V Devlaminck
email: vincent.devlaminck@univ-lille.fr

J M Charbois

State-generating Matrix for differential Mueller matrices of depolarizing optical media

INTRODUCTION

It is now well known [START_REF] Jones | A new calculus for the treatment of optical systems. VII properties of the N-matrices[END_REF][START_REF] Go | Optical Activity of Anisotropic Solutions[END_REF][START_REF] Azzam | Propagation of partially polarized light through anisotropic media with or without depolarization: A differential 4x4 matrix calculus[END_REF] that a medium whose optical properties are functions of the thickness z of the traversed medium, are governed by a differential equation:

𝑑𝑴(𝑧) 𝑑𝑧 = 𝒎(𝑧). 𝑴(𝑧) (1)
relating the Mueller matrix M(z) at a distance z into the medium to its spatial derivative along the light propagation. For a non-depolarizing medium, this equation is completely defined by the knowledge of the initial condition and the differential Mueller matrix m(z) since this matrix is a deterministic one. This differential matrix (noted mND(z) in this case) can be written using 6 parameters [START_REF] Jensen | Modulation techniques in polarization spectroscopy[END_REF][START_REF] Arteaga | Characterization of homogenous depolarizing media based on Mueller matrix differential decomposition[END_REF]. According to the notation convention used, the parameters occupy different positions in the matrix. It is worth noticing that the matrix previously described in [START_REF] Jensen | Modulation techniques in polarization spectroscopy[END_REF] used the standard quantum notation rather than the usual optical notation adopted in [START_REF] Arteaga | Characterization of homogenous depolarizing media based on Mueller matrix differential decomposition[END_REF] and in this paper. With this convention:

𝒎 𝑁𝐷 (𝑧) = [ 0 -𝐿𝐷 -𝐿𝐷 0 -𝐿𝐷′ 𝐶𝐷 𝐶𝐵 𝐿𝐵′ -𝐿𝐷′ -𝐶𝐵 𝐶𝐷 -𝐿𝐵′ 0 -𝐿𝐵 𝐿𝐵 0 ] (2)
where linear dichroism (LD) and birefringence (LB), 45°-linear dichroism (LD') and birefringence (LB'), and circular dichroism (CD) and birefringence (CB) are the six phenomenological elementary polarization properties fully characterizing the medium. Generally these parameters are functions of the thickness z of the traversed medium. The most elementary situation is given for homogeneous medium where these parameters are proportional to z. The relations between these parameters (but also between the different notations used by the first authors on this subject [START_REF] Jones | A new calculus for the treatment of optical systems. VII properties of the N-matrices[END_REF][START_REF] Go | Optical Activity of Anisotropic Solutions[END_REF][START_REF] Hofrichter | Jerusalem Symposia on Quantum Chemistry and Biochemistry V[END_REF]) and the corresponding physical effects are given in [START_REF] Schellman | Optical spectroscopie of oriented molecules[END_REF] for instance.

The underlying structure of this differential matrix is related to the 6 generators of the Lie algebra of SO [START_REF] Azzam | Propagation of partially polarized light through anisotropic media with or without depolarization: A differential 4x4 matrix calculus[END_REF][START_REF] Jones | A new calculus for the treatment of optical systems. VII properties of the N-matrices[END_REF] (the proper orthochronous Lorentz group) that can be dissociated in terms of more familiar quantities: 3 rotation generators J1, J2, J3 and 3 boost generators K1, K2 and K3 (these generators and their relations are described in Appendix A of [START_REF] Devlaminck | Physical model of differential Mueller matrix for depolarizing uniform media[END_REF]):

𝒎 𝑁𝐷 (𝑧) = -𝐿𝐷(𝑧). 𝑲 1 -𝐿𝐷 ′ (𝑧). 𝑲 2 + 𝐶𝐷(𝑧). 𝑲 3 + 𝐿𝐵(𝑧). 𝑱 1 + 𝐿𝐵 ′ (𝑧). 𝑱 2 -𝐶𝐵(𝑧). 𝑱 3

The Lie algebra of the Lorentz group can be reduced to the direct product of two sub-algebras (the algebra of SU( 2), the group of two-dimensional unitary matrices with unit determinant) by transforming its generators {𝑱 1 , 𝑱 2 , 𝑱 3 , 𝑲 1 , 𝑲 2 , 𝑲 3 } to a new basis [START_REF] Tung | Group Theory in Physics[END_REF]:

{ 𝑴 𝑚 = 𝑲 𝑚 +𝑖𝑱 𝑚 √2 𝑵 𝑚 = 𝑲 𝑚 -𝑖𝑱 𝑚 √2 𝑤𝑖𝑡ℎ 𝑚 = 1,2,3 (4)
The factor √2 is used in order to define some unitary transformations that will be necessary above. A straightforward calculation shows that: [𝑴 𝑚 , 𝑵 𝑛 ] = 0 where [A,B] = AB-BA is the commutator operator. This property of commutativity is not verified for J and K generators. It is also obvious that 𝑵 𝑚 = 𝑴 𝑚 * . It is useful to define a general retardation for each of the birefringence-dichroism pairs (usually referred to as spectroscopic notations); thus:

{ 𝐿 = -(𝐿𝐷 + 𝑖𝐿𝐵) 𝐿 ′ = -(𝐿𝐷 ′ + 𝑖𝐿𝐵 ′ ) 𝐶 = 𝐶𝐷 + 𝑖𝐶𝐵 (5)
With these notations, Eq. ( 3) becomes:

𝒎 𝑁𝐷 (𝑧) = 𝐿 √2 𝑴 1 + 𝐿′ √2 𝑴 2 + 𝐶 √2 𝑴 3 + 𝐿 * √2 𝑵 1 + 𝐿′ * √2 𝑵 2 + 𝐶 * √2 𝑵 3 (6)
Where * stands for the complex conjugate.

Eq. ( 6) is exactly the Eqs. ( 62) and (63) in the paper of Sheppard [START_REF] Sheppard | Coherency and differential Mueller matrices for polarizing media[END_REF] where the expression of m(z) is derived from the Dirac matrices (termed ij in [START_REF] Sheppard | Coherency and differential Mueller matrices for polarizing media[END_REF]) and can be related -at least in the case of homogeneous medium -to the Z matrix recently introduced by Kuntman [START_REF] Kuntman | Vector and matrix states for Mueller matrices of nondepolarizing optical media[END_REF]. The factor 1/√2 in Eq. ( 6) is necessary taking into account the factor 1/√2 in the definition of the matrices Mm and Nm given Eq. ( 4). The product of these two factors gives the present factor 1/2 in the ij matrices (Fig. 1 in Sheppard [START_REF] Sheppard | Coherency and differential Mueller matrices for polarizing media[END_REF]).

DEPOLARIZING MEDIUM

Depolarizing differential Mueller matrix

For depolarizing medium the differential Mueller matrix must be considered [START_REF] Devlaminck | Physical model of differential Mueller matrix for depolarizing uniform media[END_REF][START_REF] Devlaminck | Depolarizing differential Mueller matrix of homogeneous media under Gaussian fluctuation hypothesis[END_REF][START_REF] Ossikovski | Statistical meaning of the differential Mueller matrix of depolarizing homogeneous media[END_REF] as a matrix of random processes generated by the spatial fluctuations of the birefringence and dichroism components.

Eq. ( 1) is then a stochastic differential equation. This equation is now ccompletely defined by the knowledge of the initial condition (which can be random in the most general case) and the statistical laws associated to the differential matrix m(z) From a statistical point of view, Eq. ( 6) shows that we have to deal with three complex random processes ( L , L ', C ) and it is well known [START_REF] Amblard | Statistics for complex variables and signals[END_REF] that with such processes both the complex process and its conjugate must be considered in order to extract all the statistical information. It is obvious when we notice that 𝐿𝐷 = -(𝐿 + 𝐿 * )/2 and 𝐿𝐵 = -(𝐿 -𝐿 * )/2𝑖 for instance. M(z) is then seen as an average value of non-depolarizing Mueller matrices MND(z) associated to their differential matrices mND(z) but random [START_REF] Devlaminck | Depolarizing differential Mueller matrix of homogeneous media under Gaussian fluctuation hypothesis[END_REF][START_REF] Charbois | Stochastic model for the differential Mueller matrix of stationary and nonstationary turbid media[END_REF]:

𝑴(𝑧) = 〈𝑍 ⃖ 𝑒𝑥𝑝 (∫ 𝒎 𝑁𝐷 (𝑠)𝑑𝑠 𝑧 0 )〉 = 𝑍 ⃖ 𝑒𝑥𝑝 (∑ ∫ 𝒄 𝑛 (𝑠)𝑑𝑠 𝑧 0 +∞ 𝑛=1 ) (7)
The mean value is denoted by angular brackets, c (n) stands for the cumulant of n order and 𝑍 ⃖ is a space-ordering operator introduced by Fox [START_REF] Fox | Gaussian stochastic processes in physics[END_REF] that causes the factors in a product to appear in the same sequence as in the medium of interest. This space-ordering operator is made necessary since the matrices mND(z1) and mND(z2) do not commute for z1 z2 in general.

Complex random processes

Since we are dealing with three complex random processes (L , L ', C), we introduce the complex 6x6 matrix [START_REF] Van Den Bos | The multivariate complex normal distribution-A generalization[END_REF][START_REF] Schreier | Second-order analysis of improper complex random vectors and processes[END_REF] in order to build complex vectors:

𝑻 6 = 1 √2 [ 𝑰 3 -𝑖𝑰 3 𝑰 3 +𝑖𝑰 3 ] ( 8
)
where I3 is the identity matrix of dimension 3. It obvious that T6 is Hermitian (meaning 𝑻 6 𝑻 6 𝐻 = 𝑰 6 = 𝑻 6 𝐻 𝑻 6 where H stands for the Hermitian conjugate).

From the real and imaginary parts of ( L , L ', C ), we define a vector of dimension 6x1:

𝒓 = [ 𝑫 𝑇 𝐵 𝑇 ] 𝑇 = [ 𝑫 𝑩 ] (9)
With:

𝑫 = [ -𝐿𝐷 -𝐿𝐷′ 𝐶𝐷 ] 𝑩 = [ -𝐿𝐵 -𝐿𝐵′ 𝐶𝐵 ] (10)
We also build two vectors of dimension 1x3 where Km and Jn (Mm and Nn respectively) are considered as the components of K and J (M and N respectively) vectors:

𝑲 = [𝑲 1 𝑲 2 𝑲 3 ] 𝑱 = [𝑱 1 𝑱 2 𝑱 𝟑 ] (11) 𝑴 = [𝑴 1 𝑴 2 𝑴 3 ] 𝑴 * = 𝑵 = [𝑵 1 𝑵 2 𝑵 3 ]
From Eq. ( 3) and with these previously introduced notations, mND(z) is given by:

𝒎 𝑁𝐷 (𝑧) = [𝑲 𝑱]. [ 𝑫 𝑩 ] = [𝑲 𝑱]. 𝑻 6 𝐻 . 𝑻 6 . [ 𝑫 𝑩 ] = [𝑴 * 𝑴]. [ 𝒑 𝒑 * ] = 𝒎 𝑁𝐷 (𝒑(𝑧), 𝒑 * (𝑧) ) ( 12
)
where p is a 1x3 complex vector.

The differential Mueller matrix clearly appears as a function of the 6x1 complex vectors:

 = [ 𝒑 𝒑 * ] = 𝑻 6 . [ 𝑫 𝑩 ] = 1 √2 [ 𝑫 -𝑖𝑩 𝑫 + 𝑖𝑩 ] (13)
From Eq. ( 7) under a second order approximation hypothesis, the differential matrix m(z) is given by:

𝒎(𝑧) = 𝑚 1 (𝑧) + 𝑚 2 (𝑧) = [𝑴 * 𝑴]. [ 𝒑 𝒑 * ] = 𝒎 𝑁𝐷 (𝒑(𝑧), 𝒑 * (𝑧) ) (14)
From Eq. ( 12) and the definition of first and second order cumulants [START_REF] Devlaminck | Depolarizing differential Mueller matrix of homogeneous media under Gaussian fluctuation hypothesis[END_REF] we have:

𝑚 1 (𝑧) = ∫ 𝑑𝑧 1 𝑧 0 [𝑴 * 𝑴]. [ 〈𝒑〉 〈𝒑 * 〉 ] (15)
And

𝑚 2 (𝑧) = ∫ 𝑑𝑧 2 ∫ 𝑑𝑧 1 [𝑴 * 𝑴] 𝑧 2 0 𝑧 0 𝐶𝑜𝑣((𝑧 2 ), (𝑧 1 )) [ 𝑴 𝑻 𝑴 * 𝑻 ] (16)
Where:

𝐶𝑜𝑣((𝑧 2 ), (𝑧 1 )) = [  𝑝𝑝 𝐶 𝑝𝑝 𝐶 𝑝𝑝 *  𝑝𝑝 * ] (17)
Stands for the covariance matrix between the complex vector  at z1 and z2 respectively with:

 𝑝𝑝 = 〈𝒑(𝑧 1 )𝒑 𝐻 (𝑧 2 )〉 = 1 2 { 𝐷𝐷 (𝑧 1 , 𝑧 2 ) +  𝐵𝐵 (𝑧 1 , 𝑧 2 ) + 𝑖. [ 𝐷𝐵 (𝑧 1 , 𝑧 2 ) - 𝐵𝐷 (𝑧 1 , 𝑧 2 )]} (18)
and: 18) and ( 19) is the classical covariance matrix between a real vector X at z1 and a real vector Y at z2.

𝑪 𝑝𝑝 = 〈𝒑(𝑧 1 )𝒑 𝑇 (𝑧 2 )〉 = 1 2 { 𝐷𝐷 (𝑧 1 , 𝑧 2 ) - 𝐵𝐵 (𝑧 1 , 𝑧 2 ) -𝑖. [ 𝐷𝐵 (𝑧 1 , 𝑧 2 ) +  𝐵𝐷 (𝑧 1 , 𝑧 2 )]} (19)  𝑋𝑌 (𝑧 1 , 𝑧 2 ) = 〈𝑿(𝑧 1 )𝒀 𝑇 (𝑧 2 )〉 in Eqs. (
It is worth noticing that describing the second-order behavior of complex variable (or process) p needs the knowledge of  𝑝𝑝 and 𝑪 𝑝𝑝  𝑝𝑝 is the classical covariance matrix between the complex vector p at z1 and z2 respectively and 𝑪 𝑝𝑝 is commonly referred to under the name of complementary covariance matrix [START_REF] Schreier | Second-order analysis of improper complex random vectors and processes[END_REF].

With Eq. ( 17) and the conjugation property between N and M, Eqs. ( 15) -( 16) become respectively:

{ 𝒎 1 (𝑧) = ∫ 𝑑𝑧 1 [ 𝟏 (𝑧) +  1 * (𝑧)] 𝑧 0 𝒎 2 (𝑧) = ∫ 𝑑𝑧 2 ∫ 𝑑𝑧 1 [ ( 2 (𝑧) +  2 * (𝑧))] 𝑧 2 0 𝑧 0 (20)
with

{  1 (𝑧) = (𝑴 * . 〈𝒑〉)  2 (𝑧) = 𝑵 𝑝𝑝 𝑴 𝑻 + 𝑵𝑪 𝑝𝑝 𝑵 𝑻 (21)
𝑵 𝑝𝑝 𝑴 𝑻 is termed [START_REF] Amblard | Statistics for complex variables and signals[END_REF] the complex cumulant of order (1,1) and 𝑵𝑪 𝑝𝑝 𝑵 𝑻 is the complex cumulant of order (2,0).

Using

-on the one hand the decomposition of the differential Mueller matrix proposed by Ossikovski [START_REF] Ossikovski | Differential matrix formalism for depolarizing anisotropic media[END_REF] into its nondepolarizing and depolarizing part (verifying the Minkowski antisymmetric and symmetric property respectively), -on the other hand the commutation property of Mi et Nj matrices, is straightforward to verify that:

-𝒎 1 (𝑧) gives a non-depolarizing contribution -[ 1 (𝑧),  1 * (𝑧)] = 0 (the  1 matrix and its conjugate commutes)

Decomposing 𝒎 1 (𝑧) according to Eq. ( 20) is thus exactly the decomposition of non-depolarizing differential Mueller matrix according to the Z matrix or state-generating matrix recently introduced by Kuntman [START_REF] Kuntman | Vector and matrix states for Mueller matrices of nondepolarizing optical media[END_REF] and also analyzed by Sheppard [START_REF] Sheppard | Coherency and differential Mueller matrices for polarizing media[END_REF].

State-generating matrix for depolarizing differential Mueller matrix

Let us now consider the decomposition of 𝒎 2 (𝑧). In the general case, this second order term breaks down into a depolarizing contribution and a non-depolarizing one. The existence of this latter contribution can be related to the statistical properties of the underlying processes.

From the standpoint of statistical processes it is therefore justified to regard the complex processes L, L' and C of Eq.

(5) and their conjugates simultaneously. From Eq. ( 14) it is equivalent to consider the corresponding  𝑝𝑝 and 𝑪 𝑝𝑝 at least in a second-order approach.

Considering this new parametrization is nevertheless more interesting for analyzing the depolarizing and nondepolarizing effects present in 𝒎 2 (𝑧).. We first introduce the following notation for the covariance matrices:

 𝑋𝑌 (𝑧 1 , 𝑧 2 ) = 〈𝑿(𝑧 1 )𝒀 𝑇 (𝑧 2 )〉 = [ 𝑥1𝑦1 𝑥1𝑦2 𝑥1𝑦3 𝑥2𝑦1 𝑥2𝑦2 𝑥2𝑦3 𝑥3𝑦1 𝑥3𝑦2 𝑥3𝑦3 ] ( 22
)
Where X, Y = D, B and the convention that xi is evaluated at z1 and yj at z2.

The general expression of 𝑵 𝑝𝑝 𝑴 𝑻 + 𝑵𝑪 𝑝𝑝 𝑵 𝑻 is given in Appendix A using this new notation. It is worth noticing that the cross-correlation between same nature (linear, 45°-linear or circular) birefringence and dichroism terms only appear on imaginary part of the diagonal entries of  2 (𝑧) matrix (see Eq. (A1)). Since 𝒎 2 (𝑧) is the summation of  2 (𝑧) with its conjugate, all these cross-correlations are not preserved in the 𝒎 2 (𝑧) expression. We find the same phenomenon as for the global phase in the non-depolarizing case as described by Kuntman [START_REF] Kuntman | Vector and matrix states for Mueller matrices of nondepolarizing optical media[END_REF].

Obviously the fact that this information is not explicitly allowed in 𝒎 2 (𝑧) does not mean that it does not affect the depolarizing behavior of the medium.

An example of this phenomenon can be found in [START_REF] Charbois | Stochastic model for the differential Mueller matrix of stationary and nonstationary turbid media[END_REF], where the existence of a correlation between birefringence and dichroism terms causes the appearance of damped oscillations in the depolarization curve.

It is well known [START_REF] Ollila | On the circularity of complex random variable[END_REF] that the different configurations of the 𝑪 𝑝𝑝 term lead to different statistical properties of the associated complex processes. These different configurations for 𝑪 𝑝𝑝 can be straightforwardly related to the existence of a non-depolarizing contribution coming from the second-order terms which adds to the non-depolarizing term coming from the mean values of the processes given by Eq. [START_REF] Charbois | Stochastic model for the differential Mueller matrix of stationary and nonstationary turbid media[END_REF].

According to the different nature of the complementary covariance matrix, 𝑀𝐴𝑃(𝒎 2 (𝑧)) = If 𝑪 𝑝𝑝 is real:

 𝐷𝐵 (𝑧 1 , 𝑧 2 ) = - 𝐵𝐷 (𝑧 1 , 𝑧 2 ) gives : 𝑀𝐴𝑃(𝒎 2 (𝑧)) = [ 0 0 0 0 0 0 𝐶𝐵 𝐿𝐵′ 0 -𝐶𝐵 0 -𝐿𝐵′ 0 -𝐿𝐵 𝐿𝐵 0 ] (23)
With:

𝐶𝐵 = 𝑑1𝑑2 -𝑑2𝑑1 + 𝑏2𝑏1 -𝑏1𝑏2 2 𝐿𝐵 = 𝑑3𝑑2 -𝑑2𝑑3 + 𝑏2𝑏3 -𝑏3𝑏2 𝐿𝐵 ′ = 𝑑1𝑑3 -𝑑3𝑑1 + 𝑏3𝑏1 -𝑏1𝑏3 2 (24)
A non-depolarizing contribution is generated by the second order cumulant with birefringence terms related to the existence of correlations between the different components of dichroism vector or birefringence vector respectively.

If 𝑪 𝑝𝑝 is purely imaginary:

 𝐷𝐷 (𝑧 1 , 𝑧 2 ) =  𝐵𝐵 (𝑧 1 , 𝑧 2 ) gives: 𝑀𝐴𝑃(𝒎 2 (𝑧)) = [ 0 -𝐿𝐷 -𝐿𝐷 0 -𝐿𝐷′ 𝐶𝐷 0 0 -𝐿𝐷′ 0 𝐶𝐷 0 0 0 0 0 ] (25)
with:

𝐿𝐷 = 𝑑2𝑏3 -𝑏3𝑑2 + 𝑏2𝑑3 -𝑑3𝑏2 2
𝐿𝐷 ′ = 𝑑3𝑏1 -𝑏1𝑑3 + 𝑏3𝑑1 -𝑑1𝑏3 2 𝐶𝐷 = 𝑑2𝑏1 -𝑏1𝑑2 + 𝑏2𝑑1 -𝑑1𝑏2 2 (26)
A non-depolarizing contribution is generated by the second order cumulant with dichroism terms related to the existence of correlations between the different components of dichroism vector and birefringence vector. These effects of correlation are very similar to artifacts reported by Jansen [START_REF] Jensen | Modulation techniques in polarization spectroscopy[END_REF] with the modulation techniques in polarization spectroscopy. With this method of measurement, the artifacts result from coupling of instrumental birefringence or dichroism with birefringence or dichroism of the sample. This combined effect on light from artifact and sample is given through a matrix product between the Mueller matrix corresponding to the artefacts and the Mueller matrix of the sample. Jansen noted for instance that circular dichroism of the sample may thereby couple with linear birefringence of the instrument to give a signal interpreted as linear dichroism.

Similarly, as a second order terms, the considered cumulants (or correlation terms) lead to matrix products between matrices associated to the dichroism vector and the birefringence vector.

Eq. ( 26) and Eq. ( 24) show how the non-depolarizing contributions are generated by the second order cumulants and how ultimately the existence of a depolarization effect can affect the knowledge of the non-depolarizing part of the Mueller matrix of such a medium.

If 𝑪 𝑝𝑝 =0 𝑀𝐴𝑃(𝒎 2 (𝑧)) = 0 and no supplementary contribution from the second-order terms is added to the nondepolarizing term coming from the mean values of the processes.

Obviously in the case where 𝑪 𝑝𝑝 is complex, we find both the previous contributions of birefringence and dichroism.

It is therefore straightforward to decompose the non-depolarizing (mND) and depolarizing (mD) contribution of the differential Mueller matrix [START_REF] Ossikovski | Differential matrix formalism for depolarizing anisotropic media[END_REF] into complex conjugate matrices:

{ 𝒎 𝑁𝐷 (𝑧) = ∫ 𝑑𝑧 1 [ 𝑁𝐷 (𝑧) +  𝑁𝐷 * (𝑧)] 𝑧 0 𝒎 𝐷 (𝑧) = ∫ 𝑑𝑧 2 ∫ 𝑑𝑧 1 [ ( 𝐷 (𝑧) +  𝐷 * (𝑧))] 𝑧 2 0 𝑧 0 (27) With: {  𝑁𝐷 (𝑧) =  1 (𝑧) + 𝑀𝐴𝑃[ 2 (𝑧)]  𝐷 (𝑧) = 𝑀𝑆𝑃[ 2 (𝑧)] (28)
Where 𝑀𝑆𝑃(𝑿) = 1 2

[𝑿 + 𝑮 . 𝑿 𝑻 . 𝑮] stands for the Minkowski Symmetric Part of X.

The decomposition of 𝒎 𝐷 (𝑧)can be regarded as the corresponding Z matrix or state-generating matrix for the depolarizing term of the differential Mueller matrix.

It is worth noticing that  𝑁𝐷 (𝑧) and  𝑁𝐷 * (𝑧) do not commute now without supplementary hypothesis. Upon substituting Eq. ( 18), Eq. ( 19) and Eq. ( 21) into the computation of Eq. (29), it is also possible to verify that the matrices  𝐷 (𝑧) and  𝑫 * (𝑧) do not commute.

Stationnarity and parity hypothesis.

With a second order stationary hypothesis for the processes -meaning that  𝑋𝑌 (𝑧 1 , 𝑧 2 ) =  𝑋𝑌 (𝑧 2 -𝑧 1 ) =  𝑋𝑌 ( )- Eq. ( 18) and Eq. ( 19) become:

{  𝑝𝑝 () = 1 2 { 𝐷𝐷 () +  𝐵𝐵 () + 𝑖. [ 𝐵𝐷 𝑻 (-) - 𝐵𝐷 ()]} 𝑪 𝑝𝑝 () = 1 2 { 𝐷𝐷 () - 𝐵𝐵 () -𝑖. [ 𝐵𝐷 𝑻 (-) +  𝐵𝐷 ()]} (29)
Under a stationnarity hypothesis and a parity hypothesis (being even) for the correlation functions, it is straightforward to demonstrate (see Eq. ( 24) and Eq. ( 2628) ) that the supplementary contribution from the secondorder terms to the non-depolarizing term vanishes and we have:

𝑀𝐴𝑃[ 2 (𝑧)] = 0  𝐷 (𝑧) =  2 (𝑧) = 𝑵 𝑝𝑝 𝑴 𝑻 + 𝑵𝑪 𝑝𝑝 𝑵 𝑻 (30)
It is worth noticing that in this case 𝑵𝑪 𝑝𝑝 𝑵 𝑻 the complex cumulant of order (2,0) (its symmetric (𝑵𝑪 𝑝𝑝 𝑵 𝑻 ) * the cumulant of order (0,2) respectively) has a very simple expression (complex conjugate expression respectively) given by:

[𝒅𝟏𝒅𝟏 + 𝒅𝟐𝒅𝟐 + 𝒅𝟑𝒅𝟑 -𝑰(𝒃𝟏𝒅𝟏 + 𝒃𝟐𝒅𝟐 + 𝒃𝟑𝒅𝟑)]. 𝑰 𝟒 (31)
Where I4 stands for the identity matrix of order 4. So, on one hand, all the correlation terms contribute as an isotropic absorption, and the orher hand, the cross-correlations contribute as a global phase but this global phase information is not allowed in m(z). Both these terms evolve as z-functions.

An example of such hypothesis (stationnarity and parity hypothesis) is found for instance in [START_REF] Ossikovski | Statistical meaning of the differential Mueller matrix of depolarizing homogeneous media[END_REF] for homogeneous medium where a zero order approximation of the correlation functions is made. This zero order approximation means that the correlation is assumed to be locally constant and is valid on a much greater length than the amplitude of the fluctuations is low since a sufficient decay has to be ensured by the condition α. ζ ≪ 1 (where α formally denotes the amplitude of the fluctuations and ζ the correlation distance) to have vanishing higher orders of the correlation function (see [START_REF] Devlaminck | Depolarizing differential Mueller matrix of homogeneous media under Gaussian fluctuation hypothesis[END_REF] for details). Such hypothesis are then valid for very thin samples or for thicker samples but with smaller fluctuations (thus a lower depolarization since the level of depolarization is obviously a function of the amplitude of the fluctuations but in a nontrivial way) and as long as it can be considered that the correlation is constant.

We then have the following relations between the covariance matrices:

 𝐷𝐵 () =  𝐵𝐷 𝑻 (-) =  𝐵𝐷 𝑻 () =  𝐵𝐷 𝑻 (0) (32)
Which leads to a general expression of  𝐷 (𝑧) given in Appendix B.

In the rest of this article  𝐷𝐵 () and  𝐵𝐷 () respectively, will be denoted by  𝐷𝐵 and  𝐵𝐷 respectively since in this case the covariance matrices no longer depend on the  variable. This result leads to a z-quadratic solution (z is the thickness of the medium) for the depolarizing part of the differential matrix obtained as double integration of constant terms (see Eq.( 16)).

In the same way, stationarity imposes constant mean values. This result leads to a z-linear solution for the nondepolarizing part of the differential matrix. It is worth noticing that this z-linear solution for the non-depolarizing differential Mueller matrix is not necessarily valid if the processes are not stationary since a contribution of second order is added to the first order cumulant contribution.

Eventually, if an approximation to the zero order of the correlation functions of the processes -that are then stationary -is assumed, it is possible to verify that the matrices  𝑁𝐷 (𝑧) =  1 (𝑧) and  𝑁𝐷 * (𝑧) commute and it is also possible to verify that the matrices  𝐷 (𝑧) =  2 (𝑧) and  𝑫 * (𝑧) commute.

On the other hand, it is not possible to hope that  𝑁𝐷 (𝑧) +  𝐷 (𝑧)and its conjugate commute in the general case because it would require that 𝒎 𝑁𝐷 (𝑧) and 𝒎 𝐷 (𝑧) commute what is wrong most of the time.

Decomposing the differential Mueller matrix according to the Z matrix recently introduced by Kuntman [START_REF] Kuntman | Vector and matrix states for Mueller matrices of nondepolarizing optical media[END_REF] is therefore possible for depolarizing medium at least in the second order approximation with some hypothesis on the statistical properties of the processes. We then have 𝒁 𝐷 (𝑧) = 𝑒𝑥𝑝[ 𝐷 (𝑧)] with the expression of  𝐷 (𝑧) =  2 (𝑧) given by Eq. (21).

SCALAR COMPLEX VARIABLE CASE.

Let us considering the case where only one of the three components of the spectroscopic complex vector ( L , L' , C ) is operative. For example, let L be this component (this choice is obviously not restrictive). We then have a scalar complex variable that we assume Gaussian for example.

Fig. 1 shows an example of probability density contour when L component is the operative one. It is usual to introduce the circularity quotient [START_REF] Ollila | On the circularity of complex random variable[END_REF] as a complex coefficient defined by: From equations (33) the case  = 0 corresponds to d1b1=0, meaning no correlation between the LB and LD components.

But it is obvious that the value of  depends on the position adopted for the axes LB and LD. A rotation of the axes of an angle  /2 thus makes it possible to cancel  , i.e. to de-correlate the components. It is therefore the adopted system of representation that creates this "apparent" correlation between the physical quantities concerned.

Nevertheless this is of no consequence for the depolarization measurement. Indeed when L is the only operative component, it is possible [START_REF] Devlaminck | Depolarizing differential Mueller matrix of homogeneous media under Gaussian fluctuation hypothesis[END_REF] [START_REF] Charbois | Stochastic model for the differential Mueller matrix of stationary and nonstationary turbid media[END_REF] to show that:

𝑚 𝐷 = 𝑑𝑖𝑎𝑔(1, 0 , -𝜎 𝐿 2 , -𝜎 𝐿 2 ) (34)
where

𝜎 𝐿 2 = 𝐸(|𝐿| 2 ) = 𝜎 𝐿𝐷 2 + 𝜎 𝐿𝐵 2 .
The LD and LB variances can be expressed [START_REF] Ollila | On the circularity of complex random variable[END_REF] as:

{ 𝑑1𝑑1 = 𝜎 𝐿𝐷 2 =
with 𝛼 = 𝜓 2 and 1 , 2 stand for the eigenvalues of the 2x2 real covariance matrix associated to [LD LB] T random real vector. From Eq. (35) it is obvious to see that:

𝜎 𝐿 2 = 𝜆 1 + 𝜆 2 (36)
This result shows that  (this "apparent" correlation) is of no consequence for the depolarization measurement in an incoherent wave configuration.

On the other hand, the global phase -see Eq. (31) -related to the d1b1 term (correlation between LB and LD terms) evolves as quadratic function of z -see Eq. ( 27) -and is related to the  value, meaning the laboratory x-y axes definition of the measure. The existence of this global phase must be taken into account when processing coherent waves.

CONCLUSION

In this work, we have shown how the information contained in differentiel Mueller matrix related to a depolarizing medium can be expressed in terms of a state-generating matrix. The formulation of the problem using complex random vectors makes it possible to directly introduce the formalism of state-generating matrix in the case of differential depolarizing matrices.

Under a second order approximation hypothesis, this depolarizing state-generating matrix is directly related not only to the classical covariance matrix of the associated complex vector but also to the complementary covariance matrix. The different configurations (null, imaginary, real) for this complementary covariance matrix can be straightforwardly associated to physical consequences.

The commuting property of these state-generating matrices obtained in the case of homogeneous non-depolarizing media is preserved in the case where these media become depolarizing if an approximation to the zero order of the correlation functions of the processes are made. Such hypothesis are then valid only for very thin samples or for thicker samples but with very low depolarization.

1 2 [

 2 𝒎 2 (𝑧) -𝑮 . 𝒎 𝟐 𝑻 (𝑧) . 𝑮] the Minkowski Antisymmetric Part of 𝒎 2 (𝑧) (with G = diag(1,-1,-1,-1)) takes the following expressions:

2 LDFig. 1 .

 21 Fig. 1. Probability density contours of the scalar complex Gaussian variable L.

  𝜆 1 𝑐𝑜𝑠 2 (𝛼) + 𝜆 2 𝑠𝑖𝑛 2 (𝛼) 𝑏1𝑏1 = 𝜎 𝐿𝐵 2 = 𝜆 1 𝑠𝑖𝑛 2 (𝛼) + 𝜆 2 𝑐𝑜𝑠 2 (𝛼)

Eventually, examples of physical interpretations that can be obtained have been presented specifically for homogeneous medium. Illustrations are given when the complex vector degenerates into a complex scalar and when a Gaussian random processes hypothesis is made.

APPENDIX A

From Eq. ( 18)- [START_REF] Ossikovski | Differential matrix formalism for depolarizing anisotropic media[END_REF] and the definition of M and N, it is straightforward to show that: 𝑵 𝑝𝑝 𝑴 𝑻


L Benouaich

P Rouch

F Natta

P Thoreux

Spring jumpers vs power jumpers: ankle joint behavior in elite wushu athletes and implications for performance and injury risk

Keywords: Jumping, stretch-shortening cycle, ankle, stiffness, acrobatics

Introduction

Wushu, better known as kungfu, is the modern athletic form of Chinese martial arts, and consists of performing a routine of martial arts movements as well as acrobatic jumps. Since those acrobatic jumps are performed on a hard floor, vertical jump height plays a major role in elite performance. Wushu coaches often mention two different types of athletes: spring jumpers and power jumpers. From this empirical point of view, spring jumpers seem to jump more easily.

As in many sports, wushu jumps are performed with a run up followed by a stance phase with eccentric and concentric phases, often referred to as a 'stretch-shortening cycle' (SSC). Vertical jump height, in this case, is the result of three main components acting during the concentric phase: muscle fiber shortening, coordination between joints and recoil of elastic energy stored during the eccentric phase. Whereas the squat jump exercise (SJ) is used to evaluate the first component, the drop jump (DJ) can be used as a standard exercise to analyze the whole SSC. The overall ability of an athlete to benefit from a previous eccentric phase can be evaluated by pre-stretch augmentation (PSA), defined as the ratio of DJ and SJ performances [START_REF] Kubo | Effects of plyometric and weight training on muscle-tendon complex and jump performance[END_REF].

Ankle behavior can be characterized by the evolution of joint torque, obtained with inverse dynamics, with respect to joint angle during the different phases of the support phase, with the slope being considered as overall joint stiffness.

The aim of this study was to characterize spring and power jumpers in elite wushu athletes in terms of PSA, ankle laxity, and overall ankle stiffness during the concentric phase. The influence of the chosen jumping strategies on performance and injury risk will be also discussed.

Methods

After standardized warm-up and maximal ankle dorsiflexion angle measures [START_REF] Jones | A study to determine the reliability of an ankle dorsiflexion weight-bearing device[END_REF], 12 athletes of the French wushu team performed three repetitions of SJ and DJ, before performing 31 specific wushu jumps. Forty-two reflective markers were placed over skeletal landmarks. Kinematics were registered at 300 Hz with a 13 camera Vicon system and ground reaction forces with two AMTI force plates at 900 Hz. Marker positions were filtered with an adaptive filter, and body segment inertial parameters were personalized. Ankle, knee, and hip joint reaction moments, calculated using inverse dynamics, as well as ankle flexion angle were expressed between 0 and 100% of the support phase, and normalized by the body mass.

For SJ and DJ, performance was defined as the maximum height of the center of mass (CoM) during the flight phase minus the CoM height in the static position. Data of the best performance for each subject in SJ and DJ were kept for analysis. PSA was defined as the ratio between the maximal performance in DJ and the maximal performance in SJ, minus 1. For the DJ, plantar flexion moments were plotted vs. ankle flexion angles, and two linear regressions were performed for the two halves of the concentric phase. The two regression slopes were defined as the ankle dorsiflexion stiffness in the first half of the concentric phase (k 1 ) and the ankle dorsiflexion stiffness in the second half of the concentric phase (k 2 ) in Nm/°.

PSA results during DJ enabled the separation of the athletes into three groups of four subjects: power jumpers (low PSA), medium jumpers (medium PSA), and spring jumpers (high PSA).

Training experience and injury history were evaluated using a questionnaire. Wilcoxon tests were performed for statistical analysis, with spring jumpers as reference. jumpers (p < 0.01). This parameter is often used as an index of knee osteoarthritis risk [START_REF] Haim | Reduction in knee adduction moment via non-invasive biomechanical training: a longitudinal gait analysis study[END_REF]. In their injury history, the four spring jumpers were indeed more affected at knee level compared to power jumpers, but the small population does not allow statistical interpretation.

Conclusions

Concerning performance, absolute jump height was similar between power and spring jumpers, so their performance for acrobatic jump heights in competition would be similar. On the one hand, spring jumpers are the most efficient in terms of energy expenditure. The active strategy of power jumpers requires both high activation input by the central nervous system and high energy expenditure for muscle fiber shortening. These athletes may be more subject to a decrease in performance with fatigue. On the other hand, their larger ankle range of motion helps them in performing wushu compulsory low stances with heels on the ground.

It is difficult to know why athletes choose one strategy rather than the other. It could be linked with genetic abilities (laxity and muscle typology) as well as training habits. Power jumpers seem to practice more weight training compared to spring jumpers who are more used to specific plyometric training. It is also possible that some athletes, particularly in the medium group, do not choose the technique that best fits their capacities, explaining their lower overall performance.

For personalized training optimization and injury prevention, the athlete's jumping strategy should be taken into account, for instance, by avoiding repetitive jump training with power jumpers or by focusing on knee safety with spring jumpers.

Results and discussion

PSA was different between groups whereas performance, anthropometric data, and training experience were similar. In power and medium jumpers, k 1 was lower than in spring jumpers (p < 0.05). Maximal dorsiflexion angle was lower in power jumpers than in spring jumpers (p < 0.05). Typical flexion moment vs. flexion angle curves for spring (a) and power (b) jumpers, and their respective k 1 and k 2 are presented in Figure 1. The bilinear evolution of ankle stiffness in spring jumpers can be interpreted as two different mechanisms predominating consecutively during the concentric phase. The first and higher slope could be mainly explained by the passive shortening of previously deformed articular structures, whereas the second and lower slope could be the effect of the active shortening of the triceps surae muscle fibers. In power jumpers, the second mechanism appears to be predominant.

The higher maximal dorsiflexion angle observed in power jumpers could explain the difficulty for these athletes in the storage and recovery during recoil of elastic energy at the ankle joint, and their choice of a predominant 'active strategy' during the concentric phase, in contrast to the 'combined active-passive strategy' used by spring jumpers.

Peak valgus reaction moments, in response to external adduction moments, were higher in spring and medium

Figure 1 .

 1 Figure 1. ankle typical behavior in spring (a) and power jumpers (b) during the dJ stance phase. linear regressions (full line) represent overall ankle stiffness during the first (k 1 ) and the second half (k 2 ) of the concentric phase.

Acknowledgment

The authors would like to thank the athletes for their involvement in the study.


Keywords: WP 2, 1 D2, 1-4, pdf

. The mitigation of water stress at local scale depends not just on technological innovations, but also on the development of new integrated water management tools and decision-making practices.

 The framework is designed to assess the degree of water stress the different sectors face within a region. Our target focused on relatively small catchments in opposite to the main frameworks for describing water stress.

Introduction

We are aware that the use of indicators and its application in the water sector is already a fact and has become very important. In particular, legislations such as the implementation of the EU Water Framework Directive (WFD) have given prominence to indicators as management tools in the water sector. There are many other water indicators in use today. Most of these are single indicators of water quality, which for a long time have been the area of water research. Some indicators have been combined as measures of various dimensions of water stress, including the widely used indicators relating the quantity of water available to the amount needed for certain activities. Two well known ones are, the "Water availability per capita per year" (this defines specific thresholds relating to water shortages : 1700 m 3 /capita/year or less: water stress, 1000 m 3 /capita/year or less : water scarcity), and "Water withdrawals" (this indicates the degree of stress in a location, but does not take account of water transferred from transboundary rivers or international interbasin transfers).

There are some frameworks, like the DPSIR (Drivers Pressures States Impacts Responses) for water indicators (Jesinghaus, 1998). Arguments have been made (European Environment Agency 1999), that the DPSIR approach permits the establishment of causal relationships between human activities, their environmental impacts, and the effects of societal responses. It is certainly true in some cases, that such cause and effects can be known (e.g. the link between lead emissions to the environment, and its impact on human health), and effective policy responses have been the result (e.g. the removal of lead from petrol). A number of authors have put forward several disadvantages of the DPSIR approach. In some cases, isolated chains of indicators may not be enough to reproduce the complexity of systems, which tend to behave more like a network rather than a linear chain [START_REF] Bossel | Indicators for Sustainable Development: theory, methods, applications[END_REF], [START_REF] Rekolainen | A conceptual framework for identifying the need and role of models in the implementation of the Water Framework Directive Intl[END_REF], [START_REF] Caeiro | Sado Estuary Ecosystem: A management methodology[END_REF], [START_REF] Jorge | Measuring, monitoring and managing sustainability in Indian coastal areas: the socioeconomic dimension[END_REF].

During the process of the literature review, and building on earlier work on index development (C. [START_REF] Sullivan | Application of the Water Poverty Index at Different Scales: A Cautionary Tale[END_REF]; C. A. [START_REF] Sullivan | Derivation and Testing of the Water Poverty Index Phase 1[END_REF], it had been observed that there was some dissatisfaction in the use of indices in general. Statements are sometimes made about an index being too simple, or not comprehensive enough, with the methodology not allowing a sufficient breadth of knowledge to be included.

As a result our objective was to ease the management of water stress by providing a tool for a first assessment of the integral effect of measures taking by different sectors to combat water stress. We focused on the water system's property to: characterize the water system, to describe the water stress situation in the different water sectors, to recognize the water stressors and to properly act with bundle mitigation options against water stress.

The basis for our approach was developed within our research group (see C. A. [START_REF] Sullivan | Report on Water Stress Indicators: AquaStress Project[END_REF]). From our perspective, there are two main possibilities for the use of this tool: 1) to evaluate water stress at the local level and 2) for regional assessments.

In this paper we will apply this tool and we will present how to assess the present water stress situation in specific test sites across different water consuming sectors on a small scale. A water management tool: the Water Stress Framework (WSF)

One of the first discrepancies that aroused when we started developing the framework was the multitude of disciplines represented in our group. Interdisciplinarity can help to understand a dilemma from different paradigms but sometimes represents also a handicap for conceptions. To avoid future problems, a workshop was organized to refine the understanding of "water stress" for the different disciplines. The aim was to reach a "consilent" definition. Consilience is an attempt to bridge the culture fissure among sciences [START_REF] Wilson | Consilience: The unity of knowledge[END_REF]. The workshop helped to minimize understanding problems. The main outcome was the definition of water stress agreed upon by the different disciplines represented:

"Water stress occurs when the functions of water in the system do not reach the standards (of policies) and or perceptions (of the population) on an appropriate quantity and quality, at an appropriate scale and the adaptability for reaching those is not given"

Based on this definition, our water management tool emerges as a way of assessing those complex issues in for potentially water stressed areas. Our work is developed, along with a related index (Inter Sectoral Water Stress Index -ISWSI). As we already elucidate the water stress framework (WSF) is based on previous work done (comp. [START_REF] Sullivan | Report on Water Stress Indicators: AquaStress Project[END_REF]. Additionally for a better understanding on the structure of the WSF, there is short explanation in the annex to this paper.

Our purpose by building and using the WSF was to define the ability of the water system to exist in some preferred state and continue to deliver its services over time identifying rapidly the possible stressors. The WSF is a composite of indicators. The matrix of indicators combines five waterconsuming sectors and five categories (see figure 1).

PM Potential Marging

For regional assessments comparability across sites needed to be ensured. For this, there is a limited number of core indicators belonging to the five different categories (compare figure 1 andtable 1). These core indicators are essential for those regional assessments and the same indicators will be required for each site. To achieve this for a high number of sites only a reduced set of indicators can be used as represented in Table 1. The most important selection criteria for this set of indicators, was data availability combined with the sensitivity of the indicator to water stress. In any case and because the selection of indicators plays such an important role in the spatial applicability of the WSI, it is important to know from the beginning what the objective of the WSI calculations is. For example, if the indicators are adapted to a particular biogeographic region depending on the local conditions, the regional comparison should only be limited to those particular biogeographic regions around the world. In such a case for instance the Mediterranean subtropics with its particularities would be comparable around the world (e.g. Chilean region and Spanish region) Doc Name: Deliverable ID: WP 2.1 D2.1-4.pdf Date: 14/02/2008 Dissemination level: PU -Final Report 5

During the case study application process, autocorrelation between the indicators was also tested, which further reduced the final number of indicators to be used. For any indicator which was not available at the test site, a replacement was made. Where possible, the selection was made in consultation with the stakeholders at the site e.g. Morocco). When doing so we realized that a prerequisite for site specific evaluation is the selection of the required indicators in consultation with stakeholders. It should be noted that this should be seen as an iterative process, with refinement of this structure achievable if needed during the site process. If there are significant data gaps at a site, proxy values can be provided using expert opinion, until such time as an appropriate specific indicator becomes available.

FP6 -511231

Social and economic equity

Economy of water suppliers

Farm size dispersion Labour-related water intensity

Labour-related water intensity

Nature protection

Regarding the determination of the weightings for the calculation of the WSF, since those are a political issue and must be regarded as such, weighting should be done in consultation with stakeholders. Alternatively weights could be based on the economic relevance or on the water consumption of a sector, this choice will also depend on the priorities of the user. As it is important to ensure transparency in the interpretation of the results, in general in this paper no weights (i.e. all weights are set to 1) are applied for the first calculations.

Case studies

For the implementation of the WSF we have chosen two case studies: Sardinia and Morocco, which are both test-sites of the AquaStress project. AquaStress adopts a Case Study -stakeholder driven approach1 .

In the Sardinian case study we characterized the water stress situation of the test sites and test the sensitivity of the framework using the available data and taking into account the particularities of the test site. In here we can characterize all sectors and we adapt core indicators to the distinctiveness of the test site. In the Moroccan case study we adjust the calculation to the minimal amount of existing data and see how the WSF react.

Sardinian Case Study

The Flumendosa-Mulargia basin is located in the south-eastern part of Sardinia. The basin (1824,39 km2), which includes six interconnected reservoirs, supplies water resources in Southern Sardinia for different and conflicting uses (domestic, agriculture, industry and energy production) within the same basin and mainly for the distribution network located outside the basin in the Campidano plain. As the FP6 -511231 majority part of Mediterranean areas, it is characterised by water scarcity and quality problems due to drought. The hydrological cycle is influenced both by the geo-morphologic characteristics of the basin and by its pluviometric regime. The Flumendosa-Mulargia system is an area of particular interest for European policies and strategies because it is representative of problems due to drought typical of the Mediterranean climate. The climatic changes of the last years determined a drastic deficit between water demand and its availability, with increase of water stress phenomena in Sardinia. In particular in the last years the reservoir-system resulted in the supply of water resources considered sufficient to cover no more than 20% of the total requirements of the area. Moreover "water stress" resulted in significant problems concerning quality aspects.

The majority of the data for the Sardinian case study were drawn from the databases of Istat (the Italian statistical office), and to a lesser extent Eurostat (the database of the European Union). The spatial scale of the information contained within these databases is generally NUTS-2 or NUTS-3 level. This corresponds to the region of Sardegna and the province of Cagliari, respectively. These administrative boundaries do not correspond to the natural boundaries of the river catchment making up the test site. This is a common problem when applying information about water use and water stress in catchment studies. A potential advantage of retrieving data from the databases mentioned is that the water stress in the test site can be compared to that in other regions. In fact, the range of indicator values contained within a dataset can be used to determine the relative performance of the test site for a given indicator. While this may be not advantageous for indicators which are already normalized to percentages, it is convenient when only absolute values can be retrieved. This is for example the case with the domestic / institutional and adaptive capacity indicator 'per capita investment in water treatment'. Unfortunately, despite a fair availability of water stress indicator data for individual countries, they have only few indicators in common (Vermooten et al., 2007). This complicates comparisons of the test site with regions in similar settings throughout the entire Mediterranean area.

The dataset was extended by adding local data summarized in [START_REF] Preziosi | Report: State Of Water Stress in Different Sectors in the Test Sites Areas and Current Strategies to Cope with Water Stress[END_REF]. With respect to water use, this revealed some inconsistencies with the other data sources. The Istat data report a water use by agriculture of only 312,000 m3 (1999), while between 1997 and 2002, 210 million m3 per year was reportedly abstracted from the reservoirs, of which 100 million m3 per year for use by agriculture (Apostolaki & Assimacopoulos, 2005).

Table 2 shows the resulting indicator values for Flumendosa-Mulargia. The higher the value, the higher the inferred stress. The sectoral indicators were weighted and summed to ISWSF (see Figure 1). The environmental sector was assigned a weight of 1, while the remaining sectors were weighted according to their water use as a fraction of the total. These relative water uses were 46% for the domestic sector, 48% for agriculture, 5% for industry, and 1% for tourism. The resulting ISWSF was 0.57.

ISWSF divided by the potential margin yields the WSF. The calculated potential margin ranged from 0.02 to 0.18, and turned out to be greatly influenced by assumptions of (notably) evapotranspiration. As a consequence, the resulting WSF varies by a factor 9. It is concluded that WSF is very sensitive to the potential margin and uncertainties therein, notably at low values. Conversely, a change in the value of one single indicator does not affect the overall result to a great extent, because the effect is dampened out by the weighting procedure. This provides the WSF-approach with some degree of robustness, provided that values for a sufficient number of meaningful indicators can be retrieved.

The sensitivity of the WSF to seasonality was also assessed, by considering only the summer period (June through August) in stead of the entire year. While the overall effect on WSF remains limited to an increase of 7%, it clearly changes the sectoral contributions, as shown in the Figure 1. Agriculture and tourism become more important, illustrating the importance of carefully choosing the proper period for analysis.

FP6 -511231

6.3 -Global Change and Ecosystems

Moroccan Case Study

The Tadla region in Morocco is located about 200 km South-East of Casablanca at an altitude of 400 m and has a mean annual precipitation of 300 mm. The cropped area in Tadla is approximately 10,000 km2. The significant proportion of it is irrigated from two dams (with a total capacity of almost 2 billion cubic meters -BCM). G roundwater irrigation is also growing, but the scale of it has not been assessed. Over 12% of Morocco national production of both citrus and olives are produced in Tadla irrigated area, together with some 23% of nationally produced sugar beet. 82% of the fields have an area less than 5 ha and occupy 34% of the total area, while 3% of fields have an area more than 20 ha and occupy 34% of the total area. Tadla presents a variety of water stress problems which concern surface water scarcity, groundwater resources depletion, water pollution and more. Moreover, some national strategies for water stress mitigation are currently undergoing in the Tadla as is the case of the whole of other large scale irrigated perimeters in Morocco.

The calculations of the WSF in the Moroccan case study were done in the same way as explained in the Sardinian case study. One of the main disparities to the test site in Sardinia was the different sectors that play a role in this test site: the domestic and the agricultural sector. The Tadla test site also represents one of this test sites where difficulties getting data aroused as the main constraint for calculating the WSF. As table 4 shows, we were only able to get data for the domestic and the agricultural sector because the other water consuming sectors are not developed. Regarding the environment, as in many developing countries, data at the local level are not available.

The most important issue in Tadla case study concerns water availability for irrigation. At the beginning of the creation of the perimeter, the irrigated area was very small compared to the hydraulic capacity of the system. This lead to an excessive use of water and to an artificial recharge of the superficial aquifer. In some places of the perimeter it has even been necessary to drain lands. With the development of the irrigated area, artificial groundwater recharge also reduces proportionally to the increase in the irrigation efficiency (lower consumption per hectare). In the same time, the aquifer did allowed to irrigate when there where restriction in the water allocation for the surface water. Surface and groundwater where then complementary resources. But drought become structural and the use of groundwater where the only solution to maintain the quantity of water at the field level. This leads to an increase in the investments in borehole and to groundwater depletion. Groundwater does not play any more its initial role of insurance against occasional drought. The main problem comes from the reduction of the water allocation to the canal system that seems to be permanent now and induce behaviours consisting in drilling deeper and deeper, risking to challenge domestic consumption. Regarding the domestic sector, because of the rapid urban population increase and the fact that it is too costly to develop public infrastructure in rural area, only 15% of the population is connected to an effective sewage system. [START_REF] Loubier | Report: State Of Water Stress in Different Sectors in the Test Sites Areas and Current Strategies to Cope with Water Stress[END_REF] (compare table 4). Differently as in Sardinia, no alternative indicators to the core indicators were used. Additionally the core indicators were approved by local stakeholders through cross-checking of every indicator and its statement with the understanding of the stakeholders. No weight was given to the sectors and no seasonality was included due to the fact that water needs do not changes considerably with seasonality.

In figure 3 we see that the results of the calculation in the case of Tadla do reflect the described problematic. Those results can be used to evaluate the relative water stress interdependency to the double sectoral water stressors -domestic and agriculture -and its interactions. We can see that while the main stressors in the agricultural sector are in the category of "society and equity" and "institutional and adaptive capacity", in the domestic sector the category of "infrastructure" embodies the main deficiencies of the sub-system.

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6.3 -Global Change and Ecosystems In regard to the domestic sector, for example, the figure 3 shows the reality of the area where the insufficiency of the liquid sanitation network is one of the main causes of the development of diseases like viral hepatitis, cholera, typhoid and diarrhea (comp. http://www.matee.gov.ma/onem/monographie_centre.htm) . inclusion of indicators on mitigation options, and their success, in this dataset, a normalisation of WSF to make the approach easier to communicate. A definition of the biogeographic regions for cross-regional comparison A comparison of weights given by different stakeholder groups could be also a good demonstration tool for a same test-site. Although this was out of the scope of this paper, weights can changes the perceptions of water stress for an area. Therefore we recommend the exercise for the future to test the WSF against different weights given by different stakeholder groups

In all we consider the WSF a good instrument for fast assessment for policy purposes in small catchments because due to the structure of the framework by sector it is easier to recognize the source of the change and the source of water stress.

Conclusion

The main challenge of European water policy is centered in the good ecological and chemical status of its waters. The European Informal Council of Environment Ministers declared that an integrated approach is needed to address the water scarcity and droughts challenge. They suggest an approach that integrates supply, use and demand. With the WSF we demonstrate a tool that has the capacity to be used for this approach.

The problems of water availability in our case studies do not differ that much from the factual problems we are suffering in Europe but also around the world. Water stress results not only from water scarcity but also from water contamination or like in Morocco from mismanagement of water resources. Despite the predictions about losses of the water supply due for example to exaggerated leakage, one of the main problems that we are facing is the delivery of enough water to the main water consumer sector, e.g., the agricultural sector.

A first step for solving the problems of "getting enough water" for the different water consuming sectors is to detached the sectors demands and supplies as we have done with our WSF and to be able in that way to address the sub-problems belonging to a particular sector and to a particular subtheme of a sector. Only so we will be able to address with the suitable mitigation option the problem this sector is suffering. Gustavson et al.(1999) made the remark that "greater focus is required on modeling frameworks that can use incomplete data sets or qualitative information". Adding to their affirmation, when using the WSF following issues should be taken into account:

Our framework is more appropriate for qualitative modeling of different policy alternatives, rather than for predictions based on specific indicators. Data requirements at the local level imply a high dependency in time and money resources. Often the only possibility to get data is to find estimations based on higher levels of aggregated data. Contrary to the current in use frameworks for assessing water stress, we focus on a small number of indicators with their particular indicators-clusters: water quantity, water quality, institutional and adaptive capacity, infrastructure and social and economic equity (comp. table 1). There are some examples in other river basin that reveal that a large number of indicators do not automatically improves the consistency of models to provide policy guidance. (comp.(Gustavson et al., 1999) Concluding and after having reflected on our case studies, we can add that the WSF and other frameworks for assessing water stress are far from representing or predicting the reality of the water consuming sectors under the uncertainty attached to the element "water". Even less when the policies FP6 -511231 dominating the water consuming sectors (e.g. common agricultural policy) are unpredictable in their consequences to the water household. Therefore it is inappropriate to create a framework for assessing water stress that would just make predictions.

In this regard the WSF reaches our expectations: to evaluate the relative water stress of the case studies to multiple sectoral water stressors and to identify in which particular sector, under which particular category the stressors are located. In that way the possibility to address the water scarcity problems by their sources are higher due to the differentiation than in a single number index.

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Annex Table 5: out of the D2.1-3 (online available)

The Water Stress Index (WSI) is generated through a combination of two parts: the Integrated Secto ral Water Stress Index (ISWSI), which captures the level of water stress in the different sectors, and the Potential Margin (PM), which is an assessment of the available water resource supply. The potential margin indicates the degree of dependency on local and imported water, and the safety margins remaining available. By combining these two together, the WSI is derived as follow:

PM ISWSI WSI = [ 1 ]
The two components of the WSI thus incorporate the demand and the supply elements of the water situation, and are represented respectively by the ISWSI and Potential Margin (PM).

First component of the WSI

The Integrated Sectoral Water Stress Index, (ISWSI) shows the level of water stress across the different sectors and the type of stress associated with each sector. The major anthropogenic sectors to be considered within water management decisions are: domestic, agricultural, industrial, and tourism (which also includes services). In addition to these four sectors the environment is included, to ensure that a certain degree of water is allocated to the environment to enable ecological integrity.

For each of these five sectors the possible sources of stress have been identified and summarized in five categories: water quantity, water quality, institutional and adaptive capacity, infrastructure and social and economic equity.

The ISWSI has been constructed using is a composite index approach:

E T i Ag D E E T T I I Ag Ag D D w w w w w I w I w I w I w I w ISWSI + + + + + + + + = [2]
Where w i is the weight given to the sector i. We agreed that 1 ISWSI 0 < < I i is the water stress index for each of the component sectors, ie. Domestic (D), Agriculture (Ag), Industry (T) Tourism services (T) and the Environment (E). The indicators are first normalised to values between 0 and 1 and then aggregated to form a Secto ral Water Stress Index. Each of the sectoral water stress indices is also a composite index, made up of selected indicators (one or more for each category). Since the structure of indicators suggests that the sensitivity of the model to any one indicator will be relatively small, slight variations in any one of the indicators chosen should not create a significant impact on the overall ISWSI score.

Second component of the Water Stress Matrix: PM

The concept of the Potential Margin aims to represent how close the water resources system is approaching its natural endowment limits, including, where relevant, any imported water. The Potential Margin (PM) is calculated as a ratio between the RRR (as defined above) and the supply side of the water balance (NBE+IMP): Where: NBE = the total Natural Baseline Endo wment. This refers to all naturally available water resources for any site or area, which should include both renewable surface water and groundwater. It should take into account all the components of the balance which are naturally available due to the effective precipitation inside the test site area, as well as all the waters which enter naturally into the system: river discharge from upstream, groundwater flow (potentially usable), spring discharge, etc. IMP = imports (geographical movement of water to the site). This includes the water which is imported from outside the test site (diversions, desalinated water etc). TER = To tal Explo ited Reso urce. It represents the water which is currently withdrawn for use (consumed or eventually given back to the system) inside the test site by human activities (domestic, industrial, agricultural). The environmental water needs inside the test site should be included as well. This amount will depend on a 'political' decision, and it will affect ecological conditions locally; and downstream. Requirements to preserve environment downstream is included in the export (EXP) component. EXP = exports. This considers the volume of water which is committed to downstream users including the environment.

RRR stands for Residual Renewable Resource. This is the amount of water in the test site that is left in the local system, which has not been yet consumed by local or downstream users. This amount is obtained subtracting both TER and EXP from NBE and IMP:

Figure

  Figure 2: Sardinian Case Study Variations

  Doc Name: Deliverable ID: WP 2.1 D2.1-4.pdf Date: 14/02/2008 Dissemination level: PU -Final Report

  Global Change and Ecosystems Doc Name: Deliverable ID: WP 2.1 D2.1-4.pdf Date: 14/02/2008 Dissemination level: PU -Final Report 10

Figure 3 :

 3 Figure 3: Water Stress Characterisation for the Agricultural and Domestic Sector in Tadla

  Global Change and Ecosystems Doc Name: Deliverable ID: WP 2.1 D2.1-4.pdf Date: 14/02/2008 Dissemination level: PU -Final Report 12

  Doc Name: Deliverable ID: WP 2.1 D2.1-4.pdf Date: 14/02/2008 Dissemination level: PU -Final Report

  This measure is based on the components of a simple water balance:

  are what would be defined in Economics as supply; and TER + EXP demand. Doc Name: Deliverable ID: WP 2.1 D2.1-4.pdf Date: 14/02/2008 Dissemination level: PU -Final Report

Figure 1: Water Stress Framework Location classification

	Quantity Issues	Quantity Issues	Quantity Issues	Quantity Issues	Quantity Issues
	Quality Issues	Quality Issues	Quality Issues	Quality Issues	Quality Issues
	Institutional and	Institutional and	Institutional and	Institutional and	Institutional and
	adaptive	adaptive	adaptive	adaptive	adaptive
	capacity	capacity	capacity	capacity	capacity
	Infrastructure	Infrastructure	Infrastructure	Infrastructure	Infrastructure
	Social and	Social and	Social and	Social and	Social and
	economic equity	economic equity	economic equity	economic equity	economic equity
	Sectoral Water Stress Index (ISWSI), which captures the level of water stress in the different sectors,
	and the Po tential Margin (PM), which is an assessment of the available water resource supply (for
	more information please see annex).

The Water Stress Framework (WSF) is generated through a combination of two parts: the Integrated

ENVIRONMENT INDUSTRY (TOURISM) INDUSTRY (PRODUCTION) AGRICULTURE DOMESTIC Natural Baseline Endowment (NBE) (Quantity of water in the natural water cycle) PLUS any imported water (interbasin transfers, irrigation channels etc.) weighted and summed across ISWSI Integrated Sectoral Water Stress Index

	DEMAND
	Water Stress Outcomes
	SUPPLY

Table 1 : Example of the core indicators of the WSI

 1

	DOMES T IC	AGRICUL T URE T RY INDUS	INDUSTRY	ENVIRONMENT
					(PRODUCTION)	(TOURISM)
	Quantity Issues Drinking water	Irrigation dependability Water quality	Water use intensity Deviance from
							natural flow
	Quality Issues Quality norms	Salinity	Water treatment	Quality norms	Waste water
							polluted load
	Institutional and	Water regulation	Water saving	Recycling	Water saving	Protected areas
	adaptive			technologies		technologies
	capacity
	Infrastructure	Supply		Supply dependability	Supply interruptions	Water treatment	River
		interruptions					fragmentation

6.3 -Global Change and Ecosystems Doc Name: Deliverable ID: WP 2.1 D2.1-4.pdf Date: 14/02/2008 Dissemination level: PU -Final Report 6

Table 2 : Indicator results for the Flumendosa-Mulargia test site

 2 From Table2it follows that useful information could be found for 17 out of 25 indicators. The asterisks mark the indicators for which alternative indicators were employed in the absence of data on the core indicators. Table3below shows the associated core and alternative indicators. The usefulness of some alternative indicators may be questionable. For example the percentage companies with ISO 14001 is a proxy-indicator providing information about environmental aspects in general, rather than about water in particular.

		DOMESTIC G RICULTURE A	INDUSTRY	TOURISM ENVIRONMENT
	Quantity issues		1.00	0.17	1.00*	0.18	0.43
	Quality issues		0.00				0.34*
	Institutional	and	0.72	0.57	0.77*	NN	0.91
	adaptive capacity
	Infrastructure		0.66	0.43*	NN	0.61	0.23
	Society and equity	NN	1.00*	NN		0.71*

Table 3 : Alternative indicators

 3 Based on the values shown in Table2, the Water Stress Index was calculated in the following steps:

	Sector	Core indicator	alternative indicator
	Agriculture	Supply dependability	Ratio of potentially irrigated and irrigated area
	Agriculture	Farm size dispersion	Water use per unit labour force
	Industry	Water use intensity	Potential for alternative water use
	Industry	Recycling	Percentage companies with ISO-14001
	Environment	Waste water polluted	Diffuse eutrophication, percentage of untreated
		load	wastewater
	Environment	Nature protection	Percentage of wet ecosystems under threat

Table 4 and graphic representation: Indicators results for the Tadla test site

 4

		DOMESTIC G RICULTURE A	INDUSTRY	TOURISM ENVIRONMENT
	Quantity issues		0,00	0,52	NN	NN	NN
	Quality issues		0,00	0,00	NN	NN	NN
	Institutional	and	0,00	0,91	NN	NN	NN
	adaptive capacity
	Infrastructure		0,35	0,39	NN	NN	NN
	Society and equity	0,22	0,94	NN	NN	NN

CemOA : archive ouverte d'Irstea / Cemagref

for more information see www.aquastress.net CemOA : archive ouverte d'Irstea / Cemagref

Acknowledgements:

We thank the AquaStress Project "Mitigation of Water Stress through new Approaches to Integrating Management, Technical, Economic and Institutional Instruments" (Proposal/Contract no.: 511231-2), for the economic and scientific support.

Discussion

Despite the relative abundance of water use and water stress information in central databases such as Istat and Eurostat, additional local information and knowledge has proved to be essential to establish a realistic WSF, and to characterise the local water stress reliably. A combination of both data sources is recommended: local information to better pinpoint the water stress problem, and regional, centralized data to position the local situation in a regional or (inter)national context. Local data should be more relied on if the purpose is to transfer local experiences in combatting water stress to other, less experienced regions. Regional data may be more relied on in the context of policy making on national or supranational levels, for example by generating maps of WSF or ISWSF per administrive region. With this respect it should be noted that the ultimate value of the presented analysis is yet to be judged by the local project partners. However, the purpose of this paper is rather to share experiences in applying the WSF approach and finding the appropriate data.

Given the relative abundance of information for Italy within the Istat databases, it is expected that in other countries like Morocco more information must be collected locally. This would limit possibilities for cross-regional or -national comparisons between water stressed sites.

What is really needed to make the WSF-approach useful in practice?

A uniform, pan-European dataset of meaningful core indicators as shown in To clearly define the water resources of a specific site, the following is taken into account: boundaries of the surface and groundwater system, time variability in connection with change (e.g. climate change) and temporal representativeness (e.g. droughts and floods).