1 From e-government to the smart city -old and new issues Gilles JEANNOT, LATTS, Ecole des Ponts, Paristech.

EGPA Lausanne 2018, EGPA permanent study group 1 : e-government

1) Introduction

While the term "smart city" has become the new buzzword in urban development the whole notion continues to be highly ambiguous for a number of reasons. First, the notion of intelligence can be applied to everything that is deemed to be positive, i.e., service quality, democracy, natural resource efficiency, economic development, etc. [START_REF] Giffinger | Smart cities ranking: an effective instrument for the positioning of the cities?[END_REF]. Second, because it is being promoted by the EU and by private companies, the term "smart city" is associated with labels or classifications that rank cities using a very broad range of metrics. Branding and showcasing can therefore become more important than actual concrete measures [START_REF] Hollands | Will the real smart city please stand up? Intelligent, progressive or entrepreneurial?[END_REF]. This topic has been explored, especially post-2010 [START_REF] De | Sustainable-smartresilient-low carbon-eco-knowledge cities; making sense of a multitude of concepts promoting sustainable urbanization[END_REF], in a number of different disciplines. A first pointer may be found in IT scientific journals, reflecting efforts to apply these new technologies in an urban setting [START_REF] Batty | Smart cities of the future[END_REF]. One of the key trends involves researchers working on environmental and sustainable development issues, and we have observed a progressive shift from the themes of the "sustainable city" to the "smart city" [START_REF] De | Sustainable-smartresilient-low carbon-eco-knowledge cities; making sense of a multitude of concepts promoting sustainable urbanization[END_REF]. The term "intelligence" was also quite legitimately appropriated by specialists of the knowledge economy applied to the city and some of these researchers (Shaffers et al. 2011) developed a strong overlap between the knowledge economy and the opportunities offered by new digital technologies. It is now widely used in urban geography [START_REF] Hollands | Will the real smart city please stand up? Intelligent, progressive or entrepreneurial?[END_REF] [START_REF] Rabari | The digital skin of cities: urban theory and research in the age of the sensored and metered city, ubiquitous computing and big data[END_REF].

However, the term has not really been taken up in specialised public administration reviews. A search on "smart city" or "smart cities" in the major public administration journals revealed that, up to 2017, no articles on this topic had been published in Public administration, Administrative science quarterly, Governance, Public management review, only two articles in Public administration review, two articles in International review of administrative science, and six and nine articles respectively in Government information quarterly and Information polity, two journals specialised in new information technologies in public administration.

However, despite this paltry research output, public administration is one of the areas most affected by new digital technologies in the city and certain themes like governance comprise one of the key focuses of existing literature. This lack of enthusiasm for the term "smart city" which has flourished in other academic disciplines may be explained by its proximity to another widely-used term, namely e-government.

The objective of this programmatic article is to identify both what has already been covered by research and what opens up new research avenues by drawing upon a series of innovations deployed. Indeed, in order to avoid pitting futuristic predictions (Anthopoulos, 2017) or general principles concerning the smart city against the empirical realities of e-governance, we need to start with concrete initiatives already undertaken as part of any research into "actually existing smart cities" [START_REF] Shelton | The 'actually existing smart city[END_REF]. Because the measurement of new practices is also in its infancy, we will use an original method of identifying socio-technical processes associated with smart cities which we will apply to 20 French cities.

After this introduction (1) we will present the socio-technical processes in French cities that have been labelled "smart cities" (2) in order to describe their specific features visà-vis the practices associated with e-government (3). We will then go on to analyse these specific features in light of a few relevant pointers that have been developed in relation to e-government.

2) "Smart city"-labelled socio-technical processes in French cities

Method

Our proposed approach adheres to the practice of focusing on management applications or administrative process innovations ( [START_REF] Meeus | Introduction to Part I. Product and Process Innovation[END_REF] [START_REF] Schneider | Do attributes of innovative administrative practices influence their adoption?: An exploratory study of US local government[END_REF] initiated in the US [START_REF] Poister | Municipal management tools from 1976 to 1993: An overview and update[END_REF] that has sometimes been used in Europe (Kuhlman et al. 2008) [START_REF] Jeannot | Homogeneous National Management Policies or Autonomous Choices by Administrative Units? Inter-and Intra-Country Management Tools Use Variations in European Central Government Administrations[END_REF]). In such surveys, the general theme of changing administration management practices is tackled by segmenting practices into management tools, i.e., simple, easily quantifiable units. Obviously, we sacrifice a little of the complexity of each situation but because this allows for a certain degree of decontextualisation, comparisons may be made. This research approach is used here in an exploratory phase in order to identify the socio-technical processes currently used in cities.

Although it is declarative, we are aiming to identify practices actually deployed in French cities. But for the moment, we are not trying to measure the development of such socio-technical processes or even produce an original typology, but, more modestly, to use the list of effective innovations associated with the "smart city" to highlight a few salient characteristics that we can deploy in the "e-governance" debate.

First off, we have drawn up a list of French cities that have been promoted as "smart cities" based on a number of different sources, i.e., EU funding, public investment bank funding and several awards sponsored by businesses or newspapers. We have taken the 10 biggest French cities (Paris, Marseille, Lyon, Toulouse, Nice, Nantes, Montpelier, Strasbourg, Bordeaux, Lille-Roubaix) and those most frequently cited from among the smaller-sized cities (i.e., Besançon, Chartres, Dijon, Grenoble, Issy les Moulineaux, Montpelier, Mulhouse, Plaine Commune, Rennes, Rouen).

Working from this list, we then conducted systematic searches on their websites and other communication media for bottom-up labelling, and in projects funded and in the specialised press for top-down labelling. This gave us a list of "smart city"-labelled sociotechnical processes that was either stabilised or in the experimental phase.

Focusing only on one country leads in effect to not covering all smart city developments throughout the world [START_REF] Neirotti | Current trends in Smart City initiatives: Some stylised facts[END_REF]. Some sectors such as education are not present because they do not fall within the brief of local government. Economic circumstances also play a role given that these initiatives are often partnered by stakeholders in the private sector. The fact that France has produced several leading international urban service providers (in water, electricity, waste and sanitation) can help explain the larger proportion of initiatives in this area. Nevertheless, focusing on just one country provides an overview of everything that has been labelled "smart city" without actually defining the notion.

A certain number of these processes may be classified around traditional municipal government functions, i.e., administration, implementing democracy, network management, public service offering, public security and economic support. But original practices such as "open data" or cross-checking miscellaneous data (traditional or "big data") help to shake up this established function-based sharing by linking a number of these different components. Open data seeks to be an instrument of democracy and to support new services partnered by innovative businesses. Cross-comparison of data potentially concerns all components of municipal management including service sizing, regulation and control.

Administration

Of those innovations highlighted under the smart city label, a certain number relate directly to e-government under headings such as informing inhabitants and enhancing access to services rather than processing individual files. These are websites , "one-stop shop" telephone-based services developed by cities (e.g., Nantes dans ma poche -i.e., pocket-sized guide to Nantes) or turnkey service platforms provided by start-ups, multiservice payment cards (Bordeaux), information points (Chartres), or smart parking metres that can be used to pay for parking and obtain information.

The whole idea of smart public bodies (Meijer, Bolivar, 2016) is largely absent from official discourse although cities have bought into the notion: recruitment of a data officer (Mulhouse, Lyon), standardisation (IT urbanisation) of in-house digital offering (Lille, Bordeaux), or a comprehensive customer relations management system (CRM linked to a one-stop shop (in Issy).

Democracy

Several municipalities (particularly Mulhouse, Rennes and Paris) have linked the idea of the smart city to participative democracy and the possibility of expressing an opinion about projects. The key development vis-à-vis traditional e-government practices is to shift web-based surveys onto smartphones in order to reach a younger population.

Significant participative budgets, such as that provided by the City of Paris, have benefited from enhanced voting access for the general population.

More specifically, the use of 3D imaging helps inhabitants visualise future projects and enhances the public debate . Rennes provides a virtual representation of the entire city while Besançon provides a representation of a redeveloped district. In Strasbourg, debates are organised around on-line participatory maps (carticipe ). Some cities have deemed that civic-mindedness is contingent on having a command of these new digital practices and they offer practical training around the large public fab lab in Rennes in several community centres. Applications for flagging up problems with public amenities (holes in the street, damaged street furniture, etc.) using GPS / geo-tracking and sending a photo, together with better access have also shaken up traditional practices based on letters of complaint. Co-production also includes wikis for neighbourhoods and cities.

Regulation of urban networks

Many "smart city"-labelled features of new urban management technologies aim for maximum efficiency in the production of services or in infrastructure management. In many cases, this is directly related to resource efficiency drives and a lot of "green city" or "eco-neighbourhood" projects -mainly concerning energy, water, sanitation, waste management and road traffic -have been rebranded as "smart city" projects by putting the focus on digital technology.

The most widespread practice (because it generates immediate savings) is variable lighting tripped by the presence of pedestrians. The subject of energy consumption in private buildings was central to early innovations (Lyon confluence) and an application for public buildings has also been developed in Nantes and Rennes. Measuring how full municipal dumps are and how much individual bins weigh makes it possible to optimise waste collection route efficiency. This can also be applied to networks themselves by optimising the flow of waste in sewers as attempted in a multi-network experiment organised around the University of Lille. In the transport field, lights automatically go green when a bus arrives.

In addition to these automatic processes, "smart grids" actively get the population involved in regulation in a more original manner. As part of a resource efficiency drive or civic-mindedness campaign, inhabitants are asked to modify their consumption patterns. Smart electric grids using linky smart meters installed in households combine optimal network resource allocation with incentives to cut consumption at peak hours, thus illustrating this practice on a large scale. Households in a neighbourhood in Lyon and in Nantes were given tablet devices to help them track and optimise their electricity and water consumption. In Besançon the collection of unsorted waste is billed by weight. In Montpelier, videophones in each apartment can be used to warn of flooding.

New digital-based public services

Smart-city labelled services include both new services and revamped forms of old services. Firstly, the question of digital access is addressed, especially free wifi in various public places (Strasbourg, Mulhouse) as well as help with installing fibre-optic broadband (Dijon). There is also a major focus on mobility, providing enhanced information to commuters on public transport, finding available parking using sensors under each parking space as in Nice or through the use of look-ahead algorithms as has been done in Issy. New services include car sharing, large-scale electric vehicle programmes in Paris, bike sharing services in most cities, or more experimental initiatives involving driverless cars. Certain municipalities are expanding this dynamic to the sharing economy by overseeing car-pooling initiatives (Roubaix, Bordeaux) or making it easier to rent private car parks by the hour in the case of Issy.

Alongside the work of municipalities, a lot of effort has gone into keeping elderly people in their own homes (alerts given when toilets are not used). New digital services have also been used to provide tourist information.

Population security and control

Public security and control are less consensual issues and cities may be reticent about giving controversial activities a "smart-city" label. So, for example, despite being technologically quite similar to the socio-technical processes described previously, cars fitted with Automatic License Plate Recognition technology that can identify car owners who have not paid their parking charges are labelled "smart city" in certain cities (Chartres) but not in others (Paris).

But some municipalities are happy to link urban security and smart cities: Marseille has clearly focused its security innovations around this theme with preventive initiatives based around the automatic interpretation of video camera data.

Support for business and digital initiatives

Economic development is mainly promoted through support for the digital sector and traditional business cluster-type initiatives. Certain economically distressed cities and districts (Roubaix, Plaine commune, Besançon) have set up training programmes to promote digital literacy among young people and help tradespeople to communicate using smart phone apps or terminals. There is also a key focus on tourism with 3D heritage presentations and visitors surveys.

Open data

Open data is also linked to smart-city labelled services and has two objectives. First, greater transparency from a democratic perspective. But in concrete terms, open data means providing spreadsheets containing disaggregated data and processing this into meaningful form. Consequently, examples of disaggregated data use by the general public are relatively rare. Self production by independent associations of data on transport (openstreet maps) or noise (noisetube) using smartphone interfaces are both easier to access because they are formatted by a platform, and less likely to be sorted in a selective manner by public bodies. Cities such as Rennes have supported associations that wish to develop these alternative data capture techniques. Several cities have developed noise and pollution data capture using service vehicles to meet a specific public need (Paris, Grenoble) and public conferences/seminars have been organised in Grenoble and Lille to promote awareness of the use of such data.

The second aim of open data is to nurture start-ups providing urban services. These may include support with house buying, trip planning or managing energy retrofits in buildings. Nevertheless, the growth in these types of services has not fully lived up to the expectations of those who promoted the related law (Denis et al., 2013) and numerous cities have organised competitions to encourage the use of these data, hackathons, "geek-focused" fab labs, partnerships between start-ups and major bodies like the City of Lyon with the TUBA initiative, or large trade fairs to promote start-ups (Nice, Marseille, Paris). One recent trend involves setting up a platform that gives access to real-time data.

Cross-referencing data and big data

The accumulated data related to these various programmes raises the issue of different cities cross-referencing all of the different data.

3) Smart cities: emerging trends

All of the various label providers (i.e., financing bodies, media, municipalities) use relatively similar practices and these have a number of recurring features.

A disorderly, generally uncoordinated collection of innovations

The first impression given by the list -not due solely to the method used -is the nonintegrated character of the innovations. Both the level and approaches to development are different. Some are in the experimental phase, some restricted to a neighbourhood or a few buildings, some disseminated in a general way throughout several cities, some operated under a municipal concession arrangement, while others are part of miscellaneous partnership-type arrangements with both large and small private businesses. But no single city has an overall integrated programme for these innovations and this was borne out by a large parliamentary enquiry into this whole area [START_REF] Belot | De la smart city au territoire d'intelligence(s), rapport au premier ministre sur l'avenir des smart cities[END_REF].

Omnipresent technological dimension

A second feature of the French situation is the omnipresence of technology.

Metaphorical uses of the notion of intelligence [START_REF] Giffinger | Smart cities ranking: an effective instrument for the positioning of the cities?[END_REF] (Meijer, Bolivar, 2016)

A wider technological scope than e-government

Websites, which lie at the heart of e-governance, are still very much present however, the widespread use of the smartphone is gradually transforming this web-based interface. First, round-the-clock access is transforming uses (bus timetables won't be used in the same way in the house as in the city). Second, GPS is generating new opportunities (e.g., immediately being able to localise a problem on a road / street). The differences with e-government also lie in the Internet of Things with a new generation of microchips containing data on individuals and a whole array of sensors. As regards new data processing methods, processing and image recognition technology is gradually being taken up but deep learning-type data analytics methods are still at the proof of concept stage.

Extending administration to urban network management E-governance practices would appear to be one aspect of the smart city however the objects of the smart city are broader than those of e-government which mainly comprise administrative data (information about people, the application of rules, how services work, etc.). The examples presented combine material that belongs more in the realm of engineers than administrators with data that concerns not citizens but objects (building temperature, rain levels, the position of cars within a city, waste tonnage, etc.). Certain developments are an extension of automation processes that began more than three decades ago [START_REF] Dupuy | L'informatisation des villes[END_REF].

Other forms of citizen action

Certain ways of getting users and citizens more actively involved are an extension of web-based e-governance practices, i.e., user ability to consult general information or specific information in their file or to express their opinion. Switching information to smartphones reaches a wider public without necessarily changing these functions. But some processes involve user-citizens in urban management in a more direct way, just like "smart grids"-type apps that allow them to change their behaviour based on the information they receive. Citizens are also involved in the co-production of public goods.

They may use GPS on smartphones to flag up problems in roads/streets more effectively. They can also process and produce urban data in the course of hackathons and various forms of pro-active contribution initiatives (crowd sourcing). In France, citizens are placed at the heart of data usage because under French regulations, they must give their consent before certain data can be accessed (e.g., their GPS or wifi footprints) or before various different types of data can be cross referenced.

New ties with the private sector

Weak presence on peer to peer platforms

One last feature of the socio-technical processes identified is apparent by reading between the lines. When tackling the broad question of how digital technology affects urban practices, François [START_REF] Ménard | Penser la ville intelligente[END_REF] suggests distinguishing between three types of developments: the automated city, the "wiki" city organised around collaboration and the "uberised" city based around platforms that connect individuals to exchange services (car pooling, accommodation, taxis) (bla bla car, airbn'b, uberpop) or data (sharing GP position )(Waze). Comparing experiences actually identified against this typology shows that the bulk of the practices promoted by government departments fall into the first category, a few into the second and virtually none into the third (those few that do concern the third category are often still in the demo phase, e.g., a few car and car park sharing platforms).

4) Renewing traditional e-governance debates

We shall now revisit these specialities in light of traditional e-governance research findings.

Forms of development: Steps or phone apps

One of the most marked impressions concerning the gathering process is the nonintegrated character of the innovations highlighted and this leads us to revisit the question of "stages" of development.

Research into the deployment of e-governance initiatives is based around the idea of hierarchical stage of commitment to the use of ICTs for administrative purposes [START_REF] Layne | Developing fully functional E-government: A four stage model[END_REF] [START_REF] Lee | 10year retrospect on stage models of e-Government: A qualitative metasynthesis[END_REF]. Starting with the most basic level which is putting information on a website, three main learning paths are proposed: interaction / adaptation of each citizen, integration of the different components of the offering and ability of citizens to actively participate. The fact that most articles present the different stages in a single model table implies joint progress across the different activities. However, surveys give a more nuanced picture of the effectiveness of such a development (Noris, Reddick, 2013).

The innovations analysed represent only limited progress in the various different areas and under no circumstances may the smart city be considered a further step along the way. Individual processing of administrative files is virtually unconcerned; as regards capacity to participate in decision-making, we note that consultation processes have shifted from websites to smartphone apps; the integration of the different components of the offering has barely even begun.

Rather than an integrated model, the dominant development paradigm is an accumulation of sector-based innovations equivalent to smartphone apps [START_REF] Ménard | Penser la ville intelligente[END_REF]. Some cities like Lyon or Lille focus their development efforts on a neighbourhood but both give priority to regulation.

Comparing these practices with the models devised to analyse the development of egovernment raises important questions about the development of the smart city. Should we consider this as an incomplete form of development or a new development process? Does the fact that many developments are linked to private operators mean that it is a transition phase or an original development process?

Interoperability between administration and engineering

The fact that these disorderly processes are largely isolated once again raises the crucial issue of interoperability. If we opt for the first integration-based model, we encounter one of the key questions of e-government, namely interoperability between the different data gathered. For T Nam and T.A. [START_REF] Nam | Smart city as urban innovation: Focusing on management, policy, and context[END_REF], who are among the few researchers to have moved from e-governance to the smart city, "a smart city innovation may be classified as an interoperability arrangement". They also stress that "combination, connection and integration of systems and infrastructure" are of fundamental importance. Indeed, the failure of major IT projects is a tangible reality [START_REF] Whittaker | What went wrong? Unsuccessful information technology projects[END_REF].

The same authors believe that the organisational risks inherent in adopting egovernance practices are directly transposable to smart cities. These risks entail both balancing objectives and resources in complex institutional contexts and the fact that controlling information both internally and externally is a key factor in power politics.

Even if the technical issues involved in interoperability are resolved by new technology, the main difficulties are actually social and not technical.

The professional and human component also appears important to the success of such ventures. The possibility for a professional "intermediary" [START_REF] Khanna | The Role Of Intermediaries In Designing Information Infrastructures In Strategic Niches: The Case Of A Sustainable Mobility Infrastructure Experiment In Berlin[END_REF] of forging ties between different sectors and different levels of responsibility is a key factor in successfully coordinating different services. This issue is especially relevant in the case of data officers.

From e-democracy to responsible co-production

The new forms of proactive citizen engagement facilitated by a number of the processes studied actually reframe the whole democracy issue by strengthening co-production.

The possibility of citizens using new technologies to interact directly is one key argument in favour of smart cities as it was for e-governance. However, if we limit the whole democracy debate to ability to participate in decision-making, we encounter very few innovations among the "smart city"-labelled practices identified. This does not mean that the political dimension is absent: this is bound up less with how an opinion is expressed in decision-making than with the expression of civic responsibility within a co-production framework.

Citizens' ability to actively participate in the public service offering or regulation is an established practice that has been based around involvement in associations. The aforementioned technologies provide many more individualised possibilities for the responsible citizen wishing to co-produce public initiatives [START_REF] Linders | From e-government to we-government: Defining a typology for citizen coproduction in the age of social media[END_REF]. Everyone is being called upon to change their behaviour in favour of sustainable development (on economic or civic grounds): lowering the thermal comfort in their homes, shifting electricity use to off-peak periods, sorting waste, etc. Citizens may produce data passively (by allowing their GPS footprint to be accessed) or actively by providing information and they may become involved in processing these data in certain situations (e.g., wiki, openstreet map). They may also participate in the upkeep of public spaces in a renewed form by flagging up problems or be called on to provide services by making their parking space available for the public for example.

Shifting the political focus from participation in decision making to day-to-day coproduction of public goods helps strengthen civil society. But this still raises many questions concerning the isolation of individuals via-à-vis co production processes backed by associations, the limited proportion of the public concerned [START_REF] Linders | From e-government to we-government: Defining a typology for citizen coproduction in the age of social media[END_REF], the lack of public enthusiasm for opportunities to contribute to sustainable development (which also reopens the question of the adoption of e-governance (Carter, Bellanger, 2005)) or distortions in information or services inherent in the information gathering process [START_REF] Rabari | The digital skin of cities: urban theory and research in the age of the sensored and metered city, ubiquitous computing and big data[END_REF].

Public-private partnerships and data control

The huge variability and decisive role of private initiatives in new solutions means that a question that was previously fairly marginal in e-governance now moves to the forefront.

Subcontracting of the traditional websites that are so central to e-governance to private firms had not really been a key research focus. Public-private partnerships were tackled later when e-government functions were farmed out en bloc as part of concession arrangements in developing countries. But more recently, relations with the private sector have been re-examined with the growth in the use of social networks and concerns that exchanges between government and citizens may not remain the property of local authorities [START_REF] Mergel | A framework for interpreting social media interactions in the public sector[END_REF] [START_REF] Barns | Mine your data: open data, digital strategies and entrepreneurial governance by code[END_REF].

Data control around the use of social media has emerged as a recurring theme in the diversity of relations between municipalities and their private subcontractors. Data may be located on an external server owned by a company as is the case with certain partnerships negotiated with big corporations (IBM or CISCO) in Nice, Montpelier or Paris. This is also the case with smaller businesses that provide cloud-based software as a service. Data may also be hosted by companies responsible for day-to-day network management. Transmitting this data in real time to the public concession provider is also a major issue.

Open data and mediated transparency

Open data included in French smart city projects reopens the issue of transparency traditionally associated with e-governance.

While this may give the impression of immediate access to data as citizens are able to penetrate the municipal working environment, the opposite is actually happening: everything hinges on the mediation associated with these data as illustrated by a thesis focusing on the Lyon agglomeration [START_REF] Courmont | Open data et recomposition du gouvernement urbain: de la donnée comme instrument à la donnée comme enjeu politique[END_REF]. For a start, the available data is not as "raw" as it appears (Denis, Goeta, 2013). Management worksheets contain "unclean" data, gaps etc. Therefore, files made available to the public have actually been pre-formatted. Next, turning this data into information assumes reasonably sophisticated processing techniques. The way in which big data is handled by French municipalities illustrates the need for mediation when using these data.

Big data and privacy

One key feature of these innovations is the shift from administration to engineering.

Many of them are concerned with regulating networks and not directly with people.

Focusing on the socio-technical processes deployed in French cities and not on theoretical abstractions concerning possible uses leads to greater circumspection when analysing the frequently touted smart city based on control and surveillance and the related dangers for privacy. Compared to other e-governance sectors like taxation and health with more sensitive and complex data (Bélanger, Hiller, 2006), the whole issue of privacy appears less urgent and should be addressed more in terms of shifting uses (Zoonen, 2016). For example, weighing the net-of-recycling waste left out compared to the recycled volume could be used to shame a citizen engaged in environmentally-unfriendly practices, or filming crowd movements can sometimes be used to recognise specific individuals.

Competition for trust

The paucity of peer-to-peer developments via public platforms and the imbalance between the private and public offering of traditional public services such as urban public transport is reframing the whole question of trust as it relates to e-governance.

The theme of trust as it relates to e-governance has been tackled in two overlapping approaches. The first concerns the trust needed for users to engage in an electronic transaction with the government (Belanger, Carter, 2008) and it is a primary condition for the dissemination and use of such services. The second concerns the impact on citizens' trust in government of service quality and the image of modernity associated with e-governance [START_REF] Parent | Building citizen trust through egovernment[END_REF]. And although the links between efficiency and trust are by no means automatic (Van de [START_REF] Van De Walle | Public service performance and trust in government: the problem of causality[END_REF], the two dimensions overlap insofar as greater general trust in public bodies drives greater use of online services.

The development of private platforms offering services and data-linkage lends a new dimension to this theme. First, some of these private platforms use standard interfaces that customers are used to and expect to find in any public service compatible (Carter, Belanger, 2005) with these standards. Next, these platforms help create trust between users. By offering a chance to rate people on the platform, services such as Airbn'b act like "trust builders" and consequently appear as places of trust. There is therefore potential competition for trust. For example, several cities provide information on the opening hours of various different services such as swimming pools and a wiki that offers the same info plus comments posted by pool users -the latter will inspire much greater trust. A study conducted in France shows that citizens are reluctant to give out information such as their GPS location whereas these same citizen-consumers will gladly provide this location to private platforms. Behind this competition for trust lies the challenge to commitment to public service.

Conclusion

For the running of cities, the transition from e-governance to the "smart city" has been marked more by a series of shifts than by any major overhaul. Movements concern objects (from network administration to engineering), the technologies harnessed following the emergence of connected objects, platforms and data analytics, and the increasingly intricate relationship between public stakeholders, private operators and users/citizens. Because of these changes, applying a few of the hottest e-governance issues to new "smart city"-type practices and focusing on contrasts makes it possible to both leverage the findings of the primary research field and trace out some original research avenues.

Certain e-governance issues appear totally relevant to the broader domain of the smart city and certain findings -especially those regarding all questions of integration and coordination between services -could be transcribed directly. Nevertheless, changes and shifts observed have helped to renew certain e-governance debates, greater possibilities for interaction in services and more complex links with private sector offerings created by new perspectives, particularly around the co-production of the public service and joint trust-building.

The whole issue of smart city governance lies at the juncture of all these practical questions. The ability of municipal representatives and populations to control these developments is by no means a foregone conclusion and each of the themes tackled here is one aspect of this governance. Bibliography Alizadeh, T. ( 2017). An investigation of IBM's Smarter Cites Challenge: What do participating cities want?. Cities, 63, 70-80. Anthopoulos, L. (2017). Smart utopia VS smart reality: Learning by experience from 10 smart city cases. Cities,63,

  This is part of a longer-term strategy of pooling information around geographical information systems deployed in places like Rennes or Plaine commune. Making information accessible in shareable form also helps data circulate between different municipal services. New automatic regulation or shared user systems also generate big quantities of data, often in real time (big data). Cities can then use deep learning to combine and interpret data and come up with a cross-cutting analysis of urban practices. Many of these multi-dimensional processing techniques are still in the experimental phase: Issy has harnessed image recognition technology to other available data; Lyon tracks its own water consumption; Dijon has built a multi-sector urban network management centre supported by historical service operators; in Nice and Montpelier, IBM has provided a turnkey flood management system. Mobility in transport and public areas has been tackled in Rennes using transport mapping data, in Mulhouse by harnessing user Wifi footprints (p8), and in Paris via image processing technology.

  are fairly present in general political discourse which focuses on the importance of people vis-à-vis technology but the concrete examples referred to almost always embrace digital technology. The rare exceptions concern sustainable development-type projects backed by the state financing bank CDC or a few nontechnical economic development initiatives that nonetheless focus on digital technology or operating arrangements associated with the digital transformation such as "one-stop shops". So, behind the banner of the smart city and all the attendant rhetoric there is a practical imperative: the impact of new digital technology on the management of city life.

  Some solutions are based around complex relations with the private sector (Barreau-Serfati, 2011). Certain cities have deployed an offer structured around a partnership with a big digital operator (e.g., IBM in Nice and Montpelier, Toshiba in Lyon (Faivre d'Arcier et al., 2016)) or urban services operator (Bouygues in Dijon). Other cities have partnered innovation start-ups (Chartres), purchased turnkey solutions that are intended to be replicated in other cities (Strasbourg for wifi or non-electric vehicle sharing), or negotiated numerous agreements with big companies (Issy).

First, a big

  chunk of the data actually affected by big data practices are note linked to individuals. One of the most advanced domains is flood prevention (weather data, water height and pipeflow data). Certain traffic data based on cameras and counting can be used to manage traffic flows without ever actually recognising cars. Next, individualized data is often of poor quality: a bus or tramway travel card may reveal which line you took or maybe where you got on, but never where you got off. Security cameras can record the trips people take but are only used to recognise individuals in very specific circumstances. Generally speaking, data-linkage currently appears limited in scope when compared with what can be done with the pooling and processing of consumption data. Initiatives to harness deep learning are still mostly in the experimental phase.


Delphine Sinoquet

Frédéric Delbos

Jean Charles Gilbert

Jean Charles

A dedicated constrained optimization method for 3D reflexion tomography

Seismic reflection tomography is a method for the determination of a subsurface velocity model from the traveltimes of seismic waves reflecting on geological interfaces. From an optimization viewpoint, the problem consists in minimizing a nonlinear least-squares function measuring the mismatch between observed traveltimes and those calculated by raytracing in this model. The introduction of a priori information on the model is crucial to reduce the under-determination. The contribution of this paper is to introduce a technique able to take into account geological a priori information in the reflection tomography problem expressed as constraints in the optimization problem. This constrained optimization is based on a Gauss-Newton Sequential Quadratic Programming approach. At each Gauss-Newton step, a solution to a strictly convex quadratic optimization problem subject to linear constraints is computed thanks to an augmented Lagrangian relaxation method. Our choice for this optimization method is motivated and its original aspects are described. The efficiency of the method is shown on applications on a 2D OBC real data set and on a 3D real data set: the introduction of constraints coming both from well logs and from geological knowledge allows to reduce the under-determination of the 2 inverse problems.

Introduction

Reflection tomography allows to determine a velocity model from the traveltimes of seismic waves reflecting on geological interfaces. This inverse problem is formulated as a nonlinear least-squares function which measures the mismatch between observed traveltimes and traveltimes computed by ray tracing method. This method has been successfully applied to real data sets [START_REF] Ehinger | 3D tomographic velocity model determination for two North Sea case studies[END_REF][START_REF] Broto | Anisotropic traveltime tomography for depth consistent imaging of PP and PS data[END_REF]. Nevertheless, the under-determination of the inverse problem generally requires the introduction of additional information on the model to reduce the number of admissible models. Penalization terms modelling this information can be added to the seismic terms in the objective functions but the tuning of the penalization weights may be tricky. In this paper, we propose to handle the a priori information by the introduction of equality and inequality constraints in the optimization process. This approach allows to introduce lot of constraints of different types, provided we have at our disposal an adequate constrained optimization method. We developed an original method designed for the tomographic inverse problem which presents the following characteristics: it is a large scale problem (10000-50000 unknowns), the forward operator is nonlinear and its computation may be expensive (large number of source-receiver couples, up to 500000), the problem is ill-conditioned.

In the first part of this paper, the chosen method is motivated and its original aspects are shortly described (for further details, refer to [START_REF] Delbos | Problèmes d'optimisation de grande taille avec contraintes pour la tomographie de réflexion[END_REF]. Applications on a 2D PP/PS real data set and on a 3D PP real data set are presented in a second part.

Constrained optimization method designed for reflection tomography

The general problem in constrained reflection tomography can be formulated as , ) ( and

) ( subject to ) 2 2 1 : ( 2 2 2 u m C l e m C Rm m T T(m) f(m) Minimize I E T obs ≤ ≤ = + - = σ
where f is the objective function, m the parameters of the model, T is the nonlinear traveltime operator, R is the regularization matrix, σ is the associated regularization weight and C are the equality or inequality constraint matrices. In this last formulation, we recognize the standard regularized nonlinear least-squares problem of reflection tomography subject to linear constraints. Two main approaches can be used to solve this nonlinear optimization problem: the penalty methods including the famous Interior Points approach (IP) and the Sequential Quadratic Programming approach (SQP). The current stage of the IP method seems to require more cost function evaluation than the SQP does. This is our motivation to develop an SQP approach for solving the reflection tomography problems considering that the ray tracing solver is expensive in CPU time.

The SQP method consists in solving a sequence of quadratic problems subject to linear constraints. To be efficient the SQP method must solve each quadratic problem, called also the Tangential Quadratic Problem (TQP), in a reasonable CPU time. The TQP to be solved at the Gauss-Newton step k can be formulated as , ) . ( and ) . ( subject to : (TQP)

k k k I k k E u m C l e m C m) ( F Minimize ≤ ≤ = δ δ δ
where k F is the quadratic Gauss-Newtonian approximation of the nonlinear function f in k m and m δ is the model perturbation to be computed from k m .

Three main approaches can be used to solve this convex quadratic problem subject to linear constraints : Interior-Points approach (IP), Augmented Lagrangian approach (AL) and Active Set approach (AS). AS approach has been discarded because of its inefficiency to quickly identify the active constraints. In the present context the potential ill-conditioning of the linear system associated with IP method and the necessity to use iterative methods to solve them appear to us as a deterrent factor. Then, we believe that an AL method implemented in such a way that it does not require any matrix factorization is the appropriated method to solve the TQP.

The AL method consists in solving a sequence of quadratic problems subject to bound constraints. Each of them, called the Lagrange Problem (LP), is efficiently solved by an original method combining the three following ingredients: the projected gradient method, the active set algorithm and the conjugate gradient iterations. At each AL iteration, the Lagrange multipliers are updated thanks to the classical multiplier method of Hesteness and Powell.

The overall method explained above has been successfully tested on several reflection tomography applications (the next section presents two of them). For further details on the different optimization method used to solve our constrained reflection tomography problem, we refer the reader to [START_REF] Delbos | Problèmes d'optimisation de grande taille avec contraintes pour la tomographie de réflexion[END_REF].

Applications of constrained reflection tomography on real data sets 2D PP/PS data set:

In this section, we present an application of constrained reflection tomography to one 2D line of a 3D 4C OBC survey with PP and PS data. [START_REF] Ehinger | 3D tomographic velocity model determination for two North Sea case studies[END_REF] have already interpreted and studied this data set using an unconstrained inversion method. The velocity model is described by four velocity layers and five interfaces (cf Figure 1 left). The isotropic assumption was satisfying until the last layer (layer which contains the last two interfaces h4 and h5). By applying the anisotropic inversion methodology of [START_REF] Stopin | Reflection tomography of compression and shear modes for determination of anisotropic velocity models[END_REF] on the last layer, they obtained a model that fits the traveltimes better than any of the previous isotropic models and that, in addition, has more reliable velocity variations. The value of δ anisotropy parameter has been obtained by a trial and error approach in order to match approximately the h5 depth given by well logs. Actually, the underdetermination of the inverse problem does not allow the joint inversion of the velocities, interfaces and anisotropy parameters (especially δ parameter, [START_REF] Stopin | Reflection tomography of compression and shear modes for determination of anisotropic velocity models[END_REF]. This applied methodology is obviously not optimal. Indeed, the manual tuning of the anisotropy parameters requires lot of time: an important numbers of anisotropic inversion with different couples (η,δ) have to be performed before getting a satisfying result. Secondly, it is very hard to make this method accurate: we note a discrepancy of 150 meters for the reflector depth h5 compare to the depth given by the well data. Finally, it turned out impossible to determine the anisotropy parameter δ so that both the reflector depths of h4 and h5 given by the well logs be reasonably matched.

The solution we propose here is to compute a model using our constrained inversion method in order to fit with the reflector depths given by the well. In the left and right part of Table 1 we have respectively summed up the results of the final models obtained with the unconstrained inversion and with a constrained inversion. The final model (Figure 1 right) of the constrained inversion matches the traveltime data with the same accuracy than the result obtained by unconstrained inversion1 , and it strictly verifies the reflector depths given by well logs.

Then, the introduction of constraints at wells for the two reflectors h4 and h5 reduces the underdetermination and allows a joint inversion of velocities, interfaces and anisotropy parameters to find a model that matches the traveltime data and the well data. 3D PP data set: During the European KIMASI project, reflection tomography was applied on a 3D North Sea data set from bp [START_REF] Ehinger | 3D tomographic velocity model determination for two North Sea case studies[END_REF]. A P-velocity model was obtained thanks to a topdown layer-stripping approach where lateral and vertical velocity variations within Tertiary, Paleocene and Cretaceous (divided in two velocity layers) units have been determined sequentially (Figure 2). A strong velocity under-determination in the upper Tertiary layer was detected during the inversion process due to the large layer thickness (2.5km) and to the very poor ray aperture. Several inversions with different velocity parameterizations gave the same traveltime misfit RMS. These different tests are time consuming (each test is a whole inversion) and the reflector depths given by well data are not well retrieved, this information being not explicitly introduced in the inversion process. To obtain a model more consistent with well data, we propose to apply our developed constrained tomographic inversion. The interface depths are constrained at 5 well locations and we constrain the range of variations of the vertical velocity gradient in the Tertiary layer thanks to well measurements (Table 2).

To limit bad data fitting for deeper layers, a global inversion approach is applied. This global update is only possible if the layer thickness are constrained to avoid any non-physical intersection of interfaces.

The experiment consists then in a global inversion of 127569 traveltimes for 5960 unknowns describing 4 velocity layers and 5 interfaces, subject to 2300 constraints2 . The results are presented in Figure 3: the obtained model matches the data with the same accuracy as the model obtained by [START_REF] Ehinger | 3D tomographic velocity model determination for two North Sea case studies[END_REF] and verifies all the introduced constraints. The total number of conjugate gradient iterations for each Gauss-Newton step 3 is less than 10000 (less than twice the number of unknowns), which is a very good result for a problem with 2300 constraints.

Conclusion

Reflection tomography often requires the introduction of additional a priori information on the model in order to reduce the under-determination of the inverse problem. A nonlinear optimization method that allows the introduction of linear constraints on the model has been developed. The applications on two real data sets have shown its efficiency.

M o d el w ith o u t co n strain ts M o d el w ith co n strain ts tpal

Figure 1 :

 1 Figure 1: the left velocity models (Vp and Vs) are computed with the unconstrained inversion method; the right velocity models (Vp and Vs) are computed with the constrained inversion method.The white crosses symbolize reflector depths measured from the deviated well logs.

Figure 2 :

 2 Figure 2: Velocity model (slices along x (left) and along y (right) at one of the 5 well locations) obtained with the unconstrained reflection tomography. The RMS value of the traveltime misfits is 6.1ms.

Figure 3 :

 3 Figure 3: Velocity model (slices along x (left) and along y (right) at one of the 5 well locations) obtained with the constrained reflection tomography. The RMS value of the traveltime misfits is 6.5ms.

Z-99 A dedicated constrained optimization method for 3D reflection tomography

	D. SINOQUET*, F. DELBOS*, J.CH. GILBERT**
	*I. F.P. 1&4 av. Bois-Préau 92852 Rueil-Malmaison Cedex France
	**I.N.R.I.A. Rocquencourt BP 105 78153 Le Chesnay Cedex France

Table 1 :

 1 Inversion results of the unconstrained inversion (left) and the constrained inversion (right)

	l4	RMS (s) Delta z (m)	η	δ	l4	RMS (s) Delta z (m)	η	δ
	h4-PP	3,6	190			h4-PP	6,3	0
	h5-PP h5-PS	4,6 9,8	150	6,2%	2,0%	h5-PP h5-PS	3,9 8,1	0	8,2%	15,9%

Table 2 :

 2 Description of the equality and inequality constraints introduced in the inversion. Starting the inversion with the model obtained by unconstrained tomography[START_REF] Ehinger | 3D tomographic velocity model determination for two North Sea case studies[END_REF], we succeed to obtain a model that matches the data and the 2300 constraints.

The smooth velocity model does not allow smaller traveltime misfits than 6.3ms when h4 depth is constrained at well location. Allowing a velocity discontinuity contrast at h4 interface may help to obtain smaller misfits.

The constraints on the velocity variations and on the layer thickness are applied on a grid of 10x10x10 and 20x20 points. The total number of conjugate gradient iterations takes into account all the iterations of augmented Lagrangian method.

Acknowledgments

The authors would like to thank bp to have provided the data sets, Karine Broto and Anne Jardin for their precious comments on the applications and Carole Duffet for her crucial help.


Carole Duffet
email: carole.duffet@ifp.fr

Delphine Sinoquet
email: delphine.sinoquet@ifp.fr

QUANTIFYING UNCERTAINTIES ON THE SOLUTION MODEL OF SEISMIC TOMOGRAPHY

Keywords: Tomography, uncertainties quantification, velocity model building

Reflection tomography allows the determination of a velocity model that fits the traveltime data associated with reflections of seismic waves in the subsurface. A leastsquare formulation is used to compare the observed traveltimes and the traveltimes computed by the forward operator based on a ray tracing. This non linear optimization problem is solved classicaly by a Gauss-Newton method based on successive linearizations of the forward operator. The obtained solution is only one among many possible models. Indeed, the uncertainties on the observed traveltimes (resulting from an interpretative event picking on seismic records) and more generally the underdetermination of the inverse problem lead to uncertainties on the solution. An a posteriori uncertainty analysis is then crucial to delimit the range of possible solutions that fit, with the expected accuracy, the data and the a priori information. A linearized a posteriori analysis is possible by an analysis of the a posteriori covariance matrix, inverse of the Gauss-Newton approximation of the matrix. The computation of this matrix is generally expensive (the matrix is huge for 3D problems) and the physical interpretation of the results is difficult. Then we propose a formalism which allows to compute uncertainties on relevant geological quantities for a reduced computational time. Nevertheless, this approach is only valid in the vicinity of the solution model (linearized framework) and complex cases may require a non linear approach. An application on a 2D real data set illustrates the linearized approach to quantify uncertainties on the solution of seismic tomography and the limitations of this approach are discussed.

INTRODUCTION

Geophysical methods for imaging complex geological subsurface in petroleum exploration requires the determination of an accurate propagation velocity model. Seismic reflection tomography ( [START_REF] Bishop | Tomographic determination of velocity and depth in laterally varying media[END_REF]) turns out to be an efficient method for that: this method allows to determine a seismic velocity distribution from traveltimes data associated with the seismic waves reflecting on geological surfaces. This inverse problem is formulated as a least-square minimization problem which consists in the minimization of the mismatch between the observed traveltimes and the traveltimes computed by the forward problem (solved by a ray tracing method). The first section of this paper presents an overview of the reflection tomography method we have developed ( [START_REF] Jurado | 3D reflection tomography designed for complex structures[END_REF], [START_REF] Jurado | Fast 3D two-point raytracing for traveltime tomography[END_REF], [START_REF] Stopin | Reflection tomography of compressional and shear modes for determination of anisotropic velocity models[END_REF], [9]). Methods for the a posteriori analysis of the solution are then proposed and applied on a 2D real data set. Indeed, classically, the analysis of the solution consists only in checking the misfits between observed traveltimes and calculated traveltimes by computing characteristic values of the misfit distribution and by studying the spatial distribution of the misfits. This analysis is not sufficient: even if these quality control criteria are matched, the determined model is only one of many possible models that match the data. An uncertainty analysis should be performed to quantify the range of admissible models we can obtain from these data and the a priori information ( [START_REF] Duffet | Quantifying geological uncertainties in velocity model building[END_REF]). The different methods described here allow to access the uncertainties on the solution model thanks to the analysis of the a posteriori covariance matrix ( [START_REF] Gouveia | Resolution of seismic waveform inversion: bayes versus occam[END_REF], [START_REF] Tarantola | Inverse problem theory[END_REF]) obtained in the linearized framework. The computation of this matrix is generally expensive for 3D problems and the physical interpretation of its terms may be cumbersome. The simulation of admissible models from the a posteriori probability density function allows a more complete analysis than the sole analysis of the diagonal terms of the a posteriori covariance matrix and furnishes directly physical models. This method requires a Cholesky decomposition of the Gauss-Newton matrix which may be unfeasible for 3D realistic models. To overcome this difficulty, we propose a formalism based on linear combinations of model parameters (macro-parameters) that allows to compute uncertainties on relevant geological quantities (the average thickness of a layer for example) for a reduced computational time (the a posteriori covariance matrix is reduced to the macro-parameter space). The application of these methods on a real data set shows their effectiveness and their limitations are discussed, particularly the limitations due to the linearization: a non linear approach is proposed in order to delimit the space of admissible solutions.

THE REFLECTION TOMOGRAPHY PROBLEM

Let us first present the reflection tomography method which was developed in KIM consortium ([9]). The chosen model representation is a blocky velocity model where the velocity distribution is described by slowly varying layer velocities (also called velocity blocks) delimited by interfaces (see Figure 1 for an example). The model is thus composed of two kinds of parameters: parameters describing the velocity variations within the layers and parameters describing the geometry of the interfaces delimiting the velocity blocks1 . Moreover, anisotropy of the velocity may be modelled by two parameters η and δ (see [START_REF] Stopin | Reflection tomography of compressional and shear modes for determination of anisotropic velocity models[END_REF] for more details). This blocky representation allows the representation of complex models (e.g. presenting rapid lateral velocity variations and complex geological structures) with a reasonable number of unknowns. A strong a priori information is introduced, the number of layer being fixed: this reduces the underdetermination of the inverse problem.

The forward problem of reflection tomography consists in computing traveltimes of reflected seismic waves, given a model, an acquisition survey (locations of the sources and the receivers) and signatures (reflectors where the waves reflect). It is solved by a ray tracing method which is a high frequency approximation of the wave equation ( [START_REF] Jurado | Fast 3D two-point raytracing for traveltime tomography[END_REF]). We denote by T (m) the forward modeling operator, that is the operator that gives, for a specified reflector, the traveltimes associated with all source-receiver pairs. Reflection traveltime tomography is the corresponding inverse problem: its purpose is to adjust m such that T (m) best matches a vector of traveltimes T obs picked on seismic data. A natural formulation of this problem is the least squares formulation

T (m) -T obs 2 C -1 D + m -m prior 2 C -1 M , (1)
where

• C D is the a priori covariance operator in the data space that allows to describe errors on the data,

• m prior is an a priori model (coming, for instance, from geological knowledge and/or additionnal information resulting from well measurements),

• C M is the a priori covariance matrix in the model space that allows to describe errors on the a priori model.

In practice, building an a priori model is difficult and the a priori term, which is necessary to well pose the inverse problem, is often replaced by regularization terms made up of second derivatives of the model ( [START_REF] Delprat-Jannaud | Ill-posed and well-posed formulations of the reflection traveltime tomography problem[END_REF]) which leads to C -1 M = Q reg and m prior = 0

T (m) -T obs 2 C -1 D + 2 D 2 m 2 (2)
where a continuation technique ( [START_REF] Bube | A continuation approach to regularization for traveltime tomography[END_REF]) is applied for tuning the regularization weight : the weight is decreased until the model matches the data with the expected accuracy. The obtained model is then the smoothest model which fits the data. This methodology allows to stabilize the inversion. This large size non linear least-square problem (the objective function is not quadratic since the forward modeling operator T (m) is non linear) is solved classically by a Gauss-Newton method based on successive linearizations of the forward operator which needs the computation of the Jacobian matrix of T (J(m) = ∂T ∂m (m)) 2 . The resulting approximation of the Hessian is noted H(m) = J(m) t C -1 D J(m). The resulting quadratic approximation of the objective function ( 2)

Cn (δm) = ||J n δm -δT obs n || 2 C -1 d + 2 ||D 2 (m n + δm)|| 2 (3)
is minimized at each Gauss-Newton iteration n by a preconditionned conjugate gradient (m n is the current model, δm is the model perturbation, J n = ∂T ∂m (m n ) is the Jacobian matrix evaluated at m n , and δT obs n = T obs -T cal (m n )) ( [START_REF] Sinoquet | A dedicated constrained optimization method for 3D reflection tomography[END_REF]).

PRESENTATION OF THE APPLICATION

Let us consider a 2D PP/PS3 real data set studied by [START_REF] Broto | Anisotropic traveltime tomography for depth consistent imaging of PP and PS data[END_REF]. 45338 traveltime data were interpreted and an uncertainty of 5ms (resp. 8ms) are associated with PP data (resp. PS data). A layer-stripping approach (separate inversion of each velocity layer from the shallower layer to the deeper one) furnished the velocity model of Figure 2. The model is described by four interfaces, corresponding to the interpreted events (h1, h3, h4 et h5) which define only three layers with lateral velocity variations for the two first upper layers (v(x), there is no vertical variations) and the last layer stretching from h3 to h5 with a 2D velocity vp5(x, z) and vs5(x, z) (Figure 2). This model is composed of 4588 parameters, 592 for the interfaces and 3936 for the velocities. Anisotropy in the deepest layer is parameterized by η and δ ( [START_REF] Broto | Anisotropic traveltime tomography for depth consistent imaging of PP and PS data[END_REF]) which are assumed constant per layer (η can be seen as a measure of the velocity an-ellipticity, whereas δ controls the near vertical velocity propagation of the P-waves). For the two first layers, the inversion results are very satisfactory, with a traveltime misfit RMS (root mean square) of 3.5ms for the PPdata and of 7ms for the PS-data. For the last layer, the traveltime misfit RMS is of 3.7ms and of 5.6ms for, respectively, h4 and h5 PP-data and of 8.5ms for h5 PS-data, results which are consistent with the data uncertainties (Figure 3).

As already shown by [START_REF] Stopin | Reflection tomography of compressional and shear modes for determination of anisotropic velocity models[END_REF], it turns out that the anisotropy parameter δ is strongly undetermined from seismic data. The value of δ parameter was obtained by a trial and error approach in order to match approximatly the depth of h5 horizon given at well location. We propose then to carry on an a posteriori uncertainty analysis in order to quantify the uncertainties on this solution model: we focus on the anisotropic layer delimited by h3 and h5 (velocities vp5 and vs5).

QUANTIFYING UNCERTAINTIES ON THE SOLUTION MODEL: A LINEARIZED APPROACH

A classical approach to quantify uncertainties consists in the analysis of the Hessian matrix (or its inverse: the a posteriori covariance matrix) associated with the linearized problem (3) around the solution m ∞ . This approach is valid in the vicinity of the solution model, the size of the vicinity depending on the non-linearity of the forward map. The bi-linear form associated with the Hessian matrix measures the influence of a model perturbation δm on the quadratic cost function defined around the solution m ∞ :

C(δm) -C(0) =C(m∞) = 1 2 (J(m ∞ )δm) T C -1 D (J(m ∞ )δm) + 1 2 δm T C -1 M δm -δm T (J(m ∞ ) T C -1 D δT obs (m ∞ ) -C -1 M m ∞ ) =g(m∞)=0, gradient of C(m) vanishes at the solution m∞ = 1 2 δm T (J(m ∞ ) T C -1 D J(m ∞ ) + C -1 M )δm (4) = 1 2 δm T H(m ∞ )δm. ( 5
)
The a posteriori covariance matrix is defined by

C M = (J(m ∞ ) T C -1 D J(m ∞ ) + C -1 M ) -1 . (6)
The space of admissible models can be characterized by the contour lines

(m -m ∞ ) T C -1 M (m -m ∞ ) = constant, (7)
which are ellipsoïds of center m ∞ and correspond also to contour lines of the a posteriori Gaussian probability density function exp -

1 2 δm T C -1 M δm . (8)
The diagonal terms of C M are the uncertainties on the parameters describing the model and the off-diagonal terms are the correlations between these uncertainties. For instance, the probability that the true model parameter p i verifies -2C M i,i ≤ p i -p ∞ ≤ 2C M i,i , independently of the values of the other model parameters, is about 95%. To take into account the correlations between the parameters, we should study the 95% confidence ellipsoid. The axes of the ellipsoïds [START_REF] Gouveia | Resolution of seismic waveform inversion: bayes versus occam[END_REF] are defined by the eigenvectors of C M , the square root of the eigenvalues giving the uncertainties on the associated eigenvector.

We propose in the next sections two methods to quantify geological uncertainties on the solution model which avoid the expensive computation and the cumbersome analysis of the generally huge a posteriori covariance matrix C M .

Simulations of admissible models

The first proposed method to quantify uncertainties is the simulation of admissible models from the a posteriori probability density function (8) (pdf). The simulations furnish directly interpretable results, i.e. physical models. The method (see for instance [START_REF] Parker | Geophysical inverse theory[END_REF]) consists in random simulations of model perturbations following the pdf [START_REF] Grenié | Tomographie de temps de trajet adaptée au problème de l'imagerie 3D en exploration pétrolière: théorie, mise en oeuvre logicielle et applications[END_REF].

First, we apply the variable transformation:

δm = U δm, (9)
where U is the lower triangular matrix obtained by a Cholesky decomposition of the inverse of the a posteriori covariance matrix C -1 M = H = U T U (symmetric semi positive definite matrix). The method consists then in simulations of the uncorrelated Gaussian pdf with unit variance exp -

1 2 δm T δm . (10)
Simulations δm are transformed into correlated Gaussian simulations of vector δm by the inverse transformation of (9).

Figures 4, 5, 6, 7 and 8 show 100 simulated models obtained by this method from the solution model of Figure 2: interfaces h4 and h5, variations of velocities vp5 and vs5 along x and z directions and histograms of the anisotropy parameters η and δ. From these simulations, we observe that the highest uncertainties on the lateral velocity variations are located at the boundaries of the model, areas that are not well illuminated by the rays. Indeed, for a slice at constant z = 2.7km, we observe uncertainties of ≈ 690m/s for vp5 and of ≈ 260m/s for vs5 at the boundaries of the model and uncertainties of ≈ 450m/s for vp5 and of ≈ 140m/s for vs5 in the illuminated parts of the model. Concerning the interface depths, we observe uncertainties of ≈ 180m for reflector h4 and of ≈ 220m for reflector h5 at the boundaries and of ≈ 90m for reflector h4 and of ≈ 100m for reflector h5 elsewhere. For the anisotropy parameters η and δ, we notice uncertainties around 0.3% on η and 2% on δ. This method is quite attractive for the straightforward interpretation of the results despite its cost (cost of the Choleski decomposition): it furnishes physical models leading to small perturbations of the quadratic cost functions (3) (see Table 4.3).

X (km)

Interface H4 Z (km) 12 1 2.5

23.5

Uncertainties on geological macro-parameters

We propose here a method that allows to deal with large size models at a reasonable cost and furnishes uncertainties on chosen physical quantities. The proposed approach consists in building macro-parameters (MP) with a geophysical interest. These macroparameters are linear combinations of the inverted parameters such as the mean of the velocity variations in a zone, the slope of an interface, the average thickness of a layer, etc. [START_REF] Grenié | Tomographie de temps de trajet adaptée au problème de l'imagerie 3D en exploration pétrolière: théorie, mise en oeuvre logicielle et applications[END_REF] has introduced the notion of macro-parameter (his main motivation being to avoid numerical problems in the inversion of the complete Hessian). We propose here a generalization of his work (general definition of macro-parameters) which allows the computation of uncertainties for huge 3D problems.

We define a macro-parameter as:

P = Bp, ( 11
)
where p is the np model parameters vector, P is the n M P macro-parameters vector and B is the condensation matrix. We compute the a posteriori covariance matrix C M in the macro-parameter space:

C M = BC M B T . ( 12
)
Note that C M (a n M P × n M P matrix) is small compared to C M (a n p × n p matrix) since n M P n p . To obtain this matrix, we do not need to compute the whole inverse of the Hessian, C M = H -1 , in the parameter space. Indeed, to obtain C the MP space, we just need H -1 B T j , where B T j are the different columns of B T . We thus solve n M P linear systems, H-1 j = BH -1 B T j , that are similar to the linearized problem we solve at each Gauss-Newton iteration. Thus the computational cost for one MP is comparable to one iteration of the inversion process.

M = H-1 = BH -1 B T in X (
We applied this method on the model of Figure 2. We choose simple MP: the mean of the velocity variations and the mean of the interface depth in the illuminated part of the layer (region covered by rays) 4 . The results are listed in the Table 1. The uncertainty on anisotropy parameter δ is high: ±3.2% (if we consider twice standard deviation) and we observe high correlations between η and δ (0.93) and also correlations between anisotropy parameters and H5 depth (-0.33 and -0.36). We notice also the bad determination of the velocities: ±22% (relative value). All these results are consistent with the simulations results (section 4.1).

This method with its general formalism allows to compute uncertainties on relevant geological quantities with a reasonable computation cost and to highlight easily strong coupling between the chosen MP.

Discussions on the two methods

As already mentioned, the two methods described in sections 4.1 and 4.2 rely on the quadratic approximation (3) of the non linear cost function. In the table 4.3 we have listed the RMS of traveltime misfits for 20 simulated models. Note that the RMS traveltime misfit for the solution model is 6.2ms. Thus, there is only one model which is not acceptable (11.6ms for the model 15), the others are admissible models with a RMS traveltime misfit bounded by 7ms. The simulation method has then furnished model perturbations that correspond to small perturbations of the quadratic cost function but also to small perturbations of the non linear cost function (RMS of traveltime misfits remain small). It shows for this example, the good concordance between the quadratic cost function and the non linear one around the solution model.

Figure 9 illustrates isovalues of the probability density function (8) in the MP space, i.e. ellipsoïds defined by δm T M P C -1 M δm M P : we visualize the 2D-marginal probability density functions for several relevant MP couples, the 68% confidence ellipsoïd and the 95% confidence ellipsoïd for η and δ, vp5 and vs5, δ and h5. This representation allows to underline the correlations between the anisotropy parameters η and δ which are quite strong in this example: the 68% and the 95% confidence ellipsoïds are elongated (left part of the figure 9). On the opposite, the P and S velocities are uncorrelated: we obtain almost a circle (middle part of the figure 9). We can also observe correlations between the anisotropy parameter δ and the mean of the last reflector depth h5 (right part of the figure 9). On this Figure, we have also superimposed the anisotropy parameters η and δ and the MP vp5 and vs5, δ and MP h5 built from the 100 simulated models. This figure allows to point out the equivalence of the simulation method and the macro-parameter approach.

Note that the macro-parameter approach furnishes interesting information on the correlations of macro-parameter uncertainties and has a lower computation cost than simulations of the model parameters with the complete a posteriori covariance matrix. We could also perform gaussian simulations of the macro-parameters using the reduced a posteriori covariance matrix.

NON-LINEAR A POSTERIORI ANALYSIS

We have shown in the previous sections that a linearized uncertainty analysis allows to delimit the range of possible solution models that fit, with the expected accuracy, the data Models RMS(ms) vp5(km/s) vs5(km/s) h4(km) h5(km) η(%) δ(%) 1 6.7 and the a priori information. Nevertheless, we should keep in mind that this approach is only valid in the vicinity of the solution model (linearized framework) and complex cases may require a non linear approach.

Model 15 of table 4.3 indicates that some of the simulated models may produce unacceptable traveltimes misfits (RM S ≈ 11ms and MAX = 25ms for PP data and = 41ms for PS data). It shows the limitation of the quadratic approximation of the non linear cost function and thus the limitation of the linearized methods of uncertainty quantification. Moreover, [START_REF] Sinoquet | A dedicated constrained optimization method for 3D reflection tomography[END_REF] has shown that a tomographic inversion with constraints on the location of the interfaces at well locations furnished a very different model (model 2 of Figure 10) than the solution model of Figure 2. The two models verify the traveltime data with the expected accuracy. This second model does not belong to the range of possible models detected by the two methods we proposed: indeed, δ is equal to 15.94% whereas the expected range of values was between -8% and -1% (see for instance Figure 8). It shows

CONCLUSIONS

Reflection tomography furnishes the velocity model that best fits the traveltime data: however, this solution is only one among many admissible models. An a posteriori uncertainty analysis is crucial to delimit the range of possible solution models that would fit the data and the a priori information with the expected accuracy. In this paper, we describe two methods to perform a linearized a posteriori analysis, approach valid only in the vicinity of the solution model. The simulation method based on the analysis of the a posteriori covariance matrix (huge matrix for 3D models) is quite attractive for the straightforward interpretation of the results despite its cost. We propose a general formalism to reduce the a posteriori analysis to geological quantities of the model: we evaluate uncertainties on macro-parameters (linear combinations of model parameters) that have a geological interest. This method allows the manipulation of reduced matrices

Figure 1 :

 1 Figure 1: An exemple of a blocky subsurface model.

Figure 2 :

 2 Figure 2: Solution velocity model obtained by tomography. Left: P velocity. Right: S velocity. The RMS value of the traveltime misfits is 6.2ms. The anisotropy parameters values are: η = 6.29% and δ = -4.43%.

Figure 3 :

 3 Figure 3: Interfaces of the solution velocity model obtained by tomography and their associated traveltime misfits: impact points of rays reflected on h4 and h5 (Left: PP rays. Right: PS rays), colors represent traveltime misfits. Notice that the illuminated zone extends from 14km to 20.5km.

Figure 4 :

 4 Figure 4: Reflector h4 extracted from the 100 simulated velocity models. The bold line represents the solution interface.

Figure 5 :Figure 6 :

 56 Figure 5: Reflector h5 extracted from the 100 simulated velocity models. The bold line represents the solution interface.

Figure 7 :

 7 Figure 7: S-velocity variations along x axis at z = 2.7km (left) and along z axis at x = 16km (right) extracted from the 100 simulated velocity models.

Figure 8 :

 8 Figure 8: Histograms of the 100 simulated velocity models. Top: anisotropy parameter η. Bottom: anisotropy parameter δ. σ corresponds to the standard deviation on the anisotropy parameter η ( resp. δ) given by the macro-parameter method.(see section 4.2).

Figure 2 ,Figure 10 :

 210 Figure 10: Solution model 2 obtained by tomography with additional constraints on the interface depth at well locations. The RMS value of the traveltime misfits is 6.0ms. The anisotropy parameters values are: η = 8.27% and δ = 15.94%. The crosses represent the wells.

Table 1 :

 1 Standard deviations in bold (square roots of the diagonal terms of the a posteriori covariance matrix) associated with the MP "mean in the illuminated part of the model" and the normalized correlations between the uncertainties on the different defined MP. This uncertainty analysis is performed for the solution model of Figure2. The normalized values σ |value sol | × 100 of the standard deviations (σ) are also indicated in italic.

		Vp5	Vs5	η	δ	H4	H5
	Vp5 475.1 m/s	0.002	-0.03	-0.02	0.005	0.01
		(22%)
	Vs5	0.002	168.9 m/s -0.04	-0.03	0.005	0.01
			(22%)
	η	-0.03	-0.04	0.22% 0.93	-0.16	-0.33
				(4%)
	δ	-0.02	-0.03	0.93	1.6%	-0.17	-0.36
					(37%)
	H4	0.005	0.005	-0.16	-0.17 77.1 m	0.06
						(5%)
	H5	0.01	0.01	-0.33	-0.36	0.06	80.3 m
							(3%)

Table 2 :

 2 RMS values of the traveltime misfits, mean value of the velocities vp5 and vs5, of the interfaces h4 and h5 in illuminated region (MP defined in section 4.2 and anisotropy parameters η and δ for 20 simulated models (chosen among the 100 simulations presented in section 4.1).

  Model 1 Model 2 Model 3 mean of the P-velocity 2.849 km/s 3.623 km/s 2.895 km/s mean of the S-velocity 1.229 km/s 1.621 km/s 1.213 km/s

	η	6.29%	8.27%	6.2%
	δ	-4.4%	15.94%	2%
	mean of the depth of the reflector H4	1.838 km	1.755 km	1.659 km
	mean of the depth of the reflector H5	3.167 km	3.021 km	2.882 km

Table 3 :

 3 3 different solution models. Model 1 is the model of

In our approach ([START_REF] Jurado | 3D reflection tomography designed for complex structures[END_REF]), the subsurface model m is composed of

2D or 3D B-spline functions describing velocity variations in a layer (v(x, y)+k.z or v(x, y, z)) and 2D B-spline functions describing the interfaces (Z(x, y), Y (z, x) or X(y, z)).

The computation of the derivatives of T with respect to the model parameters is cheap, thanks to the Fermat principle ([START_REF] Bishop | Tomographic determination of velocity and depth in laterally varying media[END_REF]).

PS-wave results from the conversion at the reflector of a down-going P-wave (compressional wave) into an up-going S-wave (shear wave). In opposition to the PS-wave, the pure P mode is often called the PP-wave.

From figure3, we define the illuminated region by x ∈[14, 20.5km].

ACKNOWLEDGMENTS

The authors would like to thank bp to have provided the data set, Karine Broto and Frédéric Delbos for many fruitful discussions.

again the limitations of the linearized approach.

To perform a non linear analysis we have chosen an experimental approach which consists in performing several inversions allowing to test different geological scenarii to try to delimit the space of admissible solutions. For instance, we could test different hypothesis on the values of the anisotropy parameters for which a strong uncertainty has been detected by the linearized approach. If we introduce a constraint on the anisotropy parameter δ, such as δ > 0, we find the model 3 displayed in Figure 11: this result expands the range of admissible models (variability of δ detected by MP approach was -4.43 ± 3.2%). In the same way, we can test the variability of the anisotropy parameter η. We notice (see table 5) that the value of the anisotropy parameter η of the model 2 (Figure 10) is 8.27%, but all the simulated values of this parameter are around 6% and the variablity of η furnished by the MP is 0.44%. Nevertheless, tomographic inversion under the constraint η ≤ 4% or η ≥ 8% does not furnish a model that satisfies both these constraints and the data with the expected accuracy. We have shown that, by constrained tomography, we can test different geological hypothesis and delimit the space of admissible models and go further than the linearized approach which explores only the vicinity of the solution model which may be one local minimum around others. and thus becomes feasible in 3D. Nevertheless, we have confirmed on this real data set that this approach is only valid in the vicinity of the solution model (linearized framework) and that some cases require a non linear approach to search for local minima of the cost function. An experimental approach has consisted in solving the inverse problem under constraints (for instance constraints on anisotropy parameters) to test the validity of some geological hypothesis.