Charlotte Fourcroy

Faïz Gallouj

Clersé

Fabrice Decellas

Economics, Challenging the myth of non-materiality: energy consumption

Keywords: Service industries, Energy consumption, Materiality

This paper addresses the issue of energy consumption in service industries. Official energy statistics show that the amount of energy consumed in service industries is quite small, and this is explained in the theoretical literature by the presumed intangibility of these services. Our hypothesis, however, is that the energy consumed by services has been underestimated. After identifying and analyzing the full range of sources of energy consumption in service industries, we show that official energy statistics account for only a part of this consumption. Our work is based on a thorough rereading of the economics literature on services (especially concerning the definition of services and the specifics of the service provision process), done in light of our knowledge about energy consumption.

Introduction

In economic theory, services are often described as "non-material", "immaterial" or "intangible" [START_REF] Smith | The wealth of nations[END_REF]Say, 1803;[START_REF] De | Nouveaux principes d'économie politique ou De la richesse dans ses rapports avec la population[END_REF]. This non-materiality would explain their having smaller negative impact on the environment and lower energy consumption than does industry. Inasmuch as growth in recent decades has been largely driven by the development of services, a number of studies, particularly in the early 1990s, have argued that we are trending towards an ever more non-material society creating less environmental wear (Ettighoffer, 1992;[START_REF] Joseph | The internet economy and global warming, The Center for Energy and Climate Solutions SAY Jean-Baptiste[END_REF]. Against such conclusions, however, we would note that the most service-intensive countries are as a whole greater consumers of energy, with a larger ecological footprint, than countries whose tertiary sector is less developed (Gadrey, 2010). Given the vital stakes involved in bringing energy demand under control, primarily due to the threats of diminished fossil resources and climate change, it is important to come down on one side or the other of these opposing arguments and identify the conditions that might foster sustainability in the service economies.

In this study we consider the question of the non-materiality of services from the viewpoint of their energy consumption. The following are our hypotheses: (1) service businesses are not as non-material as has been imagined, and (2) they consume non-trivial amounts of energy, which are underestimated in the official statistics. To consider this question it is necessary to understand the forms which energy demand takes in the service sector. There are rather few theoretical studies concerned with energy consumption in the services sector. Our analyses are based on a review, guided by the question of energy consumption, of the theoretical literature on services which is concerned with their definition and an analytical breakdown of how they are provided.

The first part of the paper is devoted to a discussion of what might be called the myth of the non-materiality of services. In the second part we attempt, on the basis of a theoretical breakdown of the provision of services, to identify all the sources of direct energy consumption by services. We show that the official energy statistics account for only a part of energy consumption. By taking a life-cycle perspective on services, we have broadened our view so as to identify energy consumption indirectly related to the provision of services. The third and final part of the paper is devoted to an attempt to estimate the magnitude of neglected energy consumption, or at least, within the limited scope of this paper, some of it.

The myth of the non-materiality of services

One can trace the origin of the term non-materiality (or intangibility) of services to Adam Smith and his analysis of the accumulation of capital. In his work The Wealth of Nations (1776), Smith distinguishes two types of economic activity: productive work, like that of the worker who adds value to the materials he handles and transforms, and unproductive work which adds no value. Among the unproductive activities Smith identifies the activities we today term services. 1 Although the adjective "non-material" is never directly used, the idea of non-materiality is implicit, as the following examples illustrate: "the [unproductive] labour does not fix or realize itself in any permanent subject, or vendible commodity, which endures after that labour is past, or for which an equal quantity of labour could afterwards be procured" (ibid., p. 430), "the work of all [the services] perishes in the very instant of its production" (ibid., p. 431).

In a discussion of the accumulation of capital, Jean-Baptiste Say (1803) questions the term unproductive. Indeed, although they produce no value, the services are to Say useful and sources of visible outcomes (e.g. the cure produced by a physician). He therefore introduces the term immaterial.

Most of his contemporaries used some version of this term, which was to become the leading feature used in describing service businesses. To 19th century economists the non-materiality of a service meant that the service is consumed and produced simultaneously (Say, 1803), that its outcome does not take a material form, that it cannot be accumulated [START_REF] De | Nouveaux principes d'économie politique ou De la richesse dans ses rapports avec la population[END_REF] and that it does not produce an exchangeable value (H. Storch) 2 ).

In the 20th century, when the notion of a services sector first appeared, the adjective non-material was settled on it as well. The idea of non-materiality of services was then fleshed out by an assumed lesser usage of plant and equipment (i.e. less capital intensity). Thus Colin Clark observed that most service businesses require far less in the way of capital goods than industry or agriculture (Clark, 1940, p. 207) A number of contemporary studies, however, question the notion that non-materiality fairly characterizes services (Fuchs, 1968;Delaunay, Gadrey, 1987;[START_REF] Illeris Sven | Services and regions in Europe[END_REF]Gadrey, 1992).

Transportation and communications services are among the most capital intensive. They require the use of a great deal of equipment. Furthermore, certain services are performed upon media which are tangible goods (book and newspaper publishing, sculpture, painting, 3 etc.), while other services repair, maintain or transform tangible goods (repair and maintenance services, not to mention restaurants and food services), or move them about (transportation, wholesale and retail trade, etc.) or make them available to users (rental services). In contemporary economies, the invasiveness of information and communications technologies also casts doubt on this notion of non-materiality. All service businesses today make use of information and communications technologies (ICTs), which in some cases give a physical form to the service (a recording of a course, for example).

Be that as it may, the notion of the non-materiality of services remains a chief feature by which service operations can be distinguished from other economic activities. In their attempts to propose a positive definition of services4 , Peter Hill and Jean Gadrey preserve non-materiality as an essential characteristic of services. Thus Peter Hill (1977) 3 These activities are classified as services since it is generally acknowledged that their informational content exceeds their physical content (Illiens, 1989). According to INSEE, printing and the mass production of recordings belong in the industrial sector. person or a good5 , and Jean Gadrey stresses that a service business does not result "in the production of a good that can circulate economically apart from its medium" (Gadrey, 1992, p. 19). Other arguments can be brought in to support the continued notion of non-materiality as a relevant analytic category to describe services. Such is the case, for instance, with the analysis of the post-industrial society dominated by service businesses and founded on knowledge and information (Bell, 1973;[START_REF] Stehr | Theories of the information age[END_REF]Aoyama et Castells, 2002;Foray, 2000).

Against the current background of diminishing natural resources and environmental deterioration, the notion of non-materiality of services has often been taken to mean that services have a low impact on the environment and in particular that they consume less energy [START_REF] Illeris Sven | The nature of services[END_REF][START_REF] Rifkin | Quand tout devient service" in Rifkin Jeremy. L'âge de l'accès : la révolution de la nouvelle économie[END_REF]OCDE, 2000). This idea is corroborated by a number of empirical observations. In France's energy accounting, services in 2006 represented only 14% of total consumption [START_REF] Mairet | Déterminants de la demande d'énergie dans le secteur tertiaire en France, une analyse technico-économique[END_REF] whereas in the same year they represented 75.1% of value added and 74.7% of total domestic employment (INSEE national accounts). Thus the energy intensity of the value added by service businesses is on average much lower than by other sectors. Likewise, in France's CO2 emissions accounting (data from CITEPA6 ), the services sector represented only 7% of emissions in 2008. The idea that services, being non-material, consume less energy and exert less pressure on the environment is a highly attractive one, as it means that the increasing service orientation of an economy, which is the unstinting trend in every developed country, will automatically foster sustainability. This idea has taken hold with governments and is defended by the OECD (OECD, 2000) and the European Union in its Lisbon Strategy.

We will devote the sections that follow to demonstrating that the non-materiality of services-the presumed basis of their low environmental impact-is a myth and to examining the consequences of that in terms of energy consumption.

Finding overlooked sources of energy consumption

The task before us at this point is to propose a theoretical breakdown of service provision, which reveals the main dimensions of its materiality and therefore its main sources of energy consumption.

We distinguish between direct energy consumption, which is that required for and during the provision of services, and indirect energy consumption, which is brought about in the rest of the economy by service businesses.

Breakdown of service provision and direct energy consumption

Our theoretical breakdown of service provision is based on a rereading, from an energy viewpoint, of works that focus on the definition of services. The earliest works dealt largely with the distinction between goods and services and with the shared characteristics apparent among all services [START_REF] Smith | The wealth of nations[END_REF]Say, 1803;[START_REF] De | Nouveaux principes d'économie politique ou De la richesse dans ses rapports avec la population[END_REF]Marx, 1971;Fisher, 1939;Clark, 1940;Fourastié, 1949). Although a number of contemporary authors have attempted to work out a truly positive definition of services (Hill, 1977 and[START_REF] Joseph | The internet economy and global warming, The Center for Energy and Climate Solutions SAY Jean-Baptiste[END_REF]Gadrey, 1992) 7 none of their definitions is entirely satisfactory, and the boundaries of the services sector remain fuzzy and dependent on the conventions chosen [START_REF] Illeris Sven | Services and regions in Europe[END_REF]. Still other studies have concentrated on describing service businesses within themselves (Gadrey, 1991;Gallouj, 1999;Eiglier, Langeard, 1987). And this is the perspective we take as well.

A service rendered by a provider for one or more customers (consumer or user, payer or beneficiary) will be a combination of various operations (Gadrey, 1991;Gallouj, 1999):

 logistics and material transformation (M), which consist of transforming, moving or maintaining tangible objects;

 logistics and information processing (I), which consist of producing, entering, circulating, archiving and processing coded information;

 the mental processing of knowledge (K); 7 The definitions of services formulated by these two authors are mentioned in part in the preceding section.

 relational services (R), which are direct services performed on the customer himself.

Performing these operations results in an outcome that can be described in terms of the final features of the service (Gallouj, Weinstein, 1997;Gallouj, 2002). In the case of lodging services, for example, the service features are primarily rest, comfort, cleanliness, access to recreation, or information, etc. Weinstein, 1997 ;Gallouj, 1999 ;Gallouj, 2002).

The values prized by providers and their customers will also play a part in the provision of services.

Values and competencies, when aligned with motivation, determine the behaviour of individuals [START_REF] Mcclelland David | How motives, skills, and values determine what people do[END_REF]. The customer is at the centre of the provision of services. 9 His or her part in the production of a service can take different forms-from self-service when using an automated cash dispenser to collaboration with the provider in the case of advisory services. This participation can even be solicited in advance of the production of service; the customer must at times travel to where the service is performed. The customer's level of involvement will vary. For certain services (a plane trip, for instance) only the customer's presence is necessary; for others (such as a training program) his attention is required (Eiglier, Langeard, 1987;[START_REF] Christopher | Marketing des services[END_REF]. One portion of the service often takes place in direct contact with the customer 10 ; another portion is invisible to him, this being the internal organization of the service company (also called the back-office), consisting, for example, of the marketing, finance, human resources or logistics departments (Eiglier, Langeard, 1987).

8 These features are not always easy to identify, especially in the case of non-standardized services requiring a high level of qualification, such as intellectual and professional services (Gallouj, 2002;Gallouj, Weinstein, 1997).

9 It is even the view of Eiglier and Langeard (1987) that without the customer there is only a "potential" for service. Thus, for example, if a hotel room is not occupied for one night, there are no services. One may object, however, that it all depends on the definition of services that is used, since a part of the production of the service has taken place-the preparation of the hotel room. We shall return to this point when considering a broader definition of services. Furthermore, the participation of the customer is not necessary with respect to prevention services (defense, police, security patrols, etc.)

10 However, ICTs include the remote delivery of services.

Most of the above-mentioned technologies consume energy in order to function. These energyconsuming technologies are a major part of the information and communications technologies used in information processing operations. They include office machinery (computers, fax, telephones, photocopiers, etc.) and network and telecommunications infrastructure (servers, hubs, data centres, telephone exchanges, etc.). Today in France, almost all service businesses are equipped with computers and connected to the Internet [START_REF] Sessi | Tableau de bord des TIC et du commerce électronique, Édition décembre 2008[END_REF]; and 20% of the energy consumption in the services sector is directly attributable to the use of ICTs 11 [START_REF] Mairet | Déterminants de la demande d'énergie dans le secteur tertiaire en France, une analyse technico-économique[END_REF]. In the area of materials processing operations, moreover, certain service activities require the use of other specialized equipment. Here we are talking about such things as equipment for material handling, cooling systems, cooking (restaurants), sanitary hot water12 (swimming pools) or other particular production processes (such as laundries or bakeries).

Services, however, are not limited to the M, I, K and R operations we have just mentioned and that we propose designating "service delivery". Other activities related to the provision of services must be brought into any discussion of energy consumption.

Indeed, service deliveries necessitate interactions between provider and customer, and thus usually travel by one to the other, before the delivery. Students and teachers travel to their campus, patients to their doctor, etc. 13 This travel is a source of pollution (Gadrey, 2010; Desmarchelier and Gallouj, 2010) and energy consumption.

Service delivery also requires various preparatory activities, such as preparing the place where the service is to occur (improvements, cleaning, heating, etc.), as well as the preparation of the provider (training, lodging, etc.). This preparation does occur only in advance of the delivery; it can be extended during the delivery (heating or illuminating the space, for example). For this reason, rather than the term "preparation" we prefer to use "conditioning" to denote this type of activity. A portion of this conditioning requires energy consumption, primarily for heating, sanitary hot water, lighting and air-conditioning the premises where the service occurs. This energy consumption is not specific to services, in that it also affects industry. What is specific is the weight it has in a company's total energy balance. Heating represents half the energy consumed in the services sector. One explanation for this trait is that, as opposed to industrial operations, service operations generally use less heavy equipment needing high energy inputs. Another explanation is the importance of personal meeting and meeting places in the provision of services. A service is produced jointly; customers and providers usually interact in a common space, which has to be conditioned for the services to be provided.

11 That is, about 4% of final energy consumption in the tertiary sector.

In sum, service provision is a combination of the following three elements: delivery, conditioning and travel-each of which represents a separate, direct consumption of energy. Depending on the type of service business considered, there will be some variance in the activity level, operating cycle, need for delivered services, conditioning or travel involved. Performing these various services prompts a need for "energy services", which is to say, for example, a task, a change of place or an increase in temperature. This need for "energy services" varies as a function of a number of exogenous factors, including the outdoor climate and sources of indoor or cost-fee heat 14 . The actual energy consumption necessary to meet this need for "energy services" is, for its part, a function of the tangible property used and its technical characteristics-such as energy-using machinery and the building in which it is located-as well as how this property is used-i.e., individual behaviour. 15 The general process by which energy demand is formed is illustrated in Erreur ! Pas de séquence spécifié.

Figure 2. General Process of the Formation of Energy Demand

Not all the sources of energy consumption and related consumption that we have just identified in this analysis are accounted for in the official statistics (Erreur ! Référence non valide pour un signet.).

In the CEREN statistics 16 , the energy consumption of the services sector is limited to those associated with the buildings used by the sector, that is, to the energy consumption of the equipment used on the premises 17 of service companies (CEREN scope 18 ). Thus, no accounting is given of energy consumed 14 Indoor or cost-free heat largely derives, respectively, from other occupants or machinery in the building and sunshine.

15 This approach to energy consumption is inspired by that proposed by Château and Lapillonne, cited by Nicolas [START_REF] Mairet | Déterminants de la demande d'énergie dans le secteur tertiaire en France, une analyse technico-économique[END_REF].

16 CEREN: Centre d'Etude et de Recherches Economiques sur l'Energie [Centre for study and research into energy economics]. This is the major organization in France supplying energy statistics on the tertiary sector. 17 By service business premises we mean the entire floor area of service companies, including any parking lots or other outside areas. The premises may be owned or leased. in the course of providing the services from the operation of off-site equipment or means of transportation (for travel services, but at times for delivered services). Yet it is true that services, especially as they are produced jointly, require considerable travel and sometimes occur completely or in part outside of the service company. We have identified three sources of materiality in services that constitute as many sources of direct energy consumption: service delivery, travel and conditioning. Yet still other energy consumption is necessary for services, even if the energy in question is not directly consumed during the provision of the service. That is the topic of the following section.

Type of operation

Level of activity

Energy consumption indirectly related to the provision of services

As we have just emphasized, the operation of equipment enlisted for delivery, conditioning and travel makes up a sizable source of energy consumption. We must not forget, however, that this equipment had to be made and that its manufacture is also a source of energy consumption-energy for the design of the project, the extraction and transportation of the necessary raw materials, the processing of raw materials and fabrication of the product. Marketing this equipment, maintaining and recycling it also give rise to energy expenditures. All of this energy consumption, although indirectly related to the provision of services, cannot be excluded from a discussion of energy consumption in the tertiary sector. By including this indirect energy consumption into our analysis we are adopting a product life-cycle approach. The value of such an approach is that it gives a truer picture of the energy impact 18 CEREN defines the boundaries of the services sector in terms rather similar to those proposed by INSEE. (and more broadly, the environmental impact) of service operations. But to do so means defining the boundaries of the system we are studying.

Accordingly, in the indirect energy consumption of a given service, which we shall call "the final service", one must include the grey energy 19 of equipment used in delivery, conditioning and travel.

Producing the final service also requires other types of goods, such as buildings, furnishings, infrastructure (transportation and telecommunication) and other intermediary consumption of goods (food products, for example, in a food service business). The grey energy of these goods is tied to the production of the final service, and we choose to include it in our approach.

Apart from equipment and tangible goods, producing a final service also at times requires the intermediary consumption of other services. The marginal increase in activity of these intermediary services and the associated marginal energy consumption are tied to the production of the final service.

The energy consumption of these intermediary services is itself defined as the sum of energy consumption in the operation of their energy-using equipment and the grey energy of their equipment, furnishings, buildings, infrastructure, etc. represents the system as seen at the scale of the final service. This energy consumption indirectly related to service operations is plainly not included in the official statistics on the tertiary sector. 19 The expression grey energy or embodied energy of a good means the total energy consumption necessary for the production, manufacture and recycling of a good, i.e. the phases that precede and follow the use of a good. Lastly, we are proposing an alternative methodology to account for energy consumption, one that derives from our theoretical construction of the sources of energy consumption in the tertiary sector.

Our hope is to contribute a more realistic assessment-which is to say, a reassessment-of the amount of energy necessary to provide a service. It is not our claim, however, that we are proposing a re-allocation of energy consumption among the various economic sectors. It might become necessary to consider that some portion of services is directed towards industrial production.

Within the scope of the French tertiary sector, the energy consumption overlooked by the official statistics is as follows:

-One part of energy consumption that goes for operating service equipment:

(1) Energy consumption for the operation of means of transportation (in service travel and delivery);

(2) Energy consumption for the operation of off-site equipment used for service delivery and conditioning.

Etc.

*Production in the broad sense. This includes: the design of the project, the extraction and transportation of the necessary raw materials, the processing of raw materials and fabrication of the product.

-Energy consumption indirectly related to the provision of services:

(3) The grey energy of equipment, furnishings, buildings, infrastructure and all intermediary consumables;

(4) The energy it took to produce imported intermediary services 20 .

The total of this energy consumption is not available in the form of invoices or shipping statements.

This consumption is classified with other energy consumption not related to services-in the energy consumption of other institutional sectors. The total therefore has to be estimated. Such work generally requires using surveys of a representative sample of the population. To the traditional biases that characterize surveys 21 , one might have to add occasional lack of information or ignorance on the part of the respondents 22 . On account of such difficulties and given the available data, we restrict ourselves in this paper to estimating energy consumption for the operation of means of transportation, i.e., type 1 consumption. By demonstrating that type 1 energy consumption is not trivial compared to the official tertiary sector energy consumption figures and that it represents a significant portion of energy consumption in the transportation sector, we are confirming empirically what we had previously conceived theoretically: that the tertiary sector consumes more energy than the official energy statistics would lead one to believe.

Nor is the total amount of type 1 energy consumption available on invoices or shipping statements. To estimate its size we have made use of the findings of the 1993-1994 French national transportation and travel survey (French acronym ENTD) 23 . Although this survey is nearly 20 years old, it does allow us, by comparing the findings made about energy consumption in the tertiary sector in 1993, to estimate the approximate extent of the consumption for travel overlooked by the official statistics. A representative sample of the French population was questioned as to the trips they had made. For a few representative days (one weekday and one weekend), the respondents furnished the reasons for, the distances involved in and the means of conveyance of all of trips they made. The respondents also supplied information about themselves, including their socio-professional status. Making a few assumptions, which we lay out in Section 4.1, we have estimated the travelling done by consumers and providers of services (section 4.2). 20 Within the French tertiary sector and to avoid double counting services, intermediary services are limited to imported intermediary services. 21 Especially the bias that misunderstanding or social acceptability can throw into responses. 22 This is particularly the case with households who do not know their energy consumption by type of use. 23 The transportation survey was repeated in 2007, but the data to which we have access are incomplete and do not permit us to estimate type 1 consumption.

Assumptions

In order to estimate from the INSEE transportation survey the energy consumption created by all the travel originating in the consumption or production of services, we formulated several assumptions relative to the reasons for trips made, the business sectors involved, the combination of means used and the energy efficiency of those means. a) Reasons for travel For every trip made, the survey results show the reason for it, taken from a predefined list. We have organized this list by distinguishing those reasons we deemed to be directly related to the consumption of a service and those we deemed to be job-related (including commuting from home to work). The survey provides no information on the business sector of the individuals surveyed. We therefore relied on their socio-professional status (SPS), which was part of the questionnaire, and we posited that within a single socio-professional status the distances travelled are spread evenly among those working in the different sectors-i.e., that the share of distance travelled by service providers is proportional to the share of workers in the tertiary sector. INSEE gives the data for the intersection of socio-professional status and business sector. The data are not available for the 1993-1994 year of the survey. The closest date for which they are available is 1999. 24The data are presented in Erreur ! Référence non valide pour un signet. given on the total distance travelled and the different conveyances used, but distance is not broken down by conveyance. We have therefore made estimates. We made the assumption that for each trip the first two means of travel mentioned are the chief, if not the only, means used. We then formulated a number of assumptions as to the breakdown of distance travelled between these two means (Table 3), based on a study conducted as part of the ETHEL project. 25 The results of this study show, for example, that on average an individual who makes a trip by car and by train covers 15% of the distance by car and 85% of the distance by train. Lastly, we made assumptions as to the per unit consumption of each means of transportation. For mass transit, we made assumptions about the energy consumption per kilometre travelled and per traveller.

With regard to two-wheeled vehicles and cars, insofar as in both cases the number of people in or on the vehicle is indicated in the ENTD survey26 , we preferred to consider the energy consumption by kilometre travelled and then divide it by the number of people in or on the vehicle. For truck transport, we assumed that the driver is alone in the vehicle.

We ignored energy consumption related to travel by foot, bicycle, boat and conveyances not in the survey. (This approximation seems reasonable in that according to the findings of the ENTD 1993 survey, these last two means of transportation represent less than 6% of the kilometres travelled by the French public.)

The ENTD survey provides useful data for estimating the energy consumption of automobiles. It provides various data about vehicles owned, in particular their average consumption, in and out of the city, by type of energy consumed (diesel or petrol). We make the assumption that the weighted average of these consumption numbers (weighted by the proportion of households in the French population) is representative of the average consumption per kilometre of all cars and taxis. The results obtained are presented in Erreur ! Référence non valide pour un signet. A litre of petrol equals about 9.85 kWh and a litre of diesel, 10.58 kWh. According to the ENTD 1993 survey, 72.3% of cars on the road in France run on petrol.

The breakdown of trips between city and non-city driving can also be estimated from some of the survey data. The survey results cover total trips, as well as, at times, the number of kilometres travelled in the city, on the highway and on other roads. We made the following assumption: trips made on highways and other roads equal the non-city trips. Trips for which the kilometres travelled on each type of road are reported in the ENTD survey allow us to calculate a pattern of trips among the different geographic areas. We made the assumption that this pattern applies to the breakdown of all trips. We found that 20% of the kilometres are travelled in the city and the rest outside the city.

Finally, according to our calculations, cars and taxis in 1993 consumed on average 0.81 kWh/vehiclekilometre.

To estimate the consumption of the other means of transportation, we relied on a study conducted by the Deloitte firm for ADEME (ADEME/Deloitte 2008). This study is recent, whereas the ENTD survey dates from 1993. We therefore made assumptions regarding the improvements in energy efficiency of transportation between these two years. Between 1990 and 2009, the total energy efficiency of transportation appears to have improved by about 12%. 27 We therefore made the assumption that for each means of transportation the per unit consumption when in use decreased by 12% between the time of the ENTD survey and the ADEME/Deloitte study. In addition, our transportation categories are not completely identical to those in the study. We made assumptions to establish some correspondence between them. The per unit consumption of school and company buses (codes 41 and 42) are thought to be similar to coaches; that of trolleys and other unlisted roadway mass transit (codes 55 and 59), similar to buses; that of trains would be the average of high-speed trains, regional express trains and Paris commuter trains. 28 The per unit consumption supplied by the ADEME/Deloitte study is expressed in grams of petroleum equivalent per passenger-kilometre and we have converted that to kilowatt-hours per passenger-kilometre. Using the international energy conversion tables, 1,000 GPE equals about 11.63 kWh. For mass transit, per unit consumption is 27 http://www.odyssee-indicators.org/publications/country_profiles_PDF/fra_nl.pdf 28 These assumptions could be improved upon, especially by studying the breakdown of trips among high-speed rail, regional express lines and Parisian commuter rail. But we believe that these assumptions are satisfactory for our current purpose, which is to estimate approximate amounts of energy consumption.

presented in The breakdown of trips between city and non-city driving can also be estimated from some of the survey data. The survey results cover total trips, as well as, at times, the number of kilometres travelled in the city, on the highway and on other roads. We made the following assumption: trips made on highways and other roads equal the non-city trips. Trips for which the kilometres travelled on each type of road are reported in the ENTD survey allow us to calculate a pattern of trips among the different geographic areas. We made the assumption that this pattern applies to the breakdown of all trips. We found that 20% of the kilometres are travelled in the city and the rest outside the city. Finally, according to our calculations, cars and taxis in 1993 consumed on average 0.81 kWh/vehicle-kilometre.

. Lastly, in the ADEME/Deloitte study, the per unit consumption of two-wheeled vehicles is defined for different geographic areas (urban, peri-urban, regional and interregional). As for car travel, the ENTD survey provides us with total trips on two-wheeled vehicles, as well as, at times, the kilometres travelled on two wheels in the city, on the highway and on other roads. We make the following assumptions: travel in the city corresponds to urban travel; travel on highways, to interregional travel; travel on other roads, half to peri-urban travel and half to regional travel. Trips for which the kilometres travelled on each type of road are reported in the ENTD survey allow us to calculate a pattern of trips among the different geographic areas. We made the assumption that this pattern applies to the breakdown of all trips (Erreur ! Référence non valide pour un signet.). Finally, according to our calculations, two-wheeled vehicles in 1993 consumed on average 0.41 kWh/vehicle-kilometre.

Results of our estimates

From the assumptions made, we estimated the energy consumption for all trips related to the consumption and production of services. The energy consumption is estimated using the following formulas:

-Energy consumption due to travel by service consumers:

t t s i j j i sij C p p d                                

52

-Energy consumption due to travel by service providers, for commuting and work-related trips: In 1993, about 126 TWh of final energy appear to have been consumed for travel associated with service operations. This amount represents about 25% of the consumption in the transportation sector for that year and 70% of the consumption in the tertiary sector. If one reassigns the type 1 energy consumption, the weight of the tertiary sector in the total energy balance increases from about 10.5% to over 18%. By further updating the results of the 1993 survey based on these data 31 , we obtain an estimate of the energy consumption for travel related to service businesses on the order of 140 TWh. It thus seems a reasonable inference that the energy consumption required today for travel to consume services and to provide services is on the order of 140 to 160 TWh. If one reassigns this energy consumption to the tertiary sector, the latter's weight in the total energy balance would then almost reach 20%.

t t p i j j i j pi C p p d                                ' ' '

Conclusion

In economic theory, the supposed non-materiality of services has suggested that a highly service-oriented society would consume less energy. But the non-materiality of services is a myth. The very interactivity that is understood to be one of the essential characteristics of a service shows itself as the major source of their materiality, since it relies upon physical communications and transportation systems. In this paper we have identified various sources of materiality in the service sector, and they are all sources of energy consumption. We have also shown that services consume energy through the different stages of their provision, whether by travel and the conditioning of service premises or by the actual delivery of services. We have also shown that their production requires numerous goods (equipment, buildings, infrastructure, etc.) whose grey energy needs to be taken into account when considering the energy impact of the service sector. In the energy statistics of the tertiary sector, only a portion of the energy consumption truly required by the provision of services is recognized.

Within the limited scope of this paper, we have only estimated one of the sources of energy consumption that have been overlooked in the official statistics: that pertaining to mobility. (That is, we estimated travel by service consumers to consume the service, and the commuting travel and the interactivity of services, which is one of their major characteristics, and secondly, because it is a major source of their materiality. We estimate that the reassignment of this energy consumption has the effect of multiplying the tertiary sector's consumption by a factor of about 1.7. Efforts need to be undertaken to make estimates of the other sources of energy consumption, for the energy required by the entire tertiary sector is far greater than what we have estimated here.

We have confirmed all of our assumptions theoretically and some of them empirically, as follows: the tertiary sector is in reality far from non-material, and the provision of services calls for significant amounts of energy consumption that the official statistics underestimate due to their accounting methodology. The increasing service orientation of an economy therefore does not constitute a solution to its energy, or more broadly, environmental issues.

  defines services as a change in the condition of a 2 Heinrich Storch, 1823. The Course of Political Economy, or an Exposition of the Principles which Determine the Prosperity of Nations. Cited in Delaunay and Gadrey, 1987.

Figure 1 .

 1 Figure 1. Energy intensity of value added, by sector (Mtoe/BEUR 2000 ) in 2006

  the service (esp. their floor area)

Figure 3 .

 3 Figure 3. The Scope of Energy Consumption in the tertiary sector as assumed in CEREN statistics

Figure 4 .

 4 Figure 4. The total direct and indirect sources of energy consumption in a final service

Figure 4 .

 4 Figure 4. The total direct and indirect sources of energy consumption in a final service

  25 ETHEL:Energie, Transport, Habitat, Environnement, Localisations [= Energy, Transportation, Habitat, Environment, Locations]. A project conducted in 2003 whose findings are available at http://ethel.ishlyon.cnrs.fr/index.htm. The assumptions about multiple means of transportation are based on analyses of surveys.

  52 where: t = means of transportation; s = trips occasioned by the consumption of services; p = work-related trips (including commuting); i = individuals surveyed; i' = individuals surveyed who work in the tertiary sector; j = day of the week; C t : estimate of the average consumption of the means of transportation, per km and per passenger; d sij and d pi'j : distance travelled per trip, per individual and per day; p i and p i' : occurrence of such an individual in the French population (how representative he or she is); p j : weight given to the day the questionnaire was administered so that the results can be representative of one week;The results obtained are presented in Erreur ! Référence non valide pour un signet..

  If one places these results into the current situation, in which the consumption in the transportation sector amounts to 584 TWh and that of the tertiary sector to 225 TWh 30 , one comes up with 145 TWh and 158 TWh respectively. Furthermore, INSEE supplies an estimate of the change in travel distances between 1993 and 2007 for a few travel purposes (Erreur ! Référence non valide pour un signet.).

  work-related travel by service providers). Travel is important for two reasons: first, because it captures 31 Also in need of updating are the per unit consumption of the means of transportation and the intersection between SPS and business sector. Here we have relied on the results of the ADEME/Deloitte study and the results of the INSEE 2007 census, respectively.

  Degree of decreasing relatedness between energy consumption and provision of the final serviceGrey energy of equipment, furniture & furnishings, buildings, infrastructure, etc.

		Direct relationship		Indirect relationship
	In advance of the final service		Production* and marketing of equipment, furnishings, buildings, infrastructure and other intermediary goods consumed.	Production* and marketing of equipment, furnishings, etc.
	t 0 Provision of final service	Operation of equipment that consumes energy (delivery, conditioning, transportation)	Energy consumption of intermediary services	Operation of equipment of the intermediary service
				Recycling
			Recycling	of equipment,
			of equipment,	furnishings,
	After the		furnishings, buildings, infrastructure or other	buildings, etc.
	final		intermediary consumables.
	service

Table 1 . Reason for travel and consumption of services Reason for travel related to the consumption of services Work-related reasons for travel

 1

	Place of study (school, high school, university)	Going to a fixed, customary workplace
	Day care, nursery school	Going to an unfixed workplace (construction site,
	Shopping mall or downtown stores	business meeting, business trip, etc.)
	Neighbourhood store and other shopping	Coming home from workplace
	Medical treatment or personal care (doctor, hairdresser,
	etc.)
	Government-related errands	Other work-related reasons
	Accompanying or picking up someone at a train station,
	airport, subway stop, etc.
	Cultural and athletic events or meetings (cinema, theatre,
	concert, sports match, etc.)
	Visiting monuments and historical places	Other reasons for travel
	Playing sports (gymnastics, soccer, etc.)	Going home
	Recreational locations (parks, fairs, bowling, etc.)	Visiting family
	Going on vacation	Visiting friends
	Cafés, restaurants, bars	Going to a second home or occasional dwelling
	Going to a training program, lecture, conference, etc.	Other personal reasons
	Accompanying or picking up someone at one of the
	aforementioned places
	Returning home from one of the aforementioned places
	b) Business sectors
	With regard to work-related travel, our estimate must only include that done by individuals who work
	in service firms or organizations.

Table 2 . Intersection of socio-professional category and business sector

 2 To estimate the energy consumption from distances travelled, it is necessary to know the means of transportation used and its average consumption rate. It is not unusual for a person to use more than one means of transportation for a single trip. For each trip reported in the ENTD survey, information is

		SPS	Tertiary sector		SPS	Tertiary sector
	10 Farmers	0.0%	47 Technicians	52.6%
	21 Craftsmen	38.4%	48 Foremen	39.4%
	22 Wholesalers, Retailers	97.9%	52 Government employees	99.3%
	23 Business owners	60.9%	53 Police and military	96.9%
	31 Members of the professions	98.7%	54 Business employees	81.1%
	33 Government white-collar	98.3%	55 Wholesale/retail employees	90.2%
	34	Teachers, secondary & higher	98.9%	56 Personal services	98.7%
	35 IT, arts professionals	83.6%	62 Skilled industrial workers	16.5%
	37 Business executives	82.9%	63 Skilled craftspersons	51.9%
	38 Engineers	52.3%	64 Drivers	85.6%
	42 School teachers	99.0%	65	Skilled maintenance, warehousing workers	68.5%
	43	Healthcare intermediary professions	97.2%	67 Unskilled industrial workers	36.0%
	44 Clergy, religious vocations	98.1%	68 Unskilled craftsmen	66.4%
	45	Government intermediary workers	99.0%	69 Farm workers	14.1%
	46	Business intermediary professions	77.9%
			Source: INSEE, 1999 census of the population
		c) Multiple means of transportation

Table 3 . Assumptions about multiple means of transportation Means of transportation Breakdown of distances travelled

 3

	On foot, by bicycle and all other means of transportation	100% of the distance attributed to the other means
	Airplane and all other means	100% of the distance attributed to the airplane
	Train and car	15% of the distance attributed to the car
	Train and coach	85% of the distance attributed to the train
	Train and other means	100% of the distance attributed to the train
	Coach and car	30% of the distance attributed to the car
	Coach and bus	50% of the distance attributed to each means
	Coach and other means	100% of the distance attributed to the coach
	Car and bus	35% of the distance attributed to the car
	Car and two-wheeled vehicles	100% of the distance attributed to the car
	Car and other means	35% of the distance attributed to the car
	Bus and other means	50% of the distance attributed to each means

Source: Data from the ETHEL project, arranged to match our means of transportation d) The energy efficiency of means of transportation

Table 4 . Average energy consumption per vehicle per 100 km

 4

		City driving Outside the city
	Petrol	7.6 L	8.5 L
	Diesel	6.8 L	7.4 L
	Source: Author's calculations based on the ENTD survey results

Table 5 . Energy consumption of various means transportation

 5

	Means of transportation	Energy consumption per passenger per kilometre (kWh/passenger-km)
	High-speed rail	0.09
	Trains	0.15
	Parisian rapid transit rail system	0.09
	Coach	0.13
	Bus	0.46
	Tram	0.08
	Subway	0.09
	Airplane	0.53
	Truck	4.10
	Source: Author's calculations based on data from the ADEME/Deloitte study (2008)

Table 6 . Breakdown of two-wheeled vehicle travel by geographic area

 6

	Urban Peri-urban Regional Interregional

Table 7 . Estimate of energy consumption for travel related to service operations

 7 Sources: Calculations by the author based on the results of the ENTD 1993 survey and the assumptions established in Section 4.1.

			Trips occasioned by the	Work-related trips of service
			consumption of services	providers	Corresponding energy
			(in km)	(in km)	consumption (TWh)
	On foot and by bicycle	5,893,092,030	958,519,983	0.0
	Two-wheeled vehicles	1,476,415,676	973,440,727	0.9
	Car and taxi	29	117,731,351,689	79,858,954,616	107.9
	Truck		3,371,509	353,113,534	1.5
	Coach		13,295,234,475	1,418,965,640	1.8
	Bus		5,181,805,725	1,592,388,322	3.1
	Subway		1,816,700,221	1,942,969,762	0.3
	Parisian rapid transit rail system	889,974,273	1,301,089,674	0.2
	Tram		204,965,724	114,212,944	0.0
	High-speed rail	2,525,628,236	740,213,412	0.3
	Other trains	4,380,144,378	5,464,916,676	1.5
	Airplane		12,349,394,854	2,752,202,840	8.0
	Other		1,485,687,260	246,192,283	0.0
	Total		167,233,766,049	97,717,180,413	125.7

Table 8 . Change in distance travelled, by purpose (%)

 8

					Home to other	Other
		Work	Study	Shopping	activities	trips	Total
	Rural and thinly settled areas	26	22	29	2	0	12
	Densely populated areas	10	Not sig.	Not sig.	-10	-1	-1
	Source: Insee/SOeS/Inrets, ENTD 1993, transportation and travel survey 2007-2008

GALLOUJ Faïz, WEINSTEIN Olivier, 1997. "

  "Macroeconomics of unbalanced growth: the anatomy of urban crisis", American Economic Review, Vol. LVII, n°3, June, pp. 415-426. BELL Daniel, 1973. The coming of post-industrial society, a venture in social forecasting, Basic Books, New-York ; trad. fr. 1976. Vers la société post-industrielle, Laffont, Paris, 446p. BOURG Dominique, BUCLET Nicolas, 2005. "L'économie de la fonctionnalité. Changer la consommation dans le sens du développement durable", Futuribles, n°313, November, pp. 27-37. BUCLET Nicolas, 2005. "Vendre l'usage d'un bien plutôt que le bien lui-même : une piste pour concilier meilleurs prise en compte de l'environnement et rentabilité des entreprises", XV conférence internationale de RESER, Granada, 22-24 September 2005. "Un modèle d'analyse de la dynamique de l'innovation dans les services : le cas des services de types architecturaux", Économies et Sociétés, EGS, n°7, pp. 1973-2010. DU TERTRE Christian, 2007. "Des services aux entreprises à l'économie de la fonctionnalité : les enjeux du développement durable" in Heurgon Édith, Landrieu Josée (coord.). L'économie des services pour un développement durable. Nouvelles richesses, nouvelles solidarités, Colloque de Cerisy, Prospective, Essais & Recherches, L'Harmattan, 390p. ETTIGHOFFER Denis, 1992. L'entreprise virtuelle. Ou les nouveaux modes de travail, Editions Odile Jacob, Paris, 346p. FORAY Dominique, 2000. L'économie de la connaissance, La Découverte, Paris, 124p. FOURASTIÉ Jean, 1949. Le grand espoir du XXème siècle. Progrès technique, progrès économique, progrès social, Presses Universitaires de France, Paris, 224p.; 2 nd edition, 1963, Gallimard, 372p. FOURCROY Charlotte, 2010. Services et énergie. Formation de la demande d'énergie d'une activité de services, Mémoire de Master 2 Recherche ESII, University of Lille 1. FUCHS Victor R., 1968. The service economy, National Bureau of Economic Research, New-York, 280p. GADREY Jean, 1991. "Le service n'est pas un produit. Quelques implications pour l'analyse économique et pour la gestion", Politiques et management public, Vol. 9, n°1, March, pp. 1-24. Innovation in the service economy. The new wealth of nations, Edward Elgar, Innovation in Services", Research Policy, Vol. 26, n°4-5, pp. 537-556. After industrial society. The emerging of self-service economy, "Tangibles, intangibles and services: a new taxonomy for the classification of output", Canadian Journal ofEconomics, Vol. 32, n°2, April,

	BIBLIOGRAPHY
	ADEME/Deloitte, 2008. Efficacité énergétique, emissions de CO2 et autres émissions gazeuses
	spécifiques des modes de transport, Rapport final.
	This conclusion raises the question of the reciprocal relationship between service sector innovation and tighter energy constraints. BAUMOL William, 1967. Cheltenham, 226p.
	MacMillan, Londres, 181p.
	HILL T.P, 1977. "On goods and services", Review of Income and Wealth, Vol. 23, n°4, December,
	pp. 315-338.
	HILL T.P, 1999.

CASTELLS Manuel, AOYAMA Yuko, 2002. "Une évaluation empirique de la société informationnelle : structures de l'emploi dans les pays du

G7, 1920[START_REF] Rifkin | Quand tout devient service" in Rifkin Jeremy. L'âge de l'accès : la révolution de la nouvelle économie[END_REF]". Revue Internationale du Travail, Vol. 141, n°1-2, pp. 133-173. CLARK Colin, 1940

. The conditions of progress and security, MacMillan and co., Londres; trad. fr. 1960. Les conditions du progrès économique, Presses Universitaires de France, Paris, 500p. DELAUNAY Jean-Claude, GADREY Jean, 1987. Les enjeux de la société de services, Les Presses de Sciences Po, Paris, 333p. DESMARCHELIER Benoit, GALLOUJ Faïz, 2010. "Croissance endogène et politique environnementales : les processus de croissance et de tertiarisation des économies développées sont-ils réversibles ? Une perspective évolutionniste.", RESER conference paper, Gothenburg, Sweden. DJELLAL Faridah, GALLOUJ Faïz, 2005. EIGLIER Pierre, LANGEARD Eric, 1987. Servuction. Le marketing des services, Ediscience international, Paris, 205p. GADREY Jean, 1992. Socio-économie des services, Éditions La Découverte, Paris ; 3rd edition, 2003, 124p. GADREY Jean, 1994. "Les relations de service et l'analyse du travail des agents". Sociologie du travail, n°3, pp. 381-389. GADREY Jean, 2010. "The environmental crisis and the economics of services : the need for revolution", in DJELLAL Faridah, GALLOUJ Faïz, The handbook of innovation and services, Edward Elgar, Cheltenham. GALLOUJ Faïz, 1999. "Les trajectoire de l'innovation dans les services : vers un enrichissement des taxonomies évolutionnistes", Économies et Sociétés, Série EGS, n°1, 5, pp. 143-169. GALLOUJ Faïz, 2002. GERSHUNY Jonathan I., 1978.

IFEN, 2004. "La tertiarisation de l'économie et la réduction des émissions de CO2", Les données de l'environnement, lettre thématique mensuelle de l

'IFEN, n°95, September, 4p.

He specifically mentions the services of the State, artists, etc.

The positive definition is distinct from a residual definition, as rounding out all the other economic activities.

"A service may be defined as a change in the condition of a person, or a good belonging to some economic unit, which is brought about as the result of the activity of some other economic unit, with the prior agreement of the former person or economic unit"(Hill, 1977, p. 318). In another article(Hill, 1999), however, Hill states that non-materiality is not an exclusive feature of services and that there is also such a thing as a non-material good.

Data from CITEPA for 2008 relating to CO2 emissions, excluding LULUCF (Land Use, Land-Use Change and Forestry).

This is hot water for sanitary uses: rest rooms, food preparation, etc. It is often shortened to SHW.

The different choices as to how service is delivered and therefore of travel, as well as the thinking behind them, are discussed in Section 3.

Despite the gap in time, we believe that these data are useful, inasmuch as we are trying to estimate the approximate amount of energy consumption.

With respect to the few trips for which the number of people is not indicated (less than 1% of trips by motorbike and by car), we made the assumption that the person was alone on/in the vehicle. Since the number of trips without data is very low, this assumption will have very slight effect on the final result.

Distances travelled by car, taxi and two-wheeled vehicles appearing in the table equal the total distances travelled by all French people. To estimate the corresponding energy consumption, we first divided these distances by the number of passengers travelling by that means.

These data are from 2008 and are provided bySOeS and CEREN, respectively.

WILLIAMS Eric, TAGAMI Takashi, 2003. "Energy use in sales and distribution via e-commerce and conventional retail. A case study of the Japanese book sector", Journal of Industrial Ecology,vol.6,n°2,


N Guzina
email: guzina@wave.ce.umn.edu

Nemitz
email: nemitz@ann.jussieu.fr

Marc Bonnet ⋆ -Bojan

B Guzina

⋆⋆ -Nicolas Nemitz

Fast non-iterative methods for defect identification

Keywords: Inverse problems, topological sensitivity, fast multipole method. MOTS-CLÉS : Problèmes inverses, sensibilité topologique, méthode multipôle rapide

Fast non-iterative methods for defect identification

Introduction

The reconstruction of obstacles embedded in an acoustic medium is a challenging subject with application to e.g. sonar detection and medical imaging [START_REF] Lambert | Introduction à la diffraction inverse en acoustique et élasticité[END_REF][START_REF] Colton | Recent developments in inverse acoustic scattering theory[END_REF]. It is often formulated in terms of the minimization of a cost function featuring the experimental data and (possibly) prior information. Such cost functions are non-convex and exhibit local minima. Global search techniques, e.g. evolutionary algorithms [START_REF] Michalewicz | How to solve it: modern heuristics[END_REF] or parameter space sampling methods [START_REF] Tarantola | Inverse problem theory and methods for model parameter estimation[END_REF] require very large numbers of cost function evaluations, and hence are usually prohibitively expensive in the context of wave-based approaches. Hence, traditional minimization methods, or Newton-type algorithms for solving the observation equations, are often preferred, as they may converge within moderate iteration counts for obstacles described in terms of a small number of geometrical parameters. To optimize computational efficiency, such solution techniques are often used in conjunction with shape sensitivity techniques (Bonnet, 1995;[START_REF] Guzina | On the stress-wave imaging of cavities in a semiinfinite solid[END_REF][START_REF] Farhat | On the solution of three-dimensional inverse obstacle acoustic scattering problems by a regularized Newton method[END_REF] or level-set methods [START_REF] Litman | Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set[END_REF].

Still, the stand-alone use of gradient-based algorithms for such purposes is not always satisfactory due to their strong dependence on a reliable prior information about the geometry of the hidden object. This has prompted, over the last decade the development of alternative, non-iterative, approaches, which may be used either in isolation or as a preliminary step providing adequate initial guesses to subsequent iterative schemes. Now referred to as "sampling" or "probe" methods [START_REF] Potthast | A survey on sampling and probe methods for inverse problems[END_REF], they aim at replacing the usual optimization approach with the definition of a defect indicator function. They include approaches based on linear sampling [START_REF] Colton | A simple method for solving inverse scattering problems in the resonance region[END_REF], not pursued here, or on the concept of topological sensitivity [START_REF] Bonnet | Sounding of finite solid bodies by way of topological derivative[END_REF][START_REF] Guzina | Topological derivative for the inverse scattering of elastic waves[END_REF], or more generally on small-defect asymptotics (Ammari and Kang, 2004). This communication aims at summarizing contributions we made in this area over the last few years.

Topological sensitivity and inversion

The concept of topological derivative first appeared in [START_REF] Eschenauer | Bubble method for topology and shape optimization of structures[END_REF][START_REF] Schumacher | Topologieoptimierung von Bauteilstrukturen unter Verwendung von Lochpositionierungskriterien[END_REF] in the context of topological optimization of mechanical structures, allowing to define algorithms where "excess" material is iteratively removed until a satisfactory shape and topology is reached. Following investigations by e.g. [START_REF] Sokolowski | On the topological derivative in shape optimization[END_REF] and [START_REF] Garreau | The topological asymptotic for PDE systems: the elasticity case[END_REF] for the Laplace equation or [START_REF] Samet | The topological asymptotic for the Helmholtz equation[END_REF] and [START_REF] Pommier | The topological asymptotic for the Helmholtz equation with Dirichlet condition on the boundary of an arbitrarily shaped hole[END_REF] for the Helmholtz and Maxwell equations, we study here the usefulness of the topological sensitivity field as a defect indicator function. Our work addresses only 3-D configurations, but 2-D cases are considered by e.g. [START_REF] Feijóo | A new method in inverse scattering based on the topological derivative[END_REF] or [START_REF] Gallego | Identification of cracks and cavities using the topological sensitivity boundsary integral equation[END_REF]. This approach rests upon quantifying the sensitivity of the cost function to the nucleation of an object, of specified physical properties and small characteristic size ε, at a specified location x s in the reference medium. Letting J(ε, x s ) denote the value achieved by the featured cost function of the inverse problem when only the infinitesimal obstacle located at x s is present, the topological sensitivity T 3 (x s ) of J is (for 3D configurations involving either sound-hard or penetrable obstacles, i.e. such that either Neumann or transmission conditions are assumed on the obstacle surface) defined through the expansion

J(ε, x s ) -J(0, x s ) = ε 3 T 3 (x s ) + o(ε 3 ) [1]
Note that the value J(0, x s ) of J for the obstacle-free reference medium does not actually depend on x s . The topological sensitivity field T 3 (x s ) is then used as a defect indicator function. Heuristically, regions in the probed medium in which the nucleation of a small defect induces the strongest decrease of J are expected to be those where the field T 3 (x s ) attains its most pronounced negative values. A simple approach then consists in computing the field T 3 (x s ) over a region of space susceptible of housing defects and look for locations where T 3 attains its lowest negative values.

The topological sensitivity field T 3 (x s ) may be given an explicit expression in terms of the Green's function associated with the reference domain, its physical properties, and the type of boundary conditions assumed for the external boundary. In practice, the relevant Green's function is not known in explicit form. Another approach, better suited to numerical computations, consists in expressing T 3 (x s ) in terms of the free field u (response of the reference, i.e. defect-free, medium to the probing excitation) and the adjoint solution û (induced by an adjoint load defined in terms of the cost function). For example, the topological sensitivity field T 3 (x s ) corresponding to the nucleation of a small sound-hard spherical scatterer, under time-harmonic conditions, in a linearly acoustic reference medium characterized by wave number k is found (e.g. [START_REF] Nemitz | Topological sensitivity and FMM-accelerated BEM applied to 3D acoustic inverse scattering[END_REF]) to be given by

T 3 (x s ) = 4π 3 3 2 ∇u.∇û -k 2 uû (x s ) [2]
Expressions of T 3 (x s ) have also been established for other 3-D configurations. Such results include, for the time-harmonic case, acoustic media containing penetrable obstacles [START_REF] Guzina | Small-inclusion asymptotic of misfit functionals for inverse problems in acoustics[END_REF], elastic media with cavities [START_REF] Guzina | Topological derivative for the inverse scattering of elastic waves[END_REF][START_REF] Bonnet | Sounding of finite solid bodies by way of topological derivative[END_REF] or elastic inclusions [START_REF] Guzina | From imaging to material identification: a generalized concept of topological sensitivity[END_REF], and electromagnetic media containing penetrable inclusions (Ammari and Kang, 2004;[START_REF] Masmoudi | The topological asymptotic expansion for the Maxwell equations and some applications[END_REF]. For time domain analyses, similar formulae have been established for T 3 (x s ), in which bilinear expressions in (u, û) such as [2] become time convolution products between free and adjoint fields [START_REF] Bonnet | Topological sensitivity for 3D elastodynamic and acoustic inverse scattering in the time domain[END_REF]. It is useful to note that expressions such as [2] are formally similar to sums of a strain energy and a kinetic energy, and thus are well suited to any usual computational approach, such as the finite element method (FEM) or the boundary element method (BEM) used in our numerical experiments to date.

Topological sensitivity and fast multipole method

When the reference medium is endowed with simple (linear, piecewisehomogeneous) constitutive properties, the topological sensitivity field T 3 (x s ) may be formulated using integral representation formulae (Bonnet, 1999a), the free and adjoint fields then being first computed over the external boundary by means of a BEM. This approach is in particular well suited to dealing with unbounded reference media. As probing a 3-D region of space may necessitate large numbers of sampling points x s , such computation may nevertheless entail high CPU and memory requirement, because of the fully-populated nature of the BEM influence matrix. The Fast Multipole Method (FMM) [START_REF] Greengard | A new version of the fast multipole method for the Laplace equation in three dimensions[END_REF][START_REF] Nishimura | Fast multipole accelerated boundary integral equation methods[END_REF][START_REF] Margonari | Fast multipole method applied to the coupling of elastostatic BEM with FEM[END_REF][START_REF] Nemitz | Méthode multipôle rapide et sensibilité topologique pour l'identification approchée de défauts à partir de données de type acoustique[END_REF][START_REF] Chaillat | A fast multipole accelerated BEM for 3-D elastic wave computation[END_REF] addresses these concerns, as it provides considerable speedup and memory savings. For a frequency-domain analysis involving N BEM DOFs and M sampling points, computing the field T 3 (x s ) requires O(N LogN ) operations per GMRES iteration (for the BEM solution phase) and O(M LogM ) operations (subsequent evaluation of T 3 (x s ) at sampling points), whereas the corresponding complexities for a traditional BEM are O(N 2 ) per solver iteration (or O(N 3 ) for direct solvers) and O(M N ). The FMM allows to handle computational models with N, M = O(10 6 ) on an ordinary PC.

Numerical example. The above-summarized approach combining the topological sensitivity and the FMM [START_REF] Nemitz | Topological sensitivity and FMM-accelerated BEM applied to 3D acoustic inverse scattering[END_REF]) is now demonstrated on the following configuration: the bounded acoustic domain is the cube defined by

Ω = { |ξ i | ≤ 8a (i = 1, 2, 3) },
where a is a reference length. A simulated testing configuration is based on 30 experiments, each of which consists in applying a uniform excitation over a small region S q (1 ≤ q ≤ 30) of the external boundary S and recording the acoustic pressure over all of S (i.e. at all BE mesh nodes). The acoustic excitation is such that the wavelength is λ = 3a. Each of the six faces of the cubical domain Ω supports five excitation surfaces S q , each a disk of radius a. The cost function for the inverse problem is defined by

J (Ω ⋆ ) = 1 2 30 q=1 S |u ⋆ -u obs q | 2 dΓ ξ [3]
where u obs q is the data obtained for the q-th applied excitation, with u obs q = u true in the absence of data noise. The centroid x true of true scatterer B true to be identified is located at x true = (2a, 3a, 2a). To facilitate the graphical interpretation, a thresholded variant T (x s ) of T (x s ) is introduced according to

T3 (x s ) = T 3 (x s ) (T 3 ≤ C T min ), T3 (x s ) = 0 (T 3 > C T min ), [4]
with C = 0.25 used here. The BE meshes used for computing the free field u, the adjoint field û and the simulated data u true q are made of three-noded flat triangular elements, arranged in a regular mesh with approximately 15 nodes per wavelength. For the purposes of computing the simulated error-free data u true for each synthetic experiment, BE meshes of the true scatterer Γ true have been set up as well. Table 1 indicates the numbers of nodes and elements supported by the BE meshes.

The identification of one spherical scatterer of radius 0.4a is considered. The field T has been computed on the basis of error-free synthetic data, over a sampling grid Table 2. CPU times and (in parentheses) GMRES iteration count for computing the true, free and adjoint solutions on the boundary, and CPU times for computing the topological sensitivity over the whole sampling grid. the iso-surfaces of the field T corresponding to T = 0.55T min , computed on a truncated sampling grids such that { |ξ i | ≤ 5a (i = 1, 2, 3) }, i.e. in which sampling points whose distance to S is less than 3a have been taken out. In this central region, low negative values of T occur only in the vicinity of the actual location of the scatterer.

It should be mentioned that T has been observed to also achieve low values in zones located near edges or corners of S, where there is no scatterer.

For completeness, Table 2 provides typical CPU and iteration counts, for a cube of size 2L = 16a as above and for a cube of size 2L = 32a (the wavelength being kept at λ = 3a). It is interesting to observe that the overall CPU times for Ω(32a), which involves roughly 4 times as many nodal unknowns as Ω(16a), are about 5 times higher than those for Ω(16a) (while the expected ratio for traditional BEM would be 4 3 = 64), and that the GMRES iteration counts are only fractionally higher for Ω(32a). These observations are consistent with the almost-linear O(N LogN ) complexity per iteration predicted by theoretical analyses of the FMM [START_REF] Nishimura | Fast multipole accelerated boundary integral equation methods[END_REF][START_REF] Darve | The fast multipole method: numerical implementation[END_REF][START_REF] Sylvand | Complex industrial computations in electromagnetism using the fast multipole method[END_REF]. All computations have been performed on a Linux PC computer with one 3 GHz processor.

Similar computations featuring noisy data have also been performed. For cost functions that are quadratic with respect to measurement residuals, the adjoint field û undergoes perturbations that are proportional to the data noise. From [2], the pertur-bation induced on T 3 therefore also depends linearly on the noise level. The present approach is therefore less sensitive to noise than conventional inversion techniques.

Higher-order topological expansion

The numerical experiments performed thus far clearly establish the usefulness of the topological sensitivity field T 3 (x s ) as a defect indicator function. The information provided by T 3 (x s ) is, however, of a qualitative nature only, as approximation [1] of J at points x s such that T 3 (x s ) < 0 is not amenable to minimization with respect to the obstacle size ε. One is hence led to consider expanding J(ε, x s ) to higher order in ε for the purpose of a subsequent minimization with respect to the obstacle size. As quadratic cost functions are often used in practice, such expansion should in view of [1] be carried out to order O(ε 6 ), so as to square the O(ε 3 ) leading order in [1]. This approach has for now been considered for the scalar Helmholtz equation [START_REF] Bonnet | Inverse acoustic scattering by small-obstacle expansion of misfit function[END_REF], but can be extended to e.g. 3-D elastodynamics. The expansion of J for any centrally-symmetric sound-hard scatterer, of characteristic size ε and centered at x s , is found to have the form

J(ε, x s ) -J(0, x s ) = ε 3 T 3 (x s ) + ε 5 T 5 (x s ) + ε 6 T 6 (x s ) J6(ε,xs) +o(ε 6 ) [5]
The previously known topological derivative T 3 (x s ) and the new coefficients T 5 (x s ), T 6 (x s ) have explicit expressions (not given hre for brevity) in terms of the acoustic Green's function associated with the reference acoustic domain and the structure of external boundary conditions. Due to the symmetry assumption for the small obstacle, there is no O(ε 4 ) contribution in [5]. For cases where the Green's function is not known analytically, setting up T 3 (x s ) T 5 (x s ), T 6 (x s ) entails solving boundary integral equations for (a) the forward and adjoint acoustic fields, and (b) auxiliary solutions generated by acoustic point sources at sampling points x s . All of these solutions are governed by the same (integral) linear operator, associated with the Helmholtz equation in the reference (i.e. obstacle-free) domain, which makes the required supplementary computational work modest relative to the cost of one forward solution.

A simple approximate global search procedure. Expansions of the form [5] offer the option of minimizing the approximate polynomial expression J 6 (ε; x s ) for sampling points chosen a priori. This task is, for each sampling point, simple and computationally very light. Hence, it can conceivably be performed for locations x s spanning a fine search grid G, thereby defining an approximate global search procedure over the spatial region sampled by means of G. The best estimate of the unknown scatterer B true yielded by this procedure is defined by the location x s = x est and size ε = R est leading to an absolute minimum of J 6 (ε; x s ) over G. In practice, a partial minimization of J 6 (ε; x s ) w.r.t. ε is performed for each x s , and one defines the best estimate of the unknown scatterer over G in terms of its radius R est and center x est by Numerical results. To demonstrate the proposed approximate search procedure and thereby demonstrate the usefulness of the O(ε 6 ) expansion of J(ε; x s ), the identification of a hard scatterer embedded in an acoustic medium occupying the half-space Ω = {ξ | ξ 3 ≤ 0} is considered, as depicted in Fig. 3. A homogeneous Neumann condition is assumed on the surface x 3 = 0. Under these conditions, the relevant Green's function is, as mentioned earlier, explicitly known.

(R est , x est ) = arg min ε>0, xs∈G J 6 (ε, x s ) [6]
Three synthetic testing configurations are considered (hereinafter labelled T (1) , T (2) , T (3) ), where the square region -5a ≤ ξ 1 , ξ 2 ≤ 5a on the surface x 3 = 0 is divided into 5×5, 10×10 and 20×20 squares, respectively (a being a reference length). Sensors are placed at all vertices x m (1 ≤ m ≤ M ) of the squares thus defined, so that T (1) , T (2) , T (3) feature M = 36, 121, 441 points, respectively. Acoustic point sources of unit magnitude are placed in succession at all centers of the previously defined squares. The scattered field generated by each source is recorded at all sensors, so that T (1) , T (2) , T (3) support N = 25, 100, 400 synthetic experiments, respectively. This is a situation of limited aperture, since data is available only on part of the surface of the half-space. The identification is performed on the basis of the leastsquares cost function

J (Ω ⋆ ) = 1 2 N q=1 M m=1 u obs q (x m ) -u ⋆ q (x m ) 2 [7]
where u obs q and u ⋆ q denote the acoustic fields induced by a unit point source placed at x q and for the 'true' and 'trial' configurations Ω true = Ω \ B true and

Ω ⋆ = Ω \ B ⋆ .
The approximate global search procedure has been performed on a search grid G of 51×51×25 = 65025 regularly spaced sampling points spanning the 3-D box-shaped region defined by -10a ≤ x 1 , x 2 ≤ 10a, -10a ≤ x 3 ≤ -0.4a. The approximate cost function J 6 (ε; x s ) with coefficients T 3 (x s ), T 5 (x s ), T 6 (x s ) defined with reference Table 3. Radius estimate R est (best estimate over sampling grid G) for testing configurations T (1) , T (2) , T (3) and noise-free synthetic data: sphere (a), horizontally elongated ellipsoid (b) and vertically elongated ellipsoid (c). A distance x est sx true = 0.2a was found in all cases. to a small trial spherical scatterer , has been set up for all 65025 sampling points x s ∈ G of the search grid thus defined and using the explicit Green's function.

R est /R true -1 T (1) T (2) T (
The scatterer B true to be identified is centered at x true = (2a, 1.2a, -3a). Three geometries are considered for B true : a sphere (S) of radius 0.5a, a horizontally elongated ellipsoid (H) with semiaxes (a, 0.5a, 0.5a) and a vertically elongated ellipsoid (V) with semiaxes (0.5a, 0.5a, a). For comparison purposes, a 'true' radius R true is defined for (S) by R true = 0.5a and for (H) and (V) by R true = 2 -2/3 a, i.e. as the radius of the sphere having the same volume as (H) or (V). Three wavenumbers ka = 0.5, 1, 2 have been considered. Note that x true ∈ G: the sampling points closest to x true are located above and below x true and are separated from x true by a vertical distance 0.2a.

The obstacle radius estimate R est (best estimate over sampling grid G) obtained for the three frequencies considered and using the three testing configurations are compared to R true in table 3, for the three geometries (S), (H) and (V), using exact synthetic data. The lower-frequency case k = 0.5a is seen to yield the most accurate estimation of R true . In all cases, a distance x est sx true = 0.2a is found, i.e. the grid point x est s at which J 6 (ε; x s ) achieves its lowest value Ĵmin 6 is one of the two closest to x true featured by the sampling grid G. For the cases (H) and (V) where the shapes of the trial and true scatterers do not match, the 'equivalent radius' R true is nevertheless reasonably well estimated, and the location x est s found is also optimal among the sites allowed by the chosen search grid.

Here again, similar examples featuring simulated data noise have been considered. Numerical results for estimates (R est , x est ) were only moderately affected by the noise, except in some of the cases where the coarsest testing configuration T (1) was used.

Conclusions

The approaches summarized herein are promising in that numerical experimentations performed thus far demonstrate their ability for approximate defect identification within computational costs of the order of a few forward solutions, i.e. substantially lower than that entailed by classical gradient-based "blind" minimization schemes and far lower than that required by global search algorithms. Moreover, they perform a search of global character through either the definition of a defect indicator function or the global minimization of an approximation to the original cost function. Their computational efficiency is further enhanced by resorting to fast multipole accelerated integral formulations.

Many theoretical and implementational issues remain open. The concept of topological sensitivity as defect indicator being heuristic, further theoretical study (possibly exploiting higher-order expansions) is desirable in order to e.g. better understand the conditions under which the approach can be a priori expected to perform well. Another open issue revolves around the identification of constitutive properties of inclusions from topological sensitivity formulations, for which promising preliminary results have been obtained by [START_REF] Guzina | Small-inclusion asymptotic of misfit functionals for inverse problems in acoustics[END_REF] and [START_REF] Guzina | From imaging to material identification: a generalized concept of topological sensitivity[END_REF] but a complete analysis is still lacking. Extending the topological sensitivity to the case of vibratory measurements is also of interest, as formulations established so far are applicable only away from eigenfrequencies. Moreover, the issue of whether there are links between the present approach and that based on the distributed error in constitutive equation, which is also known from extensive numerical experiments to have the ability to (approximately) locate defects in structures [START_REF] Ladeveze | Parametric correction of finite element models using modal tests[END_REF][START_REF] Ladevèze | Application of a posteriori error estimation for structural model updating[END_REF][START_REF] Bonnet | On the estimation of the geometrical support of modelling defects using the distributed error in constitutive equation[END_REF]. Finally, from a computational standpoint, a finite element, discrete version of topological sensitivity might consist in considering perturbations of constitutive parameters (or nucleation of holes) over the scale of one finite element. Finally, applications to real data from e.g. NDT experiments sill remains to be done. Extensions to other physical contexts are currently under investigation, in particular in connection with infrared thermography.

Ammari H., Kang H., Reconstruction of small inhomogeneities from boundary measurements, Lecture Notes in Mathematics 1846, Springer-Verlag, 2004.

Bonnet M., "BIE and material differentiation applied to the formulation of obstacle inverse problems.", Engng. Anal. with Bound. Elem., vol. 15, p. 121-136, 1995. Bonnet M., Boundary Integral Equations Methods for Solids and Fluids, John Wiley and Sons, 1999a.

Figure 1 .

 1 Figure 1. Identification of spherical hard scatterer of radius 0.4a: distribution of thresholded topological sensitivity T (x s ) for sampling points x s in coordinate planes ξ 1 = x true 1 (a), x 2 = x true 2 (b) and x 3 = x true 3 (c).

Figure 2 .

 2 Figure2. Identification of spherical hard scatterer of radius 0.4a: iso-surfaces of T (x s ) for T = 0.55T min , computed on a truncated search grid such that { |ξ i | ≤ 5a (i = 1, 2, 3) }. Values of T (x s ) lower than the iso-value are inside the iso-surface.

Figure 3 .

 3 Figure 3. Identification of a spherical or ellipsoidal sound-hard scatterer in a acoustic half-space: geometry and notation.

  3) ka = 0.5 -4.18e-02 -4.20e-02 -4.20e-02 (S) ka = 1 -1.26e-01 -1.26e-01 -1.26e-01 ka = 2 -3.33e-01 -3.34e-01 -3.34e-01 ka = 0.5 -2.31e-04 9.05e-04 8.89e-04 (H) ka = 1 -1.65e-01 -1.65e-01 -1.65e-01 ka = 2 -4.22e-01 -4.22e-01 -4.22e-01 ka = 0.5 -7.42e-02 -7.51e-02 -7.51e-02 (V) ka = 1 -2.04e-01 -2.06e-01 -2.06e-01 ka = 2 -4.64e-01 -4.67e-01 -4.67e-01

Table 1 .

 1 Number of element and DOFs supported by the BE meshes.

	Cube size	Cube		Obstacle		Total
		Elements	nodes Elements nodes Elements	nodes
	2L = 16a	76800	38402	336	170	77136	38572
	2L = 32a	307200 153602	336	170	307536 153772