Jean-Luc Got
email: jlgot@univ-savoie.fr

David Amitrano

Ioannis Stefanou

Elodie Brothelande

Aline Peltier

J.-L Got

DAMAGE AND STRAIN LOCALIZATION AROUND A PRESSURIZED SHALLOW-LEVEL MAGMA RESERVOIR

Structures developing above long-term growing shallow-level magma reservoirs, such as resurgent domes, may contain information on the reservoir itself. To understand the formation of such tectonic features, we have investigated the deformation process around a shallow pressurized magma reservoir embedded in a damaging elastic volcanic edifice. Our model allows evidencing the effect of the progressive damage in producing the fault pattern associated to tectonic surface deformation. Damage is first isotropic around the cavity and constitutes a damaged zone. Then the free-surface effect appears and an anisotropic shear strain develops from the boundary of the damaged zone; it localizes on reverse faults that propagate upward to the surface. When the surface deformation is sufficient, normal faulting appears. Finally, the complete structure shows an undeformed wedge above the damaged zone. This structure is similar to those found by analogue modelling and from field geologic observations. From this model, we found a relation to estimate the reservoir radius and depth from the graben and dome widths.

From limit analysis, we deduced an analytical expression of the magma reservoir pressure which provides a better understanding of the magma pressure build-up during doming. The dip of reverse faults limiting the dome can be

inferred from the minimal pressure required to rupture the crust around the reservoir. Finally, the magma reservoir overpressure, the dip of the faults, the reservoir depth and the damaged zone radius is inferred from three parameters: the ratio ρ R computed from the dome and graben widths, the cohesion and the friction angle.

Introduction

The location of most volcanoes is related to large tectonic structures controlled by geodynamics. These structures are created by regional horizontal tectonic stresses: for instance, rifts or cordilleras are created by tensile or compressive stresses, respectively.

At the volcano scale, some remarkable structures like resurgent domes result from the pressurization of shallow-level magma reservoirs, which provides a vertical component to the stress field. Retrieving the geometrical and some physical characteristics of the pressure source from geological and geophysical surface observations, especially taking into account the faulting structure, is a challenge that requires the development of realistic models involving strain localization.

Resurgent domes are commonly observed in the central part of large active calderas.

They are found, for example, at Yellowstone, Long Valley, Lake Toba, Valles Caldera (e.g. [START_REF] Smith | Resurgent cauldrons[END_REF]), Siwi Caldera, Campi Flegrei [START_REF] Sacchi | The Neapolitan Yellow Tuff caldera offshore the Campi Flegrei: Stratal architecture and kinematic reconstruction during the last 15 ky[END_REF]. Caused by the long-term uplift of the caldera floor, they are attributed to renewed magma intrusion after the caldera collapse and subsequent pressurization of a shallow-level magma reservoir.

Most resurgent domes are elongated in shape along the same elongation direction of their hosting caldera, they are therefore mostly 2D structures.

The apex of the domes is generally occupied by one longitudinal graben (in rare cases, such as in Toba or Long Valley, domes can bear several parallel grabens). Normal faults forming this graben are well identified at the surface and have been studied at depth by geothermal drillings in the case of Redondo dome in Valles Caldera (Figure 1) or seismic imaging at Campi Flegrei [START_REF] Sacchi | The Neapolitan Yellow Tuff caldera offshore the Campi Flegrei: Stratal architecture and kinematic reconstruction during the last 15 ky[END_REF]. The nature of dome borders is D R A F T January 28, 2019, 3:18pm D R A F T less well understood, as they are not associated to outcropping faults identified at the surface. However, they represent a sharp transition between the dome flanks and the flat caldera moat, strongly suggesting that the dome lateral extension could be limited by reverse faults. This hypothesis is corroborated by the evidence of fluid circulation along borders. Multiple geyser basins (including the renown Old Faithful) concentrate along the border of Mallard Lake resurgent dome in Yellowstone caldera [START_REF] Christiansen | The quaternary and Pliocene Yellowstone plateau volcanic field of Wyoming, Idaho, and Montana, USGS[END_REF].

Electrical surveys in the Siwi caldera also evidenced probable hydrothermal circulation along the Yenkahe dome border [Brothelande et al., 2016a]. Dome borders also represent a preferential pathway for magmatic fluids, as late volcanic products seem to commonly originate from there. Examples include late cinder cones along the Yenkahe (Figure 1), late rhyolite around Long Valley resurgent dome (moat rhyolite; [START_REF] Bailey | Volcanism, structure, and geochronology of Long Valley caldera, Mono county, California[END_REF]), Redondo dome in Valles Caldera (ring domes, Figure 1; [START_REF] Smith | Resurgent cauldrons[END_REF]), and La Pacana resurgent dome [START_REF] Gardeweg | La Pacana caldera and the Atana Ignimbritea major ash-flow and resurgent caldera complex in the Andes of northern Chile[END_REF][START_REF] Lindsay | Magmatic evolution of the La Pacana caldera system, central Andes, Chile: compositional variation of two cogenetic, large-volume felsic ignimbrites[END_REF].

To understand the structural evolution leading to the building of such resurgent domes, analogue experiments have been conducted [START_REF] Sanford | Analytical and experimental study of simple geologic structures[END_REF][START_REF] Marti | Experimental studies of collapse calderas[END_REF][START_REF] Merle | Experimental modelling of thin-skinned shortening around magmatic intrusions[END_REF][START_REF] Acocella | Analogue models of collapse calderas and resurgent domes[END_REF][START_REF] Acocella | The control of overburden thickness on resurgent domes: insights from analogue models[END_REF][START_REF] Galland | Experimental modelling of shallow magma emplacement: Application to saucer-shaped intrusions[END_REF][START_REF] Brothelande | Estimation of magma depth for resurgent domes: An experimental approach[END_REF]. The models revealed the presence of reverse faults limiting the dome, and the ones using elongated sources could reproduce the development of longitudinal grabens, as observed in natural domes (Figure S1).

The question of the expansion of a cylindrical or spherical cavity has been extensively studied since the beginning of the twentieth century, first in an infinite or semi-infinite elastic medium (see, e.g., [START_REF] Jeffery | Plane stress and plane strain in bipolar coordinates[END_REF]). This has led to an improved understanding of the structures around magma reservoirs and the development of the first mechanical models

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of pressurized magmatic structures [START_REF] Anderson | The dynamics of the formation of cone sheets, ring dikes and cauldron subsidence[END_REF]; [START_REF] Sanford | Analytical and experimental study of simple geologic structures[END_REF]; [START_REF] Robson | The effect of stress on faulting and minor intrusions in the vicinity of a magma body[END_REF]; [START_REF] Phillips | The dynamic emplacement of cone sheets[END_REF]; see, e.g., [START_REF] Tibaldi | Structure of volcano plumbing systems: A review of multi-parametric effects[END_REF] for a review). However, modelling faulting around a pressurized cavity requires taking into account the plastic deformation to localize strain. In this aim, a considerable amount of work has been produced in mechanics and geomechanics to model the stresses and displacement around over-or underpressurized cavities in linear elastic, perfectly plastic materials, with various plastic flow rules (see, e.g., [START_REF] Nadai | Plasticity: Mechanics of the plastic state of matter[END_REF]; [START_REF] Hill | The theory of combined plastic and elastic deformation with particular reference to a thick tube under internal pressure[END_REF]; [START_REF] Hill | The mathematical theory of plasticity[END_REF]; [START_REF] Nadai | The theory of flow and fracture of solids[END_REF]; [START_REF] Chadwick | The quasi-static expansion of a spherical cavity in metals and ideal soils[END_REF]; [1966]; [START_REF] Vesic | Expansion of cavities in infinite soil mass[END_REF]; D'Escatha and [START_REF] D'escatha | Stabilité d'une galerie peu profonde en terrain meuble[END_REF]; [START_REF] Carter | Cavity expansion in cohesive frictional soils[END_REF]; [START_REF] Sulem | An analytical solution for time-dependent displacements in a circular tunnel[END_REF]; [START_REF] Vardoulakis | Borehole instabilities as bifurcation phenomena[END_REF]; [START_REF] Bigoni | The quasi-statica finite cavity expansion in a nonstandard elasto-plastic medium[END_REF]; [START_REF] Yu | Cavity expansion methods in geomechanics[END_REF]; see [START_REF] Bigoni | The quasi-statica finite cavity expansion in a nonstandard elasto-plastic medium[END_REF] for a historical introduction of this question).

Salencon
Numerical modelling of volcano-tectonic structures around a pressurized magma reservoir in an elasto-plastic semi-infinite medium was first made by [START_REF] Grosfils | Magma reservoir failure on the terrestrial planets: assessing the importance of gravitational loading in simple elastic models[END_REF] and Gerbault et al. [2012], and by Brothelande et al. [2016b] in the case of resurgent domes. This later model leads to phenomenological relations between source parameters and surface observables. However, elasto-plastic modelling is closely linked to mechanical instability and subject to numerical instabilities that may prevent from modelling the complete faulting geometry, especially when the magma reservoir is overpressurized and close to the free surface of a semi-infinite medium. In this work we first derive a stable representation of the faulting geometry by using continuum damage mechanics. Continuum damage mechanics is a widely used approach in mechanics and geomechanics for modelling strain localization before rupture through progressive change of mechanical properties (see, e.g., [START_REF] Kachanov | Time of the rupture process under creep conditions[END_REF]; [START_REF] Lemaitre | Mechanics of solid materials[END_REF]; [START_REF] Krajcinovic | Damage Mechanics[END_REF]; [START_REF] Lyakhovsky | Distributed damage, faulting, and friction[END_REF]; [START_REF] Main | A damage mechanics model for power-law creep and earthquake aftershock and foreshock sequences[END_REF]; de Borst [2002]; [START_REF] Turcotte | A damage model for the continuum rheology of the upper continental crust[END_REF]; [START_REF] Amitrano | Brittle creep, damage, and time to failure in rocks[END_REF]; [START_REF] Heap | Elastic moduli evolution and accompanying stress changes with increasing crack damage: implications for stress changes around fault zones and volcanoes during deformation[END_REF]).

D R A F T January 28, 2019, 3:18pm D R A F T
It has been used to model the progressive landslide failure [START_REF] Eberhardt | Numerical analysis of initiation and progressive failure in natural rock slopesThe 1991 Randa rockslide[END_REF][START_REF] Lacroix | Long-term dynamics of rockslides and damage propagation inferred from mechanical modeling[END_REF], and more recently to model the dynamics of basaltic volcanoes preeruptive deformation and seismicity [START_REF] Carrier | A damage model for volcanic edifices: Implications for edifice strength, magma pressure, and eruptive processes[END_REF][START_REF] Got | An analysis of the nonlinear magma-edifice coupling at Grimsvtn volcano (Iceland)[END_REF]. In the present work our aim was to get a complete representation of the faulting geometry around a pressurized reservoir by continuum damage mechanics, in order to use it to establish geometrical and physical relations between source parameters (reservoir overpressure, depth and radius, fault dip), surface observables (graben and dome widths) and rock mass parameters (friction angle and cohesion), by using limit analysis. Limit analysis is also a widely used method to provide analytical or numerical relations in stability problems (see, e.g., [START_REF] Davis | Plasticity and Geomechanics[END_REF] as a simple and good reference textbook on the subject)

and may be applied to volcanological mechanical questions.

Model

We investigated the damage process and its consequence for the dome growth by defining an initially axisymmetric, homogeneous, and isotropic two-dimensional model which represents a shallow-level pressurized magma reservoir in a damageable elastic half-space (Figure 2). Discretizing this model in finite elements, we compute the plane strain deformation using roller boundary conditions along the model vertical borders, and fixed boundary conditions at the bottom. Surface is free. Model was scaled to physical dimensions (Figure 2), and set to a 10 km x 10 km square, with an initial linear elastic medium containing an initial 0.5 km radius circular central cavity. A uniform pressure is applied along the boundary of the cavity all along the numerical experiment. Gravity is applied to each node, using a rock density ρ = 2700kg/m 3 , a value comprised between andesite D R A F T January 28, 2019, 3:18pm D R A F T (ρ = 2600kg/m 3 ) and basalt (ρ = 2900kg/m 3 ) rock densities in the 0-25 km depth interval (see, e.g., Christensen and Mooney [1999], Table 4).

In this modelling we use the finite element progressive damage approach of [START_REF] Amitrano | From diffuse to localized damage through elastic interaction[END_REF]; [START_REF] Amitrano | Brittle-ductile transition and associated seismicity: experimental and numerical studies and relationships with the b-value[END_REF]; [START_REF] Amitrano | Brittle creep, damage, and time to failure in rocks[END_REF], previously used in Lacroix

and [START_REF] Lacroix | Long-term dynamics of rockslides and damage propagation inferred from mechanical modeling[END_REF] and more recently in [START_REF] Riva | Damage-based time-dependent modeling of paraglacial to postglacial progressive failure of large rock slopes[END_REF]. It allows the reproduction of the progressive failure of rock by using an elastic damage model. It follows Kachanov

[1958]'s elastic damage approach in which the Young's modulus E i of each element i decreases when a failure criterion is met and a fracture occurs. Damage onset has been shown to follow a Mohr-Coulomb criterion [START_REF] Heap | The evolution of elastic moduli with increasing crack damage during cyclic stressing of a basalt from Mt. Etna volcano[END_REF], which has been chosen

as the instantaneous damage threshold by [START_REF] Amitrano | Brittle creep, damage, and time to failure in rocks[END_REF], Lacroix and

Amitrano [2013] and in the present model. This Mohr-Coulomb criterion is truncated when the normal stress is equal to the tensile strength. Progressive damage induces the progressive decrease of the Young's modulus with the number of damage events; for one damage event:

E i (n + 1) = (1 -δ) E i (n) (1)
or, for n damage events:

E i (n) = (1 -δ) n E 0 (2)
where δ is the incremental damage occurring during one damage event, and E 0 is the initial Young's modulus. Heterogeneity in the model is introduced by using a random perturbation of the Young's modulus spatial distribution.

Stress redistribution occurs around a damaged element and may induce an avalanche of damage events. The number of damaged elements is the avalanche size, which may be thought as a measure of its magnitude.

D R A F T January 28, 2019, 3:18pm D R A F T
Damage also occurs as a subcritical process, on elements where the applied shear stress is lower than the instantaneous, maximal, strength. These elements are located outside the zone of maximum damage. In that case, damage is delayed and described by a law determined from static fatigue experiments, which expresses the time to failure t f as a function of the major principal stress σ 1 and of the instantaneous compressive or tensile strength σ 0 [START_REF] Wiederhorn | Stress corrosion and static fatigue of glass[END_REF][START_REF] Das | Theory of time-dependent rupture in the earth[END_REF]:

t f = t 0 exp -b σ 1 σ 0 (3)
The time t 0 and the constant b depend on rock properties and ambient conditions [START_REF] Scholz | Static fatigue of quartz[END_REF]. They are scaling parameters for the dynamics of the rupture process. Each element i is characterized by its failure time t i without interaction between elements (initially

t i = t f )
, and the proportion of consumed lifetime [START_REF] Amitrano | Brittle creep, damage, and time to failure in rocks[END_REF],

which allows taking into account the stress history for estimating the remaining time to failure. Taking into account this subcritical crack growth process and the random part in the spatial distribution of Young's modulus perturbations avoids an instantaneous localization in a very narrow fault plane. It allows the dissipation of the elastic potential energy outside this plane and widens the damaged zone, creating heterogeneity in this zone. It contributes to regularize the strain localization process, spreading damage in time and space, and controlling its time dynamics. Therefore, in the absence of data (deformation, seismicity), the information on physical processes operating during damage is contained in the regularization process; it is the physical a priori knowledge of this process.

As a consequence, in this model, the pressure applied in the reservoir is constant, but larger than the long-term strength of the rock mass, so that earthquakes occur around the

D R A F T January 28, 2019, 3:18pm D R A F T
reservoir and weaken the volcanic edifice. This weakening occurs through the progressive decrease of the Young's modulus; it may induce strain localization when stress conditions are sufficient and remain unchanged. In this work, we will not perform a parametric study that would allow us to find phenomenological relations; we will instead search for a faulting geometry and use limit analysis to find physical analytical relations between source parameters and observables.

Results and discussion

Strain localization and spatial distribution of damage

At the beginning of the pressurization process (Figure 3 (a)), a damaged shell appears around the pressurized reservoir, and isotropic strain localization develops along more or less radial features around the reservoir. These features correspond to the logarithmic spirals that are well observed when an infinite isotropic elastic-plastic medium is plastified under the pressure applied in a hole or cylindrical cavity (see, e.g., [START_REF] Hill | The mathematical theory of plasticity[END_REF]; D'Escatha and Mandel [1974]; [START_REF] Davis | Plasticity and Geomechanics[END_REF]; [START_REF] Gerbault | Elasto-plastic and hydromechanical models of failure around an infinitely long magma chamber[END_REF]) when no spatial regularization is applied. During this initial phase, the free surface is not perturbed and the external boundary of the damaged zone is quasi-circular, as it is in the case of an infinite medium. [START_REF] Hill | The mathematical theory of plasticity[END_REF] gives an expression for the radius of the damaged zone around a pressurized reservoir in an infinite medium:

r DZ r = 2E 3Y 1 3 (4)
where r DZ is the radius of the damaged zone, r is the radius of the cavity, E is the Young's modulus and Y is the yield stress of the rock mass. Taking a value of 5 GPa for E and

D R A F T January 28, 2019, 3:18pm D R A F T
10 MPa for Y , in the order of magnitude of those for an average quality rock mass (see, e.g., [START_REF] Hoek | Practical estimates of rock mass strength[END_REF]), gives r DZ ≈ 6r, which is close to our result.

However our model is not infinite. Large superficial deformation develops after isotropic damage occurs around the reservoir (Figure 3); it appears to be a free-surface effect. It produces an anisotropic strain component that dominates the isotropic component outside the damaged zone. Free surface and damage induce the punching failure of the rock mass above the pressurized cavity. Damaged elements delimit zones where anisotropic shear strain progressively localizes, and forms a coherent and complex system of faults with reverse and normal faulting. Reverse faults Φ r progressively delimit an uplifted compartment similar to resurgent domes (see, e.g., Brothelande et al. [2016b]). These faults initiate first at depth, at the external boundary of the damaged zone, at some distance of its top, and progress towards the surface. Their average dip is about 60 • . Normal faults Φ n develop at the surface when the dome progressively builds; they delimit a central graben, which is a frequently observed feature in such structures. A characteristic undeformed wedge, delimited by faults Φ w is present above the top of the damaged zone. Though the pressure source is not located at the surface, this undeformed wedge shares strong similarities with the well-known Prandtl's wedge for shallow strip footings in geomechanics (see, e.g., [START_REF] Davis | Plasticity and Geomechanics[END_REF]). These results show that the final damage spatial distribution reproduces quite accurately the results of published geological field observations (see, e.g., [START_REF] Merle | A structural outline of the Yenkahe volcanic resurgent dome (Tanna Island, Vanuatu Arc, South Pacific)[END_REF]) and analogue modelling [START_REF] Sanford | Analytical and experimental study of simple geologic structures[END_REF], Davison et al. [2001], and, for a typical example, the Yenkahe complex [START_REF] Merle | A structural outline of the Yenkahe volcanic resurgent dome (Tanna Island, Vanuatu Arc, South Pacific)[END_REF]Brothelande et al., 2016b]). Field geological observations show that magmatic intrusions associated with resurgence are bordered by reverse faults [START_REF] Fridrich | Structural, eruptive, and intrusive evolution of the Grizzly Peak caldera, Sawatch Range, Colorado[END_REF], which can explain the abrupt transition between the flat caldera moat and the dipping layers of the dome flanks. Elements of this pattern can be found in [START_REF] Anderson | The dynamics of the formation of cone sheets, ring dikes and cauldron subsidence[END_REF], [START_REF] Gerbault | Elasto-plastic and hydromechanical models of failure around an infinitely long magma chamber[END_REF] (Φ w ), in [START_REF] Robson | The effect of stress on faulting and minor intrusions in the vicinity of a magma body[END_REF], [START_REF] Phillips | The dynamic emplacement of cone sheets[END_REF] (Φ r ), where Φ r faults are associated with cone sheets, whereas Φ w faults may prefigurate ring dykes during reservoir deflation stages.

A geometrical analysis of the spatial deformation pattern

On the external boundary of the damaged zone, the orthoradial (hoop) stress is not constant, due to the free-surface effect. This stress is tensile when the reservoir is overpressurized. The orthoradial stress on the boundary is maximum at the Jeffery's points [START_REF] Jeffery | Plane stress and plane strain in bipolar coordinates[END_REF], Figure 4). In the case where a damaged zone exists, the faults Φ r and Φ w initiate from the Jeffery's points located at the external boundary of the damaged zone.

At these points only the orthoradial or hoop stress is sufficient to rupture the sound rock.

These points may be used to determine the radius of the damaged zone. Using the distance B between the Jeffery's points, and their depth H as model parameters, we find two simple linear relations between the dome half-width W D , the graben half-width W G , which are observables, and these parameters (Figure 4):

W D = W G + 2B (5) so that B = W D -W G 2 (6) D R A F T January 28, 2019, 3:18pm D R A F T
and

W D = B + Hcotβ (7)
where β is the dip of the reverse fault φ r ; from the latter equation we deduce:

H = W D + W G 2 tanβ (8)
By the Jeffery's points we can draw an infinity of circles, one only being tangent to the straight lines issued from the point I (Figure 4). Considering that the reservoir center C is located at the intersection of the normal to these straight lines, we find the depth of this center:

h c = 1 + B H 2 H (9)
and the radius of the damaged zone:

r DZ = B 1 + B H 2 1 2 (10)
If the reservoir section is not circular, h c is the depth of the center of curvature of the damaged zone top, with the curvature taken at Jeffery's points. We notice that the ratio:

λ = H B = W D + W G W D -W G tanβ (11)
may be known only from field data, so that h c may be written:

h c = 1 + W D -W G W D + W G cotβ 2 W D + W G 2 tanβ ( 12
)
One can also express the reservoir radius r c as a function of the damaged zone radius:

r DZ = (1 + ∆) r c ( 13
)
where ∆ is the relative damage zone thickness (close to 5 in our case), so that r c may be written:

r c = W D -W G 2(1 + ∆) 1 + W D -W G W D + W G cotβ 2 1 2 (14) D R A F T
January 28, 2019, 3:18pm D R A F T

Magma reservoir pressure derived from Limit Analysis

In such a model, when the geometry of damaged zones and the displacement field are known, limit analysis theorems (see, e.g., [START_REF] Davis | Plasticity and Geomechanics[END_REF], Souloumiac et al.

[2009]) give us the possibility to compute an upper bound for the pressure in the magma reservoir. The aim of this approach is to provide an analytical expression for the magma reservoir pressure, to understand which are the control parameters of the pressurization process and what is the relation with the deformation process.

The upper bound theorem of Limit Analysis states that failure will occur if, for any kinematically admissible displacement field, the rate of work of the external forces is equal to or exceeds the rate of energy dissipation. It allows us to determine the maximum, ultimate value (upper bound) of the applied external loads that a system can bear without collapsing, which in our case corresponds to the loss of equilibrium. For collapse modes of translational type, the upper bound is equivalent to the solution found by the limit equilibrium method (see, e.g., [START_REF] Drescher | Limit load in translational failure mechanisms for associative and non-associative materials[END_REF]): the upper bound theorem expresses the energy balance of the limit load, which is equivalent to the limit equilibrium of the system on the base of the principle of virtual powers.

Results of the modelling allow us to identify the blocks that are sliding and to deduce their relative velocities (Figure 4). This type of collapse mode is of translational type and therefore the limit equilibrium method can be used as well. However, the upper bound theorem is easier to apply given the geometric complexity of the system. Applying the upper bound theorem requires computing the rate of the work of external forces (weights and pressure force) and the rate of the dissipation. In that aim we need to compute the relative velocities of the sliding blocks; they are found by constructing the hodograph

D R A F T January 28, 2019, 3:18pm D R A F T X -15
(see Appendix 1, Figure S2), taking into account the angle of dilatancy ψ. We can then express the directions of the velocities, and find the modules by projecting the Chasles's relation on each coordinates axis (see Appendix 1 for detailed calculations). Given that there is no horizontal displacement of the vertical boundaries, the horizontal forces do not work. The rate of work may therefore be found by computing the weights of the various sliding blocks (see Appendix 1):

W ext = -(2ρgHBv 01 + w 2 v 02 ) + 2P Bv 01 (15)
Applying the upper bound theorem, that is, writing that the rate of external work of the external forces is equal to the rate of internal energy dissipation D leads to find the pressure:

P = ρgH + w 2 v 02 2Bv 01 + D 2Bv 01 (16)
that may be written explicitely as (see Appendix 1 for the detailed computation of w 2 v 02 and D):

P = ρgH + 1 4 ρgB tan β λ tan β -1 2 1 - tan(β -ψ) tan(β + ψ) + 2C cos φ λ cos 2 β cos ψ + sin 2 β sin ψ sin 2β sin(β + ψ) cos(β -ψ) (17)
where λ = H/B and C is the cohesion. In the general case where a deviatoric stress exists at the depth H, the initial lithostatic pressure may be expressed as a fraction of the vertical stress:

P 0 = kρgH (18) D R A F T January 28, 2019, 3:18pm D R A F T
where 0 <= k <= 1. This can lead to derive an estimator of the overpressure:

P -kρgH ρgB = (1 -k)λ + 1 4 tan β λ tan β -1 2 1 - tan(β -ψ) tan(β + ψ) + 2κ cos φ λ cos 2 β cos ψ + sin 2 β sin ψ sin 2β sin(β + ψ) cos(β -ψ) (19)
where κ = C ρgB .

In the common case where rock friction is described by associated plasticity, the angle of dilatancy ψ is taken equal to the friction angle φ, which means that the surface rugosity alone explains the friction along fault planes. In that case, the normalized overpressure Π may be expressed as:

Π = P -kρgH ρgB = (1 -k)λ + 1 4 tan β λ tan β -1 2 1 - tan(β -φ) tan(β + φ) + 2κ cos φ λ cos 2 β cos φ + sin 2 β sin φ sin 2β sin(β + φ) cos(β -φ) (20)
Equations ( 16) and ( 20) show that overpressure may be decomposed in three contributions and allows an analysis of the pressurization and deformation process:

-a term Π 0 = (1 -k)λ controlled by the pre-existing lithostatic pressure. This term is the amount of pressure necessary to reach, from the initial lithostatic pressure, the vertical equilibrium of the rock column of height H above the reservoir considering this one isolated (ruptured) from the host rock. It causes the reservoir volume increase and isotropic damage that precede the strain localization (formation of fault planes). It corresponds to the initial pressurization phase; during this phase there is no surface deformation, no free surface effect and the state of stress around the reservoir is compressive;

-a frictional term Π f = 1 4 tan β λ tan β -1 2 1 -tan(β-φ) tan(β+φ)
which is a budget measured by the displacement of the block number 2; it corresponds to the overpressure necessary for the vertical, anisotropic, displacement and deformation along the fault plane structure D R A F T January 28, 2019, 3:18pm D R A F T that appears after strain localization. This term is controlled by the friction angle φ and it is zero when φ is null. It is also null for tan β = tan β c = λ. It results from the friction along the fault plane structure after its creation, after the weakening of the volcanic edifice and the progressive opening of the eruptive/intrusive system; this term is dominant during large (vertical) deformation processes, especially immediately before, or during, eruptions or intrusions; it tends to be dominant during the final pressurization phase, and constitutes a minimum value for the pressure;

-a dissipative term Π D = 2κ cos φ λ cos 2 β cos φ+sin 2 β sin φ sin 2β sin(β+φ) cos(β-φ) that corresponds to the (shortterm) irreversible part of the deformation. It is controlled by the C cos φ term, that is, mostly by the cohesion C since φ keeps values in a relatively narrow interval for rocks, even at the scale of the rock mass. If C is null, no effort is spent in rock decohesion and this term vanishes. In this case minimal pressure corresponds to β = β c , that is W G = 0 (no graben). Decohesion is associated to microseismicity and damage during the pre-eruptive/intrusive process; it leads progressively to strain localization when the free-surface effect appears. It precedes the phase of large displacement along the fault planes that are finally created. This term corresponds to the pressure necessary to create the fault plane structure, and to open the eruptive/intrusive system; it works during the intermediate phase between the initial and final pressurization phases described above. This is a transient contribution that decreases with cohesion and time during the preeruptive/intrusive process. In this work cohesion is used to describe the unconfined rock strength, but cohesion may also be easily related to tensile strength, which is more often used to quantify rock strength around magma reservoirs. Cohesion may eventually be D R A F T January 28, 2019, 3:18pm D R A F T low, at the scale of the rock mass, in fractured media. This term is generally considered alone to quantify the overpressure necessary for the rupture of magma reservoirs.

Therefore in this simple expression we retrieve the main phases of the deformation process, and their contribution to the magma reservoir pressure. It helps to understand how overpressure builds and may be used to describe the time evolution of this pressure, eventually from geophysical observables: seismicity and surface deformation. It highlights the multiple causes of overpressure, which may be the origin of contradictory interpretations [START_REF] Gudmundsson | Magma chambers: formation, local stresses, excess pressures, and compartments[END_REF][START_REF] Grosfils | Magma reservoir failure on the terrestrial planets: assessing the importance of gravitational loading in simple elastic models[END_REF].

Equation 20 and Figure 5 show that the overpressure Π(λ) is a parabola with a first order term in λ dependent on κ. For values of κ larger than 0.5, Π is close to a linear function of λ in the vicinity of tan β (close to 2 in Figure 5), that is, in the practical interval of interest for λ. The overpressure Π is linearly dependent on κ (equation 20).

These results may be compared to those obtained by [START_REF] Haug | Shear versus tensile failure mechanisms induced by sill intrusions -Implications for emplacement of conical and saucer-shaped intrusions[END_REF], who found that overpressure was linearly dependent on the cohesion, and decreasing as a function of B/H (inverse of λ) with a power law.

3.4. Inferring the faulting structure from λ = H B ratio, cohesion and friction angle Expression (20) also contains information on the structure itself, and on its control parameters.

For a given reservoir position and size, and given rock mechanical properties, the fault plane structure is created in such a way that minimal effort, that is minimal pressure, is required. The only parameter of the structure that controls the pressure is the angle β. Minimization of the pressure with β allows us to find the optimal angle for which

D R A F T January 28, 2019, 3:18pm D R A F T
the structure is created (Figure 6). As an example, we find that, for realistic rock mass mechanical properties (Figure 6), and λ close to 2, the pressure reaches a minimum for the optimal value β = 61 degrees, close to the value found from the finite element model (Figure 3).

From equation ( 11), we find

λ = ρ R tanβ = 1 + α 1 -α tanβ ( 21
)
where

ρ R = W D +W G W D -W G and α = W G W D .
As a consequence, α may be written:

α = λ -tanβ λ + tanβ (22)
The ratio α is between 0 and 1. From equation ( 21) we can write

tanβ λ = tanβ tanβ c = χ = 1 -α 1 + α ( 23
)
where tanβ c = H B = λ.

From equations (22,23) we infer that α is non-zero and the central graben exists if tanβ < λ = tanβ c , that is, if the dissipation effect is non-zero. Therefore α and β depend on λ and κ (Figures S4 andS5). From these results we can infer that the graben exists (α > 0) for sufficiently high values of λ and κ.

Figure 7 shows the variation of β as a function of λ, for various values of κ and φ. It

shows that the family of β = f (λ) curves has a characteristic point (λ 0 , β 0 ) independent of κ; the family is comprised between two extreme curves corresponding to κ = 0 (black line and crosses) and κ → ∞ (red circles). The geometry, showing one central graben, studied by using limit analysis only exists for β curves located below the black line, which corresponds to the limit case β c = arctan(λ). This result shows that an increased cohesion induces a decrease in β (see also Figure S5); it also shows that an increase in φ from 10 to D R A F T January 28, 2019, 3:18pm D R A F T 40 • induces a decrease of β of at most 10 • , which is reached for high cohesions and large λ (Figure S6). The characteristic values (λ 0 , β 0 ) correspond to the minimal values of λ (and β) for which the central graben exists; it is independent of the material cohesion.

Expressing the relation between β 0 and φ shows that β 0 = 3π 8 -φ 2 for 15 • < φ < 75 • degrees (Figure S7).

The Figure S8 shows that tan β -tan β 0 is a linear function of λ depending on κ and φ. A first-order approximation of this function leads to find the analytical expression (see Appendix 2 for details of the calculations):

tanβ -tanβ 0 = R(κ, φ) (λ -λ 0 ) (24)
for λ > λ 0 , where

R(κ, φ) = 1 - a 0 κ c 0 κ + d 0 1 + a 1 κ + b 1 c 1 κ + d 1 φ (25)
λ 0 = tanβ 0 = tan 3π 8 -φ 2 , and a 0 , c 0 , d 0 , a 1 , b 1 , c 1 , d 1 are constants. Equation ( 24) may be considered as the first order Taylor's series expansion in φ for large values of φ, of the solution found by optimization in β of the equation ( 20). The first-order term a 1 κ+b 1 c 1 κ+d 1 φ takes values in the [-0.25, 0.15] interval so that the dependency of β on φ is relatively moderate in equation ( 24), and mostly due to the λ 0 term. However neglecting the firstorder term leads to significant changes for the highest values of κ. Increasing the order of the expansion enlarges the interval wherein the approximation is valid, at the expense of a larger number of parameters. The optimal order is found to be the first order, using a compromise between the parameter number and the goodness of fit. Results (Figure S9)

show that β is more strongly decreasing with the cohesive term κ. Using the values issued from our computations, we can infer that, when the central graben structure exists, for realistic rock mass characteristics (φ ≈ 30 • ), β grossly varies from about 50

• for λ ≈ 1, to D R A F T January 28, 2019, 3:18pm D R A F T
about 60 • for λ ≈ 2 and 70 • for λ ≈ 6 (Figure 7). These results are coherent with those obtained from analogue models, where values of 50 degrees (λ = 0.8, [START_REF] Acocella | The control of overburden thickness on resurgent domes: insights from analogue models[END_REF]; Brothelande et al. [2016b]) and 60 degrees (λ = 2, [START_REF] Acocella | The control of overburden thickness on resurgent domes: insights from analogue models[END_REF]) were found for β.

The variation of β with λ is also coherent with the results obtained by [START_REF] Haug | Shear versus tensile failure mechanisms induced by sill intrusions -Implications for emplacement of conical and saucer-shaped intrusions[END_REF].

From tanβ c = λ = ρ R tanβ and equation ( 24), we find that

tanβ = tanβ 0 + R(κ, φ) (λ -λ 0 ) = (1 -R(κ, φ)) tanβ 0 + R(κ, φ)ρ R tanβ (26)
where ρ R = W D +W G W D -W G and λ 0 = tanβ 0 = tan 3π 8 -φ 2 ; we infer:

tanβ = 1 -R(κ, φ) 1 -ρ R R(κ, φ) tan β 0 (27) Therefore ρ R = 1, that is W G = 0, corresponds to β = β 0 when κ = 0 (and β = β c = tan -1 (λ) when κ = 0).
Equation ( 27) has to verify the condition tan

β c ≥ tan β ≥ tan β 0 for ρ R ≥ 1 (that is W G ≥ 0). The condition tan β ≥ tan β 0 is true when ρ R ≥ 1 and R (κ, φ) ≥ 0, whatever can be λ, κ, φ. The condition tan β ≤ tan β c implies that R (κ, φ) ≤ R (0, φ) = λ-λ 0 ρ R λ-λ 0 , so that equation (27) is valid when 0 ≤ R (κ, φ) ≤ λ-λ 0 ρ R λ-λ 0 . R (κ, φ) is always lower than 1 ρ R
when λ tends to infinity. As ρ R is always larger than 1, we can infer that 0 ≤ R (κ, φ) ≤ 1.

In volcanic environments, the angle β is generally not directly found from field geological measurements, as inverse faults that reach the surface are more inferred from the deformation and topography rather then directly evidenced. Equation ( 27 is compatible with those computed by [START_REF] Grosfils | Magma reservoir failure on the terrestrial planets: assessing the importance of gravitational loading in simple elastic models[END_REF]. This isotropic overpressure term induces the isotropic damage around the magma reservoir. However in real cases where inflation/deflation cycles exist and correspond to time variation of the overpressure with irreversible strain in contraction, after the first cycle isotropic damage may pre-exist before the re-pressurization of the magma reservoir. In those cases it can be questioned if any depth-dependent overpressure is needed to localize the shear strain at the boundary of the damaged zone.

Values of reservoir radius and depth have been computed for some other resurgent domes (Table 1 andFigures S13 -17) and may be compared to direct geophysical estimations.

Campi Flegrei is the most well-investigated caldera. At this site, dome and graben halfwidths are taken from [START_REF] Sacchi | The Neapolitan Yellow Tuff caldera offshore the Campi Flegrei: Stratal architecture and kinematic reconstruction during the last 15 ky[END_REF], though it is difficult to know which graben structure is actually active nowadays; variations of the graben structure may indicate that depth and radius vary with time. At Campi Flegrei, most of these studies have estimated the source location to be beneath the Pozzuoli area at a depth around 3 km below sea level (e.g. [START_REF] Trasatti | The 2004-2006 uplift episode at Campi Flegrei caldera (Italy): Constraints from SBAS-DInSAR ENVISAT data and Bayesian source inference[END_REF] and references therein; [START_REF] Bonafede | Displacement and stress fields produced by a centre of dilation and by a pressure source in a viscoelastic halfspace: application to the study of ground deformation and seismic activity at Campi Flegrei, Italy[END_REF]; Berrino et al.

[ 1984]). The order of magnitude for the depth is corroborated by the seismicity recorded during the 1982-84 uplift that clustered between 1 and 4 km below sea level [START_REF] D R A F T Ferrucci | P-SV Conversions at a Shallow Boundary Beneath Campi Flegrei Caldera (Italy): Evidence for the Magma Chamber[END_REF]. Furthermore, seismic attenuation images provided by De Siena et al. [2010] indicate the presence of a small melt pocket beneath Pozzuoli located between 3 and 3.5 km depth. All these estimations match well with ours.

At Yellowstone, several tomography surveys have imaged a very large low-velocity zone, interpreted as a giant magma reservoir. [START_REF] Miller | P and S velocity structure of the Yellowstone volcanic field from local earthquake and controlled source tomography[END_REF] Myrs ago [START_REF] Phillips | The 40 Ar/ 39 Ar age constraints on the duration of resurgence at the Valles caldera, New Mexico[END_REF]. The only direct geophysical constraint on the depth of the magma reservoir associated to resurgence is that it is greater than 3.25 km, the maximum depth of the geothermal drilling campaign that did not penetrate any intrusive rock [START_REF] Nielson | Internal geology and evolution of the Redondo dome, Valles caldera, New Mexico[END_REF].

Conclusion

In this work we have investigated the deformation process around a pressurized magma reservoir embedded in a damaging elastic volcanic edifice. It has allowed us to evidence the action of the progressive damage process, and the structure created by the damage distribution. Damage is first isotropic around the cavity and constitutes a damaged zone.

Then the free-surface effect appears and an anisotropic shear strain develops from the boundary of the damaged zone; it localizes on reverse faults that propagate upward to the surface. When the vertical surface deformation is sufficient, normal faulting appears.

Finally, the complete structure shows an undeformed wedge above the top of the damaged zone, which strongly recalls the Prandtl's wedge. This structure is very similar to what is found by analogue modelling and from field geologic observations. From this model, we found a relation for reservoir radius and depth as a function of dome and graben widths. Magma reservoir pressure is deduced from limit analysis, which allows a better understanding of the magma pressure build-up. The dip of the reverse faults is inferred from the minimization of the pressure needed to rupture the crust around the magma reservoir. From that analysis, the magma reservoir overpressure, the dip of the faults, the D R A F T January 28, 2019, 3:18pm D R A F T magma reservoir depth and the damaged zone radius may be inferred analytically from three parameters only: the ratio ρ R computed from the dome and graben widths, the cohesion, and the friction angle.

Acknowledgments. Snowshoe Mountain 5.3-5.7 0.9-1 1.4-1.5 60-65 6.5-7.5 2.6-2.7 La Pacana Purifican-Bola 5-5.5 0.4-0.6 1.15-1.3 50-55 5.5-6 2.9-3.2 Campi Flegrei Pozzuoli Bay 2.2-2.6 0.5-0.7 1.5-1.9 62-70 2.5-3.5 0.8 Table 1. Results for the ratio ρ R = W D +W G W D -W G , the fault dip angle β, the reservoir

[

  1993],[START_REF] Marti | Experimental studies of collapse calderas[END_REF],[START_REF] Merle | Experimental modelling of thin-skinned shortening around magmatic intrusions[END_REF],[START_REF] Acocella | The control of overburden thickness on resurgent domes: insights from analogue models[END_REF], Merle et al. [2013] and Brothelande and Merle [2015]): resurgent domes are structures with reverse faulting on their external boundaries and internal normal faulting (see, e.g., Acocella et al. D R A F T January 28, 2019, 3:18pm D R A F T

  ) therefore allows the estimation of β from values of ρ R , that is W D , W G measured on the field, and estimates of κ and φ determined at the scale of the rock mass. This estimation is used to compute h c and r c , and the normalized overpressure terms Π f + Π D , since λ = ρ R tan β.We used our results to compute estimates of β, h c , r DZ and Π f + Π D in the case of the D R A F T January 28, 2019, 3:18pm D R A F T resurgent dome of Yenkahe (Vanuatu). From field and map measurements, we know that (see, e.g.,Brothelande et al. [2016b]) W D = 2.3 km, W G = 0.6 km so that ρ R = 1.7; the corresponding interval of validity for the equation (27) is 0 ≤ R (κ, φ) ≤ 0.59. For reasonable values of the friction angle φ comprised between 25 • and 45 • , we find (Figure8) that 60 • ≤ β ≤ 75 • , h c comprised between 2.5 km and 4 km, and r DZ ≈ 1 km. The radius of the magma reservoir is not directly known, as it depends on the damaged zone relative thickness ∆, which is poorly known and likely to be comprised between 1 and 5. Using a value of 2 to 3 for ∆ allows us to find a magma reservoir radius of about 300 meters and a magma reservoir depth of about 2 to 3.5 km, the depth being compatible with the numerical results of Brothelande et al. [2016b]. Variations of h c and r DZ with W G and W D are presented Figure S17; they show that h c is sensitive to variations in W G more than in W D . Computation of the overpressure terms for Yenkahe within the intervals 25 • ≤ φ ≤ 40 • and the corresponding 75 • ≥ β ≥ 60 • yields 0.06 ≤ Π f ≤ 0.21 and 3.26κ ≥ Π D ≥ 1.98κ; for φ = 30 • and β = 65 • , Π f = 0.069 and Π D = 2.38κ. This result helps to quantify the fact that when the eruptive/intrusive system is closed or sealed (what happens after a long period of quiescence), the normalized overpressure required for the edifice rupture is 10κ to 50κ (0.4 ≤ κ ≤ 2) larger than the normalized overpressure needed when the edifice is already ruptured and deforms only by frictional sliding. From the former results, the overpressure involved in the crustal anisotropic deformation by the magma reservoir is close to ρgBΠ D , that is approximately 2C. For cohesion values between 0.1 and 10 MPa, this part of the overpressure varies between 0.2 and 20 MPa. Computation of the isotropic ρgBΠ 0 = (1 -k)ρgH term shows that it scales with the depth H of the magma reservoir, and reaches a value of 50 MPa in the case of the Yenkahe dome. This order of magnitude

Figure 1 .Figure 2 .Figure 3 .Figure 4 .Figure 5 .Figure 6 .

 123456 Figure 1. Simplified structural maps of (a) Redondo dome in Valles caldera (after Smith

  melting zones at about 8 km beneath the domes surface, a depth later confirmed by[START_REF] Husen | Evidence for gas and magmatic sources beneath the Yellowstone volcanic field from seismic tomographic imaging[END_REF] whereas[START_REF] Farrell | Tomography from 26 years of seismicity revealing that the spatial extent of the Yellowstone crustal magma reservoir extends well beyond the Yellowstone caldera[END_REF] found a depth of about 6 km.At Valles caldera, resurgence is no longer active and probably finished more than 1

identified probable partial D R A F T January 28, 2019, 3:18pm D R A F T

  This work was supported by grants from Université Savoie Mont Blanc (AO APS 2017) and European Research Council (ERC) under the European Union Horizon 2020 research and innovation program (Grant agreement no. 757848 CoQuake).

	Caldera	Dome	W D (km) W G (km)	ρ R β( • ) h c (km) r DZ (km)
	Siwi	Yenkahe	1.5-2.2	0.4-0.6 1.5-2.3 60-70	2-4	0.8-1
	Valles	Redondo Dome	4.5-4.9	1.6-1.8	2-2.3 65-75	9-12	1.7
	Yellowstone	Mallard Lake	4.5-5	1.3-1.7 1.7-2.2 65-75	9-12	1.9-2
	Creede
	This paper contains no data.
	D R A F T	January 28, 2019, 3:18pm	D R A F T


Clément Le Ludec

Paola Tubaro

Antonio A Casilli

How many people microwork in France? Estimating the size of a new labor force

Keywords: Microwork, digital transformation, employment, digital platforms, digital labor, algorithms

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

INTRODUCTION

The digital economy currently experiences a proliferation of "microwork" platforms, specialist services where providers perform fragmented and standardized micro-tasks to be paid on a piece-rate basis. These activities take only minutes and their reward can be as low as few cents. Often repetitive and unqualified, they consist for example in identifying or naming objects on images, transcribing invoices, translating snippets of text, moderating content (such as videos), sorting or classifying search results, responding to online surveys.

These tasks are often part of data-intensive processes and lengthy supply chains, feeding activities as varied as digitization of archives, market research, management of back-office operations and most importantly, development of artificial intelligence [START_REF] Gray | Anatomy of a crowdsourcing platformusing the example of Microworkers.com[END_REF]. Recent applications of machine learning -on which smart technologies, autonomous vehicles and virtual assistants are all based -rely on the creation and maintenance of large databases that need to be annotated, refined, labeled and more generally, augmented [START_REF] Porter | Enhancing big data in the social sciences with crowdsourcing: Data augmentation practices, techniques, and opportunities[END_REF]. Microwork is used, first, to prepare, categorize and qualify information for automatic learning algorithms; and second, to assess their performance and if necessary, to make corrections. Growing investments by technology companies confirm the importance of microwork in the current development of artificial intelligence. Giants such as Google and Microsoft either rely on existing platforms or create their own internal microwork markets.

Authors' addresses: Clément Le Ludec, MSH Paris Saclay, Paris, France, leludec. clement@gmail.com; Paola Tubaro, National Center of Scientific Research (CNRS), Laboratory of Research in Informatics, Paris-Saclay, France, paola.tubaro@lri.fr; Antonio A. Casilli, Telecom ParisTech, Paris, France, antonio.casilli@telecom-paristech.fr. This novel organization of AI-driven automation in contemporary industries does not "replace" human jobs but makes human contribution to productive processes largely inconspicuous. It supports technologies where they fail, and yet it is not a selling point that companies leverage to attract and retain large userbases. Microwork has received very little media coverage and is unfamiliar to the general public. Its invisibility is striking especially if compared to other new forms of work associated with the rise of digital platforms, such as on-demand work for urban transport applications like Uber, or for real-time delivery like Deliveroo.

Microwork platforms do not generally recruit or hire workers directly, but operate as intermediaries between client companies (usually called "requesters") who publish small tasks, and individual users (usually called "workers", "contributors", or "providers") who accept and perform the tasks. This form of digital labor pushes to the extreme the logic of workforce casualization and insecurity already noted in the context of the vast public debate and legal disputes on the status of "uberized" workers. It is therefore imperative to address this emerging phenomenon.

Strikingly, however, the size of this phenomenon is unknown: how large is the micro-workforce? Official statistics are still illequipped to capture the economics of digital platforms, and the (few) surveys conducted so far have conflated microwork with other platform-mediated activities such as transport and delivery (Uber, Deliveroo). Some insight comes from platforms themselves, whose claim is that their userbase is large: if Amazon Mechanical Turk, the most widely-known microwork service, boasted 500,000 providers ("Turkers") as early as 2014 1 , the Chinese giant Witmart alone is said to exceed 12 millions 2 . A widely cited 2015 World Bank report proposed a much more conservative estimate of 4.8 million people registered on microwork websites globally, just over 10% of whom are actually active [START_REF] Kuek | The global opportunity in online outsourcing[END_REF].

Not only are these figures very divergent, but they are also static, no serious attempt to update them having been undertaken since 2014-15. They are also somewhat misleading insofar as they take little account of the European context. This is a serious limitation from the viewpoint of national and regional policy-making: lack of even approximate estimates makes it difficult to take action and hinders initiatives in support of local microworkers.

To fill this gap, we propose an estimate of the total number of people microworking in the specific case of France. A highly industrialized country and a pioneer in information technologies, France is less documented in the (still scant) literature on microwork, and is less present in international platforms such as Amazon Mechanical Turk, partly for language reasons. France has its own microwork platforms, the most popular of which is Wirk (formerly known as Foule Factory). Its webpage displays 50,000 contributors and it has had to close its registrations [START_REF] Amar | Les plateformes collaboratives, lâĂŹemploi et la protection sociale[END_REF].

To estimate the size of the microworking population in France, we draw on publicly available sources combined with the results of DiPLab ("Digital Platform Labour"), a comprehensive study of microwork in the French-speaking world3 . We identify three modes of engagement in microwork that correspond to three pools of users of these platforms. They comprise approximately 15,000 "very active" microworkers, 50,000 "routine" microworkers and 250,000 "casual" microworkers respectively. These estimates should be interpreted as orders of magnitude rather than precise values.

These relatively high figures matter to policy-making. To the extent that they exceed the number of contributors to more highprofile platforms such as Uber and Deliveroo, they call for the attention of industry and union leaders as well as public authorities.

To the best of our knowledge, ours is the first attempt to estimate the number of microworkers in a specific country (or other geographically or institutionally limited setting) across microwork platforms. It demonstrates the need to go beyond the most highprofile cases and to take into account local conditions in addition to global trends. In this sense, our results may inspire researchers interested in re-applying our methodology to other countries.

The remainder of this paper is organized as follows. After a brief presentation of the literature and of our selection of relevant platforms (section 2), we compare and contrast different methods for assessing the size of our target population (section 3), and we give an interpretation of the results thus obtained (section 4).

WHY IS IT SO DIFFICULT TO ESTIMATE THE SIZE OF MICROWORK?

While a growing number of studies are exploring the topic of microwork, most of them have focused on one platform, the US-based Amazon Mechanical Turk, which is the oldest and by far the best known. The methods used to estimate the number of people working on this service can only partially be transposed to our questioning, because we are interested in a country as a whole, France, and we cannot therefore limit ourselves to a single platform (section 2.1).

We then proceed in stages, first by explaining our choice of platforms to be included in the analysis (2.2). In the following section (3), we will present the empirical basis for our results.

Estimating the size of Amazon Mechanical Turk micro-workforce

Two distinct scientific objectives have motivated existing research on the demographics of the Amazon Mechanical Turk workforce.

The first concerns the working conditions, the precariousness and the level of remuneration of its users, sometimes with the aim of proposing concrete actions [START_REF] Paolacci | Running experiments on Amazon Mechanical Turk[END_REF]Ross et al. 2010].

The second purports to assuring the quality of the results of scientific surveys, questionnaires and experiments for which "Turkers" are recruited by research teams in disciplines such as psychology, marketing, and cognitive sciences. This involves ensuring access to a sufficiently large "pool" of respondents [START_REF] Keith | Systems perspective of Amazon Mechanical Turk for organizational research: review and recommendations[END_REF].

Attempts to estimate the number of people working on Mechanical Turk have arisen from these dissimilar research questions, insofar as the figure of 500,000 popularized by Amazon was to be probed. A first study showed the existence of a pool of 7,300 "rather active" unique workers at any given time [START_REF] Stewart | The average laboratory samples a population of 7,300 Amazon Mechanical Turk workers[END_REF]]. In the same year, activist Kristy Milland conducted a six-week survey and counted about 30,000 workers4 . [START_REF] Difallah | Demographics and Dynamics of Mechanical Turk Workers[END_REF]] use Stewart's model and improve it, allowing for a longer observation time and taking into account the heterogeneous propensity of individuals to accept tasks. They show that about 100,000 people work on Jeff Bezos' platform, or one fifth of the advertised population.

Despite the presence of a sizable set of microworkers originating from India and smaller contingents from other countries, the first estimates of Amazon Mechanical Turk workforce paid particular attention to users from the United States -partly to serve the needs of scientific studies that need American participants. So far, no research intended to measure the total number of microworkers from a target country, all platforms combined. Moreover, no extant study dealt specifically with France and the French-speaking world so far. We need to move beyond these limitations to propose estimation methods for the different platforms operating in France.

Which platforms operate in France?

On Mechanical Turk, demand from French companies outweighs the supply of microwork from users located in France. Since 2013, the platform has been discouraging registrations outside the US and India, notably by imposing gift cards as the only means of payment. To find French microworkers, we therefore need to expand the scope of our investigation beyond the boundaries of Amazon. Recent research [START_REF] Berg | Digital Labour Platforms and the Future of Work[END_REF][START_REF] Forde | The Social Protection of Workers in the Collaborative Economy[END_REF]] covering multiple countries including France, has added CrowdFlower, Microworkers, Clickworker5 and, in the case of [START_REF] Berg | Digital Labour Platforms and the Future of Work[END_REF], Prolific, to Amazon Mechanical Turk. We include their selection of platforms except Prolific, a site specialized in surveys that exists only in English, and which does not seem to have users in France (there were no French respondents in the study of [START_REF] Berg | Digital Labour Platforms and the Future of Work[END_REF]). The case of CrowdFlower is more complex because this platform has recently changed its name to Figure Eight, and access to its tasks is now through ClixSense, which is basically a portal to other platforms. It is therefore the latter that we will retain.

In addition to these international platforms, we include Wirk, a platform that exclusively recruits its microworkers in mainland France and has established itself as a key player in the market for micro-tasks in the country. As mentioned in section 1, it recently changed its name from the originale Foule Factory ("Foule" meaning "Crowd" in French). We also include Ferpection, a French platform that recruits internationally. These services are potentially more attractive for French speakers than others like for example Microworkers, where English language is required even to sign up. Finally, we add Appen, an international platform open to recruitment in France, which lists tasks of the same nature as the others, but differs from them in that it organizes tasks in "projects" of slightly longer duration6 . It is useful to characterize this selection of platforms according to their access policies (open or limited) and the presence of at least one welcome page in French (Table 1). With regard to the first criterion, notice that Wirk has closed registration of new members but reopens it intermittently in case of surge in the demand of microwork by client firms. As for the second criterion, the presence of a French page seems to encourage access by new users.

WHICH METHOD TO ESTIMATE THE NUMBER OF FRENCH MICROWORKERS?

Several approaches can be employed to estimate the size of the French microworking population, despite its lack of visibility in public debates as well as in official statistics. It is always tempting to take into account the official figures provided by the platforms (section 3.1), although they tend to overestimate the real level of activity, as existing literature has already noted in the case of Amazon Mechanical Turk [START_REF] Stewart | The average laboratory samples a population of 7,300 Amazon Mechanical Turk workers[END_REF]]. Building up on previous studies, we then apply a "capture-recapture" model, which leads to a lower estimate of the userbase of a single platform (3.2). This second approach, however, does not scale up to the entire nationâĂŹs microworking population which may be using different platforms. Therefore, we present an original method based on website audience

3.1 The declarative method: relying on official figures from microwork platforms

Official figures published by microwork platforms (Table 2) are sometimes the only source of information. For example, some of them are featured in a report of the French General Inspectorate of Social Affairs [START_REF] Amar | Les plateformes collaboratives, lâĂŹemploi et la protection sociale[END_REF]. However as the same report points out, they call for caution. Platforms advertise the size of their userbase by counting signups without taking into account the actual level of activity, and can in this way artificially inflate the result. This effect is all the stronger as high numbers can impact potential fundraising or the attractiveness of the platform for business clients. It must be added that few platforms have systematic registration/deregistration management policies. They also differ in their degree of precision, some (such as Microworkers) giving very detailed figures and updating them regularly, others just providing approximations (such as Amazon, which never came back to its initial estimate of 500,000).

In general, international platforms calculate the number of their micro-workforce without breaking it down by country. The literature and platforms' own websites offer some approximate insight. On Amazon Mechanical Turk, French workers represent only a small fraction of the userbase [START_REF] Difallah | Demographics and Dynamics of Mechanical Turk Workers[END_REF]]. On another international platform, Microworkers, France is not among the 10 most represented countries, and remains among those where less than 20% of total transactions take place [START_REF] Gray | Anatomy of a crowdsourcing platformusing the example of Microworkers.com[END_REF]]. On the German platform Clickworker, France and other European countries (excluding Germany) account for a total of 25% of the workforce7 . ClixSense and Appen are major international players in the microwork industry, but they do not specify the national composition of their registration databases. On the other hand, French platforms present a variety of situations. If on the one hand Wirk is only accessible to residents of France, Ferpection has users signing up not only from France but also from the United Kingdom, the United States, Ireland and other countries8 .

The ecological method: the "capture-recapture" model

To address this problem, [START_REF] Stewart | The average laboratory samples a population of 7,300 Amazon Mechanical Turk workers[END_REF]] and [START_REF] Difallah | Demographics and Dynamics of Mechanical Turk Workers[END_REF]] implemented on Mechanical Turk a "capture-recapture" approach commonly used in bio-ecology and epidemiology. They posted a micro-task for a relatively long period of time, allowing repeated participation. The technique consists in "capturing" participants a first time, identifying them as having already done the task, then seeing how many of them get "recaptured" a second time. This approach is based on two assumptions: first, the population is closed (nobody left the platform during the study) and second, all subjects have the same chances of being captured. We were able to replicate this approach on the French leading microwork platform Wirk âĂŞ the only one providing interfaces and support pages entirely in French. As part of our DiPLab study (see footnote 1), we distributed a questionnaire in the form of a paid task, administered twice with a 23-day interval: the first phase allowed us to obtain 505 responses and the second 492 responses 9 . While this two-step collection was purely related to logistical issues, we saw it as an opportunity to make an estimate of the number of persons microworking for Wirk. Indeed, in the second collection, we detected 89 returning respondents -thus enabling us to apply the capture-recapture model.

As for the two assumptions required by the capture-recapture model, the first (closed population) was satisfied, as the short time interval between the two waves limited the number of those entering or exiting the platform. The so-called Lincoln-Petersen formula applied to the data thus collected, with N the total number to be estimated, n 1 the number of users captured in the first wave, n 2 the number in the second wave and m the number of recaptured participants, gives :

N = n 1 n 2 m
Based on the number of individuals observed, the results of the calculation give a population of 2792 microworkers (Table 3). This figure is certainly well below (5.6%) the platform's official declarations, but it is consistent with the domain-specific knowledge of its managers who, answering our questions about the possible duration of the data collection, were confident that "over two months, [they] can mobilize up to 3000 people on this type of task".

However, this may be an underestimation. Firstly, it is noticeable that the result N increases as the number of recaptured microworkers m decreases. Thus, given the terms of our survey (we had indeed signified that participation in the survey was allowed only once), we could have expected a much smaller number of recaptured users and therefore a larger population. Secondly, the second assumption of the model (equal chance of capture) may not apply if users display an uneven level of involvement in microwork on Wirk. When 9 Calculations after cleaning and filtering of data. facing the same issue on Amazon Mechanical Turk, [Difallah et al. 2018, p. 141] introduce a latent variable (a) that they interpret as propensity to participate: for example, it could reflect the fact that some people perform all the tasks available on a given platform, regardless of their appeal, difficulty or remuneration, while others are more selective. The initial formula is modified as follows 10 :

N * = n 1 n 2 m (1 + V ar (a) E(a) 2 ) = N (1 + V ar (a) E(a)
2 ) [START_REF] Difallah | Demographics and Dynamics of Mechanical Turk Workers[END_REF]] estimated the distribution of a based on 28 observations over two years. In our case, their parameter setting 11 would give N * = 34, 166 (Table 3), a value that still falls short of the official declarations of Wirk (50,000 as indicated above), but approximates it if we interpret it as an the order of magnitude.

However, there is no guarantee that our population shares the same characteristics as people working on Mechanical Turk. We therefore propose an alternative calculation of N * using a proxy for a, based on one of the questions asked in our survey, touching on the number of tasks carried out by each microworker through Wirk in the previous month. The answers to this question, summarized in Figure 1, show a high diversity of participation -inconsistent with the assumption of equal chances of capture 12 . Using the above formula to calculate N * , we obtain a population of 6531 microworkers on Wirk13 . Table 3 summarizes these results. The plain capture-recapture model (Lincoln-Petersen formula) produces an underestimation and 10 N ≈ N * when the variance of a is close to the mean, but N < N * otherwise. 11 Difallah et al (2018, p. 142) consider that the propensity to participate follows a beta probability distribution, and that therefore the chance to recapture the same subject n times follows a beta-binomial law with parameters α = 0.29 and β = 20.9. With these values, and a sample of S single observations, the population N * is equal to S 1-f (0|n, α , β ) . 12 As our variable was originally a categorical (ordered) one, we have first transformed it into a numerical variable (taking the value of the minimum of the interval in each case: 0, 1, 3, 6, 10) in order to infer a distribution from it, thus enabling calculation of its mean and variance. Alternative transformations of this variable into a numerical one give qualitatively similar results.

Table 3. Estimate of the Wirk population using the capture-recapture method (basic formula and corrections), based on the calculations illustrated above.

Method

Result % Declared (50,000)

Lincoln-Petersen formula 2,792 5.6%

Correction with [START_REF] Difallah | Demographics and Dynamics of Mechanical Turk Workers[END_REF] its correction (parameters of [START_REF] Difallah | Demographics and Dynamics of Mechanical Turk Workers[END_REF]) approximates the number of official signed-up worker users (Table 2). The third estimate, correcting the Lincoln-Petersen figure with a proxy from our DiPLab survey, gives an intermediate figure.

The latter estimate is more difficult to interpret based on the information provided so far. We now need to find clues to make sense of it. Also, we need to extend our analysis beyond the case of one single platform and go back to the full list of relevant microwork intermediaries that we have identified as relevant for France (section 2.2).

The panel method: measuring the audience of microwork platforms

The use of panels to measure the audience of these platforms, a method borrowed from the classical study of media, can provide another perspective on the population of microworkers in France. Since they receive too few visits via the Médiamétrie panel, a reference on website traffic in France, we use the alternative service SimilarWeb.com14 , which makes it possible to estimate the number of unique monthly visitors to microwork sites worldwide (Table 4, left column). The main difficulty in using this source is that some microwork platforms have separate interfaces for client companies (that is, their requesters) and workers. Indeed, it is the latter, namely the pages dedicated to carrying out the tasks, that we must look at. We must also exclude simple visitors from our analysis. We thus need to make the distinction explicitly and focus only on workeronly pages. Table 4 shows that the average duration of visits is relatively long (central column). This allows us to exclude the presence of simple visitors or Internet users who are just looking for information, and to argue that these sections of the platforms are almost exclusively accessed by microworkers. The right column of Table 4 provides estimates of the average number of French unique monthly visitors per platform.

Let's first comment on the result for Wirk. 6,958 unique microkworkers access it on average every month -which is very close to the figure of 6,531 that results from our corrected capturerecapture calculation detailed in section 3.2. This order of magnitude seems to correspond well to the size of the active population on this platform over a month, as it could be measured in the summer of 2018. It should also be noted that almost all of the visits originate from France, in accordance with the platform's policy. The residual cases of people signing in from outside France essentially concern French microworkers temporarily connecting from abroad âĂŤ a trend confirmed by responses to our questionnaire. Concerning the other microwork platforms offering access interfaces in French, ClixSense, Clickworker, Appen and Ferpection are more visited than Wirk. This is both due to their open registration policy and to their lifespan (at least in the case of ClixSense, created as early as 2009). Some also benefit from their role as gateways to other microwork platforms: ClixSense points towards Figure Eight, while Clickworker provides access to UHRS (footnote 2.2). To estimate the number of French microworkers on Mechanical Turk (righthand column), information from SimilarWeb.com was obviously not sufficient as the large number of observed unique visitors even exceeds the platform's own estimate of 500,000 (left column), and must therefore include surfers who are not themselves microworkers -not unlikely for a widely known platform. We thus rely on on a survey by the team led by Panos Ipeirotis at New York University 15 . Extrapolating from the size of the group of French "Turkers" that they observed, i.e. 0.25% of the total, and assuming that their sample is representative, we estimate that the number of French residents operating on Mechanical Turk is around 1,250.

Crossing methods to obtain a range of values

The calculation methods we have implemented so far provide differing estimates of the population of microworkers in France: a high estimate is the number of signed-up users advertised by the platforms themselves (11,015,829 individuals worldwide), a low estimate is the plain capture-recapture model (2,792 individuals for a single platform, Wirk) and an intermediate estimate is the audience-based measure (62,754 individuals across platforms all over France).

Each of these approaches has limitations. At least in its basic, uncorrected version, the capture-recapture model underestimates the target population, while platforms' official declarations overestimate it. The use of audience measurements brings a helpful alternative perspective with intermediate results, but it is based on proprietary techniques whose actual implementation is difficult to audit for external researchers. At this stage, we have no means to choose one single metric from among those three, all the more so as they derive from very different measurement approaches. However, we can use these three results to cross-pollinate each other and help us derive more consistent measures.

Notice, first, that in the case of Wirk alone, the two corrections to the capture-recapture model that we have proposed to compensate for the downward bias of the basic Lincoln-Petersen formula (34,166 and 6,531), are close to, respectively, the official number of registered workers provided by the platform itself (50,000) and audience figures (6,958). These results implicitly validate all these metrics, although they are all lower (to different degrees) that the total number of platform-declared microworkers.

More importantly, audience figures enable an assessment of the number of registered microworkers who are located in France. To compute their number, we need to extrapolate their frequency of connection to the platforms (calculated from Table 4) as a percentage of the total number of persons signed-up worldwide (Table 2) 16 . Table 5 shows the results of these calculations.

On this basis, we can go a step further and attempt to generalize the capture-recapture model initially used just on one platform, 15 Data from the survey "Analyzing MTurk demographics" are available at https: //github.com/ipeirotis/mturk_demographics and serve as the basis for the article by [START_REF] Difallah | Demographics and Dynamics of Mechanical Turk Workers[END_REF]. 16 Given the national scope of Wirk's activity, we consider all its registrations to be French. For Mechanical Turk we refer to the figures from P. Ipeirotis's survey mentioned above (footnote 3.3). For the other platforms, we take into account the number of visitors from France (Table 4, last column) divided by the total number of visitors (Table 4, first column), then apply the resulting percentage to the number of registrations (Table 2). For example, Ferpection has 8,116 unique visitors from France, out of a total of 28,064 visitors, or 28.92%; multiplying by its official number of registered users, equal to 50,000, we obtain 14,460. by applying its ratios (Table 3, last column) to the total number of French microworkers we have just estimated (319,096). We thus obtain a range, with a low estimate of 17,869 (5.6% of the declared total for France), a high estimate of 217,943 (with a "corrected" rate of 68.3%) and an intermediate estimate of 41,482 (with the alternative correction of 13%).

Adjusting to take into account multi-homing

We cannot rely on these three estimates until we have made an additional correction to take into account multi-homing, that is, the presence of microworkers on several platforms at the same time. If we disregard multi-homing, we may over-estimate the number of microworkers by counting the same persons multiple times. First, how widespread is this phenomenon, or put differently, how many people does it concern? To answer this question, let us turn to our DiPLab survey, in which participants were asked to indicate their possible use of different microwork platforms from a list of options. We know that of our 908 unique respondents on Wirk, 151 are also registered on at least another microwork platform. In other words, 16.6% of respondents practice multi-activity -the average usage being 1.27 platforms per person. Assuming our sample is representative, and extrapolating its behaviour to the entire French microworking population, amounts to applying the 16.6% rate to the measures previously obtained (Table 6).

Little is known of the multi-homing practices of microworkers in other contexts but, comparing our survey to studies of other online practices such as the buying and selling of goods ans services, it appears that our rate of multi-activity is low. For example, [Oxera 2015] reports an average of 2.2 platforms per user. The DiPLab survey suggests that this relative smaller rate depends largely on the specific context of microwork. Many respondents hint that entry costs of multi-homing are relatively high, both in terms of incompatibility of payment systems (for example, Wirk uses Mango Pay while Microworkers uses Paypal, Skrill or Payoneer) and in terms of limited portability of "qualifications" (unpaid tests needed to access some types of micro-tasks). Additionally, the English language used on many international platforms puts off some French microworkers, even though some of their tasks can be in French.

HOW TO INTERPRET RESULTS IN LIGHT OF MICROWORKERS' LEVEL OF ACTIVITY?

We now turn to the meaning of these different measures of the size of the French microworking population. Using the DiPlab survey, we associate our figures to characterization of users according to their level of activity. We distinguish a group of "very active" (4.1), a group of "routine" (4.2) and one of "casual" (4.3) microworkers.

Very active microworkers

Our first, lower estimate of 14,903 microworkers was initially calculated on the Wirk platform with the uncorrected capture-recapture model, subsequently extrapolated to the total number of French registrations, and finally corrected to remove duplicates. It can be interpreted as a measure of the number of users with a high level of activity. As discussed (section 3.2), the number of captured and re-captured individuals in our survey is relatively higher than expected, and corresponds to a high level of activity: more engaged workers are significantly more likely to appear in both phases of the data collection. Another reason to believe that the great majority of the respondents to our survey fall into this category isthe frequency with which they log in to look for microtasks: at least once a week for 90% of them.

Qualitative interviews conducted with a subset of respondents as part of the DiPLab project, confirm the hypothesis that there is a very active sub-population on Wirk and, by extension, on microwork platforms. This is the case for one respondent who, in addition to her primary occupation as a nurse, frequently performs "simple and quick tasks". Other users, expressing their "fear of missing out tasks", even go so far as leaving Wirk continuously open on their computers.

Routine microworkers

The measure we obtain through audience measurements from microwork platforms is much larger, with 52,337 users. It corresponds to a different, larger group of users, by definition those connected to a microwork platform (both Wirk and its competitors in France, as seen in section 3.3) at least once a month, after removing duplicates (section 3.5).

Once again, DiPLab interviews provide us with insights into this mode of engagement with microwork. A respondent whose main job is editorial manager in a communication agency, has been microworking for three years. She signed up during a period of her life in which her "lifestyle" required "an additional income", resulting in "a compulsive need to earn more money". Since then, she has been returning to microwork "from time to time", with a preference for questionnaire-based micro-tasks that she sees as an extension of her education.

In sum, the group of "routine" users corresponds to a fragmented mode of engagement in microwork, with an often-selective attitude towards tasks.

Casual microworkers

Our last estimate features 266,126 persons and corresponds to registered users as declared by platforms themselves, after removing non-French users and duplicates. It is a group of "casual" microworkers : French users who signed up with one or more microwork platforms, who may or may not have an intense level of activity. Some of them may perform less than one task per month.

The long tail of this population that connects less frequently, is more difficult to reach through surveys and may escape audience measures. Yet our interviews provide sufficient insight to portray it. We can distinguish two sub-populations. One includes users who signed up out of curiosity and quickly abandoned microwork because of the scarcity and difficulty of the tasks -two points often emphasized by our study participants. Another includes individuals who were "involved in" microwork at least once and who can, intermittently or in the future, become "routine" or even "very active" users.

One respondent, holding a day-job as administrative secretary, illustrates this situation well. Registered on Wirk in 2016, he "quickly dropped out". A few months later, he received an email indicating that the platform would delete his account due to inactivity. He then decided to resume his microwork in a relatively sustained way. He now claims he earns about 80 euros a month from micro-tasks, almost four times more than the average in our survey. Another microworker, currently seeking employment, also alternates between inactivity and intensive performance. In her working life, she has always experienced alternating periods of unemployment and periods of employment. She says she stops microwork as soon as she finds a job because she will "no longer have time", while her phases of unemployment correspond to periods of intensive activity on Wirk because she prefers to "do this rather than nothing".

Table 7 summarizes our results.

CONCLUSIONS

With this article, we have sought to launch a discussion on how to quantify the pervasiveness of new forms of work that, for the time being, are largely outside official statistics. Microwork illustrates this difficulty very clearly. Lacking the media coverage that platforms such as Uber or AirBnb receive, microwork is largely unrecognised, overshadowed by the commercial propaganda revolving around the promises of algorithms and robotic exploits [START_REF] Irani | The cultural work of microwork[END_REF]]. The voice of the individuals who perform microwork goes largely unheard. Somewhat paradoxically, microwork remains at the margin of formal economy even though it constitutes a strategic resource for big dataintensive artificial intelligence, a major component of the digital economy.

We have proposed an estimate of the size of microworking population in France. We have chosen to take into account not just one platform (as most studies that have exclusively focused on Amazon mechanical Turk so far) but a set of platforms that we have reasons to consider relevant in the country. We have combined different methods and data sources: from the public declarations of platforms themselves and the results of publicly-available audience measurement panels, to data that we collected through questionnaires and interviews as part of our DiPLab project. In this way, we have detected and qualified three groups of users according to their level of involvement: 14,903 "very active" individuals, most of whom are present on the platforms at least once a week; 52,337 "routine" users, present at least once a month, with a more fragmented level of activity and a selective choice of tasks; and 266.126 "casual" users who mostly alternate between inactivity and periods of more intensive practice of microwork, notably during phases of transition in their lives (unemployment, maternity leave, etc.).

We mentioned the limitations of the publicly available information we have used, including the opacity of audience measures -constructed by private companies for their clients and difficult to audit for independent researchers. In addition as we said, our DiPLab survey included just two waves of capture and recapture, allowing only for approximate corrections of the bias resulting from the equal-capture-chances assumption. Another limitation to our study lies in the difficulty of assessing the extent to which these three sets of microworkers, corresponding to three platform usage patterns and established through three very different estimation methods, intersect so that their populations partly overlap. Despite the questions they still raise, these figures provide extremely useful orders of magnitude.

Consequently, our estimates are critical contributions to the understanding of ongoing significant changes in the labor market in the era of digital platforms. The tens of thousands of French microworkers performing increasingly fragmented piece-based tasks can have a strong impact on the very perception of labor, its social protection and its modes and levels of remuneration in the coming years.

By way of comparison, the IGAS report we already mentioned (section 3.1) estimated the number of Uber drivers in France at around 14,000 [START_REF] Amar | Les plateformes collaboratives, lâĂŹemploi et la protection sociale[END_REF]; all platforms combined, there are around 27,000 drivers of private hire vehicles in France, according to the Ministry of transport (cited in [START_REF] Pommier | Les vrais chiffres du marchÃľ des VTC et des taxis en France[END_REF]). As for bicycle delivery companies, Deliveroo claims to have 9,300 couriers in France [START_REF] Deliveroo | Deliveroo lance Marketplace+[END_REF]]. Our estimates therefore shows that microwork has a workforce whose size is at least comparable to that of these high-profile platforms -if not higher if we factor in casual participation. It is therefore not an anecdotal phenomenon and deserves great attention from the research community, from unions and from policy-makers alike.

In France and in Europe, the current debate on social protection for digital platforms workers must take microwork into account. Research must sustain these reflections and provide elements for action, by proposing a more accurate knowledge of the social changes that come with microwork as well as of the operational functioning and business models of the digital platforms that benefit from it.

Fig. 1 .

 1 Fig. 1. Number of micro-tasks carried out on Wirk in the month preceding the survey, n = 908. Source: Authors' elaboration with DiPLab survey data.

Table 1 .

 1 Characteristics of microwork platforms included in our study

	Platform	Open sign-up French page
	Amazon Mechanical Turk No	No
	Microworkers	Yes	No
	Clickworker	Yes	Yes
	ClixSense	Yes	No
	Wirk	No	Yes
	Ferpection	Yes	Yes
	Appen	No	Yes

Source: authors' elaboration based on platforms' websites.

Table 2 .

 2 Official number of people registered at microworking sites, all countries combined.

	Platform	Registered users
	Amazon Mechanical Turk 500,000
	Microworkers	1,215,829
	Clickworker	1,200,000
	ClixSense	7,000,000
	Wirk	50,000
	Ferpection	50,000
	Appen	1,000,000
	Total	11,015,829

Source: websites of platforms included in analysis (accessed in August 2017 for Amazon Mechanical Turk; September 2018 for all other platforms).

Table 4 .

 4 Average number of unique visitors on microwork platforms (left), average duration of visits (center) and average number of monthly unique visitors from France (right).

		Average	Average	Average
		monthly	visit	monthly
	Platform	unique	duration unique
		visitors		visitors,
				France
	Worker.mturk.com	588,976	34:19	1,250
	(Amazon Mechanical
	Turk)
	Microworkers	174,808	18:22	1,835
	Workplace.click-	242,579	7:40	14,700
	worker.com
	(Clickworker)
	ClixSense	1,083,000	11:05	20,250
	Fouleurs.com (Wirk)	7,647	26:41	6,958
	Ferpection	28,064	7:10	8,116
	Appen	260,699	5:04	9,645
			Total	62,754

Source: Authors' elaboration based on information from SimilarWeb.com, accessed in September 2018. Figures are based on websites' audience tracking over the period July-September 2018.

Table 5 .

 5 Estimated number of people registered on microwork platforms and connecting from France.

	Platform	Number French
	Amazon Mechanical Turk 1,250
	Microworkers	12,766
	Clickworker	72,720
	ClixSense	130,900
	Wirk	50,000
	Ferpection	14.460
	Appen	37,000
	Total	319,096

Source: Authors' elaboration.

Table 6 .

 6 Estimate of the population of microworkers refined to take into account multi-homing.

			Result	Result
	Range	Measure	(reminder)	(minus
				multi-
				homing)
	Low	Uncorrected	17,869	14,903
		capture-
		recapture
	Intermediate	Audience	62,754	52,337
		measure
	High	Number of	319,096	266,126
		declared
		users (France)

Source: Authors' elaboration.

Table 7 .

 7 Number of microworkers in France, by level of activity.

	Group	Type of activity	Frequency	Size
	Very active Simple and quick	At least once	14,903
		micro-tasks		a week
	Routine	Selected	micro-	At least once	52,337
		tasks		a month
	Casual	Alternating periods	Less	than	266,126
		of activity and inac-	once a month
		tivity
		Source: Authors' elaboration.

As part of DiPLab, we built an inventory of microwork platforms and mobile applications being used in France, collected online questionnaire data from Wirk microworkers, and interviewed in-depth a subset of Wirk respondents together with some clients, platform managers and other stakeholders. The data collection was conducted in Spring and Summer 2018 (see Section 3.2 for more details).

The study is not published but the author has made her calculations available: Kristy Milland, 150717 Preliminary results -Mapping study, July 17, 2015, URL: https://docs.google.com/spreadsheets/d/1T3yP_ Jo4qELrwsE2NAPNs07L1AWmpAEr9vnhreGJ\discretionary{-}{}{}K0/edit#gid= 1993074859 (accessed 18 December 2018).

 5 In addition to connecting its own registered microworkers and demanders, Clickworker also provides an entry point for UHRS (Universal Human Relevance System), Microsoft's proprietary microwork platform. , Vol. 1, No. 1, Article . Publication date: January 2019.

This list is not exhaustive and there are other microwork websites and applications that are used in France. We exclude those that are partially outside the scope of our research: freelancing platforms such as Malt and 5euros.com only offer microwork tasks on an occasional basis, being generally more oriented towards projects requiring advanced skills (such as creating a logo or developing a website) over longer periods of time and with higher remuneration. We also exclude income-generating mobile applications such as BeMyEye and Roamler, which usually require physical presence in a given place, and for which we do not have sufficient data. tracking (3.3), which does not face the same limitations. We cross our different methods to improve accuracy and obtain a meaningful range of estimates (section 3.4) and finally, we refine our results by taking into account multi-activity of workers operating on several microwork platforms at the same time (3.5).

See https://www.clickworker.com/about-us/clickworker-crowd/ consulted on 3/12/2018

See http://help.ferpection.com/l/fr/les-utilisateurs/ combien-y-a-t-il-dutilisateurs-dans-la-communaute consulted on 3/12/2018 , Vol. 1, No. 1, Article . Publication date: January 2019.

Variations of this calculation, estimating distributions for each possible value in each interval, produce results of a similar order of magnitude (between 5,800 and 8,000) , Vol. 1, No. 1, Article . Publication date: January 2019.

The data used by SimilarWeb.com come from two main sources: partnerships that the site has established with Internet service providers and data that the site retrieves with an add-on on users' Internet browsers -which provides users with metrics of website traffic in exchange for their browsing data.

, Vol. 1, No. 1, Article . Publication date: January 2019.

ACKNOWLEDGMENTS

The authors would like to thank current and former DiPLab team members, notably Maxime Besenval, Odile Chagny, Marion Coville, Touhfat Mouhtare, Lise Mounier, and Manisha Venkat, for their inputs to this paper and suggestions to improve it. Any remaining errors or inconsistencies are exclusively ours.

The DiPLab research project is co-funded by Maison des Sciences de l'Homme Paris-Saclay; Force Ouvrière, a workers' union; and France Stratégie, a service of the French Prime Minister's office. We also thank Wirk for logistical and financial support when fielding our survey, and Inria for complementary funding. For more information: http://diplab.eu/