Salma Doghraji

Marc Donias

Sébastien Guillon

FREE-SHAPED POLYGONAL SEISMIC HORIZON RECONSTRUCTION USING SPACE TRANSFORMATIONS

Keywords: Seismic horizon reconstruction, faulted seismic image, partial derivative equation, space transformation, Schwarz-Christoffel transformation

In the context of faulted or occluded regions in seismic images, seismic horizon reconstruction requires the use of fast interactive approaches with respect to any bounding domain and number of passing points. The only algorithms which respect these constraints and provide a result in reasonable time in comparison with the interaction are based on the solution of a partial derivative equation either on juxtaposed quadrangular regions or using a binary mask on the whole data. While the first method requires the use of many passing points and its local nature leads to different results depending on the number and location of constraints, the second is not compatible with fast reconstruction for more than one passing point. In this paper, we propose a global and fast reconstruction method on the polygon defined by constraint points. Direct and inverse Schwarz-Christoffel space transformations lead to global reconstructions in respect to all the constraints. Experiments both on synthetic and real seismic images exhibit better results than the conventional quadrangle and mask methods.

INTRODUCTION

Seismic images are three-dimensional directional textures representing a stacking of sedimentary layers which appear as a succession of bright and dark surfaces (see figure 1). One key step of interpreting the structure of the subsoil is to reconstruct the frontiers between sedimentary layers, also known as seismic horizon. However, the interpretation process is tedious because of the data dimensionality, seismic faults due to tectonic forces causing horizon disruptions and deaf or chaotic regions. An automatic tool of seismic horizon reconstruction is needed to assist geophysicists during the interpretation stage.

Numerous surface reconstruction methods have been proposed in the literature during the last three decades. Region-growing methods can be based on the measurement of shape similarity [START_REF] Borgos | Automated geometry extraction from 3d seismic data[END_REF][START_REF] Pauget | A global approach in seismic interpretation based on cost function minimization[END_REF] or the integration of the estimated dip [START_REF] Bienati | Traveltime picking in 3-d data volume[END_REF][START_REF] Blinov | Reconstruction of 3d horizons from 3-d seismic datasets[END_REF] starting from an initial point. These methods generally drift because of error cumulation. Methods based on an overall minimization of a nonlinear Partial Derivative Equation (PDE) over the whole solution domain [START_REF] Kass | Snakes: Active contour models[END_REF][START_REF] Osher | Fronts propagating with curvature-dependent speed: algorithms based on hamiltonjacobi formulations[END_REF][START_REF] Lomask | Flattening with geological constraints[END_REF][START_REF] Wu | Horizon volumes with interpreted constraints[END_REF][START_REF] Zinck | Reconstruction d'hypersurfaces de champs de normales sous contraintes -Application à l'analyse stratigraphique des images sismiques[END_REF] are less noise sensitive. Fig. 1. Seismic horizon (in cyan) reconstructed over seismic image However, presence of faults, chaotic, noisy or missing regions, requires adding as many passing or domain points as needed. Therefore, only supervised methods allowing a fast reconstruction, which is generally carried out in the Fourier domain and requires a rectangular reconstruction domain, can be used. Moreover, a rectangular domain is not suitable in case of problematic regions. It has to be replaced by a freeshaped domain which occluds or circumvents faulted or problematic regions. Methods of weighted reconstruction can be used on a rectangular domain that surrounds the free-shaped domain by using a weighting function [START_REF] Lomask | Flattening with geological constraints[END_REF] and a particular case using a binary mask [START_REF] Zinck | Reconstruction d'hypersurfaces de champs de normales sous contraintes -Application à l'analyse stratigraphique des images sismiques[END_REF]. However, in the context of fast reconstruction schemes which are based on the Fourier transform, weighted reconstruction methods cannot be used with multiple passing points. Moreover, they do not guarantee the convergence of the solution when the normal vector field estimate is highly corrupted by discontinuities and can also introduce numerical instability as well as high computational time. In the general case of supervised methods allowing rapid and interactive consideration of multiple passing points, Zinck et al. [START_REF] Zinck | Fast seismic horizon reconstruction based on local dip transformation[END_REF] proposed an approach where the reconstruction domain is defined by the convex polygonal envelope of passing points. Each triangle from Delaunay triangulation of the domain is subdivided into three quadrangles. A partial reconstruction is then performed on each quadrangle by space transformation. This method presents drawbacks due to its local nature. Noise sensitivity depends on the dimensions of each quadrangle and the partial reconstructions derivative continuity is not guaranteed at their boundaries. Moreover, when the configuration changes by adding, changing or deleting a passing point, all the partial reconstructions need to be recomputed.

In [START_REF] Doghraji | Seismic horizon reconstruction on polygonal domains using the schwarz-christoffel transformation[END_REF], we proposed to reconstruct seismic horizons on any simply connected polygonal domain deduced from the free-shaped domain under the constraint of one passing point, using Schwarz-Christoffel (SC) transformations. In this paper, we present the generalization to seismic horizon reconstruction on any simply connected polygonal domain under any number of passing points constraints. Moreover, we discuss SC grid regularization when the SC transformations lead to over-sampled regions. The reconstruction is carried out at once on the whole domain of interest in the Fourier domain using the space transformation and does not extend it to make it convex. In this sense, it is more stable and can be qualified as global.

In the rest of this paper, we first present seismic horizon reconstruction under any kind of bijective differentiable space transformation. Then, we discuss the space transformation between a polygonal domain and rectangular domain before presenting the grid regularization transformation to improve the regularity of the grid corresponding to the SC transformation. In section 3, we compare results of our method to those of the mask and quadrangle methods both on synthetic and real seismic images.

SEISMIC HORIZON RECONSTRUCTION IN PRESENCE OF PROBLEMATIC REGIONS

The reconstruction of seismic horizons using the method of [START_REF] Lomask | Flattening with geological constraints[END_REF] is based on the solution of a non linear constrained optimization problem. It relies on the orientation field of the seismic image and refers to a Poisson PDE. The solution of the PDE in the Fourier domain can only be carried out on a hyper-rectangular domain. However, to avoid faults neighboring or occluded regions, the horizon reconstruction support does not, unless in very specific cases, have a rectangular aspect. It is most likely for the reconstruction domain to have a free polygonal shape.

Seismic horizon reconstruction under a space transformation

In a seismic image, each seismic horizon can be represented by an explicit hypersuface τ 1 defined by x N = τ 1 (x) upon the domain Ω 1 where x = (x 1 , . . . , x N -1 ) in a N -dimensional vector space. The hypersurface τ 1 is orthogonal at each point to the orientation field n [START_REF] Zinck | Reconstruction d'hypersurfaces de champs de normales sous contraintes -Application à l'analyse stratigraphique des images sismiques[END_REF] (see figure 2). An explicit hypersurface τ 1 verifies a non-linear PDE linking the gradient ∇(τ 1 ) to the vector of slopes p 1 deduced from the orientation field n [START_REF] Lomask | Flattening with geological constraints[END_REF]:

∀x 1 = (x 1 1 , . . . , x 1 N -1 ) ∈ Ω 1 , ∇τ 1 (x 1 ) = p 1 (x 1 , τ 1 (x 1 )). (1) where ∀i ∈ [1, N -1], p 1,i = - n i (x 1 ) n N (x 1 ) . ( 2
)
x 1 x 2 O Fig. 2

. Explicit hypersurface of a bidimensional orientation field

The seismic horizon is iteratively reconstructed using a non-linear constrained optimization problem:

τ 1 = argmin g∈C 2 Ω1 ∇g(x 1 ) -p 1 (x 1 , g(x 1 )) 2 . (3)
Equation 3 refers to a Poisson equation:

∆(δg(x 1 )) = -div(∇g(x 1 ) -p 1 (x 1 , g(x 1 ))). (4)
In general, the minimization of the term in 3 is done using matricial methods. But in the particular case of a rectangular domain, the solution is rapidly deduced using Fourier transforms.

In order to transform an implicit hypersurface into an explicit one or to transform a domain, a space transformation F from x 1 into x 2 can be introduced. Under the conditions of F being bijective and of class C 1 , a reconstruction is obtained [START_REF] Zinck | Ndimensional surface reconstruction from a noisy normal vector field[END_REF] using equation ( 1) expressed in the transformed space:

∀x 2 ∈ Ω 2 , ∇τ 2 (x 2 ) = p 2 (x 2 , τ 2 (x 2 )).
(

) 5
Slope vectors are deduced from the components of the orientations n j (F -1 (x M )) in the initial space:

∀i ∈ [1, N -1], p 2,i = - N j=1 ∂x 1 j ∂x 2 i (x 2 ).n j (F -1 (x 2 )) N j=1 ∂x N j ∂x 2 i (x 2 ).n j (F -1 (x 2 ))
.

In case of a chain of M bijective space transformations

F = f 1 • . . . • f M from x 1 to x M+1
, the components of the slopes vectors are expressed as:

∀i ∈ [1, N -1], p M +1,i = - n M +1 i (x M+1 ) n M +1 N (x M+1 ) , (7)
where

x i = (f M •. . .•f i ) -1 (x M+1
). The orientation vectors are expressed as:

n M+1 = J M M +1 (x M+1 )J M -1 M (x M ) . . . J 1 2 (x 2 )n(x 1 ), (8)
and introduces the terms of the Jacobian

J i i+1 of inverse space transformations f -1 i i∈[1,M ] : ∀i ∈ [1, M ], J i i+1 =      ∂x i 1 ∂x i+1 1 . . . ∂x i N ∂x i+1 1 . . . . . . . . . ∂x i 1 ∂x i+1 N . . . ∂x i N ∂x i+1 N      . (9)
In the following section we present the expression of the bijective transformation between the polygon defined by the constraints and the rectangular domain for fast PDE solution.

Transformation between a polygon and a rectangle

The Riemann mapping theorem guarantees the existence of a biholomorphic mapping f from any non-empty simply connected subset of C onto the unit disc D. This implies that f is angle-preserving and is, therefore, a conformal transformation. The SC transformation which was independently discovered by the german mathematicians Christoffel and Schwarz in late 1860s defines an analytic expression of this conformal transformation [START_REF] Driscoll | Schwarz-Christoffel Mapping[END_REF]. Ever since, it has been used in the solution of Laplacian PDE in studies such as Brownian motion [START_REF] Trefethen | Conformal mapping solution of laplace's equation on a polygon with oblique derivative boundary conditions[END_REF] and wind flow [START_REF] Waters | Modeling huddling penguins[END_REF].

The transformation of the polygonal into the rectangular domain can be seen as the compose of two SC transformations which is also a conformal transformation. The first is the inverse SC transformation f -1 P from the polygon P into the unit disc D and the second is the direct SC transformation f R of D into the rectangle R:

F = f -1 P • f R . ( 10
) -β 1 π -β 5 π -β 6 π -β 4 π -β 3 π -β 2 π w 1 w 2 w 3
w 4 w 5 w 6 P Fig. 3. Polygon P defined by its vertices (in cyan) and exterior angles Let P be a polygon defined by its vertices w 1 , w 2 , . . . , w N (see figure 3). Each vertex w k has an exterior angle of -β k π.

Note that

N k=1 β k = -2.
The SC transformation formula defines the conformal transformation of the unit disc D into the polygon P as fol- 5. Transformation of a rectangle into a polygon without regularization and with regularization h D based on the SC transformation of the polygon (in dotted frame) lows:

f P f R f -1 P f -1 P f P f -1 P f P h D h -1 D Fig.
∀z ∈ D, f P (z) = a + c z 0 N i=1 (1 - s z k ) β k ds, (11)
where a, c ∈ C and {z k } k= [1,N ] are the pre-images of the vertices of P and are called prevertices. The main task in the definition of the SC transformation is the estimation of the previous N + 2 parameters. This is known as the SC parameters problem and is at the heart of the numerical computation of the SC transformation especially as the number of vertices gets higher. However, Trefthen [START_REF] Driscoll | Schwarz-Christoffel Mapping[END_REF] proposed a fast method of numerical solution based on side-length equations after fixing the values of a and c as a = f P (w 1 ) and c = f P (w N ):

z k+1 z k f P (z)dz z 2 z1 f P (z)dz = |w k+1 -w k | |w 2 -w 1 | , k = 1, 2, . . . , N -1 (12)
which is solved using the Gauss-Newton method and a Broyden jacobian update. Integration is computed using Gauss-Jacobi method.

The inverse SC transformation corresponds to the computation of f -1 P (w) which is obtained from the differentiation and inversion of w = f P (z):

dw dz = c N i=1 (1 - s z k ) β k ⇒ dz dw = 1 c N i=1 (1 - s z k ) -β k . ( 13
)
This equation is solved numerically using the Newton iterative method starting from a rough approximation z of the solution obtained from a Runge-Kutta initialization.

Figure 4 shows the SC transformations between an Lshaped polygon P and a rectangle R under the space transformation F. The SC transformations of the rectangle lead to a highly irregular grid on the polygonal domain. Orientations at some points of the initial grid in the polygon P are not used while others have multiple contributions. This leads to variable reconstruction precision across the domain.

Grid regularization

An intuitive solution would be to increase the number of sampling points. For instance, if the reconstruction error using a grid of N P points is ε, then 4 × N P points should be considered to achieve an ε 2 error. In general, to achieve an ε 2 n , n ∈ N error, 2 2n × N P should be considered. This leads to a very high computational complexity compared to the reconstruction accuracy improvement.

Therefore, we introduce an intermediate space transformation h D between the direct and inverse SC transformation in order to regulate the grid without losing the domain geometry or changing the resolution grid size N P .

Let us consider the polar coordinates (ρ in , θ in ) of the grid corresponding to the transformation f -1 P of the regular grid of P (see figure 5). The objective is to estimate the parameters of a transformation that regularizes the distribution of the disk D in the polar domain without altering the image of the polygon vertices. Therefore h D : (ρ in , θ in ) → (ρ out , θ out ).

θ out = θ in + N k=1 u(β k )N ∫ (β k , σ θ k ), (14)
where N ∫ (µ, σ) is a folded normal-distribution of mean µ and standard-deviation σ and the function u is defined by analyzing the angular distribution θ in . Noting β N +1 = β 1 , the function u can be written:

u(β k ) =      1 if θ in is increasing on [β k , β k+1 ] -1 if θ in is decreasing on [β k , β k+1 ] 0 else (15)
In the same manner, the radial distribution is changed when the angular distribution is, unless the radius is equal to 1:

ρ out = ρ in + N k=1 |u(β k )|N (ρ k , σ ρ k ), (16)
where N (µ, σ) is a normal-distribution of mean µ and standard-deviation σ, ρ k = 1+ρ k min 2

, ρ kmin is the minimal radius corresponding to the angles

β out = β in on [β k , β k+1 ] and σ ρ k = 1-ρ k
3 . This grid regularization method proves to be efficient, however its implementation can be labourious because of the empiral choice of parameters u(β k ), σ θ k , σ ρ k which are relative to the number of vertices N .

Computational complexity

Table 1 shows SC transforms computational complexity for a grid of N P points. The direct transform has roughly a numerical complexity of O(N P ). The inverse transform is first based on the computation of the direct transform and the inversion has a complexity of O(N P × N ).

Stage Method Complexity

Parameter problem Gauss-Jacobi integration N P Broyden Jacobian update √ N P Integration Gauss-Jacobi integration N P Inversion Runge-kutta initialization N P Newton iteration N P × N Table 1. Computational complexity of the steps of direct and inverse SC transformations

The angular and radial regularization have O(N ) complexity. The SC direct and inverse transformations are computed once to estimate the terms x i , i ∈ [2, M + 1]. The Jacobians of the direct and inverse SC transformations are computed by differentiation. Both transformations are evaluated 4 times for that matter. The reconstruction has a N P log(N P ) computational complexity. The proposed method has, therefore, N p (5(N + 1) + log N p ) complexity.

This complexity is to compare to the quadrangle method for which the reconstruction has

3(N -2) k=1 N Q k log(N Q k )
where N Q k is the number of grid points in the k th quadrangle. For each quadrangle the space transformation and corresponding Jacobian need to be computed. They have linear complexity.

Overall, both methods have comparable computational execution time in practice.

RESULTS

Results on a synthetic seismic image

First, results are presented on a synthetic 300 × 300 × 300 image with high variations (see figure 6). The theoretical fault appears in red and some theoretical horizons are presented in blue.

The reconstruction using the proposed method has a polygonal domain (see figure 7) taken in order to circumvent the fault. The mask for the method of [START_REF] Zinck | Reconstruction d'hypersurfaces de champs de normales sous contraintes -Application à l'analyse stratigraphique des images sismiques[END_REF] is defined by setting to 0 the contribution of external points to the polygon in the corresponding rectangular domain.

This results in a shifted horizon using the mask method since the normals are disturbed by the presence of the fault. Even though the reconstruction is less accurate in regions where the SC transformation results in a larger sampling, the Fig. 6. Faulted synthetic seismic volume, fault model (in red) and some horizons (in blue) Fig. 7. Top: polygonal reconstruction domain defined by the vertices (in blue) projected on the analytic surface to reconstruct. Bottom: faulted seismic horizon reconstructions -theoretical (in blue), using the mask method (in green) and the proposed method (in indigo) -on the 20 th x 2 -section proposed method leads to better results. The mean quadratic error term compared to the analytic horizons is 0.18 pixels using the proposed method when the binary mask method results in a mean quadratic error of 0.97 pixels.

Results on real seismic images

In this section, we consider a 329 × 376 × 195 faulted seismic image with regions of occlusion (see figure 8). For the remainder of this paper, each voxel corresponds to a physical dimension of 10 meters in each direction. Since both the faults and the occluded regions are quasi-vertical, we reconstruct the seismic horizon on the domain in figure 8 and one passing point on coordinates (194,208,90). We compare the results of our method to those of the mask method with the same passing point and binary mask defined by the polygon of figure 8. 9 show that the reconstructions using the PDE method without and with mask method lead to surfaces that drift away from the observed horizon after the faulted region when our method leads to good results all over the reconstructed region. This is due to the fact that even though the same points are used for the mask and the proposed methods, they do not have the same neighboring. The highest reconstruction differences are located on the sections crossed by faults with a mean absolute difference of 2.43 pixels compared with the mask method and 3.16 pixels without mask.

In the rest of this section, we compare our method to the quadrangle method for a 400 × 350 × 400 real seismic image which contains faults and complex trench structures (figure 10). Fig. 9. Faulted seismic horizon reconstructions using the classic PDE method without mask (in pink), with mask (in green) and the proposed method (in blue) on the 172 th x 2 -section The reconstruction domain could be defined automatically by detecting fault position using orientation coherence, gradient magnitude or even trace similarity, as some authors proposed in the litterature [START_REF] Marfurt | 3-d seismic attributes using a semblance-based coherency algorithm[END_REF][START_REF] Randen | Automatic extraction of fault surfaces from three-dimensional seismic data[END_REF][START_REF] Gibson | Fault surface detection in 3-d seismic data[END_REF][START_REF] Hale | Methods to compute fault images, extract fault surfaces, and estimate fault throws from 3d seismic images[END_REF] combined with fault slip estimation as in [START_REF] Wu | 3d seismic image processing for faults[END_REF]. In this section, we suppose that the fault location is already known and that the reconstruction domain is defined by passing points of figure 10 which form a non-convex polygon of 9 vertices. The orientation field is estimated using the conventional structure tensor method with standard-deviations of 1 and 2 for the gradient field estimation and autocorrelation matrix computation respectively. Reconstructions are obtained after 40 iterations in the Fourier domain. For the proposed method, the reconstruction has 410×320 sampling points. The obtained quadrangles using Delaunay triangulation are presented in figure 11. The non-convex domain is transformed into a convex one by adding the region in red which leads to additional partial reconstructions. The sampling grid is imposed by the largest side of the quadrangle which leads to dense but non-homogeneous resolution along partial reconstructions (see figure 12). Figure 13 shows the results of the reconstruction using our quasi-global approach compared with the quadrangle method on different inlines. It appears that the local aspect of the quadrangle method leads to reconstructions that drift away from the observed horizon. SC transformations lead to undersampled regions (to the left of the polygonal domain). This affects reconstruction accuracy in these regions. Our method with grid regularization leads to better horizon reconstruction and corrects reconstruction errors in under-sampled regions with the SC transformations.

The absolute difference between our method and the quadrangle method highlights up to 11 pixels difference locally and a mean absolute difference of 2.48 pixels on this domain.

CONCLUSION

We proposed in this paper a fast and global method for seismic horizon reconstruction on any simply connected polygonal domain defined by passing points constraints. Our approach is based on an overall minimization of a partial derivative equation combined with direct and inverse Schwarz-Christoffel transformations beside a bijective grid regularization transformation. Results both on synthetic and real seismic images show better horizon reconstruction when compared with the mask and quadrangle method.

A possible improvement could be carried out by automatic detection and correction of regions where the grid is not regular enough by adopting a regularity criteria and using gradient descent methods.

The simply connected aspect of the domain implies that the fault can be circumvented by a simply connected polygon and that the fault does not bisect the reconstruction domain. If the fault does not intersect any edge of the reconstruction domain, the corresponding polygon would be doubly connected. We address this limitation in [START_REF] Doghraji | Caractérisation de la géométrie locale et globale de textures directionnelles par reconstruction d'hypersurfaces et transformations d'espace -Application à l'analyse stratigraphique des images sismiques[END_REF] using the Schwarz-Christoffel transformation expression in case of a doubly connected polygon. In case the domain is multiply connected, one solution would be to use the expression of the Schwarz-Christoffel transformation for multiply connected domains. Although computational complexity of estimating the parameters of this transformation is large, numerical estimation is likely to be possible in a reasonable amount of time within the relatively near future.

Fig. 4 .

 4 Fig. 4. Transformation of a rectangle R into an L-shaped polygon P using SC transformations

Fig. 8 .

 8 Fig. 8. Faulted seismic image with occluded and faulted regions and polygonal domain

Fig. 10 .

 10 Fig. 10. Seismic image sections and seismic horizon reconstructed on the polygonal domain defined by passing points (in cyan). x 2 -sections of figure 13 (in dotted lines) .

Fig. 11 .Fig. 12 .

 1112 Fig. 11. Quadrangles deduced from Delaunay triangulation of passing points (in blue) -quadrangles inside concave polygone (in green) and outside (in red) -

Fig. 13 .

 13 Fig. 13. Reconstructed seismic horizons using the quadrangle method (in green) and our method without (in indigo) and with (in blue) regularization on 135 th (a), 170 th (b) and 200 th (c) x 2 -sections

AKNOWLEDGEMENT

The authors would like to thank TOTAL company for funding this research and providing seismic data.


Sylvain Kubler
email: s.kubler@univ-lorraine.fr

Éric Rondeau
email: eric.rondeau@univ-lorraine.fr

Jean-Philippe Georges
email: jean-philippe.georges@univ-lorraine.fr

Phoebe Lembi Mutua
email: phoebe-lembi.mutua5@etu.univ-lorraine.fr

Marta Chinnici
email: marta.chinnici@enea.it

Benefit-Cost model for comparing data center performance from a Biomimicry perspective

Keywords: Green computing, Green networking, Sustainability, Multiple criteria decision-making, Analytic hierarchy process (AHP), Biomimicry

Data centers are estimated to have the fastest growing carbon footprint from across the whole information and communication technology (ICT) sector. Evaluating the performance of data centers in terms of energy efficiency and sustainability is becoming an increasingly important matter for organizations and governments (e.g., for regulation or reputation purposes). It nonetheless remains difficult to achieve such evaluation, as data centers imply to take into consideration a wide range of dimensions and stakeholders. Even though a wide range of sustainability performance indicators exist in the literature, there is still a lack of frameworks to help data center stakeholders (spanning from data center owners, governmental regulators to engineers/field operators) to evaluate and understand how a data center performs in terms of sustainable development/behavior. Our research work proposes such a framework, whose originality lies in the combination of state-of-the-art sustainability metrics with the biomimicry commandments of eco-mature system, which enables holistic sustainability assessment of data centres. From a theoretical perspective, the proposed model is designed based on a benefit-cost analysis using the Analytic Hierarchy Process (AHP) technique. This approach allows data center stakeholders for specifying their own preferences and/or expertise in the comparison process, whose practicability is demonstrated in this paper considering three data center candidates, which are respectively located in France, Germany and Sweden.

Introduction

The rapid growth of data centers is driven by the increasing need for fast and efficient data processing and storage services [START_REF] Dandres | Consideration of marginal electricity 1116 in real-time minimization of distributed data centre emissions[END_REF][START_REF] Belkhir | Assessing ict global emissions foot-1119 print: Trends to 2040 & recommendations[END_REF]. In a context where data centers are becoming the power plants of the information age, one of the greatest challenges is the exponential rise in energy consumption, as they must operate 24/24 hours and 7/7 days [START_REF] Kurkjian | Meeting the needs of 24/7 data centers[END_REF]. At the time of writing, the ICT sector contributes to 2% of the global carbon emissions with data centers accountable for 14% of the total ICT footprint [START_REF] Mills | The cloud begins with coal-an overview of the 1124 electricity used by the global digital ecosystem[END_REF]. It therefore becomes crucial that the next generation of data centers integrates solutions that can lower the total cost of ownership, while decreasing the complexity of management [START_REF] Reddy | Metrics for sustainable data centers[END_REF].

Towards this goal, data center owners and operators should put in place measures to evaluate the performance of their facilities/services [START_REF] Fiandrino | Per-1130 formance and energy efficiency metrics for communication 1131 systems of cloud computing data centers[END_REF][START_REF] Ni | A review of air conditioning energy perfor-1134 mance in data centers[END_REF]. However, How to provide data center stakeholders with a comprehensible and holistic model for comparing the overall sustainability of their data center ? Objective of this paper M X , k: Refers to a domain-specific metric, X referring to the application domain and k to the k th metric that domain-related experts are acquainted with.

Figure 1: A need of comprehensible schemes/models to help non-expert to evaluate and compare data center performance gories of stakeholders, with different areas and levels 42 of expertise, can make sense of all these metrics at 43 once? This motivation question, which is also stressed 44 by other studies [START_REF] Garimella | Technolog-1165 ical drivers in data centers and telecom systems: Multiscale 1166 thermal, electrical, and energy management[END_REF][START_REF] Arianyan | Novel energy and sla 1169 efficient resource management heuristics for consolidation of 1170 virtual machines in cloud data centers[END_REF][START_REF] Tseng | Dynamic resource prediction and allocation for cloud 1174 data center using the multiobjective genetic algorithm[END_REF], is at the origin of this re- Problem tackled Sustainability-related criteria [START_REF] Abdel-Basset | Nmcda: A frame-1199 work for evaluating cloud computing services[END_REF] 2018 Cloud service selection - [START_REF] Jatoth | Selcloud: 1202 a hybrid multi-criteria decision-making model for selection of 1203 cloud services[END_REF] 2018 Cloud service selection - [START_REF] Jatoth | Evaluating the ef-1205 ficiency of cloud services using modified data envelopment 1206 analysis and modified super-efficiency data envelopment anal-1207 ysis[END_REF] 2017 Cloud service selection - [START_REF] Liu | Decision making for the se-1209 lection of cloud vendor: An improved approach under group 1210 decision-making with integrated weights and objective/sub-1211 jective attributes[END_REF] 2016 Cloud service selection - [START_REF] Lee | A hybrid multi-criteria decision-making 1214 model for a cloud service selection problem using bsc, fuzzy 1215 delphi method and fuzzy ahp[END_REF] 2016 Cloud service selection - [START_REF] Boutkhoum | Selec-1218 tion problem of cloud solution for big data accessing: fuzzy 1219 ahp-promethee as a proposed methodology[END_REF] 2016 Cloud service selection - [START_REF] Xu | A non-parametric data envelopment 1222 analysis approach for cloud services evaluation[END_REF] 2015 Cloud service selection - [32] 2015 Site selection PUE [START_REF] Liu | Ontology-based service matching in cloud computing[END_REF] 2014 Cloud service selection - [START_REF] Kwon | A decision-making model to choose a cloud service using fuzzy ahp[END_REF] 2013 Cloud service selection - [START_REF] Daim | Site selection for a data centre -a multi-criteria decision-making model[END_REF] 2013 Site selection Power availability, Water availability, Laws related to Urban Planning [START_REF] Covas | Multicriteria decision analysis for sustainable data centers location[END_REF] 2013 Site selection Renewable Energy sources, Free cooling, Local environmental impact, Local pollution [START_REF] Alabool | Trust-based service selection in public cloud computing using fuzzy modified vikor method[END_REF] 2013 Cloud service selection Sustainability [START_REF] Su | Improving cloud computing service in fuzzy environment-combining fuzzy danp and fuzzy vikor with a new hybrid fmcdm model[END_REF] 2012 Cloud service selection - [START_REF] Karim | An end-to-end qos mapping ap-1250 proach for cloud service selection[END_REF] 2013 Cloud service selection - [START_REF] Garg | A framework for ranking 1253 of cloud computing services[END_REF] 2013 Cloud service selection DCiE, PUE, DPPE [START_REF] Silas | Efficient service selec-1256 tion middleware using electre methodology for cloud environ-1257 ments[END_REF] 2012 Cloud service selection cover their best suited services. Recently, Abdel-Basset et al. [START_REF] Abdel-Basset | Nmcda: A frame-1199 work for evaluating cloud computing services[END_REF] presented an original approach in the sense that the authors do not only focus on evaluating the quality of cloud services, but also on improving the service quality by creating a competition between cloud providers. This competition is based on their level of security, performance, accessibility, scalability and adaptability.

Overall, the above literature review reveals that even though the sustainability dimension is covered in a few studies (cf., last column of Table 1), it is too often omitted, or, when considered, the sustainability metrics remain limited. For example, only DCiE (Data Center Infrastructure Efficience), DPPE (Data Center Performance per Energy) and PUE are considered in [START_REF] Garg | A framework for ranking 1253 of cloud computing services[END_REF], while only the latter is considered in [32].

Other studies, such as [START_REF] Alabool | Trust-based service selection in public cloud computing using fuzzy modified vikor method[END_REF], state that the sustainability is covered but without providing any detail about what sustainability is referring to. Another aspect evidenced by this review is that most of the proposed models have focused on cloud service and site selection so far, but none has ever proposed a comprehensible and holistic model that helps non-expert users in comparing the overall sustainability of data centers.

In this paper, we seek to contribute to proposing such a model, as will be presented in the next section.

Sustainability metrics

Over the decades, a wide range of sustainability metrics for data centers and cloud computing solutions have been introduced [START_REF] Grishina | Energy-oriented analysis of hpc cluster queues: Emerging 1271 metrics for sustainable data center[END_REF][START_REF] Grishina | Dc energy data measurement and analysis for productivity and waste energy assessment[END_REF][START_REF] Lykou | A new methodology toward effectively assessing data center sustainability[END_REF][START_REF] Riekstin | A survey on metrics and measurement tools for sustainable distributed cloud networks[END_REF][START_REF] Whitehead | Assessing 1139 the environmental impact of data centres part 1: Background, 1140 energy use and metrics[END_REF][START_REF] Beitelmal | Servers and data centers energy performance metrics[END_REF].

At the time of writing, and to the best of our knowl-edge, the most extensive study of sustainability met-262 rics is given by Reddy et al. [START_REF] Reddy | Metrics for sustainable data centers[END_REF], who gathered and re- • Recycling: evaluates waste resource utilization efficiency of energy and materials in data centers;

• IT Performance: evaluates the extent to which the system efficiently executes the IT tasks, which can be related to the (i) Network, (ii) Storage, or (iii) Security;

• Water: this is an important resource for consideration when designing, identifying ideal location for a data center. Energy and Water are the main metrics to be controlled in a data center [START_REF] Ristic | The water footprint of data centers[END_REF];

• Financial: increased resource utilization can contribute to high energy costs and maintenance of IT equipment. Organizations should therefore assess its performance for saving money.

Table 3 highlights, for each category and subcategories, what metrics must be minimized or maximized in order to "Benefit" the efficiency of data centers, while highlighting -through the "Cost" categorization -whether this can lead to negative efficiency impacts. Indeed, many contradictions may come up when studying ecological systems, for example: (i) the electricity produced by nuclear power plants must be limited when considering the GEC (Green Energy)

metric, but at the same time should be increased with regard to CUE (Carbon), or (ii) the water usage must be mitigated when only analysing WUE (Water), but should be increased when looking at GEC (Green). In the following, we seek to specify the extent to which a given category may positively or negatively impact (benefit-cost) on each of the commandments.

To this end, we propose to first model the impact that one biomimicry commandment may have on the others, as given in Figure 3(a). Let us take the modelling related to the "Energy" category, as given in

515

To serve this purpose, a two-hierarchy approach is 516 adopted, as proposed in [START_REF] Anagnostopoulos | A fuzzy multicriteria 1177 benefit-cost approach for irrigation projects evaluation[END_REF]. Concretely, two dis- Let (C) z×z be the pairwise comparison matrix at level 1, as formalized in Eq 1, where c i j (i, j ∈ C) is supposed to reflect how many more times commandment i is preferred -or deemed more important by the data center stakeholder -over commandment j. In our study, the stakeholder's evaluation is carried out based on the Saaty's scale: {1, 3, 5, 7, 9}, where c i j = 1 means that C i and C j are of equal importance and c i j = 9 means that C i is strongly preferred over

C j . (C) z×z =                  C 1 C 2 . . . C z C 1 c 11 c 12 . . . c 1z C 2 c 21 c 22 . . . c 2z . . . . . . . . . . . . . . . C z c z1 c z2 . . . c zz                  (1)
The

E C i = z j=1 c i j 1 z z i=1 z j=1 c i j 1 z (2)
E (C) z×z = E (C) z×z ,1 , E (C) z×z ,2 , . . . , E (C) z×z ,z T (3)

Pairwise comparisons at Level 2

Pairwise comparisons at level 2 of the Benefit and Cost hierarchies must be carried out as well. As explained in Table 4, pairwise comparison matrices between the metric categories with regard to the set of commandments are created based on the weights specified in Table 3. For example, the pairwise comparison matrix regarding C 10 , which is denoted by

P C 10 3×3
, is generated as in Eq. 4 3 (cf., weights highlighted in bold/red correspond to the ones specified in 3 Note: All consistency rations (CR) of the pairwise comparison matrices of level 2 are < 0.1.

Table 3). The resulting priority vectors are denoted by 591 E (PC m ) y×y .

592

P C 10 3×3 =             
Carbon Green Recycling Carbon 1 .

3 9 3 3 Green 1 c 12 1 9 3 Recycling 1 c 13 1 c 23 1              (4
A M P k , j w×w =                          A 1 . . . A w A 1 1 . . . A M P k , j 1 A M P k , j w . . . . . . . . . . . . A w A M P k , j z A M P k , j 1 . . . 1                          (5)

Alternative ranking

All the priority vectors previously computed must now be aggregated in order to obtain the final score of each alternative with respect to the Benefit and the Cost objectives (scores being respectively denoted by

S B

A l and S C A l ). To this end, the weighted sum method is applied, as given in Eq. 6, where f (i) and f (m) refer to the mapping functions that depend on the size of vectors E (PC m ) y×y and E (MP k ) x×x respectively.

S {B;C}

A l = f (i) j=1 f (m) i=1 z m=1 E A M P k , j i w×w , j • E (MP k ) x×x ,i • E (PC m ) y×y ,m • E (C) z×z ,m (6)
The Benefit/Cost score, denoted by S A l in Eq. 7, is carried out by dividing the two scores, which is used as basis for ranking all data center candidates in terms of sustainable development/behavior. 3) [START_REF] Li | To monitor or not: Lessons from deploying wireless sensor networks in data centers[END_REF][START_REF] Liu | Green data center with iot sensing and cloud-assisted smart temperature control system[END_REF][START_REF] Hong | A synergy of the wireless sensor network and the data center system[END_REF][START_REF] Rodriguez | Wireless sensor network for data-center en-1345 vironmental monitoring[END_REF]. Nonetheless, as the primary objective of our research is not to provide an in-depth comparison study of existing data centers, but rather to present the theoretical foundation of our approach, we made use of an online calculator -Data Center Tradeoff Tools of Schneider Electric 5 -to generate the datasets related to the data center candidates. However, this tool only allows us to generate a limited number of metrics compared with the one integrated to our model (cf., Figure 4). The subset of metrics for which data could be generated is detailed in Table 6. 5 https://www.schneider-electric.com/en/work/solutions/forbusiness/data-centers-and-networks/trade-off-tools/, last access: May 2019. Given these scenarios, all metric values regarding the three data center and virtualization candidates were generated, as given in Table 6. The following section details an example of how a the weights are generated and aggregated in our approach.

S A l = S B A l S C A l (7)

Stakeholder preference & Computational steps

In this scenario, we assume that stakeholders at levels 1 and 3 do not want to prioritize one commandment/metric criterion over others. It is important to understand that this preference set is only meant to illustrate the functioning of our approach and could be easily adapted, if needed, depending on the stakeholders' needs/interests. Given the set of metrics considered in our scenario (cf., Table 6), along with the benefit/cost analysis detailed in section 3.2, the resulting Benefit and Cost hierarchies are depicted in Figures 7(a) and 7(b) respectively. In order to ease the understanding of how weights are generated/computed and aggregated in our approach, we propose to detail in the following the computational steps regarding the PUE metric from the Benefit hierarchy (cf., red/dashed lines in Figure 7(a)).

Level 1 (L1):

As it is assumed that all commandments are considered of the same importance by the stakeholder, this results in the pairwise comparison matrix and priority vector given in Eq. 8; 750

C =                                   C 3 C 4 C 5 C 6,8 C 7 C 10 C 3 1 1 1 1 1 1 C 4 1 1 1 1 1 1 C 5 1 1 1 1 1 1 C 6,8 1 1 1 1 1 1 C 7 1 1 1 1 1 1 C 10 1 1 1 1 1 1                                   E C = C 3 C 4 C 5 C 6,8 C 7 C 10 1/6 1/6 1/6 1/6 1/6 1/6 (8)
P C 3 =                       
Ener Carb Gree Fina Ener 1 1 755 764

                       E P C 3 = Ener Carb
M Ener =               PUE DCiE SPUE PUE 1 1 1 DCiE 1 1 1 SPUE 1 1 1               E M Ener = PUE DCiE SPUE 1/3 1/
A PUE =                Fr Ge Sw
Fr 1

E A PUE [Fr] • E M Ener [PUE] • E P C 3 [PUE] • E C [C 3 ] =(0.336 • 1/3 • 0.281 • 1/6) + (0.336 • 1/3 • 0.2 • 1/6) + (0.336 • 1/3 • 0.295 • 1/6) + (0.336 • 1/3 • 0.166 • 1/6) =0.0176 (

Lack of real-life datasets

Even though the primary message of this research is not to present an in-depth comparison analysis of existing data centers over the globe, but rather present the theoretical foundation of our approach, a next step would be to carry out such comparison analysis considering real-life datasets from data centers. As was discussed in section 4.1.1, this poses a two-fold challenge: (i) to be able to access backend systems of data center organizations; and (ii) to support and/or incentivize organizations to implement/monitor new sustainability metrics in their facilities.

Future research

Beyond addressing the limitations raised in the previous section, we believe that the theoretical model proposed in this paper is fully applicable to other green ICT sectors including core network infrastructures, internet providers, ICT in enterprise, or still to smart applications such as smart cities, smart transport, Industry 4.0, etc. One perspective of this research is thus to keep refining the modelling in translating other metrics in term of eco-mature system lessons. To this end, and as previously discussed in section 2.1, we believe that our model could be either re-used as part of a more generic LCA model covering the economical and social dimensions, too, or be refined with other performance indicators defined in LCA standards.

Acknowledgment

The research reported here was supported and funded by the PERCCOM Erasmus Mundus Program of the European Union (PERCCOM-FPA 2013-0231). The authors would like to express their gratitude to all the partner institutions, sponsors, and researchers involved in the PERCCOM program [START_REF] Klimova | An international master's program in green 1391 ict as a contribution to sustainable development[END_REF]. ). The formula to compute these two scores is given in Eq. 6.

Appendix A. Sustainable data center metrics

S A l

Overall score of a given alternative (data center) A l , which is the division of the Benefit and Cost aggregated scores (cf., Eq. 7). This overall score is used to achieve the final ranking of the compared data center candidates.

250-255.

[32] H. A. Ounifi, M. Ouhimmou, M. Paquet, J. Momtecinos, Data centre localization for internet services, in: 11ème Congrès International de Génie Informatique, 2015.

Figure 3 (

 3 Figure 3(b). It can be stated that metrics under this

510 3 . 2 .

 32 Benefit/Cost Analysis 511 The objective of the proposed comparison model is 512 to help assessing and identifying the overall most sus-513 tainable data center among a set of candidates, whose 514 results must be expressed in non-technical terms.

Figure 3 :

 3 Figure 3: Relationships between biomimicry commandments & Extent to which a given metric category may impact on those commandments

569

  along with the aggregation process to obtain the final 570 ranking of the data center alternatives. 571 3.2.1. Pairwise comparisons at Level 1 572

Figure 5 :

 5 Figure 5: Data center candidates compared in the case study

656 4 . 1 .

 41 Data center settings, Stakeholder preference & how datasets related to the data 659 center candidates have been collected. Section 4.1.2 660 presents both the stakeholder preferences specified in 661 the pairwise comparison process and the computa-662 tional steps performed based on our approach.

1 .

 1 Data center-related datasets It is not an easy task for academics to be able to access/collect data center-related datasets. The reason for this is two-fold: (i) due to a lack of transparency and impossibility of accessing backend systems of data center organizations; (ii) some of the state-of-the-art metrics are not necessarily implemented/monitored by data center organizations. While the first difficulty can be partially overcome/influenced by government regulatory bodies [60, 61, 62], the second one requires further sensor network deployments in data centers to be able to compute all the state-ofthe-art metrics integrated to our model (cf., Table

Figure 6 :• 696 ••Figure 7 :

 66967 Figure 6: Screenshot of Schneider Electric online calculator

740 2 .

 2 Level 2 (L2): Eq. 9 details the pairwise compar-741 ison matrix related to commandment C 3 , along 742 with the resulting priority vector (please refer to 743 section 3.2.2 to obtain more details on how the 744 pairwise comparison matrix is created). Eq. 10 745 to 12 only provide the priority vectors resulting 746 from the pairwise comparison matrix related to 747 commandments C 3 , C 4 , C 6,8 and C 7 (the ones 748 needed to compute the red/dashed part high-749 lighted in Figures 7(a)).

790 4 . 2 . 1 .•Figure 8 :Figure 9 :Figure 8

 421898 Figures 8(a) and 8(b) provide insight into the over-

873 4 . 2 . 3 .Figure 10 :

 42310 Figure 10: Benefit/Cost analysis of the three Virtualization configuration candidates: Pre-V vs. Post-V vs. Post-V-R

10181054 5 . 3 . 2 .

 532 et al.[START_REF] Zanghelini | How multi-1180 criteria decision analysis (mcda) is aiding life cycle assess-1181 ment (lca) in results interpretation[END_REF] who point out three complex factors in 1019 the ideal environmental decision-making: (i) uncer-1020 tainty & vagueness: difficulty lies in the quantity of 1021 data, the multiple unit types, judgmental values to be 1022 applied and the uncertainty of background and fore-1023 ground data; (ii) subjectivity: because personal judg-1024 ment vary on which topics are most important; and the 1025 (iii) multi-stakeholders involvement: because it must 1026 be considered in ideal decision-making. Given these 1027 factors, Petrillo et al. [74] explain that providing a real 1028 and substantial application of sustainability through 1029 the measurement and comparability of results, in a 1030 way that satisfies the principles of sustainability of 1031 all the stakeholders is one of the biggest challenge 1032 (knowing that uncertainties vary depending on who 1033 we ask). There is still research to be done in this area. 1034 Although AHP is among the most popular tech-1035 niques for dealing with MCDA, it does not allow 1036 us to tackle uncertainty and vagueness in the expert 1037 judgments/preferences, while it could be arguable to 1038 ask for precise pairwise comparison values because 1039 these coefficients are arguably imprecisely known by 1040 experts [75]. To overcome this, AHP could fur-1041 ther be combined with Fuzzy Logic, also known as 1042 FAHP (Fuzzy AHP) [43], even though some scholars, 1043 such as Dubois [76], argue that fuzzy sets have of-1044 ten been incorporated into existing methods (e.g., into 1045 AHP, PROMETHEE, ELECTRE) without clear ben-1046 efits. The main reason behind such a criticism lies in 1047 the difficulty of successfully satisfying the transitiv-1048 ity and reciprocal axioms [76, 77]. Other approaches 1049 than FAHP could also be explored to handle uncer-1050 tainty and vagueness, such as the Statistical Value 1051 Chain (SVC) [78], which is designed to help evalu-1052 ating the process of decision support from a statistical 1053 viewpoint. Misuse of the analysis? 1055 One may raise the question: "would it be pos-1056 sible to misuse the analysis by manipulating either 1057 data or analysis procedure?" A straightforward an-1058 swer would be "yes", but any multi-attribute decision-1059 making (MADM) technique that takes into consider-1060 ation (as inputs) human judgments/preferences, as is 1061 the case with AHP, can lead to risks of misusing/-1062 manipulating the input judgments in order to obtain 1063 the desired alternative ranking (different sets of prefer-1064 ences resulting in different ranking results). However, 1065 this is more a problem of integrity than a limitation of 1066 AHP. Although making transparent the pairwise comparison judgments/preferences would help to not mislead anyone, this problem is out of scope of this study.

Table 1 :

 1 Research work in which MCDM techniques have been applied for data center performance evaluation and comparison purposes

	45
	search work. Figure 1 provides a graphical represen-

191 nance, certification, SLA, security and sustainability. 192 Hybrid frameworks (i.e., combining distinct MCDM 193 techniques) have been proposed, as in [38, 29, 28], 194 in which criteria related to financial dynamics (ROI, 195 maintenance costs. . . ), management practices (risk 196 sharing, reogranization/. . . ) and performance (service 197 availability, accuracy, security. . . ) are taken into con-198 sideration. For cloud service selection, Boutkhoum et 199 al. [30] combine Fuzzy AHP and PROMETHEE, but 200 with a special focus on ranking the most suitable cloud 201 computing alternative to accommodate Big Data from 202 an e-governance, security and business continuity per-203 spective. Xu et al. [31] propose a nonparametric DEA 204 method to evaluate cloud services based on values re-205 lated to price/hour, virtual core, compute units, mem-206 ory and disk. Silas et al. [41] present a methodology 207 for selecting the best middleware services in cloud 208 computing environments by employing the ELEC-209 TRE technique; criteria such as flexibility, time, ser-210 vice cost, scalability, trust and capability are included.

Table 2 :

 2 The ten biomimicry commandments (lessons) introduced by Benyus [45] Commandment (lesson) Examples of interpretation in the context of data centers C 1 Use waste as a resource Considering data center (IT) equipment wastes as resources for recycling company. Reusing data center heat for heating building, swimming pools, etc. close to data centers.

		C 2 C 3 C 4 C 5 C 6 C 7 C 8 C 9 C 10	Diversify and cooperate to fully use the habitat Gather and use energy efficiently Optimize rather than maximize Use materials sparingly Don't foul their nests Don't draw down resources Remain in balance with biosphere Run on information Shop locally	Cooperating with local companies (ecopark); for example, reusing data center heating for heat-ing building. Mitigating data center energy consumption. Using Energy produced locally. Offering IT quality of service in respect to customer requirements. Increasing (IT) equipment lifetime. Mitigating (IT) equipment use (e.g., virtualization). Reducing harmful emissions in the environment (carbon in air. . . ). Avoiding electronic wastes. Increasing green energy use. Mitigating (IT) equipment use to avoid earth resource depletion. In our research, the idea of lessons 6 and 8 are similar and are merged accordingly. In our research, the idea of lessons 2 and 9 are similar and are merged accordingly. Using local energy for avoiding energy transportation lost. Using air and water for cooling. Manufacturing data centers with local companies.
	294	model that overcome the above limitations. The ap-proach advocated here is to use biomimicry.

295 2.4. Biomimicry & Lifes' principles 296

The biomimicry, which means "imitation of life", 297 is defined by Oxford dictionary as: "The design and 298 production of materials, structures, and systems that 299 are modelled on biological entities and processes".

300

The majority of application domains inspired by the 301 nature are materials and locomotion

[START_REF] Lurie-Luke | Product and technology innovation: What can biomimicry inspire?[END_REF]

and con-302 sists in mainly copying physiology, morphology, and 303 anatomy of vegetal and animal world for optimiz-304 ing or making more efficient the structure of materi-305 als. In computer sciences, many heuristics for opti-306 mizing data processing/transport/storage solutions are 307 commonly used and are based on natural behavioral 308 observations, such as ant colony, bat, natural evolu-309 tion (genetic algorithm), machine learning (e.g., neu-310 ral networks). However, optimizing an artefact in 311 copying biologic entity does not guarantee that the 312 optimized artefact will have less negative impact on 313 the nature (e.g., if the genetic algorithm fitness func-314 tion seeks to maximize the pollution, the solution will 315 not be environmentally friendly even if the optimi-316 sation algorithm copies a natural evolution process).

317

Another idea initially suggested by J. Benyus

[START_REF] Benyus | Biomimicry: Innovation inspired by nature[END_REF]

is 318 that the life on earth has evolved a set of strategies 319 that have sustained over 3.8 billion years and these 320 strategies could be interesting to apply when design-321 ing an artefact. The simple expectation is the fol-322 lowing: if the artefact is developed along with life 323 strategies, the artefact is likely to be environment-324 friendly. Originally, J. Benyus identified a set of ten 421 tailed in section 3.1; 422 ii) Benefit-Cost analysis: the metrics, associated 423 categories, and biomimicry commandments are 424 then combined/structured in the form of a benefit-425 cost analysis using the AHP method. This step is 426 described in section 3.2.

427 3.1. Metric Categorization & Translation 428 As a first step, the sustainability metrics listed in 429 Table A.7 are categorized in terms of benefit and cost 430 implications from a biomimicry standpoint [56, 57]. 431 Concretely, we ask ourselves whether a metric relates 432 to the energy, carbon, recycling or other categories? 433 and whether the metric maximization or minimization 434 results in a benefit or cost from a sustainability view-435 point? In this study, the metric categorization is based 436 on the categories identified in [5], namely: 437 • Energy: evaluates the energy efficiency of the 438 system's overall useful work done in comparison 439 to the energy consumed; 440 • Materials: evaluates IT equipment efficiency; 441 • Cooling: evaluates the cooling system efficiency; 442 • Green: evaluates the amount of energy of the 443 data center that comes from clean sources; 444 • Carbon: partly evaluates carbon emissions (only 445 associated with Co2/Wh); 446

Table 3 :

 3 Metric categorization from a "Benefit" and "Cost" perspective C 2,9 C 3 C 4 C 5 C 6,8 C 7 C 10

	Impact on biomimicry commandments

Table 3 ; 2

 32

	Cost hierarchies. These two scores, respectively denoted by S B A l and S C A l , are divided to get the fi-nal Benefit/Cost score, which serves as basis to rank data center candidates in terms of sustain-able development/behavior.
	As a next step, pairwise comparisons between all elements of a given level of the AHP hierarchy must be carried out by one or more stakeholders of our ap-proach/tool. Table 4 provides a brief insight into what stakeholders could be concerned by such a pairwise comparison process. Since the proposed model is in-tended to provide a holistic view of how sustainable

. Level 3 & 2: those metrics are clustered according to the category they belong to, as defined in Table

3

(e.g., Energy, Material, etc.). This is denoted by M P i , where P i refers to the category to which a given set of metrics belongs to; 3. Level 1: those categories have been, in turn, clustered depending on what biomimicry commandment(s) their impact on. For example, the Energy category has beneficial implications on C 3 , C 4 , C 6,8 , C 7 (no cost implication), which results in the bold/red linkage highlighted in Figure 4(a); 4. Level 0: aggregated scores of all data center candidates are computed for both the Benefit and a data center is, non-technical stakeholders (e.g., executives, regulators) are more likely to communicate preferences at level 1 (e.g., to place particular emphasis on C 5 to know which data center performs the best 560 PUE over other energy metrics). Level 2 corresponds 561 to the expertise needed to match the biomimicry com-562 mandments with the sustainability metric categories, 563 which corresponds to expertise. Although a first ex-564 pertise is provided in Table 3, this does not prevent 565 another biomimicry expert from modifying, if needed, 566 the proposed expertise. Sections 3.2.1 to 3.2.5 detail 567 the theoretical basis of the pairwise comparison pro-568 cess performed at levels 1, 2, 3 and 4 respectively,

Table 4 :

 4 priority vector, denoted by E (C) z×z must then be Stakeholders and respective involvement in the comparison process AHP Biomimicry experts It corresponds to the expertise needed to model the extent to which the categories of metrics impact on the biomimicry commandments. This paper provides a first expertise in Section 3.1 and Table3, although this does not prevent other biomimicry experts from modifying it.

	573
	574 575	computed. To this end, the geometric mean method proposed by Crawford and Williams [59] is applied, as formalized in Eq. 2 in this study. Even though not
	580
	8

576

detailed, it should be noted that the consistency ratio 577 (CR) of any pairwise comparison matrix must be mea-578 sured, whose inconsistency is regarded as acceptable 579 if CR ≤ 10%

[START_REF] Saaty | The Analytic Hierarchy Process[END_REF]

.

  ) stakeholder deems the recycling of material as more important than the reuse of data center heat extraction/dissipation to warm up buildings. Pairwise comparison matrices are denoted by M P k x×x and the associated priority vector denoted by E (MP k ) x×x .

	3.2.4. Pairwise comparisons at Level 4
	Finally, pairwise comparisons at level 4 are carried out between the set of alternatives (i.e., data center candidates), which is denoted by A. One difference with the previous pairwise comparisons is that, while before they were reflecting either the stakeholder pref-erences/expertise, they are now performed based upon measurable/supervised system parameters. In other words, all data center candidates are evaluated with respect to all metrics based on automated mecha-nisms. Pairwise comparison matrices are denoted by A M P k , j
	593
	594 595 596 597 598	It should be noted that we do not expect that data center stakeholders perform pairwise comparison at this level, even though it does not prevent an expert from modifying the impact analysis proposed in Sec-tion 3.1 and Table 3.
	599	3.2.3. Pairwise comparisons at Level 3
		Pairwise comparisons at level 3 (i.e., between met-

600

rics of a given category P k ) must be performed by 601 stakeholders (cf., Table

4

). For example, regarding the 602 Recycling category, one may want to put more empha-603 sis on the "MRR" metric at the expense of "ERF" be-604 cause the w×w , as in Eq. 5, j referring to the j th metric of category P k , and l to the l th data center candidate. The resulting priority vectors are denoted by E A M P k , j i w×w

Table 6 :

 6 Values of metrics related to the three use case scenarios: TC, CW, V

	PUE (Ratio) CUE (Co2/Wh) GEC (%) WUE (L/kWh) WUEs (L/kWh) DCiE (%)	Formula Total Facility Power IT Equipment Power Total Carbon Emissions Total IT Equipment Energy Green Energy used in the Data center Total Data Center Source Energy Annual Water Usage IT Equipment Energy Annual Site Water Use+Annual Srce Water Use IT Equipment Energy 1 PUE	Traditional Cooling (TC) Fr Ge Sw 1.560 1.550 1.540 0.096 0.785 0.039 0.190 0.300 0.580 0.000 0.000 0.000 10.083 3.897 29.356 0.640 0.650 0.650	Virtualization (V) Pre-V Post-V Post-V-R 2.278 1.720 2.117 0.140 0.106 0.130 0.190 0.190 0.190 0.137 0.169 0.169 10.083 10.181 10.083 0.440 0.580 0.470
	OPEX ($)	Energy Cost × Consumed Energy	1151K 1807K	1086K	1197K	733K	902K
	SPUE (Ratio) REL (Faults/h)	Total Facility Power Server Power number of failures or outages Total surviving hours	N/A N/A	N/A N/A	N/A N/A	4.556 0.001	4.473 0.001	5.518 0.0005

Table 5 :

 5 Country-related data from IEA4

		country	Hydro	Wind	Biomass	Solar CSP	Geothermal	Solar PV	Nuclear	Natural Gaz	Oil	Coal
	Energy (%)	Fr Ge Sw 41 10 7 0 0 11 4 2 2 0 3 12 9 6 0	/ 73 6 / 13 13 1 0 / 40 1 0	2 43 1
	Carbon (kg/kWh) (10 -3

  1. This is important to be noted because

	960 961 962 963 964 965 966 967 968 969 970 971 972	AHP capabilities remains an open research question [69, 70], our Benefit-Cost model is, to the best of our knowledge, the first one that explores the possi-bility of combining state-of-the-art sustainability met-rics with the biomimicry principles in order to enable holistic assessment of the sustainability performance of data centres. One key interest and novelty of such a combination lies in the fact that the biomimicry com-specified in Table 3). derlying model (except complementing the expertise ture/new metrics without having to change the un-tion that makes it possible the integration of any fu-mandments provide a universal language/representa-	1012 1013 1014 1015 1016 1017	biomimicry commandments and state-of-the-art sus-tainability metrics. However, several limitations of this approach should be noted, which are respectively discussed through sections 5.3.1, 5.3.2 and 5.3.3. 5.3.1. Dealing with uncertainties The path towards sustainable development is not a straightforward process, as explained by Zanghelini
	879 880 881 882 883 884 885 886 887 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011	the key message to be conveyed in this paper is that we do not aim to carry out an in-depth comparison study of existing data centers over the globe, but rather to present the theoretical foundation of an innovative approach that could help data center stakeholders to compare the overall sustainability of data centers in a holistic and non-technical manner. This message is important as our experiments are limited in the fol-lowing respect: with a limited number of metrics compared with the exhaustive list considered in the theoretical model of our approach (cf., section 3). Such limitations have direct impact on the relevance of our comments/findings when discussing the data center performance results. This should therefore be taken with caution, or at least understood by the reader. 5. Conclusions, implications, limitations and fu-ture research 5.1. Conclusions The importance of data centers for society has changed, as public life, economy and society depend to a large extend on the proper functioning of data cen-ters. From different parts of society, the sustainability of data centers is questioned, thus raising questions about the "data center equation" of "people, planet, profit". While it is common place to think that making data centers sustainable comes down to implementing technical measures, it actually goes well beyond the walls of the data center, touching upon economic and political questions. As a consequence, it is imperative to develop and propose holistic models that help data center stakeholders, spanning from data center own-ers, governmental regulators to engineers/field oper-ators, to evaluate and understand how a data center Second, the proposed approach also contributes to move towards increasing professional and public awareness of how data centers behave in terms of sus-tainability, both from a biomimicry viewpoint and/or when looking at specific metric categories (e.g., from an energy, financial, green or still water viewpoint). This could be helpful for organizations and regula-tory institutions to establish strategic plans to both im-prove current data center practices to meet the United Nations Sustainable Development Goals 6 (SDGs) and contribute to the welfare of the society. In this re-spect, and from a theoretical implication perspective, our approach could serve as a basis to further address the Arrow's impossibility theorem [71], as it has been proved that the combination of the AHP and geomet-ric mean methods (approach adopted in this paper) contributes to find a social welfare function that sat-isfies the four conditions: (i) Pareto optimality, (ii) Independence from irrelevant alternatives, (iii) unre-stricted domain, and (iv) Non-dictatorship [72, 73]. Third, our approach supports, to a certain extent, an efficient solution to tackle environmental exter-nalities. Indeed, different overconsumption (e.g. in terms of energy) will result in an increase of the costs (i.e., negative externality) in our model; for instance, satisfying commandment C 3 (mitigating energy con-sumption and local energy production) would be a true leverage for a private cooperation to minimize that effect (the cost will be here the incentive). The same may apply for commandment C 1 (use waste as ressource). In addition, society will intervene in the markets and apply new taxes (for instance related to Co2) and may even define temporary closures for in-stance in case of (carbon in) air pollution, leading to a loss of income and usage effectiveness. Command-ment C 6 will aim at repelling this risk. 5.3. Limitations In this research work, AHP is used as underly-ing technique for combining, in a holistic way, the 6 https://www.un.org/sustainabledevelopment, last access: May 2019.	918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959	performs in terms of sustainable development and be-havior. While a large number of sustainability metrics exist in the literature, there is still a lack of frame-works that make it possible to reduce the complexity of the performance evaluation analysis. In order to fill this gap in literature, this pa-per presents an innovative approach based on a benefit-cost analysis using the Analytic Hierarchy Process (AHP). The originality lies in the combina-tion of state-of-the-art sustainability metrics with the biomimicry commandments of eco-mature system de-fined by J. Benyus [45], and the possibility for data center stakeholders to analyze and compare perfor-mance of data centers in a more or less holistic man-ner (depending on their needs, focus, expertise). For example, one data center stakeholder may want to pri-oritize the biomimicry commandment C 6 when debat-ing on pollution aspects (data centers would, indeed, be ranked with emphasis on the extent to which the pollute locally), while another one could be provided with a more in-depth analysis of how a data center per-forms with respect to the energy, water, green or still financial dimensions. The practicability of our model is demonstrated considering three data center candi-dates, which are respectively located in France, Ger-many and Sweden. However, it is important to note that the key contribution and message to be conveyed in this paper is about the theoretical foundation of the proposed approach, rather than on comparing existing data centers over the world. 5.2. Implications This research presents three main theoretical impli-cations. First, it has implications in the the field of LCA (as was previously discussed in section 2.1), and par-ticularly of CBA (Cost-Benefit Analysis) [22]. Even though our approach differs from traditional CBA analyses (which mostly focus on economical assess-ment), many studies have considered AHP in order to cope with the CBA's weakness in reflecting stakehold-ers' knowledge in the evaluation process of projects [67, 68]. Although the assessment of CBA versus

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• we could not collect real datasets from existing 889 data centers; 890 • the online calculator enabled us to work only

Table A .

 A 7: Metrics for Sustainable data centers reported in[START_REF] Reddy | Metrics for sustainable data centers[END_REF]

				Continued from previous column
			Metric	Definition
	Metric ACPR AEUF APC ATR BJC CADE Capacity Capacity Definition Average Comparisons Per Rule Air Economizer Utilization Factor Adaptability Power Curve Corporate Average Data Center Efficiency Bits per Joule Capacity Application Transaction Rate	1107	Continued on next column Concurrent Connections Connection Establishment Rate Communication Network Energy Efficiency Co2 Savings Coefficient of Performance Ensemble Compute Power Efficiency Carbon Usage Effectiveness DCAdapt Data Center Compute Efficiency Data Center Cooling System Efficiency Data Center Energy Productivity DC-FVER Data Center Fixed to Variable Energy Ratio CC CER CNEE Co2s CoP CPE CUE DCA DCcE DCCSE DCeP DCiE Data Center Infrastructure Efficiency DCLD Data Center Lighting Density DCPD Data Center Power Density DCPE Data Center Performance Efficiency DCSSF Data Center Cooling System Sizing Factor DeD Defense Depth DeP Detection Performance DH-UE Deployed Hardware Utilization Efficiency DH-UR Deployed Hardware Utilization Ratio DPPE Data Center Performance Per Energy DTE Data Transmission Exposure DWPE Data Center Workload Power Efficiency ECR-VL Energy Consumption Rating Variable Load EDE Electronics Disposal Efficiency EER Energy Efficiency Ratio EES Energy ExpenseS ERE Energy Reuse Effectiveness ERF Energy Reuse Factor EWR GEC GUF H-POM HSE HTTP IAS IPFH IPt ITEE ITEU ITH Lat LSP MemU MRR NetT NPUE OSE OSWE OpEx PDE PEs pPUE PpW PUE PUEs RC RCD REL RSmax RT Response Time Maximum Relative Size Reliability Rogue Change Days Reachability Count Power Usage Effectiveness Scalability Power Usage Effectiveness Performance per Watt partial Power Usage Effectiveness Primary Energy Savings Power Density Efficiency Operational Expenditure Operating System Workload Efficiency Overall Storage Efficiency Network Power Usage Effectiveness Network Traffic per Kilowatt-Hour Material Recycling Ratio Memory Usage Low-cost Storage Percentage Lat Illegal Traffic Handling IT Equipment Utilization IT Equipment Energy IP throughput IP Fragmentation Handling Interface Accessibility Surface HTTP Transfer Rate HVAC System Effectiveness IT Hardware Power Overhead Multiplier Grid Utilization Factor Green Energy Coefficient Energy Wasted Ratio
	Continued on next column

Table B .

 B 8: List of acronyms and variables used throughout the paper Set of biomimicry commandments. Elements of C are denoted by {C 1 , .., C i , .., C z } and the associated pairwise comparison matrix is denoted by (C) z×z , as given in Eq. 1. P, P Cm P refers to the set of categories given in Table 3 (i.e., Energy, Materials, Cooling, etc.). Elements of P are denoted by {P 1 , .., P k , .., P u }. P Cm refers to a subset of P indicating what categories impact on, and thus are linked with, a given commandment C m . For example, P C 10 ={Cooling, Carbon, Green, Recycling} for the Benefit hierarchy, and P C 10 ={Water} for the Cost hierarchy (cf., Table 3). Elements of P Cm are denoted by {P Cm,1 , .., P Cm,i , .., P Cm,y } and the associated pairwise comparison matrix is denoted by P Cm y×y . M P k Set of metrics that belongs to category P k (cf., Table 3). Elements of M P k being denoted by {M P k ,1 , .., M P k ,i , .., M P k ,x } and the associated pairwise comparison matrix is denoted by M P k x×x . A Set of alternative (data center) candidates. Elements of A being denoted by {A 1 , .., A l , ..A w }. All candidates must be evaluated with respect to all metrics and categories. The corresponding pairwise comparison matrix is denoted by A M P k , j w×w , with j the j th metric of category P k and l the l th data center candidate. E p Priority/Eigen vector (EV) of a pairwise comparison matrix p. Elements of E p are denoted by {E p,1 , .., E p,i , .., E p,v }. In our study, p corresponds to one of the above-specified pairwise comparison matrices, namely (C) z×z , P Cm y×y , etc., whose corresponding eigenvectors are denoted by E (C) z×z , E (PC m ) y×y , etc. Aggregated scores of a given alternative (data center) A l with regard to the Benefit hierarchy (score denoted by S B

		Acronym	Full form
	Abbreviation	AHP -FAHP CBA COP CR Solar CSP-PV DEA ELECTRE ICT / IT LCA MADM MCDA MCDM PROMETHEE Preference Ranking Organization Method for Enrichment Evaluations Analytic Hierarchy Process -Fuzzy Analytic Hierarchy Process Cost-Benefit Analysis Conference of the Parties Consistency Ratio Concentrated Solar Power -Solar Photovoltaic Data Envelopment Analysis ÉLimination Et Choix Traduisant la R Éalité Information (and Communication) Technologies Life Cycle Assessment Multi-Attribute Decision Making Multi-Criteria Decision Analysis Multi-Criteria Decision Making QoS Quality of Service Pre-V Pre-Virtualization Post-V(-R) Post-Virtualization (Post-V); Post-Virtualization with Redundancy (Post-V-R) SDGs Sustainable Development Goals SLA Service Level Agreement SVC Statistical Value Chain TOPSIS Technique for Order of Preference by Similarity to Ideal Solution VIKOR VlseKriterijumska Optimizacija I Kompromisno Resenje (means: multicriteria optimization and compromise solution)
		C
	Mathematical Variables
		S {B;C} A l	and the Cost hierarchy (score denoted by S C A l	A l	)

https://bio-sis.net/life-principles/, last access: May 2019

https://www.iea.org, last access: May 2019.