our ability to evaluate and achieve climate C goals. With increasing availability of data and computational resources, there are many opportunities for the SOC modelling community to enrich its offering and to keep up with evolving methodologies, which would significantly increase transparency of the underpinning science and modelling practice. A number of recent actions are ongoing under the guidance of international initiatives such as the European Joint Programme (EJP) on Soil (https://projects.au.dk/ejpsoil).

Started in 2020, the EJP-Soil is undertaking a detailed inventory of models and all available data sources (e.g. world soil maps, satellite images, downscaled weather data), and appears as an ideal arena to facilitate the exchange of information and to further explore SOC model developments and practice. 2).

169x169mm (150 x 150 DPI) Fig. 3. Temporal changes of soil organic carbon (SOC, Mg C ha-1) observations (Observed, purple square) and simulations: blind (Blind, blue) simulations (26 models); three calibration scenarios, Generic (16 models, pink), Mixed (17 models, green) and Specific (17 models, grey) at all sites ( as in Table 2). Lines represent the multi-model median (MMM) of the simulations and shaded area represents the simulation envelope 1) versus observations (coloured symbols represent sites as in Fig. 1).

Fig. 6. Multi-year, multi-site comparison of individual model simulation of SOC (Mg C ha-1): multi-model medians (MMM) from Specific scenario simulations (17 models as in Table 1) versus observations (coloured symbols represent sites as in Fig. 1). 3).

152x169mm (300 x 300 DPI)

  texture provides evidence of variability in soil physical properties, with a gradient of intermediate situations between the sandy loam of Askov (Denmark) and the clay loam of Ultuna (Sweden). Water relations (precipitation minus reference evapotranspiration) indicate positive climatic water balance for the two North Atlantic sites only (Askov in Denmark and Rothamsted in the United Kingdom). Mean annual temperatures vary from ~6 °C in the Sweden and Russian sites (Ultuna and Kursk, respectively) to near 11 °C in the two French sites (Grignon and Versailles). Annual air temperature amplitudes -from about 14 °C in Rothamsted to near 30 °C in Kursk -indicatethat the study sites span a broad thermal gradient (Fig.1b), which likely leads to different soil thermodynamics (e.g.Zhu et al., 2019). Two widely used metrics (aridity index and frequency of heatwaves;[START_REF] Sándor | Multimodel simulation of soil temperature, soil water content and biomass in Euro-Mediterranean grasslands: Uncertainties and ensemble performance[END_REF]Sándor et al., , 2018a, b) , b) were also calculated to complete the climatic analysis of study sites (Fig.A, supplementary material).

e

  Estimates of aboveground (ANPP) and total (TNPP) net primary productivity based on the precipitation levels of each site, as provided by Del[START_REF] Del Grosso | Global potential net primary production predicted from vegetation class, precipitation, and temperature[END_REF] for non-tree dominated systems.Model simulations were carried out independently by each modelling team (which included model developers and users, and field experts of soil C dynamics) on commonly formatted data using their own approaches and technical background. Harmonising calibration techniques was out of scope of the inter-comparison exercise. The SOC outputs from each model were compared to data from the study sites before and after calibration. Calibration mostly focussed on parameters related to substrate use, C partitioning among pools and decomposition processes. However, rate equations for C pools often required the calibration of a large number of parameters, which are at the core of key processes responsible for differences among models in the understanding and interpretation of SOC processes (number of pools and type of decomposition kinetics used to represent C turnover). For the uncalibrated (blind test, Bln) simulations, the models were run for each site using the available data of weather, soil texture and bulk density (model inputs), and the initial SOC values, with no parameter adjustment other than initialisation based on historical management and land use. With this information, Bln reflects the ability of the models to simulate SOC decomposition after plant inputs has stopped, using the original parameter settings and calibration, simply by removing their components related to new C inputs. At this stage, default values were mostly used for all decomposition rates. C-pool fraction sizes were adjusted based only on C-input estimates from the information on land use prior to the establishment of the barefallow treatments.After the blind simulations were completed, SOC measurements taken during the barefallow period were supplied to each modelling group for the calibration work. Details on management (tillage), which may have influenced the SOC dynamics before the bare-fallow treatment, were also provided to improve the initialisation process. It was requested that each modelling group adjust soil parameters to improve the simulations based on the observed data, using whatever techniques they normally use, and to document the changes. At this stage, models were split into two categories: a) with spin-up (SP) and b) without spin-up (NS). Both SP and NS models require an initial estimate for SOC content and/or an adjustment of parameters towards balancing the split between soil C pools. The two classes of models work in the same way using information about plant residues and root growth that provide the C substrate for SOC dynamics simulations. NS-type models (e.g. DNDC and RothC) use the initial measured SOC value, where estimates of C inputs in the background of model runs are obtained with various methods (e.g.

  (absence of fit of the regression line) to 1 (perfect fit of the regression line): the closer the values are to 1

  We first focussed on the quantification of model-data discrepancies and then assessed the uncertainty of the individual models in comparison with the multi-model ensemble. The modelling teams provided deterministic model simulation results according to the protocol established, which meant that: 1) one run was provided for each site; 2) the spread of model results due to parameter uncertainty was not specifically addressed. The latter would have dramatically increased the range of model outputs used within the study and would have confounded the uncertainty in calibrated

Fig. 4

 4 Fig. 4 shows a high variability in the multi-model spread of responses at different sites. The results

  here)Model residuals displayed one versus the other can help establish relationships by exploring the correlation of residuals from different modelling scenarios, both among them and with external drivers. Residuals of blind test and calibration scenarios calculated from MMM (Fig.8) and individual models (Figs. B1-26 in the supplementary material) were correlated with the mean annual climate indicators such as the precipitations, maximum temperatures and temperature amplitudes. When considering the MMM, residuals of Bln were strongly correlated with Gen (r=0.90) and with Mix (r=0.59) residuals, but less with Spe (r=0.25) residuals, indicating a higher similarity of the first three approaches, while residuals of Spe were more correlated with those of Mix (r=0.65) than of Gen (r=0.39).The most prominent effect of annual climate indicators was found at the blind test stage, whose residuals were negatively correlated with precipitation (r=-0.17) and positively correlated with Tmax (r=0.41). Combinations of high maximum air temperature and low precipitation values may thus generate greater errors in blind SOC simulations. Calibration scenario Gen did not show significant correlations to climate indicators. However, calibration scenario Spe and Gen had opposite correlations. The annual precipitation positively correlated with Spe residuals (r=0.26) and with scenario Mix (r=0.15). Annual maximum temperature and scenario Spe negatively correlated (r=-0.10). These correlations with climate indicators hint that the site-specific calibration (scenario Spe) is more sensitive to precipitation than to maximum temperatures. On the contrary, Bln and Gen simulation residuals showed greater sensitivity to maximum temperatures.Residuals of individual models were approximately equally influenced by precipitation and temperature drivers, but with differences among models and scenarios (Figs. B1-26 in the supplementary material). In most of the cases, model residuals were positively correlated with annual maximum temperatures and negatively correlated with annual precipitation totals (e.g. M03, M09, M18, M22 for Bln). In some cases, e.g. M09 (Fig.B8in the supplement), the correlations among SOC residuals for different scenarios were both positive and negative (r values ranged from -0.043 to 0.36), and even the effect of climate indicators were different (e.g. for Tmax, r values ranged from -0.096 to 0.65). In other cases, e.g. M25 (Fig.B18in the supplement), SOC residuals were more similar to each other (r-values 0.17-0.80) and the effect of precipitation and temperature drivers was often important (with r>0.4). It is interesting in this respect that the Spe residuals had near-zero correlations with climatic drivers, showing a lesser influence of these factors on model results with this scenario, whereas the Bln scenario showed some correlations with Tamp (r=0.13), Tmax (r=-0.44) and precipitation (r=0.40). For M25, Gen scenario residuals (Fig.B18in the supplement) appeared unrelated with precipitation (r-value near zero), but not with temperature amplitude (r=0.50) and maximum air temperature (r=-0.56).

  (i) whether and to what extent an ensemble of models performs better than single models, (ii) the minimum ensemble size that is required to reduce the error below a given threshold, and (iii) the set of data required to prepare and substantiate ensemble estimates. This study presents a framework for interpretation of model performance and uncertainties obtained with a set of process-based biogeochemical models (individually and in an ensemble) simulating soil C contents in bare-fallow experimental systems at a variety of European sites. One of the features of SOC modelling today is the huge amount and variety of models available. Although our analysis did not take into account all sources of uncertainty (e.g. the influence of the unique choices made by modellers), it enabled the integration of several modelling teams into an ensemble protocol. Classifying and comparing different approaches have revealed great model diversity, and is the basis for the development of dedicated ensemble protocols. In this model inter-comparison, the need to accommodate challenges experienced by modellers (including C pools of different nature, and optional initialisation and calibration procedures) was reflected in the co-creation (with modellers and data providers) of alternative calibration scenarios (Mix, Gen, Spe). As far as we are aware, no previous multi-model inter-comparison studies have examined differences in such calibration scenarios or differences between models with or without spin-up.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Location (a) and characterisation of the study sites (b) with respect to mean annual temperature (°C)and mean annual temperature range (°C). Details about study sites are in Table2.558x254mm (150 x 150 DPI)

Fig. 4 .

 4 Fig. 4. Soil organic carbon (SOC, Mg C ha-1) at each site (as in Table2), for blind simulations (Blind, (26 models), three calibration scenarios (Mixed, 17 models; Specific and Generic, 16 models) and observations (Observed). For each boxplot, black horizontal lines show the multi-model median. Boxes delimit the 25th and 75th percentiles. Whiskers are 10th and 90th percentiles. Dots indicate outliers.

Fig. 5 .

 5 Fig. 5. Multi-year, multi-site comparison of individual model simulation of SOC (Mg C ha-1): multi-model medians (MMM) from blind simulations (26 models as in Table1) versus observations (coloured symbols represent sites as in Fig.1).

Fig. 7 .

 7 Fig. 7. Standardized model residuals ( (MMM-O)/〖sd〗_obs ) over time for blind (Blind) simulations and calibration scenarios Mixed, Specific and Generic at each site. 381x190mm (150 x 150 DPI)

Table 1 .

 1 The process-based simulation models used. Model names were anonymised in the

	reporting of simulation results using model codes from M01 to M34, from the initial list of 34
	models, the order of models not being identical to that used in the table.

Model name Version C pools a Spin-up URL or contact for documentation/description References

	Page 15 of 80				Global Change Biology
	YASSO15	Daily 15	5	Yes	https://en.ilmatieteenlaitos.fi/yasso	Tuomi et al. (2009)
		DayCent
		August 2014
	Exp		1	None	-
						Lorenzo	Menichetti
						(lorenzo.menichetti@slu.se)
	Exp + inert	4.5 2013	2	None	-
					martin.bolinder@slu.se
	AMG DNDC ICBM	2 CAN …	2 to 3 2	None None Yes (10 years recommended)	https://www6.hautsdefrance.inra.fr/agroimpact/Nos-dispositifs-outils/Modeles-et-outils-d-aide-a-la-http://www.dndc.sr.unh.edu https://www.slu.se	Andriulo et al. (1999); Saffih-Hdadi and Mary Andrén and Kätterer (1997); (2020) Li et al. (2012); Smith et al. Andrén et al. (2008)
				Yes,	decision/AMG-et-SIMEOS-AMG/AMG-model-description	(2008); Clivot et al. (2019) Nendel et al. (2011); Specka
	MONICA	2.0.2	7	None 20 years prior to beginning of the None	http://monica.agrosystem-models.com	Jones et al. (2003); Porter et et al. (2016); Stella et al.
	DSSAT	Apsim 7.9-…		experiment to estimate the	http://dssat.net	al. (2009); Gijsman et al. (2019)
	APSIM ORCHIDEE	r4044 2.0	3 3	Simulation from start of climate record (no additional simulation proportions of carbon in each organic matter pool Yes	http://www.apsim.info https://vesg.ipsl.upmc.fr/thredds/fileServer/IPSLFS/orchidee/ DOXYGEN/webdoc_2425/annotated.html	Keating et al. (2003); (2002); White et al. (2011); Thorp et al. (2012) Krinner et al. (2005)
		RothC10N		period)	https://www.abdn.ac.uk/staffpages/uploads/soi450/ECOSSE%	Holzworth et al. (2014)
	ECOSSE RothC	7.10 r4158 5.0.1 26.3	4 to 5	Yes None None	20User%20manual%20310810.pdf https://www.rothamsted.ac.uk/rothamsted-carbon-model-rothc	Smith et al. (2007, 2010a, Coleman and Jenkinson b); Bell et al. (2010) (1999); Farina et al. (2013)
		1.0 (but
	CANDY_CIPS ESOC1 STICS	always implemented 1.0 9.0	4 2 to 4	None Yes None	https://www.ufz.de/export/data/2/95948_CANDY_MANUAL. pdf https://doi.org/10.5281/zenodo.3539484 http://www6.paca.inra.fr/stics	Brisson et al. (1998, 2003, Kuka, (2005); Kuka et al. (2007) Moyano et al. (2018) 2008); Coucheney et al.
		in newest			fmoyano@uni-goettingen.de	(2015)

Table 2 .

 2 Long-term bare-fallow experimental sites. Table A in the supplementary material contains the summary description of the experimental sites.

	Experimental sites (country)

Experimental period N. of data/replicates

		Mean annual precipitation total (mm)	890	584	482	723	457	608
		Mean annual cumulative evaporation
			578	662	602	630	546	668
		(mm) c
		Mean annual air temperature (°C)	7.4	10.7	6.2	9.4	6.0	10.7
		Mean annual air temperature range
			17.6	16.8	29.8	14.4	22.8	16.7
		(°C) d
	Vegetation	ANPP (g C m -2 yr -1 )	1.7	1.1	0.9	1.3	0.9	1.2
	(historical
		TNPP (g C m -2 yr -1 )	3.3	2.2	1.7	2.5	1.7	2.2
	period) e
			30/4, 29/4	11/6	6/0	14/4	18/4	9/6
		Initial/final carbon stocks (Mg C ha -1 )	52.1/36.4	41.7/25.4	100.3/79.4	71.7/28.6	42.5/26.9	65.5/22.7
			Dfb (humid		Dfb (humid		Dfb (humid
	Climate a	Climate type b		Cfb (oceanic)		Cfb (oceanic)		Cfb (oceanic)
			continental)		continental		continental

a Climatic analysis was performed on longer periods than the experimental periods:

Table 3 .

 3 Modelling approaches and simulation scenarios for spin-up and no spin-up models (Gen: The term 'generic', which refers to calibration, here means 'ubiquitous' or 'universal', since the aim of any model is to work well under all conditions, without the need to adjust decomposition coefficients. In this case, the model correctly represents the main processes and integrates the main factors to accurately simulate the C cycle. The 'specific' calibration, which aims at improving the model performance, implicitly suggests an incomplete knowledge of the SOC turnover. The 'specific' calibration allow exploring the spatial variability of model parameters, but this amplitude (which is not discussed or reported here) may indicate the extend of degree of the knowledge gap in soil processes (i.e. model parameters might need a huge adjustment across sites) Twenty-six modelling teams participated in the blind test. At calibration stage, 17 teams completed scenarios Spe and Mix, and 16 the scenario Gen. Some model packages are set to restrict access to individual parameter values, which did not allow users to carry out some site-specific scenarios (Mix and Spe). The same outputs were obtained with some models (e.g. RothC, DNDC), which run blind and generic simulations with non-specific information like the previous land-use type (arable crop or grassland) and the historical climate. When results from the blind test were exactly equal to outputs from Gen scenario, they were not included for further analysis. Estimated and observed SOC values (Mg C ha -1 ) were compared at blind test and for each calibration

	Calibration
	scenarios a

generic; Mix: mixed; Spe: specific). a scenario. The agreement between simulations and observations was evaluated by the inspection of time series graphs and, numerically, through a set of performance metrics (Table

4

) combining difference-and correlation-based metrics (e.g. De Jager et al.

Table 4

 4

. Model performance metrics (P, predicted value; O, observed value; n, number of P/O pairs; i, each of P/O pairs; , mean of observed values; , average of the differences between O D predicted and observed values; S D , standard deviation of the differences between estimated and observed values).

  of the manuscript. M. Abdalla, J. Álvaro-Fuentes, M. A. Bolinder, L. Brilli, H. Clivot, M. De northern high latitudes. Nature Communications, 10, 3172. https://doi.org/10.1038/s41467-Spe and Gen as in Table3) across sites. Red (italic) and blue (bold) numbers indicate the worst and best performances by metric, respectively.

	Page 67 of												Global Change Biology
				Appendix F
			Bln Spe Mix	019-11103-1 Individual and multi-model ensemble (MMM) performance metrics (as in Table 4) for blind (Bln) 0.88 0.97 0.84 0.93 0.90 0.89 0.90 0.97 0.93 0.97 0.71 0.94 0.85 0.79 0.99 0.89 0.97 6.5 3.4 NA 5.0 3.2 NA NA 3.8 NA 3.8 8.2 NA NA 6.7 4.4 5.0 NA NA 0.90 NA 0.87 0.89 NA NA 0.89 NA 0.99 NA NA 0.55 -0.72 0.98 0.31 0.93	0.73 14.5 0.34	0.95 4.1 0.99	0.91 6.2 0.93	0.95 14.9 NA	0.59 46.2 -0.78	0.95 5.5 0.83	0.52 8.7 0.85	0.85 NA NA	0.93 NA 0.79	0.98 3.2 0.97
	Performance P(t)	metric	Gen Scenario Mix Bln Gen Spe	M01 NA NA ~0.00 ~0.00 ~0.00 ~0.00 ~0.00 ~0.00 ~0.00 ~0.00 ~0.00 ~0.00 M02 M03 M04 M05 M06 M07 M09 M12 M13 ~0.00 NA NA NA NA NA ~0.00 NA 0.13 0.97 NA 0.96 0.97 NA NA 0.97 NA ~1.00 and calibration scenarios (Mix, Model M16 M18 M19 M20 0.17 NA NA ~0.00 ~0.00 M22 NA NA 0.89 0.69 ~1.00 0.64 0.02 ~0.00 ~0.00 0.31 NA 0.71 NA NA NA NA NA 0.93 NA 0.99 0.97 NA NA 0.66 0.96 0.97 0.99 NA 0.98 0.99 NA NA 0.99 NA 0.99 0.94 NA NA 0.96 0.98	M23 NA 0.79 ~0.00 ~0.00 M24 0.08 0.98 NA 0.97 0.98 NA	M25 0.04 0.81 ~0.00 0.53 0.87	M26 ~0.00 ~0.00 M27 ~1.00 0.98 0.45 0.05 0.95 0.97 0.99 0.97	M28 0.06 NA ~0.00 ~0.00 M29 ~0.00 ~0.00 ~0.00 M30 M31 0.76 0.96 0.96 0.13 ~0.00 ~0.00 M32 NA NA 0.97 0.81 0.97 0.23 NA 0.82 -0.76 0.97 0.94 NA	M34 ~0.00 0.93 0.01 0.94 NA	~0.00 MMM 0.99 ~0.00 0.98 0.99
	R 2		Spe	0.99	~1.00	NA	~1.00 ~1.00	NA	NA	~1.00	NA	~1.00	0.99	NA	NA	0.99	~1.00	0.99	NA	0.97	~1.00	0.99	0.95	0.76	0.99	0.98	NA	NA	~1.00
			Mix	NA	~0.00	NA	~0.00 ~0.00	NA	NA	0.55	NA	0.31	NA	NA	~0.00 ~0.00	0.76	~0.00	0.54	~0.00	0.31	~0.00	NA	~0.00	0.24	~0.00	NA	~0.00	0.49
	Bln RRMSE (%)	0.73	0.92	0.67	0.83	0.79	0.86	0.76	0.89	0.83	0.90	0.33	0.81	0.69	0.63	0.95	0.76	0.92	0.41	0.86	0.76	0.92	0.21	0.82	0.35	0.57	0.80	0.94
			Bln Spe	24.1 0.46	10.9 0.99	28.0 NA	18.6 0.06	21.9 0.03	21.9 NA	23.1 NA	12.5 0.85	17.7 NA	11.8 0.34	28.6 ~0.00	15.5 NA	27.2 NA	33.1 0.12	7.9 0.93	25.4 ~0.00	11.0 NA	36.6 ~0.00	14.0 ~1.00	24.0 0.29	14.4 ~0.00 ~0.00 ~0.00 48.4 16.3	69.1 0.68	27.7 NA	16.3 NA	10.4 0.83
			Gen	NA	0.39	NA	NA	NA	NA	NA	0.79	NA	0.97	0.90	NA	NA	0.56	0.87	NA	0.89	0.09	0.86	0.93	0.91	0.82	0.93	~0.00	NA	0.85	0.95
	EF
			Gen	NA	30.8	NA	NA	NA	NA	NA	17.9	NA	5.7	11.5	NA	NA	51.3	14.0	NA	12.1	49.4	14.5	12.7	10.9	37.9	12.4	92.1	NA	15.8	10.6
			Mix Bln	NA 0.52	0.91 0.90	NA 0.49	0.90 0.72	0.91 0.60	NA 0.60	NA 0.56	0.89 0.87	NA 0.74	0.99 0.88	NA 0.33	NA 0.80	0.83 0.39	0.41 0.09	0.98 0.95	0.56 0.47	0.94 0.90	0.49 -0.11	0.99 0.84	0.95 0.53	NA 0.83	0.91 -0.93	0.84 0.78	0.87 -2.95	NA 0.37	0.82 0.78	0.97 0.91
			Mix	NA	11.0	NA	12.6	11.5	NA	NA	11.7	NA	3.8	NA	NA	23.3	45.6	4.4	29.0	8.9	33.0	4.2	9.4	NA	46.5	14.4	13.4	NA	15.9	7.2
			Spe Gen	0.97 NA	0.99 0.22	NA NA	0.98 NA	0.99 NA	NA NA	NA NA	0.99 0.73	NA NA	0.99 0.97	0.96 0.89	NA NA	NA NA	0.96 -1.17	0.98 0.84	0.99 NA	NA 0.88	0.91 -0.49	0.99 0.83	0.97 0.87	0.88 0.90	0.93 -0.19	0.98 0.87	0.94 -6.00	NA NA	NA 0.79	0.99 0.93
	d

ACKNOWLEDGEMENTS

This study was supported by the project "C and N models inter-comparison and improvement to assess management options for GHG mitigation in agro-systems worldwide" (CN-MIP, 2014-2017), which received funding by a multi-partner call on agricultural greenhouse gas research of the Joint Programming Initiative 'FACCE' through national financing bodies. S. Recous, R. Farina, L. Brilli, G. Bellocchi and L. Bechini received mobility funding by way of the French-Italian GALILEO programme (CLIMSOC project). The authors acknowledge particularly the data holders for the Long Term Bare-Fallows, who made their data available and provided additional information on the sites: V. Romanenkov, B.T. Christensen, T. Kätterer, S. Houot, F. van Oort, A. Mc Donald, as well as P. Barré. The input of B. Guenet and C. Chenu contributes to the ANR "Investissements d'avenir" programme with the reference CLAND ANR-16-CONV-0003. The input of P. Smith and C. Chenu contributes to the CIRCASA project, which received funding from the European Union's Horizon 2020 Research and Innovation Programme under grant agreement no 774378 and the projects: DEVIL (NE/M021327/1) and Soils-R-GRREAT (NE/P019455/1). The input of B. Grant and W. Smith was funded by Science and Technology Branch, Agriculture and Agri-Food Canada, under the scope of project J-001793. The input of A. Taghizadeh-Toosi was funded by Ministry of Environment and Food of Denmark as part of the SINKS2 project. The input of M. Abdalla contributes to the SUPER-G project, which received funding from the European Union's Horizon 2020 Research and Innovation Programme under grant agreement no 774124.

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request and permission of the third parties (i.e. the data holders for the Long Term Bare-Fallows, V. Romanenkov, B.T. Christensen, T. Kätterer, S. Houot, F. van Oort, A. Mc Donald, as well as P. Barré).

AUTHOR CONTRIBUTIONS

R. Farina, R. Sándor and G. Bellocchi coordinated the study, contributed to its design, conducted the analysis of data and produced the first draft of the manuscript. P. Smith, C. Chenu, F. Ehrhardt, M. A. Bolinder, C. Nendel and J.-F. Soussana contributed to the design of the study and the writing Confalonieri, R., Orlando, F., Paleari, L., Stella, T., Gilardelli, C., Movedi, E., ... Acutis, M. (2016) (coloured symbols represent sites as in Fig. 1).

Appendix B

Multi-year, multi-site comparison of individual model simulation of SOC (Mg C ha -1 ): multimodel medians (MMM) from Gen scenario simulations (16 models) versus observations.

(coloured symbols represent sites as in Fig. 1).

Appendix


Pierre-Hugo Vial

Paul Magron

Thomas Oberlin

Cédric Févotte

Phase retrieval with Bregman divergences: Application to audio signal recovery

Phase retrieval aims to recover a signal from magnitude or power spectra measurements. It is often addressed by considering a minimization problem involving a quadratic cost function. We propose a different formulation based on Bregman divergences, which encompass divergences that are appropriate for audio signal processing applications. We derive a fast gradient algorithm to solve this problem.

Introduction

Reconstructing data from phaseless measurements is at the core of many signal processing problems in various domains such as X-ray crystallography [START_REF] Harrison | Phase problem in crystallography[END_REF], optics [START_REF] Walther | The question of phase retrieval in optics[END_REF] and astronomy [START_REF] Fienup | Phase retrieval and image reconstruction for astronomy[END_REF]. The reconstruction task is usually treated as an optimization problem involving a quadratic cost function, which is minimized by, e.g., gradient descent [START_REF] Candes | Phase retrieval via Wirtinger flow: Theory and algorithms[END_REF], alternating projections [START_REF] Gerchberg | A practical algorithm for the determination of phase from image and diffraction plane pictures[END_REF], majorization-minimization [START_REF] Qiu | PRIME: Phase retrieval via majorization-minimization[END_REF], and alternating direction method of multipliers [START_REF] Liang | Phase retrieval via the alternating direction method of multipliers[END_REF][START_REF] Wen | Alternating direction methods for classical and ptychographic phase retrieval[END_REF].

In particular, many audio signal processing applications such as speech enhancement or source separation operate on the magnitude of the short-time Fourier transform (STFT), also called spectrogram. Recovering the missing phase from a modified spectrogram is then required in order to synthesize timedomain signals. The Griffin-Lim algorithm (GLA) [START_REF] Griffin | Signal Estimation from Modified Short-Time Fourier Transform[END_REF], a version of the alternating projections algorithm proposed by Gerchberg and Saxton [START_REF] Gerchberg | A practical algorithm for the determination of phase from image and diffraction plane pictures[END_REF] adapted to STFT measurements, is one of the most popular techniques in the literature and can be considered as a baseline for audio signal recovery. Improvements of GLA include an accelerated version (Fast GLA) [START_REF] Perraudin | A fast Griffin-Lim algorithm[END_REF] and real-time purposed versions [START_REF] Beauregard | An efficient algorithm for real-time spectrogram inversion[END_REF][START_REF] Zhu | Real-time iterative spectrum inversion with look-ahead[END_REF].

In this work, we propose to replace the quadratic loss in the phase retrieval (PR) optimization problem by a general Bregman divergence. It encompasses divergences such as βdivergences [START_REF] Hennequin | Beta-divergence as a subclass of Bregman divergence[END_REF], with special cases such as Kullback-Leibler and Itakura-Saito that are acknowledged for enabling better performance in audio spectral decomposition [START_REF] Févotte | Nonnegative matrix factorization with the Itakura-Saito divergence: With application to music analysis[END_REF][START_REF] Smaragdis | Static and dynamic source separation using nonnegative factorizations: A unified view[END_REF]. The resulting objective is minimized by accelerated gradient descent, with an efficient computation of the gradient involving STFT and inverse STFT. We illustrate the potential of this new approach for some well-chosen divergences on speech signals.

Phase retrieval

Phase retrieval. PR consists in reconstructing a signal x 0 ∈ C L from nonnegative measurements r ≈ |Ax 0 | d ∈ R K , where the power is applied element-wise and A ∈ C K×L is the measurement matrix. Usually, r corresponds to magnitude (d = 1) or power (d = 2) measurements. PR is often formulated as the following optimization problem:

min x∈R L r -|Ax| d 2 . (1)
Short-time Fourier transform. In this work, the measurement operator is the STFT. For every signal x ∈ R L , the STFT coefficient at time n ∈ {0, . . . , N -1} and frequency m ∈ {0, . . . , M -1} is the dot product

(Ax)[m + nM ] = x, a * m+nM , (2)
where

a m+nM [l] = w a (l-nH)e -2iπ m M l ∀l ∈ {0, . . . , L-1},
with w a the analysis window and H the hop size. The inverse STFT operator B is defined for every sequence of complex coefficients ỹ ∈ C M N as

(Bỹ)[l] = m,n ỹ[m + nM ]b * m+nM [l], (3)
where b m+nM [l] = w s (l-nH)e -2iπ m M l ∀l ∈ {0, . . . , L-1}.
Under duality assumptions for w a , w s and H, one can show that B = A H [START_REF] Portnoff | Time-frequency representation of digital signals and systems based on short-time Fourier analysis[END_REF]. In the following, the vectorized timefrequency coefficients are denoted with a tilde, i.e., x = Ax. Griffin-Lim algorithm. GLA [START_REF] Griffin | Signal Estimation from Modified Short-Time Fourier Transform[END_REF] solves [START_REF] Beauregard | An efficient algorithm for real-time spectrogram inversion[END_REF] in the STFT domain by alternating projections on M, the set of coefficients that satisfies the magnitude constraints, and C, the set of consistent coefficients:

P M (x) = r x |x| and P C (x) = AA H x, (4)
where denotes the element-wise product. GLA is proved to converge to a critical point of the measure minimized in (1) when d = 1. Like the Gerchberg-Saxton algorithm (GSA) [START_REF] Gerchberg | A practical algorithm for the determination of phase from image and diffraction plane pictures[END_REF],

GLA is an alternating projection algorithm. They however differ as GSA considers the DFT as the measurement matrix and adds a supplementary constraint on the signal to make the solution unique. In GLA, this constraint is not necessary as uniqueness is brought by the redundancy of STFT.

Algorithm

We propose a generalization of (1) to the family of Bregman divergences. The problem writes min

x∈R L J(x) := D ψ ( r | |Ax| d ), (5)
where D ψ is the Bregman divergence associated to ψ :

D ψ ( x | y ) = ψ(x) -ψ(y) -∇ψ(y), x -y . (6)
As illustrated by the Table 1, many divergences or distances can be expressed as a Bregman divergence.

Divergence

Generating function ψ(x)

2 x 2 2 Kullback-Leibler x log x Itakura-Saito -log x β-divergence x β β(β-1) -x β-1 + 1 β Table 1: Examples of usual Bregman divergences
Similarly to [START_REF] Candes | Phase retrieval via Wirtinger flow: Theory and algorithms[END_REF], we propose a gradient algorithm to minimize the criterion in [START_REF] Fienup | Phase retrieval and image reconstruction for astronomy[END_REF]. As the latter is not holomorphic, we use the Wirtinger formalism in order to express the gradient, which yields:

∇J(x) = A H (Ax) d 2 |Ax| d-2 (7)
∇ 2 ψ(|Ax| d ) T (|Ax| d -r) * .
We consider a simple gradient algorithm with a constant step in order to compare the reconstruction performances while minimizing ( 5) with different divergences. One can note that in the particular case ψ = • 2 2 , (1) and ( 5) are equivalent. Moreover, when d = 1, the gradient writes

∇J(x) = x -A H r Ax |Ax| . (8)
Thus, when ψ = • 2 2 , the gradient algorithm with a step size equal to 1 is equivalent to GLA.

Experiments

Protocol. We perform experiments on 10 utterances of the Lib-riSpeech [START_REF] Panayotov | Librispeech: an asr corpus based on public domain audio books[END_REF] speech database. All are monochannel, cropped to 2-seconds duration and sampled at 22050 Hz. The STFT uses 512 samples-long Hann windows and 50 % overlap. As phase retrieval is usually performed on modified measurements for many audio tasks, we simulate that situation by adding a Gaussian white noise at various signal-to-noise ratios (SNR) to the temporal signal. We use spectral subtraction [START_REF] Boll | Suppression of acoustic noise in speech using spectral subtraction[END_REF] in the STFT domain to produce the nonnegative measurements r.

Our gradient algorithm uses a constant step size equal to 1. As in Fast GLA [START_REF] Perraudin | A fast Griffin-Lim algorithm[END_REF], a Nesterov-like acceleration scheme is used, with an acceleration parameter equal to 0.99. Several divergences were considered, among which we present the following, as they yielded the best performances. We propose 2 distance and β-divergence with β = 0.5 for the magnitude problem (d = 1), and Kullback-Leibler divergence for the power problem (d = 2). Finally, we also studied a variant of the latter, which consists in minimizing D KL (|Ax| d | r), thus exploiting the non-symmetry of the divergence (which we do not detail here for brevity). The resulting algorithm is denoted "KL left" and uses a step size equal to 0.4. As comparison baselines, we also test GLA and Fast GLA. All algorithms use 1000 iterations. Performance criteria. The reconstruction quality is evaluated with the spectral convergence metric defined as

E SC (r, x) = r 1/d -|Ax| 2 r 1/d 2 . ( 9
)
We also consider the short-term objective intelligibility (STOI) measure [START_REF] Taal | A short-time objective intelligibility measure for timefrequency weighted noisy speech[END_REF] in order to compare the performance of the different algorithms on the speech signal from a perceptual point of view. Results. Figure 1 presents the mean results in terms of spectral convergence for several levels of input SNR. The proposed gradient algorithms outperform GLA and Fast GLA in terms of spectral convergence in some situations. In particular, the "KL left" formulation seems more robust to the input perturbation.

Figure 2 shows the STOI score obtained with the different algorithms. All algorithms show similar intelligibility performance when no noise is added. When the SNR lowers, the intelligibility decreases. Surprisingly, GLA is the less affected by this phenomenon while its performances in terms of spectral convergence are among the weakest. Finding a good performance measure both in terms of spectral approximation and perceptual audio quality still seems to be an open question and makes the interpretation of the observed results a complex task.

Conclusion

We have addressed the problem of phase retrieval with the Bregman divergences in the context of audio signal recovery. We derived a gradient algorithm and made an explicit link with GLA when the cost function is quadratic. In future work, we will consider more advanced gradient schemes with adaptive step size and extend our study to musical signals.

Figure 1 :Figure 2 :

 12 Figure 1: Mean spectral convergence values for various input SNR. A lower value corresponds to a better spectrogram approximation.

* This work is supported by the European Research Council (ERC FACTORY-CoG-6681839).


domain: chim sdv

text:
Xavier Le Guével

Benjamin Musnier

Karl Wegner

Maxime Henry

Agnes Desroches-Castan

Ute Resch-Genger

Sabine Bailly

Yves Usson

Véronique Josserand

Jean-Luc Coll

Surface chemistry-mediated metal nanoclusters for in vivo shortwave infrared imaging

come

Surface chemistry-mediated metal nanoclusters for in vivo shortwave infrared imaging

In vivo optical imaging is nowadays widely used for cancer diagnosis and imageguided therapy with various devices that have already entered into the clinic. However, current fluorescence imaging is worth improving in spatial and temporal resolution in deep tissues. A novel approach relies on collecting optical signal from the second near-infrared window between 900 to 1700 nm called Shortwave InfraRed (SWIR). The use of this wavelength range should considerably reduce light scattering and absorption by blood and tissues in vivo improving sensitivity and spatial resolution in depth [1][2][3]. There is a high need to develop bright and safe contrast agents in this spectral window especially with emission at longer wavelength (>1250nm). In this context we develop gold nanoclusters (Au NCs) with different surface chemistry to enhance and red-shift their photoluminescence to the SWIR windows [3][4]. We evaluated then the impact of light scattering effects on spatial resolution in different shortwave infrared (SWIR) sub-regions by analyzing two SWIR emissive phantoms made i) polydimethylsiloxane (PDMS)-Au NCs composite covered with mice skin, and ii) capillary tubes filled with Au NCs at different depth in intralipids. We demonstrated the significant improvement of spatial resolution and reduced scattering collecting photons in the second SWIR windows between 1500 and 1700 nm and confirmed these results in mice with the observation of the blood network few mm deeper in mice [5]. New imaging processing methods are also necessary to enable sharp and detail reconstruction deeper in tissues. To address a clinical need for non-invasive highresolution imaging of vascular and lymphatic networks at cellular level, we propose the use of a new bright SWIR emitting contrast agent based on ultra-small gold nanoclusters in pathological animal models presenting vascular disorders. We report also new imaging methods based on a series of Monte Carlo restoration (MCR) and on segmental analyses to provide a highly detailed mapping on the vasculature of mice. Imaging processing based on MCR enables more than one order improvement of spatial resolution with reduced scattering to distinguish non-invasively blood vessels deeper than 4mm. Segmentation analyses showed an increase of vessels complexity in transgenic mice KO BMP9 which confirmed previous ex vivo studies performed at the cellular level [6].

In conclusion, we presented new a method of high resolution, non-invasive, in vivo vascular imaging obtained using water-soluble and bright SWIR-emitting gold nanoclusters presenting an anisotropic surface charge combined with SWIR detection and Monte Carlo processing of the images. We applied this approach to quantify vessel complexity in mice presenting vascular disorders.

Acknowledgement

Xavier Le Guevel would like to thank l'agence pour la recherche contre le cancer ARC (RAC17012CCA) and Plan cancer (C18038CS) fundings.