Hossein A Moghaddam

Antti Tölli

Luca Sanguinetti

Merouane Debbah

Deterministic Equivalent for Max-Min SINR over Random User Locations

The max-min signal-to-interference-plus-noise ratio (SINR) problem is considered in a coordinated network wherein L base stations (BSs) each equipped with N antennas serve in total K single-antenna users that are uniformly distributed in the network. We conduct the analysis in the asymptotic regime in which N and K grow large to compute a deterministic approximation for the max-min SINR. The results are independent from fast-fading and users' locations and thus allow one to determine the optimal max-min SINR given basic system parameters such as cell radius, K, N and pathloss exponent. The provided framework can be utilized for analyzing the problem without the need to run system level simulations and for finding the optimal N , K, resource allocation and BS placement. Numerical results are used to validate the analytical results in a finite system regime and to evaluate the effects of system parameters on the system performance.

I. INTRODUCTION

Coordinated multi point transmission (CoMP) has been studied extensively in literature as an efficient method to handle inter-cell interference (ICI) [START_REF] Dahrouj | Coordinated beamforming for the multicell multi-antenna wireless system[END_REF]. Due to fast fading channels, the impact of different system parameters has been largely investigated via Monte Carlo simulations. However, the advent of large dimensional networks (as envisioned in 5G networks) have spurred renewed interest in random matrix theory (RMT) [START_REF] Couillet | Random Matrix Methods for Wireless Communications[END_REF] for the analysis and design of wireless communication systems. Even though the use of RMT allows one to remove the functional dependence of performance metrics from fast fading channels, it still requires a deterministic placement of user equipment terminals (UEs). In this work, we aim at computing asymptotic approximations of performance metrics that are independent of fading and UE locations. A similar approach is followed in [START_REF] Hoydis | Analysis of multicell cooperation with random user locations via deterministic equivalents[END_REF] but for a simple one dimensional network in which only two base stations (BS) along with UEs are located on a line. An alternative approach to tackle such a problem is stochastic geometry as in [START_REF] Haenggi | Stochastic geometry and random graphs for the analysis and design of wireless networks[END_REF]- [START_REF] Huang | A stochastic-geometry approach to coverage in cellular networks with multi-cell cooperation[END_REF]. Despite being a very powerful tool, stochastic geometry has not yet led to tractable results considering advanced joint multi-cell processing [START_REF] Hoydis | Analysis of multicell cooperation with random user locations via deterministic equivalents[END_REF].

Coordinated multi-cell resource allocation is generally formulated as an optimization problem in which a desired network utility is maximized subject to some requirements This research was supported by the Academy of Finland (Decision no. 279101 and 284590). The research of L. Sanguinetti and M. Debbah has been supported by the ERC Starting Grant 305123 MORE. [START_REF] Dahrouj | Coordinated beamforming for the multicell multi-antenna wireless system[END_REF], [START_REF] Tölli | Decentralized minimum power multi-cell beamforming with limited backhaul signaling[END_REF]. In this work, we consider the max-min signal-tointerference-and-noise ratio (SINR) problem, or equivalently, SINR balancing problem subject to a given total power constraint. This problem is closely related to the total power minimization problem subject to a given minimum SINR constraints. In [START_REF] Wiesel | Linear precoding via conic optimization for fixed mimo receivers[END_REF], the authors proved that for multiple-inputsingle-output (MISO) downlink systems the two optimization problems are dual of each other. Both problems have been addressed extensively in the literature (see for example [START_REF] Dahrouj | Coordinated beamforming for the multicell multi-antenna wireless system[END_REF], [START_REF] Tölli | Decentralized minimum power multi-cell beamforming with limited backhaul signaling[END_REF], [START_REF]Linear precoding via conic optimization for fixed MIMO receivers[END_REF]- [START_REF] Pennanen | Multi-cell beamforming with decentralized coordination in cognitive and cellular networks[END_REF]). The max-min SINR problem was first addressed in [START_REF] Yang | Optimal downlink power assignment for smart antenna systems[END_REF] using an extended coupling matrix approach. The works in [START_REF] Wiesel | Linear precoding via conic optimization for fixed mimo receivers[END_REF], [START_REF] Huang | Distributed multicell beamforming with limited intercell coordination[END_REF] reformulate the problem based on conic programming. An approach based on nonlinear Perron-Frobenius theory [START_REF] Krause | Concave perron-frobenius theory and applications[END_REF] is proposed in [START_REF] Tan | Maximizing sum rate and minimizing MSE on multiuser downlink: Optimality, fast algorithms and equivalence via max-min SINR[END_REF] where the extension to MIMO downlink transmissions is given in [START_REF] Cai | A unified analysis of maxmin weighted SINR for mimo downlink system[END_REF]. The work in [START_REF] Tölli | Linear multiuser MIMO transceiver design with quality of service and per-antenna power constraints[END_REF] considers the common rate maximization (or rate balancing) problem for the multi-antenna receiver case with per-antenna power constraint.

One of the first works considering large system analysis for linear receiver design is given by [START_REF] Tse | Linear multiuser receivers: Effective interference, effective bandwidth and user capacity[END_REF]. A single-cell setting under the assumption of Rayleigh fading is considered in [START_REF] Sanguinetti | Optimal linear precoding in multi-user MIMO systems: A large system analysis[END_REF], [START_REF] Sanguinetti | Large system analysis of the energy consumption distribution in multi-user MIMO systems with mobility[END_REF] whereas the multi-cell case is investigated in [START_REF] Lakshminaryana | Asymptotic analysis of distributed multi-cell beamforming[END_REF]- [START_REF] Sanguinetti | Large system analysis of base station cooperation for power minimization[END_REF]. Under a more restrictive assumption, the authors in [START_REF] Zakhour | Base station cooperation on the downlink: Large system analysis[END_REF] provide an asymptotic analysis in a simple case where UEs in a network with two cells are required to be served with the same target rate. The analysis is extended in [START_REF] Zakhour | Min-max power allocation in cellular networks with coordinated beamforming[END_REF] for the minimization of the maximum power problem with homogeneous channel setting. The works in [START_REF] Wagner | Large system analysis of linear precoding in correlated MISO broadcast channels under limited feedback[END_REF], [START_REF] Zhang | Large system analysis of cooperative multi-cell downlink transmission via regularized channel inversion with imperfect CSIT[END_REF] study the sum rate performance of zero-forcing (ZF) and regularized ZF precoding in large MISO broadcast systems. The work in [START_REF] Huang | Joint beamforming and power control in coordinated multicell: Max-min duality, effective network and large system transition[END_REF] studies joint beamforming and power control in a coordinated multi-cell downlink system employing the nonlinear Perron-Frobenius theory and RMT. The recent work in [START_REF] Sifaou | Max-min SINR in large-scale single-cell MU-MIMO: asymptotic analysis and low complexity transceivers[END_REF] focuses on the downlink and uplink of large-scale single-cell multiuser MIMO systems and aim at reducing the complexity of the linear SINR balancing precoder.

In this work, we utilize RMT to analyze multi-cell cooperative systems that account for channel fading and random UE locations. In particular, we do not assume a deterministic placement of user terminals but we average over random UE locations. As mentioned before, the max-min SINR and power minimization problems are closely related. The max-min SINR problem can be solved using a bisection method [START_REF] Boyd | Convex Optimization[END_REF] where at each iteration an instance of the power minimization problem is solved. Keeping this in mind, we first derive an asymptotic approximation for the total power consumption (given a set of SINR targets) as a function of SINR targets and basic system parameters such pathloss exponent, number of UEs and antennas and cell radios. Interpreting the total power consumption as power budget, we then get a relation between users' SINRs and basic system parameters which allows to determine the optimal SINR assignment. In the single-cell case, the solution to the max-min SINR problem is computed in closed form. For the general multi-cell case, the optimal SINR assignment is determined by using a bisection method. Note that the problem solved at each iteration is much simpler as compared to available methods that rely on convex optimization tools. Moreover, the resulting SINRs need to be recomputed only when the basic system parameters change. The results allow one to find approximations of the average system performance irrespective of fading and UE locations and to answer questions related to power budgeting, BS deployment strategies and network dimensioning. Moreover, the developed framework allows to study the aforementioned problems without the need to run Monte Carlo simulations.

The paper is organized as follows. The problem formulation is presented in Section II. The asymptotic analysis is provided in Section III and it is validated by means of numerical results in Section IV.

II. SYSTEM MODEL AND PROBLEM FORMULATION

Consider the downlink of a multi-cell multi-user MIMO system composed of L cells where each BS has N antennas. A total number of K single-antenna UEs are randomly dropped in the coverage area. We assume that each UE is assigned to a single BS while being interfered by the other BSs. We denote the set of

γk DL = p k h H b k ,k v k 2 j∈U b \k p j h H b,k v j 2 + σ 2 , ∀k ∈ U. (1)
Under the assumption of Gaussian inputs with zero mean and unit variance, the max-min SINR problem can be formulated as [START_REF] Yang | Optimal downlink power assignment for smart antenna systems[END_REF]:

max {v k ,p} min k∈U γDL k (2) s.t. p 1 ≤ P TX v k = 1, ∀k ∈ U
The vector {p} contains {p k ∀k ∈ U} values and the power available at BSs is upper bounded by a sum power constraint {P TX }. The thermal noise power σ2 is normalized to unity throughout the work without loss of generality.

We now review the well-known iterative bisection solution for SINR balancing problem that is essentially based on the link between the SINR balancing and the power minimization under SINR constraints underlined in [START_REF] Dahrouj | Coordinated beamforming for the multicell multi-antenna wireless system[END_REF]. The idea behind the bisection algorithm is to solve a sequence of power minimization problems with fixed user specific SINR requirements imposed on the system at each step of the algorithm. While searching for the optimal SINR assignments, the bisection algorithm is stopped when the optimized sum power is within an -distance from the power budget P TX .

The power minimization problem that need to be solved at each bisection iteration is given as [START_REF] Dahrouj | Coordinated beamforming for the multicell multi-antenna wireless system[END_REF]

min {p,v k } b∈B k∈U b p k (3)
s.t. γDL k ≥ γ k , ∀k ∈ U
where γ k is a fixed SINR target set by the outer bisection algorithm. The above problem can be solved using different approaches based on iterative uplink-downlink duality [START_REF] Dahrouj | Coordinated beamforming for the multicell multi-antenna wireless system[END_REF], on the centralized SOCP formulation [START_REF] Tölli | Decentralized minimum power multi-cell beamforming with limited backhaul signaling[END_REF] or using decentralized dual or primal decomposition techniques [START_REF] Tölli | Decentralized minimum power multi-cell beamforming with limited backhaul signaling[END_REF], [START_REF] Pennanen | Multi-cell beamforming with decentralized coordination in cognitive and cellular networks[END_REF]. The solution of interest in this work is uplink-downlink duality approach where the dual problem for (3) is formulated as [START_REF] Dahrouj | Coordinated beamforming for the multicell multi-antenna wireless system[END_REF] min

{v k },{λ k } b∈B k∈U b λ k N σ 2 (4) s.t. γUL k ≥ γ k , ∀k ∈ U
where γUL k is the dual UL SINR expression, and {λ k /N } correspond to the Lagrange multipliers associated to the SINR constraints in (3) and can be thought as the solutions to the above dual UL power minimization problem. 2 An immediate consequence from duality is that under the given constraints both UL and DL have the same total power consumption. Moreover, the SINR targets can be achieved by the same set of beamforming vectors.

The solution to uplink problem is known to be given using a fixed point iteration formula as [START_REF] Dahrouj | Coordinated beamforming for the multicell multi-antenna wireless system[END_REF]

λ k (t + 1) = λ k (t) γ k γUL k (t) (5)
where t is the iteration index and γUL k (t) is given by

γUL k (t) = λ k (t)h H b k ,k   j∈U \k λ j (t)h b k ,j h H b k ,j + N I N   -1 h b k ,k (6)
which corresponds to the SINR at iteration t for given fixed uplink powers {λ k (t)/N }. The optimal {v k } correspond to a set of normalized minimum mean square error (MMSE) receivers:

v k = ṽk ṽk , ṽk =   j∈U \k λ j h b k ,j h H b k ,j + N I N   -1 h b k ,k . (7)
The SINR formulation in ( 6) tends to a deterministic value while system dimensions (N and K) grow large which in turn allows a deterministic expression for uplink powers in [START_REF] Andrews | A tractable approach to coverage and rate in cellular networks[END_REF].

In doing so, we perform large system analysis in the next section to get an accurate approximation for the solutions of the problems in hand.

III. LARGE SYSTEM ANALYSIS

We exploit the theory of large system analysis to compute first the so-called deterministic equivalents of uplink powers in the minimum power beamforming problem. We then utilize these results to find approximations of the optimal SINR assignment in the finite system regime. In doing so, we assume the following grow rate of the system dimensions:

Assumption 1. As N → ∞, 0 < lim inf K N ≤ lim sup K N < ∞.
Theorem 1. Under Assumptions 1, the power terms in (4) converge asymptotically to λk as λ k -λk → 0 almost surely

λk = 1 η b k γ k a 2 b k ,k (8)
and

η b k =   1 - 1 N j∈U 1 1 + 1 γj η b j η b k a 2 b j ,j a 2 b k ,j    (9)
Proof: We formally prove λ k -λk N →∞ ----→ 0 via a contradiction argument as in [START_REF] Sanguinetti | Large system analysis of base station cooperation for power minimization[END_REF], [START_REF] Asgharimoghaddam | Decentralized multi-cell beamforming with QoS guarantees via large system analysis[END_REF] where a sketch of the proof is presented in Appendix A.

The above results give the deterministic equivalents for uplink powers irrespective of channel fading. However, the results still depend on users' locations. Thus, we proceed further by taking advantage of randomness of large scale pathgains. Based on these results, we first give the corresponding deterministic equivalent for η b in concise form irrespective of fading and users' locations. To this end, we make the following assumption on the network setting: Assumption 2. Consider a honeycomb-structured cellular network with Rayleigh fading channel and UEs distributed within the cell radius R of each cell. Assume also that the target SINRs {γ k } are taken from 0 ≤ γ min ≤ γ k ≤ γ max < ∞ with a given probability distribution.

Then, using the Fubini theorem [START_REF] Couillet | Random Matrix Methods for Wireless Communications[END_REF], an asymptotic approximation of η b k in (9) can be computed as follows:

Lemma 1. If Assumptions 1 -2 hold true, then η b -η → 0 almost surely with η = 1 - K b N (ϑ 1 + ϑ 2 ) (10)
where Proof: The proof is given in appendix B. Under uniform user distribution and SINR assignment, we simply have

ϑ 1 = γmax γmin γ 1 + γ F γ (γ)dγ ϑ 2 = b =b π -π γmax γmin R 1 1 1 + a 2 (r) γa 2 (θ,r ) F θ (θ)F r (r)F γ (γ)drdγdθ (
F γ (γ) = 1 γmax-γmin , F θ (θ) = 1
2π and F r (r) = 1 R . However, other settings are also possible by plugging corresponding probability distribution functions into the lemma. The quantity η in Lemma 1 can be plugged into (8) to obtain a deterministic equivalent for λk that depends only on its own path gain value. Given the pathloss dependent uplink powers λk /N , we can derive the deterministic approximation for total power consumption as in the following corollary.

Corollary 1. If Assumptions 1 -2 hold true and a b k ,k is modeled as a 2 b k ,k = 1 r ρ with ρ being the pathloss exponent, the optimal total transmit power for serving K = LK b UEs in the network, under uniform user distribution and SINR assignment, can be approximated by

P t = L η K b N χ = L 1 -K b N (ϑ 1 + ϑ 2 ) K b N χ (12)
where χ =

γ 2 max -γ 2 min 2(γmax-γmin) R ρ+1 -1 R(ρ+1) σ 2 . Proof:
The total power consumption is simply given by

P t = k λ k
N that converges to an expected value (due to strong law of large numbers) given by integration of (8) with respect to r and γ, when a 2 b k ,k = 1 r ρ and η is plugged into (8), and when F γ (γ) = 1 γmax-γmin and F r (r) = 1 R . Now, we have an approximation for total power consumption in asymptotic regime that is given as a function of SINR constraints and basic system parameters. In general, it is difficult to inversely present the SINRs as a function of total power. However, under a simplified single cell setting with γ k = γ min = γ max = γ, we have the optimal SINRs in closed form.

Corollary 2. If Assumptions 1 -2 hold true and a b k ,k is modeled as a 2 b k ,k = 1 r ρ with ρ being the pathloss exponent, the optimal SINR assignment γ in a single cell setting is approximated as

γ = -b + √ b 2 + 4a 2a a = K b N R ρ+1 -1 R(ρ + 1)
σ 2 P TX , b = a + K b N -1. (13)
Proof: Assigning F γ (γ) = δ(γ), where δ(γ) denotes the Dirac delta function, allows a closed form expression for integration in [START_REF] Pennanen | Decentralized coordinated downlink beamforming via primal decomposition[END_REF] which in turn gives an equation based on [START_REF] Shen | Distributed robust multicell coordinated beamforming with imperfect CSI: An ADMM approach[END_REF] whose solution results the optimal SINR assignment.

In the generic multi-cell setting, we can search for the optimal SINR assignment using bisection method. We define the problem as the search for the range of supported SINRs [γ min , γ max ] with a given power budget P TX . The feasibility of the problem for a given range can be easily checked using [START_REF] Shen | Distributed robust multicell coordinated beamforming with imperfect CSI: An ADMM approach[END_REF]. Let us first define an SINR range parameter as γ range . In the bisection algorithm detailed below, we look for the maximum feasible SINR values γ max with a given range γ range . The proposed method is presented in Algorithm 1. Algorithm 1 SINR balancing in general multi-cell scenario.

1: Initialize γ top with sufficiently large value, set γ bottom = 0, and set γ max = γtop-γbottom 2

. Let > 0 be the desired accuracy. 2: loop 3:

Set γ min = max(0, γ max -γ range ).

4:

Get the total power consumption P t from Corollary 1 and return γ max if P TX -P t ≤ . end if 8: end loop Compared to available methods in literature, this algorithm is particularly simple as it does not require solving any optimization problem. Instead, only a single equation is evaluated at each iteration. Moreover, the resulting SINRs are independent of fading and user locations and need to be computed only once for a given set of system parameters.

To conclude this section we present a feasibility condition as the final result that is determined by parameter η. Notice that η in [START_REF] Bjornson | Optimal multiuser transmit beamforming: A difficult problem with a simple solution structure [lecture notes[END_REF] and consequently power is positive and finite provided that K b N (ϑ 1 +ϑ 2 ) < 1. Therefore, a set of SINR targets can be satisfied depending on the sign of η value. Later in numerical results, we investigate the effects of system parameters on the optimal SINR assignment and feasibility condition.

IV. NUMERICAL ANALYSIS

We consider a network with L = 7 cells and assume that UEs are equally distributed among cells. The pathloss function is modeled as a b,k = (d 0 /d b,k ) 2.5 where {d b,k } represents the distance between BS b and UE k and d 0 = 1 m is the reference In order to evaluate the accuracy of the closed form SINR approximation in Corollary 2, a single cell setting is considered where users are dropped randomly on the coverage area and assigned with the same SINR target (γ max = γ min ) to be maximized given the power budget P TX = 10 mW. For each random drop, uplink powers are calculated from Theorem 1, and the resulting sum power k λk /N is used in the bisection search to find the corresponding max-min SINR γ (and the rate calculated as log 2 (1 + γ)) values. The drop specific SINR (rate) values are then averaged over 1000 drops.

Fig. 1 shows the convergence of the averaged max-min rate to the deterministic equivalent given by (13) while system dimensions (total number of users K and number of antennas per BS N ) increase. The small gap in the case with N = 10 indicates the fast convergence rate of the deterministic equivalents. The color bars depict the density of actual max-min rates for random users' drops. It can be seen that the drop specific max-min rates are mainly focused around the deterministic equivalents. Moreover, it can be observed that the variance of actual max-min rates is decreased while the dimensions are increased. Basically the same behavior can be observed also in the multi-cell scenario depicted in Fig. 2. In this case, the deterministic equivalents for max-min rates are found using bisection method (Alg. 1) with relatively high spatial loading ratio K/N = 4.

Next, the effect of system parameters on the deterministic equivalents for max-min rates in multi-cell scenario is investigated. All users are assigned to be served with the same max-min rate (γ max = γ min ) while the total power budget is varied from 10 mW up to 5 W for different values of path-loss exponent ρ. Table I shows the resulted rates. The system parameters dictate the rates via the terms (ϑ 1 + ϑ 2 ) and χ in [START_REF] Shen | Distributed robust multicell coordinated beamforming with imperfect CSI: An ADMM approach[END_REF], where (ϑ 1 + ϑ 2 ) accounts for the effect of interference while χ depicts the signal energy attenuation. In general, increasing pathloss exponent decreases the supported rates as depicted in Table I. As an exception, in the case with P TX = 5 W, a rise in pathloss exponent from 2.5 up to 3.5 results in an increase in the rate. This is due to the dominance of inter-cell interference where larger pathloss reduces the level of interference and results in a higher max-min rate. Another observation is the saturation of the supported maxmin rate as the power budget is increased. In the case with ρ = 2.5, for example, increasing power budget beyond 100 mW improves the rate only marginally as the system becomes interference limited.

As mentioned before, the parameter η gives a feasibility condition for a given set of system parameters. In order to study the effect of system parameters on the feasibility condition, we determine the required ratio of K/N for a given target rate, pathloss exponent and cell radius directly from [START_REF] Shen | Distributed robust multicell coordinated beamforming with imperfect CSI: An ADMM approach[END_REF]. As expected, Table II shows that a smaller ratio is required for satisfying increasing target rates. For very large target rates, every BS should have enough degrees of freedom (asymptotically one degree per user) to handle the increased inter-cell interference. This can be already observed in the table as the rate target is increased. However, higher spatial loading K/N can be allowed for smaller target rates as the interference from more distant cells is reduced. Another observation is that the variation in cell radius does not alter the ratio K/N . This is simply due to the fact that the increasing cell size is compensated with a larger power budget for given V. CONCLUSIONS Large system analysis of max-min SINR problem was considered in multi-cell multi-user network with coordinated beamforming. The results allow one to determine optimal max-min SINR assignment irrespective of users' locations and channel fading given basic system parameters such as cell radius, K, N and pathloss exponent. The derived deterministic equivalents can be utilized for system dimensioning without the need to run system level simulations, i.e, finding the optimal N , K, resource allocation and BS placement.

APPENDIX A PROOF OF THEOREM 1

We begin with rewriting (5) as follows

γ k λ k = 1 N h H b k ,k Σ \k b k + I N -1 h b k ,k (14)
where Theorem 1] we get η b as in the corollary. We still need one more step to show the derived λk with independence assumption is a deterministic equivalent for λ k which can be proved via a contradiction argument as in [START_REF] Sanguinetti | Large system analysis of base station cooperation for power minimization[END_REF], [START_REF] Asgharimoghaddam | Decentralized multi-cell beamforming with QoS guarantees via large system analysis[END_REF] APPENDIX B PROOF OF COROLLARY 1 First we evaluate the error due to plugging 1 1+τj instead of 1 1+τj ςj in (9) as 1 1 + τ j -1 1 + τ j ς j = τ j (ς j -1) (1 + τ j ς j )(1 + τ j ) = ι j [START_REF] Huang | Distributed multicell beamforming with limited intercell coordination[END_REF] where τ j a 2 b j ,j γj a 2 b,j and ς j η bj /η b . Now, writing each term inside the summation of (9) as 1 1+τj + ι j , the error terms {ι j } are averaged out by the summation, and hence, converge to E{ι} due to strong law of large number (SLLN). Bounding |E{ι}| using some inequalities, we get |E{ι}| ≤ E{η b } = η η = 1 (see [START_REF] Kendall | Kendall's Advanced Theory of Statistics[END_REF]), yields equality (c). Thus, we can conclude that the error due to using 1 1+τj instead of 1 1+τj ςj in (9) disappears asymptotically. Now, η b is given in a non-iterative form and we can proceed by averaging channel gain values. Figure 3 shows the position of a user in the serving BS with respect to neighbor cells. Since pathloss depends on distance between the user and the BS, we can average η b in (9) with respect to random entities r and θ, thanks to the strong law of large numbers.

Σ \k b k = j∈U \k λj N h b k ,j h H b k ,j .

  K b UEs served by BS b as U b and the index of the BS associated to UE k as b k . The set of all UEs is represented by U whereas B collects all BS indexes. Under this convention, we define h b,k ∈ C N as the channel from BS b to UE k assuming that h b,k = a 1/2 b,k z b,k where z b,k ∼ CN (0, I N ) represents the small-scale fading with a 2 b,k representing the large scale channel gain (or pathloss) from BS b to UE k. Denoting by v k ∈ C N and p k the normalized precoding vector and transmit power of UE k from its intended BS b k , respectively, the received SINR at user k is given by 1 :

  [START_REF] Pennanen | Decentralized coordinated downlink beamforming via primal decomposition[END_REF] with F x (x) denoting the probability distribution function of random variable x. The number of users per cell K b is assumed to be the same for all BSs. Also, we have r = r 2 + 4d 2 b + 4rd b cos θ, where, d b denotes the distance between BSs b and b.

5 : 6 : 2 ,

 562 if P t > P TX then Set γ top = γ max and γ max = γtop-γbottom else γ bottom = γ max and γ max =

Fig. 1 :

 1 Fig. 1: The rate (b/s/Hz) versus the number of antennas in a single cell scenario with N/K=1, P TX =10 mW, ρ = 2.5 and R = 500 m.

Fig. 2 :

 2 Fig. 2: The rate (b/s/Hz) versus the number of antennas in a multi-cell scenario with K/N=4, P TX =1W, ρ = 2.5 and R = 500 m.

1 h b k ,k - a 2 bN 1 N

 121 Due to dependency of variables {λ j } on channel vectors, it is not possible to apply trace lemma [2, Theorem 3.4] directly. Lets assume for a moment that {λ j } values are independent from channel vectors. Then, using trace lemma [2, Theorem 3.4] yields1 N h H b k ,k Σ \k b k + I N k ,k N T r (Σ \k b k + I N ) -1→ 0 almost surely. Furthermore, applying rank-1 perturbation lemma [2, Theorem 3.9] and using (14) yields γ k λ k -T r (Σ b k + I N ) -1 → 0 almost surely. Finally, from well known random matrix theory results[START_REF] Couillet | Random Matrix Methods for Wireless Communications[END_REF], we have T r(Σ b k + I N ) -1 -η b k N →∞----→ 0 almost surely, where η b k is the Stieltjes transform of Σ b k and at the end from[28,

Fig. 3 :≈

 3 Fig. 3: Location of users in the network in polar coordinate

TABLE I :

 I Maxmin rate vs. power budget and pathloss exponent, R=500 m

	Pbud	10mW 100mW	1W	5W
	ρ = 2.5	2.4	2.69	2.726	2.73
	ρ = 3.5	0.14	0.85	2.32	3.24
	ρ = 4	0.014	0.14	0.54	1.37
			Rates (b/s/Hz/UE)

TABLE II :

 II The feasibility condition versus system parameters

	Target rate (b/s/Hz/UE)	1	2.6 3.4 4.4 5.7
	R=100m	8.5	3.7 2.7	2.2	1.7
	R=1000m	8.5	3.7 2.7	2.2	1.7
			K/N
	target rates.

The objective can be straightforwardly generalized to a weighted minimum SINR maximization where the SINR values are weighted by arbitrary user specific priority factors.

Note that the Lagrange multipliers λ k are scaled by the factor N to ensure that the total power of system is finite, when N grows large.


Jean-Luc Dupouey

Bernard Amiaud

Nathalie Breda

Sandrine Chauchard

Catherine Massonnet

Researcher Stéphan

Ponton Researcher

Yves Lefevre

Vincent Badeau

Patrick Behr

François Geremia

Cyrielle Visine

Yoran Bornot

Maxime Burst

AKNOWLEDGEMENTS

I should first like to thank very much my trainee supervisor, Pierre Montpied, for his availability and pedagogy throughout my internship. His knowledge in ecology, botanic (more particularly mosses) and data analysis allow me to improve my knowledge in forest ecology and to acquire a solid experience in research. More, his constant cheerfulness made my study [START_REF] Husson | Les hommes et la forêt en Lorraine[END_REF]……………………9 Figure 3: Soil carbon content in "coupes affouagères" based on the soil carbon content in the "quart de réserve" (line indicates equality)……………………………………………...……10 Figure 4: Old map (1770) and IGN map of Franconville communal forest………………….11 Figure 5: General plot schema………………………………………………………………..11 Figure 6: Factorial plane 1-2 from correspondence analysis of sites dispersion……………..15 Figure 7: Factorial plane 1-2 from correspondence analysis of sites dispersion according to studies…………………………………………………………………………………………15 Figure 8: Factorial plane 1-2 from correspondence analysis with treatment effect…….…….16 Figure 9: Factorial plane 1-2 from correspondence analysis of sites dispersion…………......17

LIST OF TABLES

INTERNSHIP PRESENTATION

The French National Institute for Agronomic Research (INRA) produces scientific knowledge and works for economic and social innovation in the areas of food, agriculture and the environment. To face climate change, INRA must deal with major issues: prepare worldwide food availability and security by 2050, reduce greenhouse gas emissions from agriculture, and promote an alternative agricultural and forestry practices.

INRA centre from Nancy-Lorraine contains 16 research unities including EEF research unit (Forest Ecology and Eco-physiology) which develops research programs on interactions between environmental factors, ecosystem function, tree growth and on spatial-temporal species distribution.

My internship master is a project proposed by the phyto-ecology team from EEF (Figure 1). The work of this team is to improve the understanding of past and current dynamics of temperate forests. This involves time-and space-integrated studies of the changes of the environment that affect the health, fertility and biodiversity of forest stands.

My internship belongs to the main axis research "Impact of forest history on their current functioning".

INTRODUCTION

Nowadays, fossil energy depletion and global warming get our society on the way to renewable energy: one of these is forest biomass.

Biomass exportation presents a dilemma: it provides a solution for the two major energetic and climatic crises, but it involves land use changes [START_REF] Landmann | BIO 2 : Biomasse et biodiversité forestière. Augmentation de l'utilisation de la biomasse forestière : implications pour la biodiversité et les ressources naturelles[END_REF]. These changes can affect forests ecosystems for the long term which evolve slowly. This context has already been encountered in the past when forest was a vital energy source for local population (firewood) and economic activities (forges, salt works, naval construction...etc.).

A number of studies have already analysed short term biomass exportation effect, but very few focused on the long term effects in changes intensity of past forest management. The case of the "quart de reserve" in north-eastern France is a known situation about past forest management.

During the 18 th and 19 th century, most of the communal forests in the Lorraine region were divided into two parts. One called "coupes affouagères" (CA), was devoted to firewood production. The other one, called "quart de reserve" (QR) was not cut for firewood but kept as a wood reserve for unforeseen expenses and timber production [START_REF] Rochel | Aménagement, mises en réserve et exploitations dans les bois communaux de Lorraine au XVIIIe siècle. La forêt et ses marges[END_REF].

The "Quart de reserve" regulation is a documented historical situation by former management registers and available maps. Using this opportunity, we tested the hypothesis that: higher and more regular biomass removals in CA made long term differences (1740-2014) with QR in forest ecosystem properties. This report is a follow-up to a similar study conducted in 2010 [START_REF] Feiss | Impact du type de traitement sylvicole ancien sur l'état actuel des forêts : le cas du quart en réserve[END_REF] where 34 sites were surveyed on 3 substrates (marly, calcareous and acidic). It appeared that differences between CA and QR depended strongly on the type of substrate. Thus, we decided in our study to focus on marls only. On this substrate, differences in total organic matter content had been observed by [START_REF] Feiss | Impact du type de traitement sylvicole ancien sur l'état actuel des forêts : le cas du quart en réserve[END_REF] but for 14 sites only.

LITTERATURE SYNTHESIS

Forest ecosystem modification by human actions: state of knowledge

During the last 4000-5000 years humans have been important agents affecting landscape composition and dynamics and biological diversity in Europe (Bengtsson et al., 2000). To maintain ecosystem functions in the present forests, it's necessary to understand short and long term forest ecosystem modifications generated by previous human management practices.

Short term impacts of intensive forest biomass exportation

Soil

Several studies showed that some biomass compartments (branches, twigs and foliage) are richer in nutrients. So exportation of these compartments appears as a mineral exportation which can have impacts on soil fertility. More particularly on poor soils which can be acidified (Cacot et al., 2004;Huttl and Schaaf, 1995;Landmann et al., 2009, Ranger and[START_REF] Ranger | Effets prévisibles de l'intensification de la production et des récoltes sur la fertilité des sols de forêt. Le cycle biologique en forêt[END_REF]. Fertility is influenced by soil carbon content (C): organic matter improves soil properties and is a substrate for soil biota that perform decomposition and nutrient cycling [START_REF] Thiffault | Effects of forest biomass harvesting on soil productivity in boreal and temperate forests -A review[END_REF]. A meta analysis showed significant effects of whole-tree harvesting with a decrease of (-6%) for soil C and N [START_REF] Johnson | Effects of forest management on soil C and N storage: meta analysis[END_REF]. A more recent study reinforces this observation with a harvest-induced loss (-30% on average) for C stored in forest floor [START_REF] Nave | Harvest impacts on soil carbon storage in temperate forests[END_REF].

Vegetation

A number of forest species are threatened because of the habitat quality reduction and fragmentation or even habitat loss as a result of forest management activities. The intensification of biomass exportation generally prevents the emergence of latter stages of forest natural cycle. These latter stages are composed by old trees micro-habitats and dead wood which favour a high richness of specialist taxa [START_REF] Hartmann | Negative or positive effects of plantation and intensive forestry on biodiversity : A matter of scale and perspective[END_REF][START_REF] Landmann | BIO 2 : Biomasse et biodiversité forestière. Augmentation de l'utilisation de la biomasse forestière : implications pour la biodiversité et les ressources naturelles[END_REF]Stizia et al., 2012). Forest management can also have positive effects on primarily generalist species (easy dispersion, less sensitive to fragmentation and perturbation, light tolerant), more particularly vascular plants (Bengtsson et al., 2000;[START_REF] Feiss | Impact du type de traitement sylvicole ancien sur l'état actuel des forêts : le cas du quart en réserve[END_REF][START_REF] Paillet | Biodiversity differences between managed and unmanaged forests: meta-analysis of species richness in Europe[END_REF]. As regards slash maintaining, this one is generally favourable to floristic richness but not favourable to vascular plants which have a limited propagation capacity (Decocq et al., 2004;[START_REF] Landmann | BIO 2 : Biomasse et biodiversité forestière. Augmentation de l'utilisation de la biomasse forestière : implications pour la biodiversité et les ressources naturelles[END_REF].

Long term impacts of past land use

Former land state

In historical ecology, a lot of studies focused on the effects of past land use (ancient forest versus former agricultural land). For about two centuries, the French forest progress rapidly with nearly half of its surface was formerly agricultural lands. A distinction is visible in the current landscape: pre-existing forests at the minimum forest (ancient forests) are opposed to recent forests, installed subsequently (Dupouey et al., 2002 a). Ancient forest species were identified as associated to the interiors of ancient forests. Several studies have demonstrated a significant difference in the response of ancient forest plant species compared with other forest plant species for a variety of ecological characteristics (Ellenberg indicators, strategies, life form and phytosociology). Recent forest species are efficient colonists while ancient forest species possess a limited habitat range outside woods. This limitation could be explained by colonization difficulty (seed dispersal, production and inter specific competition) [START_REF] Hermy | An ecological comparison between ancient and other forest plant species of Europe, and the implications for forest conservation[END_REF][START_REF] Peterken | Historical factors affecting the number and distribution of vascular plant species in the woodlands of central Lincolnshire[END_REF][START_REF] Sciama | Floristic and ecological differences between recent and ancient forests growing on non-acidic soils[END_REF].

Furthermore, former agricultural land is linked to long term changes of chemical and structural soil properties which have a key role in the forest structure and composition (Dupouey et al., 2002 b). Superficial horizons of former arable lands display lower organic carbon content and higher base saturation content. Former cultures are enriched in nitrogen caused by nitrogen cycle modifications, which generate the presence of nitrophilous species [START_REF] Dupouey | Identifier les facteurs historiques de vulnérabilité dans la relation sylviculture-biodiversité. Projet FORGECO[END_REF][START_REF] Koerner | Influence des anciennes pratiques agricoles sur la végétation et les sols des forêts reboisées dans le massif vosgien[END_REF].

Former forest practices

Access to historical information is valued to understand historical trajectories and ecosystem processes. Numerous studies have illustrated the case of litter collecting (former practice) which has a strong impact on soil fertility, vegetation composition and tree growth (Burgi and Gimmi, 2007;[START_REF] Gimmi | Land-use and climate change effects in forest compositional trajectories in a dry Central-Alpine valley[END_REF][START_REF] Huffel | Influence de l'enlèvement de la couverture du sol sur la végétation[END_REF]. Experimental studies identified nutrient depletion and reduced acid neutralizing capacity as the most important effects of excessive biomass removal in forest soils [START_REF] Glatzel | The nitrogen status of Austrian forest ecosystems as influenced by atmospheric deposition, biomass harvesting and lateral organomass exchange[END_REF][START_REF] Glatzel | The impact of historic land use and modern forestry on nutrient relations of Central European forest ecosystems[END_REF] and detected long recovery times after abandonment of the practice [START_REF] Glatzel | Historic forest use and its possible implication to recently accelerated tree growth in Central Europe[END_REF]Huttl and Schaaf, 1995). [START_REF] Gimmi | Soil carbon pools in Swiss forests show legacy effects from historic forest litter raking[END_REF] have shown using a biogeochemical model that after 310 years of litter raking soil carbon pools should be reduced by an average of 17 %.

More, a phyto-sociological study in Lorraine forests has identified clearly the influence of past silviculture and actual stand structure, between high forest and coppice with standards, on part of both the ligneous and herbaceous flora (Becker et al., 1979).

To our knowledge no study has been carried out on long term impacts of firewood extraction; historical context of past silviculture in Lorraine is well documented to consider the implementation of the following study.

Historical context of "Lorraine" communal forests

Most of the communal forests in Lorraine are divided into compartments inherited from the former supervision of the State. During the 16 th century, deforestation pressure had led royal order to require the imposition of a third woodland area being kept in reserve in 1561 and then a quarter in 1573. Is in 1669 thanks to the Colbert's reformation that reserve quarter notion is generalized: local communities will have to save a forest quarter for extraordinary cuts while the remainder of the forest was exploited for firewood (Figure 2) [START_REF] Husson | Les hommes et la forêt en Lorraine[END_REF][START_REF] Rochel | Aménagement, mises en réserve et exploitations dans les bois communaux de Lorraine au XVIIIe siècle. La forêt et ses marges[END_REF]. At that time, Lorraine was not yet part of the France kingdom. However, in the 18th century, the Dukes of Lorraine adopted the measure of the reserve and applied it to the woodlands of the duchy [START_REF] Dupouey | Identifier les facteurs historiques de vulnérabilité dans la relation sylviculture-biodiversité. Projet FORGECO[END_REF]. Former registers specify that the "quart de reserve" had to be located far from forest edges and villages, to avoid any pillaging (Degron, 1999 a). Spatial organisation of communal forests was often accompanied by a first forest surveying and mapping (See Annex 1:"Survey maps" 1740 -1790). The forestry code reformation of 1827 has not changed the reserve quarter regulation but during the 19 th century extraordinary cut permissions increased considerably (Degron, 1999 b). From 1880, QR was divided caused by repetition of extraordinary cut (See Annex 1: Subdivision plan for the "quart de reserve"). In the early 20th century, the coppice and coppice with standards, are accused of producing low-quality wood and be expensive in labour. High forest conversion has been established to ensure the demand of timber and the effective division between QR/CA disappeared. It's in 1969 that QR obligations disappear by a public finance law.

First results from a previous study

First study results [START_REF] Feiss | Impact du type de traitement sylvicole ancien sur l'état actuel des forêts : le cas du quart en réserve[END_REF] showed slight effects of former management: the "Quart de reserve" contained more large wood and a lower floristic richness. Organic matter rate showed, only on marly substrates (14 sites), a significant gap between CA and QR with a loss of 14% in CA (Figure 3) [START_REF] Dupouey | Identifier les facteurs historiques de vulnérabilité dans la relation sylviculture-biodiversité. Projet FORGECO[END_REF]. However, this observation was made on 14 pairs of plots only, a too small number to securely conclude at a true difference between CA and QR. The interest of my internship is to extend and to strengthen our belief that silviculture historical effects are in relationship with soil fertility.

Hypotheses raised in the present study

As seen previously, short term impacts from high biomass exportation are well known: to cause a soil fertility loss and to foster species which are opportunistic in open and disturbed areas. Nowadays, reserve quarter regulation represents past ecological footprint from high biomass exportation period. Using the opportunity of this documented historical situation, we tested the following hypotheses: H1: High wood extraction in "coupes affouagères" has resulted in loss of mineral (cations, acidity) and organic fertility (soil carbon and nitrogen content).

H2: Disturbances, more frequent in "coupes affouagères", resulted in colonization by species of disturbed habitats (nitrophilous and heliophilous).

MATERIAL AND METHODS

Study area

The study was conducted in communal forests of the « Plateau Lorrain » in the "Meurthe et Moselle" department, France. The « Plateau Lorrain » is a large natural area relatively flat, clayey with a degraded oceanic climate with continental influence. Geologically, the "Plateau Lorrain" corresponds to Triassic and Liassic aureoles from the eastern part of the Parisian Basin. The typical substrate is marl from Keuper series. The vegetation consists of a temperate deciduous forest composed of oaks on clayey soils and beech forests on loamy and drained soils .The two main species are accompanied by noble hardwoods on rich soils (Brêthes, 1976;IGN, 2014).

Sites selection by map analysis and field prospection

The "Plateau Lorrain" was chosen in our study for its dominant marl substrate and easy access to archives of QR old maps (Figure 4). The "Plateau Lorrain" from "Meurthe et Moselle" has 202 communes with forests. According to the availability of old maps, kept in the ONF archives, we were able to find 120 communes presenting at least one map with a QR representation. The selection work by mapping analysis was achieved through the QGIS software and free access maps (Geoportail, BRGM). Communes for which it was been impossible to retrieve or to relocate the QR were excluded. To observe differences due to past practices forestry, a homogeneous condition is required between each treatment, according to following criteria: a same geological substrate, a similar topography and a homogeneous forest stand according to aerial photography (See Annex 2: Criteria maps). This first phase allowed us to retain 36 potential sites for the field prospection. Then, during field trip potential sites were selected by several criteria: a homogenous stand with the same recent management and a same disturbance degree (roads, thinning…). Finally, we selected 20 sites (See Annex 3: Study sites localisation).

Sampling plan

On each sampling site, a pair of plots was established, one in the "Quart de reserve" and one in the "Coupes affouagères". The plots were established at a minimum distance of 50 m from the boundary between QR and CA. All apparent disturbances were avoided when locating the plot (recent cuttings, paths…).

In each plot, we installed, sampled and inventoried (See Annex 4: Soil survey protocol) : -2 concentric circles ( dendrometric protocol); one with a 9m radius (254 m²), in which were recorded all stems greater than 2.5 cm diameter and another one with a 17.8 m radius (1000 m²), where all stems greater than 22.5 cm diameter were recorded.

-A 400 m² square plot (floristic protocol).

-9 topsoil samples (A horizon) with cylinders (281 cm3) at 9 points in the plot (8 at the corners of the 400m ² plot and 1 at the centre).

Dendrometric protocol

Stems more than 2.5 cm diameter were measured in the smallest circle and those of more than 22.5 cm were inventoried in the largest circle (Figure 5). For each tree was noted: species, circumference at 1.30 m (in cm), status (DO: dominant, codo: co-dominant, do: suppressed, us: understorey) and origin (S: seed or C: clump) (See Annex 5: Dendrometric survey)

Floristic protocol

The floristic survey was conducted in the 400 m² plot (Figure 5); all species were identified and their cover-abundance was estimated using a Braun-Blanquet coefficient like scale (See Annex 6: Floristic survey). Species were separately noted in the following strata:

-the high tree stratum (AH) comprising dominant and co-dominant trees -the low tree stratum (AB) comprising trees suppressed -the shrub stratum (a) comprising species between 0, 3 and 7 m height -the herbaceous stratum (h) comprising all vascular species below 0, 3 m height -the moss stratum (m) for ground growing mosses Data were directly collected in a data base by digital tablet data-entry.

Soil protocol

Humus and soil profile description

In the centre plot, a soil pit of 0.60 m was described by horizon according to visual soil properties. In addition a soil auger observation was realized for deeper horizons. Further, the most representative humus of the plot was identified (See Annex 7: Site description -Soil survey).

Soil sample

For soil analysis, we sampled 9 soil surface cylinders (0 to 5 cm) in the plot. During this manipulation, cylinder must be vertically pushed after humus removal (in flat conditions for a complete filling). These samples will allow us to estimate soil carbon stocks (bulk density x high organic carbon) and to make near infrared and medium infrared reflectance spectroscopy analysis (See Annex 8: Soil laboratory protocol). From each soil pit, 2 soil textural samples (~500g) were collected for sieve analysis: one between 5 and 10 cm and another one between 40 and 50 cm.

Soil analysis: Reflectance spectroscopy in near-IR and medium-IR (NIRS-MIRS) & chemical analysis

Since first results have showed a significant difference for the soil carbon content between CA and QR, we were used a rapid and cheap method (NIRS-MIRS) which has proven its efficiency to discriminate among functional SOM fractions (Cecillon et al., 2010). Near and medium -infrared spectra depend on the number and type of chemical bonds in the material being analysed. The wide variety of these bonds gives rise to 'fingerprints' of soil samples [START_REF] Ertlen | Discriminating between organic matter in soil from grass and forest by near-infrared spectroscopy[END_REF]. So our purpose was to compare the NIR & MIR spectra of soils from QR treatment with those of soils from CA treatment. For the physico-chemical analyses, soil samples will be sent to the Arras INRA centre and results should be received not before September 2014.

Statistical analysis

In this study, all data (soil, floristic and dendrometric) are analysed to test whether differences exist between "coupes affouagères" and "quart de reserve" modalities.

For univariate analyses, a paired t test was used to compare means according to each treatment. Each sample observation is linked to a homologous observation of a second sample. For multivariate analyses, linear models were built: the explanatory variable is the historical management (QR vs CA) treated as fixed effect, covariates are site (random effect) and study (2010 vs 2014, fixed effect). For this model, we supposed that residues follow a normal distribution of same variance. Then, analyses of variance analyse the amount of variance that is contributed to a sample by different factors.

Stand analysis

Circumference data allowed us to calculate relative total basal area according to each treatment, by wood categories (PB: small wood < 70 cm; BM: medium wood >= 70 cm to < 150 cm; GB: big wood >= 150 cm to < 212 cm and TGB: very big wood >=212 cm) and by species. Comparisons between modalities (CA vs QR) are done on relative total basal area means by a paired t test.

Floristic analysis

Species richness

Species richness indicates the species number present in a considered space. Species richness for all floristic species was compared for each treatment (CA/QR) with a paired student test.

Correspondence analysis

The contingency table of presence/absence for sites x species was analysed by correspondence analysis. This multivariate analysis allowed visualizing and summarizing the distribution of sites or species according to qualitative variables (site effect, treatment effect and study effect). To verify the significance of each effect, we fitted a linear model of the correspondence analysis according to different components and we tested effects previously mentioned in an analysis of variance.

Probability test of relative frequencies for each species

According to our knowledge, we do not know an exact test of frequencies comparison, like Fisher's exact test, which could be applied for structured data by pairs in each site. For each species, we applied a test on the ratio between relative frequencies in the two different treatments. Only species observed in at least 4 plots were studied. This test was based on 4 modalities to create a data set codification (Table 1):

Situation

Probability formula Species absent in the site P(!CA ∩ !QR) = (0,0) Species presents in CA, not in QR P(CA ∩ !QR) = (1,0) Species presents in QR, not in CA P(!CA ∩ QR) = (0,1) Species present in both CA and QR P( CA ∩ QR ) = (1,1) With the null hypothesis:

( ) ( ) ( ) ( )
Rejection of the null hypothesis indicates that one of two treatments is favourable for one species. In this test the set of observations (data set codification) can be assumed to be from an independent and identically distributed population. An empirical distribution can be drawn by constructing 9999 resamples of the observed dataset, each of which is obtained by random sampling with replacement from the original dataset (bootstrap technique). Then, we applied a confidence interval (Min, Q2.5, Q50, Q97.5, Max) on this data set. Finally, we eliminated species which were only present in one treatment (Q50 = 0 | Inf) and species which accepted the null hypothesis (Q2.5<=1 & Q97.5>=1).

Ellenberg and Julve coefficients

Soil analysis

Soil properties

To compare soil depth mean, oxidation depth apparition mean and bulk density mean between CA and QR, we have realised a set of paired t-test on these different soil properties.

NIRS-MIRS analysis

Infrared absorption spectra are composed by absorption band intensities at different frequencies. Absorption band intensities are very different to analyse it a standard normal distribution was applied. Then, the first derivative of spectra data was calculated. For data quality:

-Each sample has been analysed by 3 machine acquisition repetitions (to reduce the machine shift effect) then a mean by repetition was calculated.

-CO 2 effect was deleted in removing absorption band corresponding to this compound (MIRS: 1615 to 1640 and NIRS: 3945 to 3986).

-Site effect has been removed; we have centred data by site mean.

NIRS and MIRS data were analysed with a principal component analysis (PCA) to summarize similarities between plots and liaisons between variables. Then, the former management (QR vs CA) was tested with an analysis of variance for each principal component. Finally, significant principal component were used to visualize site distribution according to treatment effect.

All statistical analyses were conducted with R software on the Rstudio interface (http://www.rstudio.com/).

RESULTS

Studies aggregation and sites selection

A correspondence analysis of 20 sites (for species seen in at least 2 plots) from our study and 34 sites from the 2010 study presents sites dispersion by geological substrate (Figure 6). We can see that marly plots (green sites) gather, they are therefore chosen for the analysis selection. This selection associates 14 sites from the 2010 study including 12 marly sites and 2 from other substrates. However, the correspondence analysis from the selection (34 sites) shows a separation between the previous and the present studies, along the second axis (Figure 7). previous study contains Carex remota) and oak seedlings identification (Quercus petraea / Quercus robur).

-The observation period: species were observed during mid-April to late June for the previous study and only during May for our study.

Stand analysis

Comparisons of mean values of the total basal area shows no significant effect from former treatment (51.5 m²/ha for CA and 50.3 m²/ha for QR). This is also the case for basal area percentage of wood diameter categories (Table 2). For species composition (Carpinus betulus, Quercus petraea, Quercus robur, Fagus sylvatica, others), we do not observe any significant effect from treatment on the basal area. Trees species richness does not show any significant difference between "coupes affouagères" and "quart de réserve". More, a variance analysis on treatment effect does not show any preferences from species for a treatment. We have observed 2 to 6 species per plot with a median value of 3 species.

Flora analysis

For all species of all strata from 34 sites, we have observed 137 taxa with 121 taxa for QR treatment and 118 taxa for CA treatment. And by plots, we have sampled 9 to 50 taxa with a median value of 28.5 taxa. Species richness does not show a significant result comparing the two treatments.

Correspondence analysis

A correspondence analysis of the 68 plots (for species observed in at least 5 plots) shows a significant effect from former management (CA or QR) on the herbaceous composition (80 taxa), and only on this stratum (Figure 8). A model of analyse of variance, taken into account site effect then treatment effect, on the axis 1 of the correspondence analysis allow us to conclude that this axis is defined by a strong site effect with a p-value < 10 -11 and a smaller treatment effect with a p-value < 0.05. The axis 1 from correspondence analysis represents 11.5% of the total inertia of the dataset. And the axis 2 represents 9.2% of the total inertia of the dataset.

Sites positions according to these axes are probably due to ecological gradients (Figure 9).

The axis 2 distinguishes sites by studies and is also probably due to a pH gradient: the upper part of the axis contains acidic environment (thick silt) while the lower part contains calcareous environment (Woëvre). The axis 1 is probably due to soil types with the green lot which represents heavy clay soils (Pélosols-RP2008 or Epistagnic Regosols-WRB2006 / Planosols-WRB2006).

For species dispersion (Figure 10), the axis 1 formation is mainly represented by the left part of the axis with: Phyteuma spicatum (PHYspi), Brachypodium sylvaticum (BRAsyl), Carex flacca (CARfla), Cardamine pratensis (CARpra), Ranonculus auricomus (RANaur), Viburnum opulus (VIBopu), Ligustrum vulgare (LIGvul) and Primula elatior (PRIela). While the right part of the axis is less marked with: Galeopsis tetrahit (GALtet), Corylus avellana (CORave), Melica uniflora (MELuni), Milium effusum (MILeff), Quercus petraea (QUEpet) and Prunus avium (PRUavi). Species are especially differentiated by the site effect (axis 1with a p-value <10 -10 and axis 2 with a p-value < 10 -4 ) and a little by the treatment effect (axis 1with a p-value <0.05 and axis 2 with a p-value <0.10) with "coupes affouagères" which would be represented by the left part of the axis 1 and the "quart de reserve" by the other. Furthermore, an intra-site correspondence analysis (wca R function) has been realized in order to mitigate site effects but this one does not show any significant difference between CA and QR.

Probability test of relative frequencies for each species

The test of relative frequencies in the two different treatments shows us that 4 species display a significant difference between CA and QR: Ajuga reptans and Rosa arvensis for the "quart de reserve" and Carex flacca and Viburnum opulus for "coupes affouagères" (Table 3).

Ellenberg's and Julve's coefficients

In our study, means coefficients for each species were calculated with indicator values and former treatments were compared through a paired t-test (with a table of presence/absence).

Soil analysis

Soil properties

A set of paired student test were realized on different criteria between QR and CA:

-Soil depth is homogenous with a median value of 100 cm.

-Oxidation depth apparition is homogenous with a median value of 21 cm.

Bulk density

A paired student test on bulk density mean values from our study (20 sites) for the surface horizon (9 cylinders x 0-5 cm) indicates no significant effect of treatment (Figure 11). These bulk density values will serve to calculate organic matter stocks once we have received carbon content results.

NIRS -MIRS analysis

The analysis of variance of principal component analysis for each principal component shows us 16 principal components significantly different between the CA treatment and the QR treatment ( Factorial planes from the MIRS principal component analysis show us a clear separation between CA and QR (Figure 12).

DISCUSSION -CONCLUSION

Second study results are consistent with those of the first study. Indeed, we observe relatively less pronounced effects of the former forest management. There is a difference in soil composition between the two treatments which according to the first study is due to loss of organic matter in "coupes affouagères". The "coupes affouagères" tends to have preferential plant species to disturbed and open environments while the "quart de reserve" tends to have plant species associated to more closed and stable environments. Stands tend to have slightly more very large wood in "quart de reserve" compared to "coupes affouagères".

Difference of soils organic matter

Authors from the first study have observed a decline in the organic matter content, with an average loss of 14% in "coupes affouagères" compared to the "quart de reserve" [START_REF] Dupouey | Identifier les facteurs historiques de vulnérabilité dans la relation sylviculture-biodiversité. Projet FORGECO[END_REF]. By increasing the sampling of 14 pairs to 34 pairs on marl substrate, we want to consolidate this result here. Classical chemical analysis of soils which we collected (absorbent complex, pH, organic matter, phosphorous) could not be performed that after the end of our training period (by INRA Arras). However, to get some results quickly, we used an innovative technique in the domain: the analysis by infrared spectroscopy. We compared spectra, of medium and near-infrared, from soils subjected to regular biomass export (coupes affouagères) with those of considered as undisturbed soils (quart de réserve).

Principal component analysis of derived of standardized NIR-MIR spectra clearly distinguishes the type of treatment between the two floors. We observe this phenomenon on several principal components. This means that for the portions of the spectra corresponding to these principal components, some chemical components or different proportions of these components in the soil organic matter differ between "coupes affouagères" and the "quart de reserve". These preliminary results are encouraging and indicate that MIR and NIR spectroscopy has real potential to discriminate the state of soil organic matter according to silvicultural past. However, at this stage it is not interpretable. To choose pertinent principal components in both studies, our perspectives are to estimate the prediction error: several resampling techniques are possible. Firstly, the sample must be subdivided in two parts: one to build the model and another one to evaluate model performances. The cross-validation will allow us to directly estimate the error with a prediction model while the bootstrap will allow us to reduce the variance of the estimator increasing replications. Results obtained by standard analyses methods (INRA Arras), are needed to calibrate and validate the NIR and MIR absorption data. This second step may allow us to identify the chemical compounds responsible for soil discrimination observed. For example perhaps is it related to a simple difference of total organic matter content?

In this first phase, the results are consistent with our hypothesis: silvicultural treatments induced differential evolution of organic fertility. This is probably due to the fact that biomass exports were much more frequent and intense in "coupes affouagères" compared to the "quart de reserve". Forest ecosystems are therefore able in the long term in our context (about 150 years) to lose some of their fertility under intensive export of biomass. Carbon stocks in forest soils are slow to change, but their quantity is limited and highly dependent on natural flows of chemical elements and organic matter (ADEME, 2014). Our results confirm that an increase in demand for wood energy should be accompanied by recommendations for preserving mineral and organic forest soil fertility.

Colonization of plant species of disturbed habitats

The extension of the sampling on marly area allows us to see a significant difference of plant communities between "coupes affouagères" and "quarts de réserve". The correspondence analysis of the herbaceous strata shows a first axis of inertia defined by a classical gradient of soil acidity and that explains the site effect highly significant on this axis. When this effect is designated cofactor, we see a trend of the silvicultural past effect: "coupes affouagères" include species from disturbed habitats which prefer high nitrogen content and hydrophilic soils, while the "quart de réserve" has species from stable environments associated to slightly acidic soils.

A probability test of relative frequencies of each species according to former forest management has showed that:

-The "coupes affouagères" favour the presence of Carex flacca and Viburnum opulus; these species are heliophilous and neutroclines. These are species that we find in disturbed environments such as edges, roads and logging [START_REF] Rameau | Flore forestière française : guide écologique illustré[END_REF].

-The "quart de reserve" favours the presence of Ajuga reptans and Rosa arvensis; we do not explain these results since these specie prefer nitrogen-rich environments [START_REF] Rameau | Flore forestière française : guide écologique illustré[END_REF].

Contrary to what might be expected, species richness is not higher in "coupes affouagères".

Our second hypothesis is partially validated, repeated cuts in "coupes affouagères" bring about a colonization of forest species subjected to disturbance. Our results confirm what was observed previously by [START_REF] Saur | Une méthode locale de conversion des Taillis-sous-Futaie en Futaie[END_REF]: the old cuts contain herbaceous species in hydrophilic nature. However, we must consider that from the mid-18th century, all communal forests were converted from coppice-with-standard to high forest. Becker (1979) observed a number of species preferentially associated to stands under conversion (Phyteuma spicatum, Cardamine pratensis, Ranonculus auricomus) that they tend to prefer "coupes affouagères" in our flora analyses.

Preservation of very large wood in the reserve

The role of the "quart de réserve" was to keep a timber capital susceptible to be exploited for extraordinary cuts. Today, we see a historical memory with a trend to have more very large wood (circumference> = 212 cm) in the "quart de reserve" compared to "coupes affouagères". Comparisons of total basal area by diameter classes or by species show no significant differences between "coupes affouagères" and "quart de reserve"; this is due to criteria selection of our sites. The choice of uniform stands between the two treatments was part of these criteria. In addition, the conversion to high forest of former "coupes affouagères" favoured the presence of large timber and homogenized stand structure on all forests areas.

Limitations of the study

All the results presented above show low impacts on forest ecosystem due to extraction of firewood in "coupes affouagères". The main limitation of our study of historical ecology is the information source of the old silvicultural treatment. Indeed it has been difficult to find the localisation plans of communal forests in the 18th and 19th centuries to develop our study. But we have no certainty about the rates and frequencies of sampling biomass under both management methods.

According to several studies in historical biogeography, regulation of the "quart de reserve" was not fully respected:

-From 1840 to 1880, the number of authorisations of deemed extraordinary cuts increased significantly in "quarts de reserve", going from 4 to 20 per year (Degron, 1999).

-A recent study looked at the records of timber hammerings (115 cups) of Lorraine communal forests. First results show that the population of the reserve is rich in large wood but permitted exploitations not leave more than a poor density of high forests [START_REF] Rochel | Aménagement, mises en réserve et exploitations dans les bois communaux de Lorraine au XVIIIe siècle. La forêt et ses marges[END_REF].

More, real duration of these silvicultural practices is not known for studied forests.

Regarding floristic surveys, they have often been made in poor habitats in our study. This does not favour the appearance of a difference in species richness between "coupes affouagères" and the "quart de reserve". Furthermore, aggregation of data flora between the previous study and the present study clearly shows an observervation bias. Several intercalibration exercises on the quality of observation during floristic surveys indicate differences there are sometimes important between surveys of observers. These differences are mainly due to: inaccuracies allocation to a stratum, problems of determination for herbaceous species and mosses, the number of observers and the survey duration (Camaret et al., 2004).

We could talk about a final classical limit: a number of samples too low (68), increasing repetitions we can more easily highlight the differences between former silvicultural managements. But their extent would remain low.

Prospects

These first results are not sufficient to conclude that "coupes affouagères" have only a minimal impact on the organic soil fertility and vegetation. To confirm that there has maybe a net loss of organic matter in the first centimetres of the soil, we have several perspectives. First, the spectra calibration will be an essential step to know compounds and their amounts, which differ according to the treatment effect. Second, chemical analyses of new collected soils will test whether the observed trend in the previous study confirms.

Samples have also been made at different depths: one between 5 and 10cm and a second between 40 and 50 cm. The preparation and analysis of these samples will test the site homogeneity according to soil texture. The sample of 40-50 cm will test the homogeneity of the substrate. In particular, it can be tested if thickness limes is even, or if bias appear between the two treatments.

For more results, a future study could be conducted in similar environments on the impact of salt mine in Lorraine on the forest ecosystem. With the increasing development of salt mines during the Middle-Age, forests have been affected by a strong need for wood energy to extract salt [START_REF] Degron | Continuités et ruptures dans la gestion des bois communaux au XIXe siècle. Quelques exemples lorrains[END_REF]. The study would compare the wood previously destined to saline work, widely over-exploited during the 18 th century with those spared from these samples.

Finally, it would be interesting to organize a project that would combine researchers and historians. This project would allow us to acquire detailed information on the amount of wood really realized by the former treatment information, through long and meticulous work of historians, of research and analysis of timber hammering records of communal forests. Then, forest researchers could sample and analyse forests for which samples and dates of former silvicultural treatments were known. BÜRGI, M and GIMMI, U, 2007, Three objectives of historical ecology: the case of litter collecting in Central European forests. Landscape Ecology.2007. Vol. 22, no. S1, p. 77-87. CACOT, E, CHARNET, F, RANGER, J and VIEBAN, S, 2004, Impact du prélèvement des rémanents en forêt. Fiches information forêt, Afocel. 2004. No. 1, Fiche no. 686, 6

BIBLIOGRAPHIC

Soil and humus description:

In the principal plot, a pit profile of 0, 60 m is described with information horizon referring to soil properties. In complement a soil auger observation is realized for depth horizons: The more representative humus of the plot is described:

-Oln : organic layer with new litter ( very little transformation) -Olt : organic layer with transition litter ( loose material) -Olv : organic layer with old litter ( bleached, softened and stuck) -OF: fragmented organic layer ( fragments and organic matter particle) -OH : organic layer with humification processes -Earthworms castings ( 0 : nothing, 1 : low, 2 : medium, 3 : high) Soil sample:

For soil analysis, we sampled 9 soil surface cylinders (0 to 5 cm) in the plot. During this manipulation, cylinder must be vertically pushed after humus removal (in flat conditions for a complete filling). From each soil pit, 2 soil textural samples (~500g) were collected for sieve and organic matter analysis: one between 5 and 10 cm and another one between 40 and 50 cm.

Dendrometric protocol:

Stems more than 2.5 cm diameter were measured in the smallest circle and those of more than 22.5 cm were inventoried in the largest circle (Figure 3). For each tree was noted: species, circumference at 1.30 m (in cm), status (DO: dominant, codo: co-dominant, do: suppressed, us: understorey) and origin (S: seed or C: clump)

Florisitc protocol:

The floristic survey was conducted in the 400 m² plot; all species were identified and their cover-abundance was estimated using a Braun-Blanquet coefficient like scale. Species were separately noted in the following strata: -the high tree stratum (AH) comprising dominant trees -the low tree stratum (AB) comprising trees suppressed -the shrub stratum (a) comprising ligneous species between 0, 3 and 7 m height -the herbaceous stratum (h) comprising all vascular species below 0, 3 m height -the moss stratum (m) for ground growing mosses

Figure 10 :

 10 Figure2: Scene harvest firewood during the 20th century[START_REF] Husson | Les hommes et la forêt en Lorraine[END_REF]……………………9 Figure3: Soil carbon content in "coupes affouagères" based on the soil carbon content in the "quart de réserve" (line indicates equality)……………………………………………...……10 Figure4: Old map (1770) and IGN map of Franconville communal forest………………….11 Figure5: General plot schema………………………………………………………………..11 Figure6: Factorial plane 1-2 from correspondence analysis of sites dispersion……………..15 Figure7: Factorial plane 1-2 from correspondence analysis of sites dispersion according to studies…………………………………………………………………………………………15 Figure8: Factorial plane 1-2 from correspondence analysis with treatment effect…….…….16 Figure9: Factorial plane 1-2 from correspondence analysis of sites dispersion…………......17 Figure10: Factorial plane 1-2 from correspondence analysis of species repartition…………17 Figure11: Mean values of bulk density between CA and QR………………………….……19 Figure12: Principal component analysis for MIRS data………………………………..……20

Figure 1 :

 1 Figure 1: Organization chart

Figure 2 :

 2 Figure 2 : Scene harvest firewood during the 20 th century[START_REF] Husson | Les hommes et la forêt en Lorraine[END_REF]

Figure 3 :

 3 Figure 3: Soil carbon content in "coupes affouagères" based on the soil carbon content in the "quart de réserve" (line indicates equality).

Figure 4 :

 4 Figure 4: Old map (1770) and IGN map of Franconville communal forest.

Figure 5 :

 5 Figure 5 : General plot schema

  Ellenberg and Julve indicator values express the ecological behaviour of species along a scale of 9 levels. Indicator values represent environmental factors. In our study we have tested the following indicator values: Light, Temperature, Continental, Edaphic humidity//philippe.julve.pagesperso-orange.fr/). Means indicator value by sample, for environment factors previously mentioned were compared between CA and QR by a paired t test.

Figure 6 :

 6 Figure 6: Factorial plane 1-2 from correspondence analysis of sites dispersion. All sites dispersion for top right graphic. Zoom of marly sites dispersion for general graphic. (Legend; red=calcareous, green=marly and blue=acidic).

  : the previous study chosen open habitats where the flora expresses more while the present study chosen more closed habitats. -An observation bias for Carex identification (the present study contains especially Carex umbrosa and Carex pilulifera while the

Figure 7 :

 7 Figure 7: Factorial plane 1-2 from correspondence analysis of sites dispersion according to studies (Legend; red= 2010 study, blue = 2014 study).

Figure 8 :

 8 Figure 8: Factorial plane 1-2 from correspondence analysis with treatment effect (red = QR, blue = CA).

Figure 10 :

 10 Figure 10: Factorial plane 1-2 from correspondence analysis of species repartition.

Figure 9 :

 9 Figure 9: Factorial plane 1-2 from correspondence analysis of sites dispersion.

Figure 11 :

 11 Figure 11: Mean values of bulk density between CA and QR.

Figure 12 :

 12 Figure 12: Left figure, factorial plane 8-12 of principal component analysis from 20 sites of 2014 study. Right figure, factorial plane 6-8 of principal component analysis from 14 sites of 2010 study.

  LIST ADEME, 2014, Carbone organique des sols : L'énergie de l'agro-écologie, une solution pour le climat, Collectivités territoriales et monde agricole : connaître et agris, ADEME, 30 p. BECKER, M, 1979, Influence du traitement sylvicole sur la flore forestière : cas de la futaie et du taillis-sous-futaie. Vegetatio. Vol.40, no.3, p. 155-161. BENGTSSON, J, NILSSON, SG, FRANC, A and MENOZZI, P, 2000, Biodiversity, disturbances, ecosystem function and management of European forests. 2000. P. 39-50. BERTHES, Alain, 1976, CATALOGUE DES STATIONS FORESTIERES DU PLATEAU LORRAIN. ONF & INRA. 1976.

  p. CAMARET, S, BOURJOT, L, DOBREMEZ J.-F (coordinateurs), 2004, Suivi de la composition floristique des placettes du réseau (1994/95-2000) et élaboration d'un programme d'assurance qualité intensif. Office National des Forêts, Direction Technique, 86 p. CECILLON, L. and BRUN, J.-J., 2007, Near-infrared spectroscopy (NIRS): a practical tool for the assessment of soil carbon and nitrogen budget. COST action 639: greenhouse -gas budget of soils under changing climate and land use. p. 103-110. DECOCQ, G, AUBERT, M, DUPONT, F, ALARD, D, SAGUEZ, R, WATTEZ-FRANGER, A, FOUCAULT, B, DELELIS-DUSOLLIER, A and BARDAT, J, 2004, Plant diversity in a ANNEX 1: Topographic map of Franconville's forest (1770) and subdivision map of the "quart de réserve" of the Franconville forest (1886).

  IGN map, Ordnance map, Geologic map and aerial photography -Franconville's forest.

-

  Horizon : Name or number -Depth horizon : top to floor in cm -Transition : net or progressive / linear or sinuous (n,p / l,s) -Color : horizon color with Munsell code (wet) -Texture : Silt (L) , clay (A) and sand (S) -Structure : lumpy, particulate, polyhedral or compact ( grum, part, poly or comp) -HCL test (0 : absent, 1 : reaction) -Rooting : abundance in % Rooting type ( VT : very thin <0,2 cm, T : thin 0,2-0,5 cm, M : medium 0,5-2 cm, L : large > 2 cm ) -Coarse elements : abundance in % Coarse elements type (G : gravel 0,2-2 cm , SS : small stones 2-6 cm, S : stones 6-20 cm, BS : block of stone > 20cm ) -Hydromorphy : abundance in % Hydromorphy type(Deco : decoloration, Ox :oxidation stains, Conc : concretion stains)

Table 1 :

 1 Occurrence modalities of species according to treatment……………..……………13

Table 2 :

 2 Wood categories percentages according to former treatment……………………….16

Table 3 :

 3 Quantiles logarithm values, number and ratio logarithm value for each specie...…..18 Table 4: Ellenberg's and Julve's coefficients according to former treatment……………..…18 Table 5: P-value of MIRS and NIRS principal components according to studies…………...19

	LIST OF FIGURES
	Figure 1: Organization chart…………………………………………………………………...6
	Figure 2: Scene harvest firewood during the 20th century

Table 1 : Occurrence modalities of species according to treatment.

 1

Table 2 : Wood categories percentages according to former treatment.

 2

	Paired t-test according to former treatment ( mean values in percentage)
	Wood Category	CA %	QR %	p-value
	Total basal area	50.6	49.4	0.56
	PB (small wood)	15.2	13.7	0.48
	BM ( medium wood)	32	32.6	0.85
	GB (big wood)	38.2	31.1	0.26
	TGB (very big wood)	21.8	28.9	0.28

Table 4 : Ellenberg's and Julve's coefficients according to former treatment.

 4

	Specie	Min	Q2.5	Q50	Q97.5	Max	Number Ratio
	CARfla	-Inf	-1.79	-0.69	-0.18	0	10	0.5
	VIBopu	-Inf	-1.28	-0.56	-0.09	0.4	15	0.6
	ROSarv	-0.1	0.04	0.26	0.54	0.85	27	1.3
	AJUrep	0	0.51	1.39 Inf	Inf	8	4

Not any difference was observed between « coupes affouagères » and « quart de réserve » (Table

4

).

Table 3 : Quantiles logarithm values, number and ratio logarithm value for each specie.

 3

Table 5

 5

		).
		2014 Study			2010 Study
	MIRS	NIRS	MIRS	NIRS
	Principal	p-value Principal	p-value	Principal	p-value Principal	p-value
	component		component		component		component
	Axis 8 Axis 12	< 0.001 < 10 -7	Axis 2 Axis 4	< 0.01 < 0.001	Axis 3 Axis 5	< 0.05 < 0.05	Axis 3 Axis 4	<0.001 <0.05
			Axis 5 Axis 7	< 0.05 < 0.01	Axis 6 Axis 8	< 0.05 < 10 -15	Axis 6 Axis 7	<0.05 < 10 -4
			Axis 9	< 0.05
			Axis 10	< 0.001

Table 5 : P-value of MIRS and NIRS principal components according to studies.

 5

BULK DENSITY

In laboratory, the sample is sieved (2mm) and the fine fraction is dried and weighted.

For coarse and root elements, volume is calculated by water displacement based on Archimedes principle.

The density of the sample is the ratio of the mass of dry sieved soil at the exact volume of the cylinders taken less the volume of coarse elements and roots. Its bulk density is expressed in g/cm3.

NIRS-MIRS (Near Infrared and medium infrared reflectance spectroscopy)

In laboratory, soil samples are dry-sieved (2 mm), ground (using a soil crusher with rotary rings during 1 minute) to obtain a homogeneous powder and stored in polypropylene boxes. The day before analysis, powders are oven-dried at 60°C during the night to homogenize moisture and kept in a desiccator before analysis.

From 1 g to 25 g of soil is required for NIR-MIR analysis, depending on the NIR-MIR analyser. Spectra consist of absorbance (absorbance = log [1 / reflectance]) values from 1000 to 2500 nm (Cécillon et al., 2010). All our soil samples were analysed by Fourier transform infrared spectroscopy (FTIR) with a diffuse reflection (Figure 1) (Spectrometer: BRUKER / Spectroscopy software: OPUS). Each spectrum is the average of 16 scans with a spectral resolution of 4 cm -1 .

ABSTRACT

Nowadays, fossil energy depletion and global warming get our society on the way to renewable energetic strategies: one of these is forest biomass. In north-eastern France, most of the communal forests were divided into two parts during the 18 th and 19 th century. One called "coupes affouagères" (CA), was devoted to firewood production. The other one, called "quart de réserve" (QR) was kept as a wood reserve. Using this opportunity, we tested the hypothesis that: higher and more regular biomass removals in CA made long term differences with QR in forest ecosystem properties. To observe differences, we studied stand structure, vegetal communities and soil organic matter. Results show relatively less pronounced effects of the former silvicultural treatment. There is a difference in soil fertility between the two treatments with a loss of organic matter in CA. We observe a trend in "coupes affouagères" to be colonizing by plant species of disturbed habitats. And stand structure tends to have slightly more very large wood in the QR. To conclude, an increase in demand for wood energy should be accompanied by recommendations for preserving mineral and organic forest soil fertility.

Impacts à long terme de la gestion forestière sur l'écosystème forestier : le cas du "Quart de réserve" dans le Nord-Est de la France RESUME De nos jours, l'épuisement des énergies fossiles et le réchauffement climatique ont amené notre société à utiliser des énergies renouvelables : l'une d'entre elle est la biomasse forestière. Dans le Nord-Est de la France, la plupart des forêts communales ont été divisées en deux parties pendant le 18 ème et le 19 ème siècle. L'une d'elle nommée « coupes affouagères »(CA), été dévolue à la production de bois de chauffage. La seconde, appelée « quart de réserve »(QR) été maintenue en tant que réserve de bois. Face à cette opportunité, nous avons testé l'hypothèse suivante : des prélèvements réguliers et intenses de biomasse dans les coupes affouagères ont provoqué sur le long terme (1740-2014) des différences avec le quart de réserve pour certaines propriétés de l'écosystème forestier. Pour observer ces différences, nous avons étudié la structure du peuplement, les communautés végétales et la matière organique du sol. L'ensemble des résultats montrent des effets peu prononcés selon l'ancien traitement sylvicole. Il existe une différence dans la composition du sol entre les deux traitements. Nous observons une tendance dans les anciennes coupes à être colonisées par des espèces de milieux perturbés. Et les peuplements forestiers ont tendance à avoir légèrement plus de très gros bois dans leur quart de réserve. Pour conclure, la demande croissante en bois énergie devrait être accompagnée de recommandations pour préserver la fertilité minérale et organique des sols.