Alexandre Gohin

Hervé Guyomard
email: hguyomar@roazhon.ima.fr

Chantal Le Mouël

Promoting multifunctionality while minimizing trade distortion effects: the relative merits of traditional policy instruments

Keywords: Multifunctionality, trade distortion, agricultural policy, internai support, WTO

In relation to the growing debate around multifunctionality, trus paper attempts to classify alternative measures of agricultural income support according to their ability in achieving three policy objectives (supporting agricultural income, promoting positive extemalities and reducing negative ones) as well as to their induced tracte distortion effects. Four income support programs are considered: a production-linked payment program, a land-based payment program and two decoupled payment programs. Thcir effectiveness as regards to the three policy objectives and their relatives induce trade distortion effects are examined on an equal cost/support basis through a conceptual framework that allows for free entry in the sector and the land price to adjust endogenously.

Analytical results show clearly that no program uniformly dominates others. They also allow to identify the key parameters that have a substantial bearing on the relative merits of these programs.

Introduction

Although there is still considerable confusion within WTO (World Trade Organisation) Member States about what is really meant by the term multifunctionality, ail countries agree that agricultural production provides food and non-food outputs. Sorne non-food outputs are not valued by market transactions and hence can be under produced relative to what society may desire. Multifunctionality proponents claim then that production-linked payments are necessary to obtain socially desired non-food benefits because of jointness relationships between agricultural production and non-food benefits. They argue that countries should have more flexibility in the domestic policy design relative to w hat is currently provided by the provisions of the URAA (Uruguay Round Agreement on Agriculture) green box. On the other band, multifunctionality opponents argue that the green box provides sufficient flexibility to address non-food benefits, i.e., non-trade concems, with the least distortions on trade. For these countries, mainly the United States (US) and the Cairns Group (CG), multifunctionality is not a sufficient basis for continuing to pursue production-linked policies, i.e., trade distorting policies according to the URAA classification of support poli ci es. In thcir view, nonfood benefits are better addressed through specific instruments directly linked to public goods and/or positive extemalities. 2As noted by [START_REF] Bohman | The use and abuse of multifunctionality[END_REF], the WTO does not makejudgements about countries' agricultural policy objectives under the condition that policy instruments implemented to realize these objectives have no, or at most minimal, trade distortion effects or effects on production (Annex 2 of the URAA).

In this paper and in very general terms, we attempt to classify alternative "traditional" programs of agricultural income support according to their ability in achieving "traditional" goals of agricultural policies and to their induced trade distortion effects.

Four income support programs (or instruments) are retained: a production-linked payment program, a land-based payment program and two decoupled payment programs, with or without the requirement to rnaintain land in agricultural use. Three agricultural policy goals are considered: supporting agricultural income, promoting positive exter-nalities and/or public goods provision and correcting negative extemalities (pollution). 3 However, as the WTO discipline requires domestic policy programs to have no, or at most minimal, trade distortion effects or effects on production, the WTO rule of "the minimal induced trade distortion" is considered as a fourth "policy objective".

We develop a conceptual framework that allows for evaluating the impact of the four aforementioned programs on endogenous and various target variables. The mode! is an equilibrium displacement mode! with three equilibrium equations. The first one is the equilibrium condition in the agricultural output market, with aggregate supply equal to total demand (i.e., domestic demand and exports). The second one is the equilibrium condition in the land market, with aggregate supply equal to aggregate demand, both accounting for trade of land between farmers [START_REF] Leathers | The market for land and the impact of farm programs on farm numbers[END_REF]. The third one is the entry/exit condition permitting the number of fim1S to be endogenously determined within the mode!. Hence, the model's endogenous variables are the agricultural output price, the land price and the number of fanners. Knowing the effects of each considered program on these endogenous variables, the mode! also allows to determine their respective effects on various target variables such as the agricultural income, variable input (mainly fertilizers and pesticides) quantities used per hectare, yields per hectare and agricultural output exports.

One may notice that our analysis is not concemed with efficient or optimal policies. It is interested in examining the extent to which the four retained programs achieve the objectives of income support, positive extemality promotion and negative extemality reduction, whi le detennining the trade distortions arising from the use of these instruments. For that purpose, the impacts of each program regarding the four considered policy objectives are assessed through four target variables or "policy indicators". 4 The ability of each pro gram to achieve the policy objective of supporting agricultural income is evaluated through its respective impact on individual fanners' profit.

The effect of the four considered programs on agricultural commodity exports is retained as a measure of their induced trade distortion effects. The policy objective of 3 We do not consider the issue of price and/or income stabilization as we use a static analysis framework without risk and uncertainty.

4 Obviously, linking one policy objective to one synthetic indicator rests on rather restrictive assumptions.

Hence, it must be kept in mind ail along the paper that retained indicators are imperfect. They represent a simple but restrictive mean to evaluate the extent to which one program achieves one policy objective.

promoting the provision of non-food and non-market benefits is simply related to the number of farmers. In fact, following [START_REF] Hueth | The goals of US agricultural policy: a mechanism design approach[END_REF], we assume that (most) non-food or non-market benefits are directly linked to the number of (high marginal cost) farmers. This is of course a very restrictive assumption. However it allows us to simply represent the multi-product nature of non-food benefits by considering that the society perceives a connection "between the existence of relatively high-cost farm operations and the preservation or sustainability of rural communities" (Hueth, 2000, p. 22). Finally, the policy objective of reducing negative extemalities is directly linked to the use of variable inputs per hectare or to yields per hectare. In other words, we assume that negative externalities arise from an "excessive" use of variable inputs (mainly fertilizers and pesticides), so that decreasing this use contributes to reduce negative extemalities.

The paper is organised as follows. Section 2 presents the analytical framework.

In section 3, comparative static results are derived and the effects of the four alternative incarne support programs on endogenous variables as well as on policy objectives related target variables are analysed. In section 4, these effects are compared on an equal cost/support basis that makes possible to classify policy instruments according to their ability to achieve the four considered agricultural policy objectives. Section 5 concludes.

Analytical framework

A potential agricultural producer n has an initial endowment in land equal to li

•

He faces perfectly elastic supplies for ail factors of production, including land, and takes their prices as given. However the agricultural industry experiences a rising supply curve for land. It is thus an increasing-cost industry [START_REF] Hughes | The comparative statics of the competitive increasing-cost industry[END_REF]. Land can be acquired/let through rentai only. The buying/selling price of land is assumed to be adequately approximated by the discounted sum of future rentai values so that a prediction about the direction of the renta] price is equivalent to a prediction about the direction of the buying/selling price [START_REF] Leathers | The market for land and the impact of farm programs on farm numbers[END_REF].

There are N potential agricultural producers and N is large. In order to simplify presentation and analysis, we assume that they have the same production fonction.

Agricultura/ producer beltavior

The profit-maximizing pro gram of a potential agricultural producer n may be defined as (the index 11 is omitted):

(1) where y denotes the agricultural output, x an aggregate variable input,/ the amount of land used for production and nf the family labor. The price of output is p, the production subsidy is sp, the price of the variable input is w, the rentai price of land is r, the land subsidy is ,, the decoupled subsidy with mandatory production is mo and the decoupled subsidy without mandatory production is mno. The production function y = f(x, l, nf) is assumed well defined for ail non-negative variable input, land and family labor quantities. It is everywhere twice-continuously differentiable, non-negative, non-decreasing and concave. Pro gram (1) defines a profit function n(p + sp, w, r -t, ,if) which is well defined for ail positive prices, everywhere twice-continuously differentiable, nonnegative, increasing, linearly homogeneous and convex in prices, increasing and concave in family labor [START_REF] Diewert | Applications of duality theory[END_REF]. In program (1), r(l-li) represents the cost of renting additional land at price r per unit (in that case, r(l-li) ~ O) or the eamings from leasing part or ail of initial land endowment, also at price,. per unit (in that case, The individual supply function for the agricultural commodity is an increasing function of the output price, the production subsidy and the land subsidy. It is decreasing in the variable input price and the rentai price of land, and it does not depend on decoupled subsidies with or without mandatory production (equation 2). The individual derived demand fonction for land is an increasing fonction of the output price, the production subsidy and the land subsidy. It is decreasing in the rentai price of land, and it does not depend on decoupled subsidies with or without mandatory production. Impacts of changes in the variable input price on land demand depend on (Marshallian) substitution and complementarity relationships between production factors x and / ( equation 3a).

System equilibrium equations

The mode! involves three equilibrium equations. The first one is the equilibrium condition in the output market. It requires the aggregate suppl y of the agricultural commodity to equal the aggregate demand (i.e., domestic demand and exports). The second one is the equilibrium condition in the land market, which implies that there is zero excess supply in this market. The third equilibrium equation corresponds to the entry-exit condition.

The equilibrium equation in the agricultural commodity market may be written as:

(4) As a result, a potential farmer with initial endowment li will be indifferent between fam1ing and not fam1ing if:

(7) 1r(p+sp , w, r-t, 1if) + 111o= PA From (7), one notes that the entry-exit decision does not depend on the initial land endowment li. However this latter has a proportional impact, for a given land rentai price, on total profit a farmer can eam by entering and producing (see program 1). One also notes that the entry-exit decision does not depend on the decoupled subsidy without mandatory production (111110 ), but on the decoupled subsidy with mandatory production (1110 ). At this stage, it is useful to exp Iain the working of both types of decoupled subsidies. Without loss of generality, let us assume that the initial situation corresponds to a no support regime with Ki famiers who produce. Let us now assume that the govemment seeks to support farmers' incomes by means of a decoupled subsidy without mandatory production. Equation (6) implies that the latter is granted to the Ki farmers only, even if some of them decide to go out and not to produce in the new situation, but not to new entrants. Let us now assume that the income support instrnment is a decoupled subsidy with mandatory production. Equation ( 6) shows then that the latter is granted to any former who decides to produce in this new regime, but not to farmers who produced in the initial s ituation and now prefer to go out and lease out all their land endowment.

These assumptions allow us to write the entry and exit conditions in only one equation, i.e., equation ( 7).

The solution of equations ( 4), ( 5) and ( 7) gives the equilibrium price of the agricultural commodity p , the equilibrium rentai price of land r, and the equilibrium number of farn1s, K , composing the industry. The analysis of farm programs proceeds then by totally differentiating (4), ( 5) and ( 7), and solving the resulting system.

The four considered altemative/complementary incarne suppo1t instruments are the decoupled subsidy without mandatory production mno, the decoupled subsidy with mandatory production 1110, the land subsidy t, and the production subsidy sp. These four instruments are prirnarily designed to support fa1mers ' incarnes, but they do have impacts as regards to other policy objectives. Then, we assume that the govemrnent seeks to achieve four different policy obj ectives: to maintain/increase the individual earnings of persans engaged in agriculture (policy objective n° 1), to maintain/increase the provision of positive extemalities and/or public goods (policy obj ective n° 2), to reduce the negative environrn ental consequences of an excessive use of potentially polluting inputs (policy objective 11° 3) and to minimise the trade distortion induced by domestic prograrns (policy objective 11° 4).

As previously underlined, the ability of each incarne support program to achieve each of the four aforementioned policy objectives is evaluated through four target variab les. The target variable associated with policy objective n° 1 is the individual total profit of fanners (denoted PRO) . We consider that a program inducing an increase (a decrease) in the individual total profit of farmers contributes positively (negatively) to the policy objective of supporting agricultural incarne. The target variable associated with policy objective n° 4 is the agricultural commodity exports (denoted X). We consider that a program resulting in an increase (decrease) in agricultural exports goes against (favours) the policy objective of preserving the compatibility of domestic programs with WTO rules. The target variable associated with policy objective n° 3 is individual yields per hectare (denoted in for individual intensification). 5 We admit that a program inducing a decrease (an increase) in yields per hectare contributes positively (negatively) to the policy objective ofreducing negative externalities arising from intens ification of agricultural production. Finally, the target variable associated with policy objective n° 2 is the number of farmers. The preservation of a "large" number of relatively small fam ily-style faims is generally viewed as more effective to the sustainability and well-being of rural communities than a "small" number ofrelatively large fa rms (European Commission, 1999;[START_REF] Hueth | The goals of US agricultural policy: a mechanism design approach[END_REF]. In a more general but cumbersome ftamework where potential farmers have different abi lities (a higher ability carre-5 Yields have been retained instead of the aggregate variable input use per hectare because the comparative static results are far easier to derive for yields per hectare than for the variable input quantity used per hectare.

sponding to lower marginal costs), any increase in the number of fanns/fanners means that relatively high-cost fanners choose to enter and produce [START_REF] Leathers | The market for land and the impact of farm programs on farm numbers[END_REF][START_REF] Guyomard | Instruments de soutien interne des revenus agricoles et effets de distorsion sur les échanges: Un modèle avec libre entrée et endogénéisation du prix de la terre[END_REF]. One can reasonably assume that the society derives non-market benefits from the production of these relatively high-cost farmers by valuing their production beyond its market value [START_REF] Hueth | The goals of US agricultural policy: a mechanism design approach[END_REF]. Under this assumption, the policy objective of an increased number offanns/farmers may be viewed as a reduced fo rm of a more general policy objective, i.e., ensuring the fulfillment of the positive multifunctional role of agriculture. Since the sam e conclusions are derived from both the "general" m ode! (fam1ers with different abilities) and the "simplified" mode! used in this paper (farmers with identical abilities), one can interpret any increase in the number of farrns/fanners as an increased supply of public goods/positive extemalities produced by agriculture.

Impacts of alternative agricultural income support programs

We directly derive the comparative static effects of the four considered policy instruments o n the price of the agricultural output, the rentai price of land and the number of fanns/farrners. Then, using these comparative static results, we may obtain the effects of policy instmments on farmers' total profit (from ( 1)), on yields per hectare (from ( 2) and (3a)) and on exports (from the export aggregate demand fonction DE(p) ).

Details of calculations are provided in Appendix.

The decoupled subsidy witlwut mandatory production

Since the decoupled subsidy without mandatory production (mno) does not enter the three equilibrium equations ( 4), ( 5) and ( 7), it has no impact on the system but producers' incomes. In particular, it has no impact on the number of farmers, the intensification lev el (yields per hectare) and the volume of agricultural commodity exports.

The decoupled subsidy witli mandatory production

The effects of a decoupled subsidy with mandatory production (mo) on the endogenous and target variables are given by: where e; = à logy(p +sp, w,l, nf) I à logl is the restricted Marshallian s upply elasticity of output y with respect to land quantity l, while e;" = à logl(y, w, r -1,nf) I a log y is the restricted Hicksian derived demand elasticity of land l with respect to output level y .

Hence, the decoupled subsidy with mandatory production has only three unambiguous effects: a positive effect on the number of farms/fanners ( equation 10), a negative effect on the fanners' output supply ( equation 12) and a negative effect on the farmers' land demand (equation 13). The impacts of the decoupled subsidy with mandatory production on ail other variables are indetenninate and depend closely on the levels of e' , and e; relative to one.

The effect of the decoupled subsidy with mandatory production on the price of the agricultural commodity (equation 8) is unambiguously negative when the restricted Marshallian suppl y elasticity of output with respect to land quantity ( e~.) is lower than unity. When this e lasticity is strictly greater than one, the effect can become positive if the positive impact of the first right-hand side tenn of (8) outweighs the negative impact of the second right-hand side tem1 of (8). The effect on the rentai price of land ( equation 9) is unambiguously pos itive when the restricted Hicksian derived demand elasticity of land w ith respect to output level ( e;') is lower than unity. When this elasticity is strictly greater than one, the effect can become negative if the negative impact of the first righthand side te rm of (9) outweighs the positive impact of the second right-hand side term of (9). On may notice that convexity in prices of the profit fonction defined b y pro gram

(1) irnplies that the product of elasticities e~ .. e; is always smaller than one. It follows that the decoupled subsidy with mandatory production cannot simultaneously increase the output price and decrease the land price.

6

The impact of the decoupled subsidy with mandatory production on total profit PRO (equation 11) is proportional to the initial land endowment of the former. It is positive (respectively, negative) when the renta! price ofland increases (respectively, decreases).

The decoup]ed subsidy with mandatory production w ill unambiguously reduce the farn1er's output supply (equation 12) and the farrner's land demand (equation 13), but its effect on intensification (yields per hectare) is indeterminate (equation 14). It will be positive when E{ ;,,: 1 and E~, :;; 1, negative when E{ :;; 1 and E~, ;,,: 1, and indetenninate when E{ :;; 1 and E~. :;; 1.

The impact of the decoupled subsidy with mandatory production on agricultural cornmodity exports (equation 15) is indeterminate. It is positive (respectively, negative)

when the price of the agricultural commodity decreases (respectively, increases).

The main results of the comparative static analysis are summarized in Table 1 (for the four income support instruments). As regards the four policy objectives considered in this papcr, the only unambiguous cffcct of the dccouplcd subsidy with mandatory production is to increase the n umber of farms/farmers. Hence, one may conclude that the decoupled subsidy with mandatory production contributes positively to policy objective n° 2 (positive externality provision). For the three other policy objectives, the effects caimot be predicted with theory alone. The total profit of a farrner can decrease (a necessary, but not sufficient, condition for this being that the restricted Hicksiai1 derived demand elasticity ofland with respect to output is strictly greater than one), and agricultural commodity exports can decrease as well (a necessary, but not sufficient, condition for this being that the restricted Marshallian supply elasticity of output with respect to land is strictly greater than one). The impact of the decoupled subsidy with mandatory production on yields per hectare is indeterminate, depending on orders of magnitude relative to one of the two aforementioned elasticities.

The ambiguous effects of the decoupled subsidy with mandatory production may be explained as follows. Ali other things being equal, the decoupled subsidy favors the entry of new producers into fanning, creating subsequently excess supply in the output market and excess demand in the land market. A new equilibrium of our economy may be obtained through either an output price decrease and an increase in the rentai price of land ( < : , ; 1 and E; :,; 1 ), an output price increase compensated by a higher increase in the rentai price of land ( < > 1 and E ; :,; 1 ), or an output price decrease sufficient to cape with a decrease in the rentai price of land ( < : , ; 1 and E ; > 1 ). These price adjustments obviously reduce the incentives for potential producers to enter the sector, but never outweigh the initial positive effect of the decoupled subsidy on the number of farmers.

[Insert Table l]

The land subsidy

The effects of a land subsidy (t) on the endogenous and target variables are similarly derived from the total differentiation of equations ( 4), ( 5) and ( 7). We obtain:

( Thus, the land subsidy will unambiguously decrease the output price ( equation 16), increase the rentai price of land ( equation 17), decrease yields per hectare ( equation 19), increase the total profit of a former ( equation 20) and increase the exports of the agricultural commodity (equation 21). However, its effect on the number of farn1s/fanners is indetenninate ( equation 18). When the restricted Hicksian derived demand elasticity of land with respect to output is lower than one, the effect of the land subsidy is to increase the number of farmers. When this elasticity is greater than one, the effect is ambiguous and can become negative if the negative impact of the first right-hand side tenn of the square brackets in (18) outweighs the positive impact of the second right-hand side term of the square brackets in (18).

Hence, regarding the four considered policy objectives, one may conclude that the land subsidy contributes positively to policy objectives n° 1 (agricultural income support) and n° 3 (negative extemality reduction), but at the expense of increasing trade distortion effects (policy objective n° 4). Its contribution to policy objective n° 2 (positive externality provision) is ambiguous. It may be positive or negative, depending closely on the level of EJ' relative to one.

Sorne further results can be wrung out of equations ( 16) to ( 21 Hence, in the particular case where the land supply and dcmand coming from the rest of the economy co1Tespond to fixed amounts, the only non-zero effects of the land subsidy are to rai se the rentai price of land by the same an1ount (equation 23) and to increase the total profit of a fam1er proportionally to his initial endowment in land ( equation 26). Equations ( 24), ( 25) and ( 27) show that, in this particular case, the land subsidy has no impact on the number of farms/farmers (equation 24), neither on the exports of the agricultural commodity (equation 27).

This last result suggests that a land subsidy may be considered as a decoupled income support instrument, provided that there are restrictions on eligible land (and hence, payments) through the use on an aggregate base area. 7 In other words, this result shows that a policy instrument that does not fully conform to al! green box eligibility 7 Of course, this result is contingent to the mode! used, i.e., a static framework without risk and uncertainty and considering only one aggregate output.

criteria (as defined in points 1 and 6 for income support measures in Annex 2 of the URAA) may nevertheless has minimal distortion effects on trade. 8

From a European Union (EU) perspective, the situation depicted in the above particular case coITesponds to the cuITent situation in the Common Market Organisation (CMO) for cereals, oilseeds and protein crops (hereafter COP crops), if we consider COP crops as one aggregate only. Therefore, from a WTO negotiation point of view, the EU cou Id rightfully argue that the area payments in force in the EU COP sector have (at least at the aggregate level) minimal trade distortion effects (so could be considered as decoupled), although they do not fully conform to URAA decoupling criteria. 9

The production subsidy

The effects of a production subsidy (sp) on the endogenous and target variables may be written as: Then, our analytical framework leads to find common results regarding the effects of a production subsidy. It unambiguously decreases the output price ( equation 8The land subsidy considered in this paper does not conform to criterion (iv) and (v) of point 6 of Annex 2, i.e., respectively, the payment will not be related to or based on the factors of production employed and no production shall be required to receive the payment.

9 Obviously, this does not mean that the current EU area payments to COP crops have no effect on the domestic aggregated supply of COP crops with respect to a free market situation. This would be the case only if the current EU base area for COP crops is not larger than the total area which would be devoted to COP crops in a non inte rventionist regime. The URAA however does not, at least explicitly, strictly constrain to take the free trade situation as the reference situation. For a discussion on the eligibility to the green box of EU compensatory payments granted in both the COP and the beef sectors, see for example [START_REF] Gohin | The Agenda 2000 CAP reform in the WTO context: distortion effects of area compensatory payments and set-aside requirements[END_REF]. The only ambiguous effect of the production subsidy relates to the number of farmers. Equation (30) shows that the production subsidy will unambiguously increase the number of fanners when the restricted Marshallian suppl y elasticity of output with respect to land is lower than one. When this elasticity is greater than one, the impact is indeterminate and can become negative when the positive effect of the first right-hand side term of the square brackets in ( 30) is outweighed by the negative effect of the second right-hand side terni of the square brackets in (30).

One may notice that when both the domestic demand DD and the foreign demand DE are price inelastic, the impacts of the production subsidy on the rentai price of land, the number of farmers, the intensification level and the total profit of a farmer are small.

Finally, as regards to the four considered policy objectives, one may conclude that the production subsidy contributes positively to policy objective n° 1 (agricultural incarne support), but at the expense of increasing trade distortion effects (policy objective n° 4). It contributes negatively to policy objective n° 3 (negative extemality reduction) while its contribution to policy objective n° 2 (positive extemality provision) is ambiguous. lt may be positive or negative, depending closely on the level of c~, relative to one.

Previously presented comparative static results indicate whether, or under which conditions, each income support program contributes positively or negatively to the four considered agricultural policy objectives. However they do not allow to classify these support programs according to their relative ability to achieve these policy objectives.

Such a classification becomes possible if the effects of each program on policy objectives related target variables are compared on an equal cost/support basis.

Equal cost/support comparison of alternative agricultural income support programs

In this section, we assume that the initial situation corresponds to the free trade equilibrium.

10 In other words we consider that the four income support programs are initially not in force. Such an assumption makes the analysis easier. Furthermore it implies that the comparison of the effects of instruments on an equal budget cost basis is equivalent to a cornparison on an equal total agricultural income support basis.

Secondly, we consider only three of the four previous agricultural incarne support programs: the decoupled subsidy with mandatory production (ma), the land subsidy (t) and the production subsidy (sp ). As the only non-zero effect of the decoupled subsidy without mandatory production (mno) is to raise, by the same amount, the farmers' individual profit, results of the comparison analysis with other instruments are qui te obvious.

In a first step, we determine the differences between effects induced by each pair of instruments for the three endogenous variables and the four policy objectives related variables, for a constant budget cost/income support. Then, we examine the signs of these differences. Results of this first step are reported in Table 2. In a second step, the three programs are classified according to their relative ability to achieve the four considered agricultural policy objectives. Results of this second step are synthesized in Table 3.

Comparison of the effects of alternative agricultural income support programs

The differences between the effects induced by each pair of programs for endogenous and target variables are provided in Table 2.

[Insert Table 2]

As previously shown, when a program has an ambiguous effect on a variable, the sign of this effect is always closely related to the order of magnitude relative to one of, either E ~. (the restricted Marshallian supply elasticity of output with respect to land 10 Tn the previous sections, we did not specify the status of the initial market situation because derived comparative static results are valid whether the initial situation corresponds to the free trnde equilibrium or not.

quantity), E{ (the restricted Hicksian derived demand elasticity of land with respect to output level), or both. It is thus, not surprising that when the difference between the effects induced by each pair of instruments on one variable is ambiguous, its sign depends always directly on the level of one or both these elasticities relative to one.

The decoupled subsidy with mandatory production (mo} vs the land subsidy (t} Panel 2.a. shows that, for an equal budget cost, the decoupled subsidy with mandatory production leads unambiguously to a greater increase in the number of farms/farmers than the land subsidy. For ail other considered variables, the signs of the differences between the impacts of both instruments depend exclusively on the level of the restricted Marshallian suppl y elasticity of output with respect to land quantity ( c; ) relative to one.

If this elasticity is lower than unity, then the decoupled subsidy with mandatory production leads to a higher decrease (in absolute tenns) in the price of the agricultural output than the one induced by the land subsidy. In that case, the decoupled subsidy with mandatory production leads to a greater positive trade distortion effect than the land subsidy. In the opposite case (i.e., c; > I ), the positive trade distortion effect induced by the land subsidy is always greater than the trade distortion resulting from the application of the decoupled subsidy. 11

Similar results may be derived for the rentai price of land and the fanners' individual profit. When the restricted Marshallian elasticity c; is lower than one, the increases in the renta! price of land and then in the fanners' individual profit generated by the land subsidy outperform those obtained with the decoupled subsidy.12 In the oppo-11 When C: > 1 , the decoupled subsidy with mandatory production may lead either to a decrease or an increase in the price of the agricultural output. In case of a decrease, this latter will be lower (in absolute terms) than the one induced by the land subsidy. Hence, agricultural conm1odity exports will raise more with the land subsidy than with the decoupled subsidy. In case of an increase, exports of the agricultural commodity will decrease with the decoupled subsidy while raising with the land subsidy. Therefore, in both cases, the positive trade distortion effect resulting from the land subsidy application will be higher than the trade distortion effect (positive or negative) induced by the decoupled subsidy.

site case (i.e., E' . > J ), the implementation of the decoupled subsidy leads to higher increases in both the rentai price of land and the farmers' individual profit than the land b .d 13 SU SI y.

Finally, once again, simi lar conclusions arise when comparing the effects of both instruments on yields per hectare. When the restricted Marshallian elasticity < is lower than one, the decrease in the level of intensification resulting from the implementation of the land subsidy outperfom1s the one observed with the decoupled subsidy. In the opposite case (i.e., c' . > 1 ), the decoupled subsidy leads to a decrease in the level of intensification, decrease which is higher (in absolute tenns) than the one induced by the land subsidy.

The decoupled subsidy with mandatory production (,no) vs the production subsidy (sp)

As in the previous case, Panel 2.b. indicates that, for an equal budget cost, the decoupled subsidy with mandatory production leads unambiguously to a greater increase in the number of farms/farmers than the land subsidy. However, for all other considered variables, the signs of the differences between the impacts ofboth instruments now depend exclusively on the level of the restricted Hicksian derived demand elasticity of land with respect to output quantity ( i;; ) relative to one.

The following results apply when this elasticity is greater than unity. The decoupled subsidy necessarily leads to a decrease in the price of the agricultural output, decrease which is greater (in absolute terms) than the one induced by the production subsidy. Therefore, the decoupled subsidy generates a positive trade distortion effect that is higher than the one induced by the production subsidy. The increase in the rentai price of land and then in the fam1ers' individual profit generated by the production subsidy is always greater than the change observed in both variables (which may be positive or the restricted Hicksian derived demand elasticity of land with respect to output level ( i;; ) is lower than one, and negative otherwise.

13 Let's remind that i;'. i;; is always smaller than one. Therefore, when i;' .. is greater than one, i;; is necessarily lower than one. In other words, when i; ' . > 1 , the decoupled subsidy leads necessarily to increase the rentai price of land and, consequently, the farmers' individual profit. And these increases are greater than the ones induced by the land subsidy.

negative) with the decoupled subsidy. Finally, the decoupled subsidy induces an increase in the level of intensification, increase which is greater than the one resulting from the production subsidy implementation.

One observes opposite results when the restricted Hicksian elasticity E t is lower than one.

The land subsidy (t) vs the production subsidy (sp) As shown in the previous section, the land subsidy and the production subsidy both lead unambiguously to a decrease in the price of the agricultural output. However, Panel 2.c. suggests that, for an equal budget cost, the land subsidy induces a lower output price reduction (in absolute terms) than the production subsidy. Therefore, on an equal cost/support basis, the positive trade distortion effect generated by the land subsidy is always lower than the one resulting from the production subsidy.

In the same way, it has been shown in section 3 that both instruments raise the rentai price of land and, consequently, the fam1ers' individual profit. However, on an equal cost/support basis, the increase observed in both variables is always greater with the land subsidy than with the production suhsidy.

The comparison of the impacts of the land and the production subsidies on the level of intensification is qui te obvious since the former induces a decrease in yields per hectare while the latter makes this indicator to increase.

Finally, Panel 2.c. reveals that, contrary to the two previous pairs of instruments, the only ambiguous result regarding the comparison, on an equal cost/support basis, of the impacts of the land and the production subsidies relates to their relative effects on the number of farms/fanners. When the restricted Hicksian elasticity E ; is lower than one while the restricted Marshallian elasticity < is greater than one, then the increase in the number of farms/farmers induced by the land subsidy is always higher than the change observed in the same variable with the production subsidy (change which may be positive or negative). One observed the opposite situation when the restricted Hicksian elasticity E ; is greater than one while the restricted Marshallian elasticity < is lower than one. When both elasticities are lower than one, both instruments make the number of farms/ fanners to increase. But the sign of the difference between their relative impacts remains ambiguous.

Classification of the alternative income support programs accordù,g to their ability to achieve the four policy objectives

Based on the results reported in Table 2, we are in a good position to classify the programs with respect to their ability to achieve policy objectives. Table 3 reports the obtained classification, for each of the three possible sets of conditions, with grade l for the most effective program and grade 3 for the worst effective one. Bence, on an equal cost/support basis, the following results arise: i) for ail possible sets of conditions on the levels relative to one of the Marshallian and Hicksian elasticities, E' , and E; , the decoupled subsidy with mandatory production (mo)

is the most effective instrument as regards to policy objective n° 2, i.e., under our hypotheses, for promoting the provision of positive extemalities and/or public goods;

ii) when 8' , > 1 and E; s 1 , the decoupled subsidy (mo) is the most effective instrument as regards to the four policy objectives. In order words, this decoupled subsidy program is the most effective instrument for simultaneously supporting agricultural income, prornoting positive extemalities and reducing negative extemalities, while generating minimal distortion effects on trade;

iii) for the two other sets of conditions (i.e., E ; s 1 and E ; s 1 or 8' .-~ I and E; > I ), the decoupled subsidy is never the most effective instrument as regards to policy objectives 11° 1, 3 and 4. lt is always dominated by, at least, the land subsidy; iv) when E' . s I and E; s 1 or 8' . s 1 and E; > 1, the land subsidy (t) is the most effective program as regards to policy objectives n° l, 3 and 4. In other words, the land subsidy is more effective than other instruments in supporting agricultural incarne and reducing negative extemalities, while inducing minimal tracte distortion effects; v) whatever the possible set of conditions on the levels relative to one of the Marshallian and Hicksian elasticities, 8' . and E ; , the production subsidy (sp) is never the most effective instrument as regards to the four considered policy objectives. When < > 1 and E ; s l , it ranks last for ail policy objectives. When 8' . s I and E ; s 1, it ranks last for all policy objectives, but n° 2 (where the ranking between the land and the production subsidies is indetem1inate). When < s 1 and E ; > 1 , the production subsidy dominates the decoupled subsidy for ail policy objectives, but 11° 2.

From a policy perspective, our theoretical framework allows us to state that, on an equal cost/support basis, and except specific conditions, no program uniformly dominates others for achieving simultaneously the four considered policy objectives. 14 In other words and in accordance with the "targeting mie", no instrument does allow to achieve effectively simultaneously several policy objectives. Thus, a govemment consideri ng a specific instrument necessarily faces trade-off between objectives. In the same vein, a government pursuing different policy objectives rnay be well-advised to mobilize various policy instruments.

From a WTO negotiation perspective, our production subsidy program would ce11ainly be qualified as a amber box measure while both other instruments would likely be considered as green or blue box measures. Our theoretical framework then suggests that amber-box measures are not likely to be the most effective instruments in promoting multifunctionality, provided the definition of this notion in this paper. By way of consequence, promoting multifunctionality does not appear as an undeniable justification for claiming the continuation of am ber box measures in future WTO negotiations.

On the other band, determining which green or blue box measure promotes most effectively multifunctionality while minimizing tracte distortion effects is nota trivial matter. This depends on conditions that cannot be predicted by theory alone. To this regards, our results put emphasis on the key role of both the restricted Marshallian supply elasticity of output with respect to land quantity and Hicksian derived demand elasticity of land with respect to output quantity.

Conclusion

This paper develops a theoretical framework in order to analyze the ability of traditional programs of agricultural incarne support in achieving potentially conflicting/complementary policy objectives. This frarnework, which allows for free entry into the agricultural sector and land price endogeneity, show that attempts to evaluate the relative merits of various agricultural policies should take into account the impacts that these policies have on both individual producers (impact at the individual margin) and the number of producers (impact at the collective margin). For sorne instruments and some policy goals, impacts rnay be contrary to intuition or to results derived frorn a mode! with a fixed number of firms and/or an exogenous price of fannland. Moreover, 14 Specific conditions corresponds here to the case where E; > 1 and E; :0: 1 . Indeed in that case, the decoupled subsidy with mandatory production dominates both other instruments as regards to the four considered policy objectives.

this framework allows to identify the two key pararneters that have a substantial bearing on the relative effectiveness of various instruments. The next step on the research agenda will obviously be the empirical evaluation ofthese crucial parameters.

Many research directions represent important avenues for further study. For instance, the mode! is very stylized with only one output, and one single and crude indicator for negative extemalities as well as for positive externalities. This is valuable for conceptual understanding of the importance of entry-exit decisions and land market characteristics, but specific policy problems should be analyzed for particular agricultural industries with more carefully specified technologies and indicators.

  (8) detM .(dp l dmo)=-Kn,,.np(l-r.~)-(Sl,. -Dl,.)np (9) <let M .(dr I dmo) = Kn "" (-n,.)(1 -r.n-(DDP + DEµ)(-n,. ) ( 10) det M .(dK / r/1110)

  16) detM.(dp l dt)= -(SI,. -Dl,.)rr.p(-rr.,) < 0 (18) detM.(dK ! dt) = (SI,. -Dl,.)[Krr.pp(-n,.)(1-E{) -(DDP +DEp)(-rr.,.)] (19) l ' .detM.(din l dt)=-(SI -Dl)rr. [rr. rr. ' +rr. (-rr.)' +2rr. rr. (-rr.dPRO I dt = !i.(dr I dt)> 0 (21) dX I dt = DE,,.(dp l dt)> 0

  ) when SI,. = Dl,. = o . Using the developed expression of detM provided in Appendix, we obtain: (22) dp / dt = 0 (23) dr / dt = 1 (24) dK / dt = 0 (25) din / dt = 0 (26) dPRO / dt= li (27) dX I dt = 0

(

  28) det M .(dp I dsp) = -K[n,.rn; + TC pp(-nr)2 + 2n prn p(-n,.)]-(Sl,. -Dl,.)n; < 0 (29) det M.(dr / dsp) = (-DDP -DE µ)nµ(-n,.) > 0 (30) detM.(dK l dsp)=(-DDP -DEµ)[(S1,. -Dl,.)n" +Kn,.,.n p(I -E~,)] (31) / 2 . det M .(din / dsp) = (-DD,, -DE,, )(-n, )[n" n;, + n,,,,(-n,Y + 2n,,,n ,, (-n,)] > 0 (32) detM.(dPRO ! rüp)=li.(dr l dsp) >0 (33) det M .(dX / dsp) = DE,, .(dp l tùp) > 0

  28), increases the renta] price ofland (equation 29), increases the intensification level ( equation 31 ), increases the total profit of a farmer ( equation 32) and increases the exports of the agricultural commodity (equation 33).

Table 1 .

 1 The comparative static analysis of alternative income support policies Impact of a decoupled subsidy without mandatory production (when E{ ~ 1 and E~, s 1 ; -when E{ s 1 and E' ,. ~ l ; +/-when E{ s 1 and E 1 < 1 ' . = 8\og y( p + sp, w,I, 11/') / à log/, and E ; = à logl( y , w,r -t, nf) / ôlog y .Table 2. Equal cost/support comparison of the effects of each pair of instruments on the various endogenous and target variablesPanel 2.a. 1110 vs t Differences between effects induced by pairs of instruments dp I dp i =-K_rr.,.rr,, .(1 -e;) 1 .,,1t ,, -n'.,,.) K(SI, -D/, )rr,,rr ,' )-1t;(DDP+DEP)+rr:, cs1, -DI, ) = li.(_±_I -~I ) dt c dmo ,• d!Jc dX I = DE.( dp 1dpi dt , '' dmo ,. dt c) Panel 2.b. 1110 vs sp Differences between effects induced by pairs of instmments dp 1 _ dp 1 = K rr ,,, .rr; .(1 -e; )

Table 3 . Ranking of equal cost/support instruments according to the four policy objectives

 3 Panel 2.c. t vs spDi fferences between effects induced by pairs of instruments dp i dt c dp 1 == K. rr"" .rr! + rr,, .. rr:, -2rr,,rr ,rrp, == -K. rr,., .rr; + rr ,,.n;, -2rr,, n,rr,,,.

	dmo c	dsp c	• rr ,, .det M
	dr 1 dmo c	dr 1 = -K. rr,.,.rr, (1-E; ) dsp ,. detM
	dK	dK I = K	2 (rr ,,,,rr ,,.-rr ;,,. ) + K(Sl,-Dl,)rr ,.,,	K1t ,,,1t,(DD 1 , + DE,.)
	d1110 ,.	dsp ,.		det M		det M	1t p detM
	din 1 dmo ,.	din 1 1 d5p _=( l -E, ).rr,,,,.rr , . ,	-rr; l rr ,,.(DD,, + DE,, )+rr,, .(SI, -Dl,) detM./ 2
	dPRO I r/1110 I'	dPRO I = li.(_±_I -d5p , d1110 c	dr 1 -) dsp c
	dX 1 dmo ,	dX 1 = DE ( dp 1 -dsp, ,, • dmo c	dp 1 ) c/5p ('

) + + if e' . :s: 1 -if e' > 1 ) -if e' < 1 ) -+ if e' . > 1 + if e' :s: 1 ) -if e; > 1

For a more complete discussion on this opposition, see[START_REF] Lankoski | Multifunctional character of agriculture: differences in views between countries[END_REF]. See also[START_REF] Paarlberg | Multifunctionality and agricultural trade negotiations[END_REF].

If the output price increases, then the land price increases too. And if the land price decreases, then the output price decreases too.

Let's remind that the change in the rentai price of land and, consequently, in the farmers' individual profit resulting from the application of the decoupled subsidy may be positive or negative. It is positive if

Appendix

Details of calculations

Total differentiation of equations ( 4), ( 5) and ( 7 The detenninant of M, detM, is positive since it can be written as:

( 1) We now illustrate how the analysis proceeds on the example of the decoupled subsidy with mandatory production. From (S2) we immediately obtain:

dK dmo with r , TC;

.

Hence:

These three equations may equivalently be written as: The impact of 1110 on y and I may be written as, respectively: = (1 / det M ) .,.11.,.P)1t" -11. PP (SI,. -Dl,. )1t P + 1t pr (-DD P -DE µ)(-1t,. )) < 0 29 .. ( a9)

= (1 / del M ) 11 [Krc ",.n,.,. + SI,. )n" ] -n,.,.[-(Kn "" -DDP -DEP )rc,. + Krc,pn"])

= (1 / det M )[-K (rr ""n,.,. -rr,pn ,,,. )(-n,.) + n,.,, (SI,. -Dl,. )n,, -rr,.,.(-DD,, -DE,, )(-rr,.)] < 0

From (a8) and (a9), we readily deri ve the impact of 1110 on intensification i11 (yields per hectare):


V Tournat

V E Gusev

Nonlinear effects for coda-type elastic waves in stressed granular media

Keywords:

Experimental results and their interpretations are presented on the nonlinear acoustic effects of multiple scattered elastic waves in unconsolidated granular media. Short wave packets with a central frequency higher than the so-called cut-off frequency of the medium are emitted at one side of the statically stressed slab of glass beads and received at the other side after multiple scattering and nonlinear interactions. Typical signals are strongly distorted compared to their initially radiated shape both due to nonlinearity and scattering. It is shown that acoustic waves with a deformation amplitude much lower than the mean static deformation of the contacts in the medium can modify the elastic properties of the medium, especially for the weak contact skeleton part. This addresses the problem of reproducibility of granular structures during and after acoustic excitation, which is necessary to understand in the non destructive testing of the elastic properties of granular media by acoustic methods. Coda signal analysis is shown to be a powerful time-resolved tool to monitor slight modifications in the elastic response of an unconsolidated granular structure.

I. INTRODUCTION

Unconsolidated granular materials are known to exhibit a high complexity in their macroscopic behaviors. This gives rise to numerous fundamental and applied processes that are currently intensively studied, such as avalanches, dune formation, compaction etc [1,2,3,4]. Most of the studied physical processes concerning the unconsolidated granular matter are obviously related to the possibility for the grains to move relative to each other, from microscopic (much smaller than a grain diameter) to macroscopic (much larger than a grain diameter) rearrangements. While large movements of grains have been observed and modelled for the processes where the granular material behavior may be fluid-like (even complex and not classical), like avalanches, microscopic relative movements are less intuitive and less studied. They are however the only possible relative movements in solid-like granular media where a static stress is applied and the grains are confined in a given volume.

In this context, the acoustic waves, known to be highly sensitive to the contact stresses [5,6,7,8], can be useful for the monitoring of these small relative movements [9], impossible to detect by other existing methods. They could even be useful for the generation of these small movements. Among the several existing methods for the grain movements or contact stresses monitoring (carbon paper experiment, photo-elasticity, image processing from CCD camera, X-ray tomography or photo-imaging in iso-index configuration [1,2,10,11]), the acoustic methods are, to the best of our knowledge, the only ones sensitive to the contact stresses inside the volume of a 3-D non transparent medium. One of the main advantage of acoustic methods is also the temporal resolution of the measurement, which could decrease down to the wave period (one millisecond at one kHz), and is particularly interesting in transient processes like avalanches for instance. However, there is currently a lack of understanding of the acoustic transport properties through granular media, which makes these methods mostly qualitative.

Several previous works [7,9,12,13] have shown the following qualitative features for the acoustic wave propagation through stressed and non cohesive granular media. The propagation of a low amplitude (with a strain much lower than the average static strain of the contacts) and low frequency (with a wavelength much larger than the grain diameter) wave can be linear and is mostly governed by the averaged properties of the medium (average density, average static stress, coordination number) [14,15]. The propagation of a high amplitude (with a strain lower than the average static strain of the contacts but which can be comparable) and low frequency wave is nonlinear and governed partly by the so-called weak contacts (with a static strain lower than the average one) [7,16,17]. This is a consequence of the fact that individual stress-strain relationships for each contact have higher values of nonlinear parameters when the pre-stress is smaller. For acoustic waves with wavelengths comparable to the bead diameter, however, these simple features are not straightforward, and the transport properties determination remains a fundamental problem. How to describe the wave scattering in a network of beads with geometrical and contact disorder? What is the role of the so-called force chains in the transport properties of short wavelength acoustic waves? Can the weakest contacts make the multiple scattering process nonlinear?

A linear diffusion approach has been recently applied to the analysis of coda-type signals in confined granular media [8]. This is one of the first attempt to understand the transport properties of short wavelength elastic waves in such media at the laboratory scale and to extract parameters such as a characteristic time of absorption τ a and a diffusivity D. This approach is based on previous works in elastic wave diffusion carried out for slightly different media (solid rods in water [18], chaotic cavities [19], or glass bead in water [20] for instance) and on a different scale [21,22].

In this article, we report some experimental results of nonlinear acoustic wave propagation, with initially radiated wavelengths close to the bead diameter. In some well-chosen experimental configurations, it is possible to observe coda-type signals, together with a nonlinear signal of low frequency, which is identified here as a selfdemodulated signal. For some widely encountered configurations, linear and nonlinear wave processes of multiple scattering, self-demodulation and self-action are observed [START_REF] Tournat | AIP Conf. Proc., ISNA17th[END_REF][START_REF] Tournat | Proc. of the 19th ICA[END_REF]. The last nonlinear process of self-action of the multiple scattered field is observed even for moderate excitation amplitudes (deformations much lower than the average static strain of the contacts). The application of some of the presented effects to the characterization of granular media is finally discussed.

II. EXPERIMENTS

As a preamble, carrying quantitative experiments on acoustic wave propagation in unconsolidated granular media is a hard task for several reasons that are described in the following.

First, in order to excite the modes of propagation inside the solid frame, i.e. through the elastic beads and their contacts, and not through the fluid saturating phase, it is necessary to put the piezo-transducer in direct contact with the medium itself (or in contact with a solid which is itself in direct contact with the medium). Other types of transducers for the excitation of acoustic waves, like laser generation of ultrasound for instance, have not been applied to our knowledge, certainly because of the high efficiency needed in these strongly attenuating media. The application of a mechanical stress with a contact transducer has several consequences on the possible experiments. What can be done usually in liquids for acoustic metrology, i.e. changing the emitterreceiver distance in order to measure the wave velocity for instance, is extremely difficult in unconsolidated granular media, because the contact network is modified by any variation of external conditions.

The second difficulty in carrying quantitative experiments is the experimentally observed strong influence of the applied static stress on the transmission coefficient es-

Ð ×× × ´d = 2 mmµ D = 20 cm Ë Ö Ñ ØØ Ö ÄÓÒ ØÙ Ò Ð Ö Ú Ö ËØ Ø ×ØÖ ×× ÀÈ ÐØ Ö • ÑÔÐ Ø ÓÒ ÒØ Ö ÓÒ 100 kHz Ñ ØØ × Ò Ð L = 1 -18 cm P 0 ≃ 30 -500 kP a Ä Ý Ö Ó ÐÙ × FIG. 1: Schematics of the experimental configuration.
pecially for short wavelength acoustic waves. This result is discussed in the following. The consequence is the impossibility to perform quantitative reproducible and precise experiments if ones modify the applied static stress on the sample, even if it is measured to be equal after a back and forth modification. Moreover, if one would be able to apply exactly the same stress, the memory effects due for instance to the hysteresis in the quasistatic stress-strain relationship, take place [START_REF] Guyer | [END_REF]26,27]. In an attempt to control carefully the granular preparation, the acoustics of compaction has been recently studied [28]. However, it allows to perform experiments only for weakly stressed granular packings (typically from 75 P a to 3 kP a), far from the stress necessary for the observation of the signals described here.

The third difficulty, is related to the fragility of such media [1,4]. Due to the inhomogeneity of the individual contact loads [10,11,14,29], some of the contacts are weakly loaded, which make them able to clap, slip or slide under the action of a small amplitude strain wave. In contrast, the so-called strong contacts, are supposed to take part in the force chains which constitute the elastic skeleton of the bead arrangement. Previous works show by simple analytical formulae or experimentally that the weaker the contact the higher is its acoustic nonlinearity [7]. It is thus expected that the major role in the linear acoustic properties is played by the strong contacts and the force chains while the nonlinear acoustic properties are more controlled by the weak contacts. There exist few experimental configurations (in particular when short wavelengths are used) when these roles cannot be qualitatively separated like that, and either weak contacts influence importantly the linear properties [30] or strong contacts are involved in the observed nonlinear effects.

A. Experimental setup

In Fig. 1, a schematic representation of the experimental setup is shown. A wave packet made of a sine wave at 100 kHz modulated in amplitude by a Gaussian function is high-pass filtered over 40 kHz (in order to avoid any direct low-frequency excitation), amplified and launched in the medium with a shear piezo-transducer. Due to multiple scattering, mode conversions (including dilatancy), the acoustic energy is rapidly distributed between longitudinal and shear vibrations in the medium. The use of a shear emitter has the advantage of avoiding the direct excitation of a pressure wave in the air saturating the beads. In the following results, two types of Gaussian function widths for the initially launched wave packets have been used, and two types of surface treatment for the shear transducer (the first type is the raw transducer membrane with asperities estimated to be of characteristic scale ∼ 50 µm and the second type is a transducer with a glued single layer of 2 mm diameter glass beads, the same beads as in the medium itself). The container diameter is D = 20 cm and the propagation length ranges between 1 cm and 18 cm. The applied static uniaxial stress is measured with a force sensor at one end of the container and ranges from 30 kP a to 500 kP a. A longitudinal piezo-electric receiver is placed at the bottom of the container. This transducer has been chosen because of its high sensitivity over a wide-frequency band, allowing for a large range in the acoustic amplitude measurements. In all the experiments presented in the following, the estimated average contact strain ε 0 (of the typical range 3 -7 × 10 -4 for a 200 kP a applied stress) is always greater than the maximum acoustic strain ε max ≃ 10 -5 .

When using this setup, it is possible to modify the emitted acoustic frequency, the acoustic amplitude, the applied static stress, the propagation distance. But importantly, it is better for a quantitative insight in the wave propagation phenomena, to modify parameters of the setup that change the smallest number of medium (or wave) parameters. For instance, changing the frequency or the static stress modifies the wave scattering, absorption, dispersion and nonlinearity. Changing the distance, changes irreversibly and in an uncontrolled way the granular medium state, as it is necessary to unload completely the medium and add or remove some granular matter. The convenient parameter to modify is the wave excitation amplitude, which can be accurately measured, and is reproducible. The medium is a priori not irreversibly modified.

In the following several experimental results are presented and their implications on the granular material elasticity is discussed. The parameters of the experiment may be different, they are consequently recalled for each group of results when necessary.

B. Multiple scattering and nonlinearity

The signals presented in Fig. 2 are typical experimental signals that can be observed when the initial acoustic wavelength is of the order or less than the bead size and for a static stress larger than ∼ 100 kP a). For an estimation of the wavelength, one can use the elastic parameters of glass and the stress-strain relationship of an average loaded contact in an effective medium theory. However, it is important to keep in mind that this definition neglects wave velocity dispersion which can be important when the spatial scale of the beads coincides with the wavelength. As an estimation, the wave velocity in a disordered three-dimensional packing of 2 mm glass beads with an applied static stress of 200 kP a is ∼ 300±40 m/s, which gives a wavelength of ∼ 3 ± 0.4 mm at 100 kHz. This means that for a wave packet centered on 100 kHz, submitted to a 200 kP a static stress, the wavelength is of the order of the bead diameter, and consequently, strong scattering occurs. The typical experimental signals received in this case are strongly distorted due to multiple scattering, lasting from 10 to 100 times longer than the emitted signal (see Fig. 2). This feature is widely observed in seismics [21] or in multiple scattering experiments, and has been observed in roughly the same conditions in glass bead assemblage [12]. However, we found another feature in this case, not mentioned previously in the literature: for different excitation amplitudes, the normalized signals in Fig. 2 are different. This relative amplitude and shape dependence is the signature of a nonlinear process. One obvious difference between the traces is the emergence, for an increasing excitation amplitude, of a signal at earlier times, relatively lower in frequency than the 100 kHz coda-type signal. We verified in several other experiments [7,13] that this LF signal is the self-demodulated contribution associated with the rectification (demodulation) of the initial HF wave packet. Its nonlinear nature is evidenced by its nonlinear amplitude dependence on the excitation amplitude. This is an important difference compared to results [8,12], where the very same types of signals were observed (for instance for ε a = 5 × 10 -6 in Fig. 2). In [12], the LF contribution is identified as the linear coherent part of the propagated initial pulse, and the HF contribution is interpreted as the multiply scattered signal part. A possible explanation for such a different observation (mainly in the amplitude dynamics behavior and physical nature of the LF signal) is the difference in the spectral width of the emitted signal spectra. In Ref. [12], the signal spectrum width is relatively wide, which may allow for the direct radiation of the linear coherent LF component in the medium. Then, the observation may be interpreted as a strong frequency dependent transmission in the medium (due to absorption and scattering), which is able to modify importantly the transport character of different frequencies (propagative as in an effective medium, multiply scattered, diffusive...).

In the present case however, when the emitted spectrum is sharper, the direct radiation of the LF signal is negligible (and in any case is avoided by a high-pass filter in the setup). This LF signal is generated in the medium itself, by nonlinear processes of frequency mixing [7], widely described in the literature [START_REF] Novikov | Nonlinear Underwater Acoustics[END_REF][START_REF] Westervelt | [END_REF]33,34,35]. This can be seen in Fig. 3, where the relative amplitudes of the LF and the HF contributions are modified by increasing the excitation amplitude.

In Fig. 4, the received temporal signals are presented for three source-receiver distances, with the same applied static stress 300 kP a (measured with a precision of ∼ 3 % which is not sufficient to perform precise quantitative measurements). An important qualitative feature observed on these normalized (by their maximum) traces is the relative increase of the LF contribution compared to the HF one with distance. This feature is usually observed for the nonlinear self-demodulation process [START_REF] Novikov | Nonlinear Underwater Acoustics[END_REF], where over a sufficient distance, only the LF contribution is detectable, due to the difference in attenuation of the HF and the LF waves. In this experiment, the LF signal absolute energy is measured to be roughly constant with distance, while the received HF signal energy is decreasing drastically. The former is a result of the competition between nonlinear effects, which pump low frequencies through frequency-down-conversion processes, and linear attenuation.

In Fig. 5, the total energy of different signals is plotted as a function of the applied static stress for a propagation distance D = 12.5 cm and an excitation amplitude ε a ≃ ε max /50, where ε max ≃ 10 -5 . Three signal energies are defined here: the energy of the HF contribution E HF , the energy of the LF contribution E nℓ LF and the energy of a directly radiated LF signal E ℓ LF (in this case, a LF pulse with the same frequency content as the nonlinearly demodulated pulse is directly excited by the emitter and received after linear propagation). First, comparing the received energies E HF and E ℓ LF evolution as a function of the applied static stress, an important slope difference is clearly visible: E ℓ LF dependence on stress is less than to a power 1 and E HF dependence is close to a power 4 of the stress. In order to explain this difference in the evolution of the transmitted energy as a function of the applied static stress, one should recall that two dominant processes of sound attenuation may play a role: the scattering and the absorption due to linear processes. Considering that the scattering of the HF ∼ 100 kHz, is much more important than for the LF ∼ 10 kHz, the observed difference may be attributed to the strong stress dependent scattering. This is consistent with the simple idea that by increasing the static stress, contacts are created which adds new paths (or strengthen existing paths) of acoustic energy transmission. Moreover, waves following force chains with a given stress may change from an evanescent character to a propagative one with increasing stress, because the cut-off frequency (sometimes called the Einstein frequency) is increased [30,36,37]. It is then possible to ask if this scattering which strongly depends on static stress depends also on the acoustic amplitude of excitation. Is the acoustic wave able to switch dynamically propagation paths or force chains? The aim of the next section is to analyze this opportunity and its possible manifestations.

The competition between nonlinear effects and attenuation is also visible in Fig. 5 when comparing the dynamics of E ℓ LF and E nℓ LF . For signals having the same low frequency propagating through the same medium but generated either directly by a transducer or by nonlinear effects inside the medium itself, the energy dynamics as a function of the applied static stress is different. It can be assumed that the linear attenuation varies in the same way for both signals while the nonlinearity of the medium and the HF attenuation both playing a role in the generation of the LF signal [START_REF] Novikov | Nonlinear Underwater Acoustics[END_REF] may be strongly affected. These modifications only affect the nonlinearly generated signal. The E nℓ LF dynamics as a function of the applied static stress is almost constant, while the E ℓ LF is increasing. It means that the efficiency of the nonlinear interactions leading to the self-demodulated signal is decreasing with increasing static stress. Two main processes can lead to a diminishing of the nonlinear interaction efficiency: the HF attenuation increase (the amplitude of the nonlinear sources diminishes) and the intrinsic nonlinearity of the medium decrease. As the HF attenuation (probed by the dynamics of E HF ) is strongly diminishing with increasing static stress, it means that the nonlinearity of the medium is decreasing with increasing static stress. This is consistent with estimates based on the Hertzian contact nonlinearity and experimental observations [7,13]. Now, we analyze the temporal features of the HF signals, namely the codas. Coda features can be analyzed by performing an envelope detection (for instance by integrating the square field over a sliding window). This may cause problems for quantitative analysis because one has a degree of freedom for the size of the sliding window. In principle, the diffusion approximation is shown to be valid for the configurational average intensity I of the incoherent field under some conditions [38]. The incoherent field can be defined as the contribution which cancels out with averaging over different realizations of the disorder in the medium [38,39]. The average intensity I is usually accessed experimentally by the use of an intensity sensor, which is large compared to the wavelength (a photo-multiplier in optics for instance). For elastic waves however, the piezo-electric transducers are sensitive to the field itself. It is thus necessary, in order to record I , to record wave field signals, then to calculate the corresponding intensities, and finally to average the different obtained intensities over different configurations of the disorder. However, when the receiving transducer has a sensitive surface with dimensions larger than the wavelength, a spatial averaging is performed on the field. In this operation, the incoherent contribution is diminished compared to the coherent one. Consequently, even if most of the energy is incoherent at the receiver location, there could be some situations where the detected signal is composed of coherent and incoherent parts of comparable importance.

In unconsolidated granular media, the configurational averaging can be performed by several means, which may not be equivalent. One could use arrays of transducers for the excitation and detection of the multiple scattered waves, each transducer being small compared to the wavelength. Then without modifying the medium, each emitter -receiver couple, for the same distance through the medium could provide a signal to be treated. Otherwise, with one emitter and one receiver, one could modify the structure of the medium (geometrical and contact force disorder) by keeping constant as much as possible the macroscopic external conditions [8]. These two means can allow in some limits for an estimation of the diffusivity and the absorption characteristic time of the diffusion equation. The first method necessitates small enough transducers to avoid averaging of the field, which causes problems of coupling with the medium when the wavelength is of the order of the bead diameter and the transducer size. Some of the transducers may not be in efficient contact with beads. The second method, due to the memory effects [27,40] or the drastic influence of the external parameters on the elastic properties (see Fig. 5 for instance), does not guarantee to be in the same experimental conditions from one realization of the disorder to another. Small perturbations or local modifications of the contacts forces in the medium are consequently diffi- cult to probe using averaged quantities, obviously more adequate for an estimation of average properties of the medium.

C. Analysis of the amplitude dependent codas

In this section, we investigate fast dynamic modifications of the medium submitted to an acoustic wave, as well as slow dynamic modifications or permanent modifications of the medium. The experimental configuration is slightly different from the one in the previous section. Some glass beads have been glued to the shear emitting transducer surface, in order to couple better to the medium and to have a better signal to noise ratio.

The analysis of the obtained spectra indicates that it is impossible to detect neither the second nor the third harmonic component, certainly due to the strong observed attenuation above 100 kHz. However, the selfdemodulated wave is quite easily generated and detected because of its lower attenuation. In the following, the HF coda contribution and this LF self-demodulated contribution are separated by post-processing filtering in order to analyze in details the different amplitude dependent processes.

In Fig. 6(a), typical signals that are received by a sensitive wide-band longitudinal transducer (with a sensitive surface of 4.5 cm in diameter) are presented. The experimental protocol elaborated in order to analyze the amplitude dependent effects of acoustic wave transport is presented in Fig. 7(d). It contains 59 successive excitation amplitudes. The 10 first amplitudes, as well as the last 10 amplitudes, are identical and relatively low (the acoustic strain is ε a = ε 1-10 ≃ 10 -7 ). The odd numbered amplitudes 11-49 are gradually increasing up to a maximum amplitude of ε a ≃ 10 -5 , while the even amplitudes 10-50 are equal to the minimum probe amplitude ε a ≃ 10 -7 . This allows to compare the response of the medium for different excitation amplitudes, but also, to visualize the medium modifications after being excited by a strong wave by using the weak probe wave. The experiment time for the acquisition of one signal of this protocol is close to 30 s. In Fig. 6, signals are numbered according to their excitation amplitude in the protocol, the first signal of the protocol being denoted by s 1 . The excitation signal is a sine wave modulated in amplitude with a Gaussian function of 6% full width at half maximum (FWHM) in the frequency domain, which, at the central frequency f 0 = 100 kHz gives a ∼ 0.2 ms duration pulse at half the maximum. This duration is roughly the time necessary for the wave to travel from the emitter to the receiver for the 6 cm distance.

In Fig. 6(a), received signals over the full available frequency band (from 1 kHz to 300 kHz) are presented for several excitation amplitudes of the protocole: ε 10 , ε 11 , ε 19 , ε 20 . The signal structure is quite complicated and shows several packets typically 4-5, having a shape close to the emitted signal. It is important to note here that the time dependent intensity of these signals is much different from a diffusive type enveloppe, even if the average wavelength is of the order of the bead radius. Another important feature is the emergence, at a sufficiently high excitation amplitude of a relatively low frequency (LF) signal (see for instance signal s 19 corresponding to the amplitude ε 19 in Fig. 6(a)). Due to its amplitude dependent character (this signal is not visible for lower excitation amplitudes), this LF signal is nonlinear, and has been identified in previous works as being a selfdemodulated contribution [7,13].

In the following, we analyze in details the high frequency part of the signals, and for quantitative comparison, the raw signals are high-pass filtered (Fig. 6(b)). These coda-type signals have comparable time durations to those of Fig. 2, but their envelope characteristic frequency is lower. This is a consequence of the narrower Gaussian spectrum of excitation. However, despite of the difference in coda shape, the characteristics of wave transport are similar. The most important characteristics of the signal evolution with excitation amplitude is the fact that their temporal shape is strongly modified, even for very moderate changes in the excitation amplitude. This can be seen by comparing the shapes s 10 and s 11 in Fig. 6(b). Of course, by increasing further the excitation amplitude, the distortion is further increased (signal s 19 ). When the excitation amplitude is decreased down to the lowest amplitude of the protocole, i.e. to the probe amplitude ε 1 , the shape modification of s 20 compared to s 1 is barely noticeable. In order to monitor quantitatively the modifications of the signal shapes due to amplitude dependent effects, we make use of the following parameter R m,n defined from the inter-correlation function C n,m (t) = T 0 s n (t)s m (t -τ )dτ , where T is the observation window duration, by

R n,m = C n,m (0) C n,n (0)C m,m (0) . ( 1
)
The energy of signal s n (respectively s m ) is proportional to C nn (0) (respectively C mm (0)). The normalization by C nn (0)C mm (0) makes consequently the parameter R n,m independent of the actual energies of s n and s m , which is important in our case where signals with different amplitudes are compared. The parameter is only dependent on the relative shapes of the signals and can be seen as the level of resemblance of two signals, equal to 1 if the two signals are exactly the same and progressively diminishing when the signals differ.

In Fig. 7(a), the parameter R n,1 is plotted for each signal s 1 to s 59 of the experimental protocol. It is clearly visible that the parameter R 11,1 goes under a value of 0.9 while the signals s 2 to s 10 exhibit a very accurate correlation with s 1 (R 2-10,1 ≃ 1). When the amplitude is increased further, the parameter R n,1 decreases drastically to 0 or less, which shows that a strong field modification occured (the negative values of the parameter R m,n do not have a particular meaning, except when the value reaches -1, which means that the signal phases are inverted). Concerning the even signals generated with a low excitation amplitude, their resemblance parameter remains practically equals to 1 up to the 30th signal of the protocol. This means that no modification of the granular medium elasticity is detectable by this acoustic probing. In contrary, when higher acoustic amplitudes are generated in the medium, the level of resemblance of the probe wave with the first signals of the protocol begins to fall down progressively, until the typical value of 0.6 after the maximum excitation strain ε 49 . The acoustical properties of the medium have been modified by the strong wave action. A possible interpretation of this experimental observation could be found in the temperature effects on the linear and nonlinear hysteretic prop- erties of the medium [41]. The wave action on the contacts increases the average temperature and consequently modifies the linear and nonlinear contact properties for some characteristic time which can be of the order of several acoustic periods. Elastic memory effects at the level of the contacts themselves (stick-slip motion for Hertz-Mindlin contacts for instance [42]) or at the mesoscopic scale [27] could also contribute. When, at the end of the protocol, only the probe amplitude is generated (signals s 50 to s 59 ), the parameter R n,1 tends to 1 with time, the acoustic properties of the medium are progressively recovered in a slow healing process. The parameter R n,n-1 in Fig. 7(b) shows the level of resemblance of two successive signals. It is interesting to see that in this healing process at the end of the protocol, two successive signals are practically identical. The process of healing is slowly cumulative. This effect can be seen as a slow dynamic effect observed for elastic waves in other materials too, such as sandstones and cracked solids [43,44]: the strong wave action modifies the acoustic properties of the medium, which are slowly recovered after some time, long compared to the wave period. The probing of this slow process is sometimes a powerful indication of damage (associated with the presence of internal contacts) in materials [45].

The last parameter plotted in Fig. 7(c) is the characteristic time T e corresponding to the arrival time of half the total energy of the signal. It exhibits a quite clear amplitude dependence and has the same qualitative behavior as the parameter R n,1 . Until the 30th amplitude of the protocol (ε 33 ≃ 0.3 × 10 -5 ), the characteristic time T e of the probe wave (even numbers of signals) remains unchanged, while the signals s 11 to s 31 exhibit a diminishing characteristic time T e . It means that either the wave packet is propagating faster (nonlinear hardening of the medium), either the signal energy is preferentially attenuated at later times. Due to the strong nonlinear attenuation observed in the temporal signals of Fig. 6(b), the second process seems more adequate to explain the T e dependence on amplitude. This is visible in Fig. 6 In order to analyze these amplitude dependent effects on the signal energy attenuation, it is adequate now to focus on the energetical properties of the signals. In Fig. 8(d), the experimental amplitude protocol is recalled. The total energies of the high-pass filtered signals C HF n,n (0) are normalized by a quantity proportional to the excitation energy ε 2 n both for the low excitation amplitudes of the protocol (Fig. 8(a)) and for the higher excitation amplitudes (Fig. 8(b)). The parts comprised between the dashed-lines compare the same range of normalized energy, from normalized values 0.8 to 1.1 roughly, that are the extreme values taken by the signals excited with the weak probe amplitude. The lowest value observed in Fig. 8(a) corresponds to an excitation amplitude range where the parameter R n,1 begins to decrease from 1 to a lower value. It is important to note that for the normalized energies associated with the higher excitation amplitudes, in Fig. 8(b), all the values are below the region delimited by the dashed lines, which means that energy absorption is always stronger for ε a > ε 1 even when the acoustic response of the medium has been modified by a strong wave action (even signals s 40 to s 50 for instance). The general behavior of the normalized energy for increasing excitation amplitudes is decreasing and monotonous: the higher the excitation amplitude is, the higher the energy absorption is. Its associated energy is consequently deviating from the ε 2 n law. The nonlinear energy of the LF demodulated part is increasing monotonously with the excitation amplitude. The LF energy exceeds the HF energy of the signal from the excitation amplitude ε 41 , as seen in Fig. 8(c) when the LF energy is above the region delimited by the dashed lines.

In order to analyze more precisely the nonlinear selfaction process for the HF components and the LF energy nonlinear increase process, their normalized energies are plotted as a function of the excitation amplitude in log-log scale, and compared to power laws in Fig. 9. In Fig. 9(a), the normalized HF energy is plotted as a function of the excitation amplitude. This is the same quantity as that of Fig. 8(b). For the lowest excitation amplitudes, the normalized C HF n,n (0)/ε 2 n is decreasing close to a power law ε -0.8 a , which means that the energy C HF n,n (0) is increasing with a ε 1.2 a dependence. In a perfectly linear system, it would be a ε 2 a dependence. Note that the overall behavior is close to a ε 1.25 a dependence, which is a dramatic deviation from the linear law ε 2 a and also a very strong self-action compared to usually observed saturation processes in rocks for instance [START_REF] White | Underground sound[END_REF]. For higher excitation amplitudes, a small plateau is observed, denoting a local comportment law close to the linear one ε 2 a , even if the process leading to this locally linear behavior may be due to combined effects of different nonlinear processes: self-transparency, slow-dynamics, conditioning... For the highest excitation amplitudes, the dependence on amplitude is the strongest, approaching a ε 0.5 a law for C HF n,n (0). This means that if the excitation energy is multiplied by a factor 4 for instance, the received energy is multiplied by a factor √ 2. This would be an even stronger effect for the late part of the signal considering the fact that most of the nonlinearly attenuated energy is the one arriving at the latest time in the signal, according to Fig. 6 and Fig. 7. Surprisingly, the largest amplitudes of the signal are not strongly nonlinearly attenuated but the late contributions are. This peculiar observation is discussed in section III.

In Fig. 9(b), the normalized LF part energy C LF n,n (0)/ε 2 n is plotted in log-log scale as a function of the excitation amplitude. A quite clear power law ε 1.27 a is found for this NL process of frequency-down conversion (selfdemodulation), which corresponds to a LF energy proportional to ε 3.27 a . The classical NL behavior observed in homogeneous media with a quadratic nonlinearity is ε 4 a for the LF energy [START_REF] Novikov | Nonlinear Underwater Acoustics[END_REF]. The observed dependence corresponds to a ε 1.63 a power law for the field itself, to be compared to the ε 2 a law classically observed. This dependence is also closely observed with another processing method applied to the LF part: the normalized peak-to-peak amplitude of the LF contribution ε LF n /ε 2 n is plotted in Fig. 9(c) in log-log scale. A power law behavior ε -0.31 a can be observed, that is a law ε 1.69 a for the dynamics of ε LF n . If compared to the behavior obtained with C LF n,n (0) ∝ ε 3.27 a , taking into account that C LF n,n (0) ∝ (ε LH n ) 2 , this dynamics corresponds to a ε 1.69×2 a = ε 3.38 a law. This is an intermediate dependence between the classical one ε 4 a and the Hertzian clapping limit ε 3 a . The latter dependence law has been observed earlier in different experimental conditions in [7]. Note that for symmetry reasons, the Hertz-Mindlin hysteretic nonlinearity is not able to predict the effect of self-demodulation observed here at the leading order of the nonlinear approximation, and may not contribute [START_REF] Zarembo | [END_REF]48].

III. SUMMARY OF THE RESULTS AND DISCUSSIONS

Different experimental observations have been presented for the first time and the main results can be summarized as the following:

• observation of the coexistence of a nonlinear selfdemodulated LF wave together with a HF codatype wave (Fig. 2),

• a drastic increase of the HF coda energy with static stress, compared to the dependence of the LF wave energy (Fig. 4), • a strong shape dependence of the coda on the acoustic excitation amplitude (Fig. 5 and Fig. 6), and especially a strong nonlinear attenuation of the coda wave at the latest arrival times,

ε LF n /ε 2 n C LF n,n (0)/ε 2 n C HF n,n (
• almost no variation in the acoustic properties of the medium after moderate acoustic excitations up to an excitation acoustic strain of ε a ∼ 0.3×10 -5 , corresponding to signal number 31 (this represents a ratio of dynamic over static deformation of ε a /ε 0 ≃ 5 × 10 -3 ),

• dynamic modifications of the acoustic properties (nonlinear self-action, fast dynamics) even at low excitation levels ε a ∼ 2×10 -7 corresponding to sig-nal number 11 (this represents a ratio of dynamic over static deformation of ε a /ε 0 ≃ 3 × 10 -4 ),

• modification of the acoustic properties after large excitation levels (slow dynamics and conditioning effects), followed by a slow recovery.

Having in mind the ray acoustic approximation in this experiment of multiple scattering of elastic waves, the first part of the coda signals can be associated with short propagation paths while the last part of the coda signals may be associated with longer propagation paths. Note that the lengths of the long propagation paths can exceed the container size by a factor of 10, which means that reflexions on the container sides may occur. In principle, nonlinear attenuation should be more visible on the long propagation paths than on the short ones because the distance for the accumulation of this nonlinear process is larger [START_REF] Novikov | Nonlinear Underwater Acoustics[END_REF][START_REF] Zarembo | [END_REF]. This explains the stronger effect of nonlinear attenuation on the last part of coda signals, that experienced longer propagation paths. The physical nature of this nonlinear attenuation process remains undetermined. Nonlinear scattering of the elastic waves due to opening or closing of the weakest contacts and therefore dynamic switching of propagation paths under the wave action could explain the nonlinear attenuation. This is in agreement with the drastic dependence of the HF coda wave energy transmission on the applied static stress. Nonlinear absorption of the elastic waves at the contacts by stick-slip [42] or thermo-elasticity [43] processes could also contribute. Recent numerical simulations of the acoustic propagation through unconsolidated granular packings [49,50] could provide an interesting insight in the processes involved in the experimentally observed nonlinear attenuation.

For both nonlinear scattering and nonlinear absorption, the weakest contacts are supposed to be responsible for the main contribution, especially for such small acoustic excitation strains as ε a ≃ 10 -7 . Coda signals are shown to be particularly sensitive to these weak contacts, certainly due to the fact that energy is well distributed over the entire medium, including weak contacts, after few milliseconds (coda signals have durations larger than 2 ms, which corresponds to propagation paths much larger than the emitter -receiver distance). Moreover, the wavelength being of the order of the bead diameter (λ ∼ d), small features as small as the beads and their contacts are resolved by the acoustic wave. This is not the case for long wavelength propagation (λ ≫ d), which is known to be less sensitive to the weak contacts in the medium and more influenced by the average properties of the medium [7,9].

The modification of the acoustic properties after the application of a strong excitation, is commonly related to the effect of conditioning and slow dynamics [26,43,44,51]. This addresses the problem of the acous-tic wave influence on the granular medium state and memory. Under a given threshold, the medium is apparently not modified (under excitation amplitude number 31, ε a ≃ 0.3 10 -5 which corresponds to a ratio of dynamic over static deformation of ε a /ε 0 ≃ 5 × 10 -3 ), while over some excitation amplitude, the medium exhibits memory of the past acoustic excitation. Coda signal analysis is shown to be a powerful tool to monitor slight modifications in the acoustic response of an unconsolidated granular structure as a function of time, the resolution (including some time averaging) being of the order of the second.

IV. CONCLUSIONS

In this work, experimental results on the nonlinear transport of short wavelength acoustic waves (of the order of the bead diameter) are reported. It is shown that the amplitude dependence of the codas is strong, and contains some information on the strength of the propagation paths in the medium. An amplitude dependent attenuation is visible on the characteristic time T e , which diminishes with the excitation amplitude: most of the nonlinear self-action effects take place at the latest arrival times of the coda signals. This shows that the field which encountered the larger number of scattering events, or which does not follow the strongest force chains is more influenced by the effects of nonlinear dissipation (stick-slip, sliding, or thermo-elastic losses) and nonlinear scattering (closing or opening of the weak contacts for instance).

Among the possible applications of such experimental results is the non destructive testing (NDT) of granular structures with nonlinear acoustic methods, in particular with multiple scattered waves (coda signals), in contrast to the widely used and recent nonlinear methods employing coherent wave interactions [52,53]. For this purpose, it would be of interest to make use in the future of the coda wave interferometry technique [54,55,56].

It has been shown that the acoustic properties of the medium can be modified temporary by a strong acoustic excitation. In the context of NDT, this work show the range of acoustic amplitudes to use (or the time necessary to recover the initial state) in order to stay in a regime where the acoustic waves do not modify the elastic response of the medium.

= 5 × 10 - 7 ε a = 2 × 10 - 6 ε a = 5 × 10 - 6 FIG. 2 :

 5107210651062 FIG.2: Experimental temporal signals that are typically received for different excitation amplitudes. The applied static stress is ∼ 300 kP a and the emitter -receiver distance is D = 10.5 cm.

Frequency= 5 × 10 - 7 ε a = 2 × 10 - 6 ε a = 5 × 10 - 6 FIG. 3 :

 5107210651063 FIG.3: Experimental spectra corresponding to temporal signals of Fig.2. The spectrum of the emitted signal with a 22% FWHM (full width at half maximum) is superimposed on the top curve.

FIG. 4 :

 4 FIG.4: Signals received after propagation through different thicknesses of granular medium, for identical static pressure of 300 kPa and excitation amplitude εa = 5 × 10 -6 .

FIG. 5 :

 5 FIG.5: Signal energy dynamics as a function of the applied static pressure. The HF (EHF ) and LF (E nℓ LF ) contributions are separated by filtering. The energy of a reference LF (E ℓ LF ) linear signal is also plotted for comparison.

ε 20 = ε 10 ε 19 = 6ε 10 ε 11 = 2ε 10 ε 10 ≃ 10 -7 ε 10 ≃ 10 - 7 ε 11 = 2ε 10 ε 19 = 6ε 10 ε 20 = ε 10 FIG. 6 :

 2010191011101010101071110191020106 FIG. 6: Typical temporal signals (normalized by the excitation amplitude) that are received at a distance of 6 cm from the emitter for different excitation amplitudes. The emitted signal is a Gaussian wave packet centered on 100 kHz with a full width at half maximum (FWHM) of 6% in frequency. a) Row signals, b) High-pass filtered signals.

FIG. 7 :

 7 FIG. 7: (a) Resemblance parameter Rn,1 between the signal number n and the signal number 1 of the protocol (d). (b) Resemblance parameter Rn,n-1 between the signal number n and the signal number n -1 of the protocol (d). (c) Arrival time Te of half the signal energy as a function of the signal number in the protocol (d). (d) Protocol for the successive excitation amplitudes of the experiment.

  (b) when comparing signals s 10 and s 11 or s 10 and s 19 . No shift in time is detectable while the last part of the signals are strongly attenuated relatively to their first part.

2 nFIG. 8 :

 28 FIG. 8: (a) Energy of the high-pass filtered signals C HF n,n (0) normalized by a quantity proportional to the excitation energy ε 2 n for the low excitation amplitudes of the protocol (d). (b) Energy of the high-pass filtered signals for the higher excitation amplitudes of the protocol. (c) Energy of the low-pass filtered signals for the higher excitation amplitudes of the protocol. (d) Protocol for the successive excitation amplitudes of the experiment.

0)/ε 2 nFIG. 9 :

 29 FIG. 9: (a) Energy of the high-pass filtered signals C HF n,n (0) normalized by a quantity proportional to the excitation energy ε 2 n as a function of the excitation amplitude in log-log scale. (b) Energy of the low-pass filtered signals C LF n,n (0) normalized by a quantity proportional to the excitation energy ε 2 n as a function of the excitation amplitude in log-log scale (c) Peak-to-peak amplitude of the self-demodulated signal normalized by ε 2 n in log-log scale.

Acknowledgments

This work is supported by ANR project "grANuLar" NT05-3 41989. The authors would like to thank E. Brasseur, E. Egon and P. Collas for their technical help on the experimental setup.