Matthieu Crozet
email: crozet@univ-paris1.fr

Thierry Mayer

Daniel Mirza

Lionel Fontagné

Hugo Pilkington

Do migrants believe in market potential?

Keywords: Increasing Returns, Market Potential, Agglomeration, Migrations, Economic Geography Rendements croissants, Potentiel Marchand, Agglomeration, Migrations, Economie géographique

New economic geography models predict migration flows from peripheral regions toward central ones.

, we estimate this function and thereby all the parameters of the standard economic geography model. However Hanson estimated an equation of wage determination, whereas we study the market potential through a much more central relation of the model. We estimate the worker motion law on inter-regional migrations data for five European countries. The migrant flow received by a region increases with its market potential, which confirms the global relevance of the economic geography model. Moreover main parameters estimates are consistent with the theoretical framework.

Introduction

During the last decades, spatial economics has known an appreciable revival, notably thanks to [START_REF] Krugman | Integration and the Competitiveness of Peripheral Industry[END_REF], and Krugman [1991-b]. These works, constituting the so-called new economic geography, describe the mechanics which lead to the spatial agglomeration of increasing returns sectors, formalizing the intuitions originally developed by economists such as [START_REF] Perroux | note sur la notion de pôle de croissance[END_REF] and Hirshman [1958] and giving a complete framework to the principle of market potential function originally defined by both [START_REF] Isard | Location Theory and Trade Theory: Short-Run Analysis[END_REF] and [START_REF] Harris | The Market as a Factor in The Localization of Industry in the United States[END_REF] (See Fujita, [START_REF] Fujita | The Spatial Economy: Cities, Regions and International Trade[END_REF] and [START_REF] Fujita | Economics of Agglomeration[END_REF] for good surveys of the state of theoretical knowledge).

As explained by Fujita et al. [1999], these theories have not generated yet an important empirical literature and "we clearely need much more such [ empirical] works, as closely tied to the theoretical models as possible... " (p. 347). The present article has for objective to participate to fill this gap, presenting a direct validation of the central relation of new economic geography models. From a multiregional framework, where monopolistically competitive firms choose their location freely, we can estimate the motion law of inter-regional migrants within European countries. This equation is a fundamental condition for observing agglomeration dynamics in the model. Thus, we verify the global relevance of the theoretical framework, and estimate its key parameters. As economic geography models present multiple equilibria, depending on the values of those parameters, such an estimation is a useful tool for simulation exercises.

Scale economies and the presence of a significant transport cost encourage firms to locate in the largest market, creating a cumulative causation: firms want to be close to large markets, as so workers who can pay lesser transport costs on consuming goods in such locations. Thus, access to market appears to be essential for both firms and workers. Both are attracted to high market potential regions, as this function enters in the definition of demand addressed to firms -driving firms' motions -and of real wages -driving migrations 1 .

The empirical literature explicitly based on these models is recent and relatively scarce. Of course, some fundamental studies prove the relevance of agglomeration processes, guided by demand linkage or externalities in city formation or industrial agglomeration, but few of them test explicitly the new economic geography framework and the relevance of the market potential function. Nevertheless, some studies on international trade and FDI location provide more or less direct empirical verifications.

As market potential is linked to the price index function, it appears to be implicitly a determinant of the "home market effect"2 which sates that, in an increasing return industry, a one percent increase in local demand leads to a more than one percent increase in employment and production. This effect can explain sectoral agglomeration. Testing the home market effect can thus be considered as a test of an economic geography model and of the relevance of market potential. [START_REF] Davis | Does Economic Geography Matter for International Specialization ?[END_REF] found little evidence of home market effect, but, taking into account a more precise market access variable, closer to the market potential function, [START_REF] Davis | Economic Geography and Regional Production Structure: an Empirical Investigation[END_REF] showed that Krugman's [1980] model explains a significant part of the regional structure of specialization. In the same way, Feenstra et al. [1998] find a global relevance of the home market effect hypothesis for differentiated goods. [START_REF] Head | Armington Vs. Krugman: A Test of Two Trade Models[END_REF] inter-sectoral results, obtained from US and Canadian data, support Krugman's [1980] model. Finally, [START_REF] Brülhart | Home-Biased Demand and Iternational Specialisation: a Test of Trade Theories[END_REF], showed that, taking into account home-biased demand, the Dixit-Stiglitz-Krugman pattern of international trade is relevant for almost two thirds of the total industrial output.

Many studies also proved the importance of local demand in a firm's location choice. Unfortunately, most of them simply introduced local income as a proxy of the demand addressed to firms, whereas the true variable should be the market potential. Nevertheless, [START_REF] Henderson | Industrial development in Cities[END_REF], [START_REF] Friedman | What Attracts Foreign Multinational Corporation[END_REF] and [START_REF] Mayer | Comportement Stratégiques de Localisation des Firmes Multinationales[END_REF] used explicit market potential proxies in empirical studies of multinationals location choices, and gave confirmation that it can explain industrial concentration.

Those studies, as with all gravity-like estimation of trade flows, are only indirect testing of economic geography models. [START_REF] Hanson | Market Potential, Increasing Returns, and Geographic Concentration[END_REF], gave a much more explicit one, estimating the influence of the complete market potential function in wage determination. This study, which largely inspires ours, presents very encouraging results and succeeds in calibrating [START_REF] Krugman | A Dynamic Spatial Model[END_REF] model. But, as he is testing an equilibrium relation, one can relate Hanson's test back to a regional wage determination equation, a law of geographic propagation of local shocks on wages, which is finaly far from the agglomeration formation story. Using approximately the same model, we estimate the market potential function through a more fundamental equation of the model, which directly refers to agglomeration processes.

A simple model of economic geography

We present a multi-regional model of economic geography, based on [START_REF] Krugman | A Dynamic Spatial Model[END_REF]. As in a two regions model (Krugman [1991]), the motion law of mobile workers is a central key of the transitory dynamic of the model; agglomeration occurs when people move from small and relatively distant regions (ie low market potential regions) to more central ones.

To test this fundamental law we have to modify the original model in a more realistic way. As [START_REF] Thomas | Increasing Returns, Congestion Costs and the Geographic Concentration of Firms[END_REF] and [START_REF] Hanson | Market Potential, Increasing Returns, and Geographic Concentration[END_REF], we introduce a nontraded good. This specification involves less drastic final equilibria (the peripheral regions always keep a part of non agricultural activity) and because, in real world, non-traded good represent a significant part of consumer price index, it leads to more realistic migration dynamics. This last point is the most important here : as we focus on transitional dynamics, the nature of final equilibrium is not a dominant question compared to the credibility of the equation we have to estimate. This is also why, unlike these two authors, we do not replace the agricultural sector by this new one but keep both. Moreover, the nontraded sector has to be large enough to improve the realism of the theoretical framework.

In this order, and unlike [START_REF] Hanson | Market Potential, Increasing Returns, and Geographic Concentration[END_REF], we consider that all the regional production of services is nontraded3 .

Production and price indexes

The economy is divided in R regions. Each of them produce three kinds of goods : freely traded agricultural goods (Z), non-tradable services (Y ) and costly tradable industrial goods (X).

We consider two factors of production: Immobile and mobile labor. Immobile workers are specific to the agricultural sector and uniformly distributed across locations. Mobile workers are employed indifferently in the two other sectors and choose their location by comparing utilities across regions.

The agricultural good is homogeneous and produced under constant returns to scale. Good Z in chosen as a numeraire (p Z ≡ 1). We note L Z it the agricultural population present in region i at instant t.

Z it = L Z it = L Z t , ∀i ∈ R (1)
The two others factors are horizontally differentiated. The production of each variety needs a fixed cost which introduces increasing returns. The cost functions for an industrial firm m and a service firm m are:

L X mt = ε x + β x x mt (2) L Y m t = ε y + β y y m t ( 3
)
Where ε x , ε y , β x and β y , are respectively fixed and variable costs in manufacturing and services; L X mt and L Y m t are mobile labor used by firms m and m to produce quantities x mt and y m t of industrial and service goods. We note L it the non-agricultural population of region i

(L it = L X it + L Y it
). The consumers share the same Cobb-Douglas utility function.

U it = C µ Xit C φ Y it C 1-µ-φ Zit (4)
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Cahiers de la MSE -2000.30

C θit is the quantity consumed of good θ (θ = x, y, z) in region i. Consumers demand all varieties of the industrial good whatever their region of origin, and all varieties of non-tradable services produced locally.

C xit = nxt 1 c (m) σx-1 σx it dm σx σx-1 (5) C yit = n yit 1 c (m )
σy -1 σy it dm σy σy -1 [START_REF] Eaton | Technology, Geography, and Trade[END_REF] Where n x denotes the total number of industrial varieties and n yi the number of service varieties in region i.

The profit maximization of manufacture and service producers leads them to set prices as the product of a constant sector-specific mark-up and their marginal costs. If w it is the nonagricultural wage in region i, and p θit is the fob price of a variety produced in this region (θ = x, y) then:

p xit = σ x σ x -1 β x w it ( 7
)
p yit = σ y σ y -1 β y w it (8)
Then, free entry in each sector drives to a zero-profit equilibrium which allows us to define the number of manufacture and service firms in each region i:

n xit = L X it ε x σ x ( 9
)
n yit = L Y it ε y σ y
We said that the industrial good is costly tradable, the transport cost takes the form of an iceberg cost such as a producer located in i has to send τ ij > 1 units of good to ensure that one unit gets to the region j of destination. It is logically an increasing function of the bilateral distance d ij , and its specification follows that of [START_REF] Hummels | Toward a Geography of Trade Costs[END_REF]:

τ ij = Bd δ ij ∀i, j ∈ [1, R] ,
δ > 0 and B > 0 [START_REF] Feenstra | Understanding the Home Market Effect and the Gravity Equation: The Role of Differentiating Goods[END_REF] Recalling that the agricultural good is freely tradable and its price normalized to one, we can calculate, from equations (3), ( 4), ( 5), ( 7) and ( 8), the real wage in region i.

ω it = w it q φ yit q µ xit 1 1-φ-µ (11)
where q xit is the price index for manufactures and q yit the one for services:

q xit = k=R k=1 n xkt 0 Bd δ ij p xkt 1-σx dk 1 1-σx (12) = k=R k=1 n xkt Bd δ ij p xkt 1-σx 1 1-σx q yit = n yit 0 p 1-σs yt di 1 1-σs = n 1 1-σs yit p yt ( 13
)
Equation [START_REF] Fujita | The Spatial Economy: Cities, Regions and International Trade[END_REF] represents the price index of manufactures, which can be shown to be close to Harris' market potential function [START_REF] Krugman | Space: The Final Frontier[END_REF]).

The original, market potential function represents the potential demand addressed to a firm, depending on its location; so its main determinant is the one defined by Harris: the sum of regional purchasing powers negatively weighted by bilateral distances. Because, in reality, producers have to share the global demand with the other local competitors, multi-regional economic geography models, such as [START_REF] Krugman | A Dynamic Spatial Model[END_REF], express the market potential as this weighted sum (the pure market potential function) corrected by a competition effect represented by the regional price index [START_REF] Fujita | The Spatial Economy: Cities, Regions and International Trade[END_REF]. Nevertheless, this price index, reveals logically the same structure than the original weigthed sum proposed by Harris. Indeed, the price index expresses that manufactures are less expensive in regions which import only a small part of their consumption and where imports are provided by close regions 4 . As in these larger and more "central" regions the demand adressed to a single firm should be more important than in the periphery, they also benefit from a higher market potential.

Because firms look for locations where demand is important and mobile workers look for low consumption price areas, the similarity beetween the price index and the pure market potential function is a basis of the cumulative causation constituting the engine of the agglomeration dynamic.

The service price index [START_REF] Fujita | Economics of Agglomeration[END_REF], is constructed on the same basis as the precedent. It represents also a kind of market potential for service goods, but as these goods are not traded, this potential simply equals the home market size.

Real wage and migration choices

From Bilateral migrations...

The non-agricultural workers choose their location in order to maximize their utility. They will move from a region j to a region i if the indirect utility in i is superior to the one in j. In the theoretical framework developed by Krugman [1991] and [START_REF] Krugman | A Dynamic Spatial Model[END_REF] potential migrants only compare real wages to choose their location. Such assumption implicitly supposes that migration has no direct influence on the migrants' income and utility. Yet, trans-regional migrations, as international ones, involve pecuniary and moral mobility costs which affect people's income and satisfaction [START_REF] Shields | The emergence of migration theory and suggested new direction[END_REF]). We assume that these costs are proportional to the indirect utility in the destination region, taking into account all permanent moral and financial costs affecting migrant's utility, directly or via its income5 .

We suppose this cost increases with the distance between the regions of departure and destination. Because a large part of this cost is moral, we also make the assumption that it varies between individuals (as done by [START_REF] Faini | Increasing Returns, Migrations and Convergence[END_REF]).

For each non-agricultural worker l j from region j, the cost of migration to i is:

A ji (l j ) = (d ij ) λ Θ l j ( 14
)
Where λ is a strictly positive coefficient and Θ

l j = l j L j , with l j ∈ [1, L j ] (l j is uniformly distributed on [1, L j ]).
From equations ( 7) and ( 9) to ( 14) we get l j 's relative indirect utility:

u it (l j ) u jt = ω it A ji (l j ) ω jt (15)
Moreover, as all individuals are workers, we do not study migrations driven by amenity demand, but only by spatial job opportunity search. Consequently, the migrant must take also into account the probability of finding a job in the destination region. We note ρ it such a probability.

Then the condition for an individual from region j to move to region i is:

u it (l j ) ρ it > u jt ( 16
)
Using the definition of the migration cost, this condition becomes:

l j L j < ω it ω jt (d ij ) -λ ρ it
We note l * jt the marginal worker who verifies the equality

l * jt L jt = ω it ω jt (d ij ) -λ ρ it ; so l * jt
L jt represents the share of the mobile population of j who will choose to migrate to region i. Then migration flow between the two locations is:

migr jit L jt = ω it ω jt (d ij ) -λ ρ it ( 17
)
10

Cahiers de la MSE -2000.30 = w it w jt 1-φ L Y it L Y jt φ σy -1      k=R k=1 L X it d δ ij w kt 1-σx k=R k=1 L X jt d δ ij w kt 1-σx      µ σx-1 (d ij ) -λ ρ it ...

to multilateral migrations

The equation [START_REF] Head | Armington Vs. Krugman: A Test of Two Trade Models[END_REF], representing the migratory flow between two regions, is based on an economic geography model which describes very succinctly migration choices. Taking into account mobility cost, the potential migrants look for the best working conditions: Highest real wages and best job opportunities. Even if we catch here the main motivations of workers motions, this description cannot explain two-way bilateral flows nor why all migrants from one region do not move to the same one but disperse all over national space6 . Thus, estimating directly the equation ( 17) expressed in logarithms would involve trying to explain two different positive flows (from j to i and i to j) by several explicative variables which are almost exactly the opposite from each other. To overcome this difficulty we express the motion law presented by equation [START_REF] Head | Armington Vs. Krugman: A Test of Two Trade Models[END_REF] in relative terms: considering the total migrants flow from a region j, we can express the part received by a region i.

migr jit i =j migr ji t = ω it ω jt (d ijt ) -λ ρ it i =j ω i t ω jt (d i jt ) -λ ρ i t or equivalently: log    migr jit i =j migr ji t    = log   w 1-φ it L Y it φ σy -1 k=R k=1 L X it d δ ij w kt 1-σx µ σx-1 (d ij ) -λ ρ it   -a jt (18) With a jt = log i =j ω i t (d i j ) -λ ρ i t
The equation [START_REF] Head | Non-Europe : The Magnitude and Causes of Market Fragmentation in the EU[END_REF] represents the basic law of motion of the economy which will be estimated by non linear least squares.

In relatively small and highly controled countries, such as European ones, we can expect wages to not significantly differ between regions. Then we estimate the same relation supposing the unicity of nominal wages accross locations. This hypothesis can also be easily justified theoretically by making the assumption of free labor mobility between agriculture and the increasing returns sectors. The basic structure of the model is not changed, but, considering equation ( 1), wage will be normalized to 1 in the whole economy, as long as each region still produces constant return goods. Studying the transition path to the equilibrium, this modification doesn't affect our results. In this case, equation ( 18) simply becomes:

log    migr jit i =j migr ji t    = log   L Y it φ σy -1 k=R k=1 L X it d δ ij 1-σx µ σx-1 (d ij ) -λ ρ it   -a jt ( 19
)
3 Econometric strategy

Toward an estimable equation

To be correctly estimated, equations ( 18) and ( 19) need some adjustments.

First, in a large part, time dimension is fully implicit in the original model. We must lag the dependant variable to avoid problems of endogeneity between migration flows and regional work forces, and to take into account the time for migrants to observe relative job conditions and make their decisions.

In addition, national space is perfectly continuous and internal migrations do no support any cross-border cost. In order to capture non-significant short distance migrations, which do not reflect an explicit workers' willingness to move to another economic center when involving two administrative regions, we introduce in the migration cost a dummy of regional contiguity. F ij equals 0 when regions i and j have an common border, and 1 otherwise. Then the migration cost becomes :

[d ij (1 + bF ij )] -λ .
Every thing equal, a large region should receive more migrants than a smaller one. This bias has to be controlled insofar as European regions, even in the same country, can have very different sizes 7 . The size bias is captured introducing in the equation the logarithm of the surface of destination region (log(S i )).

The variables a jt represent the trickiest issue. Even if their complete form are estimable as the i-specific part of the two equations, they introduce too many determinants to ensure the robustness of the econometric result. As they only vary along time and j dimensions; we treat them as j-specific fixed effects and their time dimension is captured by a trend introduced in the equation.

Then, choosing unemployment rates as proxy of the inverse of job finding probability, the equations ( 18) and ( 19) become:

log    M igr ji,t k =j M igr jk,t    = µ σ x -1 log k=R k=1 L X k,t-1 w it (d ik ) δ 1-σx (20)
+α 1 log L Y i,t-1 + α 2 log (E i,t-1 ) + (1 -φ) log (w it ) -λ log (d ij (1 + bF ij )) + α 4 log (S i ) + a j + α 5 log (trend) + u ijt And log    M igr ji,t k =j M igr jk,t    = µ σ x -1 log k=R k=1 L X k,t-1 (d ik ) δ(1-σx) + α 1 log L Y i,t-1 + α 2 log (E i,t-1 ) (21)
-λ log (d ij (1 + bF ij )) + α 4 log (S i ) + a j + α 5 log (trend) + u ijt
The dependent variable is the part of migrants from region j who choose to settle in region i (ie the "migration share" of region i). The first explicative variable is the manufacture market potential. The second term of the right side of equations ( 20) and ( 21), is the home market size 7 For exemple, we find among german NUTS 1, Bayern (70554 km 2 ) and Bremen (404 km 2 ).

13

Cahiers de la MSE -2000.30

of services.

Econometric method and data

Whatever the models, the parameters to estimate and the expected values are the same and presented in table [START_REF] Bordas | The Economics of Immigration[END_REF].

Equations ( 20) and ( 21) are estimated by gaussian non linear least squares on three dimensional panel data 8 . This kind of regression, can lead to very weak results, and we have to ensure robustness of the regressions.

Thus, we perform regressions using other econometric methods, such as maximum of likelihood.As it leads to very similar results to the one obtained by least squares, it is not presented it here.

Above all, we have to avoid introducing collinearity within parameters, source of singularity in the Hessian matrix. Each parameter must be individually attached, at least, to a single variable and not appear as a combination of other parameters. Consequently, we can't perform robust simultaneous estimations of φ and σ y neither of µ, δ and σ x 9 . Then, we simply estimate φ σy-1 and (1 -φ) as two independent coefficients (α 1 and α 3 ). For the second group of tied parameters, the issue is more complex: µ and σ x can't be estimated together, neither δ and σ x since σ x doesn't appear independently. The most reliable solution to overcome this difficulty is to set down µ to an exogenous value. Then the main source of instability (ie µ σx-1 ) simply disappears, and, since estimation of σ x is more robust, the interdependence between δ and σ x is reduced enough to keep explicitly in the model the restriction it involves on these two parameters.

According to sectorial distribution of final household consumption data, we then perform estimations setting down µ = 0.4 10 .

Finally robustness of the regressions is enforced by performing White consistent estimations in order to control for a possible presence of heteroscedasticity.

Table 1: Coefficients and expected values

Coefficients Expected values

σ x 1 < σ x µ 0 < µ < 1 δ 0 < δ α 1 = φ σy-1 0 < α 1 < 1 α 2 α 2 < 0 α 3 = 1 -φ 0 < α 3 < 1 λ 0 < λ b 0 < b α 4 0 < α 4 a j 0 < a j
All data, except for distances, where provided by Eurostat Regio database.

As the tested equations deal with the transitionary path, the model should be expressed without introducing the long term factor market equation [START_REF] Greenwood | Research on International Migration in the United States: a Survey[END_REF]. In this case, manufacture and service employment levels should simply be replaced by sectoral productions. We chose anyway the employment specification because long series on sectoral value added are not availlable for all countries. Then L K i,t (K = X, Y ), is the annual total employment in sector K, and wages are mean remuneration by worker in both manufacture and service sectors.

Regio provides also complete matrix of annual bilateral migrations between regions. These precious informations have three main shortcomings. First, bilateral migration data only concern internal mobility, and we have no information, at the regional level, about people who cross a European national border. Second, the spatial aggregation level is quite high: NUTS1 or 2, depending of countries 11 . Finally, time dimension is not exactly consistent with other data and from one country to another. These three difficulties do not allow us to pool together all the countries. We would lose too many observations to perform non pertinent estimations since the data only concerns internal migrations 12 . Above all, we had to drop many countries. Small countries, when data focuses on relatively large regions, such as Sweden, Finland and Portugal, provide too few observations. In 11 See Data appendix 12 Anyway, the importance of border effects will lead to few relevance of international migrations as one compares them to internal ones (Helliwell [1997]).

the same way, we have too little French data: for this country migrations are only available for periods of seven years; the last period starts in 1982, and main industrial data series begins in the early eighties.

We finally kept 5 countries : Germany, Spain, Italy, the Netherlands and Great Britain.

German data covers 10 western NUTS 1 regions during the period 1983-1992 (900 observations 13 ). In order to avoid any bias in the distance measurement, we dropped islands from the panel; in consequence, Spain represents only 15 NUTS 2 regions between 1983 and 1993 (2310 observations) and Italy 18 NUTS 2 regions for 1983-1993 (3366 observations). Dutch data covers the period 1988-1994 for 10 NUTS 2 regions (630 observations). British available migration data covers a period 1980-1985 for 10 NUTS 1 regions. As regional unemployment data starts in 1983, keeping unemployment in the estimations would have lead to reduce too drastically the available observations (180). So, we performed the estimations for this country without this variable, increasing the nomber of observations to 540.

We also need all bilateral distances between the region of a same country (d ij ∀i = j) to estimate the manufacture market potential and to proxy the migration cost.

Intra national goods shipment, in our relatively small countries, use mainly roads. As we dropped out all islands and Ulster from the panel, we can expect bilateral distances between two regions to be correctly proxied by the quickest route between the two main regional cities estimated for truck mean speed on the different kinds of roads. This information was acquired individually from an electronic road atlas. This tool gives us kilometric distances by most rapid ways and travel times. Of course, the two measures are strongly correlated, and the mean speed on different domestic travels do not differ substantially (even if it differs for a country to an other one). Travel times are more precise indicators of transport costs, but we chose to keep kilometric distances to have a variable consistent with internal distances. This measure which takes into consideration the reality of geographical space (mountains, lakes and density and quality of road infrastructures) is anyway a better proxy than Great Circle distances generally used in economic geography or international trade studies. patterns and introduice multicollinearity in the model. Consequently, equation [START_REF] Herzog | Migration as Spatial Job-Search: A Survey of Empirical Findings[END_REF] appears to be less stable than equation [START_REF] Hirsman | The Strategy of Economic Development[END_REF] even if it has nearly the same behavior: the model including wages has trouble converging for the total sample of Italian regions, and fails to converge for the Netherlands 14 .

Nethertheless, whatever the model, most parameters, and above all those defining the price index, converge toward consistent values: distance and unemployment rates remaining equal, migrants will choose in priority to move toward the region which has the most important market potential.

Moreover, the most encouraging result lies in the global consistency of the results from one country to another. Indeed, it is per se a source of satisfaction to observe roughly the same behaviors in countries which could highly differ from one another, in size, wealth, labor market structure and geographic distribution of economic activities.

Migration costs, and migration patterns

The two determinants of migration costs (λ and b) are strictly positive in all countries; migration flows decrease significantly with inter-regional distances.

The coefficient on distance, λ, varies between 0 and 1, strengthening the hypothesis of concavity of the migration cost function 15 . It is relatively small in Great Britain and Italy, suggesting an important degree of worker mobility in theses countries, compared to Dutch, or German ones.

But, one cannot interpret λ independently of the observed values of b. Indeed, Italy has also a very high b coefficient (around 4.5 against values comprise beetween 0.5 and 1.7 for other countries) which suggests that a large part of Italian migrations takes place among nearby regions.

Finally, "long-distance" migration cost, which can be represented as (1 + b) λ , appears to be much larger for Italy: it is the highest (1.9), just after Spain (2.15). This result, which mixes a high degree of mobility with a low tendency to migrate toward very distant regions reveals the particularity of the Italian migration pattern. One can justify the observed low long distance mobility degree, in saying that Spanish and Italian labor market are more controled than, say, British one, which doesn't incite people to migrate, and that Spain, and to a lesser extend Italy, have also very pronounced regional identities which are natural barriers to internal migrations. But the opposition beetwen a high degree of mobility and an important cost of long distance migration, can be explained more generally by the confrontation of two kinds of migration dynamics.

First, the observation of a small degree of mobility in Italy is consistent with a gravity-like pattern of migration: as with geographical proximity, economic proximity favors inter-regional exchanges, including migrations; the more homogenous the economy, the more important the migration flows should be, because labor markets are more unified, and it is, in consequence, easier to find a job in another region without changing activity [START_REF] Shields | The emergence of migration theory and suggested new direction[END_REF], [START_REF] Faini | An Empirical Puzzle: Falling Migration and Growing Unemployment Differentials among Italian Regions[END_REF]). One will then observe a relatively small value of λ in highly polarized countries. In the same way, if the economic center, as the periphery, is split in different regions, we should observe a relatively important part of small distance migrations, most movements taking place within the core or within the periphery (ie high b). In opposite, agglomeration dynamics represented by equation ( 17) lead to expect more important migrants flows within very polarized countries than in more homogenous ones. As Italy and Spain present a more polarized geographic structure, it should be logical to observe relatively more long distance migrations resulting from centripetal workers motions who leave peripherical regions to reach economic centers (ie a small λ and a small b). For very polarized countries, the estimated model then has difficulties making the distinction beetween these two different migration flows: short distance flows resulting from gravity-like behaviors and from polarization dynamics within small geographic areas, and center-periphery long distance flows resulting from polarization at the national level.

The table [START_REF] Greenwood | Research on International Migration in the United States: a Survey[END_REF] shows that Italy has the second smallest migration rates (just after Spain). But, at the same time, its ratio of total migration rate by non-contiguous migration rate, which can be considered as a rough indicator of migrant's propensity to move toward distant regions 16 , is also the smallest (1.6 against 1.7 for Spain, 3.3 for Netherland, 3 for Germany and 2.5 for Great Britain). These stilized facts tend to prove that Italians, and to a lesser extend the Spanish, are globaly not very mobile, even if, in the same time, they don't hesitate to move far from their region of origin 17 . Then the model, fitted for Italy, express the important South-North migrations (the strictly center-periphery pattern) by a low value of λ. But the South-South gravity-like migrations, which can't result from a small mobility degree neither from the price index expressions, is catched as a bias in favor of short distance migrations (ie a high b). This intuition is confirmed by results obtained on Spain and Great Britain which are the two other polarized countries. Estimations of λ and b reveal similarities with those obtained for Italy; but as their geographic structure is less manichean, the bias in favor of short distance migration is much less important.

Dropping Southern Italian regions 18 as destination locations from the panel, the regressions for this country leads to more robust and consistent results (see table 5). Dropping South-South migrations and North-South migrations (which one can expect to square to non-economic motivations) we obtain values of λ of 0.82 and of b between 0.87 and 0.86. Italians' mobility degree is now less important, but much less biased in favour of short distances migrations.

Wages and unemployment

Coefficients on wage and unemployment are less persuasive than those determining market potential functions.

If the coefficient on nominal wages (α 3 ) is significantly positive for Italy and Great Britain, it is non significantly different from zero for Spain and highly negative for Germany. For the two first countries the coefficient is consistent with the model, as they lie between zero and one, even if they suggest not very consistent values of service share in consumption φ (0.7 for Italy and 0.1 for Great Britain). For the two other countries, it implies that migrants' decisions are not influenced by nominal wages, or tend to prefer low wages locations. This paradoxical result is however consistent with the correlation analysis presented in table [START_REF] Faini | An Empirical Puzzle: Falling Migration and Growing Unemployment Differentials among Italian Regions[END_REF] and proves the weakness of the theoretical framework in presence of labor market imperfections. The same kind of nonintuitive result appears with the coefficient on unemployment rate (α 2 ). which is positive in all countries but Germany. Such results are commonly observed in internal migration researches, as one doesn't introduce information about migrant's qualifications and personal wages nor regional labor market structures (Cf. [START_REF] Greenwood | Research on International Migration in the United States: a Survey[END_REF], [START_REF] Herzog | Migration as Spatial Job-Search: A Survey of Empirical Findings[END_REF] and [START_REF] Bordas | The Economics of Immigration[END_REF] for reviews and Decressin & Fatas [1995], Mac [START_REF] Kormick | Regional unemployment and labor mobility in the UK[END_REF] and [START_REF] Faini | An Empirical Puzzle: Falling Migration and Growing Unemployment Differentials among Italian Regions[END_REF] for a study of the weakness of wages and unemployment rate consequences on migration decisions within European countries).

Indeed, unemployment can be seen, as we have done, as a proxy of the job finding probability. But as employment is often higher in a large urbanized region than in a low-density rural one, it also could reflect the size and dynamism of the regional labor market. People don't leave low unemployment region for higher employment ones, but small and tight labor markets to larger ones. Moreover, it is a logical and established fact that people's degree of mobility increases with their own human capital stock; thus, average wages and unemployment rates are not the accurate data one should use to explain these highly skilled migrants' decisions.

Elasticities of substitution, transport cost and market potential.

The manufacture elasticities of substitution, σ x , are always strictly positive, and, as required by the theory, significantly superior to one for every country but Great Britain. Estimation of this parameter varies, for equation [START_REF] Herzog | Migration as Spatial Job-Search: A Survey of Empirical Findings[END_REF], between 1.6 (Spain) and 3.1 (Germany) (except for Italy) and, for equation [START_REF] Hirsman | The Strategy of Economic Development[END_REF], between 2.0 and 4.7 19 These values are significantly lower than those observed by [START_REF] Hanson | Market Potential, Increasing Returns, and Geographic Concentration[END_REF] and other related recent studies as Hummels [1997], [START_REF] Eaton | Technology, Geography, and Trade[END_REF] and [START_REF] Head | Armington Vs. Krugman: A Test of Two Trade Models[END_REF] 20 . Whereas it would be undue to hope a perfect concordance between studies which differs in many points, this difference with Hanson is quite surprising: we would expect opposite results because we 19 One can note that all estimated values of σ x verify the no black hole condition ( σx-1 σx > µ) which ensure the existence of a stable decentralized equilibrium. In other words, product differentiation doesn't seem to be so important that competition effect, as centrifugal force, could be less important than centripetal forces whatever the value of transport costs. 20 Hanson estimations of market potential lead to elasticities of substitution beetween 5 and 11. Hummels estimated a manufacture-specific elasticity of substitution whereas Hanson's one lies on whole regional production except for housing services. Aggregating together more goods, Hanson should observe a more important product differentiation, so relatively small elasticities of substitution.

However, in our study, the estimated level of σ x is directly dependent of the chosen value of µ; choosing a µ superior to 0.4 should have lead to an higher σ x . Indeed Hanson's estimation of µ is 0.9; the ratio µ σ-1 is then comprised between 0.1 and 0.19, which is comparable to our results. Above all, the values we observe are, to a large extent, consistent with those observed in more traditional international trade studies such as Crozet & Erkel-Rousse [1999], Erkel-Rousse & [START_REF] Mirza | Import Price-Elasticities: Reconsidering the Evidence[END_REF] or [START_REF] Grossman | Import Competition from Developed and Developing Countries[END_REF].

Estimated values of δ are also consistent from a model to an other and have always a strictly positive value. They are globally high (arround 1.8 on average) 21 . These costs also vary in a quite important range between countries: from more than 3 in Germany to 0.3 in Spain and around one for the two other countries, leading, for equation [START_REF] Hirsman | The Strategy of Economic Development[END_REF], to values of the coefficient on transport cost δ (1 -σ x ) varying from -11.9 (Germany) to -0.3 (Spain). A high δ (1 -σ x ) denotes that regional price index, resulting from inter-regional frictional trade, is relatively less dependent on the economic activity of the surrounded regions, or, in other words that regional market potentials (as perceived by migrants) are essentialy influenced by the local level of production, and finaly do not differ so much from one region to another. The low δ (1 -σ x ) observed for Spain can be explained by the high level of specialization of Spanish regions, which involves a larger relevance of the market potential function in migrations decisions.

As for services, coefficients α 1 converge toward significantly positive values comprised between 0.5 and 0.97. Recalling that α 1 = φ σy-1 , one can derive from α 1 an indirect estimation of service elasticity of substitution, plugging in credible value of the part of services in consumption. With φ = 0.4 22 , we get smaller values of elasticities of substitution than those specific to manufactures: 21 It is largely more important than the 0.2 avanced by [START_REF] Hummels | Toward a Geography of Trade Costs[END_REF], but any comparison will be hazardous: Hummels estimated the relation between distances and real trade costs, when we only mesure the influence of distant levels of employment on migration. Moreover the distance we observe are very short compared to Hummels' ones who study inter-continental trade; it is then logical to observe larger unit trade costs as we expect increasing returns in the freight industry. 22 Which leaves a part for agricultural goods of (1 -µ -φ) = 0.2 they vary between 1.4 for Italy to 1.8 for the Netherlands which is fortunately above one.

Conclusion

We provided estimations of workers' law of motion which ensure the dynamics of agglomeration in new economic geography models. These models suggest that agglomeration occurs because firms and workers' location choices are driven by a comparable determinant expressing the regional market potentential. Indeed, a region which enjoys a high market potential will attract firms ensuring them a large demand, and will draw migrants by a low price index. If competition effects and wage differentials are not too important, both will progressively settle in central regions, creating a self-reinforcing process of agglomeration.

From data on internal annual migration flows for five European countries, we showed that, although European wokers don't seem to be highly mobile, the amount of migrants received by a region increases with its industrial and service employment levels and its proximity to other large markets. This result confirms the global relevance of the models of agglomeration. Moreover, as we estimate directely the motion equation of the model, we obtain a calibration of the market potential function which can't be done from a simple gravity-like migration model. All parameters are consistent with the theory and barely comparable to those obtained by international trade studies. Whatever the country, migrants seem to share the same behavior, and estimated parameters are more or less comparable.

A Data appendix

• Germany (NUTS 1): Baden-Wurttenberg, Bayern, Bremen, Hamburg, Hessen, Niedersachsen, Nordrhein-Westfalen, Rheinland-Pfalz, Saarland, Schleswig-Holstein.

• Italy (NUTS 2): Piemonte, Valle d'Aoste, Liguria, Lombardia, Trentino-Alto Adige, Veneto, Friuli-Venezia Giulia, Emilia-Romagna, Toscana, Umbria, Marche, Lazio, Abruzzo, Molise, Campania, Puglia, Basilicata, Calabria.

• Spain (NUTS 2): Galicia, Asturias, Cantabria, Pais Vasco, Navarra, Rioja, Aragon, Madrid, Castilla-Leon, Castilla-La Mancha, Extremaduras, Catalunia, Comunidad Valenciana, Andalucia, Murcia.

• Netherland (NUTS 2):Groningen, Friesland, Drenthe, Overijssel, Gelderland, Flevoland, Utrech, Noord-Holland, Zuid-Holland, Zeeland.

• Great Britain (NUTS 1): North, Yorkshire and Humberside, East Midlands, East Anglia, South East, South West, West Midlands, North West, Wales, Scotland.

B Regressions results

  [1997], and[START_REF] Head | Armington Vs. Krugman: A Test of Two Trade Models[END_REF] both study international trade flows taking in account barriers to trade; Hummels observes a trade price elasticity value around 7.6, Eaton & Kortum around 8.3 and Head & Ries around 6.4.

Table 2 :

 2 Standards errors in parenthesis; # , ## = superiority to 1 at 5% and 10% significance levels.

		Germany -µ = 0.4 -White consistent non linear least squares
	σ x	δ	α 1	α 2	α 3	λ	b	α 4
	Equation 1
	3.081 #	5.967 *	1.097 *	-0.845 * -2.652 * 0.913 *	0.897 *	0.733 *
	(0.4160)	(0.5521) (0.042) (0.0989) (0.343) (0.0417) (0.1562) (0.0892)
		nb. obs. = 900 -R 2 = 0.8704 -MSE = 0.2147
	Equation 2
	4.693 #	3.228 *	0.785 *	-0.225 *	-	0.886 *	0.954 *	0.586 *
	(1.2415)	(0.5393) (0.0313) (0.0774)	-	(0.0412) (0.1667) (0.0797)
		nb. obs = 900 -R 2 = 0.8725 -MSE = 0.2110

* , * * , * * * = significance at 1, 5 and 10% levels.

Table 4 :

 4 Italy -µ = 0.4 -White consistent non linear least squares

	σ x	δ	α 1	α 2	α 3	λ	b	α 4
				Equation 1
	9.467 #	1.619 *	0.871 *	0.052	0.304 * * 0.262 *	17.540 *	0.2725 *
	(4.296) (0.1908) (0.0248) (0.0327) (0.119) (0.0231) (6.6155)	(0.0358)
			nb. obs. = 3366 -R 2 = 0.8135 -MSE = 0.30302
				Equation 2
	3.314 #	3.868 *	0.964 *	-0.030	-	0.383 *	4.457 *	0.031
	(0.3194) (0.1635) (0.0190) (0.0317)	-	(0.0246) (1.0168)	(0.0349)
			nb. obs = 3366 -R 2 = 0.8245 -MSE = 0.2507

Table 5 :

 5 Italy -Southtern regions dropped -µ = 0.4 -White consistent non linear least squares Standards errors in parenthesis; # , ## = superiority to 1 at 5% and 10% significance levels.

	σ x	δ	α 1	α 2	α 3	λ	b	α 4
				Equation 1
	2.940 #	2.499 *	0.914 *	0.003	0.322 * * 0.821 *	0.875 *	0.027
	(0.4136) (0.255) (0.025) (0.0445) (0.162) (0.0368) (0.1280)	(0.0376)
			nb. obs. = 2057 -R 2 = 0.8601 -MSE = 0.2363
				Equation 2
	3.027 #	2.455 *	0.921 *	-0.018	-	0.828 *	0.861 *	0.022
	(0.4603) (0.2303) (0.022) (0.0417)	-	(0.0369) (0.1256)	(0.0357)
			nb. obs = 2057 -R 2 = 0.8595 -MSE = 0.2317

* , * * , * * * = significance at 1, 5 and 10% levels.

Table 6 :

 6 Netherland -µ = 0.4 -White consistent non linear least squares Standards errors in parenthesis; # , ## = superiority to 1 at 5% and 10% significance levels.

	σ x	δ	α 1	α 2	α 3	λ	b	α 4
				Equation 1
					-
				Equation 2
	4.519 #	0.962 *	0.500 *	0.3451 *	-	1.037 *	0.490 *	0.347 *
	(1.4725) (0.2620) (0.0231) (0.0845) -(0.0420) (0.0737)	(0.0626)
			nb. obs = 630 -R 2 = 0.9077 -MSE = 0.0932

* , * * , * * * = significance at 1, 5 and 10% levels.

Table 7 :

 7 Great Britain -µ = 0.4 -White consistent non linear least squares Standards errors in parenthesis; # , ## = superiority to 1 at 5% and 10% significance levels.

	σ x	δ	α 1	α 2	α 3	λ	b	α 4
					Equation 1
	2.286 * *	1.186 *	0.570 *	-	0.9028 * * 0.454 *	1.489 *	1.011 *
	(1.1681) (0.2709) (0.0352)	-	(0.3928) (0.035) (0.340)	(0.1017)
			nb. obs. = 540 -R 2 = 0.8883 -MSE = 0.0677
					Equation 2
	2.535 * *	1.009 *	0.562 *	-	-	0.448 *	1.559 *	1.098
	(1.3081) (0.2265) (0.0349)	-	-	(0.0353) (0.3603)	(0.0948)
			nb. obs = 540 -R 2 = 0.8869 -MSE = 0.0683

* , * * , * * * = significance at 1, 5 and 10% levels.

Table 8 :

 8 Correlation table with the migration share variable

		DE	ES	IT	NL	GB
	Non-agri. employment 0.7069	0.6823	0.8235	0.6193	0.7074
		(0.0001) (0.0001) (0.0001) (0.0001) (0.0001)
	Unemployment rate	-0.4055 0.2245	0.0896 -0.0667	-
		(0.0001) (0.0001) (0.0001) (0.0428)	-
	Non-agricultural wage -0.0584 0.0507	0.1293	0.0169	0.1231
		(0.0798) (0.0148) (0.0001) (0.6075) (0.0432)
	Distance	-0.4345 -0.3958 -0.3427 -0.7303 -0.4795
		(0.0001) (0.0001) (0.0001) (0.0001) (0.0001)
	Density	-0.4213 0.3021	0.7436	0.5559	0.5593

Table 9 :

 9 Inter-regional migration rates Period Mean Std Dev Growth rate

	Italy	total	82/95 0.24% 0.0002	-23.94%
		long dist. 82/95 0.15% 0.0001	-23.76%
	Germany	total	75/93 0.54% 0.0006	-29.37%
		long dist. 75/93 0.18% 0.0002	-28.28%
	Spain	total	80/94 0.19% 0.0006	50.01%
		long dist. 80/94 0.11% 0.0003	32.75%
	Netherland total	75/95 1.17% 0.0037	15.9%
		long dist. 75/95 0.35% 0.0017	74.80%
	GB	total	80/86 1.50% 0.0009	11.4%
		long dist. 80/86 0.61% 0.0003	8.25%

Long distance migrations are motions toward a non-contiguous region

Venable [1994Venable [ , 1996] ] andKrugman & Venables [1995] showed that industrial agglomeration can occur without migration. The final demand linkage which is the main centripetal force inKrugman [1991[START_REF] Krugman | A Dynamic Spatial Model[END_REF] is replaced by intermediate demand linkage among vertically linked industries. 3 Cahiers de la MSE -2000.30

Krugman [1980]

In Hanson's framework housing services are the only nontraded goods; he underlined himself this point as being a weakness.

Cahiers de la MSE -2000.[START_REF] Kormick | Regional unemployment and labor mobility in the UK[END_REF]

Even if the industrial concentration leads to higher wages and prices in these locations.

Moving house expenses, and other sunk costs of migration should appear as negative additional terms of the real wage equation. As potential migrants have an infinite horizon, such costs can be neglected with no loss of generality.9Cahiers de la MSE -2000.[START_REF] Kormick | Regional unemployment and labor mobility in the UK[END_REF]

See Mueser [1997] for a modelisation of this issue.

Year, departure region (j) and destination region (i).

The couple λ -b is also tied, but, as b is a parameter affect to an dummy, collinearity test we performed showed it is much less damaging than the one observed within the two other groups.

In order to test for robustness of the estimations, we also drove regressions with some other values of µ. The model stays the same and parameters evolve to comparable values: a higher µ simply leads to a higher σ x and, since we haven't removed any parameter restrictions, to a smaller δ. Other parameters remain almost unchanged.

10 departure regions * 9 destination regions * 10 years

That is why we don't present this result here, even if the estimated coefficients are reliable to those observed for other countries and for equation[START_REF] Hirsman | The Strategy of Economic Development[END_REF].

The coefficient for the Netherlands is the only one superior to 1 (around 1.04), which can be explained by the very small size of this country.

It is indeed a rough indicator because it depends of the mean size of the regions and the geographical division of the country (as Italy is a long and narrow country, one can expect its regions to have, everything else being equal, less neighbors than Spanish or German ones).

[START_REF] Faini | An Empirical Puzzle: Falling Migration and Growing Unemployment Differentials among Italian Regions[END_REF] showed how much southern Italians' mobility has reduced during the last twenty years whereas regional divergence increased, even if they still keep a relatively important willingness to move.

Lazio, Abruzzo-Molise, Campania and Sud

(0.0001) (0.0001 (0.0001) (0.0001) (0.0001)

Apart from inter-regional distances, we must find a proxy of internal distances d ii . Indeed the market potential term includes, as for goods consumed abroad, the transport cost charged on manufactures produced and consumed in the same region. The theoretical framework modelises inter-regional space, but considers implicitly each region as a point driving internal iceberg trade cost to unity. [START_REF] Wei | Intra-National Versus International Trade: How Stubborn Are Nations in Global Integration[END_REF] showed how much matters the correction of this simplification in empirical studies of flows between different geographic areas. However, we can't use the same kind of data as the one presented above without introducing an important bias due to the choice of a second regional center. Therefore we adopt [START_REF] Head | Non-Europe : The Magnitude and Causes of Market Fragmentation in the EU[END_REF] grounded specification of distance from oneself: approximating a region to a disk in which consumers are uniformly distributed and production is concentrated in the center, the average distance between a producer and a consumer is two thirds of the radius. Thus, if S i denotes the regional area, internal distance becomes

π . This specification is not as precise than the one used for inter-regional distances and can be affected by two different biases. First, most consumers and producers are, in reality, located in or around the same urban centers and so they are closer from each other than expected in the disk simplification; the real internal distance should be shorter than the one we'll use. But on the other hand, it measures "as the crow flies" distances, necessarily shorter than real ones. The two biases play opposite roles, and, all things considered, we can expect this distance to be barely consistent with inter-regional ones.

Results

Estimations results for the five countries are reported in tables (2) to [START_REF] Faini | Increasing Returns, Migrations and Convergence[END_REF]. By measure of commodity, we do not present the coefficients on fixed effects a j ; in accordance with the predictions of the model, they are significantly negative in all regressions and superior to one in absolute value. Table [START_REF] Faini | An Empirical Puzzle: Falling Migration and Growing Unemployment Differentials among Italian Regions[END_REF] presents correlations between the dependent variable (the migrations share) and the main variables of the model. Table [START_REF] Greenwood | Research on International Migration in the United States: a Survey[END_REF] gives some basic statistics on internal migrations patterns.

As explained above, market labor structure in European countries has lead to a relatively important internal equalization of nominal wages. Wages are not good determinants of migration


Matthieu Diaz
email: matthieu.diaz@ens-paris-saclay.fr

Pierre-Étienne Charbonnel

Ludovic Chamoin

P.-É Charbonnel

Towards an automated physics-regularized model updating strategy applied to the monitoring of civil engineering structures in low-frequency dynamics

Keywords: Data assimilation, Kalman filtering, modified Constitutive Relation Error, Low-frequency dynamics, Shaking-table experiments, Earthquake Engineering

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

EACS 2022 Towards an automated physics-regularized model updating strategy applied to the monitoring of civil engineering structures in low-frequency dynamics

As most of modern systems are now equipped with numerous sensing devices, the integration of experimental data in model-based approaches provides new techniques for monitoring the occurrence and propagation of structural damage in a structural health monitoring context. In particular, the dialogue between physics-based models and experimental data has become of major importance in the last two decades with the emergence of new paradigms such as Digital Twins [START_REF] Wagg | Digital Twins: State-of-the-Art and Future Directions for Modeling and Simulation in Engineering Dynamics Applications[END_REF] or Dynamic Data-Driven Application Systems (DDDAS) [START_REF] Chamoin | Merging advanced sensing techniques and simulation tools for future structural health monitoring technologies[END_REF][START_REF] Darema | Dynamic Data Driven Applications Systems: A New Paradigm for Application Simulations and Measurements[END_REF]. The latter consists in establishing a close online interaction between models and measurements with a dual objective: (i) exploiting model predictions for decision-making operations on the experimental system such as model-based control and, (ii) updating the numerical model using data assimilated in real-time.

For earthquake engineering applications, the CEA/TAMARIS facility carries out seismic tests using shakingtables moved by high-power hydraulic actuators. However, controlling the hydraulic actuators of the shakingtables is still a challenging task that emphasizes the need of establishing a DDDAS framework. Indeed, the modal signature of damaging specimens can suddenly change under seismic loading conditions [START_REF] Charbonnel | Fuzzy-driven strategy for fully automated modal analysis: Application to the SMART2013 shaking-table test campaign[END_REF]. As modal signature is the key input-feature for linear control law design, the initial control strategy may become inadequate when damage occurs, leading to unstable experiments.

In this context, this work aims at contributing to the integration of Finite Element (FE) models in control loop of shaking-tables by resorting to a dynamical data-to-model interaction. In this talk, we will essentially focus on the identification/update loop of FIG. 1 from sparse and noisy measurements acquired in low-frequency dynamics. A new numerical framework is presented for data assimilation able to operate in real-time, robust to measurement noise as well as easily handleable by non-expert users. Considering the model updating problem itself, low signal-to-noise ratio measurements from seismic inputs may be hard to integrate for correcting FE models properly. Alternatively to standard deterministic or stochastic model updating techniques, a strategy based on the minimization of a modified Constitutive Relation Error functional (mCRE) has been recently tailored to low-frequency dynamics in [START_REF] Diaz | Robust energy-based model updating framework for random processes in dynamics: application to shaking-table experiments[END_REF]. Built as a Hermitian data-to-model distance written in the frequency domain enriched with a CRE residual accounting for model bias, the mCRE functional has shown extended robustness to measurement noise and the ability to process strongly nonlinear datasets (due to occurence of damage) with reasonably low computational effort. Besides, the mCREbased model updating algorithm has been made independent of any expert user's a priori knowledge: automated procedures permit to define all mCRE internal parameters and the model parametrization itself is adaptively tuned to identify damage at best according to the available measurements.

In order to adapt this model updating algorithm to online structural monitoring applications, following the strategy initiated in [START_REF] Marchand | Real-time updating of structural mechanics models using kalman filtering, modified constitutive relation error, and proper generalized[END_REF], we integrate the mCRE as new observation operator within a dual Unscented Kalman filter, leading to a new Modified Dual Kalman Filter algorithm (MDKF). It differs from classical nonlinear Kalman filtering techniques for parameter estimation as the metric of the measurement update equation allows to confront indirectly model predictions with measurements, meaning that optimal parameters are sought as mCRE minimizers. Although this new framework seems more complex to operate, general guidelines based on physical meaning and engineering judgement are provided regarding the calibration of internal tuning parameters.

In particular, noise covariance error matrices can be quickly and efficiently chosen.

After being validated using synthetic data, the methodology successfully processed actual measurements of the SMART2013 shaking-table test campaign in a real-time context. In the latter, the possibility to perform real-time monitoring of complex structures undergoing nonlinear phenomena is emphasized (see FIG. 2). The good correlation with former identification results [START_REF] Charbonnel | Fuzzy-driven strategy for fully automated modal analysis: Application to the SMART2013 shaking-table test campaign[END_REF][START_REF] Diaz | Robust energy-based model updating framework for random processes in dynamics: application to shaking-table experiments[END_REF] illustrates the relevance of this new approach for online structural monitoring and suggests promising use of MDKF for adaptive control design purposes.

Figure 2: SMART2013 -Tracking the first eigenfrequency of the specimen using MDKF. Former reference identification results are given for comparison purposes: [START_REF] Diaz | Robust energy-based model updating framework for random processes in dynamics: application to shaking-table experiments[END_REF] in blue dots, [START_REF] Charbonnel | Fuzzy-driven strategy for fully automated modal analysis: Application to the SMART2013 shaking-table test campaign[END_REF] in red dots. An accelerometer measurement is also shown to relate inputs with structural state evolution.

Figure 1 :

 1 Figure 1: DDDAS framework for the control of shaking-tables -Introduction of a numerical model in the feedback loop for adaptative control law design.

Acknowledgments EDF and CEA are deepfully acknowledged for giving access to the SMART2013 database.