José De Sousa
email: jdesousa@univ-paris1.fr.

Laetitia Duval

Michael Landesman

Olivier Lamotte

Mouhoub El

Jean-François Mouhoud

Hillel Nivet

Claudia Rapoport

Fabien Senik

François- Charles Tripier

Wolff

Geographic distance and remittances in Romania: out of sight, out of mind? 1

Keywords: International migration, remittances, bilateral data, Romania. JEL classification codes: F24, J61, O15

We analyse the role of geographic distance for bilateral remittances. We use a new data set on bilateral remittance flows from OECD countries to Romania over the period 2005-2009. Contrasting with existing literature, we find that remittances increase with distance but in a non-linear way.

Introduction

In this paper, we use a new data set on bilateral remittance flows to analyse the impact of geographic distance on remittances. It is now recognized that migration decisions are made in a family context [START_REF] Borjas | The economic analysis of immigration[END_REF], and that remittances are a central element of familial arrangements [START_REF] Rapoport | The economics of migrants' remittances[END_REF]. In this context, the migrant's distance from its family (stayed at home) may have an impact on remittances. This impact has not been much explored in the literature. 2 It appears to be ambiguous. Distance can cause remittances to rise or fall. Remittances might decrease with distance in the three following cases: (1) if remittances contain an altruistic component and "if one admits that altruism is solvable in distance" [START_REF] Rapoport | The economics of migrants' remittances[END_REF]; (2) if remote migration increases strategic behaviours. Increasing distance from family may reduce the enforcement of any familial arrangement agreed before migration. Among others, distance renders migrant's resources imperfectly observable [START_REF] Rapoport | The economics of migrants' remittances[END_REF]. Finally (3), remittances may decrease with distance if the latter is assumed to be a proxy for transfer costs [START_REF] Lueth | Determinants of bilateral remittance flows[END_REF][START_REF] Frankel | Are bilateral remittances countercyclical?[END_REF]. 3 Two arguments can be put forward to understand why remittances might increase with distance. The first argument is related to the type of data we observe. We mainly observe official remittances, transiting through formal channels, such as banks or money transfer companies. Thus, remittances may increase with geographic distance if long-distance migrants tend to remit in a more formal way than short-distance migrants. This is all the more expected if the former return home less frequently. They have fewer opportunities to make transfers in-kind or to carry themselves the money. The second argument is related to the 2 See [START_REF] Adams | The Determinants of international remittances in developing countries[END_REF] and [START_REF] Freund | Remittances, transaction costs, and informality[END_REF] for recent contributions on the determinants of aggregate remittances in developing countries. 3 Later on, we will challenge this assumption of a transfer cost effect. direct costs of emigration (e.g. transporting persons and household goods or obtaining visa) 4 and the fact that individuals in developing countries face bidding liquidity constraints to emigrate. The literature documents that informal familial arrangements may occur to alleviate such constraints. The family provides implicit loans to finance the costs of emigration. Then, migrants send back money home partly to reimburse costs of emigration (see [START_REF] Johnson | Urban-rural income transfers in Kenya: An estimated-remittances function[END_REF][START_REF] Lucas | Motivations to remit: evidence from Botswana[END_REF][START_REF] Poirine | A theory of remittances as an implicit family loan arrangement[END_REF][START_REF] Stark | Migration, remittances, and the family[END_REF][START_REF] Ilahi | Guestworker migration, remittances and the extended family: evidence from Pakistan[END_REF]. As long as the costs of emigration increase with remote destinations [START_REF] Mayda | International migration: A panel data analysis of the determinants of bilateral flows[END_REF], 5 remittances are expected to increase with the distance to the sending country.

The sign of the impact of distance on remittances is thus an empirical question. To answer this question, we use a new data set of the National Bank of Romania. This data set breaks down inflows of Romanian remittances by sending country. This bilateral dimension allows us to focus on the role of the geographic distance between Romania and the sending countries.

Romania is, for various reasons, a relevant receiving country. First, it is a recent country of massive emigration. Second, Romania is currently in the top-10 receiving countries of remittances in the world. Finally, Romania joined the European Union (EU) on January 1, 2007, but is still considered as a middle-income country. Poverty persists and acts as a push factor of migration (see World Bank, 2003).

This paper makes several contributions to the literature. First, we explore the role of geographic distance and find that long-distance migrants tend to remit more than shortdistance migrants. But, this effect tends to be at diminishing rates and specific to some groups of countries. Second, we confirm the significant effect of economic size, as well as financial and labour market factors. Third, using the time series dimension of data, from 2005 to 2009, we document a negative impact of the recent financial crises on remittances to Romania. 4 Borjas (1999: 1711) also points out two indirect costs: "forgone earnings (for example, the opportunity cost of a post-migration unemployment spell), and psychic costs (for example, the disutility associated with leaving behind family ties and social networks)". 5 Mayda (2009) shows cogently that, among the variables affecting the costs of emigration, geographic distance appears to be the most important one.

The rest of the paper is as follows. In the next section, we briefly review some stylised facts about the Romanian international migration and remittances. In section 3, we describe our bilateral data set and discuss very recent contributions in relation to the type of data we use. In section 4, we design our empirical model. In section 5, we expose the results. Finally, we conclude in section 6.

Facts and issues about migration and remittances

In this section, we review some stylised facts about the Romanian migration and remittances.

First, the Romanian transition, at the beginning of the 1990s, triggered a new and large emigration. In 2007, the stock of legal Romanian emigrants reached 1.2 million and 5.7% of population [START_REF] Ratha | Migration and remittances factbook[END_REF]. 6 This emigration follows some specific stages. Second, Romania is currently in the top-10 receiving countries of remittances in the world.

A massive emigration

Since the beginning of transition, Romanian international emigration could be subdivided into three stages (Diminescu, 2003). The first stage (1989)(1990)(1991)(1992)(1993)(1994) follows the opening of borders.

Romanian migrants headed mainly to short-distance country such as Hungary or Serbia. The second stage (1995)(1996)(1997)(1998)(1999)(2000)(2001) extends migration to Western European countries (e.g. Germany, Austria or France), and Mediterranean countries (e.g. Greece, Italy or Spain). This is a mean to face the deterioration of economic and social consequences of transition. For illustration, the unemployment rate rose from 3.0% in 1991 to over 10% from 1993 onwards. Moreover, 28.9% of the Romanian population is living below the national poverty line in 2002 and 10.2% live in severe poverty (defined as those with insufficient means to purchase a minimum caloric intake each day) (World Bank, 2003: 18). The third stage corresponds to the normalization of Romanian emigration in Europe. From 2002 to 2007, Romanians were 6 Evaluating illegal migration is an issue but stock data are still more reliable than flows. "There is a high turnover among illegal migrants, and many of them tend to be regularised after some time" (Docquier and Rapoport (2009: 4). allowed to stay without visa in the Schengen area for a maximum of 90 days 7 . After the EU accession in 2007, Romania enjoys the free movement of persons, but still not the free movement of workers.8

A top-10 remittance receiving country

Remittances to Romania increased from US $96 million in 2000 to 9.3 billion in 2008. For comparison, they represent 5.5% of GDP and about 60% of foreign direct investment inflows.

Romania is currently in the top-10 receiving countries of remittances in the world (see Figure 1).

Bilateral data on remittances

7 The Schengen agreement has been signed in 1995 between Belgium, France, Germany, Luxembourg and the Netherlands. It removed border controls between the participating countries. 8 All EU-27 members are required to implement Schengen, except Bulgaria, Cyprus, Ireland, the United Kingdom and Romania. Removal of the restrictions on the movement of Romanian workers into the EU is expected from 2012.

Data on bilateral remittances come from the National Bank of Romania. They are collected via (1) banks reports for amounts received in banks accounts, (2) reports of the money transfer companies such as Western Union and Money Gram and (3) reports of the National Post Office for amounts sent via postal orders. 9 We identify recorded flows to Romania from 18 sending countries: Austria, Belgium, Canada, Cyprus, Denmark, France, Germany, Greece, Ireland, Israel, Italy, the Netherlands, Portugal, Spain, Switzerland, Turkey, the United Kingdom and the United States. Table 1 reports the main remittance-sending countries in our sample. We also provide their bilateral geographic distance with Romania. The average distance in our sample is about 2450 kms. The most remote countries are the United States (7986 kms) and Canada (7422 kms), while Greece (741 kms) and Turkey (746 kms) are the closest. Data constraints are relatively strong in the literature. Almost all papers do not identify the the remittance-sending country. The first studies using bilateral data work with a tiny number of observations [START_REF] Lianos | Factors determining migrant remittances: the case of Greece[END_REF][START_REF] Straubhaar | The determinants of workers' remittances: The case of Turkey[END_REF]. The authors find evidence for altruism on the belief that bilateral remittances increase with the difference between sending and receiving countries' GDPs. We may wonder however whether such a difference is a good indicator to capture altruism motives (see [START_REF] Rapoport | The economics of migrants' remittances[END_REF]. 10 Lianos (1997) works on Greek inflows of remittances: 31 observations from Germany (1961( -1991( ), 11 from Belgium (1981( -1991) ) and 12 from Sweden (1980)(1981)(1982)(1983)(1984)(1985)(1986)(1987)(1988)(1989)(1990)(1991). [START_REF] Straubhaar | The determinants of workers' remittances: The case of Turkey[END_REF] uses a time series of 19 remittance flows from Germany to Turkey (1963Turkey ( -1982)).

Related to international organisms, such as the IMF [START_REF] Lueth | Determinants of bilateral remittance flows[END_REF]

A specification of bilateral remittances

Having introduced our data set on bilateral aggregate remittances, we now investigate a broader issue: the determinants of bilateral remittance flows. According to the theoretical and empirical literature, the main determinants of aggregate remittances are related to (a) economic size, (b) financial environment and (c) labour market (see Appendix 1 for a short review of the literature). Our basic regression accounts for these factors. Recall that in our sample the receiving country is always Romania. Thus, we discard receiving country-variable controls. These variables only have time-series variation, captured by allowing for time specific effects in remittances:

ln(Remittances) it = β 0 + β 1 ln(Distance) i + β 2 ln(GDP) it + β 3 ln(ExChange) it + β 4 ln(Unemploy) it + RegDum i + YearDum t + ε it , (1)
where i and t indicate the remittance-sending country and time, respectively. The dependant variable (Remittances) it is the value of bilateral remittance flows from the sending country i to Romania from the first quarter of 2005 to the third quarter of 2009. The explanatory variables are defined as follows: (Distance) i is the bilateral distance between i and Romania; (GDP) i is the quarterly Gross Domestic Product of country i; (ExChange) i denotes the quarterly nominal exchange rate of country i facing the Romanian Lei. (Unemploy) i is the quarterly unemployment rate of country i. (RegDum) i and (YearDum) t are regional and year dummies respectively.12 ε it represents the usual error term capturing unobserved factors and mismeasurements of the remittance level. We use logs on both sides of the equation to reduce the potential skewness of the distribution and to interpret the estimates in elasticity terms. The coefficient of interest to us is β 1 , the elasticity of remittances to distance.

Details about the definition and source of regressors are provided in Appendix 2. In Appendix 3, we provide some summary statistics for the variables. We first comment on the cross section estimates. In columns (1) to (3), the adjusted Rsquared shows that our regressors explain about 60% of the variation in log of bilateral remittances. We first estimate equation (1), without the distance variable (col. 1). As expected, the economic size variable exhibits a positive effect on remittances. Holding other factors constant, a 1% increase in sending country GDP increases remittances by about 0.7% on average. In addition, we find a significant negative impact of the bilateral exchange rate. It suggests that an appreciation of the sending's currency vis-à-vis the Lei yields a substitution effect: migrants remit less, while keeping unchanged the purchasing power of the amount transferred. 13 Moreover, we find a negative estimate of the unemployment rate. This is expected since an unemployment rise increases macroeconomic instability, causes significant loss of income and reduces the migrant's probability to be employed. The results on the regional dummies are worth mentioning. They establish a clear and statistically significant ranking: both European regions remit more that North American countries (the base group).

Estimates of bilateral determinants of remittances

But, within Europe, region 3 (i.e. Cyprus, Greece, Ireland, Israel, Italy, Portugal, Spain or Turkey) tends to remit more than region 2 (i.e. Austria, Belgium, Denmark, France, Germany, the Netherlands, Switzerland or the United Kingdom). 14 Estimated coefficients of year dummies also reveal some interesting patterns. Compared to 2005 (the base year), we observe a significant increase in remittances in 2006 and 2007, no difference in 2008 and a decrease in 2009. This slowdown could be the result of the actual global financial crisis, a phenomenon acknowledged by [START_REF] Ratha | Migration and remittances factbook[END_REF] in developing countries.

In column (2), we investigate the impact of distance on remittances and estimate equation ( 1). We find a statistically and economically significant positive effect of distance on remittances. Ceteris paribus, a 1% increase in distance to Romania leads to a 0.5% increase in bilateral remittances on average. This result conflicts with the negative elasticity found in [START_REF] Lueth | Determinants of bilateral remittance flows[END_REF] and [START_REF] Frankel | Are bilateral remittances countercyclical?[END_REF]. They interpret their result as a transfer cost effect. However, this interpretation is puzzling. The cost of transferring money appears unrelated to geographic distance. As an illustration, consider a US immigrant who wants to remit US $200 from the USA abroad. This transfer will cost US $17 to Colombia for a capital-to-capital distance of 3845 kms; US $3 to Mexico, for a roughly similar distance (3038 kms), and US $4 to Philippines for a much larger distance (13,794 kms). [START_REF] Ratha | South-south migration and remittances[END_REF] also find evidence of asymmetric bilateral transfer costs: the cost of remitting US $200 from a 13 See Appendix 1 for a discussion of the potential income effect of the bilateral exchange rate. 14 The Wald statistic of the difference between the European regions 2 and 3 is 117.29, with a p-value lower than 0.01. developed country to a developing country is significantly much lower than in the opposite direction. 15We document instead a positive distance elasticity of remittances but this result is not strictly comparable to [START_REF] Lueth | Determinants of bilateral remittance flows[END_REF] and [START_REF] Frankel | Are bilateral remittances countercyclical?[END_REF]. The time span and the country coverage of their sample are quite different from ours (see above). This difference may drive the reverse result. Beyond this, how can we interpret our positive distance effect? One possible interpretation is related to the loan repayment hypothesis which has been documented so far using household surveys. The costs of emigration increase with distance [START_REF] Mayda | International migration: A panel data analysis of the determinants of bilateral flows[END_REF]. Thus, a larger distance might imply a larger family loan to finance the move and then larger remittances to repay back the loan. A second interpretation is related to the type of data we observe. As previously stated, we mainly observe official remittances. Thus, if short-distance migrants return home more frequently, they have more opportunities to remit through informal channels, i.e. to make transfers in-kind or to carry themselves the money. As a result, this may lead to a sample selection bias: remittances increase with geographic distance as long as remote migrants remit in a more formal way than close ones.

Before investigating this issue, it is worth mentioning that the difference between the European regions and North America increases by controlling for the effect of distance. This might be explained by the relative remoteness of North American countries compared to Europe for Romanian emigrants. In order to see if the distance effect is driven by the most remote destinations, we drop the USA and Canada from the sample. However, as shown in column (3), the distance elasticity remains positive, highly significant (p<0.01) and with the same magnitude than in column (2).

In columns (4) to (6), we use panel data estimators. They allow us to exploit the time series dimension of the data and to control for sending country specific effects (beyond the regional characteristics). This is all the more important since the distance effect might be driven by country characteristics that we cannot observe, such as the opportunity to remit through informal channels. This opportunity may be different from one sending-country to another: e.g. travelling to Romania to carry back the money is more or less expensive and easy across countries. Travel costs depend on distance but also on travel time and transportation infrastructures. Omission to control for these factors may lead to the above mentioned sample selection bias.

The properties of the panel data estimators may depend significantly on whether the sending country effects are specified as random variables or as fixed parameters. As a first approximation, we may assume that the sending country specific effects are uncorrelated with the explanatory variables. The appropriate specification is thus a random effects (RE) model, estimated with the generalized least squares. The estimates are reported in column (4) of Table 2. The distance elasticity remains positive and significant, while its magnitude is larger than using cross section estimates (col. 2). However, the possibility that the sending country effects are correlated with some explanatory variables is so prevalent that it could make little sense to rule out such correlation a priori. In this case, the estimators are generally inconsistent. In contrast, the fixed effects estimator (FE) allows for arbitrary correlation between the sending country effects and the explanatory variables in any time period. The FE estimates are depicted in column (5) of Table 2.

In order to assess the appropriateness of fixed effects and random effects estimation, we implement a Hausman test. The Hausman test, presented at the bottom of column (5), compares the estimable coefficients of the time-varying regressors. The test statistic is quite low which suggests that the country sending effects are uncorrelated with the other variables in the model. Thus, between the two panel data estimators considered, the RE model appears to be more appropriate. Moreover, a caveat of the FE estimator is its inability to estimate the coefficient of our variable of interest: the geographic distance, which is time-invariant. To alleviate this problem we consider a particular random effects model: the [START_REF] Mundlak | On the pooling of time series and cross section data[END_REF] approach to estimating panel data. The Mundlak's aproach posits that the sending country specific effect can be projected upon the group means of the time-varying variables. As a consequence, adding the means of the time-varying variables (i.e. GDP, exchange rate and uemployment) to the equation picks up the correlation between the country effects and the explanatory variables (see [START_REF] Wooldridge | Econometric Analysis of Cross Section and Panel Data[END_REF]. The results are reported in column (6) of Table 2. As expected, the Mundlak RE model gives fairly similar estimates of the time varying variables compared to FE (col. 5). An attractive feature of this model is that it gives an estimate of time-invariant variables. Thus, it confirms the positive distance elasticity of remittances.

In summary, the cross section and panel estimates reveal a positive and significant effect of distance. This effect does not seem to be driven by sending country specific characteristics.

We find instead that the panel estimates (col. 4 and 6) of distance are larger in magnitude than the cross section estimates (col. 2). However, due to the magnitude of the standard errors, there is not much statistically significant difference between the cross and the panel estimates of distance.

In Table 3, we account for potential non linear effects of distance on remittances.

Estimation in column (1) replaces distance in logarithmic form with distance as a natural number and the square of distance. Estimates of both variables are statistically significant. This estimated equation implies that distance has a diminishing effect on remittances. This result holds when dropping North American countries from the sample (col. 2). In column (3), we allow for different slopes of distance. Thus we recast our model. We add two interactions between distance (in level) and both European regions and keep distance and the regional dummies separately. How do we interpret the coefficients on the distance variables? The estimated distance semi-elasticity for North American countries is negative -1.172 (p<0.01).

For region 2, the distance elasticity is positive (-1.172 + 1.173 = 0.001), as for region 3 (-1.172 + 1.190 = 0.018). Differences of European regions with North America are not economically large but statistically significant and positive. These results confirm that the distance effect is highly non-linear: the positive effect of distance on remittances appears to be driven by the regions 2 and 3. Estimates of the intercept and the other covariates are not reported. For presentation purposes the distance variable has been rescaled to units of 100 kms. In column (2), we have dropped the North American countries from the sample.

Conclusion

Recent literature addresses a relevant policy question: what causes different developing countries to receive different levels of remittances? Using a new data set on bilateral remittance flows we further explore this question. We wonder whether bilateral geographic distance to the remittance-sending country matters. Over the period 2005-2009, we find that long-distance migrants tend to remit more than short-distance migrants. This result holds using cross section and panel data estimators. But, this effect tends to be at diminishing rates and specific to some groups of countries. Moreover, we confirm the significant effect of economic size, as well as financial and labour market factors. Finally, using the time series dimension of data, from 2005 to 2009, we document a negative impact of the recent financial crises on remittances to Romania. rate of unemployment reduces the migrant's probability to be employed and therefore the probability to remit. In fact, in all the OECD countries, except Italy and Greece, unemployment affects immigrants especially (OECD, 2006).

d. Regional dummies:

We introduce regional dummies to control for group characteristics. Using the OECD (2006) outlook on international migration, we identify three relatively homogeneous groups of countries. Among others, they present similarities in the way they deal with the international transit of persons.

Region 1 is the base group against which comparisons are made. It includes Canada and the United States. They are immigration countries long ago and tend to attract and keep in priority skilled migrants (OECD, 2006: 194). Their structure of immigration differs from European countries (Docquier et al., 2009). They are also located on a different continent.

As acknowledged by the OECD outlook (2006: 80), we split the European countries into two distinct groups (region 2 and region 3). Region 2 regroups the following Western European countries: Austria, Belgium, Denmark, France, Germany, the Netherlands, Switzerland and the United Kingdom. They promoted a mass immigration since the post-war period until the seventies. Then, they adopted more restrictive immigration policies. In contrast, region 3 presents a different pattern of migration. This group involves Cyprus, Greece, Ireland, Israel, Italy, Portugal, Spain and Turkey. Broadly speaking, from the postwar period until the seventies, these countries were labour-sending countries. Then, they became more developed and after the adoption of restrictive policies in the older immigration countries, they became labour-receiving countries. Moreover, most of these countries are also the ones where language is very similar to Romanian. This eases the integration of the Romanian immigrants. 16

Figure 1 .

 1 Figure 1. Top Remittance-receiving countries, US $ Billion (2007)

  Data are quite recent and on a quarterly frequency. Available data cover2005, 2006, 2007, 2008 and the first three quarters of 2009. Figure2provides a simple map on the geography of remittances to Romania over the whole period.

Figure 2 .

 2 Figure 2. Remittance-sending countries to Romania, US $ Million (2005-2009)

  10 Two recent papers, done independently and concurrently to ours, work with larger samples.[START_REF] Lueth | Determinants of bilateral remittance flows[END_REF] use a sample of 11 receiving countries. Each one has recorded flows from about 16 sending countries and different period of time. The authors find evidence that remittances follow a gravity type pattern: bilateral remittances increase with the sending and receiving countries' GDP and decrease with geographic distance (see below).[START_REF] Schiopu | Determinants of workers' remittances-evidence from the European Neighbouring Region[END_REF] work with a sample of 21 European sending countries and 7 European receiving countries, over the period2000-2005.

Table 1 . The main remittance sending countries in our sample (2005-2009)

 1

	Country	Remittances to Romania (US $	Distance from Romania
		Million)	(km)
	1 Italy	473.79	1139
	2 Spain	286.29	2477
	3 United States	148.81	7656
	4 United Kingdom 67.32	2097
	5 Germany	49.56	1621
	6 Greece	42.59	741
	7 France	26.06	1875
	8 Austria	16.73	859
	9 Portugal	15.48	2978
	10 Ireland	15.40	2541

Note: Average remittances per semester in US $ million. Geographic distance from capital to capital. Source: Remittance (National Bank of Romania) and distance (CEPII).

  and the European Commission[START_REF] Schiopu | Determinants of workers' remittances-evidence from the European Neighbouring Region[END_REF], both papers have built large data sets, which is a valuable contribution. 11 Using a large sample of observations introduces more variability on remittance patterns and allows for more general results, but faces a potential drawback. Remittances are recorded in very different ways among receiving countries, due to a lack of international harmonization in the data collection. This heterogeneity undermines the scope of the results. Working on one receiving country (here Romania) and various remitters reduces the size of the sample but avoids the previous drawback. Remittances are recorded in a more homogeneous way.

Table 2

 2

reports the estimation of equation (1). We use Ordinary Least Squares (OLS) in columns (1) to (3) and panel data estimators from column (4) to (6).

Table 2 . Bilateral remittances determinants

 2

	Column:	(1)	(2)	(3)	(4)	(5)	(6)
	Dependent variable a :	ln(R)	ln(R)	ln(R)	ln(R)	ln(R)	ln(R)
	Estimation method:	OLS	OLS	OLS	RE	FE	Mundlak RE
	ln(GDP) i	0.716***	0.727***	0.651***	0.587***	0.661	0.678*
		(15.20)	(16.13)	(13.25)	(5.34)	(1.60)	(1.66)
	ln(ExChange) ir	-0.750***	-0.689***	-0.716***	-1.017**	-2.072**	-2.108**
		(3.97)	(3.85)	(4.05)	(2.31)	(2.50)	(2.46)
	ln(Unemployment) i	-0.480**	-0.587***	-0.364*	-0.742*	-0.833**	-0.820**
		(2.24)	(2.74)	(1.65)	(1.92)	(2.43)	(2.34)
	ln(Distance) ir		0.496***	0.484***	0.736**	(dropped)	0.796**
			(2.66)	(2.61)	(2.13)		(2.15)
	(Region 2 dummy) i	0.589***	1.374***		1.416	(dropped)	1.782
		(2.92)	(4.06)		(1.59)		(1.64)
	(Region 3 dummy) i	2.356***	3.291***	1.782***	3.284***	(dropped)	3.709***
		(8.64)	(8.53)	(10.88)	(3.50)		(3.03)
	2006 dummy	0.423***	0.375***	0.392***	0.271***	0.256***	0.264***
		(3.08)	(2.96)	(2.79)	(3.52)	(2.88)	(3.18)
	2007 dummy	0.617***	0.578***	0.624***	0.461***	0.346**	0.348**
		(4.41)	(4.24)	(4.39)	(4.79)	(2.12)	(2.14)
	2008 dummy	-0.282	-0.290*	-0.181	-0.258**	-0.392*	-0.403*
		(1.60)	(1.72)	(0.97)	(2.12)	(1.83)	(1.83)
	2009 dummy	-1.072***	-1.058***	-1.020***	-0.961***	-0.986***	-0.997***
		(4.34)	(4.36)	(3.89)	(4.96)	(4.77)	(4.71)
	Obs. Nb.	309	309	271	309	309	309
	Adj. R²	0.62	0.63	0.61	-	-	-
	Hausman Test
	Chi-2;d.l.; prob.					4.87; 7; 0.67
	Notes: a dependent variable: R means Remittances. Robust t statistics are in parentheses with
	*significant at 10% level; **significant at 5% level; ***significant at 1% level. Constant and means
	of the time-varying variables are not reported. In column (3), we have dropped the North American
	countries from the sample.

Table 3 . Distance

 3

	Column:	(1)	(2)	(3)
	Dependent variable a :	ln(R)	ln(R)	ln(R)
	Estimation method:	OLS	OLS	OLS
	(...)	(…)	(…)	(...)
	Distance	0.360***	0.359***	-1.172***
		(7.27)	(6.00)	(4.95)
	Distance 2	-0.010***	-0.010***
		(7.59)	(6.17)
	Region2Xdistance			1.173***
				(4.98)
	Region3Xdistance			1.190***
				(4.99)
	Regional dummies	yes	yes	yes
	Years dummies	yes	yes	yes
	Obs. Nb.	309	271	309
	Adj. R²	0.69	0.66	0.64
	Notes: a R means Remittances. Robust t statistics are in parentheses with ***significant at 1% level.

The National Bank of Romania estimates that around 40 per cent of remittances are coming through informal channels.

For instance, Frankel (2009) uses the Lueth and Ruiz-Arranz's sample to study the countercyclical effects of remittances.

Regional dummies are defined in Appendix 2 (TableA.1). We control for country specific effects by using panel data techniques (see below). We have also introduced quarter dummies into the regressions but their estimates were statistically insignificant.

Moreover, Ratha and Shaw (2007) document higher transfer costs between developing countries. Thus, the costs of transferring money seem more related to the lack of financial development in the labour-sending country[START_REF] Freund | Remittances, transaction costs, and informality[END_REF], than to the geographic distance.

We are grateful to one referee for this comment.

We argue that among others the way these groups deal with international migration may affect remittances. Restrictive immigration policies aim to prevent irregular immigration.

Since illegal Romanian migration to Europe, as a whole, seems easier than to North America, we expect a higher stock of illegal migrants in Europe and thus larger remittances (given that illegal migrants transfer money trough legal means).

According to the literature, the main determinants of aggregate remittances are related to the following factors: (a) economic size, (b) financial environment and (c) labour market. We briefly review the literature on their impact. Then, we discuss the expected effects of regional dummies.

a. Economic size:

The empirical literature is unanimous on the effect of economic size. Remittances are an increasing function of the aggregate income of the sending country i, proxied by GDP [START_REF] Elbadawi | Determinants of expatriate workers' remittances in North America and Europe[END_REF][START_REF] Lianos | Factors determining migrant remittances: the case of Greece[END_REF]. This is consistent with the patterns displayed using simple descriptive statistics. In 2007, the United States and Western Europe account for almost two-third of remittance flows to developing countries [START_REF] Ratha | Migration and remittances factbook[END_REF].

b. Financial environment:

Financial variables, such as exchange rates, are considered as important determinants of remittances [START_REF] Adams | The Determinants of international remittances in developing countries[END_REF][START_REF] El-Sakka | The macroeconomic determinants of emigrant remittances[END_REF][START_REF] Elbadawi | Determinants of expatriate workers' remittances in North America and Europe[END_REF][START_REF] Lianos | Factors determining migrant remittances: the case of Greece[END_REF]. Due to the bilateral nature of our sample, we consider the effect of the bilateral exchange rate. A variation of (ExChange) i affects the purchasing power of remittances and leads to an ambiguous effect. Consider an appreciation of the sending's currency vis-à-vis the Lei. We get to two opposite effects: (1) an income effect: migrants remit more to benefit from an increasing purchasing power; (2) a substitution effect: migrants remit less, while keeping unchanged the purchasing power of the amount transferred.

c. Labour market:

The literature documents unambiguously that an increase in the unemployment of the sending country i negatively impacts on the volume of remittances. Three complementary explanations are at hand. First, a rise in unemployment causes significant losses of income which reduce remittances. Second, this rise increases the migrant's uncertainty about future incomes, and may incite migrants to decrease their remittances in anticipation. Finally, a high

Appendix 2. Data construction


Kristin Flornes

Alex Grossmann

Matthias Holschneider

Bruno Torrésani

WAVELETS ON DISCRETE FIELDS

HAL is

Introduction

Wavelet analysis can be seen today from essentially two different points of view. A first approach is connected with the theory of coherent states of quantum physics, and can be formulated in terms of group representation theory. The second approach basically consists in an "algorithmic" version of Littlewood-Paley analysis, and yields fast algorithms for computing numerically the wavelet transform.

The connection between these two approaches is quite difficult to handle, in particular because the first one is a "continuous approach" and the second is essentially discrete. We discuss here an intermediate approach, based on the square-integrable representations of some finite groups, that could help to pave the way between the continuous and discrete aspects (we won't discuss here the purely discrete case, in which the signals also take values in a discrete field).

We consider the set ℓ 2 (Z Z p ) of finite energy sequences on Z Z p , the class of integers modulo p. When p is a prime number, Z Z p has the structure of a number field, and a discrete affine group can be defined. The canonical action of this group on ℓ 2 (Z Z p ) is square-integrable, and systems of wavelets can be associated with it. Moreover, it is easy to see that in such a case, there is an associated "fast algorithm" for the computation of the corresponding wavelet transform.

The connection with multiresolution structures is made by considering possible deformations of the action of the affine group on ℓ 2 (Z Z p ). In particular, the deformation of the dilation operator directly yields an algorithmic structure similar to that associated with quadrature mirror filters.

The paper is organized as follows. After some algebraic preliminaries (section 2), and general results on square-integrable group representations and associated systems of wavelets (section 3), we describe the wavelets on ℓ 2 (Z Z p ) and give some numerical illustrations (section 4). Then we study the possible deformations of the dilation operators and describe the corresponding wavelet systems (section 5), and give some conclusions.

2 Algebraic Background

Group Representations

Our discussion is based on a group theoretical approach to wavelets. In this section we recall the definition of a square integrable representation. Basic concepts like a group, a field and a representation of a group on a Hilbert space are assumed to be known. For the reader who is not familiar with group theory, we refer to [START_REF] Kirillov | Eléments de la Théorie des Représentations Editions Mir[END_REF] for an introduction.

Our presentation is based on the theory of square-integrable group representations, the definition of which we recall here for convenience.

Definition 1 Let g → U (g) be a strongly continuous unitary representation of a locally compact separable group G in a Hilbert space H(U ). The representation U is said to be square integrable if

• U is irreducible.
• There exists at least one ψ ∈ H(U ) such that

c ψ = G | ψ, U (g)ψ | 2 dµ(g) < ∞ ( 1
)
where µ is the left invariant measure. Such a ψ is called an "admissible vector".

Number Theory

For the construction of wavelets on a finite field, we shall need some basic concepts from number theory. More precisely, we describe here some algebraic properties needed to introduce the affine group over Z Z p , referring to [START_REF] Hecke | Lectures on the theory of algebraic numbers[END_REF] for more details. Two integers a and b which have the same remainder modulo n are said to belong to the same residue class modulo n. We write

a ≡ b (mod n) (a is congruent to b mod n) (2)
which is equivalent to n divides a -b. The different remainders mod n are the numbers 0, 1, 2, . . . , |n| -1.

Theorem 1 Let the integers a, b, n be given.The congruence

ak + b ≡ 0 (mod n) (3)
has exactly one solution mod n if a and n are relatively prime.

Theorem 2 (Fermat) For each a relatively prime to n a ϕ(n) ≡ 1 (mod n)

where ϕ(n) is the number of residue classes relatively prime to n (Euler's ϕ function).

If n is a prime number p, then ϕ(p) = p -1 and we have the congruence

a p-1 ≡ 1 (mod p) (4)
The set of remainders mod p is a field. We denote this field by the symbol Z Z p . The set Z Z * p contains the nonzero elements of Z Z p .

The main consequence for our purpose of this discussion is that if p is any prime number, the set Z Z p × Z Z * p inherits a group structure, via the group operation

(b, a) • (b ′ , a ′ ) = (b + ab ′ , aa ′ ) (5)
We will denote by G p the corresponding group, called the affine group on Z Z p .

General Theorems

In this section we present two useful theorems. The first one was presented by Grossmann, Morlet and Paul in [START_REF] Grossmann | Transforms associated to square integrable group representation[END_REF]. They introduced the wavelet transform in terms of square integrable representation theory, and obtained an isometric correspondence between the function and it's transform. The second theorem and the following proposition state that such an isometry also exists even if the representation is not irreducible. Let G be a locally compact group and U a square integrable representation of this group on the Hilbert space H(U ). Following Grossmann, Morlet and Paul, we define the transform T f of f ∈ H(U ) by

T f (g) = 1 √ c ψ f, U (g)ψ g ∈ G ( 6
)
where ψ is an admissible vector. c ψ is the constant defined by [START_REF] Cohen | Ondelettes et traitement du signal numérique Thèse de doctorat[END_REF].

Theorem 3 The correspondence f → T f is isometric from H(U ) into L 2 (G, dµ(g)), that is for every f, h ∈ H(U ) we have G T f (g)T h (g) dµ(g) = f, h (7)
Proof: The proof, which is a direct consequence of Schur's lemma, can be found in [START_REF] Grossmann | Transforms associated to square integrable group representation[END_REF] for instance (as a part of a much stronger result).

Theorem 3 is the classical result that is at the basis of continuous wavelet analysis. The irreducibility assumption can be weakened as follows.

Theorem 4 Let U be a unitary representation of a locally compact, separable group G on the Hilbert space H(U ). If U can be decomposed into disjoint square integrable components U i then

G | f, U (g)ψ | 2 dµ(g) = i c P i ψ P i f 2 (8)
where P i is the orthogonal projection operator from H into H i , µ is the left invariant measure and ψ is an admissible vector. The constant c P i ψ is given by the formula

c P i ψ = 1 P i ψ 2 H i G | P i ψ, U i (g)P i ψ | 2 dµ(g) (9)
Proof: consider the L 2 -norm of the Schur coefficients:

G | f, U (g)ψ | 2 dµ(g) = i,j G P i f, U i (g)P i ψ U j (g)P j ψ, P j f dµ(g) = i P i f, A ii f + i =j P i f, A ij P j f
It can be easily shown that the operator A ij verifies the covariance equation

U i (g)A ij = A ij U j (g) ∀g ∈ G (10)
Using Schur's lemma we see that A ij is equal to zero when i = j and A ij is a multiple of the identity operator in

H i when i = j. G | f, U (g)ψ | 2 dµ(g) = i G | P i f, U i (g)P i ψ | 2 dµ(g) = i P i f, A ii P i f = i c P i ψ P i f 2
where the constant c P i ψ is given by equation 9.

✷

We can now state the following proposition.

Proposition 1 If the constants c P i ψ = c ψ are the same for all the subspaces

H i then the corre- spondence f → f, U (g)ψ is an isometry up to a constant factor from H(U ) into L 2 (G, dµ(g)). G | f, U (g)ψ | 2 dµ(g) = i c P i ψ f 2 = c ψ i P i f 2 = c ψ P i f 2 (11)
The case in which some of the components are unitarily equivalent can be handled in the same way. Notice also that the proposition, applied to the affine group of IR provides wavelet decompositions of L 2 (IR), considered as the direct sum of its two irreducible subrepresentations H 2 ± (IR).

Wavelets on Z Z p

Let p be a prime number. In the section 2.2, we considered the field Z Z p and introduced the corresponding affine group G p . We are now going to construct wavelets on Z Z p and analyze sequences in the Hilbert space ℓ 2 (Z Z p ).

Notation

Let us first specify our conventions and introduce for convenience some useful operators.

Operators:

Fourier transform:

F f (k) = f (k) = p-1 n=0 e -2πink/p f (n) (12)
Inverse Fourier transform:

F -1 f (n) = f (n) = 1 p p-1 k=0 e 2πikn/p f (k) (13)
Notice that the Fourier transform can be defined on Z Z N for every integer N . Translation:

T b f (n) = f (n -b) (14)
Dilation:

D a f (n) = f (a -1 n) (15)
Notice that in order to make sense to a -1 , it is necessary that p is a prime number.

Modulation:

E b f (n) = ω bn f (n) ( 16
)
where ω = e 2πi/p . All these operators are well defined on Z Z p . The following properties can be easily verified

1. F T b = E -b F 2. F D a = D a -1 F 3. F T b D a = E -b D a -1 F Scalar product: f, h = p-1 n=0 f (n)h(n) ( 17
)
where h is the complex conjugate of h. (We use the notation h or h * for the complex conjugate of h).

The Parseval identity reads:

F f, F h = p f, h (18)
Convolution product:

f * g(n) = p-1 j=0 f (n -j)g(j) (19)

The Affine Group

We consider the group of affine transformations of

Z Z p n ∈ Z Z p → an + b (20)
This group can be defined as the set

G p = {(b, a) | b ∈ Z Z p , a ∈ Z Z * p } with the multiplication (b 1 , a 1 ) (b 2 , a 2 ) = (b 1 + a 1 b 2 , a 1 a 2 ) (21)
The identity in the group is the element (0, 1). The inverse (b, a) -1 = (b ′ , a ′ ) has to fulfill the congruences

aa ′ ≡ 1 (mod p) (22) b + ab ′ ≡ 0 (mod p) ⇔ b ′ ≡ -a ′ b (mod p) (23)
We reserve the symbol a -1 for the integer a ′ that satisfies (22).

(b, a) -1 = (-a -1 b, a -1 ) (24)

We will be concerned with the following representation of the group G p in ℓ 2 (Z Z p ).

Definition 1

The unitary representation π π :

G p → U [ℓ 2 (Z Z p )] (25)
where U [ℓ 2 (Z Z p )] is the set of unitary operators on ℓ 2 (Z Z p ), is defined by

π(g)f (n) = f (a -1 (n -b)) = f (b,a) (n) (26)
Using the operators of translation and dilation defined by ( 14) and (15), π can be written as

π(b, a) = T b D a (27)
Translations and dilations of a constant sequence do not change the sequence. The space of constant sequences is therefore an invariant subspace of π. It is then natural to work on spaces of sequences modulo a constant.

Let E = {f ∈ ℓ 2 (Z Z p ) | p-1 n=0 f (n) = 0}.
Lemma 1 The representation π restricted to the subspace E is irreducible.

Proof: This follows from classical arguments see e.g. [START_REF] Kirillov | Eléments de la Théorie des Représentations Editions Mir[END_REF].

Following the usual convention we call a sequence ψ in ℓ 2 (Z Z p ) which satisfies certain additional conditions stated later in theorems 5 and 6 an "analyzing wavelet". The family of sequences ψ (b,a) generated from translations and dilations of such an analyzing wavelet will be called "wavelets".

Wavelet Transform

We are now in position to introduce the wavelet transform on ℓ 2 and state its general properties.

Definition 2

The wavelet transform T f associated with the analyzing wavelet ψ is the map

T f : ℓ 2 (Z Z p ) → ℓ 2 (G p ) T f (b, a) = f, ψ (b,a) = p-1 n=0 f (n)ψ (b,a) (n) (28)
We can express the wavelet transform in terms of the Fourier transforms of f and ψ by using Parseval's identity and the relation

F T b D a = E -b D a -1 F T f (b, a) = f, ψ (b,a) = 1 p f , E -b D a -1 ψ = 1 p-1 n=0 f (n)e 2πibn/p ψ(an) * (29)
As in the continuous case the wavelet transform can be inverted on its range as shown by the two following theorems (corresponding respectively to the irreducible representation of G p on E and the reducible one on ℓ 2 ).

Theorem 5 Let ψ ∈ ℓ 2 (Z Z p ) be such that ψ ∈ E ⊂ ℓ 2 (Z Z p ) where E = {f ∈ ℓ 2 (Z Z p ) | p-1 n=0 f (n) = 0} (30)
then the mapping f → T f is isometric up to a constant factor c ψ from E into ℓ 2 (G p ). There exits an inversion formula

f (n) = 1 c ψ (b,a)∈Gp T f (b, a)ψ (b,a) (n) ( 31
)
where the constant c ψ is given by the formula

c ψ = p ψ 2 ℓ 2 (Z Zp) (32)
Proof: The theorem follows from theorem 3. It can also be proved by direct calculation.

Theorem 6 Let ψ ∈ ℓ 2 (Z Z p ) be such that (p -1)| ψ(0)| 2 = p-1 k=1 | ψ(k)| 2 (33)
then the mapping f → T f is isometric up to a constant factor c ψ from ℓ 2 (Z Z p ) into ℓ 2 (G p ). We an inversion formula

f (n) = 1 c ψ (b,a)∈Gp T f (b, a)ψ (b,a) (n) ( 34
)
where the constant is given by

c ψ = (p -1)| ψ(0)| 2 (35)
Proof: We observe first that the sequence space ℓ 2 (Z Z p ) can be written as a sum of the two subspaces

ℓ 2 (Z Z p ) ∼ = I C ⊕ E
Using equation ( 9) to calculate the constants c P i ψ in the subspaces, we find that condition (33) states that these constants are equal. The theorem then follows from proposition 1.

It is also very easy to check that the discrete wavelet transform on E shares with the continuous wavelet transform the usual properties such as for example covariance with respect to dilations and translations, and existence of reproducing kernels in the image of E by the transform.

Theorems 5 and 6 can also easily be modified in the following way. It is always possible to use for the reconstruction a wavelet different than the analysis wavelet. This remark, that was used in a number of different contexts, is a consequence of the redundancy of the wavelet transform.

Some practical remarks and illustrations

We make a few remarks concerning the transform and its interpretation before presenting some illustrations. We will focus on the version of wavelet analysis provided by theorem 5, which is closer to what we are used to.

• Localization of the wavelet. By a change of variable the wavelet transform can be written as

T f (b, a) = p-1 n=0 f (n)ψ(a -1 (n -b)) = p-1 n=0 f (an + b)ψ(n) (36)
Suppose that ψ is concentrated somewhere e.g. that it vanishes numerically outside a certain interval I ⊂ Z Z p . We can speed up our calculations by using the formula

T f (b, a) = n∈I f (an + b)ψ(n) (37)
Moreover, it is worth noticing that the D a dilation used here does not change the number of nonvanishing coefficients of the wavelet ψ (in some sense it may be called a dilation "with holes"). Then the previous formula is suitable for a fast implementation.

• Reconstruction of signals which do not have mean zero.

Let f ∈ ℓ 2 (Z Z p ) and s(n) = f (n)+k
where k is a constant. T f = T s if the analyzing wavelet ψ belongs to the set E. Suppose

p-1 n=0 f (n) = c = 0.
With the reconstruction formula of theorem 5 we actually reconstruct f (n) -c p . From a practical point of view, this means that the mean-value of an analyzed signal has to be stored before doing its wavelet decomposition. Moreover, as stressed before, the "with holes" dilation appears clearly here.

b) The wavelet is real-valued with only three nonzero values and p-1 n=0 ψ(n) = 0. The signal is a sine, and the corresponding wavelet transform has a structure similar to the continuous one (see e.g. [START_REF] Grossmann | Reading and understanding continuous wavelet transforms in "Wavelets: time-frequency methods and phase space[END_REF]).

Figure 2 The signal is an exponential f 1 (n) = e 3•2πk/p . The wavelet is also complex-valued ψ(n) = e ik 1 n e -k 2 n 2 , and it is centered around a certain frequency ω 0 . We have that

|T f (b, a)| = | ψ(ωa)|
where ω is the frequency of the signal. The modulus |T f (b, a)| has a maximum every time the parameter a is such that ωa ≡ ω 0 (mod p). This explains the periodicity (with respect to scale) that we observe.

Figure 3 The signal is a superposition of the two exponentials with different frequencies and the wavelet is the same as in figure 2. In the image with the phase we have replaced the signal and the wavelet by their Fourier transforms.

Pseudo-dilation and multiresolution structure

In the preceding section we analyzed sequences and considered a purely discrete wavelet transform. We are now going to look at the correspondence between a continuous signal and it's sampled form. Following the philosophy of the algorithm "à trous" presented in [START_REF] Holschneider | A real-time algorithm for signal analysis with the help of wavelet transform in "Wavelets: time-frequency methods and phase space[END_REF], we introduce interpolating filters. Which changes need to be made to our transform if we want to interpret a sequence as a sampling of a continuous function ? In the continuous case the dilation operator Da is defined as

Da f (x) = 1 a f ( x a ) f ∈ L 2 (IR) (38)
Suppose that the continuous wavelet g(x) oscillates in the interval -2, 2 and is zero outside. D2 g(x) will then oscillate in the interval -4, 4 . When sampled by one, the function is left with seven nonzero values. Looking at figure 1, image 1 we observe that the discrete dilation operator D a preserves the number of nonzero values of the wavelet. If we want to interpret the discrete case as a sampled version of the continuous case, our dilation does not posses the desired properties.

Pseudo-dilations

These observations lead us to introduce a pseudodilation D a which not only dilates a sequence, but also "fills the holes". Let then K a be a sequence of linear operators (labeled by the dilation parameters a) acting on ℓ 2 , and set

D a = K a D a (39)
The possible pseudo-dilation operators are constrained by the following fundamental lemma Lemma 2 The operators σ(b, a) = T b D a form a representation of G p if and only if the K a operator is a convolution operator

D a = F a * D a ( 40
)
with a filter F a that satisfies the compatibility relations

F aa ′ = F a * D a F a ′ (41)
or equivalently in the Fourier space

Faa ′ (k) = Fa (k) Fa ′ (ak) ( 42
)
Proof: The first point is that K a has to commute with translations, i.e. must be a convolution operator. Moreover, we impose that the pseudodilation possesses the composition property of a true dilation, that is to say

D aa ′ = D a D a ′ (43)
For the filters this means that we have to impose

F aa ′ = F a * D a F a ′ ⇔ Faa ′ (k) = Fa (k) Fa ′ (ak) (44)
which proves the theorem. The filters that satisfy equation 44 will be called compatible filters.

It is also interesting to notice that the theorem also holds in the ℓ 2 (Z Z) context, i.e. in the case where all products aa ′ and ak in equation 44 are products in Z Z and not modulo p.

Remarks:

1. It is important to notice that the representation of G p is not unitary, so that it is not possible to use directly the general results of section 3 to get inversion formulas for the new transform. We shall come back to this inversion problem later on.

2. Consider for simplicity the case of ℓ 2 (Z Z) (the case of ℓ 2 (Z Z p ) is completely similar), and equation 44 with only powers of 2 as scale parameters. Then we have

D 2 j ψ = F 2 j * D 2 j ψ = D 2 j-1 F 2 * F 2 j-1 * D 2 j ψ (45)
so that the new wavelet transform of a sequence f reads

T f (b, 2 j ) = D 2 j ψ * ( F2 j * f )(b) = D 2 j ψ * (D 2 j-1 F2 * ( F2 j-1 * f ))(b) (as usual, h(n) = h(-n) *
). Then we recover here the same algorithmic structure as the one that appears with multiresolution analysis. The computation of the wavelet coefficients at different scales (here of the form 2 j ) can be performed through a pyramidal algorithm involving only (truely) dilated copies of two filters: F 2 (that stands for the low-pass filter) and ψ (band-pass filter). Moreover, since the invoved dilation is the true one (i.e. D 2 ) and not the pseudo dilation D 2 , all such filters are of constant length, so that we still have a "fast" algorithm.

3. The same remark applies for arbitrary scale parameters: assuming that a family of filters satisfying equation 44 has been specified, we have

T f (b, aa ′ ) = D aa ′ ψ * ( Faa ′ * f ) (46) = D aa ′ ψ * D a Fa ′ * ( Fa * f ) (47)
Then (assuming the existence of appropriate filters) there exists a pyramidal algorithm for the computation of the transform for arbitrary value of the scale. This generalizes the usual pyramidal algorithms.

4. Now the main question is that of the existence of filters satisfying equation 44. The problem may be simplified by using decompositions of integers into prime factors. If F q is known for any prime q, F a may be defined for any a by using equation 44. The only compatibility equations that remain are the ones involving the F q with q prime numbers.

5. As a consequence, the only filters used in the algorithm are filters of constant length, dilated copies (with holes) of the F q filters with prime qs.

Existence of appropriate filters

It turns out that equations 44 are in fact very restrictive, because of the requirement that they must hold for any a and a ′ in the field of reference. In the case of Z Z p for instance, this amounts to solve (p -1) 3 non linear equations with (p -1) 2 variables. In other words, the F a filters with different values of a must satisfy some compatibility relations.

The characterization of such filters is a difficult problem (which can actually be identified with a group cohomology problem), and we do not know whether the answer is already known.

Here is a possible strategy for the construction of such filters in the case of ℓ 2 (Z Z). We shall see that in some cases it will provide explicit examples.

1. Consider a candidate for the dyadic filter F 2 (for example one of those used in the usual multiresolution analyses).

2. Let (as in the usual multiresolution scheme)

φ(k) = ∞ j=1 F2 (2 -j k) (48)
Under some well known assumptions (see e.g. [START_REF] Cohen | Ondelettes et traitement du signal numérique Thèse de doctorat[END_REF] for example) the infinite product will converge to a L 2 function.

3. For a > 2 we define the other filters Fa as the quotients

Fa (k) = µ(a) φ(ak) φ(k) ( 49
)
where µ is any multiplyier of Z Z * (i.e. any function defined on Z Z * such that for any a, a ′ ∈ Z Z * , µ(aa ′ ) = µ(a)µ(a ′ )). Clearly, such functions satisfy equation 44:

Faa ′ (k) = µ(aa ′ ) φ(aa ′ k) φ(k) = µ(a ′ ) φ(aa ′ k) φ(ak) µ(a) φ(ak) φ(k) = Fa ′ (ak) Fa (k) (50)
At this point there still are open questions concerning the possible use of the above defined filters in our scheme. Namely, the Fa functions must be 2π-periodic functions. Another important question is that of the length of the filters. We shall provide examples of compactly supported filters, associated with B-spline scaling functions.

Another question is that of the existence of such filters in the ℓ 2 context. We shall see that the B-spline type filters also work in the finite case.

Examples of filters

There is a simple family of examples for which the above method provides filters satisfying equation 44. It is the case of B-spline filters. They correspond to scaling functions φ (see equation 48) which are powers of sin(πk)/πk. Consider for instance the simplest candidate, that is the filter that interpolates linearly (up to a factor only depending on a): where δ q (n) = δ n,q . In the Fourier space we have

F a (n) = a   δ 0 + 1 a a-1 q=1 (a -q) (δ q + δ -q )   (51)
F a (k) = φ(ak) φ(k) = sin aπk/p sin πk/p 2 (52)
and it is obvious to check that such filters fulfill the compatibility relations. Thus the only filters appearing in the algorithm are given by

Fq (k) = sin ( qπk p ) sin ( πk p ) 2 (53)
and they are all finite length filters.

This example generalizes directly to any even power of sin ( qπk p ) sin ( πk p )

, yielding B-spline interpolation of various orders. In the case of odd powers, it is necessary to change a little bit the formulas, and consider

Fq (k) = 1 -e 2iπqk p 1 -e 2iπk p 2N +1 (54)
to keep the compatibility in Z Z p . We do not know by now other examples of compatible filters. As an illustration, let us give the example of the equivalent of figure 1 with the dilation replaced by the pseudo-dilation. One easily sees that the pseudo-dilation actually does the job it was introduced for, i.e. it "fills the holes".

Reconstruction

As we have seen previously, the pseudo-dilation does not yield a unitary representation of the affine group G p , and then it it not possible to use directly the general results on square-integrable representations to get an inversion formula for the new wavelet transform. Nevertheless, it is still possible to use the freedom given by the redundancy of the transform, and to look for a reconstruction wavelet χ, so that f = p This condition is equivalent to a matrix equation that can at least be solved numerically. Thus, for a given analyzing wavelet and an given family of filters, one can obtain a reconstruction wavelet. It is important to notice that since we used the σ(b, a) action of G p , the reconstruction algorithm is also a pyramidal algorithm.

      | F1 (1)| 2 | F2 (1)| 2 • • • | Fp-1 (1)| 2 | F2 -1 (2)| 2 | F1 (2)| 2 • • • | F2 -1 (p-1) (

Conclusions

We have developed in this paper a wavelet decomposition formalism adapted to the case of periodic sampled signals (with the assumption that the number of samples is a prime number). The problem was formulated as an algebraic approach to wavelet decompositions of ℓ 2 (Z Z p ), in terms of square-integrable representations of the corresponding affine group. Surprisingly enough, the study of the possible deformations of the dilation operators (which appears to be unconvenient if the analyzed sequence is to be interpreted as a sampled continuously defined signal) led us to an "algorithmic structure" quite similar to the multiresolution structure, with associated low-pass and band-pass filters. Moreover, we were able to provide explicit examples of such low-pass and band-pass filters, namely some filters associated with B-spline approximations. The problem of classification of all possible such filters amounts to a group-cohomology problem.

It is now very likely that a similar analysis (in the case where p is replaced by some power of two, and the possible scales are also restricted to powers of two) could (at least partly) fill the gap between the group-theoretic and the Littlewood-Paley approaches to wavelets.

•

  Graphical conventions. The modulus of the wavelet transform T f (b, a) is represented with gray levels inside the big box. The dilation parameter a varies vertically (a = 1 at the top and a = p -1 at the bottom). The translation parameter b varies horizontally from right to left (from b = 0 to b = p -1). The small box above the transform contains the real part of the signal and the two boxes below contain the real parts of the wavelet and the reconstructed signal. When there is a curve on the side of the wavelet transform, it indicates the energy e of the transform for the different scales e(a) = p-1 b=0 |T f (b, a)| 2 .

Figure 1 a)

 1 Figure 1 a) The wavelet is real-valued with five nonzero values and zero integral. The signal f vanishes everywhere except at n = 26 where f (26) = 1.0. This gives us that T f (b, a) = ψ(a -1 (26 -b)). "What disappears on one side reappears on the other".Moreover, as stressed before, the "with holes" dilation appears clearly here.

Figure 1 :

 1 Figure 1: Modulus of the wavelet transform of elementary functions (p = 53).

Figure 2 :Figure 3 :

 23 Figure 2: Sine waves (p = 71)

Figure 4 :

 4 Figure 4: The modulus of the wavelet transform. p = 53.

b=1 p- 1 a=1 1 a=1

 11 f, σ(b, a) • ψ σ(b, a) • χ = p b=1 p-1 a=1 T f (b, a)F a * D a χ(n -b) (55) A sequence χ(n) such that equation 55 is fulfilled will be called a reconstruction wavelet. A direct computation shows that Lemma 3 The function χ(n) is a reconstruction wavelet if and only if it satisfies p-Fa (k) 2 ψ(ak) * χ(ak) = 1 ∀k = 1, ...p -1 (56)

1 )

 1 2)| 2 . . . . . . . . . | F(p-1) -1 (p -1)| 2 | F(p-1) -1 2 (p -1)| 2 • • • | F1 (p -1)| 2 * χ(1) ψ(2) * χ(2) . . . ψ(p -1) * χ(p -1)


J.M. Innocent * and B. Torrésani † CPT, CNRS-Luminy, Case 907, 13288 Marseille Cedex 09, France Abstract: Gravitational waves generated by coalescing binary systems of neutron stars or black holes are expected to behave like chirps, i.e. amplitude and frequency modulated signals, buried into strongly correlated noise, with very low signal to noise ratio. This note presents a wavelet based algorithm for on line processing and detection of such signals, from interferometric detectors which are currently being constructed, and discusses a few examples. The details of the method and more complete simulations can be found in [START_REF] Innocent | A Multiresolution Strategy for Detecting Gravitational Waves Generated by Binary Coalescence[END_REF].

The detection of gravitational waves, predicted by general relativity but never observed so far, is a major challenge in today's experimental physics. Several projects are currently being developed, among which one may quote the LIGO [1], GEO [START_REF] Danzmann | GEO600, Proposal for a 600 m Laser-Interferometric Gravitational Wave Antenna[END_REF] and VIRGO [2] laser interferometric detectors.

Among the potential sources for gravitational waves, the most promising are presumably pulsars, and coalescing binary systems of black holes or neutron stars. Such systems are expected to produce "simple" signals, for which available models are considered reliable. We focus here on the case of coalescing binary systems. The corresponding gravitational waves take the form of chirps, i.e. amplitude and frequency modulated signals.

The detection problem may be formulated as follows. The signal takes the form

f (x) = h θ (x -x 0 ) + n(x) (1)
where h θ (x) is the reference signal (given in good approximation by various post-Newtonian theories [START_REF] Thorne | Gravitational Radiation, in 300 Years of Gravitation[END_REF]), depending on a set of parameters to be estimated, represented generically by θ, x 0 is the date of coalescence (to be estimated as well) and n(x) is the experimental detector noise.

The classical method to such a detection problem is provided by matched filters [START_REF] Thorne | Gravitational Radiation, in 300 Years of Gravitation[END_REF]. However, the matched filter applies to the case where the parameters θ of the signal are known. Since the parameters are not known, one has to construct filters for sufficiently many choices of the parameters, which leads to a very large number of filters.

This note describes an alternative approach based on wavelet transform. Since the detection problem actually amounts to that of estimating a shift in a time-scale domain, wavelet methods are a natural tool. In addition, it is important for detection purpose to keep scaling and translation covariance properties of the time-scale representation, we focus on continuous wavelet transform. The interested reader is refered to [START_REF] Innocent | A Multiresolution Strategy for Detecting Gravitational Waves Generated by Binary Coalescence[END_REF] for more details.

I. SIGNAL AND NOISE MODELS

The signal measured at the detector is a strain, denoted by h(x), and is the superposition of a gravity wave signal, and an additive detector noise.

A. Signal

Consider a system of two stars of masses m 1 and m 2 respectively, at distance R from the observer. Following the conventional use, we introduce the chirp mass

M = (m 1 m 2 ) 3 /(m 1 + m 2 )
1/5 . In the Newtonian approximation, the expected reference signal is given by h

(x) = Ah F (x 0 -x), where h F (x) = Ax α Θ(x) cos Φ - 2π β + 1 F x β+1 , (2)
with Θ(x) the Heaviside step function, α = -1/4 and β = -3/8. Here, x 0 is the coalescence date, Φ is a global phase, and A and F are related to physical parameters by F = C F M -5/8 and A = ǫC A 1 R M 5/4 for some numerical constants C A and C F , and a constant 0 ≤ ǫ ≤ 1 depending on the relative orientation between the source and the detector.

The expected signal is then a chirp, to which local amplitude A(x 0 -x) α and frequencies ν(x) = F (x 0 -x) β may be associated. An example is displayed in Figure 1.

More general post-Newtonian approximations are described in [START_REF] Thorne | Gravitational Radiation, in 300 Years of Gravitation[END_REF] for example. The detector noise is the superposition of several contributions, and is generally modeled as a weakly stationary process, with power spectral density S(ν) = E(2πν) of the form

S(ν) = S 1 ν -5 +S 2 ν -1 +S 3 1 + ν 500Hz 2 + 1 ν ∞ n=1 K (ν 2 -ν 2 n ) 2 -φ 2 ν 4 n . (3)
Here, S 1 , S 2 , S 3 , K, φ 2 are constants depending on the experimental setup. The spectral density used in our simulations is displayed in Figure 2.

Frequency (Hz)

Spectral Density (1/Hz)

II. A DETECTION ALGORITHM

A. Prewhitening wavelet transform

Our main tool will be a version of continuous wavelet transform adapted to the problem as follows. Let ψ(x) be an integrable and square-integrable function, such that ψ(k) = 0 ∀k ≤ 0 and that 0 < c ψ :

= ∞ 0 | ψ(k)| 2 dk
k < ∞ Let n(x) be a weakly stationary process, and let C be the associated covariance operator:

Cf (x) = 1 2π e ikx E(k) f (k)dk . (4)
The usual continuous wavelet transform of

f (x) ∈ L 2 (IR) is defined by T f (b, a) = f, ψ (b,a) , where ψ (b,a) (x) = (1/ √ a)ψ((x -b)/a).
We define the corresponding prewhitening wavelet transform of any f (x) ∈ L 2 (IR) as follows

W f (b, a) = C -1/2 f, ψ (b,a) = √ a 2π e ikb f (k) ψ(ak) √ E(k) dk . (5)
Clearly, given a stationary process n(x) of spectral density E(k), we have

V (a) := IE{|W n (b, a)| 2 } = ||ψ|| 2 .
In practice, we are given a signal of the form f (x) = f 0 (x) + n(x), where n(x) is as before. Introducing the new representation,

M (b, a) = |W f (b, a)| 2 -V (a) , (6)
we easily obtain that IE{M (b, a)} = |W f0 (b, a)| 2 .

B. Transform of binary coalescence signals

Let us come back to the detection problem. We stick to the Newtonian situation. If ψ(x) = A ψ (x) exp{iφ ψ (x)} is an analytic wavelet with linear phase φ ψ (x) = ω 0 x, it follows from the analysis in [START_REF] Delprat | [END_REF] that the standard continuous wavelet transform is localized in a neighborhood of a curve in the time-scale plane, called the ridge of the wavelet transform, of equation

b = b (F,x0) (a) = x 0 - 2πaF ω 0 8/3 . ( 7
)
We assume that the prewhitening wavelet transform has the same localization properties (see [START_REF] Innocent | A Multiresolution Strategy for Detecting Gravitational Waves Generated by Binary Coalescence[END_REF] for more details).

C. Line Integrals

Consider now the realistic situation

f (x) = Ah F (x 0 -x) + n(x) , (8)
where n(x) is a weakly stationary stochastic process with spectral density E(k) given in (3), modeling the detector noise. Given a candidate (τ, γ) for the pair (x 0 , F ), we consider the corresponding candidate b (τ,γ) (see Eq. ( 7)), and the line integral

L τ (γ) = M (b (τ,γ) (a), a) da a , (9)
with M (b, a) as in [START_REF] Innocent | A Multiresolution Strategy for Detecting Gravitational Waves Generated by Binary Coalescence[END_REF]. The algorithm we propose is based on the following maximization problem max (τ,γ) L τ (γ) .

(10)

D. The Maximization Algorithm

The maximization problem (10) amounts to a search in a two-dimensional space, parametrized by the pair (x 0 , F ). We now describe an algorithm compatible with a possible real time implementation.

We first fix a threshold, hereafter denoted by THRESHOLD for the line integral (9), the choice of which is based upon an a priori knowledge of the noise, and some a priori estimates on the false alarm or non detection probabilities. For a given date τ , set

F (τ ) = arg max γ L τ (γ) , (11)
Â(τ ) = max γ L τ (γ) , (12)
approach relies on a two-step optimization:

1. Solve the problem (12) with respect to γ.

Store the values F (τ ) and Â(τ ).

• IF Â(τ ) ≥ THRESHOLD: mark τ as a possible date for an event.

The maximization is carried out by standard maximization methods (an adapted version of Brent's method in our implementation)

III. NUMERICAL EXAMPLES

We present here some of the numerical results obtained with the algorithm described above, in three typical situations. A detailed analysis and more examples can be found in [START_REF] Innocent | A Multiresolution Strategy for Detecting Gravitational Waves Generated by Binary Coalescence[END_REF], together with an analysis of the statistics of detection.

The numerical tests have been performed on signals generated following the lines of Section I. The continuous wavelet transform is discretized in the form T f (b m , a n ) with b m = mν s (ν s being the sampling frequency) and a n = 2a n 0 , with a 0 = 2 1/8 and n = 0, . . . 39 (for ν s = 2 kHz, this corresponds to the frequency range 30 Hz --500 Hz). Output of the algorithm in the case m1 = m2 = 10M⊙ and R = 1.2Gpc.

We display in Figure 3 (resp. Figure 4, resp. Figure 5) the function Â(τ ) introduced in Eq (12), for the case of a binary system formed of two neutron stars of 1.4 solar masses each (resp. 1 and 10 solar masses, resp. 10 solar masses each), coalescing at a distance of 150 M pc (resp. 250M pc, resp. 1.2Gpc). In each case, the coalescence date was set to x 0 = 15.1sec.. The sampling frequency was set to ν s = 2kHz (resp. ν s = 1kHz). A significant peak is clearly seen, and in the three considered cases, the peak is located precisely at the correct date τ = x 0 .

A more detailed statistical analysis shows that in the three considered situations, the peak value Â(x 0 ) is far above the 0.1% critical value of the distribution of  in the noise alone reference.

The algorithm can also be used for estimating the frequency parameter F when an event has been detected. In the three considered examples, we obtained F = 134.6 while the true value was F = 133.47 (resp. F = 85.26 for F = 85.97, resp. F = 39.59 for F = 39.05), i.e. a precision of the order of 1%.

IV. CONCLUSIONS

In this note, we have summarized a wavelet based algorithm for on line processing of interferometric detector signals and binary coalescence detection. The results in [START_REF] Innocent | A Multiresolution Strategy for Detecting Gravitational Waves Generated by Binary Coalescence[END_REF] and those shown here prove that this approach is a reliable one. In addition, the computational cost is low enough to allow real time processing of data on standard Unix workstations. For these reasons, we believe that this approach should be considered seriously for on line detection of binary coalescence, as well as for the corresponding parameter estimation problems.

The details of the method are given in [START_REF] Innocent | A Multiresolution Strategy for Detecting Gravitational Waves Generated by Binary Coalescence[END_REF], together with results of simulations and statistical analysis of the performances of the method.

2 FIG. 1 .

 21 FIG. 1. Last second of gravitational wave generated by the collapse of a binary with m1 = 10M⊙, m2 = 1M⊙, coalescing at a distance R = 200M pc. The amplitudes are to be multiplied by 10 -21 .

FIG. 2 .

 2 FIG.2. Expected power spectral density of interferometric detector noise.

FIG. 3 .FIG. 4 .

 34 FIG. 3.Output of the algorithm in the case m1 = m2 = 1.4M⊙ and R = 150M pc.


Luc Aquilina

Daren Gooddy

Jean-Raynald De Dreuzy

Dissolved gases in groundwater and groundwater dating methods: how useful for hydrogeological modeling? -Foreword to the special issue

Introduction

One of the major challenges in hydrogeology is the basic spatial lack of continuous data as this is restricted to available boreholes. Basic information is provided by the measurements of physical parameters (e.g. porosity and permeability) at the well-scale. Information related to a larger part of the aquifer are provided for example, by continuous measurement of piezometric levels or spring flow. Temporal information is extremely powerful in order to characterize aquifer functioning globally. Groundwater dating has thus been developed in order to provide time-constraints in aquifer studies.

For young groundwater (< 50 yrs) tritium measurements were used following nuclear tests during the 1950s and 1960s. Tritium accumulated in the atmosphere and so provided a useful environmental tracer for water originating from this era. Although several monitoring networks from International Atomic Energy Agency provide precise records, seasonal and spatial variations in the tritium precipitation content limit the precision of groundwater dating using this method. Furthermore, tritium levels have decreased gradually and now precipitation concentrations are close to natural background. Groundwater dating using chlorofluorocarbons (CFCs) and more recently sulfur hexafluoride (SF 6 ) were also widely developed during the last decades [START_REF] Busenberg | Use of chlorofluorocarbons (CCl 3 F and CCl 2 F 2 ) as hydrologic tracers and age-dating tools: Example -The alluvium and terrace system of central Oklahoma[END_REF][START_REF] Busenberg | Dating young groundwater with sulfur hexafluoride: Natural and anthropogenic sources of sulfur hexafluoride[END_REF][START_REF] Cook | Chlorofluorocarbons as tracers of groundwater transport in a shallow silty sand aquifer[END_REF]. CFC concentrations have been rapidly increasing in the atmosphere during the second half of the 20 th century and their input history is well known with limited spatial variations. Compared with other groundwater dating techniques, CFCs can be rapidly determined, they require no sophisticated sampling equipment and analysis is relatively simple [START_REF] Gooddy | Using chlorofluorocarbons (CFCs) and sulphur hexafluoride (SF 6 ) to characterise groundwater movement and residence time in a lowland Chalk catchment[END_REF]. Consequently, for more than a decade, a number of laboratories have routinely analyzed groundwaters for these compounds. However, several processes such as contamination, degradation and sorption may modify the CFC concentrations in the soils and aquifers so restricting their use as potential tracers. Furthermore, the prohibition of CFCs by the Montreal protocol (1987) has led to a slow decline of their atmospheric concentrations. This decline increases the uncertainty of their use for modern groundwater. SF 6 is not subject to the Montreal protocol and its continued release from industrial processes has led to a rapid increase in atmospheric concentrations. Although it has even lower concentrations than the CFCs, it is now analyzed by several laboratories. The interest in SF 6 has been limited by potential rock production [START_REF] Friedrich | Factors controlling terrigenic SF 6 in young groundwater of the Odenwald region (Germany)[END_REF]. 85 Kr has also been used as a potential tracer with similar characteristics to CFC although not prone to their associated problems [START_REF] Cook | Recent advances in dating young groundwater: Chlorofluorocarbons, 3 H/ 3 He and 85 Kr[END_REF][START_REF] Lehmann | 81 Kr and 85 Kr in groundwater, Milk River aquifer, Alberta, Canada[END_REF]. The development of new tracers such as 39 Ar, 81 Kr, [START_REF] Loosli | Noble gas radioisotopes: 37 Ar, 85 Kr, 39 Ar, 81 Kr[END_REF][START_REF] Edmunds | Noble gas, CFC and other geochemical evidence for the age and origin of the Bath Thermal Waters[END_REF], SF 5 CF 3 and CFC-13 [START_REF] Busenberg | Dating groundwater with trifluoromethyl sulfurpentafluoride (SF 5 CF 3 , sulfur hexafluoride (SF 6 ), CF 3 Cl (CFC-13) and CF 2 Cl 2 (CFC-12)[END_REF] continues. In recent years and following on from tritium measurements, the tritiumhelium ( 3 H/ 3 He) method has also been used [START_REF] Solomon | A validation of the 3H/3He method for determining groundwater recharge[END_REF]. It is based on the measurement of the 3 He produced from the radioactive decay of tritium contained in groundwater. Although it is often preferred to CFCs by hydrogeologist, it implies the precise determination of the atmospheric, terrigenic and mantle-derived helium contributions which might be challenging [START_REF] Poreda | Tritium and Helium-Isotopes as Hydrologic Tracers in a Shallow Unconfined Aquifer[END_REF][START_REF] Schlosser | Tritium He-3 Dating of Shallow Groundwater[END_REF].

Groundwater dating has been used in various type of sub-surface environment, although heterogeneous aquifers such as karstic or fractured aquifers have been less intensively investigated [START_REF] Ayraud | Compartmentalization of physical and chemical properties in hard-rock aquifers deduced from chemical and groundwater age analyses[END_REF]. In young groundwater systems, a combination of groundwater dating techniques has been helpful to investigate important environmental issues such as sources, fate and transport of contaminants, and groundwater surface water interactions for resource management.

"Old" groundwaters can be more problematic (Philips and Castro, 2003). First, there are fewer tools that have the capability to date these groundwater and each tracer has a relatively narrow residence-time range. Second, deep and ancient fluids represent highly complex mixtures including water transported by advection and solutes provided from the rock matrix through dissolution and diffusion processes [START_REF] Aquilina | Relation of present saline fluid with paleomigration of basinal brines at the basement/sediment interface (Southeast basin -France)[END_REF]. In addition very old groundwater may also mix with various amounts of modern groundwater, sometimes through the process of sampling, leading to wide ranges of groundwater residence-times in a single sample [START_REF] Alvarado | Constraining the age distribution of highly mixed groundwater using 39 Ar: A multiple environmental tracer ( 3 H/ 3 He, 85 Kr, 39 Ar and 14 C) study in the semi confined Fontainebleau Sands Aquifer (France)[END_REF]. Carbon-14 has been used for decades but water-rock interactions with carbonate rocks can make it difficult to interpret. Chlorine-36, radiogenic helium and 81 Kr have also been used to investigate groundwater in the range of 10 3 to 10 6 years [START_REF] Marine | The use of naturally occurring helium to estimate groundwater velocities for studies of geologic storage of radioactive waste[END_REF] although the use of an appropriate technique often requires an a priori knowledge of the groundwater age.

Beside groundwater dating advancements, the integration of groundwater "ages" in hydrogeological modeling has been developing in parallel. Modeling has prompted some questioning as to the meaning of groundwater dating. Although the concept of "age" appears straightforward and very meaningful to end users, it requires assumptions about hydrogeological functioning. Indeed, the "age" determined from the atmospheric concentration deduced from the groundwater concentration represents a "piston-flow" residence-time. Piston flow is however a highly unlikely model of aquifer functioning. The use of lumped models in order to interpret tracer concentrations has become widely used [START_REF] Clark | Environmental Isotopes in Hydrology[END_REF]. This is a first raw interpretation of tracer concentrations and provides a framework to establish a groundwater flow model. However, mixing at various scales may require much more complex models (Leray et al., 2013). There is still research needed to analyze the relationships between the distribution of tracers in aquifers and the hydrogeological parameters.

As groundwater dating remains an important issue for hydrogeological sciences, two recent workshops have been devoted to groundwater dating [START_REF] Osenbrück | Selected papers from the international workshop G-DAT 2008: groundwater dating using environmental tracers[END_REF]. The second of these was organized in Rennes, western France (October 15-17, 2012). Before the G-DAT meeting, an inter-laboratory comparison exercise was organized. Dissolved gases analyses represent technical difficulties as they require extremely low analytical thresholds. Detailed sampling protocols have been published (IAEA, 2006(IAEA, , 2011;;Busenberg andPlummer, 1992, 2000). The International Atomic Energy Agency (IAEA) regularly organizes inter-laboratory comparison of the analytical technics ("TRIC", [START_REF] Gröning | Eigth IAEA Interlaboratory Comparison on the Determination of Low-Level Tritium Activities in Water (TRIC2008)[END_REF]. However, different laboratories use various methods for sampling which may induce significant uncertainties between analyses. The aim of the 2012 G-DAT inter-comparison exercise was thus to compare the sampling and analytical protocols from various laboratories.

The inter-comparison exercise was designed by a scientific committee. The idea was to first use a rather simple and homogeneous aquifer and secondly a more complex and heterogeneous one. The first sampling exercise (February 2012) took place in the Fontainebleau Miocene sand aquifer and Dogger limestone deep aquifer. Directly following the G-DAT meeting (October 2012) the second sampling exercise was carried out in a fractured aquifer [START_REF] Ayraud | Physical, biogeochemical and isotopic processes related to heterogeneity of a shallow crystalline rock aquifer[END_REF]. In both cases, a surface shallow aquifer and a deeper aquifer were sampled. Special attention was given to ensure the homogeneity of the groundwater sampled by all of the teams.

The G-DAT meeting was an opportunity to present the results of the inter-comparison exercise on one side and to bring together geochemists and modelers. This special issue "Dissolved gases in groundwater and groundwater dating methods: how useful for hydrogeological modeling?" reflects this meeting between geochemists and modelers as there are three subsections in the special issue. The first section is devoted to the inter-comparison exercise. It includes both the results of the noble gas and tritium inter-comparison (Visser et al., this issue) and those of the CFC and SF 6 inter-comparison (Labasque et al., this issue). A paper presenting the inter-comparison design and the site choice which have been the subject of discussions has been presented previously (Labasque et al., 2014). The second section is devoted to contemporary application of groundwater dating methods. It includes multi tracer studies in Austria [START_REF] Kralik | Using 18 O/ 2 H, 3 H/ 3 He, 85 Kr and CFCs to determine mean residence times and water origin in the Grazer and Leibnitzer Feld groundwater bodies[END_REF] and Italy [START_REF] Mayer | A multi-tracer study of groundwater origin and transit-time in the aquifers of the Venice region (Italy)[END_REF], and papers investigating pesticide transport [START_REF] Akesson | Modelling pesticide transport in a shallow groundwater catchment using tritium and helium-3 data[END_REF], helium fluxes [START_REF] Mahara | Effects of terrigenic He components on tritium-helium dating: A case study of shallow groundwater in the Saijo Basin[END_REF] as well as karstic systems [START_REF] Delbart | Investigation of young water components in karst aquifer systems using SF 6 -CFC-3 H/ 3 He-85 Kr-39 Ar and stable isotope components[END_REF]. The third section is devoted to the integration of groundwater dating data in hydrogeologic models in order to investigate the physical meaning of residence time/age concepts. It includes a paper advocating that the term "groundwater age" is misleading [START_REF] Suckow | The Age of Groundwater -definitions, models and why we do not need this term[END_REF], two papers illustrating the use of Bayesian modeling (Massoudieh et al., 2014a,b) and two papers devoted to 36 Cl use in paleoclimate studies [START_REF] Petersen | Quantifying paleorecharge in the Continental Intercalaire (CI) aquifer by a Monte-Carlo inversion approach of 36 Cl/Cl data[END_REF][START_REF] Rebeix | 36 Cl deposition rate reconstruction from -12 000 BP until present : a study based on groundwater records[END_REF].

Acknowledgements

The G-DAT meeting and the inter-comparison exercise beneficiated from the efforts of all the teams that participated in the exercises, although the conditions were not always ideal. We want to thank all the participants for their help and their enthusiasm. G-DAT also benefitted from the financial support of International Atomic Energy Agency, Université Rennes 1, CNRS, Observatoire des Sciences de l'Univers de Rennes (OSUR) Geosciences Rennes laboratory, the National Hydrogeological network H+, the Ministère de l'Enseignement Supérieur et de la Recherche, the Région Bretagne, the Syndicat Mixte de Production d'eau Potable du Bassin Rennais (SMPBR) and Véolia Group. Many people at Université Rennes 1 helped with the meeting and field organization. Although it is not possible to mention them all, they are all gratefully thanked.