L Métivier

R Brossier

J Virieux

S Operto

The truncated Newton method for Full Waveform Inversion

Full Waveform Inversion (FWI) is a promising seismic imaging method. It aims at computing quantitative estimates of the subsurface parameters (bulk wave velocity, shear wave velocity, rock density) from local measurements of the seismic wavefield. Based on a particular wave propagation engine for wavefield estimation, it consists in minimizing iteratively the distance between the predicted wavefield at the receivers and the recorded data. This amounts to solving a strongly nonlinear large scale inverse problem. This minimization is generally performed using gradient-based methods. We investigate the possibility of applying the truncated Newton (TrN) method to this problem. This is done through the development of general second-order adjoint state formulas that yield an efficient algorithm to compute Hessianvector products, and the design of an adaptive stopping criterion for the inner conjugate gradient (CG) iterations. Numerical results demonstrate the interest of using the TrN method when multi-scattered waves dominate the recorded data.

Introduction

Full Waveform Inversion (FWI) computes quantitative estimates of the subsurface parameters, such as the bulk wave velocity, the shear wave velocity, or the rock density. Applications range from the localisation of natural resources, such as oil and gas, to reservoir and storage monitoring, and seismic risk prevention. The seismic data is collected through the so-called seismic experiment: several sources and receivers are located at the ground surface or in wells, and the wavefield generated by these sources is recorded locally by the receivers. Based on a wave propagation modeling (from the simplest acoustic equation to more sophisticated anisotropic visco-elastic dynamics), the FWI method consists in minimizing a misfit function, which measures a distance between the data predicted by the forward problem and the recorded data. We consider the general wave equation

S(p)u = ϕ, (1)
where the subsurface parameters are denoted by p ∈ R m (model space), the linear forward problem operator corresponding to the two-way wave equation 1 is denoted by S(p), the source vector is denoted by ϕ, and the wavefield vector is denoted by u. These notations are general and can be applied either in the time domain or in the frequency domain. The FWI problem is expressed as the nonlinear least-square problem

min p f (p) = 1 2 Ns s=1 R s u s (p) -d s 2 , (2)
where d s and u s (p) are respectively the recorded dataset and the solution of the forward problem associated with the source ϕ s , N s is the total number of sources, and R s is a restriction operator that maps the wavefield u s to the receiver locations.

The minimization of f (p) is based on the local Newton approach. A sequence p k is computed from an initial guess p 0 using the update formula

p k+1 = p k + γ k ∆p k , (3)
where the scalar γ k is computed through a globalization method (linesearch, trust-region) and ∆p k is the solution of

∇ 2 f (p k )∆p k = -∇f (p k ), (4)
where the Hessian operator is denoted by ∇ 2 f (p k ). The computation of the gradient ∇f (p) can be performed efficiently with the adjoint-state method [START_REF] Plessix | A review of the adjoint-state method for computing the gradient of a functional with geophysical applications[END_REF]. Because of the large-scale aspect of the FWI problem (even for 2D applications, the problem often involves hundred thousands of unknowns parameters and discrete data), explicit computation and storage of ∇ 2 f (p), a fortiori (∇ 2 f (p)) -1 , is prohibitive. Therefore, standard methods use an approximation Q k of the inverse Hessian. Different choices can be made. For instance, choosing simply the identity matrix yields the steepest-descent method, which may converge slowly. A more sophisticated and efficient choice is the l-BFGS approximation [START_REF] Byrd | A limited memory algorithm for bound constrained optimization[END_REF]. This method estimates the effect of ∇ 2 f (p k ) -1 on the gradient. At iteration k, this approximation is based on finite differences of the l previous values of the gradient. However, Pratt et al [START_REF] Pratt | Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion[END_REF] emphasize the crucial role of the inverse Hessian operator in FWI: its acts as a defocusing filter, improving the resolution of the subsurface parameter estimation, and can help to remove artifacts on the model update related to the presence of large amplitude double-scattered waves. Accounting more accurately for effects of the inverse Hessian operator may thus yield significant improvements. As a consequence, we present in this study an implementation of a matrix-free truncated Newton (TrN) method [START_REF] Nash | A survey of truncated Newton methods[END_REF] for FWI. Instead of using an approximation of the inverse Hessian operator, this method partially solves the linear system (4) using a matrix-free conjugate gradient (CG) algorithm. This requires the capability of computing efficiently Hessian-vector products. This can be achieved using the second-order adjoint state formulas we present in the second section. In addition, an efficient adaptive stopping critertion for the CG iterations must be designed, in order to prevent from oversolving the equation ( 4), which would generate prohibitive computation costs. The Eisenstat stopping criterion [START_REF] Eisenstat | Choosing the forcing terms in an inexact newton method[END_REF] fulfills this requirement. Within this framework, numerical examples presented in the third section demonstrate that the computation cost of the l-BFGS method and the TrN method are comparable, and that the presence of large amplitude multi-scattered waves prevents the l-BFGS method from converging while the TrN method provides significantly more reliable results.

Method

Efficient computation of Hessian-vector products

In the following, we consider that N s = 1 and we drop index s. Formulas for N s > 1 are obtained straightforwardly by summation. We define the function g v (p) such that

g v (p) = (∇f (p), v) = R J † R † (Ru(p) -d), v , (5)
where u(p) is the solution of (1), J(p) = ∂ p u(p), R denotes the real part operator, and † the adjoint operator. We have ∇g v (p) = ∇ 2 f (p)v. The Lagrangian function associated with the functional g v (p) is

L v (p, u, α, λ, µ) = R R † (Ru -d), α + R (S(p)u -ϕ, µ) + R   S(p)α + m j=1 v j ∂ p j S(p)u, λ   , (6)
where the first term is related to the function g v (p), the second term to the wave equation ( 1) and the third term to the first derivatives of (1). For u and α such that

S(p) u = ϕ, S(p) α = - m j=1 v j ∂ p j S(p) u, (7)
we have

∇g v (p) = ∂ p L v (p, u, α, λ, µ) + ∂ u L v (p, u, α, λ, µ) ∂ p u(p) + ∂ α L v (p, u, α, λ, µ) ∂ p α(p). (8)
We define λ and µ such that

∂ u L v p, u, α, λ, µ = 0, ∂ α L v p, u, α, λ, µ = 0. ( 9
)
We have

S(p) † µ = -R † R α - m j=1 v j ∂ p j S(p) † λ, S(p) † λ = -R † (R u -d), (10) and (
∇ 2 f (p)v) i = R   ((∂ p i S(p)) u, µ) + (∂ p i S(p)) α, λ + m j=1 v j ∂ p j ∂ p i S(p) u, λ   , i = 1, . . . , m.
(11) Since computation of ∇f through the adjoint state method already requires the computation of λ and u [START_REF] Plessix | A review of the adjoint-state method for computing the gradient of a functional with geophysical applications[END_REF], the computation of one matrix vector product ∇ 2 f (p)v requires to solve one additional forward problem for α and one additional adjoint problem for µ, as reported in [START_REF] Epanomeritakis | A Newton-CG method for large-scale threedimensional elastic full waveform seismic inversion[END_REF][START_REF] Fichtner | Hessian kernels of seismic data functionals based upon adjoint techniques[END_REF]. For N s > 1, the overall computation cost is multiplied by N s .

CG stopping criterion

An efficient implementation of the truncated Newton method requires the definition of an adaptive stopping criterion for the CG criterion. Newton methods are based on the sequential minimization of local quadratic approximations

q k (∆p k ) = f (p k ) + (∇f (p k ), ∆p k ) + (∇ 2 f (p k )∆p k , ∆p k ). ( 12
)
The accuracy required to solve the system (4) should reflect the accuracy of these local approximations to prevent from oversolving. This is achieved using the Eisenstat stopping criterion [START_REF] Eisenstat | Choosing the forcing terms in an inexact newton method[END_REF]:

H(p k )∆p k + ∇f (p k ) ≤ η k ∇f (p k ) , (13)
where the forcing term η k measures the distance between the misfit gradient and its first order development

η k = ∇f (p k ) -∇f (p k-1 ) -H(p k-1 )∆p k-1 ∇f (p k-1 ) . ( 14
)
Moreover, far from the solution, the Hessian operator should be indefinite. An additional stopping criterion is thus introduced: as soon as a negative curvature direction is computed during the resolution of the system (4), the CG iterations are stopped. The convergence curves (fig 2) are plotted as a function of the total number of the forwardproblem resolutions. As expected, the steepest-descent algorithm converges very slowly, while the l-BFGS and the TrN method provide faster convergence. Note that the l-BFGS method is the fastest.

A near-surface imaging test case

We consider next a near-surface seismic imaging problem. Detecting and correctly imaging two concrete structures buried in the subsurface at few meters depth (fig. 4) is a challenge. The depth of investigation is limited to 3 m, the width of the exact model is 15 m. The discretization step is set to 0.15 m. We use a full acquisition system: four lines of sources/receivers are located on each side of the domain. We compute 9 datasets, from 100 Hz to 300 Hz each 25 Hz.

The very high velocity contrast between the background (300 m.s -1 ) and the concrete foundations (4000 m.s -1 ) generates energetic reflections. In addition, the close distance between the two structures is responsible for important multiple scattering. This is illustrated in figure 3 where two datasets in the time domain are presented 1 , computed using the exact model and the homogeneous background model. The background model correctly predicts only the first-arrival waves. The signal below the first arrival on the left figure, corresponding to the multi-scattered waves, is not predicted. Starting from this background model, we invert simultaneously the 9 datasets, using the l-BFGS algorithm, and the TrN method. The convergence curves and the corresponding results are presented in figures 2 and 4 (the steepest-descent result is not presented since it is very similar to the result provided by the l-BFGS method). While the steepest-descent and the l-BFGS method stop after few iterations, trapped in a local minimum, the TrN method further reduces the misfit function. Why in this case the TrN method performs better than the l-BFGS method? Consider the Hessian operator expression:

∇ 2 f (p) = Ns s=1 J † s (p)R † s R s J s (p) + Ns s=1 R † s (R s u s (p) -d s ) ∂ pp u s (p). ( 15
)
The first term of the Hessian operator is positive definite by construction, while the second one is indefinite, related to the double-scattered wavefield ∂ pp u s (p) and the residuals R s u s (p) -d s .

When single scattered waves dominate the data, gradient-based methods converge quickly from the first iterations. Therefore, the residuals decrease, and the Hessian operator tends to become positive definite. The l-BFGS methods builds a symmetric definite positive approximation of the inverse Hessian along the minimization process, and therefore improves the convergence speed of standard gradient methods. Since the Marmousi II model does not present high velocity contrasts, the amplitude of the multiscattered wavefield for this test case is weak, and the l-BFGS method is efficient. When multiscattered waves cannot be neglected, as in our second test case, gradient-based methods face difficulties to fit the data, since the gradient direction only accounts for single scattered waves. Therefore, for the first iterations, the l-BFGS estimation only slowly decreases the residuals as the inverse Hessian approximation is close from the identity. As a consequence, the true Hessian operator stays indefinite while the l-BFGFS method build a definite positive approximation of its inverse. This explains why the l-BFGS method fails in this case. Conversely, the TrN method better accounts for the Hessian operator and is able to converge.

Conclusion and perspectives

Second-order adjoint formulas and the Eisenstat stopping criterion [START_REF] Eisenstat | Choosing the forcing terms in an inexact newton method[END_REF] yield an efficient implementation of the TrN method. Within this framework, the comparison of the TrN method with the l-BFGS method demonstrates the interest of using the TrN method when the seismic data is dominated by multi-scattered waves. In this case, the second-order information embedded in the Hessian operator, which is better accounted for by the TrN method, is crucial. The authors now look forward for the application of the TrN method to more general context, such as multiparameter inversion and elastic FWI. The possibility of computing the inverse Hessian locally is also investigated in order to compute the posterior covariance matrix and provides an uncertainty estimation of the parameter reconstruction. Finally, the use of matrix-free preconditionners in order to speed-up the convergence of the CG algorithm will also be investigated.

Figure 1 .

 1 Figure 1. l-BFGS inversion results. Exact model (left), initial model (middle), estimated model (right).

Figure 2 .

 2 Figure 2. Convergence curves for the Marmousi II test case (left), for the near-surface imaging test case (right). N: truncated Newton method, LB: l-BFGS, ST: steepest-descent

Figure 3 .

 3 Figure 3. Dataset associated with the exact model (left), data associated with the initial homogeneous domain (right)

Figure 4 .

 4 Figure 4. Pressure wave velocities. Exact model (left), l-BFGS estimation (middle), TrN estimation (right).

Note that the operator S(p) depends non-linearly on parameter p

These two datasets are generated using a Ricker source located at the surface between the two concrete structures at x = 6.5 m, centered on the frequency

Hz.

Acknowledgments

This research is funded by the SEISCOPE consortium sponsored by BP, CGG-VERITAS, ENI, EXXON MOBIL, PETROBRAS, SAUDI ARAMCO, SHELL, STATOIL and TOTAL. The linear systems were solved with the MUMPS package. This work was performed by accessing to the high-performance computing facilities of CIMENT (Université de Grenoble, France) and to the HPC resources of GENCI-CINES under Grant 2011-046091.


Anaëlle Simonneau
email: anaelle.simonneau@univ-orleans.fr

Elise Doyen
email: elise.doyen@univ-fcomte.fr

Emmanuel Chapron
email: emmanuel.chapron@univ-orleans.fr

Laurent Millet
email: laurent.millet@univ-fcomte.fr

Boris Vannière
email: boris.vanniere@univ-fcomte.fr

Christian Di Giovanni
email: giovanni@univ-orleans.fr

Nicolas Bossard
email: nicolas.bossard@univ-orleans.fr

K Tachikawa

E Bard
email: bard@cerege.fr

Patrick Albéric
email: patrick.alberic@univ-orleans.fr

Géhco

Holocene land-use evolution and associated soil erosion in the French Prealps inferred from Lake Paladru sediments and archaeological evidences

Keywords: French Prealps, Holocene, lake sediment, organic geochemistry, land-use, soil erosion

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

Introduction

To understand potential future relationships between environment, human societies and global change, it is essential to identify the past interactions between human land-use and climate fluctuations [START_REF] Hurtt | The underpinnings of land-use history: three centuries of global gridded land-use transitions, wood-harvest activity, and resulting secondary lands[END_REF]. [START_REF] Bosco | Soil erosion in the Alpine area: risk assessment and climate change[END_REF] have also highlighted a probable future increase of soil erosion in the Alps. Soil erosion can result from climatic and/or anthropogenic impacts since it is mainly controlled by changes of the vegetation cover (Le [START_REF] Bissonnais | Mapping erosion risk for cultivated soil in France[END_REF]. The discrimination between climate and human-induced environmental changes remains however complex and needs to be addressed by accurate proxies supported by historical or archaeological evidences. Numerous studies have demonstrated that natural deposits, as lacustrine sedimentary sequences, can provide continuous high-resolution records of climate changes and anthropogenic environmental disturbances [START_REF] Dearing | Climate-human-environment interactions: resolving our past[END_REF][START_REF] Jacob | Millet cultivation history in the French Alps as evidence by a sedimentary molecule[END_REF][START_REF] Ariztegui | Natural and human-induced environmental change in southern Albania for the last 300 years -Constraints from the Lake Butrint sedimentary record[END_REF].

Climate or human imprints are classically highlighted in lacustrine sediments by chemical analysis [START_REF] Guiguet-Covex | Changes in erosion patterns during the Holocene in a currently treeless subalpine catchment inferred from lake sediment geochemistry (Lake Anterne, 2063 m a.s.l., NW French Alps): The role of climate and human activities[END_REF] or microfossils identification [START_REF] Tinner | Climatic change and contemporaneous land-use phases north and south of the Alps 2300 BC to 800 AD[END_REF], but can also be detected by organic geochemistry [START_REF] Noël | Human impact and soil erosion during the last 5000 yrs as recored in lacustrine sedimentary organic matte rat Lac d'Annecy, the French Alps[END_REF] and molecular markers [START_REF] Jacob | Millet cultivation history in the French Alps as evidence by a sedimentary molecule[END_REF]. Lake Paladru has been investigated by numerous subaquatic archaeological surveys along littoral environments [START_REF] Colardelle | Les habitats du lac de Paladru (Isère) dans leur environnement[END_REF][START_REF] Borel | Microlaminations lacustres : variations bathymétriques et climatiques avant l'An mil sur le site de Colletière[END_REF][START_REF] Brochier | Les variations paléoenvironnementales de 1000 avant à 1000 après J.C. et la question des « optima » climatiques de l'Antiquité tardive et du Moyen-âge sur le piedmont des Alpes du nord à Colletière[END_REF] and is therefore particularly suitable to investigate human/climate interactions. These studies have clearly identified three phases of lake-dwellings on the littoral environments of Lake Paladru, during the Late Neolithic period, the Middle Age and the Modern times and suggested possible others occupation phases during the Iron Age and the Roman period.

In this paper, we studied a Holocene lacustrine sequence retrieved from the central basin of this glacial lake together with soils and fluvial sediments from the related watershed, in order to track the impact of both climate and human activities on the environment. The present study combines physical and chemical analysis with geochemical and petrographic characterization of the organic matter. Continuous measurements are used to describe and identify the sedimentary units of the sequence whereas the punctual organic geochemistry analyses attempt (i) to qualify and quantify the allochthonous from the autochthonous compounds with specific attention on the soil and higher-plants debris supply and (ii) to detect specific agricultural practices within the watershed of the lake. Confronting continuous sedimentological records of minerogenic supply, past upper vegetation disturbance, soil erosion and molecular markers with historical and archaeological evidences, we finally attempted to quantify the consequences of Holocene human-induced disturbances in this part of the Alpine foreland.

Materials and methods

Sampling site

Lake Paladru (45°27'N-5°32'E) is located in the French Prealps at 492 m a.s.l (Figure 1). It is a small basin of glacial origin (3.73km², 5km long, 0.7km wide, 36m deep). The catchment area of 55km² is characterized by hilly slopes formed by the Miocene molasses of the alpine belt and by glacial or fluvial deposits from the Würm period covered by modern alluvial deposits [START_REF] Bravard | Carte géologique détaillée de la France au 1/50000[END_REF]. The Surand and the Courbon Rivers are the two tributaries flowing into Lake Paladru, on its western and northern edges, respectively. South of the lake, the Fure River constitutes the outlet (Figure 1). Numerous archaeological sites have been identified by subaquatic investigations in this lake. They are dated from the Late Neolithic period (2750-2730 BC, [START_REF] Bocquet | The Neolithic or Wood Age[END_REF][START_REF] Colardelle | Les habitats du lac de Paladru (Isère) dans leur environnement[END_REF] to the Medieval times (AD1006-1032, [START_REF] Colardelle | Les habitats du lac de Paladru (Isère) dans leur environnement[END_REF]; Figure 1). According to [START_REF] Brochier | Les variations paléoenvironnementales de 1000 avant à 1000 après J.C. et la question des « optima » climatiques de l'Antiquité tardive et du Moyen-âge sur le piedmont des Alpes du nord à Colletière[END_REF], each abrupt end of these lake-dwellings can be associated with rapid rise in the water lake-level.

Figure 1

In 2008, the sedimentary infill of the lake was imaged by high-resolution seismic profiling with a subbottom profiler of 12 kHz used from an inflatable boat and coupled with GPS navigation (Figure 1). 26km of profiles were realized and used to select a suitable location for the coring site PAL09 (45°27.478'N-5°32.348'E, Figures 1) in the central basin of the lake (36m deep). As shown in Figure 1, this core (12m-long) was retrieved in November 2009 with an UWITEC coring device operated from a platform, in area free from mass wasting deposits (MWD) producing chaotic to transparent lense-shaped bodies.

In November 2009, 23 carbonated pedological profiles, rich in carbonated gravels, and 3 river beds samples were collected at different altitudes and under different vegetation covers within the watershed of Lake Paladru (Figure 1). They are representative of the forested (11km²), grassy (5km²) and cultivated areas (35km²). Three types of soils were identified: (i) well developed Stagnosol or fluvic-Stagnosol in humid pastures and swamp areas, characterized by silty to sandy layers marked by stagnic colour pattern above 50cm deep; (ii) Colluvic Regosols downward the slopes, constituted of clayey to silty layers and resulting from the accumulation of material and (iii) well developed (50 to 120cm thick) calcaric to hypereutric Cambisols throughout the rest of the watershed under forest, scrub or cultivated areas.

Age-depth model

The age-depth model of core PAL09 was established based on radiometric markers ( 210 Pb, 137 Cs) measured every five millimetres for the 16 first upper centimetres and on 18 AMS 14 C radiocarbon dates performed at the Radiocarbon Laboratory in Poznan and at the Laboratoire de Mesure du Carbone 14 in Gif-sur-Yvette (Table 1, Figure 2B) on terrestrial vegetal remains sampled within the six upper meters. Calibrated ages were computed using the calibration curve of Intcal09 [START_REF] Reimer | IntCal09 and Marine09 radiocarbon age calibration curves, 0-50,000 years cal BP[END_REF] and the resulting age-depth model was interpolated by a cubic spline curve using the -clam‖ model developed by [START_REF] Blaauw | Methods and code for 'classical' age-modelling of radiocarbon sequences[END_REF].

Table 1 Figure 2

Mineral and organic signals

Non-destructive laboratory descriptions of core PAL09 were supported by physical analyses of the magnetic susceptibility (MS, Figure 2A) acquired at each centimetre using a Geotek Multi Sensor Core Logger System and high-resolution elemental analyses performed with the X-ray fluorescence core scanner (ITRAX, Cox Analytical Systems, [START_REF] Croudace | ITRAX: description and evaluation of a new multi-function X-ray core scanner[END_REF]. The relative abundance of elements, such as Ti, was measured each 5mm using a chromium tube as the X-ray source at 30kV and 12mA, with 15s of counting time. A total of 70 samples from the first six meters of core PAL09 and 34 samples from soils and river beds were selected for destructive geochemical analyses. Bulk inorganic carbon contents were measured with a LECO-CNS-2000 (Figure 2A, [START_REF] Weliky | Problems with accurate carbon measurements in marine sediments and particulate matter in seawater: A new approach[END_REF]. Bulk geochemical study of the organic matter was obtained by Rock-Eval pyrolysis to characterize the organic content of natural samples by thermal cracking. The Total Organic Carbon (TOC) and the Hydrogen Index (HI) are the two main parameters used to characterize soil organic matter [START_REF] Di Giovanni | Geochemical characterization of soil organic matter and variability of a post glacial detrital organic supply (Chaillexon lake, France)[END_REF][START_REF] Sebag | Palynofacies as useful tool to stydy origins and trandfers of particulate organic matter in recent terrestrial environments: Synopsis and prospects[END_REF] or to identify the organic matter origin in lacustrine environments [START_REF] Talbot | Hydrogen index and carbon isotopes of lacustrine organic matter as lake level indicators. alaeogeography[END_REF]. The TOC represents the amount of organic carbon in samples (%) and the S2 (or the HI, HI=S2/TOC in mgHC.g -1 TOC) corresponds to the chemical quality of the organic compounds. Its decrease highlights the oxidation of the organic matter in the sediment or a contribution of terrestrial material [START_REF] Ramanampisoa | Primary control of paleoproduction on organic matter preservation and accumulation in the Kimmeridge rocks of Yorkshire (UK)[END_REF]. Quantitative organic petrography was done on the same set of samples. It consists of the optical identification and quantification of the organic constituents present in a constant volume of sediment (1cm 3 ) or in one gram of soil, after elimination of carbonate and silicate phases by hydrochloric and hydrofluoric treatment [START_REF] Graz | Quantitative palynofacies analysis as a new tool to study transfers of fossil organic matter in recent terrestrial environments[END_REF]. Components are characterized by their optical properties (colour and reflectance), their forms (amorphous or figurative) and their origins (algal, phytoclastic or fossil, [START_REF] Tyson | Sedimentary Organic Matter: Organic Facies and Palynofacies[END_REF]. In addition, 40 samples from core PAL09 were selected for lipid analyses. Following [START_REF] Jacob | Millet cultivation history in the French Alps as evidence by a sedimentary molecule[END_REF], they were ultrasonically extracted using a mixture of dichloromethane: isopropanol (2: 1 v/v) and separated into neutral, acidic and polar compounds on aminopropyl-bounded silica. The concentrations of specific molecular markers such as pentacyclic triterpene methyl ethers (PTMEs) were estimated by gas chromatography-mass spectrometry. [START_REF] Jacob | Millet cultivation history in the French Alps as evidence by a sedimentary molecule[END_REF]2009) demonstrated that the sedimentary miliacin was the only PTME specific of the broomcorn millet and that its detection in lacustrine sediments by gas chromatography indicated that this PTME was produced in significant amount in the catchment. Since the first occurrence of miliacin was dated from 3700 cal BP in Lake Le Bourget lacustrine sediments [START_REF] Jacob | Millet cultivation history in the French Alps as evidence by a sedimentary molecule[END_REF], we focused our analyses of Lake Paladru sediments on the time-interval 4500-2000 cal BP, in order to document if such practice was regional and synchronous.

Results

Lake Paladru sediment infill and chronology

The substratum morphology of Lake Paladru has not been imaged by the seismic profiles and suggests that the sediment infill reaches more than 40 meters (Figure 1). The lake is composed of one main 36-m deep subbasin. In the central part, the rare windows of acoustic penetration show a well stratified basin infill, characterized by parallel highamplitude reflections. At the coring site, four seismic units (SU) have been identified (Figure 1). From the bottom to the top, SU1 is defined by high-amplitude and high frequency reflections, SU2 by lower-amplitude and lower frequency reflections, SU3 by an acoustically transparent unit interrupted by only one continuous reflection and SU4 by high-amplitude and low frequency reflections.

For the Holocene age-depth model of PAL09, two dates have been rejected (Figure 2B, Table 1) for being too old and potentially reworked. The Holocene period thus covered the 445 upper centimetres (Figure 2B). Modern age-depth model at site PAL09 is based on two distinct peaks in 137 Cs at 8.5 and 4cm below lake floor (Figure 2B), interpreted as corresponding to nuclear-weapons testing at AD1963 [START_REF] Beck | Estimated of fallout in the continental U.S. from Nevada weapons testing based on gummed-film monitoring data[END_REF] and the Chernobyl accident at AD1986 [START_REF] Anspaugh | The Global Impact of the Chernobyl Reactor Accident[END_REF], respectively. This is supported by dates estimated using the 210 Pb constant-flux, constant-supply model [START_REF] Appleby | The calculation of lead-210 dates assuming a constant rate of supply of unsupported 210 Pb to the sediment[END_REF]. MS, carbonate contents and TOC led to define five sedimentary units (U1 to U5, from base to top, respectively, Figure 2A). U1 is defined by the highest MS values of the sequence (20.10 -5 SI) and by the lowest calcium carbonate and organic contents (20% and <0.5%, respectively). The transition with the second unit U2 occurred around 13720±140 cal BP (Figure 2B) and matches the transition between SU1 and SU2. U2 is characterized by fluctuations of MS (from 17 to 3.10 -5 SI) values and higher calcium carbonates and organic amount than in U1 (>50%, 0.5<TOC<1%, respectively). The U2 mean accumulation rate is equal to 0.3mm.a -1 . Around 7360±140 cal BP, the sedimentation shifted to unit U3 defined by low and stable values of MS (around 1.10 -5 SI), high carbonates content (>50%), increasing values of TOC (from 0.8 to 1.2%) and a mean accumulation rate equals to 0.3mm.a -1 . One atypical light-coloured thin layer with a sharp base, dated from 5190±50 cal BP, is identified by eye-description and matches the acoustic reflection observed through SU3 on seismic profiles. Around 2370±120 cal BP, the accumulation rate increased to 0.7mm.yr -1 reflecting a change in the sedimentary infill that corresponds to the transition towards U4 which is characterized by increasing values of MS (>5.10 -5 SI) but constant carbonate and organic contents (50% and 1.2%, respectively). Higher MS values through U4 (>6.10 -5 SI) highlight some dark-coloured thin layers, whereas lower values (<5.10 -5 SI) correspond to light-coloured carbonate rich layers. Finally, U5 is characterized by lower values of MS (2.10 -5 SI), lower carbonate contents (40%) but the highest values of TOC (1.7%). Even if this last sedimentary unit is too thin to be recognized on seismic profiles, its onset in AD1970±3 suggests that U5 marked an important modern modification in the environment since the accumulation rate reached 1.8mm.a -1 .

Specific organic signals of soils, river-beds and lacustrine sediments

Soils and river beds samples (dark squares and grey triangles, respectively, Figure 3A) are characterized by various TOC contents (from 0.04 to 12.14%) and HI values essentially inferior to 300 mgHC.g -1 TOC (from 337 to 118 mgHC.g -1 TOC) typical of the terrestrial pole described by [START_REF] Millet | Response of littoral chironomid communities and organic matter to late glacial lake-level, vegetation and climate changes at Lago dell'Accesa (Tuscany, Italy)[END_REF]. Two groups of organic components are identified in soils (whatever the layer) and river-beds (Figure 3B): (i) non-pollen microfossils with colloidal red amorphous particles defined by diffuse external limits and without internal structures (rAP), ligno-cellulosic fragments (LCF) and opaque particles without high reflectance and (ii) pollen microfossils represented by spores and pollens. The rAP are similar to the ones observed in soils or lacustrine environments by [START_REF] Sebag | Palynofacies as useful tool to stydy origins and trandfers of particulate organic matter in recent terrestrial environments: Synopsis and prospects[END_REF] or [START_REF] Noël | Human impact and soil erosion during the last 5000 yrs as recored in lacustrine sedimentary organic matte rat Lac d'Annecy, the French Alps[END_REF] and correlated to pedological origin. They are preferentially delivered by runoff process occurring on grassy areas whereas LCF come from runoff on forested topsoils [START_REF] Graz | Quantitative palynofacies analysis as a new tool to study transfers of fossil organic matter in recent terrestrial environments[END_REF]. Besides, topsoil layers are always characterized by TOC superior to 2% and by the highest HI values (Figure 3A) corresponding to the preponderance of higher-plants debris (LCF) which form the litter on the surface of each topsoil [START_REF] Sebag | Palynofacies as useful tool to stydy origins and trandfers of particulate organic matter in recent terrestrial environments: Synopsis and prospects[END_REF]. This fresh organic matter is also abundant in fluvial sediments (Figure 3A), suggesting that it can be easily transported from the catchment to the lake by rivers.

Figure 3

Lacustrine sediments (Figure 3A) are defined by various TOC contents (from 0.2 to 12%) and HI values fluctuating between 750 and 300 mgHC.g -1 TOC that is between the algal pole [START_REF] Talbot | Hydrogen index and carbon isotopes of lacustrine organic matter as lake level indicators. alaeogeography[END_REF]) and the terrestrial one [START_REF] Millet | Response of littoral chironomid communities and organic matter to late glacial lake-level, vegetation and climate changes at Lago dell'Accesa (Tuscany, Italy)[END_REF]. It corresponds to a mixture of autochthonous and allochthonous material in the background lacustrine sediment. As previously presented, measurements performed in the sedimentary unit U5 are characterized by the highest organic carbon content (1.7%) and by the highest HI values (Figure 3A). Quantitative organic petrography analyses have demonstrated that all lacustrine samples only differ from the watershed ones by the presence of grey amorphous particles (gAP, Figure 3B) corresponding to products of algal growth in the lake waters [START_REF] Patience | Organic fluxes and early diagenesis in the lacustrine environment[END_REF]. Since the gAP content (Figure 4A) and HI values (Figure 4B) are correlated (R=0.86, p<0.001), HI therefore reflects the algal contribution into Lake Paladru sediments. The abundances of rAP and LCF within the lacustrine sediment are depicted in Figures 4C and4D, respectively. Both particles are transported by sheet runoff processes affecting the entire surface of the catchment since slopes are below 30% [START_REF] Cerdà | The influence of geomorphological position and vegetation cover on the erosional and hydrological processes on a Mediterranean hillslope[END_REF].

Discussion: Human-or climate-induced Holocene soil erosion and upper-vegetation disturbances

The evolution of the sedimentation accumulation rate (Figure 4E) reflects both autochthonous (Figure 4A) and allochthonous (Figures 4C and4D) contributions from the water column and from drainage-basin, respectively. Periods of higher accumulation rates are characterized by synchronous higher phytoplankton production and enhanced soil erosion (Figure 4). Following [START_REF] Reynolds | Understanding the assembly of phytoplankton in relation to the trophic spectrum: where are we now?[END_REF], soil organic fraction supply can promote nutrients loadings and favour algal productivity. Lakes can in addition be considered as integrative sinks both of past large scales environmental changes induced by climate and of local anthropogenic activities usually restricted to the vicinity of the villages [START_REF] Bouby | Economie agraire à la fin de l'âge du Bronze sur les bords du lac du Bourget (Savoie, France)[END_REF]. The organic and the minerogenic composition of the lacustrine sediment have been used to discuss whether soil destabilizations are caused by Holocene human land-use or climate (c.f. [START_REF] Noël | Human impact and soil erosion during the last 5000 yrs as recored in lacustrine sedimentary organic matte rat Lac d'Annecy, the French Alps[END_REF][START_REF] Chapron | Rhone river flood deposits in Lake Le Bourget: a proxy for Holocene environmental changes in the NW Alps[END_REF][START_REF] Arnaud | Lake Bourget regional erosion patterns reconstruction reveals Holocene NW European Alps soil evolution and paleohydrology[END_REF]. Indeed, the rAP/LCF ratio (Figure 4F) corresponds to the respective contribution of less vegetated versus forested topsoil layers in the fluxes of terrestrial organic matter transported to the lake by runoff [START_REF] Di Giovanni | Seasonal variability and threshold effects of the organic detrital sedimentation in lakes: imbalances between organic records and climatic fluctuations (Chaillexon basin, Doubs, France)[END_REF]. However, disturbed areas affected by human land-use, such as cultivated places, are preferentially eroded [START_REF] Duchaufour | Pédogenèse et classification 2 e édition[END_REF] and can thereby artificially and significantly increase the ratio rAP/LCF even if the catchment is forest-covered.

Inversely, a drop in the ratio rAP/LCF can result from intensive land-used, such as ploughing, affecting deeper soil layers where the ratio rAP/LCF is constant whatever the vegetation cover [START_REF] Graz | Quantitative palynofacies analysis as a new tool to study transfers of fossil organic matter in recent terrestrial environments[END_REF] and not from a reforestation. Such limits highlight the necessity to compare the ratio rAP/LCF with indirect measurements, such as minerogenic supply (titanium for example, Figure 4H) which points to enhanced physical weathering of alumino-silicates within soil layers that can be due to climate or more intense land-use [START_REF] Massa | A 2500 year record of natural and anthropogenic soil erosion in South Greenland[END_REF].

Figure 4

Since 13720±140 cal BP, Lake Paladru is characterized above SU1 by a drastic drop of clastic supply (documented both by high sediment MS and Ti supply), an increase of calcium carbonate production in the water column and a rise in TOC (Figure 2A) that favoured the onset of a lacustrine drape (Figure 1). During the Early Holocene period, autochthonous carbonates and algae production dominated the sedimentation, while the Mid-Late Holocene is characterized since ca. 5200 cal BP by both a progressive and irregular rise in minerogenic (MS, Ti) and organic (rAP) soil supply (Figures 2A, 4C and4H) and a gentle decrease in upper vegetation debris inputs (LCF, Figure 4D) suggesting a drop in the forested cover. Besides, this change could match first regional human land-use (Berger et al., 2007a).

The evolution of rAP/LCF is afterwards reflecting a mainly forested watershed during all the Holocene period (Figure 4F). Seven main phases of higher accumulation rates (Figure 4E) are recorded (grey strips, Figure 4) and correspond to periods of enhanced erosion of the soil organic fraction (Figure 4C) and higher landscape disturbances (Figure 4F). They are dated from 8400-7900, 6000-4800, 4500-3200, 2700-2050 1600-1100 cal BP, AD1250-1850 and after AD1970. LCF particles show, in addition, exceptional high concentrations around 5650 and 5000 cal BP.

8400-7900 cal BP: climate-induced disturbance during the Early Holocene.

The time interval between 8400 and 7900 cal BP is well known as a colder and more humid specific period through the North Atlantic regions and the Alps [START_REF] Von Granfenstein | A Mid-European Decadal Isotope-Climate Record from 15,500 to 5000 Years[END_REF]. This climatic change involved an increase in regional humidity in Western Europe [START_REF] Magny | Early and late Holocene water-level fluctuations of Lake Annecy, France: sediment and pollen evidence and climatic implications[END_REF] and modified the upper vegetation cover in the Alpine region [START_REF] Haas | Synchronous Holocene climatic oscillations recorded on the Swiss Plateau and at timberline in the Alps[END_REF]. It seems that the -8200 cal BP climatic deterioration‖ identified by von [START_REF] Von Granfenstein | A Mid-European Decadal Isotope-Climate Record from 15,500 to 5000 Years[END_REF] in the Northern Alps foreland also slightly influenced the vegetation cover of the drainage-basin of Lake Paladru since higher values of the rAP/LCF ratio are recorded (Figure 4F). It also favoured a weak increase of soil erosion (Figure 4C) in the drainage basin of Lake Paladru which induced higher algal production (Figure 4A). Similar trends and processes are described during this period by [START_REF] Hede | Multiproxy evidence for terrestrial and aquatic ecosystem responses during the 8.2 ka cold event as recorded at Højby Sø, Denmark[END_REF] in Northern Europe. 4.2. 6000-5500 and 5200-4800 cal BP: various disturbances during the Neoglacial transition.

From 6000 to 4800 cal BP, algal production as well as soil particles and upper vegetation debris contributions increased at site PAL09 (Figures 4A,4C,4D). This could result from the transition to a more humid climate which did not influence the vegetal covered but enhanced inwash of soil nutrients, favouring algal productivity. This increase in soil erosion could in particular result from the higher runoff documented at the onset of the Neoglacial period [START_REF] Wanner | Structure and origin of Holocene cold events[END_REF], characterized in the Alps by changes in moisture conditions [START_REF] Haas | Synchronous Holocene climatic oscillations recorded on the Swiss Plateau and at timberline in the Alps[END_REF]. In agreement with paleoenvironmental reconstructions from [START_REF] Bocquet | The Neolithic or Wood Age[END_REF], the rAP/LCF ratio indicates that the catchment of Lake Paladru was forested and not affected by landscape disturbances (Figure 4F). At the regional scale, and more specifically in the Rhône valley floodplains, at Lake Le Bourget and at the Bourgoin swamp (Figure 1), Late Neolithic sites are in addition documented [START_REF] Marguet | Element for a settlement history of North French Alps between 10000 and 2700 years BP[END_REF], Berger, 2007a[START_REF] Gauthier | Bronze Age at Lake Bourget (NW Alps, France): Vegetation, human impact and climatic change[END_REF][START_REF] Bernigaud | Evolution des anthroposystèmes des marais de Bourgoin-la-Verpillière (Isère) du Néolithique final à l'Antiquité tardive (3000BC-AD600) : archéologie du paysage et de l'environnement[END_REF]. Human settlements and woodland exploitation were also firmly established by 4950 cal BP on the southern shore of Lake Paladru (Figures 1 and4, [START_REF] Bocquet | The Neolithic or Wood Age[END_REF], suggesting that human activities could impact their environments in this part of the Prealps. Such practices may have induced disturbances within a catchment and could have contributed to the higher erosion of soil particles detected in core PAL09. However, since no significant landscape disturbance has been detected by the rAP/LCF ratio (Figure 4F), it is very likely that Neolithic woodland exploitation were restricted to the vicinity of the villages notably located close to the lake outlet. It suggests that lacustrine sedimentation at site PAL09 is mainly sensitive to environmental changes upstream from its tributaries (i.e. the northern parts of the drainagebasin). This further indicates that around Lake Paladru, the Late Neolithic period was more driven by progressive regional hydrological changes [START_REF] Magny | Holocene climate variability as reflected by mid-European lake-level fluctuations and its probable impact on prehistoric human settlements[END_REF][START_REF] Arnaud | 7200 years of Rhone river flooding activity in Lake Le Bourget, France: a high-resolution sediment record of NW Alps hydrology[END_REF][START_REF] Chapron | Rhone river flood deposits in Lake Le Bourget: a proxy for Holocene environmental changes in the NW Alps[END_REF] rather than human influences.

4300-3200 cal BP: Bronze Age agrarian activities and associated consequences.

A phase of progressive watershed disturbance and higher soil fluxes started in 4300 cal BP and reached its maximum in 3300 cal BP (Figures 4F and 4C). This time interval is documented as being a period of warmer and drier climate marked by two phases of reduced glacial activity in the Western French Alps [START_REF] Guyard | High-altitude varves records of abrupt environmental changes and mining activity over the last 4000 years in the Western French Alps (Lake Bramant, Grandes Rousses massif)[END_REF] allowing the development of high-altitude upper vegetation [START_REF] Carozza | Glacial fluctuations and exploitation of copper resources in high moutain during the Late Neolithic and Bronze Age in the French Alps (2500-1500 BC). Mining in European History and its impact on environment and human societies[END_REF]. Such climatic conditions could not explain the progressive increase of soil particles erosion (Figure 4C and increase in rAP/LCF ratio, Figure 4F) suggesting enhanced runoff processes and a wetter climate. During the same time lapse, Alpine human occupations are well documented through the Rhone valley nearby Lake Paladru (Figure 1, Berger et al., 2007b;[START_REF] Bernigaud | Evolution des anthroposystèmes des marais de Bourgoin-la-Verpillière (Isère) du Néolithique final à l'Antiquité tardive (3000BC-AD600) : archéologie du paysage et de l'environnement[END_REF] but only suggested by [START_REF] Brochier | Les variations paléoenvironnementales de 1000 avant à 1000 après J.C. et la question des « optima » climatiques de l'Antiquité tardive et du Moyen-âge sur le piedmont des Alpes du nord à Colletière[END_REF] around Lake Paladru. The first occurrence of miliacin in the sedimentary infill of Lake Paladru is however detected and dated in 3500 cal BP (Figure 4G) and reached its highest concentration in 3200 cal BP. Miliacin occurrence in Lake Paladru is thus almost synchronous (i) with the beginning of the millet cultivation around Lake Le Bourget in 3700 cal BP (Figure 1, [START_REF] Jacob | Millet cultivation history in the French Alps as evidence by a sedimentary molecule[END_REF] and (ii) with the extensions of agricultural territories and the development of sites with storage structures in the region during the second half of the Early Bronze Age (Berger et al., 2007b). It suggests that despite the absence of clear archaeological evidence of lake-dwelling along littoral environments, agrarian activities were developed within the drainage-basin of Lake Paladru during the Bronze Age period. The progressive increase of the ratio rAP/LCF since 4500 cal BP seems therefore to correspond both to the expansion of landscape opening for cereals cultivation and to the first rise of human-induced soil erosion.

The abrupt decrease in miliacin concentration in 3100 cal BP at site PAL09 is synchronous with the decrease of the ratio rAP/LCF (Figure 4F) reflecting the drop in soil particles erosion (Figure 4C). This abrupt stop coincided with the quasi absence of human activities in littoral environments of Lake Paladru [START_REF] Brochier | Les variations paléoenvironnementales de 1000 avant à 1000 après J.C. et la question des « optima » climatiques de l'Antiquité tardive et du Moyen-âge sur le piedmont des Alpes du nord à Colletière[END_REF]) and with regional abandonment of farmland during the Late Bronze Age period (Berger et al., 2007b).

Furthermore, this time-interval is marked by regional colder and more humid periods in the French Alps [START_REF] Arnaud | 7200 years of Rhone river flooding activity in Lake Le Bourget, France: a high-resolution sediment record of NW Alps hydrology[END_REF][START_REF] Chapron | Rhone river flood deposits in Lake Le Bourget: a proxy for Holocene environmental changes in the NW Alps[END_REF][START_REF] Guyard | High-altitude varves records of abrupt environmental changes and mining activity over the last 4000 years in the Western French Alps (Lake Bramant, Grandes Rousses massif)[END_REF][START_REF] Debret | North western Alps Holocene paleohydrology recorded by flooding activity in Lake Le Bourget, France[END_REF]. The synchronicity between the lowering of human impacts at Lake Paladru and the climatic pejoration thus supports the interpretations of [START_REF] Magny | Holocene climate variability as reflected by mid-European lake-level fluctuations and its probable impact on prehistoric human settlements[END_REF] and [START_REF] Jacob | Millet cultivation history in the French Alps as evidence by a sedimentary molecule[END_REF] that regional climate-related cultural changes favoured the abandonment of lake-dwellings sites in Prealps during the Late Bronze Age.

2700-2050 cal BP: Iron Age agrarian activities and drastic soil erosion.

The 2700-2050 cal BP time interval is a second phase of watershed disturbance (Figure 4F) associated with millet cultivation (Figure 4G). As previously, this period is synchronous with the nearby Lake Le Bourget sedimentary record [START_REF] Jacob | Millet cultivation history in the French Alps as evidence by a sedimentary molecule[END_REF], confirming a regional agrarian pattern in the French Prealps, itself synchronous with regional anthropogenic pressures (Berger et al., 2007b). This period of higher soil erosion (Figure 4C) is in addition marked by the occurrence of first traces of titanium at PAL09 coring site (Figure 4H). The simultaneous presence of organic and minerogenic pedological markers seems to indicate that until 2700 cal BP, soil disturbance was not enough intense to generate physical After 2050 cal BP, miliacine content in lacustrine sediments were not analysed, but lower soil erosion and lower rAP/LCF ratio are recorded within the sediment until 1600 cal BP (Figures 4C, 4F and4F) and contrast with the higher titanium supply (Figure 4H). If some peaks of titanium could be related to flood events induced by heavy rainfalls or storm events, enhanced titanium supply at site PAL09 can hardly be explained by wetter climate, since the time interval 2050-1600 cal BP corresponds to a sharp drop in lake-level in the Alpine region [START_REF] Magny | Holocene climate variability as reflected by mid-European lake-level fluctuations and its probable impact on prehistoric human settlements[END_REF]. The minerogenic supply during this period could therefore essentially result from the progressive destabilization of soil deeper layers initiated during the Iron Age in the drainage basin. It suggests an evolution of human land-use to intensive disturbances, such as deeper ploughing, which could remobilize deeper soil layers richer in minerogenic compounds and explain the diminution of the soil organic matter supply (Figure 4C). This interpretation is in agreement with Berger et al. (2007a) who documented both humaninduced extensive weakening of the soil systems and increasing minerogenic suspension load in the fluvial systems of the Rhone watershed, between 2050 and 1850 cal BP at the regional scale. In addition, the dilution of soil organic nutrients loading by minerogenic inputs induced a lowering of the algal production in Lake Paladru during this period (Figure 4A). . Between AD350 and AD850, core PAL09 recorded both enhanced organic and minerogenic pedological supply (Figures 4C,4H) which increased the rAP/LCF ratio (Figure 4F). This period matches well-documented human activities within the Prealps [START_REF] Noël | Human impact and soil erosion during the last 5000 yrs as recored in lacustrine sedimentary organic matte rat Lac d'Annecy, the French Alps[END_REF]Berger et al., 2007a) and notably within the catchment of Lake Paladru where [START_REF] Brochier | Les variations paléoenvironnementales de 1000 avant à 1000 après J.C. et la question des « optima » climatiques de l'Antiquité tardive et du Moyen-âge sur le piedmont des Alpes du nord à Colletière[END_REF] and [START_REF] Borel | Microlaminations lacustres : variations bathymétriques et climatiques avant l'An mil sur le site de Colletière[END_REF] described enhanced detritism phases and land openness interpreted as resulting from the development of medieval human communities. Indeed, the new increase in soil organic supply suggests an extension of human land-use and soil erosion to previously preserved territories. Such new landscape instabilities induced drastic modifications in Lake Paladru trophic levels since the algae production was multiplied by more than three (Figure 4B). This phase correspond with a very high detritic period associated with a plurisecular Rhone river floods increase in the Lake Le Bourget [START_REF] Chapron | Rhone river flood deposits in Lake Le Bourget: a proxy for Holocene environmental changes in the NW Alps[END_REF][START_REF] Arnaud | 7200 years of Rhone river flooding activity in Lake Le Bourget, France: a high-resolution sediment record of NW Alps hydrology[END_REF][START_REF] Debret | North western Alps Holocene paleohydrology recorded by flooding activity in Lake Le Bourget, France[END_REF] and with a torrential activity of the Rhone river and its tributaries in the upper Rhone valley.

950 cal BP (AD1000) in Lake Paladru

Between AD850-1200, lower soil supply and associated algal productivity are recorded in core PAL09 (Figures 4C,4H,4A) suggesting a relative stability of the landscape, in agreement with a regional pattern documented by Berger et al. (2007a) between AD900-1300/1500. [START_REF] Borel | Microlaminations lacustres : variations bathymétriques et climatiques avant l'An mil sur le site de Colletière[END_REF] and [START_REF] Brochier | Les variations paléoenvironnementales de 1000 avant à 1000 après J.C. et la question des « optima » climatiques de l'Antiquité tardive et du Moyen-âge sur le piedmont des Alpes du nord à Colletière[END_REF] largely documented a period of lower lake-level allowing the development of lake-dwellings at AD1000 (Figure 1) along the littoral environments of Lake Paladru. This drier period suggests lower runoff process and could therefore explain the lower terrestrial fluxes recorded at core PAL09. Intensive humaninduced soil destabilizations within the drainage basin were however described between AD1000-1040 [START_REF] Borel | Microlaminations lacustres : variations bathymétriques et climatiques avant l'An mil sur le site de Colletière[END_REF][START_REF] Brochier | Les variations paléoenvironnementales de 1000 avant à 1000 après J.C. et la question des « optima » climatiques de l'Antiquité tardive et du Moyen-âge sur le piedmont des Alpes du nord à Colletière[END_REF] and identified as the cause of the abrupt end of the Colletière village, in AD1040 [START_REF] Brochier | Le milieu lacustre. in: Les habitats du lac Paladru (Isère) dans leur environnement. La formation d'un terroir au Xle siècle[END_REF]. If no higher organic terrestrial supply are detected in core PAL09, the increasing contribution of titanium around AD1000 (Figure 4H) could attest of the cumulative effects of human-induced deep soil destabilization around Lake Paladru and an increase of runoff processes due to large-scale climate deterioration phases documented in the Alps by [START_REF] Wanner | Structure and origin of Holocene cold events[END_REF]. This interpretation is in agreement with the conclusions from [START_REF] Brochier | Les variations paléoenvironnementales de 1000 avant à 1000 après J.C. et la question des « optima » climatiques de l'Antiquité tardive et du Moyen-âge sur le piedmont des Alpes du nord à Colletière[END_REF] who suggested that both climate and human activities during this period could have affected the geomorphology of the lake outlet and induced a rapid rise of Lake Paladru level leading to the desertion of the Colletière village.

AD1250-1850: Little Ice Age and historical human imprints in Lake Paladru sediments

After AD1250, organic and minerogenic terrestrial markers increased (Figures 4C,4F,4H) suggesting a new phase of enhanced soil erosion favouring algal productivity (Figure 4A). This significant environmental change could result both from (i) a regional imprint of a wetter and colder climate during the Little Ice Age [START_REF] Chapron | Climatic variatbility in the northwestern Alps, France, as evidence by 600 years of terrigenous sedimentation in Lake Le Bourget[END_REF][START_REF] Chapron | Reconstructing recent environmental changes from proglacial lake sediments in the Western Alps (Lake Blanc Huez, 2543 m a.s.l. Grandes Rousses Massif, France)[END_REF][START_REF] Magny | Quantitative estimates of temperature and precipitation changes over the last millenium from pollen and lake-level data at Lake Joux, Swiss Jura Mountains[END_REF][START_REF] Wanner | Structure and origin of Holocene cold events[END_REF]) which probably contributed to the increase in soil erosion, and (ii) significant human pressures in the catchment area of Lake Paladru. According to [START_REF] Colardelle | Les habitats du lac de Paladru (Isère) dans leur environnement[END_REF], the historical ecclesiastical activities within Lake Paladru watershed have consisted of intensive disturbances resulting from the development of agriculture and fish breeding in artificial reservoir lakes (Figure 1). It seems therefore that such activities deeply affected the vegetation cover and favoured enhanced terrestrial fluxes to the lake.

Post AD 1850: Modern human-induced disturbances.

Algal production in Lake Paladru became maximal during the Modern time after AD1970 (Figure 4A).These modern drastic environmental changes confirm that human activities and in particular modern agriculture practices are not limited to erosion processes but also deeply affected the trophic state of Lake Paladru [START_REF] Touchart | La machine lacustre: l'exemple du Léman[END_REF].

Conclusions

Past interactions between human land-use and climate fluctuations in Lake Paladru are documented using the minerogenic and organic signals of lacustrine sediment with archaeological data. The study of the sedimentary organic matter allows in particular discriminating, identifying and quantifying organic components originating from the autochthonous production from those derived from runoff processes on soils. During the Early-Mid Holocene (10000-5200 cal BP), lacustrine sediment was essentially made of algae and autochthonous carbonates and the catchment was largely forested. The abrupt climatic event from 8200 cal BP induced a decrease of the upper vegetation cover and higher soil erosion reflecting an abrupt colder and more humid episode. After 5200 cal BP, the enhanced soil erosion resulted of climatic deterioration phases typical of the Neoglacial period, and/or higher human pressures within the watershed of Lake Paladru already documented by archaeologists or historical writings during the Late Neolithic, the Middle Age and the Modern period. Previously unknown phases of human activities are identified during the Bronze, the Iron and the Roman periods. Each of them was associated with enhanced soil weathering characterized by higher terrestrial organic and minerogenic supply, resulting from human-induced land openness and/or deep soil destabilization induced by modification of human land-use. Molecular markers, such as the miliacin, brought further information about past regional agrarian practices and agricultural dynamics through the Prealps, in particular considering the broomcorn millet. It particularly attests of the human presence around Lake Paladru during the Bronze Age and the Iron Age and suggests that during these two periods, human did not live on littoral sites but more within the watershed.

Figure captions

Figure 1 General localisation and geomorphological setting of Lake Paladru in the western French Prealps (left upper panel). Numerous archaeological sites (white starts) are documented by subaquatic archaeological surveys on the lake shore. The location of the longitudinal seismic profile (P1, lower panel) is given on the bathymetric map. Core PAL09 was taken in the central basin in an area free from any mass wasting deposits (MWD) remoulding the basin fill and preventing the penetration of the acoustic signal. The five first upper meters of core PAL09 presented in this study allow calibrating four different seismic units (SU1 to SU4).

Figure 2

The Master core PAL09 (A) is divided into 5 sedimentary units (from U1 to U5, from base to top) defined by variations of the magnetic susceptibility and calcium carbonate or total organic carbon (TOC) contents. Calcium carbonate amounts are calculated from the inorganic carbon measurements. The age depth model of core PAL09 (B) is supported by radiometric measurements (window) and by 15 radiocarbon ages (white leaves) detailed in Table 1.

Figure 3

Rock-Eval pyrolysis results (A) are represented on a S2 (mg HC) versus Total Organic Carbon (TOC, %) diagram for soil (black squares), river-beds (grey triangles) and lacustrine (white diamonds) samples. Black full lines 750 and 300 mgHC.g -1 TOC represent the aquatic and the terrestrial domains, respectively. The main components of the organic fraction identified and quantified by quantitative organic petrography are presented (B). Red Amorphous Particles (rAP) and ligno-cellulosic fragments (LCF) are found in soil, river-beds and lacustrine samples whereas grey Amorphous Particles (gAP) are only found in lacustrine samples.

Figure 4

Synthetic presentation of Lake Paladru data collected at core PAL09. The algal content quantified by quantitative organic petrography (A) is compared to the hydrogen index (HI) delivered by Rock-Eval pyrolysis (B), the soil particles (rAP) content quantified by quantitative organic petrography (C), the ligno-cellulosic debris (LCF) amount quantified by quantitative organic petrography (D), the accumulation rate expressed in cm.yr -1 (E), the ratio

  weathering. During the Iron Age (Hallstatt and La Tene periods), soil organic fluxes were indeed two times superior to the ones recorded during the first phase of millet cultivation (Figure 4C). It could indicate the intensification of both soil erosion and human land-use. It seems thus very likely that the first phase of exploitation, in the Mid-Late Bronze Age, only affected the superficial soil layers rich in organic fraction (rAP), while during the Iron Age, soil erosion in similar areas (or different land-use) have reached deeper layers richer in the mineral fraction. The convergence of ploughing practices intensification and of a global climatic degradation (van Geel and Rensen 1996) could thus explain this abrupt change in soil erosion process.

Acknowledgements

We acknowledge the financial support of the ANR Pygmalion and the ANR PalHydroMil.

We thank ISTO members (Disnar JR, Jacob J, Le Milbeau C, Boscardin R, Hatton M), Gratuze B (IRAMAT Orléans) and Garcia M (CEREGE, Aix-en-Provence) for their valuable advices during analyses. Ledoux G., Fanget B, Graz Y, Mallet E and Stock A. are also acknowledged for their help during field work campaigns. The Artemis radiocarbon committee (INSU/CNRS) and the Saclay team are thanked for radiocarbon dating. A.

Simonneau benefits from a PhD grant provided by the Region Centre. We would also like to warmly thank Verdel E. for logistical support and fruitful scientific discussions on human activities near Lake Paladru, and Lovato M. as well as Matheron G. for their contributions to maintain and protect Lake Paladru. We also thank the editor and two anonymous reviewers for their comments and suggestions that improved the manuscript.

rAP/LCF (F), the occurrence of miliacin molecular biomarker in Lake Paladru sediments (the black arrow symbolised the first occurrence of miliacine in Lake Le Bourget sediments, from [START_REF] Jacob | Millet cultivation history in the French Alps as evidence by a sedimentary molecule[END_REF] (G) and the titanium content (H) measured by XRF core scanning.

Table 1

Radiocarbon dates obtained from Lake Paladru sediment sequence PAL09. Age calibration was done using sets by [START_REF] Reimer | IntCal09 and Marine09 radiocarbon age calibration curves, 0-50,000 years cal BP[END_REF]. The two dates in italic are the rejected dates (see the text for more details).


Philippe Roudot

Charles Kervrann

LIFETIME ESTIMATION OF MOVING VESICLES IN FREQUENCY-DOMAIN FLUORESCENCE LIFETIME IMAGING MICROSCOPY

Keywords: FLIM, frequency-domain, phase modulation, particle tracking, template matching, M-estimator

We propose a framework for correcting the effect of vesicles motion in frequency domain FLIM imaging. Estimation of movement and lifetime are decoupled and alternatively performed. Robust M-estimation is involved to improve the accuracy of our estimate. Our method has been evaluated with both simulated and real samples.

INTRODUCTION

Fluorescence lifetime imaging microscopy (FLIM) is now a widely spread imaging technique for sensing fluorophore environment in a living biological sample (like pH, ions...). Fluorescence lifetime (i.e. the average time a fluorophore stays in excited state before relaxing to its ground state possibly emitting a photon) is particularly useful to detect the Förster resonance energy transfer (FRET) which can allow to quantify spatial proximity between neighboring molecules [START_REF] Hink | Imaging proteinprotein interactions in living cells[END_REF]. Such in vivo techniques are known to be more precise than in vitro biochemical methods. Now the former gives a much better analysis of the global phenomenon than a sample-by-sample FLIM measure carried out by an operator. To overcome that problem automatization of FLIM sample analysis is required.

In comparison to the well-established time-correlated single-photon counting (TCSPC) technique, frequency-domain methods are known to be faster [START_REF] Suhling | Time-resolved fluorescence microscopy[END_REF]. By using this technique Munster and Gadella [START_REF] Van Munster | Suppression of photobleaching-induced artifacts in frequency-domain flim by permutation of the recording order[END_REF] showed that photo-bleaching artifacts are easily negligible by permuting the frame recording order. Nevertheless, since it relies on intensity quantification over an image series, it is sensitive to sample movement that may lead to additional intensity variations [START_REF] Suhling | Time-resolved fluorescence microscopy[END_REF]. To address this problem we propose a framework that exploits the intensity model of the frequency-domain FLIM output to jointly estimate trajectories and lifetimes of vesicles.

To our knowledge, no previous work have been done on this specific topic. Fourier transform or sine fitting is often recommanded in the case of static molecules [START_REF] Spring | Image analysis for denoising fullfield frequency-domain fluorescence lifetime images[END_REF][START_REF] Lakowicz | Principles of fluorescence spectroscopy[END_REF][START_REF] Van Munster | Suppression of photobleaching-induced artifacts in frequency-domain flim by permutation of the recording order[END_REF]. However, a wealth of methods have already been proposed to track molecules in fluorescence imaging. Most of them rely on particle linking after spot detection in each frame [START_REF] Meijering | Tracking in molecular bioimaging[END_REF]. Spot detection can be performed using Gaussian mixture model fitting [START_REF] Thomann | Automatic fluorescent tag localization ii: improvement in super-resolution by relative tracking[END_REF], multi-scale wavelet analysis, top-hat filtering or local curvature analysis (see [START_REF] Smal | Quantitative comparison of spot detection methods in fluorescence microscopy[END_REF]). Linking is then achieved using scoring classification techniques, particle filtering or/and connectivity graph approach [START_REF] Thomann | Automatic fluorescent tag localization ii: improvement in super-resolution by relative tracking[END_REF]. Other approaches also propose template cross-correlation [START_REF] Bonneau | A multiple target approach for single quantum dot tracking[END_REF] or Gaussian fitting [START_REF] Anderson | Tracking of cell surface receptors by fluorescence digital imaging microscopy using a charge-coupled device camera. lowdensity lipoprotein and influenza virus receptor mobility at 4 degrees c[END_REF]. The tracker of our method is inspired from this former technique.

In this paper we propose a statistical framework to address our specific problem. In a nutshell, our method tries to combine lifetime estimation and robust M-estimator tracking in a efficient and fast way. Estimation of movement and lifetime are decoupled and alternatively performed. Indeed, lifetime parameter estimation relies on an accurate estimation of vesicle locus, and the sinusoidal model that rules intensity variation temporally is exploited to enhance tracker performance. Spot detection is performed on the first frame. Actually, we do not need sophisticated spot detection scheme since our estimator is able to detect objects which are not conform with our model. This saves computations for applications.

The remainder of this paper is organized as follows: Section 2 briefly presents the physical process related to frequency-domain FLIM measurements. In Section 3, we present our procedure and we justify our vesicle template. We describe the procedure devoted to estimate movement and lifetime. Section 4 presents our promising results.

FLUORESCENCE LIFETIME MEASUREMENT BY HOMODYNE DETECTION

Frequency-domain FLIM images have a very specific intensity footprint which needs to be described for a better understanding of our approach. To that end, this Section briefly presents the theoretical basis for lifetime measurement in the frequency domain by homodyne detection (see [START_REF] Lakowicz | Principles of fluorescence spectroscopy[END_REF] for a more detailed description). Given a fluorescence sample excited with a single laser pulse, the impulse response R(t) can be described by a sum of exponential function parameterized by decay times: these lifetimes represent the average time between its excitation time and return time to the ground state of each molecule present in the sample. For the sake of simplicity, we assume the impulse response to be described by a single exponential decay throughout the paper :

R(t) = R 0 e -t τ (1)
with R 0 the intensity at time t = 0 and τ the decay time.

Whereas the excitation source in time-domain methods is a laser pulse (like with TCPSC), the frequency-domain source is a sinusoidally modulated excitation E(t) defined as:

E(t) = C E + A E sin(ω E t + φ E ) (2)
where ω E denotes the radial frequency of the excitation signal, C E the offset, A E the amplitude and Φ E is the phase delay. Hence, the emitted fluorescent signal is defined as :

F (t) = E(t) * I(t) = C F + A F sin(ω E t + φ E -arctan(ω E τ )) (3)
where * is the convolution operator. It turns out the measure of the phase delay φ E -arctan(ω E τ ) of F (t) allows us to recover τ . Nevertheless the experimental frequency in such experiment (> 1 MHz) is known to be too high to allow us to directly measure the phase when acquiring images. Therefore the signal is phase-modulated with K ∈ IN sinusoidal signals M k (t) at the same frequency ω E (homodyne detection) as :

M k (t) = C G + A G sin(ω E t + φ M0 + 2πk K ), k ∈ [1, K] (4)
Thanks to the low pass effect of the CCD detector, the higher frequencies of the phase modulated signal F (t)G k (t) are negligible. Therefore we obtain K time-independent signals described as a function of k:

S θ (k) = G CCD (F (t)G k (t)) = C S + A S cos( 2πk K + Φ τ ), k ∈ [1, K] (5)
where

C S = C F C M , A S = A F A M , Φ τ is the phase φ M0 - φ E + arctan(ω E τ ) and θ = (C S , A S , Φ τ ) T
. Now, the phase Φ τ can be retrieved using sine fitting or Fourier decomposition techniques thanks to the lower frequency of S θ (k) . The system phase delay φ G0φ E is calibrated by recording a reference frame sequence of a fluorescent sample with known lifetime. Finally, as we monitor the frequency ω E , τ comes straightforwardly. Fig. 2 shows an example of lifetime computation using the Fourier transform on the stack presented in Fig. 1 with K = 12. This example highlights the problem of moving vesicles which create unrealistic values.

MEASURING LIFETIME IN LIVE-CELL IMAGING

The method presented above has been successfully applied on immobilized samples [START_REF] Suhling | Time-resolved fluorescence microscopy[END_REF]. However considering moving Fig. 2. On the left: Lifetime map using Fourier decomposition.

Black and white spots are abberant values due to organels movement. On the right: reconstruction using our method.

vesicles is crucial when studying energy transfer. This study focus especially on moving sub-resolution vesicles. In this section we propose a model to describe the spatial and temporal variations of vesicles through the stack of K frames.

In the following we use time to refer to the order of phasemodulated frames, even if the order of phase-modulation can be chosen as pseudo-random [START_REF] Van Munster | Suppression of photobleaching-induced artifacts in frequency-domain flim by permutation of the recording order[END_REF] so that the order is not necessarily chronological. To alleviate complexity while taking advantage of parameter inter-correlation we experimented an iterative procedure where the lifetime parameter and the movement parameter are decoupled.

Modeling spatial and temporal variations

Several degradations were taken into account to model the vesicle intensity. In our approach the microscope point spread function (PSF), corresponding to a Bessel function, is approximated as a Gaussian function. Secondly, as the confocal equipment is not rigorously stigmatic the signal is composed by a background and a pinpoint vesicle before microscope diffraction. The background also follows the model (5) due to cytoplasmic fluorescence. Finally the background is assumed to be smooth over the PSF support. It comes the following expression for intensity on frame k at position x:

I(k)(x) = S θ b (k)(x)+ S θ0 (k)σ σ (x-x 0 (k))+ ε(k)(x) (6)
where S θ0 (k) is the vesicle mean intensity parameterized by

θ 0 = (C S0 , A S0 , Φ τ0 ) T as in (5), S θ b (k)(x) is the back- ground signal parameterized by θ b (x) = (C S b (x), A S b (x), Φ τ b (x)) T , σ σ (x)
is a Gaussian function of variance σ 2 and centered on the origin which describes the shape of the vesicle, x 0 (k) is the position of the vesicle on frame k, ε is the noise introduced by the system. This set of parameters will be estimated in our procedure, though θ b (x) needs to be estimated only on the vicinity of a vesicle.

Iterated uncoupled parameter estimation

This section presents the parameter estimation procedure leading to a robust estimation of the intensity decay time τ . In a first step, we need to estimate the displacement of each individual vesicle. It turns out that movement estimation is more efficient using a prediction given by the intensity model vesicles. Hence we propose a three steps procedure: sinusoid parameters prediction, robust tracking, lifetime estimation.

Sinusoid parameter prediction

We consider a small number of frame K -∈ IN (we choose K -= 3 in samples where K = 12) to predict θ 0 and we consider a simpler model without the background influence:

I(k)(x) = S θ0 (k)σ σ (x-x 0 (k))+ε(k)(x), k ∈ [1, K -] (7)
Vesicles are first tracked using a simple Gaussian fitting [START_REF] Anderson | Tracking of cell surface receptors by fluorescence digital imaging microscopy using a charge-coupled device camera. lowdensity lipoprotein and influenza virus receptor mobility at 4 degrees c[END_REF] over a square 7×7 window W . Before tracking, spot are detected using derivative of Gaussian filtering. This process provides us an estimate x - 0 (k) and σ -. Thus we can estimate θ 0 by minimizing a robust error function:

θ - 0 = argmin θ0 k∈[1,K -] x∈W (x - 0 (k)) ρ • I(k)(x) -S θ0 (k)σ σ -(x -x - 0 (k)) (8)
where W (x - 0 (k)) is a 7×7 window centered on the estimate x - 0 (k). The function ρ is an influence function chosen with a scale parameter couple • = (λ, σ) to weight down outliers during the minimization process. Experimentations show that the Leclerc influence function:

ρ (λ,σ) (r) = 1 + e -r 2 λσ 2 (9)
gives more stable results even if no significant differences are noticeable with other M-estimator (Tukey Bi-weighted or German-McClure among others). On each frame, variance of pseudo-residual is estimated using a robust least trimmed square estimator [START_REF] Rousseeuw | Robust regression and outlier detection[END_REF], the maximum is kept for σ2 .

C S0 , and A S0 are initialized using respectively the mean and the range of {

I(x)(k) σ σ -(x-x - 0 (k)) |k ∈ [1, K -], x ∈ W (x - 0 )} and Φ τ0 = φ M0 -φ E + arctan(ω E τ ref )
with τ ref set to the fluophore lifetime measured without FRET (e.g. 2.5 ns for the EGFP presents in our experimental samples).

Robust track

Now x 0 (k) is estimated for each frame, taking advantage of θ - 0 . Tracking is performed frame-by-frame using the same robust error function with different variables:

(x 0 (k), σ) = argmin (x0(k),σ) x∈W (x0(k)) ρ • I(k)(x) -S θ - 0 (k)σ σ (x -x 0 (k)) , k ∈ [1, K] (10)
As in Section 3.2.1 the most appropriate cost function is again the Leclerc estimator. Though the least median of square robust variance estimate [START_REF] Rousseeuw | Robust regression and outlier detection[END_REF]:

σ = 1.4826 median i∈[1,|W (x0(k)|] |r i -median i∈[1,|W (x0(k)|] |r i || (11)
where {r i } are residuals of an ordinary least square estimation, performed better than least trimmed square estimator. Initialization is performed with σ = σ -and x 0 (k) = x0 (k -1), x 0 (0) = x0 (0) is determined by spot detection.

Lifetime estimation

In this step θ 0 is estimated again taking advantage of our tracker estimate x0 (k) for k ∈ [1, K]. As the measure must be as precise as possible, we do not neglect the influence of the background. Therefore we estimate θ b (x)

for x ∈ ∪ k∈[1,K]
W (x 0 (k)) then subtract S b (k)(x) from the image. Background estimation relies on cytoplasmic autofluorescence described by S b (k)(x) in [START_REF] Lakowicz | Principles of fluorescence spectroscopy[END_REF]. Similarly to [START_REF] Boulanger | Estimation of dynamic background for fluorescence video-microscopy[END_REF] we use an asymmetric M-estimator to discriminate background over vesicles. Though this last implementation does not handle still particles: if a vesicle does not move, there is no way to interpolate the underlying background. To that end we add the neighboring pixels influence in the M-estimation which gives the following robust error function to minimize:

θb (x) = argmin θ b w∈W (x) k∈[1,K] ρ • I(k)(w) -(C S b (x) + A S b (x) cos( 2πk K + Φ τ b (x))) , x ∈ ∪ k∈[1,K] W (x 0 (k)) (12)
where W (x) is a 7×7 window centered on x. As in [START_REF] Boulanger | Estimation of dynamic background for fluorescence video-microscopy[END_REF] a Leclerc estimator is chosen as influence function with • = (λ, σ) chosen assymetricaly for positive and negative residuals . Over-weighting the negative residual disadvantage the pixel belonging to a spot, as the results θ b describes the intensity of the background. Fig. 3 highlights the results.

After background subtraction, θ 0 is estimated as in Section 3.2.1 over K frames instead of K -. It comes the parameter Φ τ0 and subsequently τ

0 = tan(Φτ 0 -φ M 0 +φ E ) ω E
. We could iterate on estimation of x 0 and θ 0 though a single step already gives satisfying results.

RESULTS AND DISCUSSION

This section presents the results of our experimentation conducted on both synthetic and real sample.

Our simulations reproduce our model. Background is simulated with a real vesicle-less sample and cytoplasmic fluorescence is simulated with constant parameter θ b on the background. Vesicles intensities follow [START_REF] Thomann | Automatic fluorescent tag localization ii: improvement in super-resolution by relative tracking[END_REF] and movement is limited to a two pixels range. A mixed Gaussian noise is then applied to simulate measurement impact. Fig. 4(a) shows that our lifetime estimation method outperform the classic one on tracked vesicles (Section 3.2.1) while Fig. 4(b) compares our tracker with a basic Gaussian fitting (Section 3.2.2).

Uncontrolled vesicles interactions on experimental samples makes the generation of ground truth of lifetime impossible. Nevertheless we compared tracking accuracy. Track has been hand measured using I. Smal's MJtrack [START_REF] Smal | Quantitative comparison of spot detection methods in fluorescence microscopy[END_REF] on various confocal FLIM measurement samples provided courtesy of Institut Curie. Fig. 4 shows that tracking performance are lower on the darkest frames, preliminary investigations led us to focus on the noise model on those frames in future works. Fig. 2 shows an example of reconstruction.

Our approach Gaussian fit

CONCLUSION

Our method allows to retrieve lifetime of moving vesicle in an efficient and fast way on frequency-domain FLIM measurement. This study shows the importance of taking the intensity model into account while tracking vesicles. More tracker such as linking method will be quantitatively compared to assess the differences in performance of our procedure which is much less demanding in the spot detection area.

Fig. 1 .

 1 Fig. 1. An example of frequency-domain FLIM measurement with K = 6 phase-modulation signal.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Background estimation on experimental sample, diagram shows both profile in highlighted region.

Fig. 5 .

 5 Fig. 5. Comparing tracks of 30 vesicles of a real sample.


Author-produced version of the article published in Journal of Irrigation and Drainage Engineering-ASCE, 2012,138(1), 9-15.

The original publication is available at http://ascelibrary.org; doi/abs/10.1061/(ASCE) IR.1943-4774.0000384