Julien Lahoudere
email: julien.lahoudere@obs.u-bordeaux1.fr

Philippe Paillou

François Demontoux

Gilles Ruffié

Influence de la disposition d'antennes pour l'étude de paramètres géophysiques. Application à la mission spatiale ExoMars

La conception et la modélisation des instruments et systèmes sont des étapes importantes dans la préparation d'une mission spatiale. Un objectif de la mission européenne ExoMars est d'envoyer un rover afin de sonder et d'analyser le proche sous-sol de la planète Mars. Notre travail est de simuler un système radar, à bord du rover, qui guidera le système de forage chargé de prélever des échantillons. Pour cela, une étude complète de l'interaction du radar avec son environnement martien est nécessaire, dans des conditions idéales puis réelles, en tenant compte de paramètres tels les caractéristiques diélectriques du sol martien. Dans un premier temps, nous avons étudié l'influence de l'éloignement des antennes ainsi que leur inclinaison sur le signal de diffusion. Nous présenterons aussi brièvement un travail préliminaire concernant les paramètres de rugosité liés à la présence d'inhomogénéité de surface ainsi que d'éléments diffuseurs dans le sous-sol mais aussi une structure schématique du rover se déplaçant sur la surface martienne.

I. Introduction

Approuvé par le Conseil de Recherche de l'Union Européenne et le Conseil de l'Agence Spatiale Européenne en 2001, le programme Aurora de l'ESA fait partie de la stratégie européenne de l'exploration spatiale. Les objectifs principaux sont l'exploration du Système Solaire, stimuler le développement de nouvelles technologies et inspirer aux jeunes d'Europe un plus grand intérêt pour la science et la technologie au travers de deux thématiques. La première consiste à créer puis appliquer un plan à long terme pour l'exploration robotique et humaine de Mars. La seconde repose sur la recherche de la vie sur d'autres mondes du système solaire au moyen d'outils sophistiqués.

La première mission de ce programme, baptisée ExoMars, a pour objectif d'envoyer sur Mars un véhicule de type rover en 2013. Celui-ci sera équipé d'une charge utile scientifique qui analysera le sous-sol de Mars, à la recherche de traces de vie passée et/ou présente. Une foreuse permettra l'accès à des échantillons du sous-sol de Mars jusqu'à une profondeur de 2 mètres : c'est ce qui constitue la grande originalité d'ExoMars. En effet, la subsurface martienne reste encore inconnue et semble le meilleur endroit pour abriter des conditions favorables à la vie. Cette foreuse sera guidée par un système radar sondeur UHF, l'instrument WISDOM, qui permettra également de sonder le sous-sol de Mars jusqu'à quelques mètres de profondeur pour détecter de l'eau et étudier les structures géologiques.

Le Laboratoire d'Astrophysique de Bordeaux (LAB) est partie prenante dans la réalisation du radar sondeur WISDOM et dans l'analyse de ses données. Pour cela, il réalise des modélisations géo-électriques et électromagnétiques du sol martien avec le laboratoire IMS (caractérisation électromagnétique d'échantillons de roches et modélisation numérique). Des modèles géo-électriques des premiers mètres du sous-sol de Mars ont ainsi été conçus : ils vont permettre de spécifier les données techniques du radar et de développer des méthodes d'inversion des données. Nous disposons de modèles électromagnétiques analytiques (type IEM, GOM, SPM) et numériques (type FDTD et éléments finis) pour simuler le comportement d'une onde radar dans le sous-sol de Mars, mais aussi les interactions possibles avec la structure du rover. Ce type de simulation numérique sera nécessaire pour calibrer et interpréter les futures données de l'instrument WISDOM.

Dans un premier temps, nous présenterons les conditions d'étude et les outils utiles à la modélisation du système d'antennes. Par la suite, nous montrerons quels résultats ont été obtenus par la simulation. Nous caractériserons le couplage et les meilleures configurations du système d'antennes. Avant de conclure, nous présenterons les directions retenues pour intégrer aux modèles la rugosité de surface et les caractéristiques diélectriques du sol et du sous sol martien.

II. Présentation et exploitation du modèle numérique

Le rover embarquera un système radar bistatique pour l'exploration du sous sol de Mars, utilisant des antennes de type vivaldi. Celles-ci permettront d'effectuer des mesures sur une bande de fréquence allant de 500 MHz à 3GHz pour une incidence de 10° lorsque le rover se trouve sur un sol plat et à environ 50 cm du sol martien. Un objectif de notre travail est de concevoir un modèle numérique permettant de simuler le déplacement du radar. Dans un premier temps, nous avons utilisé comme antenne un simple cornet dont la bande passante varie de 900 MHz à 2.7 GHz, antenne que nous possédons au laboratoire. Par la suite, cette antenne sera remplacée par l'antenne réelle qui est en cours de réalisation à l'université de Dresde.

Deux antennes à proximité l'une de l'autre produisent un couplage qu'il faut déterminer. Le logiciel HFSS (High Frequency Structure Simulator) de la société ANSOFT a été utilisé pour quantifier les perturbations intrinsèques à ce système afin d'en tenir compte et de les minimiser. Le modèle créé permet d'observer un effet en fonction de la distance entre les antennes et leur inclinaison relative. Le modèle est composé de deux antennes dans un espace fini aux bords absorbants ne possédant aucun élément diffuseur, comme nous pouvons le voir sur la figure 1. Nous plaçons ensuite une plaque métallique réfléchissante face au système pour déterminer les paramètres S 21 . Nous pouvons ainsi évaluer quelle configuration d'antennes offre les meilleurs résultats. Une deuxième étape consistera à intégrer dans le modèle une surface diélectrique à la place de la surface métallique pour représenter le sol martien. Nous devrons prendre en compte la rugosité de surface, les inclusions (roches, eau, évaporites, …) mais aussi la structure du rover. L'objectif est de recréer une géologie « martienne » et un contexte réalistes. La modélisation de ces phénomènes permettra d'obtenir les coefficients de réflexion dans la gamme de fréquence désirée. Le dernier aspect à prendre en compte sera le déplacement du rover sur cette géologie (effets de topographie).

III.Premiers résultats des simulations

Une étude préliminaire a été effectuée pour caractériser une antenne cornet isolée puis au-dessus d'une surface réfléchissante. Après quoi, nous nous sommes intéressés au couplage de deux antennes. À l'aide du logiciel HFSS, nous avons fait varier la distance inter-antenne de 450 à 900 mm et l'angle d'inclinaison de 0 à 35°. Nous avons travaillé à la fréquence centrale d'émission du cornet, soit 1,4 GHz. Nous remarquons sur la figure 2 que l'augmentation de la distance entre les deux antennes entraîne une atténuation du paramètre S 21 , alors que l'augmentation de l'angle entre les antennes donne l'effet inverse. A l'aide de ce graphe et de captures de champ par le logiciel, nous avons donc constaté que le couplage entre les antennes s'amplifie lorsque l'inclinaison de chaque antenne devient supérieure à environ 25°. Cela est dû au fait que les lobes d'émission et de réception des antennes commencent à se recouvrir. Néanmoins, les amplitudes du paramètre S21 étant inférieures à -30dB, le couplage reste faible. L'utilisation d'un court-circuit va nous permettre d'étudier la diffusion suivant l'angle et la distance pour optimiser le paramètre S 21 . La disposition la plus adaptée correspond à une distance comprise entre 450 et 600 mm et une inclinaison d'antenne entre 10 et 25°, afin de recueillir un maximum de signal diffusé. En effet, nous voyons sur la figure 3 que le signal diminue en intensité avec la distance mais augmente à partir d'une distance d'approximativement 600 mm pour des grands angles. Comme nous l'avons vu précédemment, les effets du couplage deviennent notables pour des angles supérieurs à 25°, les configurations à grands angles doivent être évitées au maximum.

IV. Travaux en cours et à venir

La prise en compte d'éléments de surface et de subsurface dans les modèles va entraîner une complexification du signal. Il s'agit ici de comprendre les effets de chacune des contributions sur le signal rétrodiffusé total. En effet, en fonction des paramètres de rugosité de surface par exemple, le signal ne sera pas le même. Notre objectif est de modéliser l'allure du signal suivant les paramètres de surface et de subsurface afin de retrouver des signatures caractéristiques dans le signal mesuré. Créer une surface rugueuse suivant un profil mathématique s'est avéré être difficile avec le logiciel HFSS, nous la modéliserons donc en plaçant de manière aléatoire des volumes dont les dimensions et la proportion surfacique sont définies. En effet, la distribution des roches sur la surface de Mars peut être décrite en fonction de leur taille à l'aide d'une décroissance exponentielle F k (D) = ke -q(k)D . Il s'agit en fait de la fraction cumulée de surface recouverte par des roches de diamètre supérieur à D. Le facteur de décroissance caractéristique q(k) dépend de la proportion k de la surface totale. Cette description est valable pour des roches de diamètres inférieurs à 20 cm.

Nous nous intéresserons également aux effets dépolarisants de ces éléments diffuseurs. Pour se faire, la démarche consistera à modéliser la rotation d'une antenne sur son axe de symétrie et étudier la diffusion d'un signal polarisé linéairement. L'intérêt réside dans la compréhension des phénomènes à l'origine de la dépolarisation afin de retirer des informations importantes sur les inhomogénéités de surface et les contrastes diélectriques présents dans le sous-sol. Cette étude pourrait être à l'origine de l'élaboration d'un modèle d'inversion permettant de retrouver ces paramètres.

Une autre phase de notre travail consistera à modéliser la structure du rover et son déplacement sur divers sols. La démarche consistera à faire varier la permittivité du sol modélisé suivant une loi polynomiale en faisant apparaître des éléments diffuseurs volumiques (dans la subsurface). Il faudra tenir compte de l'impact des différents éléments (panneaux solaires, roues, modules scientifiques) présents sur le rover comme le montre la figure 4.

V. Conclusion

Nos premiers résultats concernant l'étude de la disposition des antennes nous offrent des perspectives intéressantes quant à l'utilisation du logiciel HFSS pour modéliser l'instrument WISDOM. La suite de l'étude permettra de prendre en compte la présence d'éléments diffuseurs à la surface et dans le sous-sol de Mars. Un modèle d'inversion va pouvoir ensuite être construit à partir de ce travail. La structure du rover sera intégrée à notre modèle car elle peut être à l'origine d'une perturbation du signal. Les réflexions parasites peuvent conduire à des difficultés d'interprétation du signal et peuvent perturber de l'électronique embarquée. Pour corriger ce problème, une étude sera réalisée pour optimiser la disposition des antennes et envisager d'autres solutions (filtre temporel, élément absorbant dans la structure du rover).

[1]Site de l'ESA : http://www.esa.int/SPECIALS/Aurora/index.html [2]PAILLOU P., LASNE Y., HEGGY E., MALEZIEUX J.M., RUFFIE G., "A study of P-band synthetic aperture radar applicability and performance for Mars exploration: Imaging subsurface geology and detecting shallow moisture", J.G. R., 2006R., , vol111n°E6, 10.1029/2005JE002528 [3]/2005JE002528 [3]HEGGY E., PAILLOU P., COSTARD F., MANGOLD N., RUFFIE G., DEMONTOUX F., GRANDJEAN G., MALEZIEUX J.M., "Local geoelectrical models of the Martian subsurface for shallow groundwater detection using sounding radars", J.G. R., 2003R., , vol108n°E4,10.1029/2002JE1871 [4]/2002JE1871 [4]HEGGY E., P. PAILLOU, F. COSTARD , N.MANGOLD, RUFFIE G., DEMONTOUX F.,J.GRANDJEAN , J.-M. MALEZIEUX, "Models of the Martian subsurface for radar detection of shallow ground water", J.G.R., 2002 [5]GOLOMBEK et al, "Rock size-frequency distributions on Mars and implications for Mars exploration Rover landing safety and operations", J. G.R., 2003, vol108n°E12 [6]LASNE Y., Thèse de doctorat, "Imagerie radar basse fréquence pour l'exploration des zones arides terrestres et martiennes : détection de l'humidité du sous-sol et cartographie de la paléohydrologie", Université Toulouse III, 2005

Figure 1 :

 1 Figure 1 : Modèle d'un système d'antennes.

Figure 2 :

 2 Figure 2 : Variation du paramètre S21 en fonction de l'inclinaison pour différentes distances à la fréquence 1,4 GHz.

Figure 3 :

 3 Figure 3 : Variation du paramètre S21 en fonction de l'inclinaison pour différentes distances sur sol réfléchissant à la fréquence 1,4 GHz.

Figure 4 :

 4 Figure 4 : Prototype schématique du rover intégré aux simulations.


Philippe Woloszyn
email: philippe.woloszyn@cerma.archi.fr

Acoustic diffraction patterns from regular to fractal structures: application to the Sierpinski carpet

. First part of this work relates the properties of far-field Fraunhofer region diffraction in wave acoustics for characterizing reflection on a periodic regular indented plane. Diffusion prediction of a self-similar structure, the Sierpinski carpet, is then developed through the computation of its spatial Fourier transform. Scattering intensity computation results show that after propagation of a coherent plane wave through the structure, the resulting acoustical field displays fractal properties itself, showing a self-similar structure of the reflected signal. This computational approach of the Sierpinski carpet's scattering properties will lead us to develop, in a near future, an acoustic angular scattering measurement process, applied to a Sierpinski tetrahedron 3-D scale model.

Introduction

The aim of this work is to define an acoustic diffraction pattern computation model from varying geometry complexities, of regular or fractal structures [START_REF] Woloszyn | Is Fractal Estimation of a Geometry worth for Acoustics ?[END_REF]. To do this, we'll develop twice analytical and numerical approaches of the far-field diffraction models, based on electromagnetism diffraction analogy [START_REF] Beckmann | The Scattering of Electromagnetic Waves from Rough Surfaces[END_REF]. Founding this approach, Bragg's law and Fraunhofer's model describe the field conditions for constructive and desctructive interferences, which is produced from strong diffraction. Considering diffusive structures as families of parallel planes running in different directions, each plane acts like a slightly reflective mirror, reflecting a tiny fraction of the incident acoustic wave. When in phase, those reflections lead to diffraction interferences production [START_REF] Rayleigh | The theory of Sound[END_REF].

Diffraction of a wave by a periodic structure is due to phase differences that result in constructive and destructive interference. This phenomenon can occur when waves pass through a periodic structure if the repeat distance is similar to the wavelength of the waves. Observation of diffraction patterns when acoustic waves pass through complex structures is here completed through the nature of those waves, involved diffraction process.

Diffraction patterns are most obvious when the incoming waves are coherent, which means that the phase of the sinusoidal pattern of the acoustic fields is deterministic. Coherence has two domains: spatial and temporal. Consequently, knowledge of the field phase at some point in space and/or time determines the phase at other points in space and/or time. For spatially coherent sound, the phase difference ∆φ ≡ φ1 -φ2 of the field measured at two different points in space at the same time separated by a vector distance ∆r remains constant for all times. If the phase difference measured at the same location at two different times separated by ∆t≡t1-t2 is the same for all points in space, therefore, the sound is temporally coherent.

Wave coherence

The Fresnel (near-field) diffraction region occurs when the distance between the structure and the reception plane is small compared to the size of the structure. Fresnel diffraction pattern involve an hemispheric wavefront to reach the reflective structure, which means uncoherent incoming waves defining a non-deterministic acoustic field into twice spatial and temporal domains [START_REF] Hamilton | Sound beams[END_REF].

Otherwise, if viewed at a large distance compared to the extent of the object, the sound may be accurately modeled as plane waves with different wavefront tilts. This occurs in the Fraunhofer diffraction region as seen Figure 1: The diffraction length, known as the Rayleigh distance Dr, marks the transition between the near-field and the far-field regions of the waves. (

) 1
Considering the mean path D of the reflecting structure, most of the reflected energy of wavelength λ is diffracted through an angle of the order α = λ/D radians from its original propagation direction. When the wave have travelled a distance r from the plane, about half of the reflected energy will have left the plane structure by the geometric shadow if D/R = α. Consequently, the majority of the propagating energy in the "far field region" at a distance greater than the Rayleigh distance will be diffracted energy with polar radiation pattern. Physically, the equations for Fraunhofer and Bragg diffraction are similar and embody the same functional dependence on the distance r of the structure, incident wavelength λ, and scattering angle α.

Interferences production

Within wave acoustics, interferences are described through the expression for the sum of two sinusoidal temporal oscillations of the same amplitude A and different frequencies f1 =2π/ω1 = c/λ1 and f2 =2π/ω2 = c/λ2, with λ, wavelength of the incoming wave :

      -       + = + t t 2 2) 1 ( cos . 2 2) 1 ( cos 2A 2t] [ cos 1t] [ cos A ω ω ω ω ω ω . (2)
In order to generalize waves travelling in any direction, we introduce the 3-D wavevector:

λ π 2 k k k kz] , ky [kx, 2 z 2 y 2 x = + + = = k , (3)
pointing in the direction of travel of the wave. Thus, the length of the wavevector is proportional to 1/λ. Thus, the equation for a travelling wave in 3-D space becomes:

.t] - [k.r cos A .t] - z k y k x [k cos A t] [r, f t] z, y, [x, f z y x ω ω = + + = = .
(4) A 3-D wavefront can exhibit a periodic variation in the phase φ [r, t] = kr-ωt, even when the wave is monotone as: ω1=ω2 =ω → λ1=λ2=λ. If sound from a single source is reflecting into two indents at different heights, Rayleigh's principle indicates that the sound through the two indents will "spread" and recombine. When viewed at a single location, the two sound "rays" with the same wavelength will recombine with different wavevectors such that |k1|=|k2|= |k|.

Rayleigh's principle and Bragg's Law

Rayleigh's principle proposes a phase calculation between two acoustic rays, which takes the source incidence angle into account [START_REF] Rayleigh | The theory of Sound[END_REF]. The path difference ∆d between two rays with wavelength λ and incidence angle α regarding a surface with depth Λ provides the following phase grating calculation between the two rays:

∆α = ∆d (2π / λ) = cos α (4 π Λ / λ) , (5)
with the path difference ∆d=2Λcosα.. For a weak path difference ∆d, rays are coherent and the acoustic wave is specularly reflected. Increasing ∆d interferes with rays, till ∆d = π, so that no energy is displayed in the specular direction : sound energy is then diffused. Rayleigh criterion defines the limit between specular and diffuse behavior of an incident source, corresponding to the structure depth indentations as : Λ<l/8cosα.. The specular reflection zone is defined in the lower part of the curves, taking the frequency and the angle of the incident wave into account. In interferences conditions, the propagation directions specified by the unit vectors v(d) = (φ d , 0, γ d ) for a given regular plane division, repeating ν times a spatial unit of depth Λ are defined as the characteristic directions of scattering:

α ν λ α sin 2 sin - Λ = p d , (6)
where α d is the grazing diffraction angle made by vectors v(d) = (φ d , 0, γ d ) in the direction 0x, with φ d = cos α d and γ d = sin α d , , and p the diffraction order. For Λ = 0, equality between incident and reflected angle remain true, providing specular conditions specification. We can note here that, ignoring the specular component sinα, the first term of the previous equation can be compared to Bragg's Law, leading to the following expression:

, 2 sin Λ = λ α p ( 7
)
where the integer p is the diffraction order, Λ, the distance between two reflection planes, λ, the wavelength of the incident beam and α its incidence angle.

Moreover, the two-dimensional polar response of a given indented surface can be expressed through the diffraction orders (p, q), taking the angles of incidence and diffraction into account:

λ α α ν sin sin 2 2 + Λ = + d q p . ( 8
)

Interference geometry.

The interference conditions can be expressed by defining the phases of the incident wave vector k 0 and the diffracted vector k, which both have an amplitude equal to the reciprocical of the wavelength. In order to calculate the phase of the diffracted wave, taking the path lengths difference ∆d = 2νΛ cosα into account, we will consider the difference between the path of the sound wave along the incident beam k 0 r and its path along the diffracted beam kr (figure 5). By expressing this path length difference ∆d = k 0 r -kr, the overall diffraction phase as: all points with the same value of sr are lying on a plane perpendicular to vector s, allowing the same diffraction phase (figure 6). In order to define reflection systems geometries in terms of their scattering functions, we'll define the incident sound wave in the z-direction as equation [START_REF] Woloszyn | Angular characterization of the urban frontages diffusivity factor, 17 th International Congress of Acoustics[END_REF].

r k k ) ( 2 0 - = ∆ π ϕ (9)
For a singly periodic structure which bellows uniform indentations of amplitude Λ, the two dimensional acoustic field satisfies the following Fourier series [6]:

) 2 ( 0 ) ( ) ( r j e k G k F πν + Λ - ∞ ∑ = (10)
Each of the terms of the series in previous equation is a spatial harmonic with period ν. The density-density correlation function 0 G is a main parameter of the scattering intensity of the structure. For an incident plane wave, equation [START_REF] Woloszyn | Angular characterization of the urban frontages diffusivity factor, 17 th International Congress of Acoustics[END_REF] provides result of scattering from a periodic structure as an infinite sum of plane scattered waves, mostly harmonics.

After reflection on the plane structure, amplitude and phase are found to be independent of the precise waveform of the incident wave. Moreover, the diffractance function at distance r of the structure g(r) is a characteristic property of the structure, defining the interfering behaviour of the scattered wave. However, g(r), the angular distribution function along a characteristic direction of scattering informs on the wave disturbance at a distance r from the object as:

)) ( ) ( ( ² 1 ) ( 0 r r r ρ m G m g - = (11)

Fraunhofer diffraction pattern

The mathematical relationship between the shape and size of the wave output relative to that of the spatial structure input is a Fourier transform [START_REF] Hollander | Dynamic Structure Factor in a Random Diffusion Model[END_REF]. As the Fraunhofer diffraction pattern of the structure is related to the Fourier Transform of the diffractance function g(r), at large distances from the sound source, the far field or Fraunhofer diffraction region involves the diffracted sound pattern as variable in proportion to the reciprocal of the structure dimensions. Then, for the same energy input pattern f , the one-dimensional diffraction pattern in the Fraunhofer region has the angular distribution form:

α λ πα α d z r i f r g       - ∝ ∫ ∞ 2 exp ) ( ) ( (12)
The effect of incident plane wave scattering is the creation of far-field secondary waves. These waves, arising from each point on the structure, travel in all directions giving rise to complex distributions of amplitude and phase. , where a perfectly flat angular distribution function signs the specular field behaviour. This diffractance function g(r) defines the scattering intensity of the structure for a defined angle as:

dr G r I i iqr q ∫ - = ) ( ) ( 0 r (14)
This function describes secondary acoustic waves leaving the scattering structure at a specified angle and arriving at a particular point, creating scattering sidelobes. The complex amplitude at this point is obtained by adding the individual contributions from primary and secondary sources of amplitude g(r), involving the different path lengths of the travel to the reception point. Thus, the contribution from the element dr at a distance r is specified through the following one-dimensional intensity distribution, which is a major parameter of the Fraunhofer diffraction pattern:

2 ) exp( ). ( ) ( ∫ ∞ - = dr f qr i r g r I . ( 15
)
The expression may be compared with the one-dimensional Fourier Transform of the diffractance function g(r) as:

∫ ∞ - = dr r i r g k g r ) exp( ). ( ) ( ν (16)
where ν=q/f is the spatial period of the indented structure. Taking equations ( 15) and ( 16) into account, the measured intensity I(r) and the square of the modulus of the structure diffractance Fourier transform are identical:

2 2 2 2 2 sin ∏         = = i i i n kr kr F g I π π ( 17
)
This identity allows the expression of the scattering intensity I as a function of the structure factor of the indented plane, implying the characteristic interference function sinx/x, for a frequency f = 1/2πk as shown figure 8: The structure factor Fn is the Fourier transform of the scatterers of equal strength on all points at distance r of the diffraction plane. Continous expression of this previous equation involves structure angular distribution as:

∫ = space r r n dr ik k g r F ) 2 exp( ) ( ) ( π (18)
As shown equation ( 18), the diffraction pattern is defined as the Fourier transform of the structure angular density.

Application to the Sierpinski carpet structure

In this section, we will apply the Fraunhofer diffusion formalism to a well-known two-dimensional structure, the Sierpinski carpet, in order to explore it behaviour from a wave acoustical solicitation. To do so, we will express its Fourier transform trough introduction of an iterative term into the structure factor.

Iterative structure factor

As the structure factor of the Fourier Transform of a multiplicative signal of level n can be written as either a product of n periodic functions H i (r)( i = 1, 2, . . . , n) with frequencies D i-1 , or a product of n scaled replicas of the structure factor of the first level H 1 (u) [START_REF] Alieva | Paraxial diffraction on structures generated by multiplicative iterative procedures[END_REF]:

) ( ) ( ) ( 1 1 1 1 - = = ∏ ∏ = = i n i n i i n rD H r H r F (19)
Thus, the level n one-dimensional structure factor, constructed through an iterative procedure, can be defined as:

) ( ) ( ) ( ) ( 1 1 0 1 1 1 0 1 - - = - - = - = - * = ∑ ∑ i n n i i n n i n D r H f D r r H f r H ν ν δ (20)
When the initiator is a square and the generator f1 its bottom-right quadratic subtraction complement, the Sierpinski carpet set is then obtained through the following iterative procedure: where ν i is the spatial period of the structure at iteration level i=n, with scaling coefficients Di-1 for i = 1, . . . , n.. The structure factor Fn(r) is a complex function, as the initiator as well the generator matrix can take complex values. It is know that such as multiplicative cascades for real non-negative values of the generator matrix produce regular fractals [START_REF]Umberto Marini Bettolo Marconi and Alberto Petri Domain growth on self-similar structures[END_REF]. We can observe a similarity in the construction of the multiplicative reflection structure itself and its Fourier Transform. Indeed, the structure factors in both cases are constructed as a product of n scaled replicas of the periodic signals with scaling coefficients Di-1.

Results and discussion

As an indicator of the indentation frequency, the Fourier transform discriminates clearly the structure of the reflection surface, revealing the spatial occurrences of the roughness peaks. Fourier transform of the structure shows quasiperiodic behaviour of the structure factor, at every iteration if the shape. In addition, we can note that when iteration levels goes to infinite, the fractal structure becomes invariant for its Fourier transform. The angular scattering distribution function is defined through the structure factor computation of the surface, and indicates the fractal scattering behaviour for a particular direction of the incident wave: We can easily discriminate an anomalous quasiperiodic behaviour of the scattering angular and spatial factors, even when the characteristic Fraunhofer diffraction region pattern remains obvious.

The angular density function remains quasi-periodic at any distance of the carpet, signing a quasi-perfect diffusive structure. The angular distribution of propagating energy will then no longer depend on the distance from the plane structure. Moreover, the spatial scattering distribution function offers a tridimensional representation of the scattering distribution function, showing the angular and distance distribution's 3-D map of the surface's scatterers density: Results shows a high spatial diffractance persistence for high-level iterated Sierpinski carpet. This behaviour traduces an important diffusion capacity of the Sierpinski structure at high iteration levels.

Spatial scattering distribution function data also provide scattering intensity crossdistribution plot, as presented for the three iteration levels of the carpet: As expected, the Sierpinki carpet far field scattering intensity cross-distribution shows self-similar diffraction patterns for several spatial frequency bands: after propagation of a coherent plane wave through this fractal grating, the computed acoustic field has fractal properties.

This diffraction pattern may be interpreted as the spatial filter response of the analysed structure. The location of mainlobe peak tells in which direction we get maximum response of the spatial structure from an acoustical solicitation. The mainlobe (highest peak) is then similar to the pass-band in a spatial filter, which diffracts acoustic energy in these directions.

It has been shown that diffusion capacity of a complex plane can be directly computed from the Fourier transform of its spatial structure. Applied to a fractal plane structure, the resulting spatial intensity distribution reveals the self similar behaviour of the scattered acoustic field angular distribution. Moreover, this Fraunhofer diffraction region characterisation allows scattering prediction for many types of spatial configurations, under condition of interference acoustic field, that means spatial / temporal frequencies ratio.

This powerful computation tool can also be applied to all types of complex geometries, including architectural (concert hall diffusers) and urban (façade indentations) configurations, in order to predict their far-field diffusion behaviour [START_REF] Woloszyn | Angular characterization of the urban frontages diffusivity factor, 17 th International Congress of Acoustics[END_REF].

Fig. 1 .

 1 Fig. 1. Fresnel (near-field) and Fraunhofer (far-field) diffraction regions

Fig. 2 .

 2 Fig. 2. (x, z) incoming wave vector on a regularly indented plane

Fig. 3 .

 3 Fig. 3. Interference geometry into an indent

125Fig. 4 .

 4 Fig. 4. Rayleigh criterion: limits between specularity and diffusion, as a function of frequency, incidence angle, and structure intation depths.

Fig. 5 .

 5 Fig. 5. Phase geometry Considering rcosα the component of r in the direction of the diffraction vector s,

Fig. 6 .

 6 Fig. 6. Diffraction geometryConsequently, as the length of the diffraction vector s is equal to the spatial amplitude Λ (indentation depth), sr is a periodic function in the spatial domain,

Fig. 7 .

 7 Fig. 7. Typical angular distribution function. The first shell represents the main density function at a distance r of the structure. Diffractance formulation (12) implies the density-density correlation function ) ( ) (

Fig. 8 .

 8 Fig. 8. Typical Fraunhofer diffraction patterns

Fig. 9 .

 9 Fig. 9. Sierpinski triangle Generator, and 8 th & 16 th iteration levels. Alternatively, if Hn(r) in the form (20) is introduced in the Fourier integral, then the structure factor can be represented by the following Fourier series: [ ] ) sin( ) 2 sin( exp ) / 2 exp( ) ( 1 0

Fig. 10 .

 10 Fig. 10. Space Fourier Transform of the Sierpinski carpet Generator, 8 th & 16 th iteration

  distribution leads to define the Fraunhofer diffraction patterns of each structure iteration level figure11:

Fig. 11 .

 11 Fig. 11. Sierpinski carpets scattering angular distribution functions at π/2rd for ν=λ (structure distance vs. power density function)

Fig. 12 .

 12 Fig. 12. Sierpinski carpets spatial scattering distribution functions (structure distance vs. diffraction angle vs. power density function)

Fig. 13 .

 13 Fig. 13. Sierpinski carpets scattering intensity cross-distributions (diffraction angle vs. power density function)


H Hossam
email: hany.hossam@cerma.archi.fr.

El Dien

P Woloszyn

Acoustical effect evaluation of balconies' ceiling form in protecting buildings facades against traffic noise. (IAQ T5S4)

This study presents the evaluation of acoustic performance off tall building facades closed to roadway due to one of its balconies configurations, namely ceiling, with an inclined form in terms of traffic noise reduction. Three inclined angles are tested (5°,10°, and 15°) with different balcony depths ( 2m, and 3m) by using the Pyramid Tracing model (DISIAPYR) developed by A. Farina and scale model measurements. The results in terms of A-weighted sound pressure level reduction are expressed into the balcony back wall and in free field. The protection level, defined as the difference in noise levels before and after inserting the proposed ceiling form, has been used to assess the reduction offered by that configuration. A maximum reduction due to using these forms is obtained at higher floors.

Introduction

A balcony is an extension of an internal floor with access by means of windows or doors to an external environment. It provides view and weather protection and is a familiar element in buildings. Particularly, it can provide an acoustical protection by means of its elements forms.

Most of the previous studies investigated sound levels on balconies by using experimental models as the work of Mohsen, and Oldham [START_REF] Mohsen | Traffic noise reduction due to the screening effect of balconies on a building façade[END_REF], numerical models of Hothersall [START_REF] Hothersall | Numerical modelling of the sound field near a tall building with balconies near a road[END_REF], and Li [START_REF] Li | A simple formula for evaluating the acoustic effect of balconies in protecting dwelling against road traffic noise[END_REF], and the field measurements analyzed by May [START_REF] May | Freeway noise and high-rise balconies[END_REF]), Gilbert [START_REF] Gilbert | Une étude sur la protection des habitations et technique du bâtiment[END_REF]; and Li [START_REF] Li | A simple formula for evaluating the acoustic effect of balconies in protecting dwelling against road traffic noise[END_REF]. These studies carried out a study of traffic noise reduction in the context of treatment to the sound source, in the path between the sound source and the receiver, and at the reception point.

An alternative approach concerning the self-protecting buildings was discussed by Mohsen and Oldham by means of changing balcony depth [START_REF] Mohsen | Traffic noise reduction due to the screening effect of balconies on a building façade[END_REF]. Another architectural concept was presented by Hossam [START_REF] El Dien | Balcony form; an approach to reduce sound pressure level into the building façade[END_REF][START_REF] El Dien | Balcony form; an approach to reduce sound pressure level into the building façade[END_REF][START_REF] El Dien | Prediction of the sound field into high-rise building facades due to its balcony ceiling form[END_REF], by using an inclined parapet to increase the shielding effect to protect the balcony back wall from the traffic noise nuisance in a free field.

In the present study, we aim to predict the effect of one of the balcony configurations, namely ceiling, on the traffic noise reduction into building facades.

Inclined ceilings with different angles are tested by 3D numerical model of Pyramid tracing (DISIPYR) developed by Farina [START_REF] Farina | Computer code based on experimental results for acoustical mapping of urban areas[END_REF], and in free field conditions. The results will be expressed in terms of sound pressure level reduction, in A-weighted scale, into the balcony back wall and for a high rise building.

Geometric parameters

Acoustic simulation programs (CITYMAP&DISIAPYR) are used to model the outdoor sound propagation environment within the balcony configurations and in three dimensions. This algorithm allows the simulation of outdoor sound propagation with a complex urban form and the evaluation of sound passing through sound insulating panels, taking into account the edge diffraction over the boundaries and the scattering of sound from the edges of finite surfaces, and the surface diffusion coefficient [START_REF] Farina | Computer code based on experimental results for acoustical mapping of urban areas[END_REF].

As shown in Fig. 1, the simulation has been carried out with the following variables:

1. The horizontal projection depth (W = 2, and 3 meters). 2. The balcony's ceiling inclined angle (Ø = 5°, 10°, and 15°).

And we have fixed the following parameters:

Prediction Methodology

In three dimensions, the model is equivalent to an infinite coherent line source parallel to the building facade. All the surfaces in the model are initially defined as specular surfaces (concrete with absorption coefficient 0.07 at 1K.Hz). The acoustic performance of the balcony ceiling with an inclined shape has been assessed by means of a numerical simulation. Three inclined angles have been tested (5°, 10°, and 15°).

The basic concept of inclining the balcony ceiling is to decrease the power of reflection and diffuse energy components by changing the direction of reflected rays away from the openings located at the balcony back wall as shown in Fig. 2. In the case of the inclined ceiling, we can notice that the protected surface over the balcony back wall increases by comparison of the protected surface gained from the classical form as a consequence of ceiling reflected area.

Firstly, the sound pressure reference level (SPL ref ), was calculated over the balcony back wall with an inclined angle Ø equals to 0°.

The protection level is defined as the difference in noise level at an assessment point with and without changing ceiling inclined angle, and is calculated according to the following equation:

L p = SPL ref -SPL Ø dB(A)
Where SPL ref is the reference level and SPL Ø is the sound pressure level calculated for a given ceiling angle at the same point.

Experimental system

Scale model

A 1:10 scale model of building façade with 2.50m height and 1.60m length (Fig. 3) was carried out. Building façade was simulated by varnished wood (absorption coefficient 0.06 at 1K.Hz). in order to satisfy the laws of acoustic similarity for the sound absorption by the building façade.

Acquisition system

The experimental results were obtained using a measurement exploiting maximum-length sequence stimulus (MLS).

Sound source

Line source was carried out with 8 tweeters AUDAX (TW 034X0), filtered for obtaining a frequency response from 1.6 KHz to 25 KHz. As shown in Fig. 5, the tweeters were headed towards a quadratic residue diffuser in order to obtain a quasi quarter-cylindrical propagation form in front of the building facade. By this diffusing system, the line source directivity is made quasi-uniform in the aperture angle which we interest. The sequence of the quadratic residue diffuser is calculated following the Schröder's modulo formula (Schröder.1979).

Reception Points Tweeters

Quadratic residue diffuser Fig. 5. Line source with a quadratic residue diffuser.

Simulation results

Balcony with 2m depth

The acoustic performance of inclined ceilings for a closed balcony with 2m depth was tested. Fig. 6 shows the average of sound level reduction obtained by three angles for that configuration. The average reduction obtained at the lower levels is negligible, while it increases at the higher floors. The reduction is negative from the 1 st to 5 th floor while it increases at the 6 th and 7 th floors. Generally, the angle equal to 15° has a greater effect at higher levels and allows a protection level from 0.5 to 3 dB (A). In Fig. 7, we display the results for measurements and predictions of the balcony protection level for a balcony with 2 m depth. We show the comparison results of the balcony with inclined ceiling angle equal to 15°. As demonstrated in the following figure, balconies with that configuration can give better protection on higher floors (Fig. 7 -a) at all the assessment points than those of lower floors (Fig. 7 -c). This reasonable because the path difference of a diffracted ray (diffracted at the parapet or floor of a balcony) and a direct ray increases with the relative height of a balcony to a road. While we notice at the 4 th floor (Fig. 7 -b) the higher points provide more protection level that the lower points. This reasonable because of the negative effect of the inclined ceiling form at lower points as explained in Fig. 2, where the ceiling surface rereflect the rays towards those lower points. So the windows may be the best architectural solution in that case where its surfaces will be protected by that inclined configuration.

Balcony with 3m depth

By using the same method, we found that the acoustic performance of inclined ceilings for a closed balcony with 3 m depth provides more reduction at lower levels than that obtained from the previous configurations and also for higher levels. The average reduction obtained at the lower floors is more effective than the previous depth, and also for the higher floors. Reduction values are negative at the 2 nd floor and begin to increase from the 3 rd floor to reach its maximum value at the 6 th floor (Fig. 8). In general, the average protection gained from that configuration is from 0.5 to 4 dB (A). As shown in Fig. 8, the inclined ceiling with 5° angle is less effective at lower levels, while its maximum effect appears at the higher levels. Furthermore, the effects of 10° and 15° angles are greater at higher levels. Generally, the maximum protection level can be obtained by an inclined ceiling with 5°. As the balcony with 2m depth, balconies with that configuration can give better protection on higher floors (Fig. 9 -a) at all the assessment points than those of lower floors (Fig. 9 -c). This reasonable because the effect of direct and reflected rays. While we notice at the 4th floor (Fig. 7-b) the higher points provide more protection level that the lower points for the same reason we explained before. So the windows may be the best architectural solution in that case where its surfaces will be protected by that inclined configuration.

Conclusion

In this paper the Pyramid Tracing model and experimental measurements were used to predict the performance of a balcony ceiling with inclined form. The average noise reduction provided by that configuration with reflective surface was about 0.5 -4 dB (A). This investigation showed that the inclined ceiling form can dominate the sound reflected from its bottom surface. Moreover, it can change the reflected ray direction away from a balcony back wall. Reduction obtained increases with the increase of floor level and balcony depth. On the other hand, it decreases with the increase of the ceiling inclined angle at higher floors while it increases with the increase of the ceiling inclined angles at lower floors. Generally, the predicted results showed that a balcony ceiling form can be employed to protect weak points on the facades, for high rise buildings, against traffic noise propagation.

In addition, the results of this investigation will hopefully provide practical information to the architect who wishes to design facades which can be described as self protected with respect to the external acoustic environment. It will be seen that a slight modification of existing building envelope design can provide additional sound protection without compromising other environmental requirements. Furthermore, the numerical and experimental results provide a validation of the geometrical empirical equations which we obtained (Hossam El-dien and Woloszyn 2004).

  1. Sound source (infinite line source). 2. Distance to road (8.00 m). 3. Front wall height (1.00m). 4. Balcony's length (5.00m). 5. Number of floors (8 floors).

  Fig. 1. Geometric parameters.

Fig. 2 .

 2 Fig. 2. Schematic representation of the reception surface and its image before and after changing balcony ceiling form

Fig. 3 .

 3 Fig. 3. Scale model dimensions.

Fig. 4 .

 4 Fig. 4. Reception points (6 reception points at the centre of the facade) and microphone position.4.3. Sound source

Fig. 6 .

 6 Fig. 6. Average of protection offered by inclined ceilings of a balcony with 2m depth, experimental results (EX) and simulation results (S).

Fig. 7 .

 7 Fig. 7. Protection values provided at the centre of the balcony back wall with 2m depth, numerical simulation (dashed), and empirical equation (lines) at (a) 2nd floor, (b) 4th floor, and (c) 6th floor for an angle Ø equal to 15°.

Fig. 8 .

 8 Fig. 8. Average of protection offered by inclined ceilings of a balcony with 3m depth, experimental results (EX) and simulation results (S).

Fig. 9 .

 9 Fig.9. Protection values provided at the centre of the balcony back wall with 3m depth, numerical simulation (dashed), and empirical equation (lines) at (a) 2nd floor, (b) 4th floor, and (c) 6th floor for an angle Ø equal to 10°.