B G Giraud
email: giraud@spht.saclay.cea.fr

K Katō
email: kato@nucl.sci.hokudai.ac.jp

A Ohnishi
email: ohnishi@nucl.sci.hokudai.ac.jp

Complex Scaled Spectrum Completeness for Coupled Channels

The Complex Scaling Method (CSM) provides scattering wave functions which regularize resonances and suggest a resolution of the identity in terms of such resonances, completed by the bound states and a smoothed continuum. But, in the case of inelastic scattering with many channels, the existence of such a resolution under complex scaling is still debated. Taking advantage of results obtained earlier for the two channel case, this paper proposes a representation in which the convergence of a resolution of the identity can be more easily tested. The representation is valid for any finite number of coupled channels for inelastic scattering without rearrangement.

I. INTRODUCTION, NOTATIONS

As is well known, the CSM converts the description of resonances by non-integrable Gamow states into one by square integrable states while leaving the discrete spectrum unchanged [1]. Cuts describing the continuum are rotated, however, but this may be advantageous, since they are thus disentangled when their thresholds differ from one another. (We are not interested, in this paper, in the case of channels with identical thresholds.) It is then expected that the continuum corresponding to such rotated cuts makes a much smoother contribution to the calculation of collision amplitudes, level densities, strength functions and sum rules [2] [3], since narrow resonant processes have been assumed to be peeled out explicitly by the CSM. The CSM Hamiltonian, unfortunately, is not hermitian any more, and it is not obvious that a resolution of the identity in terms of its bound states, resonances and presumably damped continuum is possible. For the one channel case, convincing arguments have been advanced a long time ago [4] to prove that this resolution exists. More recently [5], a detailed investigation of the case of two channels, coupled by straightforward potentials, generated a contour integration of the usual Green's function which provided the identity resolution. The task was made reasonably easy by the small complication of the Riemann surface in that case. The purpose of the present paper is to capitalize on the methods used for that two channel case and attempt a generalization to any finite number of channels, despite the more complicated nature of the relevant Riemann surface. We shall assume, naturally, that there already exists, derived from single poles and usual cuts, a resolution of the identity for the initial Hamiltonian, before its modification by complex scaling. Our problematics would be meaningless otherwise.

Several earlier studies, in particular by [6] [7], have been concerned with a description of resonances with square integrable states, without complex scaling. They did not restrict to the consideration of just simple poles of the S-matrix and investigated how one might, as rigorously as possible, define initial wave packets for the description of decaying states; the non purely exponential nature of their decays received a detailed attention, via the analysis of their time dependent evolutions. The present paper, however, will be content with a Gamow definition of resonances, by means of simple poles; our aim is just to generate a resolution of the identity, with time independent states extending to asymptotic regions. For earlier searches of a complete basis of states, including resonances, but within a compact interaction volume, we may refer to the review by [8] of R-matrix methods and their extensions. See also [START_REF] Robson | Nuclear Spectroscopy and Reactions[END_REF] and in particular the comparison of "class B" and "class D" theories.

In this paper, we shall again assume that all potentials V in (r) driving the channels and their couplings are local and so short ranged, Gaussian-like for instance, that the 2N Jost solutions of the N coupled equation system,

-ψ ′′ ij (k j , r) + N n=1 e 2iθ V in (e iθ r) + ℓ i (ℓ i + 1) r 2 -k 2 i δ in ψ nj (k j , r) = 0, i, j = 1, ..., N, (1)
exist and are analytical in the whole complex domain of all the momenta k j . The radius r runs from 0 to +∞, obviously, and the number N of channels is taken as finite. As an additional technicality we also assume, naturally, that the products V in ψ nj do not diverge for r → 0 when singular solutions of Eqs.(1) are considered.

We select the threshold of the lowest channel as the origin of the complex energy plane, hence E ≡ k 2 1 . The other channels with their physical thresholds E * j , which are real and positive numbers, now define channel momenta according to,

E j = k 2 j = E -e 2iθ E * j .
Notice that, given a real number E * j defining a physical threshold, the usual complex scaling where p 2 becomes e -2iθ p 2 and r becomes e iθ r does not change E * j and rotates the corresponding cut by angle -2θ. But here, we have a slightly different representation, because the Hamiltonian has been multiplied by e 2iθ . Hence kinetic operators in our Hamiltonian H, see Eqs. (1), are just -d 2 /dr 2 , every cut rotates back into being "horizontal"and starts from e 2iθ E * . For time dependent studies. it will make sense to scale time, conjugate of energy, by a factor e -2iθ . This will prevent those resonant wave packets, the energies of which have a positive imaginary part as eigenvalues of H, from exploding when t → +∞. Also in this paper no rearrangement is allowed, channels are defined by just internal excitations of the projectile and/or the target, hence all reduced masses are equal. Finally we exclude from this paper the consideration of abnormal thresholds; we shall only discuss the case of "square root thresholds". This is generic enough.

It is understood here and from now on that a first subscript, such as i or n, denotes the component of each wave ψ in channel i or n, then that any superscript, ±, or second subscript, j, denotes the boundary condition which defines ψ. For a Jost solution f ± .j , the boundary condition that we choose is "asymptotic flux e ±i(kj r-ℓjπ/2) in channel j and no asymptotic flux in the other channels". It is well known that for r → 0, the components of such Jost solutions are proportional to (k i r) -ℓi (2ℓ i -1)!!. For a regular solution ϕ .j , the boundary condition that we choose sits at r = 0 and reads, "lim r→0 (k i r) -ℓi-1 ϕ ij (r) = 0 ∀i = j, while, for i = j, then lim r→0 (k j r) -ℓj -1 ϕ jj (r) = 1/(2ℓ j + 1)!!.

Following Newton [START_REF] Newton | Scattering Theory of Waves and Particles[END_REF], it is convenient, given E and r, to set the column vectors ϕ .j into a matrix Φ(E, r) of regular solutions and the Jost solutions f + .j (resp. f - .j ) into a similar matrix f + (E, r) (resp. f -). It is also convenient to notice that Φ, viewed as a function of the k j 's as if these were independent momenta, is even under any reversal of a k j into -k j . Such is not the case for f + ; analytic continuations in either energy or momenta planes can introduce one (or several) f - .j 's into f + . For our oncoming argument we must use the Wronskian matrix with matrix elements the Wronskians W (f + .m , ϕ .n ) of the Jost solutions f + .m with the regular ones ϕ .n . This, for s waves, is the transposed of f + at r = 0,

W(E) = f + (E, 0), (2)
and for other angular momenta is only a slight modification of f + (E, 0). (Rather than just f + (E, 0) one must use limits of products (k i r) ℓi f + ij /(2ℓ i -1)!! at r = 0, explicitly, but we will disregard this technicality.) The Green's function G is then found as,

G(E, r, r ′ ) = Φ(E, r) [W(E)] -1 f + (E, r ′ ) if r < r ′ , G(E, r, r ′ ) = f + (E, r) [ W(E)] -1 Φ(E, r ′ ) if r > r ′ . (3)
Here each tilde ˜means transposition; we refer to [START_REF] Newton | Scattering Theory of Waves and Particles[END_REF] or to Appendix A of [5] for the derivation of such formulae for G. Despite different formulae whether r > r ′ or r < r ′ , and the lack of hermiticity, G is symmetric, namely G(r, r ′ ) = G(r ′ , r).

It will be noticed that the CSM, as we describe it by the system of Eqs.(1), locates thresholds on a segment of the complex E plane with slope 2θ, extending from E = 0 to e 2iθ E * N , and that the channel cuts are rotated back into being "horizontal". Conversely, bound states lie on a negative semiaxis rotated by 2θ and resonances are rotated by 2θ as well. This slight change of representation changes nothing to the physics, obviously. For trivial technical reasons [5], we normalize energy units so that E * N = 4. Also we shall use a short notation, k ≡ k 1 and K ≡ k N . We show in Figure 1 the cut energy plane in an illustrative, four channel situation when θ = π/6, E * 2 = 1.5 and E * 3 = 3.5. Equipped with this slightly unwieldy formalism, we can now investigate whether there exists a representation, and an integration contour, such that the traditional integral, I = dE G(E, r, r ′ ), calculated in two different ways, generates a resolution of the identity. This question of a representation and a contour is the subject of Section II, the main part of our argument. Additional considerations on the two ways of calculating this integral make the subject of Section III. A discussion and conclusion are proposed in Section IV.

II. REPRESENTATIONS AND CONTOURS

A. Energy plane

From Fig. 1 it is intuitive that one could start, for instance, from +∞ along the lower rim of the lowest channel cut, return to the origin, E = 0, proceed to +∞ again on the upper rim, then join there the lower rim of the second cut, return to the threshold of this second cut, go to e 2iθ E * 2 + ∞ along the upper rim, join the third cut lower rim at infinity, etc., until arriving at e 2iθ E * N + ∞ along the upper rim of the highest channel. Then the contour would be closing at infinity by means of an almost complete circle, counterclockwise, terminating at the starting point, namely at +∞ on the lower rim of the lowest channel.

Along such a contour, it would be necessary to investigate the behaviors of the ingredients f + , W and Φ of G. Furthermore, information is needed about the singularities of G inside the contour; indeed, residues of simple poles are essential for a calculation of dE G(E) by Cauchy's theorem; one also needs reasons why no singularities higher than simple poles occur.

The representation discussed in the next subsection makes easier the needed investigation, for it opens two of the cuts and limits the discussion to situations where all momenta have semipositive imaginary parts, ℑk j ≥ 0.

B. Pseudomomentum plane

A generalization from [5], where there were two channels only, the present "P representation" consists in joining the upper rim of the lowest cut and the lower rim of the highest cut, and in opening both cuts, by rational formulae,

k = P + Q 2 /P, K = P -Q 2 /P, (4)
where Q = e iθ makes a short notation for our scaling of energies such that E * N = 4 and k 2 -K 2 = 4Q 2 . Trivially, P is the average (k + K)/2 of k and K. The point is, despite an obvious failure to open additional cuts, P also give the "dominant" part of any other momentum when ℑP → +∞. Indeed, when |P | is large, say |P | >> 2, then an asymptotic value can be defined for k j , j = 1, j = N, according to the rule,

k j ≡ (k 2 -Q 2 E * j ) 1 2 = (P 2 + 2Q 2 -Q 2 E * j + Q 4 /P 2 ) 1 2 = P + Q 2 (1 -E * j /2)/P + O(P -2 ). ( 5
)
Thus the semicircle at infinity in the upper P plane corresponds to ℑk j > 0, ∀j. This is of critical value for the zoology of our Jost functions and it is expected that this semicircle properly closes the integration contour under design.

Set now P = x + iy and short notations c = cos 2θ and s = sin 2θ. A trivial calculation separates the real and imaginary parts of the (complex) energies driving each channel,

(x 2 + y 2 ) 2 ℜ(k 2 j ) = [(x 2 + y 2 + s)(x + y) + (x -y)c] [(x 2 + y 2 -s)(x -y) + (x + y)c] -E * j (x 2 + y 2 ) 2 c, (6)
and

(x 2 + y 2 ) 2 ℑ(k 2 j ) = 2[(x 2 + y 2 )x + xc + ys] [(x 2 + y 2 )y + xs -yc] -E * j (x 2 + y 2 ) 2 s. ( 7
)
and it is trivial to recover the images, in this new representation, of the cuts displayed in Fig. 1. Polar coordinates, with P = pe iη , can be also be used to decribe the j-th cut from Eq.( 7) by, Results are shown in Figure 2 for the same special case as Fig. 1. As in [5], the lowest channel is represented by the heavy, shoulder shaped line, that starts from -∞ on the real P axis, bends up, then backs into the origin P = 0, where it terminates with a slope 2θ. Along the curve, k is real and runs from -∞ to +∞, covering both rims of the initial cut. The threshold k = 0 is represented by P = iQ = e i(θ+ 1 2 π) . Partner points where k ↔ -k obtain under the symmetric transformation P ↔ -Q 2 /P. In the same way, for the highest channel, K runs with real values along the heavy dashed line, from -∞ at P = 0 to +∞ at the end of the positive ℜP semiaxis, via K = 0 for P = Q. The transform, P ↔ Q 2 /P, makes partners with opposite values of K.

p 2 sin 2η + sin(4θ -2η) p 2 = (E * j -2) sin 2θ. (8)
The other cuts remain cuts. Their thresholds lie on the image, shown as a dotted line again, of the segment already pointed out at the stage of Fig. 1. Because both ℜ(k 2 j ) and ℑ(k 2 j ) vanish for such points, it is easy to eliminate E * j between the right hand sides of Eqs.(6,7) and obtain the condition for such a locus,

x 2 + y 2 = 1, (9)
a very simple result indeed. With |P | = 1, the positions of the thresholds are easy to obtain. The special cases j = 1 and j = N give the argument η ≡ ArgP as η = θ + π/2 and η = θ, respectively. This was already known from [5].

The function sin 2η + sin(4θ -2η), see Eq.( 8), decreases monotonically when η increases from θ to θ + π/2, hence a unique solution for each E * j , and an obvious symmetry about θ + π/4 corresponding to the symmetry about E * j = 2. Then each intermediate cut generates, from Eq.( 7), an image which joins its threshold to the origin P = 0, while k j , a real number along this image, runs from 0 to ±∞, according to the rim. The image lies between the heavy full and dashed lines, and, being pinched between them at P = 0, also reaches the origin with slope 2θ. While the pinching makes numerics slightly difficult, it is easy to verify analytically from Eqs.(6,7) that infinitesimally away from both rims of such an intermediate cut, but inside the wedge created by the heavy line curves, ℑk j remains positive. To illustrate our full control of the various ℑk j 's provided by this P representation, whether inside the wedge or near the positive infinity semicircle, we show in Figure 3 the cut corresponding to E * 2 , and its continuation beyond threshold.

"semicircle". Simultaneously, we generate rims of the cut, and beyond again below threshold. To generate rims, we use Eq.( 7), or as well Eq.( 8), with E * 2 replaced by E * 2 -0.2 and E * 2 + 0.2 for the lower and upper rim, respectively. (The choice ±0.2 was made for graphical convenience, but we tested much smaller intervals, naturally.) The dots represent P = 0, where the channel energy is infinite, and the threshold, where it vanishes by definition. Like the cut, the rims are pinched by the wedge.

Then we show in Figure 4 the trajectory of k 3 when P follows this cut from P = 0, to the threshold and beyond. Notice that, E 2 being real along the line, then the imaginary part of

E 3 = E 2 + e 2iθ (E * 2 -E * 3
) is obviously negative. This does not prevent a choice of k 3 with ℑk 3 > 0, generating the leftmost trajectory in Fig. 4. Simultaneously, we show the trajectories of k 2 from both rims of the same cut. The left hand side (when seen in Fig. 3) rim induces ℜk 2 → -∞ when P → 0, with an infinitesimally positive ℑk 2 . Conversely the right hand side rim induces ℜk 2 → +∞ when P → 0, with still an infinitesimally positive ℑk 2 . When we go from either rim towards the upper semicircle at infinity, this induces ℑk 2 → +∞, as expected. The rims can be connected by any small path, see the bar above the threshold in Fig. 3, and the values of k 2 along the rims can be smoothly matched, see the curved bar in Fig. 4, the trajectory of k 2 when P follows the bar in Fig. 3. Generalizations to every k j in every part of the wedge are trivial.

C. Contour

To synthetize this Section, the P representation defines a physical sheet similar to the physical sheet of the energy plane. The region of interest is that region above the two curves which open the cuts for the lowest and the highest channels, while cuts remain for the intermediate channels. All momenta inside the wedge, and all the way to the upper semicircle at infinity, can be defined with positive imaginary parts. A contour can be found, following all cuts and closing at infinity in the upper plane. real axis, follow the "opener curve" which corresponds to the lowest channel, all the way to P = 0. From there, follow the lower rim of the cut corresponding to the second channel, back to its threshold, then turn around the threshold to follow its upper rim, down to P = 0. In turn, follow the lower rim of each intermediate channel, then its upper rim. After bouncing N -1 times at P = 0, follow the "opener curve" corresponding to the upper channel, until P → +∞ on the real axis. Then close the contour by means of the upper semicircle at infinity. In the next Section, we shall investigate what happens to the integral, I = dE G(E, r, r ′ ), when considered along this contour in the P plane.

III. THREE CONTRIBUTIONS TO THE GREEN'S FUNCTION INTEGRAL A. Upper semicircle

At infinity in this upper P plane, the integration weight, dE = 2 P -Q 4 /P 3 dP, boils down to 2P dP. All the N distinct Jost solutions boil down to exp i(P r -1 2 ℓ j π) in their respective "flux channel j", while vanishing in the other channels. At the same time, the N distinct regular solutions similarly boil down to sin P r -1 2 ℓ j π /P in their respective flux channel and vanish in the other channels. The Wronskian matrix boils down to the N -dimensional unit matrix.

Assume now r > r ′ , for instance, and thus consider the second of Eqs.(3). The product f + [ W] -1 Φ boils down to a diagonal matrix. Its j-th diagonal element reads, sc 2 dP e i(P r-ℓjπ/2) sin P r ′ -

ℓ j π 2 , (10)
and can be easily calculated by reducing the semicircle back to the real P axis. The result does not depend on j,

-i -∞ ∞ dP e i(P r-1 2 ℓj π) exp iP r ′ -iℓ j π 2 -exp iℓ j π 2 -iP r ′ = 2iπ[δ(r + r ′ ) -δ(r -r ′ )]. (11)
It is trivial to verify that the same result is obtained if r < r ′ . Furthermore the term δ(r + r ′ ) cancels out in the space of regular radial waves. Hence the contribution I sc of the semicircle makes nothing but the multichannel identity, multiplied by (-2iπ). Notice that, differing from [2], this identity is not multiplied by a factor depending on θ, since for us the ends of the semicircle, -∞ and +∞, both lie on the real P axis.

B. Continuum

It makes no difference here whether we consider the contribution of one of the "opener line" or that of one of the intermediate cuts. For in both cases we group partner terms. Such partners either come from a transform P ↔ ±Q 2 /P or from opposite rims of the intermediate cut under consideration. What is important to notice is that momenta retain their finite and positive imaginary parts and do not change when we compare two partner points, except that momentum specific to the opener line or the cut. For that momentum, which is real, "partnership" means k j ↔ -k j , with still an infinitesimal positive imaginary part. Keeping in mind that Φ is even under such a momentum flip, the contribution of such a continuum thus reads, if r > r ′ for instance,

I j = ∞ 0 2k j dk j D j (E, r) Φ(E, r ′ ), (12)
where D j (E, r) represents the following difference between partners,

D j (E, r) = f + (E, r) [ W(E)] -1 -f + (-k j , r) [ W(-k j )] -1 , (13)
a discontinuity across the cut. The notation used here takes advantage of the fact that dE = 2k j dk j , and that k j is a convenient label along the line or the cut. The first term, f + (E, r) [ W(E)] -1 , in the right hand side of Eq.( 13) clearly comes from the upper rim. The notation that we use for the second term, f + (-k j , r) [ W(-k j )] -1 , indicates that, because of analytic continuation in the physical sheet around the threshold, one Jost solution f - .j now makes the j-th be a rank one dyadic. An elementary proof of this dyadic result was given in Appendix C of [5]. Nothing changes in the argument if r < r ′ .

As a consequence of the dyadic nature of D j , and of the symmetry G(E, r, r ′ ) = G(E, r ′ , r), hence of the same symmetry for discontinuities across cuts, there exists as a column vector a solution φ .j of Eqs.( 1) that is able to represent symmetrically both D j (E, r) Φ(E, r ′ ) and Φ(E, r) Dj (E, r ′ ) in a self dual way as an outer product,

I j = ∞ 0 2k j dk j φ .j (E, r) φ.j (E, r ′ ) D(E) . ( 14
)
This solution belongs to the set of regular solutions, naturally, because of the regularity of G at both r = 0, and r ′ = 0, illustrated by the presence of Φ in Eqs. (3). The exact natures of this φ .j and of the "normalizing" denominator D are discussed in the Appendix.

At this stage, the full integral along the full contour thus gives the sum of the multichannel identity and "pseudoprojectors on the continuum", one pseudoprojector for each channel,

i 2π dE G(E, r, r ′ ) =    δ(r -r ′ ) 0 ... 0 0 δ(r -r ′ ) ... 0 . . . . 0 0 ... δ(r -r ′ )    + i π N j=1 ∞ 0 k j dk j φ .j (E, r) φ.j (E, r ′ ) D(E) . ( 15
)
The next subsection shows what happens if the same integral is evaluated by means of the Cauchy theorem.

C. Residues at poles

We assumed that, before complex scaling, namely for θ = 0, there existed an identity resolution in terms of unscaled bound states and unscaled scattering states. In other words we assumed that the corresponding, unscaled G(E) shows only isolated, simple poles, besides the physical cuts. Such poles can be on the real E axis of the physical sheet, describing bound states, or away from this axis, then describing resonances or antiresonances. The point is, now, that the CSM cannot change the nature of such poles [1]. Within our description by Eqs.(1), the CSM just rotates such poles by 2θ in the energy representation, along circular arcs, concentric around E = 0. In the P representation, the images of such arcs are also concentric arcs, with angular extension θ only. This is trivially seen from the equation which, for each initial position ε of a pole, defines those values of P which represent e 2iθ ε,

P + e 2iθ /P 2 = e 2iθ ε. (16)
Indeed, θ disappears from this equation if one sets P = e iθ P 0 , where P 0 solves for the initial position ε. It can be concluded that only simple poles will be found when a finite θ is used for our CSM. Notice, incidentally, that for ε real and negative (bound states), the P representation will align poles along the axis with polar angle θ + π/2, further than the circle with radius 1 that we found as the locus of thresholds. There will be no such alignment for resonances.

For the calculation of I by Cauchy's theorem, poles are not due to either f or Φ, since these, as functions of E or P, are regular. Only the divergence of W -1 can create poles. The situations of interest are those when the roots of the determinant, det W, are located inside the integration contour. We know that such is the case for the bound states. Depending upon θ, some resonances may also rotate into the domain. It is already known that only simple, isolated poles occur. The only question to solve is, what is the residue of G at such a pole.

Residues of G at its poles will now be obtained from derivatives d/dE. That is equivalent to a calculation in the P representation, anyhow, and slighly easier. We shall use short notations in which the dependence of Φ, f + , W, upon r, and/or r ′ and/or E will be most often understood. However, at those energies E ν where a pole occurs, we use an explicit subscript ν to specify that such quantities Φ, ... , W are evaluated at E ν .

Poles occur because of W -1 . Hence, we must only find the residue,

R ν = lim E→Eν (E -E ν ) W -1 (E) , (17)
˜ + + At a (simple!) root E ν of det W(E)
, there is necessarily one, and just one, null right eigenvector Λ ν of W. Similarly there is one, and just one, null left eigenvector Λ ′ ν . We write them as columns and normalize them by the condition,

Λ′ ν Λ ν = 1. (18)
Then the divergent part of W -1 in a neighborhood of E ν is nothing but the truncation,

W -1 tr = Λ ν Λ′ ν Λ′ ν W(E) Λ ν , ( 19
)
where there is an explicit dependence on E in the denominator. This denominator, a number, vanishes at E = E ν .

As a matrix element of W it is nothing but the Wronskian of the following two waves, F ≡ f + Λ ′ ν and ξ ≡ Φ Λ ν . The former, F, is irregular, the latter, ξ, is regular. While Λ ν and Λ ′ ν do not depend on E, since they were defined at E = E ν , both F and ξ depend on E, via f + and Φ. When their Wronskian vanishes, F and ξ become proportional to each other, and there exits a number c such that F ν = c ξ ν . This special wave is both a mixture of regular solutions and a mixture of Jost solutions, with positive imaginary parts in the momenta driving all Jost solutions. Therefore it decreases exponentially in all channels when r → ∞ and it is square integrable as well as regular. As expected it represents either a bound state or a regularized resonance.

According to Eqs.(17,19), the residue under study comes from just the reciprocal of the derivative of the Wronskian of F and ξ,

R ν = Λ ν Λ′ ν d Λ′ ν W(E) Λ ν /dE | E=Eν . (20)
In short, we must calculate the derivative of a Wronskian with respect to the energy, d Λ′ ν W(E) Λ ν /dE. To help manipulations with Wronskians, define an operator matrix U with matrix elements the CSM potentials, completed by the centrifugal barriers and the thresholds,

U ij = e 2iθ U ij e iθ r + δ ij e 2iθ E * j + ℓ j (ℓ j + 1) r 2 . (21)
Then elementary, but slightly tedious manipulations, which are already described in [START_REF] Newton | Scattering Theory of Waves and Particles[END_REF] or in Appendix B of [5], give the remarquably simple result,

d Λ′ ν W(E) Λ ν /dE | E=Eν = -c ∞ 0 dr ξ(E ν , r) ξ(E ν , r). (22)
Then the constant c cancels out between this and the numerators of f + ν Rν Φν and Φ ν R ν f + ν , which make the same, symmetric formula anyway, whether r > r ′ or r < r ′ , since

F ν = c ξ ν .
Summing upon all such residues obtained at roots E ν of det W above the "opener" curves in the P upper half-plane, the contour integral reads,

I(r, r ′ ) = -2 i π ν Φ(E ν , r) Λ ν Λν Φ(E ν , r ′ ) ∞ 0 dr ′′ Λν Φ(E ν , r ′′ ) Φ(E ν , r ′′ ) Λ ν . (23)
Here we state again that the column vector Λ ν is the null, right-hand side eigenvector of W(E ν ), namely W(E ν ) Λ ν = 0, then the column vector Φ(E ν ) Λ ν of wave functions is the wave function of the bound state or resonance, and the denominator plays the role of a "Euclidean-like square norm". This denominator is non vanishing; this corresponds to the hypothesis of single, isolated poles. All these are labeled by ν, a discrete index, or as well by P ν , an isolated root of W if viewed as a function of P.

D. Completeness

Since the three contributions I sc , j I j and I are obviously related by I sc + j I j = I, it is trivial to equate i 2π I sc , i and i 2π j I j , the latter term making the pseudoprojector upon the continuum for all channels. Naturally, in practical calculations, a cutoff and some amount of discretization will be necessary to integrate such continuum terms, but the P representation provides a suitable frame for testing the convergence of such a resolution for sum rules, level densities and similar observables. Notice that, because of the use of complex, self dual bras and kets in the resolution, such cutoff and discretization manipulations may generate spurious imaginary parts for the expectation values of hermitian observables. For a discussion and possible interpretation of imaginary parts in individual matrix elements, we refer to [START_REF] Berggren | [END_REF]. But, when summed upon all discrete and integral terms provided by the resolution, such imaginary parts must add up to a negligible, spurious noise compared to the real parts. This requested cancellation makes one more criterion to validate numerical operations.

IV. DISCUSSION AND CONCLUSION

Once again we used the ABC theorems [1] to locate the discrete spectrum at trivially rotated positions deduced from the discrete spectrum of an initial, hermitian Hamiltonian. The topological similitude provided by the CSM rotation warrants that, as long as there are no double poles or higher singularities with the initial Hamiltonian, the same will be true with the CSM Hamiltonian.

Then it was not very difficult to find a representation which allows a suitable contour integration of the Green's function. There was still a slightly complicated Riemann surface to handle, for the number of cuts was reduced to N -2 only [12], but we took great care, including a few numerical, illustrative examples, to show that all cuts in the new representation are well understood, all thresholds are easily located, all complex momenta to be used for proofs have positive imaginary parts in a physical domain of a suitable sheet, and in general that all technicalities are sound. This proof of the CSM completeness for N channels is restricted to a finite number of well separated channels, normal square root threshold singularities, in a purely inelastic situation, without rearrangement, and with short ranged forces. The case of long range forces makes a more difficult question, indeed [13] [14]. But our restrictions still allow a large class of practical problems, and for instance in nuclear physics, a very large number of collective resonances can be described by the coupled channel equations that we studied.

W u =    a b c d e f g h i j k l m n o p    . (24)
where, for instance, b is the Wronskian of f + .1 with ϕ .2 and o is the Wronskian of f + .4 with ϕ .3 . The inverse of W u reads, trivially,

W -1 u = (det u ) -1    a ′ e ′ i ′ m ′ b ′ f ′ j ′ n ′ c ′ g ′ k ′ o ′ d ′ h ′ l ′ p ′    , (25)
where det u is the determinant of W u and the prime symbols denote the corresponding cofactors. For the lower rim of the second cut a substitution occurs for the second row of W u , hence the lower rim Wronskian matrix reads,

W l =    a b c d q r s t i j k l m n o p    , (26)
where, for instance, t is the Wronskian of f - .2 with ϕ .4 . Accordingly the inverse matrix becomes,

W -1 l = (det l ) -1    a ′′ e ′ i ′′ m ′′ b ′′ f ′ j ′′ n ′′ c ′′ g ′ k ′′ o ′′ d ′′ h ′ l ′′ p ′′    , (27)
where doubleprime symbols denote new cofactors, but the cofactors of {q, r, s, t} are the same as those of {e, f, g, h}.

Again with a transparent, condensed notation, we set, for the upper and lower rim, respectively,

f u =    A B C D E F G H I J K L M N O P    , f l =    A B C D Q R S T I J K L M N O P    . (28
, f + 42 }, while {Q, R, S, T } ≡ {f - 12 , f - 22 , f - 32 , f - 42 }.
For r < r ′ , the discontinuity to be studied corresponds to the transposed of Eq.( 13), and reads, in a condensed notation,

D2 (r ′ ) = W -1 u f u (r ′ ) -W -1 l f l (r ′ ), (29)
The subscript 2 for the cut and the r ′ dependence will be now understood and we shall use trivial identities to analyze

D = W -1 l + W -1 l (∆W) W -1 u f u -W -1 l ( f u + ∆ f ) = W -1 l (∆W) W -1 u f u -W -1 l ∆ f , (30)
where ∆W = W l -W u and ∆ f = f lf u . The point is, both modifications ∆ are just substitutions for second rows; they boil down to dyadics,

∆W =    0 1 0    ⊗ [q -e r -f s -g t -h] , ∆ f =    0 1 0    ⊗ [Q -E R -F S -G T -H] . (31)
(Our use of the tensor product symbol ⊗ is actually superfluous; we just want to stress the matrix product of a column by a row.) The next point is, then, that a global dyadic form for D emerges,

D = W -1 l    0 1 0 0    ⊗ [q -e r -f s -g t -h] W -1 u f u -[Q -E R -F S -G T -H] . (32)
Furthermore, from the very definition of matrix inversion, we see that

[e f g h] W -1 u = [0 1 0 0] , (33) hence
[-e -f -g -h] W -1 u f u = -[E F G H] , (34)
and D simplifies into

D = (det l ) -1    e ′ f ′ g ′ h ′    ⊗ [q r s t] W -1 u f u -[Q R S T ] . (35)
For r < r ′ the complete discontinuity Φ(r) D(r ′ ) of G(r, r ′ ) thus reads

det l det u Φ(r) D(r ′ ) = φ(r) Ξ(r ′ ), (36)
with

φ = e ′ ϕ .1 + f ′ ϕ .2 + g ′ ϕ .3 + h ′ ϕ .4 , (37) and Ξ
= (qa ′ + rb ′ + sc ′ + td ′ ) f + .1 + (qe ′ + rf ′ + sg ′ + th ′ ) f + .2 + ... + (qm ′ + rn ′ + so ′ + tp ′ ) f + .4 -det u f - .2 . (38)
Both φ and Ξ are column vectors and relate to the second cut, hence they should actually read φ .2 and Ξ .2 in a notation compatible with Eqs. (14,15). We omitted such subscripts, for the sake of conciseness.

It may be convenient to take advantage of the cofactor nature of all the coefficients a ′ , ... p ′ . This gives indeed the formal, but condensed formula,

can themselves be interpreted, after keeping track of signs, as cofactors for the last column of the determinant shown

In the 2N -dimensional space of solutions, it is known that the 2N Jost solutions and the N regular ones are related by a formula such as,

f -= Φ W -1 w + f + w -1 W -W -1 w, ( 42
)
where W is the same as W u , while W -is the analog of W if one replaces each f + .m by its partner f - .m . Then w is a diagonal matrix, defined from the Wronskians W (f + .m , f - .n ) = -2ik m δ mn . It will be noticed from Eq.( 42) that, if we expand an f - .m on the basis spanned by all the ϕ .n and all the f + .n , the regular components of f - .m are provided by the m-th column of the matrix product W -1 w.

It is known that Ξ always belongs to the subspace of N regular solutions. In our illustrative example where N = 4 and we studied the second cut, our Ξ, according to Eq.( 38), is a superposition of five solutions, namely all the f + .n and one f - .n only, f - .2 . After an expansion of f - .2 on the basis spanned by the ϕ .n and the f + .n , all its irregular components must cancel out those preexisting irregular components of Ξ seen from Eq.( 38). (For the sake of rigor, we verified, by brute force calculations when N = 2, 3 and 4, that the components f +

.n do vanish out.) Thus we may consider the regular components only, coming from just f - .2 . The weight of f - .2 is, according to Eq.( 38), -det u . We must therefore find the second column of,

-det u W -1 w = -    a ′ e ′ i ′ m ′ b ′ f ′ j ′ n ′ c ′ g ′ k ′ o ′ d ′ h ′ l ′ p ′       -2ik 1 0 0 0 0 -2ik 2 0 0 0 0 -2ik 3 0 0 0 0 -2ik 4    , (43)
hence the final result, Ξ = 2ik 2 (e ′ ϕ .1 + f ′ ϕ .2 + g ′ ϕ .3 + h ′ ϕ .4 ) = 2ik 2 φ.

(44)

The generalization, Ξ .j = 2ik j φ .j is obvious. For any channel number N and any j-th cut, both φ and Ξ correspond to the j-th column of W -1 , hence to the cofactors of the j-th row of W. There is no need here to specify W u or W l , because the relevant cofactors are the same on both rims of the cut. The fact that Ξ and φ are the same except for the factor 2ik j gives the same, symmetrical result whether r is larger or smaller than r ′ . And the denominator present in Eqs. (14,15) reads, when all factors are collected,

D(E) = det u det l 2ik j . ( 45
)
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FIG. 1 .

 1 FIG. 1. E-plane. Physical cuts for a four channel case when θ = π/6, E * 2 = 1.5, E * 3 = 3.5 and E * 4 = 4. Lowest channel, heavy full lines, highest channel, heavy dashed lines, intermediate channels, lighter full lines. The dotted segment with slope π/3 is the locus of thresholds (big dots) in this representation.

FIG. 2 .

 2 FIG. 2. P plane. Cuts for the same four channel case, θ = π/6, E * 2 = 1.5, E * 3 = 3.5 and E * 4 = 4. Opened cut for lowest channel, heavy full line. Opened cut for highest channel, heavy dashed line. Intermediate channel cuts, not open, lighter full lines. The dotted segment is the locus of thresholds (big dots) in this P representation.

FIG. 3 .

 3 FIG.3. P plane. Again θ = π/6. Cut for the channel defined by E * 2 = 1.5. The center line, between dots, is the cut. Cut then continued for negative energies in the channel. Additional lines, lower rim (leftmost curve) and upper rim (rightmost curve), respectively. Both rims extended below threshold. Heavy line bar, connection between extended rims.

FIG. 4 .

 4 FIG. 4. k2, k3 planes. Still θ = π/6, E * 2 = 1.5 and E * 3 = 3.5. Leftmost curve, trajectory of k3 when P follows the central line of Fig. 3. Intermediate curve, trajectory of k2 for extended lower rim, see leftmost curve in Fig. 3. Rightmost curve, trajectory of k2 induced by extended upper rim, see rightmost curve in Fig. 3. Heavy line curved bar, connection trajectory for k2 when P turns around the threshold, below it.

  all coefficients such as,qa ′ + rb ′ + sc ′ + td ′ = det .. , qm ′ + rn ′ + so ′ + tp ′ = det

Acknowledgment: B.G.G. thanks the Hokkaido University for its hospitality during part of this work.

Appendix

We give here in some detail a description of that regular solution φ .j which accounts for the discontinuity of the Green's function across a cut. For the sake of pedagogy, we set the channel number to be N = 4 and shall consider only what happens for, e.g., the second cut. Generalizations are obvious and left as an exercise for the interested reader. In a condensed notation, we write the upper rim Wronskian matrix as,


Michel Galinier

François Roubille

Philippe Berdagué

Gilles Brierre

Philippe Cantie

Patrick Dary

Jean-Marc Ferradou

Olivier Fondard

Jean Philippe Labarre

Telemonitoring versus standard of care in heart failure: a randomised multicentre trial


S Marchand

P Keckhut

K H Fricke

U Blum

H Schets

S Pal

D Wardle

V Cuomo

G Pappalardo

A Hauchecorne

GOMOS VALIDATION AIR-DENSITY AND TEMPERATURE PROFILES

Lidar system and radio-soundings were used to validate both air-density and low and high-resolution temperature profiles of the GOMOS instrument onboard ENVISAT-1 satellite. No bias greater than 7% was found for the air-density comparisons with lidar and there is a very strong global agreement. The conditions of the occultations have a large impact on the comparisons concerning the temperature profile. The "dark limb" measurement conditions show great agreement between 35 and 60 km with no bias greater than 5% whereas the "bright limb" measurement conditions show poor results. The High Resolution temperature shows a very interesting potential.

INTRODUCTION

With the limited and late availability of Envisat data, only four teams of the groundbased subgroup have begun to validate the GOMOS air-density and temperature products. The locations of the stations cover the North hemisphere from mid latitudes (40°N with radio-soundings in Italy) to high latitudes (68°N with Lidar in Norway). The instruments used in this validation are Lidars (at Toronto, Observatoire de Haute Provence and Esrange) and radio-soundings (at Uccle station in Belgium and Tito Scalo in Italy) (see the Tab. 1 for exhaustive comments). The products which are validated here concerned ENVISAT level 2 files : GOM_NL__2P*. They were retrieved from the ACRI-ST server on request, and generated with GOPR_LV2_5.3.

Two temperature variables are available in the GOMOS level2 files:

• The first one is the (low-resolution) temperature (T) ; this temperature is determined from the air-density and it is a composite of three independent sources :-O 2 concentration ; Rayleigh scaterring ; climatologic model ARLETTY (A. Hauchecorne SA/ACRI-ST) which is based on ancillary data from the European meteorological model ECMWF and the atmospheric model MSIS.

• The second one is High-Resolution Temperature (HRTP) ; this product is retrieved from the scintillation observed by the two GOMOS photometers.

AIR DENSITY

Two teams have been able to perform air-density comparisons : AOID 153 at Toronto with Dr. Pal and AOID 222 at Esrange with Dr. Fricke.

For the 25/07/2002 and 25/10/2002 at Toronto, the three GOMOS occultations (Fig. 1 and Fig. 2) shows a very good global agreement but there are still some fluctuations between the GOMOS profiles and the Lidar one's. Another site at higher latitude (Esrange, see Fig. 3) for the 15th of August with a distance between the GOMOS measurement and the Lidar position of about 184 km shows a similar result. No significant deviations are reported according to the error bars -errors are the combined GOMOS and Lidar errors. A normalisation is done for both Lidar and GOMOS densities at the altitude of 35 km, that explain the perfect agreement between the two measurements at this altitude. At the top of the profiles the deviation is about 5%. The statistical comparison performed over Esrange shows the profiles of the deviation of the differences between GOMOS and Lidar (Fig. 4). These lidar data are, in contrast to those performed at Toronto, relative density profiles, which implies, that these data can not validate the absolute values, but the profile shape.

The results are summarized in histograms covering the altitudes from 30 to 70 km in altitude bands of 5 km width. 38 occultations and 18 Lidar measurements are used for these results from 18/07/2002 to 23/08/2002. For each histogram, the altitude range, the number of data points and the mean value of the deviations are given. The number of points varies because of the changing altitude grid in each GOMOS profiles and because the Lidar profile reach different maximum altitudes.

As mentioned previously, the very good agreement and small width of the histograms in the two lowest altitude range (30-35 km and 35-40 km) is artificial because of the normalization at 35 km height. Due to the large error stated for the GOMOS data in the PDS files, the general agreement can include deviations among the GOMOS and Lidar profiles up to 7%. At the higher altitudes, GOMOS density data oscillate about the Lidar data with amplitude of up to 2.6% near the stratopause. The largest random deviations among the GOMOS and Lidar data reach up to 7%.

TEMPERATURE

The five teams have been able to perform preliminary temperature validation: AOID 153 Mv=1.17, T=9700 K for orbits no. 2091 and no. 2092).

Fig. 6 temperature comparison for AOID 153 (star ID 84, Mv=2.40, T=4500 K for orbit no. 3407).

For the first two occultations at Toronto in July (Fig. 5), we can note a good general agreement between the low resolution temperature and the Lidar measurement. However, there is a problem concerning the calculation of the tropopause altitude. It seems to be not an altitude shift problem because the shapes of the profiles at the top altitude are in agreement. These two occultations are in dark limb conditions. For the third occultation (Fig. 6) at Toronto in October, we can see a large difference between the profiles and again a deviation for the tropopause altitude. But this occultation is in bright limb condition so the data are certainly noisy and the model is probably the most important contribution in the retrievals data. In this period, the planetary waves activity start to increase and hence induce a large daily and interannual variability. The differences here are the same than the differences between Lidar and model observed during this period. So it can suggest that the Lidar measurement is valid and the GOMOS product is mainly composed by model inputs.

Over Esrange in August for low-resolution temperatures (Fig. 7), we can split the altitude range in four domains:

• Under 35 km :

The 10% bias may be caused by stratospheric background aerosol at higher altitude than usual. So the bias here will be incriminated to the Lidar measurement.

• Between 35 km and 40 km

There's no bias.

• Between 40 km and 60 km

There's no significant bias and mean bias are smaller than 5% in this domain.

• Beyond 60 km

The 10% bias cannot be solely due to initialisation of the Lidar as lidar data smaller than 5% are considered. While some other instrumental causes can be suggested such as the extraction of the noise due to background emission, the GOMOS statistical errors are approximately twice as large. On Fig. 8 are represented the relative deviations for 18 Lidar measurements and 38 GOMOS occultations in histograms covering the altitudes from 30 to 70 km in altitude bands of 5 km width. We can notice here, on a statistical basis, a similar vertical behaviour than observed on the case shown in Fig. 7 with the largest deviations below 35 km and above 60 km and a good agreement with no bias larger than 5% between these two altitudes.

At the Uccle site (Belgium) we can notice in Fig. 10 a good agreement for a few cases but we can also notice some anomalous fluctuations for others. Time delay and then dark or bright limb conditions don't seem to be influent in these cases. On Fig. 11 is represented the mean relative differences (ptu-GOMOS) between 15 km and 35 km altitude range for 10 occultations and 9 radio-soundings for the period 10/10/2002 to 25/11/2002. The bias for this altitude domain is about 0.7%. The deviation can be as large as 1.7% but the mean deviation from the bias is about 0.3%.%.This agreement is again verified at the Tito Scalo station from 20 up to 35 km of height range as we can see an example on Fig. 9 . All these results seem to be very good but we should point out the fact that in this low altitude range, the GOMOS data are probably model data. These excellent bias and deviation could be probably due to a comparison between the radiosounding and the model which is probably not fully independent to the radio-soundings Esrange performed some successive orbits. It is clear from the range of temperature values that the GOMOS HRTP are incompatible with the range of temperatures encountered in the stratosphere. In this case the occultations were in bright limb conditions and may explained why the retrieval algorithm has failed. We suggest that an internal consistency check should be implemented as part of the GOMOS data processing. For another site at lower latitude (OHP) and for the 25 th August, we can see on Fig. 13 that the small-scale altitude structures are well represented. In this case the measurement is done in dark limb conditions and the process produce very good results. All the Lidar, the ECMWF and HRTP profile are in good agreement. Here the time delay between the occultation and the mean time of the LIDAR measurement is about three hours and the distance between mean tangent position and OHP is about 300 km. This single example shows that HRTP has a great potential to retrieve temperature profile as to detect the small-scale structures. We will have to smooth these retrievals to have information concerning the bias (if any) between HRTP and Lidar measurements however we will experienced some difficulties to validate the finest structures of the GOMOS profile, which will only be possible with very close collocations in time and in space registration.

Fig. 13 HRTP for AOID 360 (star ID 81, Mv=2.36, T=28000 K).

CONCLUSIONS

At this time, the first conclusions of the air density and temperature are:

• The difference of retrieval performance between dark and bright limb conditions are clearly observed ;

• No bias larger than 10% are reported for the low-resolution temperature and bias under 1% in the altitude range [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] km have been calculated. The first bias estimated with lidar above 30 km seems to be very small but a large dispersion is observed. The climatologic values includes in the GOMOS retrieval pobably contributes to these results for a large part. The knowledge of the model contribution in the low-resolution temperature is missing. We suggest to ESA to add this information in the level 2 products and to discriminate the share of each contribution to the retrievals ; • HRTP appears to have a great potential. An example shows a very good agreement with the other data sources with this GOMOS products but it clearly appears that very short time and distance collocations will be required to have a real possibility to make some good validation for the finest altitude scale structures ;

• Validation must continue and we must collect more collocations to produce more statistical comparisons for a better understanding of these products.

  Fig.1density comparison for AOID 153 (star ID 18, Mv=1.17, T=9700 K for orbits no. 2091 and no. 2092).

Fig. 2

 2 Fig. 2 density comparison for AOID 153 (star ID 84, Mv=2.40, T=4500 K for orbit no. 3407).

Fig. 3

 3 Fig. 3 density comparison for AOID 222 (star ID 155, Mv=2.90, T=6600 K for orbit no. 2400).

Fig. 4

 4 Fig. 4 histograms of density deviations for AOID 222 (38 occultations, 18 Lidar measurements).

  Fig. 5 temperature comparison for AOID 153 (star ID, 18Mv=1.17, T=9700 K for orbits no. 2091 and no. 2092).

Fig. 7

 7 Fig. 7 temperature comparison for AOID 222 (star ID 155, Mv=2.90, T=6600 K for orbit no. 2400).

Fig. 8

 8 Fig. 8 histograms of temperature deviations for AOID 222.

  Fig. 9 temperature comparison for AOID 9083.

Fig. 10

 10 Fig. 10 temperature relative differences for AOID 300.

Fig. 11

 11 Fig. 11 temperature comparison for AOID 300

Fig. 12

 12 Fig. 12 HRTP for AOID 222 (star ID 52, Mv=2.04, T=4500 K for orbits no. 2475 and 2476) and (star ID 142, Mv=2.85, T=8900 K for orbits no. 2515 and 2516).

  The AOID involved in the air-density and temperature validation.

	TEAM	PI	INSTRUMENTS	PRODUCTS
			40.36°N ; 15.44°E
		Tab. 1
	TEAM	PI	INSTRUMENTS	PRODUCTS
	AOID 153	S. Pal	Lidar (Toronto -Canada)	density, T
			43.66°N ; 79.4°W
	AOID 222	K. H. Fricke	lidar (Esrange -Sweden)	density, T, HRTP
			67.9°N ; 20.1°E
	AOID 300	D. De Muer	Radio-soundings (Uccle -Belgium)	T
			50.8°N ; 4.35°E
	AOID 360	P. Keckhut	Lidar (OHP -France)	HRTP
			43.94°N ; 5.71°E
	AOID 9083	V. Cuomo	Radio-soundings (Tito Scalo -Italy)	T

Proc. of Envisat Validation Workshop, Frascati, Italy, 9 -13 December 2002 (ESA SP-531, August 2003)

ACKNOWLEDGEMENTS

The entire staff of Esrange is acknowledged for providing the largest contribution to the GOMOS temperature validation data. Operations at Toronto were coordinated by Hans Fast (MSC) and the measurements were carried out by Richard Berman ( Spectral Applied Research), Richard L. Mittermeier (MSC) and Jonathan Davies ( MSC). The OHP and UCCLE Staffs are also acknowledged. The contribution from the Bonn University was supported by the DLR-Raumfahrt. The Toronto Lidar measurement program is supported by the Meteorological Service of Canada (MSC) and the financial support for the ACVT activity is provided by the Canadian Space Agency. The Service d'Aéronomie in France was supported by the French Space Agency CNES. The OHP and Toronto lidar profiles used in this publication were obtained as part of the Network for the Detection of Stratospheric Change (NDSC) and are publicly available after their final validation (see http://www.ndsc.ncep.noaa.gov).


Djamé Seddah

Farah Essaidi

Amal Fethi

Matthieu Futeral

Benjamin Muller

Pedro Javier Ortiz Suárez

Benoît Sagot

Abhishek Srivastava

Joakim Nivre

Mitchell Abrams

Željko Agić

Lars Ahrenberg

Lene Antonsen

Maria Jesus

Gashaw Arutie

Masayuki Asahara

Luma Ateyah

Mohammed Attia

Aitziber Atutxa

Liesbeth Augustinus

Elena Badmaeva

Miguel Ballesteros

Esha Banerjee

Sebastian Bank

Verginica Barbu Mititelu

John Bauer

Sandra Bellato

Kepa Bengoetxea

Riyaz Ahmad Bhat

Erica Biagetti

Eckhard Bick

Rogier Blok- Land

Victoria Bobicev

Carl Börstell

Cristina Bosco

Gosse Bouma

Sam Bowman

Adri- Ane Boyd

Aljoscha Burchardt

Marie Candito

Bernard Caron

Gauthier Caron

Gülşen Ce- Biroğlu Eryiğit

Giuseppe G A Celano

Savas Cetin

Fabricio Chalub

Jinho Choi

Yongseok Cho

Jayeol Chun

Silvie Cinková

Aurélie Col- Lomb

Çağrı Çöltekin

Miriam Connor

Marine Courtin

Elizabeth Davidson

Marie-Catherine De Marneffe

Valeria De Paiva

Arantza Diaz De Ilarraza

Carly Dickerson

Peter Dirix

Kaja Dobrovoljc

Timothy Dozat

Kira Droganova

Puneet Dwivedi

Marhaba Eli

Ali Elkahky

Binyam Ephrem

Tomaž Erjavec

Aline Etienne

Richárd Farkas

Hector Fernandez

Jen- Nifer Foster

Cláudia Freitas

Katarína Gaj- Došová

Daniel Galbraith

Marcos Garcia

Moa Gärdenfors

Kim Gerdes

Filip Ginter

Iakes Goenaga

Koldo Gojenola

Memduh Gökırmak

Yoav Goldberg

Xavier Gómez Guinovart

Berta Gonzáles Saavedra

Matias Grioni

Normunds Grūzītis

Bruno Guillaume

Céline Guillot- Barbance

Nizar Habash

Jan Hajič

Linh Hà Mỹ

Na-Rae Han

Kim Harris

Dag Haug

Barbora Hladká

Jaroslava Hlaváčová

Florinel Hociung

Petter Hohle

Jena Hwang

Radu Ion

Elena Irimia

Tomáš Jelínek

Anders Johannsen

Fredrik Jørgensen

Hüner Kaşıkara

Sylvain Kahane

Hiroshi Kanayama

Jenna Kan- Erva

Tolga Kayadelen

Václava Kettnerová

Jesse Kirchner

Natalia Kotsyba

Simon Krek

Sookyoung Kwak

Veronika Laippala

Lorenzo Lambertino

Tatiana Lando

Septina Dian Larasati

Alexei Lavrentiev

Phương John Lee

Lê Hồng

Alessandro Lenci

Saran Lertpra- Dit

Cheuk Herman Leung

Ying Li

Josie Li

Keying Li

Kyungtae Lim

Nikola Ljubešić

Olga Loginova

Olga Lyashevskaya

Teresa Lynn

Vivien Macketanz

Aibek Makazhanov

Michael Mandl

Christopher Manning

Ruli Ma- Nurung

Cătălina Mărănduc

David Mareček

Héctor Katrin Marheinecke

Martínez Alonso

André Martins

Jan Mašek

Yuji Matsumoto

Ryan Mcdonald

Gustavo Mendonça

Niko Miekka

Anna Missilä

Cătălin Mititelu

Yusuke Miyao

Simonetta Montemagni

Amir More

Laura Moreno Romero

Shinsuke Mori

Bjartur Mortensen

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Kadri Muis- Chnek

Yugo Murawaki

Kaili Müürisep

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Building a User-Generated Content North-African Arabizi Treebank: Tackling Hell

We introduce the first treebank for a romanized user-generated content variety of Algerian, a North-African Arabic dialect known for its frequent usage of code-switching. Made of 1500 sentences, fully annotated in morpho-syntax and Universal Dependency syntax, with full translation at both the word and the sentence levels, this treebank is made freely available. It is supplemented with 50k unlabeled sentences collected from Common Crawl and webcrawled data using intensive data-mining techniques. Preliminary experiments demonstrate its usefulness for POS tagging and dependency parsing. We believe that what we present in this paper is useful beyond the low-resource language community. This is the first time that enough unlabeled and annotated data is provided for an emerging user-generated content dialectal language with rich morphology and code switching, making it an challenging testbed for most recent NLP approaches.

Introduction

Until the rise of fully unsupervised techniques that would free our field from its addiction to annotated data, the question of building useful data sets for under-resourced languages at a reasonable cost is still crucial. Whether the lack of labeled data originates from being a minority language status, its almost oral-only nature or simply its programmed political disappearance, geopolitical events are a factor highlighting a language deficiency in terms of natural language processing resources that can have an important societal impact. Events such as the Haïti crisis in 2010 [START_REF] Munro | Crowdsourced translation for emergency response in haiti: the global collaboration of local knowledge[END_REF] and the current Algerian revolts [START_REF] Nossiter | Algeria protests grow against president bouteflika, ailing and out of sight[END_REF] 1 are massively reflected on social media, yet often in languages or dialects that are poorly re-sourced, namely Haitian Creole and Algerian dialectal Arabic in these cases. No readily available parsing and machine translations systems are available for such languages. Taking as an example the Arabic dialects spoken in North-Africa, mostly from Morocco to Tunisia, sometimes called Maghribi, sometimes Darija, these idioms notoriously contain various degrees of code-switching with languages of former colonial powers such as French, Spanish, and, to a much lesser extent, Italian, depending on the area of usage [START_REF] Habash | Introduction to Arabic Natural Language Processing[END_REF][START_REF] Cotterell | An Algerian Arabic-French code-switched corpus[END_REF][START_REF] Saadane | A conventional orthography for Algerian arabic[END_REF]. They share Modern Standard Arabic (MSA) as their matrix language [START_REF] Myers-Scotton | Common and uncommon ground: Social and structural factors in codeswitching[END_REF], and of course present a rich morphology. In conjunction with the resource scarcity issue, the codeswitching variability displayed by these languages challenges most standard NLP pipelines, if not all. What makes these dialects especially interesting is their widespread use in user-generated content found on social media platforms, where they are generally written using a romanized version of the Arabic script, called Arabizi, which is neither standardized nor formalized. The absence of standardization for this script adds another layer of variation in addition to well-known user generated content idiosyncrasies, making the processing of this kind of text an even more challenging task.

In this work, we present a new data set of about 1500 sentences randomly sampled from the romanized Algerian dialectal Arabic corpus of [START_REF] Cotterell | An Algerian Arabic-French code-switched corpus[END_REF] and from a small corpus of lyrics coming from Algerian dialectal Arabic Hip-Hop and Raï music genre that had the advantage of having already available translations and of being representative of Algerian vernacular urban youth language. We manually annotated this data set with morpho-syntactic information (partsof-speech and morphological features), together with glosses and code-switching labels at the word level, as well as sentence-level translations. Furthermore, we added an additional manual annotation layer following the Universal Dependencies annotation scheme (Nivre et al., 2018), making of this corpus, to the best of our knowledge, the first user-generated content treebank in romanized dialectal Arabic. This treebank contains 36% of French tokens, making it a valuable resource to measure and study the impact of code-switching on NLP tools. We supplement this annotated corpus with about 50k unlabeled sentences extracted from both Common Crawl and additional web crawled data, making of this data set an important milestone in North-African dialectal Arabic NLP. This corpus is made freely available under a Creative Commons license. 2

The Language

As stated by [START_REF] Habash | Introduction to Arabic Natural Language Processing[END_REF], Arabic languages are often classified into three categories : (i) Classical Arabic, as found in the Qur'an and related canonical texts, (ii) Modern Standard Arabic, the official language of the vast majority of Arabic speaking countries and (iii) Dialectal Arabic, whose instances exhibit so much variations that they are not mutually understandable across geographically distant regions. As space is missing for an exhaustive description of Arabic language variations, we refer the reader to [START_REF] Habash | Introduction to Arabic Natural Language Processing[END_REF], [START_REF] Samih | Dialectal Arabic processing Using Deep Learning[END_REF] and especially to [START_REF] Saadane | A conventional orthography for Algerian arabic[END_REF] for a thorough account of Algerian dialectal Arabic, which is the focus of this work. In short, the key properties of North-African dialectal Arabic are:

• It is a Semitic language, non codified, mostly spoken; • It has a rich-inflexion system, which qualifies this dialect as a morphologically-rich language [START_REF] Tsarfaty | Statistical parsing of morphologically rich languages (spmrl): what, how and whither[END_REF], even though [START_REF] Saadane | A conventional orthography for Algerian arabic[END_REF] write that many properties present in Classical Arabic are absent from this dialect (e.g. it has simplified nominal and verbal case systems); • It displays a high degree of variability at all levels: spelling and transliteration conventions, phonology, morphology, lexicon; • It exhibits a high degree of code-switching; due to historical reasons and cultural influence of French in the media circles, the Algerian dialect, as well as Tunisian and Morocco, is known for its heavy use of French words. As stated above, this dialect is mostly spoken and has even been dubbed with disdain as a Creole language by the higher levels of the Algerian political hierarchy. 3 Still, its usage is ubiquitous in the society and, by extension, in social media usergenerated content. Interestingly, the lack of Arabic support in input devices led to the rise of a romanized written form of this dialect, which makes use of alphanumeric letters as additional graphemes to represent phonemes that the Latin script does not naturally cover. Not limited to North-African dialectal Arabic, this non-standard "transliteration" concurrently emerged all over the Arabic-speaking world, and is often called Arabizi. Whether or not written in Arabizi, the inter-dialectal divergences between all Arabic dialects remain.

The following list highlights some of the main properties of Arabizi compared to MSA written in the Arabic script.

• Unlike in MSA written in the Arabic script, where short vowels are marked using optional diacritics, all vowels are explicitly written; • Digits are used to cope with Arabic phonemes that have no counterpart in the Latin script; for instance, the digit "3" is often used to denote the ayin consonant, because it is graphically similar to its rendition in Arabic script; • No norms exist, resulting in a high degree of variability between people writing in Arabizi. From now on, we will call NArabizi the Algerian dialect of Arabic when written in Arabizi, thereby simultaneously referring to the language variety and to the script itself. Table 1 presents several examples of lexical variation within NArabizi. Interestingly, this variability also affects the codeswitched vocabulary, which is mostly French in the case of NArabizi. A typical example of NArabizi that also exhibits code-switching with nonstandard French spelling can be seen in Example 1.

(

Corpus

As other North-African Arabic dialects, NArabizi is a resource-poor language, with, to the best of our knowledge, only one available corpus developed by [START_REF] Cotterell | An Algerian Arabic-French code-switched corpus[END_REF] for language identification purposes.

Data Collection

Cotterell et al. ( 2014)'s corpus was collected in 2012 from an Algerian newspaper's web forums and covers a wide range of topics (from discussion about football events to politics). We collected the 9973 raw sentences from its GitHub repository4 and sampled about 1300 sentences. In addition, because they were available with translations in French and English, we included lyrics from a few dozen recent popular songs of various genres (Raï, hip-hop, etc.), leading to an additional set of 200 sentences. These 1500 sentences form the core of our NArabizi treebank annotation project. In order to make our corpus usable by modern, resource-hungry natural language processing techniques, we also used data-driven language identification models to extract NArabizi samples among the whole collection of the Common-Crawl-based OSCAR corpora [START_REF] Javier | Asynchronous Pipeline for Processing Huge Corpora on Medium to Low Resource Infrastructures[END_REF] as well as 2 millions sentences of additional crawled webdata, resulting in 50k NArabizi sentences of high quality, to date the largest corpus of this language. This makes this collection a valuable test bed for low-resource NLP research.

Annotation Layers

Our NArabizi treebank contains 5 annotations layers: (i) tokenization, (ii) morphology, (iii) codeswitching identification, (iv) syntax and (v) translation.

Tokenization Following [START_REF] Seddah | The French Social Media Bank: a Treebank of Noisy User Generated Content[END_REF] and their work on the French Social Media Bank, we decided to apply a light tokenization process where we manually tokenized only the obvious cases of wrongly detached punctuations and "missing whitespaces" (i.e. cases where two words are contracted into one token).5

Morphological Analysis This layer consists of two sets of part-of-speech tags, one following the Universal POS tagset [START_REF] Petrov | A universal part-of-speech tagset[END_REF] and the other the FTB-cc tagset extended to deal with user-generated content [START_REF] Seddah | The French Social Media Bank: a Treebank of Noisy User Generated Content[END_REF]. In cases of word contractions, we followed their guidelines and used multiple POS as in cetait (`itwas')/PRON+VERB/CLS+V. In addition, we added several morphological features following the Universal Dependency annotation scheme (Nivre et al., 2018), namely gender, number, tense and verbal mood. Note that instead of adding lemmas, we included French glosses for two reasons: firstly for practical reasons, as they helped manual corrections done by non-native speakers of NArabizi, and secondly because of the non-formalized nature of this language, which makes lemmatization very hard, almost akin to etymological research as in the case of garjouma/the throat which can either originate from French gorge or be of Amazigh root.

Code-Switching identification Unlike other works in user-generated content for minority languages [START_REF] Lynn | Codeswitching in irish tweets: A preliminary analysis[END_REF], we do not distinguish between inter-and intra-sentential code-switching and consider word-level codemixing as lexical borrowing.

We annotate code-switching at the word level with information about the source language, regardless of the canonical-ness of spelling.

Syntactic Annotations Here again we follow the Universal Dependencies 2.2 annotation scheme (Nivre et al., 2018). When facing sequences of French words with regular French syntax, we followed the UD French guidelines; otherwise, we followed the UD Arabic guidelines, following the Prague Arabic Dependency UD Treebank.

Translation Layer Our final layer is made up for sentence-level translations in French. It shall be noted that the validation of these translations often led to massive rewording, as the annotators came from different regions of Algeria and could diverge in their interpretations of a given sentence. A sample of 200 sentences was blindly translated (without access to the morpho-syntactic analysis) in order to favor further research on the fluency of machine translation for this dialect.

All annotations layers are displayed in Figure 1.

Extending Our Data Set With Noisy Unlabeled Data

The need for more data has never been more striking as they are needed for important tasks such as handling lexical sparseness issues via word embeddings, lexicon acquisition, domain adaptation via self-training, or fine-tuning pre-trained language models, its modern incarnation. The trouble with NArabizi is that it is a spoken language whose presence can be mostly found in informal texts such as social media. More importantly, the Arabizi transliteration process is also used by other Arabic dialects, making the data collection a needle in a haystack search task. We therefore present in this section the process we used to mine an additional set of 50k

First method: SVM-based classifier

Using keywords-based web scrapping tools, we collected a raw corpus of 4 million sentences, called CrawlWeb, that in fine contained a mixture of French, English, Spanish, MSA and Arabizi texts. Since we are only interested in NArabizi, we designed a classifier to extract proper sentences from that raw corpus. The corpus we used as gold standard is made of 9k sentences of attested NArabizi from our original corpus and 18k of French and English tweets. Using language identification [START_REF] Lui | langid.py: An off-the-shelf language identification tool[END_REF], we convert each sentence from the gold-standard corpus to a feature vector containing language-identification scores and use it as input to a SVM classifier with a classical 80/10/10 split. With a precision and recall score of 94%, we filtered out 173k code-mixed sentences out of the CrawlWeb corpus. Preliminary experiments showed promising initial results, but further analysis pointed out a high level of noise in this initial set, both in terms of erroneous language identification and on the amount of remnant ASCII artifacts that could not easily be removed without impacting the valid NArabizi sentences.

Second method: Neural-based classification

The objectives of this method are twofold: (i) selecting data from CommonCrawl using a neural classifier and (ii) using this data set to intersect the data collected with the previous method. The idea is to ensure the quality of the final resulting unlabeled corpus.

Given the large number of noisy data in Com-monCrawl, a "noise" class is added to the language classification model and is built according to several heuristics. 6 That "noisy" class corpus is made of 40k sentences randomly selected among the result of the application of these rules to a short, 10M-sentence sample of CommonCrawl. We then trained a classifier using Fasttext [START_REF] Joulin | Fasttext. zip: Compressing text classification models[END_REF] on 102 languages, 40k sentences each, extracted from the CommonCrawl-based, languageclassifed OSCAR corpus, to which we added the 9k sentences of the NArabizi original corpus and the "noise" class. The final dataset is composed of 4,090,432 sentences and is split into 80% train, 10% development and 10% test sets. The classifier consists in a linear classifier (here logistic regression) fed with the average of the n-gram embeddings. n-grams are useful in this case as they enables the model to capture specific sequences of NArabizi characters such as lah, llah, 3a, 9a, etc. We choose to embed 2-to 5-grams. These parameters lead to precision and recall scores of 97% on the NArabizi test set.

After an intensive post-processing step (cf. Appendix A.2), this process results in a dataset of 13,667 sentences extracted from half the Common-Crawl corpus. 7 To evaluate the quality of the resulting data set, we randomly picked 3 times 100 sentences, and genuine NArabizi sentences were manually identified, which allowed us to assess the accuracy of our corpus as reaching 97%. Table 2 presents the results of the evaluation of the two classification methods performed on both the development and test sets of the original NArabizi corpus. 8 Results show that the fastText classifier and its n-gram features is more precise than its nonneural counterpart and its language-id feature vectors.

Corpus intersection

When applied to the CrawlWeb corpus, the Fasttext model extracted 44,797 unique Arabizi sentences while the SVM model extracted 83,295 unique Arabizi sentences. The intersection of both extractions amounts to 39,003 Arabizi sentences (with a 99% precision). This means that 44,292 sentences were classified as Arabizi by the SVM model and not by Fasttext. Among them, by random sampling, it can be stated approximately that 55% are indeed NArabizi. Mistakes are misclassified sentences (Spanish and English sentences, for instance) or sentences with only "noise" (such as symbols). 5,794 sentences were classified as NArabizi by the Fasttext model and not SVM. Among them, by random sampling, it can be stated that approximately 60% are indeed Arabizi. Errors are long sentences with only figures and numbers or sentences with many symbols (e.g. " { O3 } " or "!!!! !!!!"). In order to ensure that the collected corpus contains as little non-NArabizi data as possible, we only release the intersection of the data we classified, to which we add the original NArabizi corpus [START_REF] Cotterell | An Algerian Arabic-French code-switched corpus[END_REF] (after having removed the annotated data we extracted from it). Table 3

Pre-annotation Tool Development via Noisy Transliteration of an Arabic UD Treebank

In order to speed up the annotation process of our data, we decided to create a pre-annotation morphosyntactic and syntactic annotator trained on quasi-synthetic data obtained by "transliterating" a pre-existing Arabic (MSA) treebank, the Prague Arabic Dependency Treebank (PADT), into the NArabizi Latin script, together with data from the French GSD UD treebank. Both are taken from the UD treebank collection (Nivre et al., 2018). Before it can be used as training data, the PADT needs to first be transformed into a form similar to NArabizi. Since the PADT corpus is a collection of MSA sentences with no diacritics, it is impossible to directly "transliterate" into NArabizi. We first diacritized it, in order to add short-vowel information, and then "translitterated" it into an Arabizi-like corpus. We describe this process in this Section. The results of the pseudo-NArabizi parser trained on the "translitterated" corpus are then presented in Section 6.2.

Random diacritics

As vowels are always written in Arabizi, the PADT corpus needs to be diacritized before transliteration. Using an equiprobable distribution, diacritics were added randomly, and the text then transliterated using the probabil-ity distributions we describe below. The BLEU score [START_REF] Papineni | Bleu: a method for automatic evaluation of machine translation[END_REF] of this method on the small parallel corpus provides a baseline of 0.31.

Proper diacritization

Using the Farasa software [START_REF] Abdelali | Farasa: A fast and furious segmenter for arabic[END_REF], PADT sentences are diacritized with 81% precision rate, 9 then tokens aligned with corresponding diacritized words. The text is then transliterated the same way as before.

The BLEU score of this version is 0.60. An example showing how this system visibly improves the transliteration can be seen in the "Prop. Diac." output in Example 2.

(2) Source: berlin tarfoudhou 7oussoul charika amrikia 3ala ro5sat tasni3 dabbabat "léopard" al almania Trans.: Berlin refuses to authorize an American firm to produce the "Leopard" German tank.

Random diac.: brouliyani trfidh 7iswla chiroukou amiyirikyoui 3alia rou5soui tasaniya3i dhabouaboui louyiwibiaridha alalmaanouyou Proper diac.: birlin tarfoudhou 7ousolou charikatin 2amiriqiatin 3alaa rou5sati tasni3i dabbatin " lyuberid " el2almeniati Transliteration Once diacritized, the corpus can be properly transliterated. Arabic letters are either consonant sounds or long vowels, each one may have several different transliterations in NArabizi, depending on the writer's age, accent, education and first learned Western language. For example, the letter ث 10 can be transliterated as "t" or "th". A probability must be assigned for each possibility, and to make it as close as possible to what is produced by NArabizi speakers, a small parallel corpus of PADT sentences and their transliteration 9 Other diacritization systems have better performances [START_REF] Belinkov | Arabic diacritization with recurrent neural networks[END_REF] but are either not maintained with the proper python packages, or come with a fee.

10 Theh, U+062B.

by ten NArabizi speakers was assembled, and then each letter aligned with all its possible matches to get probability distributions.

Usability

In this section we describe preliminary experiments on part-of-speech tagging and statistical dependency parsing that show promising results while highlighting the expected difficulty of processing a low-resource language with a high level of code-switching and multiple sources of variability.

POS Tagging

The baseline POS tagger we used is alVWTagger,11 a feature-based statistical POS tagger, which ranked 3rd at the 2017 CoNLL multilingual parsing shared task [START_REF] Daniel Zeman | CoNLL 2017 shared task: Multilingual parsing from raw text to universal dependencies[END_REF]. It is briefly described in (de La [START_REF] De | The ParisNLP entry at the ConLL UD shared task 2017: A tale of a #Parsing-Tragedy[END_REF]. In short, it is a left-to-right tagger that relies on a set of carefully manually designed features, including features extracted from an external lexicon, when available, and a linear model trained using the Vowpal Wabbit framework. 12 In our case, we simply created an "external" lexicon by extracting the content of the training set. It contributes to improving the POS accuracy because it provides the tagger with (ambiguous, partial) additional information about words in the right context of the current word.

Early Parsing experiments

As stated earlier in this paper, NArabizi contains a high-level of code-switching with French and is closely related to MSA. We described in Section 5 how we built a mixed treebank based on the French GSD UD treebank and our Arabizi version of the Prague Arabic Dependency Treebank. We trained the UDPipe parser [START_REF] Straka | Tokenizing, pos tagging, lemmatizing and parsing ud 2.0 with udpipe[END_REF] on various treebanks obtained by combining different proportions of the French GSD and our PADT-based pseudo-Arabizi treebank. We ran these parsers with already annotated gold partsof-speech. 6), this first parser did not performed very well when evaluated on Arabizi dev . This performance level proved insufficient to speed up the annotation task. We therefore manually annotated 300 more NArabizi sentences (Arabizi train300 ), to be used as additional training data. When added to MIX train , parsing performance did improve, yet not to a sufficient extent, especially in terms of Labeled Attachement Score (LAS). It turned out that training UDPipe on these 300 manually annotated NArabizi sentences only (Arabizi train300 ) produced better scores, resulting in a parser that we did use as a pre-annotation tool in a constant bootstrap process to speed up the annotation of the remaining sentences.

Discussion

How interleaved are French and NArabizi? As stated before, NArabizi takes its root in Classical Arabic and in multiple sources of integration of French, MSA and Berber, the Amazigh language. As the NArabizi treebank contains more than 36% of French words, it is of interest to use recent methods of visualization to see how interleaved it is with some of its source languages. To this end, we extract words embeddings using fastText [START_REF] Joulin | Fasttext. zip: Compressing text classification models[END_REF] from a corpus made of the "translitterated" PADT described in Section 5, the French UD GSD and NArabizi original corpus [START_REF] Cotterell | An Algerian Arabic-French code-switched corpus[END_REF]. Two-dimensional representations of the resulting embeddings space for 300 selected words are shown in Figure 2 for embeddings of size 50 and 100. We notice that the overall shapes of both representations are very similar, apart from a non significant x-axis reversal. On the first components, increasing the embedding size does not provide more information.

We also see that French and transliterated Arabic words are clearly separated into two clusters of low standard deviation, while NArabizi words are very spread out. Some fall within the French cluster, they correspond to French words present in this Algerian dialect. Others are in the middle of the Arabic cluster, these are the purely Arabic words of the dialect. Between the two, there are Amazigh words (rak, mech), arabized French words (tomobile < French automobile), Arabic words whose Berber pronunciation has resulted in an unexpected NArabizi rendering (nta instead of expected enta 'you', mchit instead of expected machayt 'to go-2SING').

What is the Impact of Code-Switching in POS-tagging performance? Given the large degree of interleaving between French and NArabizi, it is interesting to assess the impact of the French vocabulary on the performance of a POS-tagger trained on French data only. For these experiments, we use the StanfordNLP neural tagger [START_REF] Qi | Universal dependency parsing from scratch[END_REF], which ranked 1st in POS tagging at the 2018 UD shared task, trained on the UD French ParTUT treebank, using French fastText vectors [START_REF] Mikolov | Advances in pre-training distributed word representations[END_REF]. In order to perform a meaningful evaluation, we split the NArabizi training set into 4 buckets of approximately 25% of it size in tokens, with a increasing proportion of identified NArabizi tokens. Results in Table 7 show a clear drop of performance between the sentences that contain more code-switching (59.55% of UPOS accuracy) and those with none (16.84%). This suggests that low-resource languages with a highlevel of code-switching such as NArabizi can benefit from NLP models trained on the secondary language. The level of performance to expect from these cross-language approaches is yet to be determined. 2016), we provide here the cost figures of this annotation campaign. We do not include the salaries of the permanent staff, nor do we include the overhead. These figures are meant as an indication of the effort needed to create an annotated data set from scratch. It shall be noted that even though the interannotator agreement gave us early indications on the difficulty of the tasks, it also acted as a metric in terms of language variability among annotators. None of them come from the same part of North-Africa and none of them has the same familiarity with the topics discussed in the web-forums we annotated. We had to constantly re-annotate sentences and update the guidelines every time new idiosyncrasies were encountered and most importantly accepted as such by the annotators. Compared to what was reported in [START_REF] Héctor | From Noisy Questions to Minecraft Texts: Annotation Challenges in Extreme Syntax Scenarios[END_REF], the figures are here much higher (about 5 times higher), because unlike their work on French treebanks, we could not use preexisting guidelines for this language and because we could not keep the same team all along the project, so that new members had to be trained almost from scratch or to work on totally different layers.

Related Work

Research on Arabic dialects is quite extensive. Space is lacking to describe it exhaustively. In relation to our work regarding North-African dialect, we refer to the work of [START_REF] Samih | Dialectal Arabic processing Using Deep Learning[END_REF] who along his PhD covered an large range of topics regarding the dialect spoken specifically in Morocco and generally regarding language identification [START_REF] Samih | Multilingual code-switching identification via LSTM recurrent neural networks[END_REF] in code-switching scenario for various Arabic dialects [START_REF] Attia | POS tagging for improving code-switching identification in Arabic[END_REF]. Unlike NArabizi dialects, the resource situation for Arabic dialects in canonical written form can hardly be qualified as scarce given the amount of resources produced by the Linguistic Data Consortium regarding these languages, see [START_REF] Diab | LDC Arabic treebanks and associated corpora: Data divisions manual[END_REF] for details on those corpora. These data have been extensively covered in various NLP aspects by the former members of the Columbia Arabic NLP team, among which Mona Diab, Nizar Habash, and Owen Rambow, in their respective subsequent lines of works. Many small to medium scale linguistics resources, such as morphological lexicons or bilingual dictionaries have been produced [START_REF] Shoufan | Natural language processing for dialectical Arabic: A survey[END_REF]. Recently, in addition to the release of a small-range parallel corpus for some Arabic dialects [START_REF] Bouamor | A multidialectal parallel corpus of Arabic[END_REF], a larger corpus collection was released, covering 25 city dialects in the travel domain [START_REF] Bouamor | The MADAR Arabic dialect corpus and lexicon[END_REF].

Regarding the specific NLP modeling challenges of processing Arabic-based languages, as part of the morphologically-rich languages, recent advances in joint models have been addressed by [START_REF] Zalmout | Joint diacritization, lemmatization, normalization, and fine-grained morphological tagging[END_REF] that recently efficiently adapted a neural architecture to perform joint word segmentation, lemmatization, morphological analysis and POS tagging on an Arabic dialect. Recent works on cross-language learning using the whole massively multilingual pre-trained language models artillery have started to emerge [START_REF] Srivastava | Unsupervised Learning for Handling Code-Mixed Data: A Case Study on POS Tagging of North-African Arabizi Dialect[END_REF]. If successful, such models could help to alleviate the resource scarcity issue that plagues low-resources languages in the more-than-ever data hungry modern NLP.

Conclusion

We introduced the first treebank for an Arabic dialect spoken in North-Africa and written in romanized form, NArabizi. More over, being made of user-generated content, this treebank covers a large variety of language variation among native speakers and displays a high level of codeswitching. Annotated with 4 standard morphosyntactic layers, two of them following the Universal Dependency annotation scheme, and provided with translation to French as well as glosses and word language identification, we believe that this corpus will be useful for the community at large, both for linguistic purposes and as training data for resource-scarce NLP in a high-variability scenario. In addition to the annotated data, we provide around 1 million tokens (over 46k sentences) of unlabeled NArabizi content, resulting in the largest dataset available for this dialect. Our corpora are freely available 14 under the CC-BY-SA license and the NArabizi treebank is also released as part of the Universal Dependencies project.

Figure 1 :

 1 Figure 1: Annotation example: "mouto ya les égyptiens rana fi la coupe de monde m3a wladna erajala les vrais algeriens" [Die Egyptians! We are at the world cup with our children of Algeria, the real men!]. Code-switching is highlighted in blue.

Figure 9 :Figure 9 :

 99 Figure 9: Word embbeddings de 300 mots (100 arabe translittéré, 100 français, 100 arabizi) calculés avec l'algorithme FastText (jaune : français, bleu : arabe, rouge : arabizi) 29

Table 1 :

 1 Examples of lexical variation in NArabizi

	Gloss Attested forms	Lang
	why	wa3lach w3alh 3alach 3lache NArabizi
	all	ekl kal kolach koulli kol	NArabizi
	many beaucoup boucoup bcp	French

2 http://almanach-treebanks.fr/NArabizi

Table 2 :

 2 F1-scores of both language classification models on the Arabizi class.

		F1-Score Arabizi
	Fasttext	0.97
	SVM Classifier	0.94

  provides quantitative information about our corpora.

	Dataset	#Sentences #Tokens
	Original source data	9,372	203k
	Manually Annotated	1,434	22k
	Unlabeled NArabizi	46,941	1.02M

Table 3 :

 3 Corpus statistics

Table 4 :

 4 BLEU score of both transliteration systems.

	System	BLEU score
	Random diacritization	0.31
	Proper diacritization	0.60

Table 5 :

 5 POS tagging results.

	13
		Dev		Test
	All	OOV	all	OOV
	OOV %	32.28		32.75
	UPOS (a) 78.74 55.85 80.37 57.42
	MFEATS (b) 88.10 70.04 87.17 69.12
	(a)+(b) 72.61 40.94 73.87 43.50

  The best scores were obtained with a model trained on a mix 30% of pseudo-Arabizi and 70% of French, which we call the MIX treebank, totaling 5,955 training sentences. We split this treebank into training, development and test sets, called MIX train/dev/test , following a 80/10/10 split. We used a very small manually annotated NArabizi development dataset of 200 NArabizi sentences, called Arabizi dev , to evaluate our parser. As shown in Table 6 (line "Mix"), despite good results on MIX's development and training sets, MIX dev and MIX test respectively (see Table

.67 89.42 87

  .69 89.44 39.28 51.52 MIX+Arabizi 300 87.42 89.20 87.44 89.22 55.54 65.36 Arabizi 300 39.11 49.62 39.14 49.65 63.

	Training corpus	Dev	Test	Test Arabizi
	LAS UAS LAS UAS LAS UAS
	MIX 87

03 71.21Table 6 :

 6 Results of UDPipe (trained 100 epochs) on the preliminary test set.

Table 7 :

 7 POS tagging Performance with regard to codemix proportion trained on UD French Partut treebank

	% of NArabizi per sent. <60	60-78 78-100	100
	bucket set size (sent.)	322	286	283	276
	StanfordNLP (French)	59.55 35.93	25.41 16.84
	8 Treebanking Costs

Following Martínez Alonso et al. (

https://www.nytimes.com/2019/03/01/world/ africa/algeria-protests-bouteflika.html

https://www.lesoirdalgerie.com/articles/ 2010/02/17/article.php?sid=95823&cid=2

https://github.com/ryancotterell/arabic_ dialect_annotation

We corrected in average one tokenization error (less frequently two) per sentence on the web forum parts. We noticed a high degree of variance. Some users displayed this behavior much more than others. This led some of our annotators to believe it resulted from an ill-functioning input device.

These heuristics are presented in the Appendix for reproducibility.

Due to computing power limitation, we were not able to run our selection on the whole CommonCrawl.

Note that the precision and recall are slightly different in both methods, but the rounding at the second decimal made them equal.

Note that we performed a set of baseline experiments with UDPipe 2.0[START_REF] Straka | Tokenizing, pos tagging, lemmatizing and parsing ud 2.0 with udpipe[END_REF] as well on a previous version of this data set. It reached only 73.7 of UPOS on the test set.

12 https://github.com/VowpalWabbit/vowpal_ wabbit/

wiki13 Without this endogenous lexicon extraction step, the tagger performed slightly worse, although the difference is small.

Acknowledgments

The work was partially funded by the French Research Agency projects ParSiTi (ANR-16-CE33-0021), SoSweet (ANR15-CE38-0011-01) and by the French Ministry of Industry and Ministry of Foreign Affairs via the PHC Maimonide France-Israel cooperation programme, as well as by the Sagot's chair in the PRAIRIE institute funded by the French national agency ANR as part of the "Investissements d'avenir" programme under the reference ANR-19-P3IA-0001.

A Appendix

A.1 Class-noise for our language classifier Given the large number of noisy data in Common Crawl, a class noise is added to the classification model we presented section 4.2 and is built according to the following empirical rules :

• If the word "url" appears more than two times, the sentence is added to class noise. • If the sentence has more than four " [ " , the sentence is added to class noise. • The same rule as above works for " ", " ", " { " or " } " symbols. • If the word "http" appears more than two times, the sentence is added to class noise. • If more than two "@" character, the sentence is added to class noise in order to capture sentences with email address or tweets with only mentioned people. • If the phrase "WARC-Refers-To" appears, the sentence is added to class noise.

A.2 Post-processing steps

• Get unique sentences (about 20K sentences).

• The model is likely to classify as Arabizi sentences which contain any letter repeated a lot of times in a row (e.g. "iiiiiiii..", "uuuuuuu..", "ffffff..."). These sentences are deleted from the dataset. • Due to the n-gram embeddings, "lah" is considered as a marker of Arabizi, so a lot of sentences containing "blah" are classified as Arabizi. If this phrase appears more than 5 times, the sentence is deleted. • Figures and numbers are widespread in Arabizi (particularly "3" and "9"), so the model classifies too many sentences which contains only numbers. Therefore, sentences which have more "number of figures" characters than 80% of the number of characters (excluded figures) are deleted.