Mohamed El

Hedi Arouri
email: mohamed.arouri@u-clermont1.fr

Amine Lahiani
email: amine.lahiani@univ-orleans.fr

Khuong Duc

Nguyen
email: dnguyen@iscparis.com

World gold prices and stock returns in China: insights for hedging and diversification strategies

Keywords: Stock markets, gold prices, diversification and hedging effectiveness, GARCH models JEL classification: G12, F3, Q43

In this paper we make use of several multivariate GARCH models (CCC-, DCC-, BEKK-, diagonal BEKK-, and VAR-GARCH) to investigate both return and volatility spillovers between world gold prices and stock market in China over the period from March 22, 2004 through March 31, 2011. We also analyze the optimal weights and hedge ratios for gold-stock portfolio holdings and show how empirical results can be used to build effective diversification and hedging strategy. Our results show evidence of significant return and volatility cross effects between gold prices and stock prices in China. In particular, past gold returns play a crucial role in explaining the dynamics of conditional return and volatility of Chinese stock market and should thus be accounted for when forecasting future stock returns. Our portfolio analysis suggests that adding gold to a portfolio of Chinese stocks improves its risk-adjusted return and that gold risk exposures can be effectively hedged in portfolios of stocks over time. Finally, we show that the VAR-GARCH model performs better than the other multivariate GARCH models.

Introduction

The purpose of the present study is threefold. Our first objective is to examine the dynamic return and volatility transmission using a bivariate VAR-GARCH model for gold and stock markets in China. The empirical model is particularly advantageous in that it allows simultaneous shock transmission in the conditional returns and volatilities. We then use the model's results to compute and analyze the optimal weights and hedge ratios for gold-stock portfolio holdings. We finally assess the diversification and hedging effectiveness of gold in China based on different competing multivariate GARCH-based models.

The literature on gold and other precious metal markets has recently regained particular attention from finance researchers and practitioners. Such a tendency can straightforwardly be explained by the investors' willingness to produce a hedge to diversify away the increasing risk in the stock markets through investing in other asset classes. Indeed, the high volatility and widespread contagion caused by successive financial turbulences and crises over the last decades have prompted investors to consider alternative investment instruments as a part of diversified portfolios of stocks. Not only oil asset but also major precious metals including gold emerged as natural desirable asset classes eligible for portfolio diversification because they offer different volatilities and returns of lower correlations with stocks, at both sector and market levels (Arouri and Nguyen, 2010;[START_REF] Daskalaki | Should investors include commodities in their portfolio after all? New evidence[END_REF]. It is now common that when risk aversion mounts due, for example, to increasing instability and uncertainty in stock markets or to long swings in the price of oil, most investors are directed towards the metals markets and gold in particular, being viewed as the refugee or safe haven asset in time of crises.

It is thus not surprising that the existing literature on the gold asset is part of the one examining the price dynamics, stochastic properties and roles of the commodities markets in portfolio management. A number of studies have questioned the responses of precious metal prices to changes in international institutional and macroeconomic factors (e.g., Kaufman and Winters, 1989;[START_REF] Christie-David | Do macroeconomic news releases affect gold and silver prices[END_REF][START_REF] Heemskerk | Do international commodity prices drive natural resource booms? An empirical analysis of small-scale gold mining in Suriname[END_REF][START_REF] Ciner | On the long-run relationship between gold and silver prices: A note[END_REF][START_REF] Batten | The macroeconomic determinants of volatility in precious metals markets[END_REF]. [START_REF] Sjaastad | The price of gold and the exchange rate[END_REF] find, for example, that fluctuations of floating exchange rates of major currencies, following the breakdown of the Bretton Woods currency arrangements, have led to price instability in the world gold market over the period from January 1982 to December 1990. For their part, [START_REF] Batten | The macroeconomic determinants of volatility in precious metals markets[END_REF] document the sensitivity of precious metals volatility to macroeconomic factors, but with different degrees. The overall results suggest that precious metals are too distinct to be considered a single asset class. In particular, gold volatility can be empirically explained by monetary variables. Gold also seems to be highly sensitive to exchange rate and inflation, which implies that it can offer the best hedge during inflationary pressures and exchange rate fluctuations and an optimal portfolio of precious metals that minimizes risk should be dominated by gold [START_REF] Hammoudeh | Risk management of precious metals[END_REF].

More recent studies have rather looked at the issues of gold price volatility modeling and information transmission between precious metals and other commodities in order to draw the implications of the estimated results for portfolio diversification and hedging strategies involving precious metals. 1 For example, [START_REF] Hammoudeh | Metal volatility in presence of oil and interest rate shocks[END_REF] make use of various GARCH-based models to examine the properties of conditional volatility for three important metals (gold, silver, and copper) while controlling for shocks from world oil prices (WTI) and three-month US Treasury bill interest rate. Using daily three-month futures prices of the three metals, they find that conditional volatility of gold and silver is more persistent, but less sensitive to leverage effects than that of copper. This finding leads to suggest, on the one hand, the importance of accurate volatility modeling especially when gold and silver are used as underlying assets in financial derivatives contracts, and on the other hand the valuable contribution of these two metals in down markets and crisis times. [START_REF] Hammoudeh | Precious metals-exchange rate volatility transmissions and hedging strategies[END_REF] document, from a multivariate VARMA-GARCH model, weak volatility spillovers across precious metals, but strong sensitivity of metal volatility to exchange rate variability. They further point out the role of gold as a hedge against exchange rate risk when optimal weights and hedge ratios are computed.

While understanding the dynamic interactions between gold price changes and stock markets is a crucial element for portfolio designs, risk management and asset pricing, these interactions have only been examined very recently by, to the best of our knowledge, two studies. [START_REF] Baur | Is gold a hedge or a safe haven? An analysis of stocks, bonds and gold[END_REF] show that gold serves as a safe haven for stocks in the US, the UK, and Germany especially after extreme negative shocks affecting stock markets. Gold is also a hedge for stocks in the US and the UK. [START_REF] Baur | Is gold a safe haven? International evidence[END_REF] test whether gold represents a safe haven against stocks of major emerging and developing countries, and show that over the period 1979 to 2009 gold is both a hedge and a safe haven for major European stock markets and the US but not for Australia, Canada, Japan and large emerging markets such as the BRIC countries. Furthermore, gold is found as a strong safe haven for most developed markets during severe episodes of the recent financial crisis.

Our study extends the existing literature into return and volatility spillover between gold and stock markets, portfolio designs in the presence of both gold and stocks, and choice of appropriate models for modeling gold-stock interactions. Specifically, we provide a thorough analysis of how shocks and volatility are transmitted from world gold market to Chinese stock market and from Chinese stock market to world gold market. This research is of particular interest for several reasons. First, none of the previous studies has considered the volatility spillover between gold and stock markets while a better understanding of the transmission mechanisms between them helps building accurate stock valuation models and accurate forecasts of the volatility of both markets. Empirical results from volatility spillover analysis also permit to address several important issues such as hedging strategies, optimal portfolio allocation, and derivatives management with respect to the uncertainties associated with gold price fluctuations. Note that the works of [START_REF] Baur | Is gold a hedge or a safe haven? An analysis of stocks, bonds and gold[END_REF] and [START_REF] Baur | Is gold a safe haven? International evidence[END_REF] only provide grounds for understanding the gold-stock return relationships. Second, the linkages between world gold prices and Chinese stock market have never been examined despite the increasing role of China in the world economy. According to the IMF statistics, China showed, over the last decade, a very high economic growth rate ranging from 8.3% (2001) and 14.2% (2007). This strong economic performance made China the second largest economy in 2010 with respect to the total GDP measured at purchasing power parity. In the meantime, the study of [START_REF] Lai | The role of Chinese stock market in global stock markets: A safe haven or a hedge?[END_REF] suggests that the Chinese stock market is not only a safe haven but also a hedge for global investors' portfolios, in particular during times of financial turbulence. Additionally, we note the continuing surge in China's gold demand in recent years. The World Gold Council's report on May 17, 2012 pointed out that China's investment demand for physical gold is more than doubled from 40.7 tons a year ago to 90.9 tons in the first quarter of 2012. Soaring international market prices, high uncertainties in stock and property markets, and rising inflation expectations are undoubtedly candidate factors that have led to the robust growth of the domestic gold investment market in China.

Like anywhere in the world, gold may thus be seen as a good alternative investment and the best hedge to inflation. Finally, our empirical analysis relies on the multivariate specification of the vector autoregressive -generalized autoregressive conditional heteroskedasticity model (VAR-GARCH) developed by [START_REF] Ling | Asymptotic theory for a vector ARMA-GARCH model[END_REF]. This model offers the possibility to explore the conditional volatility dynamics of the return series as well as the conditional cross effects and volatility spillover between them. It also provides meaningful estimates of the model's parameters with fewer computational complications than other multivariate GARCH specifications, such as the full-factor GARCH model. Furthermore, the findings can be used to analyze the diversification and hedging effectiveness across gold asset and stock market. Some papers have taken the VAR-GARCH approach to investigate the volatility spillover and hedging strategies between Gulf Arab equity sectors [START_REF] Hammoudeh | Shock and volatility spillovers among equity sectors of the Gulf Arab stock markets[END_REF], between previous metals and exchange rates [START_REF] Hammoudeh | Precious metals-exchange rate volatility transmissions and hedging strategies[END_REF], between crude oil spot and futures returns of the Brent and WTI oil price benchmarks [START_REF] Chang | Crude oil hedging strategies using dynamic multivariate GARCH[END_REF], and between oil and stock markets [START_REF] Arouri | Volatility spillovers between oil prices and stock sector returns: Implications for portfolio management[END_REF]. These studies commonly suggest the suitability of the VAR-GARCH model for capturing the dynamic linkages between different asset markets as well as building optimal portfolios. Overall, we find evidence of significant volatility cross effects between world gold price and stock market in China over the period 2004-2011. In particular, past gold shocks are found to play a crucial role in explaining the time-varying patterns of conditional volatility of Chinese stock returns and should thus be accounted for when making volatility forecasts of future stock returns. On the other hand, our portfolio analysis suggests that adding the gold asset to a well-diversified portfolio of Chinese stocks improves its risk-adjusted performance and that stock risk exposures can be effectively hedged using gold. Moreover, we show that the VAR-GARCH model performs better than four alternative multivariate volatility models (CCC-, DCC-, scalar and diagonal BEKK-GARCH) since it provides higher risk-adjusted return performance and hedging effectiveness.

The remainder of this article is structured as follows. Section 2 introduces our empirical methodology. Section 3 presents the sample data and their stochastic properties. Section 4 discusses the empirical findings and shows their implications on optimal portfolio designs and risk management. Section 5 provides some concluding remarks.

Empirical method

Since their apparition, the ARCH/GARCH-family models have received a particular attention from researchers and practitioners as far as the issue of volatility modeling and forecasting of macroeconomic and financial variables is addressed. When the objective is to investigate volatility interdependence and spillovers between different time-series, multivariate GARCH specifications such as the CCC-GARCH model of [START_REF] Bollerslev | Modelling the coherence in short-run nominal exchange rates: a multivariate generalized ARCH approach[END_REF], the BEKK-GARCH model of [START_REF] Engle | Multivariate simultaneous generalized ARCH[END_REF] or the DCC-GARCH model of [START_REF] Engle | Dynamic conditional correlation: a simple class of multivariate GARCH models[END_REF] should be more relevant than univariate models. Empirical results reported in [START_REF] Hassan | Multivariate GARCH model of sector volatility transmission[END_REF], [START_REF] Agnolucci | Volatility in crude oil futures: a comparison of the predictive ability of GARCH and implied volatility models[END_REF], and [START_REF] Kang | Forecasting volatility of crude oil markets[END_REF], among others, confirm effectively the superiority of these models and show that they satisfactorily capture the stylized facts of commodities' conditional volatility as well as the dynamics of volatility interactions.

Given that we attempt to examine the shock and volatility spillover between gold prices and stock market in China as well as to derive the implications of the results on optimal weights and hedge ratios for gold-stock portfolio holdings, the abovementioned multivariate GARCH models are naturally suitable for our research question. However, these models are involved with an excessive number of parameters when the number of variables in the system is important. They can also encounter convergence problems during the estimation process, especially when additional exogenous variables are introduced into the conditional mean and variance equations. These reasons lead us to employ the VAR-GARCH model developed by [START_REF] Ling | Asymptotic theory for a vector ARMA-GARCH model[END_REF]. This model typically combines a multivariate GARCH process and a VAR model. Its main advantage rests on its flexibility to model the conditional mean crosseffects and conditional volatility transmission between the series under consideration with fewer computational complexities than other multivariate volatility models. The ability of the VAR-GARCH model to capture cross-market volatility interactions has been confirmed by recent research [START_REF] Chang | Crude oil hedging strategies using dynamic multivariate GARCH[END_REF][START_REF] Arouri | Volatility spillovers between oil prices and stock sector returns: Implications for portfolio management[END_REF].

In this study, we use the bivariate VAR-GARCH as baseline model to shed light on the diversification and hedging effectiveness of a portfolio involving the gold asset and stocks in China. We also consider four competing bivariate volatility models (CCC-GARCH, DCC-GARCH, diagonal BEKK-GARCH and scalar BEKK-GARCH) for comparison purpose.

Bivariate VAR(1)-GARCH(1,1)

We use the VAR(1)-GARCH(1,1) model to accommodate the return and volatility interdependencies between gold and stock markets. 2 The conditional mean of our empirical model is specified as follows

   = + Φ + = - t t t t t t H R R η ε ε µ 2 / 1 1 (1)
where µ is the vector of constant terms in the VAR.

) , ( ′ = g t s t t r r R
is the vector of returns on the stock market index and the gold price index, respectively. Φ refers to a (2×2) matrix of 2 Using the univariate and bivariate AIC and BIC information criteria to choose the optimal lag length of the univariate GARCH(p,q) process and then that of the bivariate VAR-GARCH model for the pair of gold and stock markets, our results select one lag for both conditional mean and variance equations. They thus lead us to choose the bivariate VAR(1)-GARCH(1,1) specification.

2 1 2 1 1 2 2 2 1 2 1 1 2 1 2 ) ( ) ( g t s g t s s t s s t s s s t h h C h - - - - × + × + × + × + = ε α β ε α β (2) 2 1 2 1 1 2 2 2 1 2 1 1 2 1 2 ) ( ) ( s t g s t g g t g g t g g g t h h C h - - - - × + × + × + × + = ε α β ε α β (3)
It is obvious from Eq. ( 2) and Eq. ( 3) that the volatility transmission across the gold and stock markets over time comes from two sources: (i) the cross values of error terms,

        = 2 1 2 2 2 2 2 1 g g s s A α α α α and         = 2 1 2 2 2 2 2 1 g g s s B β β β β
Let ρ be the constant conditional correlation, the conditional covariance between stock and gold returns is modelled as:

g t s t sg t h h h × × = ρ (4)
According to the above specifications, our empirical model simultaneously allows longrun volatility persistence as well as shock and volatility transmissions between gold and stock markets under consideration. Even though the assumption of constant conditional correlation may be viewed as restrictive given changing economic conditions, the extension to dynamic conditional correlation is not possible for instance because the statistical properties of such a model have not been analyzed at the theoretical level yet. We estimate the parameters of the above bivariate model by quasi-maximum likelihood estimation (QMLE), which is robust to any departure from normality conditions. See [START_REF] Ling | Asymptotic theory for a vector ARMA-GARCH model[END_REF] for further details on necessary and sufficient conditions regarding the consistency of QMLE for the model used.

Competing bivariate GARCH(1,1) models

We define again ) , ( ′ = g t s t t r r R
as the vector of the returns on stock market index and gold price index, and

        = g t sg t sg t s t t h h h h H
as the conditional variance-covariance matrix of stock and gold returns. We specify the conditional mean of the bivariate AR(1)-GARCH(1,1) as follows [START_REF] Engle | Multivariate simultaneous generalized ARCH[END_REF], the CCC-GARCH of [START_REF] Bollerslev | Modelling the coherence in short-run nominal exchange rates: a multivariate generalized ARCH approach[END_REF] and the DCC-GARCH of [START_REF] Engle | Dynamic conditional correlation: a simple class of multivariate GARCH models[END_REF].

   = + Φ + = - t t t t t t H R R η ε ε µ 2 / 1 1 (
The full BEKK-GARCH which imposes positive definiteness restrictions for t H is given by

B H B A A C C H t t t t 1 1 1 ' ' ' ' - - - + + = ε ε (6)
where C is a (n×n) upper triangular matrix, A and B are (n×n) coefficient matrices. C C ' is the decomposition of the intercept matrix. Each element

th j i ) , (
in t H depends on the corre- The diagonal parameterization of the BEKK-GARCH(1,1) is obtained by imposing the diagonality of the parameter matrices A and B. The advantage of this diagonalization is that the model is guaranteed to be positive definite and requires the estimation of fewer parameters than the full-rank parameterization. Consequently, the conditional variance and covariance processes take the following forms:

sponding th j i ) , ( element in ) ( ' 1 1 - - t t ε ε
       + + = + + = + + = - - - - - - - sg t g s g t s t g s sg sg t g t g g t g g g t s t s s t s s s t h C h h C h h C h 1 1 1 1 2 2 1 2 1 2 2 1 2 ) ( ) ( β β ε ε α α β ε α β ε α (7)
where s t h and g t h are the conditional variances of s t r and g t r respectively. Eq. ( 7) indicates that direct volatility transmission between gold and stock returns is not possible because the conditional volatility of each market depends only on its own past shocks and past volatility.

The diagonal BEKK-GARCH model is covariance stationary under the following conditions:

2 2 < + S S β α , 1 2 2 < + g g β α
, and 1 < +

g s g s β β α α .
The scalar BEKK-GARCH is more restricted than the diagonal BEKK-GARCH. It is derived from the full-rank BEKK-GARCH by imposing that the parameter matrices A and B are both equal to the product between a scalar and an identity matrix. Its formal representation is given by

1 2 1 1 2 ' - - - + + = t t t t H b a C C H ε ε (8)
The CCC-MGARCH model of [START_REF] Bollerslev | Modelling the coherence in short-run nominal exchange rates: a multivariate generalized ARCH approach[END_REF] and the DCC-GARCH model of [START_REF] Engle | Dynamic conditional correlation: a simple class of multivariate GARCH models[END_REF] belong to another class of multivariate GARCH processes where the time-varying conditional covariances are parameterized to be proportional to the product of the corresponding conditional standard deviations. Here, the objective is to obtain an intuitive and meaningful economic interpretation of the conditional correlation coefficients.

Applying to our research question, the variance-covariance matrix of the bivariate CCC-GARCH(1,1) model is defined as follows:

t t t PD D H = (9)
where ) , (

g t s t t h h diag D = ,
       = + + = + + = - - - - g t s t sg t g t g g t g g g t s t s s t s s s t h h h h C h h C h ρ β ε α β ε α 1 2 1 1 2 1 ) ( ) ( (10
) [START_REF] Bollerslev | Modelling the coherence in short-run nominal exchange rates: a multivariate generalized ARCH approach[END_REF] shows that the positiveness of the coefficients associated with ARCH and GARCH terms is indeed not required to obtain a positive definite matrix P. The model is covariance stationary when the roots of

( ) ( ) 2 / 1 2 / 1 ) ( ) ( - - = t t t t Q diag Q Q diag P (11) where ) ( ij t t q Q =
is a (2×2) symmetric positive definite matrix given by ( )

1 1 1 1 - - - + ′ + - - = t t t t Q Q Q β η αη β α (12)
In Eq. ( 12), α and β are non-negative scalars such that 1 <

+ β α

, and Q is a (2×2) matrix of unconditional correlations of the standardized errors t η . The conditional variances are specified as univariate GARCH(1,1) processes, similar to those of the CCC-GARCH model. [START_REF] Engle | Dynamic conditional correlation: a simple class of multivariate GARCH models[END_REF] indicates that the specification of the dynamic conditional correlation structure presents no obstacle to model estimation.

Overall, the bivariate GARCH-type models presented above do not explicitly allow for shock and volatility cross-effects between gold and stock market returns, as compared to the VAR-GARCH model. It is thus expected that our benchmark model provides a better description of the dynamic linkages between markets of interest and allows for more accurate decisions regarding portfolio diversification and hedging effectiveness. Table 1 presents selected descriptive statistics for the return series. The gold market provided higher daily returns than the Chinese stock market over our study period. The severe impacts of the recent global financial crisis 2007-2009 seem to be the main reason that explains the underperformance of the Chinese stock market as well as the performance of the world gold market. The unconditional volatility, as measured by standard deviations, is substantially higher for the stock market than for the gold market. As a result, gold was a better investment with higher risk-adjusted returns. This finding thus suggests that gold might be a good hedge for portfolios of stocks, especially in times of crises and bear market periods. 3 We also estimated our empirical models with gold spot, also traded at COMEX in New York, but the results remain intact. These results can be made entirely available under request addressed to corresponding author. Note finally that the correlation between gold spot and gold futures is 0.853 over the study period. The table reports the basic statistics of return series as well as their stochastic properties. Jarque-Bera is the empirical statistics of the Jarque-Bera test for normality based on skewness and excess kurtosis. Q(15) and Q 2 (15) are the empirical statistics of the Ljung-Box tests for autocorrelations applied to return and squared return series. ARCH refers to the empirical statistics of the [START_REF] Engle | Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation[END_REF] test for conditional heteroscedasticity. ADF and PP are the empirical statistics of the Augmented Dickey-Fuller and Phillips-Perron unit root tests. We present the model with a constant and a trend, but the results are similar when models without a constant and/or a trend are used. ++ and +++ indicate the rejection of the null hypotheses of normality, no autocorrelation, no ARCH effects, and unit root at the 5% and 1% levels, respectively.

Data and preliminary analysis

With negative skewness coefficients and kurtosis coefficients above three, the distributions of Chinese stock market and gold returns are typically asymmetric, and the probability of observing large negative returns is higher than that of a normal distribution. The Jarque-Bera test statistics clearly confirm the rejection of the null hypothesis of normality for both return series. The results from the Ljung-Box test indicate evidence of autocorrelation in return and squared return series for both gold and stock markets. The ARCH test for conditional heteroscedasticity provides strong evidence of ARCH effects in both return series, which consequently suggests the usefulness and suitability of GARCH-type models for examining volatility dynamics and transmission of gold and stock markets in China. Finally, two commonlyused unit root tests, Augmented Dickey-Fuller (ADF) test and Phillips-Perron (PP) test, are performed in order to examine the stationarity property of the series considered. The results indicate that return series are stationary, and thus they can be straightforwardly used for further analysis.

Results and portfolio implications

This section discusses the estimation results of our bivariate VAR(1)-GARCH(1,1) model for gold returns and stock market returns in China. We particularly focus on the extent of volatility transmission between the markets considered. The competing models (CCC-GARCH, DCC-GARCH, diagonal BEKK-GARCH, and scalar BEKK-GARCH) are also estimated, but the results are not shown here in order to conserve space. Finally, we compare the diversification and hedging effectiveness across models.

Volatility cross-effects in gold and stock markets in China

Table 2 presents the estimation results of our bivariate VAR(1)-GARCH(1,1) model, together with statistical tests applied to standardized residuals. Taking a close look at the estimates of the mean equations, we find that own autoregressive parameters are not significant for both markets under consideration. The one-period lagged gold returns are found to significantly affect the current returns of Chinese stock market at the 1% level. This implies that one can predict Chinese stock returns by using past gold returns over time. Some recent papers also document that Chinese stock market is not weak-form efficient almost all the time (see, e.g., [START_REF] Groenwold | The Chinese Stock Market: Efficiency, Predictability and Profitability[END_REF][START_REF] Gao | Calendar effects in Chinese stock market[END_REF]. Inversely, past stock market returns in China has no impact on gold returns. Notes: Stock(-1), Gold(-1), and ρ are the one-period lagged returns of Chinese stock market and gold market, and conditional constant correlation, respectively. Const(µ) and Const(v) are the constant terms in the conditional mean and variance equations. * , ** , and *** indicate significance at the 10%, 5%, and 1% levels, respectively. The AIC and BIC criteria measure the relative goodness of fit of the estimated model. JB, Q(15), Q²(15), and ARCH(15) refer to the empirical statistics of the Jarque-Bera test for normality, the Ljung-Box tests for autocorrelations of order 15 applied to standardized residuals and squared standardized residuals, and the [START_REF] Engle | Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation[END_REF] test for conditional heteroscedasticity of order 15, respectively. + , ++ , and +++ indicate the rejection of the null hypothesis of associated statistical tests at the 10%, 5%, and 1% levels, respectively.

As to the coefficients associated with of ARCH and GARCH terms, which capture shock and volatility independence in the conditional variance equations, we find common patterns for both gold returns and stock returns in China. First, these coefficients appear to be highly significant. Second, the estimated conditional volatility series do not change very rapidly under the effects of return innovations given the small size of ARCH coefficients.

They tend instead to develop gradually over time with respect to substantial effects of past volatility, advocated by the large values of GARCH coefficients. Accordingly, investors and fund managers seeking profit from trading gold and Chinese stocks may consider active investment strategies based on volatility persistence and current market trends. It would be advisable, for example, to increase the amount of portfolio investment if markets are actually rising and to decrease it if they are falling, all while keeping in mind that the viability of such strategies depends on the stability and the strength of performance between successive periods.

The extent of volatility transmission between gold market and Chinese stock market is particularly interesting. Our findings show that past gold shocks play a crucial role in explaining the time-dynamics of conditional volatility of stock returns in China and should thus be accounted for when making volatility forecasts of future stock returns. Moreover, the conditional volatility of gold returns is significantly affected by unpredicted changes in the returns on the MSCI China index, ( ) 2 1 s t-ε . A shock to Chinese stock market, regardless of its sign, thus implies an increase in the volatility of gold returns. On the other hand, past volatility of Chinese stock market, represented by s t h 1 -, has no significant effects on the gold return volatility. As for the opposite direction, the impact of past gold volatility, g t h 1 -, on the conditional volatility of Chinese stock market is also statistically insignificant.

Turning out to the estimates of the constant conditional correlation (CCC), we find that Chinese stock market returns are positively correlated with gold returns, but the cross-market correlation is relatively weak (0.137). One may expect, in view of this finding, substantial gains from having both assets in the same portfolio. Moreover, it is obvious that the significant volatility cross-effects we show previously require portfolio managers and investors to quantify the optimal weights and hedging ratios to properly deal with the comovement between gold and Chinese stock returns.

Lastly, the results of diagnostic tests based on standardized residuals show that the departure from normality and the serial correlation are greatly reduced and that there are no ARCH effects, as compared to test statistics we report for the returns in Table 1. Thus, the bivariate VAR(1)-GARCH (1,1) model offers a flexible way to capture the joint dynamics of gold returns and Chinese stock market returns.

Optimal portfolio designs and hedging ratios in the presence of gold

We now show the implications of our previous results on gold-stock optimal portfolio. To this end, we consider a hedged portfolio made up of gold and Chinese stock market index in which an investor seeks to protect himself from exposure to stock price movements by investing in the gold asset. In practice, the investor's objective is to minimize the risk of his gold-stock portfolio while keeping the same expected returns. Following [START_REF] Kroner | Modeling asymmetric movements of asset prices[END_REF] we presented in Section 2. When short selling is not allowed, the mean-variance portfolio optimization approach imposes the following constraints on the optimal weight of gold: From Eq. ( 13), the proportion of wealth that the investor put on the Chinese stock market index is

     > ≤ ≤ < = 1 , 1 1 0 , 0 ,
gs t w - 1 .
Now if the objective of our investor is to optimally hedge the risk of his investment in stock markets, he should take an appropriate position on the gold market so that it minimizes the risk of the hedged portfolio. Concretely, a long position (buying) of one dollar on the stock segment must be hedged by a short position (selling) of sg t β dollars on the gold asset.

Following [START_REF] Kroner | Time-varying distributions and dynamic hedging with foreign currency futures[END_REF], the optimal hedge ratio sg t β can be expressed as The average values of realized optimal weights sg t w and optimal hedge ratios sg t β are reported in Table 3. A glance at the coefficients shows that the optimal weight for the gold asset in the hedged portfolios is 0.330 according to our VAR(1)-GARCH(1,1) model and varies from 0.322 to 0.333 with respect to competing bivariate GARCH-based models. The highest optimal weight for gold of 0.333 is obtained for the CCC-GARCH and DCC-GARCH models, while the scalar BEKK-GARCH model provides the lowest one (0.322). Overall, our results show that, to minimize the risk without lowering the expected return of the gold-stock portfolio, the investor operating in Chinese stock markets should hold more stocks than gold. This finding can be effectively explained by the low correlation of gold returns with stock returns in China, and then simply adding a small portion of gold to a diversified portfolio of Chinese stocks will reduce substantially its overall risk for a given level of expected return. Notes: The table reports average optimal weight of gold and hedge ratios for an gold-stock portfolio using conditional variances and covariance estimated from our benchmark model and four competitive volatility-spillover models: VAR(1)-GARCH(1,1), diagonal BEKK-GARCH(1,1), DCC-GARCH(1,1), CCC-GARCH(1,1), and scalar BEKK-GARCH(1,1). For all models considered, the conditional mean equations contain a constant term and an autoregressive component.

The average optimal hedge ratios, gs β , are generally low and do not exceed 0.2. This finding has several important implications for short hedgers. First, the low hedge ratios suggest that stock investment risk can simply be hedged by taking short positions in the gold futures markets. The largest ratio is obtained from the VAR-GARCH model (0.197), followed closely by the DCC-GARCH and the CCC-GARCH. This implies that one dollar long (buy) in the Chinese stock market index should require the investor to go short 19.7 cents in the gold futures market. The lowest hedge ratio is obtained in the case of the scalar BEEK-GARCH model (0.178), meaning that the investor must sell the smallest quantity of gold in the gold futures market. Altogether, our findings for optimal hedge ratios support with the view that the gold asset should be an integral part of a diversified portfolio of stocks and help reduce the risk of the hedged portfolio, particularly in periods of crisis.

Diversification and hedging effectiveness

In subsection 4.2 we show, based on the estimations of selected bivariate GARCH models, that adding the gold asset in the diversified portfolio of stocks leads to improve its riskadjusted performance. One may, however, wonder how effective are the diversification and the hedging associated with the gold asset? For this reason, we propose to run portfolio simulations to empirically examine the diversification and hedging effectiveness of gold.

These portfolio simulations are based on our optimal portfolio weights and hedging ratios.

More precisely, we use the estimates of our VAR-GARCH and four competing bivariate GARCH models to build two distinct portfolios: a portfolio composed of Chinese stocks only (PF I) and a gold-stock portfolio with the optimal weights calculated in subsection 4.2 (PF II).

For each model, the effectiveness of the portfolio diversification is evaluated by comparing the realized risk and return characteristics of the considered portfolios (PF I vs. PF II). In the highest mean, standard deviation, and risk-return trade-off. PF I is a portfolio of 100% stocks. PF II is a weighted gold-stock portfolio in which the weights are given by the optimal weights reported in Table 4.

The results from portfolio simulations in Table 4 show that adding the gold asset to the diversified portfolio of stocks improves its risk-adjusted return ratios regardless of the models used. More interestingly, our benchmark VAR-GARCH model provides the best risk-adjusted return ratio, closely followed by the DCC and CCC-GARCH models. 14). Numbers in boldface indicate the hedged portfolio with lowest variance and the highest variance reduction. PF I is a portfolio of 100% stocks. PF II is a weighted gold-stock portfolio in which the weights are given by the optimal weights reported in Table 4.

Table 5 presents the hedging effectiveness (HE) ratios for the models we consider. The results show that hedging strategies involving stocks and gold assets lead to reduce considerably the portfolio's risk (variance), as compared to the risk of the portfolio made up of stocks only. The variance reduction ranges from 47.48% for the scalar BEKK-GARCH model to 49.04% for the VAR-GARCH model. This finding, consistent with the analysis of portfolio designs and diversification, thus consolidates the dominant position of the VAR-GARCH model as it helps reducing the largest part of the initial portfolio's variance. The diagonal and scalar BEKK-GARCH models are the worst models in terms of portfolio's variance reduction. [START_REF] Chang | Crude oil hedging strategies using dynamic multivariate GARCH[END_REF] reach similar conclusion when they use the same models to calculate optimal portfolio weights and optimal hedge ratios for the crude oil spot and futures returns of two major crude oil benchmarks, Brent and WTI.

Conclusion

In this paper, we investigate the extent of volatility transmission, portfolio designs, and hedging effectiveness in gold and stock markets in China. We employ the recent VAR-GARCH modeling approach, which allows for volatility spillover cross effects, we find significant volatility transmission between Chinese stock market and world gold market. The examination of optimal weights and hedge ratios suggests that optimal portfolios should have stocks outweigh gold assets and that the stock investment risk can be hedged with relatively low hedging costs by taking a short position in the gold futures markets. In particular, we show that optimally hedged gold-stock portfolios outperform traditional portfolios of stocks regardless of bivariate volatility models, and that our baseline VAR-GARCH model is the best performing model in terms of both diversification and hedging effectiveness. On the whole, gold assets can be considered a dynamic and valuable asset class that helps to improve the risk-adjusted performance of a well-diversified portfolio of stocks.

  capture the impact of direct effects of shock transmission, and (ii) the cross values of lagged conditional volatilities, g t h 1 -and s t h 1 -, which directly account for the risk transfer between markets. The stationary condition requires that the roots of the equation the unit circle, where L is a lag polynomial, I 2 is a (2×2) identity matrix, and

  , past shocks and volatility are allowed to directly spill over from a market to another, and they are captured by coefficients of A and B matrices. Since the BEKK-GARCH model suffers from the curve of dimensionality and may encounter convergence problem, researchers often use its two restricted versions: the diagonal BEKK-GARCH and the scalar BEKK-GARCH.

  proposes the DCC-GARCH(1,1) model that allows the conditional correlation matrix to vary over time as follows

  Our sample data consist of daily time series of the Chinese stock market and 3-month gold futures prices (GOLD3M) over the period fromMarch 22, 2004 to March 31, 2011. The MSCI China Index which is a free-float weighted equity index developed with a base value of 100 as of December 31 1992 by Morgan Stanley Capital International is used to represent the overall performance of the Chinese stock market. The gold futures, widely traded at COMEX (Commodity Exchange) in New York, are used to represent the world gold market fluctuations. Their prices are measured in US dollars per troy ounce. 3 Data were extracted from MSCI and Bloomberg databases. We compute the return series by taking the differences in the logarithm of two consecutive prices.

Figure

  Figure 1. Stock and gold price dynamics

  the

Table 1 . Descriptive statistics and stochastic properties of return series

 1

		MSCI China	Gold
	Mean (%)	0.057	0.067
	Maximum (%)	14.044	8.625
	Minimum (%)	-12.836	-7.574
	Standard deviation	2.010	1.261
	Skewness	-0.044	-0.131
	Kurtosis	9.459	7.202
	Jarque-Bera	3187.202 +++	1354.089 +++
	Q(15)	37.346 +++	43.728 +++
	Q 2 (15)	1870.689 +++	378.195 +++
	ARCH(15)	38.061 +++	40.761 +++
	ADF	-42.002 +++	-41.760 +++
	PP	-42.002 +++	-41.785 +++
	Notes:

Table 2 . Estimates of VAR(1)-GARCH(1,1) model for gold returns and stock market returns in China

 2

		MSCI China		Gold
	Const(µ)	0.084 ***		0.067 ***
		(0.032)		(0.024)
	Stock(-1)	0.023		0.009
		(0.025)		(0.015)
	Gold(-1)	0.240 ***		0.001
		(0.028)		(0.026)
	Const(v)	0.138 ***		0.091 ***
		(0.040)		(0.016)
	2 1 ) t-( s ε	0.295 *** (0.018)		0.050 ** (0.023)
	2 1 ) t-( g ε	0.105 * (0.055)		0.184 *** (0.015)
	s t h 1 -	0.947 *** (0.006)		-0.0002 (7.238)
	g t h 1 -ρ	0.107 (0.088)	0.137 ***	0.977 *** (0.004)
			(0.022)
	LogL		-6257.688
	AIC		6.872
	SIC		6.923
	JB	59.333 +++		668.278
	Q(15)	30.572 ++		12.733
	Q²(15)	19.937		42.675 +++
	ARCH(15)	14.665		13.168

  , we determine the optimal holding weight of gold in a one-dollar portfolio of

	stock/gold at time t, denoted by gs t w , as follows
	gs t w	=	s t h	sg t h h t sg + -2 s t h -	t h	g	(13)
	where g t h , s t h and sg t h refer respectively to the conditional volatility of the gold returns, the
	conditional volatility of the Chinese stock market index returns and the conditional covariance
	between gold and stock returns at time t. These series are estimated from our benchmark
	bivariate VAR(1)-GARCH(1,1) model and four competing bivariate GARCH-based models

  0

		if	gs t w
	gs t w	gs t w	if	gs t w
		if	gs t w

Table 3 . Optimal weights and hedge ratios for portfolio of gold and stocks in China

 3

			VAR-GARCH CCC-GARCH DCC-GARCH	Scalar BEKK-GARCH	Diagonal BEKK-GARCH
	w	gs	0.330	0.333	0.333	0.322	0.325
	β	gs	0.197	0.192	0.194	0.178	0.188

  meantime, the effectiveness of hedging across constructed portfolios can be assessed by examining the realized hedging errors (HE) which are determined as follows(Ku et al., 2007)

		HE	=	   	-unheged Var unheged Var hedged Var	   	(15)
	where	hedged Var	refers to the variance of the hedged portfolio's returns (PF II), and

unhedged

Var the variance of the unhedged portfolio's returns (PF I). Eq. (

15

) indicates that the higher the HE ratio, the more effective the hedging in terms of portfolios' variance reduction. The model that generates the highest HE ratio is the best one to be used for building gold-stock hedging strategy.

Table 4 . Portfolio designs and diversification in presence of the gold asset

 4 This table compares the realized risk-adjusted returns, measured by relating each portfolio's mean to its standard deviation, of different portfolios. Figures in boldface indicate

	Mean	Standard deviation	Realized risk-adjusted returns (×100)

Table 5 . Hedging effectiveness

 5 This table reports the portfolio variance and hedge effectiveness ratios, computed using Equation (

	Variance (%)	Hedge effectiveness (%)

See Arouri et al. (2012) for a detailed discussion of this literature.


Didier Henrion

Jamal Daafouz

Mathieu Claeys

Optimal switching control design for polynomial systems: an LMI approach

We propose a new LMI approach to the design of optimal switching sequences for polynomial dynamical systems with state constraints. We formulate the switching design problem as an optimal control problem which is then relaxed to a linear programming (LP) problem in the space of occupation measures. This infinitedimensional LP can be solved numerically and approximately with a hierarchy of convex finite-dimensional LMIs. In contrast with most of the existing work on LMI methods, we have a guarantee of global optimality, in the sense that we obtain an asympotically converging (i.e. with vanishing conservatism) hierarchy of lower bounds on the achievable performance. We also explain how to construct an almost optimal switching sequence.

Introduction

A switched system is a particular class of a hybrid system that consists of a set of dynamical subsystems, one of which is active at any instant of time, and a policy for activating and deactivating the subsystems. One may encounter such dynamical systems in a wide variety of application domains such as automotive industry, power systems, aircraft and traffic control, and more generally the area of embedded systems. Switched systems have been the concern of many researchers and many results are available for stability analysis and control design. They put in evidence the important fact that it is possible to orchestrate the subsystems through an adequate switching strategy in order to impose global stability. Interested readers may refer to the survey papers [START_REF] Decarlo | Perspectives and results on the stability and stabilizability of hybrid systems[END_REF][START_REF] Liberzon | Basic problems in stability and design of switched systems[END_REF][START_REF] Shorten | Stability criteria for switched and hybrid systems[END_REF][START_REF] Lin | Stability and stabilizability of switched linear systems: A survey of recent results[END_REF] and the interesting and useful books [START_REF] Liberzon | Switching in systems and control[END_REF][START_REF] Sun | Switched linear systems: control and design[END_REF] and the references therein.

In this context, switching plays a major role for stability and performance properties. Indeed, switched systems are generally controlled by switched controllers and the control signal is intrinsically discontinuous. As far as optimality is concerned, several results are also available in two main different contexts:

• the first category of methods exploits necessary optimality conditions, in the form of Pontryagin's maximum principle (the so-called indirect approaches), or through a large nonlinear discretization of the problem (the so-called direct approaches), see [START_REF] Bengea | Optimal control of switching systems[END_REF][START_REF] Branicky | A unified framework for hybrid control: model and optimal control theory[END_REF][START_REF] Cassandras | Optimal control of a class of hybrid systems[END_REF][START_REF] Hedlund | Optimal control of hybrid systems[END_REF][START_REF] Piccoli | Hybrid systems and optimal control[END_REF][START_REF] Riedinger | Time optimal control of hybrid systems[END_REF][START_REF] Riedinger | An optimal control approach for hybrid systems[END_REF][START_REF] Seatzu | Optimal control of continuous-time switched affine systems[END_REF][START_REF] Sussmann | A maximum principle for hybrid optimization[END_REF][START_REF] Shaikh | On the hybrid optimal control problem: theory and algorithms[END_REF][START_REF] Xu | Results and perspectives on computational methods for optimal control of switched systems[END_REF] for details. Therefore only local optimality can be guaranteed for general nonlinear problems, even when discretization can be properly controlled;

• the second category collects extensions of the performance indexes H 2 and H ∞ originally developped for linear time invariant systems without switching, and use the flexibility of Lyapunov's approach, see for instance [START_REF] Geromel | Dynamic output feedback control of switched linear systems[END_REF][START_REF] Deaecto | Dynamic output feedback H ∞ control of switched linear systems[END_REF] and references therein.

Even for linear switched systems, the proposed results are based on nonconvex optimization problems (e.g. bilinear matrix inequality conditions) difficult to solve directly. Sufficient linear matrix inequality (LMI) design conditions may be obtained, but at the price of introducing a conservatism (pessimism) which is hard, if not impossible, to evaluate. Since the computation of this optimal strategy is a difficult task, a suboptimal solution is of interest only when it is proved to be consistent, meaning that it imposes to the switched system a performance not worse than the one produced by each isolated subsystem [START_REF] Geromel | Suboptimal switching control consistency analysis for switched linear systems[END_REF].

Despite the interest of these existing approaches, the optimal control problem is not completely solved for switched systems and new strategies are more than welcome, as computationally viable design techniques are missing. In this paper, we consider the problem of designing optimal switching rules in the case of polynomial switched dynamical systems. Classically, we formulate the optimal control switching problem as an optimal control problem with controls being functions of time valued in {0, 1}, and we relax it into a control problem with controls being functions of time valued in [0, 1]. In contrast with existing approaches following this relaxation strategy, see e.g. [START_REF] Bengea | Optimal control of switching systems[END_REF][START_REF] Riedinger | Time optimal control of hybrid systems[END_REF], relying on Pontryagin's maximum principle, our aim is to apply the approach of [START_REF] Lasserre | Nonlinear optimal control via occupation measures and LMI relaxations[END_REF], which consists in modeling control and trajectory functions as occupation measures. This allows for a convex linear programming (LP) formulation of the optimal control problem. This infinitedimensional LP can be solved numerically and approximately with a hierarchy of convex finite-dimensional LMIs. On the one hand, our approach follows the optimal control modeling framework. On the other hand, it exploits the flexibility and computational efficiency of the convex LMI framework. In contrast with most of the existing work on LMI methods, we have a guarantee of global optimality, in the sense that we obtain an asympotically converging (i.e. with vanishing conservatism) hierarchy of lower bounds on the achievable performance.

The paper is organized as follows: in Section 2 we state the optimal switching problem to be solved, and we propose an alternative, relaxed formulation allowing chattering of the trajectories. In Section 3 we introduce occupation measures as a device to linearize the optimal control problem into an LP problem in the cone of nonnegative measures. In Section 4 we explain how to solve the resulting infinite-dimensional LP with a converging hierarchy of finite-dimensional LMI problems. As explained in Section 5, an approximate optimal switching sequence can be extracted from the solutions of the LMI problem, and this is illustrated with classical examples in Section 6. Finally, the paper ends with a sketch of further research lines.

Optimal switching problem

Consider the optimal control problem

p * = inf T 0 l σ(t) (t, x(t))dt s.t. ẋ(t) = f σ(t) (t, x(t)), σ(t) ∈ {1, 2, . . . , m} x(0) ∈ X 0 , x(T ) ∈ X T x(t) ∈ X, t ∈ [0, T ] (1)
with given polynomial velocity field f σ(t) ∈ R[t, x] n and given polynomial Lagrangian l σ(t) ∈ R[t, x] indexed by an integer-valued signal σ : [0, T ] → {1, 2, . . . , m}. System state x(t) belongs to a given compact semialgebraic set X ⊂ R n for all t ∈ [0, T ] and the initial state x(0) resp. final state x(T ) are constrained to a given compact semialgebraic set X 0 ⊂ X resp. X T ⊂ X. In problem (1) the infimum is w.r.t. sequence σ and terminal time T .

In this paper, for the sake of simplicity, we assume that the terminal time T is finite, that is, we do not consider the asymptotic behavior. Typically, if a solution to problem (1) is expected for a very large or infinite terminal time, we must reformulate the problem by relaxing the state constraints.

Optimal control problem (1) can then be equivalently written as

p * = inf u T 0 m k=1 l k (t, x(t))u k (t)dt s.t. dx(t) = m k=1 f k (t, x(t))u k (t)dt x(0) ∈ X 0 , x(T ) ∈ X T x(t) ∈ X, u(t) ∈ U, t ∈ [0, T ] (2)
where the infimum is with respect to a time-varying vector u(t) which belongs for all t ∈ [0, T ] to the (nonconvex) discrete set U := {(1, 0, . . . , 0), (0, 1, . . . , 0), . . . , (0, 0, . . . , 1)} ⊂ R m .

In general the infimum in problem (2) is not attained (see our numerical examples later on) and the problem is relaxed to

p * R = inf T 0 m k=1 l k (t, x(t))u k (t)dt s.t. dx(t) = m k=1 f k (t, x(t))u k (t)dt x(0) ∈ X 0 , x(T ) ∈ X T x(t) ∈ X, u(t) ∈ conv U, t ∈ [0, T ] (3)
where the minimization is now with respect to a time-varying vector u(t) which belongs for all t ∈ [0, T ] to the (convex) simplex

conv U = {u ∈ R m : m k=1 u k = 1, u k ≥ 0, k = 1, . . . , m}.
In [START_REF] Bengea | Optimal control of switching systems[END_REF], problem (3) is called the embedding of problem (1), and it is proved that the set of trajectories of problem (1) is dense (w.r.t. the uniform norm in the space of continuous functions of time) in the set of trajectories of embedded problem [START_REF] Branicky | A unified framework for hybrid control: model and optimal control theory[END_REF]. Note however that these authors consider the more general problem of switching design in the presence of additional bounded controls in each individual dynamics. To cope with chattering effects due to the simultaneous presence of controls and (initial and terminal) state constraints, they have to introduce a further relaxation of the embedded control problem. In this paper, we do not have controls in the dynamics, and the only design parameter is the switching sequence.

An equivalent way of writing the dynamics in problem ( 3) is via a differential inclusion

ẋ(t) ∈ conv {f 1 (t, x(t)), . . . , f m (t, x(t))}. (4)
By this it is meant that at time t, the state velocity ẋ(t) can be any convex combination of the vector fields f k (t, x(t)), k = 1, . . . , m, see e.g. [4, Section 3.1] for a tutorial introduction. For this reason, problem (3) is also sometimes called the convexification of problem [START_REF] Batenkov | Complete algebraic reconstruction of piecewise-smooth functions from Fourier data[END_REF].

Since problem ( 3) is a relaxation of problem (1), it holds p * R ≤ p * . For most of the physically relevant problems, and especially when the state constraints in problem (1) are not overly stringent, it actually holds that p * R = p * . For a discussion about the cases for which p * R < p * , please refer to [16, Appendix C] and references therein.

Occupation measures

Given an initial condition x 0 ∈ X 0 and an admissible control u(t), denote by x(t|x 0 , u), t ∈ [0, T ], the corresponding admissible trajectory, an absolutely continuous function of time with values in X. Define the occupation measure

µ(A × B|x 0 , u) := T 0 I A×B (t, x(t|x 0 , u))dt
for all subsets A × B in the Borel σ-algebra of subsets of [0, T ] × X, where I A (x) is the indicator function of set A, equal to one if x ∈ A, and zero otherwise. We write x(t|x 0 , u) resp. µ(dt, dx|x 0 , u) to emphasize the dependence of x resp. µ on initial condition x 0 and control u, but for conciseness we also use the notation x(t) resp. µ(dt, dx). The occupation measure can be disintegrated into

µ(A × B) = A ξ(B|t)ω(dt)
where ξ(dx|t) is the distribution of x ∈ R n , conditional on t, and ω(dt) is the marginal w.r.t. time t, which models the control action as a measure on [0, T ]. The conditional ξ is a stochastic kernel, in the sense that for all t ∈ [0, T ], ξ(.|t) is a probability measure on X, and for every B in the Borel σ-algebra of subsets of X, ξ(B|.) is a Borel measurable function on [0, T ]. An equivalent definition is ξ(B|t) = I B (x(t)) = δ x(t) (B) where δ is the Dirac measure. The occupation measure encodes the system trajectory, and the value T 0 µ(dt, B) = µ([0, T ]×B) is equal to the total time spent by the trajectory in set B ⊂ X. Note also that time integration of any smooth test function v : [0, T ] × X → R along a trajectory becomes a time and space integration against µ, i.e.

T 0 v(t, x(t))dt = T 0 X v(t, x)µ(dt, dx) = vµ.
In optimal control problem (3), we associate an occupation measure

µ k (dt, dx) = ξ k (dx|t)ω k (t)
for each system mode k = 1, . . . , m, so that globally

m k=1 µ k = µ
is the occupation measure of a system trajectory subject to switching. The marginal ω k is the control, modeled as a measure which is absolutely continuous w.r.t. the Lebesgue measure, i.e. such that

X µ k (dt, dx) = ω k (dt) = u k (t)dt
for some measurable control function u k (t), k = 1, . . . , m. The system dynamics in problem (3) can then be expressed as

dx(t) = m k=1 f k (x(t))ω k (dt) (5)
where the controls are now measures ω k . To enforce that u(t) ∈ conv U for almost all times t ∈ [0, T ], we add the constraint

m k=1 ω k (dt) = I [0,T ] (t)dt (6)
where the right hand side is the Lebesgue measure, or uniform measure on [0, T ].

Given a smooth test function v : [0, T ] × X → R and an admissible trajectory x(t) with occupation measure µ(dt, dx), it holds

T 0 dv(t, x(t)) = v(T, x(T )) -v(0, x(0)) = T 0 ∂v ∂t (t, x(t)) + k grad v(t, x(t))f k (t, x(t))u k (t) dt = T 0 X ∂v ∂t (t, x)µ(dt, dx)+ k grad v(t, x)f k (t, x)µ k (dt, dx)) = k ∂v ∂t µ k + grad vf k µ k .
Now, consider that the initial state is not a single vector x 0 but a random vector whose distribution is ruled by a probability measure µ 0 , so that at time t the state x(t) is also modeled by a probability measure µ t (.) := ξ(.|t), not necessarily equal to δ x(t) . The interpretation is that µ t (B) is the probability that the state x(t) belongs to a set B ⊂ X. Optimal control problem (3) can then be formulated as a linear programming (LP) problem:

p * M = inf k l k µ k s.t. vµ T -vµ 0 = k ∂v ∂t µ k + grad vf k µ k k wµ k = w ∀v ∈ C 1 ([0, T ] × X), ∀w ∈ C 1 ([0, T ]) (7)
where the infimum is w.r.t. measures

µ 0 ∈ M + (X 0 ), µ T ∈ M + (X T ), µ k ∈ M + ([0, T ] × X), k = 1, .
. . , m with M + (A) denoting the cone of finite nonnegative measures supported on A, identified as the dual of the cone of nonnegative continuous functions supported on A.

It follows readily that p * ≥ p * R ≥ p * M , and under some additional assumptions it should be possible to prove that p * R = p * M and that the marginal densities u k extracted from solutions µ k of problem [START_REF] Claeys | Measures and LMI for impulsive optimal control with applications to space rendezvous problems[END_REF] are optimal for problem (2) and hence problem [START_REF] Batenkov | Complete algebraic reconstruction of piecewise-smooth functions from Fourier data[END_REF]. We leave the rigorous statement and its proof for an extended version of this paper.

Note that the use of relaxations and LP formulations of optimal control problems (on ordinary differential equations and partial differential equations) is classical, and can be traced back to the work by L. C. Young, Filippov, and then Warga and Gamkrelidze, amongst many others. For more details and a historical survey, see e.g. [START_REF] Fattorini | Infinite dimensional optimization and control theory[END_REF]Part III].

Solving the LP on measures

To summarize, we have formulated our relaxed optimal switching control problem (3) as the convex LP [START_REF] Claeys | Measures and LMI for impulsive optimal control with applications to space rendezvous problems[END_REF] in the space of measures. This can be seen as an extension of the approach of [START_REF] Lasserre | Nonlinear optimal control via occupation measures and LMI relaxations[END_REF] which was originally designed for classical optimal control problems. Alternatively, this can also be understood as an application of the approach of [START_REF] Claeys | Measures and LMI for impulsive optimal control with applications to space rendezvous problems[END_REF] where the control measures are restricted to be absolutely continuous w.r.t. time. Indeed, absolute continuity of the control measures is enforced by relation [START_REF] Cassandras | Optimal control of a class of hybrid systems[END_REF]. The infinite-dimensional LP on measures [START_REF] Claeys | Measures and LMI for impulsive optimal control with applications to space rendezvous problems[END_REF] can be solved approximately by a hierarchy of finite-dimensional linear matrix inequality (LMI) problems, see [START_REF] Lasserre | Nonlinear optimal control via occupation measures and LMI relaxations[END_REF][START_REF] Claeys | Measures and LMI for impulsive optimal control with applications to space rendezvous problems[END_REF][START_REF] Henrion | Convex computation of the region of attraction of polynomial control systems[END_REF] for details (not reproduced here). The main idea behind the hierarchy is to manipulate each measure via its moments truncated to degree 2d, where d is a relaxation order, and to use necessary LMI conditions for a vector to contain moments of a measure. The hierarchy then consists of LMI problems of increasing sizes, and it provides a sequence of lower bounds p * d ≤ p * which is monotonically increasing, i.e. p * d ≤ p * d+1 and asymptotically converging, i.e. lim d→∞ p * d = p * . The number of variables N d at the LMI relaxation of order d grows linearly in m (the number of modes), and polynomially in d, but the exponent is a linear function of n (the number of states). In practice, given the current state-of-the-art in general-purpose LMI solvers and personal computers, we can expect an LMI problem to be solved in a matter of a few minutes provided the problem is reasonably well-conditioned and N d ≤ 5000.

Optimal switching sequence

Let

y k,α := T 0 X t α µ k (dt, dx) = T 0 t α ω k (dt), α = 0, 1, . . .
denote the moments of measure ω k , k = 1, . . . , m + 1. Solving the LMI relaxation of order d yields real numbers {y d k,α } α=0,1,...,2d which are approximations to y k,α . In particular, the zero order moment of each measure µ k is an approximation of its mass, and hence of the time t k = µ k ∈ [0, T ] spent by an optimal switching sequence on mode k. At each LMI relaxation d, it holds m+1 k=1 y d k,0 = 1 for all d, and lim d→∞ y d k,0 = t k , so that in practice, it is expected that good approximations of t k are obtained at relatively small relaxation orders.

The (approximate) higher order moments {y d k,α } α=1,...,2d allow to recover (approximately) the densities u k (t) of each measure ω k (dt), for k = 1, . . . , m+1. The problem of recovering a density from its moments is a well-studied inverse problem of numerical analysis. Since we expect in many cases the density to be piecewise constant, with possible discontinuities here corresponding to commutations between system modes, we propose the following strategy.

Let us assume that we have the moments

y α := T 0 t α ω(dt)
of a (nonnegative) measure with piecewise constant density

ω(dt) = u(t)dt := N k=1 u k I [t k-1 ,t k ] (t)dt
such that the boundary values are zero, i.e. u 0 = 0 and u N +1 = 0. The Radon-Nikodym derivative of this measure reads

u (dt) = N k=1 (u k+1 -u k )δ t k (dt)
where δ t k denotes the Dirac measure at t = t k . Let

y α := T 0 t α u (dt) = N k=1 (u k+1 -u k )t α k
denote the moments of the (signed) derivative measure. By integration by parts it holds y α = -αy α-1 , α = 0, 1, 2, . . . which shows that the moments of u can be obtained readily from the moments of u. Since u is a sum of N Dirac measures, the moment matrix of u is a (signed) sum of N rank-one moment matrices, and the atoms t k as well as the weights u k+1 -u k , k = 1, . . . , N can be obtained readily from an eigenvalue decomposition of the moment matrix as explained e.g. in [START_REF] Lasserre | Moments, positive polynomials and their applications[END_REF]Section 4.3].

More generally, the reader interested in numerical methods for reconstructing a measure from the knowledge of its moments is referred to [START_REF] Henrion | Mean squared error minimization for inverse moment problems[END_REF] and references therein, as well as to the recent works [START_REF] Castro | Exact reconstruction using Beurling minimal extrapolation[END_REF][START_REF] Batenkov | Complete algebraic reconstruction of piecewise-smooth functions from Fourier data[END_REF][START_REF] Candès | Towards a mathematical theory of superresolution[END_REF] which deal with the problem of reconstructing a piecewisesmooth function from its Fourier coefficients. To make the connection between moments and Fourier coefficients, let us just mention that the moments y α = t α u(t)dt of a smooth density u(t) are (up to scaling) the Taylor coefficients of the Fourier transform û(s) := e -2πist u(t)dt = α

(-2πi) α α!
y α s α . If the y α are given, then û(s) is given by its Taylor series, and the density u(t) is recovered with the inverse Fourier transform u(t) = e 2πist û(s)ds. Numerically, an approximate density can be obtained by applying the inverse fast Fourier transform to the (suitably scaled) sequence {y d α } α=0,1,...,2d of moments.

6 Examples

First example

Consider the scalar (n = 1) optimal control problem (1):

p * = inf 1 0 x 2 (t)dt s.t. ẋ(t) = a σ(t) x(t) x(0) = 1 2 , x(1) ∈ [-1, 1] x(t) ∈ [-1, 1], ∀t ∈ [0, 1]
where the infimum is w.r.t. to a switching sequence σ : [0, 1] → {1, 2} and

a 1 := -1, a 2 := 1.
In Table 1 we report the lower bounds p * d on the optimal value p * obtained by solving LMI relaxations of increasing orders d, rounded to 5 significant digits. We also indicate the number of variables (i.e. total number of moments) of each LMI problem, as well as the zeroth order moment of each occupation measure (recall that these are approximations of the time spent on each mode). We observe that the values of the lower bounds and the masses stabilize quickly.

In this simple case, it is easy to obtain analytically the optimal switching sequence: it consists of driving the state from x(0) = 1 2 to x( 12 ) = 0 with the first mode, i.e. u 1 (t) = 1, u 2 (t) = 0 for t ∈ [0, 1 2 [, and then chattering between the first and second mode with equal proportion so as to keep x(t) = 0, i.e.

u 1 (t) = 1 2 , u 2 (t) = 1 2 for t ∈] 1 2 , 1]. It follows that the infimum is equal to p * = 1/2 0 1 2 -t 2 dt = 1 24 ≈ 4.1667 • 10 -2 .
Because of chattering, the infimum in problem (1) is not attained by an admissible switching sequence. It is however attained in the convexified problem (3). The optimal moments can be obtained analytically

y 1,α = 1 2 0 t α dt + 1 2 1 1 2 t α dt = 2 + 2 -α 4 + 4α , y 2,α = 1 2 1 1 2 t α dt = 2 -2 -α 4 + 4α
and they can be compared with the following moment vectors obtained numerically at the 7th LMI relaxation: We observe that the approximate moments y 7 k closely match the optimal moments y k , so that the approximate control law u k extracted from y 7 k will be almost optimal.

Second example

We revisit the double integrator example with state constraint studied in [START_REF] Lasserre | Nonlinear optimal control via occupation measures and LMI relaxations[END_REF], formulated as the following optimal switching problem:

p * = inf T s.t. ẋ(t) = f σ(t) (x(t)) x(0) = [1, 1], x(T ) = [0, 0] x 2 (t) ≥ -1, ∀t ∈ [0, T ]
where the infimum is w.r.t. to a switching sequence σ : [0, T ] → {1, 2} with free terminal time T ≥ 0 and affine dynamics

f 1 := x 2 -1 , f 2 := x 2 1 .
We know from [START_REF] Lasserre | Nonlinear optimal control via occupation measures and LMI relaxations[END_REF] that the optimal sequence consists of starting with mode 1, i.e. u 1 (t) = 1, u 2 (t) = 0 for t ∈ [0, 2], then chattering with equal proportion between mode 1 and 2, i.e. u 1 (t) = u 2 (t) = 1 2 for t ∈ [2, 5 2 ] and then eventually driving the state to the origin with mode 2, i.e. u 1 (t) = 0, u 2 (t) = 1 for t ∈ [ 5 2 , 7 2 ]. Here too the infimum p * = 7 2 is not attained for problem [START_REF] Batenkov | Complete algebraic reconstruction of piecewise-smooth functions from Fourier data[END_REF], whereas it is attained with the above controls for problem [START_REF] Branicky | A unified framework for hybrid control: model and optimal control theory[END_REF].

In Table 1 we report the lower bounds p * d on the optimal value p * obtained by solving LMI relaxations of increasing orders d, rounded to 5 significant digits. We also indicate the number of variables (i.e. total number of moments) of each LMI problem, as well as the zeroth order moment of each occupation measure (recall that these are approximations of the time spent on each mode). We observe that the values of the lower bounds and the masses stabilize quickly to the optimal values p * = 7 2 , y 1,0 = 5 2 , y 2,0 = 5 4 . The optimal switching sequence, which corresponds here to control measures ω k (dt) with piecewise constant densities, is obtained numerically as explained in Section 5, by considering the moments of the (weak) derivative of control measures.

d p * d N d y d 1,0 y d 2,0 1

Third example

Consider the optimal control problem (1):

p * = inf ∞ 0 x(t) 2 2 dt s.t. ẋ(t) = A σ(t) x(t) x(0) = [0, -1]
where the infimum is w.r.t. to a switching sequence σ : [0, ∞) → {1, 2} and

A 1 := -1 2 1 -3 , A 2 := -2 -2 1 -1 .
Since our framework cannot directly accomodate infinite-horizon problems, we introduce a terminal condition x(T ) 2 2 ≤ 10 -6 so that terminal time T is finite. It means that the switching sequence should drive the state in a small ball around the origin.

In Table 3 number of variables (i.e. total number of moments) of each LMI problem, as well as the zeroth order moment of each occupation measure (recall that these are approximations of the time spent on each mode). We observe that the values of the lower bounds stabilize quickly. In Figure 1 we plot an almost optimal trajectory inferred from the moments of the occu-pation measure, for an LMI relaxation of order d = 8. The trajectory consists in starting from x = (-1, 0) with mode 1 during 0.065 time units, and then chattering between mode 1 and mode 2 with respective proportions 49.3/50.7 until x reaches the neighborhood of the origin x(T ) 2 2 ≤ 10 -6 for T = 3.84. This trajectory is slightly suboptimal, as it yields a cost of 0.24351, slightly bigger than the guaranteed lower bound of 0.24347 on the best achievable cost obtained by the LMI relxation. It follows that this trajectory is very close to optimality. For comparison with available suboptimal solutions, using [13, Theorem 1], the so-called min switching strategy yields with piecewise quadratic Lyapunov functions a suboptimal trajectory with a cost of 0.24948.

Conclusion

In this paper we address the problem of designing an optimal switching sequence for a hybrid system with polynomial Lagrangian (objective function) and polynomial vector fields (dynamics). With the help of occupation measures, we relax the problem from (control) functions with values in {0, 1} to (control) measures which are absolutely continuous w.r.t. time and summing up to one. This allows for a convex linear programming (LP) formulation of the optimal control problem that can be solved numerically which a classical hierarchy of finite-dimensional convex linear matrix inequality (LMI) relaxations.

We can think of two simple extensions of our approach:

• Open-loop versus closed-loop. In problem (1) the control signal is the switching sequence σ(t) which is a function of time: this is an open-loop control, similarly to what was proposed in [START_REF] Claeys | Measures and LMI for impulsive optimal control with applications to space rendezvous problems[END_REF] for impulsive control design. In addition, constrain the switching sequence to be an explicit or implicit function of the state, i.e. σ(x(t)), a closed-loop control signal. In this case, each occupation measure will be explicitly depending on time, state and control, and it will disintegrate as µ(dt, dx, du) = ξ(dt | t, u)ω(du | t)dt, and we should follow the framework described originally in [START_REF] Lasserre | Nonlinear optimal control via occupation measures and LMI relaxations[END_REF].

• Switching and impulsive control. We may also combine switching control and impulsive control if we extend the system dynamics (5) to

dx(t) = m k=1 f k (x(t))ω k (dt) + p j=1 g j (t)τ j (dt)
where g j are given continuous vector functions of time and τ j are signed measures to be found, jointly with the switching measures ω k . Whereas switching control measures ω k are restricted by [START_REF] Cassandras | Optimal control of a class of hybrid systems[END_REF] to be absolutely continuous w.r.t. the Lebesgue measure of time, impulsive control measure τ j can concentrate in time. For example, for a dynamical system dx(t) = g(t)τ (dt), a Dirac measure τ (dt) = δ s enforces at time t = s a state jump x + (s) = x -(s) + g(s). In this case, to avoid trivial solutions, the objective function should penalize the total variation of the impulsive control measures, see [START_REF] Claeys | Measures and LMI for impulsive optimal control with applications to space rendezvous problems[END_REF].

• Removing the states in the occupation measures. In the case that all dynamics f k (t, x), k = 1, . . . , m are affine in x, we can numerically integrate the state trajectory and approximate the arcs by polynomials of time. It follows that the occupation measures µ k (dt, dx), once integrated, do not depend on x anymore. They depend on time t only. We can then use finite-dimensional LMI conditions which are necessary and sufficient for a vector to contain the moments of a univariate measure, there is no need to construct a hierarchy of LMI relaxations. There is however still a hierarchy of LMI problems to be solved, now indexed by the degree of the polynomial approximation of the arcs of the state trajectory. To cope with high degree univariate polynomials, alternative bases than monomials are recommended (e.g. Chebyshev polynomials), see [START_REF] Claeys | Moment LMI approach to LTV impulsive control[END_REF] for more details.

y 7 1 =

 1 [0.74996 0.31246 0.18746 0.13277 0.10308 • • • ] , y 1 = [0.75000 0.31250 0.18750 0.13281 0.10313 • • • ] , y 7 2 = [0.25004 0.18754 0.14588 0.11723 0.096919 • • • ] , y 2 = [0.25000 0.18750 0.14583 0.11719 0.096875 • • • ] .

Figure 1 :

 1 Figure 1: Suboptimal trajectory starting at source point x = (-1, 0) with mode 1, then chattering between modes 1 and 2 to reach the target, a neighborhood of the origin.

Table 1 :

 1 Lower bounds p * d on the optimal value p * obtained by solving LMI relaxations of increasing orders d; N d is the number of variables in the LMI problem; y d k,0 is the approximate time spent on each mode k = 1, 2.

	1 -5.9672 • 10 -9 18 0.74056 0.25944
	2 4.1001 • 10 -2	45 0.75170 0.24830
	3 4.1649 • 10 -2	84 0.74632 0.25368
	4 4.1666 • 10 -2 135 0.74918 0.25082
	5 4.1667 • 10 -2 198 0.74974 0.25026
	6 4.1667 • 10 -2 273 0.74990 0.25010
	7 4.1667 • 10 -2 360 0.74996 0.25004

Table 2 :

 2 Lower bounds p * d on the optimal value p * obtained by solving LMI relaxations of increasing orders d; N d is the number of variables in the LMI problem; y d k,0 is the approximate time spent on each mode k = 1, 2.

	2.5000 30 1.7500 0.75000
	2 3.2015 105 2.1008 1.1008
	3 3.4876 252 2.2438 1.2438
	4 3.4967 495 2.2484 1.2484
	5 3.4988 858 2.2494 1.2494
	6 3.4993 1365 2.2496 1.2497
	7 3.4996 2040 2.2498 1.2498

Table 3 :

 3 we report the lower bounds p * d on the optimal value p * obtained by solving LMI relaxations of increasing orders d, rounded to 5 significant digits. We also indicate the Lower bounds p * d on the optimal value p * obtained by solving LMI relaxations of increasing orders d; N d is the number of variables in the LMI problem; y d k,0 is the approximate time spent on each mode k = 1, 2.

	d	p * d	N d	y d 1,0	y d 2,0
	1 0.24294 30 1.4252 1.4489
	2 0.24340 105 2.0639 1.9237
	3 0.24347 252 1.9639 1.8922
	4 0.24347 495 1.9537 1.8904
	5 0.24347 858 1.9572 1.8872
	6 0.24347 1365 1.9677 1.8940
	7 0.24347 2040 1.9669 1.8928

CNRS; LAAS; 7 avenue du colonel Roche, F-31077 Toulouse; France. henrion@laas.fr

Université de Toulouse; UPS, INSA, INP, ISAE; UT1, UTM, LAAS; F-31077 Toulouse; France

Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, CZ-16626 Prague, Czech Republic

Université de Lorraine, CRAN, CNRS, IUF, 2 avenue de la fort de Haye,

Vandoeuvre cedex, France. jamal.daafouz@univ-lorraine.fr

Acknowledgments

This work benefited from discussions with Milan Korda, Jean-Bernard Lasserre and Luca Zaccarian.


Philippe Négrel
email: p.negrel@brgm.fr

Romain Millot

Emmanuelle Petelet-Giraud

Eline Malcuit

Agnès Brenot

Water Rock Interaction [WRI

Keywords: Groundwater, Eocene sands water body, Water-rock interactions

This study focuses on the geochemistry of groundwater in large Eocene aquifers sedimentary system with the main goals of investigating the influence of rock weathering and quantifying the inputs from each of several end-members (i.e silicate, carbonate, evaporite) by using an inverse method approach of budget equations. We report here the results obtained on 42 groundwater samples collected in a large aquifer (The) in the Adour-Garonne district (SW France). Mass budget equations solved using an inverse method allowed to estimate the contribution of the different sources among which, carbonate inputs dominate, evaporites often plays the second role, and silicate inputs may be important.

Introduction

This study is part of ongoing research based mainly on the major element composition of groundwater from the sedimentary Eocene aquifer system in the Adour-Garonne district of southwest France. This aquifer covers an area of 116 000 km² and represents 1/5 of the French territory [START_REF] Ph | Heterogeneities and interconnections in groundwater: coupled B, Li and stable isotope variations in a large aquifer system ( France)[END_REF]. It is limited by the Massif Central to the east, by the Armorican Massif to the north, by the Pyrenees to the south and by the Atlantic Ocean to the west (Fig. 1). The Eocene aquifer system constitutes a series of major aquifers used for drinking water, agriculture irrigation, thermo-mineral water resource and gas storage. The Eocene aquifer, composed of sandy Tertiary sediments alternating with carbonate deposits, is a multi-layer aquifer being artesian to the west of the district and confined with piezometric levels around 250 m depth to the east of the district. The Eocene aquifer system is characterized by a high permeability and a thickness of several tens to a hundred meters, and is made up of at least five aquifers: Paleocene (P), Eocene Infra-Molassic Sands (IMS), Early Eocene (EI), Middle Eocene (EM), Late Eocene (ES). The deposit sequences characterizing the Eocene aquifer system are progradational westward, from detrital deposits to carbonates with evaporites domes. Groundwater recharge may occur to the east at the edge of the Massif Central, to the south at the edge of the Pyrenees and by inflow from the Paleocene aquifer. The mineralogy of the sand deposits is very simple, consisting mainly of quartz augmented with calcite and at places dolomite and K-feldspars.

The potentiometric surface is characterized by a trough in the north of the Aquitaine Basin due to extensive long-term groundwater abstraction. The historical decline of the potentiometric surface has been roughly one meter per year in the center of the basin. In the south, this surface shows a decline of the water table in the IMS Eocene and in the Paleocene aquifer, which is less of a decline than in the north (around half meter per year). In some areas of Eocene or Paleocene outcrops a rainfall influence inducing direct aquifer recharge can be seen in the water table evolution (P-4). The main flows in the ISM aquifer are from East to North-West and from the North Pyrenean Front to the North and from the South-East to the North-West for the Paleocene flow. Anthropogenic activities such as agricultural practices does not influence the chemical composition (no nitrates, residence time of the water > 5ka). This study complements previous investigations using isotopes [START_REF] Ph | Heterogeneities and interconnections in groundwater: coupled B, Li and stable isotope variations in a large aquifer system ( France)[END_REF][START_REF] Ph | Multi isotopic and geochemical constraints of interconnection and heterogeneities of water bodies in the Adour-Garonne district (SW France) -The CARISMEAU research project[END_REF][START_REF] Ph | Use of isotopes for groundwater characterisation and monitoring[END_REF] that revealed three main weathering endmembers (evaporate dissolution, weathering of silicate minerals, and input from carbonates) plus rainwater input. This study aims at quantifying the inputs from each of these end-members by applying the inverse technique to optimize the estimation of the different sources [START_REF] Ph | Erosion sources determined from inversion of major, trace element ratios and strontium isotopic ratios in riverwater: the Congo Basin case[END_REF]. A total of 42 groundwater samples were collected from borewells in the Eocene sands water body (Fig. 1, [START_REF] Ph | Heterogeneities and interconnections in groundwater: coupled B, Li and stable isotope variations in a large aquifer system ( France)[END_REF][START_REF] Ph | Multi isotopic and geochemical constraints of interconnection and heterogeneities of water bodies in the Adour-Garonne district (SW France) -The CARISMEAU research project[END_REF]). The water samples were chemically analysed by analysed by ionic chromatography (Cl, SO 4 , NO 3 ), induced coupled plasma atomic emission spectrometry (Ca, Na, Mg, K) and inductively coupled plasma mass spectrometry (Sr). Philippe Négrel/ Procedia Earth and Planetary Science 00 (2011) 000-000

Geochemical data

Salinity of the groundwater and water types

The Total Dissolved Solids (TDS, the total mineralization of water calculated by summing cations and anions) have large variations between 74 (P-3) and 2513 mg/l (SIM-8). Variations within the same aquifer are also significant, from 74 up to 1253 mg/l in the Paleocene aquifer; from 255 up to 1033 mg/l in the middle Eocene aquifer and from 230 up to 2513 mg/l in the Eocene infra-molassic sand (IMS). The highest mineralization is generally encountered in water draining evaporite formations (gypsum and/or halite) as evidenced in the southern part of the aquifer. The spatial distribution of the TDS values indicates several places having salinities > 1000 mg/l, the highest being observed in the southeast part of the basin. Waters from the Eocene sands aquifer are mostly calcium bicarbonate type but sodium bicarbonate, sodium chloride and sodium sulphate waters can be observed in several locations. For example, in the Eocene infra-molassic sand, the calcium bicarbonate type is predominantly observed, changing eastward to sodium chloride and then to sodium bicarbonate. These water types reflect the geology of the basin, from silicates to carbonates and evaporites deposits (west and north of the basin).

Contribution of rain and rock weathering to ground water chemical compositions

For a given lithology, depending on the residence time, water fluxes, water-rock interactions and possible dilution/evaporation effects, a range of concentrations in dissolved element concentrations are reached. In order to compensate for the variable influence of these effects, concentrations of dissolved elements are commonly normalized to Na concentrations [START_REF] Ph | Erosion sources determined from inversion of major, trace element ratios and strontium isotopic ratios in riverwater: the Congo Basin case[END_REF][START_REF] Gaillardet | Chemical and physical denudation in the Amazon River Basin[END_REF]. Mg/Na versus Ca/Na molar ratios of groundwater are reported in Fig. 2a in which mixing between two end-members is represented by a straight line, as illustrated for the calculated relationship between the carbonate and evaporate endmembers. In addition to the clear trend observed between Mg/Na and Ca/Na molar ratios, some sampling points (SIM-7, 8 and SIM-13) cannot be explained by the general mixing scheme. The main trend observed in Fig. 2 corresponds to the evolution between halite dissolution and partial silicate dissolution (low Mg/Na and Ca/Na molar ratios) and carbonate dissolution (high Mg/Na and Ca/Na molar ratios). Cyclic salts [START_REF] Gaillardet | Chemical and physical denudation in the Amazon River Basin[END_REF] inputs as represented by seawater play a weak role in the origin of the groundwater compositions [START_REF] Ph | Use of isotopes for groundwater characterisation and monitoring[END_REF]. The 4 samples plotting outside the mixing field may be impacted by carbonate precipitation as Ca contents are among the lowest (<0.4 mmol L -1 ) and groundwaters are oversaturated with respect to calcite and dolomite [START_REF] Ph | Multi isotopic and geochemical constraints of interconnection and heterogeneities of water bodies in the Adour-Garonne district (SW France) -The CARISMEAU research project[END_REF][START_REF] Ph | Use of isotopes for groundwater characterisation and monitoring[END_REF]. As a complementary approach to determining chemical sources, the Sr isotope and Ca/Sr compositions of the groundwaters allow additional constraint on the different end-members as shown by Négrel et al. [START_REF] Ph | Use of isotopes for groundwater characterisation and monitoring[END_REF]. Sr isotopes are particularly useful for identifying the weathering of evaporite formations (as their 87 Sr/ 86 Sr ratio of around 0.7075 is that of Eocene marine evaporites), the weathering of carbonates (as the increase in the Ca/Sr ratio suggests) and the weathering of silicate (as the more radiogenic value of around 0.715 refers to). Variations of boron isotopes confirm this mixing by coupling with the observed lithologies in the boreholes plus the rainwater inputs [START_REF] Ph | Heterogeneities and interconnections in groundwater: coupled B, Li and stable isotope variations in a large aquifer system ( France)[END_REF] (δ 11 B around 0‰ for silicate, 20‰ for carbonate and evapotites ([B] around 10 and 2000 µg/L, respectively).

Having constrained the end-members, we formalize the mixing model for the major ions, constrain the source reservoirs that correspond to the identified end members in the groundwater, and we test this model using an inversion scheme similar to that presented by Négrel et al. [START_REF] Ph | Erosion sources determined from inversion of major, trace element ratios and strontium isotopic ratios in riverwater: the Congo Basin case[END_REF]. Details on the mixing model equations, the inverse method of resolution and choice of the end-member ratios are given in [START_REF] Ph | Erosion sources determined from inversion of major, trace element ratios and strontium isotopic ratios in riverwater: the Congo Basin case[END_REF] and [START_REF] Gaillardet | Chemical and physical denudation in the Amazon River Basin[END_REF]. The model presented above was applied to the groundwaters and allowed to calculate the proportions of major ions (Na, Ca, Mg…) derived from rain and from silicate, carbonate and evaporite weathering. From the mass balance point of view, the global contribution of each end-member was obtained by summing the individual contribution of each element. The inversion results are illustrated in Fig 2b for some selected groundwater samples identified in Fig. 2a: EM9 from the mid Eocene, SIM 2, 4, 15 from the Eocene Infra-Molassic Sands aquifer; carbonate inputs dominate the groundwater chemistry with regards to major ions, particularly for SIM2 (>90%) and evaporite often plays the second major role as in EM9 (around 27%). Rainwater inputs are always negligible. Silicate inputs may be important as in SIM15 (around 11%). This work highlights the potential use of the inversion scheme previously defined and largely used in river basins [START_REF] Ph | Erosion sources determined from inversion of major, trace element ratios and strontium isotopic ratios in riverwater: the Congo Basin case[END_REF][START_REF] Gaillardet | Chemical and physical denudation in the Amazon River Basin[END_REF]. The inverse method constituted of several mass budget equations and solved using the best set of values for the parameters among those that verify the equations is particularly adapted for the purpose of this study where we want to extract the rock components (carbonate, evapoorites and silicate) from the groundwater.. Further work will concern extension of the technique to the whole Aquitanian Basin, as well as other large basin like the Parisian Basin.

Fig. 1 :

 1 Fig. 1: Hydrogeologic framework of the Eocene aquifer in the Bassin d'Aquitaine, groundwater sampling points.

Fig. 2 :

 2 Fig. 2: (a) Mixing diagrams using Na-normalized molar ratios in the groundwaters; (b) illustration of the contribution of the endmembers for selected groundwater samples

Acknowledgements

This work was financially supported within the scope of a research partnership between BRGM, the Adour Garonne Water Agency and the Regional Council Gironde through the CARISMEAU project (http//carismeau.brgm.fr).