William Latham
email: latham@udel.edu.

Christian Le Bas
email: christian.lebas@univ-lyon2.fr

Dmitry Volodin
email: volodin@udel.edu

Value of invention, prolific inventor productivity and mobility: evidence from five countries, 1975-2002 1

Keywords: inventeur prolifique, mobilité, productivité, valeur des inventions

The aim of this paper is to provide new insights into (1) the determinants of the value of inventions and (2) the role that mobility plays in the behavior of prolific inventors, whom we identify based on the number of patents exceeding a threshold of productivity. We examine mobility in two dimensions: from firm to firm (inter-firm) and from one technical field to another. We exploit data on patents filed by inventors from five countries (France, the UK, Germany, the US and Japan) in the US Patent and Trademark office during the period from 1975 to 2002. From our regressions we obtain a rich set of results. In particular we show that: (1) as predicted by evolutionary theory, inventor productivity is a positive determinant of invention value, (2) inter-firm mobility is a consistently positive determinant of productivity and (3) technological mobility is a negative determinant. The last implies that the more specialized an inventor is, the higher his productivity is.

Key words: prolific inventor, mobility, productivity, value of invention Le but de ce papier est de fournir de nouveaux éclairages sur (1) les déterminants de la valeur des inventions, (2) le rôle que pourrait jouer la mobilité sur le comportement des inventeurs prolifiques (que nous définissons comme les inventeurs ayant franchi un seuil en termes de productivité). Nous examinons la mobilité entre firmes et entre champs technologiques. Nous exploitons une base de données de brevets déposés à l"US patent office dans la période 1975-2002 et concernant 5 pays (France, Royaume-Uni, Allemagne, USA et Japon). Nous obtenons de 4 régressions pour chacun des pays un riche ensemble de résultats. On montre que : (1) ainsi que le prédit la théorie évolutionniste la productivité de l"inventeur est un déterminant positif de la valeur des inventions (2) dans l"équation de productivité des inventeurs la mobilité inter firme est toujours positive alors que la mobilité technologique est significativement négative. Cela signifie que plus un inventeur est spécialisé (en termes de champs technologiques) plus il est productif en termes d"inventions.

Introduction: The importance of highly productive individuals in the process of technological change.

The aim of this paper is to provide new insights into the determinants of the value of invention. Our basic assumption is that individuals must be considered in this creative process because innovation is not simply a product of firms and organizations, it requires individual creativity. Furthermore we focus our analysis on prolific inventors because they contribute so hugely to national invention totals (Le [START_REF] Bas | Prolific inventors: Who are they and where do they locate? Evidence from a five countries us patenting data set[END_REF] and tend to produce inventions that have more economic value [START_REF] Gambardella | The value of patents[END_REF][START_REF] Gay | Collective knowledge, prolific inventors and the value of inventions: An empirical study of French, German and British owned U.S. patents, 1975-1998[END_REF].

Recent studies have shown how individuals within the firm help it to cope with technological change. The role that individuals play in firm imitation, adaptation and innovation can be understood by drawing on results and analyses provided by the sociology of science. That literature has generated conflicting viewpoints regarding the importance of "key" very productive individuals in the development of scientific knowledge (noted in particular by [START_REF] Hess | The Formation of Dynamic Capabilities[END_REF]; see also Latham and Le Bas, 2011). In innovation theory prior research has demonstrated that certain individuals are very important to a firm"s innovation activity [START_REF] Lacetera | Do firms change capabilities by hiring new people? A study of the adoption of science-based drug discovery[END_REF][START_REF] Rothaermel | Building dynamic capabilities: Innovation driven by individual, firm, and network level effects[END_REF][START_REF] Zucker | Labor Mobility from Academe to Commerce[END_REF]. A similar idea is frequently found in strategic management studies: particular individuals are often crucial resources contributing to a firm"s competitive advantage [START_REF] Tushman | External communication and project performance: An investigation into the role of gatekeepers[END_REF][START_REF] Coff | Human assets and management dilemmas: Coping with hazards on the road to resource-based theory[END_REF]. The role of key individuals or star employees has been documented by academics (Ernst et al., 2000 5 ; Tushman, 1977; Zucker, et al., 2002a) and goes back to the famous Lotka-Price Law of Scientific Knowledge Distribution.

In general the productivity and value of important individuals is related to the degree to which the individuals are connected to both internal and external sources of knowledge [START_REF] Allen | Managing the Flow of Technology[END_REF][START_REF] Tushman | External communication and project performance: An investigation into the role of gatekeepers[END_REF][START_REF] Zucker | Individual action and the demand for institutions: Star scientists and institutional transformation[END_REF]. [START_REF] Hess | The Formation of Dynamic Capabilities[END_REF] have tried to develop a framework emphasizing that the firm"s dynamic capabilities, as elaborated by [START_REF] Teece | Dynamic capabilities and strategic management[END_REF], are dependent on the individuals within the firm and on their roles in technological innovation.

As far as prolific inventors are concerned we propose that they act as "knowledge integrators" [START_REF] Gay | Collective knowledge, prolific inventors and the value of inventions: An empirical study of French, German and British owned U.S. patents, 1975-1998[END_REF]. As knowledge workers, they play a prominent role in the design, development and integration of pieces of knowledge within a department of research in their invention team or network. Prolific inventors and their engineering knowledge are essential.

They increase the rate at which individuals and organizations learn and consequently achieve sustainable competitive advantages. Prolific inventors are innovation "champions". Through their professional mobility they can be viewed as "knowledge translators" or "knowledge brokers" (Brown and Duguit, 1999) between firms, organizations and communities. They help transfer pieces of knowledge through the different communities with which they interact at one or at several different points of time. Thus it seems to us that prolific inventors are basically "science gatekeepers". Gatekeepers are individuals with specific characteristics. [START_REF] Tushman | External communication and project performance: An investigation into the role of gatekeepers[END_REF] provide a well-known description of gatekeepers as "boundary spanning individuals who can be an important linking mechanism between organizations and their external environments."6 They explain the role of gatekeepers in the transfer of information to a single R&D division by comparing the performance of project teams with and without gatekeepers. Gatekeepers perform a linking role only for projects performing tasks that are locally oriented (meaning development-oriented tasks). [START_REF] Allen | Technology transfer as a function of position in the spectrum from research through development to technical services[END_REF] analyze how research and development projects effectively acquire new technology. Projects show higher levels of performance when all project team members maintain high levels of communication with colleagues outside their organization. Innovation projects show higher performance when external communications are monopolized by one or a few project members: in short the gatekeepers. Allen et al.: The gatekeeper seems to be an offspring of technological needs and organizational conditions. Because technological problems are defined in local terms, reflecting the firm's interests and strategy as well as its value system, most technologists have difficulty in communicating effectively with outsiders about those problems. Fortunately, however, often there appear to be a few individuals who maintain consistent ongoing contact outside their organizations, who understand the way in which outsiders differ in perspective from their own organizational colleagues, and who are able to translate between the two systems. (p. 703) Allen et al. also note that the gatekeeper"s "role is of greater importance when the technology is ... sophisticated". Parichuri (2009) has shown that the relative importance of these inventors varies with the firm"s location within inter-firm networks.

In previous papers we have presented arguments justifying the choice of prolific inventors (Le Bas et al., 2009). In this paper we test if some inventors" characteristics (their inventive productivity, their mobility) impact the value of their inventions. As in previous papers (e.g., Le Bas et al., 2009) we identify prolific inventors as those who have been issued at least 15 patents by the US Patent andTrademark Office between 1975 and[START_REF] Zucker | Labor Mobility from Academe to Commerce[END_REF]. Previous studies of prolific (or "key") inventors have focused more on the firms in which they work or on the industries in which the firms operate. Narin and Breitzman"s (1995) seminal work on the topic is based on an analysis of only four firms 7 in a single sector and a recent paper by [START_REF] Pilkington | Defining key inventors: A comparison of fuel cell and nanotechnology industries[END_REF] uses data for only two industries. We have adopted a different perspective in our analysis. We use a very large data set which permits comparisons across thousands of inventors in thousands of firms in the five largest countries in terms of technological activities (France, the UK, Germany, the US and Japan) to estimate the determinants of inventive productivity at the individual level, including the effect of inventor mobility on inventor productivity.

The paper is organized as follows: Section 1delineates our theoretical framework and suggests hypotheses to be tested, Section 2 presents the definitions of variables and some related measurement issues, Section 3 describes the empirical model, estimation issues are discussed in Section 4 and the discussions of our results is in Section 5. Conclusions and suggestions for further research are set out in Section 6.

Section 1. An evolutionary framework: the laws of inventor productivity and value of invention.

The point of view developed here finds its roots in the analysis of the growth of knowledge by recombination first systematically described by [START_REF] Weitzman | Hybridizing growth theory[END_REF] and [START_REF] Antonelli | Localized Technological Change[END_REF]. In his approach the production of new knowledge is a process that cannot be modeled in general by 7 In defense of Narin and Breitzman"s use of individual firms we note that Ernst, Leptein and Vitt (2000) recommend that, "because of variations in the propensity to patent across firms and industries, key inventors should be identified within their company"s organizational framework only." analogy with the "discovery of new oil fields". Instead new knowledge is often produced by a recombination of scattered existing bits of knowledge. [START_REF] Weitzman | Hybridizing growth theory[END_REF] supports his view with the following examples: "The idea of an "electric light" is itself a hybrid, the first practical example of which was made in 1879, between the idea of "artificial illumination" and the idea of "electricity." The idea of an "electricity production and distribution network" was conceived by Edison in the 1880's as an explicit combination of the idea of "electricity" with the idea of a "gas distribution system," where electricity is essentially substituted for gas (Weitzman, 1996: 209). His basic idea is that the expression of human imagination is "recombinatoric in essence."

We find this same important concept in [START_REF] Fleming | Exploring the tail of creativity: an evolutionary model of breakthrough invention[END_REF], for whom the same mechanisms of creativity apply both in science and in technology. They start their analysis with a psychological model first elaborated by [START_REF] Simonton | Origins of Genius: Darwinian Perspectives on Creativity[END_REF]. Inventors generate new ideas through combinatorial thought trials subject to psychological and social selection processes [START_REF] Fleming | Breakthroughs and the 'long tail' of innovation[END_REF]. They note that individuals who simultaneously juxtapose, combine, and evaluate a stream of uncombined inputs will be more creative. Generative creativity is the assembly or rearrangement of existing components into new combinations. The more the inventor tries recombinant actions, the more he/she increases the likelihood of a productive hit. As a consequence we hypothesize a correlation between an inventor"s total output and the likelihood that he/she finds inventions with high impact. "A one-hit wonder is very unlikely......The most prolific inventor is the one most likely to invent a breakthrough" (Fleming and Szigety, 2006: 340) 8 .

The important output of such an analysis with respect to [START_REF] Weitzman | Hybridizing growth theory[END_REF] model is that it predicts a relation between inventor productivity and the economic value of new bits of knowledge produced. The new bits of knowledge may be embodied in inventions. Thus one can examine the behavior of inventors and the value of their inventions to find evidence of the Weitzman model"s validity. This reasoning leads us to our first testable hypothesis: 8 Fleming and Szigety (2006) make an inventory of the factors (technological and social-psychological variables) that have an influence on "the second moment of the creative outcome distribution" and consequently also affect the propensity to create breakthroughs. For example, among the important variables that have an expected positive impact on the variance of the distribution are: diversity of collaborators, dissolution of collaborative relationship, and changes of creative fields: as has been noted by many researchers, an inventor cannot invent alone, he/she invents collectively and within an "ecological context." As a consequence there are organizational influences on the evolution of the distribution of inventive behavior as well. [START_REF] Fleming | Breakthroughs and the 'long tail' of innovation[END_REF] finds empirical results in favour of this thesis.

H1. The more productive an inventor is, the more valuable his inventions will be on average.

Recent strands of the literature dealing with invention value have proposed the strategic importance of inventor mobility as linked to invention value. The scale, determinants and effects of inventor mobility have been analyzed recently by [START_REF] Hoisl | Tracing mobile inventors -The causality between inventor mobility and inventor productivity[END_REF]2009), [START_REF] Schankerman | Software Patents, Inventors and Mobility[END_REF][START_REF] Trajtenberg | The Names Game: Using Inventors Patent Data in Economic Research, Power-point Presentation[END_REF] among others. Hoisl, (2007) using European patents (a survey of 3049 German inventors), finds that an increase in inventor productivity (number of patents per inventor) decreases the number of moves from firm to firm. She tests the causality of the productivity of inventors on inventor mobility and finds that more productive inventors are not more mobile from firm to firm. For Hoisl, a move increases productivity (number of patents) but an increase in productivity decreases the probability of observing a move. Hoisl has investigated the differences in gains from a move between high and lower performing inventors. This point is particularly crucial for us because we want to assess the role of prolific inventors" mobility on their performance. [START_REF] Hoisl | Does mobility increase the productivity of inventors?[END_REF] finds that inventors at the upper end of the performance distribution (our prolific inventors) are better able to benefit from a move to draw level with or to overtake non-movers in the post-move period. Whereas at the bottom of the performance distribution a higher level of education has a positive impact on inventive performance, education does not matter significantly at the upper end of the performance distribution. [START_REF] Schankerman | Software Patents, Inventors and Mobility[END_REF] have studied the mobility of inventors using patents in the software industry in the US. Their findings are in accord with Hoisl"s: they show that the very productive inventors have a decreasing probability of moving between assignees as their careers progress [START_REF] Schankerman | Software Patents, Inventors and Mobility[END_REF]26). As far as value of inventions is concerned, [START_REF] Trajtenberg | The Names Game: Using Inventors Patent Data in Economic Research, Power-point Presentation[END_REF] showed that inter-firm mobility is related to inventors" patents that are more technologically focused (more concentrated in technological categories) and those having more value (i.e. more cited). He pointed out that the Israeli inventors who tend to move more frequently both across countries and between assignees have the most highlycited patents. But he concludes that there exists an endogeneity problem: we cannot determine whether it is the (high) value of invention that provokes the move or if it is the learning effect due to the move that tends to increase the invention's value. [START_REF] Schankerman | Software Patents, Inventors and Mobility[END_REF] discuss the issue of inventor mobility in the framework of an inventor-employer matching process in the software industry. Asymmetric information between employer and employee about the value of an invention should be a relevant incentive for a move. They finally argue: "we did not find support in the data that mobility is a matching process between the inventor and his employer, and that the quality of the inventor's patents increases after a move. If any, there seem to be some short term costs of mobility, which seem lower when moving to a larger firm."

We extend these studies by considering prolific inventors, the source, as we will show, of most innovation in the five countries, using several indicators of mobility as well as indicators of productivity and the value of inventions. In this paper we focus only on inter-firm mobility and technological mobility.9

Our model of knowledge creation through recombination takes into account the empirical evidence reviewed as follows: We suppose some bits of primary knowledge exist in the first time period. Through the process of recombination, some new bits of knowledge are created in the next time period. Note that the bits of previous knowledge that enter into the process of recombination continue to exist as useful knowledge. Significantly, the process of recombination does not stop in the second time period. It goes on. In this way, in the next time periods we will have additional new bits produced. The recombination process becomes more complex: it combines bits of recomposed knowledge with bits of "raw" knowledge. We think that the new elements of knowledge have more value than the primary bits of knowledge.

The knowledge that is used in recombination is both explicit and tacit. Codified explicit knowledge circulates, in general, through publications. The part that is tacit in nature is important in the process of recombination. Its structure depends on local factors such as firm organization, core technological competences of a region, etc.) and its diffusion is generally through the mobility of experts including inventors.

Inventors work in firms that are part of industries that are located in particular regions. We hypothesize that the set of bits of "primary knowledge" existing at the period of time under observation are differ according to these "places" (firms and industries as well as geographic regions). It may be that some bits or elements are common across places, but some are different. By moving (from firm to firm or from region to region) the inventor can get new bits of ideas that enter into the evolving processes of recombination. Before moving he might decide that he has exhausted the opportunities for successful recombinations in his current location. By moving he may find new fields for hybridizing or new avenues for creating new economically valuable inventions. He retains the bits of knowledge he had accumulated at the previous place but now works in a new knowledge environment. His potential for recombinations has become higher. Moreover due to social ties knowledge interactions persist even after formerly co-located individuals are separated after moves [START_REF] References Agrawal | Gone But Not Forgotten: Labor Flows, Knowledge Spillovers and Enduring Social Capital[END_REF].

Inventor mobility has always been recognized as a key mechanism for transferring tacit knowledge from one place to another between firms, industries, regions or countries (see [START_REF] References Agrawal | Gone But Not Forgotten: Labor Flows, Knowledge Spillovers and Enduring Social Capital[END_REF]. But a move is also a way to learn more, or to learn more quickly. In this sense mobility does not simply transfer knowledge from place to place as a spillover, but also increases the capacity to solve problems and basically increases the human capital of knowledge. Mobility as a mean for knowledge diffusion and extension matches the knowledge "reuse" approach of [START_REF] Langlois | Knowledge, consumption, and endogenous growth[END_REF], a type of increasing economy of scale at the core of economic growth process. Thus we expect that mobility affects: (1) the productivity of prolific inventors, as measured by their average number of inventions per year over their active inventive lives, and (2) the value of their inventions, measured as the number of citations a patent receives in the years after its application is filed. 10As a consequence we hypothesize: H2. Following a move (from firm to firm, or from region to region) an inventor"s productivity increases.

A corollary is: H2". An inventor who moves a lot is more productive than an inventor who moves less.

Variables: Definition and Measurement Issues

Measuring "prolificness."

The distribution of number of patents by inventors is clearly not "normal", in fact it is highly skewed, with most inventors having few inventions and a few inventors having many inventions (Latham and Le Bas, 2011). Other research cited above established that prolific inventors produce more valuable patents (as measured by citations).

We focus on these prolific inventors as the ones most valuable in contributing to economic development and seek to understand the determinants of their mobility. No theory leads to a clear delineation of the number of patents needed to qualify an inventor as "prolific." We use 15 patents to identify prolificness, a number that takes us far to the right in the distribution of numbers of patents by an inventor. Our models are robust to variations in the threshold for identification of prolificness: we have estimated our models based on larger and smaller thresholds and do not find dramatic changes as the threshold for identification changes between 10 and 20 patents, so the use of 15 is adopted without concern for that selection affecting the results significantly. 11 2.2 Accounting for inventor career effects. In our dataset we observe that there are some inventors with careers of patenting that span many years and others whose fifteen or more patents are all produced in a very short period. To account for this variation we measure the duration of an inventor"s career (the number of years from the inventor"s first patent application to the last) and adjust other effects for duration. We use duration to compute productivity, value, and inter-firm mobility on a per year of career basis. Some investigators (e.g., Hoisl: 2006[START_REF] Hoisl | Tracing mobile inventors -The causality between inventor mobility and inventor productivity[END_REF][START_REF] Schankerman | Software Patents, Inventors and Mobility[END_REF] have tracked the numbers of patents and/or the numbers of citations that an inventor has prior to a move from one firm to another. Moves are assumed to be based only on past performance. We adopt a different approach, essentially assuming that the number of inventions that an inventor eventually produces is a measure of the potential that the inventor has always had. We assume that employers make rational (mostly accurate) predictions about the future productivity of inventors when they are hired. This assumption allows us to compute single measures of productivity or average citations per patent for each inventor.

Measuring inventor productivity.

The simplest measure of an inventor"s productivity is the number of patents he obtains (patent grants) over a career. We adjust this for the career length to obtain the average number of patents per year as our productivity measure. 12 We add to the simple average a measure of the dispersion of patenting activity over the inventor"s 11 Latham and Le [START_REF] Bas | Prolific inventors: Who are they and where do they locate? Evidence from a five countries us patenting data set[END_REF] provide additional justifications for the use of the 15 inventor threshold to identify prolific inventors.

12 Alternatives might include the number of patent applications, instead of grants, or the number of design and utility patents. If we could track every inventor's applications, we would use the applications count together with the patent count. However, we do not have the applications data. Fortunately the number of patents per year is intuitively appealing, easily understood and computed and has been widely used by others, so it is our choice.

career. The measure we use in our analysis is the inverse of dispersion; it is the Herfindahl-Hirschman Index for the time pattern of the number of patents in each year. We might have chosen the n-year concentration ratio instead, but the HHI more appropriately gives extra weight to years of higher concentration.

2.4

Measuring the value of an inventor's inventions. Ideally one would measure the value of a patent through market-based metrics such as the actual prices paid for them or the increase in the value of a firm in response to the acquisition of a patent. However, such measures are not available except for limited numbers of patents. The research literature on patents has, in the absence of any other measures for large patent data sets, accepted the number of citations a patent receives as a good proxy for the value of that patent. [START_REF] Harhoff | Institutionalized incentives for ingenuity, patent value and the German Employees" Inventor Act[END_REF] have shown that the number of citations is, in fact, a good proxy for the value of a patent. Therefore the value of all of an inventor"s patents can be measured as the total number of citations his patents have received. An inventor"s value can alternatively be viewed as his total number of citations, his average number of citations per patent, his average number of citations per year or his average number of citations per patent per year. The total number of citations fits with the concept of an inventor"s potential but, just as with productivity it needs to be corrected for the duration of a career so the number of citations per year is a good measure. It is our primary measure of value but we also consider the number of citations per patent and the number of citations per patent per year.

Measuring inter-firm mobility.

The simplest way of identifying inter-firm mobility (a move within the industrial structure from one firm to another) is to count the number of firms for which an inventor has worked and assume that the number of moves is this number minus one. 13 However, this approach does not allow for the movement away from a firm and a 13 We assume that the assignee listed on a patent is an inventor's employer. [START_REF] Hoisl | Tracing mobile inventors -The causality between inventor mobility and inventor productivity[END_REF] concluded that, at least in Germany, it is safe to assume that the inventor worked for the assignee stated in the patent document. In our data we have assignee names which we equate with firm names. Changes in assignees may be the result of a merger of firms or a change in the name of a firm or movement from one subsidiary of the same firm to another. Such changes would be inter-assignee and not inter-firm. We hope to address these issues in future work.

subsequent return to it. Nor does such a measure consider the temporal pattern of the inventor"s association with different firms. Another type of measure that might have been used is a measure of concentration, either the percentage of patents at n firms with the highest percentage (an n-firm concentration ratio) or a Herfindahl-Hirschman Index that accounts for the variability in the distribution of patents across firms. However, these measures also fail to consider the sequential temporal pattern in any way (in the same way that a simple count of the number of firms also does not consider the pattern).

Still another way to measure inter-firm mobility is to list an inventor"s patents chronologically and to count a move each time the assignee of the patent changes. Such a count results in the maximum possible measure of the number of moves that an inventor makes. Under this definition a single inventor in our dataset would be said to have moved 53 times. Such a high level of mobility may correctly represent the inventor, but certain patterns in a list of assignees seem to call that definition into question. For example suppose that Inventor # 1 assigns his first patent to firm A, the second to firm B, and the third to firm A, the fourth to firm B and so forth through the assignment of the tenth patent to firm B. Inventor # 2 assigns her first five patents to firm A and the next five to firm B. Inventor # 1 will be counted as having 9 moves while Inventor # 2 will have only 1 move. Surely this result does not adequately capture a strong sort of mobility well. In attempting to deal with this problem we have measured moves in several alternative ways. In the alternatives we consider whether or not the inventor returned to a prior assignee within some specified period of time. If so, we do not consider the temporary or transient change in assignee to be an indication of a strong variety of mobility. We consider a two year persistence of a change to qualify for a move. We know the numbers of moves measured under this definition will be smaller than under the first definition. In our various procedures we have attempted to avoid identifying what Hoisl (2007) identified as "false mobility." In results not reported here we have used several of these alternative definitions of mobility and have found, surprisingly, that our results are not sensitive to the definition of mobility.

Measuring technological mobility.

By Technological mobility of inventors refers to the fact that an inventor"s patents are not all related to the same technological field (or class) over a career. Technological mobility witnesses to the fact that an inventor worked on industrial research projects having different technological cores. We interpret this as evidence of "movement" from one particular technological field to another, or simply as "mobility." We could instead, and perhaps more appropriately, refer to an inventor"s "level of technological specialization". In the interpretation of our results we can say that an inventor with low technological mobility is very specialized (as far as his technological competences are concerned), and conversely high mobility implies a lower level of specialization. Another term that we can use in this context is "intellectual mobility," which emphasizes the mental capacity of individuals to change the focus of their thinking as their inventive activities change from one technological field to another.

Among the possible measures of technological mobility of inventors that we considered were a count of the number of different technological fields in which an inventor has worked and the number of changes from one technological field to another. These measures are similar to the inter-firm mobility measures discussed in the preceding paragraph. In the case of technologies, though, the temporal pattern of an inventor"s patenting in different fields does not seem to be of as much interest as the temporal pattern of inter-firm mobility.

Consequently we determined that a concentration measure would be good. We considered technology concentration ratios for the single highest concentration field and for the top 2 or other numbers. However, the Herfindahl-Hirschman Index (HHI) for technological fields appeals because of its greater emphasis (through the squaring of each field"s percentage) on higher concentrations. We implemented the HHI at the level of six broad technological classes. 14 Patenting in more than one of the 36 more detailed technology fields (as defined in the NBER data) was observed to be very frequent and moves within the broader categories were also observed to be very frequent. Thus it seemed that the moves between the six broad technology classes might be more indicative of real technological mobility by inventors. We also identify the broad field within which each inventor has the highest concentration of patents so that we can examine whether or not inventors who concentrate in each of these (2002). For inventors whose entire inventive career falls within this span of years, there is no problem of bias from omitted years of activity before or after the sample period. However, for inventors who were already active prior to the sample or who remained active after the sample period, the truncation problem may be significant. All of our measures such as career duration, citations and number of patents could be underestimated if the sample truncates the careers of inventors. We have estimated our equations only for individuals whose patenting careers seem to fall wholly within the sample (those who have no inventions prior to 1979 and none after 1997). On the basis of this analysis, which showed no significant differences from the analysis including all inventors, we have not excluded for possible truncation bias any individuals. While there are some individuals whose patenting careers have been truncated, we are confident that our results have not been significantly affected by their inclusion in the analysis.

Models

In this paper we test a number of hypotheses about the relationships between and among inventors" productivity, their mobility in both the technological and the inter-firm dimensions, and the values their patents create. The following four equations express our conceptual empirical framework for testing these hypotheses. The parallel specifications of the equations are the result primarily of the limitations of our data. For example, while we are well-aware that there are both theories and empirical studies of productivity that highlight the roles of inventors" education and training, the capital available to them, the nature of the rewards system and the role of institutional constraints such as retirement ages and the nature of the patent system, we do not have those variables available to us. Similarly for both mobility and the value of patents many other variables have been suggested in theory and in other empirical studies. Consequently our work is not in the framework of those that attempt to propose and test comprehensive theories of the determinants of productivity, mobility, or value. Instead ours is a partial approach. We examine the ways in which productivity, mobility and value influence each other given our limited range of knowledge about other variables. While we can include a few other variables as controls, such as the temporal pattern of an inventor"s career, we essentially assume that all of the omitted variables can safely be "held constant" for our analysis.

Table 1 lists the variables that we have available for the analysis. For each of the conceptual variables included in the four equations, Table 1 shows that there are alternative specifications that can be used for most of our variables. We do not have economic theory to guide either our selections of functional forms of the equations (log, linear, etc.) or our selection of various ways of specifying the particular variables that we will use. In the end we rely heavily on the empirical results in deciding which specifications to report. We have estimated and tested many alternative specifications. The combinations of alternative ways of specifying the equations lead to large numbers of possible equation specifications that fit within our structural framework. 16

16 For example, in the productivity equation, we have tested three different dependent variables: (1) number of patents, (2) (number of patents)/(career duration), and (3) (number of patents)/(number of years with 1 or more patents). In addition we have estimated the equation with the number of patents as the dependent variable with both Poisson and negative binomial distributions. So there are four different dependent variables. We have also tested alternative specifications for various variables: three for citations, two for the inventor"s patenting pattern over time, three for technical mobility, two for inter-firm mobility, and two for career duration . So, just for the productivity equation we have tested 4 x 3 x 2 x 3 x 2 x 2 = 288 equations for 5 countries = 1440 equations.

In this paper we focus on prolific inventors. Table 2 shows some characteristics of our inventors including the significance of the prolific inventors in terms of the proportion of patents for which they are responsible in each country. The data in Table 2 indicate that there is probably a misidentification of prolific inventors to a greater extent in the Japanese data than in the other countries. The famous "names" problem [START_REF] Trajtenberg | The Names Game: Using Inventors Patent Data in Economic Research, Power-point Presentation[END_REF]) of identifying inventors is compounded with Japanese names. We believe that many of the prolific inventors identified in the Japanese data are likely multiple inventors, the similarity of whose names led them to be identified as a single inventor. For this reason, the Japanese results ought to be viewed with a great deal of caution. It is interesting that, in spite of the names problem, the estimated coefficients for the Japanese models are frequently quite similar to those of the other countries. Beyond the obvious names problem with the Japanese data, we are aware that the "names" problem also manifests itself differentially but systematically across other countries as well. This fact provides a good justification for estimating each country"s equations separately below.

Table 3 provides summary statistics for the data. Again there are systematic differences across the countries, especially for Japan. We re-emphasize that care must be taken in making comparisons across countries. It is important not to attribute to differential behaviors or institutions effects that are actually based simply on different degrees in the precision with which prolific inventors can be identified.

Empirical Estimation Issues

All of our estimates use robust standard errors so that we do not have to be concerned about the effects of heteroscedasticity and, since our observations are inventors" careers, we do not have serial correlation problems with our estimates.

The dependent variable for our productivity equation is the number of patents per year of an inventor"s career within our dataset. The values are technically truncated at 0, but have enough variability and range so that we are able to estimate the parameters with ordinary least squares. Similarly the dependent variable for inter-firm mobility, moves per year, allows us to use OLS. Our dependent variable for technical mobility is (1-a technical concentration ratio), which is strictly limited to the range from zero to one. Consequently we use censored normal (Tobit) analysis to obtain our parameter estimates. The dependent variable in our value equation is the number of citations an inventor"s patents receive subsequent to their issue, clearly an integer count measure. We estimate the parameters assuming both Poisson and negative binomial distributions. We find little difference in the estimates and no evidence of overdispersion so we report only the Poisson distribution results.

The parallel specifications of the equations also may indicate simultaneity in the nature of the relationships among the variables. We are aware that our results might be influenced by simultaneous equations bias. We have performed some analysis not reported in the paper using instrumental variables. Preliminary analysis indicates that some parameters may be significantly affected. However, in the current equation structure with the data that we have available, there are few additional variables to use as instruments. We hope explore the simultaneity further in the future along with implementing systematic tests for exogeneity of the independent variables.

The issue of the "right" specification when there are many potential alternatives based on alternative specifications of both the dependent and independent variables led us to estimate all of the basic 20 relationships (4 equations for reach of the five countries) many different ways. Of course, in such an exercise the hope is that all coefficients will be substantially unchanged under alternative specifications so that one can declare the models to be robust to alternative specifications of the variables. We have found a lot of stability but cannot yet declare that ours is the most stable of all possible specifications. The most stable relationships are those between productivity and value in both the value and productivity equations: their coefficients never change signs over hundreds of alternative equation specifications across all five countries. Productivity is also very stable in the technical mobility equations across all the countries. Every other variable has some degree of instability.

Discussion of the Results

Tables 4 through 7 present estimation results for each of the four equations. For each country we estimate equations for productivity, mobility (in both the inter-firm and technological dimensions), and the value of inventions (as measured by patent citations).

across the countries: value is insignificant in the UK, France and Germany, significant and negative in the US and Japan (but the coefficients are very small).

Determinants of inventor productivity.

The regressions all have relatively high Rsquared values. Inter-firm mobility is consistently positive and significant, technological mobility (Hoisl's measure) is consistently negative and significant. Value (citations per patent) is again inconsistent in effect: insignificant in UK and France, negative in Germany and the US, Positive in Japan.

Determinants of the value of inventor's inventions.

Inventor productivity is consistently positive and significant (with very high coefficients), which confirms the findings of [START_REF] Gambardella | The value of patents[END_REF] using the PATVAL survey. They found that the characteristics of the inventor, in particular his/her past number of patents is the main determinant of the private value of inventions and more important than the characteristics of the organization in which he/she is employed. Our previous empirical research [START_REF] Gay | Collective knowledge, prolific inventors and the value of inventions: An empirical study of French, German and British owned U.S. patents, 1975-1998[END_REF] has also confirmed this remarkable result. It seems in fact there is no (so-called) "inventor dilemma" regarding a choice between patent quality and patent quantity [START_REF] Mariani | Stracking" or "picking" patents?The inventors" choice between quality and quantity[END_REF].

Note that the effect of technological mobility is negative in all countries and significant in Germany, the US, and Japan. Inter-firm mobility affects value positively in four countries and negatively in Japan. It is significant in all countries except France.

Nonlinear effects.

The strictly linear specifications we have reported are, of course, highly restrictive. Yet they allow us to test our hypotheses about the relationships among mobility, productivity and value for productive inventors and have yielded results that make sense and also can be related meaningfully to other results in the literature. It is still possible that nonlinearities in the relationships might change our conclusions. We have explored some potential such nonlinearities. One straightforward way to examine nonlinearities might be to simply estimate the relationships in logs. However, two of the major variables of interest, numbers of citations, and inter-firm moves, both have a significant number of zero observations so that some form of adjustment is required to use full log specifications. Of course, partial log specifications might also be used, but instead we have taken two alternative approaches: (1) use of nonlinear terms in the equations and (2) use of interaction terms.

First we have tested to see whether or not the effects of one variable on the other variables are linear by adding a squared term to the equations. We chose technological mobility as the one to test first. For the inter-firm mobility equations, in which the technical mobility coefficient is positive and significant for each of the countries, the squared term is negative and it is significant for all except for France. This result shows a nonlinear effect: the effects of technical mobility are moderated as technical mobility increases. However, neither the signs nor the significance of technological mobility or any of the other variables are changed by this effect, so we need not deal with this nonlinearity. The effects of technological mobility in the productivity equations are also moderated by the squared terms which have the opposite signs from the linear terms and are significant for Germany, the US and Japan. Again, the other results are not changed by the addition of the nonlinear term. In the value equations, the results of adding the squared technological mobility terms are mixed: not significant for the UK, the US and Germany, significant and of opposite sign for France (the moderating effect again), and for Japan the sign of technological mobility changes to positive and is significant and the squared term has a negative sign and is significant. Once again the rest of the results are basically unchanged.

Another form of nonlinearity can arise from interactions among the independent variables.

For example, the effect of technological mobility might be influenced by the length of an inventor"s career. We have explored this possibility to a limited extent by introducing some interaction terms in the equations. Introduction of a technological mobility-career length variable does not fundamentally alter our results. In the inter-firm mobility equations the interaction term is positive for all five countries and is significant for all but France. This seems to indicate that the interaction should be added to the equations. For the US and Japan other results are not affected; for both France and Germany, the addition of the interaction term causes both the career duration and technological mobility variables to become insignificant, perhaps indicating that the effect can be captured either in the interaction or the individual terms but not in both. In the productivity equations, the interaction term was insignificant without affecting other results in four of the countries. For France, duration became negative and significant and the interaction term was positive and significant. In the value equations, the interaction term varied in significance and in sign, never affected the career duration variable and sometimes changed the sign or the significance of the technological mobility variable. On balance our conclusion is that the particular interaction term that we used does not make a major contribution to the analysis. It is possible that some interactions that we have not explored would yield different results. It does seem that care must be exercised in introducing such terms. Positive, very significant effects, but weak in comparison with the negative effect of intellectual mobility.

Effect is stronger for the USA than for the European countries.

Coefficients are always negative and always significant (except for France 8%)

Effect is much stronger for Japan and USA compared to that for UK and France

Value of invention (number of citations)

Positive, very significant effects (except for France, where there is no significant effect), but significantly negative for Japan.

Significant and negative for USA, Japan, Germany.

Not significant (but a negative coefficient) for France and UK.

Al of the reported results are after controlling for patenting time concentration, career duration, career time gap, and technological field of highest concentration for each inventor.

Conclusions

In this paper, prolific inventors are at the core of the study. Through "purely descriptive regressions" (a term of Trajtenberg) for the five largest countries in terms of innovative activities, we investigate the main determinants of inventor productivity, mobility and invention value. Our results extend previous research focused on the analysis of the complex relationships between (prolific) inventors" productivity and mobility and the value of their inventions. We investigate two forms mobility: inter-firm and technological. Our analysis of technological mobility (technological specialization at the inventor level) is the first empirical study of a large sample of inventors.

Our results are somewhat ambiguous because all the countries do not show the same trends: a. Our inter-firm mobility equations tell us either (1) that the value of patents does not affect inventor mobility (for France and US) or (2) that inventors having low-value inventions are more mobile (Japan). The UK provides an exception, because for the UK, the relation is positive and significant. Our results are not really similar to those obtained for the US by Trajtenberg (2006). Exploring the relationship between the indicators of patent "quality" (including his measures of Citations Received, Generality, Originality, and Number of Claims) and mobility, Trajtenberg (2006) concludes that inter-firm moves are positively related to quality as measured by these four indicators.

b. In the productivity equation the variable inter-firm mobility is everywhere highly significantly positive. Our hypothesis H2" An inventor moving a lot is more productive than an inventor moving less is confirmed. By contrast, the coefficient related to technological mobility is significantly negative. It means that when the inventor"s specialization becomes more important (technological mobility decreases) inventor productivity increases. That is a finding in favour of specialization as a factor pushing inventive productivity (more specialized inventor"s are more productive).

c. In line with other recent studies, there is a clear and highly-significant relation that emerges from the invention value equation for all five countries. Inventor productivity is a determinant of invention value at the inventor level (but a lot of variety across countries exists as far as the size of the coefficient is concerned). As a consequence our H,. The more productive an inventor is, the more valuable his inventions will be on average, is well confirmed. In our patent value equations the coefficients for inter-firm mobility are positive in four countries (not Japan) and significant in all except France.

We find it particularly significant that, in general, the results are similar for the five countries with the notable exception for patent value both when it appears as an independent variable in the equations explaining inter-firm mobility, technological mobility and productivity and when it appears as the dependent variable. These results may indicate that countries differ in their responses to and determinants of patent values because of differences in their institutional arrangements, especially their approaches to national systems of innovation.

This last point appears very important in what it delineates a possible research agenda for the future. We feel that our work to date, on several countries and over a relatively long time period, might provide a basis, in terms of data, estimates, and trends in inventor characteristics (productivity, mobility, career) for constructing a fresh analysis of national innovation system.

14 15 2. 7

 157 Our data come from the National Bureau of Economic Research"s compilations of data extracted from the US Patent and Trademark Office files for 1975-2002 (Hall et al., 2001 describe the methodology). The NBER data assign each patent to one of six broad technology classes; the assignment is based only on the first US Patent Class listed in the patent). The broad technology classes are: (1) Chemical, (2) Computers & Communications, (3) Drugs & Medical, (4) Electrical & Electronic, (5) Mechanical, and (6) Others .categories have significantly different behavior with respect to mobility, productivity and value. The truncation problem. Our patent data begin in one year(1975) and end in another

1 )

 1 Inter-firm Mobility = f(Technological Mobility, Productivity, Value of Patents, Temporal Patenting Pattern, Technical Field, Country) 2) Technological Mobility = f(Inter-firm Mobility, Productivity, Value of Patents, Temporal Patenting Pattern, Technical Field, Country) 15 While the definition of a prolific inventor might arguably be different for each different technology class because of differences in the nature of inventive activity in different industries, the we cannot implement such a procedure at the inventor level because inventors move from technology to technology.3) Productivity = f(Technological Mobility, Inter-firm Mobility, Value of Patents, Temporal Patenting Pattern, Technical Field, Country) 4) Value of Patents = f(Technological Mobility, Inter-firm Mobility, Productivity, Temporal Patenting Pattern, Technical Field, Country)

Table 1 . List of Variables Variable Definition or formula to calculate

 1

	Observation Units	Observations are individual inventors
	Career Measures
	Career_1st_year	Year of first patent application (for a patent eventually granted)
	Career_Last_Year	Year of last patent application
	Career_Duration	Year of last patent application -Year of first patent application + 1
	Career_ Prod_Years	The number of years with at least one application
	Career_Max_Gap	The maximum number of years between two consecutive applications
	Productivity Measures
	Patents_Number	Number of patents
	Patents_per_Year	Patents_Number/Career_Duration
	Patents_per_Prod_Years	Patents_Number/Career_ Prod_Years
	Value Measures
	Citations_Number	Sum of all citations for the inventor's patents
	Citations_per_Patent	Citations_Number/Patents_Number
	Citations_per_Pat_per_Yr	Citations_Number/Patents_Number/Career_Duration
	Patenting Pattern
	Patent_Time_Conc	Time Concentration = Share of patents in the year with most patents
	Patent_Time_HHI	Herfindahl-Hirschman Index (= Sum of squared shares) for patents per year
	Patent_Time_Skew	Skewness of patents per year distribution (NA if it cannot be calculated)
	Patent_Time_Kurt	Kurtosis of patents per year distribution (NA if it cannot be calculated)
	Technical Field ID
	Tech_Field_Dom	Dominant field for inventor's patents (among 30 two digit fields)
	Tech_Field_2	Second dominant field if the number of patents in the top 2 is the same
	Tech_Field_3	Third dominant field if the number of patents in the top 3 is the same
	Tech_Field_4	Fourth dominant field if the number of patents in the top 4 is the same
	Tech_Field_Wts	Weight used to calculate "fractional allocations" of patents to fields
	Technical Mobility
	Tech_Cat_Mobility_Conc	1 -Share of inventions in the dominant category at the six-technical-category level
		1 -Herfindahl-Hirschman Index (= Sum of squared shares) for the six technical
	Tech_Cat_Mobility_HHI	category distribution
	Tech_Cat_Moves	Number of moves from one of the six technical categories to another
	Tech_Field_Moves	Number of moves from one of the 30 technical fields to another
	Interfirm Mobility
	Assignee_Moves	Number of times the inventor changed assignees in the sequence of his patents
		Number of times the inventor changed assignees with more than 2 years since the
	Assignee_Moves_Adj	last observed move

Table 3 . Summary Statistics for Inventors from Five Countries for 1980-2002

 3

									All
		All Prolific Inventors	UK	France Germany	US	Japan	Inventors
		Mean Std. Dev. Mean Mean	Mean	Mean	Mean	Mean
	CAREER_DURATION	18.77	7.12	17.58	17.33	18.60	17.55	20.68	4.90
	CAREER_PROD_YEARS	12.12	4.83	10.85	11.34	12.14	10.91	13.92	2.50
	CAREER_TIME_GAP	4.53	3.24	4.49	4.18	4.29	4.58	4.56	2.27
	PATENTS_NUMBER	32.02	35.38	24.29	26.36	29.69	26.85	40.57	3.84
	PATENTS_PER_YEAR	1.87	1.70	1.64	1.71	1.78	1.84	1.94	0.97
	PATENTS_PER_PROD_YEARS	2.58	1.67	2.35	2.36	2.44	2.58	2.63	1.23
	CITATIONS_NUMBER	209.16	271.96 130.78	119.90	134.26	206.98	241.19	23.82
	CITATIONS_PER_PATENT	6.62	5.04	5.36	4.58	4.55	7.66	5.90	5.81
	PATENT_TIME_CONC	0.23	0.10	0.24	0.23	0.22	0.25	0.20	0.73
	PATENT_TIME_HHI	0.14	0.07	0.15	0.14	0.13	0.15	0.12	0.70
	PATENT_TIME_SKEW	-0.31	0.93	-0.13	-0.12	-0.08	-0.29	-0.43	-0.11
	PATENT_TIME_KURT	0.42	2.53	0.34	0.24	0.14	0.52	0.36	0.28
	TECH_CAT_CONC	0.68	0.22	0.75	0.78	0.76	0.73	0.58	0.89
	TECH_CAT_HHI	0.58	0.24	0.66	0.69	0.66	0.63	0.46	0.87
	TECH_CAT_MOVES	2.29	1.47	1.55	1.42	1.77	1.89	3.09	0.42
	TECH_FIELD_MOVES	5.38	4.62	3.04	2.96	3.95	4.05	7.89	0.75
	FIRM_MOVES	10.23	25.11	4.81	4.84	5.21	5.27	19.10	0.85
	FIRM_MOVES_ADJ	5.15	10.93	2.66	2.49	2.80	2.78	9.37	0.49
	TECH_CAT_1	0.27	0.44	0.27	0.31	0.42	0.26	0.24	0.18
	TECH_CAT_2	0.15	0.36	0.09	0.08	0.03	0.15	0.19	0.14
	TECH_CAT_3	0.13	0.34	0.33	0.27	0.16	0.17	0.06	0.10
	TECH_CAT_4	0.17	0.38	0.10	0.11	0.09	0.17	0.20	0.16
	TECH_CAT_5	0.17	0.38	0.12	0.13	0.20	0.13	0.23	0.20
	TECH_CAT_6	0.10	0.31	0.09	0.09	0.10	0.12	0.08	0.22

Table 8 . Inter-firm and Technological Mobility Effects Effect on Inter-firm mobility technological mobility Productivity (patents per year)

 8

According to Ernst's definition, Key inventors are those who are both "industrious" and "talented." In his classification of inventors, an inventor with a great number of inventions could be a key inventor (if his/her patents are very valuable) or could be an industrious inventor, if the quality of inventions is not as high.

[START_REF] Hess | The Formation of Dynamic Capabilities[END_REF] note that gatekeepers serve to bridge organizational and environmental boundaries in order to act as an information filter by evaluating and streamlining knowledge flows.

We use the terms technical and technological interchangeablyLatham et al. (2011) present preliminary analysis of the scale of inventor geographic mobility for three European countries.

[START_REF] Harhoff | Institutionalized incentives for ingenuity, patent value and the German Employees" Inventor Act[END_REF] have shown that the number of citations is a good proxy for the value of a patent.

countries. This finding is in accordance with Hoisl"s (2007) results for Germany, but our results extend it to the other countries as well. Value is an insignificant determinant in France and the US, negative and significant in Germany and Japan and positive and significant in UK. These last results are interesting. They might be interpreted to mean that there is a global inter-firm mobility process in relation to productivity and technical mobility but that each country has its own process with respect to value. We know that countries differ through their institutional arrangements. Of course additional analysis will be necessary before we have definitive insights.

Determinants of technological mobility.

Here we innovate in the sense that ours is the first analysis (to our knowledge) of technical mobility. For that reason we cannot refer to prior studies for guidance or for insights into the interpretation of our results. A significant finding is that inventor productivity is significant and negatively related to technical mobility in all five countries (in France the significance level is only 10.2%). This means that the very productive prolific inventors (sometimes called "stars inventors") are less intellectually (technologically) mobile. To put it simply: they are more technologically specialized. This seems to confirm, for individual inventors a long-standing observation: the more specialized you are and you remain, the more productive you continue to be. The effect of inter-firm mobility is positive and significant (but very small) in all countries. The last result suggests that the two types of mobility are not only correlated, but deeply linked in the inventor"s career as well. As far as value of inventions is concerned, again we find divergent trends


L A Inza

J I Mars

J P Métaxian

G S O'brien

O Macedo

Localization with multicomponent seismic array

Seismo-volcano source localization is essential to improve our understanding of volcano systems. The lack of clear seismic wave phases prohibits the use of classical location methods. Seismic antennas composed of one-component (1C) seismometers provide a good estimate of the back-azimuth of the waveeld. The depth estimation, on the other hand, is difcult or impossible to determine. In order to determine the source location parameters (back-azimuth and depth), we extend the 1C seismic antenna approach to 3Cs. This communication discusses a highresolution location method using a 3C array survey (3C-MUSIC algorithm) with data from two seismic antennas installed on an andesitic volcano in Peru (Ubinas volcano). After introducing the 3C MUSIC processing, we evaluate the robustness of the location method on a full waveeld 3D synthetic dataset generated using a digital elevation model of Ubinas volcano and an homogeneous velocity model. Results show that the back-azimuth determined using the 3C array has a smaller error than a 1C array. Only the 3C method allows the recovery of the source depths. Finally, we applied the 3C-MUSIC to two seismic events recorded in 2009. Therefore, extending 1C arrays to 3C arrays in volcano monitoring allows a more accurate determination of the source epicenter and now an estimate for the depth.

I. INTRODUCTION

Source location in term of back-azimuth and depth determination is a fundamental goal in volcano monitoring. Locating seismic events (LP) is necessary to improve the knowledge of the dynamics and magmatic systems, but it is extremely difficult to do it by classical methods (as phase picking or cross-correlation technique [START_REF] De Barros | Source geometry from exceptionally high resolution long period event observations at mt etna during the 2008 eruption[END_REF]). Here we focus on the multicomponent array method for locating these events. Dense one-component (1C) array methods based on time delays between close sensors, have been used by several authors ( [START_REF] La Rocca | Probabilistic source location of explosion quakes at stromboli volcano estimated with double array data[END_REF], [START_REF] Métaxian | Locating sources of volcanic tremor and emergent events by seismic triangulation[END_REF] ) and allow a good estimation of the back-azimuth of the wavefield. Unfortunately, the depth estimation is poorly resolved because incidence angle is very difficult to determine. To overcome this problem, triaxial sensors (3C) can be used. In this work we focus on the improvement in back-azimuth resolution and depth determination obtained by 3C rather than 1C seismometers. In our study two 3C arrays of 12 threecomponent broadband seismometers were installed on Ubinas volcano, Peru, in March 2009 in order to determine the backazimuth and depth of the seismo-volcano sources. Ubinas volcano, situated in the Central Volcanic Zone with a caldera floor lying approximately 5100 m above sea level (Fig. 1a), is considered as the most active Peruvian volcano during the last 500 years, threatening 3,500 people living on the edge of the Ubinas valley. At the time of the experiment, the eruption was characterized by almost permanent ash emissions. Source location of the LP events with a higher resolution compared to the 1C dense array methods, particularly in the depth determination, is the main objective of this work. To achieve it and before working with real wavefield dataset, we performed several numerical simulations of seismic wave propagation using a 3D digital elevation map and heterogeneous velocity model determined from a tomography study [START_REF] Monteiller | An efficient algorithm for double-difference tomography and location in heterogeneous media, with an application to the kilauea volcano[END_REF]. Sources were placed at 8 different depths below the crater and receivers are situated at the same positions as the two field experimental cross-shaped arrays (Fig. 1b). New 3C-MUSIC and 1C-Music [START_REF] Schmidt | Multiple emitter location and signal parameter estimation[END_REF] are applied to the synthetic data to determine the backazimuth, the incidence angle and the depth for both arrays and all sources. This procedure is after applied to the real data recorded in March 2009.

II. PROCESSING APPROACH

We consider 𝑃 sources travelling from different directions, and impinging at the antenna of 𝑁 triaxial sensors (𝑃 < 𝑁). Let 𝑤 𝑛 (𝑡), the signal recorded by one component of the the 𝑛 𝑡ℎ sensor in the time-space domain as.

𝑤 𝑛 (𝑡) = 𝑃 ∑ 𝑝=1 𝑠 𝑝 (𝑡 -𝜏 𝑛,𝑝 ) + 𝑏 𝑛 (𝑡); 𝑛 = 0, 1, ..., 𝑁 -1 (1)
where, 𝑠 𝑝 (𝑡) is the 𝑝 𝑡ℎ source signal, 𝜏 𝑛,𝑝 is the relative propagation time delay of the 𝑝 𝑡ℎ source for the 𝑛 𝑡ℎ sensor. The noise 𝑏 𝑛 (𝑡) is usually assumed to be Gaussian, uncorrelated with the sources and both temporally and spatially white with variance 𝜎 2 𝐵 . The corresponding relative propagation time delay is defined as 𝜏 𝑛,𝑝 = d n ⋅ u(𝜃 𝑝 , 𝜙 𝑝 ). where, d 𝑛 is the relative position vector of sensor "𝑛" with respect to the first sensor located at (0,0,0) and u(𝜃 𝑝 , 𝜙 𝑝 ) is the slowness vector indicating the direction of the 𝑝 𝑡ℎ source. The antenna output of the equation (1) for narrowband signals can be represented in matrix form as:

W(f o ) = A(f o )S(f o ) + B(f o ) (2)
where, the 𝑁 × 𝑃 matrix A(f o ) is the "array response matrix" or "steering matrix" [START_REF] Miron | High resolution vector-sensor array processing based on biquaternions[END_REF] and is given in equations (3). The 𝑃 sources are represented by a 𝑃 × 1 matrix as a

S(f o ) = [𝑆 1 (𝑓 𝑜 ), ..., 𝑆 𝑃 (𝑓 𝑜 )] 𝑇 . A(f o ) = [a(𝜃 1 , 𝜙 1 ), a(𝜃 2 , 𝜙 2 ), ..., a(𝜃 𝑃 , 𝜙 𝑃 )] (3)
𝑝 (𝜃 𝑝 , 𝜙 𝑝 ) = 1 √ 𝑁 [ 1 𝑒 -2𝜋𝑗(d2⋅k𝑝) .... 𝑒 -2𝜋𝑗(d𝑁 ⋅k𝑝) ] 𝑇
with k 𝑝 the wavenumber vector. Usually, A(f o ) is a full rank matrix assuming that the array manifolds a 𝑝 (𝜃 𝑝 , 𝜙 𝑝 ) with different path directions are independent. MUltiple SIgnal Classification (MUSIC) [START_REF] Schmidt | Multiple emitter location and signal parameter estimation[END_REF] is an eigen structure subspace analysis method, that is widely used in geophysics, particularly to enhance the signal-to-noise ratio in order to estimate the direction of arrival of multiple waves impinging the array [START_REF] Paulus | Vector-sensor array processing for polarization parameters and doa estimation[END_REF], [START_REF] Wong | Self-initiating music direction finding & polarization estimation in spatio-polarizational beamspace[END_REF]. To extend 1C-MUSIC to 3C, a data window for all components of 3C sensors array is selected, corresponding to the first arrival signal of LP event. The data model for every bin sampled is given as:

W 𝑚 = [Xn 𝑚 Yn 𝑚 Zn 𝑚 ] (4)
where, 𝑋𝑛, 𝑌 𝑛 and 𝑍𝑛 represent the data vector of each component of the sensor for the sample 𝑚. Then the crossspectral correlation estimation over 𝑀 frequencies bins is defined as:

Γ𝑊 = 𝑀 ∑ 𝑚=1 𝜉 { W 𝑚 W H 𝑚 } (5)
𝜉 denote the mathematical expectation operator and H is the conjugate transpose. The cross-spectral matrix Γ𝑊 can be writen as [START_REF] Paulus | Vector-sensor array processing for polarization parameters and doa estimation[END_REF], [START_REF] Paulus | Wideband spectral matrix filtering for multicomponent sensors array[END_REF], [START_REF] Miron | Vector-sensor music for polarized seismic sources localization[END_REF].

Γ𝑊 = 𝑨 𝜉{S S 𝐻 } 𝑨 𝐻 + 𝜉{B B 𝐻 } (6) Γ𝑊 (𝑓 𝑜 ) = 𝑨 (𝑓 𝑜 ) Γ𝑠 (𝑓 𝑜 ) 𝑨 𝐻 (𝑓 𝑜 ) + 𝜎 2 𝐵 (𝑓 𝑜 ) I
where, Γ𝑠 (𝑓 𝑜 ) = 𝜉{S S 𝐻 } is the cross-spectral matrix of the source, I is the identity matrix and 𝜎 2 𝐵 is the variance of the noise. The eigenstructure of 𝑁 × 𝑁 cross-correlation matrix Γ𝑊 solves the eigenvalues 𝜆 𝑛 and eigenvectors v n respectively (N=number of sensors). The 𝑃 largest eigenvalues correspond to the signal subspace, while 𝑁 -𝑃 eigenvalues correspond to the noise subspace. The 3C-MUSIC estimator is given in the equation ( 7) by 𝑀 3𝐶 which computes the azimuth and incidence angles.

𝑀 3𝐶 (𝜃, 𝜙) = 1 A 𝐻 (𝜃, 𝜙) Π ⊥ A (𝜃, 𝜙) (7)
where,

Π ⊥ = ∑ 𝑁 𝑛=𝑝+1 v 𝑛 v 𝐻
𝑛 is the noise sub-space projector.

III. APPLICATION: SYNTHETIC AND REAL DATASET

We deployed on Ubinas volcano between May and July 2009, two cross-shaped antennas (NUBI and WUBI) with 12 3C seismometers each (Fig. 1a.). Continuous recordings were acquired with a sampling frequency of 100 Hz. Distances between seismometers was set to approximately 50 meters. Seventeen explosion earthquakes and 450 of LP events were recorded during the experiment.

A. Synthetic Data Analysis

In order to test the accuracy of the location methods, a 3D discrete numerical elastic lattice method [START_REF] O'brien | A 3d discrete numerical elastic lattice method for seismic wave propagation in heterogeneous media with topography[END_REF] coupled with a digital elevation map of Ubinas topography and the 2009 experimental array locations, was used to create a full waveform synthetic dataset. In order to compare the 3C and 1C methods, we analysed the full synthetic dataset for the 3C and only the vertical components for the 1C. Same analysis for both arrays and for all eight sources are done. The crossspectral matrix was then calculated by using 32 windows around the dominant peak frequency. Comparison between 1C and 3C-MUSIC are shown in the Fig. 2a and2b for NUBI and WUBI antennas respectively. The back-azimuths from the 3C-MUSIC correspond to the model values for both antennas with a resolution of ±3 ∘ . 1C-MUSIC gives equivalent results, but with higher errors (±6 ∘ ). The incident angle varies with depth when it is well determined with the 3C-MUSIC. Knowing the distance from the centre of the antennas and the hypocentre of the sources, depth resolution can be deduced for each antenna (500 m for NUBI and 400 m for WUBI). On the other hand, the 1C-MUSIC analysis does not allow the depth to be determined. The theoretical velocity has an accuracy of ±150 m/s using the 3C-MUSIC algorithm. The 1C-MUSIC measures higher velocities at NUBI and lower velocities at WUBI. For NUBI antenna, the 1C-MUSIC analysis gives depths close to the model values for the superficial sources (1, 2 and 3) while depths for sources 4, 5, 6, 7 and 8 are far away from the model values. None of incident angle of sources can be distinguished with the 1C-MUSIC analysis. For WUBI antenna, the 1C-MUSIC analysis gives higher incident angles than the model values. Errors are larger than the 3C algorithm. The 1C-MUSIC analysis does not give reliable solutions for any of the synthetic source depths. In summary, incident angles obtained by the 3C-MUSIC algorithm are close to the theoretical values for both antennas, whereas those obtained with the 1C-MUSIC are not reliable. In order to represent the results for the depth of sources, the PDF of the source position is derived from the different PDF's of the back-azimuth and the incident angle [START_REF] Métaxian | Locating sources of volcanic tremor and emergent events by seismic triangulation[END_REF]. Hence, 3C-MUSIC allows to locate the synthetic sources in depth and in all the cases. 1C analysis can not give this results. Figure 3 shows in details the results obtained for each synthetic source with the 2 antennas NUBI and WUBI ( 1) Real Data Analysis: We discuss the performance of 3C-MUSIC method on 2 real seismic events recorded at Ubinas volcano recorded on the 2 antennas (an explosion earthquake signal and an LP event). Fig. 4b andc gives all information in time and frequency at NUBI antenna. Results of the source position for the LP event and the explosion earthquake are showed in Fig. 5. source PDF for the LP event is situated 3000 m below the bottom of the crater at the altitude of 2240 m with a radius of 730 m. The source area of the explosion is situated 150 m West and 1000 m below the bottom of the crater at an altitude of 4200 m, the radius R is 660 m.

IV. CONCLUSION

We have presented a source localization method (3C-MUSIC) based on the use of 3C arrays and, compared it with 1C-MUSIC. All results in terms of back-azimuth, incident angles, velocities and depth is estimated with higher precision with 3C-MUSIC than 1C MUSIC in synthetic case. In a real The vertical dash lines in d) and e) indicate the time window selected for the processing. f) and g) Averaged energy spectrum calculated for all the receivers and all the components, where the vertical dash lines represent the frequency windows used for the cross spectral matrix calculation experiment, we located a depth of the LP event and an explosion earthquake using the 3C-MUSIC. We conclude that 3C-MUSIC provides realistic estimates of the depth of volcanic sources, unlike the 1C-MUSIC or other antenna methods based on time delays measurements. Given the performance of the 3C-MUSIC algorithm, we will apply it to other explosions and LP events recorded at Ubinas during the experiment to better characterize the eruptive dynamics of this volcano. Further studies must take into account the bandwidth of the signal with wideband 3C MUSIC or Quaternion MUSIC processing [START_REF] Hobiger | Musique: A quaternion-based array processing technique for surface wave polarization analysis[END_REF].

Fig. 1 .

 1 Fig. 1. a) Map of Ubinas volcano showing the location of the north (NUBI) and the north-west (WUBI) antennas, b) East-West profile through the crater showing altitude and depths of the synthetic sources for the 3D synthetic dataset.

Fig. 2 .

 2 Fig. 2. Results obtained for the synthetic data calculated for the eight sources. Open triangles and open stars represent results obtained with the 3C-MUSIC and 1C-MUSIC, respectively. Sources are numbered as in figure 1b. a) Backazimuth, incidence angle and apparent velocity for NUBI antenna. b) Backazimuth, incidence angle and apparent velocity for WUBI antenna. Dash lines represent the values. The filled circles represent the real data results

  Fig 3 left) and with a third synthetic antenna added and laying at Est part (Fig 3 right). The horizontal position is well determined for all sources. Looking at the vertical views and comparing with exact position given in Fig 1b, the estimated source position is well for the 2 highest sources which are very close together (first and second subplots). Uncertainty is larger for the source 3 in case of 2 antennas but position is better estimated in case of 3 antennas. The PDF of the source position gives a good solution for sources 4 and 5. These sources 4 and 5 can be clearly differentiated from sources 1, 2 and 3. It is difficult to differentiate sources 7 and 8 in 2 antennas, but adding a third antenna allows to separate them.

Fig. 3 . 8

 38 Fig. 3. 8 Synthetic sources localizations with: 2 antennas NUBI and WUBI (left part) and 3 antennas (right part). Exact position is given by stars point related to fig 1b.

Fig. 4 .

 4 Fig. 4. a) Z component seismogram of the LP event and explosion earthquake recorded by the central station of NUBI antenna. The time and date of the first sample is indicated in the upper left of the record. Shaded zones represent part of the signals enlarged in b) and c). d) and e) Z component of the LP event and the explosion earthquake recorded by 10 stations of NUBI antenna.The vertical dash lines in d) and e) indicate the time window selected for the processing. f) and g) Averaged energy spectrum calculated for all the receivers and all the components, where the vertical dash lines represent the frequency windows used for the cross spectral matrix calculation

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