Nicolas Misdariis
email: nicolas.misdariis@ircam.fr

Patrick Susini

Olivier Houix

Rivas Roque

Clement Cerles

Eric Lebel

Alice Tetienne

Aliette Duquesne

Misdariis Nicolas

Susini Patrick

Houix Olivier

Cerles Clement

Tetienne Alice

Duquesne Aliette

Mapping Sound Properties and Oenological Characters by a Collaborative Sound Design Approach -Towards an Augmented Experience

Keywords: sound design, codesign, taste, methodology, tools

published or not. The documents may come

Introduction

The present project comes within the broad scope of crossmodal correspondences, i.e. the synesthetic associations that people tend to operate between different sensory modalities. In the literature, several studies aimed at describing or investigating the psychological mechanisms and the rationale of such associations [START_REF] Spence | Crossmodal correspondences: A tutorial review[END_REF]. The global aim of most of them is finally to understand how the percept in one modality can be interpreted or altered by sensory information given in another modality.

In the auditory domain, examples addressing that issue can be described, among others, with the following questions: what is the sound of a big/small or sharp/rounded objects? (e.g., the 'booba-kiki' effect studied by McCormick [START_REF] Mccormick | Sound to Meaning Mappings in the Bouba-Kiki Effect[END_REF]). How does blue, red or yellow sound (e.g., the music/color correspondence, especially formalized by Kandinsky [START_REF] Kandinsky | Du spirituel dans l'art[END_REF])? To what musical timbre a given flavor can be associated with (e.g., bitter paired with the French horn and sweet with the clarinet [START_REF] Crisinel | As bitter as a trombone: Synesthetic correspondences in nonsynesthetes between tastes/ flavors and musical notes[END_REF] [START_REF] Crisinel | A sweet sound? Exploring implicit associations between basic tastes and pitch[END_REF]). Or, quoting Charles Spence, can you taste the music? In other words, how music can influence the experience of taste, and change the personal emotional state?

In that domain, the "complex tasting experience that is drinking wine" [START_REF] Spence | Wine and music (II): can you taste the music? Modulating the experience of wine through music and sound[END_REF] is especially focused. Not long ago, some works studied different interactions between what we drink and what we hear, among which the effect of different music styles on basic sensations (fresh, powerful, soft) during the taste of wine [START_REF] North | The effect of background music on the taste of wine[END_REF], or the perceptual and cognitive mechanisms underlying sensory modulations due to cross-modality [START_REF] Spence | Wine and music (I): on the crossmodal matching of wine and music[END_REF].

In that scope, we recently conducted a long-term project (2017-18) with a famous French Champagne producer: Maison Krug. The goal of the project was to realize a mapping between sound properties and oenological characters in order to guide a sound design process and create an augmented multisensory tasting experience. This experience associates sound pieces with different types of wine coming from different regions. The challenge here was to understand the oenological concepts and transcribe them into sonic properties used, afterwards, for sound composition.

To do that, we worked in collaboration with members of the Krug winemaking team (Eric Lebel, the Cellar Master or Chef de Caves, and Alice Tetienne, Winemaker) and a music composer (Roque Rivas). Moreover, the project aimed at being implemented in a dedicated room -La Salle des 400 vins -where a specific multi-channel sound diffusion system has been designed and installed in order to render spatial properties of the sound production (see Fig. 1 and Sec. 4.2 for details). The main claim of this work concerns the concept of semantic transformation and one dedicated mean to achieve it, the collaborative design approach.

Semantic transformation is a concept that has been initially formalized in the visual domain by Karjalainen and Snelders [START_REF] Karjalainen | Designing visual recognition for the brand[END_REF]. It is a translation operation that addresses the issue of transcription of intentions. It relies on the association between words attached to given intentions (e.g. Brand identity) and words able to deliver design insights. Usually, semantic transformation are supported by mediations tools [START_REF] Boujut | Intermediary objects as a means to foster co-operation in engineering design[END_REF] like moodboards, card sets, etc. and are implemented within methodological frameworks.

Collaborative design -or codesign -can precisely be a relevant method to implement semantic transformation. It is a creative methodology based on a participatory approach that started to emerge in the late nineties [11][12]. It starts from the assumption that end-users are the experts of their own activity, so that they should be actively involved in the design process [START_REF] Darras | Design du codesign Le rôle de la communication dans le design participatif[END_REF]. It is applied in several domains going from engineering to education, through design or arts [START_REF] Côté | Utilisation du Codesign dans la formation d'ingénieurs: exemple de projets en acoustique[END_REF].

In the sound design domain, semantic transformation and codesign have recently been studied and applied in a long-term research within a sound branding issue. In that frame, tools and methods were developed to convey sound identity and build corporate sounds [START_REF] Carron | Designing sound identity: providing new communication tools for building brands corporate sound[END_REF] [START_REF] Carron | Méthodes et Outils pour Définir et Véhiculer une Identité Sonore[END_REF]. That work aimed at making a link between the semantic definition of a Brand (Brand-words) and semantic descriptions of sound characteristics (sound-words), in order to provide sound design recommendations.

The present study is directly inspired from this process. It tends, this time, to implement a semantic transformation between oenological identities (wine-words) and sound-words, in order to give insight to sound design composition.

The article relates the workflow implemented to conduct this project. In a first stage, two expert groups (wine and sound) learnt from each other and passed on their respective knowledge and skills (Sec. 2). In a second stage, a codesign process is implemented and resulted in mapping strategies between wine-words and soundwords (Sec. 3). In a third stage, sound-words are transformed into composed sound pieces to illustrate oenological characters and transcend the tasting experience (Sec. 4). Then, after a conclusion, we open to perspectives, especially regarding evaluation, and finally reposition the whole project into a global conceptual framework.

Expertise Sharing

During the first stage, wine and sound experts learnt from each others. This stage was motivated by the participatory methodology implemented in the project. In fact, as the protagonists involved ought to work together within a collaborative framework, it appeared necessary to share a common expertise and language to speak about wines and sounds in order to elaborate efficient recommendations for the sound creation.

Speaking about wine

Vocabulary used to describe wine characteristics is quite huge. This is first due to the fact that wine tasting involves several sensory modalities corresponding to different sensory operations: we look at wine (sight), we smell it (smell) and we taste it (taste).

Each step brings specific information on wine. For instance, sight informs on color, intensity or viscosity (superficial aspects), smell informs again on intensity but also on flavors, and taste informs on mouth flavors (aroma, bouquet, etc.), balance or length in mouth. Each of these dimensions gets specific terminology and represents a semantic profile by itself. The visual analysis uses words like clear/blurry, brilliant/dull, fluid/thick, pale/intense, etc., together with all the shades of red (purple, burgundy, ruby, …) or white (colourless, yellow, golden, …) colors. The olfactory analysis uses words like closed/opened, poor/strong, etc. together with all the families of odors. The gustatory analysis uses words like soft/nervous, bitter, flexible/heavy, fleshy, velvet, short/long, etc. together with all the families of flavors.

This massively polymorphic character of wine comes mainly from the fact that it results from complex mechanisms (terroir, soil composition, sunshine, fermentation, conservation, etc.) occurring all through the production of the liquid that will become, in fine, a wine or a Champagne. Precisely for Champagne -and especially in the Krug traditional process -this complexity is amplified by two elements: i) at early stage of production, a Champagne is a blend of several elementary wines -called 'vins clairs' -that the Chef de Caves mixes together to build the Cuvées of the year; ii) vinification, and especially effervescence (formation of bubbles), takes at least seven years to be completed, a period during which the liquid inside the bottle goes on evolving according to its oenological nature.

On that basis, it was really ambitious and utopic to learn how to speak and taste about wine in the frame of the project -also considering that it takes a life of learning and practice to become a professional winemaker! Nevertheless, the Krug team made the task easy by, first, opening the doors of an internal tasting session and, second, delivering a simplified (but relevant) nomenclature of their oenological references.

The tasting session was a regular meeting of the Krug team (5 winemakers) dealing with the characterization of 15 yearly samples of 'vins clairs'. This kind of session is done twice a year: from October to December, after the grape harvest, and from February to March, before the Champagne creation. It is done for all the 'vins clairs' collected in that particular year, each of them corresponding to a specific grape variety ('cepage'), vineyard ('cru'), and even parcel ('parcelle'). All the tasting notes are registered in a repository document (the 'Krug black book') that also compiles notes from previous years. It is used during the blending process as a reminder of tasting notes (gustatory sensations) that have been previously produced.

The simplified nomenclature concerned the gross regions of growing included in the overall certified Champagne Region (East of France, around the city of Reims). In fact, after the first steps of the project, it was assumed with the Chef de Caves that this level of description was a good compromise between relevance (regions with indeed specific characters) and feasibility (number of regions compatible with sound design capability and sensitivity). Thus, 10 regions were defined (e.g. 'Montagne Reims Nord', 'Cote des Blancs', etc.) and specified with words in a 3-class typology: i) oenological cursors, i.e. six words that forms a standard grid for the oenological tasting at Krug (e.g., 'structure' or 'expression'); ii) additional terms, i.e. words that can be freely added by anyone in the winemaking team (e.g., 'roundness' or 'liveliness'); ii) additional marks, i.e. words that rather correspond to metaphoric associations or affective evocations. For that latter category, it is worth noticing that musical/instrumental metaphor were often used (e.g., violin, trumpet, marimba, etc.) Outputs of this first section was then a table of 10 regions, each described by a group of words (20 in average) structured in 3 categories going from standard dimensions to free metaphors or associations, and a sound design team -including the composer R. Rivas -that was less novice about the semantic world and process of wine tasting.

Speaking about sounds

The second section consisted in the dual approach: learn novices in sonic issues (the Krug team) how to listen to sound and speak about them.

For that, we started from the research undertaken by [START_REF] Carron | Méthodes et Outils pour Définir et Véhiculer une Identité Sonore[END_REF] in the domain of sound branding and from which the present study is inherited [START_REF] Carron | Méthodes et Outils pour Définir et Véhiculer une Identité Sonore[END_REF]. In fact, based on an analysis of several articles related to sound semantic description, Carron et al. (2017) proposed a list of common words related to sound features independent of the meaning, the process that produced the sound, or its location. Words are related to the sound itself, its acoustical characteristics and timbre features rather than illustrative analogies. Then, a lexicon -called afterwards SpeaK -that includes a list of 37 words was developed as an application displaying each word with a definition and sound examples in different categories (musical instruments, voices, environmental sounds, etc.). Within Carron's work, this operational tool was used as a training environment before a sound indexing task but also as a support for codesign sessions [START_REF] Carron | Speaking about sounds: a tool for communication on sound features[END_REF].

In the present work, the SpeaK tool was precisely used to introduce the wine experts (Krug team) to the world of sounds -and related words. Beforehand, the tool was improved in the light of the collaboration with the composer R. Rivas. A preliminary working session was organized in order to refine, and if needed extend, the lexicon.

This session actually gathered two composers (R. Rivas and Frederic Le Bel), three researchers (the 3 first authors of the article) and the sound designer who initially composed sound examples for the first version of SpeaK (Thomas Rotureau). During the session, the 22 semantic scales of the lexicon (15 bipolar scales + 7 single words) were methodically discussed with regard to the precision of the definition and the relevance of the sound examples. According to the latter point, the two composers were previously asked to prepare and bring for this session, alternative examples able to complement or improve the existing ones.

The global outcome of this session was a new release of SpeaK with 3 new elements: i) a simpler and user-friendly interface; ii) a 5-class generic structure of the sound examples (music, voice, environment, sound effects and basic synthesis); iii) if need be, new sound examples able to improve the quality of the illustration. This current version of SpeaK was then used during a training session with the Krug team directly involved in the project: two members of the winemaking team -among whom the 'Chef de Caves' -, and the international marketing and communication director who also contributed to the collaborative process throughout the project.

The learning stage was inspired from Carron's experimental approach [START_REF] Carron | Méthodes et Outils pour Définir et Véhiculer une Identité Sonore[END_REF], and especially used the same training sound corpus. After some adjustments, the learning test included 4 exercises. An ownership period (45 min.) by a free browsing inside the lexicon. A first individual task (20 min.) that consisted in choosing one sound among five (5-AFC) with regard to a given semantic attribute -e.g. "choose the sound that is the most fluctuant". A second individual task (20 min.) that consisted in selecting one attribute among 10 words (a reduced list of the 37 words) with regard to a given sound. Answers from these two tasks were collectively discussed in order to share everyone's view. Finally, a third collective task (20 min.) consisted in the free description of a sound with 3 to 5 words from the lexicon. This last task allows the participants to agree on the perceptive qualities associated with a term in the lexicon.

Outputs of this second second learning section was, firstly, an adopted tool to support the semantic description of sounds, that will be used thereafter supporting the codesign session (Sec. 3), and secondly, a group of wine experts provided with expert language on sound which should refine perceptive capacities, enriches sensory exploration, and facilitates information selection, identification and comparison.

Collaborative Design

Then, after having investigated successively the wine and sound world (and words), we implemented a collaborative sound design (codesign) approach that basically consisted in going from the wine-words to the sound-words with a methodological process. This was done to give insights to the sound designer, so that he would be able to create and arrange the most relevant sound matters and forms.

Apparatus

The codesign environment was also transposed from Carron's work [START_REF] Carron | Méthodes et Outils pour Définir et Véhiculer une Identité Sonore[END_REF] and was formerly inspired by specific design approaches like Kansei [START_REF] Gentner | Kansei Cards: A Visual Tool Supporting the Investigation; Discussion; and Representation of the Kansei-Related Intentions of a Product to be Designed[END_REF]. Discussions were mediated by series of cardsets and the area of reflexion was materialized with a board, by analogy with a standard board game or role play. Supporting that, the SpeaK lexicon played the role of help to which anyone can refer during the session.

Three series of cardsets were built. They corresponds to the 10 wine regions, the 44 wine-words extracted from Krug terminology (6 oenological cursors + 38 additional terms) and the 37 sound-words of the lexicon, organized in 3 gross categories (General, Morphology, Timbre). Cardsets got a color chart to make their handling easier. Moreover, two things were added to facilitate the session proceedings: blank cards to possibly introduce new terms, and a trash bin to put aside non relevant words.

The board was configured with concentric circles considering that the bull's-eye (middle) was the region and its oenological description, and that the rim could be used to organise, and in case hierarchise, the corresponding sound properties (Fig. 2).

Protocol

After a warm-up stage (SpeaK browsing and exercise), the codesign session went as follows: one region is chosen, the Krug team fixed its oenological properties and placed the corresponding cards on the board. Then, semantic transformation is processed by discussing and placing cards around the target. At someone's request, examples or definition attached to a particular sound-word may be given by means of the lexicon. After a while, the instrumental metaphor is delivered to revitalize the debates and launch a new round of card handling. When all the participants agreed, the semantic portrait of the region is finalized and fixed (Fig. 2).

The processing of one region took approximately 30 minutes and systematically involved the same 7 persons. One of them, from the research team, played the role of mediator who opened the discussion and led or chased up the dialogue. The whole codesign stage (for 10 regions) occurred in 2 consecutive half-days.

At the very end, the semantic portraits were reconsidered to adjust descriptions in light of global coherence. Besides, few words were instinctively added to reach affective or emotional character of each region, such as 'serious', 'happy', 'warm', 'shy', etc. This additional contribution came out of the initial methodological frame, and was motivated by the composer who felt the need to collect more sensitive and complex dimensions than the basic sound properties given by the lexicon (Fig. 2).

Outcomes

The main outcome of the codesign stage was 10 semantic portraits respectively of the 10 oenological regions (Figure 2 gives an example on 'Vitryat'). From that, a synthesis was done, by trying to highlight global coherence and local differences among regions. For example, we tended to reveal global characteristics related to the Montagne Reims that were common to the 3 sub-areas (Reims Nord, Sud and Ouest), and local differences to discriminate between these same areas.

A reflexive look at this approach can also form another outcome of this stage. In fact, methodological elements can be usefully extracted from this experiment. They especially concern the role and the status of the mediator, the position of the composer -as the ultimate sound expert and potential session leader -or the use of extended sound examples, especially from musical / instrumental databases, able to feed the discussion and enlighten or consolidate raw ideas. These statements may certainly help us to improve the collaborative sound design approach which seems to be a rather specific practice within the general frame of codesign (see also Sec. 5.1).

Composition and Implementation

After the apprenticeship and codesign parts (Sec. 2 & 3), the third link of this project was mainly dedicated to composition and implementation of sonic transcriptions of concepts and words that have been handled, up to then. This part addresses two main issues: an artistic one dealing with the way to create sound sequences on the basis of words describing basic properties of sound or emotions; and a methodological one dealing with the way to transpose the work done in studio to the location where the sound design pieces were intended to be playeda fortiori if the sound diffusion system is technologically complex and massively multi-channels.

From description to composition

The composition started from all the semantic tags produced during the codesign session, plus the composer's personal notes. During the composition process (nearly 2 months), R. Rivas was also fed by some listening feedbacks (from the working team) that guided him for semantic interpretations and Krug's expectations on aesthetics.

The transcription work -from sound-words to sounds -is a challenge that contains a non-negligeable part of artistic intuition and, here, that sometimes forced the composer to read into root notionswine-words or evocations -in a personal manner. Nevertheless, he tried to rationalize his approach by translating sound-words into the three fundamentals sonic dimensions: frequency, time and space.

Frequency and time were generally informed by the basic SpeaK vocabulary and gave rise to specific spectral contents or temporal envelopes. For instance, notions like 'low/medium/high', 'crescendo/decrescendo', 'brilliant/dull' or 'fluctuating' get nearly direct correspondences in the acoustic domain. Here, the main difficulty came from the need to combine several basic notions, make them physically compatible and musically relevant. Back to root notions, as evoked above, terms like 'complexity' (oenological cursor), or 'lemony' (additional term) can respectively be related to a dense multi-layer mix of several sound textures or high-pitched and rough elementslike the sound of the harmon trumpet mute that Miles Davis used to play.

On this point, the notion 'natural/artificial' used for some portraits also drove R. Rivas for the choice of specific sound synthesis paradigms. But for that, he also relied on the instrumental metaphor attached to each region by the Krug team (e.g., violin, marimba, horn, etc.). These essential references gave the color of each piece while staying subtile and integrated by means of sound transformations or complex mixing. Space was also fully used by R. Rivas to get another degree-of-freedom in the transcription process. In fact, thanks to the multi-channel device attached to the project, a real 3D-soundscape could have been designed in order to convey sound properties like 'close/far' or illustrate complex notions like 'aerial' or 'powerful'.

These spatial effects were able to be developed either in the frontal plane of the device (18-loudseaker wall) or in the surrounding space of the room by mainly dealing with opening, localisation, dynamic trajectories (Fig. 3), or in case, immersivity and envelopment.

From studio work to location specific mastering

The specificity of this project lies also in the sound diffusion system it is associated with. This is a 32-loudspeaker system, split in 3 parts: 3 lines of 6 speakers associated with the '400 wines wall', 6 loudspeakers placed in circle around the audience, 6 speakers integrated in the ceiling and 2-subwoofers (Fig. 1). This device -developped by Amadeus1 upon design specifications made by the Ircam/EAC team -, offers a rather unique configuration for diffusing the sound pieces created in the project. But, added to specific acoustic conditions of the tasting room (glass or plaster walls, tiles floor), it also addresses sound engineering issues related to the mastering practice This being, in situ setting sessions were set. They mainly consisted in: i) tuning the the audio mix quality, potentially altered by resonances or reflections due to room acoustics or electroacoustic response; ii) adjusting the 3D-soundscape with regard to the room dimensions and behavior. These operations were directly done in the audio sessions made in the studio and resulted in a multi-channel bounce for each of the 10 pieces, dedicated to be played back by a direct-to-disk device. This part of the work points out a crucial practice in sound design: the location specific mastering. In fact, mastering is the final step of standard music production that aims at optimizing listening conditions in as many diffusion systems as possible. Sound design practices are commonly faced at specific mastering issues because of the diversity of sound devices usually used in this domain -from few (cheap) buzzers or loudspeakers placed in non conventional rooms (e.g. an automotive cockpit) to a lot of diffusion sources placed in large hall, such as a museum or a commercial hall 2 . One way to deal with this problem is to use audio simulation strategies based on 3D impulse response (IR) measurements that -by a deconvolution process -are able to render the effect of a given source diffused in the given room.

Presently, this virtual approach was not implemented for sake of time and project phasing. Instead, the composer and sound engineers (Clement Cerles and Colin Lardier), worked in studio either in a standard monitoring device or with a pseudo-'400 wines wall'. This system was built on purpose with similar unit sound devices and complied with the real volumic dimensions (wall's area, room's volume, etc.). In addition, as mentioned above, location specific mastering sessions were set and resulted in several modifications (EQs, internal balance of the mix, panoramic, tight filtering, etc.) that significantly improved the listening quality of the sound pieces.

Conclusions and Perspectives

To conclude, we conducted a long term project (nearly 2 years) that led us into the unexplored territory of wine industry and Champagne know-how. The main goal of the project was to design sound pieces informed by the knowledge of semantic correspondences between wine and sound worlds.

The research and creation team was formed with researchers in auditory perception and sound design, a composer and a sound engineer who also played the role of a computer music designer [START_REF] Zattra | A questionnaire-based investigation of the skills and roles of Computer Music Designers[END_REF]. The industrial collaboration was mainly interfaced with the Krug winemaking team and the marketing/communication department.

The project implemented the concept of semantic transformation and unfolded a participatory approach within a collaborative design (codesign) process. Moreover, it leaned on a methodological tool -previously developed and improved in the present frame: a sound lexicon (SpeaK) built as a dictionary collecting the major words used to describe sound properties, together with definitions and sound examples.

Within this frame, the project got four main stages in order to successively learn how to speak about wines, speak about sounds and collectively implement the semantic transformation from wine-words to sound-words. The fourth stage was dedicated to sound design (creation) and aimed at translating sound-words into sounds and musical composition that finally resulted in ten 1-min sound pieces diffused by a multi-channel sound device placed in a dedicated room: the Krug tasting room.

Perspectives

In the light of its originality and complexity, the Krug's project brings into front an emblematic approach. As a research process, it leads to open perspectives that should be further investigated in order to complement the project outputs. These perspectives mainly concern two components: codesign methodology and evaluation. As a sound design research, this project may also contribute to enhance our knowledge on the discipline and be part of a conceptual framework called sciences of sound design [START_REF] Misdariis | Sciences du design sonore. Approche intégrée du design sonore au sein de la recherche en design[END_REF].

The codesign methodology applied to sound design seems to be quite encouraging and promising. As in its first implementation [START_REF] Carron | Méthodes et Outils pour Définir et Véhiculer une Identité Sonore[END_REF], the lexicon that supported the approach confirmed to be an efficient and relevant tool able to help communication and understanding on sounds. Nevertheless, the current codesign implementation showed some weakness that should be investigated and, may be, improved.

For instance, whereas the preliminary training exercises appeared to be unmissable, some uncertainties arose according the relevancy of the sound examples dedicated to these exercises and especially their ability to express just one basic property. Attention must be paid on the selection of these sounds and their polysemic content.

This precise issue is also addressed to the lexicon itself. In fact, on behalf of its rather 'encyclopedic' status, SpeaK must provide irrevocable and unequivocal specimens of sound examples illustrating sound attributes. This effort goes through the re-design of most of all examples from all categories. This work has yet started with the voice category by means of a recording campaign conducted by the composer R. Rivas; it will soon produce high quality and controlled vocal samples. Indeed, concerning SpeaK, a more conceptual issue appeared during the Krug experiment: the fact that the list of pre-defined words were not sufficient enough to describe an oenological identity and that the composer needed high level descriptions (emotions, evocations, character) to be able to translate ideas into composition.

Finally, and more globally, we observed the fact that the experimental apparatus (board, cardsets, lexicon) prevented from describing dynamic changes of a semantic portrait as it often occurs in sound perception, but also wine tasting! This may force us to imagine new paradigms that would also consider the temporal dimension of sounds which is not equally relevant in other domains (e.g. graphics). On the other hand, whereas the PDS research group used to promote a 3-step sound design model (Analysis, Creation, Validation) [START_REF] Susini | Sound design: an applied, experimental framework to study the perception of everyday sounds[END_REF], the evaluation stage is, right now, rather completely missing from the project proceedings -except few informal (and positive) feedbacks from the first tasting sessions at Krug. This point addresses an interesting and controversial issue that the project itself could help to investigate.

In fact, this asks the following fundamental questions: why should we evaluate and how can we proceed an evaluation? A rational answer to the first question could be: to verify the match between solutions and specifications, or to ensure the usefulness, usability, and desirability of the user experience produced by the solutions [START_REF] Robare | Sound in Product Design[END_REF]. As for evaluation procedures, they should to be inspired, as usual, by the experimental psychology discipline with physiological, perceptual or cognitive measurements.

But, transposed to the current use case, the previous rationale appears to be more complex to argue and implement. In fact, in that case, the main specification to evaluate could be the perception of the wine characters into the sound composition. In other words, does one recover the basic oenological attributes of a region into a 1-min sound piece experience? Or more globally, is the semantic transformation finally valid, i.e. does it help the composer to create a relevant sound content and the listener to recognize the wine identity that intended to be illustrated? Or, alternatively, what does all of this bring to the tasting experience? All these questions address methodological issues in terms of experiment (what / how to measure?) but also in terms of contextualization (how to put the participants into controlled -and ethically acceptable -tasting conditions?). These issues form a work-in-progress reflexion that we hope to investigate and implement in a near future. By listing all these outcomes and perspectives, we can observe that the Krug project brings considerable knowledge on the sound design discipline itself, its protagonists, its process and even its production. In that way, and even if it initially targeted a direct application -sound pieces composition for a tasting experience -, this project could finally be seen as a potential research project implementing a research-throughdesign approach that aims at producing knowledge instead of only solutions [START_REF] Findeli | La recherche-projet en design et la question de la question de recherche: essai de clarification conceptuelle[END_REF].

This precisely comes into the conceptual frame we recently tried to make emerge and promote, in accordance with the three research loci of Nigel Cross' formalization on design research: people, process and products [START_REF] Cross | Designerly ways of knowing[END_REF]. In fact, transposed to the discipline of sound design, we look at laying the foundations of sciences of sound design that will investigate simultaneously the character of the sound designer, the sound design process, tools or methods and the status of the designed sound, i.e. what sound design produces in fine [START_REF] Misdariis | Sciences du design sonore. Approche intégrée du design sonore au sein de la recherche en design[END_REF]. Then, to some degree, we can expect that the present project may have helped -and will help -to inform this approach and, by quoting [START_REF] Cross | Designerly ways of knowing: Design discipline versus design science[END_REF] [START_REF] Cross | Designerly ways of knowing: Design discipline versus design science[END_REF] give some elements to answer to the seminal question: is there a designerly way of knowing, thinking and acting in sound design?

Acknowledgments. The authors -namely the Ircam group -are sincerely thankful to Maison KRUG (Maggie Henriquez, and the last three authors of this paper) for their fruitful collaboration and flawless welcome. All authors would also like to thanks Cyril Beros, Jeremie Henrot (Ircam PROD), Olivier Warusfel (Ircam-STMS EAC), Gaetan Byk (Amadeus) and Emmanuelle Zoll for their contributions to the study.

Fig. 1 .

 1 Fig. 1. Krug tasting room (Reims) equipped with a 32-loudspeaker system (right, ©O.Warusfel), and the "400 wines wall" representing the 400 'vins clairs' held in the Krug wine library (left, ©P.Susini)

Fig. 2 .

 2 Fig. 2.Illustrative elements of the codesign sessions. The environment was formed of cardsets representing the different semantic spaces and board with concentric circles (left). At the end of each round, a semantic portrait of an oenological region is formed by associating wine-wordset the center -and sound-words -in concentric circles. In each portrait, the musical metaphor is recalled and some additional affective or emotional words are added (right). (©N.Misdariis)

Fig. 3 .

 3 Fig. 3. Depiction of spatial sketches as stated by the composer R. Rivas in the composition phase. The background plane represents the '400 wines wall' and its 18-loudspeaker device and the drawings represent sound localisation or trajectories designed inside this grid.

http://amadeusaudio.fr/en/

https://www.ircam.fr/article/detail/mastering-hors-du-studio-trois-experts-en-design-sonore-decryptentdes-nouvelles-pratiques-1/


Olivier Massol
email: olivier.massol@ifp.fr

Frederic Lantz

Michael Martens

Élodie Sentenac

A cost function for the natural gas transmission industry: further considerations

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Introduction

As far as engineering economics is concerned, the year 2009 corresponds to a special anniversary: 60 years ago, a promising PhD student named Hollis B. Chenery 1 published a seminal article that illustrated how the production function of microeconomic theory can be rewritten with engineering variables [START_REF] Chenery | Engineering production functions[END_REF]. His goal was to present a rigorous analysis of the cost function of an industry whose total production function consists of several processes which can be combined in varying proportions to produce a given output. As an illustration, he provided an illuminating case study based on the natural gas transmission industry. In this industry, the combination of processes is such that it is possible to increase output by adding more compressors while keeping the same pipeline or, in the long-run, varying simultaneously both parameters. As he has shown, this flexibility is at the origin of the massive scale economies observed in that industry. Despite a considerable early influence in the academic community (e.g. [START_REF] Smith | Engineering Data and Statistical Techniques in the Analysis of Production and Technological Change: Fuel Requirements of the Trucking Industry[END_REF][START_REF] Smith | The Theory of Investment and Production[END_REF]Thomson et al., 1972), the engineering approach pioneered by Chenery has gradually disappeared from the applied economic literature during the last 30 years. Moreover, it must be acknowledged that, apart rare exceptions (e.g. [START_REF] Callen | Production, Efficiency, and Welfare in the Natural Gas Transmission Industry[END_REF], most of the articles that fuelled the debate associated with the impressive regulatory reforms implemented in the gas industry did not consider this engineering approach. Given the importance of the gas sector to the energy industry, this situation is somehow frustrating. However, a recent article published in The Engineering Economist [START_REF] Yépez | A cost function for the natural gas industry[END_REF] provided a refreshing revival. In addition to the perpetuation of Chenery's method, at least three reasons can be advanced to illustrate the value added by this contribution. Firstly, this paper presents a clear and modernized version of the Chenery's methodology which includes some interesting refinements. In particular: a more recent version of the gas flow equation 2 and the possibility to deal with a non-steady elevation of the pipeline. Secondly, Dr. Yépez derived a set of formal equations that describe how total, average and marginal costs vary with output in the short run. And last but not least, Dr. Yépez provided a detailed case study based on a Mexican project. As [START_REF] Chenery | Engineering production functions[END_REF] only represents the tip of the iceberg, some complements may be needed to shed some light on Chenery's shrewd thoughts on the gas transmission industry. This article is organized as follows. In the first section, a complete review of the Chenery-Yépez's model aims at offering some eclectic complements. In the second section, an extension of that model is provided to deal with the case of seasonal variations in the volumes of gas to be transported. The influence of these variations on the optimal design of the transmission equipment is discussed. 1 During his subsequent career, Pr. Chenery's research interest moved to development economics. From 1972From to 1982, he , he was the World Bank vice president for development policy and played a key role in the definition of the Bank's strategy. 2 [START_REF] Chenery | Engineering production functions[END_REF] was based on the Weymouth equation that is not so appropriate for large gas pipelines.

Investment recommendations are then addressed in the last section to analyze the rationale of a "building ahead the demand behaviour".

-A commented review on the Chenery-Yépez approach

In this first section, three points are successively discussed: the long-run optimal decision, the behaviour of the cost functions in the short-run and an alternative formulation of the long-run optimal design.

Long-run economics

The Chenery-Yépez approach Both authors developed a rigorous model-based cost function for the gas transmission industry based on the case of a gas pipeline that runs a given distance l and transports Q , a daily flow of natural gas in MMcfd (millions of cubic feet per day) that is assumed to remain steady all along the infrastructure's lifetime. In their model, output and investment decisions are assumed to be taken separately and the estimate of output is supposed to be made prior to the investment decision. This assumption is consistent with industrial practice because in many cases, this flow is an outcome of exogenous negotiations between a natural gas producer and a group of buyers. As far as investment decision is concerned, two engineering variables have to be considered in the long-run: D the inside diameter of the pipe in inches; and the compressor horsepower per million of cubic feet of gas H .

As the firm is assumed to have perfect information on the volume of gas to be transported and on the associated revenues, the project's value is maximized by an investment program which minimizes the total cost of production over the period. Since output is assumed to remain steady all along the infrastructure lifetime, the optimum plan minimizes also the annual total costs. For a given volume to transport, the objective of the gas transmission firm is thus to find a technology-compatible combination of inputs ( ) , D H that minimizes its annual total cost.

A given combination ( )

, D H is technology-compatible if, and only if, the two following relations simultaneously hold. Firstly, for a given flow rate Q of natural gas, the frictional loss of energy through a pipe (measured as pressure drop between 1 P the initial pressure and 2 P the terminal one in psia) is a decreasing function of the diameter D of the pipe. Considering a steady flow of natural gas and assuming: (1) a friction coefficient that is a function of the diameter and (2) a constant and uniform elevation all along the pipeline, this general flow equation is relatively simple [START_REF] Yépez | A cost function for the natural gas industry[END_REF]:

1/ 2 2 8/ 3 0 1 2 1 c P Q D P l       = -        
Here, 0 c is an exogenous constant parameter (cf. [START_REF] Yépez | A cost function for the natural gas industry[END_REF] and l is the pipeline length in miles.

Secondly, the power needed to compress natural gas from a given inlet pressure 0 P to a predefined outlet pressure 1 P is given by the following empirical expression [START_REF] Yépez | A cost function for the natural gas industry[END_REF]:

( ) 1 . 1 H c R Q β = -
Where R is the pressure ratio 1 0 1 P P ≥ and both 1 c and β are dimensionless constant parameters.

Following a simplification initially proposed by Chenery, if the inlet pressure 0 P is assumed to be equal to the terminal pressure 2 P at the delivery point and if we assume that the terminal pressure 2 P corresponds to an exogenous parameter, then the two technological equations presented above can be simplified (by eliminating R ) into a unique engineering production function:

( )

2 2 2 16 / 3 0 1 , , 1 1 0 lQ H F D H Q c D c Q β   = + - + =     (1)
This production function embodies the key features of the gas transmission industry and clearly suggests the possibility of smooth continuous substitution between pipe diameter and compression horsepower. From a strict technical perspective, the combination of these two capital factors (diameter and compressor horsepower) is such that it is possible to increase output by adding more compressors while keeping the same pipeline or increasing the pipeline diameter while keeping the same design for the compressors.

As far as costs are concerned, two different elements have to be considered. Firstly, the total yearly capital and operating cost per mile is given by 3 Usually, the wall thickness of the pipe plays also an important role in the total cost of the line. But, following [START_REF] Chenery | Engineering production functions[END_REF] and [START_REF] Yépez | A cost function for the natural gas industry[END_REF], we assume that this parameter can be computed as a function of the diameter of the pipe. charge dependent on the depreciation and real interest rates so that For a rational firm, the objective is to minimize its annual costs and the Chenery-Yépez approach suggests the following cost-minimization programme:

( ) ( ) ( ) ( ) , , . s.t.
, , 0

D H D H Min LRTC D H l C D C H F D H Q = + = (2)
With the functional specifications and the numerical parameters presented in [START_REF] Yépez | A cost function for the natural gas industry[END_REF], this constrained minimization problem has a unique4 solution. The optimal mix of inputs ( )

* *
, D H can be obtained thanks to the Lagrangian method. The Lagrangian L for this constrained minimization problem is:

( ) ( ) ( ) ( ) , , = . , , D H D H l C D C H F D H Q λ λ + + L
And the optimal solution ( )

( ) ( ) ( ) ' , , = . , , 0 D F D H l C D D H Q D D λ λ ∂ ∂ + = ∂ ∂ L ( ) ( ) ( ) ' , , = , , 0 H F D H C H D H Q H H λ λ ∂ ∂ + = ∂ ∂ L Where, ( ) ' D C D (respectively ( ) ' H
C H ) is the marginal annual cost of the pipeline (respectively of the compressor station). Note that a straightforward property of that optimal combination of pipeline diameter and horsepower of the compressor station ( ) of the noteworthy differences between [START_REF] Chenery | Engineering production functions[END_REF] and [START_REF] Yépez | A cost function for the natural gas industry[END_REF]. As it will be shown below, these specifications play a non-negligible role on the economics of the cost function in both short and long-run. Thus, they deserve a short comment.

( ) ( ) ( ) ( ) LR LR D LR H LR LR LR F D H Q l C D D F C H D H Q H ∂ ∂ = ∂ ∂ A remark
Chenery used only linear specifications with constant linear coefficients, a choice that he justified as follows: "Although theoretically these parameters may be functions of the indicated engineering variables, the available data show them to be practically constant over the range which has been experienced in practice." (Chenery, 1949, p. 518). In his contribution, Dr. Yépez preferred concave functional forms for a basic description of the data (source, number of observations, etc.) and a discussion on the regression results (t-statistics, R 2 , etc.)... Nevertheless, the intrinsic importance of this remark can be toned down since publicly available information on cost data for that industry is rather scarce. Given Dr Yépez's position, his estimates can be viewed as reliable and up-to-date sources. Thus, it is almost certain that many practitioners and/or gas economists prefer his publicly available results to many a lesser man's studies.

Scale economies, some empirical evidences

In his discussion on the long-run economics of the natural gas transportation industry, Dr. Yépez noticed the presence of significant economies of scale. His affirmation is inspired by the shape of the long-run cost curves as both long-run average cost (LRAC) and long-run marginal cost (LRMC) are decreasing and the former exceeds the later. In his conclusion, he rightly underlined that this result has important implications for the design of appropriate pricing policies, since applying a long-run marginal-cost pricing do not allow the industry to break even.

To complete this plot-inspired remark, some additional results may be useful to quantify those scale economies. In another papers dedicated to investment issues, Pr. Chenery suggested that the long-run cost function of the gas pipeline has almost constant elasticity of output with respect to cost over most of its range [START_REF] Chenery | Overcapacity and the Acceleration Principle Overcapacity and the Acceleration Principle[END_REF]. Thus, the function

1 Q ψ γ
, where ψ represents a scale coefficient 5 and γ a constant, could provide an acceptable approximation of the long-run total annual cost function In this empirical sub-section, data are generated from the industry process model and depict the longrun total cost for various quantities. Based on these "pseudo data", simple relationships such as those suggested by Pr. Chenery can be statistically estimated. In fact, this method has its roots in Pr.

Griffin's set of studies published in the 1970's [START_REF] Griffin | Long-run Production Modelling With Pseudo-data: Electric Power Generation[END_REF][START_REF] Griffin | Joint Production Technology: The Case of Petrochemicals[END_REF][START_REF] Griffin | Statistical Cost Analysis Revisited[END_REF]. At that time, available computational technologies prohibited the direct inclusion of cumbersome process models in large inter-industry simulation and forecasting models. The "pseudo data" approach was seen as a computational-friendly tool able to simplify these complex engineering models into single equation cost function.

In the present study, a data set of 84 observations was generated by running several instances of Dr.

Yépez's model with the list of parameters presented in Appendix 1. In these numerical simulations, the output varied from 0.25 to 21 Bcm/year (billions cubic meter of gas per year) -i.e: 25.6 to 2147.5 MMcfd -by regular steps of 0.25 Bcm/year. Here, cost data are thus drawn over a sufficiently wide range that represents usual operation conditions in the natural gas transmission industry. Thanks to the usual log transformation, an Ordinary Least Square (OLS) regression was performed to estimate Chenery's specification. Table 1 presents the empirical results for this statistical cost function. These results clearly confirm the pertinence of Chenery's suggestion. The goodness of fit measure R 2 indicates an excellent explanatory power which is quite unusual for such a simple specification. The estimated values are clearly of significant as the t-statistics suggests that the probabilities attached to 5 For such a cost function, the ratio of average cost to marginal cost is constant and equal to ψ .

the real values being 0 are insignificant. Hence, the specification suggested by Pr. Chenery provides an acceptable approximation of the Long Run Total cost function. As far as the long-run scale coefficient ψ is concerned, these results clearly confirm the presence of significant scale economies: for any given output, the Long-Run Average Cost is always 63% greater that the Long-Run Marginal Cost.

This value clearly reinforces the value of Dr. Yépez's conclusion on price setting: a simple tariff set at the long-run marginal cost would systematically induce significant losses.

Short-run cost economics

The Chenery-Yépez approach

In the short-run, both [START_REF] Yépez | A cost function for the natural gas industry[END_REF] and [START_REF] Chenery | Overcapacity and the Acceleration Principle Overcapacity and the Acceleration Principle[END_REF] precise that there are some limits on the substitutability between the two factors D and H . In fact, pipeline diameter is an indivisible factor whereas continued adjustment of pumping equipment is easy to implement. Moreover, pipe costs are usually significantly greater than compression costs6 . As a result, short-run adjustments needed to serve possible additional demand are usually obtained by increasing the compression horsepower while keeping the same pipeline.

For a given pipeline whose diameter is D , the engineering production function ( 1) can be reorganized to get an expression of the compression horsepower H required in the short-run. This required compression horsepower is fully determined by Q the daily flow rate of natural gas to be transported.

In microeconomic jargon, this function is nothing but a short-run factor demand function

( ) D H g Q =
where D g is the following function:

2 2 1 2 16 / 3 0 :

. 1 1

D lQ g Q c Q c D β   + -     a (3)
As D g is a smooth, strictly increasing and strictly convex function, it is clear that a 100% increase in the quantity to be transported through a given pipeline requires a more than 100% increase in the (  )

1 H D Q C g Q - .
Adding a strictly decreasing positive pipeline average cost ( )

1 . D Q l C D - provides a
typical U-shaped curve for the Short Run Average Cost function ( ) SRAC Q . In that case, there will be an efficient level of production * Q at which the average cost reaches its minimum. Above that particular output, the firm's short-run operations will clearly exhibit decreasing returns to scale.

With the concave specification of ( ) H C H chosen by Dr. Yépez, the behaviour of the short run cost function is not so clear. As far as the variable cost is concerned, two opposite effects are at work. Firstly, a "pro-decreasing returns to scale" effect may be expected because of the both increasing and convex nature of the short-run conditional factor demand function Nevertheless, the short-run average cost function

( ) ( ) ( ) ( ) ( ) 1 . D H D SRAC Q l C D C g Q Q = +
is smooth and defined for all strictly positive output Q . With Dr. Yépez's numerical values, it is clear that ( )

lim Q SRAC Q →∞ = +∞ .
Hence, it is clear that the increasing return to scale suggested on Dr.

Yépez's Figure 2 (Yépez, 2008, p.80) is a local phenomenon that cannot be generalized to any output.

Stated differently, it simply means that with those compression cost parameters, the concavity of ( )

H
C H is not sufficient to counterbalance the "diseconomies of scale" effect induced by the convex nature of the short-run conditional factor demand function

( ) D H g Q = .
This finding is consistent with the U-shaped Short Run Average Cost curves obtained from numerical simulations based on Dr.

Yépez's numerical values (cf. Figure 1).

As will be discussed below, this finding has important implications when considering the adaptation of any already installed transmission equipment to a significant increase in the planned output.

"Looping" economics, a simple introduction

An obvious corollary emerges from these U-shaped short run average cost curves. In his paper, Dr.

Yépez rightly suggested that the cheapest way to accommodate an additional output in the short run is the addition of compression horsepower. This affirmation clearly holds for small increments in the gas flows. But, is it also the cheapest solution when considering the adaptation needed to serve a very large increment? Even without paying any attention to possible technological restrictions (e.g.: an upper limit on the pipeline's operating pressures), there can exist an economically grounded argument to choose a different solution. A sufficiently large incremental volumes to be transported can lead the transmission firm to operate in the "diseconomies of scale" region. If that increment is large enough, it can be worth comparing two alternative solutions: a continued addition of pumping equipment versus laying a second gas transmission line in parallel onto the existing one. In the natural gas industry, this "replication" is quite frequent and is called "looping".

A simple introduction to "looping" economics can be found in [START_REF] Chenery | Overcapacity and the Acceleration Principle Overcapacity and the Acceleration Principle[END_REF]. In that paper, Pr.

Chenery studied the simplistic case of a strict replication of the existing line and noted that it pays to duplicate a line when the increment Q ∆ is large enough to get:

( ) 2. 2 Q Q SRTC Q Q SRTC + ∆   + ∆ ≥    
If that inequation holds, a solution based on a doubled transmission infrastructure where each pipeline transports one half of the total flow Q Q + ∆ is less costly than those based on a single line design whose total cost is ( )

SRTC Q Q + ∆ . A
similar reasoning is also valid for the transition from two to three lines and so forth. On the assumption that the diameter of pipeline installed remains unchanged, a lower envelope curve can be exhibited. This later curve was named "intertemporal cost curve" by the late Pr. Chenery who noted that it represents the evolution of the total annual cost over time if output always increases on the assumption that the size of unit installed (here, the pipeline diameter) remains unchanged.

Figure 2 provides an illustration in the case of a 100-miles infrastructure that was initially designed to transport a daily flow rate of 383.487 MMcf of natural gas. With the numerical assumptions listed in Appendix 1, the optimal design includes pipe diameter of 27 inches and a compressor of 3,858 hp. This treatment is based on the simple case of a strict duplication of the pre-existing infrastructure. In real industrial cases, planners are free to choose a different diameter for the new pipeline to be laid. To conclude on looping, it is also interesting to underline that meshed networks are frequent in the gas transmission industry. This feature introduces a significant complexity in the identification of an optimal looping policy (e.g.: [START_REF] Andre | Optimization of Capacity Expansion Planning for Gas Transportation Networks[END_REF]. Nevertheless, this straightforward description provides an intuitive introduction to "looping" economics.

Long-run vs short-run, an alternative and simplified view

This section aims at showing that the long-run problem studied by both Chenery and Yépez can be reformulated in a simplified manner.

First, attention is focussed on the role played by pipeline diameter in the short run. Any change in that diameter has a major impact on the short-run economics of the transmission equipment since it changes the repartition of the total cost between fixed costs (pipeline related cost) and outputdependant ones (the compression costs). In a sense, D can be viewed as an index of scale that characterizes the size of a given transmission equipment.

For a given flow of gas to be transported Q , the engineering production function (1) can also be reorganized to get the compression horsepower H required in the short-run as a differentiable and strictly decreasing function of the pipeline diameter D (provided that D remains strictly positive)

and parameterized by Q the daily flow rate of natural gas to be transported:

2 2 1 2 16 / 3 0 :

. 1 1

Q lQ f D c Q c D β       + -           a (4)
The influence of the pipeline diameter D on the Short-Run Total Cost to transport a given flow rate of gas Q is described by the cost function

( ) ( ) ( ) ( ) . Q D H Q SRTC D l C D C f D = +
, parameterized by Q . Obviously, the pipeline element ( ) cost and those values varies with the pipeline diameter D . In this subsection, it will be proved that the long-run cost minimization problem can be viewed equivalently as selecting an appropriate (and unique) element in that family of short-run cost curves.

. D l C D is
Restated with simple algebra, the goal is to find the particular Short-Run Total Cost function that minimizes the annual cost of transporting a given flow Q of gas. As D is assumed to be a continuous parameter, a necessary condition for minimum annual cost is that the derivative of Short-Run Total

Cost with respect to D be zero:

( ) 0 Q dSRTC D dD = (5) Equivalently, ( ) ( ) ( ) ( ) ' ' ' . D Q H Q l C D f D C f D = - × .
As usual, an economic interpretation can be given to that expression: at the optimum, the marginal cost of the pipeline is exactly equal to the marginal compression horsepower economy.

From a technical perspective, some straightforward arguments can be given to illustrate the existence of a unique solution to equation (4). As For the interested reader, a straightforward proof of that result is provided in Appendix 2.

Clearly this approach simplifies the long run problem as it only requires solving a single variable equation. In the next sections, that simplification will be helpful to illustrate some new results. On top of that, this very simple approach provides a clear argument for presenting the Long Run Total Cost curve as the lower envelope of the Short Run Total Cost curves. Figure 3 offers an useful illustration of that assertion.

-Dealing with seasonal variations

In their papers, both Pr. Chenery and Dr. Yépez studied the case of a transmission equipment that is planned to transport a steady flow of natural gas Q all across the year. How is the short-run total annual cost impacted by seasonal changes in that flow-rate? How does this seasonal pattern influence the optimal design of that infrastructure? These are precisely some of the questions addressed in this second section.

A further distinction: expansion vs contraction costs

This preliminary subsection provides a useful piece of methodological background. Following the usual convention, Dr. Yépez defines the short-run as a period of time in which the quantity of at least one input is fixed (here the diameter D ) and the quantities of the other input H can vary. As in classic microeconomic theory, the long run corresponds to an hypothetical situation in which the quantities of all inputs can vary. Aside from that usual distinction, a further distinction could be valuable to analyze the short-run economics of that industry: that between expansion and contraction costs. That distinction pioneered by the "French marginalist school" relies on the asymmetry between plant expansion and plant contraction since some fixed costs to be incurred in case of expansion cannot be recouped in case of contraction (Cf. [START_REF] Dreze | Some Post-war Contributions of French Economists to Theory and Public Policy: With Special Emphasis on Problems of Resource Allocation[END_REF] for a comprehensive survey). In fact, Chenery himself also perceived the pertinence of that further distinction: "The functions resulting from fixing the pipe size (scale of plant) and varying other factors (amount of pumping equipment, etc.) will be called "intermediate" cost functions. It is possible to move along the intermediate cost curves only as long as demand is expanding. A contraction of demand will involve a movement along a "plant" curve (...) where the only important variable is fuel consumption." (Chenery, 1952, p. 4).

Dr. Yépez's framework provides an interesting starting point to implement that distinction in the natural gas industry. On the assumption that the pipeline diameter remains unchanged in the short-run, the output-variable element corresponds to the total annual cost of the compressor stations that includes the capital costs and the operation and maintenance costs. Given that any expansion requires the installation of an additional compression capacity whose annual cost is fully captured in this shortrun cost function, it is clear that Dr. Yépez's short-run total cost function describes the variation of annual cost if output always increases. But in case of a sudden and temporary drop in output, rigidity would probably be observed in the downward adjustment of the compression capacity. Stated differently, it means that there are few chances for such a restriction in output to be accompanied with an instantaneous premature scrapping of the excess in compression capacity.

With those remarks in mind and using the previous notations, a simple reformulation can now be proposed to distinguish expansion and contraction costs. For a gas transmission infrastructure that transports a given flow 0 Q with a pipeline whose diameter is D and an adapted compression capacity ( )

0 0 D H g Q = , the
4 0 0 Q C H Q .
To summarize, the short-run total expansion cost function SRTEC of that infrastructure is given by:

( ) ( ) ( ) ( ) ( ) ( ) 3 4 0 : . . D C D D SRTEC Q SRTC Q l C D C g Q C g Q Q Q α = + + ∀ ≥ a
Whereas the short-run total contraction cost function SRTCC is:

( ) ( ) ( ) 3 0 4 0 0 0 : . . D C Q SRTCC Q l C D C H C H Q Q Q α + + ∀ < a
Hence, the short-run total cost function takes either one or the other expression depending on whether Q the output to be served, is larger or not than the initial level 0 Q . As far as short-run marginal cost is concerned, we can notice a discontinuity for the particular output 0 Q Q = . For this output, the right derivative, i.

.

C D D d C g Q C g Q dQ α   + 
 , gives the marginal expansion cost whereas the left derivative, i.e.

( )

4 0 0 1 C H Q
, corresponds to the marginal contraction cost. Explicit reference must thus be made to one of these concepts when considering practical applications of marginal costs. More generally, the possible non-equality of left-hand and right-hand short-run marginal costs at adapted capacity has already been emphasized in the literature (see for example [START_REF] Pierru | Short-run and long-run marginal costs of joint products in linear programming[END_REF] for an economic interpretation in the case of linear-programming models).

The case of seasonal variations

Obviously, gas consumption varies over time -across seasons, weeks and days. Moreover, the amplitude of these variations can be large. A typical illustration is provided by the seasonal swing observed in countries where natural gas is largely used for heating. As an example, in north-western Europe approximately two-thirds of the gas is consumed during the winter period (October-March).

Moreover, "residential users consume about 90 percent of their overall gas during the winter period.

For local gas providers, it is therefore not uncommon to have daily peaks in gas delivery in the winter amounting to more than ten times the delivery on a summer day." (Höffler and Kübler, 2007, pp. 5206-5207). In many cases, gas cannot be stored near end-users. Significant seasonal fluctuations can thus be observed in the daily flows of gas to be transported. For a transmission infrastructure, such a situation results in an average daily output that is strictly smaller than the peak flow.

To analyze the influence of these seasonal variations, we study the case of a rational firm that plans to build an equipment to transport time-varying gas flows. The firm is supposed to know precisely: t q the daily flow of gas demanded on day t , and p Q the peak flow to be transported on that infrastructure (i.e. : p t t

Q Max q =

). A one year periodicity is assumed for the gas flows (i.e.: 365 t t q q + = ). The average daily flow rate to be observed during a particular year is denoted q . A strictly less than one load factor is also assumed for that infrastructure, which means that p q Q < (the case of a 100% load factor corresponds to the steady flow case studied by both Chenery and Yépez).

Here again the firm's decision can be analyzed as an annual cost minimization problem. As in the Chenery-Yépez approach, the problem faced by the firm can be viewed as a cost-minimization programme:

{ } ( ) ( ) ( ) ( ) { } { } 3 4 , , .
. . s.t. , , 0 1,..., 365 1,...,365

p t t D C p p D H H p t t t p q Min l C D C H C H Q F D H q t H H t α + + = ∀ ∈ ≤ ∀ ∈ (6)
The objective is to minimize the annual total cost incurred to transport the daily flows ( ) { } . Given that the model at hand is fully deterministic, there is no incentive to build any extra capacity and the compression capacity to be installed is thus supposed to be equal to the minimum required to serve those peak flow p Q . Hence, the peak compression horsepower p H is given by

( ) p Q f D .
Here again, this problem can be reformulated as a single variable optimization programme: i.e. finding the unique optimal diameter * , p q Q D that minimizes the short-run total contraction cost function SRTCC to serve the average daily flow q knowing that this infrastructure must be capable to supply the peak output p Q , i.e.

( )

* , p p p Q q Q H f D = .
The optimal pipeline design, denoted * , p q Q D , minimizes the following annual cost function

( ) ( ) ( ) ( ) ( ) 3 4 ,

:

. .

p p p D C Q Q q Q p q SRTCC D l C D C f D C f D Q α + + a
. A necessary condition is that the derivative of that cost function with respect to pipeline diameter D be zero:

( ) , * , 0 p p q Q q Q dSRTCC D dD = (7)
As usual, that condition has its economic interpretation. At that optimum, the marginal increase in the pipeline cost is exactly equal to the marginal compression cost reduction.

As a benchmark, it is interesting to compare this optimal pipeline diameter * ,

p p p Q q Q q Q D f D and ( ) ( ) * * , p p p Q Q Q D f D
represents a technology-compatible choice.

Proposition 1:

Assume a gas transmission firm with a seasonal-varying output, costs and engineering production function as described above. For that firm, the long run optimal combination of input ( )

( ) * * , , , p p p Q q Q q Q D f D
involves a smaller diameter and a larger horsepower capacity than those that would have been installed to serve a steady daily flow equal to the peak value p Q .

Again, a straightforward proof of that result is provided in Appendix 2.

This result is rational: a lower load factor creates an incentive to lower the transportation cost by preferring a mix of inputs that includes more compressor horsepower (that generates flexibility) and less pipeline.

-Building ahead the demand, an irrational decision?

This section provides a discussion on the optimal investment policies that can be derived from the Chenery-Yépez approach. In his discussion on the economics of the natural gas transportation industry, Dr. Yépez suggests that: "Whatever the planned level of output, the rational firm will select a transportation system whose short-run average total cost is tangent to its LRAC at that capacity" (Yépez, 2008, p. 80). It is tempting to confront this rule with the investment decisions actually taken in the gas industry.

Preliminary remarks

Evidences drawn from the gas industry provides numerous cases of transmission infrastructures that were designed with a significantly oversized diameter. The "Yamal pipeline" -an impressive 56inches diameter infrastructure that runs from the Yamal peninsula (northern Russia) to Germany across Belarus and Poland -provides an archetypal example. Since its construction in the end-1980's, the gas flow transported on that infrastructure has never exceeded 20 Bcm/y, which is a relatively low figure compared to the initial plans [START_REF] Victor | Bypassing Ukraine: exporting Russian gas to Poland and Germany[END_REF]. Another case is given by the contemporary Nabucco project, a large gas transmission infrastructure that has been proposed for construction across the South-East Europe and Turkey to carry gas from the Caspian region to Austria and other European markets. The design chosen for this project involves a large diameter (56-inches) that looks considerably oversized for the expected flow (8 Bcm/y). In both cases, a cost-minimizing design based on the Chenery-Yépez approach would certainly suggest a smaller diameter/compression horsepower ratio. Why did the teams of skilled and experienced planners who designed those infrastructures preferred alternative solutions based on larger diameters?

This question calls for a closer examination of the model's hypotheses. More precisely, the planned infrastructure is designed to transport a steady flow Q whose value is known and expected to remain steady during the whole project's life. This assumption allows analyzing the decision in a static and deterministic framework. In the previous industrial cases, this important condition was not fulfilled.

For the "Yamal pipeline", the initial plan was based on the construction of two parallel 56-inch pipelines allowing an export potential of 67 Bcm/y. By now, only one of these two pipelines has been built with a reduced number of compressors from the original plan [START_REF] Victor | Bypassing Ukraine: exporting Russian gas to Poland and Germany[END_REF]. As far as Nabucco is concerned, a phased design has also been adopted. It relies on a single large-diameter pipeline with a compression capacity initially adjusted to transport 8 Bcm/y during the first phase.

Then, continued additions of compression capacity will ultimately allow increasing its output to 31

Bcm/y, a large flow that justifies a 56-inches diameter. In both cases, planner's decision to use an oversized pipeline diameter reflected the perceived massive -but uncertain when the pipeline design was decided -exports perspectives to the European gas markets. Stated differently, planners explicitly take into consideration the possibility to have significant, but still uncertain, expansions in the flows that will be transported during the infrastructure's lifetime.

Overcapacity, an irrational decision?

The aim of this subsection is to propose a simple extension of the Chenery-Yépez framework to take into consideration the remarks above; hence, to analyze the rationality of such a "building ahead the demand" behaviour. This discussion echoes the so-called capacity expansion problem studied in economic theory, a field also pioneered by H.B. Chenery. In his seminal contribution, [START_REF] Chenery | Overcapacity and the Acceleration Principle Overcapacity and the Acceleration Principle[END_REF] discussed the effect of technology on investment behaviour with the goal to illustrate how the simultaneous presence of growing demand and economies of scale may motivate the construction of an oversized equipment. Stated differently, Pr. Chenery proved that "a building ahead the demand" decision can be rational. Following that contribution, the optimal degree of excess capacity to be built into a new facility has motivated an admirable stream of literature with a noteworthy extension provided in [START_REF] Manne | Capacity Expansion and Probabilistic Growth[END_REF] that analyzed the case of a random-walk pattern of trends in demand (see [START_REF] Luss | Operations Research and Capacity Expansion Problems: A Survey[END_REF] for a complete survey).

In this subsection, a simple framework is used to analyze whether that "building ahead the demand" behaviour can be observed in the gas transmission industry. For expository reasons, a simple discrete time context will be used (extension to continuous version is straightforward). To do so, I focus on the case of a firm that, at date 0 t = , is considering the construction of a single-line transportation infrastructure that is expected to transport a given steady daily flow of gas 0 Q over a predefined planning horizon of Y years. Given the long durability of gas transmission equipments, it is assumed that there will be no equipment replacements during that period.

Compared to the previous models, we now consider the case of a possible future expansion of the output. Hence, there is a known date T , { } 1,..., T Y ∈ at the beginning of which the possible output expansion will be decided or not. The decision outcome is still uncertain but is assumed to be restricted to two cases: either a sudden output expansion to a known value 1

Q , with 1 0 Q Q > ; or a
status quo to 0 Q . The probability to observe a rise to 1 Q is denoted p .

If such an additional output was to be decided, the cheapest way to accommodate that additional flow would be the addition of compression horsepower. Hence, the planner's set of decision variables can be restricted to three elements: the pipeline diameter D , the initial compressor horsepower 0 H required to move 0 Q , and the compressor horsepower 1 H that could eventually be needed to transport , C H H ∆ and is equal to:

( ) ( ) ( ) ( ) ( ) 0 1 3 1 3 0 4 1 4 0 , . ; H C H H C H C H C H C H α ∆ ∆ = - + -    
where

( ) ( ) ( ) ( ) ( ) ( ) 0 1 0 1 0 0 1 , , 0 0 1 1 1 , , , . , 1 s.t. , , 0
, , 0

T D H H Min C D H H LRTC D H p C H H r F D H Q F D H Q = + ∆ + = =
Using the previous modus operandi, this problem can easily be rearranged into a single-variable objective function to be minimized. In fact, the optimal pipeline diameter computed for a probability p is called * p D and must minimize the expected total annual cost function:

( ) ( ) ( ) 0 1 : , , Q Q C D C D f D f D a
With the functional specifications chosen by Dr. Yépez, this function has a unique minimum which is the only point where the derivative of that function with respect to pipeline diameter D is equal to zero . Thus, a necessary (and in that case sufficient) condition for a minimum is:

( ) * 0 p dC D dD = (8)
That condition has its economic interpretation: at the optimal diameter the marginal increase in the pipeline cost is exactly equal to the expected marginal reduction in compression cost.

As a benchmark, a planner could find interesting to confront that outcome with the optimal diameter * 0 D that would have been selected in case of a zero probability for the sudden net increase in output.

Proposition 2:

Assume a gas transmission firm with a probability p for an sudden rise in its output at date T , with costs and engineering production function as described above. For that firm, the long run optimal combination of equipments to be installed at date 0 t = involves a pipeline diameter * p D that is larger than those, denoted * 0 D that would have been installed by a planner that do not take into consideration this possible future rise in output.

Again, a straightforward proof of that result is provided in Appendix 2.

In many countries, the level of prices charged by gas transmission firms is subject to public control and this result has important implications for the design of appropriate regulatory policies. In many cases, a rate of return regulation is implemented. This form of regulation sees costs as exogenous and observable and forms prices on the basis of observed costs and appropriate rate of return on capital.

One of the principal criticisms that has arisen for this kind of regulation is based on the so-called Averch-Johnson effect. According to [START_REF] Averch | Behavior of the Firm Under Regulatory Constraint[END_REF], the profit seeking behaviour of the regulated firm subject to rate of return regulation induces a distortion in the input choice: the optimal choice of that firm is not the cost minimising one. More precisely, the capital/labour ratio chosen by the firm subject to rate of return regulation is greater than the cost minimising capital/labour ratio for the given the level of output. Obviously, such a statement calls for a condemnation of the tendency of regulated firms to engage in excessive amounts of durable capital accumulation to expand the volume of their profits. The previous proposition suggests a rather different explanation for the firm's preference for a capital intensive technology. Here, the firm's decision to choose an input mix that includes a larger diameter/horsepower ratio (in economist's jargon: a greater capital/energy ratio) is completely independent of the regulatory environment, such a choice is entirely motivated by an anticipation of possible future output expansion.

Conclusions

Throughout this article the cost functions for the natural gas transmission industry presented in [START_REF] Chenery | Engineering production functions[END_REF] and [START_REF] Yépez | A cost function for the natural gas industry[END_REF] are discussed in the light of some the key features of that industry.

A commented review of the Chenery-Yépez methodology has enabled the derivation of some interesting insights such as an empirical quantification of the scale economies encountered in that industry and an economic analysis of "looping". Moreover, two notable extensions have been provided to deal with the case of a seasonal varying output and with those of an uncertain future output expansion. All these results have important implications for the design of appropriate regulatory policies in that industry.

In the vein of [START_REF] Chenery | Engineering production functions[END_REF] and [START_REF] Yépez | A cost function for the natural gas industry[END_REF], the technical representation of the gas transmission industry used in this paper remains simple: a radial infrastructure that transports an unidirectional flow gas over a given distance. In many countries, the development of the natural gas industry came with the construction of a large meshed transmission network and the optimal operation of these infrastructures offers a stimulating field of research for the operations research community (e.g.: [START_REF] Hansen | Optimization of pipe networks[END_REF]André et al., 2009). Nevertheless, the Chenery-Yépez method has a great merit: it offers a simple engineering-based approximation of the cost functions encountered in the natural gas transmission industry. For this reason, it is worth being considered as a valuable tool to get a better understanding of the gas pipeline economics. Moreover, it provides an useful complement to the purely statistical approach in the determination of production, cost, and factor demand relationships. A tribute must thus be paid to the late H.B. Chenery for his inspiring work.

APPENDIX 1

All the numerical simulations presented in this paper are entirely based on the following assumptions chosen for an hypothetical 100-miles long project ( l = 100 miles). The technical specifications for the gas are exactly those used in Yépez ( 2008 This formulation has been suggested by stability of pipe concerns (cf. Ruan et al., 2009, p. 3044).

Hence, the annual cost of the line per mile is ( ) Q.E.D.

Proof of Proposition 1:

The short run total cost to transport an average daily flow q on a transmission infrastructure designed to transport a peak output p Q is given by: (  )

( ) ( ) ( ) ( ) 4 , 1 . p p p Q Q q Q p q SRTCC D SRTC D C f D Q   = --      
Q Q Q Q Q p dSRTCC dC q D f D f D dD Q dD   = --      
1 1 0 0 * ' * ' * ' * ' * 0 0 3 0 0 3 0 Q Q Q Q d D f D C f D f D C f D dD ω   = - × - ×  
Let us have a look at the sign of ( ) ( )

1 0 ' * ' * 0 0 Q Q f D f D - . At
1 0 ' * ' * 0 0 0 Q Q f D f D < < , hence ( ) ( ) 1 0 ' * ' * 0 0 0 Q Q f D f D - > .
With Dr. Yépez's strictly concave and increasing specification for (  )

0 1 ' * ' * 3 0 3 0 0 Q Q C f D C f D > > .
As a result:

Q Q Q Q Q Q Q f D C f D f D C f D f D f D C f D × - × > - ×
As the right-hand side is strictly positive, we have:

( ) * 0 0 d D dD ω < .
As 4 C is also a smooth concave functions, a similar line of reasoning can also be applied to show that the derivative of ( ) ( ) ( ) ( )

1 0 * * 4 0 4 0 Q Q C f D C f D -
with respect to D is also strictly negative.

As a result, we have:

( ) * 0 0 dC D dD < .
Q.E.D.

  line per mile dependent on the diameter; L α is the fixed-cost annual percentage

α

  the sum of two smooth functions of H the horsepower: is the fixed-cost annual percentage charge dependent on the depreciation and real interest rates, the replacement value of the compressor station dependent on the installed horsepower capacity and ( ) 4 C H is the operation and maintenance cost.

  on the annual costs of the line and of the compressor stationThe specifications chosen for the annual costs functions

  a choice that he justified with an empirical argument based on the result of a personal investigation. Regarding that empirical section, a representative econometrician could regret the absence of some of the classical attributes of a typical empirical study:

  Given that Dr. Yépez's analysis differs from Pr. Chenery (as it includes a modernized version of the technological relations and a non-linear specification for the inputs costs , the validity of this approximation needs to be validated with a statistical approach.

  compression horsepower. As it will be shown below, this remark plays an important role to analyze the behaviour of the Short Run Total Cost function ( ) SRTC Q . This approach allowed Dr. Yépez to derive a single variable expression for SRTC the Short-Run Total Cost function. Here, ( ) SRTC Q is the sum of two elements: a fixed pipeline cost the annual cost of the compressor station. A U-shaped short-run average cost curve Dr. Yépez's Figure 2 (Yépez, 2008, p.80) provides an intriguing suggestion. On that figure, a couple of strictly decreasing Short Run Average Cost functions are plotted. All these curves suggest the presence of some scale economies in the short-run cost functions. But it is perilous to derive definitive conclusions from such a graph as curves have only been plotted on a limited range of output. Some complementary remarks are thus needed to discuss the short-run economics because the presence of increasing/decreasing returns to scale in the short-run is, among other factors, heavily influenced by the functional form chosen for ( ) H C H . Depending on that choice, rather different conclusions can be obtained. There are slight differences in the specifications chosen by Chenery and Yépez that deserves a discussion. When H C is an increasing convex function (which includes the linear specification used in Chenery (1949)), the Short-Run Marginal Cost function ( ) SRMC Q will be strictly increasing with respect to output and ( ) SRMC Q will always be greater than the short run average variable cost ( )

Figure 1 :

 1 Figure 1: U-shaped Short Run Average Cost curves.

  Figure 2: Short-run and intertemporal total cost curves for natural gas pipelines.

  usually a strictly increasing function of the pipeline diameter, whereas the annual compression cost is strictly decreasing with respect to D because of the technical substitutions mentioned earlier. A simple reformulation of the long-run cost minimization problem can now be proposed. Each strictly positive value of D corresponds to a unique Short-Run Total Cost function that is parameterized by a continuous parameter: Q . Hence, a family of Short-Run Total Cost functions indexed by a continuous variable D can be defined. For a given output Q , each of these function provides a short run total

  , it is clear (Rolle's theorem) that there is at least one solution to equation (4). Of course, uniqueness of that solution depends on the functional specifications chosen for both D C and H C . With Pr. Chenery's linear specifications, the function ( ) Q SRTC D is strictly convex and has thus an unique global minimum. With Dr. Yépez'sconcave specifications, convexity is no longer verified but the numerical values used in[START_REF] Yépez | A cost function for the natural gas industry[END_REF] insure a U-shaped curve with the succession of two strictly monotonic patterns: a decreasing one and an increasing one. As a result, there is also a unique minimum cost with Dr. Yépez's specification.Hence, the problem at hand is well behaved as both Dr. Yépez's and Pr. Chenery's specifications are consistent with the existence of a unique solution to equation. Let's denote * run cost minimization programme proposed byChenery and Yépez. Here comes an interesting lemma:Lemma 1: Assume a gas transmission firm with costs and engineering production function as described above. For that firm, the long run optimal combination of inputs ( ) single variable problem above.

Figure 3 :

 3 Figure 3: The Long Run Total Cost curve as the lower envelope of the Short Run Total Cost ones.

  total expansion cost incurred to serve a larger steady flow-rate of natural gas 0 Q Q ≥ is given by Dr Yépez's short-run total cost function ( ) SRTC Q that includes the extra capital expenses required for the installation of an additional pumping capacity. In case of a contraction in the operation and maintenance costs are reduced to reflect a lower annual rate of operation of the compressors. As most of these operation and maintenance costs correspond to the cost of the energy (gas or electricity) used to drive the compressor, a linear scale down rule can reasonably be posited to model those variations. These costs are thus supposed to vary in proportion to the capacity-utilisation rate 0 Q Q ; here, these variable costs are equal to ( )

  would have been chosen if the firm had to transport a steady flow p Q . As the infrastructure must be designed to transport the peak flow p Q , each of the two following combinations of inputs

1Q

  from date T to Y . total cost of an equipment designed to transport a steady daily flow 0 Q during the predefined planning horizon Y . Besides, planners have to take into consideration a possible extra compression cost in case of an output expansion. It is assumed that the increased horsepower 1 H can be obtained by installing, at the beginning of year T , more compressors in addition to the existing ones at a cost new compressors will only be used during Y T -. Evaluated at year T , the annual equivalent extra-cost required to install and operate these new compressors during Y T -

Hα∆

  is the fixed-cost annual percentage charge dependent on the depreciation and real interest rates to operate those additional compressors duringY T -. In fact, H α ∆ is obtained by computing the constant annual outlay stream that has an expected present value equal to that of all future cost outlay over the horizonY T -. In most real cases, we have neutral planner is supposed to minimize the expected total annual cost of that infrastructure H subject to the usual engineering equations:

  dimensionless constants have the following values: 0 c = 0.742 and 1 c = 183.2[START_REF] Yépez | A cost function for the natural gas industry[END_REF]. The annual cost parameters (in US Dollars) are also those presented in[START_REF] Yépez | A cost function for the natural gas industry[END_REF]: pipe thickness τ is assumed to be a linear function of D the inside diameter:

  short-run total contraction cost. The derivative of that second function with respect to D is equal to:

QD

  strictly decreasing function and 4 C is an increasing one, the right-hand side of this equation is strictly positive which means that the short-run total contraction cost is locally strictly increasing with D in the neighbourhood of the particular diameter * function with a unique minimum, it indicates that the optimal diameter * , larger compression capacity than those chosen in case of a steady demand equal to p As in the proof of Proposition 1, the sketch of proof relies on an evaluation of the sign of the derivative of the expected total annual cost function with respect to pipeline diameter D evaluated for the particular diameter * 0 D . As C is a smooth function with a unique minimum, this information allows to conclude on the relative size of * 0 satisfies condition (8) in the particular case of 0 p = . In case of 0 p > , the derivative of C with respect to D is equal to:With the engineering relation at hand, it is clear that pipeline and compression are two substitutes: any increase in the pipeline diameter reduces the horsepower capacity required to move the gas. strictly negative which gives:

  this stage, it is worth studying the influence of the parameter Q on

Table 1 : Empirical results obtained from an OLS regression on Chenery's specification. The numbers in parentheses are standard errors of coefficients.

 1

	( log LRTC	)	=	12.34312	+	0.613467	( ) log Q
				(0.000984)		(0.000145)
			R 2 = 0.999995 ; S.E. of regression = 0.001218.

With the functional forms chosen in his 2008 article, Dr. Yépez notes that the second-order conditions for the existence and uniqueness of a minimum are satisfied.

For example: in the data set described above, the share of the pipeline-related costs is always greater than

% of the longrun total cost.