Valérie Cayol

Olivier Bodart

Farshid Dabaghi

Oliver Bodart

A new

Ω Γ N Γ D Γ D Γ D Γ C n Figure 1
. Volcanic domain Ω and its boundaries the solid Ω. The Cauchy stress σ(u) and strain ε(u) are given by σ(u) = λTr ε(u) I R 3 + 2µε(u) and ε(u) = 1 2 (∇u + ∇u T ),

where (λ, µ) are the Lamé coefficients of the material, I R 3 the identity matrix, and Tr(•) the matrix trace.

We also assume the presence of a crack in the volcano, which is represented mathematically by a 2-D surface Γ C ⊂ Ω on which a traction or pressure force will be exerted (see also Figure 1). In a volcanic context, Γ C might represent a magma-filled crack on a fault. As already said in the introduction, the shape and position of the crack are supposed to be known in this work. The deformation field u of the solid is supposed to satisfy the following elastostatic system:

-div σ(u) = f in Ω, (1)
u = 0 in Γ D , (2)
σ(u) • n = 0 on Γ N , (3)
σ(u) • n ± = t ± on Γ C . (4)
The first equation is the equilibrium law describing the (linear) elastic behaviour of the material.

Condition (2) describes the fact that the displacement vanishes on the underground boundary of the solid. In equation (3), and in the sequel, n denotes the unit normal vector on the boundary of Ω, oriented externally. This condition describes the free movement of the ground part of the volcano.

Finally, equation ( 4) describes the force acting on the crack Γ C . The vectors n + and n -= -n + are unit normal vectors on the crack Γ C (see Figure 2). The vector functions t ± (x) denote the force exerted on the crack. They are such that t + = -t -. When this force acts normally to the crack, that is t ± (x) = p(x)n ± , the force is called a pressure, otherwise in general it is called a traction force.

This traction force is the actual unknown of our problem. From the mathematical point of view it is supposed to be quadratically integrable as well as its gradient.

In order to derive a finite element approximation of the state equations ( 1)-( 4), we need a weak formulation of the system. This is done by multiplying the equation (1) by a regular vector field v such that v = 0, and then integrating by parts over Ω. Using the conditions (2)-(4) then yields:

Ω σ(u) : ε(v) dΩ = Ω f • v dΩ + Γ C (t ± ) • v dΓ C , (5)
where : is the double inner product for matrices, i.e. for A = (a ij ) i,j=1,2,3 and B = (b ij ) i,j=1,2,3 , Notice that this approach allow us to deal with situations where the elastic material is nonhomogeneous and anisotropic, that is when the Lamé coefficients are not constant.

A : B = 3 i=1 3 j=1 a ij b ij . Γ + D Ω - Ω + Γ + N Γ - D Γ - D Γ 0 Γ + D Γ - N Γ 0 n + Γ C n -

Finite element model

The direct problem (1)-(4) efficiently will have to be solved iteratively during the inversion procedure. Therefore we need an efficient algorithm to do so. To this aim, we use a domain decomposition method. More precisely, following [START_REF] Bodart | Xfem-based fictitious domain method for linear elasticity model with crack[END_REF], the domain Ω is split into two sub-domains such that each point of the domain lies on one side of the crack or on the crack. For this purpose, we use an artificial extension Γ 0 of the crack Γ C . Assuming that Γ F = Γ C ∪ Γ 0 splits the domain into two subdomains Ω + and Ω -, we have Ω = Ω + ∪ Γ F ∪ Ω -, Γ N = Γ + N ∪ Γ - N and Γ D = Γ + D ∪ Γ - D . We define on Γ C two opposite unit outward normal vectors n + (from Ω + ) and n -(from Ω -). The global unknown u is split into u

+ = u |Ω + and u -= u |Ω -.
Therefore, the equations ( 1)-( 4) can be rewritten with u + and u -as unknowns:

-div σ(u ± ) = f ± in Ω ± , (6)
u ± = 0 on Γ D ∩ ∂Ω ± , (7)
(σ(u) • n) ± = 0 on Γ N ∩ ∂Ω ± , ( 8
) (σ(u) • n) ± = t ± on Γ C , (9)
[[u]] = 0 on Γ 0 , (10)
[[σ(u)]] • n + = 0 on Γ 0 , (11)
where [[v]] = v + -v -denotes the jump of a function across Γ 0 . The equations and conditions ( 6)-( 9) are a straightforward rewriting of (1)-( 4). The two last conditions ( 10) and ( 11) are imposed to enforce the continuity of displacement and stress across Γ 0 to ensure that u = (u -, u + ) solves the original problem ( 1)-( 4).

Let us now describe the discretization of this problem via the finite element method. A lagrange multiplier λ defined on Γ 0 is introduced to enforce the continuity of the displacement across the Γ 0 .

Defining the associated lagrangian functional and expressing the associated saddle point conditions give a weak formulation for Problem (6)-( 11) which unknown is in the form X = (u + , u -, λ) (see [START_REF] Bodart | Xfem-based fictitious domain method for linear elasticity model with crack[END_REF] for details).

In order to numerically approximate the system, consider a tetrahedral mesh of Ω (as e.g. in Figure 3). Let ϕ ± i be a finite element basis defined on this mesh. Also defining a mesh of the extended crack, we build finite elements bases ψ i and θ j on Γ 0 and Γ C respectively.

Identifying the unknowns u + , u -and λ with there value on the nodes of the mesh, the discretized form of Problem ( 6)-( 11), then reads

KX = F. (12)
The matrix K of this system is built from the stiffness matrices A ± , and the constraint coupling matrices on Γ 0 denoted by B ± . More precisely, we have

K =      A + 0 B + T 0 A --B -T B + -B - 0     
, where

A ± := Ω ± σ(ϕ ± i ) : ε(ϕ ± j ) dΩ ± ij , B ± := Γ 0 ϕ ± i • ψ j dΓ 0 ij
The right hand side vector F ± is given by

F =      F + F - 0      , F ± := Ω ± f ± • ϕ ± i dΩ ± + Γ C t ± • ϕ ± i dΓ C i ,
which boils down to the algebraic formulation

F =      F + F - 0      =      M + Ω f + M - Ω f - 0      +      +M C t -M C t 0     
, where the mass matrices M ± Ω and the coupling matrix on Γ C , denoted by M C , are given by

M ± Ω ij = Ω ± ϕ ± i • ϕ ± j dΩ ± , [M c ] ij = Γ C ϕ ± i • θ j dΓ C .
Eventually, we can naturally define two matrices L Ω and L C such that we have

F = L Ω f + L C t.
The matrix K is symmetric and positive definite, which allows the use of powerful classical solvers. The implementation of this model uses the finite element library GetFem++ Renard & Pommier (http://home.gna.org/getfem). The library provides all the necessary tools to handle various types of PDEs, and links to classical powerful solvers (conjugate gradient, SuperLU, MUMPS) as well as other routines for mesh management, definition of cracks and boundaries via level set functions) which make it quite versatile and powerful. We refer the reader to [START_REF] Bodart | Xfem-based fictitious domain method for linear elasticity model with crack[END_REF] for a detailed mathematical and computational analysis of this method. See also [START_REF] Bodart | Fictitious domain method for an inverse problem in volcanoes[END_REF][START_REF] Bodart | An inverse problem in an elastic domain with a crack : a fictitious domain approach[END_REF] for a mathematical analysis of similar optimal control problems aking use of the domains decomposition technique.

INVERSION FROM FULL GROUND MEASUREMENTS

Mathematical formulation of the problem

Let us assume the displacement field of the volcano to be fully measured on the ground Γ N , and call it u d . The inversion problem consists in finding a traction vector t such that the solution of Problem

(1)-( 4) satisfies u = u d on Γ N . However this is known to be an ill-posed problem (e.g. the solution is not unique), and moreover the measurements are perturbed by random uncertainties. Therefore we adopt a least squares approach, combined with a regularization (i.e. smoothing) technique to build the inversion process.

Consider the following cost function:

J(t) = 1 2 Γ N (u -u d ) C -1 (u -u d ) dΓ N + α 0 2 Γ C |t| 2 dΓ C + α 1 2 Γ C |∇t| 2 dΓ C , (13)
where C denotes the covariance operator of the measurements uncertainties (see e.g. [START_REF] Tarantola | Inverse Problem Theory and Methods for Model Parameter Estimation[END_REF]),

and is assumed to be positive definite, denotes the transpose operator, and finally, the non-negative constants α 0 and α 1 are the Tikhonov regularization (or smoothing) parameters.

We aim at minimizing this cost function over the space of feasible traction vectors. This space consists in quadratically integrable vectors, which gradient is also quadratically integrable.

In [START_REF] Bodart | Fictitious domain method for an inverse problem in volcanoes[END_REF][START_REF] Bodart | An inverse problem in an elastic domain with a crack : a fictitious domain approach[END_REF], a simplified version of this problem is mathematically studied. The cost function only features the first smoothing term. As it will be shown later on, our tests have proved the necessity of smoothing the gradient, in a practical framework, to obtain a physically admissible solution t. The optimal choices for the values of α 0 and α 1 will be discussed further.

To derive the optimality conditions for the Problem (13), we introduce the following Lagrangian:

L(u, t, φ) = J(t) - Ω (σ(u) : ε(φ) -f ) • φ dΩ + Γ C t ± • φ dΓ C ,
where φ is the Lagrange multiplier for the constraint (1)-( 4) in the weak form (5). When ∇ u L = 0 and ∇ φ L = 0, we have ∇ t L = ∇J. Cancelling ∇ φ L gives problem (1)-( 4). Cancelling ∇ u L and performing integration by parts yields that φ is the solution of

           -div σ(φ) = 0 in Ω, φ = 0 in Γ D , σ(φ) • n = C -1 (u -u d ) on Γ N , (14)
called the adjoint state system. Finally, computing ∇ u L, for u and φ satisfying (1)-( 4) and ( 14),

gives:

∇J(t) = α 0 t + α 1 ∇t + (φ • n ± ). ( 15
)
We refer the reader to [START_REF] Bodart | Fictitious domain method for an inverse problem in volcanoes[END_REF][START_REF] Bodart | An inverse problem in an elastic domain with a crack : a fictitious domain approach[END_REF], for further details.

After splitting the computational domain Ω, the cost function J in the optimization problem (13) becomes:

J(t) = 1 2 Γ + N (u + -u + d ) C -1 (u + -u + d ) dΓ + N + 1 2 Γ - N (u --u - d ) C -1 (u --u - d ) dΓ - N + α 0 2 Γ C |t| 2 dΓ C + α 1 2 Γ C |∇t| 2 dΓ C . (16)
The previous optimality conditions rewrite naturally from this new formulation.

The Discrete Problem

The next step is to adapt the cost function, the adjoint state and the gradient of the cost function to the discrete system (12). To this aim, it has to be noticed that all terms in (16) are symmetric.

Discretizing the two first terms will involve the introduction of mass matrices on Γ ± N . On needs to keep the symmetry of the discrete form. Since the covariance matrix C is positive definite, it is also the same for its inverse and we can write

C = QDQ -1 , C -1 = QD -1 Q -1 ,
where Q is the matrix which columns are the eigenvectors of C and D the diagonal matrix of the eigenvalues of C. This diagonalized form allows us to write

C -1 = C -1 2 C -1 2 , where C -1 2 = QD -1 2 Q -1 .
In view of this, we rewrite the continuous cost function J as

J(t) = 1 2 Γ + N (u + -u + d ) C -1 2 C -1 2 (u + -u + d ) dΓ + N + 1 2 Γ - N (u --u - d ) C -1 2 C -1 2 (u --u - d ) dΓ - N + α 0 2 Γ C |t| 2 dΓ C + α 1 2 Γ C |∇t| 2 dΓ C . ( 17
)
As in Section 2.2, we will identify the previous functions with their values at the mesh nodes.

Define a set of finite element basis functions χ ± i (with 3 degrees of freedom at each mesh node) defined on Γ ± N . Then, define the ground mass matrix

M G =   M + G 0 0 M - G   , (18)
with

M ± G ij = Γ N χ ± i • χ ± j dΓ N .
Let us also define the matrices M F 0 and M F 1 as

[M F 0 ] ij = Γ C θ i • θ j dΓ C , [M F 1 ] ij = Γ C ∇θ i • ∇θ j dΓ C .
Recall that θ j are the finite element basis functions defined on Γ C .

Finally, in view of the domain decomposition method, the mesh nodes located on the ground will be on either side of the crack. Therefore, the measured displacement and the restriction to the ground nodes of the computed displacement (denoted u G ) can be written as

u d =   u + d u - d   , u G =   u + G u - G   .
Consider then the reduction matrix O R such that

O R X = u G ,
where X is the solution of ( 12).

The discrete version of the cost function J defined by ( 17) is then

J d (t) = 1 2 (O R X -u d ) T C -1 2 M G C -1 2 (O R X -u d ) + α 0 2 (t M F 0 t) + α 1 2 (t M F 1 t). (19)
The optimality conditions for the minimizer of the discrete cost function J d are obtained via the characterization of the saddle point of the following (discrete) Lagrangian function:

L d (X, t, φ) = J d (t) -KX -(L Ω f + L C t), φ .
Computing the partial derivative of L d and then cancelling them, lead to the discrete counterpart of the adjoint problem (14) as:

Kφ = O R C -1 2 M G C -1 2 (O R X -u d ), which rewrites, denoting u G = O R X, Kφ = O R C -1 2 M G C -1 2 (u G -u d ), ( 20
)
where K is the (symmetric) matrix of System ( 12). The gradient of J d at any point t is then:

∇J d (t) = α 0 M F 0 t + α 1 M F 1 t + L C φ. ( 21
)

Practical minimization algorithm

In a previous work (see [START_REF] Bodart | An inverse problem in an elastic domain with a crack : a fictitious domain approach[END_REF]), we presented two methods to minimize a simpler version of our cost function J d : the conjugate gradient algorithm and a quasi-Newton method (low storage BFGS). These two techniques are natural choices due to the quadratic structure of the cost function.

They were studied in depth in terms of computational performance and accuracy. The BFGS method appeared to be converging faster when the mesh of the domain Ω is rather coarse. When the mesh becomes finer, the number of unknowns in the problem increases and the two methods tend to give similar results. In the framework of applications which we are interested in, we aim at fast processes, that is we will most of the time consider meshes that are rather coarse except in the neighborhood of the crack Γ C . Therefore we will use a BFGS algorithm (see [START_REF] Broyden | The convergence of a class of double-rank minimization algorithms 1. general considerations[END_REF]; [START_REF] Fletcher | A new approach to variable metric algorithms[END_REF]; [START_REF] Goldfarb | A family of variable-metric methods derived by variational means[END_REF]; [START_REF] Shanno | Conditioning of quasi-newton methods for function minimization[END_REF]) in our numerical tests. It involves the construction of a sequence of approximation of the inverse of the Hessian matrix of the cost function J d . We use a limited storage version of the algorithm called L-BFGS (see [START_REF] Nocedal | Updating quasi-newton matrices with limited storage[END_REF]). The method is presented in Algorithm 1, with the necessary adaptations to our problem.

As usual with this type of methods sequences are built and will be denoted as follows:

• t k : traction force vector,

• X k : solution of the state equation ( 12),

• u k : displacement field on the ground,

• φ k : solution of the adjoint state equation ( 20),

• g k : gradient of the cost function J d at point t k ,

• d k , w k : displacement directions for the optimization,

• ρ k : optimal displacement step.

The number k ≥ 0 represents the iteration number.

Notice that the underlying quadratic form in the cost function allow to compute explicitely the optimal step size at each iteration, via formula (28). The detailed computation is presented in the Appendix.

In the algorithm, the domain decomposition technique is used to solve the discrete state and adjoint state equations ( 12) and ( 20). Notice that the matrix K is symmetric and does not change during the iterative process, which means that the computation cost due to matrix factorization is reduced.

The algorithm iterates until the gradient of J d becomes sufficiently small, as precised in the algorithm below.

Algorithm 1 Algorithm to minimize J d Require: t 0 , u d and > 0

k ← 0 Solve (12) with t = t 0 → X 0 u 0 ← O R X 0 Solve (20) with u = u 0 → φ 0 Initial gradient: g 0 ← α 0 M F 0 t 0 + α 1 M F 1 t 0 + L C φ 0 Initial direction: d 0 ← -g 0 H 0 ← I C while g k+1 Γ C < g 0 Γ C do Solve (12) with t = t k → W k w k ← O R W k Compute step size ρ k by (28) with u = u k , t = t k , d = d k , w = w k t k+1 = t k + ρ k d k u k+1 = u k + ρ k w k Solve (20) with u = u k+1 → φ k+1
g k+1 in (15) with t k+1 and φ k+1

Compute H k+1 via formula (30)

d k+1 = -H k+1 g k+1 k ← k + 1
end while

Numerical tests and validation with synthetic data

The numerical application is performed by a synthetic test via simulation of a flat volcano as follows:

A semi-infinite elastic domain at center with radius 100 km and extending down to [0, 20] km (see 3(b) (the yellow patches) and t ± = (0, 0, 0) on the rest of the fracture (the blue patches). In some numerical experiments, this traction is interpreted as a pressure p ± = 1.5 MPa applied on the yellow zone.

Finding a robust and efficient method to compute appropriate regularization parameters α 0 and α 1

for given mesh and noisy data, is not straightforward. The admissible solution t is dependent on the chosen α 0 and α 1 and it is crucial to obtain a good approximation of the solution to the optimization problem. For choosing the optimal parameters, a series of tests is organized by using the graphical tools. Indeed, three important criteria are compared by varying α 0 and α 1 as follows: we set α 0 ∈ {1.0E -12, 1.0E -11, . . . , 1.0E0} and α 1 ∈ {1.0E -5, 1.0E -4, . . . , 1.0E5}. Then we compare: A relative ground error defined by:

E u = Γ N |u -u d | 2 dΓ N Γ N |u d | 2 dΓ N × 100.
A relative error of traction t on the fracture source, defined by:

E t = |t -t exact | 2 |t exact | 2 dΓ C × 100.
Note that an inevitable numerical error is already produced to compute the traction t on the fracture source. Using the non-conformal mesh to implement the fracture by level set method, is the reason of this error which can be reduced by using a finer mesh around the fracture.

finally, the iteration number for a given = 1.0E -14 in Algorithm 1 to satisfying the convergence criteria

Γ C |g k+1 | 2 dΓ C Γ C |g 0 | 2 dΓ C < .
The numerical tests presented in Figure 4, show that, α 1 = 1 can be chosen as an acceptable parameter. Moreover, it seems, the above criteria are less sensitive to α 0 between 1.0E -7 and 1.0E -4.

Accordingly, we employ L-curve, another graphical tool, to trade off between two criteria: the misfit

u -u d = Γ N |u -u d | 2 dΓ N 1 2 , ( 22
)
and the norm of the traction or smoothing

t = Γ C |t| 2 dΓ C ) 1 2 . ( 23
) (b) (a) (c) (d)
Figure 5. L-curves used to find α 1 representing the best compromise between the data fit (equation ( 22)) and the smoothing (equation ( 23)) in the cost function (equation ( 13)). The fracture is located at -0. The L-curves are depicted by setting α 0 = 1.0E -7 and varying α 1 . Closest point to the origin shows the best compromise between misfit and smoothing, and it will be the acceptable α 1 . In particular case when we aim to find the pressure, the traction t may replaced by the pressure p. In Figure 5, for the above fracture and mesh, two L-curves corresponding to both pressure p and traction t are illustrated.

Fracture pressure corresponding to α 1 = 1.0E1 as an appropriate choice is presented in Figure 5(c). As shown in Figures 6 and7, in 3D realistic volcanoes where we interested in large-scale problem, the values around the 1.0E1 are yet acceptable. Normal and tangential tractions are depicted in Figure 7. Note that we suppose (0, 0, -1) as the unit normal vector n.

We then aim to consider the impact of the fracture depth to the approximated admissible solution.

To do that, the numerical experiments are repeated for the fracture at -0.9 km of depth. The best combination for α 1 = 1.0E -1 and α 0 between 1.0E -7 and 1.0E -4, in red box are shown in Figure 8. The L-curves depicted for p and t in Figure 9(a, b) show that the closest point to the origin for the 0.9 km is α 1 = 1.0E0. The normal stress projected on the fracture presented in Figure 9(c) 1, indicates that the number of iterations is augmented for the greater values of α 1 .

As seen in the numerical simulations, we may trust to the L-curves results to find an appropriate α 1 for more complicated geometry of fracture and mesh, which is clearly help us to computational cost. Table 1. Comparison between the number of iterations for two depths -0.3 and -0.9 km and when the unknowns are pressure p and traction t for α 0 = 1.0E-7.

TAKING INTO ACCOUNT THE EARTH-SATELLITE DIRECTIONS

Mathematical framework and new optimization problem

Before going further, let us first remark that the previous section can easily be adapted to the case where the measurements of the displacement field are performed only on a part of the ground Γ N ⊂ Γ N . The cost function to be minimized becomes

J(t) = 1 2 Γ N (u -u d ) C -1 (u -u d ) dΓ N + α 0 2 Γ C |t| 2 dΓ C + α 1 2 Γ C |∇t| 2 dΓ C .
Rewriting the optimality conditions and adapting the algorithms is straightforward. This is what is actually implemented in our software.

However, assuming the displacement field to be known in cartesian coordinates is not realistic for the applications. In this section we aim at adapting our work to the case where measurements are provides by satellite radar interferometry (as e.g. inFukushima et al. (????)). We will first state precisely what measurements are made and then adapt our methods to this new framework.

Let u d = (u x , u y , u z ) be a displacement field on Γ N , written in cartesian coordinates. A satellite will aim at the ground to measure a displacement, and the resulting measurement will be in the form p u d where the aiming direction (also called earth-satellite direction) p = (p x , p y , p z ) is a unit vector oriented from the ground to the satellite. Let then N ≥ 1 be the number of satellites, each associated to a direction p i = (p ix , p iy , p iz ) for i = 1, . . . , N . We can build a matrix P : R 3 → R N as

P = (p 1 , p 2 , • • • , p N )
, so that the actual measurement is

R d = Pu d .
The vector field R d ∈ R N is defined on Γ N , and the matrix P also operates on the solutions of System

(1)-( 4). Hence, given the field R d , our new optimization problem consists now in finding the traction vector t minimizing the following cost function:

J(t) = 1 2 Γ N (Pu -R d ) C -1 2 C -1 2 (Pu -R d ) dΓ N + α 0 2 Γ C |t| 2 dΓ C + α 1 2 Γ C |∇t| 2 dΓ C , ( 24
)
where α 0 > 0 and α 1 > 0 are the regularization parameters. The optimality conditions for this problem are derived in the same way than in Section 3.1. We will not go into details about this and focus on the discrete version of the problem.

Using the definitions of Sections 2.2 and 3.2, and in view of ( 19) and ( 24) the discrete cost function becomes

J d (t) = 1 2 (PO R X -R d ) T C -1 2 M G C -1 2 (PO R X -R d ) + α 0 2 (t M F 0 t) + α 1 2 (t M F 1 t). ( 25
)
where X is the solution of the discrete system (12). This finite dimensional problem then boils down to finding the saddle point of the following Lagrangian

L d (X, t, φ) = J d (t) -KX -(L Ω f + L C t), φ .
Computing the partial derivative of L d and then cancelling them, leads to the discrete adjoint problem ( 14):

Kφ = O R P C -1 2 M G C -1 2 (PO R X -R d ), (26)
and the gradient of J d is:

∇J d = α 0 M F 0 t + α 1 M F 1 t + L C φ. (27)
The minimization Algorithm 1 can then be adapted to the new adjoint state and gradient to obtain the numerical approximation of the traction t on the crack by using the surface measurements provided by radar interferometry r d .

Applications to synthetic test

The numerical applications here are proceeded in three different steps: First, we aim to the adaptation of the theoretical results in Earth-Satellite directions. For the sake of simplicity, we assume the covariance is an identity matrix. Next, in order to reduce the observed data, the identity covariance matrix is reduced to the nodal mesh points. We then consider a problem with dense covariance matrix adapted to a limited number of observed data.

The vector directions are chosen by using InSAR satellite directions, listed in Table 2. The numerical tests are performed with one to four radar looks to confirm the theoretical results. We set the same configuration as previous section for the problems in Cartesian coordinate. The horizontal circular fracture beneath the flat topography located at 0.3 km (see Figure 3). We still take α 0 = 10 -7

and by depicting the L-curves, we find the best α 1 with the best compromise between misfit and the traction norm t or pressure p (see Figure 10) Normal and tangential tractions projected on the fracture corresponding to the best α 1 = 1 are presented in Section supplementary Material, Figure ??. A comparison between the iteration numbers listed in Table 3 for the observed synthetic data provided Table 2. InSAR Unit vector directions by different number of radar looks, show that the use of more radar looks leads to a decrease in the number of iteration.

In a more realistic scenario, the number of observed data are limited. In the previous numerical experiments, we assumed a synthetic u d in P2 finite elements. The degrees of freedom's nodes of a classical P2 Lagrange elements on a 3D tetrahedron, are located on the vertices or nodal mesh points and the midpoints of the edges. In realistic volcano phenomenons (three-dimensional elasticity problem), using the P2 finite element guarantees a good approximated solution. However, we aim to reduce the synthetic u d to nodal mesh points on the ground surface as a P1 finite elements. On the other side, to conserve the P2 finite elements for the elasticity problem, we should conserve the dimension of the covariance C -1 2 and the mass matrix M G in a P2 elements. Therefore, from implementation Table 3. Comparison between the number of iterations taking into account the earth-satellite directions with identity covariance matrix in InSAR unit vector directions. The fracture is located at -0.3 km, when solving for pressure p and traction t. Here, we set α 0 = 1.0E-7.

point of view, the degrees of freedom's related to the midpoints of the edges in C -1 2 and M G are set to zero. Despite the increase in the iterations number, the numerical experiments presented still show very satisfactory results (see Section Supplementary Material Figure ??).

Another step to approach the reality is using the dense covariance matrices for limited number of data, obtained by InSAR and cGNSS.

To do that, we are using the DEfvolc interfaces The dense covariance matrices are provided by DefVolc, pre and post-processor software. We started by creating the synthetic u d and the dense covariance matrix in S4 radar look. The results obtained in Figure 11 and first row of Table 4, confirm the adaptability of our method to the dense covariance matrices. In reality, most of the time, the atmospheric contribution caused masked and noisy InSAR data. Therefore, some numerical experiments is performed by creating masked and noisy synthetic u d and covariance matrix. The numerical experiments are presented in Figure 12. The L-curves are depicted for synthetic u d projected to S4 an S6 directions. First without any mask and noise in the data and then different possible cases by adding the mask and noise to the synthetic data. We summarized appropriate choices for α 1 and the number of iterations to achieve the convergence with these α 1 , for different cases of synthetic u d , in Table 4.

Despite of choosing a greater α 1 , for the masked data, the number of iteration is reduced. However, for the noisy data the convergence of the inversion process is much more harder. Moreover the vector size of the synthetic u d for each case are listed in Table 4. 1.e-4, (j) with noise and mask and variance 1.e-4 Here, we set α 0 = 1.0E -7. Table 5. Comparison between the number of iterations for fracture at 0.3 km depth and when the unknowns are the traction t for different radar looks with dense covariance matrix for noise produced by different variance.

Noise influence

APPENDIX

The optimal step size

This section is dedicated to some mathematical details for interested readers. First part is concerned to compute the optimal step size. To simplify the presentation, let us introduce c N defined on Γ C c N (w, w) = Γ N w C -1 w dΓ N , so that the cost functional in (13) becomes

J(t) := 1 2 c N (u -u d , u -u d ) + α 0 2 Γ C |t| 2 dΓ C + α 1 2 Γ C |∇t| 2 dΓ C .
Then the directional derivative of J in the direction of a given d reads

∂J ∂t (t), d α 0 ,α 1 = c N (u -u d , w) + α 0 Γ C (t • d) dΓ C + α 1 Γ C (∇t • ∇d) dΓ C ,
where w is the solution to (5) with t = d. Therefore, we may compute the optimal step size ρ, with a search direction d by solving ∂J ∂t (t + ρd), d α 0 ,α 1 = 0,

+ α 0 Γ C (t • d) dΓ C + α 1 Γ C (∇t • ∇d) dΓ C = 0.
The optimal step size is therefore

ρ * = - c N (u -u d , w) + α 0 Γ C (t • d) dΓ C + α 1 Γ C (∇t • ∇d) dΓ C c N (w, w) + α 0 Γ C (d • d) dΓ C + α 1 Γ C (∇d • ∇d) dΓ C . ( 28

L-BFGS Update

The second part is concerned to L-BFGS Update in minimization Algorithm 1

H k+1 = (I -θ k s k y k )H k (I -θ k y k s k ) + θ k s k s k , (30)
with

s k = t k+1 -t k , y k = g k+1 -g k , θ k = 1 y k s k .

Figure 2 .

 2 Figure 2. Splitting a volcanic cracked domain by domain decomposition method

Figure 3 Figure 3 .

 33 Figure3(a)), with the elasticity parameters: the Young modulus E = 5000.0 MPa and Poisson ratio ν = 0.25. A horizontal circular fracture is defined with radius r = 1 km and located at (x, y, z) = (0, 0, 0.3) km in the topo flat. The mesh generation and level set implementation is explained in[START_REF] Bodart | Fictitious domain method for an inverse problem in volcanoes[END_REF][START_REF] Bodart | An inverse problem in an elastic domain with a crack : a fictitious domain approach[END_REF]. The ground surface is free to move and the other sides is fixed i.e. satisfy the boundary conditions on Γ N and Γ D respectively. We generated a synthetic solution u d associated to a discontinuous traction t ± = (0, 0, ±1.5) MPa, applied on a part of the fracture Γ C , as shown in Figure

Figure 4 .

 4 Figure 4. Systematic exploration of the smoothing parameters α 0 and α 1 for minimizing the cost function in equation (13). The source is located at -0.3 km depth beneath the flat topography. The acceptable combination of smoothing parameters is found by comparing (a) relative ground error, (b) relative error of traction on the disk and (c) iteration number. The source has 291 unknowns. The best compromises are indicated by the red boxes.

  3 km. (a) Lcurve when solving for pressure p. The larger points indicate the best compromise. (b) L-curve when solving for traction t. The larger points indicate the best compromise. (c) Fracture pressure corresponding to the best α 1 = 1.0E1. (d) Normal and (e) tangential tractions projected on the fracture corresponding to the best α 1 = 1.0E1. Here, we set α 0 = 1.0E -7.

Figure 6 .

 6 Figure 6. Fracture pressure located at -0.3 km, corresponding to (a) α 1 = 1.0E-1, (b) α 1 = 1.0E0 and (c) α 1 = 1.0E2

Figure 7 .Figure 8 .Figure 9 .

 789 Figure 7. Normal and tangential tractions projected on the fracture located at -0.3 km, corresponding to (a) α 1 = 1.0E0, (b) α 1 = 1.0E1 and (c) α 1 = 1.0E2. The directions of the tangetial stress are shown by red vectors.

Figure 10

 10 Figure 10. L-curves used to find α 1 representing the best compromise between the data fit (equation (22)) and the smoothing (equation (23)) in the cost function (equation (24)) taking into account the earth-satellite directions with identity covariance matrix. The fracture is located at -0.3 km, when solving for pressure p and traction t. The larger points indicate the best compromise. (a) in S4 (b) in S4 and S6 (c) in S4, S6 and TSXA (d) in S4, S6, TSXA and TSXD directions. Here, we set α 0 = 1.0E -7.

Figure 11 .

 11 Figure11. L-curves used to find α 1 representing the best compromise between the data fit (equation (22)) and the smoothing (equation (23)) in the cost function (equation (24)) taking into account the earth-satellite directions with the dense covariance matrix in S4 radar look direction. The fracture is located at -0.3 km, when solving for the traction t. The larger points indicate the best compromise. (b) Normal (c) tangential tractions projected on the fracture corresponding to the best α 1 = 1.0E1. Here, we set α 0 = 1.0E -7.

Figure 12 .

 12 Figure 12. L-curves used to find α 1 representing the best compromise between the data fit (equation (22)) and the smoothing (equation (23)) in the cost function (equation (24)) taking into account the earth-satellite directions with the dense covariance matrix. The fracture is located at -0.3 km, when solving for the traction t. The larger points indicate the best compromise. The normal and tangential traction are presented for the appropriate α 1 . The synthetic data u d is projected in S4 and S6 radar looks. (a) without any noise and mask, (d)with mask, (g) with noise (j) with noise and mask. Here, we set α 0 = 1.0E -7.

Figure 13 .

 13 Figure 13. L-curves used to find α 1 representing the best compromise between the data fit (equation (22)) and the smoothing (equation (23)) in the cost function (equation (24)) taking into account the earth-satellite directions with the dense covariance matrix to consider the noise effect to the solutions. The fracture is located at -0.3 km, when solving for the traction t. The larger points indicate the best compromise. The normal and tangential traction are presented for the appropriate α 1 . The synthetic data u d is projected in S4 and S6 radar looks. (a) with noise and variance 1.e-3 (d) with noise and mask and variance 1.e-3, (g) with noise and variance

  that isc N (u + ρwu d , w) + α 0 Γ C (t + ρd) • d dΓ C + α 1 Γ C (∇t + ρ∇d) • ∇d dΓ C = 0, which gives ρ c N (w, w) + α 0 Γ C (d • d) dΓ C + α 1 Γ C (∇d • ∇d) dΓ C + c N (u -u d , w)

) 5. 2

 2 The optimal step size for Earth-Satellite directionThe optimal step size for Earth-Satellite direction as previous, is still obtained by solving∂J ∂t (t + ρd), d α 0 ,α 1 = 0 that is c N (Pu + ρPwr d , Pw) + α 0 Γ C (t + ρd) • d dΓ C + α 1 Γ C (∇t + ρ∇d) • ∇d dΓ C = 0,where w is still the solution to (5) with t = d. This gives the optimal step sizeρ * = -c N (Pu -r d , Pw) + α 0 Γ C (t • d) dΓ C + α 1 Γ C (∇t • ∇d) dΓ C c N (Pw, Pw)

Table 4 .

 4 Comparison between the number of iterations for fracture at 0.3 km depth and when the unknowns are

	Radar look	α 1	Number of iteration Vector size u d
	S4	1.0E1	98	1616
	S4 S6	1.0E1	102	3232
	S4 S6 with mask	1.0E2	56	1087
	S4 S6 with noise	1.0E2.5	344	3232
	S4 S6 with mask and noise	1.0E3	306	1087

the traction t for different radar looks with dense covariance matrix.

of the volcano and free to move. The subset Γ D is an artificial boundary defined in order to work on a finite size object and is assumed to to satisfy a 0 displacement condition. We assume that the elastic solid occupying Ω is subject to a body force field f . In the sequel, boldface letters will be used to denote (usually 3-dimensional) vector fields. Plain letters will represent scalar quantities.We denote by x = (x, y, z) the generic point in R 3 , and by u = u(x) the displacement field of

ACKNOWLEDGMENTS CONCLUSION


Metalepsis (revised version; uploaded 13 July

2016)

In its narratological sense, metalepsis, first identified by Genette, is a deliberate transgression between the world of the telling and the world of the told: "any intrusion by the extradiegetic narrator or narratee into the diegetic universe (or by diegetic characters into a metadiegetic universe, etc.), or the inverse […], produces an effect of strangeness that is either comical […] or fantastic" ([1972] 1980: 234-35). After reviewing a few examples, Genette observes that " [a]ll of these games, by the intensity of their effects, demonstrate the importance of the boundary they tax their ingenuity to overstep, in defiance of verisimilitude-a boundary that is precisely the narrating (or the performance) itself: a shifting but sacred frontier [or boundary] between two worlds, the world in which one tells, the world of which one tells. […] The most troubling thing about metalepsis indeed lies in this unacceptable and insistent hypothesis that the extradiegetic is perhaps always diegetic and that the narrator and his narratees-you and I-perhaps belong to some narrative" (236, original emphasis). Described as "taking hold of (telling) by changing level" (235, n. 51), narrative metalepsis combines the principle of narrative levels (Pier → Narrative Levels [1]) with "author's metalepsis," a narrative figure with roots in the trope of metalepsis. Narrative metalepsis constitutes a "deliberate transgression of the threshold of embedding […]: when an author (or his reader) introduces himself into the fictive action of the narrative or when a character in that fiction intrudes into the extradiegetic existence of the author or reader, such intrusions disturb, to say the least, the distinction between levels," producing an effect of "humor" or of "the fantastic" or "some mixture of the two […], unless it functions as a figure of the creative imagination" (Genette [1983] 1988: 88).

These definitions, which remain foundational, providing the basis for a narrative category which, up to the early 1970s, had never been properly formulated, have been expanded, amended and refined by subsequent research, partly by Genette himself in his book Métalepses (2004), an exploration of the phenomenon not only in narrative fiction but also in theater, film, television, painting and photography.

Definition

John Pier Created: 11. June 2011Revised: 14. July 2016 These developments have come about with the realization that metalepsis is not a mere localized stylistic device or oddity, but also that it occurs in various forms, thus calling for the elaboration of typologies, that it can be found in media other than language and is indeed a phenomenon which is not inherently bound by or restricted to narrative, and that its effects are not exclusively anti-illusionistic. A survey of the literature suggests that the criteria for determining the occurrence of metalepsis and the conditions of its extension are the focus of as much if not more attention than the various definitions that have been set forth.

In addition to Genette's "transgression" of levels or to [START_REF] Wagner | Glissements et déphasages: note sur la métalepse narrative[END_REF] "sliding" between levels, metalepsis has been characterized as "undermining the separation between narration and story" (Rimmon-Kenan [1983] 2002: 93); as a "strange loop" in the structure of narrative levels or a "short circuit" between the "fictional world and the ontological level occupied by the author" (McHale 1987: 119, 213); as a "narrative short circuit" causing "a sudden collapse of the narrative system" (Wolf 1993: 358); or as producing a "disruptive effect on the fabric of narrative" (Malina 2002: 1). Being the "embryo" or "outline" (esquisse) of a fiction, metalepsis triggers "a playful simulation of belief" (Genette 2004: 17, 25).

As can be seen from the diversity of these characterizations (among others), current research cannot be neatly classified into clearly identified paradigms.

Nevertheless, three partially overlapping conceptions do seem to stand out, all deriving more or less directly from the definitions of narrative metalepsis listed above, although with little reference to its connection with the trope of metalepsis:

Rhetorical vs. ontological metalepsis

This distinction is contained implicitly in Genette's definitions and examples but is not systematically elaborated.

Transmedial dimensions

This approach expands investigations to include non-verbal and plurimedial manifestations of the phenomenon. A type of metareference, metalepsis, particularly in its ontological form, possesses a potential for self-reference and thus for laying bare the fictionality of the work in which it appears.

Metalepsis as paradox

Recentering on the original definition of narrative metalepsis, this approach insists not on the rhetorical/ontological distinction but on the logically paradoxical movements between at least two hierarchically 2 Explication distinct text-internal narrative levels.

On the whole, discussions support the idea that metalepsis appears only in fictional contexts. Essentially, it functions with varying dosages of three parameters: (a) illusion of contemporaneousness between the time of the telling and the time of the told; (b) transgressive merging of two or more levels; (c) doubling of the narrator/narratee axis with the author/reader axis. These features are illustrated by Balzac's "While the venerable churchman climbs the ramps of Angoulême, it is not useless to explain… "-a "minimal" metalepsis (cf. Pier 2005: 249-50) which, being incipiently transgressive, leaps the boundary between narrator and extradiegetic narratee on the communicative plane and puts story time on hold while the narrator, in a relative cohabitation with the character, intervenes with a metanarrative comment, demonstrating the latent metaleptic quality of narrative embedding in general. This example leads to the idea that fictional narrative is by nature metaleptic, that it is bound to the paradox of "a current presentation of the past" (Bessière 2005), that it betrays "at least the potential for narrative metalepsis" (Nelles 1997: 152).

It is important to bear in mind that although metalepsis has its roots in ancient rhetoric, narrative metalepsis is a recent concept in the history of poetics, with the practice itself, under different denominations, or none at all, reaching back to antiquity in both literary and visual forms, as copiously demonstrated by a recent anthology edited by Eisen and von Möllendorff (2013). The fact that metalepsis can now be theorized and applied according to definable criteria has opened up avenues of historical research that extend beyond the corpus of modernist and postmodernist works habitually taken into consideration in the study of the concept and the practice.

The etymology of metalepsis is disputed, but its sense can readily be grasped from the word's Latin equivalent-transumptio: "assuming one thing for another."

Metalepsis has a complex history in that it has been regarded either as a variety of metonymy, a particular form of synonymy, or both. As metonymy, it has been identified (a) in simple form as an expression of the consequent understood as the antecedent or vice versa and (b) as a chain of associations ("a few ears of corn" for "a few years," the transfer of sense implying "a few harvests" and "a few summers"). Another possibility is to regard metalepsis in terms of an overlap between synonymy and homonymy in such a way as not to respect the semantic 3 History of the Concept and its Study

The Rhetorical Background

demarcation between distinct signifiers, resulting in the use of an inappropriate synonym: cano ("sing") is a synonym of canto ("sing") and canto ("repeat") a synonym of dico ("relate"); therefore, cano is a synonym of dico (cf. Lausberg [1960] 1973: § 571;Morier 1961;Burkhardt 2001;Meyer-Minnemann 2005: 140-43;Roussin 2005: 41-4).

From the perspective of narrative theory, two positions derive from the rhetoric of metalepsis. Genette (2004: 7-16), drawing on the first of the two types above, notes that metalepsis shares with metaphor and metonymy the principle of transfer of sense, and he considers it (following Dumarsais) a metonymy of the simple type; he then expands it (with Fontanier) beyond the single word to include an entire proposition. Metalepsis of antecedent and consequent, he argues, is implicitly metalepsis of cause for effect or effect for cause. From such causal relations he forges the notion of author's metalepsis whereby an author "is represented or represents himself as producing what, in the final analysis, he only relates" (Fontanier). He also draws attention to the proximity for the two rhetoricians of metalepsis and hypotyposis (a figure in which the copy is treated, illusorily, as though it were the original, as in a present-tense description), but particularly to the fact that, with metalepsis, the narrator transgresses not merely the threshold of narrative but that of representation, resulting in a "reduced metadiegetic" or "pseudodiegetic" narrative in which, due to the lack of metadiegetic relay, the secondary narrator effectively takes the place of the primary narrator (see also Genette [1972Genette [ ] 1980: 236-37: 236-37; a more radical form is "heterodiegesis," which "gathers in one single universe the world of production, fiction and reception"; Rabau 2005: 60).

There have also been proposals to refer narrative metalepsis back to metalepsis as use of an inappropriate synonym, notably by Meyer-Minnemann (2005) and [START_REF] Schlickers | Inversions, transgressions, paradoxes et bizzareries. La métalepse dans les littératures espagnole et française[END_REF] (see also Nelles 1997: 152-57). The emphasis here is not on authorial metalepsis as a type of metonymy, but on the transgression of boundaries, of which there are two main types: one at discourse level, with breaching of the "mehere-now" of enunciation (in verbis transgression), the other at story level, with violation of the coordinates of the enunciate (in corpore transgression) (see § 3.2.1 below).

Recent research has taken a somewhat different view of the rhetorical heritage of narrative metalepsis. Thus, Nauta (2013a), re-examining the sources of metalepsis from antiquity to Dumarsais and Fontanier, delineates two strains, one concerned with allusion (following Quintilian), the other with narrative (metalepsis as metonymy of the preceding and the following). The latter, he maintains, is a trope in its own right, "operating on an expression signifying the act of representing a situation or action, in which such an expression is substituted by one signifying the act of creating or causing that situation or action" (477)-a conception which is close to the narratological definition: "any intrusion by the extradiegetic narrator or narratee into the diegetic universe (or by diegetic characters into a metadiegetic universe, etc.), or the inverse" (Genette [1972(Genette [ ] 1980: 234-35: 234-35). Yet, in his more recent work on the topic, Genette advocated a special case of metonymy in which cause is substituted for effect or effect for cause, proposing, accordingly, "to restrict the term 'metalepsis' from now on to a manipulation-at least figural, but sometimes fictional […]-of this particular causal relation which, in one way or another, connects the author to his work or, more broadly, the producer of a representation to this representation itself" (Genette 2004: 13-4). But from Nauta's rhetorical perspective, "manipulation" of causal relation is not substitution of cause and effect, and it is, moreover, inconsistent with metalepsis as a reflexive relationship between narrative levels (2013a: 479-80; this position is also rallied to by Klimek 2010: 34-7).

The connection between the metalepsis of ancient grammar and rhetoric and narrative metalepsis is "rather tenuous," as Nauta observes. Nevertheless, important work has been undertaken in the study of the metaleptic features of ancient literatures such as de Jong's (2009) seminal discussion of apostrophe, the blending of narrative voices and other techniques in Greek texts, Baumann's (2013) study of metalepsis in ancient ekphrasis or Nauta's (2013b) considerations on metalepsis and metapoetics in Latin poetry, but also Cornils ' (2005) essay on the metaleptic effects of evidentia, a specific form of phantasia (characterized by a persuasive function) in the Acts of the Apostles, to mention only a few sources. One major finding of these studies is that, unlike modern practices, metalepsis in ancient literatures is a serious technique which is used not for comic or anti-illusionistic effects, but rather as a means for increasing the narrator's authority and intensifying the credibility of the narrative. This suggests the need for further work on the rhetorical dimension of metalepsis, possibly in conjunction with pragmatics and the theory of argumentation.

One widely acknowledged group of theories, originally formulated by [START_REF] Ryan | Metaleptic Machines[END_REF]

2006), consists in breaking metalepsis down into rhetorical (Genette) and ontological (McHale) forms. This represents an extension of Ryan's theory of illocutionary and ontological boundaries, frames and stacks (cf. Pier → Narrative Levels [1], § 3.2.3) in so far as it incorporates the transgression of boundaries which, in principle, are

Principal Approaches

Rhetorical vs. Ontological Metalepsis

inviolable in narrative. The distinction remains implicit in Genette, she notes, although his more recent "figural" vs. "fictional" metalepsis corresponds roughly to her own. The rhetorical variety "opens a small window that allows a quick glance across levels, but the window closes after a few sentences, and the operation ends up reasserting the existence of the boundaries" while the ontological type "opens a passage between levels that results in their interpenetration, or mutual contamination" (207). Taking a cue from McHale (1987: chap. 8), Ryan defines ontological metalepsis in accordance with Hofstadter's (1979: 10, 621) Strange Loops and Tangled Hierarchies, and she further comments on the connection of the violation of narrative hierarchies with similar phenomena in logic, mathematics, language and science. As for Ryan's rhetorical metalepsis, Klimek (2010: 65) finds this inappropriate, and, referring instead to Cohn's ([2005] 2012) metalepsis at the discourse level and metalepsis at the story level, she proposes the term "discourse metalepsis." Ryan's distinction has been further broken down by Fludernik (2003). On scrutinizing Genette's narrative metalepsis, she concludes that this is an umbrella term which A related group of theories, less focused on the rhetorical/ ontological divide, emphasizes what [START_REF] Wagner | Glissements et déphasages: note sur la métalepse narrative[END_REF] has termed "metaleptic movements." Wagner divides these movements into three varieties: (a) from a higher to a lower level (extra-to intradiegetic or, jumping a level, intra-to metadiegetic; also intra-to metadiegetic: an author intervenes in her fiction); (b) from a lower to a higher level, proceeding in the opposite direction, as when a character transgresses the extradiegetic boundary; (c) "auto-intertextuality" between diegeses of the same level, thus confronting parallel heterogeneous fictive universes. He also takes up the question, largely neglected, of the compositional distribution of metalepses: their location, amplitude and frequency can have a significant impact on the strategy and readability of a narrative (on this point, see also Häsner 2001: 40-3). Two comments, however. First, although he does not use the term, Wagner implicitly adopts Bal's "hypodiegetic" inversion levels, (a) being an ascending transgression for Genette and (b) a descending transgression (the latter dubbed "antimetalepsis" by Genette 2004: 27). Second, the metaleptic status of (c), later called "horizontal" metalepsis (dubbed "lateral" metalepsis by [START_REF] Thoss | When Storyworlds Collide: Metalepsis in Popular Fiction, Film and Comics[END_REF]"horizontal transmigration" by Alber &Bell 2012: 169), has been contested, notably by Klimek ( 2010: 67-8;[START_REF] Hofer | Metalepsis in Live Performance: Holographic Projections of the Cartoon Band 'Gorillaz' as a Means of Metalepsis[END_REF]cf. Hanebeck 2014: 63-4), who considers this feature to be akin to intertextuality or quotation. The phenomenon has been studied under the name of "transfictionality," defined by Saint-Gelais as "the phenomenon by which at least two texts, by the same author or not, jointly relate to a same fiction, be it by taking over characters, extending a prior plot or sharing a fictional universe" (2011: 7). The disputed status of these "horizontal" movements has recently been revisited by Lavocat (2016: 498-99), who proposes the notions of "transfictional metalepsis" and "self-referential author metalepsis." Klimek herself is among those to subscribe to descending and ascending metalepsis (see § 3.2.3 below). But mention must also be made at this point of the model elaborated by Meyer-Minnemann (2005) and [START_REF] Schlickers | Inversions, transgressions, paradoxes et bizzareries. La métalepse dans les littératures espagnole et française[END_REF]. Taking a cue from Genette, this model provides for metalepsis of enunciation (in verbis, at discourse level) and metalepsis of the enunciate (in corpore, at story level), where each functions either vertically (bottom-up or top-down) or horizontally, without change of level (dubbed "perilepsis" by Prince 2006: 628). To take only a few illustrations: Bovary. In this system metalepsis is seen as producing an effect of strangeness, either comical or fantastic, but it is not considered a figure of fictionality in Genette's (2004) sense. The Meyer-Minnemann/Schlickers model of metalepsis forms part of a larger theory of "paradoxical narration" in which devices are employed either to cancel out boundaries (syllepsis, epanalepsis, the latter type including mise en abyme) or to transgress boundaries (metalepsis, hyperlepsis, the latter equivalent to pseudodiegesis: metadiegetic narrative presented as though it were diegetic). For an exhaustive typology following these parameters, identifying forty-one subtypes, see Lang (2006).

In an earlier model, Nelles (1997: 152-57) differentiates metalepsis as being either "unmarked" (occurring at discourse level) or "distinctly marked" (occurring at story level). The latter divides into "intrametalepsis" (movement from the embedding to the embedded level) and "extrametalepsis" (movement in the opposite direction), with each type possessing analeptic and proleptic forms on the temporal plane (for the related notions of "inward" vs. "outward" metalepsis, see Malina 2002: 46-50).

Rather than the rhetorical (discourse) vs. ontological (story) distinction, Nelles, invoking epistemic (vertical) and ontological (horizontal) embedding, adopts epistemological or verbal metalepsis (knowledge of the other world) as opposed to ontological or modal metalepsis (physical penetration of the other world). However, this latter pair partly reduplicates and contradicts the other distinctions while the classification as a whole leaves little room for the transgressive or paradoxical nature of metalepsis.

In a proposal that partly cuts across the above models, Pier (2005: 252-53) sets descending metalepsis off from ascending metalepsis. The former, which occurs in Fludernik's authorial and narratorial (type 1 ontological) varieties, follows an intrametaleptic movement while the latter, found in the lectorial (type 2 ontological) variety, involves an extrametaleptic movement; as for discourse (or minimal) metalepsis, it remains poised, sometimes precariously, between the two movements. Moreover, intrametaleptic movements mark an affinity between narrator and narratee, and extrametaleptic movements an affinity between character and narratee. Finally, these movements pertain both in external metalepsis (between the extradiegetic and the intradiegetic levels) and in internal metalepsis (occurring between two levels within the story itself; cf. Cohn [2005] 2012).

In a recent critical overview, Lavocat (2016) identifies two major approaches to metalepsis, one favoring the boundaries of fiction (Genette), the other the erasure of these boundaries (McHale). Examining the issues through the possible worlds approach to fiction and emphasizing the relations between (narrative) worlds rather than hierarchical levels, she observes the tendency in the latter approach, which is focused largely on ontological metalepsis, to stress the fusion of worlds, thereby effacing the boundary between fact and fiction, a tendency favored, in part, by expanding study of the phenomenon to the various media. Lavocat insists, however, on the intrafictional quality of metalepsis-not its relation with the "real"-thus on playing with the boundaries, with reality-in-fiction and fiction-in-fiction, rather than suppressing them. Transgressive by its paradoxical nature, metalepsis owes its power to its degree of literalness, and on this basis she proposes, in place of the rhetorical/ontological divide, three "degrees" of metalepsis: (a) passage from one level of embedding to another through an act of enunciation or its equivalent in other media; (b) (non-)fictional representation of the author or reader/spectator, but at a level different from that of the characters; (c) authors, reader/spectators and characters in same world.

Originally, metalepsis was formulated within the scope of language-based narratives, and its study was largely reserved to works of high culture and the avant-garde. Rather quickly, however, it was realized that the phenomenon also extends to other media as well as to works of popular culture, particularly those involving plurimedial and/or non-narrative forms of representation. Examples can be found in Genette (2004) and in Pier and Schaeffer, eds. (2005) but also in Kukkonen and Klimek, eds. (2011), not to mention a host of other publications too numerous to mention here. (On transmediality, see Ryan → Narration in Various Media [2], Ryan ed. 2004.) An important step toward a specifically transmedial conception of metalepsis was taken by Wolf (2005) who, looking at examples from drama, film, comics and painting, laid the foundations for "exporting" metalepsis to media other than language. Four features are singled out that enable metalepsis to occur beyond verbal media: (a) it is found within artefacts/performances that represent possible worlds (cf. Ryan 1991: esp. chap. 9), but has no essential link with narrativity; (b) existence within these artefacts/ performances of distinct levels or possible (sub)worlds that differ from one another with reference to "reality" vs. "fiction" (the latter combining "fictio" as artefact and "fictum" as "invention without direct reference to reality"; Wolf 1993: 38-9)

; (c) actual transgression between or confusion of (sub)worlds; (d) paradoxical nature of the transgression with reference

to a "natural" or conventional belief in the inviolability of these (sub)worlds in "normal" life and fiction. On this basis, metalepsis, in any medium, is defined as " a usually intentional paradoxical transgression of, or confusion between, (onto)logically distinct (sub)worlds and/or levels that exist, or are referred to, within representations of possible worlds" (Wolf 2005: 91, original emphasis;for a commentary, cf. Hanebeck 2014: 66). Note, however, should be made of the fact that this definition (as is the case with the partial redefinition in Wolf 2009: 50) is heavily weighted in favor of ontological (i.e. story level) metalepsis involving impossible physical transgressions, and that although rhetorical metalepsis is included in the discussion, the different types of metaleptic movements mentioned in the previous section are not taken into account. Also introduced is epistemological metalepsis, the "impossible" knowledge characters might have of their fictional status, the effect of which is to reflect the metareferential nature of metalepsis, although metareferential potential remains highest in the ontological form, laying bare the fictionality of the work . Following up on Wolf is Thoss's transmedial definition of metalepsis as "a paradoxical transgression of the line that separates the inside from the outside of a storyworld" (2015: 4) together with three "prototypes": (a) transgressions between a storyworld and another imaginary world;

Transmedial Dimensions

(b) transgressions between a storyworld and reality; (c) transgressions between a storyworld and the discourse that represents it.

Affirming the indisputably transmedial quality of metalepsis, Hanebeck, following a different line of reasoning, focuses on how this phenomenon resists the narrative structuration of (hermeneutic) understanding by denying the logical and pragmatic

rules that govern the act of narration. "Metalepsis," he observes, "occurs when and if a recipient of a (narrative) representation feels that the logic of acts of (narrative) representation are violated or negated in such a way that the 'natural' spatial, temporal and hierarchical relationships between the domain(s) of the signifier and the domain(s) of the signifier no longer apply" (Hanebeck 2014: 69). Building on existing typologies (Nelles 1997;Fludernik 2003;Klimek 2010;[START_REF] Ryan | Metaleptic Machines[END_REF] 37). On this basis, he proposes a "tree structure" of metaleptic types in which "figurative" metalepsis subsumes epistemological metalepsis and rhetorical metalepsis, on the one hand, and where ontological metalepsis breaks down into "recursive" and "immersive" metalepsis, on the other, each with a series of subtypes .

Exploring the transmedial dimensions of metalepsis poses the challenge, as the above proposals show, of rethinking narrative metalepsis so as to accommodate the features of visual and performance media, for which the language-based storydiscourse distinction is not well adapted. One option is of course to address the issues through ontological reconceptualization. Another possibility is to take into consideration so-called media affordances, i.e. how the various media influence and shape the forms of representation, but also how, in the different media environments, metalepsis interacts with representation. This is the avenue chosen by the contributors to Kukkonen and Klimek, eds. (2011), a collection of essays on metalepsis in media-rich artifacts drawn from popular culture. In her introductory essay, Kukkonen (2011b) identifies the essential terms of metalepsis as worlds, boundaries and transgressions along with their types, effects and functions; she also provides a "basic matrix of types" applicable across media which allows for various combinations of the direction (ascending or descending) and mode (rhetorical or ontological) of metalepsis as well as for horizontal or intertextual metalepsis-a matrix that overarches the various models developed in the volume.

On the basis of an exhaustive typology developed out of this matrix, Limoges (2011) demonstrates the strong potential of animation film for illusionistic extradiegetic transgressions, both ascending and descending. This is unlike comics, where the "gutter" between panels that governs the page layout offers possibilities of foregrounding such that a character might lift the corner of a page to hide an object in the image (or throw it out), thus highlighting the production process through ontological metalepsis (Kukkonen 2011a). Klimek (2011: 26-7) observes that if metalepsis in the performing arts has a potential for spilling over into the audience's "real" world, this is not the case in narrative fictions, where it can occur only between levels within the artifact (on metalepsis in film, theater, the visual arts and picture books, see Klimek 2010: 73 ff.). Where Klimek considers horizontal "intertextual" transgressions not to be metaleptic, Feyersinger ( 2011), studying trans-world "crossovers" in TV series and spinoffs in which characters and situations are carried over from one show to another, sees crossovers and metalepses as two poles along a spectrum of world-connecting devices that share certain elements and effects. As shown by these and other essays, technical innovations brought in by the mass media and, more recently, by the digital technologies, have contributed significantly to the use of metalepsis and to the diversity of metaleptic effects in the popular culture corpus.

At the heart of metalepsis is transgression of the "sacred boundary" between the world of the telling and the world of the told. In the logic of representation, levels of existence are distinct, and their violation constitutes a paradox. In literary theory such paradox is often understood in the everyday sense of a statement contrary to received opinion or belief, something "unnatural." In the technically logical sense, however, paradox is an issue that arises in self-reference, as illustrated by the liar's paradox, where the principle that a proposition cannot be both true and false at the same time is contradicted (Epimenides, a Cretan, says "All Cretans are liars")-a mind-bender also conveyed visually by the Möbius strip, Klein's bottle and Escher's drawings. Hofstadter (1979) examines various manifestations of this paradox in modern mathematics and science, even providing a recursive dialogue (103-26) that illustrates the problem of metalepsis, although the term appears nowhere in the book.

It is important to note that paradox has been integrated into the poetics of postmodernist fiction, a type of writing which, according to McHale (1987), "foregrounds ontological issues of text and world" (27). Adopting an ontology taken from possible worlds theory (33-6), McHale recasts Genette's narrative levels in terms of ontological levels, and he goes on to describe metalepsis as "the ontological dimension of recursive embedding" (120). Metalepsis is characterized, on the one hand, as a "short circuit" between the "fictional world and the ontological

Metalepsis as Paradox

level occupied by the author" (213), a special case which, as observed by Klimek ( 2010: 57), corresponds to Genette's author's metalepsis. On the other hand, the violation of narrative levels in more complex forms of metalepsis is identified with the "Strange Loop," a phenomenon that occurs "whenever, by moving upwards (or downwards) through the levels of some hierarchical system, we unexpectedly find ourselves right back where we started," and also with a subcategory of the Strange Loop, the "Tangled Hierarchy": "when what you presume are clean hierarchical levels take you by surprise and fold back in a hierarchy-violating way" (Hofstadter 1979: 10, 691;qtd. in McHale 1987: 119). Conceptually speaking, however, short circuits and strange loops/tangled hierarchies are not of the same order.

In a refinement of this model, Wolf (1993: 349-72), considering the forms of disturbance of mimetic illusion caused by the failure to observe ontological boundaries, sets the "contamination" of extra-fictional reality with textually produced fiction off from that of inner-fictional boundaries. Unlike in McHale (1987), where metalepsis, short circuit and strange loop are employed synonymously, here it is only the latter, inner-fictional form that gives rise to metalepsis, also called "narrative short circuit" by Wolf, a metafictional technique whose effect is to trigger "a sudden collapse of the narrative system" (358). Narrative short circuits appear punctually either (a) between the extradiegetic and the intradiegetic levels or (b) between the intradiegetic and one or more hypodiegetic levels, although no distinction is made between descending and ascending metalepsis as discussed in the previous section. To these simple forms of metalepsis is added a complex form in which the previous two types are combined, setting in motion a recurrent Möbiusstrip-like contamination of levels, as would be the case of a first-person narrator confronted with her own fictionality on reading a text about herself.

It is against the backdrop of a critical discussion of McHale, Wolf and other authors

that Klimek sets out a theory of metalepsis, not in terms of ontology but rather of paradox. All metalepses, she argues, are paradoxical, but not all forms of paradox (e.g. temporal, spatial) are metaleptic. The "short circuit" metaphor is rejected and with it the idea that metalepsis "collapses the narrative system," thereby systematically disrupting aesthetic illusion. Klimek's conception is in fact closely aligned with Genette's original definition with which, not surprisingly, the expansion of metalepsis from "figure" to "fiction" (cf. "All fictions are woven through with metalepses"; Genette 2004: 131) is judged incompatible (Klimek 2010: 36).

The typology of metalepsis developed out of these considerations makes no reference to the rhetorical vs. ontological paradigm or to the reality vs. fiction divide evoked by many of the transmedial approaches. Rather, three major types are identified: (1) descending metalepses, passing (a) from extradiegesis to intra-or hypodiegesis, or (b) from intradiegesis to hypodiegesis; (2) ascending metalepses, going in the opposite direction; (3) complex forms including (a) Möbius-strip narratives in which ( 1) and ( 2) fold recursively onto one another, the intradiegesis turning out to be the extradiegesis and vice versa, and (b) tangled heterarchy, where the representing and the represented are not hierarchically ordered (in computer science heterarchy is "a structure in which there is no single 'highest level '";Hofstadter 1979: 134) (Klimek 2010: 69-72;[START_REF] Hofer | Metalepsis in Live Performance: Holographic Projections of the Cartoon Band 'Gorillaz' as a Means of Metalepsis[END_REF].

It will be noted that with the introduction of complex forms this typology rules out horizontal metalepsis (e.g. [START_REF] Wagner | Glissements et déphasages: note sur la métalepse narrative[END_REF]Meyer-Minnemann 2005;[START_REF] Schlickers | Inversions, transgressions, paradoxes et bizzareries. La métalepse dans les littératures espagnole et française[END_REF]Lang 2006;[START_REF] Thoss | When Storyworlds Collide: Metalepsis in Popular Fiction, Film and Comics[END_REF]. This is due to the fact that the representation of parallel worlds belonging to the same level entails no transgression between the world of the telling and the world of the told (Klimek 2010: 68). Moreover, the complex forms, although compatible with Genette's original treatment of metalepsis, were not foreseen by him, or in any case they were nearly ruled out ([1983] 1988: 88).

Finally, underlying Klimek's system is an explicit theory of metareference which incorporates paradox: (a) gradated metareference demanding a strict separation of sign levels; (a.1) infinite metareference, a gradated and never-ending circular repetition; (a.2) recursive metareference, e.g. mirror within a mirror; (b) paradoxical metareference, as in Escher's Drawing Hands (cf. Fricke 2003, 2011: 256-57;Klimek 2010: 51, 330 ff.).

As research on metalepsis has advanced, so too has reflection on the conditions, diversity and nuances of its effects. Noted early on for the strangeness of its comic or fantastic effects or the mixing of humor and the fantastic and also as something "troubling," metalepsis has been characterized as "a figure of the creative imagination" (see § 1 above). Between its deconstructive "mutinous nature as a narrative device that disrupts narrative structure" (Malina 2002: 132) and its immersive qualities there lies a store of positions on these issues.

For a starter, it is more likely that metalepsis will be encountered in the baroque, in romanticism and in postmodernism than in classicism or realism, and also that it will be employed in the comic and fantastic genres more readily than in tragedy or in lyric poetry (Pier & Schaeffer 2005: 10-1). Moreover, the effects will vary widely according to the media and combinations of media in which metaleptic devices are employed (e.g. Wolf 2005;Kukkonen & Klimek eds. 2011).

The anti-illusionistic quality of metalepsis has never been called into question. Even so, there remains the thorny question of knowing under what conditions it is illusionbreaking or illusion-building. Metalepsis has been described by Wolf as a radically

Effects

disruptive metafictional device that prevents immersion and aesthetic illusion [START_REF] Wolf | Ästhetische Illusion und Illusionsdurchbrechung in der Erzählkunst: Theorie und Geschichte mit Schwerpunkt auf englischem illusionsstörenden Erzählen[END_REF](Wolf : 356 ff., 2005: 103;: 103;Wolf → Illusion (Aesthetic) [3]). But in consideration of his work on metareference, he has more recently come to the view that similar metaleptic devices may, subject to "filter factors" such as the intracompositional makeup of the work, generic frames and habituation, produce different effects and possibly contribute to immersion: "the feeling of experientially participating in a representation" (Wolf 2013: 121). [START_REF] Schaeffer | Métalepse et immersion fictionnelle[END_REF] takes a different view of the matter. From a cognitive perspective, metalepsis, as a representational technique, is not incompatible with immersion but serves, rather, as an "emblem" of the "split state" of immersion: "the dynamics of immersion involves metaleptic mental operations in the most literal sense of the term" (333; for a critique, cf. Wolf 2013: 121, n. 14;on metalepsis and "double-scope" cognitive blending, see Feyersinger 2012). Klimek (2010), focusing on the device itself, looks at the issues in the context of descending and ascending metalepses. The former, both as production (cf.

author's metalepsis) and as reception (cf. reader immersion), tends toward aesthetic illusion whereas the latter (when for instance a character bursts out of the fiction) postulates a higher and purely fictitious reality .

It is also possible to consider the effects of metalepsis through the lens of defamiliarization. Metalepsis was never identified as such by the Russian formalists, but it can be associated with one of their key concepts: "laying bare the device."

Rather than a rhetorical figure, the violation of ontological boundaries or a paradox, and rather than culminating in the collapse of narrative categories or in the breaking of mimetic illusion, metalepsis conceived as laying bare the device enters the work's composition via sjužet construction: more even than digressions, parallelisms, etc., it highlights the artificial relations between "form" and "material," between sjužet and fabula, and thus supports the idea that art is "made" of devices. These principles were set out particularly in Šklovskij ([1921] 1990). This famous essay discusses the digressions and various techniques employed in Tristram Shandy for conflating narration and action in a conspicuous way so as to defamiliarize the objects of perception in the process of sjužet construction, compelling the reader to a heightened awareness of the constructedness of narrative (cf. Schmid 2005Schmid , [2005Schmid ] 2010: 176-79): 176-79).

Mise en abyme is founded on a relation of similarity between the embedded and embedding stories-"simple," "infinite" or "aporetic" reduplication or reflection, according to Dällenbach (1977)-rather than on transgression. Although both phenomena are dependent on levels, they must not be confused. Even so, there is a significant coincidence between the aporetic form ("fragment supposedly including

Related Concept: Mise en abyme

the work in which it is included"; 51), or what Cohn ([2005] 2012) termed "pure mise en abyme," and metalepsis. The two are bound together by the troubling effect produced on the reader by the "unacceptable and insistent hypothesis that the extradiegetic is perhaps always diegetic and that the narrator and his narratees-you and I-perhaps belong to the same narrative" (Genette [1972(Genette [ ] 1980: 236): 236). Such a mise en abyme, triggering a sense of vertigo, is the product of a Möbius-strip-like metalepsis, or paradoxical iteration occurring in the system of metareference (cf. Fricke 2003(cf. Fricke , 2011: 257): 257).

More than a rhetorical flourish, metalepsis raises the question of the porosity of Alber, [START_REF] Thon | Zur Metalepse im Film[END_REF]Alice Bell (2012). "Ontological Metalepsis andUnnatural Narratology." Journal of Narrative Theory 42.2, 66-92. Baumann, Mario (2013). "Der Betrachter im Bild. Metalepsen in antiken Epphrasen."

U. E. Eisen & P. von Möllendorff (eds.). Über die Grenze. Metalepse in Text-und Bildmedien des Altertums. Berlin: de Gruyter, 257-91. Bessière, Jean (2005). "Récit de fiction, transition discursive, présentation actuelle du récit, ou que le récit de fiction est toujours métaleptique." J. Pier & J.-M. Schaeffer (eds.). Métalepses. Entorses au pacte de la représentation. Paris: Éd. de l 'EHESS, 4 Topics for Further Investigation 5 Bibliography

Works Cited

  contains an implicit four-term typology: (a) authorial metalepsis (Virgil "has Dido die"): a metafictional strategy that undermines mimetic illusion, foregrounding the inventedness of the story; (b) narratorial or type 1 ontological metalepsis (in Eliot's Adam Bede, the narrator invites the narratee to accompany him to Reverend Irwine's study): transgression from the extradiegetic to the intradiegetic level is illusionary, drawing a fine line between the reader's immersion and lifting of the mimetic illusion; (c) lectorial or type 2 ontological metalepsis (in a story by Cortázar, the reader of a novel is [almost] killed by a character in that novel): implication of the narratee on the story level or passage of a character from an embedded to an embedding level (also occurs in second-person narration); (d) rhetorical or discourse metalepsis (simultaneity of time of the telling/time of the told; cf. Pier 2005: 249-50 on "minimal" metalepsis).

  (a) vertical metalepsis of enunciation (top-down) obtains when an extradiegetic narrator transgresses the intradiegetic boundary; (b) horizontal metalepsis of enunciation occurs with the juxtaposition of two communicative situations at the same level; (c) with transgression of the diegetic, ontological, spatial or temporal order there occurs a vertical metalepsis of the enunciate; (d) horizontal metalepsis of the enunciate is produced when, say, Woody Allen enters the world of Madame

  2006), Hanebeck goes on to single out four "scales" along which diegetic universes, represented through any medium, may either remain distinct or collapse: (a) temporal dimensions; (b) spatial dimensions; (c) worlds ("the holistically structured networks of meaning and relatedness") which coincide with diegetic levels; (d) hierarchical relations (

  levels and boundaries in narratives and in other cultural representations, but not their dissolution. Research in recent years has expanded the scope of the phenomenon considerably and contributed to significant refinement of scholarly understanding of its workings and modalities. Among topics requiring additional study are the following: (a) relative weight of local vs. global effects of metalepsis; (b) metalepsis and fictionality (breaking/intensification of mimetic illusion, immersion, etc.); (c) the role of metalepsis in trans-/intermediality with regard to multimedia and to popular culture; (d) metalepsis and related practices in historical poetics going back to ancient narrative as well as a historical inventory of artistic movements and corpuses employing these devices; (e) the rhetorical potential of metalepsis.

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