Armelle Brun
email: armelle.brun@loria.fr

Hala Skaf-Molli

Anne Boyer
email: anne.boyer@loria.fr

RAISING UP ANNOTATIONS IN PEDAGOGICAL RESOURCES BY HUMAN-COMPUTER COLLABORATION

Huge repositories of pedagogical resources such as the French initiative of digital thematic libraries are now accessible by both students and professors. This high quantity of data makes their access difficult to the students, as they cannot get easily pertinent pedagogical resources that fit their needs. One way to make this access easier is to add annotations to these resources and to exploit these annotations to find pertinent answers. Of course, these annotations can be semantic. Semantic wikis are a new approach that automatically processes semantic annotations and that can be used to find the adequate resources given the requests of students. However, annotating semantically resources is not an easy task for human. Despite of their high potential, semantic wikis suffer from a lack of human provided semantic annotations, resulting in a loss of their efficiency. We propose a system (called HCA) that suggests automatically computed annotations to users in semantic wikis. In this paper users are students and professors. Users only have to validate, complete, modify, refuse or ignore these suggested annotations. Therefore, the annotation task becomes easier, and we assume that more users will provide annotations, leading to an improvement of the system and a facilitated access to pertinent pedagogical resources. The HCA system is based on collaborative filtering recommender systems, it does not exploit the content of the pages but the usage made on these pages by the students and the professors. The resulting semantic wikis contain several kinds of annotations, with different status: human, computer or humancomputed provided annotations.

Introduction

Web 2.0 encourages the production of new pedagogical resources and the collaboration of students and professors to increase the quality of available resources by using for instance, wikis and blogs. This results in a huge number of resources broadly available on the Internet via dedicated repositories, such as the French initiative of digital thematic libraries [START_REF] Boyer | Providing French Universities with a Powerful Environment to Support Elearning development[END_REF]. However, these facilities lead to a too huge number of pedagogical resources accessible. Therefore students, and professors as well, may be lost in this sea of resources, they cannot easily find the resources they search.

Of course, these pedagogical resources are often annotated or tagged by the professors who produce them. However, these annotations are often different from those the students or other professors use in their requests. Students and professors may never find pertinent resources. Moreover, these annotations are put a priori and may not reflect the real usage of the resources. The objective of this paper is to learn about the real usage made on these resources to provide a posteriori annotations that reflect better the reality of usage of these resources, compared to the a priori annotations. As a result, the use of these annotations makes the access to pertinent resources easier.

Semantic wikis are one of the most promising approaches to automatically process information. Compared to classical wikis, they allow users to add semantic annotations in the wiki pages. In this paper the wiki pages will be pedagogical resources such as lessons or exercises. When using semantic wikis, users do not only collaborate for writing the content of the wiki pages but also for writing the semantic annotations that will allow a better and easier usage of these pages. It will then be possible to answer user queries by exploiting semantic annotations. However, adding semantic annotations is not an easy task and is time consuming. In the frame of repositories of pedagogical resources, the consequence is that only few users annotate pages semantically. As it is not easy to motivate students or professors to provide with annotations, many existing semantic wikis contain only few annotations; their potential added value is thus decreased. One possible solution is to propose a system that suggests pertinent annotations to users (students and professors). This paper proposes a new kind of collaboration between users and machine to produce pertinent, useful and reliable annotations in semantic wikis. The production of these annotations is based on a recommender system. Many works about recommender systems [START_REF] Goldberg | Using collaborative filtering to weave an information tapestry[END_REF][START_REF] Adomavicius | Toward the next generation of recommender systems: A survey of the state-of-the-art[END_REF][START_REF] Pazzani | The Adaptive Web, chapter Content-Based Recommendation Systems[END_REF] and semantic wikis [START_REF] Kr Otzsch | Semantic wikipedia[END_REF][START_REF] Buffa | Sweetwiki: A semantic wiki[END_REF] can be found in the literature. However, few existing works [START_REF] Durao | Tag-based recommendation in kiwi[END_REF] propose to use recommender systems in the frame of semantic wikis. This paper proposes to suggest new annotations to users by mining the automatically collected observations about the real usages of wiki pages. These annotations represent better the real interest of users for the pedagogical resources and will provide additional information about them. These "computer recommended" annotations are suggested to students and professors who can validate (and/or complete), modify, refuse or ignore them. Based on these suggested annotations, the task of annotating is made easier as they do not have to create the annotations from the scratches, they only have to decide about proposed annotations.

The first section gives a brief overview of the proposed system. The following section summarizes the required background in term of semantic wikis and recommender systems. Then we describe how we use recommender systems to automatically determine possible additional semantic annotations. It presents also how the users and machines collaborate to provide reliable semantic annotations. The last section concludes and points out the perspectives of this work.

Overview of the System

The efficiency of semantic wikis relies among others on the quantity and the quality of the available semantic annotations. As semantic wikis often face a lack of user provided annotations, the question that we raise in this paper is: how can we encourage and support users of the repository of pedagogical resources to provide semantic annotations?

This paper addresses this question by introducing the machine as a partner of the collaborative process of annotation in semantic wikis. The community of users becomes a community gathering students, professors and computers working together to produce semantic annotations. Students, professors and computers do not have equivalent roles within the community: computers can only suggest new semantic annotations but they do not have any influence on the final decision to keep, modify or discard them. In contrast, students and professors can add new semantic annotations and participate to the final decision of integrating or not a given annotation provided by computers. The tool we propose can be viewed as a semi-automatic annotation tool. Concretely, a specific recommender suggests possible annotations to students and professors. Each of them can decide to ignore, accept (and/or complete) or refuse them. If agreed, the status of the annotation will change to become a Human Computer Annotation. If refused, the annotation will be discarded but memorized by the system as Refused Annotation to avoid to suggest it again. The validated annotations will be added to the usual annotations resulting from human collaboration. An annotation can thus have four different status:

1. Human Annotation (HA): this annotation results from human collaboration; 2. Computer Annotation (CA): this annotation is suggested by the recommender but not reviewed by humans;

3. Human Computer Annotation (HCA): this is a computer annotation that has been validated (and/or completed) by the users;

4. Refused Annotation (RA): this is a computer annotation that has been discarded by humans.

Both HA and HCA annotations are used to support navigation and answer semantic queries, the CA annotations can only be used to support navigation when no other annotation is available.

This work uses collaborative recommendation based on usage mining to compute annotations. The recommender exploits the usages by the students and professors of the wiki pages, here pedagogical resources, to determine pertinent suggestions of annotations. It exploits implicit collaboration between users of the wiki: we consider that students or professors who share usages collaborate implicitly. This approach does not take into account the content of the pedagogical resources to perform recommendations, only their usage is considered.

The system integrates three levels of collaborations.

• Level 1: Explicit human collaboration when users write annotations, HA.

• Level 2: Implicit human collaboration when the system suggests CA.

• Level 3: Human-computer collaboration when users validate and complete CA that become HCA.

Suppose we apply Human-computer collaboration to a repository about ''Computer Science''. Let two pedagogical resources be "C programming" and "Principles of compilation", these two resources are not directly linked by semantic annotations. The "C programming" page may contain the following semantic annotation: "DedicatedTo" as "Beginners". However, "C programming" and "Principles of compilation" may be often altogether as students who want to write a C program may be interested in knowing how programs are compiled. Therefore, the semantic wiki pages of these two pedagogical resources should be semantically linked. As many students interested in "C programming" often also consult the "Principles of compilation", our HCA system will automatically discover this link, based on the usage made by students and it this link will be recommended to users as a CA annotation. The semantic annotations will be enriched and the usability of the semantic wiki will be improved, students will access easier pertinent resources.

Background

This section presents backgrounds about semantic wikis and recommender systems.

Semantic wikis

Semantic Wikis are an extension of wiki systems, they embed semantic annotations in the wiki content. These annotations allow to better organize and structure the wiki contents. They guide the users from informal knowledge contained in texts to more formal structures. Many semantic wikis are being developed such as Semantic MediaWiki (SMW) [START_REF] Kr Otzsch | Semantic wikipedia[END_REF], SweetWiki [START_REF] Buffa | Sweetwiki: A semantic wiki[END_REF] and Swooki [START_REF] Skaf-Molli | Peer-to-peer semantic wikis[END_REF]. In SMW, links between wiki pages are typed. For instance, a link between the wiki pages ''France'' and ''Paris'' may be annotated by a user as ''has Capital''. Annotations express semantic relationships between wiki pages. They are usually written in a formal syntax so they are processed automatically by machines and are exploited by semantic queries. In semantic wikis, semantic annotations are added by users, they can be viewed as Human Annotations, they correspond to the Level 1 of collaboration as presented in the previous section. Semantic wikis, as classical wikis, suffer from scalability, availability and performance problems and they do not support offline works. To overcome these limitations, peer to peer extensions [START_REF] Skaf-Molli | Peer-to-peer semantic wikis[END_REF] [13] for semantic wiki have been proposed . SWooki [START_REF] Skaf-Molli | Peer-to-peer semantic wikis[END_REF] is a peer to peer (P2P) semantic wiki that follows the same annotation principles as SWM. It is a P2P network of autonomous semantic wiki servers, every server hosts a copy of all semantic wiki pages and the semantic data. Every peer can autonomously offer all the services of a semantic wiki server. When a peer updates its local copy of data, it generates the corresponding operation. This operation 1) is executed immediately against the local replica of the peer, 2) it is broadcasted to all other peers, 3) it is received by the other peers, 4) and it is integrated to their local replica. If needed, the integration process merges this modification with concurrent ones, generated either locally or received from a remote server.

Recommender systems

Recommender Systems provide personalization to users to cope with the well-known problem of overload of information [START_REF] Adomavicius | Toward the next generation of recommender systems: A survey of the state-of-the-art[END_REF]. Among the possible approaches in recommender systems are content-based [START_REF] Pazzani | The Adaptive Web, chapter Content-Based Recommendation Systems[END_REF] and collaborative filtering approaches [START_REF] Goldberg | Using collaborative filtering to weave an information tapestry[END_REF]. The first approach uses the content of the resources to compute recommendations for users, leading to accurate recommendations. However, the content of all types of resources cannot be automatically analyzed (videos, audio, etc.), thus this analysis often requires human interventions. Moreover, only resources directly linked to those the user has consulted can be suggested: no "novelty" can be recommended to users, users may thus be frustrated. Collaborative filtering (CF) approaches do not take into account the content of the resources. They only consider the usage of these resources to compute recommendations. The usage can be the consultation made by users, the votes given by users, etc. A CF-based recommender system exploits the traces of usage to deduce information about the resources. CF-based recommenders can either compute similarities between resources [START_REF] Sarwar | Item-based collaborative filtering recommendation algorithms[END_REF] or exploit data mining techniques to learn relationships between the resources [START_REF] Rahhal | Multi-synchronous Collaborative Semantic Wikis[END_REF]. As in content-based recommender systems, given a user, his previously consulted resources are used and are linked/compared to all possible resources. The comparison is no more made in terms of content but on the similarities or relationships computed between resources, based on their usage. This approach allows to recommend "original'' resources: resources that are not semantically linked to the past resources consulted by the user (but that are similar in terms of usage) can be recommended.

Recommender Systems for Annotation Suggestion

Existing recommender systems for Semantic Wikis directly transpose recommenders to suggest wiki pages to users as in [START_REF] Durao | Tag-based recommendation in kiwi[END_REF]. In our work, we go a step further by suggesting annotations to wiki pages based on usage traces. We use CF-based recommender systems to provide automatically pages with additional annotations. We exploit the usage of wiki pages: which users have consulted which wiki pages and which page(s) is(are) frequently consulted after a given page? To deduce the links/relationships between pages, by using approaches similar to those presented in [START_REF] Sarwar | Item-based collaborative filtering recommendation algorithms[END_REF][START_REF] Adomavicius | Toward the next generation of recommender systems: A survey of the state-of-the-art[END_REF]. Given the relationships between pages of pedagogical resources and the annotations given by users (HA), the recommender system will suggest additional annotations. These annotations can be made on pages that either already have HA or not. Suggested annotations correspond to the implicit human collaboration level (level 2).

Suggestion of Semantic Computer Annotations

We propose two algorithms to compute annotations to suggest to students about a given page Pj. The first one is similar to those used in item-based approaches and classification of pages based approach [START_REF] O'connor | Clustering items for collaborative filtering[END_REF], the second one is based on data mining techniques for recommendations [START_REF] Mobasher | Data Mining for Web Personalization, chapter 3[END_REF].

Item-based approach. This algorithm first computes a similarity matrix of wiki pages [START_REF] Sarwar | Item-based collaborative filtering recommendation algorithms[END_REF]. This matrix is computed based on the traces of usage of the wiki pages. This approach is based on the hypothesis that two similar pages may have similar semantic annotations. Thus, given two similar pages or pedagogical resources, the recommender suggests the semantic annotations of the first one to the second one and suggests the semantic annotations of the second one to the first one. These suggested annotations are called CA. Figure 1 presents how the recommender computes candidate annotations. Given a page Pj, the recommender searches the pages that are similar to Pj, those with a high similarity value in the similarity matrix. The recommender collects the HA from the pages similar to Pj. The suggested CA to Pj can be computed by several policies:

1. The set of HA from the similar pages are suggested. The recommender may propose too many annotations, and some of them may be not pertinent.

2. The set of HA present in at least n similar pages are suggested. An appropriate value of n has to be fixed. This policy proposes a lower number of annotations and more "reliable'' as they are present in several pages. However, according to the value of n, some pages may have no annotation.

3. Given a HA, the similarity values of the pages containing that HA are summed up. The HA with a sum of similarity values above a given threshold are suggested. This policy is more accurate than the second one as the similarity values are considered. Classification-based approach. In this approach, given a set of pages, the recommender first computes a classification of these pages to create classes of "similar'' pages (see Figure 2). As in the item-based approach, the classification is based on the usage of the pages by the users, not on the content of the pages. Then, given a page Pj, the recommender searches its corresponding class, and exploits all the pages in this class. All the policies presented in the item-based approach can be used to compute annotations. The set of pages used to compute annotations is predefined (the pages in the class) whereas it is dynamically computed in the item-based approach, that can take time.

Data Mining based approach. It exploits data mining techniques to extract information about the usage of the resources. They study the sequences of consultation of resources by using, for example, association rules or Markov models to discover frequent patterns [START_REF] Bonnin | A low-order markov model integrating long-distance histories for collaborative recommender systems[END_REF]. The algorithm is based on the hypothesis: if two or more pages are frequently consulted in sequence, then the links used to traverse these pages are useful and it is important to annotate semantically these links. The recommender suggests CA to the users to annotate semantically the frequently passed links. The provided annotations are not semantic annotations, they are just annotations, as the type of the annotation cannot be automatically discovered by the system. The recommender suggests that an annotation at a given place should be important. It can also specify the label of the annotation. For instance, suppose in the example given in the previous section, there is no semantic annotation between the pages "C programming" and " Principles of compilation", the recommender can learn that this link is highly passed, thus annotating semantically this link may be useful. The recommender suggests to the users the link "C programming". The user can accept, modify, refuse or ignore this suggested annotation and type it with [ApplicationDomain], for example. The annotation task is made easier as the system suggests to the user where useful annotations should be, the user is guided.

The HCA approach

As presented in the previous sections, each semantic wiki page, i.e. pedagogical resource, has three sources of annotations: first, those added by humans (HA), they correspond to the explicit human collaboration, in our framework these annotations are proposed by students and professors; second, those suggested by the recommender system (CA), they come from implicit human/student interaction. These annotations are a new kind of annotations, they do not exist in classical semantic wikis, they will be used to encourage the users to annotate semantically pages. Third, those reviewed by the students, they result from the Human computer collaboration (HCA).

To integrate these annotations to a peer to peer semantic wiki, several questions are raised: how the recommender suggests annotations (CA) to students? how to make the students validate (and/or complete), modify, refuse or ignore a CA? How to make the other students know these annotations? The way we propose to answer these questions is presented in Figure 3.

When new annotations for pedagogical resources are suggested by the recommender, they are broadcasted to all the local replica with the status Computer Annotation (CA). They are then proposed to the students in a pop-up box, for example, to be differentiated from Human Annotations HA.

A student (a user) can choose among four possible operations : 1) Ignore the CA annotation. 2) Accept it, 3) modify (and) type it or 4) refuse it. If he does nothing, the annotations remain with a CA status. If the annotations are typed (from item-based or classification-based approaches), he can either validate, modify or refuse each of them. If the annotations have no type (from data mining approach), he can complete (and thus validate), or modify each of them. When a validation, completion or modification action is made, the local replica of the user is modified and the CA is updated to a HCA. The information about this action is broadcasted to the other local replica and the annotation is suggested to users as a HCA.Let us notice that when a user does not agree on a received HCA, he can modify it as in the case of classical HA in semantic wikis. If the user refuses the annotation, it is discarded from the set of CA and is included in the set of RA and is also broadcasted to other users. This set is used by the recommender to avoid re-suggesting annotations that have already been refused by users. These RA can however be re-suggested in the case the content of the wiki page has been highly modified.

Conclusion

This paper proposes an original system to encourage semantic annotations in semantic wikis of repositories of pedagogical resources. This ongoing work is based on the observation that users do not often semantically annotate pages in semantic wikis, as this task is not easy. This leads to semantic wikis with few annotations, they are therefore less efficient than they could be and students and professors cannot find resources that fit their requests. Our HCA system suggests to users annotations on the wiki pages. To annotate pages semantically, the users can either create semantic annotations as in a normal semantic wiki or can use the annotations suggested by the system. The HCA system exploits the usage the students and the professors make of the wiki pages and is based on classical collaborative filtering recommender systems; it uses item-based, classification-based and data mining-based approaches. These approaches learn the similarities/links, in terms of usage, between the wiki pages. The HCA system exploits both these similarities/links and the semantic annotations provided by humans that are present on the pages to suggest annotations to wiki pages. These annotations are not directly stored in the pages, they are suggested to users that can validate, modify or refuse these annotations. The resulting annotations in the semantic wiki are based on several levels of collaboration: explicit human collaboration when writing HA, implicit human collaboration to suggest CA and human-machine collaboration to produce HCA.

We are currently conducting experimentations and user studies to validate our approach; the preliminary results are very encouraging. As a future work, we intend to refine the way the HCA are obtained; we aim at not making a CA become directly a HCA when a user validates or modifies it, or being a RA if a user refuses it, but by waiting to have a consensus about the actions of the users. This approach, that will be based on a server of traces, will have the advantage that annotations will not change constantly.

Figure 1 :

 1 Figure 1: Annotation suggestion of HCA with the item-based approach

Figure 2 :

 2 Figure 2: Automatic suggestion of HCA with the classification-based approach

Figure 3 :

 3 Figure 3: The way CA are proposed, validated and propagated in a P2P semantic wiki


Rémi Carles
email: remi.carles@math.cnrs.fr

On Schrödinger equations with modified dispersion

Keywords: 2010 Mathematics Subject Classification. Primary 35Q55; Secondary 35A01, 35B33, 35B45

We consider the nonlinear Schrödinger equation with a modified spatial dispersion, given either by an homogeneous Fourier multiplier, or by a bounded Fourier multiplier. Arguments based on ordinary differential equations yield ill-posedness results which are sometimes sharp. When the Fourier multiplier is bounded, we infer that no Strichartztype estimate improving on Sobolev embedding is available. Finally, we show that when the symbol is bounded, the Cauchy problem may be ill-posed in the case of critical regularity, with arbitrarily small initial data. The same is true when the symbol is homogeneous of degree one, where scaling arguments may not even give the right critical value. This index corresponds to the one given by scaling arguments: if u solves (1.1), then for any Λ > 0, u Λ : (t, x) → Λ m/(2σ) u(Λ m t, Λx) solves the same equation. The value of s

Introduction

For (t, x) ∈ R + × R d , we consider (1.1) i∂ t u + P (D)u = λ|u| 2σ u ; u |t=0 = u 0 , where D = -i∇ x , P : R d → R, λ ∈ R and σ > 0. Since the Fourier multiplier P is real-valued, the free flow (λ = 0) generates a unitary group on Ḣs (R d ), s ∈ R: D) .

S(t) = e -itP (
We consider two cases:

• P is homogeneous, P (µξ) = µ m P (ξ), for all ξ ∈ R d , µ > 0, with m 1.

• P is bounded, P ∈ L ∞ (R d ). The first case includes the standard Schrödinger operator (P (ξ) = -|ξ| 2 ), and the fourthorder Schrödinger operator (P (ξ) = |ξ| 4 ), studied for instance in [START_REF] Ben-Artzi | Dispersion estimates for fourth order Schrödinger equations[END_REF][START_REF] Pausader | Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case[END_REF][START_REF]The cubic fourth-order Schrödinger equation[END_REF]. Smoothing effects and dispersive estimates have been established for rather general Fourier multipliers P in [START_REF] Saut | Uniform decay estimates for a class of oscillatory integrals and applications[END_REF][START_REF] Kenig | Oscillatory integrals and regularity of dispersive equations[END_REF]. The case d = 1, P (ξ) = ξ 2j+1 , j ∈ N, has been studied initially in [START_REF]Higher-order nonlinear dispersive equations[END_REF]; the case d = 2, with P a polynomial of degree m = 3 has been studied in [START_REF]Dispersion estimates for third order equations in two dimensions[END_REF], revealing different dispersive phenomena according to the precise structure of P . The case m ∈ 2N, with P elliptic and ∇ 2 P non-degenerate outside {ξ = 0}, appears as a particular case of the framework in [START_REF] Cui | Well-posedness of higher-order nonlinear Schrödinger equations in Sobolev spaces H s (R n ) and applications[END_REF]. It is shown there that if s > max(0, s 0 ), then the Cauchy problem (1.1) is locally well-posed in H s (R d ), where (1.2)

s 0 = d 2 - m 2σ .
for which the Ḣs (R d )-norm is invariant under u → u Λ is s = s 0 . We will see in §1.3 that this scaling argument may not yield the sharp Sobolev regularity: if m = 1, the Cauchy problem (1.1) may be strongly ill-posed in H s (R d ) for all s d/2.

In [START_REF] Cui | Well-posedness of higher-order nonlinear Schrödinger equations in Sobolev spaces H s (R n ) and applications[END_REF], the proof of local well-posedness uses dispersive and Strichartz estimates for S, established in [START_REF] Cui | Pointwise estimates for a class of oscillatory integrals and related L p -L q estimates[END_REF] for d = 1, and in [START_REF]Pointwise estimates for oscillatory integrals and related L p -L q estimates. II. Multidimensional case[END_REF] for d 2. Note that in the case d = 1, dispersive and Strichartz estimates for S are proved in [START_REF] Hayashi | Asymptotic properties of solutions to dispersive equation of Schrödinger type[END_REF] for P (ξ) = |ξ| m and m 2 (not necessarily an integer). By resuming arguments similar to those presented in [START_REF]Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations[END_REF][START_REF] Christ | Ill-posedness for nonlinear Schrödinger and wave equations[END_REF], we prove that in this framework, the index s 0 is sharp, in the sense that the nonlinear flow map fails to be uniformly continuous at the origin in H s (R d ), if s < s 0 . This property has been established in [START_REF]The cubic fourth-order Schrödinger equation[END_REF] for the case P (ξ) = |ξ| 4 with (d, σ) = (3, 1). However, the index s 0 may not correspond to the critical Sobolev regularity (see §1.3).

The second case, where P is bounded, is motivated by the results presented in [START_REF] Debussche | Modified energy for split-step methods applied to the linear Schrödinger equation[END_REF], where P (ξ) = -1 h arctan(h|ξ| 2 ) is considered to construct numerical approximations of the solution to the linear Schrödinger equation

i∂ t u + ∆u = V (x)u,
and 0 < h ≪ 1 denotes the time step. We will see below that in such a framework, no Strichartz estimate is available, even if one is ready to pay some loss of derivative. Another example of bounded symbol one may think of is

i∂ t u + ∆ (1 -∆) -1 u = λ|u| 2σ u.
In these two examples, P is elliptic. We will see however that no Strichartz estimate (better than Sobolev embedding) is available there, and that the critical regularity is s c = d/2.

Norm inflation.

Our result in this direction is:

Theorem 1.1. Let d 1, λ ∈ R, σ > 0.
Assume that either σ is an integer, or that there exists an integer r such that 2σ r > d/2. 1. Suppose that P is m-homogeneous, with m 1, and denote s 0 = d/2 -m/(2σ). Suppose that s 0 > 0 and let 0 < s < s 0 . There exists a family (u h 0 ) 0<h 1 in S(R d ) with u h 0 H s (R d ) → 0 as h → 0, and a solution u h to (1.1) and 0 < t h → 0, such that:

u h (t h ) H s (R d ) → +∞ as h → 0.
2. If P is bounded, then the above conclusion remains true for any 0 < s < d/2. Theorem 1.1 is proved in §2, by adapting the ordinary differential equation mechanism used in, e.g., [START_REF]Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations[END_REF][START_REF] Christ | Ill-posedness for nonlinear Schrödinger and wave equations[END_REF]. However, the critical Sobolev regularity s c may satisfy s c > s 0 (see §1.3): in view of [START_REF] Cui | Well-posedness of higher-order nonlinear Schrödinger equations in Sobolev spaces H s (R n ) and applications[END_REF] and Theorem 1.1, we have s c = s 0 at least when P (ξ) = µ|ξ| m , µ ∈ R \ {0}, m ∈ 2N \ {0}. We also refer to [START_REF] Saut | Dispersive blow-up II. Schrödinger-type equations, optical and oceanic rogue waves[END_REF] where a different result concerning the lack of well-posedness is established for a broad variety of dispersive equations, even in the linear case.

Absence of Strichartz estimates.

In this paragraph, we focus our discussion on the case where P is bounded. For s > d/2, H s (R d ) being an algebra, local well-posedness in H s (R d ) is straightforward, provided that the nonlinearity is sufficiently smooth (see e.g. [START_REF] Tao | Nonlinear dispersive equations[END_REF]). Therefore, the critical threshold is s c = d/2, and from Theorem 1.1, no dispersive property is present to decrease this number. More precisely, no Strichartz estimate is available for S(•), even if one is ready to pay some loss of regularity which is not worse than the result provided by Sobolev embedding. To state this property precisely, we recall the standard definition. Definition 1.2. A pair (p, q) = (2, ∞) is admissible if p 2, q 2, and

2 p = d 1 2 - 1 q .
By Sobolev embedding, for all (p, q) (not necessarily admissible) with 2 q < ∞, there exists C > 0 such that for all u 0 ∈ H d/2-d/q (R d ), and all finite time interval I,

S(•)u 0 L p (I;L q (R d )) C S(•)u 0 L p (I;H d/2-d/q (R d )) C u 0 L p (I;H d/2-d/q (R d )) = C|I| 1/p u 0 H d/2-d/q (R d ) .
When P is bounded, this estimate cannot be improved:

Corollary 1.3. Let d 1, and P ∈ L ∞ (R d ; R).
Suppose that there exist an admissible pair (p, q), an index k ∈ R, a time interval I ∋ 0, |I| > 0, and a constant C > 0 such that

S(•)u 0 L p (I;L q (R d )) C u 0 H k (R d ) , ∀u 0 ∈ H k (R d ). Then necessarily, k 2/p = d/2 -d/q.
The fact that no standard Strichartz estimate (with no loss) is available for S(•) is rather clear, since the dispersion relation is given by τ = P (ξ), and defines a characteristic variety which is bounded in τ . However, one could expect the existence of Strichartz estimates with loss of regularity, in the same fashion as in [START_REF] Anton | Strichartz inequalities for Lipschitz metrics on manifolds and nonlinear Schrödinger equation on domains[END_REF][START_REF] Blair | On Strichartz estimates for Schrödinger operators in compact manifolds with boundary[END_REF][START_REF] Burq | Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds[END_REF] (where the geometric framework -the space variable belongs to a compact manifold -rules out the existence of the standard dispersive properties). The fact that this is not so is a rather direct consequence of Theorem 1.1 (where σ > 0 is arbitrary), and of the argument given in [START_REF] Burq | Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds[END_REF] to prove Proposition 3.1. It may seem surprising to prove Corollary 1.3 as a consequence of a nonlinear analysis; we insist on the fact that the proof of Theorem 1.1 is rather simple, and the deduction of Corollary 1.3 involves another nonlinear result, whose proof is also quite short (see Proposition 3.2 below).

Critical cases.

In [START_REF] Cazenave | The Cauchy problem for the critical nonlinear Schrödinger equation in H s[END_REF], local well-posedness in H sc (R d ) for small data is established for Equation (1.1) in the case P (ξ) = -|ξ| 2 , where

s c = d 2 - 1 σ
coincides with s 0 in that case, since m = 2. In [START_REF] Nakamura | Nonlinear Schrödinger equations in the Sobolev space of critical order[END_REF], local well-posedness in

H d/2 (R d )
for small data is established for the same operator, with nonlinearities which are allowed to grow exponentially. In these two papers, the proof uses Strichartz estimates (in Besov spaces). On the other hand, when P is bounded, the Cauchy problem may be ill-posed in H d/2 (R d ), even for nonlinearities growing algebraically. Moreover, when P is mhomogeneous with m = 1, the critical Sobolev regularity may not be s 0 , but s c = d/2 > s 0 , with ill-posedness for s = s c .

Proposition 1.4. Let λ ∈ R \ {0}, σ > 0.
Assume that either σ is an integer, or that there exists an integer r such that 2σ r > d/2. In either of the two cases:

• P (ξ) = c • ξ, c ∈ R d , or • P is constant, for all δ > 0, there exists u 0 ∈ H d/2 (R d ) with u 0 H d/2 (R d ) δ such that (1.1) has a unique solution u ∈ C(R + ; D ′ (R d ))
, and for any t > 0, u(t,

•) ∈ H d/2 (R d ).
For comparison with other results, note that in the first case, P is not elliptic if d 2. In the second case, P = 0 is elliptic, but ∇ 2 P = 0 is obviously degenerate.

Unlike what happens in the presence of Strichartz estimates ( [START_REF] Cazenave | The Cauchy problem for the critical nonlinear Schrödinger equation in H s[END_REF][START_REF] Nakamura | Nonlinear Schrödinger equations in the Sobolev space of critical order[END_REF]), this result yields examples where local well-posedness fails in the critical case s = s c , even for small data. We prove Proposition 1.4 in §4: we present the cases d = 2, σ > 0, and d 1, σ = 1/2, only, for the convenience of the exposition, but the argument can be extended to any space dimension, up to more intricate computations.

In the case m = 1 (at least), the mere assumption that P is m-homogeneous is not enough to characterize the critical Sobolev regularity in (1.1), or the existence of Strichartz estimates. Indeed, the symbol P (ξ) = |ξ| corresponds to the wave equation, for which Strichartz estimates are available when d 2, and so s c < d/2. See e.g. [START_REF] Ginibre | Generalized Strichartz inequalities for the wave equation[END_REF]. This suggests that also when m > 1, the value s 0 may not be sharp when P is not proportional to |ξ| m , but for instance of the form |ξ| m-1 c • ξ, c ∈ R d , or more generally when P is not elliptic; see also [START_REF]Dispersion estimates for third order equations in two dimensions[END_REF][START_REF] Saut | Uniform decay estimates for a class of oscillatory integrals and applications[END_REF][START_REF] Kenig | Oscillatory integrals and regularity of dispersive equations[END_REF] for remarks in this direction.

Proof of Theorem 1.1

The proof of Theorem 1.1 proceeds along the same lines as in [START_REF]Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations[END_REF][START_REF] Christ | Ill-posedness for nonlinear Schrödinger and wave equations[END_REF] (see also [START_REF] Tao | Nonlinear dispersive equations[END_REF]). Fix s as in Theorem 1.1. Consider initial data of the form

u h 0 (x) = h s-d/2 κ h a 0 x h , with 0 < h ≪ 1, a 0 (x) = e -|x| 2
, and

κ h = log 1 h -θ
for some θ > 0 to be fixed later. We have

u h 0 H s (R d ) -→ h→0 0. Introduce the scaling ψ(τ, y) = h d/2-s u h h 2+α τ, hy ,
for some α to be precised later:

ψ(τ ) Ḣs (R d ) = u h h 2+α τ Ḣs (R d ) .
Denote ε = h 2σ(d/2-s)-2-α . The function ψ solves the Cauchy problem (2.1)

iε∂ τ ψ + h 2σ(d/2-s) P h -1 D y ψ = λ|ψ| 2σ ψ; ψ |τ =0 = κ h a 0 .

Choice of α.

When P is m-homogeneous, Equation (2.1) simplifies to

iε∂ τ ψ + h 2σ(d/2-s)-m P (D y )ψ = λ|ψ| 2σ ψ; ψ |τ =0 = κ h a 0 .
For ω > 0, we set

2 + α = 1 m + ω (m -1 + ω) 2σ d 2 -s + m ,
in which case we have: ε = h 2σ(s0-s)/(m+ω) . Therefore, ε → 0 as h → 0 since s < s 0 . We also compute

h 2σ(d/2-s)-m = ε m+ω . When P is bounded, we consider 2 + α = σ(d/2 -s). Therefore, 2σ d 2 -s -2 -α > 0 (hence ε → 0 as h → 0)
, and 2 + α > 0.

The ODE approximation. Introduce the solution to

iε∂ τ ϕ = λ|ϕ| 2σ ϕ; ϕ |τ =0 = κ h a 0 .
It is given by

ϕ(τ, y) = κ h a 0 (y) exp -iλ τ ε κ h 2σ |a 0 (y)| 2σ .
Since a 0 is a Gaussian, ϕ ∈ C ∞ (R × R d ) regardless of σ > 0, and for any r 0,

(2.2) ϕ(τ ) H r (R d ) κ h 1+2σr τ ε r + κ h .
Lemma 2.1. Let r > d/2 be an integer, with in addition r 2σ if σ ∈ N. In either of the cases of Theorem 1.1, we can find δ > 0 independent of θ > 0 such that

sup 0 τ ε(log 1 ε ) δ ψ(τ ) -ϕ(τ ) H r (R d ) -→ ε→0 0.
PROOF. Denote by w = ψ -ϕ the error. It solves

iε∂ τ w + h 2σ(d/2-s) P h -1 D y w = h 2σ(d/2-s) P h -1 D y ϕ + λ |w + ϕ| 2σ (w + ϕ) -|ϕ| 2σ ϕ ,
with w |τ =0 = 0. Using the facts that P is real-valued, z → |z| 2σ z is sufficiently smooth, and H r (R d ) is an algebra, we find

w(τ ) H r (R d ) 1 ε τ 0 h 2σ(d/2-s) P h -1 D y ϕ(τ ′ ) H r (R d ) dτ ′ + 1 ε τ 0 w(τ ′ ) 2σ H r (R d ) + ϕ(τ ′ ) 2σ H r (R d ) w(τ ′ ) H r (R d ) dτ ′ .
In the case where P is m-homogeneous, we have

h 2σ(d/2-s) P h -1 D y ϕ(τ ′ ) H r (R d ) h 2σ(d/2-s)-m ϕ(τ ′ ) H r+m (R d ) h 2σ(s0-s) ϕ(τ ′ ) H r+m (R d ) .
In the case where P is bounded, we have

h 2σ(d/2-s) P h -1 D y ϕ(τ ′ ) H r (R d ) h 2σ(d/2-s) ϕ(τ ′ ) H r (R d ) h 2σ(s0-s) ϕ(τ ′ ) H r (R d ) ,
where we set s 0 = d/2 in this case.

In both cases, we check that there exists β > 0 (independent of θ) such that

h 2σ(s0-s) = ε 1+β .
It is given by the formula

β = 2σ(s 0 -d/2) + 2 + α 2σ(d/2 -s) -2 -α .
In the homogeneous case, this formula becomes β = m -1 + ω, and when P is bounded, β = 1. Therefore, in view of (2.2) and since κ h 1, there exist β, γ > 0 such that

h 2σ(d/2-s) P h -1 D y ϕ(τ ′ ) H r (R d ) ε 1+β τ ε γ + 1 .
So long as w(τ ) H r (R d ) 1, with τ as above, we infer from (2.2):

w(τ ) H r (R d ) τ 0 ε β τ ′ ε γ + 1 dτ ′ + 1 ε τ 0 1 + τ ′ ε r w(τ ′ ) H r (R d ) dτ ′ .
Gronwall's Lemma yields

w(τ ) H r (R d ) ε β τ ε γ + 1 e Cτ /ε+C(τ /ε) r+1 ε β e 2Cτ /ε+C(τ /ε) r+1 .
By choosing δ > 0 sufficiently small, the right hand side is controlled by, say,

ε β/2 , for all 0 τ ε log 1 ε δ . The condition w(τ ) H r (R d )
1 is verified for such times τ , provided that ε is sufficiently small, and the lemma follows.

Conclusion.

Let r > d/2 as in Lemma 2.1. With δ > 0 given by Lemma 2.1, we have, since s < d/2:

u h h 2+α ε log 1 ε δ H s (R d ) ϕ ε log 1 ε δ H s (R d ) -C ϕ ε log 1 ε δ -ψ ε log 1 ε δ H r (R d )
.

On the other hand, similar to (2.2), we have:

ϕ(τ ) H s (R d ) κ h 1+2σs τ ε s -Cκ h ,
and so,

ϕ ε log 1 ε δ H s (R d ) C log 1 ε sδ-θ-2σθs
-o(1).

For θ > 0 sufficiently small, sδ -θ -2σθs > 0, and Lemma 2.1 yields

u h h 2+α ε log 1 ε δ H s (R d ) -→ h→0 +∞.
Theorem 1.1 follows, with

t h = h 2+α ε log 1 ε δ = Ch 2σ(d/2-s) log 1 h δ -→ h→0 0.

Proof of Corollary 1.3

We argue by contradiction, by using a slight generalization of [7, Proposition 3.1].

Definition 3.1 (From [19]). Let s ∈ R. The Cauchy problem (1.1) is well posed in H s (R d ) if, for all bounded subset B ⊂ H s (R d ), there exist T > 0 and a Banach space X T continuously embedded into C([0, T ]; H s (R d )) such that for all u 0 ∈ H s (R d ), (1.1)
has a unique solution u ∈ X T , and the mapping u 0 → u is uniformly continuous from

(B, • H s ) to C([0, T ]; H s (R d )).
Proposition 3.2. Let d 1, P : R d → R. Suppose that there exist an admissible pair (p, q), an index k < 2/p = d/2 -d/q, T 0 > 0, and a constant C > 0 such that

(3.1) S(•)u 0 L p ([0,T0];L q (R d )) C u 0 H k (R d ) , ∀u 0 ∈ H k (R d ).
Then for all

k + d q < s < d 2 , 0 < σ < p 2 ,
the Cauchy problem for (1.1) is well posed in H s (R d ).

Since in Theorem 1.1, we can always consider σ = 1, Theorem 1.1 and Proposition 3.2 imply Corollary 1.3 in the non-endpoint case p > 2. The endpoint case then follows by interpolation with the case (p, q) = (∞, 2): if an endpoint Strichartz estimate (with some loss) was available, then an non-endpoint would be as well.

PROOF. For 0 < T T 0 , introduce

X T = C [0, T ]; H s (R d ) ∩ L p [0, T ]; W ℓ,q (R d ) ,
where ℓ = s -k. By assumption, ℓ > d/q, so we have

X T ⊂ L p [0, T ]; L ∞ (R d ) .
This space is equipped with the norm

u XT = sup 0 t T u(t) H s (R d ) + (1 -∆) ℓ/2 u L p ([0,T ];L q (R d ))
.

We construct the solution to (1.1) by a fixed point argument. Set

Φ(u)(t) = S(t)u 0 -iλ t 0 S(t -τ ) |u(τ )| 2σ u(τ ) dτ.
We prove that for T ∈]0, T 0 ] sufficiently small, Φ is a contraction on some ball of X T centered at the origin. In view of (3.1) and Minkowski inequality,

Φ(u) XT u 0 H s (R d ) + t 0 |u(τ )| 2σ u(τ ) H s (R d ) dτ u 0 H s (R d ) + t 0 u(τ ) 2σ L ∞ (R d ) u(τ ) H s (R d ) dτ u 0 H s (R d ) + T γ u 2σ L p ([0,T ];L ∞ (R d )) u L ∞ ([0,T ];H s (R d )) , with γ = 1 -2σ/p > 0. Therefore, Φ(u) XT C |u 0 H s (R d ) + CT γ u 2σ+1 XT .
Similarly,

Φ(u) -Φ(v) XT CT γ u 2σ XT + v 2σ XT u -v XT .
This yields the local well-posedness result stated in Proposition 3.2.

Ill-posedness

The key remark is that all the cases of Proposition 1.4 boil down to an ordinary differential equation mechanism. Denote by v the solution to

i∂ t v = λ|v| 2σ v; v |t=0 = u 0 . When P (ξ) = c • ξ, we have u(t, x) = v (t, x + ct) ,
and when P (ξ) = c, we have u(t, x) = v(t, x)e ict , so it suffices to prove Proposition 1.4 in the case P = 0. For fixed x ∈ R d , we have

(4.1) v(t, x) = u 0 (x)e -iλt|u0(x)| 2σ .
The idea is then that

H d/2 (R d ) is not an algebra. Consider (4.2) u 0 (x) = δ × log 1 |x| α χ |x| 2 , x ∈ R d , with χ ∈ C ∞ 0 (R), χ = 1 near the origin, and suppχ ⊂] -1, 1[. We compute ∇v(t, x) = e -iλt|u0(x)| 2σ ∇u 0 (x) -2iσλt|u 0 (x)| 2σ ∇u 0 (x).
We split the proof into three cases: for d = 2, the proof is straightforward, for d 4 even, the proof is similar but we omit the details of computations, and for d odd, we simply sketch the argument.

Case d = 2. First, u 0 ∈ H 1 (R 2 ) provided that α < 1/2. Now Proposition 1.4 follows if we can choose α < 1/2 so that |u 0 | 2σ ∇u 0 ∈ L 2 (R 2 ).
Near the origin, we have, leaving out the constants,

|u 0 (x)| 2σ ∇u 0 (x) 2 ≈ 1 |x| 2 (log |x|) 4ασ+2α-2 .
The right hand side fails to be in L 1 loc (R 2 ) if we impose 4ασ + 2α 1. So Proposition 1.4 follows, with 1 4σ + 2 α < 1 2 .

Case d 4 even. The argument is the same as in the case d = 2, with more computations that we simply sketch. We check by induction that for k 1, there exist coefficients (β jk ) 1 j k such that near the origin,

∂ k r u 0 (x) = 1 r k k j=1 β jk log 1 |x| α-j , with β 1k = (-1) k-1 (k -1)!α.
Therefore, the asymptotic behavior of ∂ k r u 0 near the origin is given by:

∂ k r u 0 (x) ∼ r→0 (-1) k-1 (k -1)! α r k log 1 |x| α-1
.

Like in the case d = 2, u 0 ∈ H d/2 (R d ) provided that α < 1/2. We compute, for t > 0, and as x → 0:

(4.3) |∂ k r v(t, x)| = 1 r k c k (t) log 1 |x| γ k + O log 1 |x| γ k -ω k ,
with c k (t) > 0, ω k > 0, and γ k = max ((2σk + 1)α -k, (2σ + 1)α -1) .

For t > 0, v(t, •) ∈ H d/2 (R d ) if, for k = d/2, the first term in (4.3) is almost in L 2 loc (R d ), but not quite: we choose α so that 2γ d/2 = -1. We find (like for d = 2)

α = 1 4σ + 2 ,
which is consistent with the requirement α < 1/2. Thus, the first term in (4.3) is not in L 2 loc (R d ) due to a logarithmic divergence, while the remainder term is in L 2 loc (R d ), since ω k > 0.

The case when d is odd. We keep u 0 of the same form as in even dimensions, since we have found a value for α which does not depend on d even: When d = 1, we check that u 0 ∈ H 1/2 (R). We can also check that for t > 0, v(t, •) ∈ H 1/2 (R).

When d 3, we compute ∂ k r u 0 and ∂ k r v in the same fashion as above, and check that ∇ [d/2] u 0 ∈ H 1/2 (R d ), and for t > 0, ∇ [d/2] v(t, •) ∈ H 1/2 (R d ).

We leave out the details, since the technicalities are more involved than in the even dimensional case, and we believe that proving Proposition 1.4 in details is not worth such an effort.

u 0

 0 (x) = δ × log 1 |x| 1/(4σ+2) χ |x| 2 , x ∈ R d . Recall the characterization of H s (R d ) when s ∈]0, 1[: a function f ∈ L 2 (R d ) belongs to H s (R d ), s ∈]0, 1[, if and only if R d ×R d |f (x) -f (x + y)| 2 |y| d+2sdxdy < ∞.

Acknowledgement. The author is grateful to Valeria Banica for valuable comments on this work.

This work was supported by the French ANR project R.A.S.


Ariane Ducellier

Hiroshi Kawase

Shinichi Matsushima

VELOCITY STRUCTURE INVERSIONS FROM HORIZONTAL TO VERTICAL (H/V) SPECTRAL RATIOS OF EARTHQUAKE MOTIONS

In this study, the illumination of an observation site from the source is assumed to be produced only by incident plane waves. The average of normalized ground motion spectral densities will then depends only on depth and a one dimensional description of wave propagation for a diffuse field of ground motions can be applied. Thus, the imaginary part of the Green function at the free surface when source and receiver are both at the same point is proportional to the square of the absolute value of the corresponding transfer function for a plane, vertically incident wave with unit amplitude. Average strong motion H/V spectral ratio of observed data at a particular site can then be computed and compared to the ratio of theoretical horizontal and vertical transfer functions corresponding to the 1D soil structure of the considered site. A series of inversions of underground 1D structure from the bedrock to the surface from observations sites in the Kyoto prefecture, Japan, is then carried out with a Genetic Algorithm code, following the proposed theory for earthquake H/V ratios.

INTRODUCTION

Site effects are of great importance in seismology and earthquake engineering because they modify the frequency content of the incident wave and lead to an amplification of the wave propagating inside the site (e.g. soft sedimentary basins). As a result, the damage pattern of the engineering structures is often linked to the subsurface geology. The deployment of vertical arrays has permitted detailed wave propagation analysis of one dimensional soil columns, leading to obtain quantitative physical parameters, such as shear-wave velocity, pressure-wave velocity and damping factors, through different inversion techniques. However, as vertical arrays are not always available, it would be quite useful to be able to invert soil columns parameters with only seismic signal at the surface.

In this study, strong ground motion in elastic layered media is considered. All sites are assumed to be "sufficiently" flat layered sites and we restrict ourselves to earthquake sources with small epicentral distances and deep enough, such that surface waves are not dominant or do not appear yet. The surface waves can then be neglected and the illumination can be conceived as produced by incident plane waves. In that case, the average of normalized ground motion spectral densities will depend only on depth and a one dimensional description of wave propagation for a diffuse field of ground motions can be applied. Thus, the imaginary part of the Green function at the free surface when source and receiver are both at the same point is proportional to the square of the absolute value of the corresponding transfer function for a plane, vertically incident wave with unit amplitude.

Average strong motion H/V spectral ratio of observed data at a particular site can then be computed and compared to the ratio of theoretical horizontal and vertical transfer functions corresponding to the 1D soil structure of the considered site. A series of inversions of underground 1D structure from the bedrock to the surface from observations sites in the Kyoto prefecture, Japan, is then carried out with a Genetic Algorithm code, following the proposed theory for earthquake H/V ratios.

AVERAGE HORIZONTAL TO VERTICAL SPECTRAL RATIO

All sites considered in this study are supposed to be "sufficiently" flat layered sites, such that 1D seismic wave propagation can be assumed (as small as 7 km from Table 1, we cannot say that it is deep enough). Moreover, we keep only a short window of the actual earthquake ground motions observed at the sites of interest. The time window chosen lasts 20 s and begins just before the arrival of S-wave. As long as these conditions are respected, the local structure around an observation site will be illuminated mostly by plane body waves from the source, including multiple scattering waves, so that the expression for the average horizontal to vertical (H/V) spectral ratio can be obtained as Equation (1).

ܪሺ߱ሻ ܸሺ߱ሻ ൌ ඨ ܩ݉൫ܫ ଵଵ ଵ ሺ,ݔ ,ݔ ߱ሻ൯ ܩ݉൫ܫ ଶଶ ଵ ሺ,ݔ ,ݔ ߱ሻ൯ ܩ݉൫ܫ ଷଷ ଵ ሺ,ݔ ,ݔ ߱ሻ൯
where ݉ܫ ቀܩ ଵ ሺ,ݔ ,ݔ ߱ሻቁ is the ij th component of the imaginary part of the one dimensional Green function between source x and receiver x at frequency ω.

As we consider only a one dimensional Green function, we can write; ܩ ଵଵ ଵ ሺ,ݔ ,ݔ ߱ሻ ൌ ܩ ଶଶ ଵ ሺ,ݔ ,ݔ ߱ሻ and Equation (1) becomes:

ܪሺ߱ሻ ܸሺ߱ሻ ൌ ඨ ܩ݉൫ܫ2 ଵଵ ଵ ሺ,ݔ ,ݔ ߱ሻ൯ ܩ݉൫ܫ ଷଷ ଵ ሺ,ݔ ,ݔ ߱ሻ൯
By summing motions of a large enough number of seismic events, we can assume that the ground motion is spatially homogeneous in a statistical sense. Therefore, the motion will depend only on depth and we can apply a one dimensional description of wave propagation for a diffuse field of ground motions. [START_REF] Claerbout | Synthesis of a layered medium from its acoustic transmission response[END_REF] established the relationship between reflection response and autocorrelation of surface motion for one dimensional layered media. [START_REF] Kawase | The Optimal use of Horizontal-to-Vertical Spectral Ratios of Earthquake Motions for Velocity Inversions Based on Diffuse Field Theory for Plane Waves[END_REF] derived a corollary of Claerbout's result for a plane, vertically incident wave with unit amplitude propagating into multiple horizontal layers overlaying a half space; and found that the imaginary part of the Green function at the free surface for the surface source is equal to the square of the absolute value of the corresponding transfer function, divided by four times the frequency and the halfspace impedance:

ܩ݉ቀܫ ଵ ሺ0, ߱ሻቁ ൌ 1 4߱ߩ ு ܿ ு ܨܶ| ሺ0, ߱ሻ| ଶ
where ܨܶ ሺ,ݖ ߱ሻ is the i th component of the transfer function at depth z and frequency ω, ρ H is the mass density and c H is either P-or S-wave velocity of the half-space, depending whether we consider the vertical or the horizontal component of the transfer function. By writing Equation (3) for both horizontal and vertical motions, [START_REF] Kawase | The Optimal use of Horizontal-to-Vertical Spectral Ratios of Earthquake Motions for Velocity Inversions Based on Diffuse Field Theory for Plane Waves[END_REF] showed that the H/V spectral ratio can be written as:

ܪሺ0, ߱ሻ ܸሺ0, ߱ሻ ൌ ඨ 2ߙ ு ߚ ு ܨܶ| ଵ ሺ0, ߱ሻ| ܨܶ| ଷ ሺ0, ߱ሻ|
where α H and β H are P-and S-wave velocities of the half-space.

We can then compare the ratio of transfer functions from Equation (4) obtained with a theoretical 1D soil column and the observed ratio of horizontal and vertical square power spectral densities averaged on several seismograms. The ratio of transfer function will be then considered as a characteristic function associated with each set of soil layers properties and the H/V ratio as the objective function for the inversion of the soil column. The inversion is performed using a Genetic Algorithm code [START_REF] De Martin | Influence of the Nonlinear Behavior of Soft Soil on Strong Ground Motions[END_REF]De Martin et al., 2010), in which the horizontal and vertical transfer functions are computed with the Thomson-Haskell propagator matrix method [START_REF] Thomson | Transmission of elastic waves through a stratified solid[END_REF][START_REF] Haskell | The dispersion of surface waves in multilayered media[END_REF].

EARTHQUAKE DATA

In this study, we focus on soil column inversions performed at three sites from K-NET (KYT008 and KYT009) and KiK-net (KYTH04) networks. We selected 32 seismograms between 1997 and 2009, with epicenter less than 200 km from the observation sites and with PGA superior to 3 cm/s², so as to avoid having too much noise. Dates, depths, magnitudes, distances from observation site and PGA of all seismograms used in this study are given in Table 1. After eliminating accelerograms not recorded or with PGA inferior to 3 cm/s², there are 28 earthquakes left for station KYT008, 13 for station KYT009 and 19 for stations KYTH04. [START_REF] Kawase | The Optimal use of Horizontal-to-Vertical Spectral Ratios of Earthquake Motions for Velocity Inversions Based on Diffuse Field Theory for Plane Waves[END_REF] showed that the summation of synthetic waveforms converges quite fast to the theoretical prediction, such that we do not need many ground motions to get convergent ratios. Therefore, we assume that we have enough recorded earthquake motions to have a good quality of the observed H/V ratio. This assumption should however be investigated more thoroughly. Indeed, as can be seen from the localizations of the earthquakes and three observations sites shown in Fig. 1, the spatial homogeneity of the ground motion does not seem to have been reached. Particularly, it appears that there are more earthquakes located south and east of the observation sites than north and west.

INVERSION OF THREE SOIL COLUMNS

Inversions of soil columns at observation sites in the Kyoto and Osaka, prefectures, Japan were carried out. We show here the results obtained for three soil columns with a relatively good agreement between the objective function and the best ratio of transfer functions found at the end of the inversion.

First observation site: K-NET station KYT008

Information about soil layers properties was provided by previous studies [START_REF] Kawase | Amplification characteristics of K-NET, KiK-net and JMA Shindokei network sites based on the spectral inversion technique[END_REF][START_REF] Kawase | Site effects derived from spectral inversion method for K-NET, KiK-net and JMA strong-motion network with special reference to soil nonlinearity in high PGA records[END_REF], and was used to establish an initial velocity log of the soil column. Initial values of S-wave velocity (Vs 0 ), P-wave velocity (Vp 0 ) and thickness (TH 0 ) of each layer are given in Table 2. Some soil layers parameters have only little influence on the corresponding theoretical horizontal and vertical transfer functions, such that any value of these parameters will give nearly the same ratio of transfer functions. Hence, these parameters will not be easy to invert as many possible values could provide a good fit between the observed H/V ratio and the theoretical ratio of horizontal and vertical transfer functions. Therefore, we choose to invert only around ten parameters for each station, selecting those which seem to be the most influential. The other parameters are set constant and are given the initial value. By restricting the search domain of the Genetic Algorithm, we aim to obtain a unique solution. Moreover, to avoid local minima and check whether the solution found is unique, ten independent inversions with the same input parameters are carried out and we keep the ten best velocity logs obtained for each inversion.

The range intervals of variation of the inverted parameters are indicated in Table 2. It should be noted that the initial value of an inverted parameter has no influence on the result of the inversion; it only helps choosing the interval of variation of this parameter. As the inversion is performed with a Genetic Algorithm, the inverted parameters can only take discrete values. In this study, we divide the intervals of variation into fifteen equal parts, such that the inverted parameters can take sixteen different values.

It should be noted that only the ratio between P-and S-wave velocities of the half-space can be inverted by this method, as can be deduced from Equation (4). For simplification, we choose to invert only S-wave velocity of the half-space and set P-wave velocity constant. However, any couple of P-and S-wave velocities with the same ratio between both velocities can be considered as a solution.

Results of the inversion are shown in Fig. 2. On the left panel (a), we can see the objective H/V ratio computed from earthquakes data in black plain line, the ratio of horizontal and vertical transfer functions corresponding to the initial velocity log in black dotted line and all the best ratios of horizontal and vertical transfer functions corresponding to the best velocity logs found by the ten inversions in grey lines. The center panel (b) shows the corresponding S-wave velocity and P-wave velocity logs. The black line is the initial log and the grey lines are the best logs obtained by the ten inversions, corresponding to the best ratios on the left panel. The right panel (c) is a detail of the center panel (b) and shows the velocity logs of the first 30 meters.

Fig. 2. Results of the inversion for observation site KYT008.

There is a good agreement between the theoretical ratio of horizontal and vertical transfer functions obtained from the inversion and the objective H/V ratio obtained from earthquakes data, particularly for the higher frequencies (higher than 4 Hz). The agreement is less good for the lower frequencies. All the independent inversions give the same value of P-wave velocity, thickness and S-wave velocity of the shallow layers, which indicates a good convergence of the inversion. However, the S-wave velocity of the deepest layers is not well resolved. Indeed, a large range of different values can be obtained from the different independent inversions. The effect of the deepest layers on the ratio of transfer functions can be seen especially at the lower frequencies. Meanwhile, at low frequency, the fit between the observed H/V ratio and the ratio of transfer functions is not very good. It seems that, in the domain of search chosen for this inversion, no actual good solution can be found, such that the algorithm hesitates to choose the best one among several solutions.

A quick parametric study was carried out to check which parameters could have the biggest influence on the theoretical ratio of horizontal and vertical transfer functions. Figure 3 shows the variations of the theoretical ratio when each parameter is changed one at a time. The first row corresponds to the S-wave velocities of the nine soil layers and the bedrock; the second row corresponds to the P-wave velocities of the nine soil layers; the third row corresponds to the thicknesses of the nine soil layers. In each figure, the black line represents the theoretical ratio of horizontal and vertical transfer functions in function of frequency corresponding to the initial velocity log. The two grey lines correspond to the theoretical ratio when only one parameter varies and takes either the minimal value, either the maximal value of the intervals of variation given in Table 2.

Fig. 3. Variations of theoretical H/V ratio for observation site KYT008 when changing a single parameter (S-wave velocity, P-wave velocity or thickness of each layer) at once.

As can be seen from Fig. 3, the S-wave velocities of layers 1, 2 and 6 and the thickness of layer 1 seem to have the strongest influence on the theoretical H/V ratio. The solution obtained from the inversion for these parameters can then be considered as rather robust. The other inverted parameters (S-wave velocities of layers 8 and 9, P-wave velocities of layers 8 and 9 and thickness of layer 6) have only a slight influence on the theoretical H/V ratio, such that the robustness of the solution from the inversion is more uncertain for these parameters.

Finally, the S-wave velocity of the bedrock does not seem to have any influence on the theoretical H/V ratio, which explains why the algorithm has difficulty to find a solution for this parameter.

Second observation site: K-NET station KYT009

Initial value of S-wave velocity (Vs 0 ), P-wave velocity (Vp 0 ) and thickness (TH 0 ) of each layer are given in Table 3. The ranges of variation of the inverted parameters are also indicated in Table 3. The agreement between the theoretical ratio of horizontal and vertical transfer functions obtained from the inversion and the objective H/V ratio obtained from earthquakes data is satisfactory for the whole range of frequencies where the inversion is performed. An amplitude peak is found at the same frequency but with a slightly different form. As in the previous case, all the independent inversions give the same value of P-wave velocity, thickness and S-wave velocity of the shallow layers, which indicates a good convergence of the inversion. However, the S-wave and P-wave velocities of the deepest layers are not well resolved. Indeed, a large range of different values can be obtained from the different independent inversions. This could again be explained by the difficulty to fit the ratio at low frequency.

The results of the parametric study are shown in Fig. 5. The first row corresponds to the Swave velocities of the ten soil layers and the bedrock; the second row corresponds to the Pwave velocities of the ten soil layers; the third row corresponds to the thicknesses of the ten soil layers. In each figure, the black line represents the theoretical ratio of horizontal and vertical transfer functions in function of frequency corresponding to the initial velocity log.

The two grey lines correspond to the theoretical ratio when only one parameter varies and takes either the minimal value, either the maximal value of the intervals of variation given in Table 3.

Fig. 5. Variations of theoretical H/V ratio for observation site KYT009 when changing a single parameter (S-wave velocity, P-wave velocity or thickness of each layer) at once.

As can be seen from Fig. 5, the S-wave velocities of layers 1, 2 and 7, the P-wave velocity of layer 7 and the thickness of layer 1 seem to have the strongest influence on the theoretical H/V ratio. The solution obtained from the inversion for these parameters can then be considered as rather robust. The other inverted parameters (P-wave velocities of layers 1, 8 and 10) have only a slight influence on the theoretical H/V ratio, such that the robustness of the solution from the inversion is more uncertain for these parameters. Finally, the S-wave velocities of the tenth layer and the bedrock do not seem to have any influence on the theoretical H/V ratio, which explains why the algorithm has difficulty to find a solution for these parameters.

Third observation site: KiK-net station KYTH04

Initial value of S-wave velocity (Vs 0 ), P-wave velocity (Vp 0 ) and thickness (TH 0 ) of each layer are given in Table 4. The intervals of variation of the inverted parameters are also indicated in Table 4.

Fig. 6. Results of the inversion for observation site KYTH04.

There is a good agreement between the theoretical ratio of horizontal and vertical transfer functions obtained from the inversion and the objective H/V ratio obtained from earthquakes data for the whole range of frequencies where the inversion is performed. All the independent inversions give nearly the same values of S-wave velocity, which indicates a good convergence of the inversion. However, the P-wave velocities are not well resolved. Indeed, a large range of different values can be obtained from the different independent inversions. It seems that the P-wave velocity of the soil layers has less influence on the ratio of transfer functions, such that this parameter is more difficult to invert.

The results of the parametric study are shown in Fig. 7. The first row corresponds to the Swave velocities of the seven soil layers and the bedrock; the second row corresponds to the P-wave velocities of the seven soil layers; the third row corresponds to the thicknesses of the seven soil layers. In each figure, the black line represents the theoretical ratio of horizontal and vertical transfer functions in function of frequency corresponding to the initial velocity log. The two grey lines correspond to the theoretical ratio when only one parameter varies and takes either the minimal value, either the maximal value of the intervals of variation given in Table 4. As can be seen from Fig. 7, the S-wave velocities of layers 2, 3, 4, 5 and 7 seem to have the strongest influence on the theoretical H/V ratio. The solution obtained from the inversion for these parameters can then be considered as rather robust. The other inverted parameters (Pwave velocities of layers 3, 4, 6 and 7) have only a slight influence on the theoretical H/V ratio, such that the robustness of the solution from the inversion is more uncertain for these parameters. Finally, the S-wave velocity of the bedrock seems to have a slight influence on the theoretical H/V ratio, which explains why the algorithm can more easily invert this parameter for this observation site than for the two other observation sites.

DISCUSSION

The velocity structure obtained from the inversion with the genetic Algorithm for the three observation sites taken as example gave theoretical predictions of the H/V ratio close to the observed H/V ratio. However, further investigation should be made to assert the precision of the H/V ratio. Particularly, we do not know whether the number of seismograms selected in this study is sufficient or not. Moreover, the spatial homogeneity in a statistical sense of the seismic events used in this study should be asserted. Uncertainty in the computation of the H/V ratio from observed earthquakes data should be evaluated, in order to establish the uncertainty with which the parameters of the soil column are inverted.

We encountered more difficulties when trying to invert the parameters of the deepest soil layers. Supplementary inversions could be made with only the deepest layers parameters varying and while fitting the ratio of transfer functions against the observed H/V ratio only for the lower frequencies. However, this could be done only if we ascertain that the observed H/V ratio is well evaluated also for the low frequencies.

Only some parameters were inverted, such that the velocity logs obtained by the inversion are not the unique ones which could explain the observed H/V ratio. This should be taken into account while comparing the results of the inversion with the method exposed in this study to results of other inversions with different methods.

CONCLUSION

Horizontal to vertical spectral ratios of earthquake motions were used to perform inversions of 1D velocity structure from the bedrock to the surface with a Genetic Algorithm code for three observations sites in the Kyoto, prefecture, Japan. A good agreement was found between observed H/V ratio and their theoretical prediction using the best velocity logs obtained from the inversion. However, the deepest layers remained difficult to be inverted. Moreover, the precision with which the observed H/V ratio is computed should be established, to evaluate the precision of the inversion.

Fig. 1 .

 1 Fig. 1. Localization of earthquakes and observation sites.

  inversion are shown in Fig.4. On the left panel (a), we can see the objective H/V ratio computed from earthquakes data in black plain line, the ratio of horizontal and vertical transfer functions corresponding to the initial velocity log in black dotted line and all the best ratios of horizontal and vertical transfer functions corresponding to the best velocity logs found by the ten inversions in grey lines. The center panel (b) shows the corresponding S-wave velocity and P-wave velocity logs. The black line is the initial log and the grey lines are the best logs obtained by the ten inversions, corresponding to the best ratios on the left panel. The right panel (c) is a detail of the center panel (b) and shows the velocity logs of the first 30 meters.

Fig. 4 .

 4 Fig. 4. Results of the inversion for observation site KYT009.

  inversion are shown in Fig.6. On the left panel (a), we can see the objective H/V ratio computed from earthquakes data in black plain line, the ratio of horizontal and vertical transfer functions corresponding to the initial velocity log in black dotted line and all the best ratios of horizontal and vertical transfer functions corresponding to the best velocity logs found by the ten inversions in grey lines. The center panel (b) shows the corresponding S-wave velocity and P-wave velocity logs. The black line is the initial log and the grey lines are the best logs obtained by the ten inversions, corresponding to the best ratios on the left panel. The right panel (c) is a detail of the center panel (b) and shows the velocity logs of the first 30 meters.

Fig. 7 .

 7 Fig. 7. Variations of theoretical H/V ratio for observation site KYTH04 when changing a single parameter (S-wave velocity, P-wave velocity or thickness of each layer) at once.

Table 1 .

 1 Characteristics of seismmograms used in this study

	Date	Depth (km)	Magnitude	Distance from site (km) KYT008 KYT009 KYTH04 KYT008 KYT009 KYTH04 PGA (cm/s²)
	2009/02/18 -06:47	9	5.2	94.1		82.4	6.81		8.647
	2008/08/30 -18:28	15	4.2	45.7	43.33	32.7	9.70	8.53	14.843
	2008/08/08 -04:35	15	4.2	44.8		32.7	5.26		8.295
	2007/04/15 -12:19	16	5.4	106.0	92.55	94.1	8.21	9.64	8.521
	2007/01/22 -02:16	13	4.5	100.4		88.2	7.03		6.147
	2006/06/03 -00:48	7	4.1	32.7	32.09	19.8	15.85	6.59	42.071
	2006/02/16 -23:10	14	4.4	104.6		91.3	8.26		5.996
	2005/12/24 -11:02	43	4.8			117.0			5.325
	2004/12/01 -23:30	13	4.0	44.9	31.26	35.1	5.63	7.51	7.706

Table 2 .

 2 Initial values and intervals of variation of soil column parameters.

	Layer number	Vs 0 (m/s)	Variations of Vs	(m/s) Vp 0	Variations of Vp	(m) TH 0	Variation of TH

Table 3 .

 3 Initial values and intervals of variation of soil column parameters.

	Layer number	Vs 0 (m/s)	Variations of Vs	(m/s) Vp 0	Variations of Vp	(m) TH 0	Variation of TH
	1	60.00	0.50 * Vs 0 -1.25 * Vs 0	150.00	0.50 * Vp 0 -1.25 * Vp 0	1.80	0.50 * TH 0 -1.25 * TH 0
	2	60.00	Vs 0 0.50 * Vs 0 -1.25 *	150.00	Not inverted	0.20	Not inverted
	3	400.00	Not inverted	1400.00	Not inverted	1.00	Not inverted
	4	400.00	Not inverted	1400.00	Not inverted	0.60	Not inverted
	5	400.00	Not inverted	1400.00	Not inverted	2.40	Not inverted
	6	850.00	Not inverted	3410.00	Not inverted	0.95	Not inverted
	7	850.00	Vs 0 0.50 * Vs 0 -1.25 *	Vp 0 3410.00 0.50 * Vp 0 -1.25 *	113.05	Not inverted
	8	2226.19	Not inverted	Vp 0 4172.46 0.50 * Vp 0 -1.25 *	620.00	Not inverted
	9	3250.94	Not inverted	5689.09	Not inverted	630.00	Not inverted
	10	3265.14 0.50 * Vs 0 -1.25 * 5710.11 0.50 * Vp 0 -1.25 * 480.00	Not inverted

Table 4 .

 4 Initial values and intervals of variation of soil column parameters.

	Layer number	Vs 0 (m/s)	Variations of Vs	Vp 0 (m/s)	Variations of Vp	TH 0 (m)	Variation of TH
	1	220.00	Not inverted	470.00	Not inverted	1.00	Not inverted
	2	1050.00	Vs 0 0.50 * Vs 0 -1.25 *	3160.00	Not inverted	11.00	Not inverted
	3	1380.00	Vs 0 0.50 * Vs 0 -1.25 *	3160.00	Vp 0 0.50 * Vp 0 -1.25 *	22.00	Not inverted
	4	1680.00	Vs 0 0.50 * Vs 0 -1.25 *	3690.00	Vp 0 0.50 * Vp 0 -1.25 *	20.00	Not inverted
	5	1960.00	0.50 * Vs 0 -1.25 *	5290.00	Not inverted	76.00	Not inverted

ACKNOWLEDGEMENTS

Part of this work was carried out when one of the authors (AD) was visiting the Disaster Prevention Research Institute (DPRI) of Kyoto University, at Uji, Kyoto, Japan, supported by the CARNOT Institute funding program. Free distribution of strong motion data by K-NET and KiK-net networks, operated by NIED, Japan, was fully appreciated.


Aude Bouchier

Khaldoun Saleh

Pierre-Henri Merrer

Olivier Llopis

P.-H Merrer

Phase Noise Reduction of Narrow Linewidth Optical Fibre-Ring Based Microwave Oscillators : Modelling and Experimental Results

published or not. The documents may come   L'archive ouverte pluridisciplinaire

The utilisation of optical carriers is of strong interest for microwave applications such as microwave generation [1]. For example, optical resonators can replace microwave resonators and allow to overcome the performance of traditional microwave oscillators in terms of compactness and quality factor, which will strongly reduce the phase noise of the overall microwave oscillator. Whispering gallery modes resonators have been studied and used in this aim [START_REF] Gorodetsky | Ultimate Q of optical microsphere resonator[END_REF][START_REF] Armani | Ultrahigh-Q toroid microcavity on a chip[END_REF]. Whereas they exhibit very interesting characteristics in terms of quality factor, the long-term stability of the resulting microwave oscillator is reduced because of the mechanical instability induced by the fibre-to-resonator coupling systems. An interesting alternative consists in using optical fibres based ring resonators [START_REF] Merrer | High-Q optical resonators for stabilization of high spectral purity microwave oscillators[END_REF]. This kind of set-up presents easily a high stability in time in terms of light coupling, while having great potential for exhibiting high-Q values. It is composed of a single-mode fibre with a length between 10 and 20 m, and two low losses fibered couplers. The choice of the fibre length and the couplers ratios are critical for the optical quality factor of the resonator. A first study of an optoelectronic oscillator (OEO) based on this fibre ring (Fig. 1) has shown that, despite a very high Q-factor of the resonator (about 2.10 8 at 1.55 µm for a length of 10 m), the phase noise of the OEO can be high (Fig. 2) and has to be decreased [START_REF] Merrer | High-Q optical resonators for stabilization of high spectral purity microwave oscillators[END_REF]. We have thus based our optimisation of resonant fibre rings on a theoretical study of the OEO phase noise to understand the influence of the resonator on the noise of the oscillator. This study concerns mainly the phase noise due to the white frequency noise induced by our resonators and is comparable to the one proposed for purely microwave oscillators by J. Everard [START_REF] Everard | Low noise oscillators[END_REF]. Comparisons between modelling and experiments, performed on both the resonator and oscillator noise, have allowed us to determine with accuracy the physical parameters of our first fibre ring, mainly the transmission and losses of its couplers and the intrinsic losses of the ring. An in-depth study of the influence of the couplers and ring losses on the noise performance allowed to conclude that the best performance in terms of optical quality factor does not lead automatically to the lowest phase noise. A global optimum has thus to be found. In the frame of this work, we thus have proposed a new resonant fibre ring design thanks to this analysis. Its quality factor is 5.10 9 at 1.55 µm, for a length of 20 m. We have thus built a new OEO with it. Its phase noise has considerably been improved (Fig. 2). We can observe on this spectrum that the main component of the OEO phase noise is the flicker-like noise. The next step of this study is to understand the origin of this noise in this case and how to reduce it.

1Fig. 2

 2 photodiode


Sylvain Guignot

Nour-Eddine Menad

F Diot

N Kanari

Maurice Save

Philippe Wavrer

Françoise Bodénan

Fabien Thomas

Eric Silvente

Marie-Thérèse Lecler

Valorization of Automotive Shredder Residues in metallurgical furnaces

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Valorization of Automotive Shredder Residues in metallurgical furnaces

Project REFORBA Menad N. * (1) , Save M.

(1) , Wavrer P.

(1) , Bodenan F.

(1) , Guignot S.

(1) , Kleihauser J.

(2) , Buffoni C.

(2) , Russo P.

(3) , Quatravaux T.

(3) , Kanari N. According to the directive n°2000/53/CE 85 % by weight of an end-of-life vehicle (EVL) must be recycled. The current state of practice comprises four steps. Liquids from battery, gearboxes casing, brake fluid, coolant and air conditioning circuit are first collected. Tyres, catalysts, windscreens and windows are then manually removed, prior to the shredding of the car. A magnetic separation is processed on the shredded materials, to recover valuable ferrous components. The remaining part (Automotive Shredder Residue -ASR) undergoes successive separation steps, leading among others to two potentially valuable fractions, one mostly composed of plastics (P 1 ) and the other of iron oxides (P 2 ). Currently, P 2 and P 1 are respectively landfilled or used as a fuel source in various processes. In the blast furnace (BF) and the electric arc furnace (EAF) routes, P 1 could be used as substitute for coal or coke, and P 2 could replace part of the ore, provided some elements (Cl, Cu, Zn) are removed. Chlorinated compounds (PVC) can release toxics (dioxin) or corrosive metal chlorides, detrimental to the installations and the environment. Copper can spoil the cast-iron produced, and zinc favors the development of scabs that modify the flow configuration inside the BF. The whole objective of REFORBA is to assess the possibility of using "purified" fractions P 1 and P 2 in the BF and the EAF, as reducing agents or "iron ore". This would provide steelmakers with raw materials cheaper than coke. As additional potential benefits the amount of CO 2 generated in the processes and the volume of landfilled shredded residues would be lowered. Representative fractions of P 1 and P 2 are obtained by sampling an industrial line of treatment. Comprehensive chemical characterization (ICP-AES, SEM, FTIR, thermal analyses) is performed to know the global content and localization of materials containing chemicals to remove. Sorting operations (grinding, screening, magnetic, eddy current, and sink float separations) are then performed on P 1 and P 2 , until the modeled BF global heat and flow profiles match the classical configuration. The possibility of using P 1 in place of C and CO is evaluated by a lab-scale thermochemical characterization of the iron oxides reduction, and of the quality of the cast-iron processed with some percent of P 2 in the ore feed. This will give estimations of P 1 and P 2 nominal contents to use in the BF, and assess the feasibility of a pilot-scale demonstration.

(1) BRGM (2) Recylux (3) ArcelorMittal (4) Laboratoire Environnement et Minéralurgie -UMR 7569 (5) INRS * Corresponding author: n.menad@brgm.fr