P L Combettes

L Condat

J.-C Pesquet

B C Vũ

A FORWARD-BACKWARD VIEW OF SOME PRIMAL-DUAL OPTIMIZATION METHODS IN IMAGE RECOVERY

Keywords: convex optimization, duality, parallel computing, proximal algorithm, variational methods, image recovery

A wide array of image recovery problems can be abstracted into the problem of minimizing a sum of composite convex functions in a Hilbert space. To solve such problems, primal-dual proximal approaches have been developed which provide efficient solutions to large-scale optimization problems. The objective of this paper is to show that a number of existing algorithms can be derived from a general form of the forward-backward algorithm applied in a suitable product space. Our approach also allows us to develop useful extensions of existing algorithms by introducing a variable metric. An illustration to image restoration is provided.

INTRODUCTION

Many image recovery problems can be formulated in Hilbert spaces H and (Gi) 1 i m as structured optimization problems of the form

minimize x∈H m i=1 gi(Lix), (1)
where, for every i ∈ {1, . . . , m}, gi is a proper lower semicontinuous convex function from Gi to ]-∞, +∞] and Li is a bounded linear operator from H to Gi. For example, the functions (gi • Li) 1 i m may model data fidelity terms, smooth or nonsmooth measures of regularity, or hard constraints on the solution. In recent years, many algorithms have been developed to solve such a problem by taking advantage of recent advances in convex optimization, especially in the development of proximal tools (see [START_REF]Proximal splitting methods in signal processing[END_REF][START_REF] Sra | Optimization for Machine Learning[END_REF] and the references therein). In image processing, however, solving such a problem still poses a number of conceptual and numerical challenges. First of all, one often looks for methods which have the ability to split the problem by activating each of the functions through elementary processing steps which can be computed in parallel. This makes it possible to reduce the complexity of the original problem and to benefit from existing parallel computing architectures. Secondly, it is often useful to design algorithms which can exploit, in a flexible manner, the structure of the problem. In particular, some of the functions may be Lipschitz differentiable in which case they should be exploited through their gradient rather than through their proximity operator, which is usually harder to implement (examples of proximity operators with closed-form expression can be found in [START_REF] Chaux | A variational formulation for frame-based inverse problems[END_REF][START_REF]Proximal splitting methods in signal processing[END_REF]). In some problems, the functions (gi) 1 i m can be expressed as the infimal convolution of simpler functions (see [START_REF] Combettes | Systems of structured monotone inclusions: duality, algorithms, and applications[END_REF] and the references therein). Last but not least, in image recovery, the operators (Li) 1 i m may be of very large size so that their inversions are costly (e.g., in reconstruction problems). Finding algorithms which do not require to perform inversions of these operators is thus of paramount importance. Note that all the existing convex optimization algorithms do not have these desirable properties. For example, the Alternating Direction Method of Multipliers (ADMM) [START_REF] Fortin | Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems[END_REF][START_REF] Figueiredo | Deconvolution of Poissonian images using variable splitting and augmented Lagrangian optimization[END_REF][START_REF] Goldstein | The split Bregman method for ℓ1regularized problems[END_REF] requires a stringent assumption of invertibility of the involved linear operator. Parallel versions of ADMM [START_REF] Setzer | Deblurring Poissonian images by split Bregman techniques[END_REF] and related Parallel Proximal Algorithm (PPXA) [START_REF] Combettes | A proximal decomposition method for solving convex variational inverse problems[END_REF][START_REF] Pesquet | A parallel inertial proximal optimization method[END_REF] usually necessitate a linear inversion to be performed at each iteration. Also, early primal-dual algorithms [START_REF] Briceño-Arias | A monotone + skew splitting model for composite monotone inclusions in duality[END_REF][START_REF] Chambolle | A first-order primal-dual algorithm for convex problems with applications to imaging[END_REF][START_REF] Chen | A proximal-based decomposition method for convex minimization problems[END_REF][START_REF] Combettes | Dualization of signal recovery problems[END_REF][START_REF] Esser | A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science[END_REF][START_REF] He | Convergence analysis of primal-dual algorithms for a saddle-point problem: from contraction perspective[END_REF] did not make it possible to handle smooth functions through their gradients. Only recently, have primal-dual methods been proposed with this feature. Such work was initiated in [START_REF]Primal-dual splitting algorithm for solving inclusions with mixtures of composite, Lipschitzian, and parallel-sum type monotone operators[END_REF] in the line of [START_REF] Briceño-Arias | A monotone + skew splitting model for composite monotone inclusions in duality[END_REF] and subsequent developments can be found in [START_REF] Becker | An algorithm for splitting parallel sums of linearly composed monotone operators, with applications to signal recovery[END_REF][START_REF] Bot | Convergence analysis for a primal-dual monotone + skew splitting algorithm with applications to total variation minimization[END_REF][START_REF] Chen | A primal-dual fixed point algorithm for convex separable minimization with applications to image restoration[END_REF][START_REF] Combettes | Systems of structured monotone inclusions: duality, algorithms, and applications[END_REF][START_REF] Condat | A primal-dual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms[END_REF][START_REF] Repetti | A penalized weighted least squares approach for restoring data corrupted with signal-dependent noise[END_REF][START_REF] Vũ | A splitting algorithm for dual monotone inclusions involving cocoercive operators[END_REF]. As will be seen in the present paper, another advantage of these approaches is that they can be coupled with variable metric strategies which can potentially accelerate their convergence.

In Section 2, we provide some background on convex analysis and monotone operator theory. In Section 3, we introduce a general form of the forward-backward algorithm which uses a variable metric. This algorithm is employed in Section 4 to develop a versatile family of primal-dual proximal methods. Several particular instances of this framework are discussed. Finally, we provide illustrating numerical results in Section 5.

NOTATION AND BACKGROUND

Monotone operator theory [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert Spaces[END_REF] provides a both insightful and elegant framework for dealing with convex optimization problems and developing new solution algorithms that could not be devised using purely variational tools. We summarize a number of related concepts that will be needed.

Throughout, H, G, and (Gi) 1 i m are real Hilbert spaces. We denote the scalar product of a Hilbert space by • | • and the associated norm by • . The symbol ⇀ denotes weak convergence, 1and Id denotes the identity operator. We denote by B (H, G) the space of bounded linear operators from H to G, we set S (H) = L ∈ B (H, H) | L = L * , where L * denotes the adjoint of L. The Loewner partial ordering on S (H) is denoted by . For every α ∈ [0, +∞[, we set Pα(H) = U ∈ S (H) | U α Id , and we denote by √ U the square root of U ∈ Pα(H). Moreover, for every U ∈ Pα(H) and α > 0, we define the norm x U = U x | x . We denote by G = G1 ⊕ • • • ⊕ Gm the Hilbert direct sum of the Hilbert spaces (Gi) 1 i m , i.e., their product space equipped with the scalar product : (x, y) → m i=1 xi | yi where x = (xi) 1 i m and y = (yi) 1 i m denote generic elements in G.

Let A : H → 2 H be a set-valued operator. We denote by gra

A = (x, u) ∈ H × H | u ∈ Ax the graph of A, by zer A = x ∈ H | 0 ∈ Ax the set of zeros of A, and by ran A = u ∈ H | (∃ x ∈ H) u ∈ Ax its range. The inverse of A is A -1 : H → 2 H : u → x ∈ H | u ∈ Ax , and the resolvent of A is JA = (Id +A) -1 . Moreover, A is monotone if (∀(x, y) ∈ H × H)(∀(u, v) ∈ Ax × Ay) x -y | u -v 0, (2)
and maximally monotone if it is monotone and there exists no monotone operator B :

H → 2 H such that gra A ⊂ gra B and A = B. An operator B : H → H is β-cocoercive for some β ∈ ]0, +∞[ if (∀x ∈ H)(∀y ∈ H) x -y | Bx -By β Bx -By 2 . ( 3
)
The conjugate of a function f :

H → ]-∞, +∞] is f * : H → [-∞, +∞] : u → sup x∈H x | u -f (x) , (4)
and the infimal convolution of f with g :

H → ]-∞, +∞] is f g : H → [-∞, +∞] : x → inf y∈H f (y) + g(x -y) . (5)
The class of lower semicontinuous convex functions

f : H → ]-∞, +∞] such that dom f = x ∈ H | f (x) < +∞ = ∅ is denoted by Γ0(H). If f ∈ Γ0(H), then f * ∈ Γ0(H)
and the subdifferential of f is the maximally monotone operator

∂f : H → 2 H x → u ∈ H | (∀y ∈ H) y -x | u + f (x) f (y) . ( 6
)
Let U ∈ Pα(H) for some α ∈ ]0, +∞[. The proximity operator of f ∈ Γ0(H) relative to the metric induced by U is [22, Section XV.4]

prox U f : H → H : x → argmin y∈H f (y) + 1 2 x -y 2 U . (7)
When U = Id, we retrieve the standard definition of the proximity operator [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert Spaces[END_REF][START_REF] Moreau | Proximité et dualité dans un espace hilbertien[END_REF]. Let C be a nonempty subset of H. The indicator function of C is defined on H as

ιC : x → 0, if x ∈ C; +∞, if x / ∈ C. (8)
Finally, ℓ 1 + (N) denotes the set of summable sequences in [0, +∞[.

A GENERAL FORM OF FORWARD-BACKWARD ALGORITHM

Optimization problems can often be reduced to finding a zero of a sum of two maximally monotone operators A and B acting on H. When B is cocoercive (see (3)), a useful algorithm to solve this problem is the forward-backward algorithm, which can be formulated in a general form involving a variable metric as shown in the next result. In the next section, we shall see how a judicious use of this result allows us to derive a variety of flexible convex optimization algorithms.

A VARIABLE METRIC PRIMAL-DUAL METHOD

Formulation

A wide array of optimization problems encountered in image processing are instances of the following one, which was first investigated in [START_REF]Primal-dual splitting algorithm for solving inclusions with mixtures of composite, Lipschitzian, and parallel-sum type monotone operators[END_REF] and can be viewed as a more structured version of the minimization problem in (1): Problem 4.1 Let z ∈ H, let m be a strictly positive integer, let f ∈ Γ0(H), and let h : H → R be convex and differentiable with a Lipschitzian gradient. For every i ∈ {1, . . . , m}, let ri ∈ Gi, let gi ∈ Γ0(Gi), let ℓi ∈ Γ0(Gi) be strongly convex, 2 and suppose that 0 = Li ∈ B (H, Gi). Suppose that

z ∈ ran ∂f + m i=1 L * i (∂gi ∂ℓi)(Li • -ri) + ∇h . (11)
Consider the problem

minimize x∈H f (x) + m i=1 (gi ℓi)(Lix -ri) + h(x) -x | z , (12)
and the dual problem

minimize v 1 ∈G 1 ,...,vm∈Gm f * h * z - m i=1 L * i vi + m i=1 g * i (vi) + ℓ * i (vi) + vi | ri . ( 13
)
Note that in the special case when ℓi = ι {0} , gi ℓi reduces to gi in [START_REF]Proximal splitting methods in signal processing[END_REF].

Let us now examine how Problem 4.1 can be reformulated from the standpoint of monotone operators. To this end, let us define g ∈ Γ0(G), ℓ ∈ Γ0(G) and L ∈ B (H, G) by

g : v → m i=1 gi(vi), ℓ : v → m i=1 ℓi(vi)
and L : x → (L1x, . . . , Lmx).

Let us now introduce the product space K = H⊕G and the operators

A : K → 2 K (x, v) → (∂f (x) -z + L * v) × (-Lx + ∂g * (v) + r) (15)
and

B : K → K (x, v) → ∇h(x), ∇ℓ * (v) . (16)
The operator A can be shown to be maximally monotone,whereas B is cocoercive. A key observation in this context is that, if there exists (x, v) ∈ K such that (x, v) ∈ zer(A + B), then (x, v) is a pair of primal-dual solutions to Problem 4.1 [START_REF]Primal-dual splitting algorithm for solving inclusions with mixtures of composite, Lipschitzian, and parallel-sum type monotone operators[END_REF]. This connection with the construction for a zero of A + B makes it possible to apply a forward-backward algorithm as discussed in Section 3, by using a linear operator V n ∈ B (K, K) to change the metric at each iteration n. Depending on the form of this operator various algorithms can be obtained.

A first class of primal-dual algorithms

Let α ∈ ]0, +∞[, let (Un) n∈N be a sequence in Pα(H) such that (∀n ∈ N) Un+1 Un. For every i ∈ {1, . . . , m}, let (Ui,n) n∈N be a sequence in Pα(Gi) such that (∀n ∈ N) Ui,n+1 Ui,n. A first possible choice for (V n)n∈N is given by

(∀n ∈ N) V -1 n : (x, v) → (U -1 n x -L * v, -Lx + U -1 n v) (17)
where

U n : G → G : (v1, . . . , vm) → (U1,nv1, . . . , Um,nvm). ( 18
)
The following result constitutes a direct extension of [14, Example 6.4]: Proposition 4.2 Let x0 ∈ H, and let (an) n∈N and (cn) n∈N be absolutely summable sequences in H. For every i ∈ {1, . . . , m}, let vi,0 ∈ Gi, let (bi,n) n∈N and (di,n) n∈N be absolutely summable sequences in Gi. For every n ∈ N,

let µn ∈ ]0, +∞[ be a Lipschitz constant of U 1/2 n • ∇h • U 1/2 n and, for every i ∈ {1, . . . , m}, let νi,n ∈ ]0, +∞[ be a Lipschitz constant of U 1/2 i,n • ∇ℓ * i • U 1/2 i,n . Let (λn) n∈N be a sequence in ]0, 1] such that inf n∈N λn > 0. For every n ∈ N, set δn = m i=1 Ui,nLi √ Un 2 -1/2 -1, ( 19
)
and suppose that

inf n∈N δn (1 + δn) max{µn, ν1,n, . . . , νm,n} > 1 2 . ( 20
)

Set

For n = 0, 1, . . .

                  pn = prox U -1 n f xn -Un m i=1 L * i vi,n + ∇h(xn) +cn -z + an yn = 2pn -xn xn+1 = xn + λn(pn -xn) For i = 1, . . . , m       qi,n = prox U -1 i,n g * i vi,n + Ui,n Liyn -∇ℓ * i (vi,n) -di,n -ri + bi,n vi,n+1 = vi,n + λn(qi,n -vi,n). ( 21
)
Then (xn) n∈N converges weakly to a solution to [START_REF]Proximal splitting methods in signal processing[END_REF], for every i ∈ {1, . . . , m} (vi,n) n∈N converges weakly to some vi ∈ Gi, and (v1, . . . , vm) is a solution to [START_REF]Primal-dual splitting algorithm for solving inclusions with mixtures of composite, Lipschitzian, and parallel-sum type monotone operators[END_REF].

In the special case when Un ≡ τ Id with τ ∈ ]0, +∞[ and, for every i ∈ {1, . . . , m}, Ui,n ≡ σi Id with σi ∈ ]0, +∞[, we recover the parallel algorithm proposed in [START_REF] Vũ | A splitting algorithm for dual monotone inclusions involving cocoercive operators[END_REF]. Variants of this algorithm where, for every i ∈ {1, . . . , m}, ℓi = ι {0} are also investigated in [START_REF] Condat | A primal-dual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms[END_REF]. In this case, less restrictive assumptions on the choice of (τ, σ1, . . . , σm) can be made. Note that this algorithm itself can be viewed as a generalization of the algorithm which constitutes the main topic of [START_REF] Chambolle | A first-order primal-dual algorithm for convex problems with applications to imaging[END_REF][START_REF] Esser | A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science[END_REF][START_REF] He | Convergence analysis of primal-dual algorithms for a saddle-point problem: from contraction perspective[END_REF] (designated by some authors as PDHG). A preconditioned version of this algorithm was proposed in [START_REF] Pock | Diagonal preconditioning for first order primal-dual algorithms in convex optimization[END_REF] corresponding to the case when m = 1, (∀n ∈ N) Un and U1,n are constant matrices, and no error term is taken into account. Algorithm (21) when, for every n ∈ N, λn ≡ 1, Un and (Ui,n) 1 i m are diagonal matrices, h = 0, and (∀i ∈ {1, . . . , m}) ℓi = ι {0} appears also to be closely related to the adaptive method in [START_REF] Goldstein | Adaptive primal-dual hybrid gradient methods for saddle-point problems[END_REF].

A second class of primal-dual algorithms

Let α ∈ ]0, +∞[, let (Un) n∈N be a sequence in Pα(H) such that (∀n ∈ N) Un+1 Un. For every i ∈ {1, . . . , m}, let (Ui,n) n∈N be a sequence in Pα(Gi) such that (∀n ∈ N) Ui,n+1 Ui,n. A second possible choice for (V n)n∈N is given by the following diagonal form:

(∀n ∈ N) V -1 n : (x, v) → U -1 n x, ( U -1 n -LUnL * )v (22)
where U n is given by [START_REF] Fortin | Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems[END_REF]. The following result can then be deduced from Theorem 3.1. Its proof is skipped due to the lack of space. Proposition 4.3 Let x0 ∈ H, and let (cn) n∈N be an absolutely summable sequence in H. For every i ∈ {1, . . . , m}, let vi,0 ∈ Gi, let (bi,n) n∈N and (di,n) n∈N be absolutely summable sequences in Gi. For every n ∈ N, let µn ∈ ]0, +∞[ be a Lipschitz constant of U

1/2 n •∇h•U 1/2 n
and, for every i ∈ {1, . . . , m}, let νi,n ∈ ]0, +∞[ be a Lipschitz constant of U

1/2 i,n • ∇ℓ * i • U 1/2 i,n . Let (λn) n∈N be a se- quence in ]0, 1] such that inf n∈N λn > 0. For every n ∈ N, set ζn = 1 - m i=1 Ui,nLi √ Un 2 (23

Set

For n = 0, 1, . . .

                sn = xn -Un(∇h(xn) + cn -z) yn = sn -Un m i=1 L * i vi,n For i = 1, . . . , m       qi,n = prox U -1 i,n g * i vi,n + Ui,n Liyn -∇ℓ * i (vi,n) -di,n -ri + bi,n vi,n+1 = vi,n + λn(qi,n -vi,n). pn = sn -Un m i=1 L * i qi,n xn+1 = xn + λn(pn -xn). ( 25
)
Assume that f = 0. Then (xn) n∈N converges weakly to a solution to [START_REF]Proximal splitting methods in signal processing[END_REF], for every i ∈ {1, . . . , m} (vi,n) n∈N converges weakly to some vi ∈ Gi, and (v1, . . . , vm) is a solution to [START_REF]Primal-dual splitting algorithm for solving inclusions with mixtures of composite, Lipschitzian, and parallel-sum type monotone operators[END_REF].

The algorithm proposed in [START_REF] Loris | On a generalization of the iterative soft-thresholding algorithm for the case of non-separable penalty[END_REF][START_REF] Chen | A primal-dual fixed point algorithm for convex separable minimization with applications to image restoration[END_REF] is a special case of the previous one, in the absence of errors, when m = 1, H and G1 are finite dimensional spaces, ℓ1 = ι {0} , Un ≡ τ Id with τ ∈ ]0, +∞[, U1,n ≡ σ Id with σ ∈ ]0, +∞[, and no relaxation (λn ≡ 1) or a constant one (λn ≡ κ < 1) is performed.

APPLICATION TO IMAGE RESTORATION

We illustrate the flexibility of the proposed primal-dual algorithms on an image recovery example. Two observed images w1 and w2 of the same scene x ∈ R N (N = 256 2 ) are available (see Fig. 1(a)-(c)). The first one is corrupted with a noise with a variance θ 2 1 = 576, while the second one has been degraded by a linear operator H ∈ R N×N (7 × 7 uniform blur) and a noise with variance θ 2 2 = 25. The noise components are mutually statistically independent, additive, zero-mean, white, and Gaussian distributed. Note that this kind of multivariate restoration problem is encountered in some push-broom satellite imaging systems.

An estimate x of x is computed as a solution to [START_REF]Proximal splitting methods in signal processing[END_REF] where m = 2, z = 0, r1 = 0, r2 = 0,

h = 1 θ 2 1 • -w1 2 + 1 θ 2 2 H • -w2 2 , (26)
g1 = ι [0,255] N , g2 = κ • 1,2, (27)
f = 0, ℓ1 = ℓ2 = ι {0} (28)
where the second function in [START_REF] Repetti | A penalized weighted least squares approach for restoring data corrupted with signal-dependent noise[END_REF] denotes the ℓ1,2-norm and κ ∈ ]0, +∞[. In addition, L1 = Id and L2 = [G ⊤ 1 , G ⊤ 2 ] ⊤ where G1 ∈ R N×N and G N×N 2 are horizontal and vertical discrete gradient operators. Function g1 introduces some a priori constraint on the range values in the target image, while function g2 • L2 corresponds to a classical total variation regularization. The minimization problem is solved numerically by using Algorithm [START_REF] Pesquet | A parallel inertial proximal optimization method[END_REF] with λn ≡ 1. In a first experiment, standard choices of the algorithm parameters are made by setting Un ≡ τ Id, U1,n ≡ σ1 Id, and U2,n = σ2 Id with (τ, σ1, σ2) ∈ ]0, +∞[ 3 . In a second experiment, a more sophisticated choice of the metric is made. The operators (Un) n∈N , (U1,n) n∈N and (U2,n) n∈N are still chosen diagonal and constant in order to facilitate the implementation of the algorithm, but the diagonal values are optimized in an empirical manner. A similar strategy was applied in [START_REF] Pock | Diagonal preconditioning for first order primal-dual algorithms in convex optimization[END_REF] in the case of Algorithm [START_REF] He | Convergence analysis of primal-dual algorithms for a saddle-point problem: from contraction perspective[END_REF]. The regularization parameter κ has been set so as to get the highest value of the resulting signal-to-noise ratio (SNR).

The restored image is displayed in Fig. 1(d). Fig. 2 shows the convergence profile of the algorithm. We plot the evolution of the normalized Euclidean distance (in log scale) between the iterates and x in terms of computational time (Matlab R2011b codes running on a single-core Intel i7-2620M CPU@2.7 GHz with 8 GB of RAM). An approximation of x obtained after 5000 iterations is used. This result illustrates the fact that an appropriate choice of the metric may be beneficial in terms of speed of convergence.

(

  ∀n ∈ N) yn = xn -γnVn(Bxn + bn) xn+1 = xn + λn Jγ n VnA (yn) + an -xn . (10) Then xn ⇀ x for some x ∈ Z. At iteration n, variables an and bn model numerical errors possibly arising when applying Jγ n VnA or B. Note also that, if B is µ-cocoercive with µ ∈ ]0, +∞[, one can choose β = µ(sup n∈N Vn ) -1 , which allows us to retrieve [14, Theorem 4.1].

) 2 .Fig. 1 .

 21 Fig. 1. Original image x (a), noisy image w1 (SNR = 5.87 dB) (b), blurred image w2 (SNR = 16.63 dB) (c), and restored image x (SNR = 21.61 dB) (d).

Fig. 2 .

 2 Fig. 2. Normalized norm of the error on the iterate vs computation time (in seconds) for Experiment 1 (blue, dash dot line) and Experiment 2 (red, continuous line).

  Theorem 3.1 Let α ∈ ]0, +∞[, let β ∈ ]0, +∞[, let A : H → 2 H be maximally monotone, and let B : H → H be cocoercive. Let (ηn) n∈N ∈ ℓ 1 + (N), and let (Vn) n∈N be a sequence in Pα(H) such Let (λn) n∈N be a sequence in ]0, 1] such that inf n∈N λn > 0 and let (γn) n∈N be a sequence in ]0, 2β[ such that inf n∈N γn > 0 and sup n∈N γn < 2β. Let x0 ∈ H, and let (an) n∈N and (bn) n∈N be absolutely summable sequences in H. Suppose that Z = zer(A + B) = ∅, and set

	that
				sup n∈N Vn < +∞ (∀n ∈ N) (1 + ηn)Vn+1 Vn	(9)
	and V	1/2 n	BV	1/2 n	is β-cocoercive.

In a finite-dimensional space, weak convergence is equivalent to strong convergence.

For every i ∈ {1, . . . , m}, ℓ i is ν -1 i -strongly convex with ν i ∈ ]0, +∞[ if and only if ℓ * i is ν i -Lipschitz differentiable[START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert Spaces[END_REF] Theorem 18.15].

This work was supported by the CNRS MASTODONS project (grant 2013 MesureHD).


Philippe Calcagno
email: p.calcagno@brgm.fr

Aurélie Chamaret

Nicolas Chauvin

Sunseare Gabalda

Santiago Gabillard

Antonio Guillen

David Michea

Fabien Pasquier

Joachim Pouderoux

Yannick Vigier

Vincent Bouchot

Thermo2Pro: A tool to Help Professionals of the Geothermal Sector for Exploration in Deep Basin Context in France -Development, Update, and Perspectives

Keywords: deep geothermal exploration, information system, web tool, sedimentary basin, dissemination

The understanding of geology and petrophysical properties is necessary to develop deep geothermal projects. However, this knowledge must be shared within the French professional geothermal community to make the exploration process more efficient. Thermo2Pro is designed to fill the gap between the information derived from R&D projects and the authorities dealing with the deep geothermal exploration in basin context. Contracting authorities, local authorities, and design offices have been gathered in a user committee involved in the development of the project since its inception.

The main vector for Thermo2Pro is a Web application where the user accesses the information by simulating vertical or horizontal sections, or boreholes. The information is stored in 3D regular grids derived from scientific projects. The functionalities of Thermo2Pro have been demonstrated through information related to the Trias aquifer of the Paris basin acquired and inferred in the scope of the CLASTIQ-2 project. In that area, the available parameters were mainly a model of lithology, an estimation of porosity, and a model of temperature.

The initial Thermo2Pro project was dedicated to the development of the Web tool and its demonstration. The list of parameters available in Thermo2Pro is being expanded through the TRANS-CLAS project, by adding new information from the Paris basin and other areas such as Alsace and the French Massif Central. In addition, dissemination of data from the Dogger reservoir in the Paris basin is under study.

INTRODUCTION

France is one of the pioneers for geothermal heating exploitation in the world but efforts must continue to double geothermal heat networks by 2020 ("Grenelle de l'environnement" target). In the Paris basin, the carbonate aquifer of the Dogger is operated between 1500 and 2000 m with temperatures between 55 and 85°C. The density of installations reached its limits, particularly in the Val de Marne and to a lesser extent, the Seine-Saint-Denis area.

On the other hand, cold water re-injected into the ground may gradually alter the production areas in the short term. Indeed, the extension of the "cold bubble" around the injection wells may result in a breakthrough in thermal production wells. These issues are discussed within the framework of the Dogger Management project [START_REF] Goyénèche | Mise en oeuvre d'un système de gestion de la ressource géothermique du Dogger de la région Ile-de-France : Conception et réalisation de la base de données, collecte de l'ensemble des données historiques[END_REF][START_REF] Goyénèche | Mise en oeuvre d'un système de gestion de la ressource géothermique du Dogger de la région Ile-de-France[END_REF][START_REF] Hamm | Mise en oeuvre de la gestion de la ressource géothermique du Dogger de la région Ile-de-France dans le cadre du dispositif d'acquisition et de stockage des données -Rapport final de la phase 3[END_REF]2013). In parallel, it is essential to look for other potential geothermal aquifers such as the reservoir of the Trias or the Lusitanian carbonate reservoir (Fig. 1) Trias is a deep reservoir that has been explored mainly for its oil potential (only 3 geothermal exploration doublets have been drilled). The Trias reservoir is the aim of the scientific project CLASTIQ-2 [START_REF] Bouchot | CLASTIQ-2 : projet de recherche sur les ressources géothermales des réservoirs clastiques en France[END_REF] following the CLASTIQ-1 project [START_REF] Bouchot | Projet CLASTIQ : CLAyed sandSTone In Question[END_REF].

The Lusitanian aquifer, shallower than the Dogger one, can also be a target for new geothermal operations, especially in areas where the Dogger is overexploited. The Lusitanian reservoir would feed new urban heat networks designed for low consumption buildings where a lower temperature resource would be sufficient (around 50°C).

As described above, there is a need is for a better understanding of the deep geothermal aquifers in sedimentary basins. Various scientific projects are dedicated to these aquifers. However, this knowledge will be relevant only if it is useful to the geothermal industry professionals. Indeed, the regional and local authorities, and design offices need to understand the favourable areas for exploration. So far, there is no tool to forward this knowledge to professionals of the deep geothermal sector.

CONTEXT

The exploitation of the geothermal resource in deep sedimentary basin is managed by the non-geoscience professionals of the sector, such as decision makers (government and/or local authorities), contracting authorities and design offices.

Scientific projects

Several projects led by ADEME and BRGM have been focusing on deep aquifers during the last years. They aim is to acquire a better knowledge of the deep geothermal reservoirs in sedimentary basins in France. Below are two examples of such scientific projects.

The Dogger Management project [START_REF] Goyénèche | Mise en oeuvre d'un système de gestion de la ressource géothermique du Dogger de la région Ile-de-France : Conception et réalisation de la base de données, collecte de l'ensemble des données historiques[END_REF][START_REF] Goyénèche | Mise en oeuvre d'un système de gestion de la ressource géothermique du Dogger de la région Ile-de-France[END_REF][START_REF] Hamm | Mise en oeuvre de la gestion de la ressource géothermique du Dogger de la région Ile-de-France dans le cadre du dispositif d'acquisition et de stockage des données -Rapport final de la phase 3[END_REF]2013) focuses on the exploitation of the Dogger aquifer in the Paris Basin. It essentially aims at designing and supplying a database from operators and design offices. This database is used to model the thermal breakthrough and "cold bubbles". The objective is to inform decision makers dealing with permit applications rehabilitation processes. The knowledge of "cold bubbles" -due to injection -distribution is one of the criteria to optimize the selection of spots to be developed or renovated.

COPGEN [START_REF] Genter | Méthodologie d'inventaire du potentiel géothermique basse et moyenne énergies en Limagne (projet COPGEN[END_REF] and CLASTIQ-1 [START_REF] Bouchot | Projet CLASTIQ : CLAyed sandSTone In Question[END_REF] and CLASTIQ-2 [START_REF] Bouchot | CLASTIQ-2 : projet de recherche sur les ressources géothermales des réservoirs clastiques en France[END_REF] projects search for geothermal resources in French clastic reservoirs in France. They aim at improving the understanding of silicoclastic formations, such as Trias (CLASTIQ-2), in deep sedimentary basin mainly from the existing oil bore holes data.

The geological knowledge produced by these projects is scale-dependent.

At regional scale:

 3D geometric models of reservoirs (Alsace, Paris basin, Limagne, …) including porosity and permeability estimation.  3D temperature models at the basin scale (Alsace, Bresse, Limagne, Paris basin, South-East basin).

 Quantification of the geothermal potential (Paris basin, Bresse, Limagne, Alsace).

At local scale:

 Spatial evolution of transmissivity in the Dogger aquifer (Val de Marne, Seine-Saint-Denis).  Model of the "cold bubbles" extension resulting from fluid injection.  Brine flow simulation for a geothermal doublet (Dogger and Trias).

Technical Committee for geothermal exploitation in deep aquifers

The Technical Committee for geothermal exploitation in deep aquifers (CTGAP), created in 2013, follows previous initiatives in response to a request from the deep geothermal professionals of the Ile-de-France area (Goyeneche et al., 2009). The design of deep geothermal operations in such context implements complex techniques and require high budgets. The CTGAP's mission is to help the professionals of the geothermal sector by capitalizing knowledge and know-how, by facilitating the improvement of techniques, and by disseminating information.

CTGAP is a natural vector to link the knowledge inferred from scientific projects with the professionals of the geothermal sector.

European context

The European Community (EC) context is favourable to the development of tools for knowledge dissemination for deep geothermal energy.

At the R&D level, the EC Joint Programme on Geothermal Energy of the European Energy Research Alliance (EERA-JPGE, www.eera-set.eu/index.php?index=22) is involved in the medium term to the development of databases and tools to assist in the exploration of geothermal resources across Europe. This work is mainly conducted within the JPGE's "Resource assessment" subprogramme.

At the agencies and government level, the EC 7th FP Geothermal European Research Area Network (GEO ERA-NET, www.geothermaleranet.eu) coordinates national initiatives with a special focus on a European geothermal database.

State of the art for dissemination

In France, the main national tool to disseminate information about geothermal energy is dedicated to shallow aquifers. It aims at promoting geothermal energy among the general public and contracting authorities. This tool has been developed since 2005 in partnership with ADEME and BRGM over several French regions. This tool is based on an atlas of groundwater resources that can be exploited for geothermal energy in a shallow slice between 0 and 100 m deep. The Atlas proposes a Geographic Information System (GIS, Fig. 2) available online via the "Geothermie perspectives" website www.geothermie-perspectives.fr.

Although the process of dissemination is comparable to the needs expressed above, this tool is not designed to meet the needs of the professionals who work on the use of deep geothermal resources.

In Europe, the dissemination tools dedicated to geothermal deep sedimentary basins are few. Among them there is GeotIS developed in Germany by LBEG and LIAG [START_REF] Pester | GeotIS -the Geothermal Information System for Germany[END_REF], and ThermoGIS, developed by TNO in the Netherlands [START_REF] Kramers | Direct heat resource assessment and subsurface information systems for geothermal aquifers; the Dutch perspective[END_REF].

GeotIS (www.geotis.de) is mainly an electronic geothermal atlas. It displays thematic maps and socio-economic information.

Although it covers the whole German territory, information is mainly provided in basin areas: Rhine graben, molasse basin, and North German basin. 3D geological models are available in some locations.

ThermoGIS (www.thermogis.nl, Fig. 3) is a decision support tool for geothermal exploitation of the Netherlands territory down to 1500 to 4000 m deep. The GIS-type user interface displays maps of geological formations thickness, petrophysical parameters, positions of boreholes, as well as uncertainties associated with these data. This tool also proposes economic simulations for exploitation by a geothermal doublet based on transmissivity modelling.

OBJECTIVES

Thermo2Pro [START_REF] Calcagno | Thermo2Pro -Outil d'Aide à l'Exploration pour la Géothermie Profonde des Bassins Sédimentaires à l'Usage des Professionnels[END_REF]) is a project started in 2011 and dedicated to the French territory in the context described above.

The two main objectives of this project are: (i) to make professionals of the geothermal sector able to access the knowledge from scientific projects by developing a dedicated tool that will be demonstrated through a pilot site, and (ii) to promote the geothermal sector by gathering the players throughout the project and by allowing them to participate to the development of the tool.

The following describe in more details how these objectives are tackled. The next chapters focus on the Thermo2Pro tools and data management.

Develop a tool to favour geothermal exploration

The aim of Thermo2Pro is the development of a Web-based tool to provide relevant knowledge to professionals working at geothermal exploration in deep sedimentary basins. The information displayed by Thermo2Pro is intended to give indications prior to a feasibility study.

The targeted information is (i) the geometry of the aquifer through maps and sections, and (ii) the estimation of reservoir properties (temperature, permeability, transmissivity, etc.) This material will be derived from 3-dimensional models and simulations performed in scientific projects (see above).

On top of that, uncertainties need to be associated with the information displayed as much as possible to help users with the interpretation. This could be, for instance, an indication of the most favourable and the most unfavourable values of a given parameter at a given position. In addition, indication of the processes that led to the information is to be given to ensure traceability. If possible, methodologies and softwares used in the process as well as references will be mentioned.

Demonstration area

Thermo2Pro's capabilities were demonstrated using information from the Trias in an area of the Paris basin. Suitable knowledge acquired in this demonstration area during the project CLASTIQ-2 [START_REF] Bouchot | CLASTIQ-2 : projet de recherche sur les ressources géothermales des réservoirs clastiques en France[END_REF] was used. The main geothermal parameters were:

 Lithology Geological model including 10 sequences interpreted from borehole logs.  Temperature Interpolated from borehole measurements (profiles and/or Bottom Hole Temperature).

 Porosity

Interpolated from logging electric measurements.  Permeability Computed using Phi.K law. Two qualitative results, optimistic and pessimistic, were provided.

 Transmissivity

Computed using permeability and geological formations thickness. Two qualitative results, optimistic and pessimistic, were provided.

Promote the geothermal sector

In France, the geothermal energy sector is still under development and needs visibility. Thermo2Pro is likely to contribute to the development of the sector by facilitating access to technical information to a large number of users. In addition, the project is an opportunity to disseminate information about on-going research on alternative aquifers to the professionals of the sector who mainly deal with the Dogger aquifer. The User Committee of Thermo2Pro is the devoted place for such an exchange.

USER COMMITTEE

Thermo2Pro is developed in close cooperation with professionals via the CTGAP. A User Committee has been established to define the needs and issues and to guide the development of the tool. The committee is also involved in the tests and the validation process.

The User Committee is composed of about twenty people representing key professionals involved in deep geothermal resource: contracting authorities, state and local authorities, operators, and design offices. They meet on a regular basis to follow the developments of the project.

THERMO2PRO TOOLS

Dissemination of the information is achieved through the Thermo2Pro Website that presents the project and proposes a cartographic application to make queries on the available geothermal parameters.

Web site

Thermo2Pro Website is implemented in HTML 4 (HyperText Markup Language). Webpages are presented with CSS2 (Cascading Style Sheet). They are based on a content management system developed with the SPIP software. This makes easy administration and updating via a back-office area of the site from any Web browser.

The Website is reachable at the following address: www.thermo2pro.fr. It presents the project and updated news (Fig. 4). It is also the portal to access the available information on geothermal parameters using a cartographic tool.

Cartographic application

The cartographic tool is a Web interface that makes the user able to visualize and check the geothermal parameters (geology, temperature, porosity, etc.).

The 1st step consists in locating the zone of interest (Fig. 5). This is achieved using a GIS type design where various layers can be displayed such as geographic maps and cartographic data on a given area. In addition to the layers embedded in the cartographic tool, extra information can be reached and displayed via the Internet using the WMS protocol (Web Map Service). In a 2nd step, the geothermal parameters are visualized using 3 main queries:

 Horizontal section. The user chooses the altitude of the section.  Vertical section. The user defines the origin and the extremity of the section on a map and sets the vertical extension.  Borehole. The user defines the location of the borehole collar on a map and sets the vertical extension. An option allows to make an inclined borehole by giving its azimuth and dip.

This process allows the user to spot interesting values of parameters -or combination of parameters -at depth. A picking functionality on the image of the resulting section or borehole returns the value of the parameters (Fig. 6).

Figure 6: Result of a query on a vertical section.

In addition, a report of the queries and results can be automatically generated, printed, and saved.

DATA MANAGEMENT

The sections or boreholes simulated during the queries are computed using the information about geothermal parameters that is stored in 3D regular grids.

3D grids for 2D and 1D queries

As described above, geographical queries are performed in Thermo2Pro on 2D (vertical or horizontal sections) or 1D (borehole) profiles. The visualization of parameter on these profiles is computed via the information stored in 3D regular grids (Fig. 7). Parameter management is one of the main originality of Thermo2Pro. Working with 3D grids allows to retrieve information at depth and not only on a map. In addition, 3D regular grids have been chosen to overcome specific formats considering the variety of parameters and consequently the variety of tools that produce the information about these parameters. A 3D regular grid is a quite generic object that is also suitable for 2D information, e.g. a temperature map (2D) can be stored in a 3D grid.

A 3D regular grid covers the volume where parameters are available (values of temperature, porosity, etc.) A given cell of the grid hosts the value of each parameter available at the 3D location of the cell. The 3D grids are described in an eXtensible Markup Language (XML) file. Such a file contains the topology of the 3D grid, the values of the parameters, color scales and metadata.

Implementation of the query service

Following the request of the user, an image is computed on the fly and displayed on the cartographic client. A scalar grid is associated to the image for picking the value of the parameter at any location of the picture. Both the image and the scalar grid are calculated from the 3D grid containing the values of the parameters.

The process of querying the 3D grid is made through a Web Processing Service (WPS) defined by the Open Geospatial Consortium (OGC) and based on a WPS 52th North server developed in Java.

Computations are done by a stand-alone program developed in C++. This program loads the grids, processes them and generates images or probing results using VTK (Visualization Tool Kit), an open-source library dedicated to 2D and 3D scientific data processing and visualization developed by Kitware.

CONCLUSION, UPDATE AND PERSPECTIVES

The first stage of Thermo2Pro was achieved in 2013 by developing the basic functionalities dedicated to the dissemination [START_REF] Calcagno | Thermo2Pro -Outil d'Aide à l'Exploration pour la Géothermie Profonde des Bassins Sédimentaires à l'Usage des Professionnels[END_REF]. These functionalities of the Thermo2Pro's Webtool web demonstrated on a part of the Trias in the Paris basin area. However, the tools of Thermo2Pro are designed to host information from other aquifers and other basins, as long as they are stored in 3D grids. The plan for Thermo2Pro is to be the cradle for accessing such information on the French territory. In addition, the first stage of Thermo2Pro ended with recommendations for future developments and improvements of Thermo2Pro in relation with the User Committee work.

Following these recommendations, a new project was launched in 2013: TRANS-CLAS. This is a BRGM+ADEME ongoing project for making available more information via Thermo2Pro than the existing demonstrator. The goal is to disseminate information from clastic environments in the Paris basin, the Rhine graben, and the Limagne d'Allier basin.

Furthermore, a strong expectation has been expressed by the User Committee of Thermo2Pro to include data from the Dogger aquifer that is exploited for more than 30 years in the Paris basin. This objective raises two main issues. On one hand, displaying information from the Dogger aquifer through Thermo2Pro will require some adaptations because data are in 2 dimensions. On the other hand, models describing the parameters, including "cold bubbles" spread need to be updated on a regular basis to provide comprehensive and up to date information to the French professionals of the geothermal sector. This work is currently under study.

Figure 1 :

 1 Figure 1: Geological time scale and the main aquifers of the Paris basin.

Figure 2 :

 2 Figure 2: GIS for the geothermal potential of shallow aquifers in France, dedicated to the general public (ADEME-BRGM). Zoom on the "région Centre". Colours indicate a qualitative geothermal potential combining the estimated flow rate and depth of the aquifer: High (blue), Medium (green) and Low (red).

Figure 3 :

 3 Figure 3: ThermoGIS (The Netherlands).

Figure 4 :

 4 Figure 4: Thermo2Pro Website homepage at www.thermo2pro.fr.

Figure 5 :

 5 Figure 5: Cartographic page for queries.

Figure 7 :

 7 Figure 7: Parameters are stored in 3D grids and displayed as 2D or 1D grids (images) in sections and boreholes.

ACKNOWLEDGEMENTS

This work is supported by ADEME (French Agency for Energy and Environment): contracts n°1005C0154 (Thermo2Pro) and 1305C0061 (TRANS-CLAS).

Thermo2Pro and TRANS-CLAS are co-financed by the ADEME (French Environment and Energy Management Agency) and BRGM (French Geological Survey).


Philippe Calcagno
email: p.calcagno@brgm.fr

Vincent Bouchot

Isabelle Thinon

Bernard Bourgine

3D Fault Model of the Bouillante Geothermal Province Interpreted from Onshore and Offshore Structural Knowledge (French West Indies)

Keywords: 3D modelling, fault modelling, onshore and offshore combination, geothermal energy, geothermal exploration, Bouillante

The Bouillante area is known for its high-temperature geothermal field which produce electricity in a 15 MWe installed power plant (Guadeloupe Island, French West Indies). This zone is a complex geodynamic area. It stands at the convergence of the major tectonic and volcanic structures of the inner arc of the Lesser Antilles. The two main regional fault systems joining at the Bouillante geothermal field are the NW-SE Montserrat-Bouillante (MB) fault system, and the ESE-WNW to E-W Bouillante-Capesterre (BC) fault system.

Two other factors make rather difficult the interpretation of the geological structuration. Firstly, Bouillante is located on the West coast of the Guadeloupe Island. It means that data and observations both from the Island and from the sea are needed. Secondly, a wet tropical climate prevails in that region. That is why exposures suitable for direct structural measurements are almost only found on the seaside where coastal erosion has stripped away the soil.

The available data show that, on one hand, faults observed on the field mainly elongate along the E-W direction. On the other hand, offshore structures interpreted from marine seismic lines show a larger range of directions. In order to achieve a coherent interpretation of the Bouillante geothermal area, a 3D fault model is completed from onshore and offshore structural knowledge. A 15 km x 16 km zone crossing the island coastline is investigated down to 2 km depth.

The fault network constructed reveals a hierarchy in the family of structures and highlights the prevalence of the NNW-SSE direction, associated with secondary NE-SW-trending structures, and the E-W direction. On a geographical point of view, the modelled faults are gathered in 3 clusters along the coast. The 3D model facilitates a coherent interpretation of the faults in the Bouillante province but this model can also be a cradle for hydro dynamic simulation.

INTRODUCTION

The Bouillante geothermal province area is located in the French West Indies, on the West of the Guadeloupe Basse-Terre Island (Fig. 1). That zone is known for its high-temperature geothermal field (e.g. [START_REF] Bouchot | Assessment of the Bouillante geothermal field (Guadeloupe, French West Indies): toward a conceptual model of the high temperature geothermal system[END_REF] which produce electricity at the 15 MWe installed power plant in Bouillante (Fig. 1b). Our study aims at better understand the fault structuration of the Bouillante geothermal province. This is a prerequisite for a better knowledge of the Bouillante geothermal field understanding. At the scale of the geothermal province, onshore and offshore data are required to interpret in a coherent way the faults in the area because of its location both inland and at sea. Previous works aimed at describing the structural geology on land (e.g. [START_REF] Mathieu | L'impact des mouvements décrochants sur la structure des volcans : une étude de cas des volcans de la Guadeloupe, de Maderas et du Mont Cameroun[END_REF] and at sea (e.g. [START_REF] Thinon | Study of the Bouillante Bay (West Basse-Terre Island shelf): contribution of geophysical surveys to understanding of the structural context of Guadeloupe (French West Indies -Lesser Antilles)[END_REF]. We propose to merge data and knowledge from field work and from marine exploration to produce a model in 3 dimensions representing the faults down to 2 km depth at the scale of the 250 km² Bouillante geothermal province. A detailed version of this work can be found in [START_REF] Calcagno | A New 3D Fault Model of the Bouillante Geothermal Province Combining Onshore and Offshore Structural Knowledge (French West Indies)[END_REF].

STRUCTURAL SETTING

The geodynamic context of the West Indies shows that the fore arc is developed on ~250 km wide in E-W between the subduction zone and the western edge of the volcanic arc of the Lesser Antilles (grey area on Fig. 1a). From North to South, the motion ranges from sinistral extensional shear to dextral oblique thrusting (Fig. 1a).

The Bouillante geothermal province is a regional key geodynamic area where the major tectonic and volcanic structures of the inner arc of the Lesser Antilles join. The Bouillante geothermal field is located at the junction of two regional fault systems (Fig. 1b). The first one, located at sea, is the NW-SE Montserrat -Bouillante (MB) fault system. The second one, interpreted on the Island, is the ESE-WNW to E-W Bouillante -Capesterre (BC) fault system. The trajectory of this hypothetical fault is extracted from the geological, morphological and structural map of southern Basse-Terre realized by [START_REF] Feuillet | Arc parallel extension and localization of volcanic complexes in Guadeloupe, Lesser Antilles[END_REF].

A third fault system, described by [START_REF] Thinon | Study of the Bouillante Bay (West Basse-Terre Island shelf): contribution of geophysical surveys to understanding of the structural context of Guadeloupe (French West Indies -Lesser Antilles)[END_REF], is a N160°E-trending escarpment which limits the shelf of the Bouillante sector. This escarpment is interpreted as a major sinistral strike-slip system, playing the role of transfer zone between the MB and Les Saintes (LS) fault systems. This discontinuity belongs to the western limit of a zone of sinistral extensional shear, developed on 250 km wide in E-W between the subduction zone and the western edge of the volcanic arc of the Lesser Antilles (Fig. 1a).

Onshore structural knowledge

The studied area is located on the West side of the volcanic Basse-Terre Island where the prevailing climate is wet tropical. This specific geographic location constrains the acquisition of structural data. Exposures suitable for structural measurements are mainly found only on the seaside where coastal erosion has stripped away the soil (Fig. 2). As soon as one moves away from the coast, the volcanic island is covered by dense tropical forest on very steep relief. Given a rainfall that is 10 times higher than at the coast, this forest zone is the site of intense supergene alteration that impedes ground observation, except where ravines generated by torrents have been cut into the weathering profile. Nevertheless, the hilly areas where topographic regular structures are sharp can provide indirect information on fault traces [START_REF] Feuillet | Arc parallel extension and localization of volcanic complexes in Guadeloupe, Lesser Antilles[END_REF].

From a recent synthesis of existing structural interpretation (Bouchot et al., 2008) and acquisition of new structural measurements by the authors (Fig. 2), it appears that most of the measured faults have a trend E-W to ESE-WNW, with a dominant normal throw showed by the evidence of normal offset of volcanic layer on each side of the fault surface with striae and drag fold in some places. Moreover, a very limited second family, consisting of only two faults visible onshore, trends NE-SW with a strong dextral strikeslip component, according to tectoglyphs such as striae and tension gashes.

Offshore structural knowledge

Three main surveys were conducted in the Bouillante Bay area in 1998 and 2003 by BRGM ((Bathybou98 and Geoberyx03 cruises; [START_REF] Guennoc | Levé bathymétrique le long de la côte ouest de l'île de Basse-Terre, Guadeloupe. Apports à la compréhension du contexte structural du champ géothermique de Bouillante[END_REF][START_REF] Thinon | Study of the Bouillante Bay (West Basse-Terre Island shelf): contribution of geophysical surveys to understanding of the structural context of Guadeloupe (French West Indies -Lesser Antilles)[END_REF]. Firstly, the bathymetry was investigated with a shallow water multibeam echosounder (MBES). The depth of the seabed is a reference data for the 3D model, as much as the topography. It reveals the shape of the bay and highlights major directions such as the the E-W-trending Marsolle escarpment in the northern part of the Bouillante Bay.

Secondly, a very high resolution (VHR) reflection seismic was acquired. A dense network, about 180 km long, shows the tectonic relations between the sedimentary cover and the volcanic basement (Fig. 3). The acoustic basement is interpreted as being the volcanic basement of Basse-Terre Island. Vertical throws of faults appears to be a few meters to tens of meters at least but the horizontal displacement is not interpretable and the faults within the acoustic basement are not visible. Thirdly, a high resolution (HR) magnetic survey was performed and processed (e.g. [START_REF] Truffert | Using MAGIS for Geothermal Application -Guadeloupe Archipelago in French West Indies[END_REF]. The conbination of the geophysical data provides a good image of the geometry of the structures. However, no geological data are available to constrain the age neither the type of formations, nor the dips of the faults.

The interpretation of these three surveys shows that the main structures elongate along NNW-SSE and E-W directions but some NW-SE and NE-SW features also occur [START_REF] Thinon | Study of the Bouillante Bay (West Basse-Terre Island shelf): contribution of geophysical surveys to understanding of the structural context of Guadeloupe (French West Indies -Lesser Antilles)[END_REF]. For instance, the main N160°E-trending escarpment is interpreted as a major sinistral strike-slip system, playing the role of transfer zone between the NW-SE MB and LS fault systems. Furthermore, E-W major structures are observed crossing the NNW-SSE-and NW-SE-striking faults.

3D FAULT MODEL

A 15 km x 16 km zone (see Fig. 1b for location) is modelled down to 2 km below sea level in the local "Guadeloupe, St. Anne -UTM20 -IGN", WGS84 projection system. In that zone, faults are interpreted by taking into account both offshore and onshore geological knowledge.

Data

Onshore, data consist in strike and dip measurements of fault measured during field work. Due to outcrops accessibility, these data are located on the coast. Inland, some structures are interpreted from previous works (e.g. [START_REF] Feuillet | Arc parallel extension and localization of volcanic complexes in Guadeloupe, Lesser Antilles[END_REF]. Offshore, location and dip of the faults are derived from the interpretation of 52 Geoberyx03 seismic profiles [START_REF] Thinon | Study of the Bouillante Bay (West Basse-Terre Island shelf): contribution of geophysical surveys to understanding of the structural context of Guadeloupe (French West Indies -Lesser Antilles)[END_REF]. The magnetic lineaments, interpreted as markers such as faults or dykes are clues for the existing orientations of structures on the shelf [START_REF] Henkel | Magnetic features of fracture zones[END_REF]. The information on the age and nature of geological formations are derived from geological surveys at land.

The top of the 3D model is bounded by a Digital Elevation Model (DEM). This DEM combines information from the land and from the sea. The IGN data (Institut Géographique National, France) were used on the Island. Regarding bathymetry, two sets of data were taken into account: the deep water multibeam echosounder of the AGUADOMAR cruise (1998-1999, IPG, Deplus et al., 2001) and the shallow water multibeam echosounder of the Bathybou98 cruise (1998, BRGM, [START_REF] Guennoc | Levé bathymétrique le long de la côte ouest de l'île de Basse-Terre, Guadeloupe. Apports à la compréhension du contexte structural du champ géothermique de Bouillante[END_REF]. Then, a coherent 50 m resolution DEM covering the whole modelled area was interpolated with the GDM (1) geostatistics software [START_REF] Bourgine | Tools and methods for constructing 3D geological models in the urban environment. The Paris case[END_REF]

Methodology

It was shown in the previous section that the pre-existing knowledge describes structures interpreted separately onshore and offshore. The present study doesn't consist in trying to connect these features but in interpreting them in a coherent way over our sea and land area. To build a consistent structural 3D interpretation, the geometry of a fault is computed using a set of data merging marine and inland locations of this fault (e.g. the Bouillante fault on Fig. 4). Faults are interpolated using the potential field interpolation and geological rules methods developed in BRGM [START_REF] Lajaunie | Foliation fields and 3D cartography in geology; principles of a method based on potential interpolation[END_REF][START_REF] Calcagno | Geological modelling from field data and geological knowledge, Part I -Modelling method coupling 3D potential-field interpolation and geological rules[END_REF] and implemented in the 3D GeoModeller (2) software.

Results

The 3D interpretation of the faults displays coherent structures gathering onshore and offshore knowledge (Fig. 5). Most of the faults belong to one of the two main families: The Montserrat -Bouillante -Les Saintes (MBLS) fault system, and the E-W-to ESE-WNW-striking fault system. In more detail, the main set of the offshore faults is striking ESE-WNW, and mainly dipping SSW. Two others sets are striking NNW-SSE and NE-SW with opposite dipping. In contrast, the main set of the onshore faults is striking from E-W to ESE-WNW and dipping north or south as a horst and graben pattern. Taking into account the whole set of onshore and offshore faults, it appears that at the model scale, faults are mainly striking from ESE-WNW to NNW-SSW and mainly dipping SSW to WSW.

3.3.1 The MBLS fault system: the case of the NNW-SSE-trending section around Bouillante The MBLS fault system is considered as a sinistral strike slip fault zone recognized over 300 km long between St Kitts and Les Saintes north to south (Fig. 1b). We combine the NW-SE MB fault system between Montserrat and Directeur Volcano seamount [START_REF] Duperret | Étude structurale de la zone Montserrat -Marie-Galante (Petites Antilles) : Analyse des levés sismiques de la campagne FAGUAD[END_REF], and the NNW-SSE MB fault system between Malendure and Coreil and the NW-SE LS fault system [START_REF] Deplus | Submarine evidence for large-scale debris avalanches in the Lesser Antilles Arc[END_REF] to model the MBLS fault system. This fault system is generally oriented NW-SE, but forms a NNW-SSE-trending virgation along the Northern part of Basse-Terre volcanic island (Fig. 5). This virgation is characterised by a sub-continuous 30 km long fault zone which strikes N160°E and dips west. In contrast, on both end of the virgation, the NW-SE-striking corridors consist of east-dipping ESE-WNW-striking oblique fault. This "en echelon" oblique normal faults network is interpreted as an indicator of a sinistral transtension (Fig. 1) and the NNW-SSE section of the MBLS fault zone as a N160°E sinistral strike slip fault (Thinon et al., 2010).

In addition, a network of several parallel faults, well developed offshore, striking NE-SW, and 3 km long is linked to the offshore N160°E-striking faults. The onshore expression of these faults is very limited (Fig. 5). Identified only twice on the coast (Bouillante and Malendure outcrops), the NE-SW-striking faults show a dextral movement with a weak normal component. We interpret these short faults as a dextral conjugate network, grafted onto the main N160°E-striking sinistral fault, as a pinnate pattern. In the 3D fault model, some NW-SE-striking faults were also modelled mainly on the insular slope (2 dark blue faults) and on the external shelf (2 blue faults). They are located offshore the Bouillante Bay (Fig. 5). They are limited and sometimes shifted by the E-W and NE-SW faults. The NW-SE faults are also present at North of Malendure, near Pointe Noire town (see Malendure cluster below).

The E-W-to ESE-WNW-striking fault system

Onshore and on the shelf, the main fracturing strikes E-W to ESE-WNW. These structures can be gathered into 3 clusters of faults from North to South: Malendure, Bouillante, and Coreil (purple dashed ellipses on Fig. 5).

The Malendure cluster is located at the junction between the NW-SE section and the NNW-SSE section of the MB fault system, in the Malendure area (Fig. 1b and Fig. 5). On the shelf, the vertical offsets of two NW-SE-striking faults are low. Onshore, these faults are correlated to discrete fracture zones, controlling reddish hydrothermal alteration. The E-W-to ESE-WNW-striking normal faults do not seem connected to a regional E-W fault but they are well developed between the NW-SE and the NNW-SSE sections of the MBLS fault system. At north of the Malendure cluster, the NW-SE faults observed on coast have been correlated to the faults observed on the seismic profiles. The low density of marine data prevents from specifying the tectonic network and the relationships between the NW-SE Pointe-Noire faults and the EW faults of Malendure.

In the Bouillante cluster, the roughly EW-striking, steeply dipping, normal faults network forms a horst and graben pattern. However, on the shelf, i.e. approaching the main N160°E-striking fault system, the E-W faults are discontinuous and stopped on the NE-SW-striking faults (Fig. 5). Evidence of intense deformation, focused on the Bouillante cluster, is a strong argument to defend the existence of the major fault zone of BC fault system [START_REF] Feuillet | Arc parallel extension and localization of volcanic complexes in Guadeloupe, Lesser Antilles[END_REF]. The steeply south dipping E-W-striking Bouillante fault appears as a main member of the regional E-W fault, on which are associated the other E-W-striking normal faults whose dips are opposed (Fig. 5). The fracturing cluster of Bouillante is interpreted as the junction zone between the major E-W fault zone through Bouillante town, and the major sinistral-(normal) N160°E strike-slip faults on which were grafted dextral-(normal) NE-SW strike slip fault (Fig. 5).

The Coreil cluster highlights two sets of offshore faults: a NE-SW-striking faults set and an ESE-WNW-striking faults set (Fig. 5) and few indicators of deformation exist onshore. The set of 4 short NE-SW-striking faults (Fig. 5) are interpreted as a conjugate strike slip network, grafted onto the main N160°E-striking sinistral fault. Between the Coreil and Bouillante clusters, the 3D model highlights an area without observed faults, which coincides with a strong sub-circular magnetic anomaly and a convex morphology of the shelf. These observations could mark the occurrence of young volcanic magnetic product -hiding the faults -as suggested by youngest K/Ar age obtained around Bouillante, on the Muscade volcanic rock (479 ± 16 ky; [START_REF] Sanjuan | Travaux de recherche liés au projet GHEDOM-ADEME[END_REF].

CONCLUSION AND PERSPECTIVES

Merging onshore and offshore data improves the geological interpretation of the MBLS fault system. The resulting 3D fault model helps to better understand the regional structural geology of the Bouillante geothermal province. Data from the sea and data from the land are used simultaneously to build a coherent interpretation across the coast (e.g. Fig. 4 for the Bouillante fault). In addition, the geological knowledge from the land benefits to the comprehension of the structures at sea and reciprocally. This methodology is significantly more powerful than trying to connect two separate interpretations, one onshore and the other offshore.

The interpretation presented in previous works (e.g. [START_REF] Thinon | Study of the Bouillante Bay (West Basse-Terre Island shelf): contribution of geophysical surveys to understanding of the structural context of Guadeloupe (French West Indies -Lesser Antilles)[END_REF] is refined. In particular, our methodology highlights NE-SW structures linked to the MBLS fault system. Such structures define, limit and shift compartments where the E-W to ESE-WNW faults elongate. The marine seismic interpretation combined to the field observation is a guide to define the faults hierarchy by showing that the NE-SW-striking faults cut the E-W to ESE-WNW ones. This example demonstrates how the methodology allows to build a fault network giving clues for geometry and kinematics pattern.

The 3D fault model provides indications for future field work. It was not possible to connect some E-W structures interpreted offshore to observation onshore and vice versa. However places were these modelled E-W faults cross the coast will be interesting spots to investigate. Some NE-SW structures modelled offshore would affect inland terrains if prolonged. In that way this 3D fault model is a tool to point out probable areas where to explore for such structures onshore. In particular, the clusters of faults identified from the fault model could be a guide for exploration. They emphasize areas where permeability is a priori favourable, making them potential geothermal fields. In addition, the geometry of the fault model will be used to constrain hydrothermal simulations. Such simulations will help to understand how fluids circulate within these promising permeable areas.

Another step is to fill the fault model with relevant geological formations at the regional scale. Outcrops correlated to the seismic units would help to set the geometry of the geological formations. Furthermore, integrating deep refraction seismic data would help to extend the 3D model at depth in order to describe deeper geological structures. Then, geophysical data, e.g. gravity and magnetics, would be additional information to validate and refine the model by combining its geometry to geophysical data in a forward or an inversion process.

Figure 1 :

 1 Figure 1: (a) Geodynamic context of the West Indies (modified from Feuillet et al., 2002). NAM: North American plate. SAM: South American plate. (b) Regional structural setting of the Guadeloupe Island (modified from Thinon et al., 2010). Thick square: location of the 3D fault model. BCF: Bouillante-Capesterre fault system.

Figure 2 :

 2 Figure 2: Outcrops are quite rare and minly located on the seaside. Here is a picture of the N95°E-striking, dipping 70°S normal Machette -Pointe à Sel fault located in the Anse Machette area.

Figure 3 :

 3 Figure 3: A Geoberyx03 VHR seismic profile (Thinon et al., 2010) showing the sedimentary cover (UP, UC, RB, UB), the acoustic basement (US), and the major faults. RB: surficial formation. Various seismic units are noted (names starting by U). Location of the marine seismic section on the left hand side (West Guadeloupe offshore).

Figure 4 :

 4 Figure 4: Example of onshore and offshore knowledge combination: The Bouillante fault (yellow surface) is constructed from onshore field observation (dark blue point), onshore interpretation (red points) and offshore seismic profiles interpretation (light blue points). View from WSW. The coastline (white crosses) separates onshore and offshore domains. See Fig. 1b for modelled area location.

Figure 5 :

 5 Figure 5: The whole 3D fault interpretation combines onshore and offshore knowledge. The coastline (white crosses) separates onshore and offshore domains. Two main families of faults are modelled: the MBLS fault system (green and blue faults); the E-W to ESE-WNW-striking fault system (yellow faults). Three clusters are identified (purple dashed ellipses). See Fig. 1b for modelled area location. View from SE.

ACKNOWLEDGEMENTS

This work is supported by ADEME (French Agency for Energy and Environment): contract 0805C0044 (GHEMOD) and contract 0805C0039 (GEO3BOU).

The AGUADOMAR bathymetric data are provided by the Institut de Physique du Globe de Paris.

FOOTNOTES

(1) GDM is a commercial software developed by BRGM. For further information visit: http://gdm.brgm.fr/?lang=en.

(2) 3D GeoModeller is a commercial software developed by BRGM and Intrepid Geophysics. For further information visit: http://www.geomodeller.com.


Philippe Calcagno
email: p.calcagno@brgm.fr

3D GeoModelling for a Democratic Geothermal Interpretation

Keywords: 3D Geomodelling, geothermal exploration, interdisciplinary interpretation, data integration, consensual interpretation, geology, gravimetry, magnetotellurics, geochemistry, conceptual model

Geothermal exploration aims at locating favourable areas for exploitation. To reach this goal, various disciplines are implemented. Among the most common ones are geology, geophysics and geochemistry. Data are generally acquired in the field, such as geological observations, gravimetric surveys or thermal sources sampling. These data are interpreted to characterize the geometry, and the properties of the studied zone. They provide separate but complementary information to understand the area explored. However, combining geological, geophysical and geochemical interpretations is not an easy task.

In such a context, one of the main difficulties lies in how to mix all the information to infer a coherent geothermal conceptual model. Merging them in the same space can help their combination to lead to a consistent understanding. First of all, this methodology allows to check the location of separate figures and to ensure their coherence. Moreover, it makes possible to build an overall interpretation based on various information. GeoModelling in 3-dimensions is an interesting candidate for this job because it allows to input materials from various origins to achieve an interpretation of the geothermal area.

GeoModelling provides a common platform for interpretation during the exploration phase of a geothermal project. The final model can be completed through successive stages bringing new information at each step. For instance, a preliminary 3D geomodel can be based on very rough data from bibliography, even before any field work dedicated to the exploration. In a second time, geological data can be observed on the ground and incorporated in the model to refine the interpretation. The process can be continued using a gravimetric survey to improve the model at depth. Then, magnetotelluric resistivity can be injected in the model to infer possible fluid occurrence. Finally, location and properties of geothermal springs can be displayed in the 3D model to complete the interpretation. This kind of interdisciplinary workflow leads to a coherent geomodel filled by geology, geophysics and geochemistry.

Making a 3D geomodel by associating complementary interpretations is an interesting perspective but giving the experts of each discipline the opportunity to interact in a democratic process is even more powerful. Indeed, geological, geophysical, and geochemical interpretations are not disconnected. Even if a preliminary work has to be carried out separately by each discipline, the interpretation coming from one can be fed by the others instead of putting them one after the other in a sequential workflow. To do so, the methodology has to be object oriented, where the central object is the 3D geomodel. In this light, the 3D model benefits from a common interpretation implemented jointly by geologists, geophysicists and geochemists. In other words, they can compare, connect, discuss, and adapt their own approaches for a mutual result in a GeoModelling environment. At the end, the conceptual model is not a conglomerate of distinct interpretations but a consensus shared by the contributors.

The methodology described above is illustrated with two examples. These case-studies show how 3D GeoModelling is helpful to infer a democratic interpretation during the exploration phase. Beyond the interpretation for exploration, the geomodel can be enhanced during the next phases, when new data are acquired, to provide an up-to-date image of the investigated region. Such a 3D model can also be used to mesh the geometry of the zone and to compute dynamic simulations.

INTRODUCTION

The main objective of geothermal exploration is to target favourable zones for successful drilling in the scope of future resource exploitation. Drilling is the only way to access directly the subsurface structure and parameters, at least along the borehole (lithology, temperature, permeability, etc.) However, considering the high investment needed for a drilling campaign, various techniques are used to investigate the subsurface prior to any drilling (Flovenz et al., 2012). The major ones for interpreting geological structures of the underground are geological field observation, exhumed analogues and seismic or gravimetric measurements. Parameters such as temperature or permeability can be inferred from hydrogeology, geochemistry or geophysical electric methods investigations. Each of the methods described above gives a piece of knowledge. Gathering these pieces is a key for a comprehensive interpretation [START_REF] Noorollahi | GIS integration model for geothermal exploration and well siting[END_REF][START_REF] Shako | Geo-Scientific Data Integration to Evaluate Geothermal Potential using GIS[END_REF][START_REF] Witter | Integrated 3D Geophysical Inversion and Geological Modelling for Improved Geothermal Exploration and Drillhole Targeting[END_REF][START_REF] Higuchi | Integration of Geothermal Exploration Data and Numerical Simulation Data using GIS in a Hot Spring Area[END_REF][START_REF] Calcagno | Estimation of the deep geothermal potential within the Tertiary Limagne basin (French Massif Central): An integrated 3D geological and thermal approach[END_REF]. That is why, as much as possible, data coming from various disciplines are combined to enhance the understanding of a geothermal area. But often the more available are the sources of data; the more complex is the interpretation. In this paper we propose a collaborative way to combine multi-sources data to achieve an interpretation shared by the specialists of each discipline. The principle is illustrated by two case-studies.

DEMOCRATIC INTERPRETATION

The word "democratic" is used in this paper for synthetizing two major ideas. On one hand, the geothermal interpretation benefits from the input of multiple scientific fields. On the other hand, these disciplines collaborate for a cooperative and cohesive interpretation of the geothermal area. Such a shared interpretation will be easier to produce if the experts have a common platform to help them for exchanging.

3D GeoModelling

3D geological modelling is dedicated to the representation of the solid Earth using surface and underground data in a computer aided process [START_REF] Houlding | 3D Geoscience Modeling; Computer Techniques for Geological Characterization[END_REF][START_REF] Mallet | Geomodeling[END_REF][START_REF] Wu | An effective method for 3D geological modeling with multi-source integration[END_REF]. Numerous packages implement several methodologies to achieve 3D geological models. Indeed, 3D geomodelling is based on integrating and combining data and aims at representing 3-dimensional structures. In that sense, 3D geomodelling is a good candidate to help geothermal explorers.

We focus here on building 3D static models that represent a georeferenced interpretation of the subsurface structure. Data such as geological measurements on the field, gravimetric and seismic surveys can be combined to infer the geological formations and their shapes. The resulting structural framework can be the cradle for locating and associating results from electric methods and geochemistry to enhance the interpretation and design a conceptual model of the geothermal area.

Principle for a democratic interpretation

As stated above, 3D geomodelling enable multi-disciplinary data combination. In the meantime, we look for a way helping the experts to collaborate and -even more important -to interact. Building a 3D geomodel allows to shape and to visualize a current knowledge and to update it when new data or interpretation are available. The classical protocol followed to aggregate data is the workflow procedure [START_REF] Maxelon | A workflow to facilitate three-dimensional geometrical modelling of complex poly-deformed geological units[END_REF][START_REF] Moretti | Integrated exploration workflow in the south Middle Magdalena Valley (Colombia)[END_REF]). Let's take an example where four scientific fields contribute in the exploration phase: geology, gravimetry, magnetotellurics, and geochemistry. Typically, in a workflow, the contributions from each of the scientific fields are added one after the other in a sequential process to complete the final model (Fig. 1). Associating data in a workflow methodology satisfies the forecasted objective to combine multi-discipline contributions. However, scientific fields are used chronologically and independently; making difficult common reasonings. Moreover, the closer to the final model a scientific field is, the bigger impact it may have, i.e. "the one who is right is the last one who spoke".

Merging the contributions from the fields instead of simply aggregating them is a step forward. The interpretation of a given discipline may benefit from the interpretation of another discipline, with no chronological preference. For instance, the geological interpretation could benefit from the magnetotellurics interpretation or vice versa. In that case, the geomodel is no more only the final result of the integration process but the geomodel is the main and central object of the interpretation process (Fig. 2). The democratic interpretation (Fig. 2) is an iterative and interactive practice where making the geomodel is the way for crossinterpretation of the scientific disciplines. It requires an interactive geomodel generation platform to achieve such a cooperation in real time.

This principle is illustrated in the two following case-studies.

CASE-STUDIES

These case-studies regard the geothermal exploration phase. A multidisciplinary approach is used to complete a consensual geothermal interpretation. In both cases, 3D geomodels are used to make easier the interaction between the disciplines involved in the geothermal exploration.

Interpretation combining knowledge from geological field and seismic marine: The 3D fault model of the Bouillante geothermal province

The Bouillante area is located on the West side of the Guadeloupe Island (French West Indies); a region well known for geothermal resources. This sector is a key geodynamic area (e.g. [START_REF] Bouysse | Reconnaissance des structures tectoniques et volcaniques sous-marines de l'arc récent des Petites Antilles[END_REF][START_REF] Feuillet | Arc parallel extension and localization of volcanic complexes in Guadeloupe, Lesser Antilles[END_REF] at the junction of two regional fault systems: i) a major submarine N160°E strike-slip fault belonging to the normal-sinistral Montserrat-Bouillante-Les Saintes system, only detected offshore [START_REF] Thinon | Study of the Bouillante Bay (West Basse-Terre Island shelf): contribution of geophysical surveys to understanding of the structural context of Guadeloupe (French West Indies -Lesser Antilles)[END_REF], and ii) the western end of the interpretative ESE-WNW Bouillante-Capesterre normal fault which is probably a major fault of the E-W Marie-Galante graben system (Fig. 3a). At the junction, faults observed on the field mainly elongate along the E-W direction [START_REF] Bouchot | Assessment of the Bouillante geothermal field (Guadeloupe, French West Indies): toward a conceptual model of the high temperature geothermal system[END_REF] whereas offshore structures interpreted from marine seismic lines shows a larger range of directions [START_REF] Thinon | Study of the Bouillante Bay (West Basse-Terre Island shelf): contribution of geophysical surveys to understanding of the structural context of Guadeloupe (French West Indies -Lesser Antilles)[END_REF].

Onshore, data consist in faults observed and measured during field work (Fig. 3a). Due to outcrops poor accessibility, these data are mainly located on the coast where sea erosion is intense. Inland, some structures are interpreted from previous works mainly based on topography (e.g. [START_REF] Feuillet | Arc parallel extension and localization of volcanic complexes in Guadeloupe, Lesser Antilles[END_REF]. Offshore, location and dip of the faults are derived from the acoustic basement offset interpreted in 52 seismic profiles representing about 180 km long in total (Fig. 3b). The challenge is to mix knowledge from the sea and from the Island to infer a coherent structural model of the faults at the scale of the Bouillante geothermal province that includes both inland and sea domains. To achieve this goal, marine and ground geologists worked conjointly, applying the following methodology:

-E-W structures observed onshore were as much as possible a guide to interpret and to connect E-W structures offshore. -Faults crossing the coastline are identified both offshore and onshore to gather a unique set of data to model geometry (e.g. the Bouillante fault on Fig. 4a)

Finally, a coherent fault interpretation of the zone combining data and observations from the sea and from the ground is built through a 15 km x 16 km 3D model down to 2 km vertical extension (Fig. 4b). The Bouillante geomodel is achieved using the potential field interpolation and geological rules methods developed in BRGM [START_REF] Lajaunie | Foliation fields and 3D cartography in geology; principles of a method based on potential interpolation[END_REF][START_REF] Calcagno | Geological modelling from field data and geological knowledge, Part I -Modelling method coupling 3D potential-field interpolation and geological rules[END_REF] and implemented in the 3D GeoModeller (1) software. The 3D fault model of the Bouillante geothermal province presented here (Fig. 4b) is the final state of numerous iterations. Preliminary models were constructed to test various hypotheses for interpreting faults. These intermediary models were modified, rejected or refined conjointly by the experts but they are not shown here. Such an iterative process lead to the final model presented above. Even if the methodology is deterministic -i.e. a single model is presented -the evolution of the interpretation is intrinsic to the final model.

This work is fully detailed in [START_REF] Calcagno | A New 3D Fault Model of the Bouillante Geothermal Province Combining Onshore and Offshore Structural Knowledge (French West Indies)[END_REF] where the 3D interpretation of the faults is presented and discussed as well as the implication for exploring the Bouillante geothermal province.

Inter-disciplinary mutualisation: geology, gravimetry, magnetotellurics, and geochemistry

Note: This case-study refers to a confidential exploration project; let's call it the x-area. No mention of location or values of the parameters are displayed in the text and figures of this section. The main geological body in the x-area is a granitic formation. This case-study is used to illustrate a methodology that can be reproduced elsewhere.

The exploration of the x-area starts by a bibliographic compilation. Since the very beginning, a preliminary 3D geomodel is built to gather information on geological structures (Fig. 5a). This step allows to display the existing information in the same framework where the bibliographic data are checked and turned into a coherent interpretation. This preliminary 3D geomodel is used to prepare the fieldworks stage by helping to target interesting places for investigation.

Then, the acquisition on the field allows to enhance the previous geological interpretation. These new field data imply changes in the whole 3D model. Even if they are measured on the ground, they also impact the interpretation at depth. On the x-area, the location and dip of faults are revised, a contact aureole is observed around the granite body, springs are located, and an altered zone is identified and interpreted in the underground (Fig. 5b).

In the next step, a gravimetric survey is completed on the zone to obtain a Bouguer anomaly map of the x-area. Densities are associated to the geological formations of the geomodel. Then, the Bouguer anomaly is combined with the geological bodies of the geomodel in an inversion process based on the Monte Carlo algorithm [START_REF] Guillen | Geological modelling from field data and geological knowledge, Part II, Modelling validation using gravity and magnetic data inversion[END_REF]. The result of the inversion contributes to refine the 3D interpretation in a common work done by geologists and geophysicists, while keeping coherence with the observations of the previous stage (Fig. 5c).

A last survey is performed on the x-area to measure the magnetotelluric signal. Afterwards, data are turned into a 3D resistivity interpretation and input in the geomodel. Here, the cooperative interpretation derived from geology and gravimetry is upgraded with the resistivity contribution. On top of that, information from springs' fluid analysis helps the final interpretation. All the material is shared by geologists, geophysicists and geochemists. A conceptual model forecasting the hydrothermal behaviour of the x-area is collectively achieved by the experts (Fig. 5d). The interpretations and hypothesis made by the experts in their own fields were gathered, pooled, and confronted to lead to a consensual interpretation. The successive 3D geomodels of the x-area (Fig. 5) were completed using the potential field interpolation, the geological rules, and the gravity data inversion methodologies developed in BRGM [START_REF] Lajaunie | Foliation fields and 3D cartography in geology; principles of a method based on potential interpolation[END_REF][START_REF] Calcagno | Geological modelling from field data and geological knowledge, Part I -Modelling method coupling 3D potential-field interpolation and geological rules[END_REF][START_REF] Guillen | Geological modelling from field data and geological knowledge, Part II, Modelling validation using gravity and magnetic data inversion[END_REF]. They are implemented in the 3D GeoModeller (1) software.

CONCLUSION AND PERSPECTIVES

Several disciplines are involved in the exploration phase of a geothermal project. The more they can interact, the better is the final interpretation and the better constrained is the conceptual model of the geothermal area. It was shown here that achieving a 3D

geomodel is a good way to gather and share most of the information in a common platform. Moreover, starting this geomodel since the very beginning of the exploration allows to use it as a framework for planning field work and updating the interpretation of the area.

Beyond the multi-disciplinary integration, it has been demonstrated the added-value of experts working together to revise their own interpretation depending on the ones of the others. Instead of a sequential workflow process where the scientific fields are quite disconnected, it has been introduced the democratic interpretation concept where disciplines interact. The democratic interpretation leads to a cooperative interpretation validated by all the experts. Two examples -the Bouillante geothermal province and the so called x-area -have been used to illustrate how the mutual work on a shared 3D geomodel enhances the interpretation in the exploration phase.

In this light, the experts and their interaction are even more important than the data themselves. The extra time spent on the collaborative effort is awarded by a conceptual model shared and validated by the community. On a tool point of view, the approach requires a flexible and interactive platform to enhance in real-time the geomodel during common work sessions.

Other disciplines than the ones cited above can be integrated in the process; hydrogeology for instance. Even if some do not directly contribute to the design of the 3D geomodel, they can use it as knowledge support. In addition, a current model can be enhanced and turned into an up-to-date interpretation if new data are available after the exploration phase. Another perspective is to use such static 3D geomodel describing the inter-disciplinary interpretation for dynamic simulation purpose.

Figure 1 :

 1 Figure 1: Workflow is the classical way for combining data. The 3D geomodel is the final result of sequential interpretations.

Figure 2 :

 2 Figure 2: The 3D geomodel is the central part of the democratic interpretation. Scientific fields cooperate and interact to enhance the interpretation.

Figure 3 :

 3 Figure 3: Example of the onshore and offshore knowledge in the Bouillante geothermal province. a. Picture of the Anse Machette inland outcrop at the north end of the Bouillante bay (Bouchot et al., 2010); view from W, see box in b. for location. Box: Regional structural setting of the Guadeloupe Island (modified from Thinon et al., 2010). Thick square: location of the 3D fault model. BCF: Bouillante-Capesterre fault system. b. A Geoberyx03 very high resolution marine seismic profile opposite to the Bouillante bay; see box on the left hand side (Thinon et al., 2010).

Figure 4 :

 4 Figure 4: Two views of the fault model achieved for the Bouillante geothermal province (see Fig. 3a for location). The coastline (white crosses) separates onshore and offshore domains. a. The Bouillante fault (yellow surface) is constructed from onshore field observation (dark blue point), onshore interpretation (red points) and offshore seismic profiles interpretation (light blue points). View from WSW. b. The final 3D interpretation reveals three main families of faults (yellow, green, and blue). View from SSE.

Figure 5 :

 5 Figure 5: The x-area is modelled in 3-dimensions. The progress in the interpretation is illustrated here in the same vertical cross-section. The main geological feature is a granitic body (purple). Light blue dots locate the springs observed in the field. Stars show the mutual involvement of experts and data: geology (green), gravimetry (white), magnetotellurics (red), geochemistry (blue). a. The preliminary 3D geomodel is built from bibliography knowledge. b. The geological data acquired in the field are used to refine the interpretation. Fault dips are changed, an altered area is interpreted in the middle of the section (red border and lighter colours), and springs are located. c. The gravimetric survey inversion leads to shorten the "nose" of the granite, and to interpret a new basement (brown). d. Resistivity computed from a magnetotelluric 3D survey is superimposed to the geological/gravimetric interpretation (coloured curves) to infer a conceptual model of the x-area, shared by all the experts.

ACKNOWLEDGEMENTS

This work is supported by IMAGE (http://www.image-fp7.eu). This project has received funding from the European Union's Seventh Programme for research, technological development and demonstration under grant agreement No: 608553.

The work regarding the Bouillante geothermal province was supported by ADEME (French Agency for Energy and Environment): contract 0805C0044 (GHEMOD) and contract 0805C0039 (GEO3BOU).

FOOTNOTE

(1) 3D GeoModeller is a commercial software developed by BRGM and Intrepid Geophysics. For further information visit: http://www.geomodeller.com.