Brigitte Juanals

Jean-Luc Minel

Version Auteur

Keywords: Standardization, text mining, ontologies. 1. INTROUCTION

Many organizations are in charge of global security management. This paper outlines and argues for the construction of a theoretical and methodological framework in order to critically assess the new technopolitics currently being developed in the field of global security and which are materialized in standards. The main purpose is to design both a methodology and specific text mining tools to investigate these standards. These tools will be implemented in a platform designed to provide cartographic representations of standards and to assist the navigation of an end-user through a corpus of standards.

The research developed in this paper relates to the industrial standardization of security and risks, located at the intersection of global security and international standardization. The technical and political aspects of these two areas give rise to technopolitics, which we contend will play a leading role in the field of industrial regulation in the XXIst century, causing major industrial, economic and geopolitical impacts.

Many organizations are in charge of global security management. To address this issue, ISO (International Standards Organization), the main international organization for technical standardization, has launched a set of studies in the wake of the ANSI program "Homeland Security Standard Panel" (law of 2002, November 25), subsequently adopted by European and national standardization organizations. This paper outlines and argues for the construction of a theoretical and methodological framework in order to critically assess the new technopolitics currently being developed in the field of global security and which are materialized in standards. The main purpose is to design both a methodology and specific text mining tools to investigate standards. These tools are designed as heuristics which enable reformulations, semiotic transfer (texts to diagrams), and comparison between texts. This work is part of a scientific research project called "NOTSEG"1 which stands for "Standardization and Global Security, The formulation in Standardization of the global security concept". The partnership includes academic and industrial partners: two research centres, MoDyCo from the University of Paris Ouest Nanterre La Défense -CNRS, CQP2I from the Technological University of Compiègne, AFNOR, the French national standardization body, and the company Sector SA, specialized in decision-making and studies in the area of technological and organisational risks.

The NOTSEG project mission is to draw up the cartography of existing normalization frameworks in the field of security and crisis management. The project comprises several successive stages, namely: i) inventorying the list of standards to be studied; ii) analysing existing standardization frameworks; iii) studying and selecting variables; iv) analysing and monitoring a corpus of standards using text mining tools. The methodology of these tools is the topic of this paper.

Recent evolutions in standards are the core of our analysis. The processes of international industrial standardization apply not only to the artefacts of technical devices (in technical standards), but also, since the year 2000, to organizational methods and the evaluation process of these devices, including state regulations, especially in standards of management. Domains such as "business continuity" and "resilience" now possess standards of 'security management". The rest of this paper is organised as follows. Section 2 explains the theoretical approach and issues. Section 3 details our hypothesis and methodological approach. Section 4 describes the experimentation in progress, section 5 details the platform specifications, section 6 presents a use case, and finally section 6 presents the conclusion.

THEORETICAL APPROACH AND ISSUES

Texts on industrial standards are here considered as means of validating and communicating technical choices, knowledge, and professional practices, as manifested in their cultural and industrial contexts within the wider international and economic situation. We aim to identify, in a specific sector of activity, the conditions and mechanisms that are conducive to new standards contributing to focusing and communicating certain practices, processes, modes of organization and socio-technical arrangements, on the European or international level. In the second step, it will be necessary to investigate whether these new standards really play a role in the evolution of the abovementioned elements. If this is the case, changes in stakeholders' knowledge and professional practices will then be considered.

In international standardization, the process of establishing each standard is dependent on the conditions of production goals and modalities of use anticipated by the standard. The development of standards implies the analysis, case by case, of the means of action and coordination of activities by organizations and communities [START_REF] Thévenot | Un gouvernement par les normes. Pratiques et politiques des formations d'information[END_REF].

In what way do standards contribute to the communication of knowledge in a specific sector of activity? Standards encode knowledge, modes of organization (such as the « Plan-Do-Check-Act model », a widely used corporate model that structures the management process), approaches (to the nature and evaluation of risks, for instance), or procedures that need rethinking.

In the writing process of standards, our aim is to investigate, beyond the institutional display of a "consensual" operation, a pragmatic dimension in a given sector of activity (its technical committee), the goals which govern their construction, the categories of stakeholders involved, the motivations which influence choices (in terms of writing the contents), the participants in workgroups, and the adjustments that take place between stakeholders from different cultures and languages. We wish to point out that these stakeholders have professional practices in the same domain which can be similar, different, complementary but seldom conflicting.

What conceptual and organizational approaches will be foregrounded (acknowledged) in international standards? Will the different concepts and national practices combine to form new standards? This seems highly unlikely as concepts and practices differ, making them to a large extent incompatible. It can be assumed that the standards which are the most widely acknowledged and implemented on the international or national level will be used to develop the international standard in the domain in question. In the development process, competition may arise between influential national standards and standards backed by lobbies (such as ASIS in business continuity activity, for instance). For this reason, following the example of technical standards, corporate standards, called "management standards" are used by consultants in the domain and by companies aiming to apply for certification of their activities. Lastly, the intertextuality and performative dimension of standards, which play a crucial role in their application, remain largely unexplored.

The formatting constraints imposed by standards create the conditions of a « performative » [START_REF] Austin | Quand dire, c'est faire (original title: How to Do Things with Words)[END_REF] that is to say a form of action is present, both intrinsically and extrinsically, in the writing process of these formats.

Hence the importance of:

-monitoring standards during the writing process; -perusing and comparing standards closely related with the domain under the process of standardisation within different technical committees; -confronting and attempting to ensure agreement on the definition of terms and concepts. These three issues are generic and trans-sectorial. In the NOTSEG project, we have chosen BCA (Business Continuity Activity) as the field of experimentation for our methodology but the tools designed are expected to be applicable to other fields.

HYPOTHESIS AND METHODOLOGICAL APPROACH

Theoretically, research in progress deals with the relation between media flow of knowledge and natural language processing, in particular text mining tools.

The qualitative analysis of textual data enables data to be processed in such a way as to reveal the heterogeneity of large text corpora. Information traceability places communities (in terms of professional cultures, lobbies, etc.) and memory at the heart of knowledge flow, thus highlighting the interest of a socio-cognitive and political approach to these issues. Grasping and elucidating the diversity of points of view (political, economical, institutional, national, etc.) of stakeholders, who express their opinions in various spatio-temporal and cultural registers, is a tremendous task.

Our proposal is to combine natural language processing and linguistics with the sociology of science, in order to produce a fruitful analysis of corpora [START_REF]Analyses textuelles en sociologie, logiciels, méthodes, usages[END_REF], since the heterogeneity of the texts imposes complex computing and epistemological processing.

Using concepts from text linguistics [START_REF] Adam | La linguistique textuelle : Introduction à l'analyse textuelle des discours[END_REF] is mandatory if one is to identify enunciative polyphony and to build classifications or cartographies. When combined with insights from the sociology of science [START_REF] Latour | Changer de société. Refaire de la sociologie (Titre original : Re-assembling The Social[END_REF] concerning stakeholders, their institutions and industrial or scientific policies, these representations could provide relevant categorizations of knowledge flows or controversies 2 .

In the context of the NOTSEG project, we focus on the role of text mining tools which can provide a new way to apprehend the complexity involved in comparing large numbers of texts. This kind of approach combines linguistic engineering, knowledge engineering and knowledge communication [START_REF] Juanals | La circulation médiatique des savoirs dans les sociétés contemporaines[END_REF].

In the first stage, we aim to check a methodological hypothesis concerning the comprehensive analysis of a corpus of standards by using qualitative text mining tools. More precisely, we attempt to identify the diversity of stakeholders and their institutions by analysing the tracks they have left in the text.

Characterizing stakeholders profiles.

2 Massachusetts Institute of Technology, « Mapping Controversies »,[www.demoscience.org/resources/index.php] Presently, we have identified several profiles of stakeholders who are likely to provide, prescribe or use standards. A diverse range of practitioners is involved: -editors of standards: their role is to edit standards by comparing them with other existing standards on closely related topics produced by other NGOs or national, European, or international bodies.

-prescribers and their representatives (companies): these are officials (employees or consultants) in charge of making use of a standard in organizations (public or private) and monitoring the impact of these standards on legislation and regulations.

-stakeholders involved in making use of these documents and in the development of the corresponding operations.

-end users and their representatives.

Socio-organizational modelling of actors

This modelling step aims at making the connection between standards and their socio-organizational context based on criteria related to geographical and institutional origins and on working group participants. This categorization has been implemented thanks to the work of AFNOR which carried out a cartography of the people involved.

Institutions include international standardization bodies, European or national standardization bodies, representatives of States (Ministries), NGOs, private actors (from industry and French or European think tanks). Participants in working groups -the editor and expert-members -are also taken into account, as are editors and expert-members belonging to companies. Figure 1 shows the different elements of this modelling in the form of a UML diagram of classes.

Construction of lexical networks: contrastive glossary of lexical or semantic variations

Undertaking semantic and pragmatic analysis should make it possible to build a dynamic glossary (in English and French) in the field of security -limited first to a sub-field of security which accounts for semantic variations of terms and variations in their uses. This glossary is composed of a list of terms systematically defined in standards in the section "terms and definitions".

This work does not aim at imposing a particular point of view or a unified vision of the field investigated. Conducted according to a comparative approach, this semantic and pragmatic analysis of key-terms in the studied standards aims at revealing zones of convergence, divergence, and crosschecking, including controversies, based on criteria which have to be identified. This glossary should account for semantic variations in terms and variations in their contexts of use. To achieve this aim, several possible methods of work are being investigated:

-the hypothesis of a semantic corpus search with the help of text analysis software (for instance, the list of terms defined in the section "terms and definitions" of the standard).

-expert evaluation concerning the identification of key-terms (corresponding to an interpretative choice of terms), -a combination of the two approaches.

Concerning the semantic search hypothesis, the techniques and practices related to "clusterisation" and to graphicparticularly cartographic -representations which derive from it, could enable the construction of networks of categories and concepts, or networks of co-citations to identify actor networks.

Centered on the comprehensive analysis of textual data, computerized qualitative analysis tools can account for the studied corpus in a detailed way and yield a representation in the form of a graph (Lexico3, for instance). The construction of networks of associated words is among the techniques implemented by these tools.

In national, European and international standards, the principle is to analyze documents by comparing language expressions at the level of words, nominal groups, phrases, and paragraphs.

The objective is to understand the meaning of identical language expressions in the international standard and in one or several national or European standards, or, on the contrary, their absence or their relative scarcity.

This work is an attempt to go beyond a debate of opinions and to identify a method which provides metrics for comparing texts, i.e. which makes use of statistical results calculated on the basis of the presence of linguistic markers. In other words, it is possible to conduct an analysis in terms of frequency of occurrence or to display the terms which frequently co-occur in a given standard, for instance, thereby displaying the context of use of a key-word and describing its semantic environment.

Construction of networks of terms and concepts using formal languages RDF/OWL

First, we propose to put at the user's disposal tools which offer a synthesis of the standard's contents in a cartographic or textual form. These new representations are designed to help the appropriation of concepts, ways of thinking, ways of organizing or defining modes of operation, these elements being connected to their socio-organizational context. This work will lead to the construction of a plurality of graphs -each of them representing a point of view -which will take into account the main "concepts" of national, European and international standards.

The originality of this approach is the construction of local networks of concepts, very close of the notion local ontologies, which represent points of view corresponding to semantic worlds of standards studied individually. But while an ontology is often used to produce a unified representation of a field, here it is used to express the point of view of a category of stakeholders and a sector of activity.

The methodology consists in constructing a conceptual and lexical representation for each standard, described for example in terms of concepts, terms, organization, practices and economic competition. This work implies the design of a sociotechnical device (section 5) with the following characteristics:

-This device, which makes it possible to group or contrast concepts and terms, takes the form of ontology described with RDF, SKOS and OWL.

-It is a formal language for the description of concepts and their inter-relations. These networks of terms and concepts is built according to a pragmatic approach, both linguistic and non-linguistic, introducing parameters such as actors, editors, committees, nationalities or contexts (industrial or cultural).

-This technical device uses the semantic variations present in terms associated to texts.

Although an ontology is always a more or less objective representation, the construction of networks of terms and concepts is an essential step in building a conceptual representation.

Comparative analysis of local networks concepts

Then, a comparative analysis of the network of concepts built will be conducted in order to build bridges between them. This analysis will rely on a device which allows navigation between different linguistic expressions of the same concept in all the texts. Some visual support will be provided, with a set of colours for instance, to assist the user.

The goal is to launch the construction of interactive cartographies which facilitate cross-comprehension and provide comparative information about texts of standards and their associated context.

These cartographies will cast light on overlapping, similar, different or convergent text areas between national or international standards.

Another goal is to construct a general ontology from these local ontologies. The ability to identify an abstract concept related to local concepts and their linguistic descriptions in texts of standards would be very useful for monitoring standards.

EXPERIMENTATION

In order to assess our methodology, we have undertaken experimentation in the field of BCA. This experimentation comprises the following steps :

1. Corpus selection: selection of standards in a more restricted field than that of information security. We have selected BCA;

2. socio-organizational inventory of the universe of standardization, in connection with national bodies (AFNOR);

3. Building a glossary: Comparative analysis of texts of standards;

4. Building the conceptual map and local ontologies; setting up a referential of terms with the help of experts; identifying words which are important for the description of professional knowledge or practices;

5. Specification of the platform.

In this paper, only steps 1 and 5 are described.

Corpus selection

PLATFORM SPECIFICATION

One of goals of the NOTSEG project is to specify a platform dedicated to the management of a large corpus of standards, between 15 and 100 texts, applied to a engineering field, such as for instance BCA. This platform, designed to provide a common environment, will be used by two kinds of end-users: first, by consultants and editors of standards to help them during the writing process of a new standard (cf. section 3.1); second, by corporate departments in charge of tackling the implementation of standards in order to comply with national or international regulations (cf. 3.2). Two main ideas underlie our approach: one is to provide cartographic representations of standards, the other to assist the navigation of an end-user through the corpus of standards.

Considering that there is no universal representation independent of the goals and the organizational context, we aim to provide both graphic and textual representations, and several tools enabling comparison between several standards. It must be emphasized that all the representations are interconnected and that the platform provides specific interfaces allowing the end-user to navigate between them. Furthermore, this navigation will be assisted by applying specific knowledge based on the NaviText model [START_REF] Couto | NaviTexte, a Text Navigation Tool[END_REF].

Textual and Graphic Representation

The glossary of the standardized domain is the main textual tool. For each term, semantic and usage variations in the selected corpus are provided and enriched links (see section 5.2 below) can be followed; at any moment, the textual contexts of the same term in two standards can be compared by accessing them in one or several standards. This very simple tool is extremely useful to preserve conceptual coherence during the writing process of a new standard by using the same word to refer to an identical concept or on the contrary by choosing a new word to highlight the creation of a new concept. Graphic representations complete the glossary. As explained in section 3.5, we consider that conceptual maps (or local ontologies) provide a useful level of abstraction, while at the same time keeping and foregrounding the relations between concepts and qualifying their semantics.

For example, relations could be linked to the different phases of the PDCA cycle which governs all the standards.

Assisted Navigation

One of drawbacks of printed standards is that few tools exist for navigating through them, such as indexes and tables of contents. Thanks to digitization, a wide range of possibilities are presently available. The main point is the granularity of the objects accessed. In the NOTSEG platform, the basic granularity is the word. From a structural point of view, words are included in one or more sentences, which are included in a paragraph, which are included in a section and so on. From a text point of view, the frequency and context of a word, and cooccurrence networks, can be computed and linked. From a semantic point of view, the definition of a term in the standard, and the semantics of verbs which co-occur with specific terms, can be automatically annotated with metadata. Finally, from a pragmatic point of view, organizational metadata can be manually added in order to highlight the influence of certain lobbies on the definition of concepts.

All these data and metadata will be used by navigational knowledge [START_REF] Couto | A Linguistic and Navigational Knowledge Approach to Text navigation[END_REF] to afford different means of circulating through the standards. This kind of tool should enhance the intelligibility and comprehension of standards for engineers responsible for implementing them in companies.

Technical issues

The platform must be interoperable and in line with international standards of knowledge representation (RDF, OWL), and offer API in order to cooperate with other software. Two paths have been checked. First, Semantic Turkey (ST) designed by Tore Vergata University [START_REF] Pazienza | Let`s talk about our "being": A linguistic-based ontology framework for coordinating agents[END_REF], which is a "a Firefox based Knowledge Management and Acquisition Platform for the Semantic Web". The main asset of ST is its ability to combine texts and several ontologies and to provide tools to keep track of concepts and their usages in different texts (see figure 2). Furthermore, an experimentation in the legal domain is in progress at the Artificial Intelligence Department of Roma 2.

Second, Protegé designed by Stanford University (http://protege.stanford.edu/doc/) is used worldwide and provides a sturdy background to implement the ontology. Furthermore, an endpoint SPARQL is included to store data and metadata.

After several attempts we decided to build the platform KONTRAST [START_REF] Lafréchoux | création d'un glossaire contrastif à partir d'un corpus de normes internationales[END_REF] relying on Protégée.

The Platform KONTRAST KONTRAST has been designed in order to be able to represent several notional and terminological systems in a unique base and rely on formal languages such RDF, SKOS and OWL. One of the deliverables of NOTSEG project is an XML structured glossary 3 . KONTRAST was automatically populated from the XLM glossary (cf. supra). XSLT templates were written to transform terms and relations into RDF/OWL triples. To ensure consistency, different references of the same entity within the glossary (for example the name of the standard) should have the same unique identifier (URI) in KONTRAST. In spite of the thoroughness of technical writers, some morphological variants occurred in the corpus; they were solved by regular expressions written with the software TextMate [START_REF] Lafréchoux | création d'un glossaire contrastif à partir d'un corpus de normes internationales[END_REF].

During this populating step, relations were automatically computed in order to bind terms which have identical, almost identical or similar definitions but which are not explicitly referenced as such. This proved necessary because in T&D sections, instead of referencing existing definitions, the definitions are frequently quoted in full; this induces certain variants, and as a result they are not identical from a graphical, morphological, and even sometimes from a syntactic, point of view. One reason is that national standard bodies do not have the same typographical and discursive conventions. In order to solve this issue, an algorithm, based on the Levenshtein function, computes the proximity between two definitions and chooses to link these definitions, depending on the computed value, with one of the following three relations :"skos:exactMatch", "skos:closeMatch", "skos:relatedMatch". All these data were then imported into Protégé4, consistency was checked with OWL tools, and ultimately validated by a linguist for the terminological part, and by an expert in the field for the socio-organizational part.

Presently, KONTRAST is an RDF/OWL base stored in a SparqlEndPoint, describing 291 Concepts associated with 633 terms and definitions, 18 standards with their Technical Committees, Working Groups, and several thousands of relations between them. In comparison, the terminological guide ISO 73:2009 which is considered as a reference in this matter describes 51 terms and definitions. Furthermore, complex queries written in Sparql can be carried out to search through the SparqlEndPoint.

Relying on the Protégée plugin "Ontograf", several iconic visualizations (cf. figures 3 and 4) have been developed in order to assist the user navigation in all these terminological and socio-organizational data.

USE CASE

KONTRAST can model all the terminological data from the texts in the corpus together with the socio-organizational data.

With around one thousand individuals (standards, terms, standard bodies), it is a powerful tool for analyzing the BCA field.

One of the assumptions formulated by the NOTSEG project [START_REF] Juanals | A Communicational Analyses of Internal Management Standards in Societal Security, Communication in a World of Norms[END_REF][START_REF] Juanals | Writing and Monitoring in International Standardization, Theoretical Choices and Methodological Tools[END_REF] is that the identification of quotations or borrowing of parts of definitions within the texts of standards can be used to track lobbying networks. Tracing these inheritances back to their source may make it possible to identify clusters of similar standards and thus to clarify the terminological choices which governed the writing of these standards. Exploring relations through KONTRAST is the means we have been using to track down these networks of influence. Let us give an example: an expert suspects the terminological influence of a national standard, NS, over an international one, IS. One way to confirm this suspicion is to search KONTRAST for a couple of terms (t1, t2) with definitions related by: "skos:exactMatch" or "skos:closeMatch", t1 coming from NS, t2 coming from IS under the condition that the publication year of NS is earlier than that of IS. It is very straightforward to write this query in SPARQL language [START_REF] Juanals | Writing and Monitoring in International Standardization, Theoretical Choices and Methodological Tools[END_REF].

We consider that two categories of users could be interested by KONTRAST. First, users we call "Readers": they will use enduser functionalities designed to navigate through the ontological terminological glossary, using iconic and graphic visualizations. Second, users we call 'Explorers": they will use the Sparql language to rummage through the data, searching for complex links to check a hypothesis or to compute and quantify terminological facts. They could also use some reasoners (Fact++, Hermit, etc.) from Protege or other platforms to perform consistency or abstracting checking in order to advance the terminological stabilization of an emerging field.

CONCLUSIONS

We have described a methodology based on hypotheses of how to combine natural language processing tools with a sociology of science approach.

Figure 1 :

 1 Figure 1: Draft representation of the textual, conceptual and semantic context (UML formalism)

Figure 2 :

 2 Figure 2: A screenshot from Semantic Turkey [9]

Figure 3 :

 3 Figure 3: Visualization of a term, its context of usage and its relations with others terms using the same written form

Figure 4 :

 4 Figure 4: Visualization of all terms with the written form « consequence » and their relations.

The NOTSEG Project (www.notseg.fr) is funded by the French National Research Agency (ANR) over a 3-year period (2009-2012).

Malik, M., Construction d'un glossaire contrastif, delivery for ANR-NOTSEG, 2011

We have presented the specifications of a platform which is dedicated to editors, prescribers or end-users in the field of standardization and work in progress in the domain of BCA. This specific study on the texts of BCA standards will be used for a more generic task, as the empirical core of a widerranging, prospectively designed inquiry.

The first step is to identify the main concepts and recommendations (operational, behavioral, technical, etc.) in which these formats are embodied. In the second step, this work on texts, related to the socio-organizational context of their production and communication, should provide information to identify or critically assess the new technopolitics already developed or under development in the field of security and crisis management.

Finally, this work will provide the opportunity to begin exploring, in the industrial domain, ways of thinking about the culture of security and risks which are presently the domain of engineering.


Marc Moniez
email: moniez@lal.in2p3.fr

Positioning and orienting a static cylindrical radio-reflector for wide field surveys

Keywords: Instrumentation, interferometers -Cosmology, large-scale structure of Universe -dark energy -Radio lines, galaxies

Positioning and orienting a static cylindrical radio-reflector for wide field surveys

Introduction

Several baryonic oscillation (BAO) radio projects [START_REF] Peterson | The Hubble Sphere Hydrogen Survey[END_REF][START_REF] Ansari | 21 cm observation of large-scale structures at z ∼ 1: Instrument sensitivity and foreground subtraction[END_REF]3 plan to operate a series of parallel static reflectors of large parabolic cylinder shape, to map the 21cm HI emission line. The sky will transit over the acceptance lobe, which is defined by an angular sector of aperture ∆ centered around a vertical plane (see Fig. 1).

Δ

Figure 1: The reflector and its field of view, defined as the angular sector ∆ that is focalised within the antenna's acceptance.

The projection on the sky of this angular sector can be seen on Fig. 2. In this paper, I produce the exact calculation of the transit time of a celestial object, as a function of its declination. In section 5, I use the results of the calculation to compare the performances of various radio-telescope latitudes (France, Morocco, South Africa, equator) and configurations (orientation). In particular, we show that the North-South orientation usually considered may not be 1 the optimal one for high-z BAO studies that need large exposure times, but not necessarilly the largest possible field of view. Another possible use of the exact expression for the transit time is the production of exact coefficients for Fourier Transform calculations along the cylinder axis. The celestial sphere with the projected position of the observer M0 (latitude λ), the projected orientation of the reflector (A) and the projected portion of detectable sky (detection lobe), defined as the angular sector ∆ of axis P0P 0 (in grey), where P0 and P 0 are the projections of the reflector's axis. θ/2π is the fraction of the sideral day that an object of declination δ will spend within the detection lobe.

Notations

We will use the following notations (see Fig. 2):

• λ is the observatory's latitude,

• M 0 its position on Earth.

• A is the azimuth of the reflector (with respect to the meridian).

• ∆ is the lobe's aperture. A celestial object can be detected only if it enters this lobe.

• P 0 and P 0 are the intersections of the lobe's definition planes on the celestial sphere.

P 0 P 0 M 0 define a large circle on the sphere, with ( P 0 OM 0 ) = ( M 0 OP 0 ) = π/2.

• δ is the declination of a celestial object.

The (sideral) daily exposure of an object is given by the fraction of its corresponding parallel that is included in the acceptance lobe. On Fig. 2, this exposure is given by θ/2π ×1 sideral day. 2

When λ, A and ∆ are defined, it depends only on the declination of the object. From the figure, it can be seen that the daily exposure is in general not uniform for a random choice of λ and A. The objective of this paper is to systematically study the exposure as a function of the declination for any antenna configuration, and to provide an optimization tool.

3 The stereographic projection The geometrical tool used to establish the exposure versus declination function is the stereographic projection, because of the geometrical configuration includes only circles, and mainly large circles (see Fig. 3). Fig. 2 is then projected from the South pole on the equatorial plane (Fig. 4).

The main properties of the stereographic projection that we will use are the following:

• The projection of a circle on the sphere is a circle or a straight line on the plane.

• The projection of a large circle is a circle (or a straight line) that intercepts the equator in 2 diametrally opposite points.

• The projection of a meridian is a straight line that includes the origin.

• Angles between tangents on the sphere are invariant under the projection.

• Lengths and surfaces are not invariant under the projection, but for symmetry reasons, the scaling is constant along a given parallel. The fraction of a parallel that is included in the acceptance lobe will then be invariant under the projection.

On figure 4, the thick circle (of radius 1) is the equator, the crescent is the projection of the lobe, and the blue circle (H) is the projection of the horizon of M 0 (projected on M ). We do not restrict the generality by assuming that the observatory is located in the northern hemisphere (in the reverse case, juste exchange North and South) and that 0 < A < π/2. The projection of the visible part of the sky is then given by the white (not shaded) area (that contains M (0 < x M < 1), projection of M 0 ). The two circles (C 1 ) and (C 2 ) are the projections of the large circles defining the lobe, that intercept each other at P 0 and P 0 . Since P 0 and P 0 are on the same meridian (because they are antipodic), their projections P and P are aligned with the origin. As a consequence of the angle conservation, the projection of the large circle defining the median plane of the lobe intercepts the observer's meridian projection (Ox axis) at angle A.

We define I as the center of P P segment. The horizon circle (H), projection of the large circle horizon of M 0 , contains P , P , and intersects the equator at S(0, -1) and (0, 1) with angle π/2 -λ. Its center V is aligned with IC which is the median of P P a for symmetry reasons.

a This circle is described by P and P when the angle A varies.

λ α Δ A 2 ± Δ 2 ± V (H) H ) (C (C 1 ) (C (C 2 ) (C) (C)
Figure 4: The stereographic projection of Fig. 2. This construction is drawn from point S as follows:

-H is such that ( OSH) = λ.

-M and M , projections of M0 and its antipod M 0 are the intersections of Ox with the circle of radius HS, centered on H.

-C is the intersection of the vertical line containing H and the line defined by M (or M ) and angle A (or -A).

-C is the intersection of the vertical line containing H and the line defined by M (or M ) and angle A + π/2 (or -A + π/2).

-P and P are the intersections of the line OC and the circle C of radius CM centered on C.

-I is the center of P P .

-The horizon circle (H) is centered on V, intersection of 0x with IC.

Useful relations

Our aim is to find the intersections of the projected lobe sides (C 1 ) and (C 2 ) with a given declination circle. In this subsection, we first determine OI, IP and IC that are needed to establish the equations of these circles. Fig. 5 shows some important geometrical relations involving the observer's position and its antipodic point, in the transverse view of the stereographic projection. These relations are valid for any couple of antipodic points. The following series of relations allows to extract the most pertinent parameters of the lobe projection for subsequent calculation of the exposure time.

• OS = 1 (1)
• Let H be the center of M M where M is the projection of M 0 , antipodic of M 0 :

OH = tan λ (2) M' S H M x λ M 0 M' 0 π 2 = 1 [ π 2 ] -+ λ π 4 λ 2 - 0 ( )
Figure 5: Vertical view of the stereographic sphere along the X axis, showing the relations between the projections M and M of two antipodic points M0 (at latitude λ) and M 0 . S is the south pole.

OM 2 + OS 2 = M S 2 OM 2 + OS 2 = M S 2 M S 2 + M S 2 = M M 2
The two first relations reported in the third one give

M M 2 = OM 2 + OM 2 + 2OS 2 which is also equal to M M 2 = (M O + OM ) 2 = OM 2 + OM 2 + 2.OM.OM It follows that OM × OM = OS 2 = 1.
because H is the center of the circle circumscribing the M SM right triangle (Fig. 5), implying that HS = HM and therefore ( HM S)

= ( M SH) = π/4 + λ/2 and ( OSH) = ( M SH) -( M SO) = λ.
• It also follows from this that

HM = HS = 1/ cos λ (3)
• As the horizon circle (H) intersects the equator with angle π/2 -λ, it is tangent to HS; then, taking into account that OS = 1, its center V is such that

OV = cot λ (4)
and its radius is V S = 1/ sin λ.

(5)

• Let C be the center of circle (C) projected from the lobes' median large circle. i) (C) includes M , M , P and P , then its center C belongs to the median of M M (defined by HC on Fig. 4).

ii) The angle between Ox (projection of the meridian) and the tangent of (C) at M is A, by virtue of the angle conservation. CM is then orthogonal to that tangent (see Fig. 4). It follows that :

HC = HM tan A = 1 cos λ tan A . (6)
• We also define C (bottom-left in Fig. 4) as the center of the circle projected from the large circle perpendicular to the lobe at M 0 . i) This projected circle includes M and M , then C also belongs to the median of M M defined by HC.

ii) Since this large circle is orthogonal to the lobe, its projected circle is orthogonal to (C) at M ; it follows that C M is tangent to (C) at M (see Fig. 4).

iii) Since P 0 and P 0 are the poles of this large circle, for symmetry reasons, this circle intersects the P 0 P 0 meridian with right angle. It follows that P P (aligned with O because P 0 and P 0 are on the same meridian) is perpendicular to the projected circle, and subsequently aligned with its center C (see Fig. 4). It follows that :

HC = HM. tan A = tan A cos λ . (7)
•

CC = HC + HC = 1 cos λ tan A + tan A cos λ = 1 cos λ cos A sin A . ( 8
)
• The angle α, as marked on Fig. 4 will be useful for subsequent calculations.

HC = OH tan α = tan λ tan α (using(2)). (9)
Combining with (7), one obtains tan A = sin λ tan α.

(10)

• CC 2 = CI 2 + C I 2 and C I/CI = tan α = tan A/ sin λ and (8) => IC 1 + tan 2 A/ sin 2 λ = 1 cos λ cos A sin A . (11)
Then

IC = tan λ sin A cos A 1 √ sin 2 λ + tan 2 A ( 12
)
and

IC = IC tan α = IC tan A/ sin λ = 1 cos λ cos 2 A 1 sin 2 λ + tan 2 A . ( 13
)
• Using relations ( 2) and (7) one finds:

OC = OH 2 + HC 2 = tan 2 λ + tan 2 A cos 2 λ = sin 2 λ + tan 2 A cos λ . (14)
• OI = IC -OC . Using (13) and ( 14), one obtains, after simplification:

OI = cos λ sin 2 λ + tan 2 A . ( 15
)
• P and P are the images of antipodic points, equivalently to M and M . Using Fig. 5 the relation:

OH 2 + OS 2 = HS 2 = HM 2 can be transposed as OI 2 + 1 = IP 2 . Then IP = 1 + OI 2 = 1 cos A 1 sin 2 λ + tan 2 A . ( 16
)

Expression of the exposure time

The final calculations are made in the rotated frame (XoY) as shown in Fig. 6. The parameters we will use are OI, ∆ and φ that is given by

tan φ = IC/IP = tan λ sin A (17)
(from (12) and ( 16)). The angle ∆ between the lobe's side circles is invariant under the projection, and is shown in Fig. 6. To find the fraction of a sideral day spent by an object at declination δ within the lobe acceptance, one needs to find the intersections of the declination circle (C D ) with the images of the large circles (C 1 ) of center

C 1 (X 1 , Y 1 ) and (C 2 ) of center C 2 (X 2 , Y 2 )
. The equations of (C D ) and (C 1 ) are:

X 2 + Y 2 = tan 2 (π/4 -δ/2) (18) (X -X 1 ) 2 + (Y -Y 1 ) 2 = P C 2 1 (19)
δ d e c li n ati on c ir c l e

P 0 P' P' I S' X Y C 2 C 1 C δ ϕ γ 1 θ 2 θ 1 Δ (C (C 1 ) ( C ( C2 ) (H) H ) (C D ) V γ 2
Figure 6: The projections of the detection lobe (in grey), the horizon (H, blue circle) and a declination circle (CD in red) in the rotated frame (see text). The radius of the declination circle is constructed from S and δ similarly to the construction of OM from S and λ in Fig. 5.

(19) => X 2 + Y 2 -2(XX 1 + Y Y 1 ) + X 2 1 + Y 2 1 = P C 2 1 (20) => tan 2 (π/4 -δ/2) -2(XX 1 + Y Y 1 ) + OI 2 + IC 2 1 = P C 2 1 = P I 2 + IC 2 1 (21) => tan 2 (π/4 -δ/2) -2(XX 1 + Y Y 1 ) = P I 2 -OI 2 = 1 (f rom (16)) (22)
using polar coordinates

X = R. cos θ 1 X 1 = OC 1 . cos γ 1 (23) Y = R. sin θ 1 Y 1 = OC 1 . sin γ 1 (24) (18) => R = tan(π/4 -δ/2) (positive because -π/2 < δ < π/2) and (25) (22) => tan 2 (π/4 -δ/2) -2 tan(π/4 -δ/2) × OC 1 .(cos θ 1 cos γ 1 + sin θ 1 sin γ 1 ) = 1 (26) => tan 2 (π/4 -δ/2) -2 tan(π/4 -δ/2) × OC 1 . cos(θ 1 -γ 1 ) = 1 (27) => cos(θ 1 -γ 1 ) = tan 2 (π/4 -δ/2) -1 2 tan(π/4 -δ/2) × OC 1 = -tan δ OC 1 (28)
using the formula of the half-angle tangent. γ 1 is given by:

tan γ 1 = IC 1 /OI = IP tan(φ -∆/2) OI = tan(φ -∆/2) cos λ cos A , (29)
using ( 15) and ( 16). OC 1 is given by (using also (15)):

OC 2 1 = OI 2 + IC 2 1 = OI 2 (1 + tan 2 γ 1 ) = cos 2 λ sin 2 λ + tan 2 A 1 + tan 2 (φ -∆/2) cos 2 λ cos 2 A ( 30
)
which can be written:

OC 2 1 = cos 2 λ cos 2 A + tan 2 (φ -∆/2) 1 -cos 2 λ cos 2 A . (31)
It follows

cos(θ 1 -γ 1 ) = -tan δ 1 -cos 2 λ cos 2 A cos 2 λ cos 2 A + tan 2 (φ -∆/2) . ( 32
)
The expression for the searched angles is given by (0, 1 or 2 solutions):

θ 1 = arctan tan(φ -∆/2) cos λ cos A ± arccos -tan δ 1 -cos 2 λ cos 2 A cos 2 λ cos 2 A + tan 2 (φ -∆/2) ( 33
)
where

tan φ = tan λ sin A => tan(φ -∆/2) = tan φ -tan(∆/2) 1 + tan φ tan(∆/2) = tan λ -sin A tan(∆/2) sin A + tan λ tan(∆/2) ( 34
)
Choice of determinations:

-The fact that λ > 0 and 0 < A < π/2 implies that the determination of the arctan (for angle γ 1 ) is between -π/2 and +π/2.

-As the two solutions correspond to the two determinations for the arccos, the choice of the first determination can be made between 0 and π.

Exchanging -∆ into +∆ and γ 1 into γ 2 gives the corresponding result for θ 2 .

Conditions of observability

An object with declination δ is observable if i) there is a solution for θ 1 or θ 2 , and if ii) this solution corresponds to a configuration above horizon, i.e. if the associated point on the sphere of Fig. 2 belongs to the half-sphere of pole M 0 . The stereographic projection of the limit of this half-sphere is the horizon circle (H). The intersections of (C D ) with (C 1 ) or (C 2 ) on the projection correspond to the visibility limits if they are inside (H), that contains the hatched lobe defined by P P and M .

-The first condition (existence of at least one solution) can be expressed by:

| tan δ| < cos 2 λ cos 2 A + tan 2 (φ + ∆/2) 1 -cos 2 λ cos 2 A ( 35
)
or equivalently

(1 -cos 2 λ cos 2 A) tan 2 δ < cos 2 λ cos 2 A + tan 2 (φ + ∆/2) (36)
-The second condition (visibility) is satisfied if the intersection is within the disk centered on V with radius V P = 1/ sin λ, corresponding to the inequality relation:

(X -OI) 2 + (Y + IV ) 2 < 1/ sin 2 λ, (37)
with IV = OI tan α = OI tan A/ sin λ (from Fig. 4 and (10)).

In polar coordinates (R, θ), this condition, applied to the intersection points, becomes

X 2 + Y 2 + 2(Y.IV -X.OI) + OV 2 < 1/ sin 2 λ <=> (38) tan 2 (π/4-δ/2)+2 tan(π/4-δ/2) cos λ sin 2 λ + tan 2 A ( tan A sin λ sin θ -cos θ)+cot 2 λ < 1/ sin 2 λ (39)
using ( 15), ( 4) and R = tan(π/4 -δ/2). After simplification, one gets:

( tan A tan λ sin θ -cos λ cos θ) 1 sin 2 λ + tan 2 A < 1 -tan 2 (π/4 -δ/2) 2 tan(π/4 -δ/2) = tan δ (40)
using again the half-angle tangent formula. As λ > 0, one obtains finally the condition:

tan A sin θ -sin λ cos θ < tan δ tan λ sin 2 λ + tan 2 A.

(41)

Exposure time calculation

After establishing the list of lobe-crossings that are visible (above horizon), one has to distinguish different relative configurations of the declination circle with respect to the lobe (the illustrations of the next section may help the reader at this stage) :

• No lobe-crossing (0 solution): The declination circle is completely inside or outside the lobe (shaded area). Assuming λ > 0, the daily exposure is 24 sideral hours if δ > 0 and if the North pole (projection O) is within the lobe. The pole is within the lobe if C is not between C 1 and C 2 (see Fig. 6), condition expressed by:

(tan φ -tan(φ -∆/2)) × (tan φ -tan(φ + ∆/2)) > 0. (42)
Otherwise, the exposure time is zero.

• Lobe-crossings happens and the pole is NOT in the lobe: The exposure time at a given latitude is obtained by ordering the list of 0 < θ 1 < 2π and 0 < θ 2 < 2π values that satisfy the visibility condition (41) by increasing order (θ(i), i=1 to 4 at maximum) from zero, and account for the value (θ(i + 1) -θ(i))/2π × 1 sideral day per lobe-crossing.

• Lobe-crossings happens and the pole is in the lobe: In this case the θ = 0 point of the declination circle is within the detection lobe. The list of θ 1 and θ 2 that satisfy the visibility conditions has to start with the largest value (between 0 and 2π), followed by the others by increasing order from zero. Then the exposure time is obtained by the sum of values ((θ(i + 1) -θ(i))/2π × 1 sideral day starting from i = 1.

Some particular cases

• A = 0, antenna oriented North-South (Fig. 7a). φ = π/2 and (33) simplifies into

θ 1 = arctan cot(∆/2) cos λ ± arccos   -tan δ sin λ cos 2 λ + cot 2 (∆/2)   (43)
-If δ < λ -π/2, the object is not visible.

-If λ -π/2 < δ < π/2 -λ, then the object enters the visibility lobe once per day during the exposure time

t exp = 1 day π   -arctan cot(∆/2) cos λ + arccos   -tan δ sin λ cos 2 λ + cot 2 (∆/2)     (44) Y X 0 (C (C 1 ) (C (C 2 ) P Δ M I P' P' δ d e c l i n a ti on (H) H ) Y X 0 P Δ M (C (C 1 ) (C (C 2 )
δ d e c l i n a ti on E q u at or (H) H ) using the positive determinations for the arctan and the arccos in this expression.

-If δ > π/2 -λ and tan δ <

√ cos 2 λ+cot 2 (∆/2) sin λ
, the object is circumpolar and enters the visibility lobe twice per day during the total exposure time given by:

t exp = 1 day π   2 arccos   -tan δ sin λ cos 2 λ + cot 2 (∆/2)   -π   = 1 day π   π -2 arccos   tan δ sin λ cos 2 λ + cot 2 (∆/2)     . ( 45
) -If tan δ > √ cos 2 λ+cot 2 (∆/2) sin λ
(which is close to the condition δ > π/2 -∆/2 if ∆ is small), the object is near the pole and is always in the visibility lobe.

• If A = 0 and λ = 0 (antenna on the equator), then the lobe is defined by two half-lines (Fig. 7b), and the full sky is visible with uniform daily exposures t exp = ∆/2π × 1 sideral day.

• A = π/2, antenna oriented East-West (Fig. 8a). φ = λ and (33) simplifies into

θ 1 = π/2 ± arccos -tan δ tan(λ -∆/2) (46)
From Fig. 8a we find that the visibility conditions are simply min(0, λ -∆/2) < δ < λ + ∆/2.

• λ = π/2 (antenna at the north pole).

The lobe can be seen on Fig. 8b. φ = π/2 and (33) simplifies into

θ 1 = π/2 ± arccos [-tan δ. tan(∆/2)] . ( 47
)
If δ > π/2 -∆/2 then the object is always in the lobe; if δ < π/2 -∆/2 then the daily exposure is given by

t exp = (1 -2 π arccos [tan δ tan(∆/2)]) × (1 sideral day).

Study of configurations

In this section, we study the impact of several orientations of a static cylindrical reflector on the field coverage and on the daily exposure time. BAO low-z studies should benefit from the widest coverage, favoured by the North-South (A = 0) orientation of the cylinder axis; but highz studies would need long exposures of low synchrotron background temperature fields, to allow deeper observations, a sensitivity that could be easier to reach with a different axis orientation. Fig. 9 shows the map of foreground galactic synchrotron emission 4 at ∼ 74cm b . This foreground has to be considered together with the exposure maps given below, in order to optimize the position and azimuth of the antenna. b http://lambda.gsfc.nasa.gov/product/foreground

Nançay

• For A = 0 • , 21500 square degree (52%) of the sky are covered with a daily exposure larger than 300s, and 2000 square degree (5%) are covered with an exposure larger than 1500s.

• For A = 45 • , 17800 square degree (43%) of the sky are covered with a daily exposure larger than 300s, and 2800 square degree (7%) are covered with an exposure larger than 1500s.

• For A = 90 • , 12200 square degree (30%) of the sky are covered with a daily exposure larger than 300s, and 3900 square degree (10%) are covered with an exposure larger than 1500s. RIGHT: field covered by the antenna as a function of the minimum daily exposure and the maximum synchrotron sky temperature. For example, for A = 45 • (center), the point of coordinates (15min., 30K) on the 3000 square degree curve means that 3000 square degrees of field with a sky background temperature below 30K transit during more than 15min. per sideral day in the instrumental lobe. These series of curves allows one to study the compromise between the field of view, the background level, and the exposure time.

Morocco

Fig. 11 (left) gives the exposure time for an antenna with a ∆ = 2 • lobe, located in central Morocco (latitude 33 • ) as a function of the galactic coordinates for different orientations. Fig. 11 (right) gives the field covered by the antenna with a daily exposure exceeding the abscissa-value and a sky synchrotron temperature lower than the ordinate-value.

• For A = 0 • , 28200 square degree (68%) of the sky are covered with a daily exposure larger than 300s, and 1150 square degree (3%) are covered with an exposure larger than 1500s.

• For A = 45 • , 22200 square degree (54%) of the sky are covered with a daily exposure larger than 300s, and 2100 square degree (5%) are covered with an exposure larger than 1500s.

• For A = 90 • , 9600 square degree (23%) of the sky are covered with a daily exposure larger than 300s, and 4200 square degree (10%) are covered with an exposure larger than 1500s.

South Africa

Fig. 12 shows the exposure time and the field coverage for an antenna located at Hartebeesthoek Radio Astronomy Observatory (South Africa). • For A = 0 • , 31600 square degree (77%) of the sky are covered with a daily exposure larger than 300s, and 760 square degree (2%) are covered with an exposure larger than 1500s.

• For A = 45 • , 24300 square degree (59%) of the sky are covered with a daily exposure larger than 300s, and 1600 square degree (4%) are covered with an exposure larger than 1500s.

• For A = 90 • , 8100 square degree (20%) of the sky are covered with a daily exposure larger than 300s, and 4300 square degree (10%) are covered with an exposure larger than 1500s.

Equator

Fig. 13 shows the exposure time at the equator. In this case, the exposure time is uniform within the complete observable field, whatever be the azimuth of the antenna (but the observable field varies with A, see Fig. 13). For A = 0 • , the daily exposure time is 480s on the full sky. For A = 45 • , 29000 square degree (71%) of the sky are covered with a daily exposure time of 679s.

Conclusions

The purpose of this study is to provide the exact expression of the transit time of a given celectial object within the lobe of a cylindrical reflector. Each particular case has been examined and the results have been used to analyse different telescope configurations. It is clear from this study that the groups planning to use a static setup of cylinders should seriously consider the orientation as a degree of freedom to favour either the largest field coverage with the shortest mean transit time (North-South orientation), or a smaller field coverage, but allowing a deeper survey (East-West orientation).

Figure 2 :

 2 Figure 2:The celestial sphere with the projected position of the observer M0 (latitude λ), the projected orientation of the reflector (A) and the projected portion of detectable sky (detection lobe), defined as the angular sector ∆ of axis P0P 0 (in grey), where P0 and P 0 are the projections of the reflector's axis. θ/2π is the fraction of the sideral day that an object of declination δ will spend within the detection lobe.

Figure 3 :

 3 Figure 3: Stereographic projection from the North pole.

Figure 7 :

 7 Figure 7: (a) The projected lobe when the antenna is oriented North-South (A = 0 • ). (b) The particular case of the equatorial location (λ = 0 • and A = 0 • ).

Figure 8 :

 8 Figure 8: (a) The projected lobe when the antenna is oriented East-West (A = π/2). (b) The particular case of the polar location (λ = π/2 and A = π/2).

Figure 9 :

 9 Figure 9: The Haslam map of the synchrotron galactic emission at 408 MHz (galactic coordinates).

Fig. 10 (

 10 Fig. 10 (left) gives the exposure time for an antenna with a ∆ = 2 • lobe, located at Nançay (France) as a function of the galactic coordinates for different orientations. Fig. 10 (right) gives the field covered by the antenna with a daily exposure exceeding the abscissa-value and a sky synchrotron temperature lower than the ordinate-value.

Figure 10 :

 10 Figure 10: Antenna located at Nançay latitude (France). LEFT: Sky visibility (luminosity proportional to the Exposure time) as a function of galactic coordinates. From top to bottom: antenna azimuth A = 0 • (North-South), A = 45 • and A = 90 • (East-West).RIGHT: field covered by the antenna as a function of the minimum daily exposure and the maximum synchrotron sky temperature. For example, for A = 45 • (center), the point of coordinates (15min., 30K) on the 3000 square degree curve means that 3000 square degrees of field with a sky background temperature below 30K transit during more than 15min. per sideral day in the instrumental lobe. These series of curves allows one to study the compromise between the field of view, the background level, and the exposure time.

Figure 11 :

 11 Figure 11: Antenna located in central Marocco latitude. LEFT: Sky visibility (luminosity proportional to the Exposure time) as a function of galactic coordinates. From top to bottom: antenna azimuth A = 0 • (North-South), A = 45 • and A = 90 • (East-West). RIGHT: field covered by the antenna as a function of the minimum daily exposure and the maximum synchrotron sky temperature.

Figure 12 :

 12 Figure 12: Antenna located at Hartebeesthoek Radio Astronomy Observatory latitude (South-Africa). LEFT: Sky visibility (luminosity proportional to the Exposure time) as a function of galactic coordinates. From top to bottom: antenna azimuth A = 0 • (North-South), A = 45 • and A = 90 • (East-West). RIGHT: field covered by the antenna as a function of the minimum daily exposure and the maximum synchrotron sky temperature.

Figure 13 :

 13 Figure 13: LEFT: the exposure time as a function of the antenna azimuth A (from 0 to 90 • ) and the declination (from -90 • to 90 • ) in the particular case of the equatorial location (λ = 0 • ). RIGHT: exposure time as a function of galactic coordinates when the antenna has azimuth A = 0 • (North-South, up) and A = 45 • (down).