V Martinet
email: vincent.martinet@grignon.inra.fr

J Peña-Torres

M De Lara
email: delara@cermics.enpc.fr

H Ramírez
email: hramirez@dim.uchile.cl

Risk and Sustainability: Assessing Fishery Management Strategies

Keywords: sustainability, risk, fishery economics and management, stochastic viability

We develop a theoretical framework to assess the sustainability of fishery management strategies, when the bioeconomic dynamics are marked by uncertainty and several conflicting objectives have to be accounted for. Stochastic viability ranks management strategies according to their probability to sustain economic and ecological outcomes over time. The approach is extended to build stochastic sustainable production possibility frontiers representing the trade-offs between sustainability objectives at any risk level, given the current state of the fishery. This framework is applied to a Chilean fishery faced with El Niño uncertainty.

We study the viability of effort and quota strategies when catch and biomass levels have to be sustained. We show that i) for these sustainability objectives, whatever the level of the outcomes to be sustained, quota-based management results in a better viability probability than effort-based management, and ii) the fishery's historical quota levels were not sustainable given the stock levels in the early 2000s.

Introduction

The analysis in this paper originates in real concerns related to the management of Chilean fisheries. The jack-mackerel fishery is being challenged by uncertain El Niño cycles, which increase uncertainty about the availability of the resource [START_REF] Barber | Biological consequences of El Niño[END_REF], making management of the fishery more difficult [START_REF] Costello | The value of El Niño forecasts in the management of salmon: A stochastic dynamic assessment[END_REF]. 1In addition to the usual objective of maximizing profits, current management is aimed at avoiding stock collapse. Sustainable resource management requires a framework that takes account of both economic and ecological objectives under risk and over time.

The standard economic approach to assessing the performance of fishery management strategies relies on the expected discounted utility framework [START_REF] Clark | On uncertainty renewable resource stocks: Optimal harvest policy and the value of stock surveys[END_REF][START_REF] Reed | Optimal escapement levels in stochastic and deterministic harvesting models[END_REF][START_REF] Sethi | Fishery management under multiple uncertainty[END_REF]. This approach has the great advantage of defining a unique value, the expected discounted utility of harvesting, which characterizes optimal strategies and ranks alternative management strategies. However, it has some practical limits when applied to sustainable resource management issues encompassing several dimensions and the concern for intergenerational equity. First, accounting for ecological objectives requires the definition of a multi-attribute Social Welfare Function (SWF) prior to the maximization problem. However, if uncertainties are pervasive and if the sustainability issues affect multiple and heterogeneous stakeholders, the task of agreeing on a common SWF can be extremely intricate. Second, the discounted utility framework allows for intertemporal compensation of good and bad outcomes for the system, which may raise intergenerational equity issues (particularly if the discount rate is positive).

In practice, fishery management strategies, often defined as simple "rules of thumb," are evaluated in so-called "multicriteria" frameworks [START_REF] Geromont | Development and application of management procedures for fisheries in Southern Africa[END_REF][START_REF] De Oliveira | Developing and refining a joint management procedure for the multispecies South African pelagic fisheries[END_REF][START_REF] Kell | An evaluation of the implicit management procedure used for some ICES roundfish stocks[END_REF][START_REF] Smith | Scientific tools to support the practical implementation of ecosystem-based fisheries management[END_REF]. These methods are based on simulations and do not rely on an optimization framework. They provide no common metrics for conflicting (ecological and economic) objectives and risk. Therefore, they cannot rank alternative management strategies explicitly. Thus, there is a gap in resource management between theory and practice. Developing a practical framework based on solid theoretical grounds to assess the sustainability of fishery management strategies under risk is a challenging task. This paper proposes a framework which accounts for conflicting sustainability issues and risk, and provides an explicitly ranking of alternative management strategies. This framework echoes the concept of stewardship,2 which defines sustainable resource management as a strategy that sustains economic and ecological outcomes over time, corresponding to a "satisficing" objective à la [START_REF] Simon | A behavioral model of rational choice[END_REF]. Technically, we build on the stochastic viability approach (De [START_REF] Lara | Sustainable Management of Natural Resources[END_REF]. Given a set of multidimensional indicators referring to economic or ecological outcomes, viability is defined as the ability to sustain the levels of these indicators above some thresholds characterizing sustainability objectives (e.g., minimal biomass, minimal profit). We assess fishery management strategies according to their probability of achieving these objectives jointly, and at all times, over the planning horizon.

While stochastic viability has been used in previous studies as a simulation tool to examine fishery management issues (e.g., [START_REF] Doyen | A stochastic viability approach to ecosystem-based fisheries management[END_REF], the present paper differs in two important respects, each of which constitutes theoretical novelty. First, we embed stochastic viability in a theoretical optimization framework with economic interpretations, defining a value function for our optimization problem. This value measures the ability to sustain several outcomes over time. Second, while in viability analysis the thresholds of the viability constraints are usually exogenously fixed parameters, we treat these sustainability thresholds as explicit arguments of our value function. This allows us to define and build stochastic sustainable production possibility frontiers which describe the necessary trade-offs between sustained levels of economic and ecological outcomes and risk. Such possibility sets depend on the current (over-)exploitation status of the fishery.

Our framework does not rely on an a priori representation of social preferences but can be used to reveal some of these preferences. Defining actual sustainability thresholds amounts to determining what should be sustained over time [START_REF] Martinet | Economic Theory and Sustainable Development: What can we preserve for future generations?[END_REF]. This is a social choice problem which is not addressed explicitly here. It corresponds to a generalized, multidimensional maximin problem [START_REF] Solow | Intergenerational equity and exhaustible resources[END_REF][START_REF] Martinet | A characterization of sustainability with indicators[END_REF], with low substitutability among sustainability issues, and strong aversion to intertemporal inequality on all sustainability dimensions. Stochastic sustainable production possibility frontiers can be used to inform the social choice of sustainability objectives in the fishery, and to reveal social preferences related to sustainability issues.

These theoretical novelties allow us to bridge the gap between the economic literature on optimal resource management under risk, and the practical-oriented literature on sustainable fisheries management. The viability probability provides a common metrics to aggregate the outcomes of the system with respect to the several sustainability dimensions. It can be used to rank alternative management strategies. Marginal analysis makes it possible to examine the trade-offs between sustained outcomes and risk. Thus, our approach is closer to economics than the usual multi-criteria fishery management approaches. It can be implemented if no SWF is available.

We illustrate the implications of our approach in the case of the (small pelagic) Chilean jack-mackerel fishery which is threatened by El Niño uncertainty. In particular, we compare effort-based (price-like) and quota-based (quantity-like) strategies for their ability to sustain both catch and biomass levels over time given current information on the resource stock. While the price versus quantity issue in relation to fisheries has been debated extensively from an economic point of view, to our knowledge, the analysis in this paper is the first attempt to examine this issue from a sustainable management perspective.

Section 2 highlights the differences between the fishery economics literature and the fishery management literature which were the motivation for our approach. Section 3 presents our theoretical framework to assess risk and sustainability and compare management strategies. In Section 4, we apply this framework to the Chilean jack-mackerel fishery case-study. Section 5 concludes by discussing the relevance of our results for practical fisheries management.

Background and settings

Optimality in fishery economics is usually defined as maximization of the expected discounted profit of the harvest. Depending on the type of uncertainty and economic specifications, optimal harvesting may correspond to very specific management strategies, and be hard to apply in practice. 3 Moreover, in a sustainability context, manage-3 See [START_REF] Reed | Optimal escapement levels in stochastic and deterministic harvesting models[END_REF]; [START_REF] Clark | On uncertainty renewable resource stocks: Optimal harvest policy and the value of stock surveys[END_REF]; [START_REF] Sethi | Fishery management under multiple uncertainty[END_REF]; [START_REF] Nøstbakken | Uncertainty in bioeconomic modelling[END_REF]; [START_REF] Nøstbakken | Stochastic modelling of the North Sea Herring fishery under alternative management regimes[END_REF][START_REF] Mccough | Optimally managing a stochastic renewable resource under general economic conditions[END_REF]. When responding to uncertain stock fluctuations, optimality may require strong yearly variations of the Total Allowable Catch (TAC), pulse-fishing (Da-Rocha ment objectives are often not limited to profit maximization. Ecosystem-Based Fishery Management is aimed at conserving resources and sustaining the socio-economic benefits from fishing [START_REF] Cochrane | Reconciling sustainability, economic efficiency and equity in fisheries: the one that got away?[END_REF][START_REF] Pikkitch | Ecosystem based fishery management[END_REF]. This increases the number of objectives and stakeholders [START_REF] Fletcher | The application of qualitative risk assessment methodology to prioritize issues for fisheries management[END_REF] with the result that fisheries are faced with unsustainable situations whenever one of these objectives is not met. Prioritizing social and economic objectives over ecological targets has been identified as an important reason for management failure in fisheries [START_REF] Hilborn | Defining success in fisheries and conflicts in objectives[END_REF]. Management procedures (MP) 4 should be ranked according to their capacity to yield acceptable results with respect to all sustainability objectives while being robust to uncertainties [START_REF] Charles | Living with uncertainty in fisheries: analytical methods, management priorities and the Canadian groundfishery experience[END_REF].

Extending the economic optimization approach to account for ecological objectives is a delicate exercise. In theory, one could define a multi-attribute SWF that would fully characterize social preferences over the various dimensions of interest, prior to the optimization problem. However, stakeholders may be unable to agree on a SWF. This a form of "collective" bounded rationality results in the impossibility to define a continuous representation of preferences over payoffs across various dimensions and risks.

An alternative option would be to add ecological constraints to the profit maximization problem. Note that setting the levels of these constraints is a social choice problem which should not be overlooked. In the deterministic case, the optimization problem provides the marginal cost of complying with the constraint. This information can be used in a back-and-forth process with stakeholders to adjust the constraints level and reveal preferences over economic and ecological outcomes. This feature is lost in the stochastic case, 5 where a theoretical and technical issue emerges, i.e., how to interpret and handle constraints under uncertainty. It is possible to "translate" the deterministic economic criterion into its expected value but it is more difficult to "translate" a constraint in stochastic terms. Requiring constraint satisfaction with probability one, i.e., that the optimal strategy satisfies the constraint in all possible states of the world, et al., 2013), and even fishery closure if the stock size is too small [START_REF] Nøstbakken | Regime switching in a fishery with stochatic stock and price[END_REF], whereas fishing industries favor stability of catches [START_REF] Charles | Living with uncertainty in fisheries: analytical methods, management priorities and the Canadian groundfishery experience[END_REF].

4 A MP is a set of rules which translates fishery data into a regulatory mechanism, such as TAC or maximum fishing effort (Butterworth et al., 1997). MPs have been developed (though not always implemented) for a number of fisheries since their development within the International Whaling Commission in the late 1980s [START_REF] De Oliveira | Developing and refining a joint management procedure for the multispecies South African pelagic fisheries[END_REF]. 5 It will be seen that our framework provides somewhat similar information to support the choice of sustainability constraints in the stochastic case.

usually restricts decisions to the extent that the optimization problem loses its interest.

Another possibility would be accepting a risk of constraint violation. This amounts to considering the performance of the system with respect to the ecological constraint, by providing a measure of the risk of violating it. There are then two outcomes for each strategy: the expected economic profit, and the ecological risk.

This last option, in fact, is close to the Management Strategy Evaluation (MSE) approach. 6 MSE relies on simulations to compare the performance of given management strategies against the conflicting objectives of limiting risk to the resource, reducing TAC variation over time, and increasing average catches. The results are usually represented graphically, in a map of "mean catch -risk to the resource" (see, e.g., [START_REF] Smith | Scientific tools to support the practical implementation of ecosystem-based fisheries management[END_REF]. Fig. 1 displays the results for the Chilean jack-mackerel fishery. "Ideal" management strategies present low risk to the resource and high mean catches, and are depicted in the South-East of the figure. Since there is no common metrics between objectives, the two performances cannot be aggregated, and non-dominated strategies cannot be ranked. 7

The problem lies mainly in the fact that the economic and ecological objectives are not treated in the same way: the former is to maximize an outcome while the latter is to satisfy a constraint. The economics approach to risk is usually to define preferences characterizing value (i.e., to aggregate economic and ecological outcomes in a SWF) and to account for risk by computing expectation of value. 8 The MSE approach compares the expected economic value with the ecological risk (probability to overshoot a given ecological threshold). The ecological objective is defined separately from economic value, which makes it difficult to aggregate the two outcomes.

Thus, assessing the sustainability of resource management strategies under risk is difficult when there is no SWF describing the preferences related to different issues.

To address this challenge, we propose a theoretical framework that reflects the concept 6 Various scientific tools, mainly in "multicriteria" frameworks, have been developed to support sustainable fisheries management [START_REF] Smith | Scientific tools to support the practical implementation of ecosystem-based fisheries management[END_REF]. MSE is the most developed (Butterworth et al., 1997;[START_REF] Charles | Living with uncertainty in fisheries: analytical methods, management priorities and the Canadian groundfishery experience[END_REF][START_REF] Geromont | Development and application of management procedures for fisheries in Southern Africa[END_REF][START_REF] Sainsbury | Design of operational management strategies for achieving fishery ecosystem objectives[END_REF][START_REF] De Oliveira | Developing and refining a joint management procedure for the multispecies South African pelagic fisheries[END_REF][START_REF] Kell | An evaluation of the implicit management procedure used for some ICES roundfish stocks[END_REF].

7 Moreover, the MSE approach provides no information on the opportunity cost of the ecological constraint or the marginal gains from relaxing its level. in terms of risk to the resource (measured as the probability that the stock falls below 20% of the pre-exploitation spawning stock biomass) and expected mean annual catches (used as a proxy variable for the economic objective). Adapted from [START_REF] Yepes | Dinámica poblacional del jurel: Reclutamiento asociado a factores ambientales y sus efectos sobre la captura[END_REF].

of stewardship. We assume that intertemporal equity requires the economic and ecological performance of the system to be sustained over time. These conditions can be represented by constraints on (ecological and economic) indicators, which should be maintained above some thresholds at all times. This issue is addressed in a stochastic viability framework which defines the (maximal) probability of satisfying jointly several viability constraints over time in dynamic, uncertain models. Any management strategy satisfies these viability constraints with some probability. This viability probability provides a common metrics to assess and rank alternative strategies.

This approach treats all the relevant sustainability objectives as minimal outcomes to be sustained over time. Defining the viability thresholds as arguments of the stochastic viability value function, we build stochastic sustainable production possibility frontiers, which exhibit the necessary trade-offs between the targeted sustained outcomes and risk. These frontiers can be used in the social choice of sustainability objectives.

A metrics for risk and sustainability

Let us formalize the decision problem in a general framework. The model and method described below are appropriate for setting up any stochastic viability analysis, and therefore can be applied to a variety of resource management situations or to environmental problems with stocks of pollutants. We provide examples based on the fisheries case.

Modeling framework

Dynamic system Consider a resource harvesting model, which accounts for dynamics, uncertainty and exploitation decisions. The model is described by the following discrete-time control dynamic system

x(t + 1) = G t, x(t), c(t), ω(t) , t = t 0 , . . . , T -1 , x(t 0 ) = x 0 , (1) where
• the time index t is discrete, belonging to T = {t 0 , . . . , T } ⊂ N; the time period [t, t + 1[ is a year for example; t 0 is the initial time ; T is the finite time;

• the state vector x(t) ∈ X ⊂ R n could be a vector of abundance-at-age for one or for several species; It could also represent abundances at different spatial patches or include capital stocks (e.g., fishing vessels);

• the control vector c(t) ∈ C ⊂ R p could denote catches or harvesting effort;

• ω(t) ∈ W ⊂ R q denotes a vector of uncertainty which affects the dynamics at time t (e.g., recruitment or mortality uncertainties in a dynamic population model, climate fluctuations or trends, unknown technical progress, price uncertainty);

• G : T × X × C × W → X represents the dynamics of the system. It could be one of the numerous dynamic population models, such as logistic or age-class models;

it could also include capital accumulation dynamics;

• x 0 ∈ X is the given initial state for the initial time t 0 . It is supposed to be known.

The notation c(•) means a control trajectory c(•) = c(t 0 ), . . . , c(T ) whereas x(•) =

x(t 0 ), . . . , x(T ) denotes a state trajectory.

Probability distributions over scenarios A scenario is a sequence of uncertainty vectors denoted by ω(•) = (ω(t 0 ), . . . , ω(T -1)). We define the set of all possible scenarios as

Ω = W T -t 0 . ( 2
)
We assume that the set of scenarios Ω is equipped with a probability distribution P. 9

Formally, this probability P could be either an objective probability derived from a statistical model using real world data (as in our case study in Section 4), or a subjective probability representing the decision-maker's beliefs.

Stochastic Viability

Sustainability objectives described with indicators and thresholds Consider

K real-valued functions I k : T × X × C → R, for k = 1, . . . ,
I k t, x(t), c(t) ≥ τ k , ∀k = 1, . . . , K , ∀t = t 0 , . . . , T . (3)
In the viability framework, a trajectory that does not satisfy one (or more) of the constraints at some time is not viable. At a given time period, the violation of some of the sustainability constraints is not compensated by good outcomes in 9 Technically, the probability P is defined over the Borel σ-algebra of Ω. In what follows, we assume proper measurability assumptions for all the functions we consider. 10 We consider sustainability "goods," for which an ad-hoc indicator is defined. This indicator is then constrained to be above a certain threshold. For "bads," such as pollution (e.g., CO 2 concentration), one can take their negative value as an indicator.

other sustainability dimensions. Violation of the sustainability constraints at some time periods is not compensated by good outcomes at other time periods. 11 The requirement to satisfy all constraints at all times reflects the idea that sustainability has to encompass ecological and economic issues in an intergenerational equity perspective.

In a stochastic framework, it is generally impossible to satisfy the constraints for all scenarios ω(•). We use the term viable scenarios to refer to the uncertainty scenarios where all viability constraints are satisfied at all times under a given strategy.

Viable scenarios associated with a management strategy For any management strategy c, initial state x 0 , and initial time t 0 , we define the set of viable scenarios as:

Ω c,t 0 ,x 0 =                  ω(•) ∈ Ω x(t 0 ) = x 0 x(t + 1) = G t, x(t), c(t), ω(t) c(t) = c t, x (t)
I k t, x(t), c(t) ≥ τ k , k = 1, . . . , K t = t 0 , . . . , T                  . (4)
For a given strategy c and a given scenario ω(•), the dynamics (1) produces a state trajectory x(•) and a control trajectory c(•) once the strategy c(t) = c(t, x(t)) is applied.

Therefore, a viable scenario ω(•) ∈ Ω c,t 0 ,x 0 is one where the state and control trajectory

x(•), c(•) driven by the strategy c satisfies the constraints (3).

In the ideal case where a strategy c exists such that Ω c,t 0 ,x 0 coincides with Ω, viability can be achieved for all scenarios by applying this strategy. If this is not the case, since Ω is equipped with a probability P, we can measure the likelihood that a strategy c will meet the objectives by the probability of associated viable scenarios,

P [Ω c,t 0 ,x 0 ],
which is called the viability probability associated with the management strategy c, the initial time t 0 , and the initial state x 0 .

Management strategy assessment by stochastic viability For any given set of sustainability thresholds τ 1 , . . . , τ K , a management strategy can be assessed by its 11 For given sustainability thresholds, there are no trade-offs, either among sustainability issues or among time periods. All trade-offs occur when the thresholds are defined [START_REF] Martinet | A characterization of sustainability with indicators[END_REF][START_REF] Martinet | Economic Theory and Sustainable Development: What can we preserve for future generations?[END_REF]. We emphasize how our framework can be used to support the definition of the thresholds.

viability probability. To stress the dependency on thresholds, we introduce the notation

Π( c, τ 1 , . . . , τ K ) = P                  ω(•) ∈ Ω x(t 0 ) = x 0 x(t + 1) = G t, x(t), c(t), ω(t) c(t) = c t, x(t) I k t, x(t), c(t) ≥ τ k , k = 1, . . . , K t = t 0 , . . . , T                  . ( 5
)
This viability probability is a common metrics to evaluate the consistency of a given strategy and sustainability objectives. The higher this probability, the lower the risk of violating the sustainability constraints.

Note that, as in the case of expected discounted utility, stochastic viability analysis depends on the probability distribution P. In particular, since we are dealing with intertemporal issues, we need to be cautious about how P captures temporal dependencies among uncertainties (e.g., independent random variables, Markov chains, or time series). Investigating the sensitivity of the results to the probability distribution is beyond the scope of this paper.

Ranking of management strategies

The stochastic viability approach ranks strategies according to their viability probability. A management strategy c is "more viable" than another strategy if the corresponding set of viable scenarios has a higher probability. A most viable strategy c (τ 1 , . . . , τ K ) is one that maximizes the viability probability Π( c, τ 1 , . . . , τ K ) for a given set of sustainability thresholds τ 1 , . . . , τ K over all possible strategies c ∈ C.

Theoretical extension to the stochastic viability framework

This paper is original in treating the viability thresholds as arguments of the viability probability. This defines a value function for our sustainability problem.

A "value function" for sustained outcomes The maximal viability probability

Π (τ 1 , . . . , τ K ) = max c∈C Π( c, τ 1 , . . . , τ K ) (6)
is the highest probability that objectives (τ 1 , . . . , τ K ) are sustained. It is the value function of the stochastic viability optimization problem. This value function depends on the threshold levels. We use this value function to describe the trade-offs among sustainability objectives.

Stochastic sustainable production possibility frontiers When the maximal viability probability function Π (τ 1 , . . . , τ K ) varies smoothly with respect to the threshold levels (as generally the case when the probability distribution P has a smooth density), the marginal variation of viability probability with respect to the threshold level τ k is

∂ ∂τ k Π (τ 1 , . . . , τ K ).
This represents the marginal cost, in terms of viability probability, of increasing the level of this constraint. It provides information on the difficulty of sustaining the corresponding outcome over time, given other sustainability objectives.

The value function ( 6) can be used to build stochastic sustainable production possibility frontiers exhibiting the trade-offs among sustained levels of outcomes and viability probability. In particular, for any confidence level π ∈ [0, 1], it is possible to define the threshold levels τ 1 , . . . , τ K at which Π (τ 1 , . . . , τ K ) = π. The marginal rate of substitution between thresholds τ i and τ j along the corresponding iso-value viability probability curve is then defined by

∂Π (τ 1 , . . . , τ K )/∂τ i ∂Π (τ 1 , . . . , τ K )/∂τ j = ∂τ j ∂τ i |Π (τ 1 ,...,τ K )=π (7)
This rate measures the necessary trade-offs between the two sustainability objectives, at a given risk level, i.e., how much one objective must be reduced to increase the other without changing the viability probability.

Suboptimal cases Our framework can be used also if it is not possible to identify an optimal strategy (e.g., because it cannot be computed). In a second-best setting, it is possible to consider subsets of strategies C ⊂ C and define the associated (sub-optimal) viability probability:

Π(τ 1 , . . . , τ K ) = max c∈ C Π( c, τ 1 , . . . , τ K ) (8)
While we recognize the pitfalls involved in such comparisons with an ad hoc reduced number of management strategies, this provides an analytical tool for comparing and ranking realistic management strategies according to a well-defined yardstick that is based on the corresponding viability probability. This ranking exercise could be used to inform stakeholders in the discussion of given strategies with management relevance (e.g., effort-based or quota-based strategies). The viability probability of the strategies then provides a metrics for ranking them. In particular, by letting sustainability thresholds vary, it is possible to define within which range of sustainability threshold levels one type of strategy performs better than another.

A case-study: the Chilean jack-mackerel fishery

We model the Chilean jack-mackerel fishery and use it as a case-study to apply the stochastic viability approach, and in particular, the theoretical extensions described in the previous section.

Description of the fishery and management issues

The jack-mackerel fishery has been the largest fishery in Chile for many years, in terms of both annual catch and economic value. 12 Like other small pelagic stocks, jackmackerel stocks are affected by the recurrences of El Niño in uncertain cycles. Since the late 1990s, the fishery has been managed under a yearly-defined TAC and closed entry, taking particular account of the stability of catch levels over time. Additionally, since the mid-2000s, the jack-mackerel fishery has pioneered (in Chile) the inclusion of biology-related risk indicators in its management practices. 13 These indicators provide additional information for the policy decision making process, with the underlying objective of capping biological (collapse) risk; however, they are not applied within a formal framework allowing trade off of this risk against measures of economic return.

Despite its management strategies, the Chilean jack-mackerel fishery is currently in crisis.

Historical data on the jack-mackerel fishery are provided in the Appendix, Table 1.

Year 2002 appears to be a turning point for two reasons: i) biomass levels were half the peak in the late 1980s, and recruitment was half the levels in the previous five years, 14

12 Annual catch peaked at 4.4 million tons in 1995, and value generation was around US$ 400 millions of yearly sales until the 2010s.

13 SUBPESCA, the regulatory body for Chilean fisheries, started assessing the probabilities of reducing the SSB, relative to a historical base level, for various exogenously defined quota levels (see SUBPESCA (2004, p. 26-27) and IFOP (2006, p. 33-39)).

14 This was probably related to lagged effects from the very strong 1997/98 El Niño event (Peña [START_REF] Torres | Fish stock endogeneity in the harvest function: El niño effects on the Chilean Jack-Mackerel fishery[END_REF][START_REF] Torres | El niño and fishing location decisions: The Chilean straddling Jack-Mackerel stock[END_REF].

ii) the spatial distribution of the stock changed (Peña [START_REF] Torres | El niño and fishing location decisions: The Chilean straddling Jack-Mackerel stock[END_REF], moving part of the stock outside Chile's Exclusive Economic Zone (EEZ), which triggered the re-opening of an international-waters jack mackerel fishery (see Table 1 column (2) This modeling exercise has two objectives. First, we assess the sustainability of some management strategies and compare them to the fishery's historical evolution.

Second, we build stochastic sustainable production possibility frontiers for the fishery given the 2002 stock. This allows us to determine the levels of sustainable outcomes,

given the stock at the beginning of the period.

Bioeconomic model 16

Biology: We describe the dynamics of the Chilean jack-mackerel stock using an ageclass model [START_REF] Quinn | Quantitative Fish Dynamics. Biological Resource Management Series[END_REF][START_REF] Tahvonen | Economics of harvesting age-structured fish populations[END_REF] with a Ricker recruitment function. 17 Time is measured in years. The initial year is t 0 = 2002 and the final year is T = 2011. The time index t = t 0 , t 0 + 1, . . . , T represents the beginning of year t. Let A = 12 denote the maximum age group, and a ∈ {1, . . . , A} be an age class index, all expressed in years. The vector N = (N a ) a=1,...,A ∈ R A + is a vector of abundance-at-age:

15 The Chilean fishery research institute (IFOP) estimated this parameter at SSB virg = 14.3 million tons. It uses the maximum recorded SSB for this fishery (in 1988) as a proxy.

16 Data, parameters and computational details are described in the Appendix. 17 The Ricker model is frequently used for species with highly fluctuating recruitment, involving high fecundity as well as high natural mortality rates [START_REF] Begon | Population Ecology; A unified study of animals and plants[END_REF]. These two features characterize small pelagic species such as jack-mackerel.

for a = 1, . . . , A -1, N a (t) is the number of individuals aged between a -1 and a at the beginning of year t; N A (t) is the number of individuals older than A -1.

The dynamics of the form of eq. ( 1) is provided in the Appendix (eqs. 11, 13 and 14). The state vector

(A + 1-dimensional) is x(t) = N 1 (t), . . . , N A (t), SSB N (t -1) ,
where the SSB is defined by eq. ( 13). Fishing activity is represented by a fishing effort multiplier λ(t), assumed to be applied continuously during the period t. The control then is c(t) = λ(t). Total annual catches Y , measured in million tons, are given by the Baranov catch equation (eq. 12).

El Niño cycles model: The El Niño phenomenon is the result of a wide and complex system of climatic fluctuations between the ocean and the atmosphere, whose frequency and intensity are uncertain. We simulate the uncertain El Niño cycles using a model with a periodic part and an error term, to produce a cycle with random shocks.

Details are provided in the Appendix.

Economics: We make the following standard economic assumptions [START_REF] Reed | Optimal escapement levels in stochastic and deterministic harvesting models[END_REF][START_REF] Clark | On uncertainty renewable resource stocks: Optimal harvest policy and the value of stock surveys[END_REF][START_REF] Clark | Mathematical Bioeconomics[END_REF].

(a1) Demand is infinitely elastic. The harvest from this fishery goes mainly to fish meal, a commodity with high demand substitution. Therefore, this fishery is essentially a price-taking industry, and we assume that any unit harvested is sold for a given, exogenous price.

(a2) Per unit harvest costs are not dependent on harvest volume and vary with population abundance. These costs increase as the size of the population decreases.

This is equivalent to assuming that fishing effort has a constant unit cost, and that Catches Per Unit of Effort (CPUE) decrease if the stock decreases.

Under these assumptions, since the CPUE decreases when stock size falls, there is a minimal stock size below which the marginal cost of fishing effort (which is constant) is higher than the marginal revenue from fishing effort. We assume that no extra fishing effort occurs once the marginal profit is nil. This implies that fishing effort has an upper bound.

For fisheries satisfying these assumptions, price and cost levels do not have a qualitative effect on our results. The regulator usually observes prices but fishing costs are private information and depend on factors specific to fishing vessels. Thus, profit functions are difficult to estimate without strong assumptions related to fleet homogeneity. In practice, the most frequent approach is to use catches to proxy for revenue, and fishing effort related variables to proxy for costs. Since in practice quotas are defined in quantity terms, it is reasonable to focus on harvest quantities and fishing effort to proxy for revenue and fishing costs. This assumption is in line with, for example, [START_REF] Reed | Optimal escapement levels in stochastic and deterministic harvesting models[END_REF], [START_REF] Clark | On uncertainty renewable resource stocks: Optimal harvest policy and the value of stock surveys[END_REF] and [START_REF] Sethi | Fishery management under multiple uncertainty[END_REF], where the expected discounted sum of harvest rather than the expected discounted sum of profit is maximized.

Economic and biological sustainability objectives

We consider the ecological objective of sustaining the SSB above some limit defined as a percentage of SSB virg . This objective is formalized by the constraint

SSB N (t) SSB virg ≥ p , ∀t = t 0 , t 0 + 1, ..., T , (9)
where the threshold p denotes the desired minimum percentage of SSB virg to be preserved over time. In our analysis, p ∈ [0.15; 0.25], which means that the constraint on the SSB(N (t)) varies between 15% and 25% of SSB virg . 18 The constraint (9) corresponds to the following indicator and threshold: I 1 (t, x(t), c(t)) = SSB N (t) SSB virg and

τ 1 = p.
We also consider the socio-economic objective of sustaining the annual yield above a level y min :

Y (N (t), λ(t)) ≥ y min , ∀t = t 0 , t 0 + 1, ..., T . (10)
The minimum level of landings to be sustained over time (y min ) can take values from 0 to 2 million tons, corresponding to catch levels observed in this fishery in the first decade of 2000. The constraint (10) corresponds to the following indicator and threshold:

I 2 (t, x, c) = Y N, λ and τ 2 = y min . This constraint presumes that the fishery regulator aims at maintaining a minimum level of fishing activity, due possibly to socioeconomic considerations.

Viability assessment of management strategies

Using the stochastic viability approach, we compare management strategies for the Chilean jack-mackerel fishery.

Although optimization approaches provide a description of "optimal" management strategies, many fisheries are managed using much simpler tools. 19 Constant fishing effort and constant quotas are two basic management strategies. The former approach, known also as fixed fishing mortality, is based on advice from biologists and results in fluctuating harvests as stocks fluctuate. The optimal strategy may be neither of these approaches [START_REF] Hannesson | How to set catch quotas: Constant effort or constant catch[END_REF] but these rules of thumb are still frequently proposed (and indeed used sometimes) as potential management strategies in some fisheries. In the 1980s and 1990s, Chilean fisheries were de facto managed under a constant effort rule (frozen maximum effort). In 2000, a quota system was applied with a posteriori very small changes to TAC levels from year to year. For example, the management strategy applied to the jack-mackerel fishery over the studied period resembles a constant quota-type policy (see Table 1).

We focus on two different types of strategies: constant fishing effort and constant quota, both stationary over a fixed period of 10 years.

A constant effort strategy (CES) is a strategy defined by a constant effort 20 λ(t, N ) = λ. The set of all possible CES is denoted by CE ⊂ C.

A constant quota strategy (CQS) is a strategy implicitly defined by a constant quota Y . The associated fishing effort multiplier λ(t, N ) is such that Y N, λ(t, N ) = Y whenever this is possible, i.e., if the corresponding effort level is below the upper bound for fishing effort. If it is not, the actual catch level may be lower than the quota. The set of all possible CQS is denoted by CQ ⊂ C.

For each subset of strategies CE and CQ , we compute the associated maximal viability probability as a function of the two sustainability thresholds: For each pair (p, y min ) ∈ [0; 2] × [0.15; 0.25] of economic and ecological thresholds, 21 we define, within each subset of management strategies, the level of the policy instrument which re-19 E.g., [START_REF] Singh | Fisheries management with stock uncertainty and costly capital adjustment[END_REF] describe the Alaskan Pacific halibut stock as being managed by setting the yearly harvest as a fixed fraction of the exploitation biomass; this constant harvest rate rule is shown to smooth catches over time more than the optimal policy. 20 In our model, fishing mortality is proportional to fishing effort if the fishing technology is constant.

Thus, a CES is identical to the constant fishing mortality strategy depicted here.

21 Technically, we discretize the intervals.

sults in the highest viability probability (best constant quota, or best constant effort, to sustain the given objectives). The viability probability is approximated by a frequency given by Monte Carlo simulations (over 1, 000 simulations). We compute a 95% confidence interval for its value. These viability probabilities are displayed in Fig. 2.

For each strategy (left-hand panel for CES and right hand-side panel for CQS), we draw iso-probability curves over the two thresholds, for the levels of maximal viability probability {0; 0.1; 0.5; 0.9; 0.99; 1}. Both graphics in Fig. 2 (Green circle at (20,0.8) corresponding to the sustainability thresholds used for the simulations of Figure 5) viability value" of each type of strategy as a function of the sustainability thresholds (see eq. 5).

Ranking management strategies For any given pair of sustainability thresholds, we can rank the alternative management strategies using their viability probability.

This allows us to identify the levels of sustainability objectives for which a strategy is likely to perform better than the other from a viability point of view. We determine whether the confidence interval for the viability probability of one type of strategy lies strictly above the confidence interval for the other strategy. Fig. 3 depicts the strategy type with the highest viability probability for each pair (p, y min ) of biological and economic thresholds. The domain, in terms of sustainability thresholds, where CQS performs strictly better than CES is shaded black. The gray area corresponds to the threshold levels at which the performance of both policy types cannot be statistically distinguished (i.e., confidence intervals intersect). This happens only for viability probabilities close to 1, i.e., for objectives which are easily sustained. The white area corresponds to unsustainable objectives, i.e., thresholds with a viability probability close to zero.

We conclude from this analysis that, for any sustainability objective in the studied range, CQS perform better than CES to sustain catches and biomass levels. 22This dominance of quota-based strategies over effort-based strategies is not surprising given the nature of the sustainability constraints considered. To explain this, let us refer to the theoretical result in De [START_REF] Lara | Multi-criteria dynamic decision under uncertainty: A stochastic viability analysis and an application to sustainable fishery management[END_REF]. In a general framework with an application to fishery, they show that if the dynamics and viability constraints satisfy some monotonicity properties, the maximal viability probability is achieved with the feedback rule which maximizes the escapement level given that the viability constraints are satisfied at the current time. This management strategy can be interpreted as a "precautionary rule." It ensures the achievement of economic objective at the present time while maximizing the probability of economic and ecological objectives being achieved in the future. 23 When the economic constraint is a minimal (Green circle at (20,0.8) corresponding to the sustainability thresholds used for the simulations of Figure 5) catch level, the rule corresponds to a constant quota at the level of the constraint.

Since the Ricker recruitment function is non-monotonic, with a declining part for large stocks, the model studied here is not monotonic in the sense of De [START_REF] Lara | Multi-criteria dynamic decision under uncertainty: A stochastic viability analysis and an application to sustainable fishery management[END_REF]. However, the range of SSB modeled belongs to the monotonic part of the Ricker function, which means that the model behaves as if it were monotonic. As one of the viability constraint is a minimal catch level, a constant quota at this level results in the highest viability probability.

The problem of determining which of the effort-based and quota-based strategies dominates in fishery economics is a particular case of the "prices versus quantities" the viability probability is conservative, and results in the resource stock kept as "far" as possible from the biological threshold, given the economic objective. 20 debate. A management strategy based on direct control of fishing effort has similar features to tax based management [START_REF] Danielsson | Efficiency of catch and effort quotas in the presence of risk[END_REF][START_REF] Weitzman | Landing fees vs harvest quotas with uncertain fish stocks[END_REF]. By imposing a maximal fishing effort, one imposes a maximal marginal cost, which interrupts the fishing period before the open access equilibrium. Controlling the effort is similar to imposing a particular landing fee (such as a very high fee starting at some point).

Landing fees are a (relatively) better solution to control the (marginal) fishing effort (or cost) but suffer from the drawback of inability to control catch levels. Harvest quotas, on the other hand, have the advantage that they fix the total quantity of fish caught but suffer from the drawback of inability to control the possible excess effort exerted to fish down a stock that is experiencing low recruitment in the fishing period.

The related literature shows that, depending on the characteristics of the fishery (i.e., its biological dynamics and economic structure) and the type of uncertainty affecting the model (i.e., whether fish stock and/or economic returns are uncertain), either quota or effort tools may perform better in terms of discounted payoffs [START_REF] Hannesson | How to set catch quotas: Constant effort or constant catch[END_REF][START_REF] Quiggin | How to set catch quotas: A note on the superiority of constant effort rules[END_REF][START_REF] Danielsson | Efficiency of catch and effort quotas in the presence of risk[END_REF][START_REF] Jensen | Prices versus Quantities in Fisheries Models[END_REF][START_REF] Hannesson | Landing fees versus fish quotas[END_REF][START_REF] Hansen | Prices versus Quantities in fisheries models: Comment[END_REF]. In the stochastic viability framework, the result depends not only on the characteristics of the fishery under study but also on the nature of the sustainability objectives.

Stochastic sustainable production possibility frontiers Fig. 2 presents what was defined in the theoretical analysis of section 3.3 as stochastic sustainable production possibility frontiers. The lines denoting the iso-probabilities represent the tradeoffs between sustainability thresholds (p, y min ) at various viability probability levels, as characterized by eq. ( 7). For any given viability probability level, it is necessary to reduce one sustainability threshold to increase another. There is also a trade-off between the sustainability thresholds and confidence in achieving sustainability. Increasing the thresholds results in a decreased viability probability.24

These graphical representations are useful to support the social choice of sustainability objectives. They depict the trade-offs between the policy objectives represented by the sustainability thresholds, and the risk of failing to (simultaneously) achieve them.25

When no SWF can be determined prior to the evaluation of management strategies, and the interest is in sustaining ecological and economic outcomes over time, presenting the trade-offs over all possible sustainability objectives to stakeholders may help to reveal their preferences.

Discussion

We can draw some policy-oriented conclusions from the results of our analysis. The important contribution is not the finding of dominance of quota over effort strategies but the representation of the trade-offs between sustainability issues by means of stochastic sustainable production possibility frontiers.

In the early 2000s, biomass levels had been experiencing (for almost a decade) worsening status. As a consequence, our simulation results report non-viable solutions for any threshold pair with p ≥ 25%, either under CQS or CES, whatever the minimum catch threshold.

Over the period analyzed, the TAC was maintained at above 1.3 million tons; however, actual catches did not match this level. Notwithstanding the ecological constraint, Fig. 2 shows that the probability of sustaining the TAC level was not high. Even the best policy among those studied has a low viability probability (around 50%). This is illustrated in Fig. 4, which compares simulated trajectories for the best CQS and CES for sustainability thresholds (p, y min ) = (0, 1.3), to the historical data (dashed line).

The catch level of 1.3 million tons is sustained only in few scenarios (1 for CES, and 3 for CQS).

The main message to the Chilean regulator is that, notwithstanding the choice of instrument, historical quota targets were not sustainable. The information provided by our stochastic sustainable production possibility frontiers could have helped to set lower sustainability targets. For example, Fig. 5 represents simulated trajectories for the best CQS and CES for sustainability thresholds (p, y min ) = (0.2, 0.8), which are achievable with a higher probability than historical levels of quotas (see the green circle at these threshold levels on Fig. 2 and3). The viability probability for CES is quite low, close to 10%. None of the depicted trajectories are viable. The viability probability for Constant Quota Strategies is very close to one. All the depicted trajectories are viable.

However, these results should be interpreted with caution and political economy considerations should not be underestimated. One of the basic reasons for pursuing the high quota management strategy despite worsening biomass numbers, was that the

Conclusions

Many problems related to the management of natural resources, such as fisheries, are marked by dynamics and uncertainty. When there are conflicting economic, ecological and social objectives at stake, multicriteria evaluation methods that take account of uncertainty are required to rank potential management strategies. One such method is the Management Strategy Evaluation approach, which characterizes potential management strategies using a set of performance statistics. However, due to the absence of a common metrics for comparing and trading-off conflicting issues, decision-makers are devoid of tools to rank the various management strategies.

To contribute to policy-oriented decision making related to natural resources management problems, we have developed a framework based on stochastic viability. A set of constraints is used to represent the various sustainability objectives of the dynamic ecological economic system. In this framework, management strategies are ranked according to the probability that the resulting intertemporal trajectory satisfies all the objectives over the planning horizon. The viability probability ranks the various management options, defining the strategy that results in the highest viability probability.

This approach acts to complement the traditional economic approach when it is not possible to define a multi-attribute social welfare function. The objective is to maximize the probability of achieving the sustainability constraints. Stochastic viability provides a good way to model decision problems involving several stakeholders interested in sustaining the levels of various indicators. All sustainability dimensions are treated in the same way as constraints representing the minimal rights to be guaranteed to all generations. The decision-maker's preferences are expressed when sustainability thresholds are defined.

The theoretical extension to stochastic viability presented in this paper should help stakeholders to define what should be sustained. Our stochastic viability value function exhibits trade-offs between sustainability objectives (thresholds) and viability probability. Building stochastic sustainable production possibility frontiers allows the set of objectives that can be sustained with some probability to be described.

The proposed stochastic viability methodology is general, and can be applied to a wide range of problems. For example, in this paper we examined the management of a real fishery, using estimated parameters. We applied numerical techniques to examine the efficiency of effort-and quota-based management strategies for achieving sustainability objectives, defined as constraints on biological and economic indicators.

Monte Carlo simulations were run to estimate the viability probability of each policy with respect to these objectives.

The main contribution of the paper is the development of a framework which provides a common metrics to compare management strategies and to describe the tradeoffs among sustainability objectives, in a way that complements the MSE approach.

We suggest that the proposed approach fills the gap between the theoretical economics literature on optimality, and practical decision-making.

Historical data for the Chilean Jack-Mackerel fishery where M a is the natural mortality rate of individuals of age a, F a is the mortality rate of individuals of age a due to harvesting between t and t + 1, supposed to remain constant during year t (the vector (F a ) a=1,...,A is termed the exploitation pattern).

Total annual catches Y , measured in million tons, are given by the Baranov catch equation (Quinn and Deriso, 1999, p. 255-256):

Y N, λ = A a=1 a λF a λF a + M a (1 -exp (-(M a + λF a ))) N a , (12)
where ( a ) a=1,...,A are the weights at age.

The spawning stock biomass (SSB) is given by the expression

SSB(N ) := A a=1 γ a a N a , (13)
where (γ a ) a=1,...,A are the proportions of mature individuals at age a (some may be zero).

Annual recruitment is a function of the SSB with a two-year delay, i.e., depending on the spawning stock biomass of two periods earlier:28 N 1 (t + 1) = αSSB N (t -1) exp βSSB N (t -1) + w(t) ,

where {w(t)} is a random process reflecting the impact of climatic factors on the stock recruitment relationship (see below).

We use the parameter estimation proposed in [START_REF] Yepes | Dinámica poblacional del jurel: Reclutamiento asociado a factores ambientales y sus efectos sobre la captura[END_REF], which relies on official data from the Instituto de Fomento Pesquero (IFOP).29 Parameters of the Ricker recruitment function at expression (14) were estimated using linear time-series analysis.

The estimated parameters are α = e 2.39 and β = -2.2 • 10 -7 (see Yepes, 2004, p. 56).

The values for parameters M a and F a are taken from IFOP's official model for this fishery, so that M a is equal to 0.23 for all a and F a is equal to the vector of averages values of F a during 2001-2002. 30 Stochastic model Following the statistical analysis in [START_REF] Yepes | Dinámica poblacional del jurel: Reclutamiento asociado a factores ambientales y sus efectos sobre la captura[END_REF], we simulate El Niño uncertain cycles using a sinusoidal function with random shocks. 31 The random process w(t) supposed to capture the effects of the El Niño phenomenon has a periodic part and an error term, w(t) = -0.12 × niño(t) + (t), where

• the estimated error terms { (t)} correspond to (t) = 0.71 (t -1) -0.65 (t -

2) + µ(t), where {µ(t)} is a sequence of i.i.d. random variables with Normal distribution N (0; 0.18),

• niño(t) = 1 {-1.2 sin (18.19+2π(t-1951)/3.17)>0.5} is a dummy (0 or 1) variable reflecting the presence of El Niño phenomenon.

Simulation process From a theoretical point of view, it is possible to determine the strategy that maximizes the viability probability by solving the dynamic programming equation characterizing the viability problem [START_REF] Lara | Monotonicity properties for the viable control of discrete time systems[END_REF]. It is possible to obtain a closed-form solution for some problems [START_REF] Lara | Multi-criteria dynamic decision under uncertainty: A stochastic viability analysis and an application to sustainable fishery management[END_REF].

Determining optimal strategies in dynamic optimization problems under uncertainty is not easy. Optimization in the stochastic viability framework is not exceptional. In particular, the curse of dimensionality can be a serious obstacle to the computation of optimal viability strategies. Oceanic El Niño Index (ONI) as the difference in current sea surface temperature (SST) with respect to the historical average SST for the period 1971-2000. We then computed a three-month moving average series, on the basis that El Niño occurs if this average is greater than 0.5 • C for five consecutive months (see the expression of niño(t)). The ONI is modeled via a sinusoidal function whose parameters are estimated using a non-linear iterative algorithm (Yepes, 2004, p. 64), to represent the different cycles of El Niño.

produce scenarios following the distribution P. For each scenario, a given management strategy is applied. If, for the corresponding trajectory, all the viability constraints in (4) are respected in each time period over the whole planning horizon, the scenario is viable for the applied management strategy. When the number of scenarios tested is large, the frequency of viable scenarios can be used as an approximation of the viability probability.

Figure 1 :

 1 Figure 1: MSE for the Chilean Jack Mackerel fishery: performance of various management strategies

  ).Despite the changes in the biology of the stock and its exploitation pattern after 2002, the Chilean fisheries regulator decided to keep TAC levels almost constant for the Chilean fleet targeting jack mackerel within and beyond the Chilean EEZ over the period 2000 -2010 (see Table1 column (3)). Biomass levels began a monotonic decline, from 48% of virgin SSB (SSB virg ) 15 in 2002, down to 16% in 2012. The management strategy changed only in 2011, when the TAC fell by 76% between 2010 and 2011, from 1,300 to 315 k-tons; in 2013 it was around 250 k-tons. Thus, the period 2002-2011 is of particular interest for this fishery. It covers 10 years of management, which is the management horizon used by IFOP. It starts with a change in the biology of the stock, and ends with a collapse of the fishery and a change in management strategy. We model this period over a 10 year horizon, taking 2002 as the initial year of our simulation.

Figure 2 :

 2 Figure2: Maximal viability probability of effort and quota strategies (1,000 Monte-Carlo simulations). Isoprobability curves are drawn for values {0; 0.1; 0.5; 0.9; 0.99; 1}.

Figure 3 :

 3 Figure 3: Comparison of CES and CQS policy types (1,000 Monte Carlo's simulations).

Figure 4 :Figure 5 :

 45 Figure 4: Examples of trajectories under CQS and CES (5 simulations corresponding to 5 different uncertainty scenarios) for sustainability thresholds (p, y min ) = (0, 1.3), compared to historical data (dashed line). The yield threshold is represented by a horizontal (green) line. Catch levels equal the threshold level if constant quota trajectories are feasible. Viable trajectories are in blue. Non-viable trajectories are in red.

  (b) The Chilean fleet's TAC in column (3) is binding for catches within and beyond the Chilean EEZ. The first year to which TAC was applied in this fishery was 1999; the policy was resumed in 2001 (for more details see Gomez-Lobo et al. (2011)). (c) To deduce the Chilean fleet's (implicit) fishing effort multiplier (λ) in column (4), we replaced the annual catch Y (N, λ) by its real historical values (column 1) in the Baranov equation (12) and simulated the stock dynamics: starting from the initial vector of abundances at age (for year 2002); we then applied the stock dynamics (equation 11) while considering the deterministic version of the Ricker recruitment function (equation 14), including the deterministic effect of El Niño events (in those years when it occurred, based on the definition in footnote 31). Sources: (1)-(2), (5)-(7): IFOP (2013); (3): Subsecretaría de Pesca (Chilean Fisheries Regulator); (4): authors' own calculations Biological model We provide details of the model in §4.2. The model is age-structured, with a Ricker stock-recruitment function. Abundance dynamics are given by N a+1 (t + 1) = exp (-(M a + λ(t)F a ))N a (t) , a = 1, . . . , A -2 N A (t + 1) = exp (-(M A-1 + λ(t)F A-1 ))N A-1 (t) + exp (-(M A + λ(t)F A ))N A (t) (11)

From a practical

  point of view, it is possible to estimate the viability probability of any given strategy by means of Monte Carlo simulations. A random generator is used to 30 See Subsecretaria de Pesca, Valparaíso -Chile, SUBPESCA (2006) Pre Informe Final. Investigación Evaluation y CTP Jurel 2006. 31 Based on Chilean marine biologists advice, Yepes (2004) calculates the occurrence of the El Niño phenomenon based on National Oceanic and Atmospheric Administration (NOAA) data on sea surface temperatures measured at the region known as Niño 3.4 (120W-170W, 5N-5S). NOAA computes the

Risk to the resource Mean catch level ('000 tones) Management Strategies Evaluation

	High risk to the resource	90% 100%	-						Dominated strategies
		80%
		70%
		60%			Preferences
		50%							Non-dominated
									strategies
		40%
		30%
		20%
	Low risk to the resource	0% 10% 1 100	1 200	1 300	1 400	1 500	+	1 600	Management strategies
	Low catch level						High catch level

  Table 1 details the historical values of interest for the fishery.

		(1)	(2)	(3)	(4)	(5)	(6)	(7)
		Total Catch	Total Catch DWFNs	TAC	F. Effort Multiplier	Recruits	SSB	Total
		Chilean Fleet (beyond Chilean EEZ) Chilean Fleet	Chilean Fleet			Biomass
		(10 3 tons)	(10 3 tons)	(10 3 tons)	(implicit l value) (10 3 individuals) (10 6 individuals)	(10 3 tons)	(10 3 tons)
	1980	562	340	-		21738	10564	15973
	1981	1061	438	-		27215	10825	17114
	1982	1495	733	-		27652	10335	17861
	1983	865	849	-		25645	10432	17471
	1984	1426	1060	-		47886	10265	19017
	1985	1457	799	-		60875	10653	20827
	1986	1184	838	-		28735	12190	21942
	1987	1770	863	-		15962	13822	22698
	1988	2138	863	-		17644	14304	22534
	1999	2391	876	-		23051	13652	21673
	1990	2472	872	-		26461	12616	20751
	1991	3020	544	-		20834	11428	19708
	1992	3212	38	-		16344	10377	18002
	1993	3236	0	-		14933	9392	16140
	1994	4041	0	-		16942	7824	14545
	1995	4404	0	-		18434	5775	12596
	1996	3883	0	-		21071	4557	10378
	1997	2917	0	-		24326	3844	9345
	1998	1613	0	-		21460	4070	8862
	1999	1220	0	1902		24704	4815	9622
	2000	1235	2	-		24298	5643	10771
	2001	1650	20	1425		20597	6312	11720
	2002	1519	76	1625	0,32	12873	6848	11852
	2003	1421	158	1350	0,46	8365	7073	11559
	2004	1452	295	1475	0,45	6339	6722	10793
	2005	1431	244	1484	0,42	3112	5988	9482
	2006	1380	363	1400	0,39	5725	4934	8167
	2007	1303	439	1600	0,36	7040	3685	6812
	2008	896	405	1600	0,22	5808	2740	5348
	2009	835	372	1400	0,17	7011	1967	4364
	2010	465	240	1300	0,08	7826	1706	3586
	2011	247	61	315	0,03	7158	1910	3418
	2012	227	40	252	0,02	10892	2286	4034
	2013	242	47	250

Table 1 :

 1 (a) DWFNs: Total annual catch of Distant Water Fishing Nations' Fleets (fishing jack mackerel outside the Chilean EEZ).

In some extreme cases, recruitment uncertainties and management decisions have led to the collapse of important small pelagic stocks, such as the Peruvian anchovy in1972-1973.

As discussed in the Stern review for climatic change[START_REF] Stern | The Economics of Climate Change[END_REF].

For some types of utility functions, e.g., Constant Absolute Risk Aversion functions, preferences under risk may be represented by means of a linear function of expected (mean) profits and a simple proxy for risk such as variance of profits.

In the case of South African small pelagic fisheries (sardines and anchovies) in the late

1980s and early 1990s, the fishery regulator considered p = 0.2 when applying such biological criteria[START_REF] Butterworth | The development of a management procedure for the South African anchovy resource[END_REF].

This result is robust to the initial state of the fishery. We performed a sensitivity analysis for different initial stocks defined as multiples of the 2002 stock (from 60% to 150%).

Note that, for many fisheries, the International Council for the Exploration of the Sea (ICES) management strategy is based on a rather different strategy: the catch level is set at the highest level compatible with the biological conservation target in the following year, given a confidence interval (precautionary fishing mortality value) (De[START_REF] Lara | Is a management framework based on spawning-stock biomass indicators sustainable? A viability approach[END_REF][START_REF] Kell | An evaluation of the implicit management procedure used for some ICES roundfish stocks[END_REF]. By construction, this strategy leads the stock close to the ecological constraint, with the risk of fishery closure in the shortmedium term if the stock falls below the biological conservation threshold. The strategy maximizing

The figure could be made 3-dimensional, with the viability probability as a function of the thresholds, to emphasize these two different trade-offs.

Note that these trade-offs are between sustainability objectives, not different management strategies (as was the case for the MSE in Fig.1).

This 2-year delayed effect is due to the biological growth dynamics of the species.

Subsecretaria de Pesca, Valparaíso -Chile: Cuota Global de Captura para la Pesquería del Recurso Jurel, Año 2001 (SUBPESCA, 2000); and Instituto de Fomento Pesquero, Valparaíso -Chile: Informe Complementario Investigación CTP Jurel, 2003: Indicadores de Reclutamiento (IFOP, 2003).

Acknowledgments. We acknowledge financial support from the STIC-AmSud program (CNRS, Conicyt-Chile, INRIA and the French Ministry of Foreign Affairs) for the international research framework MIFIMA (Mathematics, Informatics and Fisheries Management). Héctor Ramírez was supported by Conicyt-Chile, under ACT project 10336, FONDECYT 1110888 and BASAL Project (Centro de Modelamiento Matemático, Universidad de Chile), and by project BIONATURE of CIRIC, INRIA-Chile. We thank Claire Nicolas (ENSTA-ParisTech student), Pauline Dochez (Polytechnique-ParisTech student) and Pedro Gajardo (Universidad Federico Santa Maria, Valparaiso, Chile) for related works, as well as Pablo Koch (Centro de Modelamiento Matemático, Universidad de Chile and INRIA Chile). We are grateful to the participants in various seminars (Rencontres de l'Environnement 2009; CIREQ 2010; UCSB Bren School 2011) and conferences (Diversitas 2009; SURED 2010; WCERE 2010; IIFET 2010), and also Florian Diekert for comments. We also thank the Editor and two anonymous referees.


A L Haj Omar

Peres Véronique

Serris Eric

Grosjean François

Kittel Jean

Ropital François

Cournil Michel

Thermogravimetric experiments coupled with acoustic emission analysis dedicated to high-temperature corrosion studies on metallic alloys

Keywords: Thermogravimetric analysis, Acoustic emission, High temperature corrosion, Waveguide, Elastic waves

High temperature corrosion of metallic alloys in industrial equipments, such as refinery and petrochemical equipments concerns several phenomena: oxidation, carburization… These phenomena can create stresses in the materials, the relaxation of which mostly produces transient elastic waves.

Several methods enable the recording and analyzing of these transient elastic waves.

Piezoelectric sensors fixed directly on the sample can record elastic waves with a low decrease in energy and frequency at ambient temperature. In case of high temperature environments, a waveguide can also be used to transmit waves from sample to sensors. For this purpose, alumina or platinum are mainly used as waveguide materials.

The goal of this study is to assign the acoustic signals to the corrosion phenomena and establish an acoustic signals database. This database will then be useful for the monitoring of industrial equipment using acoustic emission methods. For this purpose, thermogravimetric analysis (TGA) has been coupled with acoustic emission (AE) devices. Simultaneous measurements of the mass variation and of the acoustic signals emitted during the oxidation of samples of the zirconium based alloy, Zircaloy-4, at high temperature 900 °C can provide complementary information to increase the level of understanding of high temperature corrosion mechanisms. Our work focuses on a specific waveguide (WG) conception and on the transmission of elastic waves (acoustic signals) through the waveguide at high temperature. Results on experiments concerning the corrosion of zirconium alloy plates under oxidizing atmosphere are presented.

I. Introduction

Iron, nickel, cobalt alloys are broadly used as reactor or container materials in several industrial domains (refinery, petrochemical …). To be able to control the level of damage of these equipments in service represents a major challenge [START_REF] Ono | Acoustic emission, Encyclopedia of acoustics[END_REF]. Corrosion of iron based alloys becomes significant at high temperatures (400 °C -900 °C). In order to quantify the level of damage of these alloys, acoustic emission seems to be an interesting method owing to its sensitivity and its non-destructive aspect. AE does neither require ultrasonic signal exciters, nor a heavy infrastructure. The implementation cost of AE systems remains moderate.

Acoustic emission measurements can be applicable to equipment in service under various environmental conditions. This technique has been used to investigate the behavior of different materials at high temperature. Ferrer and al. [2] studied the effect of dimethyl disulfide (DMDS) on the inhibition of the attacks by metal dusting on Incolloy 800 tubes by means of acoustic emission. Acoustic emission experiments were also performed by Schulte and al. [START_REF] Schulte | [END_REF] to evaluate the sulfidation of ferritic and austenitic steels. Bennett and al. [4] mounted an acoustic emission device into a thermobalance. Works were also carried out by Schmutzler and al. [5] to study the oxidation mechanisms of Fe-Cr-Al by coupling TGA and AE. More recently, Tran and al. [6] also studied the oxidation of titanium, zirconium, nickel and chromium pure metals in a tubular furnace instrumented with an acoustic emission device.

High temperature corrosive phenomena create stresses in the materials; the relaxation of these stresses produces transient elastic waves which can be recorded and analyzed using the AE system. The study of these transient waves enables the anticipation of the degree of damage of these alloys [START_REF] Ono | Acoustic emission, Encyclopedia of acoustics[END_REF]. Several methods are used to record and analyze these transient elastic waves. At ambient temperature, piezoelectric sensors in contact with the sample can record an important number of elastic waves with a low attenuation of the signal energy [START_REF] Shuthe | The sulfidation behavior of several commercial Ferritic and Austenitic steels[END_REF][START_REF] Zhou | Thermal fatigue failure induced by delamination in thermal barrier coating[END_REF].

At high temperature, waveguides are used to transmit the waves from the sample to the sensors; the waveguide design depends on several factors, like wave propagation velocity, wave damping, waveform conservation, experimental atmosphere and temperature. Alumina and platinum are mainly used as waveguide materials [START_REF] Renusch | Measuring and modeling the TBC damage kinetics by using acoustic emission analysis[END_REF]; these materials reduce the amplitude and the energy of the signals but they conserve the waveforms [START_REF] Sikorska | The effects of waveguide material and shape on AE transmission characteristics part 1: Traditional features[END_REF].

The aim of our study is to associate the elastic waves with the corrosion phenomena in order to be able to monitor the industrial equipment using acoustic emission method. For this purpose thermogravimetric analysis has been coupled with acoustic emission devices to measure acoustic signals emitted during the corrosion of samples at high temperature [START_REF] Grosjean | Coupling of thermogravimetric and acoustic emission technics for high temperature corrosion studies[END_REF].

Simultaneous measurements of the mass variation and of the acoustic signals can give additional information to increase the level of understanding of high temperature corrosion mechanisms. This work focuses on the transmission of the acoustic signals via the waveguide at ambient temperature and at high temperature. The selection of the waveguide was applied according to several criteria; one of them is the resistance under corrosive atmosphere. For this purpose, a long waveguide was manufactured in alumina, a material which presents a chemical inertness towards the samples and the different atmospheres. In order to validate the transmission of the waves from the sample to the sensors via the alumina waveguide, two tests were performed. The first test, based on the normalized Hsu-Nielsen method, was carried out at ambient temperature. The second test was executed at high temperature using the oxidation of zirconium alloy plates under oxidizing atmosphere at 900 °C to create cracks and thus elastic waves.

II. Experimental methods

II.1 Thermogravimetric analyzer

Thermogravimetric analysis coupled with the acoustic emission principle was developed in 1977 by R.F. Hochman [START_REF] Hochman | Proc. of the symp. On Properties of High Temperatures Alloys with Emphasis on Environnemental Effects[END_REF], and then used by H.J. Grabke [START_REF] Grabke | [END_REF] and more recently by F.

Ropital [START_REF] Ropital | Proust Coupling Thermogravimetric and Acoustic Emission measurements : its application to study the inhibition of catalytic coke deposition[END_REF][START_REF]Corrosion et dégradation des matériaux métalliques compréhension des phénomènes et application dans l'industrie pétrolière et des procédés[END_REF]. The experiment consists in positioning a sample on a waveguide inside the TGA furnace and record simultaneously the mass variation and the AE signals using piezoelectric sensors placed in the cold part of the TGA.

In our study, experiments were carried out on a symmetric thermobalance (SETARAM TGA 24) at 900 °C at atmospheric pressure (1 atm) with a gas flow rate of 3 L/h. The TGA equipment is constituted of two symmetric parts; one is used for the sample, whereas the other one is used as a reference. Both parts include a furnace consisting of a graphite resistor which enables to be heated up to 1600 °C. The resistor is protected from oxidation by an internal dense alumina tube and by a steady flow of argon. Thermocouples (PtRh 6 % / PtRh 30 %) record the temperature in each furnace. Gases can be analyzed at the TGA outlet using a mass spectrometry analyzer, a humidity sensor or an oxygen sensor.

Piezoelectric sensors should be placed as close as possible to the source of the acoustic emission but in the cold part of the TGA on the top of the waveguide. The received signal is pre-amplified and subsequently processed using the acquisition system of the AE (Fig. 1).

II.2 Acoustic emission

When materials undergo local plastic deformation, cracking or volume transformation, abrupt variations of the field stress take place. This variations lead to elastic energy relaxation. The abrupt release of energy stored in a delimited source region generates high frequency elastic waves (10 -1000 kHz) called acoustic emission (AE) [START_REF] Golan | [END_REF][START_REF] Lockner | The role of acoustic emission in the study of rock fracture[END_REF]. Crack formation, grain rearrangement, friction between solid surfaces and other grain-scale motions are typical processes which can produce AE. High temperature corrosive phenomena as the growth of oxide layers or the chemical reduction of protective oxides (Al 2 O 3 , Cr 2 O 3 ) can also produce elastic waves which propagate through the bulk of the material as well as on the surface of the material. Piezoelectric sensors are used to convert these mechanical waves into electric signals.

Discontinuous acoustic emission control (AE burst) has been used during our experimental

tests. This method is applicable when elastic waves energy are located in the range of 10 -11 J to 10 -4 J. The waveform of the acoustic signals is similar to damped waves form. It is necessary to choose adequate AE burst parameters (Table 1) in order to analyze the acoustic emission signals. An AE burst starts when the signal amplitude exceeds the prescribed threshold (this threshold may be fixed or self-adjusting to the noise level). Further analysis can give more information about the recorded AE signals.

The parameters monitored (counts, amplitude, absolute energy, average frequency…) depend on the objective of each study. For example there may be a close relation between the burst amplitude and the level of damage of the material [18,19].

The main parameters used in this study are:

• Burst amplitude: the maximum amplitude reach during an AE burst (dB AE );

• Burst duration: the time difference between the first and the last threshold crossing (µs);

• Counts: the number of threshold crossing during an AE burst;

• Rise time: the time difference from the first threshold crossing to the burst amplitude (µs);

• Absolute energy: integration of the square of the signal deviation from its average:

Eabs(T) = | ²|
Absolute energy unit is attojoule (1 aJ = 10 -18 J);

• Average frequency: a calculated feature obtained from (Count) divided by (Duration), which determines an average frequency over one AE burst (kHz).

In order to record the acoustic signals emitted during the TGA experiments, a specific acoustic device has been developed (transducer, waveguide, acquisition chain) (fig. 1). AE sensors were placed inside the cold part of the balance where the temperature does not exceed 150 °C. The sensors are linked to an acquisition chain controlled by the AEwin TM software provided by the Physical Acoustics Corporation Company. Data were analyzed using Noesis TM software provided by the same company. The characteristics of the acquisition chain are given in Table 1.

II.3 Waveguide tests

Our work focuses on a specific wave guide (WG) conception and on its implementation in a TGA device. The waveguide was chosen according to the following criteria: i) chemical resistance against corrosive environments (oxygen or carbon-reductive atmospheres); ii) chemical inertness with regard to the alloy samples; iii) good transmission of the acoustic signals via the waveguide (low attenuation of the signal energy). Following these requirements, a long waveguide was designed in alumina. Its conception was limited by the internal diameter of the alumina tube (2 cm) and by the maximal weight that can be supported by the balance (10 g). Samples were placed on a specific support to optimize the contact between the sample and the surface of the waveguide.

At ambient temperature, the normalized Hsu-Nielsen test (AFNOR NF EN 1330-9) was carried out to verify the AE system. This test simulates an acoustic emission event by breaking a 0.5 mm pencil lead tip against the sample surface. This generates an intense acoustic signal, quite similar to a natural AE source that the sensors detect as a strong burst.

Generally, the lead breaks should generate amplitudes of at least 80 dB AE for a reference voltage of 1 mV and a total system gain of 80 dB AE .

The signals amplitude resulting from our test amounts to 90 dB AE . These results confirm the good transmission of the acoustic signals via the waveguide at ambient temperature.

In order to verify the transmission of elastic waves (acoustic signals) through the waveguide at high temperature, we developed a reference test using the corrosion of zirconium alloy plates (Zircaloy-4) under oxidizing atmosphere (75 % He + 20 % O 2 + 5 % Air) at high temperature (900 °C). Oxidation mechanism of this alloy in such conditions is well described [6,20,21]. The oxidation of the Zircaloy-4 is characterized by an inward oxide layer (ZrO 2 ) growth on the sample surface. The growth of ZrO 2 layer is associated with a volume increase. The Pilling-Bedworth ratio (R PB ) makes a comparison between the oxide molar volume (V Zr o ) and the corresponding metal molar volume (V Zr ).

III. Results and discussion

The experiments were performed on rectangular Zircaloy-4 specimens (4.8 mm x 4.6 mm x 0.5 mm). The chemical composition of this material is presented in Table 2.

Thermogravimetric analyses were performed at 900 °C under a mixed gas atmosphere (75 % He + 20 % O 2 + 5 % Air) introduced by two flow meters with a total gas flow rate of 3 L/h at atmospheric pressure (1 atm). The heating-up (15 K/min) to 900 °C was made under helium. Once the isothermal and isobaric conditions have been attained, the gas mixture (75 % He + 20 % O 2 + 5 % Air) was introduced for 3 hours. The cooling (15 K/min) was done under the same process gas mixture. The AE threshold has been fixed to 30 dB AE .

Blank tests without any specimen were carried out in order to check if the waveguide did not significantly react with the gas mixture. During these tests, the stability of the mass signal validates the chemical inertness of the waveguide. The acoustic signals recorded during the blank tests result from the instrumental noise (IN). The characteristics of these AE signals are

given in Table 3.

The mass variation of a Zircaloy-4 sample during the oxidation test, mass normalized with regard to the sample surface, as a function of temperature and time is presented in Figure 2.

There is a small decrease in mass during the heating up which is due to the non-symmetric aspect between the two parts of the thermobalance (fig. 1). A fluctuation of the mass variation can be observed after the introduction of the oxidant gas mixture (3500 s). A significant increase of the mass gain is observed during the high-temperature dwell under oxidizing

atmosphere. An acceleration of the mass gain rate is observed after 5000 s due to the breakaway which is a kinetic transition. The breakaway corresponds to the cracking of the primary dense thin layer of ZrO 2 leading to a free access for oxygen to the metal oxide interface. After the breakaway the oxidation process is enhanced. The mass gain corresponds to the growth of the oxide layer (ZrO 2 ) on the sample surface.

As predicted in paragraph III, this phenomenon is accompanied by acoustic emission signals. The AE activity, six times higher than the blank test activity at the end of the dwell time, increases during the growth of the oxide layer (Fig. 3). According to the graph showing the mass gain rate variation as function of time (Fig. 3), we note that the breakaway occurs The knowledge of Zircaloy-4 oxidation mechanism at 900 °C under oxidizing atmosphere allows us to assign these AE bursts to the physical phenomena which represent the source of emission of these AE bursts. Cracks have appeared during the high-temperature dwell mainly due to the compressive stresses inside the zirconia layer. Furthermore, cracks have appeared during the cooling stage due to the difference in thermal expansion coefficient between the metal and the oxide. SEM pictures of post-mortem oxidized samples (Fig. 6) indicate that cracks are located inside the thick zirconia layer parallel to the metal oxide interface or perpendicular to this interface. The large perpendicular cracks are normally related to the difference in thermal expansion coefficient appearing during the cooling.

In order to attribute the acoustic bursts to the different kinds of cracks (parallel or perpendicular cracks), a detailed AE analysis of the waveform (WF) and of the wavelet transform (WT) was carried out on numerous bursts in the dwell part of the treatment and in the cooling part. Figure 7 presents an example of this treatment. The wavelet transform analysis of AE signals recorded during the high-temperature dwell time (Fig. 7a) possesses an acoustic signature different from the AE signals recorded during the cooling (Fig. 7b). The first wavelet transform is mainly punctual while the second one is more continuous (with some spots).

Combining the wavelet transform analysis as well as the quantitative AE analysis (cumulative count number, amplitude analysis, average frequency analysis) with the SEM observation of post mortem oxidized sample, we can assign the AE bursts to the different cracks. Oxidation cracks are numerous and they are parallel to the metal oxide interface. The AE signals associated with this phenomenon are numerous and their energy is of medium order of magnitude. Cracks due to cooling are less frequent but characterized by highly energetic bursts.

IV. Conclusion

Thermogravimetric experiments coupled with acoustic emission analysis are an interesting way to improve knowledge on the corrosion of metallic materials at high temperature. In such experimental systems, an important aspect is the transmission of acoustic emission signals from the specimen to the sensors in the elevated temperature range. Furthermore, this waveguide shall not disturb the mass gain measurements.

The oxidation of the zirconium alloy induces the growth of a zirconia layer on the material surface. The ZrO 2 layer is subjected to compressive stresses which induce internal cracks. Thermogravimetric experiments coupled with acoustic emission analysis using this alumina waveguide will be used to study the metal dusting corrosion of iron based alloys under carbon rich atmosphere.

Figures:

Fig.

  value higher than unity means that the oxide layer is exposed to compressive stresses. When these stresses exceed the zirconia fracture resistance (σ > σ f ), cracks occur in the oxide layer and emit AE signals. The record of a significant number of AE signals during this control test may then validate the transmission of acoustic signals through our waveguide at high temperature.

  after 5000 s. This phenomenon is accompanied by change in the slope of the curve of the cumulative counts (C.C.) during the oxidation test. The wave transmission from the sample to the sensors via the alumina waveguide at high temperatures has been validated by this test. AE analysis of the data recorded during the oxidation test of Zircaloy-4 reveals the presence of three populations of acoustic signals. The parameters of each population are presented in Table 3. Signals assigned to the instrumental noise (IN) are characterized by a very short duration and a low counts' number, 95 % of bursts of this population are characterized by (1 count -1 µs). These signals are also recorded throughout the blank test. Bursts recorded during the dwell time and the cooling are absent during the blank test carried out without any sample. These signals are really linked to the sample oxidation process.Bursts recorded during the dwell time are characterized by mean amplitudes in the range of (32 -55) dB AE as can be seen in Figure4, a high counts' number (tens of counts per burst) and a high absolute energy(1 -2000 aJ). During cooling, bursts of very high energy are recorded. They are characterized by a counts' number in the order of magnitude of hundreds of counts per burst, a long duration (thousands of µs per burst), a high amplitude varying between 55 dB AE and 75 dB AE (as shown in Figure4), and a very high absolute energy (2 10 5 aJ). The analysis of the average frequency of the acoustics bursts during the oxidation treatment (Fig.5) shows that AE signals recorded during dwell time and cooling steps possess a low average frequency varying between 0 kHz and 200 kHz. The corresponding signals assigned to the instrumental noise are characterized by a high average frequency in the range of 200 kHz and 1000 kHz including the resonance frequency of the sensors (300 kHz).

  These numerous cracks, parallel to the metal oxide interface, are observed on post-mortem oxidized samples. During the isothermal treatment, the cracks are pointed out by numerous specifics acoustic signals of medium energy. Fewer large transversal cracks are also observed in the oxide layer on post-mortem samples. They are mainly due to the cooling, and the AE signals related to this phenomenon are characterized by a high energy and long durations. The recording of numerous bursts during the oxidation test of Zircaloy-4, and furthermore, the correlation of population of signals to different cracking phenomena validated the alumina waveguide design.

1 .Fig. 3 .Fig. 4 .Fig. 5 .Fig. 6 .

 13456 Fig.1. AE acquisition system inside the symmetric thermobalance (TGA 24)

Table .

 .

		2. Chemical analysis of zircaloy-4 samples
	Zr	Sn	Fe	Cr	Hf
	98.23 %	1.45 %	0.21 %	0.1 %	0.01 %

Tables:

Table