Mustapha Ridaoui
email: mustapha.ridaoui@univ-paris1.fr

Michel Grabisch
email: michel.grabisch@univ-paris1.fr

Choquet integral calculus on a continuous support and its applications

Keywords: Choquet integral, distorted Lebesgue measure, risk measure, OWA operator

In this paper we give representation results about the calculation of the Choquet integral of a monotone function on the nonnegative real line. Next, we represent the Choquet integral of nonmonotone functions, by construction of monotone functions from nonmonotone ones, by using the increasing and decreasing rearrangement of a nonmonotone function. Finally, this paper is completed with some applications of these results to the continuous agregation operator OWA, and to the representation of risk measures by Choquet integral.

Introduction

The notion of measure is a very important concept in mathematics, particularly for the theory of integrals. These measures are based on the property of additivity. This property has been abandoned in many areas such as in decision theory and in the theory of cooperative games. It becomes essential to define nonadditive measures, which are usually called capacities [START_REF] Choquet | Theory of capacities[END_REF] or fuzzy measures [START_REF] Sugeno | Theory of fuzzy integrals and its applications[END_REF]. A fundamental concept that uses these nonadditive measures is the Choquet integral [START_REF] Choquet | Theory of capacities[END_REF], defined as an integral with respect to a capacity. The Choquet integral is known as a nonadditive integral of a function with respect to a capacity (or nonadditive measure, or fuzzy measure). It was characterized mathematically by Schmeidler [START_REF] Schmeidler | Integral representation without additivity[END_REF], and then by Murofushi and Sugeno [START_REF] Murofushi | An interpretation of fuzzy measures and the Choquet integral as an integral with respect to a fuzzy measure[END_REF] using the concept of the capacity introduced by Choquet. Later it was used in utility theory [START_REF] Schmeidler | Subjective probability and expected utility without additivity[END_REF], leading to the so-called Choquet Expected Utility. So far, many studies have focused on the theory and the applications of the Choquet integral defined on a discrete set (Faigle and Grabisch [START_REF] Faigle | A discrete Choquet integral for ordered systems[END_REF], Grabisch and Labreuche [START_REF] Grabisch | Fuzzy measures and integrals in MCDA[END_REF]). In the discrete case, the Choquet integral of a function with respect to a capacity is easy to calculate. However, this is not the case for the Choquet integrals of functions on a continuous support. Recent developments that have been conducted on the Choquet integral of real functions [START_REF] Sugeno | A note on derivatives of functions with respect to fuzzy measures[END_REF][START_REF] Sugeno | A way to Choquet calculus[END_REF] appear to open up new horizons. This paper is in the continuation of the seminal work of Sugeno [START_REF] Sugeno | A note on derivatives of functions with respect to fuzzy measures[END_REF][START_REF] Sugeno | A way to Choquet calculus[END_REF] and results and applications already established by Narukawa and Torra [START_REF] Narukawa | Distances defined by Choquet integral[END_REF][START_REF] Narukawa | Choquet integral with respect to a symmetric fuzzy measure of a function on the real line[END_REF]. In particular, we provide on the theoretical side methods for the calculation of the Choquet integral for nonmonotone functions, first by providing an analytic calculation for functions with a single maximum or minimum, and second by providing a general method based on the increasing or decreasing rearrangement of a function. In the second part of the paper, we give some possible applications of these new methods, e.g., for computing the continuous version of OWA operators, and for computing distortion risk measures This paper is organized as follows. Section 2 is devoted to the concepts of measures and capacities, and to the presentation of the Choquet integral, and essentially focuses on definitions. In Section 3, we present the results obtained by Sugeno [START_REF] Sugeno | A way to Choquet calculus[END_REF] for calculating the Choquet integral of a monotone function on the nonnegative real line with respect to a capacity, in particular with respect to distorted Lebesgue measure. Then we represent the Choquet integral of nonmonotone functions, exploring analytical calculation methods with examples, and by using the increasing and decreasing rearrangement of a nonmonotone function to turn it into a monotone function. In section 4, we focus on the application of the results obtained from the previous sections above, on the continuous aggregation operator OWA, and we make some links between the Choquet integral in the context of a distorted probability, and concepts used in finance such as the notions of risk measure. Finally, we end this paper with some concluding remarks.

Preliminaries

In this section, we present some basic definitions and properties of measures and Choquet integral.

Additive and nonadditive measures

We recall some definitions and properties of additive measures and capacities.

Let Ω be a set, and let A a collection of subsets of Ω. A is a σ-algebra if it satisfies the following conditions:

(1) Ω ∈ A, (2) A ∈ A ⇒ A c ∈ A, (3) ∀n ∈ N, A n ∈ A ⇒ n A n ∈ A. A pair (Ω, A) is called a measurable space.
Let (Ω, A) be a measurable space. A set function µ: A → R + is a σ-additive measure if it satisfies the following conditions: [START_REF] Artzner | Coherent measures of risk[END_REF]

µ(∅) = 0, (2) µ( +∞ n=1 A i ) = +∞ n=1 µ(A i ) for every countable sequence (A n ) n∈N of A of pairwise disjoint sets (A i ∩ A j = ∅ for all i = j).
The triplet (Ω, A, µ) is called a measure space. A probability measure P on (Ω, A) is an additive measure such that µ(Ω) = 1. The triplet (Ω, A, P ) is called a probability space.

Let B be the smallest σ-algebra including all the closed intervals of R. There is a measure λ on (R, B) such that: λ([a, b]) = ba for every interval [a, b], with -∞ < a ≤ b < ∞. This measure is called the Lebesgue measure.

Consider two measurable spaces (Ω 1 , A 1 ) and

(Ω 2 , A 2 ) . A function f : Ω 1 → Ω 2 is measurable if ∀E ∈ A 2 , f -1 (E) ∈ A 1 .
Let (Ω, A, P ) be a probability space. A measurable function f from (Ω, A) to (R, B) is called a random variable.

We call a distribution function of a random variable X the function F : R → [0, 1] given by: ∀x ∈ R, F (x) = P (X ≤ x).

The notion of capacity was introduced by Choquet [START_REF] Choquet | Theory of capacities[END_REF] in his theory of capacities. A similar concept was proposed by Sugeno [START_REF] Sugeno | Theory of fuzzy integrals and its applications[END_REF] under the name "fuzzy measure", and by Denneberg [START_REF] Denneberg | Non-additive measure and integral[END_REF] under the name "nonadditive measure". Definition 1. Let (Ω, A) be a mesurable space. A set function µ: A → [0, 1] is called a capacity [START_REF] Choquet | Theory of capacities[END_REF] or fuzzy measure [START_REF] Sugeno | Theory of fuzzy integrals and its applications[END_REF] if it satisfies the following conditions: [START_REF] Artzner | Coherent measures of risk[END_REF]

µ(∅) = 0, (2) µ(A) ≤ µ(B), if A ⊆ B, A, B ∈ A.
For any capacity µ, the dual capacity µ is defined by µ(A) = 1µ(A c ) for any A ∈ A. The capacity µ is normalized if in addition µ(Ω) = 1. Definition 2. Let µ be a capacity on (Ω, A). µ is called concave or submodular, if µ(A) + µ(B) ≥ µ(A ∪ B) + µ(A ∩ B), for all A, B ∈ A. A capacity µ is called convex or supermodular, if it satisfies the previous property with reverse inequality.

Distortion measures

We call a distortion function every nondecreasing function m :

[0, 1] → [0, 1] with m(0) = 0 et m(1) = 1.
Let P be a probability measure on (Ω, A) and let m be a distortion function. The set function m • P defined by m • P (A) = m(P (A)), ∀A ∈ A is called a distorted probability. Definition 3. [START_REF] Sugeno | A way to Choquet calculus[END_REF] Consider a Lebesgue measure λ on (R, B) and m : R + → R + a differentiable nondecreasing function such that m(0

) = 0. Then µ m = m • λ is a distorted Lebesgue measure. We have µ m ([a, b]) = m(b -a).

Choquet integral

Definition 4. [START_REF] Choquet | Theory of capacities[END_REF][START_REF] Murofushi | An interpretation of fuzzy measures and the Choquet integral as an integral with respect to a fuzzy measure[END_REF] Let µ be a capacity on (Ω, A), and let f : Ω → R + be a measurable function. The Choquet integral of f with respect to µ is defined by:

(C) f dµ = +∞ 0 µ f (r)dr, where µ f (r) = µ({τ |f (τ ) ≥ r})
Let A ⊂ Ω. The Choquet integral of f with respect to µ on A is defined by [START_REF] Sugeno | A note on derivatives of functions with respect to fuzzy measures[END_REF]:

(C) A f dµ = ∞ 0 µ({τ |f (τ ) ≥ r} ∩ A)dr.
We give some well-known properties of the Choquet integral (see, e.g., Denneberg [START_REF] Denneberg | Non-additive measure and integral[END_REF]) Proposition 1. Let µ be a capacity on (Ω, A). Let f and g be two measurable functions on (Ω, A). We have the following properties:

(1) if f ≤ g then (C) f dµ ≤ (C) gdµ (2) (C) λf dµ = λ(C) f dµ, for every λ ∈ R + (3) (C) (f + c)dµ = (C) f dµ + cµ(Ω), for every constant c ∈ R (4) (C) -f dµ = -(C) f dµ Proposition 2.
Let µ be a capacity on (Ω, A). Let f and g be two measurable functions on (Ω, A).

• If µ is supermodular, then the Choquet integral with respect to µ is superadditive:

(C) (f + g)dµ ≥ (C) f dµ + (C) gdµ.
• If µ is submodular, then the Choquet integral with respect to µ is subadditive:

(C) (f + g)dµ ≤ (C) f dµ + (C) gdµ.
Proposition 3. [START_REF] Narukawa | Distances defined by Choquet integral[END_REF] Let µ be a submodular capacity on (Ω, A). Let f and g be two measurable functions on (Ω, A). The Choquet integral satisfies the following inequality:

(C) (f + g) 2 dµ 1 2 ≤ (C) f 2 dµ 1 2 + (C) g 2 dµ 1 2

Choquet integral calculus methods

In this section, we introduce methods for calculating the continuous Choquet integral with respect to distorted Lebesgue measures on the nonnegative real line. The calculus of continuous Choquet integrals was recently studied by Sugeno [START_REF] Sugeno | A note on derivatives of functions with respect to fuzzy measures[END_REF], [START_REF] Sugeno | A way to Choquet calculus[END_REF] for nonnegative monotonic functions, and mainly nondecreasing functions. Based on the results in [START_REF] Sugeno | A note on derivatives of functions with respect to fuzzy measures[END_REF], [START_REF] Sugeno | A way to Choquet calculus[END_REF], we will start by calculating the Choquet integral for nonnegative monotonic functions with respect to distorted Lebesgue measures. Next, we construct monotonic functions from nonmonotone ones for calculating the Choquet integral of nonmonotonic functions.

Let F = {f | f : R + → R + , f : measurable, derivable, monotone } be the class of measurable, differentiable, nonnegative and monotone functions. We denote by F + the nondecreasing functions of F , and by F -the nonincreasing functions of F .

In [START_REF] Sugeno | A note on derivatives of functions with respect to fuzzy measures[END_REF], [START_REF] Sugeno | A way to Choquet calculus[END_REF], Sugeno used the Laplace transform for establishing the basis of Choquet integral calculus of nondecreasing functions.

Let f be a function defined for all real numbers t ≥ 0. The Laplace transform of f is the function

F (s) = L[f (t)] defined by F (s) = ∞ 0 f (t)e -st dt,
for those s ∈ C for which the integral is defined.

The inverse Laplace transform is given by the following complex integral:

f (t) = L -1 [F (s)] = 1 2πi γ+i∞ γ-i∞ F (s)e st ds,
where γ is a real number so that the contour path of integration is in the region of convergence of F (s).

Without any assumption on f , the integral ∞ 0 f (t)e -st dt does not necessarily exist. Sufficient condition for the existence of Laplace transform are:

(i) f is a piecewise continuous function. (ii) |f (t)| ≤ Me αt , M > 0 and α ∈ R.
We list properties of the Laplace transform which will be useful in the sequel:

(i) L t 0 f dt = F (s)/s, (ii) L [df /dt] = sF (s) -f (0), (iii) L(f * g) = F (s)G(s)
, where * is the convolution product.

Choquet integral of monotonic functions

Let µ be a capacity, and consider that µ([τ, t]) is differentiable with respect to τ on [0, t], for every t > 0. Theorem 1. [START_REF] Sugeno | A note on derivatives of functions with respect to fuzzy measures[END_REF] Let g ∈ F + , then the Choquet integral of g with respect to µ on [0, t] is represented as:

(C) [0,t] gdµ = - t 0 µ ′ ([τ, t])g(τ )dτ
In particular, if µ = µ m , we have:

(C) [0,t] gdµ m = t 0 m ′ (t -τ )g(τ )dτ Remark 1. (C) [0,t] gdµ m = m ′ * g(t)
, where * is the convolution product.

The next proposition shows how to calculate the Choquet integral by using the Laplace transformation. Theorem 2. [START_REF] Sugeno | A note on derivatives of functions with respect to fuzzy measures[END_REF] Let g ∈ F + , and let µ m be a distorted Lebesgue measure. The Choquet integral of g with respect to µ m on [0, t] is given by:

(C) [0,t] gdµ m = L -1 [sM(s)G(s)]
where M(s) is the Laplace transform of m, G(s) is the Laplace transform of g, and L -1 is the inverse Laplace transform.

Example 1. Let m(t) = t , and g(t) = e t -1, then:

(C) [0,t] gdµ m (τ ) = (C) [0,t] (e τ -1)dµ m (τ ) = L -1 [sM(s)G(s)] with M(s) = L[m(t)] = 1 s 2 , et G(s) = L[g(t)] = 1 s -1 - 1 s . Then sM(s)G(s) = 1 s(s -1) - 1 s 2 , therefore: (C) [0,t]
gdµ m (τ ) = e tt -1.

Theorem 3. [START_REF] Sugeno | A note on derivatives of functions with respect to fuzzy measures[END_REF] Let µ m be a distorted Lebesgue measure, let f be a nondecreasing and continuous function with f (0) = 0, then the solution of the Choquet integral equation

f (t) = (C) [0,t] g(τ )dµ m (τ )
is given by :

g(t) = L -1 F (s) sM(s) , with g ∈ F + .
Theorem 4. [START_REF] Sugeno | A note on derivatives of functions with respect to fuzzy measures[END_REF] Let g ∈ F -, then the Choquet integral of g with respect to µ on [0, t] is represented by:

(C) [0,t] gdµ = t 0 µ ′ ([0, τ ])g(τ )dτ
In particular, if µ = µ m , we have:

(C) [0,t] gdµ m = t 0 m ′ (τ )g(τ )dτ

Choquet integral of nonmonotonic functions

In the previous section, we have shown methods for calculating the continuous Choquet integral of monotonic functions on the nonnegative real line. In this section, we will use the nondecreasing and nonincreasing rearrangements of a nonmonotonic function, for the transformation into a monotonic function. This issue was addressed by Ralescu and Sugeno [START_REF] Ralescu | Fuzzy integral representation[END_REF], Ralescu [START_REF] Ralescu | Extensions of fuzzy aggregation[END_REF], and recently studied by Sugeno [START_REF] Sugeno | A way to Choquet calculus[END_REF] and Narukawa et al. [START_REF] Narukawa | Choquet integral with respect to a symmetric fuzzy measure of a function on the real line[END_REF].

Let G be the class of nonnegative and continuous functions on [0, t], for some fixed

t ∈ R + : G = {g | g : [0, t] → R + , g continuous}.

Analytic calculation

In this section, we provide an explicit expression of the Choquet integral for functions with one maximum or minimum, that is, which are increasing then decreasing on an interval (or the converse).

Let g ∈ G. We calculate the Choquet integral of g on [0, t], with τ m ∈ [0, t] such that g = g(τ m ) = max 0≤τ ≤t g(τ ) (or g = g(τ m ) = min 0≤τ ≤t g(τ )).

We assume first that function g is nondecreasing on [0, τ m ], and nonincreasing on [τ m , t], with g(τ m ) = g (see Fig. 1).

Let

g 1 (τ ) = g(τ ) on [0, τ m ], and g 2 (τ ) = g(τ ) on [τ m , t]. If g(t) ≥ g(0) (respectively, g(t) ≤ g(0)
), for every r ∈ [g(t), g] (respectively, r ∈ [g(0), g]), there exists a unique pair (τ r , τ r ), such that: τ r = g -1 1 (r), and τ r = g -1 2 (r) (see Fig. 1).

τ g(τ ) t g(t) g(0) {g ≥ r} r τ m g τ r τ r 0 [ ] ({g ≥ r} ∩ [0, t])
Figure 1: case of a nonmonotonic function

• If g(t) ≥ g(0)
, the Choquet integral of g with respect to measure µ is:

(C) [0,t] gdµ = ∞ 0 µ({τ |g(τ ) ≥ r} ∩ [0, t])dr = g(0) 0 µ([0, t])dr + g(t) g(0) µ([g -1 1 (r), t])dr + g g(t) µ([g -1 1 (r), g -1 2 (r)])dr = µ([0, t])g(0) + g(t) g(0) µ([g -1 1 (r), t])dr + g g(t) µ([g -1 1 (r), g -1 2 (r)])dr
In particular, if µ = µ m , we have:

(C) [0,t] gdµ m = m(t)g(0) + g(t) g(0) m(t -g -1 1 (r))dr + g g(t) m(g -1 2 (r) -g -1 1 (r))dr (1)
• If g(t) ≤ g(0), the Choquet integral of g with respect to measure µ is represented by:

(C) [0,t] gdµ = µ([0, t])g(t) + g(0) g(t) µ([0, g -1 2 (r)])dr + g g(0) µ([g -1 1 (r), g -1 2 (r)])dr
In particular, if µ = µ m , we have:

(C) [0,t] gdµ m = m(t)g(t) + g(0) g(t) m(g -1 2 (r))dr + g g(0) m(g -1 2 (r) -g -1 1 (r))dr (2)
We assume now that g is nonincreasing on [0, τ m ], and nondecreasing on [τ m , t], with g = g(τ m ) = min 0≤τ ≤t g(τ ).

Proceeding similary, we find:

• If g(t) ≥ g(0), the Choquet integral of g with respect with respect to measure µ is:

(C) [0,t] gdµ = µ([0, t])g + g(0) g µ([0, g -1 1 (r)] ∪ [g -1 2 (r), t])dr + g(t) g(0) µ([g -1 2 (r), t])dr
In particular, if µ = µ m , we have:

(C) [0,t] gdµ m = m(t)g + g(0) g m(t + g -1 1 (r) -g -1 2 (r)]dr + g(t) g(0) m(t -g -1 2 (r)]dr (3)
• If g(t) ≤ g(0), the Choquet integral of g with respect with respect to measure µ is represented by:

(C) [0,t] gdµ = µ([0, t])g + g(t) g µ([0, g -1 1 (r)] ∪ [g -1 2 (r), t])dr + g(0) g(t)
µ([0, g -1 1 (r)])dr

In particular, if µ = µ m , we have:

(C) [0,t] gdµ = m(t)g + g(t) g m(t + g -1 1 (r) -g -1 2 (r))dr + g(0) g(t) m(g -1 1 (r))dr (4)
Example 2. Let m(r) = r 2 , and g(r) = r 2 -2r + 2, ∀r ∈ [0, 2].

We have g -1

1 (r) = 1 + √ r -1, g -1 2 (r) = 1 - √ r -1,
gdµ m = t 2 + t 2 -2t+2 1 (t -2 √ r -1) 2 dr + 2 t 2 -2t+2 (1 - √ r -1) 2 dr = 4 3 t 4 - 28 3 t 3 + 14t 2 -8t + 2.

Choquet integral of nonmonotonic function and rearrangement

The principle of the theory of monotone equimeasurable rearrangements of functions [START_REF] Ghossoub | Equimeasurable rearrangements with capacities[END_REF] is that, for every function f defined on the real line, there exists an nondecreasing (respectively, nonincreasing) function that has the same distribution function that the function f with respect to Lebesgue measure. This function is called the nondecreasing (respectively, nonincreasing) rearrangement of the function f . Based on the results obtained in [START_REF] Sugeno | A way to Choquet calculus[END_REF] and [START_REF] Narukawa | Choquet integral with respect to a symmetric fuzzy measure of a function on the real line[END_REF], we shall further explore the calculation of the Choquet integral of a nonmonotonic function on the nonnegative real line.

Let g ∈ G be a continuous and nonnegative function on [0, t], with τ m ∈ [0, t] such that g is nondecreasing (respectively, nonincreasing) on [0, τ m ], and nonincreasing (respectively, nondecreasing) on [τ m , t], and that g(t) ≥ g(0), with g = g(τ m ) = max 0≤τ ≤t g(τ ) (respectively, g = g(τ m ) = min 0≤τ ≤t g(τ )).

Let λ g : [g(0), g] → [0, t] (respectively, λ g : [g, g(t)] → [0, t]) be a function defined by:

λ g (r) = λ({τ |g(τ ) ≥ r})
The function λ g is continuous, and nonincreasing on [g(0), g] (respectively, [g, g(t)]).

We define a function g * : [0, t] → [g(0), g] (respectively, g * : [0, t] → [g, g(t)]) by:

g * (τ ) = λ -1 g (t -τ ), (5)
where g * is called the nondecreasing rearrangement of g on [0, t]. The function g * is nondecreasing and continuous on [0, t].

Remark 2. For all τ ∈ [0, τ t ] (respectively, τ ∈ [τ 0 , t]), where g(τ t ) = g(t) (respectively, g(τ 0 ) = g(0)), g * (τ ) = g(τ ).

If g(t) ≤ g(0), we define a function g * : [0, t] → [g(t), g] (respectively, g * : [0, t] → [g(0), g]) by:

g * (τ ) = λ -1 g (τ ) (6)
The function g * is the nonincreasing rearrangement of g on [0, t], it is continuous and nonincreasing on [0, t].

Remark 3. For all τ ∈ [τ 0 , t] (respectively, τ ∈ [τ t , t]), where g(τ 0 ) = g(0) (respectively, g(τ t ) = g(t)), g * (τ ) = g(τ ).

Proposition 4. Let g ∈ G, and let g * be a rearrangement of g on [0, t], then:

λ({τ |g * (τ ) ≥ r}) = λ({τ |g(τ ) ≥ r})
Corollary 1. Let g ∈ G, and let g * be a rearrangement of g on [0, t], then the Choquet integral of g with respect to measure µ m on [0, t] can be written as:

(C) [0,t] gdµ m = (C) [0,t] g * dµ m
Hence by Theorems 1 and 4 we find: • If g(t) ≥ g(0), the Choquet integral of g with respect to measure µ m on [0, t] becomes:

(C) [0,t] gdµ m = t 0 m ′ (t -τ )g * (τ )dτ,
with g * is given by (5).

• If g(t) ≤ g(0), the Choquet integral of g with respect to measure µ m on [0, t] becomes:

(C) [0,t] gdµ m = t 0 m ′ (τ )g * (τ )dτ,
with g * is given by [START_REF] Ghossoub | Equimeasurable rearrangements with capacities[END_REF].

Example 3. Let m(τ ) = τ 2 , and g(τ ) = 4 -(τ -2) 2 .
The function g is nonnegative, continuous, and nonmonotonic on [0, t], where t ∈]2, 4]. A function g is nondecreasing on [0, 2], and nonincreasing on [2, t]. We have: g(t) ≥ g(0), then the nondecreasing rearrangement g * of the function g on [0, t] is defined by:

g * (τ ) = g(τ ), if 0 ≤ τ ≤ 4 -t 4 -(t -τ ) 2 /4, if 4 -t ≤ τ ≤ t τ g(τ ) t τ t g(t) τ g r λ g (r) g(0) τ g * (τ ) t τ t g g(t) g(0)
Figure 2: The function g and its nondecreasing rearrangement g * on [0, t] The Choquet integral of g with respect to measure µ m on [0, t] is represented by:

(C) [0,t] gdµ m = t 0 m ′ (t -τ )g * (τ )dτ = 2 t 0 (τ -t)g * (τ )dτ = 2 4-t 0 (τ -t)g(τ )dτ + 2 t 4-t (τ -t)(4 -(t -τ ) 2 /4)dτ = 2(t 4 -8t 3 + 16t 2 -16) -1/6(t -4) 2 (7t 2 -16)

Some Applications

In this section, we review some applications of the Choquet integral on the nonnegative real line.

OWA operator on the real line

The ordered weighted averaging (OWA) operator was introduced by Yager [START_REF] Yager | On ordered weighted averaging aggregation operators in multi-criteria decision making[END_REF].

OWA operator

In this section, we consider that Ω = {1, . . . , n}, and A = 2 Ω . Definition 5. [START_REF] Yager | On ordered weighted averaging aggregation operators in multi-criteria decision making[END_REF] Let w = (w 1 , . . . , w n ), such that w i ∈ [0, 1], and n i=1 w i = 1. The ordered weighted averaging operator (OWA) with respect to w is defined for any a = (a 1 , . . . , a n ) ∈ R n + by:

OWA w (a) = n i=1 w i a σ (i) ,
where σ is a permutation on {1, . . . , n} such that a σ (1) . . . a σ (n) .

A capacity µ is said to be symmetric if µ(A) = µ(B) whenever |A| = |B|, ∀A, B ∈ A.

Proposition 5. [START_REF] Murofushi | Some quantities represented by the choquet integral[END_REF] For every OWA w operator, there existe a symmetric capacity µ given by µ({1}) = w 1 , and µ({1, . . . , i}) = w i , i = 1, . . . , n, such that:

OWA w (a) = (C) adµ,
for any a ∈ R n + .

Continuous OWA operator

In this section, we will use the Choquet integral for defining the continuous OWA operator (COWA) [START_REF] Narukawa | Aggregation operators on the real line[END_REF].

Consider a Lebesgue measure λ, and µ a capacity on ([0, 1], B). µ is said to be symmetric if λ(A) = λ(B) implies µ(A) = µ(B). Definition 6. [START_REF] Narukawa | Aggregation operators on the real line[END_REF] Let µ be a symmetric capacity, and f : R + → R + be a measurable function. The continuous OWA operator is defined by:

COWA µ (f ) = (C) f dµ.
Let m : [0, 1] → [0, 1] be a distortion function. Then a distorted Lebesgue measure µ m is a symmetric capacity. It follows that we can consider the Choquet integral with respect to µ m as a COWA operator.

Corollary 2. Let f : [0, t] → R + be a differentiable and nondecreasing function. Then we have:

COWA µm (f ) = L -1 [sM(s)F (s)], with F (s) = L[f (t)], and
M(s) = L[m(t)]
Corollary 3. Let f : [0, t] → R + be a differentiable and nonincreasing function. Then we have:

COWA µm (f ) = t 0 m ′ (t)f (τ )dτ.
Example 4. [START_REF] Narukawa | Choquet integral with respect to a symmetric fuzzy measure of a function on the real line[END_REF] Let f : [0, 1] → R + be a differentiable and nondecreasing function.

We define the sequence of functions f k by: f

1 (t) = t 0 f (τ )dτ , f k+1 (t) = t 0 f k (τ )dτ , for t ∈ [0, 1], k ∈ {1, 2, . . . , n}. Let us show that COWA µm (f ) = n!f n (t),
where m(t) = t n .

f being nondecreasing, the continuous OWA operator is given by:

(C) [0,t] f dµ m = L -1 [sM(s)F (s)],
where

F (s) = L[f (t)], et M(s) = L[m(t)] = n! s n+1 . Then (C) [0,t] f dµ m = n!L -1 F (s) s n . Note that F (s) s = L t 0 f (τ )dτ = L[f 1 (t)] = F 1 (s), hence F (s) s n = L[f n (t)]
, therefore:

COWA µm (f ) = n!L -1 [L[f n (t)]] = n!f n (t).
To complete the example, let us take the function f (x) = e x -1 of Example 1, with m(t) = t n . We have

f n (x) = e x -t n n! -t n-1 (n-1)! . . . t 2 2 -t -1 , then COWA µm (f ) = n!(e t -t -1) -t n -nt n-1 . . . (n(n -1) . . . 3)t 2 , x ∈ [0, 1].

Risk Measures and Choquet integral

Risk management is a subject of concern in finance and insurance. One of the most significant problems in managing risk is the determination of a measure that can take into account the different characteristics of the distribution of losses. For this, there are tools to quantify and predict the risk, which are called risk measures. They allow risk assessment and to compare differents risks between them. To manage these risks, several risk measurements have been proposed, each having its advantages and disadvantages. Artzner et al. [START_REF] Artzner | Coherent measures of risk[END_REF] sought to characterize what would determine whether a risk measurement is "effective". For this they introduced the notion of a coherent risk measurement.

Distortion risk measures

Distortion risk measures originated in Yaari's paper [START_REF] Yaari | The dual theory of choice under risk[END_REF], they consist in measuring the risks from the distortion of probabilities. Wang et al. [START_REF] Wang | Axiomatic characterization of insurance prices[END_REF] developed the concept of distortion risk measure by calculating expected losses from a nonlinear transformation of the cumulative probability of the risk factor. Distortion risk measures allow the production of better risk measures from the distortion of the originals. Definition 10. [START_REF] Wang | Axiomatic characterization of insurance prices[END_REF] Let g : [0, 1] → [0, 1] be a distortion function, and X ∈ X a random variable. The distortion risk measure associated with distortion function g for X is defined by:

ρ g (X) = +∞ 0 g(G X (x))dx + 0 -∞ [g(G X (x)) -1]dx, where G X = P (X > x) = 1 -P (X ≤ x) = 1 -F X (x), and F X (x) is the distribution function of X.
Several popular risk measures belong to the family of distortion risk measures. For example, the value-at-risk (VaR), the tail value-at-risk (TVaR) and the Wang distortion measure.

Remark 4. When the distortion function is concave, the distortion risk measure is also subadditive (Wang and Dhaene [START_REF] Wang | Comonotonicity, correlation order and premium principles[END_REF], Wirch and Hardy [START_REF] Wirch | Distortion risk measures: coherence and stochastic dominance[END_REF]).

If X is nonnegative, then the distortion risk measure associated with distortion function g is defined by: ρ g (X) = +∞ 0 g(G X (x))dx.

Remark 5. ρ g (X) is the Choquet integral of X with respect to the distorted probability µ g = g • P .

Remark 6. If the distortion function g is differentiable, and the distribution function F is continuous (and strictly increasing), then the distortion risk measure can be written as follows:

ρ g (X) = 1 0 F -1 X (1 -x)g ′ (x)dx. (7)
Proof. For every X ∈ X , we have: = -ρ g (-X).

Conclusion

The work presented in this paper revolves around two axes. The first axis focused on the methods of calculating the Choquet integral with respect to a Lebesgue distortion measure. The second purpose is to apply the calculation of the Choquet integral.

After introducing basic concepts of Choquet integrals, we presented calculation methods for monotone and nonmonotone functions on the positive real line.

In the case of nonmonotone functions, we used the nondecreasing and nonincreasing rearrangements of a nonmonotone function to turn it into a monotone function, in order to apply the calculation results of the Choquet integral for monotone functions.

In this paper, we used the results of calculation of Choquet integral for calculating the continuous aggregation operator OWA, and we represented the risk measures by Choquet integrals to facilitate verification of the convexity risk measures.

Many aspects of the work remain obviously to deepen. An interesting subject of study would be to try to obtain results about the calculation of the Choquet integral with respect to a general capacity, not limited to distorted Lebesgue measures.

It is also important to find a general representation for the calculation of the Choquet integral of a monotone or nonmonotone function on the real line. Using these calculations in practice would allow addressing many areas.

ρ

  g (X) = (C) -(-X)dµ g = -(C) -Xdµ g = -(C) -Xdµ g , (µ g = µ g )

Risk Measures

Let (Ω, A) be a measurable space. Denote by X the space of the measurable functions X such that X ∞ = sup w∈Ω |X(w)| is bounded:

Definition 7. [START_REF] Denuit | Mathématiques de l'assurance non-vie. Tome 1 : principes fondamentaux de théorie du risque[END_REF] A mapping ρ : X → R is called coherent risk measure if it satisfies the following conditions for all X, Y ∈ X : (1) monotonicity:

We adopt the definition that is used in the case where the random variables are interpreted as losses.

The risk measure definition given by Artzner and al. [START_REF] Artzner | Coherent measures of risk[END_REF] coincides up to the sign with the definition referred to above. However, in the definition of Artzner et al. [START_REF] Artzner | Coherent measures of risk[END_REF], the sign " + " of property ( 2) is changed to the sign " -" , and property (1) becomes:

satisfies the axioms of monotonicity and translation invariance.

Definition 9. [4]

A monetary risk measure ρ is called a convex risk measure if it satisfies the following property, ∀X, Y ∈ X , ∀λ ∈ [0, 1]:

Let (Ω, A) be a measurable space, and let µ be a capacity. We define the function ρ µ : X → R by:

In the case where the random variables are interpreted as gains, we define the function ρ µ as follows:

By the properties of Choquet integral, we know that the function ρ µ is a positive homogeneous monetary risk measure, and if µ is a submodular capacity, the function ρ µ is a convex risk measure. Indeed, by the change of variable u = F X (x), and integration by parts :

Proposition 6. Let F X be a continuous and strictly increasing distribution function of

Proof. Let X be a nonnegative random variable,then we have:

F -1 X dµ g , by theorem 1.

Remark 7. The Value at Risk (VaR) is one of the most popular risk measures due to its simplicity and intuitiveness. However it is known that the distortion function associated to this risk measure is not differentiable. Therefore our results do not apply in this case. One way to obviate this problem may be to consider a differentiable approximation to the discontinuous function that yields the VaR. This approach, which may be very interesting to study, is left for future work.

Proposition 7. For every X ∈ X , ρ g (X) = -ρ g (-X), with g(x) = 1g(1x) the dual function of g.


Candide Lissak

Etienne Edouard Cossart

Vincent Viel

Monique Fort

Benoit Carlier

Gilles Arnaud-Fassetta

Torrential activity facing global change in Southern French Alps

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

Scientific questioning

In mountainous areas, especially in large river catchments with torrential tributaries, the sediment supply significantly increases the flood impacts in the valley bottoms. Within the framework of the SAMCO (ANR) project, dealing with the potential impacts of global changes (climate, ecosystem e.g. land use, socio-economic system) on mountain hazards, we aim to a better understanding of Alpine river systems through the analysis of the sediment budgets. For estimating these budgets it implies the analysis of geomorphic processes contribution in the sediment production and transfers, and the identification of different sources, pathways and sinks. In the study area (Guil catchment, South-west France), remobilisation of sediment accumulated in hillslope storage are largely determined by geomorphic extreme events (debris flow, shallow landslides, runoff). The contribution of these processes in sediment delivery to channel is the main objective of this poster. We propose a compartment analysis modelling in order to quantifying sediments fluxes and stores using system dynamics modelling. Our study aims at analysing the temporal composition of sediment cascade in the Peynin sub-catchement which highly contributes to sediment budget of the Guil river catchment. The initial phase of our study consists in identifying sediment sources and storage through geomorphological analysis and mapping. Then a conceptual model of the cascade can be generated [START_REF] Schrott | Spatial distribution of sediment storage types and quantification of valley fill deposits in an alpine basin, Reintal, Bavarian Alps, Germany[END_REF][START_REF] Heckmann | Sediment budget and morphodynamics of an alpine talus cone on different timescales[END_REF] The Guil catchment (Queyras, Southern French Alps -317km²) is prone to catastrophic summer flood. For risk mitigation, some protection equipments were built after the 1957 event, but most of them are now poorly maintained and might be less effective in case of a new flood event. Since 1810, the Guil River valley has been affected by 19 catastrophic floods [START_REF] Arnaud-Fassetta | Hydro-geomorphic hazards and impact of man-made structures during the catastrophic flood of June 2000 in the Upper Guil catchment (Queyras, Southern French Alps)[END_REF] with serious damages to infrastructures and buildings (specific reaches and confluences with debris flow prone tributaries) (Fig. 2). Today several elements at risk are still located in the valley bottom (roads, tourist infrastructures, villages at the confluence of tributaries prone to debris flows,... i.e POSTER EGU2016-5767 -Session NH9. .

These catastrophic consequences can be explained by steep, lower order drains which carry a contrasted solid discharge, including predominantly sandy-loam particles mixed with gravels and boulders during flood events (sandstone schists, ophiolites). Abundant sediment supply by frost shattering, snow avalanche and landslides is then reworked during snowmelt or summer storm runoff events, and may result in catastrophic, very destructive floods along the main channel. -SD designed to assess the relations between the structure of a system and its behavior over time [START_REF] Ford | Modeling the environment: an introduction to system dynamics models of environmental systems[END_REF] -SD mode was implanted in Vensim ®software (Ventana Systems, 2007).

-A compartment model to simulate the part of different processes, to determine how different compartments (sediment storages) of the system are related between them and how they evolve.

-There is no hierarchical decomposition in the model development but a longitudinal organization from upslope (contributing area) to downslope (source to sink).

-A model to estimate the required time for an optimal hydro-sedimentary functioning. To understand how a specific hydro-meteorological event corresponds to a hydro-sedimentary behavior.

-Geomorphic processes are not sptialized but calibrated by specific threshold and physical parameters

H M L

Direct weighting

Potential contributing areas

Scenario 2 (High intensity hydro-meteorological event)

Sediment sources dynamics

Connectivity

-Historical data analysis shows that scree talus located in catchment head were few mobilized during the last geomrophic events -Numerous blocks were located for a long time into the channel before being mobilized during flood events The sedimentary cascade highlights (1) the minor importance of the snow avalanches into the sediment transfers and (2) the predominance of torrential stream, debris flows for the sediment delivery.

-Sedimentary signals through time show a complex response of the system, which does not seem to react directly to climate but to occasional, meteorological forcing only = we consider a decoupling precipitation / sedimentary signal -According to simulation it takes almost 80 years for an optimal reconstruction of sediment storages -The flood impact depends on the initial state of the stocks and on the intensity of a hydro-meteorological given event.

C/ Model validation

-

  . Then we provide a stock & flow modeling based on System Dynamics to modelize sedimentary signals through time considering processes and storages interactions or connections. The Upper Guil valley is characterised by asymmetrical structural slopes related to the late Alpine reverse folding of the Pennine nappes (Lemoine & Tricart, 1988) (Fig. 1) -Average relief: 1500/2000 m -Valley characterised by floodplains and gorges (Fig. 1c) and many tributaries (Fig. 1b) -Lithological context (Fig. 1a): "Schistes lustrés" belts in the inner Alps -Schists represent 90-95% of the total outcrops with local heterogenities of facies (sandstones, calschists). These are mainly impervious, tectonised, frost-sensitive bedrock -Easily shattered schists playing an important role in the torrential flow modalities -A litho-structural context favorable for sediment/ debris production and transport -During previous catastrophic event, protective measures were not sufficient, first-order catchments provided a great quantity of frost-shattered debris, mobilised as debris flows or hyperconcentrated torrential flows

Figure 1 -

 1 Figure 1 -Study area

Figure 2 -

 2 Figure 2 -Guil flood events a) Inset map of France with location of Guil river catchment and major geological units, b) Elevation map of Queyras Regional Parc with Guil catchment and major preferential flows, c) Diversity of landscapes A) Guil river before 1957 flood event, B) Guil river after 1957 flood event C)/D) Guil river during 2000 flood event

-Figure 3 -

 3 Figure 3 -Dataset & method for source areas definition

Figure 5 -

 5 Figure 4 -Results

Figure 9 -

 9 Figure 9 -Relative slope deformation at TLS 3 hotspot since 2013

Figure 6 -Figure 7 -Figure

 67 Figure 6 -Sediment cascade in Vensim software

Acknowledgements -This research was funded through the project SAMCO: "Adaptation de la société aux risques en montagne dans un contexte de changement globaleux" (2013-2017) of the Agence Nationale de la Recherche (ANR-12-SENV-0004).


Gilles Bouch

On the Atlantic façade, megaliths are a characteristic of Brittany in general, especially of the Morbihan area, but they can also be found across the Channel, in Ireland and the other Celtic countries east of the Irish Sea, and at the southernmost part of the Atlantic façade, that is Portugal. Research is in progress, but when confronted with unmovable and speechless artefacts, archaeologists may have a hard time finding an interpretation for some characteristic engravings. The history of religions can help interpret some of those artefacts, thanks to a few basic concepts on pre-Christian beliefs about the origin of life, involving deer and the role this animal might have had in our ancestors' mind, back in the Mesolithic but also in the Neolithic eras. Hopefully this animal, unnoticed until now by archaeologists, will find its due place in a world dominated by man, and will also help make things clearer between Irish and Breton megalithism. Their relationship has remained unclear in spite of Michael Herity's 1974 hypothesis1 on the Breton origin of Irish passage tombs. This origin remains unchallenged in Alison Sheridan's * Bibliothécaire adjoint spécialisé, en charge du fonds « Bretagne et pays celtiques, Breizh ha broioù keltiek »

2003 contribution2, in which, however, the emergence of passage tombs is much older than previously thought. Provided progress in archaeology makes it possible, this paper will first examine the likely origins of Breton megalithism, between Barnenez and the Morbihan. It will then examine the relationship between hunter-gatherers and agriculturists, their possible melting, their migration to Ireland around 4000 BC and, through a comparison with related cultures, some of them very far, the unexpected importance of deer in their mythologies and their rites. The paper will then focus on the figure of Anna, and suggest that she runs throughout the cultural layers that have succeeded one another in these two regions, Ireland and Brittany. This anthropomorphic Neolithic female figure, who probably replaced the zoomorphic Mesolithic one in Brittany, both of them anonymous, probably originates from Anatolia where the agriculturists came from as soon as the 7th millennium BC. She was probably given her name, Ana, by the Celts of the first or second Iron Age, at the time when their culture was dominant in the region. Of course, in Turkish Ana today means 'mother' but it is an Indo-European word. It will also be suggested that Irish Celtic sovereignty, applied to the land considered as a mother, must have been a new concept for these hunter-gatherers and agriculturists. Eventually, the paper speculates that in these two regions, Ana may be viewed today as the syncretic symbol of the female figures of each successive people and civilizations.

Mo rbih a n

The words Megalithism and Neolithic are sometimes misused, especially in regions like Brittany where the two phenomena coincide. Both terms are sometimes used indifferently for the same period of time, although Megalithism is a cultural phenomenon which occurred in the course of the Neolithic era. In other words, the Neolithic era, which originated with agriculture in the Middle East around the 12th millennium BC, encompasses to a great extent, in terms of time and 2. SKERIDAN, Alison, 2003, '2. French Connections I: spreading the marmites thinly.' in Ian Armit , Eileen Murph y, Eiméar Nelis and Derek Simpson , eds., Neolithic Settlement in Ireland and Western Britain. Oxford, Belfast, 4.

G. BOUCHERIT -Irish and Breton m e g alithis m… 1 3 place, Megalithism, the origin of which lies in Brittany and Portugal at the end of the 6th millennium or so. [Ill. 1 ] Megalithism is widely believed to have been born in the Gulf of Morbihan, maybe because of the world-renowned alignments in the Carnac area, among other famous places. Locmariaquer and Mane Vihan or the Petit Mont, for instance, are located at each end of the mouth of the Gulf of Morbihan, which means 'small sea' in Breton, where the isle of Gavrinis happens to be. However, radiocarbon dating places the earliest constructions in Barnenez, in the bay of Morlaix, on the northwest coast of Brittany, rather than in Morbihan, in 'around 4700-4500 BC […]',3 whereas 'the mounds in Morbihan were constructed around the period 4500 BC.'4 [Ill. 2] It is necessary to start from the beginning, by examining the Mesolithic era between the Paleolithic and the Neolithic, i.e. in roughly 9000-5000 BC, at Téviec and Hoëdic, situated in the bay of Quiberon in Morbihan. These two islands could not be reached by foot at the time under consideration. Collective burial places have been found in 'Mesolithic shell middens in artificial mound-like structures, which contained also burial cists and hearths associated with them'.5 This was before the beginning of the Neolithic period, which began in Western Europe when people starting from the Levant (Anatolia) around 7000 BC6 reached the Atlantic coast, bringing with them agriculture. Until then, the Mesolithic hunter-gatherers would bury their dead in a flexed position and often ornament their graves with deer antlers in sometimes magnificent style, as in the case of the woman with child of grave J in Hoëdic, possibly as a tribute to her mythical status. 'Téviec and Hoëdic also display elements which might be interpreted as nascent monumentality, as exemplified by the small tumuli erected over graves at Téviec, or the stone stele of grave K at Hoëdic.'7 It must also be observed that in the majority of cases where controlled excavation of a long mound has been undertaken, the long mound has been found to be an addition to or later extension of an earlier funerary structure of rectangular or more often circular plan.8

The idea put forward by scientists is that Morbihan monumentality might have roots in those Mesolithic hunter-gatherer burials. But the long mounds (tertres funéraires) belong in the second stage of monumental structures of the initial Neolithic phase;9 the earliest class of monumental structures in Armorica are the menhirs, which appeared as early as 4600-4500 BC at Lannec er Gadouer10 for instance:

[The long mounds] are found exclusively in the Morbihan, overlooking the Baie de Quiberon and therefore represent a distinctive regional development. It was in much the same area that the menhirs, many of them decorated, were also found.11

Regarding the chronology of menhirs, it is worth mentioning this example taken at Saint-Just in South-Central Brittany: The question now is to know how this initial Neolithic phase in the region of Morbihan developed.13 In Portugal as in Brittany, shell middens were maritime places where the ancestors were interred. In the Morbihan region, the main phase of use covered the thousand years from 5500 to 4500 BC although the sites of Téviec and Hoëdic continued to be used up to 4000 BC.14 Yet the careful examination of lithic material contributes to dismiss the possibility of an extension of Neolithic culture along the northwest Atlantic coast round the Iberian peninsula.17 The chronological argument, however, may not weigh in favour of Brittany: dates of around 4500 BC for passage graves in the valley of the Guardiana suggest a broad contemporaneity with developments in Brittany.18 Among the different theories put forward, the acculturation of indigenous Mesolithic communities through contacts with the south [in Brittany] offers a much more persuasive model than the arrival of pioneer farming communities to explain the transition to the Neolithic. 19 The coast of Vendee is also occupied by populations which bring with them the new neolithic techniques, may-be as early as 5.300 BC. Very soon, some tools are adapted by Mesolithic groups of the Pays-de-la-Loire and Brittany, who still keep their traditional techniques.20

The various traces of megalithic burial rites along the Atlantic coast reflect the different patterns of interaction between indigenous belief systems and new ideas reaching the area as part of the Neolithic package. In Brittany, After a fifth-millennium phase of long mounds and menhirs, the menhirs (or at least some of them) were pulled down and incorporated into an entirely new form of burial monument -the passage grave.21

For instance, at the beginning of the fourth millennium BC, the original earthen mound known as Mane Vihan dating from the midfifth millennium with a stele at one end was partially overlain by a stone-built cairn. Surprisingly, this new burial system with menhirs and long mounds may date back to the Mesolithic period, since a small standing stone has been erected in front of one of the burial sites at Hoëdic. Moreover, it has been shown that the cists found inside those monuments were 'often set within small circular mounds'.22 What can be seen at those sites may reveal a continuity of ritual use.23 In Morbihan, the original system was replaced by another one with stone passage graves. With this new burial system, some menhirs or steles in the open air, possibly related to fertility of the land, had to be placed underground.

B a r nenez

Cairn Barnenez is situated in the bay of Morlaix and it has been found that '[…] chambers F (4720-4040 BC) and G (4940-4330 BC) are several centuries older than the outer sections of their passages. […] These outer passages also incorporate reused slabs in their construction, some or all of which may originally have been free-standing menhirs':24 [Ill. 3] The clearest case, however, is provided by the outer capstone of passage J, which bears on its lower surface a carved motif sometimes interpreted as a head with long flowing hair. Thus the enlargement of Barnenez involved the demolition and reutilization of a series of free-standing menhirs or stone settings.25

In Barnenez, the reuse of free-standing menhirs in passage graves is thought to have happened at the beginning of the fifth millennium. It also happened in Morbihan, but several centuries later. Furthermore, it is known that 'if final Mesolithic sites are absent from the coastal fringe, it was equally found that earlier Neolithic sites were absent from the interior',26 as if Neolithic and Mesolithic populations had lived a separate yet contemporary life for a while. The length of time elapsing between the erection of the freestanding menhirs, their bringing down and eventual incorporation into passage graves is not known. It is very likely that the same process also occurred in Morbihan, probably for the same reasons, since the same stele called sometimes 'déesse-mère' ended up underground. Moreover, some features of an anthropomorphic stele similar to those found on reused slabs in Barnenez have been found in Larmor-Baden and the Moustoir, near Carnac. The idea of a so-called mother goddess in each civilisation, Mesolithic and Neolithic, is not in question.

The way she appears 'as a head with long flowing hair' does not seem to be in question either, but rather the place where she is located, that is above or under ground in a grave. But is the very concept of a goddess relevant in the fifth millennium BC?

[…] We might […] assign a notional date of about 4500 BC as the earliest probable time for the culture reconstructed from the inherited vocabulary of the Indo-European languages. […] a terminal date for the Proto-Indo-Europeans would be set not much later than 2500 BC, possibly somewhat earlier. In the broadest terms then, the Proto-Indo-Europeans were a Late Neolithic or Eneolithic society which began to diverge about 4500 to 2500 BC.27

In other words, the 'sky-god' idea was born among Proto-Indo-European breeders, around the middle of the fourth millennium BC.28 The word for 'god' is attested in the third millennium, in Indo-European, on the Pontic-Caspian steppe, and the Celtic word is derived from it:

'Dagdae from d a g -"good" and d i a "god" : -día m. "god" -dia m. "day", from *diyes, accusative *diyem, as in Latin diēs, diem. The name for day in Celtic comes obviously from the Indo-European name for the day sky and the day god […]. D-65'29 Therefore, how can a concept that did not exist at that time be used today for Mesolithic or Neolithic Western Europe? 'Great mother' for instance would be a more relevant terminology. In summary, hunter-gatherers lived in a world of spirits. Agriculturists, who are food-producing people, probably mark the mythological transition from spirits to divinities. The Celts of the Iron Age had their gods and goddesses, so did the Tuatha Dé Danann with Dagda and Brigit, for instance. As far as agriculturists are concerned, it is not surprising that they should locate the source of life underground -as this is where life that makes vegetation grow comes from and where dead people are buriedand think of earth as a mother. But why should hunter-gatherers locate the origin of life ungerground since they are indebted to animals living above ground for everyday life? And at the time of departure, their souls are supposed to escape to the otherworld sometimes in the company of an animal like a deer, for instance, called psychopomp by specialists, as can be seen in Téviec and Hoëdic. And even though the soul of their seer(esse)s is supposed to go in the upper, or even in the lower world, they don't go underground. Many other Mesolithic people of Western Europe are known to have used deer artefacts reminiscent of those found in Téviec and Hoëdic. In Star Carr (around 8700 and 8460 B.C.) on the coast of North Yorkshire, findings:

[They] include a large number of antler spear heads barbed along a single edge […] as well as 'mattock heads' manufactured from elk antler, […] and a series of red deer antler 'frontlets' which may have served either as hunting disguises or as head-dresses in some ceremonial activity. 1The Evenki from Siberia can help us understand Star Carr deer artefacts thanks to their hunting rites.

These people, formerly known as Tungus, are living in the Taiga from the river Ob'Irtysh in the west, [to] the Okhotsk seacoast and Sakhalin in the 29. LAMBERT, Pierre-Yves, ed., 1966, Lexique étymologique de l'irlandais ancien de J. Vendryes, Lettre D, CNRS ed. Translated and abridged by the author.

G. BOUCHERIT -Irish and Breton m e g alithis m… 1 9 east, [and] the river Upper Tunguska (Angara), [the] Lake Baikal and the river Amur in the south 2 .

Ethnically they belong to the Baikal or Paleo-Sibirian group, close enough to the Mongols. In the north they speak Tunguso-Manchourian languages. They are hunters and breed reindeer in the enormous territory from the Yenisei to the Okhotsk Sea. They mainly hunt equidae, for meat and fur. They also fish and raise horses and cattle in the south.

'The Evenk religion is of historical interest since it retains some extremely early archaic forms of belief. […] Among the most ancient ideas are spiritualization of all natural phenomena, personification of them, belief in an upper and lower world, belief in the soul (omi) and certain totemistic concepts. Later on these rituals were conducted by the shamans.'30

The very word 'shaman' originated among these people. These deer artefacts were used for very important ritual ceremonies: 'And in the course of their shingkelevun ritual (arranged by the Evenkis before a hunt in order to ensure success)':31 Evenki hunters, […], wore ritual costumes, with caps made from the skulls of reindeer or elks, imitating the heads of these animals.32 The hunters performed a pantomime dance, portraying the characteristic movements of the animals, and accompanied it with an improvised song addressed to the beasts.33

For example: At this point one should also make reference to the tribes situated on the other side of the Bering Straight, since this Evenki pantomime dance bears similarities with the 'dancing' of the Blackfoot decoy runner36 '[to whom] spiritual forces are believed to be transmitted during the bison drive ritual from the medicine woman […], and in turn to the bison.'37 The Blackfoot's name in the vernacular is Niitsítapi meaning 'the original people'.38 They live in the Great Plains as far north as Canada, they hunt bison and have the reputation of being good warriors. Both these American and Siberian tribes obtain their food through the mediation of a woman representing the protective spirit of the clan. Perhaps this grandmother is not very different from her possible heiress with a child in grave J at Hoëdic, in a flexed position, ornamented with deer antlers, who has already mentioned. Therefore 'the head with flowing hair' in Barnenez on the menhir formerly raised to the sky could well have been sculpted by hunting-gathering tribes of the end of the Mesolithic era, in so far as it is a female head. But the fact that this head is anthropomorphic is in contradistinction with the Mesolithic populations, who use zoomorphic symbolism in their graves. It may be evidence of the influence of agriculturists with their anthropomorphic sculptures. The Birth of the Gods and the Origins of Agriculture,39 the title of a book authored by Jacques Cauvin, makes it possible to grasp the nature of the evolution from hunting to digging in the course of which spirits made way for divinities. The difficulty with those feminine skulls from the 10th millennium in the Levant is that they are hairless. Thus one can logically opt for syncretism in Barnenez, as a provisional solution. The same can be said for cosmology when one examines the Evenk myth of the Cosmic Hunt:

G. BOUCHERIT -Irish and Breton m e g alithis m… 2 1

In this connection, of particular interest are the Evenk concepts of the cosmic elk Kheglen (Kheglun), which are identified with the constellation of the Great Bear. Conformant with these concepts, the visible sky is nothing else but the taiga of the upper world. Within it lives the cosmic elk Kheglen. During the day the elk goes into the thickets of the heavenly taiga, and therefore is not visible from the land of people. At night he comes out onto the mountain peaks, and being the most powerful of the dwellers (stars) in heaven, may be seen by people from the earth. Judging by the fact that the constellation Little bear is considered in these concepts as Kheglen's calf, we may conclude that the cosmic image of the elk is understood at the same time as that of a 'mother-elk '.40 But the agriculturists also depend on the sky, that is the weather, for their food:

The belief system manifest in the construction of the passage grave developed in the Atlantic zone, in Iberia and Brittany, in the first half of the fifth millennium. It embodied a complex cosmology that required some of the tombs to be set out so as to 'capture' the setting or rising sun on the midwinter or midsummer solstice. As part of this cosmology, linear and circular settings of standing stones were erected, the former most spectacularly in the Carnac region, and an intricate art developed involving the 'carving' and painting of the surfaces of the standing stones.41 Is it possible that a whole mythology might give sense to these megalithic representations on earth, involving for instance 'ar Wrac'h, Kornandon(ez)' and so on? The etymology 'Korr an andon', i.e. 'dwarf of the source' would fit our purpose well indeed. Barry Cunliffe thinks that:

Contact by sea [on the Atlantic façade] must have been maintained over a period of time and in all probability these disparate regions remained in contact on a continuous basis. [

I rel a nd

Let us now move on to the first place the hunters-gatherersagriculturists moved in with their passage graves, that is Ireland.

'There may well be a north-west French (specifically Breton) involvement in the appearance of the passage tomb tradition in Ireland and along Britain's Atlantic façade'.43 Here is the way Irish megalithism probably began around 4000 BC: 'at this time there seem to have been long-distance contacts between centres of power, linking Orkney with the Boyne Valley, and the Boyne valley with Brittany and Spain. Some of these movements appear as diaspora-like spreads up the Atlantic façade.'44 'It may have been in this way that the foragers of southwestern Ireland, living on the remote tip of the Dingle peninsula [Ill. 4], were able to acquire cattle and a polished stone axe around 4100 BC.'45 'In Ireland, […] a series of such monuments, boulder-built, apparently without cairns, and with kerbs less than 25 m. in diameter, are to be found on and near the coast around the north and west, from It is now necessary to have a look at the reasons why around 4000 BC there appeared 'the concept of collective burial in megalithic tombs and the associated belief systems'. First of all, it can be said that: 'These tombs do indeed lie at the beginning of the Irish passage tomb 43. SHERIDAN, Alison, 'French Connections I: spreading the marmites thinly', in: Neolithic Settlement in Ireland and Western Britain, Op. cit., p. 4. 44. Ibid., p. 5. 45. CUNLIFFE, Barry, Op. cit., p. 137. 46. SHERIDAN, Alison, Op. cit., p. 12. 47. Id. Note that Michael Herity's explanation (Irish passage graves. Dublin, 1974) is now obsolescent. He made passage grave builders (Neolithic B) arrive at the Boyne mouth around 2500 BC and Neolithic farmers (Neolithic A) at a conventional 3000 BC.

G. BOUCHERIT -Irish and Breton m e g alithis m… 2 3 sequence'.48 And as has been seen, some archaeologists 'briefly argue that the change which took place, between 4300 and 3900 BC, […] marks a fundamental shift in ideology, […]':49 the 'passage tombs, with their womb-like design and their new imagery emphasizing female referents, are seen as an expression of this new ideology. Its ascendency over the older, male-oriented order is reflected in the deliberate destruction of some of the stelae and their physical incorporation within some passage tombs.'50 This explanation leaves no room for syncretism. It could well be that both civilisations, Mesolithic and Neolithic, were female oriented but obviously not in the same manner. It should not be forgotten that in Ireland, elk and roe deer as well as red deer were absent or extremely rare during the first half of the Irish Holocene (around 11,000 years ago). Red deer were reintroduced in Ireland at the beginning of the Irish Neolithic, and also into the Orkneys as on the tiny Holm of Papa Westray North for instance, during the final stages of the Orcadian Neolithic. 'The significance of their introduction may be as much to do with ritual as economic reasons.'51 It is known that 'when neolithic man first came to Orkney, around 4000-3500 BC, it was a virgin territory'.52 But is it possible to imagine agriculturists with a deer totem? Or can it be that former hunter-gatherers worshipping ancestors turned agriculturists, for instance, or lived side by side with them? Or else, a kind of mixed population? This does not mean that sporadic clashes are excluded, of course. This female-oriented way of seeing things seems better adapted to the future situation with its different newcomers seeking for roots, as can be shown.

A n a

Let's jump a few centuries forward now. The Celts of the first or second Iron Age, otherwise called Warrior Celts, have conquered the island and the peninsula. They have settled where the former inhabitants lived, used their monuments, as narrated in sagas like the Battle of Maige Tuired (plain of the pillars), and eventually superseded the old mythology with their own. 'Ar Wrac'h' for instance in Brittany, is associated to monumentality. 'Graniol', otherwise known as Grah Niol, the name of a massive stone shaped and decorated in a passage grave in Arzon on the Rhuys peninsula in Morbihan, is interpreted as 'Gwrac'h an Heol', which makes one think of the 'Cailleach' in Ireland, even perhaps of the Old Woman of Beare.

The last sheaf of grain was often called the hag or Cailleach of the harvest, and was passed to those who had not yet finished their reaping.53 This is an agricultural context. But this female spirit puts on an impressive range of different representations depending on the cultural layer wherein she happens to be, starting from the Mesolithic to the Irish Celts of the Middle Ages: Cailleach Bhéarach, […] is primarily a nature and wilderness goddess here, inimical guardian of the fertility of the vegetable world (as in so many other texts of the animal kingdom too.) She stands in the relationship to the hero of the Divine Hag of War and Death.54 Symbolically, she covers three realms: wilderness, vegetation and war, corresponding possibly to the Mesolithic, Neolithic and Celtic eras or civilisation. It is noticeable that people in both countries seem to think of a female origin of things, at least before christianisation. However, one should not overlook the fact that: 'The Proto-Indo-Europeans, […], were a martial, semi-nomadic folk, [ Hittite therefore gives us the best way to explain Anna (or Ana57)'s Celtic origin till now.58 Killarney, Co. Kerry is in Munster where lie 'Dá Chích Anann', the two fabulous hills: 'There is a stream running between the hills. One half flows north into a small lake called Lough Nageeha and the other half flows south into the Clydagh River.'59 Most of the time the presence of water is overlooked. However, from a farmer arriving in a new country, the presence of water is essential. All the animals, sheep, cattle and goats that the Neolithic people brought with them produce milk. As a result, water can be associated with the land and with the milk the farmers get from their cattle, living from the land as if it were a mother. In Armorica, the bond between water, milk and the land is clearly evidenced around Carhaix, for instance in the Menez Are, or Aré Mountains:

There was neither a mother nor a nurse throughout the country who didn't think they had to go [to the Frout chapel] once a year to invoke or to thank the 'Milk Virgin' known as the 'Sources Mother'. Sources spring from everywhere in the small stony and sunken way which goes down to the oratory. Their water would come in mains to fill a round basin dug in the holy enclosure. […] It was the nurses' duty to clear and clean the water basin and then to throw needles in it from their blouse as a gift, and recite this short prayer: Lady of the Frout, mother and Virgin Good health to the child, and to us milk!60

Commana, in the Aré Mountains, is a good example of this possible complex. The Neolithic grave 'Ti lia ven'61 in Mougev Vihan dates back from the 3rd millennium BC.62 It was built close to the source of a watercourse -the Mougau -which flows into the Elorn and to the sea. The northern pillar of the passage grave63 has the (double) breasts of a Mother-Goddess or 'stele idol' as it is called by archaeologists. The name Commana is thought to originate from Komm, which means 'valley', which suits well 'the high place of the village between two nascent hills',64 and An(n)a.65 Finally, Anatolia is the original place of the agriculturists mentioned earlier. The worship of a stele idol buried underground which can be observed in Western Europe is certainly very different from what it was in the 7th millennium when these people left for the Balkan peninsula. The Hittite Anna -Ana in contemporary Turkish -, meaning mother, should not be put in relation with this worship in so far as there are no blood ties between them. This Indo-European name came to Anatolia with the Hittite people, long after the agriculturists had left. The Celts, as for them, probably brought this name with them and spread it in Europe where it might have replaced former names which nobody has any idea of, yet hopefully in continuity with the symbolism of their former mythology. It must be emphasized that cauldrons, which are an important element in the Celtic mythological way of life, can be found in Ireland as early as the Late Bronze Age, i.e. between 1200-1000 BC. Indeed the Galatians went to Anatolia, but this was only in the 3rd century BC. Finally, it might be interesting to mention a series of deeds that suit well the female spirit, the name of whom differs in each one of the numerous cultures she has been through: she dropped cairns onto hills in Meath66 out of her apron, was responsible for moving islands in west Kerry, built mountains from rocks carried in her creel in Scotland, and was queen of the Limerick fairies.67 Wouldn't she be at home in Brittany as well with a little imagination, among the alignments in Carnac for instance? This other example shows that she has been acculturated several times:

In that 'old time', the time before history and at the dawn of human existence, when culture and memory first come into play, a hag, we are told, a cailleach -with the implicit connotations of the divine, ancestral otherworld female elder -lived in County Kerry; in a valley called Gleann na mBiorach. Gleann na mBiorach means the Valley of the Horned Herds, herds of animals such as cattle and deer with whom the figure of the cailleach/hag is associated in Gaelic tradition, in both Ireland and Scotland.68

One can wonder the extent to which the name Cornely, quite widespread in the Carnac region, and related to horned animals such as deer or oxen, might have something to do somehow with the Gleann na mBiorach?

Conclusion

It is a well-known fact that the Neolithic package did not reach Britanny from Portugal along the Atlantic coast, and that agriculture arrived through contact with pioneers south of the Loire rather than by colonists. In addition to that, this paper has argued that the origin of menhirs is grounded in a small standing stone in a Mesolithic tomb at Hoëdic.69 However, nothing has been said about the origin of the agriculturists on the coast of the bay of Morlaix. Why not suggest that they came by sea, since they stayed on the coastal fringe in the beginning?

The study can be summarized as follows: free-standing menhirs in Barnenez, some of them decorated, were incorporated into passage chambers inside an elongated cairn, under what used to be a tumulus. Similar anthropomorphic features have been found in Larmor-Baden in the Morbihan area, where free-standing steles were found in the openair. The idea of free-standing stones may have originated among the 67. MURPHY, Gerard, 1953, 'The lament of the old woman of Beare', in: Royal Irish Academy, Vol. 55, 84. 68. Ó CRUALAOICH, Gearóid, 'Non-sovereignty Queen aspects of the Otherworld Female', in: Irish Hag Legends: the case of the Cailleach Bhéarra, in Béaloideas, Iml. 62-3, 1994-5. 69. CUNLIFFE, Barry, Op. cit., p. 133. Mesolithic population, although the archaeological evidence, compared with Téviec and Hoëdic, is lacking in Barnenez, for the time being. The fact that the 4500-4000 shift to passage graves in Morbihan came from the bay of Morlaix is now widely adhered to but the reason why it occurred is still up for debate. If tribal hunter-gatherers were as femaleoriented as the agriculturists with their culture coming ultimately from the Levant, syncretism is a possible option, especially when considering the introduction of deer in Ireland and in the Orkneys for 'ritual as [well as] economic reasons'. Later on, the Celts of the first or second Iron Age adapted their myths to this Megalithic civilisation that was already extinct at the very end of the Neolithic era.70 Ana is a good example of syncretism between Hunter-gatherers/agriculturists and Celtic warriors,71 since in Ireland her name is applied at least to one Neolithic site which symbolizes a kind of Celtic Sovereignty. Her name can also be found in Brittany in Keranna or Commana for example, as if this Celtic female name were applied to a female spirit at the root of everything that made her way successfully throughout a non exclusively male-oriented Celtic civilisation.

  Co. Down to Co. Mayo' [Ill. 5].46 This is true also for the megalithic tomb at Achnacreebeag [Ill. 6] situated on the west coast of Scotland: The Achnacreebeag monument may well have been constructed by an immigrant community (or descendants thereof) from [southern Brittany around 4000 BC]. […] The tomb type stands at the head of long and complex developmental sequences within Scotland; […] similarly simple monuments […] can be recognized elsewhere along the Atlantic coast of Britain and around the coast in the northern half of Ireland. These monuments constitute the beginning of the passage tomb tradition in these areas.47

2 4

 4 Breta gn e/Irland e : Quelles relati ons ?

G

  . BOUCHERIT -Irish and Breton m e g alithis m… 2 5

2 8

 8 Breta gn e/Irland e : Quelles relati ons ?

  70. Around 2000 BC with the arrival of the first Indo-European people called 'Bell Beakers'. 71. The Irish Celts may have come as early as the Late Bronze Age. See KOCH, John Thomas, 2007, An atlas for Celtic studies. Oxford, 136; CUNLIFFE, Barry, 2001, Facing the ocean. Oxford, 284. [Ill. 1] Chris SCARRE. Contexts of monumentalism, 2002. [Ill. 2] Chris SCARRE, Megalithic origins in Atlantic Europe, 1992. [Ill. 3] Chris SCARRE, Megalithic origins in Atlantic Europe, 1992. [Ill. 4] Peter WOODMAN and Margaret McCARTHY. Contemplating some awful(ly interesting) vistas : importing cattle and red deer into prehistoric Ireland, p. 35. 2004. [Ill. 5] Alison SHERIDAN. 2. French Connections I : spreading the marmites thinly, p.13. 2004. [Ill. 6] Alison SHERIDAN. 2. French Connections I : spreading the marmites thinly. 2004.

  […] [concerning] menhirs and tertres, […], the socket for the fallen menhir at La Croix-Saint-Pierre contained charcoal, yielding a date of 6070 ± 80 bp (5270-4740 BC) (Briard et al. 1995). The charcoal may pre-date the erection of the menhir, but the hearths associated with quartz menhirs 17, 18 and 19 in Le Moulin alignments at Saint-Just were also dated to the period 4940-4100 BC (Le Roux et al. 1989).12

  There [in Portugal] the earliest Neolithic settlements (c. 5750-5250 BC) are found in the inland limestone massifs of the Estremadura, away from the principal Mesolithic population areas focused around the Tagus and Sado shell middens. It was only during a second stage (from 5250 BC) that

	8. SCARRE, Chris, Ibid., p. 49.
	9. CUNLIFFE, Barry, Op. cit., p. 133.
	10. SCARRE, Chris, Op. cit., p. 49.
	11. CUNLIFFE, Barry, Op. cit., p. 162-163.
	12. SCARRE, Chris, Op. cit., p. 49.
	13. LE ROUX, Charles-Tanguy, 2001, Carnac, Locmariaquer et Gavrinis. Rennes: Ouest-France,
	17.
	14. CUNLIFFE, Barry, Op. cit., p. 130 and 160-161.
	G. BOUCHERIT -Irish and Breton m e g alithis m… 1 5
	Neolithic communities expanded beyond the Estremadura (and comparable
	early settlement nuclei) and absorbed or replaced the indigenous Mesolithic groups.15
	By comparison with Brittany:
	à la pointe de Kerpenhir, au sud de la presqu'île, l'étude d'une 'tourbière littorale' aujourd'hui submergée a révélé que, dès la fin du VIe millénaire
	av. J.-C., la région avait été vigoureusement défrichée et mise en culture ; c'est à ce jour le plus ancien témoignage de néolithisation reconnu en
	Bretagne.16

  26. SCARRE, Chris, Op. cit, p. 36. 27. MALL ORY, James Patrick, 1989, In search of the Indo-Europeans: Language, Archaeolog y, and Myth. London: Thames & Hudson, 127. 28. DEXTER, Miriam Robbin, 1990, Whence the goddesses. New York, Oxford, Beijing, 51.

	1 8 Breta gn e/Irland e : Quelles relati ons ?

  They appeal to […] the guardian and defender of the clan, saying : 'Grandmother bugady, intercede, help!' They entreat her, as the mistress of the taiga and animals : 'Grandmother bugady, give us animals, send us a catch!34' […] She appears as a zoomorphic being, merging completely with the image of an elk cow or a doe of a wild deer.35 30. VASILE VICH, G.M., and A.V. SMOLY AK, 1956, 'The Evenks', in LEVIN, M. G. and L. P. POTAPOV, eds., 1964, Narody Sibiri (The Peoples of Siberia), Chicago and London: University of Chicago Press, 620-647. 31. ANISIMOV, Arkadiy Fedorovich, 'Cosmological concepts of the peoples of the North', in: Micha el Henry N, ed, 1963, Studies in Siberian shamanism. Toronto, 176. 32. OKLADNIKOV, Alexey Pavlovich, 1970, Yakutia, Arctic Institute of North America, 97. 33. Id. 34. ANISIMOV, Arkadiy Fedorovich, op. cit., p. 177. 35. Id. 2 0 Breta gn e/Irland e : Quelles relati ons ?

  36. 'A young man disguised with buffalo horns and robe to decoy the herd' and lure it where it should go. Buffalo hunting on the Northern Plains' [http://nativeamericannetroots.net/ diary/889], p. 1 out of 4, 15/6/2014. 37. KEHOE, Thomas F., 'Subsistence and beyond in Upper Palaeolithic bison economy', in BRUGAL, Jean-Philip, DAVID, Francine and James G. ENLOE [et al.], 1999, 'Le bison: gibier et moyen de subsistance des Hommes du paléolithique aux Paléoindiens des Grandes Plaines', Actes du colloque international, Toulouse 6-10 juin 1995, Antibes: éd. ADPCA, 257. 38. https://fr.wikipedia.org/wiki/Confédération_des_Pieds-Noirs. -(page consultée en avril 2014) 39. CAUVIN, Jacques, Naissance des divinités, naissance de l'agriculture. Translated by Trevor Watkins, Cambridge, 2000.

  …]. From the beginning of the fourth millennium BC, the concept of collective burial in megalithic tombs and the associated belief systems spread eastwards, deeper into Europe. […]. By the mid-third millennium the Atlantic zone of Europe had become linked by corridors of communication by sea and by way of the great rivers. […] The various cultures of western Europe were now beginning to share many things.42 40. MICHAEL , Henry N., Op. cit. p. 161-162. 41. CUNLIFFE, Barry, Op. cit., p. 166. 42. Ibid., p. 166-167.

2 2 Breta gn e/Irland e : Quelles relati ons ?

  48. Id. 49. SHERIDAN, Alison, Op. cit., p. 14. 50. WOODMAN, Peter, and Margaret McCARTHY, 2005, 'Contemplating some awful(ly interesting) vistas: importing cattle and red deer into prehistoric Ireland', in: Armit , Ian, Murph y, Eileen, Nelis , Eiméar and Derek Simpson , eds, 2003, Neolithic Settlement in Ireland and Western Britain. Oxford, Belfast, 36. 51. Ibid., p. 37. 52. HEDGES, John W., 1984, Tomb of the eagles. London, 155, 201-203, 205.

  …]. Most of the goddesses worshipped by historical-era Indo-European cultures […] owe their origins to Neolithic Europe and to the near east […].'55 One example should be enough to show it: In Hittite […], which is the oldest written Indo-European language, the goddess HannaHanna meaning ancestress became 'anna' for 'mother',56 The 53. Mac LEOD, Sharon Paice, 2012, Celtic myth and religion. London, 172. 54. Ó CRUALAOICH, Gearóid, 1988, 'Continuity and adaptation in legends of Cailleach Bhéarra', in: Béaloideas, Vol. 56, 176. 55. DEXTER, Miriam Robbin, Op. cit., p. 51. 56. BENVENISTE, Émile, 1973, Le vocabulaire des institutions indo-européennes. Paris: Les éditions de Minuit, 6.

  57. MURPHY, Gerard, [1941], 1953, 'The book of the lays of Fionn', in: Duanaire Finn, part III. Dublin: Irish Texts Society, Vol. XLIII, 210. 58. BOUCHERIT, Gilles, 2013, Ana/*Danu, Between Siberia and Anatolia?, Trier, à paraître. 59. http://en.wikipedia.org/wiki/Paps_of_Anu, (page consultée en avril 2014) 60. 'Itrôn ar Frout, mamm ha gwerc'hès, / Yéched d'ar buguel ha d'emp les!' AR BRAZ, Anatole, 1994, 'Les saints Bretons d'après la tradition populaire en Cornouaille (1893-1894)', in: Magies de la Bretagne, Paris, 934. 61. LE GOFFIC, Michel, 1984, Commana et les monts d'Arrée. Rennes: Ouest-France, 7; HEMON, Roparz, liac'h f. pierre plate, mégalithe. (p. 1773), liac'hven f. pierre plate, mégalithe. (p. 1774) in Geriadur istorel ar brezhoneg. Levrenn III, 1995-1998. 62. L'HELGOUARC'H, Jean, http://fr.wikipedia.org/wiki/Commana (page consultée en mars 2013).

	2 6 Breta gn e/Irland e : Quelles relati ons ?

2 Breta gn e/Irland e : Quelles relati ons ?

RENFREW, Colin, 1974, British prehistory : a new outline. London : Duckworth, 85


Landslide Hazard Assessment and Mapping in the Guil Catchment (Queyras, Southern French Alps) From Landslide Inventory to Suscep�bility Modelling

Context.

Landslides are common natural hazards in the Southern French Alps, where they may affect human lives and cause severe damages to infrastructures. As a part of the SAMCO research project dedicated to risk evaluation in mountain areas, this study focuses on the Guil river catchment (317 km²), Queyras, to assess landslide hazard poorly studied until now. In that area, landslides are mainly occasional, low amplitude phenomena, with limited direct impacts when compared to other hazards such as floods or snow avalanches. However, when interacting with floods during extreme rainfall events, landslides may have indirect consequences of greater importance because of strong hillslope-channel connectivity along the Guil River and its tributaries (i.e. positive feedbacks). This specific morphodynamic functioning reinforces the need to have a better understanding of landslide hazards and their spatial distribution at the catchment scale to prevent local population from disasters with multi-hazard origin.

Objective. The aim of this study is to produce a landslide susceptibility mapping at 1:50 000 scale as a first step towards global estimation of landslide hazard and risk.

Introduction

Study area

Geographical setting

An Alpine catchment of 317 km², between 868 and 3357 m a.s.l.

Scattered urbanization with several villages established in the main valleys

Landslide inventory

A detailed inventory of 69 landslides

Extraction from national databases (BDMvt, RTM) Photointerpretation of aerial images Extensive �eld work Louise Roulleau 1,3 , François Bétard 1,3,* , Benoît Carlier 1,3 , Candide Lissak 2,3 , Monique Fort 1,3 , Mathieu Fressard 2 * Presenting and corresponding author: francois.betard@univ-paris-diderot.fr

Predisposing factors: spatial database construction

Slope gradient

Lithology

Sur�cial formations Land use

Slope curvature Aspect

Susceptibility modelling: logistic regression results and interpretations

Logistic regression model

A

  logistic regression model to analyze the spatial and mathematical relationships between the response variable (i.e. absence/presence of landslides) and the set of predictive variables (i.e. predisposing factors) Model evaluation by a calibration/validation strategy based on ROC curve and AUC analyses (procedure details in Roulleau, 2015) Active landslide scar above the village of Aiguilles, Queyras. March 2014. Fort M., Arnaud-Fassetta G., Bétard F., Cossart E., Geai M.L., Madelin M., Bouccara F., Carlier B., Sourdot G., Tassel A., Bletterie X., Charnay B. (2015) -Sediment dynamics and channel adjustments following torrential floods in an upper alpine valley (Guil river, Southern French Alps). In: Lollino G., Arattano M., Rinaldi M., Giustolisi O., Marechal J.-C., Grant G.E. (eds.): Engineering Geology for Society and Territory -Volume 3, IAEG XII Congress Volumes, Springer, 313-317.

Distance to streams

Data constructed from various sources (DTM, geological maps, satellite images...) and stored in a spatial database (ArcGIS©) Selection as input variables to be included in the model after statistical analysis based on χ2 test and Cramer's V coef.

A strong hillslope-channel connectivity

A final susceptibility map in 4 classes using a discretization method based on Jenks natural breaks Results show the major influence of the distance-tostreams variable on the model => con�rms the strong hillslope-channel coupling observed empirically during extreme rainfall events (Fort et al., 2015).