Francesco Ferrante
email: ferrante@laas.fr

Frédéric Gouaisbaut

Sophie Tarbouriech

Observer-based control for linear systems with quantized output*

This paper is devoted to the observer-based control design for linear systems with quantized output. To this end, the output dynamic controller issued by the Luenberger observer is considered. Thus, through appropriate sector conditions which embed the quantization error, the global ultimate boundedness stability problem is tackled for the closed-loop system. Namely, to deal with the discontinuity induced by the quantizer, the whole treatise is carried out considering Krasovskii solutions to the closed-loop system. Furthermore, the controller is designed via a convex optimization setup, providing a constructive method that can be efficiently implemented.

I. INTRODUCTION

Nowadays, controlled systems are mainly heterogeneous systems interconnected by digital communication networks. The nature of these networks may often cause delays, asynchronism, and also quantization of the transmitted data (see [START_REF] Hurtado | Control under limited information: Special issue (part i)[END_REF] and references therein). In that sense, the old paradigm of an infinite precision of the available measures should be re-interpreted. Now, upstream from the network, some devices encode the data to be transmitted through the network and then re-encoded to be sent to the actuators. All these operations are no more transparent for the control and should be taken into account in the design [START_REF] Elia | Stabilization of linear systems with limited information[END_REF]. The quantization phenomenon, which may also appears in some sensors used for position measurement of mechanical devices (optical incremental encoders), is a nonlinearity that transforms a signal with values in R into a signal with values in a discrete set. Introduced into a control loop, it often induces performance degradation, like limit cycles or chaotic phenomena [START_REF] Delchamps | Stabilizing a linear system with quantized state feedback[END_REF] and they may even destabilize the closed-loop system. Thus in recent years, many studies have been dedicated to the analysis, control and observation of systems subject to quantization in order to reduce their impact on the closed loop. Several methods have been proposed in the literature depending on the nature of the quantization. When the quantized measures are without memory (also called uniform quantization), it is usually (at least for an unstable open loop) impossible to prove convergence to the origin. We can only show the convergence to an attractive and invariant set [START_REF] Wong | Systems with finite communication bandwidth constraints. i. state estimation problems[END_REF], [START_REF] Wong | Systems with finite communication bandwidth constraints. ii. stabilization with limited information feedback[END_REF]. One first method is to use the techniques issued from the robust control framework. The system is then modeled as a nominal system submitted to an uncertainty whose size is depending on the quantization precision. Quantization errors are then encapsulated into sector conditions and classical tools like small gain theorem or Lyapunov functions coupled with ISS properties [START_REF] Coutinho | Input and output quantized feedback linear systems[END_REF] or S-procedure [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF] are used to prove the ultimate boundedness of the states [START_REF] Fu | The sector bound approach to quantized feedback control[END_REF], [START_REF] Tarbouriech | Control design for quantized linear systems with saturations[END_REF]. These approaches have been extended to the H 8 control [START_REF] Picasso | Stabilization of discrete-time quantized linear systems: An H 8 {l 1 approach[END_REF], L 2 attenuation [START_REF] Tarbouriech | L2 stability for quantized linear systems with saturations[END_REF] and robust stabilization with respect to an input delay [START_REF] Fridman | Control under quantization, saturation and delay: An LMI approach[END_REF]. In the case of quantizer with memory (see [START_REF] Liberzon | Switching in Systems and Control[END_REF] and references therein), as the precision of the quantizer may be infinite when approaching the origin, the stability properties of the closed loop can be obtained using controls based on hybrid modeling [START_REF] Liberzon | Switching in Systems and Control[END_REF], [START_REF] Ceragioli | Discontinuities and hysteresis in quantized average consensus[END_REF]. In that case, the quantization is scaled dynamically when approaching the origin. Note that in all these techniques, except in [START_REF] Liberzon | Observer-based quantized output feedback control of nonlinear systems[END_REF], it was assumed that all the states were measured which may be quite restrictive. In this paper, we focus on a linear system subject to output quantization and we aim at building an observer-based control. Following the methods of [START_REF] Liberzon | Observer-based quantized output feedback control of nonlinear systems[END_REF] and [START_REF] Brockett | Quantized feedback stabilization of linear systems[END_REF], we model the system as a nominal system subject to a quantization error, which is bounded. Afterwards, as shown in [START_REF] Fu | The sector bound approach to quantized feedback control[END_REF], this quantization error is described by a specific nonlinearity which satisfy some sector bound conditions. Then, by using a quadratic Lyapunov function for the overall system, we prove that the state converges to an ultimate bounded set surrounding the origin, for every initial condition. The obtained criterion is then expressed into an optimization procedure based on LMI. This procedure determines automatically the gain of the observer while reducing a measure of the size of the ultimate set. Furthermore, the proposed optimization setup can be also used to improve the closed-loop performance, when a quantized state-feedback is exploited to control the system. Although our work is related to the work proposed by Liberzon, the way we adopt to deal with the quantizer is different. Indeed in [START_REF] Liberzon | Observer-based quantized output feedback control of nonlinear systems[END_REF], for the quantizer an explicit definition is not provided, that is the quantizer is assumed to be a certain nonlinear map characterized by some properties, conversely in our paper the uniform quantizer is a well defined discontinuous nonlinear map and due to such a discontinuity we rely on proper concept of solution for discontinuous right-hand side differential equations. This allows even to deal with sliding motions on the quantization boundaries for the closed-loop system, which are often disregarded. Another difference with respect to the work of Liberzon is that we exhibit constructive results in the sense that the conditions arising for this paper can be efficiently implemented, thus the proposed approach represents a real synthesis tool. Concerning this latter aspect we can consider that, except for the saturation, our work represents an extension of the work in [START_REF] Tarbouriech | Control design for quantized linear systems with saturations[END_REF]. Moreover, it is worthwhile to point out that this paper provides a numerical procedure to deal with Krasovskii solutions and this represents a considerable novelty. The paper is organized as follows. Section I presents the system under consideration and the problem we intend to solve. Section III is dedicated to the main theoretical conditions solution to the stated control problem. In Section IV, an algorithm procedure is proposed in order to efficiently implement the proposed theoretical conditions. In Section V, the effectiveness of the approach is illustrated through some examples. Finally, a conclusion ends the paper.

Notation: For every complex number ω, ℜpωq stands for the real part of ω. I n denotes the identity matrix whereas 1 n denotes the vector of dimension n with all components equal to 1, furthermore 0 denotes the null matrix (equivalently the null vector) of appropriate dimensions. For a matrix A P R nˆm , A 1 and tracepAq denote the transpose and the trace of A, respectively. HepAq " A `A1 . For two symmetric matrices, A and B, A ą B means that A ´B is positive definite. In partitioned symmetric matrices, the symbol ‹ stands for symmetric blocks. The matrix diagtA 1 ; . . . ; A n u is the block-diagonal matrix having A 1 , . . . , A n as diagonal blocks. For a vector x P R n , x piq denotes the i ´th component and x 1 denotes the transpose of x, |x| is the vector given by the absolute value of each component of x, whereas signpxq is the componentwise signum function for which signp0q " 0. The set ∆Z p is the set of the p-tuples of integers multiple of ∆. The symbol x¨, ¨y denotes the standard Euclidean inner product. For a set U, IntpUq denotes the interior of U. Let A 1 . . . A m be some given sets,

Ś m i"1 A i " A 1 ˆA2 ˆ. . . A m
, where ˆstands for the standard Cartesian product.

II. PROBLEM STATEMENT

Consider the following continuous-time linear system with quantized output:

9 x " Ax `Bu y " qpCxq (1)
where x P R n , u P R m , y P R p are respectively the state, the input and the measured output of the system. A, B,C are constant matrices of appropriate dimensions. Whereas, the uniform quantizer is defined as follows:

q : $ & % R p Ñ ∆Z p z Þ Ñ ∆ signpzq Z |z| ∆ ^(2)
where ∆ is an arbitrary strictly positive number which represents the quantization error bound i.e., | qpz i q ´zi | ď ∆ for all i " 1, . . . , p. Throughout the paper, we consider that system (1) is controlled through a dynamic output feedback controller of the form:

" 9 x " A x `Bu `Lpy ´C xq u " K x (3)
where x P R n , K and L are constant matrices of appropriate dimensions to be designed. Notice that the structure of the system (3) is based on a Luenberger observer, and thus the state x P R n can be viewed as an estimate of the state x. Hence, the closed-loop system (1)-(3) can be written as

" 9 x " Ax `BK x 9 x " pA `BK ´LCq x `L qpCxq (4)
Classically by defining the error ε " x ´x and the nonlinearity Ψpvq " qpvq ´v, the closed-loop system (1)-(3) reads:

" 9
x " pA `BKqx ´BKε 9 ε " pA ´LCqε ´LΨpCxq.

(5)

Note that system (5), due the presence of the nonlinearity Ψp¨q, is described by a discontinuous right-hand side differential equation. For this, as pointed out in [START_REF] Ceragioli | Discontinuities and hysteresis in quantized average consensus[END_REF], Carathéodory solutions to system (5) may not exist, thus relying on a suitable concept of solution is needed. Especially in this paper we focus on Krasovskii solutions to system (5), the existence of such solutions is guaranteed under very mild conditions, which are naturally satisfied by the function Ψp¨q, for an exhaustive treatise about these aspects, we invite to see [START_REF] Cortès | Discontinuous dynamical systems -a tutorial on solutions, nonsmooth analysis, and stability[END_REF]. It seems to be suitable to recall that Krasovskii solutions of ( 5) are Carathéodory solutions of the following differential inclusion

" 9 x 9 ε  P " pA `BKqx ´BKε pA ´LCqε ´LK pΨpCxqq  loooooooooooooooomoooooooooooooooon W px,εq (6)
where the Krasovskii operator K p f pxqq is defined as

K p f pxqq " č δą0 cop f pBpx, δqqq.
For further details see [START_REF] Ceragioli | Discontinuous stabilization of nonlinear systems: Quantized and switching controls[END_REF] and [START_REF] Ceragioli | Discontinuities and hysteresis in quantized average consensus[END_REF]. Especially, in this case for i " 1 . . . p, the above mentioned Krasovskii operator, by continuity arguments, applied to the i-th component of the nonlinearity ΨpCxq provides the following inclusion:

K pΨ piq pC piq xqq Ď $ & % Ψ piq pC piq xq if C piq x ‰ ∆Zzt0u,
tλ signpC piq xq∆, λ P r´1, 0su otherwise, (7) Figure 1 depicts the right-hand side of the above inclusion in the scalar case. Notice also that, as shown in [START_REF] Paden | A calculus for computing Filippov's differential inclusion with application to the variable structure control of robot manipulators[END_REF] Theorem

K pΨpvqq Ď p ą i"1 K pΨpv piq qq. (8)
Moreover, it should be noticed that due to the dead-zone effect induced by the quantizer, achieving the asymptotic stability property of the origin for closed-loop system (4) (and then equivalently for system (5)), with respect to Krasovskii solution, is not possible if the matrix A is not Hurwitz. Indeed, there exists a neighborhood B of the origin such that qpCxq " 0, @x P B then system (4) behaves locally as the linear system described by:

" 9 x 9 x " " A BK 0 A `BK ´LC  " x x (9)
which is unstable if A is not Hurwitz. Nevertheless, assuming A to be Hurwitz is too much restrictive and we aim at establishing a practical stability property for the closed-loop system. To this end, we rely on the ultimate boundedness property proposed in [START_REF] Khalil | Nonlinear Systems[END_REF]. Hence, the problem we intend to solve is summarized as follows: Problem 1 (Practical stabilization): Determine the gains K and L and characterize a set S u Ă R 2n such that for every initial condition rx 1 p0q ε 1 p0qs 1 P R 2n the resulting Krasovskii solutions to closed-loop system (5) are ultimately bounded in S u .

III. MAIN RESULT In order to solve Problem 1, we firstly show some sector conditions for Ψp¨q, which will be useful in the sequel. Concerning the function Ψpvq defined in (2) the following Lemma holds:

Lemma 1: The function Ψpvq satisfies the following conditions:

Ψ 1 pvqS 1 Ψpvq ´∆2 1 1 p S 1 1 p ď 0, @v P R p (10)
Ψ 1 pvq S 2 pΨpvq `vq ď 0, @v P R p (11)
for any diagonal positive definite matrices S 1 , S 2 P R pˆp .

Proof: According to the definition of Ψpvq one gets:

Ψ 2 i pv piq q ´∆2 ď 0 ( 12
)
then taking p positive arbitrarily scalars s 1 1 , . . . , s 1 p it follows that s 1 i Ψ 2 i pv piq q ´∆2 s 1 i ď 0 for every i " 1, . . . , p. Hence by summing for i " 1, . . . , p, by defining S 1 " diagts 1 i ; . . . ; s 1 p u, from the foregoing relation, one gets [START_REF] Fridman | Control under quantization, saturation and delay: An LMI approach[END_REF]. Concerning relation [START_REF] Fu | The sector bound approach to quantized feedback control[END_REF], take p positive arbitrarily scalars s 2 1 , . . . , s 2 p and set S 2 " diagts 2 i ; . . . ; s 2 p u. Then, by noticing that for every v belonging to R p , Ψ piq pv i qpv i `Ψpiq pv i qq ď 0, for i " 1, . . . , p, by similar arguments as in the former case, it follows that (11) holds.

Remark 1: Notice that, even if the sector conditions provided by the foregoing Lemma concern the single-valued map Ψp¨q, it turns out that such conditions, as pointed out by Figure 1, can be exploited to deal with the set-valued map K pΨpvqq as well. Indeed, it can be easily shown that for every v P R p and for every i P t1, . . . , pu, every element of the set Ś p i"1 K pΨpvq piq q satisfies inequalities ( 10) and [START_REF] Fu | The sector bound approach to quantized feedback control[END_REF] and due to the inclusion reported in [START_REF] Delchamps | Stabilizing a linear system with quantized state feedback[END_REF], it follows that every element of the set K pΨpvqq satisfies [START_REF] Fridman | Control under quantization, saturation and delay: An LMI approach[END_REF] and [START_REF] Fu | The sector bound approach to quantized feedback control[END_REF] as well.

Now, considering closed-loop system (5), the following result provides a solution to Problem 1.

Proposition 1: If there exist two symmetric positive definite matrices P 1 , P 2 P R nˆn , two diagonal positive definite matrices S 1 , S 2 P R pˆp , two matrices K P R mˆn and L P R nˆp and a positive scalar τ such that:

M 1 ă 0 ( 13
)
where M 1 is defined in (15) (at the top of next page), then K, L and S u " trx 1 ε 1 s 1 P R 2n :

x 1 P 1 x `ε1 P 2 ε ď 1u (14)
are solution to Problem 1.

Proof: Consider the following quadratic Lyapunov function V px, εq " x 1 P 1 x `ε1 P 2 ε

with P 1 " P 1 1 ą 0 and P 2 " P 1 2 ą 0. Define 9 V px, εq " x∇V px, εq, f y, @ f P W px, εq where W px, εq is defined in [START_REF] Cortès | Discontinuous dynamical systems -a tutorial on solutions, nonsmooth analysis, and stability[END_REF]. We want to prove that there exist two positive scalars β1 and β2 such that

9 V px, εq ď ´β 1 x 1 x ´β 2 ε 1 ε, @rx 1 ε 1 s 1 P R 2n z IntpS u q. ( 16
)
where the set S u is defined in [START_REF] Liberzon | Switching in Systems and Control[END_REF]. By using S-procedure argument in order to prove relation ( 16) it suffices to prove that there exists a positive scalar τ such that for every rx, εs P R 2n and for every f P W px, εq 9 V px, εq ´τp1 ´x1 P 1 x ´ε1 P 2 εq ď ´β

1 x 1 x ´β 2 ε 1 ε. ( 17
)
To this end, notice that:

9

V px, εq " x 1 `pA `BKq 1 P 1 `P1 pA `BKq ˘xὲ 1 `pA ´LCq 1 P 2 `P2 pA ´LCq ˘ε2

x 1 P 1 BKε ´2ε 1 P 2 Lw, @w P K px, εq.

Moreover, since for every w P K px, εq relations ( 10) and ( 11) are satisfied, then once again according to S-procedure arguments, to show relation ( 16) it suffices to show that there exist some diagonal positive definite matrices S 1 , S 2 and a positive scalar τ such that:

9 V px, εq ´2w 1 S 2 Cx ´2w 1 S 2 w ´w1 S 1 w `11 p S 1 1 p ∆ 2 ´τp1 ´x1 P 1 x ´ε1 P 2 εq ď ´β 1 x 1 x ´β 2 ε 1 ε @rx, εs P R 2n . (19)
Now, defining ξ " rx 1 ε 1 w 1 1 1 p s 1 the right hand side of relation ( 19) can be re-written equivalently as follows:

ξ 1 M 1 ξ (20)
where M 1 is defined in [START_REF] Liberzon | Observer-based quantized output feedback control of nonlinear systems[END_REF]. Therefore, the satisfaction of relation [START_REF] Khalil | Nonlinear Systems[END_REF] implies that there exist four positive real scalars β 1 , β 2 , β 3 , β 4 such that

ξ 1 M 1 ξ ď ´ξ1 diagtβ 1 I n ; β 2 I n ; β 3 I p ; β 4 I p uξ. (21)
Then, by the virtue of definition of ξ, one has

ξ 1 M 1 ξ ď ´β 1 x 1 x ´β 2 ε 1 ε (22)
hence relation [START_REF] Ripka | Modern sensors handbook[END_REF] holds implying that inequality ( 17) is satisfied as well as inequality [START_REF] Hurtado | Control under limited information: Special issue (part i)[END_REF]. Thus according to [START_REF] Ceragioli | Discontinuous stabilization of nonlinear systems: Quantized and switching controls[END_REF] and [START_REF] Khalil | Nonlinear Systems[END_REF], it follows that the Krasovskii solutions to [START_REF] Chilali | Robust pole placement in LMI regions[END_REF] are ultimately bounded in the set S u defined in ( 14) which concludes the proof.

Classically, it appears in Proposition 1 that the gains K and L cannot be computed simultaneously through a convex problem [START_REF] Tarbouriech | Stability and stabilization of linear systems with saturating actuators[END_REF]. However, an adequate change of variable allows us to compute the gain L as described below.

Corollary 1: If there exist two symmetric positive definite matrices P 1 , P 2 P R nˆn , two diagonal positive definite matrices S 1 , S 2 P R pˆp , two matrices K P R mˆn and J P R nˆp and a positive scalar τ such that:

M 2 ă 0 ( 23
)
with M 2 defined in [START_REF] Wong | Systems with finite communication bandwidth constraints. i. state estimation problems[END_REF] (at the top of next page), then K, L " P ´1 2 J and S u defined in ( 14) are solution to Problem 1.

Proof: Setting P 2 L " J in [START_REF] Liberzon | Observer-based quantized output feedback control of nonlinear systems[END_REF], the result immediately follows from Proposition 1.

M 1 " » - - - He pP 1 pA `BKqq `τP 1 ‹ ‹ ‹ ´K1 B 1 P 1 He ppP 2 pA ´LCqq `τP 2 ‹ ‹ ´S2 C ´P2 L ´2S 2 ´S1 ‹ 0 0 0 S 1 ∆ 2 ´τI p p fi ffi ffi fl (15) M 2 " » - - - He pP 1 pA `BKqq `τP 1 ‹ ‹ ‹ ´K1 B 1 P 1 He pP 2 A ´JCq `τP 2 ‹ ‹ ´S2 C ´J ´2S 2 ´S1 ‹ 0 0 0 S 1 ∆ 2 ´τI p p fi ffi ffi fl (24)
IV. COMPUTATIONAL ISSUES Relation ( 23) becomes linear with respect to the decision variables if τ and K are fixed. In this case relation ( 23) is linear in P 1 , P 2 , S 1 , S 2 , S 3 and J.

The selection of τ is not a difficult task and it can be selected through an iterative search. In our proposal, K is supposed to be known and it allows us to compare our solution with those obtained in the static state feedback case, u " K qpxq, which is treated in [START_REF] Tarbouriech | Control design for quantized linear systems with saturations[END_REF]. In fact, thanks to the results shown in [START_REF] Tarbouriech | Control design for quantized linear systems with saturations[END_REF] and also in [START_REF] Fridman | Control under quantization, saturation and delay: An LMI approach[END_REF], one obtains a stabilizing gain K and a set S K u Ă R n in which the resulting closed-loop trajectories are ultimately bounded.

One idea is to compute the gain L (for Proposition 1 or Corollary 1) aiming at reduce the size of the projection in the x-coordinates of the set S u defined in [START_REF] Liberzon | Switching in Systems and Control[END_REF]. Another idea is computing the gain L in order to reduce the chattering behavior observed on the time evolution of the state and/or the input of the closed-loop system, due to the presence of the quantizer. In that case, the chattering phenomenon is reduced by the filtering effect of the observer [START_REF] Utkin | Sliding modes in control and optimization[END_REF]. These two aspects will be illustrated in the numerical examples. Moreover, the conditions of Proposition 1 or Corollary 1 can be used in a convex optimization setup allowing to minimize the size of the set S u defined in [START_REF] Liberzon | Switching in Systems and Control[END_REF]. Several optimality criteria can be considered, depending on the size criterion chosen for measuring the set S u .

For example, as shown in [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF], volume, minor axis and direction of interests could be a good measure of an ellipsoidal set. Among the various criteria provided by the literature, we adopt a trace maximization criterion, in order to induce, through the optimization procedure, an uniform contraction on S u . Furthermore, it is important to note that we are more specifically interested by minimizing the size of S u in the xdirection. Hence, we consider the maximization of the trace of P 1 . An optimization of the set S u can be achieved through the following algorithm:

Algorithm 1:

' Step 1 Initializing K and τ. For example K can be fixed at the value issued from the static state feedback problem.

' Step 2 Given K and τ solve the following LMI optimization problem:

min

P 1 , P 2 S 1 , S 2 ,J ´tracepP 1 q
subject to (23), P 1 ą 0, P 2 ą 0, S 1 ą 0; S 2 ą 0;

which gives L " P ´1 2 J.

' Step 3 Given K, L and τ solve the following LMI optimization problem:

min

P 1 , P 2 S 1 , S 2 ´tracepP 1 q
subject to (13), P 1 ą 0, P 2 ą 0, S 1 ą 0; S 2 ą 0;

(26)

The Step 3 is performed in order to shrink as much as possible the ultimate set S u .

Remark 2: We could add in ( 25) and (26) a constraint imposing that the projection along the x-direction of the set S u defined in ( 14) is included in the set S K u resulting of the static state feedback case. To do this, supposing that S K u is defined from a symmetric positive definite matrix P K as S K u " tx P R n : x 1 P K x ď 1u, it suffices to add the constraint P 1 ´PK ě 0, being P K given 1 .

V. NUMERICAL EXAMPLES

Example 1: Consider the system (1) defined by the following data:

A " " 0 1 1 ´1 , B " " 1 1  C " r1 0s , ∆ " 0.25. ( 27
)
By the conditions proposed in [START_REF] Tarbouriech | Control design for quantized linear systems with saturations[END_REF] one gets K " r´1.7047 ´1.0566s then by fixing τ " 0.8 the optimization scheme proposed in (25) leads to L 1 " r10.618 6.5623s.

In Figure 2, several closed-loop trajectories converging in a finite time in S u are plotted. As expected, the optimization procedure leads to a small set S u , which represents to a good estimation of the real ultimate set in which the closed-loop system trajectories are captured.

Example 2 (Furuta pendulum [START_REF]Rotary Inverted Pendulum ROTPEN[END_REF]): We consider a Furuta pendulum designed by Quanser [START_REF]Rotary Inverted Pendulum ROTPEN[END_REF]. In that case the whole state vector is measured and exploited to control the system. However, such a measure is performed by a device that, through a digital post-processing of the data delivered by an optical incremental encoder [START_REF] Ripka | Modern sensors handbook[END_REF], provides a finite resolution measure of the state. Such a phenomenon can be modeled by a uniform quantization process characterized by ∆ " 2π{1024 « 0.006. Specifically, the plant is controlled by static state feedback by means of the quantized measured state, that is:

u " K qpxq (28) 1 In order to prevent the occurrence of a too large value for the gain L (high observer gain), one can add in [START_REF] Wong | Systems with finite communication bandwidth constraints. ii. stabilization with limited information feedback[END_REF] an additional constraint, specifying a certain pole placement for the eigenvalues of the matrix A ´LC. For example, one can impose that the eigenvalues of the matrix A ´LC, denoted by λ i , belong to a strip of the complex plane as ´α ď ℜpλ i q ă 0, α ą 0 [START_REF] Chilali | Robust pole placement in LMI regions[END_REF]. where:

K " r1 ´24.9103 2.0751 ´3.6549s .

The considered plant model is nonlinear but it has been linearized around the unstable equilibrium point, providing the following linear model: where x 1 , x 2 represent respectively the base angle and the pendulum angle, whereas x 3 " 9 x 1 , x 4 " 9 x 2 . Furthermore in the studied case, we assume that the control system designer has not access to the gain K, which cannot be modified 2 . Thus in this situation one can attempt to improve the closedloop performances, exploiting the filtering action provided by controller (3) on the measured variables. To this end, we want to test our methodology considering C " I n and successively comparing it to static state feedback controller (28), which is analyzed by using conditions stated in [START_REF] Tarbouriech | Control design for quantized linear systems with saturations[END_REF]. Therefore, by fixing:

τ " 1.3 the proposed optimization procedure (25) provides the following observer gain: The projections of the set S u on the px i , x j q plane are reported in Figure 3. Clearly, designing an observer allow us to reduce the estimation of the ultimate set compared with the static state feedback case. Furthermore, Figure 4 and Figure 5 depict respectively the time-evolution of the 2 Notice that such situations are typical in control engineering practice, due to the presence of technological constraints. Hence analyzing this example is undoubtedly interesting. Namely, in the present case, the gain K results from an LQ control setup performed on linear model (29) neglecting the quantization effect. One can then observe that, effectively thanks to the filtering action of the controller (3), the high frequency oscillations affecting the state behavior is reduced in terms of magnitude as well as the chattering phenomenon affecting the control variable in the static state feedback case is alleviated (see Figure 5). That highlights the real benefits provided in using an observer-based control, even if whole the state is measurable, confirming our guess.

L " » - - 2
VI. CONCLUSION This paper proposed a methodology to design an observerbased control for a linear system with quantized output, in order to achieve the global ultimate boundedness stabilization property of the closed-loop system with respect to Krasovskii solutions. Furthermore, the observer design is performed computing the observer gain by a convex optimization setup, allowing to reduce the size of the ultimate set. Moreover, numerical solutions have shown that, even if a quantized measure of the state is available, adopting an observer-based control can improve the closed-loop system performance. This paper offers several sparks that could be studied in future works. Among these, the design of a fullorder dynamic filter, with a different structure from that one proposed in (3), should be addressed. Moreover, inspired by the work of [START_REF] Tarbouriech | Control design for quantized linear systems with saturations[END_REF], one could analyze the observer-based control design problem, for linear systems whose output results in a sandwich between saturation and quantization nonlinearity. Another open question consists of designing also the gain K by a convex optimization setup.

  (a). The function Ψpvq (b). The set-valued map K pΨpvqq

Fig. 1 .

 1 Fig. 1. Comparison between the function Ψp¨q and the associated set-valued map K pΨpvqq 1, for every v P R p , one has

Fig. 2 .

 2 Fig.2. Projection of the set S u on the px 1 , x 2 q plane (magenta) and plant convergent trajectories obtained setting xp0q " 0.

Fig. 3 .

 3 Fig.3. Projections of the set S u on the px i , x j q planes i, j P t1, 2, 3, 4u , i ‰ j: static controller (dashed) and observer-based controller (plain)

Fig. 4 .Fig. 5 .

 45 Fig.4. Closed-loop steady-state behavior, concerning the plant state x, arising from xp0q " r0 0.1 0 0s 1 , xp0q " 0. x 1 ptq (blue), x 2 ptq (magenta), x 3 ptq (black), x 4 ptq (light-blue)

*This work has been supported by ANR project LimICoS contract


Armel Menant

Laurent Jolivet

Pietro Sternai

Maxime Ducoux

Romain Augier

Aurélien Rabillard

Taras Gerya

Laurent Guillou-Frottier

Various-scale controls of complex subduction dynamics on magmatic-hydrothermal processes in eastern Mediterranean

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

A. Menant 1,2,3 , L. Jolivet 1,2,3 , P. Sternai 1,2,3 , M. Ducoux 1,2,3 , R. Augier 1,2,3 , A. Rabillard 1,2,3 , T. Gerya 4 , L. Guillou-Frottier 2,1,3 1 Univ. d'Orléans, ISTO, UMR 7327, 45071 Orléans, France (a.menant@brgm.fr) 2 BRGM, ISTO, UMR 7327, 45060 Orléans, France

B a l k a n s P i n d o s o c e a n T a u r i d e s Lycian-Tauride ophiolites

Tavsanlı-Afyon

G a v r o v o -T r i p o l i t z a

Rhodopes K ir ş e h ir

Magmatism

Serifos island

Block rotations [van Hinsbergen et al., 2005] Appearance and westward migration of alkaline volcanism [Dilek & Altunkaynak, 2009] Westward migration of plutonism from the Menderes to Cyclades with an increase of the mantle component [Altherr & Siebel, 2002] In the Cyclades, the activity of large-scale detachment systems such as the North Cycladic Detachment System (NCDS) and the West Cycladic Detachment System (WCDS) is contemporaneous with the emplacement of magmatic intrusions and ore deposits

  3 CNRS/INSU, UMR 7327, 45071 Orléans, France 4 Swiss Federal Institute of Technology (ETH), Zurich, Switzerland oceanic crust d e h y d r a t i o n o f o c e a n i c l i t h o s p h e

  calc-alkaline to shoshonitic v./p. high-K calc-alkaline v./p. medium-to high-K calc-alkaline v./p. medium-K calc-alkaline v./p. silica breccia and base metal-barite veins Using a simple large-scale model of subduction, can we reproduce these geological observations and relate them to the subduction dynamics and associated mantle flow? Since 35-30 Ma, fast slab retreat resulted in the opening of the Aegean back-arc domain. However, 15 Ma ago, a major change occurred in the dynamics of this back-arc opening. -Syn-extensional granodiorite below the WCDS -Fe-rich high temperature and medium temperature skarns

  ing co nti ne nta l pla te (E ur as ia) co nt in en t oc ea nic pla te (A ra bi a) lit ho sp he ric m an tle A st h e n o sp h e activity is observed from the Menderes massif to the Cyclades Barren periods are associated to dominant compressional or transpressional tectonics within the upper plate (e.g. continental accretion) Cu-rich periods are related to stagnant magmatic arc resulting from the dehydration of the subducted oceanic crust and the partial melting of the mantle wedge Au-rich periods are related to fast slab retreat and associated mantle flow inducing the partial melting of the lithospheric mantle or the base of the crust where Au was previously stored Possible consequence of a slab tearing event below western Anatolia and related hot asthenospheric flow Dilek Y. & S. Altunkaynak (2009), Geochemical and temporal evolution of Cenozoic magmatism in western Turkey: mantle response to collision, slab break-off, and lithospheric tearing in an orogenic belt, Spe. Pub. -Geol. Soc. London, 311, 213-233. Gerya T. V. & D. A. Yuen (2003), Characteristics-based marker-in-cell method with conservative finite-differences schemes for modeling geological flows with strongly variable transport properties, Phys. Earth Planet. Int., 140, 293-318. Richards J. P. (2011), Magmatic to hydrothermal metal fluxes in convergent and collided margins, Ore Geol. Rev., 40, 1-26. van Hinsbergen D. J. J., et al. (2005), Revision of the timing, magnitude and distribution of Neogene rotations in the western Aegean region, Tectonophysics, 396, 1-34. Altherr R. & W. Siebel (2002), I-type plutonism in a continental back-arc setting: Miocene granitoids and monzonites from the central Aegean Sea, Greece, Contrib. Mineral Petrol., 143, 397-415.

  p l e t e v i d e o o f t h e s e k i n e m a t i c r e c o n s t r u c t i o n s i s a v a i l a b l e ! V i d e o s o f t h i s m o d e l a r e a l s o a v a i l a b l e ! O r e d e p o s i t s in s u b d u c t i o n e n v i r o n m e n t M a j o r e c o n o m i c i n t e r e s t ! -5 5 % o f t h e w o r l d p r o d u c t i o n i n C u -1 6 % o f t h e w o r l d p r o d u c t i o n i n A u


Ph Négrel
email: p.negrel@brgm.fr

C Guerrot

E Petelet-Giraud

R Millot

Reinforcing the origin of volcanic rocks from the Massif Central through the isotopic composition of lead and strontium

Keywords: lead isotopes, strontium isotopes, basalts, French Massif Central volcano

The Massif Central in France hosts a large volume of Tertiary to Recent continental alkaline volcanism, with evidence of crustal contamination of the differentiated magmas. Detailed investigations on feldspathic basalts and basanite rocks from the Cantal volcano exposed in the Allanche basin by lead and strontium isotopes reveal i) the role of the mantle components Depleted Mantle (DM) and Low Velocity Component (LVC, corresponding to the European asthenospheric reservoir) end-members based on 208 Pb/ 204 Pb and 206 Pb/ 204 Pb isotopic ratios, and ii) show contamination by meta-igneous granulites from the lower crust when using 207 Pb/ 204 Pb vs. 206 Pb/ 204 Pb ratios, as only part of the samples plot along the Northern Hemisphere Reference Line (NHRL) mixing line between DM and HIMU (High U/Pb Mantle) end-members. Cross plotting of Pb-Sr isotopes rules out recycled sediment and/or igneous granulite as potential crustal contaminant for the Cantal basalts, but suggests mixing between the low-velocity component LVC and EMI (Enriched Mantle I)-pelagic sediments.

Replaced in the regional context of the Massif Central, we confirm the existence of several mantle sources and different crustal contaminations in the magmas of the Cantal volcano.

A C C E P T E D M A N U S C R I P T 1 Introduction

During the Cenozoic, Africa and Europe converged, seafloor spreading took place in the Atlantic and, as a direct result of the convergence, numerous microplates in the Mediterranean area collided. This resulted in an intense igneous activity either related to subduction or to intra-plate geodynamics [START_REF] Lustrino | The circum-Mediterranean anorogenic Cenozoic igneous province[END_REF]. Consequently, Tertiary-Quaternary volcanic fields are found throughout Southern, Western and Central Europe [START_REF] Wilson | Mafic alkaline magmatism associated with the European Cenozoic rift system[END_REF][START_REF] Lustrino | The circum-Mediterranean anorogenic Cenozoic igneous province[END_REF], corresponding to the later phases of the Alpine orogeny and the Neogene collapse of the Mediterranean and Pannonian basins [START_REF] Wilson | Mafic alkaline magmatism associated with the European Cenozoic rift system[END_REF]. The main phases of volcanic activity occurred during the Miocene and Pliocene, with Pleistocene activity restricted only to specific areas. A possible geodynamic origin has been evoked for explaining diapiric mantle upwellings beneath the European lithosphere [START_REF] Granet | Imaging a mantle plume beneath the French Massif Central[END_REF][START_REF] Kolb | Petrogenesis of mafic to felsic lavas from the Oligocene Siebengebirge volcanic field (Germany): implications for the origin of intracontinental volcanism in Central Europe[END_REF]. [START_REF] Hoernle | Seismic and geochemical evidence for large scale mantle upwelling beneath the eastern Atlantic and western and central Europe[END_REF] proved, using seismic tomography, the presence of a low S-wave-velocity anomaly, named Low Velocity Zone (LVC), extending from the eastern Atlantic to Central Europe. The LVC I interpreted as a common lithospheric mantle reservoir named European Asthenospheric Reservoir (EAR; [START_REF] Granet | Imaging a mantle plume beneath the French Massif Central[END_REF].

Here, we studied the central eastern part of the Cantal volcano, located in the French Massif Central. This area is one of the largest volcanic provinces in the country, whose and alkali-rich nepheline-normative rocks (basanite, tephrite and phonolite).

Geochronological dating and lithostratigraphic studies linked the changes recorded in chemical composition to a change in their mantle source during spreading, which occurred however during a short geological period of 3 to 4 million years. Niobium (Nb), thorium (Th) and tantalum (Ta) and their respective ratios are key elements, and changes in Nb or Th contents reflect the gradual change in chemical composition between infra-and supra-Cantal basalts, i.e. changes in the composition of the underlying mantle during the construction of the volcano. This led to the development of several models of heterogeneous mantle source to explain the Massif Central basalts [START_REF] Stettler | 87Rb -86Sr constraints on the genesis and evolution of the Cantal continental volcanic system (France)[END_REF][START_REF] Downes | Tertiary and Quaternary volcanism in the Massif Central, France[END_REF][START_REF] Granet | Imaging a mantle plume beneath the French Massif Central[END_REF][START_REF] Hoernle | Seismic and geochemical evidence for large scale mantle upwelling beneath the eastern Atlantic and western and central Europe[END_REF][START_REF] Downes | Mantle domains in the lithosphere beneath the French Massif Central: trace element and isotopic evidence from mantle clinopyroxenes[END_REF].

For several decades, the use of geochemistry comprising trace elements and/or radiogenic isotopes as natural tracers for investigating geological phenomena has been readily accepted by scientists scrutinizing processes that occur within the Earth [START_REF] Allègre | Chemical geodynamics[END_REF]. In that way, numerous Tertiary-Quaternary volcanic fields have been investigated in Southern,

Western and Central Europe [START_REF] Downes | Tertiary and Quaternary volcanism in the Massif Central, France[END_REF][START_REF] Witt-Eickschen | Lithospheric mantle evolution beneath the Eifel (Germany): constraints from Sr-Nd-Pb isotopes and trace element abundances in spinel peridotite and pyroxenite xenoliths[END_REF][START_REF] Dautria | 160 Ma of sporadic basaltic activity on the Languedoc volcanic line (Southern France): A peculiar case of lithosphereasthenosphere interplay[END_REF][START_REF] Kolb | Petrogenesis of mafic to felsic lavas from the Oligocene Siebengebirge volcanic field (Germany): implications for the origin of intracontinental volcanism in Central Europe[END_REF][START_REF] Lustrino | Heterogeneous mantle sources feeding the volcanic activity of Mt. Karacadag (SE Turkey)[END_REF][START_REF] Lustrino | Origin and evolution of Cenozoic magmatism of Sardinia (Italy). A combined isotopic (Sr-Nd-Pb-O-Hf-Os) and petrological view[END_REF]Bosch et al., 2014 and references therein). The present study investigated the Pb and Sr isotopes in rocks in the Allanche river drainage basin that drains part of the Massif Central (Fig. 1). In this study, we measured Pb and Sr isotopes to identify their sources in rock over the recent lava flows of the Cantal volcano and discuss the processes generating this large basalt area.

Geology of the studied area

A C C E P T E D M A N U S C R I P T

The Allanche basin in the Massif Central covers 160 km 2 with an elevation ranging from 1060 up to 1400 m a.s.l. (Fig. 1). The Allanche River flows through the lava plateau between the Cantal and Cézallier volcanoes [START_REF] De Goër De Hervé | La planèze de Saint-Flour, structure et stratigraphie[END_REF][START_REF] De Goër De Hervé | Notice explicative de la feuille Saint-Flour (1/50 000[END_REF][START_REF] Brousse | Le volcanisme en France et en Europe limitrophe[END_REF]. The main phase of volcanic activity within the Massif Central took place during the Late Miocene to Pliocene [START_REF] Nehlig | Notice explicative de la feuille Murat (1/50 000[END_REF], dominated by eruptions from the Cantal (11-2.5 Ma) and Mont Dore (4-0.3 Ma), and by basaltic fissure eruptions in the Cézallier, Velay, Coirons, Aubrac and Devès areas. Volcanic activity resumed in some of the outlying areas of the massif (Causses, Sioule and Limagne), the youngest volcanic activity occurring in the Vivarais (12,000 yrs BP) and the Chaine des Puys (4,000 yrs BP). More specifically, the Cantal and Cézallier volcanoes were marked by two main periods of basalt emission: the Infracantalian (13-9.5 Ma) and the Supracantalian periods (6-3 Ma). The basalt composition ranges from alkali basalt to basanite, explained by variations in the degree of partial melting of a common mantle source [START_REF] Granet | Imaging a mantle plume beneath the French Massif Central[END_REF][START_REF] Pilet | The metasomatic alternative for ocean island basalt chemical heterogeneity[END_REF].

The Allanche basin is underlain by alkaline basalt from three volcanic cycles: 8 to 5.4

Ma (Late Miocene), 5.4 to 3 Ma (Pliocene) and 240,000 to 6,000 yrs BP. The rocks in this basin comprise feldspathic basalt and basanite [START_REF] De Goër De Hervé | La planèze de Saint-Flour, structure et stratigraphie[END_REF][START_REF] De Goër De Hervé | Notice explicative de la feuille Saint-Flour (1/50 000[END_REF][START_REF] Brousse | Le volcanisme en France et en Europe limitrophe[END_REF][START_REF] Nehlig | Notice explicative de la feuille Murat (1/50 000[END_REF]. Feldspathic basalt has an

Sampling and Methods

Figure 1a shows the general location of the volcanic areas in the Massif Central, Figure 1b showing the basin configuration and the bedrock sampling points, corresponding lithologies are illustrated in Figure 1c. Sampling was primarily guided by the presence of outcrops and secondly by their accessibility along the river and the possibility to collect fresh unaltered rock. Three types of rocks were collected, feldspathic basalt, basanite (nepheline-leucite) and basanite (nepheline) according to the classification given by De Goër De [START_REF] De Goër De Hervé | Notice explicative de la feuille Saint-Flour (1/50 000[END_REF].

The collected samples (several kg) were powdered-quartered prior to analysis. Pb isotopes were determined with a Thermo-Finnigan MC-ICPMS Neptune. For isotopic measurements, the powdered sample was transferred into a Teflon container and subjected to standard dissolution procedures in hot HCl, HNO 3 , HF and HClO 4 . Chemical separation of Pb was done with an ion-exchange column and blanks for this procedure were lower than 100 pg and are negligible relative to the amount of sample analyzed. After chemical separation, the solutions were adjusted to a Pb concentration of 50 μg L -1 and, similarly, a standard solution (NBS 981) was also adjusted at 50 μg L -1 . Around 25 ng Tl was added to the solution. A bracketing procedure with the sequence blankstandard or samplecleaning solution was used for controlling the potential mass-bias instability of the instrument. [START_REF] Cocherie | Direct measurement of lead isotope ratios in low concentration environmental samples by MC-ICP-MS and multi-ion counting[END_REF].

Chemical purification of Sr was done with an ion-exchange column (Sr-Spec) before mass analysis according to [START_REF] Pin | Evaluation of a strontium-specific extraction chromatographic method for isotopic analysis in geological materials[END_REF] with a total blank of less than 1 ng for the entire chemical procedure. After chemical separation, Sr was analyzed with a Finnigan

Results

Lead and strontium isotope results obtained for seven rocks are given in Table 1. The major oxide and trace element data from [START_REF] Négrel | Natural and anthropogenic budgets of a small watershed in the Massif Central (France): Chemical and strontium isotopic characterization in water and sediments[END_REF] are also reported. The SiO 2 and K 2 O variations of the seven basaltic rocks from the basin can be evaluated through mean values and standard deviations (Table 1). SiO 2 and K 2 O contents average 46.8% ± 3.1 and 2.0% ± 0.5, respectively. The 206 Pb/ 204 Pb ratios range between 18.9 and 19.5 in the bedrock (Table 1) and the seven analyzed bedrock samples show 87 Sr/ 86 Sr ratios in the range between 0.70357 and 0.70378 (Table 1). Variations between the lead-isotope ratios ( 206 Pb/ 204 Pb, 207 Pb/ 204 Pb and 208 Pb/ 204 Pb) are illustrated in Figure 2. Variations between the 87 Sr/ 86 Sr ratios are illustrated in Figure 3 and plotted against the 1/Sr and Rb/Sr ratios. In these figures, the results of the lead and strontium isotopes obtained on the Allanche basin have been replaced

in the large frame of the volcanic rocks from the Massif Central using data from the literature.

Compared with ratios of other volcanic rocks from the Massif Central, our data lie in the range of published Pb and Sr isotope ratios. The samples from the Allanche basin display the highest Pb contents observed in the volcanic rocks of the Massif Central.

Discussion

Mantle source heterogeneity revealed by the Massif Central volcanism

Several mantle sources have been identified worldwide including depleted mantle (DM) and high U/Pb mantle (HIMU). The mantle is encompassed by just four components, or end

A C C E P T E D M A N U S C R I P T

members: depleted MORB mantle (DM), HIMU (high µ, or high 238 U/ 204 Pb), EM-I and EM-II (enriched mantle EM-type ocean-island basalt 'I' and 'II', respectively), as summarized by [START_REF] Zindler | Chemical geodynamics[END_REF]; [START_REF] Hart | Heterogeneous mantle domains: signatures, genesis and mixing chronologies[END_REF] and review in [START_REF] Stracke | Earth's heterogeneous mantle: A product of convection-driven interaction between crust and mantle[END_REF]. Considering the evolution of the oceanic lithosphere, [START_REF] Pilet | The metasomatic alternative for ocean island basalt chemical heterogeneity[END_REF] 5). This first approach shows the role of the major mantle components in the origin of volcanic rocks of the Massif Central. The rest of the samples being shifted from the NHRL line are related to another process like fractional crystallization and assimilation.

Fractional crystallization and assimilation documented by the Massif Central volcanism

The relationships between SiO 2 and K 2 O contents in the samples from the Allanche area within the context of the Massif Central volcanism are illustrated in Figure 6. The samples values fall between the fields of basanite and feldspathic basalt and are coherent with literature data (De Goër de [START_REF] De Goër De Hervé | Notice explicative de la feuille Saint-Flour (1/50 000[END_REF][START_REF] Nehlig | Notice explicative de la feuille Murat (1/50 000[END_REF], whereas enrichment in alkalis reflects the different series in the volcanic rocks from the Massif Central (Fig. 1, [START_REF] Chauvel | Nd-Sr isotope and REE geochemistry of alkali basalts from the Massif Central, France[END_REF][START_REF] Downes | Tertiary and Quaternary volcanism in the Massif Central, France[END_REF][START_REF] Downes | The nature of the lower continental crust of Europe: petrological and geochemical evidence from xenoliths[END_REF][START_REF] Wilson | Mafic alkaline magmatism associated with the European Cenozoic rift system[END_REF][START_REF] Lenoir | Nouvelles données géochronologiques, géochimiques et isotopiques sur le volcanisme du Forez : relation avec l'évolution cénozoïque du manteau du Massif Central[END_REF][START_REF] Dautria | Particularités de la contamination crustale des phonolites : exemple du Velay oriental (Massif Central)[END_REF][START_REF] Pilet | The metasomatic alternative for ocean island basalt chemical heterogeneity[END_REF][START_REF] Hamelin | A low δ7Li lower crustal component: Evidence from an alkalic intraplate volcanic series (Chaîne des Puys, French Massif Central)[END_REF]. Fractional crystallization and assimilation were shown to be an important phenomenon in the Massif Central volcanoes, particularly in the most differentiated trachyte and phonolite [START_REF] Chauvel | Nd-Sr isotope and REE geochemistry of alkali basalts from the Massif Central, France[END_REF][START_REF] Downes | Tertiary and Quaternary volcanism in the Massif Central, France[END_REF]. For the less evolved basanite and alkali basalt, fractional

A C C E P T E D M A N U S C R I P T
crystallization and assimilation cannot be ruled out [START_REF] Downes | Tertiary and Quaternary volcanism in the Massif Central, France[END_REF], as can be seen in the concentration of selected incompatible elements or in the consistency of ratios like the Rb/Sr ratio [START_REF] Chauvel | Nd-Sr isotope and REE geochemistry of alkali basalts from the Massif Central, France[END_REF].

The Tertiary volcanic rocks from the Massif Central have been widely investigated by means of the Sr-Nd-Pb isotope triptych. Investigations using Sr isotopes [START_REF] Stettler | 87Rb -86Sr constraints on the genesis and evolution of the Cantal continental volcanic system (France)[END_REF], and Sr-Nd isotopes [START_REF] Downes | Sr and Nd isotope geochemistry of coexisting alkaline magma series, Cantal, Massif Central, France[END_REF][START_REF] Chauvel | Nd-Sr isotope and REE geochemistry of alkali basalts from the Massif Central, France[END_REF][START_REF] Briot | Magma mixing versus xenocryst assimilation: The genesis of trachyandesites in Sancy volcano, Massif Central, France[END_REF][START_REF] Briot | Geochemical evolution in crustal magma reservoirs: Trace-element and Sr-Nd-O isotopic variations in two continental intraplate series at Monts Dore, Massif Central, France[END_REF][START_REF] Lenoir | Nouvelles données géochronologiques, géochimiques et isotopiques sur le volcanisme du Forez : relation avec l'évolution cénozoïque du manteau du Massif Central[END_REF][START_REF] Dautria | Particularités de la contamination crustale des phonolites : exemple du Velay oriental (Massif Central)[END_REF] indicate that the calc-alkaline and alkali magmas originated from similar magmatic sources in the lithospheric mantle and assimilated similar rocks of the continental crust. These studies showed that the rocks contain xenoliths derived from the mantle (peridotite) and the continental crust (granulite, granite gneiss, and granite). According to [START_REF] Stettler | 87Rb -86Sr constraints on the genesis and evolution of the Cantal continental volcanic system (France)[END_REF], the evolution of Rb and Sr may be interpreted in two different ways depending on whether fractional crystallization or assimilation is the predominant differentiation process. The evolution of Rb vs. Sr content in our data and those existing for the Massif Central, illustrated in Figure 7 [START_REF] Stettler | 87Rb -86Sr constraints on the genesis and evolution of the Cantal continental volcanic system (France)[END_REF][START_REF] Condomines | Chronological and Geochemical Study of Lavas from the Chaine des Puys, Massif Central, France: Evidence for Crustal Contamination[END_REF][START_REF] Chauvel | Nd-Sr isotope and REE geochemistry of alkali basalts from the Massif Central, France[END_REF][START_REF] Downes | Sr and Nd isotope geochemistry of coexisting alkaline magma series, Cantal, Massif Central, France[END_REF][START_REF] Dautria | Particularités de la contamination crustale des phonolites : exemple du Velay oriental (Massif Central)[END_REF][START_REF] Pilet | The metasomatic alternative for ocean island basalt chemical heterogeneity[END_REF][START_REF] Hamelin | A low δ7Li lower crustal component: Evidence from an alkalic intraplate volcanic series (Chaîne des Puys, French Massif Central)[END_REF] ) precluding simple linear mixing as suggested by [START_REF] Briot | Magma mixing versus xenocryst assimilation: The genesis of trachyandesites in Sancy volcano, Massif Central, France[END_REF] and indicating that the magma reservoir was not isotopically homogeneous. A significant regression can be seen when considering the relationship between the 87 Sr/ 86 Sr vs. Rb/Sr ratios (Fig. 3c andd) in our samples, the equation for which is 87 Sr/ 86 Sr = -0.00927 x Rb/Sr + 0.70423 (R²=0.57, n=7).

However, this relationship cannot be applied to the entire Massif Central (Stettler and Allègre, [START_REF] Stettler | 87Rb -86Sr constraints on the genesis and evolution of the Cantal continental volcanic system (France)[END_REF][START_REF] Condomines | Chronological and Geochemical Study of Lavas from the Chaine des Puys, Massif Central, France: Evidence for Crustal Contamination[END_REF][START_REF] Chauvel | Nd-Sr isotope and REE geochemistry of alkali basalts from the Massif Central, France[END_REF][START_REF] Downes | Sr and Nd isotope geochemistry of coexisting alkaline magma series, Cantal, Massif Central, France[END_REF][START_REF] Dautria | Particularités de la contamination crustale des phonolites : exemple du Velay oriental (Massif Central)[END_REF][START_REF] Pilet | The metasomatic alternative for ocean island basalt chemical heterogeneity[END_REF][START_REF] Hamelin | A low δ7Li lower crustal component: Evidence from an alkalic intraplate volcanic series (Chaîne des Puys, French Massif Central)[END_REF]. This suggests that contamination by the crust In the Sr-Nd diagram, the mixing plots indicate that saturated magmas assimilated up to 40% of deep-crustal granulite, whereas alkali-rich magmas assimilated less than 20% [START_REF] Faure | Origin of igneous rocks: the isotopic evidence[END_REF]. Similar to the Nd-Sr data, the history recorded in the lead-isotope compositions of the Massif Central reveals a complex sequence of events. The 207 Pb/ 204 Pb vs. 206 isotopic ratios seem to be too low [START_REF] Downes | Hercynian granodiorites and monzogranites, Massif Central, France[END_REF]. Another contaminant may be the lower crust, through meta-igneous and meta-sedimentary granulite xenoliths [START_REF] Downes | The nature of the lower continental crust of Europe: petrological and geochemical evidence from xenoliths[END_REF]; (1987) and [START_REF] Wilson | Mafic alkaline magmatism associated with the European Cenozoic rift system[END_REF]. As many authors interpret the scatter of lead-isotope values in the Massif Central as being a sign of crustal contamination, the coherent variations of Pb-isotope compositions in Figure 5 clearly agree with the assimilation of lower crust as a main contaminant. Contrary to the Chaine des Puys volcanoes, where a meta-sedimentary granulite contamination is suspected by [START_REF] Hamelin | A low δ7Li lower crustal component: Evidence from an alkalic intraplate volcanic series (Chaîne des Puys, French Massif Central)[END_REF], the best candidate for a contamination end-member of the Cantal volcanoes is meta-igneous granulite (Fig. 5).

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 10
Therefore, the presence of such volcanic rocks in the Allanche catchment is consistent with mixing between two components: one component having LVC isotope characteristics, and a second component having isotope compositions similar to meta-igneous granulite as discussed by [START_REF] Downes | Pb and O isotope systematics in granulite facies xenoliths, French Massif Central: implications for crustal processes[END_REF]. The 207 Pb/ 204 Pb and 206 Pb/ 204 Pb mixing line between these two components (Fig. 5) shows that a mixture between these end-members explains the scattering of Pb-isotope values in the volcanic rocks of the Allanche basin, which agrees with the resulting trend observed in our data. Comparing to other data from the Cantal [START_REF] Downes | Tertiary and Quaternary volcanism in the Massif Central, France[END_REF][START_REF] Wilson | Mafic alkaline magmatism associated with the European Cenozoic rift system[END_REF], our data show a higher degree of contamination.

A C C E P T E D M A N U S C R I P T

Combining lead and strontium isotope data: a better constraint of the crustal contamination of volcanic rocks from the Massif Central and evaluation of their potential mantle sources

Figure 9 shows the variation of the 87 Sr/ 86 Sr vs. 206 Pb/ 204 Pb ratios in bedrock data from the Massif Central [START_REF] Downes | Tertiary and Quaternary volcanism in the Massif Central, France[END_REF][START_REF] Wilson | Mafic alkaline magmatism associated with the European Cenozoic rift system[END_REF][START_REF] Lenoir | Nouvelles données géochronologiques, géochimiques et isotopiques sur le volcanisme du Forez : relation avec l'évolution cénozoïque du manteau du Massif Central[END_REF][START_REF] Pilet | The metasomatic alternative for ocean island basalt chemical heterogeneity[END_REF][START_REF] Hamelin | A low δ7Li lower crustal component: Evidence from an alkalic intraplate volcanic series (Chaîne des Puys, French Massif Central)[END_REF] this study) with the different end-members. The Sr isotopes reinforce the lead data and their contribution to our knowledge of the volcano and its formation processes.

The isotopic data reveal two evolutionary trends, characterized by samples from the Cantal [START_REF] Pilet | The metasomatic alternative for ocean island basalt chemical heterogeneity[END_REF] and our data) and the Chaine des Puys [START_REF] Hamelin | A low δ7Li lower crustal component: Evidence from an alkalic intraplate volcanic series (Chaîne des Puys, French Massif Central)[END_REF].

Samples from [START_REF] Pilet | The metasomatic alternative for ocean island basalt chemical heterogeneity[END_REF] curves between Sr and Pb isotopes, as it was done for mantle-crust mix [START_REF] Allègre | Isotope Geology[END_REF]. In order to illustrate the mixing in Figure 9, we used the characteristics of EMI and EMII that have been associated with pelagic and terrigenous sediments, recycled with, or without, oceanic crust [START_REF] Allègre | Isotope geodynamics[END_REF]. The Sr-Pb data for the mixing calculation are from [START_REF] Weaver | The origin of ocean island basalt end-member compositions: trace element and isotopic constraints[END_REF] for the pelagic and terrigenous sediments ( 87 Sr/ 86 Sr = 0.712; Sr content = 1500 µg g -1 206 Pb/ 204 Pb = 15.91; Pb content = 50 µg g -1 ) and [START_REF] Hoernle | Seismic and geochemical evidence for large scale mantle upwelling beneath the eastern Atlantic and western and central Europe[END_REF] for the LVC component ( 87 Sr/ 86 Sr = 0.7032; Sr content = 600 µg g -1 206 Pb/ 204 Pb = 20; Pb content = 3 µg g -1 ).

A LVC-EMI-pelagic sediments mixing curve (Fig. 9) explains all data from the Cantal, i.e. data from [START_REF] Pilet | The metasomatic alternative for ocean island basalt chemical heterogeneity[END_REF] and this study, though our data are the most EMIcontaminated found in the Cantal. This mixing curve also fits the data from the eastern volcanic provinces of the Forez [START_REF] Lenoir | Nouvelles données géochronologiques, géochimiques et isotopiques sur le volcanisme du Forez : relation avec l'évolution cénozoïque du manteau du Massif Central[END_REF]. However, sedimentary and/or igneous granulite as crustal contaminants for the Cantal basalts [START_REF] Downes | Tertiary and Quaternary volcanism in the Massif Central, France[END_REF] do not fit the data (Fig. 9). On the other hand testing such mixing between LVC and the different crustal contaminants as done in previous studies, (i.e. granulite, terrigenous sediments, crust) explains well the Chaine des Puys rocks [START_REF] Hamelin | A low δ7Li lower crustal component: Evidence from an alkalic intraplate volcanic series (Chaîne des Puys, French Massif Central)[END_REF]Fig. 9). This testifies the existence of a ca. 100-km-scale heterogeneity within the magmatic sources of the western and eastern Massif Central volcanism. Consequently, the Cantal basalts cannot be considered as homogeneous with respect to their Sr and Pb isotopic compositions, demonstrating the involvement of variable sediment contamination of their mantle sources. According to the mixing curve in Fig. 9, the degree of contamination of Cantal basalts fluctuates between 0.3 and 1.5%.

Conclusions

We present lead and strontium-isotopic ratios together with major-and trace elements in the bedrock of a small mono-lithologic basalt basin (Allanche River, 160 km²) belonging to the 206 Pb/ 204 Pb, the Massif Central rocks only partly plot along the NHRL mixing line between the mantle components (DM and HIMU end-members). Most of the regional volcanic rocks diverge from the NHRL mixing line, implying another process like fractional crystallization and assimilation (e.g. contamination) by another reservoir. This is similar to the Chaine des Puys volcanoes that were contaminated by meta-sedimentary granulite from the lower crust [START_REF] Hamelin | A low δ7Li lower crustal component: Evidence from an alkalic intraplate volcanic series (Chaîne des Puys, French Massif Central)[END_REF]. Data from the Allanche bedrock clearly define a different trend from that of the Chaine des Puys, implying another contamination source; they are consistent with previous results on the Cantal volcano and plead in favor of binary mixing between the LVC component and meta-igneous granulite from the lower crust. Finally, data from the Allanche catchment also suggest a higher degree of contamination by meta-igneous granulite than previously found by [START_REF] Wilson | Mafic alkaline magmatism associated with the European Cenozoic rift system[END_REF] Chaine des Puys, the Cantal, the Forez and the Languedoc [START_REF] Hamelin | A low δ7Li lower crustal component: Evidence from an alkalic intraplate volcanic series (Chaîne des Puys, French Massif Central)[END_REF][START_REF] Downes | Sr and Nd isotope geochemistry of coexisting alkaline magma series, Cantal, Massif Central, France[END_REF][START_REF] Wilson | Mafic alkaline magmatism associated with the European Cenozoic rift system[END_REF][START_REF] Pilet | The metasomatic alternative for ocean island basalt chemical heterogeneity[END_REF][START_REF] Dautria | 160 Ma of sporadic basaltic activity on the Languedoc volcanic line (Southern France): A peculiar case of lithosphereasthenosphere interplay[END_REF]. Regression line are indicated in the graph (see text) for Allanche bedrock. Cantal, the Forez, the Velay, the Mont Dore, the Aubrac and the Languedoc [START_REF] Hamelin | A low δ7Li lower crustal component: Evidence from an alkalic intraplate volcanic series (Chaîne des Puys, French Massif Central)[END_REF][START_REF] Downes | Sr and Nd isotope geochemistry of coexisting alkaline magma series, Cantal, Massif Central, France[END_REF][START_REF] Wilson | Mafic alkaline magmatism associated with the European Cenozoic rift system[END_REF][START_REF] Pilet | The metasomatic alternative for ocean island basalt chemical heterogeneity[END_REF][START_REF] Chauvel | Nd-Sr isotope and REE geochemistry of alkali basalts from the Massif Central, France[END_REF][START_REF] Dautria | Particularités de la contamination crustale des phonolites : exemple du Velay oriental (Massif Central)[END_REF]2010;[START_REF] Condomines | Chronological and Geochemical Study of Lavas from the Chaine des Puys, Massif Central, France: Evidence for Crustal Contamination[END_REF][START_REF] Stettler | 87Rb -86Sr constraints on the genesis and evolution of the Cantal continental volcanic system (France)[END_REF]. Regression line and grayed area corresponding to 95% confidence. (c) and (d) -Plot of the 87 Sr/ 86 Sr vs.

the Rb/Sr ratios in the volcanic rocks of the Allanche bassin compared with ratios from other rocks in the Massif Central. [START_REF] Downes | The nature of the lower continental crust of Europe: petrological and geochemical evidence from xenoliths[END_REF][START_REF] Downes | Crustal evolution of the Hercynian belt of Western Europe: Evidence from lower-crustal granulitic xenoliths (French Massif Central)[END_REF][START_REF] Downes | Hercynian granodiorites and monzogranites, Massif Central, France[END_REF]. Additional data are from the Chaine des Puys, the Cantal, the Forez, and the Languedoc [START_REF] Hamelin | A low δ7Li lower crustal component: Evidence from an alkalic intraplate volcanic series (Chaîne des Puys, French Massif Central)[END_REF][START_REF] Downes | Sr and Nd isotope geochemistry of coexisting alkaline magma series, Cantal, Massif Central, France[END_REF][START_REF] Wilson | Mafic alkaline magmatism associated with the European Cenozoic rift system[END_REF][START_REF] Pilet | The metasomatic alternative for ocean island basalt chemical heterogeneity[END_REF][START_REF] Dautria | 160 Ma of sporadic basaltic activity on the Languedoc volcanic line (Southern France): A peculiar case of lithosphereasthenosphere interplay[END_REF][START_REF] Lenoir | Nouvelles données géochronologiques, géochimiques et isotopiques sur le volcanisme du Forez : relation avec l'évolution cénozoïque du manteau du Massif Central[END_REF] [START_REF] Downes | The nature of the lower continental crust of Europe: petrological and geochemical evidence from xenoliths[END_REF][START_REF] Downes | Crustal evolution of the Hercynian belt of Western Europe: Evidence from lower-crustal granulitic xenoliths (French Massif Central)[END_REF][START_REF] Downes | Hercynian granodiorites and monzogranites, Massif Central, France[END_REF]. Additional data (Fig. 1) are from the Chaine des Puys, the Cantal, the Forez, and the Languedoc [START_REF] Hamelin | A low δ7Li lower crustal component: Evidence from an alkalic intraplate volcanic series (Chaîne des Puys, French Massif Central)[END_REF][START_REF] Downes | Sr and Nd isotope geochemistry of coexisting alkaline magma series, Cantal, Massif Central, France[END_REF][START_REF] Wilson | Mafic alkaline magmatism associated with the European Cenozoic rift system[END_REF][START_REF] Pilet | The metasomatic alternative for ocean island basalt chemical heterogeneity[END_REF][START_REF] Dautria | 160 Ma of sporadic basaltic activity on the Languedoc volcanic line (Southern France): A peculiar case of lithosphereasthenosphere interplay[END_REF]Lenoir et al., 2000, and Briot personal communication; 'undifferentiated rock samples from the Massif Central volcanoes identified as Bedrock MC').

Figure 6. Relationships between the SiO 2 and K 2 O contents of the Allanche volcanic rocks.

The Basanite and F-Basalt fields (for feldspathic basalts, see text) are reported. Bedrock with SiO 2 >50% corresponds to erratic blocks in the upstream part of the Allanche catchment [START_REF] Négrel | Natural and anthropogenic budgets of a small watershed in the Massif Central (France): Chemical and strontium isotopic characterization in water and sediments[END_REF]. Data referred to as "Bedrock geological maps" are from basaltic rocks of the Cantal volcano (De Goër de [START_REF] De Goër De Hervé | Notice explicative de la feuille Saint-Flour (1/50 000[END_REF][START_REF] Nehlig | Notice explicative de la feuille Murat (1/50 000[END_REF]. Additional data are from volcanic rocks from the Chaine des Puys, the Cantal, the Forez, the Velay, the Mont Dore, the Aubrac and the Languedoc [START_REF] Hamelin | A low δ7Li lower crustal component: Evidence from an alkalic intraplate volcanic series (Chaîne des Puys, French Massif Central)[END_REF][START_REF] Downes | Sr and Nd isotope geochemistry of coexisting alkaline magma series, Cantal, Massif Central, France[END_REF][START_REF] Wilson | Mafic alkaline magmatism associated with the European Cenozoic rift system[END_REF][START_REF] Pilet | The metasomatic alternative for ocean island basalt chemical heterogeneity[END_REF][START_REF] Chauvel | Nd-Sr isotope and REE geochemistry of alkali basalts from the Massif Central, France[END_REF][START_REF] Dautria | Particularités de la contamination crustale des phonolites : exemple du Velay oriental (Massif Central)[END_REF]2010;[START_REF] Condomines | Chronological and Geochemical Study of Lavas from the Chaine des Puys, Massif Central, France: Evidence for Crustal Contamination[END_REF][START_REF] Stettler | 87Rb -86Sr constraints on the genesis and evolution of the Cantal continental volcanic system (France)[END_REF][START_REF] Lenoir | Nouvelles données géochronologiques, géochimiques et isotopiques sur le volcanisme du Forez : relation avec l'évolution cénozoïque du manteau du Massif Central[END_REF].

Figure 7. Relationships between the Sr and Rb contents of volcanic rocks of the Massif Central; including the studied Allanche river basin and data from the Chaine des Puys, the Cantal, the Forez, the Velay, the Mont Dore, the Aubrac and the Languedoc [START_REF] Hamelin | A low δ7Li lower crustal component: Evidence from an alkalic intraplate volcanic series (Chaîne des Puys, French Massif Central)[END_REF][START_REF] Downes | Sr and Nd isotope geochemistry of coexisting alkaline magma series, Cantal, Massif Central, France[END_REF][START_REF] Wilson | Mafic alkaline magmatism associated with the European Cenozoic rift system[END_REF][START_REF] Pilet | The metasomatic alternative for ocean island basalt chemical heterogeneity[END_REF][START_REF] Chauvel | Nd-Sr isotope and REE geochemistry of alkali basalts from the Massif Central, France[END_REF][START_REF] Dautria | Particularités de la contamination crustale des phonolites : exemple du Velay oriental (Massif Central)[END_REF]2010;[START_REF] Condomines | Chronological and Geochemical Study of Lavas from the Chaine des Puys, Massif Central, France: Evidence for Crustal Contamination[END_REF][START_REF] Stettler | 87Rb -86Sr constraints on the genesis and evolution of the Cantal continental volcanic system (France)[END_REF].

Figure 8. Plot of 87 Sr/ 86 Sr vs. the SiO 2 of volcanic rocks of the Massif Central; including the studied Allanche river basin and data the Chaine des Puys, the Cantal, the Forez, the Velay, the Mont Dore, the Aubrac and the Languedoc [START_REF] Hamelin | A low δ7Li lower crustal component: Evidence from an alkalic intraplate volcanic series (Chaîne des Puys, French Massif Central)[END_REF][START_REF] Downes | Sr and Nd isotope geochemistry of coexisting alkaline magma series, Cantal, Massif Central, France[END_REF][START_REF] Wilson | Mafic alkaline magmatism associated with the European Cenozoic rift system[END_REF][START_REF] Pilet | The metasomatic alternative for ocean island basalt chemical heterogeneity[END_REF][START_REF] Chauvel | Nd-Sr isotope and REE geochemistry of alkali basalts from the Massif Central, France[END_REF][START_REF] Dautria | Particularités de la contamination crustale des phonolites : exemple du Velay oriental (Massif Central)[END_REF]2010;[START_REF] Condomines | Chronological and Geochemical Study of Lavas from the Chaine des Puys, Massif Central, France: Evidence for Crustal Contamination[END_REF][START_REF] Stettler | 87Rb -86Sr constraints on the genesis and evolution of the Cantal continental volcanic system (France)[END_REF]. Dashed line and grayed area are from the Chaine des Puys, the Cantal, the Forez, and the Languedoc [START_REF] Hamelin | A low δ7Li lower crustal component: Evidence from an alkalic intraplate volcanic series (Chaîne des Puys, French Massif Central)[END_REF][START_REF] Pilet | The metasomatic alternative for ocean island basalt chemical heterogeneity[END_REF][START_REF] Dautria | 160 Ma of sporadic basaltic activity on the Languedoc volcanic line (Southern France): A peculiar case of lithosphereasthenosphere interplay[END_REF][START_REF] Lenoir | Nouvelles données géochronologiques, géochimiques et isotopiques sur le volcanisme du Forez : relation avec l'évolution cénozoïque du manteau du Massif Central[END_REF]. Mixing curves involving crust, pelagic sediment, sedimentary rock and igneous granulite contaminations are illustrated. [START_REF] Négrel | Natural and anthropogenic budgets of a small watershed in the Massif Central (France): Chemical and strontium isotopic characterization in water and sediments[END_REF].

A C C E P T E D M A N U S C R I P T

  volcanic forms include Strombolian cones aligned along fissures (Chaine des Puys), extensive basaltic plateaus (Aubrac and Devès), and two large central volcanoes (Cantal and Mont Dore). The Cantal volcano, equivalent in size of the Etna, was active between 13 and 3 million years ago. Numerous outcrops and the diversity of rocks emitted during its activity make it an ideal study site to investigate different magmatic processes affecting the composition of the volcanic rocks. Basalts from the Cantal stratovolcano are characterized by a significant change in their chemical composition over time, comparable to that observed for A C C E P T E D M A N U S C R I P T consist of silica-saturated alkali rocks (olivine basalt, trachyandesite, trachyte, and rhyolite)

  SiO 2 content close to 46-49% and less than 5% of Na 2 O+K 2 O. The main mineral in these basalts is plagioclase (An 55-70) with normative nepheline, hypersthene and olivine. Basanite (nepheline-or leucite basalt) has a SiO 2 content of 41-45%, low Na 2 O+K 2 O (<5%), and modal or normative nepheline or leucite. The groundmass includes clinopyroxene and plagioclase (An 50-60).

  discussed that EM-I represents the final stage of metasomatic melt evolution, EM-II corresponding to the extreme fractionation stage for trace elements by metasomatic processes, while[START_REF] Jackson | Compositions of HIMU, EM1, and EM2 from global trends between radiogenic isotopes and major elements in ocean island basalts[END_REF] argued for the presence of recycled sediment or continental crust, or metasomatism by low degree melts and/or fluids. Recently,[START_REF] Willbold | Formation of enriched mantle components by recycling of upper and lower continental crust[END_REF] discussed the recycling of upper and lower continental crust, and[START_REF] Wilson | Mafic alkaline magmatism associated with the European Cenozoic rift system[END_REF] and[START_REF] Pilet | The metasomatic alternative for ocean island basalt chemical heterogeneity[END_REF] considered all end-members as influencing the volcanic rocks from the Massif Central. The correlation between the 207 Pb/ 204 Pb vs.1/Pb ratios shown in Figure2is R²=0.31, and R²=0.77 between the 208 Pb/ 204 Pb and 206 Pb/ 204 Pb vs.1/Pb ratios. These relationships reflect the existence of two endmembers with a lowering of the lead isotope ratio as the lead content increases, indicating at least two sources for the volcanic rocks. A similar set of relationships can be observed in volcanic rocks from the Chaine des Puys[START_REF] Hamelin | A low δ7Li lower crustal component: Evidence from an alkalic intraplate volcanic series (Chaîne des Puys, French Massif Central)[END_REF]. Figure4shows the variation of the 208 Pb/ 204 Pb vs.206 Pb/ 204 Pb ratios for the Allanche samples and other volcanic rocks from the Massif Central and the different end-members forming the mantle. These are HIMU and DM, and EM-I and -II, the linear mixing line defined as the Northern Hemisphere Reference Line is also represented(NHRL, Hart, 1984). All data in Figure4plot close to the NHRL, diverging only by lower 206 Pb/ 204 Pb ratios through the EM-I and II influence. Along the NHRL, the role of the low-velocity component (LVC;[START_REF] Hoernle | Seismic and geochemical evidence for large scale mantle upwelling beneath the eastern Atlantic and western and central Europe[END_REF] in the origin of the volcanic rocks is obvious. Isotopic characterization has allowed defining a common end-member in the mix along the NHRL line in Europe, the Mediterranean and the western Canaries. Several authors have summarized lead isotope ratios for this component, assigning highly radiogenic values to the LVC which was explained by[START_REF] Granet | Imaging a mantle plume beneath the French Massif Central[END_REF] as being of A C C E P T E D M A N U S C R I P T asthenospheric origin (EAR) and interpreted as an upwelling in the upper mantle by Hoernle et al. (1995). Thus, regarding 208 Pb/ 204 Pb vs. 206 Pb/ 204 Pb ratios, the volcanic rocks of the Massif Central are mainly the result of mixing between the DM and LVC components. However, the investigated samples from the Allanche area show a dominance of the DM over LVC. Such an influence was previously stated by Downes et al. (2003) regarding the mantle components in the Massif Central and by Lenoir et al. (2000) studying the evolution of the upper mantle below the Massif Central through the Forez volcanism (Fig. 1). As for Figure 4, the 207 Pb/ 204 Pb vs. 206 Pb/ 204 Pb ratios of the volcanic rocks in the Massif Central partly plot along the NHRL mixing line together with the LVC component that also plots along the NHRL line (Figure

  show two trends. The first trend relates basalt to andesite, in which Rb and Sr are positively correlated and in that way the trend reflects the differentiation of the basaltic melt. The second trend relates the first trend to rhyolite, where high Rb are associated with low Sr, and reflects evolved fractional crystallization. The 87 Sr/ 86 Sr vs. 1/Sr ratios, illustrated in Figure 3a and b, show no clear relationship (R²=0.20, n=7

  component does not have the same Rb/Sr ratio over the area or that this relationship corresponds to a different behavior of Sr-Rb during basalt crystallization. With increasing SiO 2, the differentiated samples display increasing 87 Sr/ 86 Sr shown in Figure8as previously found by[START_REF] Kolb | Petrogenesis of mafic to felsic lavas from the Oligocene Siebengebirge volcanic field (Germany): implications for the origin of intracontinental volcanism in Central Europe[END_REF] in Germany (e.g. the Siebengebirge Volcanic Field that forms part of the Central European Volcanic Province). The data of[START_REF] Kolb | Petrogenesis of mafic to felsic lavas from the Oligocene Siebengebirge volcanic field (Germany): implications for the origin of intracontinental volcanism in Central Europe[END_REF] on Figure8fit well our data. Such relationships indicate an open-system behavior during differentiation associated with crustal assimilation. This field fits well our data and those from the rest of the Massif Central volcanism. The defined trend seems to indicate that regional lower crust, represented by granulite xenoliths, might be a potential contaminant candidate.

  Pb/ 204 Pb plot (Fig.5) shows that the volcanic rocks in the Massif Central partly plot along the NHRL mixing line. Most of the reported volcanic bedrock data in the Massif Central plot away from the NHRL, again interpreted in terms of component mixing between different end-members.This reinforces the arguments developed first by[START_REF] Downes | Tertiary and Quaternary volcanism in the Massif Central, France[END_REF] and[START_REF] Downes | Pb and O isotope systematics in granulite facies xenoliths, French Massif Central: implications for crustal processes[END_REF], who suggested that crustal contamination occurred in the volcanic rocks from the Massif Central. Studying the Chaine des Puys volcanoes in the northern part of the Massif Central,[START_REF] Hamelin | A low δ7Li lower crustal component: Evidence from an alkalic intraplate volcanic series (Chaîne des Puys, French Massif Central)[END_REF] investigated such possible contaminating crustal components. One possibility they discuss is the upper continental crust (Variscan granitoids), but their Pb-

  in the Chaine des Puys volcanoes.However, the samples from the Allanche basin clearly diverge from the Chaine des Puys line as defined by[START_REF] Hamelin | A low δ7Li lower crustal component: Evidence from an alkalic intraplate volcanic series (Chaîne des Puys, French Massif Central)[END_REF], whereby a predominant feature is a larger variation of the 206 Pb/ 204 Pb ratio than the 207 Pb/ 204 Pb ratio. The data from the Allanche basin were regressed in Figure5to give a least-square line describing the general trend between the two lead-isotope ratios. The resulting trend, with a R² ~0.51 (n=7) and a slope of 0.03, agrees with some of the data from the Cantal volcano included in the trend and reported by Downes

  Fig. 9 seem to support a three-component model. When looking at the isotopic character of the rocks in relation to the mantle end-member components, the involvement of LVC and EM (I and II) can be inferred. Combining Pb and Sr isotopic data for volcanic rocks from the eastern (Forez, Lenoir et al., 2000) and western volcanic provinces (Cantal, Chaine des Puys) reveals common isotopic values similar to those of LVC, followed by a contrasting divergence. The Cantal and Forez show isotopic signatures more similar to that of enriched mantle EM-I, while the Chaine des Puys is more similar to the EM-II component. Thus, the Sr-Pb isotope diagram reveals for the Cantal basalts a variation between EMI and LVC compositions, while basalts from other areas of the Massif Central vary between the EMII and the LVC compositions. This can be modeled by calculating mixing A C C E P T E D M A N U S C R I P T

  the volcanic Massif Central province, which resulted from a complex history and multiple reservoir sources and mixing. Part of the lead-isotope data from the Allanche basin agree with literature data of Massif Central volcanic rocks that resulted from of a mixing between the well-defined mantle sources: (DM) and (LVC) end-members. The latter is of asthenospheric origin and corresponds to the European asthenospheric reservoir (EAR). The bedrock samples analyzed during this work show a strong influence of the DM component based on 208 Pb/ 204 Pb and 206 Pb/ 204 Pb isotopic ratios. Using 207 Pb/ 204 Pb vs.

  Figure captions

Figure 2 .

 2 Figure 2. Plot of 206 Pb/ 204 Pb, 207 Pb/ 204 Pb, 208 Pb/ 204 Pb vs. the 1/Pb ratio (µg. g -1 ) of volcanic rocks of the Massif Central including the studied Allanche river basin and data from the

Figure 3 .

 3 Figure 3. (a) and (b) -Plot of 87 Sr/ 86 Sr vs. 1/Sr ratio (µg. g -1 ) of volcanic rocks of the Massif Central including the studied Allanche river basin and data from the Chaine des Puys, the

Figure 4 .

 4 Figure 4. Plot of 208 Pb/ 204 Pb vs. 206 Pb/ 204 Pb ratios of the Allanche volcanic rocks. 'NHRL'corresponds to NHRL mixing line of[START_REF] Hart | A large-scale isotope anomaly in the Southern Hemisphere mantle[END_REF] defined for the northern hemisphere between the end-members DM and HIMU. Upper and lower crust end-members are also indicated (Variscan granitoids, meta-igneous granulites and meta-sedimentary granulites,

  study by[START_REF] Kolb | Petrogenesis of mafic to felsic lavas from the Oligocene Siebengebirge volcanic field (Germany): implications for the origin of intracontinental volcanism in Central Europe[END_REF] in Germany are indicated in the graph (see text).

Figure 9 .

 9 Figure 9. Plot of 87 Sr/ 86 Sr vs. 206 Pb/ 204 Pb ratios of the Allanche volcanic rocks. Mantle endmembers are indicated (HIMU, DM, EM1, EM2, LVC, see text for details). Additional data

Figure 1 A

 1 Figure 1

Figure 3 A

 3 Figure 3

Figure 4 A

 4 Figure 4

Figure 5 A

 5 Figure 5

Figure 6 A

 6 Figure 6

  for the Cantal are close to the LVC component (lowest 87 Sr/ 86 Sr ratio and the highest 206 Pb/ 204 Pb ratio) representing the EAR. On the contrary, samples from Hamelin et al. (2009) have the highest 87 Sr/ 86 Sr ratio and intermediate 206 Pb/ 204 Pb ratio. Samples from this study are characterized by a rather low 87 Sr/ 86 Sr (close to that of Pilet et al., 2005) and the lowest 206 Pb/ 204 Pb ratio. The roughly triangular positions of isotopic data in

  Plot of 207 Pb/ 204 Pb vs. 206 Pb/ 204 Pb ratios of the Allanche volcanic rocks. Mantle end-

	communication; 'undifferentiated rock samples from the Massif Central volcanoes identified
	as Bedrock MC').
	Figure 5. members are indicated (HIMU, DM, EM1, EM2, LVC, see text for details). End-members of the mantle (HIMU, DM, EM-I, EM-II, LVC) and of the upper and lower crust are indicated M A N U S C R I P T (Variscan granitoids, meta-igneous granulites and meta-sedimentary granulites, Downes,
	A C C E P T E D
	Briot personal

Table caption Table 1 .

 caption1 Analyzed bedrock samples (feldspathic basalt, basanite (nepheline-leucite), basanite (nepheline) with ages of the corresponding volcanic cycles according to De Goër De Hervé[START_REF] De Goër De Hervé | Notice explicative de la feuille Saint-Flour (1/50 000[END_REF]. Lead-isotope data from bedrock expressed as 206 Pb/ 204 Pb, 207 Pb/ 204 Pb and 208 Pb/ 204 Pb ratios, strontium isotope data expressed as 87 Sr/ 86 Sr. External reproducibility on lead-isotopic ratios is 0.1% (2) and ±10 ppm (2σ m ) for strontium isotopic ratios. Data for major elements (SiO 2 , K 2 O) and trace elements (Rb, Sr and Pb), are from

Table 1 Name Reference Description Age 206 Pb/ 204 Pb 207 Pb/ 204 Pb 208 Pb/ 204 Pb Pb

 1

	ACCEPTED MANUSCRIPT

Acknowledgements

This work was financially supported by the BRGM Research Division programs. It benefited from the collaboration of the BRGM Isotope laboratories, and M. Robert and A.M. Gallas are thanked for their help. We are grateful to Dr. H.M. Kluijver for proofreading and editing the English text. The authors are grateful to the anonymous reviewers for thoughtful revisions of the manuscript and constructive comments.

thus provides additional insight into the volcanic history of the Massif Central and illustrates the lead-strontium source fingerprinting at regional scale (160 km 2 ).

A C C E P T E