Eduard R Muslimov

Marc Ferrari

Emmanuel Hugot

Jean-Claude Bouret

Coralie Neiner

Simona Lombardo

Gerard Lemaitre

Robert Grange

Spectrographs with holographic gratings on freeform surfaces: design approach and application for the LUVOIR mission

Keywords: freeform optics, holographic grating, spectral resolution, LUVOIR mission, optical design

In the present paper we demonstrate the approach to use a holographic grating on a freeform surface for advanced spectrographs design. On the example POLLUX spectropolarimeter medium-UV channel we chow that such a grating can operate as a cross-disperser and a camera mirror at the same time. It provides the image quality high enough to reach the spectral resolving power of 126 359-133 106 between 11.5 and 195 nm, which is higher than the requirement. Also we show a possibility to use a similar element working in transmission to build an unobscured double-Schmidt spectrograph. The spectral resolving power reaches 2750 for a long slit. It is also shown that the parameters of both the gratings are feasible with the current technologies.

INTRODUCTION

Freeform optics is a rapidly emerging direction in the optical technology, which is expanding the borders of achievable optical systems performance. With use of such surfaces it becomes possible to create a wide-field fast optics with fewer number of optical elements and also create new systems with geometry, which was impossible with ordinary aspheres.

Application of a freeform optical surfaces for design of diffractive optical elements represents an attractive prospect. Combining of a complex surface with a special grooves pattern could allow to get more degrees of freedom for aberrations correction and achieve a qualitatively new level in spectroscopy, imaging and beam shaping. Such a possibility was recently considered and demonstrated by a number of authors. For example, an optical design with a blazed grating on a non-symmetrical tilted elliptical surface was developed and fabricated for the IGIS spectrograph [START_REF] Bourgenot | Towards freeform curved blazed gratings using diamond machining[END_REF] . However, in that case the grating had equally spaced straight grooves (in projection to the tangent plane). Another freeform grating was used in the Offner-type design for the ELOIS spectrometer [START_REF] De Clercq | ELOIS: an innovative spectrometer design using a free-form grating[END_REF] . It had a complex freeform shape without rotational symmetry and the grooves pattern varying in order to maintain their positioning with respect to the local surface normal. In general, application of freeform surfaces for imaging spectrographs design was considered in different studies [START_REF] Liu | Comparison of hyperspectral imaging spectrometer designs and the improvement of system performance with freeform surfaces[END_REF][START_REF] Wei | Optical design of Offner-Chrisp imaging spectrometer with freeform surfaces[END_REF] . One of such publications 5 demonstrates an Offner-Chrisp system with a grating on a general freeform surface described by Zernike fringe polynomials. Also there are ongoing research activities, which will allow to combine a freeform surface, complex grooves pattern and blazed grooves profile in a single optical element. Finally, a number of publications have shown the possibilities of application of freeform diffractive elements in adjacent fields like display technologies, or image reconstruction [START_REF] Liu | Design of a uniform-illumination binocular waveguide display with diffraction gratings and freeform optics[END_REF][START_REF] Liu | Design and fabrication of DOEs on multi-freeform surfaces via complex amplitude modulation[END_REF] .

In the present paper we consider the most general case, when a grating with curved and unequally spaced grooves is imposed over a freeform surface. It is supposed that the grating is manufactured holographically, i.e.it represents a recording of interference pattern from two coherent sources. We developed a set of tools for description of such holographic freeform gratings in Zemax. The primary application of these tools was the design of POLLUX spectropolarimeter for LUVOIR [START_REF] Bolcar | The Large UV/Optical/Infrared Surveyor (LUVOIR): Decadal Mission concept design update[END_REF] mission. And we show that using of the freeform grating allows to obtain the required image quality with the minimal number of optical surfaces. However, the possible applications are not limited by this particular case. It is demonstrated that a transmission freeform grating can help to achieve a high performance of a double-Schmidt type spectrograph.

Thus the paper is organized as follows: chapter 2 presents the ways used to describe the freeform surface shape and the grating pattern; chapter 3 demonstrates usage of such an optical element for the POLLUX instrument design with estimation of the obtained image quality; chapter 4 presents a demonstrative optical design with a transmission freeform grating with the performance assessment and section 5 contains the general conclusions on the study.

DESIGN APPROACH

To describe a holographic grating on a freeform surface one need to define the surface sag and the normals and then to set up a procedure of diffraction computation in any point of this surface. Below we describe in details the means, used to model both of the points practically in Zemax.

Freeform description

The freeform surface in our case is described by one of the orthonormal polynomials sets -Zernike or Legendre polynomials. The Zernike polynomials 10 are widely used in optical engineering. They are orthonormal over the unite circle, what simplifies the freeforms design and optimization with them. Another polynomial set chosen for this task is the Legendre polynomials. They are orthonormal over the unite square. Advantages of their use in some cases were demonstrated in a number of publications [START_REF] Ye | Comparative assessment of orthogonal polynomials for wavefront reconstruction over the square aperture[END_REF][START_REF] Ye | Review of optical freeform surface representation technique and its application[END_REF] . To facilitate the further explanations we provide sag diagrams for the 4 th to 11 th modes of each set on Figure 1 (the piston, tip and tilt are the same, so they are excluded). It is necessary to mention that a freeform surface can be described any of these polynomial sets, but due to some features of the numerical optimization process one way of the freeform description can be preferable [START_REF] Muslimov | Combining freeform optics and curved detectors for wide field imaging: a polynomial approach over squared aperture[END_REF] .

Grating description

For modelling of a holographic grating we used the general raytracing procedure [START_REF] Welford | A vector raytracing equation for hologram lenses of arbitrary shape[END_REF] . The vectorial equation used for computations is:

𝑁 ⃗ ⃗ × (𝑟 𝑖 ⃗ ⃗ -𝑟 𝑑 ⃗⃗⃗ ) = 𝑘 𝜆 𝜆 0 𝑁 ⃗ ⃗ × (𝑟 1 ⃗⃗⃗ -𝑟 2 ⃗⃗⃗ ) (1)
Here 𝑁 ⃗ ⃗ is the normal vector, 𝑟 𝑖 ⃗ ⃗ , 𝑟 𝑑 ⃗⃗⃗ are vectors of the recording rays, 𝑟 1 ⃗⃗⃗ , 𝑟 2 ⃗⃗⃗ are that of the incident and diffracted rays, k is the order of diffraction, λ0 and λ are the recording and working wavelengths respectively.

Hereafter we assume that the grating is recorded by two point sources. So the recording and operating geometry is presented on Figure 2. In this case the grating works in reflection, but for a transmission grating the equation and all the definitions remain similar except of the sign before the diffracted ray vector and inclusion of the refraction index into equation (1). Combining the given descriptions of surface shape and grooves pattern one can create a user defined surface in an optical design software like Zemax (using customized .dll library) and after that use the standard raytracing and optimization tools.

OPTICAL DESIGN WITH A REFLECTION GRATING

Design description

The primary application of the developed design tools is optical design for POLLUXa high resolution UV spectropolarimeter for LUVOIR mission. The target spectral resolving power of the instrument is 120 000 in the extended UV region 90-390 nm [START_REF] Muslimov | Optical design of UV echelle spectrograph for a next generation space mission[END_REF] . Such a high resolution can be achieved in an instrument with 3 channels, when each of them represents an echelle spectrograph. It was decided to split the channels as follows: far UV (FUV) channel operates from 90 to 124.5 nm and is fed by a flip mirror; medium UV (MUV) channel works between 118.5 and 195 nm, while near UV (NUV) channel works in the range 195-390 nm and they are separated by a dichroic splitter. It must be also mentioned, that the spectral length of a diffraction order should be 6 nm at least to avoid splitting of some broadened analytical lines.

The design solutions of the polarimetric units are different for each channel, while the spectral part is similar for all the 3 channels. The MUV channel optics general view is shown as an example on Figure 3. The entrance beam at F/20 passed through the pinhole and the polarimeter is collimated by an off-axis parabolic (OAP) mirror. Then it is dispersed by the echelle grating. The cross-disperser represents a freeform holographic grating, so it simultaneously separates the echelle's diffraction orders and focuses them on the detector. Use of such a complex element allows to correct the aberrations and achieve a high resolution and also decrease the total number of bounces and thus increase the throughput. One of the main features of this design is spatial separation of the spectral components on the cross-disperser's surface (see Figure 4, left). It means that the aberrations for this element considerably vary across the working range. So the grating properties and the surface shape should vary locally to compensate the aberrations. It also requires use of a freeform holographic grating. After the design and optimization the grating frequency equals to 212.3 1/mm and the recording sources rectangular coordinates (100.194,1913.946) and (-99.558,1936.898) in the case of use of Ar laser. The focal length of the grating acting as a camera mirror is 1200 mm and its clear aperture is 215.4x98.3 mm. The surface shape is described by the vertex sphere and 6 Zernike terms (primary coma, astigmatism and trefoil, 3 rd order spherical, astigmatism and quadrofoil). The deviation from the best-fit sphere (BFS) is shown on Figure 4, right as a colormap. The root-mean square (RMS) deviation is 2.27 µm and the peak-to-valley (PTV) value is 3.36 µm.

Performance analysis

During the design and optimization process the image quality is assessed via the spot diagrams (see Figure 5 for an example of diagrams for a single order). It is an easy way to compute an image quality estimation. One should keep in mind that the minimized function is an RMS size of the spot and that the X (horizontal) direction corresponds to the echelle dispersion direction, so the X size of the spot defines the spectral resolution. Also it should be noted that the order spectral length is 3.7 nm without the part overlapping with the adjacent orders, though the full length on the detector corresponds to 6.1 nm. However, the dimensions of the spots don't correspond to the spectral resolving power directly. In order to compute the spectral resolving power the spectrograph's instrument functions (IF) are calculated. The entrance slit width is 31.2 µm, what corresponds to 0.03" on sky. The results for the same control wavelengths in a single order are given on Figure 6. As one can see, the aberrations correction is good enough, so the FWHM (full width at half maximum) of IF equals to the slit width. The same data re-computed to spectral resolving poser units is summarized in Table 1.

Table 1. POLLUX MUV spectral resolving power summary. The spectral resolving power is higher than the required value by ~10%, that allows us to leave some margins for possible manufacturing errors and misalignments.

Wavelength. nm

In general, this modeling shows that with a freeform holographic grating it is possible to achieve the required spectrograph performance, while the grating recording scheme is feasible and the surface asphericity is small.

OPTICAL DESIGN WITH A TRANSMISSION GRATING 4.1 Design description

However, the design tools developed for Pollux can generalized and applied for other optical systems. Here as an example we consider a spectrograph with transmission freeform grating similar to the VIRUS instrument design [START_REF] Lee | Visible Integral-field Replicable Unit Spectrograph (VIRUS) optical tolerance[END_REF] . The considered design works in the visible domain 350-550 nm with F/3.1 aperture. Similarly to the prototype we start with a design using inverted Schmidt telescope as the collimator and another Schmidt telescope as the camera$ the focal distances of the both units are equal to 420 mm. In our case the corrector plates for both of the Schmidt-type parts is united with the dispersive element thus forming the transmission freeform grating. Thanks to the Schmidt camera properties the spectrograph can operate with a long entrance slit up to 50 mm. Due to use of asymmetrical freeform surface it becomes possible to introduce tilts on the spherical mirrors and avoid any central obscuration in the optical scheme. The general view of the optical design is given on Figure 7. Note that two flat folding mirrors were introduced to decrease the overall dimensions. WE also should mention that in this case the focal plane is curved with radius of 290.96 mm, though a similar result can be obtained with a flat image plane and a field flattener. The grating frequency at the vertex is 636.4 1/mm and the recording sources coordinates for a Nd:YAG laser are (403.610,2153.123) and (630.212,4027.459). Since the aperture stop is located at the grating, its clear aperture is circular with diameter of 126 mm. The optimization starts with a plane-parallel plate, but the vertex radius of curvature is set as a free parameter. For demonstrative purposes in this case we use Legendre polynomials to describe the freeform, though the surface can be also described by the Zernike modes. 9 Legendre modes, including T20 and all the modes of the 3 rd and 4 th order symmetrical with respect to the YZ plane are used. The deviation from BFS of the surface is shown om Figure 8. The BFS radius is 6644.8mm. The RMS deviation is 70.7µm and the PTV deviation is186.6µm. These values are considerably larger than the ones obtained for previous design, however they are still manufacturable with the current level of technology. µm.

Figure 8. Deviation of the transmission freefrom grating surface from the BFS in microns.

Performance analysis

As well as for the first design, we consider the image quality estimations. The spot diagrams for long slit are presented on Figure 9. One can note that the aberrations are slowly growing towards the slit edges. However, the main contribution to this spot blurring is made by astigmatism. It means that the spot size in the dispersion direction remains relatively stable and the spectral resolving power doesn't change. In order to prove and quantify the image quality we consider thee spectrograph's IF's. The IF's for the slit center are shown on Figure 10. The entrance slit width is equal to 50 µm in this case. The IF plots indicates on presence of a residual coma aberration, but the FWHM is equal to the slit width across the entire working spectral range and for all the slit points. Table 2 summarizes the spectral resolving power data. Though a direct comparison is note fully correct, we can say that the achieved values are higher than that of the prototype design.

CONCLUSIONS

In the present paper we demonstrated application of freeform holographic gratings for spectrographs design. Such an optical element provides an outstanding possibility for aberration correction, which allows to improve the performance and decrease the number of optical elements. The first application of this element is the optical scheme of POLLUX spectropolarimeter for LUVOIR mission. The freeform reflection grating works as the cross-disperser and camera simultaneously and allows to fulfill the requirements for spectral resolution and minimize the number of bounces in the scheme. The second example is a demonstrative design of a double-Schmidt spectrograph with transmission grating. The design is notable for absence of central obscuration and relatively high spectral resolving power.

Obviously, the used geometries and polynomials sets can be used in different combinations, e.g. to describe a reflective holographic grating on a Zernike-type freeform surface. This design tool and the corresponding approach to optical schemes development can be of special interest for the upcoming and future instruments intended for scientific research, especially for the astronomical instrumentation [START_REF] Thatte | The E-ELT first light spectrograph HARMONI: capabilities and modes[END_REF][START_REF] Saunders | Efficient and affordable catadioptric spectrograph designs for 4MOST and Hector[END_REF] .

Figure 1 .

 1 Figure 1. Modes of the orthonormal polynomials used for the freeform optical surfaces description.

Figure 2 .

 2 Figure 2. Definition of the holographic grating recording and operation geometry.

Figure 3 .

 3 Figure 3. Optical design of the POLLUX instrument MUV channel: leftgeneral view, rightzoom-in of the polarimetric unit.

Figure 4 .

 4 Figure 4. POLLUX MUV cross-disperser grating surface: leftfootprint diagram, rightmap of the deviation from BFS in microns.

Figure 5 .

 5 Figure 5. Sample spot diagrams of the POLLUX MUVchannel in the 40 th order.

Figure 6 .

 6 Figure 6. Sample instrument functions of the POLLUX MUVchannel in the 40 th order (the entrance slit projection width is 31.2 µm).

Figure 7 .

 7 Figure 7. Optical design of a double-Schmidt spectrograph with a transmission freeform holographic grating.

Figure 9 .

 9 Figure 9. Spot diagrams of the double-Schmidt spectrograph for a long slit.

Figure 10 .

 10 Figure 10. Instrument functions of the double-Schmidt spectrograph (the entrance slit width is 50 µm).

Reciprocal linear dispersion, nm/mm Instrument function FWHM, µm

				Spectral resolving
				power
	188.9	31	0.047	31.2	128 909
	191.9			31.2	131 008
	195.0			31.95	133 106
	150.6	40	0.036	31.2	132 673
	148.8			31.2	131 035
	146.9			31.95	126 359
	120.2	50	0,029	31.95	129 700
	119.0			31.2	131 502
	118.0			31.2	130 391

Table 2 .

 2 The double-Schmidt spectrograph spectral resolving power summary.

	Wavelength.	Slit height,	Reciprocal linear	Instrument function	Spectral resolving
	nm	mm	dispersion, nm/mm	FWHM, µm	power
	350	0	4.0	50.0	1750
	450			50.0	2250
	550			50.0	2750
	350	25	4.0	50.0	1750
	450			50.0	2250
	550			50.0	2750

* eduard.muslimov@lam.fr; phone +33 4 91 05 69 18; lam.fr

ACKNOWLEDGEMENTS

The authors acknowledge the support from the European Research council through the H2020 -ERCSTG-2015 -678777 ICARUS program. This research was partially supported by the HARMONI instrument consortium.


A Hassler

J E Martin

R Amiot

T Tacail

F Arnaud Godet

R Allain

V Balter

Calcium isotopes offer clues on resource partitioning among Cretaceous predatory dinosaurs

Keywords: Ecology, Cretaceous terrestrial ecosystems, dinosaurs, calcium isotopes, spinosaurs, palaeodiet

Large predators are overabundant in mid-Cretaceous continental dinosaur assemblages of North Africa. Such unbalanced ecosystem structure involves, among predatory dinosaurs, typical abelisaurid or carcharodontosaurid theropods co-occurring with long-snouted spinosaurids of debated ecology. Here, we report calcium isotope values from tooth enamel (expressed as δ 44/42 Ca) to investigate resource partitioning in mid-Cretaceous assemblages from Niger (Gadoufaoua) and Morocco (Kem-Kem Beds). In both assemblages, spinosaurids display a distinct isotopic signature, the most negative in our dataset. This distinct taxonomic clustering in calcium isotope values observed between spinosaurids and other predators provides unambiguous evidence for niche partitioning at the top of the trophic chains: spinosaurids foraged on aquatic environments while abelisaurid and carcharodontosaurid theropods relied almost exclusively on terrestrial resources.

1.INTRODUCTION

Dietary inference in extinct vertebrates allows addressing the structure of extinct food webs and remains critical for understanding how ecosystems were structured, and how these structures evolved with time and environmental changes. In modern ecosystems, the trophic cascades concept [START_REF] Ripple | Status and Ecological Effects of the World's Largest Carnivores[END_REF] illustrates the complex role of top carnivores in regulating, directly and indirectly, populations from subsequent trophic levels. By modulating the intensity of resource exploitation (top-down), along with productivity feedback loops (bottom-up), top predators can shape the structure of ecosystems. However, reconstructing long extinct ecosystems, such as those of the Mesozoic, remains challenging because taxa with unknown ecology and physiology dominated the scene.

In contrast to modern mammalian continental ecosystems where herbivores represent most of the biomass of vertebrates, large predatory theropod dinosaurs were overabundant in mid-Cretaceous continental biotas of North Africa. The origin and the significance of this overabundance is debated [START_REF] Mcgowan | A surfeit of theropods in the Moroccan Late Cretaceous? Comparing diversity estimates from field data and fossil shops[END_REF][START_REF] Dyke | Palaeoecology: Different dinosaur ecologies in deep time?[END_REF] but recent studies have highlighted the peculiar nature of such predator-dominated ecosystems [START_REF] Läng | Unbalanced food web in a late cretaceous dinosaur assemblage[END_REF][START_REF] Benyoucef | Overabundance of piscivorous dinosaurs (Theropoda: Spinosauridae) in the mid-Cretaceous of North Africa: The Algerian dilemma[END_REF]. These faunas include very large theropods such as abelisaurids, carcharodontosaurids, and spinosaurids, which are all considered as having high food requirements due to their elevated metabolic rates [START_REF] Amiot | Oxygen isotopes from biogenic apatites suggest widespread endothermy in Cretaceous dinosaurs[END_REF][START_REF] Amiot | Oxygen isotope evidence for semi-aquatic habits among spinosaurid theropods[END_REF]. Their high abundance relative to herbivorous dinosaurs thus raises questions about how resources could be partitioned between them. Due to their remarkable anatomy, spinosaurids have attracted attention with previous work inferring a semi-aquatic lifestyle [START_REF] Amiot | Oxygen isotope evidence for semi-aquatic habits among spinosaurid theropods[END_REF][START_REF] Ibrahim | Semiaquatic adaptations in a giant predatory dinosaur[END_REF] associated to a diverse diet [START_REF] Läng | Unbalanced food web in a late cretaceous dinosaur assemblage[END_REF][START_REF] Benyoucef | Overabundance of piscivorous dinosaurs (Theropoda: Spinosauridae) in the mid-Cretaceous of North Africa: The Algerian dilemma[END_REF][START_REF] Taquet | Une curieuse spécialisation du crâne de certains Dinosaures carnivores du Crétacé: le museau long et étroit des Spinosauridés[END_REF][START_REF] Charig | Baryonyx walkeri, a fish-eating dinosaur from the Wealden of Surrey[END_REF][START_REF] Sereno | A Long-Snouted Predatory Dinosaur from Africa and the Evolution of Spinosaurids[END_REF][START_REF] Taquet | New data on spinosaurid dinosaurs from the early Cretaceous of the Sahara[END_REF][START_REF] Evers | A reappraisal of the morphology and systematic position of the theropod dinosaur Sigilmassasaurus from the 'middle' Cretaceous of Morocco[END_REF][START_REF] Holtz | Spinosaurs as crocodiles mimics[END_REF][START_REF] Rayfield | Functional Morphology of Spinosaur 'Crocodile-Mimic[END_REF][START_REF] Buffetaut | Pterosaurs as part of a spinosaur diet[END_REF][START_REF] Hone | A Century of Spinosaurs A Review and Revision of the Spinosauridae with Comments on Their Ecology[END_REF][START_REF] Vullo | Convergent evolution of jaws between spinosaurid dinosaurs and pike conger eels[END_REF], although the respective proportions of feeding sources, whether terrestrial or aquatic, remains unknown. Indeed, if for some spinosaurids, trophic links can be assessed using direct evidence like stomach content and feeding marks [START_REF] Charig | Baryonyx walkeri, a fish-eating dinosaur from the Wealden of Surrey[END_REF][START_REF] Buffetaut | Pterosaurs as part of a spinosaur diet[END_REF], such fossils remain rare and account for a short time window.

Morphology can also provide a wealth of data concerning potential trophic links, but interpretations remain hypothetical due to the lack of modern analogues for spinosaurids and their ecosystem. The jaw morphology of spinosaurids has been repeatedly, albeit inexactly, considered analogous to that of crocodiles [START_REF] Holtz | Spinosaurs as crocodiles mimics[END_REF][START_REF] Rayfield | Functional Morphology of Spinosaur 'Crocodile-Mimic[END_REF] and has recently been compared to that of pike conger eels [START_REF] Vullo | Convergent evolution of jaws between spinosaurid dinosaurs and pike conger eels[END_REF], reflecting some uncertainty in palaeoecological inference when considering morphological evidence alone [START_REF] Thomason | Functional morphology in vertebrate paleontology[END_REF]. In this context, we investigate diet and resource partitioning among mid-Cretaceous Saharan vertebrates using Ca isotopes (δ 44/42 Ca) of tooth enamel and scale ganoine, a method that has previously proved informative both in modern [START_REF] Skulan | Calcium isotope fractionation between soft and mineralized tissues as a monitor of calcium use in vertebrates[END_REF][START_REF] Melin | Calcium and carbon stable isotope ratios as paleodietary indicators[END_REF][START_REF] Martin | Calcium isotopes reveal the trophic position of extant and fossil elasmobranchs[END_REF] and fossil vertebrate assemblages [START_REF] Martin | Calcium isotopes reveal the trophic position of extant and fossil elasmobranchs[END_REF][START_REF] Martin | Non-traditional isotope perspectives in vertebrate palaeobiology[END_REF][START_REF] Heuser | Calcium isotopes in fossil bones and teeth -Diagenetic versus biogenic origin[END_REF][START_REF] Martin | Calcium Isotopic Evidence for Vulnerable Marine Ecosystem Structure Prior to the K/Pg Extinction[END_REF]. Because Ca is almost exclusively derived from food among terrestrial vertebrates [START_REF] Skulan | Calcium isotope fractionation between soft and mineralized tissues as a monitor of calcium use in vertebrates[END_REF], the systematic change between the isotopic signature of their food relative to their mineralized tissues allows inferring trophic levels and diet. Moreover, Ca is the major constituent of the mineral phase of bone and tooth enamel, i.e. hydroxylapatite (Ca 10 (PO 4 ) 6 OH 2 ), which gives the method two main advantages. First, the diagenetic overprinting of biogenic Ca is unlikely to occur providing that secondary carbonates are absent or have been leached accordingly [START_REF] Martin | Non-traditional isotope perspectives in vertebrate palaeobiology[END_REF]. Second, minute amounts of bone or tooth enamel sample (~100 µg) is sufficient for a routine measurement of the Ca isotopic ratios. The technique is thus applicable to precious fossils with minimal damages.

Our study is focused on fossils from the Aptian-Albian fluvial deposits of Gadoufaoua (Niger, ~113 Ma ago) [START_REF] Taquet | Géologie et paléontologie du gisement de Gadoufaoua[END_REF] and of the Cenomanian Kem Kem beds (Morocco, ~100 Ma ago) [START_REF] Cavin | Vertebrate assemblages from the early Late Cretaceous of southeastern Morocco: An overview[END_REF]. In these vertebrate assemblages, crocodylomorphs, spinosaurids and non-spinosaurid theropods are overabundant relative to herbivorous dinosaurs [START_REF] Sereno | A Long-Snouted Predatory Dinosaur from Africa and the Evolution of Spinosaurids[END_REF][START_REF] Taquet | Géologie et paléontologie du gisement de Gadoufaoua[END_REF][START_REF] Cavin | Vertebrate assemblages from the early Late Cretaceous of southeastern Morocco: An overview[END_REF], making them typical examples of North African mid-Cretaceous ecosystems. Despite age differences, the two deposits share similarities in their environmental context and host a comparable diversity of vertebrates. The analyzed specimens include fish, crocodylomorphs, pterosaurs, herbivorous (sauropods, ornithopods) and predatory (theropod) dinosaurs, including spinosaurids. Fossils of the spinosaurids Suchomimus tenerensis and Cristatusaurus lapparenti [START_REF] Sereno | A Long-Snouted Predatory Dinosaur from Africa and the Evolution of Spinosaurids[END_REF][START_REF] Taquet | New data on spinosaurid dinosaurs from the early Cretaceous of the Sahara[END_REF] are found in Gadoufaoua, where predators such as the non-spinosaurid theropods Kryptops palaios and Eocarcharia dinops [START_REF] Sereno | Basal Abelisaurid and Carcharodontosaurid Theropods from the Lower Cretaceous Elrhaz Formation of Niger[END_REF] occur. In the Kem Kem beds, the spinosaurids Spinosaurus aegyptiacus and Sigilmassasaurus brevicollis [START_REF] Amiot | Oxygen isotope evidence for semi-aquatic habits among spinosaurid theropods[END_REF][START_REF] Evers | A reappraisal of the morphology and systematic position of the theropod dinosaur Sigilmassasaurus from the 'middle' Cretaceous of Morocco[END_REF][START_REF] Stromer | Das Original des Theropoden Spinosaurus aegyptiacus nov. gen., nov. spec. Abhandlungen der[END_REF] are associated with fossils of non-spinosaurid theropods such as Carcharodontosaurus saharicus [START_REF] Cavin | Vertebrate assemblages from the early Late Cretaceous of southeastern Morocco: An overview[END_REF].

METHODS (a) Sample collection

Samples comprising 73 tooth enamel, enameloid and fish scale ganoine from the early to middle Cretaceous of Morocco and Niger have been analyzed for elemental concentration and calcium isotope composition. Niger samples, totaling 7 taxa, come from the locality of Gadoufaoua belonging to the Aptian/Albian Elrhaz Formation [START_REF] Taquet | Géologie et paléontologie du gisement de Gadoufaoua[END_REF]. They consist of 8 spinosaurid theropod teeth (Suchomimus tenerensis or Cristatusaurus lapparenti), 9 nonspinosaurid theropod teeth (cf. Kryptops palaios and cf. Eocarcharia dinops), 2 teeth of the sauropod Nigersaurus taqueti, 9 teeth of the ornithopod Ouranosaurus nigeriensis, 8 teeth of the pholidosaurid crocodylomorph Sarcosuchus imperator, 3 teeth tentatively referred to the notosuchian crocodylomorph cf. Araripesuchus sp., 3 teeth tentatively referred to an indeterminate large pterosaur, 5 teeth referred to an indeterminate pycnodont fish, 2 scales of an indeterminate lepisosteiform fish, and 1 scale attributed to the lepisosteiform fish Pliodetes sp.. Moroccan samples previously analyzed for their apatite oxygen and carbon isotope compositions [START_REF] Amiot | Oxygen isotope evidence for semi-aquatic habits among spinosaurid theropods[END_REF] come from three localities (Bou Laâlou, Jebel Al Qabla, and Takemout) from the early Cenomanian Ifezouane Formation of the Kem Kem Beds [START_REF] Cavin | Vertebrate assemblages from the early Late Cretaceous of southeastern Morocco: An overview[END_REF]. They consist of 7 teeth attributed to non-spinosaurid theropods (4 teeth of Carcharodontosaurus saharicus, and 3 teeth of an indeterminate large theropod), 8 teeth referred to the spinosaurid theropods Spinosaurus aegyptiacus or Sigilmassasaurus brevicollis, 3 teeth of an indeterminate crocodylomorph, 3 teeth of an indeterminate large pterosaur, 1 scale of the amiiform fish Stromerichthys sp. and 1 tooth plate of the lungfish Neoceratodus africanus.

Here we investigate dietary and ecological aspects using the calcium isotope composition of tooth enamel, enameloid and scale ganoine (δ 44/42 Ca) rather than bone or dentine, which are porous and incorporate secondary diagenetic minerals. These biological materials are compact and resistant to diagenetic alteration [START_REF] Heuser | Calcium isotopes in fossil bones and teeth -Diagenetic versus biogenic origin[END_REF]. Sampling was performed using a handled micro-drill (8200 Dremel with tungsten steel solid carbide bit and diamond burr), a micro-drill station (MicroMill with tungsten steel solid carbide bit and diamond burr), and a scalpel made in stainless steel. Tooth enamel, enameloid and scale ganoine have been carefully separated from dentine and sediment using these tools.

(b) Analytical techniques

Fossil samples were placed in Teflon beakers and dissolved in 500µL of ultrapure nitric acid (15 N) at 130°C during 1 hour. Following this attack, samples were dried out, then dissolved in a dilute solution of ultrapure nitric acid (0.5 N).

A fraction of the solutions was measured for elemental concentrations of trace elements (Rare Earth Elements (REE), strontium (Sr) and barium (Ba)) and major elements (such as calcium (Ca), magnesium (Mg) and phosphorus (P)). These measurements were performed on an inductively coupled plasma mass spectrometer (7500 Series ICP-MS, Agilent Technologies) for trace elements, and on an inductively coupled plasma atomic emission spectrometer (ICAP 6000 Series ICP spectrometer, Thermo electron corporation) for major elements. The reliability of measurements has been controlled through a set of blanks and standards (see measures NIST-SRM1400, USGS-MAPS-4 and USGS-MAPS-5 in supplementary table).

Calcium from samples was chemically purified following previously described protocols [START_REF] Tacail | A simplified protocol for measurement of Ca isotopes in biological samples[END_REF]. Briefly, samples were dissolved in double distilled HCl and loaded onto columns filled with AG 50WX-12, separated from matrix including phosphates. Calcium, collected together with strontium, is then further processed in HNO 3 medium using Eichrom Sr-specific resin, allowing removal of Sr. The reliability of elutions have been controlled by processing and analyzing standard materials (see measures of NIST-SRM1486 in the supplementary table).

Calcium isotope abundance ratios ( 44 Ca/ 42 Ca) were measured using a Neptune Plus MC-ICP-MS (Multi-Collector Inductively Coupled Plasma Mass Spectrometer) following previously described methods [START_REF] Tacail | A simplified protocol for measurement of Ca isotopes in biological samples[END_REF][START_REF] Tacail | Precise Analysis of Calcium Stable Isotope Variations in Biological Apatites using Laser Ablation MC-ICPMS[END_REF]. After purification, Ca samples were dissolved in ultrapure 0.05N HNO3 and Ca concentration was set at 2 mg.l -1 for all samples and standards. All Ca isotope compositions are expressed using the "delta" notation defined as follows:

δ 44/42 Ca = (( 44 Ca/ 42 Ca) sample / ( 44 Ca/ 42 Ca) ICP Ca Lyon ) -1) x 1000 where ( 44 Ca/ 42 Ca) sample and ( 44 Ca/ 42 Ca) ICP Ca Lyon are the Ca isotope abundance ratios measured in sample and ICP Ca Lyon reference standard respectively. The ICP Ca Lyon standard, used as a bracketing standard, is a Specpure calcium plasma standard solution (Alfa Aesar) previously described here: [START_REF] Martin | Calcium isotopes reveal the trophic position of extant and fossil elasmobranchs[END_REF][START_REF] Martin | Non-traditional isotope perspectives in vertebrate palaeobiology[END_REF][START_REF] Tacail | A simplified protocol for measurement of Ca isotopes in biological samples[END_REF][START_REF] Tacail | Precise Analysis of Calcium Stable Isotope Variations in Biological Apatites using Laser Ablation MC-ICPMS[END_REF][START_REF] Tacail | Assessing human weaning practices with calcium isotopes in tooth enamel[END_REF]. The SRM1486 cow bone meal standard (NIST) was used as secondary standard, in order to assess accuracy of analytical procedure, including chemical purification.

(c) Monitoring of the isotopic measurement accuracy

Blanks realized during purification protocols have been analyzed with the MC-ICP-MS. All blanks contained less than 85 ng of Ca, which is 470 times less than the Ca contained in samples.

Thus this nano-pollution does not alter the isotopic measurements of our samples, considering our measurement precision.

The mass dependency curve of the isotopic measures follows the expected relation for Ca (figure 1), with a slope value of 0.509 ±0.016 (2SE, n=74) [START_REF] Martin | Calcium isotopes reveal the trophic position of extant and fossil elasmobranchs[END_REF][START_REF] Martin | Non-traditional isotope perspectives in vertebrate palaeobiology[END_REF][START_REF] Martin | Calcium Isotopic Evidence for Vulnerable Marine Ecosystem Structure Prior to the K/Pg Extinction[END_REF][START_REF] Tacail | A simplified protocol for measurement of Ca isotopes in biological samples[END_REF][START_REF] Tacail | Precise Analysis of Calcium Stable Isotope Variations in Biological Apatites using Laser Ablation MC-ICPMS[END_REF][START_REF] Tacail | Assessing human weaning practices with calcium isotopes in tooth enamel[END_REF], supporting the accuracy of the measures.

Measurements collected for the standard SRM1486 exhibit a δ 44/42 Ca of -1.00 ±0.10‰ (2SD, n=32), which is in the range of previously published data for this standard: -0.96 ±0.14‰ (2SD, n=17) [START_REF] Tacail | A simplified protocol for measurement of Ca isotopes in biological samples[END_REF], -1.04 ±0.11‰ (2SD, n=25) [START_REF] Martin | Calcium isotopes reveal the trophic position of extant and fossil elasmobranchs[END_REF], -1.03 ±0.13‰ (2SD, n=120) [START_REF] Tacail | Precise Analysis of Calcium Stable Isotope Variations in Biological Apatites using Laser Ablation MC-ICPMS[END_REF], -1.03 ±0.12‰ (2SD, n=147) [START_REF] Tacail | Assessing human weaning practices with calcium isotopes in tooth enamel[END_REF].

(d) Diet modelling

In order to perform diet inferences with Ca isotope data, we make use of our data to feed a simple diet mixing model. This model provides estimates of the relative contribution of Ca sources in the diet of predators thanks to δ 44/42 Ca. Two main parameter need to be constrained prior to using it.

First, the isotopic signature of Ca sources needs to be constrained for studied animals.

Calcium is more concentrated in mineralized tissues (bone, enamel, scale ganoine) than in soft tissues or in water. Thus, it has been shown that even a few percent of mineralized tissues in diet was sufficient to constitute the major source of Ca for organisms [START_REF] Heuser | Calcium isotopes in fossil bones and teeth -Diagenetic versus biogenic origin[END_REF]. There is fossil evidence of bone and fish scale consumption among numerous taxa of crocodylomorphs, spinosaurid and non-spinosaurid theropods [START_REF] Charig | Baryonyx walkeri, a fish-eating dinosaur from the Wealden of Surrey[END_REF][START_REF] Farlow | The fossil record of predation in dinosaurs[END_REF]. Thus, even if this consumption can be restricted to occasional events such as feeding on juveniles for bones, the high concentration of Ca in mineralized tissues supports that these tissues were most likely the main source of Ca among these taxa. Moreover, considering the heterogeneity of modern animal data on soft tissues [START_REF] Skulan | Calcium isotope fractionation between soft and mineralized tissues as a monitor of calcium use in vertebrates[END_REF][START_REF] Tacail | A simplified protocol for measurement of Ca isotopes in biological samples[END_REF][START_REF] Chu | Establishing the potential of Ca isotopes as proxy for consumption of dairy products[END_REF], and that blood can exhibit δ 44/42 Ca close to bones [START_REF] Tacail | A simplified protocol for measurement of Ca isotopes in biological samples[END_REF], this supports the use of δ 44/42 Ca values of mineralized tissues as representative of the main source of Ca for predators in our model. We have collected Ca isotope data for herbivorous dinosaurs and fishes from Gadoufaoua, which represent the most likely prey of the analyzed apex predators (crocodylomorphs, spinosaurid and non-spinosaurid theropods). Thus we have constrained the isotopic signature of the potential Ca sources to be entered in the model. Second, our model needs an estimate of the isotopic shift between Ca from food and teeth of the predators. According to data collected on a variety of extant vertebrates (chicken, horse, seal, deer, minipigs, sheep and mouse [START_REF] Skulan | Calcium isotope fractionation between soft and mineralized tissues as a monitor of calcium use in vertebrates[END_REF][START_REF] Tacail | A simplified protocol for measurement of Ca isotopes in biological samples[END_REF][START_REF] Chu | Establishing the potential of Ca isotopes as proxy for consumption of dairy products[END_REF]), the mean isotopic shift between Ca from diet to bones is 0.57 ±0.10‰ (2SE, n = 21) and is relatively homogenous among these vertebrates. Thus, considering that the studied predators are phylogenetically close (all are archosaurs) and that all of them share comparable Ca-cycles [START_REF] Skulan | Calcium isotope fractionation between soft and mineralized tissues as a monitor of calcium use in vertebrates[END_REF], we can reasonably assume that they exhibited a similar diet-bone isotopic offset. Only limited data have been collected on diet and enamel couples in modern faunas. Some studies show differences between these two shifts (±0.2‰; [START_REF] Martin | Calcium isotopes reveal the trophic position of extant and fossil elasmobranchs[END_REF][START_REF] Martin | Non-traditional isotope perspectives in vertebrate palaeobiology[END_REF][START_REF] Heuser | Calcium isotopes in fossil bones and teeth -Diagenetic versus biogenic origin[END_REF][START_REF] Tacail | A simplified protocol for measurement of Ca isotopes in biological samples[END_REF][START_REF] Clementz | New insight from old bones: stable isotope analysis of fossil mammals[END_REF]) but these studies are often conducted on mammals, whose teeth formed in early life, and thus were not representative of the adult diet for which the comparison is made. However, in human wisdom teeth, which developed along with an adult diet, the isotopic shift between enamel and food is similar to the diet-bone shift of -0.57 ±0.10‰ (2SE, n = 21) [START_REF] Tacail | Assessing human weaning practices with calcium isotopes in tooth enamel[END_REF]. Based on these results, we assume a similar diet-enamel shift among studied taxa, and use this shift to estimate the δ 44/42 Ca value of the food of predators. We thus applied this model to Sarcosuchus imperator, spinosaurid and non-spinosaurid theropods, in order to investigate the proportion of fish and herbivorous dinosaurs in the diet of these predators.

We estimate the Ca source fractions using the estimation of food δ 44/42 Ca between the first and the third quartile of each population of predators, considering that it is more representative of their general diet. Moreover, to take into account the variability of δ 44/42 Ca values among sources, these fractions are calculated as if predators with the most depleted δ 44/42 Ca values fed on fish and herbivorous dinosaurs with the most enriched δ 44/42 Ca values, and vice versa. In this way, we maximized the source fraction scattering, which sets the limit of diet inferences achievable with our model.

RESULTS

(a) Elemental concentrations

The full dataset compiling all the elemental concentrations measured in this study are reported in the supplementary table of this article.

Among important results, the mean calcium versus phosphorus ratio is of 2.3 ±0.1 (1SD).

(b) Calcium isotope ratios

The full dataset compiling all the calcium isotopic ratios measured in this study are reported in the supplementary table of this article. The δ 44/42 Ca values of vertebrates from Gadoufaoua range between -0.35 and -1.51‰ with a mean 2SD of 0.10‰. In this locality, the sauropod Nigersaurus taqueti is the taxon which exhibits the highest δ 44/42 Ca with a mean value of -0.43 ±0.11‰ (2SD, n=2), whereas the spinosaurids exhibit the lowest values with -1.30 ±0.15‰ (2SD, n=8). Among the other dinosaurs, the ornithopod Ouranosaurus nigeriensis exhibits values at -0.56 ±0.05‰ (2SD, n=9), whereas non-spinosaurid theropods exhibit values at -0.94 ±0.16‰ (2SD, n=9). The small, supposedly terrestrial crocodylomorph, cf. Araripesuchus sp., exhibits a mean value of -1.12 ±0.05‰ (2SD, n=3), whereas for the semi-aquatic crocodylomorph Sarcosuchus imperator this value is of -1.12 ±0.13‰ (2SD, n=8). For fishes mean values are of -0.74 ±0.02‰ (2SD, n=2) for indeterminate Lepisosteiformes, of -0.67‰ (n=1) for Pliodetes sp., and of -1.08 ±0.04‰ (2SD, n=5) for Pycnodontiformes. Finally, pterosaurs exhibit a mean value of -0.78 ±0.05‰ (2SD, n=3). The δ 44/42 Ca of spinosaurid theropods is significantly lower than that of non-spinosaurid theropods (Wilcoxon rank sum test: p value = 9.9 x 10 -4 ).

The δ 44/42 Ca values of Kem Kem beds vertebrates range between -0.41 and -1.56 ‰ with a mean 2SD of 0.09‰. In Kem Kem beds, the lungfish Neoceratodus africanus is the taxon which exhibits the highest δ 44/42 Ca with a value of -0.41‰ (n=1), whereas the other fish Stromerichthys sp. exhibits a low value of -1.26‰ (n=1). Spinosaurids exhibit the lowest mean value of this fauna, which is -1.39 ±0.27‰ (2SD, n=8). For large non-spinosaurid theropods (Carcharodontosaurus saharicus and indeterminate theropods) this value is of -0.64 ±0.13‰ (2SD, n=7). Semi-aquatic indeterminate crocodylomorphs exhibit a mean value of -0.66 ±0.09‰ (2SD, n=3), and indeterminate pterosaurs from the family of Anhangueridae exhibit a mean value of -0.67 ±0.17‰ (2SD, n=3). The δ 44/42 Ca of spinosaurids is significantly lower than for non-spinosaurid theropods (Wilcoxon rank sum test: p value = 3.1 x 10 -4 ).

(c) Fractions of Ca sources in diet

With our diet mixing model we estimate that the maximum Ca fraction from fish in the diet reaches 30%, 58% and 100% for non-spinosaurid theropods, Sarcosuchus imperator and spinosaurids, respectively. On the other hand, the maximum Ca fraction from herbivorous dinosaurs in the diet reaches 100% for all of these predators. The maximum Ca fraction derived from fish reaches 100% for 6 of the 8 studied spinosaurid individuals according to this model, which is possible for only two individuals of Sarcosuchus imperator and one non-spinosaurid theropod. However, the maximum Ca fraction derived from herbivorous dinosaurs is also 100% for 4 of the 8 spinosaurid individuals.

DISCUSSION (a) Diagenesis

Enamel, ganoine and to a lesser extent enameloid tissues, are highly mineralized and exhibit a lower porosity than dentine or bone. They have a limited exchange surface with surrounding fluids and sediment, and for this reason they are considered resistant to diagenesis [START_REF] Martin | Non-traditional isotope perspectives in vertebrate palaeobiology[END_REF][START_REF] Heuser | Calcium isotopes in fossil bones and teeth -Diagenetic versus biogenic origin[END_REF][START_REF] Wang | A model of fossil tooth enamel and bone diagenesis: implications for stable isotope studies and paleoenvironment reconstruction[END_REF].

Moreover, with almost 40% of their mass constituted of Ca, it is very unlikely that the Ca isotopic signature of these tissues can be entirely overprinted during early diagenetic processes.

The Ca/P ratios measured for samples (supplementary table) exhibit values of 2.3 ±0.1 (1SD), which are close to those measured in modern animals [START_REF] Balter | Were Neandertalians essentially carnivores? Sr and Ba preliminary results of the mammalian palaeobiocoenosis of Saint-Césaire[END_REF]. This supports that Ca isotope ratios from these fossils have not been extensively modified during diagenesis.

Quantifying elemental concentration of non-essential trace elements such as rare earth elements (REE) and of major elements such as Ca and phosphorus, can provide information on both the autochthony and diagenetic alteration [START_REF] Reynard | Trace elements and their isotopes in bones and teeth: Diet, environments, diagenesis, and dating of archeological and paleontological samples[END_REF]39]. Among fossils from Gadoufaoua, REE enrichment profiles (normalized to the Post Archean Australian Shale, PAAS) exhibit a similar shape with different REE enrichment intensity (figure 2A and supplementary table). They can thus reasonably be considered as autochthonous [39]. Three teeth exhibit distinct REE profile shapes (figure 2A). They have thus been identified as allochthonous fossils and have not been considered in the palaeoecological discussion. The general REE profile at Gadoufaoua is "hat shaped" according to the definition given by Reynard & Balter 2014 [38]. This shape, associated with measured La/Yb and La/Sm ratios (supplementary table), differs from the REE profiles and ratios expected after extensive recrystallization processes during diagenesis [START_REF] Reynard | Trace elements and their isotopes in bones and teeth: Diet, environments, diagenesis, and dating of archeological and paleontological samples[END_REF]40].

Among fossils from the Kem Kem beds REE enrichment profiles (normalized to the Post Archean Australian Shale, PAAS) exhibit a distinct profile between the three studied localities (figure 2B and supplementary table). However, within each locality, fossils exhibit similar REE profiles. They can thus reasonably be considered as autochthonous [39]. As for Gadoufaoua, the general "hat shaped" REE profile of the Kem Kem beds [START_REF] Reynard | Trace elements and their isotopes in bones and teeth: Diet, environments, diagenesis, and dating of archeological and paleontological samples[END_REF] and the measured La/Yb and La/Sm ratios (supplementary table) are different from those expected for diagenetic recrystallization processes [START_REF] Reynard | Trace elements and their isotopes in bones and teeth: Diet, environments, diagenesis, and dating of archeological and paleontological samples[END_REF]40]. Thus, the Ca/P ratios and the REE concentrations jointly support the hypothesis that the Ca isotopic composition of the fossils have not been extensively affected by diagenesis. Moreover, taxa from Gadoufaoua exhibit δ 44/42 Ca values consistent with their putative ecology (figure 3A). Indeed, herbivorous dinosaurs represented by Nigersaurus taqueti and Ouranosaurus nigeriensis exhibit the highest δ 44/42 Ca values of this dataset, with values ranging between -0.35‰ ±0.14 (2SD, n=3) and -0.66‰ ±0.05 (2SD, n=4), respectively. Their potential predators (the pholidosaurid crocodylomorph Sarcosuchus imperator, spinosaurid and non-spinosaurid theropods) exhibit lower δ 44/42 Ca values, below -0.70‰. This isotopic distribution between herbivores and predators follows what has been observed in modern continental faunas [START_REF] Skulan | Calcium isotope fractionation between soft and mineralized tissues as a monitor of calcium use in vertebrates[END_REF], and supports, along with the observations presented above, the biogenic nature of δ 44/42 Ca values.

(b) Resource partitioning among predators

Compared to non-spinosaurid theropods, spinosaurids exhibit distinctly lower δ 44/42 Ca values (Wilcoxon rank sum test: p-value = 9.9 x 10 -4 ), the lowest ones of the dataset (figure 3A). This distinct pattern is also observed in the Kem Kem beds of Morocco (figure 3B) with spinosaurids possessing significantly lower δ 44/42 Ca values compared to non-spinosaurid theropods (Wilcoxon rank sum test: p-value = 3.1 x 10 -4 ). Physiological homogeneity in Ca isotope fractionation from food to tooth enamel in these theropods is supported both by their phylogenetic proximity, all are archosaurs, and the observed homogeneity in modern vertebrates [START_REF] Skulan | Calcium isotope fractionation between soft and mineralized tissues as a monitor of calcium use in vertebrates[END_REF][START_REF] Tacail | A simplified protocol for measurement of Ca isotopes in biological samples[END_REF][START_REF] Chu | Establishing the potential of Ca isotopes as proxy for consumption of dairy products[END_REF]. This pattern thus highlights the distinctive ecological niche of spinosaurids relative to the other theropods.

The trophic pyramid in these fluvial palaeo-ecosystems could have been supported both by herbivorous dinosaurs and fish [START_REF] Läng | Unbalanced food web in a late cretaceous dinosaur assemblage[END_REF][START_REF] Benyoucef | Overabundance of piscivorous dinosaurs (Theropoda: Spinosauridae) in the mid-Cretaceous of North Africa: The Algerian dilemma[END_REF][START_REF] Taquet | Géologie et paléontologie du gisement de Gadoufaoua[END_REF][START_REF] Cavin | Vertebrate assemblages from the early Late Cretaceous of southeastern Morocco: An overview[END_REF], which would represent two potential primary sources of Ca for predators. In Gadoufaoua, the Ca source constituted by herbivorous dinosaurs exhibits a higher δ 44/42 Ca value than fish, which allows discriminating potential dietary sources among predators. Using the diet model described above (Method (d)) for the vertebrate assemblage of Gadoufaoua, we observed that the estimated fractions of Ca originating from herbivorous dinosaurs and fish in the diet of predators do overlap between predators. The important range of fraction between the maximum and minimum fraction estimated for each predator was expected considering the scattering of the δ 44/42 Ca data from fish, herbivorous dinosaurs and predators. However, even with this important scattering in isotopic values, the difference between spinosaurid and non-spinosaurid theropods remains statistically significant, implying distinct resource partitioning between these large predators. Here, δ 44/42 Ca data support the idea that a majority of non-spinosaurid theropods mainly fed on herbivorous dinosaurs, with less than 30% of their Ca potentially stemming from fish sources (figure 4A). The giant crocodylomorph Sarcosuchus imperator exhibits an intermediate diet, possibly composed of both fish and herbivorous dinosaurs (figure 4B). Moreover, if the diet of spinosaurid theropods potentially included a mix of fish and herbivorous dinosaurs according to the model, these data also support that a majority of spinosaurids could be exclusively piscivorous (along with one non-spinosaurid theropod and two Sarcosuchus imperator individuals), and that half of them were at least partially piscivorous (figure 4C). Nevertheless, if this clearly illustrates that spinosaurids were the predators of Gadoufaoua which had the most piscivorous-oriented diet, this also leads to the conclusion that their diet was potentially a mix between fish and herbivorous dinosaurs. Similarly, Kem Kem spinosaurids are also strongly depleted in heavy Ca, confirming a piscivorous diet in another fossil assemblage. In previous studies, several diets have been proposed for these groups of theropods. For spinosaurids, fish consumption inferred on the basis of the ecosystem structure [START_REF] Läng | Unbalanced food web in a late cretaceous dinosaur assemblage[END_REF][START_REF] Benyoucef | Overabundance of piscivorous dinosaurs (Theropoda: Spinosauridae) in the mid-Cretaceous of North Africa: The Algerian dilemma[END_REF], morphology [START_REF] Taquet | Une curieuse spécialisation du crâne de certains Dinosaures carnivores du Crétacé: le museau long et étroit des Spinosauridés[END_REF][START_REF] Sereno | A Long-Snouted Predatory Dinosaur from Africa and the Evolution of Spinosaurids[END_REF][START_REF] Evers | A reappraisal of the morphology and systematic position of the theropod dinosaur Sigilmassasaurus from the 'middle' Cretaceous of Morocco[END_REF][START_REF] Holtz | Spinosaurs as crocodiles mimics[END_REF][START_REF] Rayfield | Functional Morphology of Spinosaur 'Crocodile-Mimic[END_REF][START_REF] Hone | A Century of Spinosaurs A Review and Revision of the Spinosauridae with Comments on Their Ecology[END_REF][START_REF] Vullo | Convergent evolution of jaws between spinosaurid dinosaurs and pike conger eels[END_REF], or direct fossil evidences [START_REF] Charig | Baryonyx walkeri, a fish-eating dinosaur from the Wealden of Surrey[END_REF] is thus supported by our dataset. The consumption of terrestrial preys, which is also proposed in these papers, as well as the direct evidences related to this behavior [START_REF] Charig | Baryonyx walkeri, a fish-eating dinosaur from the Wealden of Surrey[END_REF][START_REF] Buffetaut | Pterosaurs as part of a spinosaur diet[END_REF], also remain possible according to our data. In addition, the consumption of herbivorous dinosaurs that is commonly assumed for non-spinosaurid theropods [START_REF] Farlow | The fossil record of predation in dinosaurs[END_REF], such as abelisaurids and carcharodontosaurids, is also supported by our dataset. The fraction of terrestrial preys in the diet of these theropods appears to be significantly higher than that of spinosaurids. However, whether these preys have been acquired through hunting [START_REF] Charig | Baryonyx walkeri, a fish-eating dinosaur from the Wealden of Surrey[END_REF][START_REF] Holtz | Spinosaurs as crocodiles mimics[END_REF][START_REF] Buffetaut | Pterosaurs as part of a spinosaur diet[END_REF][START_REF] Hone | A Century of Spinosaurs A Review and Revision of the Spinosauridae with Comments on Their Ecology[END_REF], scavenging [START_REF] Charig | Baryonyx walkeri, a fish-eating dinosaur from the Wealden of Surrey[END_REF][START_REF] Holtz | Spinosaurs as crocodiles mimics[END_REF][START_REF] Buffetaut | Pterosaurs as part of a spinosaur diet[END_REF] or active fishing [START_REF] Taquet | Une curieuse spécialisation du crâne de certains Dinosaures carnivores du Crétacé: le museau long et étroit des Spinosauridés[END_REF][START_REF] Charig | Baryonyx walkeri, a fish-eating dinosaur from the Wealden of Surrey[END_REF][START_REF] Evers | A reappraisal of the morphology and systematic position of the theropod dinosaur Sigilmassasaurus from the 'middle' Cretaceous of Morocco[END_REF][START_REF] Holtz | Spinosaurs as crocodiles mimics[END_REF][START_REF] Rayfield | Functional Morphology of Spinosaur 'Crocodile-Mimic[END_REF][START_REF] Hone | A Century of Spinosaurs A Review and Revision of the Spinosauridae with Comments on Their Ecology[END_REF][START_REF] Vullo | Convergent evolution of jaws between spinosaurid dinosaurs and pike conger eels[END_REF]] cannot be assessed with our dataset.

In the model, for some non-spinosaurid theropod individuals, the inferred δ 44/42 Ca signature of the diet is more positive than the source with the most positive δ 44/42 Ca values. Four nonmutually exclusive hypotheses could be proposed to explain this inconsistency. First of all, it can be due to the absence of a heavier source of Ca in our dataset. One of the possibilities is that this heavy source indeed corresponds to an unidentified terrestrial taxon at the bottom of the food chain. Alternatively, it can correspond to Nigersaurus taqueti mineralized tissues for which only two teeth have been measured, and which exhibit the highest δ 44/42 Ca values of the dataset.

Another possibility is that this source of heavy isotopes of Ca comes from the digestion of carbonated gastroliths. The ingestion of gastroliths among dinosaurs has been documented thanks to comparisons with modern archosaurs and fossil evidence, notably for spinosaurids [START_REF] Charig | Baryonyx walkeri, a fish-eating dinosaur from the Wealden of Surrey[END_REF]. Because of the heavy Ca isotope composition of carbonates [START_REF] Chu | Establishing the potential of Ca isotopes as proxy for consumption of dairy products[END_REF]41,42], this consumption could buffer the δ 44/42 Ca towards 44 Ca enriched values among studied archosaurs. This could lead to an overestimation of the fraction of herbivorous dinosaurs in diets inferred from our mixing model, which could explain the inconsistency with non-spinosaurid theropods in our model. The third hypothesis is that this inconsistency is the result of an overestimation of the diet-enamel shift, due to the lack of data collected on modern archosaurs, which can also lead to an overestimation of the fraction of herbivorous dinosaurs in diets (values potentially shifted toward heavy calcium). However, none of these hypotheses is susceptible to change the conclusions made on the resource partitioning among these predators or to the more piscivorousoriented diet of spinosaurids. Indeed, a validation of these hypotheses will imply a more piscivorous diet for all these predators, even providing more support to the piscivorous-oriented diet of spinosaurids.

CONCLUSIONS

Our data imply that the abundance of apex predators in these peculiar African mid-Cretaceous ecosystems can be sustained simultaneously by at least two main sources: fish and herbivorous dinosaurs, resulting in a general abundance of prey. The present Ca isotope dataset also supports the ecological partitioning of potential predatory competitors [START_REF] Amiot | Oxygen isotope evidence for semi-aquatic habits among spinosaurid theropods[END_REF][START_REF] Ibrahim | Semiaquatic adaptations in a giant predatory dinosaur[END_REF]. Indeed, the different ecological specializations of these predators resulted in different mixtures between carnivorous and piscivorous habits and are likely to have reduced the ecological competition between them.

Calcium isotopes support the hypothesis of spinosaurids regarded as semi-aquatic predators [START_REF] Amiot | Oxygen isotope evidence for semi-aquatic habits among spinosaurid theropods[END_REF][START_REF] Ibrahim | Semiaquatic adaptations in a giant predatory dinosaur[END_REF] and adopting a piscivorous diet, with non-spinosaurid theropods having a more terrestrial ecology and diet, feeding mainly on herbivorous dinosaurs. This ecological partitioning, combined with the abundance of prey, is consistent with the observed high abundance of predators in these faunas [START_REF] Läng | Unbalanced food web in a late cretaceous dinosaur assemblage[END_REF][START_REF] Benyoucef | Overabundance of piscivorous dinosaurs (Theropoda: Spinosauridae) in the mid-Cretaceous of North Africa: The Algerian dilemma[END_REF].

The present study underlines a complex trophic structure relying on the interaction between aquatic and terrestrial environments. One of the key points for the sustainability of these

] [%] [ppm] [ppm] [ppm] [ppm] [ppm] -[ppm]N [ppm]N [ppm]N [ppm]N [ppm]N [ppm]N [ppm]N [ppm]N [ppm]N [ppm]N [ppm]N [ppm]N [ppm]N [ppm]N --[ppm] [ppm] [ppm] [ppm]

CaptionsFigure 1 .

 1 Figure 1. Three isotopes plot: δ 43/42 Ca as a function of δ 44/42 Ca (‰, reference standard ICP Ca)

Figure 2 .

 2 Figure 2. (A) PAAS Shale-normalized REE profiles of fossil bioapatite from the Aptian-Albian

Figure 3 .

 3 Figure 3. δ 44/42 Ca data reported from Gadoufaoua (A) and Kem Kem beds faunas (B). Groups

Figure 4 .

 4 Figure 4. Mixing model of the calcium source for non-spinosaurid theropods (A) Sarcosuchus

983, p-value < 2.2e-16 δ 43 42 Ca

	REE normalised concentration [ppm]	La A	Ce	Pr Nd		Sm Eu Gd		Tb Dy Ho Er Tm Yb Lu		La B	Ce	Pr Nd		Sm Eu Gd	Tb Dy Ho Er Tm Yb Lu
		103	101	99	97	95	93	91	89	87	85	103	101	99	97	95	93	91	89	87	85
															Ionic radius [pm]
			Nigersaurus taqueti		Non-spinosaurid theropods		Crocodylomorpha	cf. Araripesuchus
			Ouranosaurus nigeriensis	Spinosauridae				Fish			Pterosauria indet.

N

Ionic radius [pm]

Acknowledgments.

The authors are grateful to the LABEX Lyon Institute of Origins (ANR-10-LABX-0066) of the Université de Lyon for financial support within the program ''Investissements d'Avenir'' (ANR-11-IDEX-0007) of the French government operated by the National Research Agency (ANR).

Consumables used in the clean lab for this study were partly paid for by J.E. Martin's DIUNIS project ("Dietary Inferences Using Novel Isotope System" -TelluS INSU 2017 -action INTERRVIE). A. Hassler thanks the Jurassic Foundation for having funded this project through the Jurassic Foundation Grant Program. We thank G. Clément for allowing us to sample fossil fish from Gadoufaoua in the collections of MNHN (Paris). For technical assistance on spectrometers, we thank E. Albalat and P. Telouk. We thank the two anonymous referees for their insightful comments on the manuscript. The animal outlines in figures 3 and 4 were retrieved from www.phylopic.org with some modifications, we thus thank their authors : Duane Raver, T. Michael Keesey, Todd Marshall, Nobu Tamura, Martin Kevil, Milton Tan, Scott Hartman, FunkMonk, Smokeybjb, Zimices, Heinrich Harder and T. Michael Keesey, for helping us to illustrate our study by having transferred their work to the public domain or under a creative commons license.

Data accessibility.

The dataset table compiling all elemental concentrations and isotopic ratio measurements is available as electronic supplementary material.

Authors' contributions.

The project was conceived by J.E.M., A.H., R.Am., T.T., R.Al. and V.B. . R.Am. and R.Al. accessed samples for analysis. A.H. and J.E.M. sampled the specimens and conducted purification in the clean lab. A.H. and F.A.G. performed elemental concentration measurements.

A.H., J.E.M. and T.T. performed MC-ICP-MS measurements. All authors discussed the results.

The first draft version of the text was written by A.H. then subsequently, J.E.M., R.Am., T.T., R.Al., F.A.G. and V.B. contributed to it.

Competing interests.

We declare we have no competing interests.


Trung-Viet Tran

Franck Schoefs

Emilio Bastidas-Arteaga

Risk-Based-Optimization of Geo-positioning of Sensors in Case of Spatial Fields of Deterioration/Properties

come   L'archive ouverte pluridisciplinaire

INTRODUCTION

The localization of weak properties or bad behavior of a structure is still a challenge for Structural Health Monitoring (SHM) design [START_REF] Okasha | Integration of structural health monitoring in a system performance based life-cycle bridge management framework[END_REF]. In case of random loading or material properties, this challenge is arduous because of the limited number of sensors and the quasi-infinite potential positions of local failures. Deterministic algorithms and specific multi-sensors systems are developed to this aim. In that case, generally, the precision of the positioning of a defect goes with a lack of sizing. However, the stochastic field is rarely a pure white noise and has a stochastic structure and probabilistic properties. These properties should be used to provide rational aid tools for optimizing the number of sensor. The paper shows that the stationary property is sufficient to find the optimal quantity of sensors and their position in view to assess accurately the shape of the auto-correlation function (see Fig. 1) that guides the position of control through Non Destructive Techniques (NDT) during the service life. If we focus on embedded sensors, these stochastic properties are less sensitive to environmental conditions (sun, humidity, …) at the surface of the structures where usual NDT tools are affected by. That will help the help the planning of NDT campaigns. The evolution of measurement with time is out of the scope of the present paper.

The paper starts with key concepts and illustration of spatial random field modeling.

In the next section, a method for optimal sensor positioning is suggested by focusing on the assessment of the correlation function. The quality is ex-pressed in terms of confidence interval on the parameters of the auto-correlation function.

SPATIAL RANDOM FIELD MODELLING

Usual approaches for spatial field modeling

The stochastic field could take several forms more or less complicated. The most simple when the degradation can be considered as homogeneous is the stationary stochastic field that can be used, for instance, to model chloride distribution (O'Connor & Kenshel 2013), concrete properties [START_REF] Bazant | Statistical Size Effect in Quasibrittle Structures: II. Nonlocal Theory[END_REF]2000a;2000b) or soil properties. In some cases, when a stochastic process is influenced by several phenomena that vary with time (sea wave for instance) or with space (concreting of a structure in several steps with heterogeneity of these steps), it can be modeled as piecewise stationary. This more sophisticated stochastic field can represent, for ex-Risk-Based-optimization of geo-positioning of sensors in case of spatial fields of deterioration/properties ABSTRACT: The localization of weak properties or bad behavior of a structure is still a very challenging task that concentrates the improvement of Non Destructive Testing (NDT) tools on more efficiency and higher structural coverage. In case of random loading or material properties, this challenge is arduous because of the limited number of measures and the quasi-infinite potential positions of local failures. The paper shows that the stationary property is useful to find the minimum quantity of NDT measurements and their position for a given quality assessment. A two stages procedure allows us (i) to quantify the properties of the ergodic, stationary field (ii) to assess the distribution of the characteristics. The paper focuses on optimization of sensor geo-positioning to reach (i): the criterion relies on the width of the confidence interval. The concept of critical upper bound spatial correlation is introduced.

ample, the variability of concreting materials by pieces or the corrosion of structures located in contiguous environments with different characteristics [START_REF] Boéro | Statistical analysis of corrosion process along French coast[END_REF][START_REF] Boéro | Timefunction reliability of harbour infrastructures from stochastic modelling of corrosion[END_REF]. That is the case for the corrosion of sheet-piles in marine environment where the corrosion process is governed by several phenomena that depend on the exposure zone E. For illustrating this case, Figure 2 presents the spatial variability of the mean and the 95% quantiles of steel thickness loss in different zones at 10, 25 and 50 years (t10, t25 and t50. It can be observed that both the mean and the 95% quantile are well defined for the considered zones -i.e., E∈[E A , E T , E L , E I , E M , E S ] limited by dashed lines: from the top to the bottom, Aerial (A), Tide (T), Low level of tide (L), Immersion (I), Mud (M) and Soil (S). this case, a piecewise stationary stochastic field could be used to have a good representation of the loss of thickness.

Usual approaches for inspection modeling

During measurement, there are many factors that influence the quality of measurements -e.g, environmental conditions, error in the placement of the sensor, error due to sensor and error induced by the operator (Bonnet et al. unpubl., Breysse et al. 2009). These factors could lead, for a given measurement, to under or overestimations of the measured parameter. If the parameter is underestimated and the owner could decide, "do nothing" when repair is required.

On the contrary, an overestimation generates a "wrong decision" where the early repair generates overcharges [START_REF] Rouhan | Probabilistic modeling of inspection results for offshore structures[END_REF][START_REF] Sheils | Investigation of the Effect of the Quality of Inspection Techniques on the Optimal Inspection Interval for Structures, Structure and Infrastructure Engineering: Maintenance, Management, Life-Cycle Design and performance (NSIE)[END_REF]. In this paper we will consider inspection as perfect: that (i) there is no bias, (ii) error is negligible or redundancy allows us to assume it as negligible.

Main assumptions for the stochastic modeling

Starting from the previous sections and to limit the study, we consider five main assumptions about the sensor and the random field modeling.

-The stochastic field is statically homogeneous, and only few information on the marginal distribution is known: the type of the unique probability density distribution. In the following applications we consider a Gaussian field; -A limited number (from 30 to 50) of discrete sensors can be placed; -Second order stationary stochastic field can piecewise or totally describe the spatial fields; -Sensors are considered as perfect as defined by [START_REF] Schoefs | Assessment of spatially dependent ROC curves for inspection of random fields of defects[END_REF].

Given a probability space (Ω, F , P), a stochastic field or process with state space Z is a collection of Z-valued random variables indexed respectively by a set s "space" or t "time". Let us denote Z(x, θ) the one-dimensional stochastic field where θ represents the randomness and x the spatial coordinate.

Z(x,θ i ) is called the i th trajectory of this field and corresponds to a given realization θ i of the field for whatever location. Z(x 1 , θ) is a random variable that is generated by θ at a given location x = x 1 . We consider here homogeneous fields only. This means that the marginal distribution of Z(x 1 , θ) does not depend on the location. A stochastic field is second order stationary if it follows three main properties:

• expectation E[Z(x, θ)] does not depend on the lo- cation x -i.e., E[Z(x, θ)] = E[Z]; • variance V[Z(x, θ)] does not depend on the loca- tion x -i.e., V[Z(x, θ)] = V[Z]; and • spatial covariance COV[Z(x, θ), Z(x, θ)] depends
only on the distance (x-x'). Thus, the second order stationary is a property restricted to the two first probabilistic moments. It can be shown that geometries of welds for ships or the spatial distribution of chloride concentration in reinforced concrete structures can be represented by stationary stochastic fields (Schoefs et al. 2011a). For instance, Figure 3 presents the correlation function of the chloride diffusion coefficient as function of the distance between two points (duratiNet project: http://www.duratinet.org, following O'Connor & Kenshel 2013). Note that if E[Z(x, θ)] is not constant with space, the signal can be centered and the properties been proved for the field (Z(x, θ) -E[Z(x, θ)]).

Several approaches can be used to represent a stochastic field Z(x, θ): Karhunen-Loève expansion, approximation by Fourier series, and approximation EOLE [START_REF] Li | Optimal discretization of random field[END_REF]. In this paper, we select a Karhunen-Loève expansion to represent the stochastic field of resistance of a structure Z(x, θ). This expansion represents a random field as a combination of orthogonal functions on a bounded interval [-a, a]:

Z(x,θ ) = µ Z +σ Z . λ i .ξ i (θ ). i=1 n ∑ f i (x) (1)
where, µ Z is the mean of the field Z, σ Z is the standard deviation of the statistically homogeneous field Z, n is number of terms in the expansion, ξ i is a set of centered Gaussian random variables and λ i and f i are, respectively, the eigenvalues and eigen functions of the correlation function ρ(Δx). It is possible to analytically determine the eigenvalues λ i and eigen functions f i for some correlation functions [START_REF] Ghanem | Stochastic Finite Elements -A Spectral Approach[END_REF]. For example, it can be determined if we assume that the field is second order stationary and we use an exponential correlation function [START_REF] Vanmarcke | Random fields: analysis and synthesis[END_REF]:

ρ(Δx = x 1 -x 2 )=exp( - Δx b ); 0 < b (2)
where b is the correlation length and Δx ∈ [-a, a].

Accounting for the stationarity of the process, the following transcendental equations can be obtained:

1 b -ω tan(ωa) = 0 ω - 1 b tan(ωa) = 0 " # $ $ % $ $ (3)
where ω is obtained by solving equation ( 3). If the solution of the second equation is noted ω*, the eigenfunctions are: 3 SENSOR POSITIONNING STRATEGY AND GOALS

⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ - + = i ω a ω a x ω i ω a ω a x ω x f i i i i i i i odd for 2 / ) 2 sin( ) sin( even for 2 / ) 2 sin(

Key ideas for sensor geo-positioning

The paper focuses on the assessment of the autocorrelation function of a stationary field. An optimal geo-positioning of sensors along a trajectory (Fig. 4) should give an accurate value of the parameters (i.e. b) with a limited number of sensors.

x(m) When looking to the usual shape of a correlation function (Fig. 1) a regular spacing 'L b = a' of N s sensors appears not optimal: it provides a dense information for distances 'a, 2a, … (N s -1)a' but none between. We get a non-homogeneous quantity of data: N s for distance 0, (N s -i) for distance ia (1<i< N s ). The objective will be to get a spacing of sensor that leads to a more homogeneous quantity of data in the zone f high correlation as defined on Figure 5. Moreover, there is a limited feed-back on the autocorrelation length on site and the uncertainty on the value of 'b' (Eq. 2) when installing the sensors is significant. These reasons lead us to suggest a non-regular spacing of the sensors.

?L b ?L b Z N s (measurements)

Definition of the Spatial Correlation Threshold

The When the uncertainty of the should of an stages inspection we suggest should provide both the parameters of the spatial correlation function and independent events that characterize the marginal distribution of Z. We considerer in this paper a onedimensional spatial field and we will apply the methodology on a set of trajectories: these trajecto-ries could be a set of 1D components (beams), or a very long 1D-component subdivided artificially or physically (expansion joint or construction joints) in a set of short components, or belong to a wall structure (steel sheet pile or concrete wall).

In view to limit the costs of the monitoring (number of sensor) we inspect a trajectory with a "sufficiently low distance L b " to assess the shape of the correlation function (2): for instance the shape parameter b. This "sufficiently low" can be seen as an Inspection Distance Threshold (ICD). Thus the non regular distances of sensor spacing L bi should satisfy: L bi ∈ ]0, IDT[ where L is the length of the trajectory.

For illustrating the methodology, we consider in the following a set of 1D-components (beams) with infinite length L ~ ∞: we don't discuss the case of components with a limited size, especially those where L< L b for which it is quite impossible to characterize the spatial variability. The IDT is defined by assuming that after a given distance, the events measured from an inspection can be assumed as statistically independent. A Spatial Correlation Threshold SCT of the spatial correlation gives this weak correlation. For instance in Figure 3 (red line) for SCT=0.4, IDT=3 meters. It is linked with IDT by the relationship:

IDT = b. ln(SCT ) (6)

Parameterization of the non-regular spacing

In view to reduce the set of potential solutions and simplify the design of the network of sensors, we suggest a parameterization. It is based on a division of the trajectory into n p pieces of same size L m (see Fig. 6) and then a subdivision of each piece into a decreasing number of pieces of same size following a series [START_REF] References Bazant | Probabilistic Nonlocal Theory for Quasibrittle Fracture Initiation and Size Effect. I: Theory[END_REF].

N S i ≈ N S 1i j " # " $ % $ ; j ∈ 1i...n p { } with N S j j ∑ = N S (7)
Where .

! " # $ denotes the floor function, N S j is the number of sensors in the j th piece and N S 1i is the initial number of sensors in the 1 st piece, the final number of inspection N S 1 f in the first section is computing following:

N S 1 f = N S -N S j j>1 ∑ (8)
For instance, for two pieces and 6 sensors, 4 are placed in the first piece and 2 in the second (Fig. 6). Figure 7 presents the case 7 sensors and 3 pieces.

x(m) Our objective is now to optimize the number of inspections in view to reach a given quality of the result.

0,5(L c 1 +L c 2 ) L c 2 =l/4 Z L c 1 =l/8 x 0 N S =6 (sensors), n=2 pieces L c 1 L c 1 X={x 0 , x 0 +l/8, x 0 +l/4, x 0 +3l/8, 5l/8, 7l/8 } N s 1 =4 N s 2 =2 L L m =l/2 L m =l/2

APPLICATION

Assumptions and inputs

We consider in this section a beam of length L=25 m with a limited number of sensors N s =30. The objective is to optimize their position in view to reach a good assessment of b (see Eq. 2). Only one trajectory is instrumented and the uncertainty due to the stochastic field is simulated by (1) with: µ Z =100 (MpA) ; σ Z =20 (MPa) ; b=1m.

We analyze the error between the theoretical value and the assessed one.

Results and inputs

We vary the number of pieces from 1 (30 sensors) to 6 (12, 6, 4, 3, 3, 2 sensors on each piece). Figure 8 presents plots the difference ( b -b) between the estimated value and the theoretical one (here b=1) for 1000 simulated trajectories. The regular spacing obtained for n p =1 shows the biggest uncertainty in the assessment of b with potential errors of 0.8 (near 100%). The cutting with 6 pieces shows a systematic underestimation of b.

Let us focus on ε b *, the normalized mean error on b presented in Equation (9). Figure 9 gives the plot of this error as a function of n p . Following this criterion, it is shown that the spacing with n p =2 is optimal.

CONCLUSION

We propose in this paper an original method for the non-regular spacing of sensors devoted to the assessment of stationary fields from embedded sensors. The method is based on the uncertainty assessment of the parameter of the auto-correlation function. Numerical simulations of Gaussian stationary stochastic illustrate the potential of the method by showing that it provides a decision aid tool when a limited number of sensors are placed in a structure.

Figure 1 .

 1 Figure 1. Auto-correlation function fitted with a parametric function.

Figure 2 .

 2 Figure 2. Mean value and 95% quantiles of steel thickness loss in each zone at time 10, 25 and 50 years.

Figure 3 .

 3 Figure 3. Spatial correlation of the chloride diffusion coefficient in concrete -data Trinity College of Dublin.

Figure 4 .

 4 Figure 4. Non-regularly spaced sensors along a trajectory.

Figure 5 .

 5 Figure 5. High correlation zone and homogeneous measurements long the trajectory.

Figure 6 .Figure 7 .

 67 Figure 6. Distribution of sensors for N s =6 and n p =2.

Figure 8 .

 8 Figure 8. Distribution of sensors for N s =7 and n p =3.

Figure 9 .

 9 Figure 9. Normalized error on b as a function of n p .

  .V. Tran, F. Schoefs, E. Bastidas-Arteaga LUNAM Université, Université de Nantes, Institute for Research in Civil and Mechanical Engineering, CNRS

T

AKNOWLEDGEMENTS

The authors would like to acknowledge the Pays de la Loire Region and the project ECND-PdL for their support and the duratiNet EC Interreg project (http://www.duratinet.org).